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# A bordism theory related to matrix Grassmannians
A.V. Ershov ershov.andrei@gmail.com
###### Abstract.
In the present paper we study a bordism theory related to pairs $(M,\,\xi),$
where $M$ is a closed smooth oriented manifold with a stably trivial normal
bundle and $\xi$ is a virtual $\mathop{\rm SU}\nolimits$-bundle of virtual
dimension 1 over $M$. The main result is the calculation of the corresponding
ring modulo torsion and the explicit description of its generators.
## Introduction
In the present paper we study the bordism theory related to pairs $(M,\,\xi),$
where $M$ is a closed smooth oriented manifold with a stably trivial normal
bundle and $\xi$ is a virtual $\mathop{\rm SU}\nolimits$-bundle of virtual
dimension 1 over $M$. The bordism is defined with the help of analogous pairs
$(W,\,\sigma)$, where $W$ is a compact smooth oriented manifold with boundary
$\partial W$ and with a stably trivial normal bundle and $\sigma$ is a virtual
$\mathop{\rm SU}\nolimits$-bundle of virtual dimension 1 over $W$, where the
boundary operator $\partial$ is defined as $\partial(W,\,\sigma)=(\partial
W,\,\sigma|_{\partial W}).$ A ring structure is induced by the product
$(M,\,\xi)\times(M^{\prime},\,\xi^{\prime}):=(M\times
M^{\prime},\,\xi\boxtimes\xi^{\prime})$.
Our main result is the calculation of the corresponding graded ring up to
torsion elements, which turns out to be the polynomial ring
$\mathbb{Q}[t_{2},\,t_{3},\,\ldots],\;\deg t_{n}=2n,$ and the explicit
description of the ring generators which have the form
$t_{n}=[S^{2n},\,\xi^{(n)}],$ where $\xi^{(n)}$ is the virtual $\mathop{\rm
SU}\nolimits$-bundle of virtual dimension 1 that is the generator in the
multiplicative group of such bundles over $S^{2n},$ and the brackets $[\,,\,]$
denote the corresponding bordism class. Of course, $S^{2n}=\partial D^{2n+1}$
but it is clear that the bundle $\xi^{(n)}$ can not be extended to the whole
ball.
Note that in contrast to “usual” bordisms, the stabilisation in our case does
not correspond to the taking of Whitney sum with trivial bundles but with the
tensor product by trivial bundles. Therefore in our case the Thom spaces are
not stabilized by usual suspension (see Section 5) and the corresponding limit
object is not a suspension spectrum.
It seems that the obtained results are closely related to [1].
## 1\. Main definitions
Consider a pair $(M,\,\xi),$ where $M$ is a closed smooth oriented manifold of
dimension $d$ with a stably trivial normal bundle and $\xi\in\mathop{\rm
KSU}\nolimits(M)$ is a virtual $\mathop{\rm SU}\nolimits$-bundle of virtual
dimension $1$ (here $\mathop{\rm KSU}\nolimits$ denotes the $K$-functor
related to $\mathop{\rm SU}\nolimits$-bundles).
Pairs $(M,\,\xi)$ and $(M^{\prime},\,\xi^{\prime}),\;\dim M^{\prime}=\dim M=d$
are called bordant if there exists a pair $(W,\,\sigma)$, where $W$ is a
compact $d+1$-dimensional oriented manifold with boundary $\partial W$ and
with a stably trivial normal bundle and $\sigma\in\mathop{\rm
KSU}\nolimits(W),\;\dim\sigma=1$ such that $\partial
W=M\bigsqcup(-M^{\prime})$ and
$\sigma|_{M}=\xi,\;\sigma|_{M^{\prime}}=\xi^{\prime}$ ($-M^{\prime}$ denotes
$M^{\prime}$ with reversed orientation).
Clearly that to be bordant is an equivalence relation111the transitivity
follows from the exactness of $\mathop{\rm KSU}\nolimits(W_{1}\cup
W_{2})\rightarrow\mathop{\rm KSU}\nolimits(W_{1})+\mathop{\rm
KSU}\nolimits(W_{2})\rightarrow\mathop{\rm KSU}\nolimits(W_{1}\cap W_{2})$ and
the corresponding equivalence classes $[M,\,\xi]$ of pairs $(M,\,\xi),\>\dim
M=d$ form an abelian group with respect to the disjoint union which we denote
by $\Omega^{d}.$ The product
$[M,\,\xi]\times[M^{\prime},\,\xi^{\prime}]:=[M\times
M^{\prime},\,\xi\boxtimes\xi^{\prime}]$ equips the direct sum
${\mathop{\oplus}\limits_{d}}\Omega^{d}$ with the structure of the graded ring
$\Omega^{*}$, (here $\boxtimes$ denotes the “exterior” tensor product of
virtual bundles).
We want to reduce the classification problem of pairs $(M,\,\xi)$ modulo
bordism to the problem of the calculation of the homotopy groups of some Thom
space. Let us briefly describe the corresponding argument.
Consider a pair $(M,\,\xi)$ as above. Let $\eta\in\mathop{\rm
KSU}\nolimits(M)$ be the inverse element for $\xi$ with respect to the tensor
product, i.e. $\xi\otimes\eta=[1],$222if $\xi=1+\widetilde{\xi},$ where
$\widetilde{\xi}\in\widetilde{\mathop{\rm KSU}\nolimits}(X),$ then
$\eta=1-\widetilde{\xi}+\widetilde{\xi}^{2}-\ldots$, but
$\widetilde{\xi}^{r}=0$ because $M$ is compact where $[n]$ denotes a trivial
$\mathbb{C}^{n}$-bundle over $M$. Let $k,\,l$ be a pair of relatively prime
positive integers, i.e. their greatest common divisor $(k,\,l)=1.$ Assume that
$d=\dim M<2\min\\{k,\,l\\}.$ Then for virtual bundles $k\xi,\>l\eta$ of
dimensions $k$ and $l$ respectively there are geometric representatives
$\xi_{k}\rightarrow M$ $\eta_{l}\rightarrow M,$ i.e. “genuine” vector bundles,
which are unique up to isomorphism. Moreover,
$\xi_{k}\otimes\eta_{l}\cong[kl]$ is a trivial bundle of dimension $kl.$ We
will show that there is a natural bijection between virtual bundles
$\xi\in\mathop{\rm KSU}\nolimits(M),\;\dim\xi=1$ and isomorphism classes of
pairs $(\xi_{k},\,\eta_{l}).$ Furthermore, such pairs are classified by so-
called matrix Grassmannian $\mathop{\rm Gr}\nolimits_{k,\,l}$ (defined below),
i.e. there is a natural one-to-one correspondence between isomorphism classes
of pairs $(\xi_{k},\,\eta_{l})$ over $M,\;\dim M<2\min\\{k,\,l\\}$ and the set
of homotopy classes $[M,\,\mathop{\rm Gr}\nolimits_{k,\,l}]$ of maps
$M\rightarrow\mathop{\rm Gr}\nolimits_{k,\,l}$. Using this result we will show
that there is a natural one-to-one correspondence between bordism classes of
pairs $(M,\,\xi),\>\dim M=d$ and homotopy groups $\pi_{d+2kl}({\rm
T}(\vartheta_{k,\,l}))$ of the Thom space of the trivial
$\mathbb{C}^{kl}$-bundle $\vartheta_{k,\,l}\rightarrow\mathop{\rm
Gr}\nolimits_{k,\,l}$.
## 2\. $\mathop{\rm SU}\nolimits$-bundles and matrix Grassmannians
In this section we recall in a suitable form some results from [3].
Recall that the matrix Grassmannian $\mathop{\rm Gr}\nolimits_{k,\,l}$ is a
space which parametrizes unital $*$-subalgebras isomorphic to
$M_{k}(\mathbb{C})$ (“$k$-subalgebras”) in a fixed algebra
$M_{kl}(\mathbb{C})$. As a homogeneous space it can be represented in the form
$\mathop{\rm PU}\nolimits(kl)/(\mathop{\rm PU}\nolimits(k)\otimes\mathop{\rm
PU}\nolimits(l))$ (here the symbol “$\otimes$” denotes the Kronecker product
of matrices). In case $(k,\,l)=1$ it can also be represented as
(1) $\mathop{\rm SU}\nolimits(kl)/(\mathop{\rm
SU}\nolimits(k)\otimes\mathop{\rm SU}\nolimits(l)).$
The tautological $M_{k}(\mathbb{C})$-bundle
${\mathcal{A}}_{k,\,l}\rightarrow{\rm Gr}_{k,\,l}$ is the subbundle of the
direct product ${\rm Gr}_{k,\,l}\times M_{kl}(\mathbb{C})$ consisting of pairs
$\\{(x,\,T)\mid x\in{\rm Gr}_{k,\,l},\>T\in M_{k,\,x}\subset
M_{kl}(\mathbb{C})\\},$ where $M_{k,\,x}$ denotes the $k$-subalgebra
corresponding to a point $x\in{\rm Gr}_{k,\,l}$. Let
$\mathcal{B}_{k,\,l}\rightarrow\mathop{\rm Gr}\nolimits_{k,\,l}$ be the
$M_{l}(\mathbb{C})$-bundle formed by fiberwise centralizers to the subbundle
${\mathcal{A}}_{k,\,l}\subset{\rm Gr}_{k,\,l}\times M_{kl}(\mathbb{C}).$
Clearly, there is the canonical trivialization
(2) ${\mathcal{A}}_{k,\,l}\otimes{\mathcal{B}}_{k,\,l}\cong{\rm
Gr}_{k,\,l}\times M_{kl}(\mathbb{C}).$
It is easy to see that ${\mathcal{A}}_{k,\,l}$ is associated (by means of the
representation $\mathop{\rm SU}\nolimits(k)\rightarrow\mathop{\rm
PU}\nolimits(k)\cong\mathop{\rm Aut}\nolimits(M_{k}(\mathbb{C}))$) with the
principal $\mathop{\rm SU}\nolimits(k)$-bundle
(3) $\mathop{\rm SU}\nolimits(k)\rightarrow\mathop{\rm
SU}\nolimits(kl)/(E_{k}\otimes\mathop{\rm
SU}\nolimits(l))\rightarrow\mathop{\rm Gr}\nolimits_{k,\,l}$
(cf. (1)), while ${\mathcal{B}}_{k,\,l}$ with the principal $\mathop{\rm
SU}\nolimits(l)$-bundle
(4) $\mathop{\rm SU}\nolimits(l)\rightarrow\mathop{\rm
SU}\nolimits(kl)/(\mathop{\rm SU}\nolimits(k)\otimes
E_{l})\rightarrow\mathop{\rm Gr}\nolimits_{k,\,l}.$
Let $\xi_{k,\,l}\rightarrow\mathop{\rm
Gr}\nolimits_{k,\,l},\,\eta_{k,\,l}\rightarrow\mathop{\rm
Gr}\nolimits_{k,\,l}$ be vector $\mathbb{C}^{k}$ and $\mathbb{C}^{l}$-bundles
associated with principal bundles (3) and (4). There are isomorphisms
${\mathcal{A}}_{k,\,l}\cong\mathop{\rm
End}\nolimits(\xi_{k,\,l}),\;{\mathcal{B}}_{k,\,l}\cong\mathop{\rm
End}\nolimits(\eta_{k,\,l})$ and the canonical trivialization
(5) $\vartheta_{k,\,l}:=\xi_{k,\,l}\otimes\eta_{k,\,l}\cong\mathop{\rm
Gr}\nolimits_{k,\,l}\times\mathbb{C}^{kl}$
which gives (2) after the application of $\mathop{\rm End}\nolimits.$
###### Proposition 1.
A map $f\colon M\rightarrow\mathop{\rm Gr}\nolimits_{k,\,l}$ is the same thing
as a triple $(\xi_{k},\,\eta_{l},\,\varphi)$ consisting of vector $\mathop{\rm
SU}\nolimits$-bundles $\xi_{k},\,\eta_{l}$ with fibers $\mathbb{C}^{k}$ and
$\mathbb{C}^{l}$ over $M$ such that $\xi_{k}\otimes\eta_{l}\cong[kl]$ and a
trivialization $\varphi\colon\xi_{k}\otimes\eta_{l}\cong
M\times\mathbb{C}^{kl}.$
Proof. For a given map $f\colon M\rightarrow\mathop{\rm Gr}\nolimits_{k,\,l}$
the triple $(\xi_{k},\,\eta_{l},\,\varphi)$ is defined as follows:
$\xi_{k}:=f^{*}(\xi_{k,\,l}),\>\eta_{l}:=f^{*}(\eta_{k,\,l})$ and the
trivialization $\varphi$ is induced by (5).
Conversely, for a given triple $(\xi_{k},\,\eta_{l},\,\varphi)$ over $M$ as in
the statement of the proposition the trivialization $\varphi$ determines the
trivialization $\mathop{\rm End}\nolimits(\varphi)\colon\mathop{\rm
End}\nolimits(\xi_{k}\otimes\eta_{l})\cong M\times M_{kl}(\mathbb{C})$ of the
bundle $\mathop{\rm End}\nolimits(\xi_{k}\otimes\eta_{l})=\mathop{\rm
End}\nolimits(\xi_{k})\otimes\mathop{\rm End}\nolimits(\eta_{l}).$ Thereby
$\mathop{\rm End}\nolimits(\xi_{k})$ can be considered as a family of unital
$k$-subalgebras in a fixed algebra $M_{kl}(\mathbb{C})$, hence we obtain the
required map $f\colon M\rightarrow\mathop{\rm
Gr}\nolimits_{k,\,l}.\quad\square$
Two triples
$(\xi_{k},\,\eta_{l},\,\varphi),\;(\xi_{k}^{\prime},\,\eta_{l}^{\prime},\,\varphi^{\prime})$
over $M$ are called equivalent if
$\xi_{k}\cong\xi_{k}^{\prime},\>\eta_{l}\cong\eta_{l}^{\prime}$ and $\varphi$
is homotopic to $\varphi^{\prime}$ in the class of trivializations.
###### Corollary 2.
There is a natural one-to-one correspondence between equivalence classes of
triples $(\xi_{k},\,\eta_{l},\,\varphi)$ over $M$ and the set
$[M,\,\mathop{\rm Gr}\nolimits_{k,\,l}]$ of homotopy classes of maps
$M\rightarrow\mathop{\rm Gr}\nolimits_{k,\,l}.$
Proof easily follows from the previous proposition.$\quad\square$
Let $\lambda_{k,\,l}\colon\mathop{\rm
Gr}\nolimits_{k,\,l}\rightarrow\mathop{\rm BSU}\nolimits(k)$ be a classifying
map for the principal $\mathop{\rm SU}\nolimits(k)$-bundle (3) (i.e. for the
vector bundle $\xi_{k,\,l}$), $\mu_{k,\,l}\colon\mathop{\rm
Gr}\nolimits_{k,\,l}\rightarrow\mathop{\rm BSU}\nolimits(l)$ a classifying map
for the principal $\mathop{\rm SU}\nolimits(l)$-bundle (4) (i.e. for the
vector bundle $\eta_{k,\,l}$).
Consider the fibration (cf. (1))
(6) $\mathop{\rm
Gr}\nolimits_{k,\,l}\stackrel{{\scriptstyle\lambda_{k,\,l}\times\mu_{k,\,l}}}{{\longrightarrow}}\mathop{\rm
BSU}\nolimits(k)\times\mathop{\rm
BSU}\nolimits(l)\stackrel{{\scriptstyle\otimes}}{{\rightarrow}}\mathop{\rm
BSU}\nolimits(kl).$
The map $\lambda_{k,\,l}\times\mu_{k,\,l}$ corresponds to the functor
$(\xi_{k},\,\eta_{l},\,\varphi)\mapsto(\xi_{k},\,\eta_{l})$ which forgets
trivialization $\varphi.$ We are going to prove that for manifolds $M$ of
dimension $\dim M<2\min\\{k,\,l\\}$ such a trivialization $\varphi$ is unique
up to homotopy (see Proposition 4). It requires some preparation.
###### Proposition 3.
If $(km,\,ln)=1$ then the embedding $\mathop{\rm
Gr}\nolimits_{k,\,l}\rightarrow\mathop{\rm Gr}\nolimits_{km,\,ln}$ induced by
a unital $*$-homomorphism $M_{kl}(\mathbb{C})\rightarrow M_{klmn}(\mathbb{C})$
induces an isomorphism of homotopy groups up to dimension $\sim
2\min\\{k,\,l\\}.$
Proof follows from the representation (1) and the sequence of homotopy groups
of the corresponding fibration, see [3].$\quad\square$
The proven proposition implies that the homotopy type of the direct limit
$\lim\limits_{\longrightarrow\atop{\\{k_{i},\,l_{i}\\}}}\mathop{\rm
Gr}\nolimits_{k_{i},\,l_{i}}$ does not depend on the choice of a sequence of
pairs $\\{k_{i},\,l_{i}\\}$ of positive integers provided
$(k_{i},\,l_{i})=1,\;k_{i}|k_{i+1},\,l_{i}|l_{i+1}\;\forall i$ and
$k_{i},\,l_{i}\rightarrow\infty$ when $i\rightarrow\infty.$ This homotopy type
we will denote by $\mathop{\rm Gr}\nolimits.$
The space $\mathop{\rm Gr}\nolimits$ has the natural $H$-space structure
induced by maps $\mathop{\rm Gr}\nolimits_{k,\,l}\times\mathop{\rm
Gr}\nolimits_{m,\,n}\rightarrow\mathop{\rm
Gr}\nolimits_{km,\,ln},\;(km,\,ln)=1$ defined by the tensor product of matrix
algebras $M_{kl}(\mathbb{C})\times M_{mn}(\mathbb{C})\rightarrow
M_{kl}(\mathbb{C})\otimes M_{mn}(\mathbb{C})\cong M_{klmn}(\mathbb{C})$. By
$\mathop{\rm Gr}\nolimits$ we will also denote this $H$-space.
Put $\mathop{\rm
BSU}\nolimits(k^{\infty}):=\lim\limits_{\longrightarrow\atop{n}}\mathop{\rm
BSU}\nolimits(k^{n}),\;\mathop{\rm
BSU}\nolimits(l^{\infty}):=\lim\limits_{\longrightarrow\atop{n}}\mathop{\rm
BSU}\nolimits(l^{n}),$ where direct limits are taken over maps induced by
tensor products. We consider these spaces as $H$-spaces with the
multiplication induced by the tensor product of the corresponding bundles.
A simple calculation with homotopy groups shows that $\mathop{\rm
Gr}\nolimits$ has the same homotopy groups as $\mathop{\rm BSU}\nolimits$ and
the maps $\lambda_{k^{\infty},\,l^{\infty}}\colon\mathop{\rm
Gr}\nolimits\rightarrow\mathop{\rm
BSU}\nolimits(k^{\infty}),\;\mu_{k^{\infty},\,l^{\infty}}\colon\mathop{\rm
Gr}\nolimits\rightarrow\mathop{\rm BSU}\nolimits(l^{\infty})$ are the
localizations over $k$ and $l$ respectively (in the sense that $k$ and $l$
become invertible). Moreover, these localizations are $H$-spaces
homomorphisms. This implies that $\mathop{\rm Gr}\nolimits$ is isomorphic to
$\mathop{\rm BSU}\nolimits_{\otimes}$ as an $H$-space (recall that the product
in $\mathop{\rm BSU}\nolimits_{\otimes}$ is induced by the tensor product of
virtual bundles of virtual dimension 1).
###### Proposition 4.
Assume that $\dim M<2\min\\{k,\,l\\}$. Then for a classifying map
$M\rightarrow\mathop{\rm BSU}\nolimits(k)\times\mathop{\rm BSU}\nolimits(l)$
of a pair $(\xi_{k},\,\eta_{l})$ such that $\xi_{k}\otimes\eta_{l}\cong[kl]$ a
lift $M\rightarrow\mathop{\rm Gr}\nolimits_{k,\,l}$ in (6) exists and is
unique up to homotopy.
Proof. Assume that a map $\bar{f}=\bar{f}_{1}\times\bar{f}_{2}\colon
M\rightarrow\mathop{\rm BSU}\nolimits(k)\times\mathop{\rm BSU}\nolimits(l)$
classifies the pair of bundles $(\xi_{k},\,\eta_{l})$ as in the proposition
statement, i.e.
$\xi_{k}=\bar{f}_{1}^{*}(\xi_{k,\,l}),\;\eta_{l}=\bar{f}_{2}^{*}(\eta_{k,\,l}),$
and moreover $\xi_{k}\otimes\eta_{l}\cong[kl]$. Then
$\otimes\circ\bar{f}\simeq*$ (see (6)) and it follows from the exactness of
(6) that there exists some lift $f\colon M\rightarrow\mathop{\rm
Gr}\nolimits_{k,\,l},$ i.e. $\lambda_{k,\,l}\circ
f\simeq\bar{f}_{1},\;\mu_{k,\,l}\circ f\simeq\bar{f}_{2}.$
In order to prove the uniqueness up to homotopy of the lift provided $\dim
M<2\min\\{k,\,l\\}$ let us use the above introduced direct limits. Recall that
$\lambda_{k^{\infty},\,l^{\infty}}\colon\mathop{\rm
Gr}\nolimits\rightarrow\mathop{\rm
BSU}\nolimits(k^{\infty}),\;\mu_{k^{\infty},\,l^{\infty}}\colon\mathop{\rm
Gr}\nolimits\rightarrow\mathop{\rm BSU}\nolimits(l^{\infty})$ are the
localizations over $k$ and $l$ respectively. Together with the condition
$(k,\,l)=1$ this implies that the map
$\lambda_{k^{\infty},\,l^{\infty}}\times\mu_{k^{\infty},\,l^{\infty}}\colon\mathop{\rm
Gr}\nolimits\rightarrow\mathop{\rm BSU}\nolimits(k^{\infty})\times\mathop{\rm
BSU}\nolimits(l^{\infty})$
(see (6)) induces an injective homomorphism of groups
$[M,\,\mathop{\rm Gr}\nolimits]\rightarrow[M,\,\mathop{\rm
BSU}\nolimits(k^{\infty})\times\mathop{\rm BSU}\nolimits(l^{\infty})]$
of homotopy classes of maps. Now using Proposition 3 we obtain the required
assertion.$\quad\square$
Recall (see the end of the previous section) that given a virtual bundle
$\xi\in\mathop{\rm KSU}\nolimits(M),\;\dim\xi=1$ and a pair
$k,\,l,\>(k,\,l)=1,\;\dim M<2\min\\{k,\,l\\}$ we can find a unique up to
isomorphism pair of geometric bundles $\xi_{k},\,\eta_{l}$ such that
$\xi_{k}\otimes\eta_{l}\cong[kl]$ which (according to the proven proposition)
defines a classifying map $f\colon M\rightarrow\mathop{\rm
Gr}\nolimits_{k,\,l}$ unique up to homotopy.
Conversely, for a given map $f\colon M\rightarrow\mathop{\rm
Gr}\nolimits_{k,\,l}$ we want to define a virtual bundle $\xi\in\mathop{\rm
KSU}\nolimits(M),\;\dim\xi=1$.
Let $m,\,n$ be another pair of positive integers such that
$(km,\,ln)=1=(k,\,m),\;\dim M<2\min\\{m,\,n\\}.$ Consider the diagram
(7) $\mathop{\rm Gr}\nolimits_{k,\,l}\stackrel{{\scriptstyle
i}}{{\rightarrow}}\mathop{\rm Gr}\nolimits_{km,\,ln}\stackrel{{\scriptstyle
j}}{{\leftarrow}}\mathop{\rm Gr}\nolimits_{m,\,n},$
where maps $i$ and $j$ are induced by matrix algebra homomorphisms. It follows
from Proposition (3) that for $f:=f_{k,\,l}$ there exists a unique up to
homotopy map $f_{m,\,n}\colon M\rightarrow\mathop{\rm Gr}\nolimits_{m,\,n}$
such that $i\circ f_{k,\,l}\simeq j\circ f_{m,\,n}\colon
M\rightarrow\mathop{\rm Gr}\nolimits_{km,\,ln}.$ Moreover,
$i^{*}(\xi_{km,\,ln})\cong\xi_{k,\,l}\otimes[m],\;j^{*}(\xi_{km,\,ln})\cong\xi_{m,\,n}\otimes[k].$
Hence for the bundle $\xi_{k}:=f^{*}_{k,\,l}(\xi_{k,\,l})$ over $M$ there
exists the bundle $\xi_{m}:=f^{*}_{m,\,n}(\xi_{m,\,n})$ such that
$\xi_{k}\otimes[m]\cong\xi_{m}\otimes[k],$ hence the relation
$m\xi_{k}=k\xi_{m}$ in the $K$-functor.
Suppose $u,\,v$ be a pair of integers such that $uk+vm=1$ (recall that we have
chosen $m$ such that $(k,\,m)=1$). Then
$\xi_{k}=uk\xi_{k}+vm\xi_{k}=uk\xi_{k}+vk\xi_{m}=k(u\xi_{k}+v\xi_{m}).$
Suppose $\xi:=u\xi_{k}+v\xi_{m},$ then $\xi\in\mathop{\rm
KSU}\nolimits(M),\;\dim\xi=1.$ Moreover,
$m\xi=um\xi_{k}+vm\xi_{m}=(uk+vm)\xi_{m}=\xi_{m}.$ Thereby to a map $f\colon
M\rightarrow\mathop{\rm Gr}\nolimits_{k,\,l}$ we assign a virtual bundle
$\xi\in\mathop{\rm KSU}\nolimits(M)$ of virtual dimension 1, and hence we have
a bijection $[M,\,\mathop{\rm
Gr}\nolimits_{k,\,l}]\stackrel{{\scriptstyle\cong}}{{\rightarrow}}1+\widetilde{\mathop{\rm
KSU}\nolimits}(M).$ It is easy to see that this bijection can be extended to
the group isomorphism $[M,\,\mathop{\rm
Gr}\nolimits]\stackrel{{\scriptstyle\cong}}{{\rightarrow}}(1+\widetilde{\mathop{\rm
KSU}\nolimits}(M))^{\times}=[M,\,\mathop{\rm BSU}\nolimits_{\otimes}]$ (this
time without any condition on $\dim M$). In particular, we again have
established the $H$-space isomorphism $\mathop{\rm
Gr}\nolimits\cong\mathop{\rm BSU}\nolimits_{\otimes}$.
In particular, we have proven the following theorem.
###### Theorem 5.
For any pair $(M,\,\xi)$ such that $\dim M<2\min\\{k,\,l\\}$ there exists a
unique up to homotopy map $f_{\xi}\colon M\rightarrow\mathop{\rm
Gr}\nolimits_{k,\,l}$ representing $\xi$ (in the sense that $\xi$ can be
uniquely restored by the pair
$f^{*}_{\xi}(\xi_{k,\,l}),\;f^{*}_{\xi}(\eta_{k,\,l})$).
Note that two pairs $(\xi_{k},\,\eta_{l})$ and $(\xi_{m},\,\eta_{n})$ provided
$(km,\,ln)=1$ correspond to the same bundle $\xi$ if
$\xi_{k}\otimes[m]\cong\xi_{m}\otimes[k],\;\eta_{l}\otimes[n]\cong\eta_{n}\otimes[l]$
(cf. (7)). In general (without assumption $(km,\,ln)=1$) we have to take the
transitive closure of this relation.
## 3\. Bordism of triples
In this section using the obtained results we replace pairs $(M,\,\xi)$ by
some triples $(M,\,\xi_{k},\,\eta_{l})$ of more geometric nature.
Let $M,\;\dim M=d$ be a smooth oriented manifold with a stably trivial normal
bundle, $f\colon M\rightarrow\mathbb{R}^{d+N}$ a smooth embedding, in addition
we assume that the trivial normal bundle $\nu\cong M\times\mathbb{R}^{N}$ is
equipped with an almost complex structure ($\Rightarrow 2\mid N$) and moreover
there is a representation $\nu\cong\xi_{k}\otimes\eta_{l}$ ($\Rightarrow
N=2kl$) as the tensor product of (complex) vector bundles $\xi_{k},\,\eta_{l}$
over $M$ of dimensions $k,\,l,\;(k,\,l)=1$ and with structural groups
$\mathop{\rm SU}\nolimits(k)$ and $\mathop{\rm SU}\nolimits(l)$ respectively.
If $d<2\min\\{k,\,l\\},$ then, according to the previous section, the pair
$(\xi_{k},\,\eta_{l})$ determines a classifying map $M\rightarrow\mathop{\rm
Gr}\nolimits_{k,\,l}$ which is unique up to homotopy. Therefore we can replace
pairs $(M,\,\xi)$ by equivalent triples $(M,\,\xi_{k},\,\eta_{l}).$
Let $W,\,\dim W=d+1$ be a compact oriented manifold with boundary $\partial W$
and with trivial normal bundle $\nu_{W}$ for an embedding $F\colon
W\rightarrow\mathbb{R}^{d+1+N}_{+},\quad F(\partial
W)\subset\mathbb{R}^{d+N}$, moreover, $\nu_{W}=\sigma_{k}\otimes\rho_{l}$ for
some vector bundles $\sigma_{k},\,\rho_{l}$ with structural groups
$\mathop{\rm SU}\nolimits(k),\,\mathop{\rm SU}\nolimits(l)$ respectively. Then
we can define a boundary operator as follows:
$\partial(W,\,\sigma_{k},\,\rho_{l})=(\partial W,\,\sigma_{k}|_{\partial
W},\,\rho_{l}|_{\partial W}).$
In particular, for $W=M\times
I,\;\sigma_{k}=\widehat{\xi}_{k}:=\pi^{*}(\xi_{k}),\;\rho_{l}=\widehat{\eta}_{l}:=\pi^{*}(\eta_{l}),$
where $\pi$ is the projection onto the first factor $M\times I\rightarrow M$
we have:
$\partial(M\times
I,\,\widehat{\xi}_{k},\,\widehat{\eta}_{l})=(M,\,\xi_{k},\,\eta_{l})\bigsqcup(-M,\,\xi_{k},\,\eta_{l}).$
Furthermore, we can define an equivalence relation: two triples are bordant if
they become isomorphic (in the natural sense) after taking the disjoint union
with boundaries. It follows from the previous section that there is a natural
one-to-one correspondence between bordism classes of triples
$(M,\,\xi_{k},\,\eta_{l})$ and bordism classes of pairs $(M,\,\xi)$ as we have
defined in Section 1.
In order to take into account the possibility of choices of pairs of bundles
$(\xi_{k},\,\eta_{l})$ of different dimensions $k,\,l,$ related to a virtual
bundle $\xi$, we have to extend the equivalence relation. It is generated by
the equivalence between $(M,\,\xi_{k},\,\eta_{l})$ and
$(M,\,\xi_{k}\otimes[m],\,\eta_{l}\otimes[n])$ provided $(km,\,ln)=1$ (cf.
Proposition 3).
## 4\. Thom spaces
Suppose we are given a triple $(M,\,\xi_{k},\,\eta_{l})$ and a bundle
$\nu\cong\xi_{k}\otimes\eta_{l}\cong M\times\mathbb{R}^{N}$ as in the previous
section. Then, according to Proposition 4, we have a unique up to homotopy map
$f\colon M\rightarrow\mathop{\rm Gr}\nolimits_{k,\,l}$ which classifies the
pair $(\xi_{k},\,\eta_{l})$. That is we have the map of trivial bundles
$\textstyle{\nu\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\vartheta_{k,\,l}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{\mathop{\rm
Gr}\nolimits_{k,\,l}}$
which is compatible with the representations
$\nu=\xi_{k}\otimes\eta_{l},\;\vartheta_{k,\,l}=\xi_{k,\,l}\otimes\eta_{k,\,l}$
in the form of tensor products (see (5)), i.e.
$f^{*}(\vartheta_{k,\,l})=f^{*}(\xi_{k,\,l})\otimes
f^{*}(\eta_{k,\,l})\cong\xi_{k}\otimes\eta_{l}=\nu,$ and it can be extended to
the map $\varphi$ of their one-point compactifications, i.e. the Thom spaces
$\varphi\colon{\rm T}(\nu)\rightarrow{\rm T}(\vartheta_{k,\,l})$. Then the
composition of the map $S^{d+N}\rightarrow{\rm T}(\nu)$ ($N=2kl$) contracting
the complement to a tubular neighborhood for the embedded manifold $M\subset
S^{d+N}$ to the base point with the map $\varphi$ defines some map
$S^{d+N}\rightarrow{\rm T}(\vartheta_{k,\,l})$. It is easy to see that a
bordism between $(M,\,\xi_{k},\,\eta_{l})$ and some other triple determines a
homotopy $S^{d+N}\times I\rightarrow{\rm T}(\vartheta_{k,\,l})$. So we can
assign some element of $\pi_{d+N}({\rm T}(\vartheta_{k,\,l}))$ to the bordism
class of a triple $(M,\,\xi_{k},\,\eta_{l})$.
The standard argument using t-regularity to the smooth submanifold
$\mathop{\rm Gr}\nolimits_{k,\,l}\subset{\rm T}(\vartheta_{k,\,l})-\\{*\\}$
(where $\\{*\\}$ is the base point of the Thom space) shows that, conversely,
we can assign the bordism class of some triple $(M,\,\xi_{k},\,\eta_{l})$ to
an element of the group $\pi_{d+N}({\rm T}(\vartheta_{k,\,l}))$.
Thus we have proven the following theorem.
###### Theorem 6.
The above described correspondence defines an isomorphism between the group of
bordism classes of triples $(M,\,\xi_{k},\,\eta_{l}),\;\dim M=d$ and the
homotopy group $\pi_{d+N}({\rm T}(\vartheta_{k,\,l}))$.
According to the previous results (concerning the relation between virtual
bundles $\xi$ of virtual dimension $1$ with pairs $(\xi_{k},\,\eta_{l})$) we
also have an isomorphism between the group of bordisms of pairs $(M,\,\xi)$ as
in Section 1 and the homotopy group $\pi_{d+N}({\rm T}(\vartheta_{k,\,l}))$.
## 5\. Stabilization
In contrast with “usual” bordism theories, in our case the stabilization is
related to the tensor product of bundles, therefore we have to use another
functor instead of the suspension.
According to the above theorem, for any element of $\pi_{d+N}({\rm
T}(\vartheta_{k,\,l})),\;N=2kl,\;d<2\min\\{k,\,l\\}$ there exists a well-
defined bordism class $[M,\,\xi_{k},\,\eta_{l}],\;\dim M=d$. Consider the
triple $(M,\,\xi_{k}\otimes[m],\,\eta_{l}\otimes[n]),\;(km,\,ln)=1$ and the
corresponding map $S^{d+2klmn}\rightarrow{\rm T}(\vartheta_{km,\,ln})$. It is
easy to see that the corresponding element $\pi_{d+2klmn}({\rm
T}(\vartheta_{km,\,ln}))$ is well defined by the bordism class of the triple
$(M,\,\xi_{k},\,\eta_{l})$, and therefore we obtain a homomorphism
$\pi_{d+N}({\rm T}(\vartheta_{k,\,l}))\rightarrow\pi_{d+2klmn}({\rm
T}(\vartheta_{km,\,ln})).$
###### Proposition 7.
If $d<2\min\\{k,\,l\\}$ then the above defined homomorphism $\pi_{d+N}({\rm
T}(\vartheta_{k,\,l}))\rightarrow\pi_{d+2klmn}({\rm T}(\vartheta_{km,\,ln}))$
is an isomorphism.
Proof. 1) Surjectivity. Using the t-regularity argument, we see that every
element of $\pi_{d+2klmn}({\rm T}(\vartheta_{km,\,ln}))$ comes from some
triple $(M,\,\xi_{km},\,\eta_{ln}),\;\dim M=d.$ Since, according to
Proposition 3 the inclusion $\mathop{\rm
Gr}\nolimits_{k,\,l}\rightarrow\mathop{\rm Gr}\nolimits_{km,\,ln}$ for
$(km,\,ln)=1$ is a homotopy equivalence up to dimension $2\min\\{k,\,l\\}$, we
see that for $d<2\min\\{k,\,l\\}$ a classifying map $M\rightarrow\mathop{\rm
Gr}\nolimits_{km,\,ln}$ for the pair $(\xi_{km},\,\eta_{ln})$ comes from some
map $M\rightarrow\mathop{\rm Gr}\nolimits_{k,\,l},$ i.e. the triple
$(M,\,\xi_{km},\,\eta_{ln})$ comes from some triple $(M,\,\xi_{k},\,\eta_{l})$
as described above.
2) Injectivity. Given a homotopy between two maps $S^{d+2klmn}\rightarrow{\rm
T}(\vartheta_{km,\,ln})$ we have the corresponding bordism given by a
$d+1$-dimensional manifold with boundary. Using the same argument as in item
1), we see that already the corresponding maps $S^{d+N}\rightarrow{\rm
T}(\vartheta_{k,\,l})$ are homotopic.$\quad\square$
###### Remark 8.
Note that for $d<2\min\\{m,\,n\\}$ we also have an isomorphism
$\pi_{d+2mn}({\rm T}(\vartheta_{m,\,n}))\rightarrow\pi_{d+2klmn}({\rm
T}(\vartheta_{km,\,ln}))$. Hence the group $\pi_{d+2mn}({\rm
T}(\vartheta_{m,\,n}))$ does not depend on the choice of $m,\,n,\;(m,\,n)=1.$
So, the bordism group $\Omega^{d}$ can be defined as the direct limit
$\lim\limits_{\longrightarrow\atop{(k,\,l)=1}}\pi_{d+2kl}({\rm
T}(\vartheta_{k,\,l}))$ which is stabilized from some dimension.
## 6\. The ring structure
Let
$(M,\,\xi_{k},\,\eta_{l}),\;(M^{\prime},\,\xi^{\prime}_{m},\,\eta^{\prime}_{n})$
be two triples as above. Then
$(M,\,\xi_{k},\,\eta_{l})\times(M^{\prime},\,\xi^{\prime}_{m},\,\eta^{\prime}_{n}):=(M\times
M^{\prime},\,\xi_{k}\boxtimes\xi^{\prime}_{m},\,\eta_{l}\boxtimes\eta^{\prime}_{n})$
is a new triple of the same kind (here “$\boxtimes$” denotes the “exterior”
tensor product of bundles).
If $f\colon S^{d+2kl}\rightarrow{\rm T}(\vartheta_{k,\,l}),\;f^{\prime}\colon
S^{d^{\prime}+2mn}\rightarrow{\rm T}(\vartheta_{m,\,n})$ classify triples
$(M,\,\xi_{k},\,\eta_{l}),\;(M^{\prime},\,\xi^{\prime}_{m},\,\eta^{\prime}_{n})$
respectively, then the triple $(M\times
M^{\prime},\,\xi_{k}\boxtimes\xi^{\prime}_{m},\,\eta_{l}\boxtimes\eta^{\prime}_{n})$
is classified by some map $S^{d+d^{\prime}+2klmn}\rightarrow{\rm
T}(\vartheta_{km,\,ln})$.
Note that the stabilization introduced in the previous section corresponds to
the product by
$(M^{\prime},\,\xi^{\prime}_{m},\,\eta^{\prime}_{n})=(\mathop{\rm
pt}\nolimits,\,\mathbb{C}^{m},\,\mathbb{C}^{n}).$
It is easy to see that the introduced product of triples defines the structure
of a graded ring on their bordism classes, moreover (because of the $H$-space
isomorphism $\mathop{\rm Gr}\nolimits\cong\mathop{\rm
BSU}\nolimits_{\otimes}$) it coincides with the one introduced in Section 1 on
the bordism group of pairs $(M,\,\xi),\;\xi\in\mathop{\rm
KSU}\nolimits(M),\quad\dim\xi=1.$
## 7\. The calculation of the ring $\Omega^{*}\otimes\mathbb{Q}$
In this section we compute the ring $\Omega^{*}\otimes\mathbb{Q}.$ First let
us formulate two obvious corollaries from the classical Theorems [2].
###### Theorem 9.
Since the trivial bundle $\vartheta_{k,\,l}\rightarrow\mathop{\rm
Gr}\nolimits_{k,\,l}$, clearly, is orientable, we have the Thom isomorphism
$H_{d}(\mathop{\rm
Gr}\nolimits_{k,\,l},\,\mathbb{Z})\stackrel{{\scriptstyle\cong}}{{\rightarrow}}H_{d+2kl}({\rm
T}(\vartheta_{k,\,l}),\,\mathbb{Z})$.
###### Theorem 10.
Since the Thom space ${\rm T}(\vartheta_{k,\,l})$ is $(2kl-1)$-connected, we
see that the Hurewicz homomorphism $\pi_{d+2kl}({\rm
T}(\vartheta_{k,\,l}))\rightarrow H_{d+2kl}({\rm
T}(\vartheta_{k,\,l}),\,\mathbb{Z})$ is a $\mathcal{C}$-isomorphism for
$d<2kl-1.$ Here $\mathcal{C}$ is the Serre class of finite abelian groups.
Since the space $\mathop{\rm Gr}\nolimits_{k,\,l}$ is homotopy equivalent to
$\mathop{\rm BSU}\nolimits$ up to dimension $\sim 2\min\\{k,\,l\\}$, we see
that in this dimensions ${\rm rk}H_{d}(\mathop{\rm
Gr}\nolimits_{k,\,l},\,\mathbb{Z})$ is equal to $0$ for $d$ odd and the number
of partitions $\frac{d}{2}$ into the sum of $2,\,3,\,4,\ldots$ for $d$ even.
Actually, we will show that
$\Omega^{*}\otimes\mathbb{Q}\cong\mathbb{Q}[t_{2},\,t_{3},\,t_{4},\ldots],$
where $\deg t_{n}=2n.$ Moreover, one can take the bordism class of the triple
$(S^{2n},\,\xi_{k},\,\eta_{l}),\;n<\min\\{k,\,l\\}$ as $t_{n}$, where
$(\xi_{k},\,\eta_{l})$ is the generator (i.e. its classifying map
$S^{2n}\rightarrow\mathop{\rm Gr}\nolimits_{k,\,l}$ represents the generator
in $\pi_{2n}(\mathop{\rm Gr}\nolimits_{k,\,l})\cong\mathbb{Z}$). In other
words, the pair $(\xi_{k},\,\eta_{l})$ corresponds to the generator
$\xi\in\mathop{\rm KSU}\nolimits(S^{2n}),\;\dim\xi=1$ according to the
correspondence described in Section 2.
In order to calculate $\Omega^{*}\otimes\mathbb{Q}$ it is sufficient to
consider only rational characteristic classes. By analogy with Pontryagin’s
theorem, we can prove the following result:
###### Theorem 11.
For a pair $(M,\,\xi)$ as in Section 1 and an arbitrary characteristic class
$\alpha(\xi)\in H^{d}(M,\,\mathbb{Q})$ of the bundle $\xi$ the characteristic
number $\langle\alpha(\xi),\,[M]\rangle\in\mathbb{Q}$ (where $[M]\in
H_{d}(M,\,\mathbb{Q})$ is the fundamental homology class of the manifold $M$)
depends only on the bordism class $[M,\,\xi]$.
Consider a pair $(M,\,\xi)$ as above and the Chern character
$ch(\xi)=1+ch_{2}(\xi)+ch_{3}(\xi)+\ldots,\;ch_{n}(\xi)\in
H^{2n}(M,\,\mathbb{Q})$ ($ch_{1}(\xi)=0$ because $\xi$ is a virtual
$\mathop{\rm SU}\nolimits$-bundle).
###### Remark 12.
If a pair $(M,\,\xi)$ corresponds to a triple $(M,\,\xi_{k},\,\eta_{l}),$ then
$ch(\xi)=\frac{ch(\xi_{k})}{k}.$ Note that
$\frac{ch(\xi_{k})}{k}=\frac{ch(\xi_{m})}{m}$ if pairs
$(\xi_{k},\,\eta_{l}),\;(\xi_{m},\,\eta_{n})$ are equivalent in the sense that
their classifying maps $M\rightarrow\mathop{\rm
Gr}\nolimits_{k,\,l},\;M\rightarrow\mathop{\rm Gr}\nolimits_{m,\,n}$ are
homotopic as maps to $\mathop{\rm Gr}\nolimits_{km,\,ln}$, see (7).
Let $\\{(S^{2n},\,\xi^{(n)})\mid n\geq 2\\}$ be the collection of pairs such
that $ch_{n}(\xi^{(n)})=\iota_{n},$ where $\iota_{n}\in
H^{2n}(S^{2n},\,\mathbb{Z})\subset H^{2n}(S^{2n},\,\mathbb{Q})$ is the
generator (recall that the Chern character takes integer values on spheres).
Then elements $\xi^{(n)}\in(1+\widetilde{\mathop{\rm
KSU}\nolimits}(S^{2n}))^{\times}$ themselves are generators (note that
$(1+\widetilde{\mathop{\rm KSU}\nolimits}(S^{2n}))^{\times}\cong\mathbb{Z}$).
Let $\xi^{(n)k}$ be the $k$’th power of the bundle $\xi^{(n)},$ then
$ch_{n}(\xi^{(n)k})=k\iota_{n}.$ (Indeed,
$\xi^{(n)}=1+\widetilde{\xi}^{(n)},\;(1+\widetilde{\xi}^{(n)})^{k}=1+k\widetilde{\xi}^{(n)},$
because $\widetilde{\xi}^{(n)2}=0$ in the ring $\widetilde{\mathop{\rm
KSU}\nolimits}(S^{2n})$).
###### Proposition 13.
$[S^{2n},\,\xi^{(n)k}]=k[S^{2n},\,\xi^{(n)}]$ in the group
$\Omega^{2n}\otimes\mathbb{Q}.$
Proof. We have: $\langle ch_{n}(\xi^{(n)k}),\,[S^{2n}]\rangle=k=k\langle
ch_{n}(\xi^{(n)}),\,[S^{2n}]\rangle$. From the other hand, $H^{*}(\mathop{\rm
BSU}\nolimits,\,\mathbb{Q})=\mathbb{Q}[ch_{2},\,ch_{3},\,\ldots]$, hence the
products of the form $ch_{n_{1}}\ldots ch_{n_{r}},\;2\leq n_{1}\leq\ldots\leq
n_{r},\;n_{1}+\ldots+n_{r}=n,\;r\geq 1$ form an additive basis of
$H^{2n}(\mathop{\rm BSU}\nolimits,\,\mathbb{Q}).$ The required assertion now
follows from Theorems 9 and 10, additivity of characteristic numbers (with
respect to the addition of bordism classes) and from the fact that
$ch_{m}(\xi^{(n)k})=0$ for $m\neq n.\quad\square$
Note that the existence of a bordism
$(S^{2n},\,\xi^{(n)k})\bigsqcup(S^{2n},\,\xi^{(n)l})\sim(S^{2n},\,\xi^{(n)k+l})$
in $\Omega^{2n}$ can be perceived from a geometric argument.
Note also that for the inverse element
$-[S^{2n},\,\xi^{(n)}]=[-S^{2n},\,\xi^{(n)}]$ we have $\langle
ch_{n}(\xi^{(n)}),\,[-S^{2n}]\rangle=-1=\langle
ch_{n}(\xi^{(n)(-1)}),\,[S^{2n}]\rangle,$ where $\xi^{(n)(-1)}$ is the inverse
element for $\xi^{(n)}$ in the group $(1+\widetilde{\mathop{\rm
KSU}\nolimits}(S^{2n}))^{\times}.$ This is connected with the existence of the
orientation-reversing diffeomorphism (for instance, the antipodal map)
$f\colon S^{2n}\rightarrow S^{2n}$ such that $\xi^{(n)(-1)}\cong
f^{*}(\xi^{(n)})$. Thus, $-[S^{2n},\,\xi^{(n)}]=[S^{2n},\,\xi^{(n)(-1)}]$.
Now we want to prove that the classes $[S^{2n_{1}}\times\ldots\times
S^{2n_{r}},\,\xi^{(n_{1})}\boxtimes\ldots\boxtimes\xi^{(n_{r})}]=[S^{2n_{1}},\,\xi^{(n_{1})}]\ldots[S^{2n_{r}},\,\xi^{(n_{r})}],\;2\leq
n_{1}\leq\ldots\leq n_{r},\;n_{1}+\ldots+n_{r}=n,\;r\geq 1$ form an additive
basis of $\Omega^{2n}\otimes\mathbb{Q}.$
###### Proposition 14.
$\langle ch_{m_{1}}\ldots
ch_{m_{s}}(\xi^{(n_{1})}\boxtimes\ldots\boxtimes\xi^{(n_{r})}),\;[S^{2n_{1}}\times\ldots\times
S^{2n_{r}}]\rangle\neq 0$ only if the partition $n_{1}\ldots n_{r}$ of $n$ is
a refinement of the partition $m_{1}\ldots m_{s}.$
Proof. We have:
(8)
$ch(\xi^{(n_{1})}\boxtimes\ldots\boxtimes\xi^{(n_{r})})=(1+\iota_{n_{1}})\otimes\ldots\otimes(1+\iota_{n_{r}}),$
whence $ch_{m}(\xi^{(n_{1})}\boxtimes\ldots\boxtimes\xi^{(n_{r})})$ is the
degree $2m$ homogeneous component of the right-hand side of (8). Multiplying
the obtained expressions, we get the required assertion.$\quad\square$
Let $p^{\prime}(n)$ be the partition number of writing $n$ as a sum of numbers
$2,\,3,\,\ldots,\,n$ (with 1 omitted).
###### Theorem 15.
(cf. [2]) $p^{\prime}(n)\times p^{\prime}(n)$-matrix consisting of
characteristic numbers
$\langle ch_{m_{1}}\ldots
ch_{m_{s}}(\xi^{(n_{1})}\boxtimes\ldots\boxtimes\xi^{(n_{r})}),\,[S^{2n_{1}}\times\ldots\times
S^{2n_{r}}]\rangle,$
where $m_{1}\ldots m_{s}$ and $n_{1}\ldots n_{r}$ runs over all partitions of
$n$ into a sum of positive integers $\neq 1$ is nonsingular.
Proof. There is a partial order on the set of partitions of $n$ defined by
refinement. Extending it to a total order, we obtain the corresponding
$p^{\prime}(n)\times p^{\prime}(n)$-matrix consisting of numbers as in the
statement of the theorem. According to the previous proposition, this matrix
is a lower triangular with zeros over the main diagonal, while its diagonal
elements
$\langle ch_{n_{1}}\ldots
ch_{n_{r}}(\xi^{(n_{1})}\boxtimes\ldots\boxtimes\xi^{(n_{r})}),\,[S^{2n_{1}}\times\ldots\times
S^{2n_{r}}]\rangle$
clearly are nonzero. Hence the asserted nonsingularity.$\quad\square$
###### Example 16.
Take $n=6$. We have $4$ partitions which we order as follows:
$(2\,2\,2),\;(3\,3),\;(2\,4),\;6.$ For
$\xi^{(2)}\boxtimes\xi^{(2)}\boxtimes\xi^{(2)}$ over $S^{4}\times S^{4}\times
S^{4}$ we have:
$ch(\xi^{(2)}\boxtimes\xi^{(2)}\boxtimes\xi^{(2)})$
$=(1+\iota_{2})\otimes(1+\iota_{2})\otimes(1+\iota_{2})=1\otimes 1\otimes
1+\iota_{2}\otimes 1\otimes 1+1\otimes\iota_{2}\otimes 1+1\otimes
1\otimes\iota_{2}+$ $+\iota_{2}\otimes\iota_{2}\otimes 1+\iota_{2}\otimes
1\otimes\iota_{2}+1\otimes\iota_{2}\otimes\iota_{2}+\iota_{2}\otimes\iota_{2}\otimes\iota_{2},$
whence
$ch_{2}(\xi^{(2)}\boxtimes\xi^{(2)}\boxtimes\xi^{(2)})=\iota_{2}\otimes
1\otimes 1+1\otimes\iota_{2}\otimes 1+1\otimes 1\otimes\iota_{2};$
$ch_{4}(\xi^{(2)}\boxtimes\xi^{(2)}\boxtimes\xi^{(2)})=\iota_{2}\otimes\iota_{2}\otimes
1+\iota_{2}\otimes 1\otimes\iota_{2}+1\otimes\iota_{2}\otimes\iota_{2};$
$ch_{6}(\xi^{(2)}\boxtimes\xi^{(2)}\boxtimes\xi^{(2)})=\iota_{2}\otimes\iota_{2}\otimes\iota_{2};$
and
$ch_{3}(\xi^{(2)}\boxtimes\xi^{(2)}\boxtimes\xi^{(2)})=0=ch_{5}(\xi^{(2)}\boxtimes\xi^{(2)}\boxtimes\xi^{(2)}).$
We have:
$ch_{2}ch_{2}ch_{2}=3!\iota_{2}\otimes\iota_{2}\otimes\iota_{2}=6\iota_{2}\otimes\iota_{2}\otimes\iota_{2};$
$ch_{2}ch_{4}=3\iota_{2}\otimes\iota_{2}\otimes\iota_{2};\quad
ch_{6}=\iota_{2}\otimes\iota_{2}\otimes\iota_{2}$
$(ch_{n}:=ch_{n}(\xi^{(2)}\boxtimes\xi^{(2)}\boxtimes\xi^{(2)})).$ Therefore
the corresponding characteristic numbers are $6,\,3,\,1$ respectively.
Reasoning in this way we obtain the following table of characteristic numbers:
(9) $\begin{array}[]{ccccc}&S^{4}\times S^{4}\times S^{4}&S^{6}\times
S^{6}&S^{4}\times S^{8}&S^{12}\\\ 2\,2\,2&6&0&0&0\\\ 3\,3&0&2&0&0\\\
2\,4&3&0&1&0\\\ 6&1&1&1&1\\\ \end{array}$
Note that, in particular, the class $[S^{2m}\times
S^{2n},\,\xi^{(m)k}\boxtimes\xi^{(n)}]$ is equal to the class $[S^{2m}\times
S^{2n},\,\xi^{(m)}\boxtimes\xi^{(n)k}]$ in $\Omega^{2(m+n)}\otimes\mathbb{Q}.$
###### Corollary 17.
The bordism classes $[S^{2n_{1}}\times\ldots\times
S^{2n_{r}},\,\xi^{(n_{1})}\boxtimes\ldots\boxtimes\xi^{(n_{r})}]=[S^{2n_{1}},\,\xi^{(n_{1})}]\ldots[S^{2n_{r}},\,\xi^{(n_{r})}],\;2\leq
n_{1}\leq\ldots\leq n_{r},\;n_{1}+\ldots+n_{r}=n,\;r\geq 1$ form an additive
basis of $\Omega^{2n}\otimes\mathbb{Q}.$
Put $t_{n}:=[S^{2n},\,\xi^{(n)}],\;\deg t_{n}=2n.$ The previous results imply
the following theorem:
###### Theorem 18.
The graded algebra $\Omega^{*}\otimes\mathbb{Q}$ is isomorphic to the
polynomial algebra $\mathbb{Q}[t_{2},\,t_{3},\,t_{4},\ldots],$ where $\deg
t_{n}=2n.$
Note that $[S^{4},\,\xi^{(2)}]$ is a nondivisible element of $\Omega^{4}$
because $ch_{2}$ on $\mathop{\rm SU}\nolimits$-bundles coincides with the
second Chern class $c_{2}$ and therefore $ch_{2}\in H^{4}(\mathop{\rm
BSU}\nolimits,\,\mathbb{Z})$, while $\langle
ch_{2}(\xi^{(2)}),\,S^{4}\rangle=1.$
## References
* [1] E.E. Floyd: Bordism groups of bundles. (In the book R.S. Palais “Seminar on the Atiyah-Singer Index Theorem”, Princeton University Press, 1965).
* [2] J.W. Milnor, J.D. Stasheff: Characteristic Classes. Princeton, New Jersey, 1974.
* [3] A.V. Ershov: Homotopy theory of bundles with fiber matrix algebra // arXiv:math/0301151v2 [math.AT]
|
arxiv-papers
| 2010-10-04T17:29:23 |
2024-09-04T02:49:13.449963
|
{
"license": "Public Domain",
"authors": "A.V. Ershov",
"submitter": "Andrey V. Ershov",
"url": "https://arxiv.org/abs/1010.0646"
}
|
1010.0713
|
arxiv-papers
| 2010-10-04T21:48:48 |
2024-09-04T02:49:13.459172
|
{
"license": "Public Domain",
"authors": "Junbai Wang, Lucas D. Ward, and Harmen J. Bussemaker",
"submitter": "Junbai Wang",
"url": "https://arxiv.org/abs/1010.0713"
}
|
|
1010.0760
|
# Thermodynamic properties of hot nuclei within
the self-consistent quasiparticle random-phase approximation
N. Quang Hung1 nqhung@riken.jp N. Dinh Dang2,3 dang@riken.jp 1) Center for
Nuclear Physics, Institute of Physics, Hanoi, Vietnam
2) Heavy-Ion Nuclear Physics Laboratory, RIKEN Nishina Center for Accelerator-
Based Science, 2-1 Hirosawa, Wako City, 351-0198 Saitama, Japan
3) Institute for Nuclear Science and Technique, Hanoi, Vietnam
###### Abstract
The thermodynamic properties of hot nuclei are described within the canonical
and microcanonical ensemble approaches. These approaches are derived based on
the solutions of the BCS and self-consistent quasiparticle random-phase
approximation at zero temperature embedded into the canonical and
microcanonical ensembles. The obtained results agree well with the recent data
extracted from experimental level densities by Oslo group for 94Mo, 98Mo,
162Dy and 172Yb nuclei.
Suggested keywords
###### pacs:
21.60.-n, 21.60.Jz, 24.60.-k, 24.10.Pa
## I INTRODUCTION
Thermodynamic properties of highly excited (hot) nuclei have been a topic of
much interest in nuclear physics. From the theoretical point of view,
thermodynamic properties of any systems can be studied by using three
principal statistical ensembles, namely the grand canonical ensemble (GCE),
canonical ensemble (CE) and microcanonical ensemble (MCE). The GCE is an
ensemble of identical systems in thermal equilibrium, which exchange their
energies and particle numbers with the external heat bath. In the CE, the
systems exchange only their energies, whereas their particle numbers are kept
to be the same for all systems. The MCE describes thermally isolated systems
with fixed energies and particle numbers. For convenience, the GCE is often
used in most of theoretical approaches, e.g. the conventional finite-
temperature BCS (FTBCS) theory BCS , and/or finite-temperature Hartree-Fock-
Bogoliubov theory HFB . These theories, however, fail to describe
thermodynamic properties of finite small systems such as atomic nuclei or
ultra-small metallic grains. The reason is that the FTBCS neglects the quantal
and thermal fluctuations, which have been shown to be very important in finite
systems Moretto ; SPA ; Zele ; MBCS ; FTBCS1 ; Ensemble . These fluctuations
smooth out the superfluid-normal (SN) phase transition, which is a typical
feature of infinite systems as predicted by the FTBCS theory.
Because an atomic nucleus is a system with the fixed particle number, the
particle-number fluctuations are obviously not allowed. The use of the GCE in
nuclear systems is therefore an approximation, which is good so long as the
effect caused by particle-number fluctuations are negligible. The CE and MCE
are often used in extending the exact solutions of the pairing Hamiltonian
Ensemble ; Sumaryada ; Exact to finite temperature, whereas the CE is
preferred in the quantum Monte-Carlo calculations at finite temperature
(FTQMC) QMC ; QMC1 . However, it is impracticable to find all the exact
eigenvalues of the pairing Hamiltonian to construct the exact partition
functions for large systems. For instance, in the half-filled doubly-folded
multilevel model (also called the Richardson model) with $N=\Omega$ with
$\Omega$ being the number of single-particle levels and $N$ \- the number of
particles, this cannot be done already for $N>$ 14 Ensemble ; Sumaryada .
Meanwhile, the FTQMC is quite time consuming and cannot be applied to heavy
nuclei unless a limited configuration space is picked up. It is worth
mentioning that the pairing Hamiltonian can also be solved exactly by using
the Richardson’s method, i.e. solving the Richardson equations. Using this
method, the lowest eigenvalues of the pairing Hamiltonian can be obtained even
for very large systems, e.g. with $N=\Omega$ = 1000 (See, e.g., Ref. Dukelsky
). Nonetheless, these lowest eigenstates (obtained after solving the
Richardson equations) are not sufficient for the construction of the exact
partition function at finite temperature since the latter should contain all
the excited states, not only the lowest ones. In principle, the CE-based
approaches can also be derived from the exact particle-number projection (PNP)
at finite temperature on top of the GCE ones FTPNP . However, this method is
rather complicated for application to realistic nuclei.
The static path plus random phase approximation (SPA + RPA) with the exact
number parity projection CSPA(p) CSPA and the latter extension of the number
projected SPA (NPSPA) NPSPA offer quite good agreement with the exact CE of
the Richardson model as well as the empirical heat capacities of heavy nuclei.
However, Ref. CSPA makes no comparison with experimental data, whereas Ref.
NPSPA uses a thermal pairing gap, which is obtained from a direct extension
of the odd-even mass difference to finite temperature. As has been pointed in
Ref. Ensemble such simple extension fails in the region of intermediate and
high temperatures. In principle, the SPA can also be used to evaluate the MCE
quantities based on the GCE ones by fixing the energy and particle number of
the system MCESPA . However this method is still quite complicated for
practical applications to realistic nuclei, especially the heavy ones. From
the experimental point of view, the CE and MCE are usually used to extract
various thermodynamic quantities of nuclear systems. This is carried out by
using the nuclear level density, which can be experimentally measured at low
excitation energy $E^{*}<$ 10 MeV. Within the CE, the measured level densities
are first extrapolated to high $E^{*}$ using the back-shifted Fermi-gas model
(BSFG). The CE partition function is then constructed making use of the
Laplace transformation of the level density. Knowing the partition function,
one can calculate all the thermodynamic quantities within the CE such as the
free energy, total energy, heat capacity and entropy. The thermodynamic
quantities of the systems obtained within the MCE are calculated via the
Boltzmann’s definition of entropy. Although several experimental data for
nuclear thermodynamic quantities extracted in this way by the Oslo group have
recently been reported Oslo ; Oslo1 ; Chankova ; Kaneko , most of present
theoretical approaches, derived within the GCE, cannot describe well these
data, which are extracted within the CE and MCE. Recently we have proposed a
method, which has allowed us to construct theoretical approaches within the CE
and MCE to describe rather well thermodynamic properties of atomic nuclei CE-
BCS . The proposed approaches are derived by solving the BCS and self-
consistent quasiparticle RPA (SCQRPA) equations with the Lipkin-Nogami (LN)
PNP for each total seniority $S$ (number of unpaired particles at zero
temperature) SCQRPA . The obtained results are then embedded into the CE and
MCE. Within the CE, the resulting approaches are called the CE-LNBCS and CE-
LNSCQRPA, whereas they are called the MCE-LNBCS and MCE-LNSCQRPA within the
MCE. The results obtained within these approaches are found in quite good
agreement with not only the exact solutions of the Richardson model but also
the experimentally extracted data for 56Fe isotope. The merit of these
approaches reside in their simplicity and feasibility in the application even
to heavy nuclei, where the exact solution is impracticable and the FTQMC is
time consuming. The goal of present article is to apply the above-mentioned
approaches to microscopically describe the recently extracted thermodynamic
quantities of 94,96Mo, 162Dy and 172Yb nuclei.
The article is organized as follows. The pairing Hamiltonian and the
derivations of the GCE-BCS, CE(MCE)-LNBCS and CE(MCE)-LNSCQRPA are presented
in Sec. II. The numerical results are analyzed and discussed in Sec. III,
whereas the conclusions are drawn in the last section.
## II FORMALISM
### II.1 Pairing Hamiltonian
The present article considers the pairing Hamiltonian
$H=\sum_{k\sigma=\pm}\epsilon_{k}a_{k\sigma}^{\dagger}a_{k\sigma}-G\sum_{kk^{\prime}}a_{k+}^{\dagger}a_{k-}^{\dagger}a_{k^{\prime}-}a_{k^{\prime}+}~{},$
(1)
where $a_{k\sigma}^{\dagger}$ and $a_{k\sigma}$ are particle creation and
destruction operators on the $k$th orbitals, respectively. The subscripts $k$
here imply the single-particle states in deformed basis. This Hamiltonian
describes a system of $N$ particles (protons or neutrons) interacting via a
monopole pairing force with constant parameter $G$. The pairing Hamiltonian
(1) can be diagonalized exactly by using the SU(2) algebra of angular momentum
Exact . At finite temperature $T\neq 0$, the exact diagonalization is done for
all total seniority or number of unpaired particles $S$ because all excited
states should be included in the exact partition function. Here $S=$ 0, 2,
$\ldots$ $N$ for even-$N$ systems, and $S=$ 1, 3, $\ldots$ $N$-1 for odd-$N$
systems. For a system of $N$ particles moving in $\Omega$ degenerated single-
particle levels, the number $n_{\rm Exact}$ of exact eigenstates ${\cal
E}_{i_{S}}^{\rm Exact}$ ($i_{S}=$1, …, $n_{\rm Exact}$) obtained within exact
diagonalization is given as
$n_{\rm Exact}=\sum_{S}{\rm C}_{S}^{\Omega}\times{\rm C}_{N_{\rm
pair}-\frac{S}{2}}^{\Omega-S}~{},$ (2)
which combinatorially increases with $N$, where $C_{n}^{m}=m!/[n!(m-n)!]$ and
$N_{\rm pair}=N/2$ Ensemble . Therefore, the exact solution at $T\neq 0$ is
impossible for large $N$ systems, e.g. $N>$ 14 for the half-filled case
($N=\Omega$), because of the huge size of the matrix to be diagonalized.
### II.2 GCE-BCS
The well-known finite-temperature BCS (FTBCS) approach to the pairing
Hamiltonian (1) is derived based on the variational procedure, which minimizes
the grand potential
$\Omega=\langle{H}\rangle-T{\cal S}-\lambda N\hskip 14.22636pt\rm{so\hskip
5.69054ptthat}\hskip 14.22636pt\delta\Omega=0~{},$ (3)
where ${\cal S}$ is the entropy of the system at temperature $T$. The chemical
potential $\lambda$ is a Lagrangian multiplier, which can be obtained from the
equation that maintain the expectation value of the particle-number operator
to be equal to the particle number $N$. The expectation value $\langle{\cal
O}\rangle$ denotes the GCE average of the operator ${\cal O}$ MBCS (the
Boltzmann’s constant $k_{B}$ is set to 1),
$\langle{\cal O}\rangle\equiv\frac{{\rm Tr}[{\cal O}e^{-\beta(H-\lambda
N)}]}{{\rm Tr}e^{-\beta(H-\lambda N)}}~{},\hskip
14.22636pt\beta=\frac{1}{T}~{}.$ (4)
The conventional FTBCS equations for the pairing gap $\Delta$ and particle
number $N$ are then given as
$\Delta=G\sum_{k}u_{k}v_{k}(1-2n_{k})~{},\hskip
14.22636ptN=2\sum_{k}\left[(1-2n_{k})v_{k}^{2}+n_{k}\right]~{},$ (5)
where the Bogoliubov’s coefficients $u_{k}$, $v_{k}$, the quasiparticle energy
$E_{k}$ and the quasiparticle occupation number $n_{k}$ have the usual form as
$u_{k}^{2}=\frac{1}{2}\left(1+\frac{\epsilon_{k}-Gv_{k}^{2}-\lambda}{E_{k}}\right)~{},\hskip
14.22636ptv_{k}^{2}=\frac{1}{2}\left(1-\frac{\epsilon_{j}-Gv_{k}^{2}-\lambda}{E_{k}}\right)~{}.$
$E_{k}=\sqrt{(\epsilon_{k}-Gv_{k}^{2}-\lambda)^{2}+\Delta^{2}}~{},\hskip
14.22636ptn_{k}=\frac{1}{1+e^{\beta E_{k}}}~{}.$ (6)
The systems of Eqs. (5) and (6) are called the GCE-BCS equations. The total
energy, heat capacity and entropy obtained within the GCE-BCS are given as
${\cal
E}=2\sum_{k}\left[(1-2n_{k})v_{k}^{2}+n_{k}\right]-\frac{\Delta^{2}}{G}-G\sum_{k}(1-2n_{k})v_{k}^{4}~{},$
$C=\frac{\partial{\cal E}}{\partial T}~{},\hskip 14.22636pt{\cal
S}=-2\sum_{k}\left[n_{k}{\rm ln}n_{k}+(1-n_{k}){\rm ln}(1-n_{k})\right]~{}.$
(7)
### II.3 CE-LNBCS
Different from the GCE-BCS, the CE-LNBCS is derived based on the solutions of
the BCS equations combined with the Lipkin-Nogami particle-number projection
(PNP) LN at $T=0$ for each total seniority $S$ of the system. When the pairs
are broken, the unpaired particles denoted with the quantum numbers $k_{S}$
block the single-particle levels $k$. As the result, these blocked single-
particle levels do not contribute to the pairing correlation. Therefore, the
Lipkin-Nogami BCS (LNBCS) equations at $T=0$ can be derived by excluding these
$k_{S}$ blocked levels. These equations are given as
$\Delta^{\rm LNBCS}(k_{S})=G\sum_{k\neq k_{S}}u_{k}v_{k},\hskip
14.22636ptN=2\sum_{k\neq k_{s}}v_{k}^{2}+S~{},$ (8)
where
$u_{k\neq
k_{S}}^{2}=\frac{1}{2}\left(1+\frac{\epsilon_{k}-Gv_{k}^{2}-\lambda(k_{S})}{E_{k}}\right)~{},\hskip
14.22636ptv_{k\neq
k_{S}}^{2}=\frac{1}{2}\left(1-\frac{\epsilon_{k}-Gv_{k}^{2}-\lambda(k_{S})}{E_{k}}\right)~{},$
(9) $E_{k\neq
k_{S}}=\sqrt{[\epsilon_{k}-Gv_{k}^{2}-\lambda(k_{S})]^{2}+[\Delta^{\rm
LNBCS}(k_{S})]^{2}}~{},$ (10)
$\lambda(k_{S})=\lambda_{1}(k_{S})+2\lambda_{2}(k_{S})(N+1)~{},\hskip
14.22636pt\lambda_{2}(k_{S})=\frac{G}{4}\frac{\sum_{k\neq
k_{S}}u_{k}^{3}v_{k}\sum_{k^{\prime}\neq
k^{\prime}_{S}}u_{k^{\prime}}v_{k^{\prime}}^{3}-\sum_{k\neq
k_{S}}u_{k}^{4}v_{k}^{4}}{(\sum_{k\neq
k_{S}}u_{k}^{2}v_{k}^{2})^{2}-\sum_{k\neq k_{S}}u_{k}^{4}v_{k}^{4}}~{}.$ (11)
As for the blocked single-particle levels, $k=k_{S}$, their occupation numbers
are always equal to $1/2$. Solving the systems of Eqs. (8) - (11), one obtains
the pairing gap $\Delta^{\rm LNBCS}(k_{S})$, quasiparticle energies $E_{k}$
and Bogoliubov’s coefficients $u_{k}$ and $v_{k}$, which correspond to each
position of unpaired particles on blocked levels $k_{S}$ at each value of the
total seniority $S$. There are $n_{\rm LNBCS}=\sum_{S}C_{S}^{\Omega}$
configurations of $k_{S}$ levels distributed amongst $\Omega$ single-particle
levels at each value of $S$, which is also the number of eigenstates obtained
within the LNBCS. The LNBCS energy (eigenvalue) ${\cal E}_{i_{S}}^{\rm LNBCS}$
for each configuration is then given as
${\cal E}_{i_{S}}^{\rm LNBCS}=2\sum_{k\neq
k_{S}}{\epsilon_{k}v_{k}^{2}}+\sum_{k_{S}}\epsilon_{k_{S}}-\frac{[\Delta^{\rm
LNBCS}(k_{S})]^{2}}{G}-G\sum_{k\neq
k_{S}}v_{k}^{4}-4\lambda_{2}(k_{S})\sum_{k\neq k_{S}}u_{k}^{2}v_{k}^{2}~{}.$
(12)
The partition function of the so-called CE-LNBCS approach is constructed by
using the LNBCS eigenvalues ${\cal E}_{i_{S}}^{\rm LNBCS}$ as CE-BCS
$Z_{\rm LNBCS}(\beta)=\sum_{S}d_{S}\sum_{i_{S}=1}^{n_{\rm
LNBCS}}{e^{-\beta{\cal E}_{i_{S}}^{\rm LNBCS}}}~{},$ (13)
where $d_{S}=2^{S}$ is the degeneracy. Knowing the partition function (13), we
can calculate all thermodynamic quantities of the system such as the free
energy $F$, entropy ${\cal S}$, total energy ${\cal E}$, and heat capacity $C$
as follows
$F=-T{\rm ln}Z(T),\hskip 14.22636pt{\cal S}=-\frac{\partial F}{\partial
T}~{},\hskip 14.22636pt{\cal E}=F+T{\cal S},\hskip
14.22636ptC=\frac{\partial{\cal E}}{\partial T}~{}.$ (14)
The pairing gap is obtained by averaging the seniority-dependent gaps
$\Delta_{i_{S}}^{\rm LNBCS}=\Delta^{\rm LNBCS}(k_{S})$ at $T=0$ in the CE by
means of the CE-LNBCS partition function (13), namely
$\Delta_{\rm CE-LNBCS}=\frac{1}{Z_{\rm
LNBCS}}\sum_{S}d_{S}\sum_{i_{S}}^{n_{\rm LNBCS}}{\Delta^{\rm
LNBCS}_{i_{S}}e^{-\beta{\cal E}_{i_{S}}^{\rm LNBCS}}}~{}.$ (15)
### II.4 CE-LNSCQRPA
As mentioned previously in sec. II.1, a complete CE partition function should
include all eigenstates. The LNBCS theory (at $T=0$) produces only the lowest
eigenstates. For instance, for even (odd) $N$ there is only one state at $S=$
0, which is the ground state. For $S>$ 0 there are also excited states in even
(odd) systems, whose total number nLNBCS is much smaller than nExact.
Consequently, the results obtained within the CE-LNBCS can be compared with
the exact ones only at low $T$ because at high $T$, higher eigenstates
(excited states), which the LNBCS theory cannot reproduce, should be included
in the CE partition function. This can be done by going beyond the
quasiparticle mean field by introducing the SCQRPA with Lipkin-Nogami PNP
(LNSCQRPA), which incorporates not only the ground states but also the pairing
vibrational excited states predicted by the QRPA SCQRPA . The derivation of
the LNSCQRPA equations has been presented in details in Refs. FTBCS1 ; SCQRPA
; Chemical , so we do not repeat it here. The LNSCQRPA formalism at $T=0$ for
each total seniority $S$ is proceeded in the same way as that of the LNBCS
described in the previous section, namely the LNSCQRPA equations are derived
only for the unblocked levels $k\neq k_{S}$, whereas the levels, blocked by
the unpaired particles $k=k_{S}$, do not contribute to the pairing
Hamiltonian. The SCQRPA equations at $T=$ 0 has been derived in Ref. SCQRPA ,
whose final form reads
$\left(\begin{array}[]{cc}A&B\\\
B&A\end{array}\right)\left(\begin{array}[]{cc}X_{k}^{\nu}\\\
Y_{k}^{\nu}\end{array}\right)=\omega_{\nu}\left(\begin{array}[]{cc}X_{k}^{\nu}\\\
-Y_{k}^{\nu}\end{array}\right)~{},$ (16)
The SCQRPA submatrices are given as
$A_{kk^{\prime}}=2\bigg{[}b_{k}+2q_{kk^{\prime}}+2\sum_{k^{\prime\prime}}q_{kk^{\prime\prime}}(1-{\cal
D}_{k^{\prime\prime}})-\frac{1}{{\cal
D}_{k}}\bigg{(}\sum_{k^{\prime\prime}}d_{kk^{\prime\prime}}\langle\bar{0}|{\cal
A}_{k^{\prime\prime}}^{\dagger}{\cal A}_{k}|\bar{0}\rangle$
$-2\sum_{k^{\prime\prime}}h_{kk^{\prime\prime}}\langle\bar{0}|{\cal
A}_{k^{\prime\prime}}{\cal
A}_{k}|\bar{0}\rangle\bigg{)}\bigg{]}\delta_{kk^{\prime}}+d_{kk^{\prime}}\sqrt{{\cal
D}_{k}{\cal D}_{k^{\prime}}}+8q_{kk^{\prime}}\frac{\langle\bar{0}|{\cal
A}_{k}^{\dagger}{\cal A}_{k^{\prime}}|\bar{0}\rangle}{\sqrt{{\cal D}_{k}{\cal
D}_{k^{\prime}}}}~{},$ (17)
$B_{kk^{\prime}}=-2\bigg{[}h_{kk^{\prime}}+\frac{1}{{\cal
D}_{k}}\bigg{(}\sum_{k^{\prime\prime}}d_{kk^{\prime\prime}}\langle\bar{0}|{\cal
A}_{k^{\prime\prime}}{\cal
A}_{k}|\bar{0}\rangle+2\sum_{k^{\prime\prime}}h_{kk^{\prime\prime}}\langle\bar{0}|{\cal
A}_{k^{\prime\prime}}^{\dagger}{\cal
A}_{k}|\bar{0}\rangle\bigg{)}\bigg{]}\delta_{kk^{\prime}}$
$+2h_{kk^{\prime}}\sqrt{{\cal D}_{k}{\cal
D}_{k^{\prime}}}+8q_{kk^{\prime}}\frac{\langle\bar{0}|{\cal A}_{k}{\cal
A}_{k^{\prime}}|\bar{0}\rangle}{\sqrt{{\cal D}_{k}{\cal D}_{k^{\prime}}}}~{},$
(18)
where $b_{k}$, $d_{kk^{\prime\prime}}$, $h_{kk^{\prime\prime}}$, and
$q_{kk^{\prime}}$ (all $k\neq k_{S}$) are functions of $u_{k}$, $v_{k}$,
$\epsilon_{k}$, $\lambda$ and $G$ as given in Eqs. (13), (15), (17) and (18)
of Ref. SCQRPA . The screening factors
$\langle\bar{0}|\mathcal{A}_{k}^{\dagger}\mathcal{A}_{k^{\prime}}|\bar{0}\rangle$
and $\langle\bar{0}|\mathcal{A}_{k}\mathcal{A}_{k^{\prime}}|\bar{0}\rangle$
with ${\cal A}^{\dagger}\equiv\alpha^{\dagger}_{k}\alpha^{\dagger}_{-k}$ being
the creation operator of two-quasiparticle pair are given in terms of the
SCQRPA amplitudes $\mathcal{X}_{k}^{\nu}$ and $\mathcal{Y}_{k}^{\nu}$ as
$\langle\bar{0}|\mathcal{A}_{k}^{\dagger}\mathcal{A}_{k^{\prime}}|\bar{0}\rangle=\sqrt{\langle{\cal
D}_{k}\rangle\langle{\cal
D}_{k^{\prime}}\rangle}\sum_{\nu}{\mathcal{Y}_{k}^{\nu}\mathcal{Y}_{k^{\prime}}^{\nu}}~{},\hskip
14.22636pt\langle\bar{0}|\mathcal{A}_{k}\mathcal{A}_{k^{\prime}}|\bar{0}\rangle=\sqrt{\langle{\cal
D}_{k}\rangle\langle{\cal
D}_{k^{\prime}}\rangle}\sum_{\nu}{\mathcal{X}_{k}^{\nu}\mathcal{Y}_{k^{\prime}}^{\nu}}~{}.$
(19)
where $\langle\bar{0}|\ldots|\bar{0}\rangle$ denotes the expectation value in
the SCQRPA ground state. The ground-state correlation factor ${\cal D}_{k}$ is
expressed in term of the backward-going amplitudes ${\cal Y}^{\nu}_{k}$ as
${\cal D}_{k}=[1+2\sum_{\nu}({\cal Y}^{\nu}_{k})^{2}]^{-1}$ with the sum
running over all the SCQRPA solutions $\nu$.
After solving the LNSCQRPA equations (8), (16) – (18) for each total seniority
$S$, we obtain a set of eigenstates, which consists of C${}^{\Omega}_{S}$
lowest eigenstates (the ground state at $S=$0 or 1), as well as higher
eigenstates (excited states) on top of these lowest ones, which come from the
solutions of the LNSCQRPA equations, whose eigenvalues are
$\omega_{\nu}^{(S)}$ ($\nu=1,\ldots\Omega-S$)111The SCQRPA has altogether
$\Omega-S+1$ solutions with positive energies. However the lowest one
corresponds to the spurious mode, whose energy is zero within the QRPA.
Therefore it is excluded in the numerical calculations.. As the result, the
total number of eigenstates obtained within the LNSCQRPA is given as
$n_{\rm LNSCQRPA}=\sum_{S}{\rm C}_{S}^{\Omega}\times(\Omega-S)~{}.$ (20)
Consequently, the so-called CE-LNSCQRPA partition function is calculated as
$Z_{\rm LNSCQRPA}(\beta)=\sum_{S}{d_{S}}\sum_{i_{S}=1}^{n_{\rm
LNSCQRPA}}e^{-\beta{\cal E}_{i_{S}}^{\rm LNSCQRPA}}~{},$ (21)
which is formally identical to the CE-LNBCS partition function (13), but the
LNBCS eigenvalues ${\cal E}_{i_{S}}^{\rm LNBCS}$ are now replaced with ${\cal
E}_{i_{S}}^{\rm LNSCQRPA}$. From this partition function, the thermodynamic
quantities obtained within the CE-LNSCQRPA are calculated in the same way as
those in Eq. (14). Although the number $n_{\rm LNSCQRPA}$ of the LNSCQRPA
eigenstates is larger than $n_{\rm LNBCS}$, it is still much smaller than
$n_{\rm Exact}$. This most important feature of the present method
tremendously reduces the computing time in numerical calculations for heavy
nuclei. As an example, we show in Table 1 the number of eigenstates and the
total executing time (the elapsed real time) for the exact diagonalization of
the pairing Hamiltonian, CE-LNBCS and CE-LNSCQRPA calculations within the
Richardson model at several values $N$ of particle number, which is taken to
be equal to number $\Omega$ of single-particle levels (the half-filled case).
This table shows that the execution time within the LNSCQRPA (LNBCS) is
shorter than that consumed by the exact diagonalization by about two (four)
orders.
Table 1: Number of eigenstates and computation time for the exact diagonalization of the pairing Hamiltonian as well as the numerical calculations within the CE-LNBCS and CE-LNSCQRPA for the doubly-folded equidistant multilevel pairing model at several values of $N=\Omega$. The computation time is estimated based on a shared large memory computer Altix 450 with 512GB memory of RIKEN Integrated Cluster of Clusters (RICC) system. | | Number of eigenstates | | | Computation time | |
---|---|---|---|---|---|---|---
$N$ | Exact | LNBCS | LNSCQRPA | Exact | LNBCS | LNSCQRPA |
$10$ | 8953 | 512 | 2560 | 1 ${\rm hr}$ | 1 ${\rm sec.}$ | 10 ${\rm sec.}$ |
$12$ | 73789 | 2048 | 12288 | 10 ${\rm hrs}$ | 10 ${\rm sec.}$ | 1 ${\rm min.}$ |
$14$ | 616227 | 8192 | 57344 | 24 ${\rm hrs}$ | 1 ${\rm min.}$ | 10 ${\rm min.}$ |
$16$ | 5196627 | 32768 | 262144 | - | 10 ${\rm min.}$ | 1 ${\rm hr}$ |
$18$ | 44152809 | 131072 | 1179648 | - | 1 ${\rm hr}$ | 3 ${\rm hrs}$ |
$20$ | 377379369 | 524288 | 5242880 | - | 3 ${\rm hrs}$ | 10 ${\rm hrs}$ |
### II.5 MCE-LNBCS, MCE-LNSCQRPA
The MCE entropy is calculated by using the Boltzmann’s definition
${\cal S}({\cal E})={\rm ln}{\cal W}({\cal E})~{},\hskip 14.22636pt{\cal
W}({\cal E})=\rho({\cal E})\delta{\cal E}~{},$ (22)
where $\rho({\cal E})$ is the density of states. In the LNBCS (LNSCQRPA)
approaches, ${\cal W}({\cal E})$ is the number of LNBCS (LNSCQRPA) eigenstates
within the energy interval (${\cal E},{\cal E}+\delta{\cal E})$ Ensemble .
Knowing the MCE entropy, one can calculate the MCE temperature as the first
derivative of the MCE entropy with respect to the excitation energy ${\cal
E}$, namely
$T=\left[\frac{\partial{\cal S}({\cal E})}{\partial{\cal E}}\right]^{-1}~{}.$
(23)
The corresponding approaches, which embed the LNBCS and LNSCQRPA eigenvalues
into the MCE, are called the MCE-LNBCS and MCE-LNSCQRPA, respectively.
### II.6 Level density
The inverse relation of Eq. (22) reads
$\rho({\cal E})=e^{{\cal S}({\cal E})}/\delta{\cal E}~{},$ (24)
which can be used to calculate the density of states $\rho({\cal E})$ from the
fitted MCE entropy.
Within the CE, the density of states $\rho({\cal E})$ is calculated by using
the method of steepest descent to find the minimum of the Laplace transform of
the partition function Ericson . As a result the density of states $\rho({\cal
E}$) at temperature $T=\beta_{0}^{-1}$, which corresponds to this minimum, is
approximated as
$\rho({\cal E})\approx{Z(\beta_{0})e^{\beta_{0}{\cal
E}}}\bigg{[}2\pi\frac{\partial^{2}{\rm
ln}Z(\beta_{0})}{\partial\beta_{0}^{2}}\bigg{]}^{-1/2}\equiv e^{{\cal S}(\cal
E)}\bigg{(}-2\pi\frac{\partial{\cal
E}}{\partial\beta_{0}}~{}\bigg{)}^{-1/2}~{},$ (25)
where $Z(\beta_{0})$, ${\cal S}({\cal E})$ and ${\cal E}$ are the CE partition
function, entropy and total excitation energy of the systems, respectively.
The density of states $\rho({\cal E})$ is obtained within the CE-LNBCS and CE-
LNSCQRPA by replacing the partition function $Z$ in Eq. (25) with that
obtained within the CE-LNBCS in Eq. (13) and CE-LNSCQRPA in Eq. (21).
At finite angular momentum $J$, in principle, the approach of LNSCQRPA plus
angular momentum, which has been proposed by us in Ref. SCQRPAM , should be
used to calculate the angular-momentum dependent level density $\rho({\cal
E},M)$ with $M$ being the $z$-projection of the total angular momentum. In
this case the former doubly-degenerated quasiparticle levels are resolved
under the constraint $M=\sum_{k}m_{k}(n_{k}^{+}-n_{k}^{-})$ with the
quasiparticle occupation numbers $n_{k}^{\pm}$, which are described by the
Fermi-Dirac distribution $n_{k}^{\pm,FD}=\\{{\rm exp}[\beta(E_{k}\mp\gamma
m_{k})]+1\\}^{-1}$ within the non-interacting quasiparticle approximation,
where $m_{k}$ is the spin-projections of the $k$th single-particle state
$|k,\pm m_{k}\rangle$, $E_{k}$ is the quasiparticle energy, and $\gamma$ is
the rotation frequency. Knowing $\rho({\cal E},M)$, one can find $\rho({\cal
E},J)=\rho({\cal E},M=J)-\rho({\cal E},M=J+1)$ in the general case, where the
total angular momentum $J$ is not aligned with the $z$-axis Bohr . The total
level density $\rho_{tot}({\cal E})$ and experimentally observed level density
$\rho_{obs}({\cal E})$, are then defined as Gilbert
$\rho_{tot}({\cal E})=\sum_{J}(2J+1)\rho({\cal E},J)~{},\hskip
14.22636pt\rho_{obs}({\cal E})=\sum_{J}\rho({\cal E},J)~{}.$ (26)
The empirical entropy ${\cal S}_{obs}({\cal E})$ is extracted from the
observed level density $\rho_{obs}({\cal E})$ in the same way as in Eq. (22),
replacing $\rho({\cal E})$ with $\rho_{obs}({\cal E})$, namely
${\cal S}_{obs}({\cal E})={\rm ln}[\rho_{obs}({\cal E})\delta{\cal E}]~{},$
(27)
Because the present article considers non-rotating nuclei at low angular
momentum, we assume that $\rho({\cal E},J)\simeq\rho({\cal
E},0)\equiv\rho({\cal E})$. Therefore, by fitting the MCE entropy ${\cal
S}({\cal E})$ in Eq. (22) to the experimentally observed entropy ${\cal
S}_{obs}({\cal E})$ in Eq. (27), i.e. ${\cal S}({\cal E})\simeq{\cal
S}_{obs}({\cal E})$, and inverting the obtained result by using Eq. (24), what
we get is actually the level density comparable to the experimentally observed
one, $\rho_{obs}({\cal E})={\rm exp}[{\cal S}({\cal E})]/\delta{\cal E}$. This
means that the density of states $\rho({\cal E})$ calculated by using Eq. (24)
or Eq. (25) without taking into account the effect of finite angular momentum
is identical to the level density like $\rho_{obs}({\cal E})$, not the total
level density $\rho_{tot}({\cal E})$, because of the absence of the factor
$(2J+1)$.
## III ANALYSIS OF NUMERICAL RESULTS
The proposed approaches are used to calculate the pairing gap, total energy,
entropy and heat capacity within the CE and MCE for a number of heavy
isotopes, namely 94,98Mo, 162Dy and 172Yb 222See, e.g. Fig. 1 of Ref. CE-BCS
and Appendix A of the present article for the accuracy of the present
approaches in comparison with the exact solutions of the Richardson model..
The single-particle energies are taken from the axially deformed Woods-Saxon
potential with the depth of the central potential WS
$V=V_{0}\left[1\pm k\frac{N-Z}{N+Z}\right]~{},$ (28)
where $V_{0}=$ 51.0 MeV, $k=$ 0.86, whereas the plus and minus signs stand for
proton ($Z$) and neutron ($N$), respectively. The radius $r_{0}$, diffuseness
$a$, and spin-orbit strength $\lambda$ are chosen to be $r_{0}=$ 1.27 fm, $a=$
0.67 fm and $\lambda=$ 35.0. The quadrupole deformation parameters $\beta_{2}$
are estimated from the experimental $B(E2;2^{+}_{1}\rightarrow 0^{+}_{1})$,
which are 0.15, 0.17, 0.281 and 0.296 for 94Mo, 98Mo, 162Dy and 172Yb,
respectively Kaneko . The pairing interaction parameters $G$ are adjusted so
that the pairing gaps for neutrons and protons obtained within the LNSCQRPA at
$T=$ 0 and $S=$ 0 reproduce the values extracted from the experimental odd-
even mass differences, namely $\Delta_{N}\simeq$ 1.2, 1.0, 0.8 and 0.8 MeV for
neutrons, and $\Delta_{Z}\simeq$ 1.4, 1.3, 0.9 and 0.9 MeV for protons in
94Mo, 98Mo, 162Dy and 172Yb, respectively.
It is well-known that pairing is significant only for the levels around the
Fermi energy. Therefore, within the CE, we apply the same prescription
proposed in Ref. QMC1 to calculate the CE partition function for medium and
heavy isotopes. According to this prescription, we calculate the LNBCS and
LNSCQRPA pairing gaps in the space spanned by 22 degenerated (proton or
neutron) single-particle levels above the doubly-magic 48Ca core for Mo
isotopes, whereas the same is done on top of the doubly-magic 132Sn core for
Dy and Yb nuclei. The obtained partition function is then combined with those
obtained within the independent-particle model (IPM) by using Eq. (15) of Ref.
QMC1 , namely
${\rm ln}Z^{\prime}_{\nu}={\rm ln}Z^{\prime}_{\nu,tr}+{\rm
ln}Z^{\prime}_{sp}-{\rm ln}Z^{\prime}_{sp,tr}~{},$ (29)
where $Z^{\prime}_{\nu,tr}\equiv Z_{\nu,tr}e^{\beta{\cal E}_{0}}$ is the
excitation partition function with respect to the ground state energy ${\cal
E}_{0}$ with $Z_{\nu,tr}$ being the CE partition function obtained within the
LNBCS [Eq. (13)] or LNSCQRPA [Eq. (21)] for 22 degenerated single-particle
levels around the Fermi energy. $Z^{\prime}_{sp}$ is the CE partition function
obtained within the IPM [See e.g. Eq. (8) of Ref. QMC1 ] for the space spanned
by the levels from the bottom to $N=$ 126 closed shell, whereas
$Z^{\prime}_{sp,tr}$ is the same partition function but for the truncated
space spanned by 22 levels around the Fermi energy.
### III.1 Results for molybdenum
Figure 1: (Color online) Pairing gaps $\Delta$ and heat capacities $C$
obtained within the CE as functions of $T$ and entropies ${\cal S}$ obtained
within the MCE as functions of $E^{*}$ for 94Mo (left panels) and 98Mo (right
panels). In (a) and (d), the solid and dash-dotted lines denote the pairing
gaps for protons and neutrons, respectively, whereas the thin and thick lines
correspond to the CE-LNBCS and CE-LNSCQRPA results, respectively. In (b) and
(e), the thin and thick lines stand for the CE-LNBCS and CE-LNSCQRPA results,
whereas the thin and thick dash-dotted lines depict the experimental results
taken from Refs. Chankova and Kaneko , respectively. Shown in (c) and (f) are
the MCE entropies obtained within the MCE-LNBCS (rectangles), MCE-LNSCQRPA
(triangles), and extracted from experimental data (circles with error bars) of
Ref. Chankova .
Shown in Fig. 1 are the pairing gaps, heat capacities and entropies for 94Mo
[Figs. 1 (a)-1 (c)] and 98Mo [Figs. 1 (d)-1 (f)] obtained within the
CE(MCE)-LNBCS and CE(MCE)-LNSCQRPA versus the experimental data from Refs.
Chankova and Kaneko . There is a clear discrepancy in the heat capacities
extracted from the same measured level density in these two papers [Figs. 1
(b) and 1 (e)]. The heat capacity, extracted in Ref. Kaneko , clearly shows a
pronounced peak at $T\sim$ 0.7 MeV for both 94Mo and 98Mo, whereas the
corresponding quantity, extracted in Ref. Chankova , shows no trace of any
peak. The source of the discrepancy comes from the difference in the scale of
excitation energy $E^{*}$, which was used for extrapolating the measured level
density before evaluating the CE partition function using the Laplace
transformation of the level density. In Ref. Chankova , the level density is
extrapolated up to $E^{*}\sim$ 40 - 50 MeV, whereas in Ref. Kaneko this is
done up to $E^{*}=$ 180 MeV. Given that all the excited states should be
included in the partition function, the energy $E^{*}\sim$ 40 - 50 MeV used in
Ref. Chankova seems to be too low, which might affect the resulting heat
capacity. As Figs. 1 (b) and 1 (e) show, the heat capacities predicted by the
CE-LNSCQRPA are much closed to those obtained in Ref. Kaneko . They are also
consistent with the FTQMC calculations for other nuclei QMC ; QMC1 . It is
important to emphasize here that quantal and thermal fluctuations within the
CE-LNBCS (LNSCQRPA) indeed smooth out the SN phase transition. As the result,
the pairing gaps [Figs. 1 (a) and (d)] obtained for protons (solid lines) and
neutrons (dash-dotted lines) within both CE-LNBCS (thin lines) and CE-LNSCQRPA
(thick lines) do not collapse at the critical temperature $T=T_{c}$ of the SN
phase transition, as predicted by the GCE-BCS, but monotonously decrease with
increasing $T$. The neutron gap in Fig. 1 (a) obtained within the CE-LNSCQRPA
for 94Mo (thick dash-dotted lines) is close to the three-point gap (dashed
lines) obtained in Ref. Kaneko by simply extrapolating the odd-even mass
formula to finite temperature. As has been pointed out in Ref. Ensemble such
naive extrapolation contains the admixture with the contribution from
uncorrelated single-particle configurations, which do not contribute to the
pairing correlation. Therefore, to avoid obviously wrong results at high $T$,
such contribution should be removed from the total energy of the system.
Nonetheless, in the low temperature region ($T<$ 1.3 MeV) as that considered
here, where the contribution of uncorrelated single-particle configurations is
expected to be small, the simple extension of the three-point odd-even mass
formula to $T\neq$ 0 can still serve as a useful indicator.
Figure 2: (Color online) Microcanonical entropy as function of $E^{*}$
obtained within the MCE-LNSCQRPA for 94Mo using various values of energy
interval $\delta{\cal E}$.
As has been discussed in Ref. CE-BCS , at low $E^{*}$ the genuine
thermodynamic observable is the MCE entropy because it is calculated directly
from the observable level density by using the Boltzmann’s definition (22).
The experimental MCE entropies for 94,98Mo are plotted in Figs. 1 (c) and 1
(f) along with the predictions by the MCE-LNBCS and MCE-LNSCQRPA. These
figures show that the MCE-LNSCQRPA results fit the available experimental data
remarkably well. It is worth mentioning that the results obtained within the
MCE-LNBCS(LNSCQRPA) are sensitive to the choice of energy interval
$\delta{\cal E}$, which is used to calculate the number of accessible states
${\cal W}({\cal E})$ in Eq. (22). Figure 2 shows the entropies obtained within
the CE-LNSCQRPA for 94Mo using several values of $\delta{\cal E}$ ranging from
0.2 MeV to 1.0 MeV. It is clear to see from this Fig. 2 that the MCE entropies
increase with increasing $\delta{\cal E}$. In this respect, we found that the
values of $\delta{\cal E}$ = 1 MeV for 94Mo and 0.7 MeV for 98Mo are
reasonable to fit the experimental data. The reason for choosing large values
of $\delta{\cal E}$ for these two nuclei comes from the deficiency of the CE-
LNSCQRPA(LNBCS), which includes only low-lying excited states.
### III.2 Results for dysprosium and ytterbium
Figure 3: (Color online) (a), (b), (e) and (f): Pairing gaps $\Delta$, heat
capacities $C$ as functions of $T$ obtained within the CE; (c), (d), (g) and
(h): Entropies ${\cal S}$ and temperatures $T$ as functions of $E^{*}$
obtained within the MCE for 162Dy (left panels) and 172Yb (right panels).
Notations are the same as those in Fig. 1. Experimental data are taken from
Ref. Oslo1 .
The results obtained for 162Dy and 172Yb are shown in Fig. 3. Similar to the
results for 94,98Mo, the CE heat capacities and MCE entropies obtained within
the CE(MCE)-LNSCQRPA for both 162Dy and 172Yb are in good agreement with the
experimental data. The neutron and proton gaps obtained within the CE-LNBCS
(LNSCQRPA) do not collapse at $T=T_{c}$ but decrease with increasing $T$ and
keep finite at high $T$ even for the two heavy nuclei considered here. The
peak in the experimental heat capacity near $T=$ 0.4 MeV is seen in 172Yb,
whereas it disappears in 162Dy. This is again due to the fact that the
measured level densities for these two nuclei are extrapolated only up to
$E^{*}=$40 MeV instead of 180 MeV as was done in Ref. Kaneko for other
nuclei. This is confirmed by the heat capacities obtained within the CE-
LNSCQRPA (thick solid lines), which clearly show a peak around $T=$ 0.4 MeV.
In Figs. 3 (d) and 3 (h), one can see that the MCE temperatures, extracted
from the experimental data (circles with error bars) by using Eq. (23),
scatter around the experimental (thick dash-dotted lines) or theoretical
(thick and thin lines) CE results. The results of calculations with the MCE-
LNBCS (squares) and MCE-LNSCQRPA (triangles) by using the same definition (23)
and $\delta{\cal E}=$ 0.5 also describe well these values. The results for MCE
entropies in Figs. 1 and 3 show the importance of the effect beyond the
quasiparticle mean field included in the self-consistent coupling QRPA
vibrations. In fact, the MCE-LNSBCS results for the entropy clearly
underestimate the experimental values. The discrepancy with the MCE-LNSCQRPA
results increases with $E^{*}$ to reach about 20% at $E^{*}=$ 20 MeV.
### III.3 Level density
The level densities obtained within the CE-LNSCQRPA using Eq. (25) and MCE-
LNSCQRPA using Eq. (24) are plotted in Fig. 4 as functions of excitation
energy $E^{*}$ in comparison with the experimental data Oslo1 ; Chankova
$\rho_{obs}({\cal E})=\rho_{0}\times{\rm exp}[{\cal S}_{obs}({\cal E})]$. In
the latter $\rho_{0}$ is a normalization factor, which should be put equal to
$1/\delta{\cal E}$ according Eq. (27). However, because of fluctuations in
level spacings, which make the entropy sensitive to $\delta{\cal E}$, the
authors of Ref. Oslo1 ; Chankova chose the values of $\rho_{0}$ to obtain
entropy ${\cal S}_{obs}=$ 0 at $T=$ 0\. In this way the value of $\rho_{0}$ is
set to 1.5 MeV-1 for 94,98Mo Chankova and 3 MeV-1 for 162Dy and 172Yb Oslo1 .
Figure 4 shows that the level densities obtained within the MCE-LNSCQRPA offer
the best fit to the experimental data for all nuclei under consideration. The
results obtained within the CE-LNSCQRPA are closer to the experimental data
for 94,98Mo at $E^{*}\leq$ 4 MeV, whereas at higher $E^{*}$ the MCE-LNSCQRPA
offers a better performance. The S shape in the MCE-LNSCQRPA level density at
low $E^{*}$ might have come from the fixed value of the energy interval
$\delta{\cal E}$, within which the levels are counted, according to the
definition (22), whereas the denominator in the definition of the CE level
density [at the right-hand side of Eq. (25)] depends on $E^{*}$. A larger
value $\delta{\cal E}$ at $E^{*}\leq$ 4 MeV would eventually increase the MCE-
LNSCQRPA level density, improving the agreement with the observed level
density in this region, but there is no physical justification for doing so.
The discrepancy between the CE-LNSCQRPA and experimental results seems to be
larger and increases with $E^{*}$ for 162Dy and 172Yb. This might be due to
the absence of the contribution of higher multipolarities such as dipole,
quadrupole etc., which are not included in the present study and may be
important for rare-earth nuclei. On the other hand, the use of SCQRPA plus
angular momentum SCQRPAM , discussed previously, may also improve the
agreement.
Figure 4: (Color online) Level densities as functions of $E^{*}$ obtained
within the CE-LNSCQRPA (solid line) and MCE-LNSCQRPA (triangles) versus the
experimental data (circles with error bars) for 94Mo (a), 98Mo (b), 162Dy (c),
and 172Yb (d).
## IV CONCLUSIONS
The present article applies the canonical and microcanonical ensembles of the
LNBCS and LNSCQRPA approaches, derived in Ref. CE-BCS , to describe the
thermodynamic properties as well as level densities of several nuclei, namely
94,98Mo, 162Dy and 172Yb. The results obtained show that the CE(MCE)-LNSCQRPA
describe quite well the recent experimental level densities and the
thermodynamic quantities extracted for these nuclei by the Oslo group Oslo ;
Oslo1 ; Chankova ; Kaneko . It confirms that the SN phase transition is
smoothed out in nuclear systems due to the effects of quantal and thermal
fluctuations leading to the nonvanishing pairing gap at finite temperature
even in heavy nuclei Moretto ; SPA ; Zele ; MBCS ; FTBCS1 ; Ensemble . The
discrepancy between the heat capacities obtained within the two different
experimental works, which extrapolate the same experimental level density to
different excitation energies, are also discussed. The heat capacities
obtained within the CE-LNBCS(LNSCQRPA) for all nuclei show a pronounced peak
at $T\sim T_{c}$, whereas the results extracted from the same experimental
data by Refs. Chankova and Kaneko show different behaviors. The better
agreement between the predictions of our approaches as well as those of the
FTQTMC and the results of Ref. Kaneko gives a strong indication to the fact
that, to construct an adequate partition function for a good description of
thermodynamic quantities, the measured level density should be extended up to
very high excitation energy $E^{*}\sim$ 180 MeV or 200 MeV. The small
differences between the CE(MCE)-LNBCS(LNSCQRPA) results and the experimental
data might be due to the absence of the contribution of higher multipolarities
such as dipole, quadrupole etc., which are not included in the present study.
In order to tackle this issue, the LNSCQRPA plus angular momentum SCQRPAM
should be used and extended to included also the multipole residual
interactions higher than the monopole pairing force. This task remains one of
the subjects of our study in the future.
###### Acknowledgements.
The numerical calculations were carried out using the FORTRAN IMSL Library by
Visual Numerics on the RIKEN Integrated Cluster of Clusters (RICC) system. A
part of this work was carried out during the stay of N.Q.H. in RIKEN under the
support by the postdoctoral grant from the Nishina Memorial Foundation and by
the Theoretical Nuclear Physics Laboratory of the RIKEN Nishina Center.
## Appendix A MCE results within the Richardson model
Figure 5: (Color online) MCE entropies and level densities as functions of
$E^{*}$ obtained within the MCE-LNBCS (squares), MCE-LNSCQRPA (triangles)
versus the exact results for the Richardson model (circles) with $N=\Omega=14$
and $G=$ 1 MeV. Results obtained by using the energy bin $\delta{\cal E}=$ 1
MeV are shown in (a) and (b), whereas those obtained by using $\delta{\cal
E}=$ 5 MeV are shown in (c) and (d). Lines connecting the squares and
triangles are drawn to guide the eye.
The CE-LNBCS and CE-LNSCQRPA has been tested within the Richardson model in
Ref. CE-BCS and the results obtained are found in very good agreement with
the exact solutions whenever the latter are available. In order to have more
convincing evidences on the accuracy of present approaches, we show in Fig. 5
the MCE entropies and level densities obtained within the MCE-LNBCS and MCE-
LNSCQRPA versus the exact ones for the Richardson model with $N=\Omega$ = 14
and $G$ = 1 MeV. Two different values of energy interval $\delta{\cal E}$,
namely $\delta{\cal E}$ = 1 MeV (left panels) and $\delta{\cal E}$ = 5 MeV
(right panels) are used in calculations. This figure shows that the MCE-
LNSCQRPA always offers the best fit to the exact results, whereas the MCE-
LNBCS underestimates the exact ones. The decreasing of the entropy as well as
level density for the case with small value of $\delta{\cal E}$ = 1 MeV shown
in Figs. 5 (a) and 5 (b) is due to the small configuration space with
$N=\Omega$ = 14 in the present case. This feature is ultimately related to the
problem of using thermodynamics in very small system with discrete energy
levels, where the temperature may decrease with increasing the excitation
energy ${\cal E}^{*}$ (See Fig. 2 of Ref. Ensemble ). This shortcoming can be
effectively overcomed by using a larger $\delta{\cal E}$ = 5 MeV. As the
result, the entropy and level density increase with increasing ${\cal E}^{*}$
as shown in the right panels of Fig. 5, although there is no physical
justification for using such a large value of $\delta{\cal E}$.
## References
* (1) J. Bardeen, L. Cooper, and Schrieffer, Phys. Rev. 108, 1175 (1957); M. Sano and S. Yamasaki, Prog. Theor. Phys. 29, 397 (1963).
* (2) K. Tanabe and K. Sugaware-Tanabe, Phys. Lett. B 97, 337 (1980); A.L. Goodman, Nucl. Phys. A 352, 30 (1981); K. Tanabe, K. Sugaware-Tanabe, and H.J. Mang, Nucl. Phys. A 357, 20 (1981); Ibid. 357, 45 (1981).
* (3) L.G. Moretto, Phys. Lett. B 40, 1 (1972); A.L. Goodman, Phys. Rev. C 29, 1887 (1984); J.L. Egido, P. Ring, S. Iwasaki, and H.J. Mang, Phys. Lett. B 154, 1 (1985).
* (4) R. Rossignoli, P. Ring and N.D. Dang, Phys. Lett. B 297, 9 (1992); N.D. Dang, P. Ring and R. Rossignoli, Phys. Rev. C 47, 606 (1993).
* (5) V. Zelevinsky, B.A. Brown, N. Frazier, and M. Horoi, Phys. Rep. 276, 85 (1996).
* (6) N. Dinh Dang and V. Zelevinsky, Phys. Rev. C 64, 064319 (2001); N. Dinh Dang and A. Arima, Phys. Rev. C 67, 014304 (2003); N.D. Dang and A. Arima, Phys. Rev. C 68, 014318 (2003); N.D. Dang, Nucl. Phys. A 784, 147 (2007).
* (7) N. Dinh Dang and N. Quang Hung, Phys. Rev. C 77, 064315 (2008).
* (8) N.Q. Hung and N.D. Dang, Phys. Rev. C 79, 054328 (2009).
* (9) T. Sumaryada and A. Volya, Phys. Rev. C 76, 024319 (2007).
* (10) R.W. Richardson, Phys. Lett. 3, 277 (1963); Ibid. 14, 325 (1965); A. Volya, B.A. Brown, and V. Zelevinsky, Phys. Lett. B 509 (2001) 37.
* (11) S. Liu and Y. Alhassid, Phys. Rev. Lett 87, 022501 (2001);
* (12) Y. Alhassid, G. F. Bertsch, and L. Fang, Phys. Rev. C 68, 044322 (2003).
* (13) J. Dukelsky, S. Pittel and G. Sierra, Rev. Mod. Phys. 76, 643 (2004).
* (14) R. Rossignoli and P. Ring, Ann. Phys. (NY) 235, 350 (1994); R. Rossignoli, P. Ring, and N. D. Dang, Phys. Lett. B297, 9 (1992); K. Tanabe and H. Nakada, Phys. Rev. C 71, 024314 (2005); H. Nakada and K. Tanabe, Phys. Rev. C 74, 061301(R) (2006).
* (15) R. Rossignoli, N. Canosa, and P. Ring, Phys. Rev. Lett 80, 1853 (1998).
* (16) K. Kaneko and A. Schiller, Phys. Rev. C 75, 044304 (2007); ibib 76, 064306 (2007).
* (17) R. Rossignoli, Phys. Rev. C 54, 1230 (1996).
* (18) E. Melby _et al._ , Phys. Rev. Lett. 83, 3150 (1999); A. Schiller _et al._ , Phys. Rev. C 63, 021306 (R) (2001); E. Algin _et al._ , Phys. Rev. C 78, 054321 (2008).
* (19) M. Guttormsen _et. al_ , Phys. Rev. C 62, 024306 (2000).
* (20) R. Chankova _et al._ , Phys. Rev. C 73, 034311 (2006).
* (21) K. Kaneko _et al._ , Phys. Rev. C 74, 024325 (2006).
* (22) N.Q. Hung and N.D. Dang, Phys. Rev. C 81, 057302(BR) (2010).
* (23) N.Q. Hung and N.D. Dang, Phys. Rev. C 76, 054302 (2007); Ibid. 77, 029905(E) (2008).
* (24) H. J. Lipkin, Ann. Phys. (NY) 9 272 (1960); Y. Nogami, Phys. Lett. 15 4 (1965).
* (25) N. Dinh Dang and N. Quang Hung, Phys. Rev. C 81, 034301 (2010).
* (26) T. Ericson, Adv. Phys. 9, 425 (1960).
* (27) N.Q. Hung and N.D. Dang, Phys. Rev. C 78, 064315 (2008).
* (28) A. Bohr and B.R. Mottelson, Nuclear structure, Vol. 1 (Benjamin, NY, 1969).
* (29) A. Gilbert and A.G.W. Cameron, Can. J. Phys. 43, 1446 (1965).
* (30) S. Cwiok et al., Comput. Phys. Commun. 46, 379 (1987).
|
arxiv-papers
| 2010-10-05T03:39:25 |
2024-09-04T02:49:13.465725
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "N. Quang Hung and N. Dinh Dang",
"submitter": "Nguyen Quang Hung",
"url": "https://arxiv.org/abs/1010.0760"
}
|
1010.0868
|
# Jacob’s ladders and some new consequences from A. Selberg’s formula
Jan Moser Department of Mathematical Analysis and Numerical Mathematics,
Comenius University, Mlynska Dolina M105, 842 48 Bratislava, SLOVAKIA
jan.mozer@fmph.uniba.sk
###### Abstract.
It is proved in this paper that the Jacob’s ladders together with the A.
Selberg’s classical formula (1942) lead to a new kind of formulae for some
short trigonometric sums. These formulae cannot be obtained in the classical
theory of A. Selberg, and all the less, in the theories of Balasubramanian,
Heath-Brown and Ivic.
###### Key words and phrases:
Riemann zeta-function
## 1\. The A. Selberg’s formula
A. Selberg has proved in 1942 the following formula
(1.1) $\int_{T}^{T+U}X^{2}(t)\left(\frac{n_{2}}{n_{1}}\right)^{it}{\rm
d}t=\sqrt{\frac{\pi}{2}}\frac{U}{\sqrt{n_{1}n_{2}}}\left(\ln\frac{P^{2}}{n_{1}n_{2}}+2c\right)+\mathcal{O}(T^{1/2}\xi^{5})$
(see [19], p. 55), where
(1.2) $\begin{split}&X(t)=\frac{1}{2}t^{1/4}e^{\frac{1}{4}\pi
t}\pi^{-\frac{s}{2}}\zeta(s),\ s=\frac{1}{2}+it,\\\ &U=T^{1/2+\epsilon},\
\xi=\left(\frac{T}{2\pi}\right)^{\epsilon/10},\ \epsilon\leq\frac{1}{10},\
P=\sqrt{\frac{T}{2\pi}}\\\ &n_{1},n_{2}\in\mathbb{N},(n_{1},n_{2})=1,\
n_{1},n_{2}\leq\xi,\end{split}$
(comp. [19], pp. 10, 18, $a=1/2+\epsilon,\ \epsilon>0$) and $c$ is the Euler’s
constant. Since (see [19], p. 10, [20], p. 79)
$Z^{2}(t)=\left|\zeta\left(\frac{1}{2}+it\right)\right|^{2}=\sqrt{\frac{2}{\pi}}X^{2}(t)\left(1+\mathcal{O}(\frac{1}{t})\right),$
i.e.
(1.3)
$X^{2}(t)=\sqrt{\frac{2}{\pi}}Z^{2}(t)\left(1+\mathcal{O}(\frac{1}{t})\right)$
where
(1.4)
$\begin{split}&Z(t)=e^{i\vartheta(t)}\zeta\left(\frac{1}{2}+it\right),\\\
&\vartheta(t)=-\frac{1}{2}t\ln\pi+\text{Im}\ln\Gamma\left(\frac{1}{4}+\frac{1}{2}it\right)=\frac{t}{2}\ln\frac{t}{2\pi}-\frac{t}{2}-\frac{\pi}{8}+\mathcal{O}(\frac{1}{t})\end{split}$
is the signal defined by the Riemann zeta-function $\zeta(s)$. Following eqs.
(1.1) and (1.3) we obtain
(1.5) $\int_{T}^{T+U}Z^{2}(t)\left(\frac{n_{2}}{n_{1}}\right)^{it}{\rm
d}t=\frac{U}{\sqrt{n_{1}n_{2}}}\left(\ln\frac{P^{2}}{n_{1}n_{2}}+2c\right)+\mathcal{O}(T^{1/2}\xi^{5})$
###### Remark 1.
If $n_{1}=n_{2}=1$ then the Hardy-Littlewood-Ingham formula
$\int_{T}^{T+U}Z^{2}(t){\rm
d}t=U\ln\frac{T}{2\pi}+2cU+\mathcal{O}(T^{1/2}\xi^{5})$
follows from the A. Selberg’s formula (1.5) (comp. [20], p. 120).
###### Remark 2.
Let us remind that the A. Selberg’s formula (1.5) played the main role in
proving the fundamental Selberg’s result
$N_{0}(T+U)-N_{0}(T)>A(\epsilon)U\ln T$
where $N_{0}$ stands for the number of zeroes of the function $\zeta(1/2+it),\
t\in(0,T]$.
In this paper it is proved that the Jacob’s ladders together with the A.
Selberg’s classical formula lead to a new kind of results for some short
trigonometric sums.
This paper is a continuation of the series of works [3] \- [18].
## 2\. The result
### 2.1.
Let us remind some notions. First of all
(2.1) $\tilde{Z}^{2}(t)=\frac{{\rm d}\varphi_{1}(t)}{{\rm d}t},\
\varphi_{1}(t)=\frac{1}{2}\varphi(t),$
where
(2.2)
$\tilde{Z}^{2}(t)=\frac{Z^{2}(t)}{2\Phi^{\prime}_{\varphi}[\varphi(t)]}=\frac{Z^{2}(t)}{\left\\{1+\mathcal{O}\left(\frac{\ln\ln
t}{\ln t}\right)\right\\}\ln t}$
(see [3], (3.9); [5], (1.3); [9], (1.1), (3.1), (3.2)) and $\varphi(t)$ is the
Jacob’s ladder, i.e. the solution of the following nonlinear integral equation
$\int_{0}^{\mu[x(T)]}Z^{2}(t)e^{-\frac{2}{x(T)}t}{\rm
d}t=\int_{0}^{T}Z^{2}(t){\rm d}t$
that was introduced in our paper [3]. Next, we have (see [1], comp. [18])
(2.3) $\begin{split}&G_{3}(x)=G_{3}(x;T,U)=\\\ &=\bigcup_{T\leq g_{2\nu}\leq
T+U}\\{t:\ g_{2\nu}(-x)\leq t\leq g_{2\nu}(x)\\},\ 0<x\leq\frac{\pi}{2},\\\
&G_{4}(y)=G_{4}(y;T,U)=\\\ &=\bigcup_{T\leq g_{2\nu+1}\leq T+U}\\{t:\
g_{2\nu+1}(-y)\leq t\leq g_{2\nu+1}(y)\\},\ 0<y\leq\frac{\pi}{2},\end{split}$
and the collection of sequences $\\{g_{\nu}(\tau)\\},\ \tau\in[-\pi,\pi],\
\nu=1,2,\dots$ is defined by the equation (see [1], [18], (6))
$\vartheta_{1}[g_{\nu}(\tau)]=\frac{\pi}{2}\nu+\frac{\tau}{2};\
g_{\nu}(0)=g_{\nu}$
where (comp. (1.4))
$\vartheta_{1}(t)=\frac{t}{2}\ln\frac{t}{2\pi}-\frac{t}{2}-\frac{\pi}{8}.$
### 2.2.
In this paper we obtain some new integrals containing the following short
trigonometric sums
$\begin{split}&\sum_{2\leq p\leq\xi}\frac{1}{\sqrt{p}}\cos(t\ln p),\
\sum_{2\leq n\leq\xi}\frac{1}{\sqrt{n}}\cos(t\ln n),\\\ &\sum_{2\leq
n\leq\xi}\frac{d(n)}{\sqrt{n}}\cos(t\ln n)\end{split}$
where $p$ is the prime, $n\in\mathbb{N}$ and $d(n)$ is the number of divisors
of $n$. In this direction, the following theorem holds true.
###### Theorem.
Let
(2.4) $G_{3}(x)=\varphi_{1}(\mathring{G}_{3}(x)),\
G_{4}(y)=\varphi_{1}(\mathring{G}_{4}(y)).$
Then we have
(2.5) $\begin{split}&\int_{\mathring{G}_{3}(x)}\left(\sum_{2\leq
p\leq\xi}\frac{1}{\sqrt{p}}\cos(\varphi_{1}(t)\ln
p)\right)Z^{2}\\{\varphi_{1}(t)\\}\tilde{Z}^{2}(t){\rm d}t\sim\\\
&\frac{2x}{\pi}U\ln P\ln\ln P,\ x\in(0,\pi/2],\\\
&\int_{\mathring{G}_{4}(y)}\left(\sum_{2\leq
p\leq\xi}\frac{1}{\sqrt{p}}\cos(\varphi_{1}(t)\ln
p)\right)Z^{2}\\{\varphi_{1}(t)\\}\tilde{Z}^{2}(t){\rm d}t\sim\\\
&\frac{2y}{\pi}U\ln P\ln\ln P,\ y\in(0,\pi/2],\end{split}$
(2.6) $\begin{split}&\int_{\mathring{G}_{3}(x)}\left(\sum_{2\leq
n\leq\xi}\frac{1}{\sqrt{n}}\cos(\varphi_{1}(t)\ln
n)\right)Z^{2}\\{\varphi_{1}(t)\\}\tilde{Z}^{2}(t){\rm d}t\sim\\\
&\frac{1}{\pi}\left\\{\left(\frac{2\epsilon}{5}-\frac{\epsilon^{2}}{50}\right)x+\frac{\epsilon^{2}}{50}\sin
x\right\\}U\ln^{2}P,\\\ &\int_{\mathring{G}_{4}(y)}\left(\sum_{2\leq
n\leq\xi}\frac{1}{\sqrt{n}}\cos(\varphi_{1}(t)\ln
n)\right)Z^{2}\\{\varphi_{1}(t)\\}\tilde{Z}^{2}(t){\rm d}t\sim\\\
&\frac{1}{\pi}\left\\{\left(\frac{2\epsilon}{5}-\frac{\epsilon^{2}}{50}\right)y-\frac{\epsilon^{2}}{50}\sin
y\right\\}U\ln^{2}P,\end{split}$
(2.7) $\begin{split}&\int_{\mathring{G}_{3}(x)}\left(\sum_{2\leq
n\leq\xi}\frac{d(n)}{\sqrt{n}}\cos(\varphi_{1}(t)\ln
n)\right)Z^{2}\\{\varphi_{1}(t)\\}\tilde{Z}^{2}(t){\rm d}t\sim\\\ &\frac{\sin
x}{2500\pi^{3}}U\ln^{4}P,\\\ &\int_{\mathring{G}_{4}(y)}\left(\sum_{2\leq
n\leq\xi}\frac{d(n)}{\sqrt{n}}\cos(\varphi_{1}(t)\ln
n)\right)Z^{2}\\{\varphi_{1}(t)\\}\tilde{Z}^{2}(t){\rm d}t\sim\\\ &-\frac{\sin
y}{2500\pi^{3}}U\ln^{2}P,\end{split}$
where
(2.8) $t-\varphi_{1}(t)\sim(1-c)\pi(t),\ t\to\infty,$
and $\pi(t)$ is the prime-counting function.
###### Remark 3.
Let $T=\varphi_{1}(\mathring{T})$, $T+U=\varphi_{1}(\widering{T+U})$, (comp.
(2.4)). Then from (2.8), similarly to [14], (1.8), we obtain
$\rho\\{[T,T+U];[\mathring{T},\widering{T+U}]\\}\sim(1-c)\pi(T);\
T+U<\mathring{T},$
where $\rho$ stands for the distance of the corresponding segments.
###### Remark 4.
The formulae (2.5) - (2.7) cannot be obtained in the classical theory of A.
Selberg, and, all the less, in the theories of Balasubramanian, Heath-Brown
and Ivic.
## 3\. New asymptotic formulae for the short trigonometric sums: their
dependence on $|\zeta\left(\frac{1}{2}+it\right)|^{2}$
We obtain, putting $x=y=\pi/2$ in (2.5)
$\begin{split}&\int_{\mathring{G}_{3}(\pi/2)\cup\mathring{G}_{4}(\pi/2)}\left(\sum_{2\leq
p\leq\xi}\frac{1}{\sqrt{p}}\cos\\{\varphi_{1}(t)\ln
p\\}\right)Z^{2}\\{\varphi_{1}(t)\\}\tilde{Z}^{2}(t){\rm d}t\sim\\\ &2U\ln
P\ln\ln P.\end{split}$
Using successively the mean-value theorem (since
$\mathring{G}_{3}(\pi/2)\cup\mathring{G}_{4}(\pi/2)$ is a segment), we have
(3.1) $\begin{split}&\sum_{2\leq
p\leq\xi}\frac{1}{\sqrt{p}}\cos\\{\varphi_{1}(\alpha_{1})\ln
p\\}\int_{\mathring{G}_{3}(\pi/2)\cup\mathring{G}_{4}(\pi/2)}Z^{2}\\{\varphi_{1}(t)\\}Z^{2}(t){\rm
d}t=\\\ &=\sum_{2\leq
p\leq\xi}\frac{1}{\sqrt{p}}\cos\\{\varphi_{1}(\alpha_{1})\ln
p\\}Z^{2}\\{\varphi_{1}(\alpha_{2})\\}\int_{\mathring{G}_{3}(\pi/2)\cup\mathring{G}_{4}(\pi/2)}\tilde{Z}^{2}(t){\rm
d}t\sim\\\ &2U\ln P\ln\ln P,\
\alpha_{1},\alpha_{2}\in\mathring{G}_{3}(\pi/2)\cup\mathring{G}_{4}(\pi/2);\
\alpha_{1}=\alpha_{1}(T,U)=\alpha_{1}(T,\epsilon),\dots.\end{split}$
Since
(3.2)
$\int_{\mathring{G}_{3}(\pi/2)\cup\mathring{G}_{4}(\pi/2)}\tilde{Z}^{2}{\rm
d}t=\left|\mathring{G}_{3}(\pi/2)\cup\mathring{G}_{4}(\pi/2)\right|$
(comp. Remark 8), and
(3.3) $m\\{\mathring{G}_{3}(x)\\}\sim\frac{x}{\pi}U,\
m\\{\mathring{G}_{4}(y)\\}\sim\frac{y}{\pi}U\ \Rightarrow\
\left|\mathring{G}_{3}(\pi/2)\cup\mathring{G}_{4}(\pi/2)\right|\sim U$
(see [2], (13), $m$ stands for the measure) then we obtain from (2.5) (see
(3.1) - (3.3)) the following
###### Corollary 1.
For every $T\geq T_{0}[\varphi_{1}]$ there are the values
$\alpha_{1}(T),\alpha_{2}(T)\in\mathring{G}_{3}(\pi/2)\cup\mathring{G}_{4}(\pi/2)$
such that
(3.4) $\sum_{2\leq
p\leq\xi}\frac{1}{\sqrt{p}}\cos\\{\varphi_{1}(\alpha_{1}(T))\ln
p\\}\sim\frac{2\ln P\ln\ln
P}{\left|\zeta\left(\frac{1}{2}+i\varphi_{1}(\alpha_{2}(T))\right)\right|^{2}},\
T\to\infty$
where
$\varphi_{1}(\alpha_{1}(T)),\varphi_{1}(\alpha_{2}(T))\in\mathring{G}_{3}(\pi/2)\cup\mathring{G}_{4}(\pi/2)$.
Similarly, we obtain from (2.6)
###### Corollary 2.
For every $T\geq T_{0}[\varphi_{1}]$ there are the values
$\alpha_{3}(T),\alpha_{4}(T)\in\mathring{G}_{3}(\pi/2)\cup\mathring{G}_{4}(\pi/2)$
such that
(3.5) $\begin{split}&\sum_{2\leq
n\leq\xi}\frac{1}{\sqrt{n}}\cos\\{\varphi_{1}(\alpha_{3}(T))\ln n\\}\sim\\\
&\sim\left(\frac{2\epsilon}{5}-\frac{\epsilon^{2}}{50}\right)\frac{\ln^{2}P}{\left|\zeta\left(\frac{1}{2}+i\varphi_{1}(\alpha_{4}(T))\right)\right|^{2}},\
T\to\infty\end{split}$
where $\varphi_{1}(\alpha_{3}(T)),\varphi_{1}(\alpha_{4}(T))\in
G_{3}(\pi/2)\cup G_{4}(\pi/2)$.
###### Remark 5.
From the asymptotic formulae (3.4), (3.5) it follows that the values of
mentioned short trigonometric sums are connected with the values of the
Riemann zeta-function $\zeta\left(\frac{1}{2}+it\right)$ for some infinite
subset of $t$.
## 4\. New asymptotic formulae on two collections of disconnected sets
$G_{3}(x),G_{4}(y)$
From (2.7), similarly to p. 3, we obtain
###### Corollary 3.
(4.1) $\begin{split}&\left.\left\langle\sum_{2\leq
n\leq\xi}\frac{d(n)}{\sqrt{n}}\cos\\{\varphi_{1}(t)\ln
n\\}\right\rangle\right|_{\mathring{G}_{3}(x)}\sim\\\
&\sim\frac{1}{2500\pi^{2}}\frac{\sin x}{x}\frac{\ln^{4}P}{\langle
Z^{2}\\{\varphi_{1}(t)\\}\rangle|_{\mathring{G}_{3}(x)}}\\\
&\left.\left\langle\sum_{2\leq
n\leq\xi}\frac{d(n)}{\sqrt{n}}\cos\\{\varphi_{1}(t)\ln
n\\}\right\rangle\right|_{\mathring{G}_{4}(y)}\sim\\\
&\sim-\frac{1}{2500\pi^{2}}\frac{\sin y}{y}\frac{\ln^{4}P}{\langle
Z^{2}\\{\varphi_{1}(t)\\}\rangle|_{\mathring{G}_{4}(y)}},\
T\to\infty\end{split}$
where $\langle(\dots)\rangle|_{\mathring{G}_{3}(x)},\dots$ denote the mean-
value of $(\dots)$ on $\mathring{G}_{3}(x),\dots$ .
###### Remark 6.
It follows from (4.1) that the short trigonometric sum
$\sum_{2\leq n\leq\xi}\frac{d(n)}{\sqrt{n}}\cos\\{t\ln n\\},\ t\geq
T_{0}[\varphi_{1}]$
has an infinitely many zeroes of the odd order.
## 5\. Law of the asymptotic equality of areas
Let
$\begin{split}&\mathring{G}_{3}^{+}(x)=\left\\{t:\ t\in\mathring{G}_{3}(x),\
\sum_{2\leq n\leq\xi}\frac{d(n)}{\sqrt{n}}\cos\\{\varphi_{1}(t)\ln
n\\}>0\right\\},\\\ &\vdots\\\ &\mathring{G}_{4}^{-}(x)=\left\\{t:\
t\in\mathring{G}_{4}(x),\ \sum_{2\leq
n\leq\xi}\frac{d(n)}{\sqrt{n}}\cos\\{\varphi_{1}(t)\ln
n\\}<0\right\\}.\end{split}$
Then we obtain from (2.7), (comp. Corollary 3 in [14])
###### Corollary 4.
(5.1)
$\begin{split}&\int_{\mathring{G}_{3}^{+}(x)\cup\mathring{G}_{4}^{+}(x)}\left(\sum_{2\leq
n\leq\xi}\frac{d(n)}{\sqrt{n}}\cos\\{\varphi_{1}(t)\ln
n\\}\right)Z^{2}\\{\varphi_{1}(t)\\}\tilde{Z}^{2}(t){\rm d}t\sim\\\
&\sim-\int_{\mathring{G}_{3}^{-}(x)\cup\mathring{G}_{4}^{-}(x)}\left(\sum_{2\leq
n\leq\xi}\frac{d(n)}{\sqrt{n}}\cos\\{\varphi_{1}(t)\ln
n\\}\right)Z^{2}\\{\varphi_{1}(t)\\}\tilde{Z}^{2}(t){\rm d}t.\end{split}$
###### Remark 7.
The formula (5.1) represents the law of the asymptotic equality of the areas
(measures) of complicated figures corresponding to the positive part and the
negative part, respectively, of the graph of the function
(5.2) $\sum_{2\leq n\leq\xi}\frac{d(n)}{\sqrt{n}}\cos\\{\varphi_{1}(t)\ln
n\\}Z^{2}\\{\varphi_{1}(t)\\}\tilde{Z}^{2}(t),\
t\in\mathring{G}_{3}(x)\cup\mathring{G}_{4}(x),$
where $x\in(0,\pi/2]$. This is one of the laws governing the _chaotic_
behaviour of the positive and negative values of the signal (5.2). This signal
is created by the complicated modulation of the fundamental signal
$Z(t)=e^{i\vartheta(t)}\zeta\left(\frac{1}{2}+it\right)$, (comp. (1.4),
(2.2)).
## 6\. Proof of the Theorem
### 6.1.
Let us remind that the following lemma holds true (see [8], (2.5); [9],
(3.3)): for every integrable function (in the Lebesgue sense) $f(x),\
x\in[\varphi_{1}(T),\varphi_{1}(T+U)]$ we have
(6.1) $\int_{T}^{T+U}f[\varphi_{1}(t)]\tilde{Z}^{2}(t){\rm
d}t=\int_{\varphi_{1}(T)}^{\varphi_{1}(T+U)}f(x){\rm d}x,\ U\in(0,T/\ln T],$
where $t-\varphi_{1}(t)\sim(1-c)\pi(t)$. In the case (comp. (2.4))
$T=\varphi_{1}(\mathring{T})$, $T+U=\varphi_{1}(\widering{T+U})$, we obtain
from (6.1) the following equality
(6.2)
$\int_{\mathring{T}}^{\widering{T+U}}f[\varphi_{1}(t)]\tilde{Z}^{2}(t){\rm
d}t=\int_{T}^{T+U}f(x){\rm d}x.$
### 6.2.
First of all, we have from (6.2), for example,
$\int_{\mathring{g}_{2\nu}(-x)}^{\mathring{g}_{2\nu}(x)}f[\varphi_{1}(t)]\tilde{Z}^{2}(t){\rm
d}t=\int_{g_{2\nu}(-x)}^{g_{2\nu}(x)}f(t){\rm d}t,$
(see (2.3). Next, in the case
$f(t)=\left(\sum_{2\leq p\leq\xi}\frac{1}{\sqrt{p}}\cos\\{\varphi_{1}(t)\ln
p\\}\right)Z^{2}\\{\varphi_{1}(t)\\}$
we have the following $\tilde{Z}^{2}$-transformation
(6.3) $\begin{split}&\int_{\mathring{G}_{3}(x)}\left(\sum_{2\leq
p\leq\xi}\frac{1}{\sqrt{p}}\cos\\{\varphi_{1}(t)\ln
p\\}\right)Z^{2}\\{\varphi_{1}(t)\\}\tilde{Z}^{2}(t){\rm d}t=\\\
&=\int_{\mathring{G}_{3}(x)}\left(\sum_{2\leq
p\leq\xi}\frac{1}{\sqrt{p}}\cos\\{\varphi_{1}(t)\ln p\\}\right)Z^{2}(t){\rm
d}t,\\\ &\int_{\mathring{G}_{4}(y)}\left(\sum_{2\leq
p\leq\xi}\frac{1}{\sqrt{p}}\cos\\{\varphi_{1}(t)\ln
p\\}\right)Z^{2}\\{\varphi_{1}(t)\\}\tilde{Z}^{2}(t){\rm d}t=\\\
&=\int_{\mathring{G}_{4}(y)}\left(\sum_{2\leq
p\leq\xi}\frac{1}{\sqrt{p}}\cos\\{\varphi_{1}(t)\ln p\\}\right)Z^{2}(t){\rm
d}t.\end{split}$
Let us remind that we have proved (see [2], (13) and Corollary 7) the
following formulae
(6.4) $\begin{split}&\int_{G_{3}(x)}\left(\sum_{2\leq
p\leq\xi}\frac{1}{\sqrt{p}}\cos\\{\varphi_{1}(t)\ln p\\}\right)Z^{2}(t){\rm
d}t\sim\frac{2x}{\pi}\ln P\ln\ln P,\\\ &\int_{G_{4}(y)}\left(\sum_{2\leq
p\leq\xi}\frac{1}{\sqrt{p}}\cos\\{\varphi_{1}(t)\ln p\\}\right)Z^{2}(t){\rm
d}t\sim\frac{2y}{\pi}\ln P\ln\ln P.\end{split}$
Now, our formulae (2.5) follow from (6.3), (6.4).
### 6.3.
Similarly, from the formulae
$\begin{split}&\int_{G_{3}(x)}\left(\sum_{2\leq
n\leq\xi}\frac{1}{\sqrt{n}}\cos\\{t\ln n\\}\right)Z^{2}(t){\rm d}t\sim\\\
&\sim\frac{x}{\pi}\left(\frac{2\epsilon}{5}-\frac{\epsilon^{2}}{50}+\frac{\epsilon^{2}}{50}\frac{\sin
x}{x}\right)U\ln^{2}P,\\\ &\int_{G_{4}(y)}\left(\sum_{2\leq
n\leq\xi}\frac{1}{\sqrt{n}}\cos\\{t\ln n\\}\right)Z^{2}(t){\rm d}t\sim\\\
&\sim\frac{y}{\pi}\left(\frac{2\epsilon}{5}-\frac{\epsilon^{2}}{50}-\frac{\epsilon^{2}}{50}\frac{\sin
y}{y}\right)U\ln^{2}P,\end{split}$
and
$\begin{split}&\int_{G_{3}(x)}\left(\sum_{2\leq
n\leq\xi}\frac{d(n)}{\sqrt{n}}\cos\\{t\ln n\\}\right)Z^{2}(t){\rm
d}t\sim\frac{\sin x}{2500\pi^{3}}U\ln^{4}P,\\\
&\int_{G_{4}(y)}\left(\sum_{2\leq n\leq\xi}\frac{d(n)}{\sqrt{n}}\cos\\{t\ln
n\\}\right)Z^{2}(t){\rm d}t\sim-\frac{\sin
y}{2500\pi^{3}}U\ln^{4}P\end{split}$
(see [2], (13) and Corollaries 8 and 9) we obtain (2.6) and (2.7),
respectively.
###### Remark 8.
The formulae of type (3.2) can be obtained from (6.2) putting $f(t)\equiv 1$.
I would like to thank Michal Demetrian for helping me with the electronic
version of this work.
## References
* [1] A. Moser, ‘New mean-value theorems for the function $|\zeta\left(\frac{1}{2}+it\right)|^{2}$‘, Acta Math. Univ. Comen., 46-47 (1985), 21-40, (in russian).
* [2] J. Moser, ‘The structure of the A. Selberg’s formula in the theory of the Riemann zeta-function‘, Acta Math. Univ. Comen., 48-49, (1986), 93-121, (in russian).
* [3] J. Moser, ‘Jacob’s ladders and the almost exact asymptotic representation of the Hardy-Littlewood integral’, (2008), arXiv:0901.3973.
* [4] J. Moser, ‘Jacob’s ladders and the tangent law for short parts of the Hardy-Littlewood integral’, (2009), arXiv:0906.0659.
* [5] J. Moser, ‘Jacob’s ladders and the multiplicative asymptotic formula for short and microscopic parts of the Hardy-Littlewood integral’, (2009), arXiv:0907.0301.
* [6] J. Moser, ‘Jacob’s ladders and the quantization of the Hardy-Littlewood integral’, (2009), arXiv:0909.3928.
* [7] J. Moser, ‘Jacob’s ladders and the first asymptotic formula for the expression of the sixth order $|\zeta(1/2+i\varphi(t)/2)|^{4}|\zeta(1/2+it)|^{2}$’, (2009), arXiv:0911.1246.
* [8] J. Moser, ‘Jacob’s ladders and the first asymptotic formula for the expression of the fifth order $Z[\varphi(t)/2+\rho_{1}]Z[\varphi(t)/2+\rho_{2}]Z[\varphi(t)/2+\rho_{3}]\hat{Z}^{2}(t)$ for the collection of disconnected sets‘, (2009), arXiv:0912.0130.
* [9] J. Moser, ‘Jacob’s ladders, the iterations of Jacob’s ladder $\varphi_{1}^{k}(t)$ and asymptotic formulae for the integrals of the products $Z^{2}[\varphi^{n}_{1}(t)]Z^{2}[\varphi^{n-1}(t)]\cdots Z^{2}[\varphi^{0}_{1}(t)]$ for arbitrary fixed $n\in\mathbb{N}$‘ (2010), arXiv:1001.1632.
* [10] J. Moser, ‘Jacob’s ladders and the asymptotic formula for the integral of the eight order expression $|\zeta(1/2+i\varphi_{2}(t))|^{4}|\zeta(1/2+it)|^{4}$‘, (2010), arXiv:1001.2114.
* [11] J. Moser, ‘Jacob’s ladders and the asymptotically approximate solutions of a nonlinear diophantine equation‘, (2010), arXiv: 1001.3019.
* [12] J. Moser, ‘Jacob’s ladders and the asymptotic formula for short and microscopic parts of the Hardy-Littlewood integral of the function $|\zeta(1/2+it)|^{4}$‘, (2010), arXiv:1001.4007.
* [13] J. Moser, ‘Jacob’s ladders and the nonlocal interaction of the function $|\zeta(1/2+it)|$ with $\arg\zeta(1/2+it)$ on the distance $\sim(1-c)\pi(t)$‘, (2010), arXiv: 1004.0169.
* [14] J. Moser, ‘Jacob’s ladders and the $\tilde{Z}^{2}$ \- transformation of polynomials in $\ln\varphi_{1}(t)$‘, (2010), arXiv: 1005.2052.
* [15] J. Moser, ‘Jacob’s ladders and the oscillations of the function $|\zeta\left(\frac{1}{2}+it\right)|^{2}$ around the main part of its mean-value; law of the almost exact equality of the corresponding areas‘, (2010), arXiv: 1006.4316
* [16] J. Moser, ‘Jacob’s ladders and the nonlocal interaction of the function $Z(t)$ with the function $\tilde{Z}^{2}(t)$ on the distance $\sim(1-c)\pi(t)$ for a collection of disconneted sets‘, (2010), arXiv: 1006.5158
* [17] J. Moser, ‘Jacob’s ladders and the $\tilde{Z}^{2}$-transformation of the orthogonal system of trigonometric functions‘, (2010), arXiv: 1007.0108.
* [18] J. Moser, ‘Jacob’s ladders and the nonlocal interaction of the function $Z^{2}(t)$ with the function $\tilde{Z}^{2}(t)$ on the distance $\sim(1-c)\pi(t)$ for the collections of disconnected sets‘, (2010), arXiv: 1007.5147.
* [19] A. Selberg, ‘On the zeroes of Riemann’s zeta-function‘, Skr. Norske vid. Akad. Oslo, 10 (1942), 1-59.
* [20] E.C. Titchmarsh, ‘The theory of the Riemann zeta-function‘, Clarendon Press, Oxford, 1951.
|
arxiv-papers
| 2010-10-05T12:42:37 |
2024-09-04T02:49:13.477513
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jan Moser",
"submitter": "Michal Demetrian",
"url": "https://arxiv.org/abs/1010.0868"
}
|
1010.0975
|
# Remote sensing and control of phase qubits
Dale Li dale.li@boulder.nist.gov Fabio C.S. da Silva Danielle A. Braje
Raymond W. Simmonds David P. Pappas National Institute of Standards and
Technology, Boulder, Colorado 80305, USA
###### Abstract
We demonstrate a remote sensing design of phase qubits by separating the
control and readout circuits from the qubit loop. This design improves
measurement reliability because the control readout chip can be fabricated
using more robust materials and can be reused to test different qubit chips.
Typical qubit measurements such as Rabi oscillations, spectroscopy, and
excited-state energy relaxation are presented.
###### pacs:
Superconducting phase qubits are one of the most promising technologies for a
scalable quantum computer.MartinisQIP Introduction and improvement of
specialized materials and structures has significantly reduced losses and
improved coherence times.Oh However, evaluation of these materials creates
challenges in the design and fabrication of qubit circuits primarily because
of variations in material composition and crystalline order.Kline The ability
to explore different materials would be greatly simplified if the control and
readout circuit to measure the qubit could be fabricated separately from the
qubit devices under investigation. The readout circuit could then be made of
well-established materials and designs, and would operate reliably independent
of materials being used for the qubits. In this letter, we developed a self-
aligning flip-chip technique to separate the qubit circuit from its readout.
The readout chip is inductively coupled to the phase qubit, and contains the
SQUID readout and the superconducting coils for microwave and dc flux control.
Previous superconducting circuits have used flip-chips to perform noise and
remote detection measurements BerggrenRemote ; MayCoil . Flip-chip
implementations of charge qubits operating as interferometers have also been
reported.BornChargeQ In addition, flip-chips have been used to separate
dissipative single-flux quantum (SFQ) circuits from the temperature-sensitive
qubit circuits.YorozuFlux Bennett et al. used a separate chip suspended above
an rf-SQUID qubit chip to obtain fast bias pulses.BennettFlip Steffen et al.
describe a SQUID-less readout scheme that reduces the number of junctions in
the qubit to one (the qubit junction itself). This scheme allows for the
multiplexing of many qubits. However, the overall performance of the system is
affected by the coupling between the microwave feed line and the qubit
circuit.DispReadout Michotte uses the flip-chip technique to separate the
microstrip line from the SQUID sensor in a microstrip-SQUID
amplifier.MicrostripFlip
Figure 1: (color online). (a) The top chip is self-aligned a distance $z$
above the bottom chip by use of sapphire spheres of diameter $D$. The top chip
pocket diameter $d$ is determined by Eq. (1) with a fixed bottom chip pocket
depth of $h$. (b) The top chip and bottom chip are separated, showing the
alignment sites and the scale of each chip. Note that the top chip is smaller
than the bottom chip to allow space for wire bonding. (c) The assembled flip-
chip.
Our flip-chip design contains the phase qubit loop on the top chip, which
self-aligns, by use of four $200\pm 2.5$ $\mu$m diameter sapphire spheres, to
the bottom chip containing the control/readout circuitry. Sapphire spheres
have a small thermal contraction coefficient, which helps to maintain proper
alignment when the sample is cooled to dilution-refrigerator temperatures. The
spheres sit in pockets etched into the silicon substrates by a deep reactive
ion etcher.
Figure 1(a) shows a cross-sectional drawing of the deeply etched cylindrical
pockets in the top and bottom chips and the self-aligning sapphire spheres.
The diameter of the top chip pocket is given by $d=2\sqrt{(D-h-z)(h+z)}$,
where $D$ is the diameter of the sapphire sphere, $h$ is the depth of the
pocket etched into the bottom chip (with etched diameter equal to $D$), and
$z$ is the desired vacuum gap size. Deep pockets in the bottom chip held the
sapphire spheres in place for reuse, while the shallower pockets in the top
chip were etched deep enough that the sapphire spheres only touch the top chip
at the edges of the pockets. Different pocket diameters for different top
chips were fabricated, giving vacuum gap sizes from 10 $\mu$m to 50 $\mu$m.
Photographs of the fabricated top and bottom chips are shown separately, with
the bottom chip wire-bonded to a test board in Fig. 1(b), and in the flip-chip
configuration in Fig. 1(c). The four positions for the sapphire spheres
facilitate a stable self-alignment, minimize wobble, and place the spheres far
away from the circuit elements. The entire flip-chip assembly is held together
under slight compression by a beryllium-copper leaf spring placed inside a
brass lid, which encloses the two chips and fastens to the circuit board.
Figure 2: (color online). (a) Flip-chip circuit drawing shows the simple qubit
circuit and the three inductively coupled control coils for microwave
excitation, DC flux bias, and DC SQUID bias, as well as the DC three-junction
SQUID for qubit readout. (b) A photograph of qubit loop near the final steps
of fabrication as patterned on the top chip. A final wiring layer connects the
junction and the via (not shown). (c) Photograph of measurement and excitation
circuitry as fabricated on the bottom chip. The dashed large rectangle
indicates where the qubit will align.
Figure 2(a) shows the circuit model for the entire phase qubit including
control and readout (C/R). The C/R circuit consists of a three-junction dc
SQUID (readout), a dc flux bias loop that applies magnetic flux to the qubit
(control), a secondary dc flux bias loop to tune the magnetic flux in the
SQUID (control), and a microwave flux loop that excites the qubit with
microwave frequencies (control). Each inductive loop utilizes a gradiometric
design to minimize both unwanted cross-coupling between coils and the effects
of shifts in background homogeneous magnetic fields by symmetric placement.
Fig. 2(b) shows a photograph of the qubit loop as patterned on the top chip.
To test this flip-chip approach, standard Al/amorphous-Al2O3/Al Josephson
junctions $13$ $\mu$m2 in area were designed and fabricated for qubit
frequencies around 7 GHz. The qubit loop was closed by an Al cross-over wire
connecting the junction and the via (not shown). Fig. 2(c) shows a photograph
of the C/R circuitry above which the qubit loop is placed (dashed rectangle)
when aligned.
For a vacuum gap size $z=20$ $\mu$m, the mutual inductance coupling terms were
calculated between pairs of coils (qubit-SQUID: 71 pH, qubit-flux bias: 5.5
pH, qubit-SQUID bias: $<$1 pH, qubit-microwave line: 5.5 pH, SQUID-SQUID bias:
2 pH, SQUID-flux bias: $<$1 pH). The qubit loop was designed with a self
inductance of 880 pH, while the SQUID was designed with a self-inductance of
341 pH. These large inductances ensured a strong measurable coupling between
the qubit chip and the C/R chip, although smaller inductances could also
provide adequate coupling, depending on the gap size.
Figure 3: (color online). Qubit steps for two different qubit chips (same
readout chip) showing different coupling. (a) $z$=$10$ $\mu$m gap size. The
steps are curved due to the large overlap coupling to the dc SQUID. A flux
quantum in the qubit is observed with the applied voltage $\Phi_{0}$=$44.6$
mV. (b) $z$=$20$ $\mu$m gap size has weaker coupling and samples just the
linear regime of the SQUID. A larger applied voltage is needed to excite a
flux quantum with $\Phi_{0}$=$766$ mV.
We tested the remote sensing and control of the phase qubit with four typical
measurements showing coherent control and reliable readout: qubit steps,
spectroscopy, Rabi oscillations, and $T_{1}$.HistFitting Additionally, the
response of the SQUID was measured as a function of the applied flux through
the SQUID bias line in order to test the C/R circuit independently of the
qubit. The SQUID bias line also provided the ability to tune the SQUID to a
sensitive, mostly linear regime.
First, we measured the qubit steps by applying a magnetic flux to the qubit
loop and measuring the corresponding value of the SQUID switching current
$I_{s}$. Here, the applied flux is measured in units of the voltage across a
10 k$\Omega$ resistor connected in series with the qubit bias coil. Figure
3(a) shows the behavior of $I_{s}$ versus the applied flux for a gap size
between the bottom and top chips of $10$ $\mu$m. The pronounced nonlinear
behavior of $I_{s}$ arises from a large field change as sensed by the SQUID at
different qubit states, which maps to a larger, less linear regime in the
SQUID response. For this gap size of $10$ $\mu$m, the voltage difference
necessary to induce a quantum of flux ($\Phi_{0}$) variation in the qubit is
44.6 mV. For an increased gap size of $20$ $\mu$m, the flux bias voltage per
flux quantum increased to 766 mV as shown in Fig. 3(b). This change in flux
bias per flux quantum corresponds to a reduction of the coupling by a factor
of 17. Furthermore, the reduction of coupling decreased the amount of qubit
flux sensed by the SQUID so that its response mapped to a more linear regime,
as shown in Fig. 3(b).
Figure 4: (color online). Data collected from $z=20$ $\mu$m gap sized flip-
chip. (a) Spectroscopy data showing the tunability of the qubit resonant
frequency as a function of the applied flux from the bottom chip. The inset
shows a zoom in of one of many splittings due to coupling with parasitic two
level systems in this qubit. (b) Rabi Oscillations in the qubit from microwave
excitation. (c) Relaxation time measurement.
Second, we measured the qubit spectroscopy for a gap size of 20 $\mu$m. The
phase qubit exhibits a tunable absorption spectrum at its transition frequency
($\omega_{01}$) between the ground and first excited state. In Fig. 4(a) the
qubit spectroscopy shows a 2 GHz range of $\omega_{01}$ values centered around
7 GHz. The visibility of only one transition line in the spectroscopy data
indicates that the qubit chip was cooled to low enough temperatures to be
operated as a qubit. The discontinuities in the spectrum are assumed to be due
to parasitic two-level systems in the large-area amorphous-Al2O3 tunnel
barrier.Simmonds2004 A zoom-in of one such discontinuity is shown in the
inset.
Third, Fig. 4(b) shows Rabi oscillations in the same qubit. This experiment is
performed by holding a constant dc flux bias in a region of the spectroscopy
with few discontinuities and applying a microwave pulse for a varied period.
Rabi oscillations demonstrate the ability for state mixing between the ground
and first excited state of the qubit. The oscillation amplitude decays due to
decoherence with a spin bath and should ideally saturate to a $50\%$
occupation probability. In the data, the saturation occurs at about a $33\%$
occupation probability. This discrepancy is due to the measurement process,
which sweeps the coupling of the qubit through many avoided crossings with
parasitic two-level systems that syphon energy from the qubit in Landau-Zener-
like transitions.Cooper2004
Fourth, Fig. 4(c) shows a longitudinal relaxation experiment in the same
qubit. In this experiment, a partially excited qubit state is prepared with a
fixed microwave pulse length of 50 ns, and the qubit state is measured as a
function of time as it decays to its ground state. Our flip-chip test used
similar design considerations, materials, and fabrication techniques as for
integrated chips so we expected the experimental data to agree with previous
results without the introduction of additional noise or loss. Though the
observed relaxation time $T_{1}=23$ ns is short, it matches reported results
for a phase qubit with a $13$ $\mu$m2 thermally oxidized amorphous Al2O3
tunnel barrier on a Silicon substrate.Cooper2004
In conclusion, we demonstrated the remote sensing and control of a phase qubit
by separating the qubit loop and the control/readout (C/R) circuit. Typical
characterization and performance measurements done in several qubit loops with
the same C/R circuit demonstrated reliability and robustness of this design.
The technique has therefore proven to be an adequate candidate for studying
the improvement of specialized materials and structures for superconducting
qubits. Other types of qubits, such as flux qubits could also potentially use
the same flip-chip technique either by direct coupling across a smaller
controlled gap, or by mediated coupling through a resonator circuit or rf-
SQUID.Shane
This research was funded in part by the Office of the Director of National
Intelligence (ODNI) and by Intelligence Advanced Research Projects Activity
(IARPA). All statements of fact, opinion or conclusions contained herein are
those of the authors and should not be construed as representing the official
views or policies of IARPA, the ODNI, or the U.S. Government. Official
contribution of the National Institute of Standards and Technology; not
subject to copyright in the United States.
## References
* (1) J.M. Martinis, Quantum Inf. Process 8, 81 (2009).
* (2) S. Oh, K. Cicak, J.S. Kline, M.A. Sillanpaa, K.D. Osborn, J.D. Whittaker, R.W. Simmonds, and D.P. Pappas, Phys. Rev. B 74, 100502(R) (2006).
* (3) J.S. Kline, H. Wang, S. Oh, J.M. Martinis, and D.P. Pappas, Supercond. Sci. Technol. 22, 015004 (2009).
* (4) K.K. Berggren, D. Nakadab, M. J. O’Hara, T.P. Orlando, E.M. Macedo, R. Slattery, T. Weir, An Integrated Superconductive Device Technology for Qubit Control (Rinton, Princeton, 2001), pp. 121-126.
* (5) T. May, E. Il’ichev, and H.-G. Meyer. Rev Sci. Instrum. 74, 1282 (2003).
* (6) D. Born, V.I. Shnyrkov, W. Krech, Th. Wagner, E. Il’ichev, M. Grajcar, U. Hubner, H.-G.Meyer, Phys. Rev. B 70, 180501(R) (2004).
* (7) S. Yorozu, T. Miyazaki, V. Semenov, Y. Nakamura, Y. Hashimoto, K. Hinode, T. Sate, Y. Kameda, and J.S. Tsai. J. Phys: Conf. Series 43, 1417 (2006).
* (8) D.A. Bennett, L. Longobardi, V. Patel, W. Chen, J.E. Lukens. Supercond. Sci. Technol. 20, S445 (2007).
* (9) M. Steffen, S. Kumar, D. DiVincenzo, G. Keefe, M. Ketchen, M.B. Rothwell, and J. Rozen. Appl. Phys. Lett. 96, 102506 (2010).
* (10) S. Michotte. Appl. Phys. Lett. 94, 122512 (2009).
* (11) J. Lisenfeld, A. Lukashenko, and A. V. Ustinova. Appl. Phys. Lett. 91, 232502 (2007).
* (12) R.W. Simmonds, K.M. Lang, D.A. Hite, S. Nam, D.P. Pappas, and John M. Martinis. Phys. Rev. Lett. 93, 077003 (2004).
* (13) K.B. Cooper, Matthias Stefen, R. McDermott, R.W. Simmonds, Seongshik Oh, D.A. Hite, D.P. Pappas, and John M. Martinis. Phys. Rev. Lett. 93 180401 (2004).
* (14) M.S. Allman, F. Altomare, J.D. Whittaker, K. Cicak, D. Li, A. Sirois, J. Strong, J.D. Teufel, and R.W. Simmonds. Phys. Rev. Lett. 104 177004 (2010).
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arxiv-papers
| 2010-10-05T18:45:44 |
2024-09-04T02:49:13.489497
|
{
"license": "Public Domain",
"authors": "Dale Li, Fabio C. S. da Silva, Danielle A. Braje, Raymond W. Simmonds,\n and David P. Pappas",
"submitter": "Dale Li",
"url": "https://arxiv.org/abs/1010.0975"
}
|
1010.1044
|
# On the Capacity of the $K$-User Cyclic Gaussian Interference Channel
††thanks: Manuscript received October 4, 2010; revised May 8, 2012; accepted
August 28, 2012. Date of current version August 30, 2012. This work was
supported by the Natural Science and Engineering Research Council (NSERC). The
material in this paper was presented in part at the 2010 IEEE Conference on
Information Science and Systems (CISS), and in part at the 2011 IEEE Symposium
of Information Theory (ISIT). ††thanks: The authors are with The Edward S.
Rogers Sr. Department of Electrical and Computer Engineering, University of
Toronto, Toronto, ON M5S 3G4 Canada (email: zhoulei@comm.utoronto.ca;
weiyu@comm.utoronto.ca). Kindly address correspondence to Lei Zhou
(zhoulei@comm.utoronto.ca). ††thanks: Copyright (c) 2012 IEEE. Personal use of
this material is permitted. However, permission to use this material for any
other purposes must be obtained from the IEEE by sending a request to pubs-
permissions@ieee.org.
Lei Zhou, Student Member, IEEE and Wei Yu, Senior Member, IEEE
###### Abstract
This paper studies the capacity region of a $K$-user cyclic Gaussian
interference channel, where the $k$th user interferes with only the $(k-1)$th
user (mod $K$) in the network. Inspired by the work of Etkin, Tse and Wang,
who derived a capacity region outer bound for the two-user Gaussian
interference channel and proved that a simple Han-Kobayashi power splitting
scheme can achieve to within one bit of the capacity region for all values of
channel parameters, this paper shows that a similar strategy also achieves the
capacity region of the $K$-user cyclic interference channel to within a
constant gap in the weak interference regime. Specifically, for the $K$-user
cyclic Gaussian interference channel, a compact representation of the Han-
Kobayashi achievable rate region using Fourier-Motzkin elimination is first
derived, a capacity region outer bound is then established. It is shown that
the Etkin-Tse-Wang power splitting strategy gives a constant gap of at most 2
bits in the weak interference regime. For the special 3-user case, this gap
can be sharpened to $1\frac{1}{2}$ bits by time-sharing of several different
strategies. The capacity result of the $K$-user cyclic Gaussian interference
channel in the strong interference regime is also given. Further, based on the
capacity results, this paper studies the generalized degrees of freedom (GDoF)
of the symmetric cyclic interference channel. It is shown that the GDoF of the
symmetric capacity is the same as that of the classic two-user interference
channel, no matter how many users are in the network.
###### Index Terms:
Approximate capacity, Han-Kobayashi, Fourier-Motzkin, K-user interference
channel, multicell processing.
## I Introduction
The interference channel models a communication scenario in which several
mutually interfering transmitter-receiver pairs share the same physical
medium. The interference channel is a useful model for many practical systems
such as the wireless network. The capacity region of the interference channel,
however, has not been completely characterized, even for the two-user Gaussian
case.
The largest known achievable rate region for the two-user interference channel
is given by Han and Kobayashi [1] using a coding scheme involving common-
private power splitting. Chong et al. [2] obtained the same rate region in a
simpler form by applying the Fourier-Motzkin algorithm together with a time-
sharing technique to the Han and Kobayashi’s rate region characterization. The
optimality of the Han-Kobayashi region for the two-user Gaussian interference
channel is still an open problem in general, except in the strong interference
regime where transmission with common information only achieves the capacity
region [1, 3, 4], and in a noisy interference regime where transmission with
private information only achieves the sum capacity [5, 6, 7].
In a breakthrough, Etkin, Tse and Wang [8] showed that the Han-Kobayashi
scheme can in fact achieve to within one bit of the capacity region for the
two-user Gaussian interference channel for all channel parameters. Their key
insight was that the interference-to-noise ratio (INR) of the private message
should be chosen to be as close to $1$ as possible in the Han-Kobayashi
scheme. They also found a new capacity region outer bound using a genie-aided
technique. In the rest of this paper, we refer this particular setting of the
private message power as the Etkin-Tse-Wang (ETW) power-splitting strategy.
The Etkin, Tse and Wang’s result applies only to the two-user interference
channel. Practical systems often have more than two transmitter-receiver
pairs, yet the generalization of Etkin, Tse and Wang’s work to the
interference channels with more than two users has proved difficult for the
following reasons. First, it appears that the Han-Kobayashi common-private
superposition coding is no longer adequate for the $K$-user interference
channel. Interference alignment types of coding scheme [9] [10] can
potentially enlarge the achievable rate region. Second, even within the Han-
Kobayashi framework, when more than two receivers are involved, multiple
common messages at each transmitter may be needed, making the optimization of
the resulting rate region difficult.
In the context of $K$-user Gaussian interference channels, sum capacity
results are available in the noisy interference regime [5, 11]. In particular,
Annapureddy et al. [5] obtained the sum capacity for the symmetric three-user
Gaussian interference channel, the one-to-many, and the many-to-one Gaussian
interference channels under the noisy interference criterion. Similarly, Shang
et al. [11] studied the fully connected $K$-user Gaussian interference channel
and showed that treating interference as noise at the receiver is sum-capacity
achieving when the transmit power and the cross channel gains are sufficiently
weak to satisfy a certain criterion. Further, achievability and outer bounds
for the three-user interference channel have also been studied in [12] and
[13]. Finally, much work has been carried out on the generalized degree of
freedom (GDoF as defined in [8]) of the $K$-user interference channel and its
variations [9, 14, 15, 16].
Figure 1: The circular array soft-handoff model
Instead of treating the general $K$-user interference channel, this paper
focuses on a cyclic Gaussian interference channel, where the $k$th user
interferes with only the $(k-1)$th user. In this case, each transmitter
interferes with only one other receiver, and each receiver suffers
interference from only one other transmitter, thereby avoiding the
difficulties mentioned earlier. For the $K$-user cyclic interference channel,
the Etkin, Tse and Wang’s coding strategy remains a natural one. The main
objective of this paper is to show that it indeed achieves to within a
constant gap of the capacity region for this cyclic model in the weak
interference regime to be defined later.
The cyclic interference channel model is motivated by the so-called modified
Wyner model, which describes the soft handoff scenario of a cellular network
[17]. The original Wyner model [18] assumes that all cells are arranged in a
linear array with the base-stations located at the center of each cell, and
where intercell interference comes from only the two adjacent cells. In the
modified Wyner model [17] cells are arranged in a circular array as shown in
Fig. 1. The mobile terminals are located along the circular array. If one
assumes that the mobile terminals always communicate with the intended base-
station to their left (or right), while only suffering from interference due
to the base-station to their right (or left), one arrives at the $K$-user
cyclic Gaussian interference channel studied in this paper. The modified Wyner
model has been extensively studied in the literature [17, 19, 20], but often
either with interference treated as noise or with the assumption of full base-
station cooperation. This paper studies the modified Wyner model without base-
station cooperation, in which case the soft-handoff problem becomes that of a
cyclic interference channel.
This paper primarily focuses on the $K$-user cyclic Gaussian interference
channel in the weak interference regime. The main contributions of this paper
are as follows. This paper first derives a compact characterization of the
Han-Kobayashi achievable rate region by applying the Fourier-Motzkin
elimination algorithm. A capacity region outer bound is then obtained. It is
shown that with the Etkin, Tse and Wang’s coding strategy, one can achieve to
within $1\frac{1}{2}$ bits of the capacity region when $K=3$ (with time-
sharing), and to within two bits of the capacity region in general in the weak
interference regime. Finally, the capacity result for the strong interference
regime is also derived.
A key part of the development involves a Fourier-Motzkin elimination procedure
on the achievable rate region of the $K$-user cyclic interference channel. To
deal with the large number of inequality constraints, an induction proof is
used. It is shown that as compared to the two-user case, where the rate region
is defined by constraints on the individual rate $R_{i}$, the sum rate
$R_{1}+R_{2}$, and the sum rate plus an individual rate $2R_{i}+R_{j}$ ($i\neq
j$), the achievable rate region for the $K$-user cyclic interference channel
is defined by an additional set of constraints on the sum rate of any
arbitrary $l$ adjacent users, where $2\leq l<K$. These four types of rate
constraints completely characterize the Han-Kobayashi region for the $K$-user
cyclic interference channel. They give rise to a total of $K^{2}+1$
constraints.
For the symmetric $K$-user cyclic channel where all direct links share the
same channel gain and all cross links share another channel gain, it is shown
that the GDoF of the symmetric capacity is not dependent on the number of
users in the network. Therefore, adding more users to a $K$-user cyclic
interference channel with symmetric channel parameters does not affect the
per-user rate.
## II Channel Model
Figure 2: $K$-user cyclic Gaussian interference channel
The $K$-user cyclic Gaussian interference channel (as depicted in Fig. 2) has
$K$ transmitter-receiver pairs. Each transmitter tries to communicate with its
intended receiver while causing interference to only one neighboring receiver.
Each receiver receives a signal intended for it and an interference signal
from only one neighboring sender plus an additive white Gaussian noise (AWGN).
As shown in Fig. 2, $X_{1},X_{2},\cdots X_{K}$ and $Y_{1},Y_{2},\cdots Y_{K}$
are the complex-valued input and output signals, respectively, and
$Z_{i}\thicksim\mathcal{CN}(0,\sigma^{2})$ is the independent and identically
distributed (i.i.d) Gaussian noise at receiver $i$. The input-output model can
be written as
$\displaystyle Y_{1}$ $\displaystyle=$ $\displaystyle
h_{1,1}X_{1}+h_{2,1}X_{2}+Z_{1},$ $\displaystyle Y_{2}$ $\displaystyle=$
$\displaystyle h_{2,2}X_{2}+h_{3,2}X_{3}+Z_{2},$ $\displaystyle\vdots$
$\displaystyle Y_{K}$ $\displaystyle=$ $\displaystyle
h_{K,K}X_{K}+h_{1,K}X_{1}+Z_{K},$
where each $X_{i}$ has a power constraint $P_{i}$ associated with it, i.e.,
$\mathbb{E}\left[|x_{i}|^{2}\right]\leq P_{i}$. Here, $h_{i,j}$ is the channel
gain from transmitter $i$ to receiver $j$.
Define the signal-to-noise and interference-to-noise ratios for each user as
follows:
$\mathsf{SNR}_{i}=\frac{|h_{i,i}|^{2}P_{i}}{\sigma^{2}}\quad\mathsf{INR}_{i}=\frac{|h_{i,i-1}|^{2}P_{i}}{\sigma^{2}},\quad
i=1,2,\cdots,K.$ (1)
The $K$-user cyclic Gaussian interference channel is said to be in the weak
interference regime if
$\mathsf{INR}_{i}\leq\mathsf{SNR}_{i},\quad\forall i=1,2,\cdots,K.$ (2)
and the strong interference regime if
$\mathsf{INR}_{i}\geq\mathsf{SNR}_{i},\quad\forall i=1,2,\cdots,K.$ (3)
Otherwise, it is said to be in the mixed interference regime. Throughout this
paper, modulo arithmetic is implicitly used on the user indices, e.g., $K+1=1$
and $1-1=K$. Note that when $K=2$, the cyclic channel reduces to the
conventional two-user interference channel.
## III Within Two Bits of the Capacity Region in the Weak Interference Regime
The generalization of Etkin, Tse and Wang’s result to the capacity region of a
general (nonsymmetric) $K$-user cyclic Gaussian interference channel is
significantly more complicated. In the two-user case, the shape of the Han-
Kobayashi achievable rate region is the union of polyhedrons (each
corresponding to a fixed input distribution) with boundaries defined by rate
constraints on $R_{1}$, $R_{2}$, $R_{1}+R_{2}$, $2R_{1}+R_{2}$ and
$2R_{2}+R_{1}$, respectively. In the multiuser case, to extend Etkin, Tse and
Wang’s result, one needs to find a similar rate region characterization for
the general $K$-user cyclic interference channel first.
A key feature of the cyclic Gaussian interference channel model is that each
transmitter sends signal to its intended receiver while causing interference
to only one of its neighboring receivers; meanwhile, each receiver receives
the intended signal plus the interfering signal from only one of its
neighboring transmitters. Using this fact and with the help of Fourier-Motzkin
elimination algorithm, this section shows that the achievable rate region of
the $K$-user cyclic Gaussian interference channel is the union of polyhedrons
with boundaries defined by rate constraints on the individual rates $R_{i}$,
the sum rate $R_{sum}$, the sum rate plus an individual rate $R_{sum}+R_{i}$
($i=1,2,\cdots,K$), and the sum rate for arbitrary $l$ adjacent users ($2\leq
l<K$). This last rate constraint on arbitrary $l$ adjacent users’ rates is new
as compared with the two-user case.
The preceding characterization together with outer bounds to be proved later
in the section allows us to prove that the capacity region of the $K$-user
cyclic Gaussian interference channel can be achieved to within a constant gap
using the ETW power-splitting strategy in the weak interference regime.
However, instead of the one-bit result for the two-user interference channel,
this section shows that one can achieve to within $1\frac{1}{2}$ bits of the
capacity region when $K=3$ (with time-sharing), and within two bits of the
capacity region for general $K$. Again, the strong interference regime is
treated later.
### III-A Achievable Rate Region
###### Theorem 1
Let $\mathcal{P}$ denote the set of probability distributions $P(\cdot)$ that
factor as
$\displaystyle P(q,w_{1},x_{1},w_{2},x_{2},\cdots,w_{K},x_{K})$ (4)
$\displaystyle=p(q)p(x_{1}w_{1}|q)p(x_{2}w_{2}|q)\cdots p(x_{K}w_{K}|q).$
For a fixed $P\in\mathcal{P}$, let $\mathcal{R}_{\mathrm{HK}}^{(K)}(P)$ be the
set of all rate tuples $(R_{1},R_{2},\cdots,R_{K})$ satisfying
$\displaystyle 0\leq R_{i}$ $\displaystyle\leq$
$\displaystyle\min\\{d_{i},a_{i}+e_{i-1}\\},$ (5)
$\displaystyle\sum_{j=m}^{m+l-1}R_{j}$ $\displaystyle\leq$
$\displaystyle\min\left\\{g_{m}+\sum_{j=m+1}^{m+l-2}e_{j}+a_{m+l-1},\right.$
(6)
$\displaystyle\left.\qquad\quad\sum_{j=m-1}^{m+l-2}e_{j}+a_{m+l-1}\right\\},$
$\displaystyle\sum_{j=1}^{K}R_{j}$ $\displaystyle\leq$
$\displaystyle\min\left\\{\sum_{j=1}^{K}e_{j},r_{1},r_{2},\cdots,r_{K}\right\\},$
(7) $\displaystyle\sum_{j=1}^{K}R_{j}+R_{i}$ $\displaystyle\leq$
$\displaystyle a_{i}+g_{i}+\sum_{j=1,j\neq i}^{K}e_{j},$ (8)
where $a_{i},d_{i},e_{i},g_{i}$ and $r_{i}$ are defined as follows:
$\displaystyle a_{i}$ $\displaystyle=$ $\displaystyle
I(Y_{i};X_{i}|W_{i},W_{i+1},Q)$ (9) $\displaystyle d_{i}$ $\displaystyle=$
$\displaystyle I(Y_{i};X_{i}|W_{i+1},Q)$ (10) $\displaystyle e_{i}$
$\displaystyle=$ $\displaystyle I(Y_{i};W_{i+1},X_{i}|W_{i},Q)$ (11)
$\displaystyle g_{i}$ $\displaystyle=$ $\displaystyle
I(Y_{i};W_{i+1},X_{i}|Q)$ (12)
$r_{i}=a_{i-1}+g_{i}+\sum_{j=1,j\notin\\{i,i-1\\}}^{K}e_{j},$ (13)
and the range of indices are $i,m=1,2,\cdots,K$ in (5) and (8),
$l=2,3,\cdots,K-1$ in (6). Define
$\mathcal{R}_{\mathrm{HK}}^{(K)}=\bigcup_{P\in\mathcal{P}}\mathcal{R}_{\mathrm{HK}}^{(K)}(P).$
(14)
Then $\mathcal{R}_{\mathrm{HK}}^{(K)}$ is an achievable rate region for the
$K$-user cyclic interference channel 111The same achievable rate region has
been found independently in [21]..
###### Proof:
The achievable rate region can be proved by the Fourier-Motzkin algorithm
together with an induction step. The proof follows the Kobayashi and Han’s
strategy [22] of eliminating a common message at each step. The details are
presented in Appendix -A. ∎
In the above achievable rate region, (5) is the constraint on the achievable
rate of an individual user, (6) is the constraint on the achievable sum rate
for any $l$ adjacent users ($2\leq l<K$), (7) is the constraint on the
achievable sum rate of all $K$ users, and (8) is the constraint on the
achievable sum rate for all $K$ users plus a repeated one. We can also think
of (5)-(8) as the sum-rate constraints for arbitrary $l$ adjacent users, where
$l=1$ for (5), $2\leq l<K$ for (6), $l=K$ for (7) and $l=K+1$ for (8).
From (5) to (8), there are a total of $K+K(K-2)+1+K=K^{2}+1$ constraints.
Together they describe the shape of the achievable rate region under a fixed
input distribution. The quadratic growth in the number of constraints as a
function of $K$ makes the Fourier-Motzkin elimination of the Han-Kobayashi
region quite complex. The proof in Appendix -A uses induction to deal with the
large number of the constraints.
As an example, for the two-user Gaussian interference channel, there are
$2^{2}+1=5$ rate constraints, corresponding to that of $R_{1}$, $R_{2}$,
$R_{1}+R_{2}$, $2R_{1}+R_{2}$ and $2R_{2}+R_{1}$, as in [1, 22, 2, 8].
Specifically, substituting $K=2$ in Theorem 1 gives us the following
achievable rate region:
$\displaystyle 0\leq R_{1}$ $\displaystyle\leq$
$\displaystyle\min\\{d_{1},a_{1}+e_{2}\\},$ (15) $\displaystyle 0\leq R_{2}$
$\displaystyle\leq$ $\displaystyle\min\\{d_{2},a_{2}+e_{1}\\},$ (16)
$\displaystyle R_{1}+R_{2}$ $\displaystyle\leq$
$\displaystyle\min\\{e_{1}+e_{2},a_{1}+g_{2},a_{2}+g_{1}\\},$ (17)
$\displaystyle 2R_{1}+R_{2}$ $\displaystyle\leq$ $\displaystyle
a_{1}+g_{1}+e_{2},$ (18) $\displaystyle 2R_{2}+R_{1}$ $\displaystyle\leq$
$\displaystyle a_{2}+g_{2}+e_{1}.$ (19)
The above region for the two-user Gaussian interference channel is exactly
that of Theorem D in [22].
### III-B Capacity Region Outer Bound
###### Theorem 2
For the $K$-user cyclic Gaussian interference channel in the weak interference
regime, the capacity region is included in the set of rate tuples
$(R_{1},R_{2},\cdots,R_{K})$ such that
$\displaystyle R_{i}$ $\displaystyle\leq$ $\displaystyle\lambda_{i},$ (20)
$\displaystyle\sum_{j=m}^{m+l-1}R_{j}$ $\displaystyle\leq$
$\displaystyle\min\left\\{\gamma_{m}+\sum_{j=m+1}^{m+l-2}\alpha_{j}+\beta_{m+l-1},\right.$
(21)
$\displaystyle\left.\qquad\mu_{m}+\sum_{j=m}^{m+l-2}\alpha_{j}+\beta_{m+l-1}\right\\},$
$\displaystyle\sum_{j=1}^{K}R_{j}$ $\displaystyle\leq$
$\displaystyle\min\left\\{\sum_{j=1}^{K}\alpha_{j},\rho_{1},\rho_{2},\cdots,\rho_{K}\right\\},$
(22) $\displaystyle\sum_{j=1}^{K}R_{j}+R_{i}$ $\displaystyle\leq$
$\displaystyle\beta_{i}+\gamma_{i}+\sum_{j=1,j\neq i}^{K}\alpha_{j},$ (23)
where the ranges of the indices $i$, $m$, $l$ are as defined in Theorem 1, and
$\displaystyle\alpha_{i}$ $\displaystyle=$
$\displaystyle\log\left(1+\mathsf{INR}_{i+1}+\frac{\mathsf{SNR}_{i}}{1+\mathsf{INR}_{i}}\right)$
(24) $\displaystyle\beta_{i}$ $\displaystyle=$
$\displaystyle\log\left(\frac{1+\mathsf{SNR}_{i}}{1+\mathsf{INR}_{i}}\right)$
(25) $\displaystyle\gamma_{i}$ $\displaystyle=$
$\displaystyle\log\left(1+\mathsf{INR}_{i+1}+\mathsf{SNR}_{i}\right)$ (26)
$\displaystyle\lambda_{i}$ $\displaystyle=$
$\displaystyle\log(1+\mathsf{SNR}_{i})$ (27) $\displaystyle\mu_{i}$
$\displaystyle=$ $\displaystyle\log(1+\mathsf{INR}_{i})$ (28)
$\displaystyle\rho_{i}$ $\displaystyle=$
$\displaystyle\beta_{i-1}+\gamma_{i}+\sum_{j=1,j\notin\\{i,i-1\\}}^{K}\alpha_{j}.$
(29)
###### Proof:
See Appendix -B. ∎
### III-C Capacity Region to Within Two Bits
###### Theorem 3
For the $K$-user cyclic Gaussian interference channel in the weak interference
regime, the fixed ETW power-splitting strategy achieves to within two bits of
the capacity region222This paper follows the definition from [8] that if a
rate tuple $(R_{1},R_{2},\cdots,R_{K})$ is achievable and
$(R_{1}+b,R_{2}+b,\cdots,R_{K}+b)$ is outside the capacity region, then
$(R_{1},R_{2},\cdots,R_{K})$ is within $b$ bits of the capacity region..
###### Proof:
Applying the ETW power-splitting strategy (i.e.,
$\mathsf{INR}_{ip}=\min(\mathsf{INR}_{i},1)$) to Theorem 1, parameters
$a_{i},d_{i},e_{i},g_{i}$ can be easily calculated as follows:
$\displaystyle a_{i}$ $\displaystyle=$
$\displaystyle\log\left(2+\mathsf{SNR}_{ip}\right)-1,$ (30) $\displaystyle
d_{i}$ $\displaystyle=$ $\displaystyle\log\left(2+\mathsf{SNR}_{i}\right)-1,$
(31) $\displaystyle e_{i}$ $\displaystyle=$
$\displaystyle\log\left(1+\mathsf{INR}_{i+1}+\mathsf{SNR}_{ip}\right)-1,$ (32)
$\displaystyle g_{i}$ $\displaystyle=$
$\displaystyle\log\left(1+\mathsf{INR}_{i+1}+\mathsf{SNR}_{i}\right)-1,$ (33)
where $\mathsf{SNR}_{ip}=|h_{i,i}|^{2}P_{ip}/\sigma^{2}$. To prove that the
achievable rate region in Theorem 1 with the above $a_{i},d_{i},e_{i},g_{i}$
is within two bits of the outer bound in Theorem 2, we show that each of the
rate constraints in (5)-(8) is within two bits of their corresponding outer
bound in (20)-(23) in the weak interference regime, i.e., the following
inequalities hold for all $i$, $m$, $l$ in the ranges defined in Theorem 1:
$\displaystyle\delta_{R_{i}}$ $\displaystyle\leq$ $\displaystyle 2,$ (34)
$\displaystyle\delta_{R_{m}+\cdots+R_{m+l-1}}$ $\displaystyle\leq$
$\displaystyle 2l,$ (35) $\displaystyle\delta_{R_{sum}}$ $\displaystyle\leq$
$\displaystyle 2K,$ (36) $\displaystyle\delta_{R_{sum}+R_{i}}$
$\displaystyle\leq$ $\displaystyle 2(K+1),$ (37)
where $\delta_{(\cdot)}$ is the difference between the achievable rate in
Theorem 1 and its corresponding outer bound in Theorem 2. The proof makes use
of a set of inequalities provided in Appendix -D.
For $\delta_{R_{i}}$, we have
$\displaystyle\delta_{R_{i}}$ $\displaystyle=$
$\displaystyle\lambda_{i}-\min\\{d_{i},a_{i}+e_{i-1}\\}$ (38) $\displaystyle=$
$\displaystyle\max\\{\lambda_{i}-d_{i},\lambda_{i}-(a_{i}+e_{i-1})\\}$
$\displaystyle\leq$ $\displaystyle 2.$
For $\delta_{R_{m}+\cdots+R_{m+l-1}}$, compare the first terms of (6) and
(21):
$\displaystyle\delta_{1}$ $\displaystyle=$
$\displaystyle\gamma_{m}+\sum_{j=m+1}^{m+l-2}\alpha_{j}+\beta_{m+l-1}-g_{m}+\sum_{j=m+1}^{m+l-2}e_{j}$
(39) $\displaystyle+a_{m+l-1}$ $\displaystyle=$
$\displaystyle(\gamma_{m}-g_{m})+\sum_{j=m+1}^{m+l-2}(\alpha_{j}-e_{j})+(\beta_{m+l-1}-a_{m+l-1})$
$\displaystyle\leq$ $\displaystyle l.$
Similarly, the difference between the second term of (6) and (21) is bounded
by
$\displaystyle\delta_{2}$ $\displaystyle=$
$\displaystyle\mu_{m}+\sum_{j=m}^{m+l-2}\alpha_{j}+\beta_{m+l-1}-\sum_{j=m-1}^{m+l-2}e_{j}+a_{m+l-1}$
(40) $\displaystyle=$
$\displaystyle(\mu_{m}-e_{m-1})+\sum_{j=m}^{m+l-2}(\alpha_{j}-e_{j})$
$\displaystyle+(\beta_{m+l-1}-a_{m+l-1})$ $\displaystyle\leq$ $\displaystyle
l+1.$
Finally, applying the fact that
$\min\\{x_{1},y_{1}\\}-\min\\{x_{2},y_{2}\\}\leq\max\\{x_{1}-x_{2},y_{1}-y_{2}\\},$
we obtain
$\delta_{R_{m}+\cdots+R_{m+l-1}}\leq\max\\{\delta_{1},\delta_{2}\\}\leq l+1.$
(41)
For $\delta_{R_{sum}}$, the difference between the first terms of (7) and (22)
is bounded by
$\displaystyle\sum_{j=1}^{K}\alpha_{j}-\sum_{j=1}^{K}e_{j}=\sum_{j=1}^{K}(\alpha_{j}-e_{j})\leq
K.$ (42)
In addition,
$\displaystyle\rho_{i}-r_{i}$ $\displaystyle=$
$\displaystyle\beta_{i-1}+\gamma_{i}+\sum_{j=1,j\notin\\{i,i-1\\}}^{K}\alpha_{j}$
(43) $\displaystyle-a_{i-1}+g_{i}+\sum_{j=1,j\notin\\{i,i-1\\}}^{K}e_{j}$
$\displaystyle=$ $\displaystyle(\beta_{i-1}-a_{i-1})+(\gamma_{i}-g_{i})$
$\displaystyle+\sum_{j=1,j\notin\\{i,i-1\\}}^{K}(\alpha_{j}-e_{j})$
$\displaystyle\leq$ $\displaystyle K$
for $i=1,2,\cdots,K$. As a result, the gap on the sum rate is bounded by
$\displaystyle\delta_{R_{sum}}$ $\displaystyle=$
$\displaystyle\min\left\\{\sum_{j=1}^{K}\alpha_{j},\rho_{1},\rho_{2},\cdots,\rho_{K}\right\\}$
(44)
$\displaystyle-\min\left\\{\sum_{j=1}^{K}e_{j},r_{1},r_{2},\cdots,r_{K}\right\\}$
$\displaystyle\leq$
$\displaystyle\max\left\\{\sum_{j=1}^{K}(\alpha_{j}-e_{j}),\rho_{1}-r_{1},\right.$
$\displaystyle\left.\rho_{2}-r_{2},\cdots,\rho_{K}-r_{K}\right\\}$
$\displaystyle\leq$ $\displaystyle K.$
For $R_{sum}+R_{i}$, we have
$\displaystyle\delta_{R_{sum}+R_{i}}$ $\displaystyle=$
$\displaystyle\beta_{i}+\gamma_{i}+\sum_{j=1,j\neq
i}^{K}\alpha_{j}-a_{i}+g_{i}+\sum_{j=1,j\neq i}^{K}e_{j}$ (45)
$\displaystyle=$
$\displaystyle(\beta_{i}-a_{i})+(\gamma_{i}-g_{i})+\sum_{j=1,j\neq
i}^{K}(\alpha_{j}-e_{j})$ $\displaystyle\leq$ $\displaystyle K+1$
Since the inequalities in (34)-(37) hold for all the ranges of $i$, $m$, and
$l$ defined in Theorem 1, this proves that the ETW power-splitting strategy
achieves to within two bits of the capacity region in the weak interference
regime. ∎
### III-D 3-User Cyclic Gaussian Interference Channel Capacity Region to
Within $1\frac{1}{2}$ Bits
Chong, Motani and Garg [2] showed that by time-sharing with marginalized
versions of the input distribution, the Han-Kobayashi region for the two-user
interference channel as stated in (15)-(19) can be further simplified by
removing the $a_{1}+e_{2}$ and $a_{2}+e_{1}$ terms from (15) and (16)
respectively. The resulting rate region without these two terms is proved to
be equivalent to the original Han-Kobayashi region (15)-(19).
This section shows that the aforementioned time-sharing technique can be
applied to the $3$-user cyclic interference channel (but not to $K\geq 4$). By
a similar time-sharing strategy, the second rate constraint on $R_{1},R_{2}$
and $R_{3}$ can be removed, and the achievable rate region can be shown to be
within $1\frac{1}{2}$ bits of the capacity region in the weak interference
regime.
###### Theorem 4
Let $\mathcal{P}_{3}$ denote the set of probability distributions
$P_{3}(\cdot)$ that factor as
$\displaystyle P_{3}(q,w_{1},x_{1},w_{2},x_{2},w_{3},x_{3})$ (46)
$\displaystyle=$ $\displaystyle
p(q)p(x_{1}w_{1}|q)p(x_{2}w_{2}|q)p(x_{3}w_{3}|q).$
For a fixed $P_{3}\in\mathcal{P}_{3}$, let $\mathcal{R}_{\textrm{HK-
TS}}^{(3)}(P_{3})$ be the set of all rate tuples $(R_{1},R_{2},R_{3})$
satisfying
$\displaystyle R_{i}$ $\displaystyle\leq$ $\displaystyle d_{i},\quad i=1,2,3,$
(47) $\displaystyle R_{1}+R_{2}$ $\displaystyle\leq$
$\displaystyle\min\\{g_{1}+a_{2},e_{3}+e_{1}+a_{2}\\},$ (48) $\displaystyle
R_{2}+R_{3}$ $\displaystyle\leq$
$\displaystyle\min\\{g_{2}+a_{3},e_{1}+e_{2}+a_{3}\\},$ (49) $\displaystyle
R_{3}+R_{1}$ $\displaystyle\leq$
$\displaystyle\min\\{g_{3}+a_{1},e_{2}+e_{3}+a_{1}\\},$ (50) $\displaystyle
R_{1}+R_{2}+R_{3}$ $\displaystyle\leq$
$\displaystyle\min\\{e_{1}+e_{2}+e_{3},a_{3}+g_{1}+e_{2},$ (51) $\displaystyle
a_{1}+g_{2}+e_{3},a_{2}+g_{3}+e_{1}\\},$ $\displaystyle 2R_{1}+R_{2}+R_{3}$
$\displaystyle\leq$ $\displaystyle a_{1}+g_{1}+e_{2}+e_{3},$ (52)
$\displaystyle R_{1}+2R_{2}+R_{3}$ $\displaystyle\leq$ $\displaystyle
a_{2}+g_{2}+e_{3}+e_{1},$ (53) $\displaystyle R_{1}+R_{2}+2R_{3}$
$\displaystyle\leq$ $\displaystyle a_{3}+g_{3}+e_{1}+e_{2},$ (54)
where $a_{i},d_{i},e_{i},g_{i}$ are as defined before. Define
$\mathcal{R}_{\textrm{HK-
TS}}^{(3)}=\bigcup_{P_{3}\in\mathcal{P}_{3}}\mathcal{R}_{\textrm{HK-
TS}}^{(3)}(P_{3}).$ (55)
Then, $\mathcal{R}_{\textrm{HK-TS}}^{(3)}$ is an achievable rate region for
the $3$-user cyclic Gaussian interference channel. Further, when $P_{3}$ is
set according to the ETW power-splitting strategy, the rate region
$R_{\textrm{HK-TS}}^{(3)}(P_{3})$ is within $1\frac{1}{2}$ bits of the
capacity region in the weak interference regime.
###### Proof:
We prove the achievability of $\mathcal{R}_{\textrm{HK-TS}}^{(3)}$ by showing
that $\mathcal{R}_{\textrm{HK-TS}}^{(3)}$ is equivalent to
$\mathcal{R}_{\textrm{HK}}^{(3)}$. First, since
$\mathcal{R}_{\textrm{HK}}^{(3)}$ contains an extra constraint on each of
$R_{1},R_{2}$ and $R_{3}$ (see (5)), it immediately follows that
$\mathcal{R}_{\textrm{HK}}^{(3)}\subseteq\mathcal{R}_{\textrm{HK-TS}}^{(3)}.$
(56)
In Appendix -C, it is shown that the inclusion also holds the other way
around. Therefore, $\mathcal{R}_{\textrm{HK}}^{(3)}=\mathcal{R}_{\textrm{HK-
TS}}^{(3)}$ and as a result, $\mathcal{R}_{\textrm{HK-TS}}^{(3)}$ is
achievable.
Applying the ETW power-splitting strategy (i.e.,
$\mathsf{INR}_{ip}=\min\\{\mathsf{INR}_{i},1\\}$ and $Q$ is fixed) to
$\mathcal{R}^{(3)}_{\textrm{HK-TS}}(P_{3})$, and following along the same line
of the proof of Theorem 3, we obtain
$\displaystyle\delta_{R_{i}}$ $\displaystyle\leq$ $\displaystyle 1,$ (57)
$\displaystyle\delta_{R_{i}+R_{i+1}}$ $\displaystyle\leq$ $\displaystyle 3,$
(58) $\displaystyle\delta_{R_{sum}}$ $\displaystyle\leq$ $\displaystyle 3,$
(59) $\displaystyle\delta_{R_{sum}+R_{i}}$ $\displaystyle\leq$ $\displaystyle
4,$ (60)
where $i=1,2,3$. It then follows that the gap to the capacity region is at
most $1\frac{1}{2}$ bits in the weak interference regime. ∎
As shown in Appendix -C, the rate region (47)-(54) is obtained by taking the
union over the achievable rate regions with input distributions
$P_{3},P_{3}^{*},P_{3}^{**}$ and $P_{3}^{***}$, where $P_{3}^{*},P_{3}^{**}$
and $P_{3}^{***}$ are the marginalized versions of $P_{3}$. Thus, to achieve
within $1\frac{1}{2}$ bits of the capacity region, one needs to time-share
among the ETW power-splitting and its three marginalized variations, rather
than using the fixed ETW’s input alone.
The key improvement of $\mathcal{R}_{\textrm{HK-TS}}^{(3)}$ over
$\mathcal{R}_{\textrm{HK}}^{(3)}$ is the removal of term $a_{i}+e_{i-1}$ in
(5) using a time-sharing technique. However, the results in Appendix -C hold
only for $K=3$. When $K\geq 4$, it is easy to verify that
$\mathcal{R}_{\textrm{HK-TS}}^{(4)}(P_{4})$ is not within the union of
$\mathcal{R}_{\textrm{HK}}^{(4)}(P_{4})$ and its marginalized variations,
i.e., $\mathcal{R}_{\textrm{HK}}^{(4)}\nsubseteq\mathcal{R}_{\textrm{HK-
TS}}^{(4)}$. Therefore, the techniques used in this paper only allow the two-
bit result to be sharpened to a $1\frac{1}{2}$-bit result for the three-user
cyclic Gaussian interference channel, but not for $K\geq 4$.
## IV Capacity Region in the Strong Interference Regime
The results so far pertain only to the weak interference regime, where
$\mathsf{SNR}_{i}\geq\mathsf{INR}_{i}$, $\forall i$. In the strong
interference regime, where $\mathsf{SNR}_{i}\leq\mathsf{INR}_{i}$, $\forall
i$, the capacity result in [1] [4] for the two-user Gaussian interference
channel can be easily extended to the $K$-user cyclic case.
###### Theorem 5
For the $K$-user cyclic Gaussian interference channel in the strong
interference regime, the capacity region is given by the set of
$(R_{1},R_{2},\cdots,R_{K})$ such that 333This capacity result was also
recently obtained in [23].
$\displaystyle\left\\{\begin{array}[]{l}R_{i}\leq\log(1+\mathsf{SNR}_{i})\\\
R_{i}+R_{i+1}\leq\log(1+\mathsf{SNR}_{i}+\mathsf{INR}_{i+1}),\end{array}\right.$
(63)
for $i=1,2,\cdots,K$. In the very strong interference regime where
$\mathsf{INR}_{i}\geq(1+\mathsf{SNR}_{i-1})\mathsf{SNR}_{i},\forall i$, the
capacity region is the set of $(R_{1},R_{2},\cdots,R_{K})$ with
$R_{i}\leq\log(1+\mathsf{SNR}_{i}),\;\;i=1,2,\cdots,K.$ (64)
###### Proof:
Achievability: It is easy to see that (63) is in fact the intersection of the
capacity regions of $K$ multiple-access channels:
$\bigcap_{i=1}^{K}\left\\{(R_{i},R_{i+1})\left|\begin{array}[]{l}R_{i}\leq\log(1+\mathsf{SNR}_{i})\\\
R_{i+1}\leq\log(1+\mathsf{INR}_{i+1})\\\
R_{i}+R_{i+1}\leq\log(1+\mathsf{SNR}_{i}+\mathsf{INR}_{i+1}).\end{array}\right.\right\\}.$
(65)
Each of these regions corresponds to that of a multiple-access channel with
$W_{i}^{n}$ and $W_{i+1}^{n}$ as inputs and $Y_{i}^{n}$ as output (with
$U_{i}^{n}=U_{i+1}^{n}=\emptyset$). Therefore, the rate region (63) can be
achieved by setting all the input signals to be common messages. This
completes the achievability part.
Converse: The converse proof follows the idea of [4]. The key ingredient is to
show that for a genie-aided Gaussian interference channel to be defined later,
in the strong interference regime, whenever a rate tuple
$(R_{1},R_{2},\cdots,R_{K})$ is achievable, i.e., $X_{i}^{n}$ is decodable at
receiver $i$, $X_{i}^{n}$ must also be decodable at $Y_{i-1}^{n}$,
$i=1,2,\cdots,K$.
The genie-aided Gaussian interference channel is defined by the Gaussian
interference channel (see Fig. 2) with genie $X_{i+2}^{n}$ given to receiver
$i$. The capacity region of the $K$-user cyclic Gaussian interference channel
must reside inside the capacity region of the genie-aided one.
Assume that a rate tuple $(R_{1},R_{2},\cdots,R_{K})$ is achievable for the
$K$-user cyclic Gaussian interference channel. In this case, after $X_{i}^{n}$
is decoded, with the knowledge of the genie $X_{i+2}^{n}$, receiver $i$ can
construct the following signal:
$\displaystyle\widetilde{Y}_{i}^{n}$ $\displaystyle=$
$\displaystyle\frac{h_{i+1,i+1}}{h_{i+1,i}}(Y_{i}^{n}-h_{i,i}X_{i}^{n})+h_{i+2,i+1}X_{i+2}^{n}$
$\displaystyle=$ $\displaystyle
h_{i+1,i+1}X_{i+1}^{n}+h_{i+2,i+1}X_{i+2}^{n}+\frac{h_{i+1,i+1}}{h_{i+1,i}}Z_{i}^{n},$
which contains the signal component of $Y_{i+1}^{n}$ but with less noise since
$|h_{i+1,i}|\geq|h_{i+1,i+1}|$ in the strong interference regime. Now, since
$X_{i+1}^{n}$ is decodable at receiver $i+1$, it must also be decodable at
receiver $i$ using the constructed $\widetilde{Y}_{i}^{n}$. Therefore,
$X_{i}^{n}$ and $X_{i+1}^{n}$ are both decodable at receiver $i$. As a result,
the achievable rate region of $(R_{i},R_{i+1})$ is bounded by the capacity
region of the multiple-access channel $(X_{i}^{n},X_{i+1}^{n},Y_{i}^{n})$,
which is shown in (65). Since (65) reduces to (63) in the strong interference
regime, we have shown that (63) is an outer bound of the $K$-user cyclic
Gaussian interference channel in the strong interference regime. This
completes the converse proof.
In the very strong interference regime where
$\mathsf{INR}_{i}\geq(1+\mathsf{SNR}_{i-1})\mathsf{SNR}_{i},\forall i$, it is
easy to verify that the second constraint in (63) is no longer active. This
results in the capacity region (64). ∎
## V Symmetric Channel and Generalized Degrees of Freedom
Consider the symmetric cyclic Gaussian interference channel, where all the
direct links from the transmitters to the receivers share the same channel
gain and all the cross links share another same channel gain. In addition, all
the input signals have the same power constraint $P$, i.e.,
$\mathbb{E}\left[|X_{i}|^{2}\right]\leq P,\forall i$.
The symmetric capacity of the $K$-user interference channel is defined as
$\displaystyle C_{sym}=\left\\{\begin{array}[]{l}\textrm{maximize \
min}\\{R_{1},R_{2},\cdots,R_{K}\\}\\\ \textrm{subject to \
}\;(R_{1},R_{2},\cdots,R_{K})\in\mathcal{R}\end{array}\right.$ (68)
where $\mathcal{R}$ is the capacity region of the $K$-user interference
channel. For the symmetric interference channel, $C_{sym}=\frac{1}{K}C_{sum}$,
where $C_{sum}$ is the sum capacity. As a direct consequence of Theorem 3 and
Theorem 5, the generalized degree of freedom of the symmetric capacity for the
symmetric cyclic channel can be derived as follows.
###### Corollary 1
For the $K$-user symmetric cyclic Gaussian interference channel,
$\displaystyle
d_{sym}=\left\\{\begin{array}[]{l}\min\left\\{\max\\{\alpha,1-\alpha\\},1-\frac{\alpha}{2}\right\\},\;\;0\leq\alpha<1\\\
\min\\{\frac{\alpha}{2},1\\},\qquad\qquad\qquad\qquad\qquad\alpha\;\geq
1\end{array}\right.$ (71)
where $d_{sym}$ is the generalized degrees of freedom of the symmetric
capacity.
Note that the above $d_{sym}$ for the $K$-user cyclic interference channel
with symmetric channel parameters is the same as that of the two-user
interference channel derived in [8].
## VI Conclusion
This paper investigates the capacity and the coding strategy for the $K$-user
cyclic Gaussian interference channel. Specifically, this paper shows that in
the weak interference regime, the ETW power-splitting strategy achieves to
within two bits of the capacity region. Further, in the special case of $K=3$
and with the help of a time-sharing technique, one can achieve to within
$1\frac{1}{2}$ bits of the capacity region in the weak interference regime.
The capacity result for the $K$-user cyclic Gaussian interference channel in
the strong interference regime is a straightforward extension of the
corresponding two-user case. However, in the mixed interference regime,
although the constant gap result may well continue to hold, the proof becomes
considerably more complicated, as different mixed scenarios need to be
enumerated and the corresponding outer bounds derived.
### -A Proof of Theorem 1
For the two-user interference channel, Kobayashi and Han [22] gave a detailed
Fourier-Motzkin elimination procedure for the achievable rate region. The
Fourier-Motzkin elimination for the $K$-user cyclic interference channel
involves $K$ elimination steps. The complexity of the process increases with
each step. Instead of manually writing down all the inequalities step by step,
this appendix uses mathematical induction to derive the final result.
This achievability proof is based on the application of coding scheme in [2]
(also referred as the multi-level coding in [24]) to the multi-user setting.
Instead of using the original code construction of [1], the following strategy
is used in which each common message $W_{i},i=1,2,\cdots,K$ serves to generate
$2^{nT_{i}}$ cloud centers $W_{i}(j),j=1,2,\cdots,2^{nT_{i}}$, each of which
is surrounded by $2^{nS_{i}}$ codewords $X_{i}(j,k),k=1,2,\cdots,2^{nS_{i}}$.
This results in achievable rate region expressions expressed in terms of
$(W_{i},X_{i},Y_{i})$ instead of $(U_{i},W_{i},Y_{i})$. For the two-user
interference channel, Chong, Motani and Garg [2, Lemma 3] made a further
simplification to the achievalbe rate region expression. They observed that in
the Han-Kobayashi scheme, the common message $W_{i}$ is only required to be
correctly decoded at the intended receiver $Y_{i}$ and an incorrectly decoded
$W_{i}$ at receiver $Y_{i-1}$ does not cause an error event. Based on this
observation, they concluded that for the multiple-access channel with input
$(U_{i},W_{i},W_{i+1})$ and output $Y_{i}$, the rate constraints on common
messages $T_{i}$, $T_{i+1}$ and $T_{i}+T_{i+1}$ are in fact irrelevant to the
decoding error probabilities and can be removed, i.e., the rates
$(S_{i},T_{i},T_{i+1})$ are constrained by only the following set of
inequalities:
$\displaystyle S_{i}$ $\displaystyle\leq$ $\displaystyle
a_{i}=I(Y_{i};X_{i}|W_{i},W_{i+1},Q)$ (72) $\displaystyle S_{i}+T_{i}$
$\displaystyle\leq$ $\displaystyle d_{i}=I(Y_{i};X_{i}|W_{i+1},Q)$ (73)
$\displaystyle S_{i}+T_{i+1}$ $\displaystyle\leq$ $\displaystyle
e_{i}=I(Y_{i};W_{i+1},X_{i}|W_{i},Q)$ (74) $\displaystyle S_{i}+T_{i}+T_{i+1}$
$\displaystyle\leq$ $\displaystyle g_{i}=I(Y_{i};W_{i+1},X_{i}|Q)$ (75)
$\displaystyle S_{i},T_{i},T_{i+1}$ $\displaystyle\geq$ $\displaystyle 0$ (76)
Now, compare the $K$-user cyclic interference channel with the two-user
interference channel, it is easy to see that in both channel models, each
receiver only sees interference from one neighboring transmitter. This makes
the decoding error probability analysis for both channel models the same.
Therefore, the set of rates $\mathcal{R}(R_{1},R_{2},\cdots,R_{K})$, where
$R_{i}=S_{i}+T_{i}$, with $(S_{i},T_{i})$ satisfy (72)-(76) for
$i=1,2,\cdots,K$, characterizes an achievable rate region for the $K$-user
cyclic interference channel.
The first step of using the Fourier-Motzkin algorithm is to eliminate all
private messages $S_{i}$ by substituting $S_{i}=R_{i}-T_{i}$ into the $K$
polymatroids (72)-(76). This results in the following $K$ polymatroids without
$S_{i}$:
$\displaystyle R_{i}-T_{i}$ $\displaystyle\leq$ $\displaystyle a_{i},$ (77)
$\displaystyle R_{i}$ $\displaystyle\leq$ $\displaystyle d_{i},$ (78)
$\displaystyle R_{i}-T_{i}+T_{i+1}$ $\displaystyle\leq$ $\displaystyle e_{i},$
(79) $\displaystyle R_{i}+T_{i+1}$ $\displaystyle\leq$ $\displaystyle g_{i},$
(80) $\displaystyle-R_{i}$ $\displaystyle\leq$ $\displaystyle 0,$ (81)
where $i=1,2,\cdots,K$.
Next, use Fourier-Motzkin algorithm to eliminate common message rates $T_{1}$,
$T_{2}$, $\cdots$, $T_{K}$ in a step-by-step process so that after $n$ steps,
common variables $(T_{1},\cdots,T_{n})$ are eliminated. The induction
hypothesis is the following $5$ different groups of inequalities, which is
assumed to be obtained at the end of the $n$th elimination step:
(a) Inequalities not including private or common variables $S_{i}$ and
$T_{i},i=1,2,\cdots,K$:
$\displaystyle R_{i}$ $\displaystyle\leq$ $\displaystyle d_{i},\quad
i=1,2,\cdots,K$ (82) $\displaystyle-R_{i}$ $\displaystyle\leq$ $\displaystyle
0,\quad i=1,2,\cdots,n$ (83) $\displaystyle R_{K}+R_{1}$ $\displaystyle\leq$
$\displaystyle g_{K}+a_{1},$ (84) $\displaystyle R_{m}$ $\displaystyle\leq$
$\displaystyle a_{m}+e_{m-1},$ (85) $\displaystyle\sum_{j=l}^{m}R_{j}$
$\displaystyle\leq$
$\displaystyle\min\left\\{g_{l}+\sum_{i=l+1}^{m-1}e_{j}+a_{m},\sum_{j=l-1}^{m-1}e_{j}+a_{m}\right\\},$
$\displaystyle\sum_{j=1}^{m}R_{j}$ $\displaystyle\leq$ $\displaystyle
g_{1}+\sum_{j=2}^{m-1}e_{j}+a_{m},$ (87) $\displaystyle\sum_{j=K}^{m}R_{j}$
$\displaystyle\leq$ $\displaystyle g_{K}+\sum_{j=1}^{m-1}e_{j}+a_{m},$ (88)
where $m=2,3,\cdots,n$ and $l=2,3,\cdots,m-1$.
(b) Inequalities including $T_{K}$ but not including $T_{n+1}$:
$\displaystyle R_{K}-T_{K}$ $\displaystyle\leq$ $\displaystyle a_{K},$ (89)
$\displaystyle-R_{K}-T_{K}$ $\displaystyle\leq$ $\displaystyle 0,$ (90)
$\displaystyle-T_{K}$ $\displaystyle\leq$ $\displaystyle 0,$ (91)
$\displaystyle\sum_{j=K}^{p}R_{j}-T_{K}$ $\displaystyle\leq$
$\displaystyle\sum_{j=K}^{p-1}e_{j}+a_{p},$ (92)
where $p=1,2,\cdots,n$.
(c) All other inequalities not including $T_{n+1}$:
$R_{n+1}+T_{n+2}\leq g_{n+1},$ (93)
and all the polymatroids in (77)-(81) indexed from $n+2$ to $K-1$.
(d) Inequalities including $T_{n+1}$ with a plus sign:
$\displaystyle T_{n+1}$ $\displaystyle\leq$ $\displaystyle e_{n},$ (94)
$\displaystyle-R_{n+1}+T_{n+1}$ $\displaystyle\leq$ $\displaystyle 0,$ (95)
$\displaystyle\sum_{j=l}^{n}R_{j}+T_{n+1}$ $\displaystyle\leq$
$\displaystyle\min\left\\{\sum_{j=l-1}^{n}e_{j},g_{l}+\sum_{j=l+1}^{n}e_{j}\right\\},$
$\displaystyle\sum_{j=1}^{n}R_{j}+T_{n+1}$ $\displaystyle\leq$ $\displaystyle
g_{1}+\sum_{j=2}^{n}e_{j},$ (97) $\displaystyle\sum_{j=K}^{n}R_{j}+T_{n+1}$
$\displaystyle\leq$ $\displaystyle g_{K}+\sum_{j=1}^{n}e_{j},$ (98)
$\displaystyle\sum_{j=K}^{n}R_{j}+T_{n+1}-T_{K}$ $\displaystyle\leq$
$\displaystyle\sum_{j=K}^{n}e_{j},$ (99)
where $l$ goes from $2$ to $n$.
(e) Inequalities including $T_{n+1}$ with a minus sign:
$\displaystyle R_{n+1}-T_{n+1}$ $\displaystyle\leq$ $\displaystyle a_{n+1},$
(100) $\displaystyle R_{n+1}-T_{n+1}+T_{n+2}$ $\displaystyle\leq$
$\displaystyle e_{n+1},$ (101) $\displaystyle-T_{n+1}$ $\displaystyle\leq$
$\displaystyle 0.$ (102)
It is easy to verify the correctness of inequalities (82)-(102) for $n=2$. We
next show that for $n<K-2$, if at the end of step $n$, the inequalities in
(82)-(102) hold, then they must also hold at the end of step $n+1$. Towards
this end, we follow the Fourier-Motzkin algorithm [22] by first adding up all
the inequalities in (94)-(99) with each of the inequalities in (100)-(102) to
eliminate $T_{n+1}$. This results in the following three groups of
inequalities:
(a) Inequalities due to (100):
$\displaystyle R_{n+1}$ $\displaystyle\leq$ $\displaystyle a_{n+1}+e_{n},$
(103) $\displaystyle 0$ $\displaystyle\leq$ $\displaystyle a_{n+1},$ (104)
$\displaystyle\sum_{j=l}^{n+1}R_{j}$ $\displaystyle\leq$
$\displaystyle\min\left\\{\sum_{j=l-1}^{n}e_{j}+a_{n+1},\right.$ (105)
$\displaystyle\left.\qquad\quad g_{l}+\sum_{j=l+1}^{n}e_{j}+a_{n+1}\right\\},$
$\displaystyle\sum_{j=1}^{n+1}R_{j}$ $\displaystyle\leq$ $\displaystyle
g_{1}+\sum_{j=2}^{n}e_{j}+a_{n+1},$ (106) $\displaystyle\sum_{j=K}^{n+1}R_{j}$
$\displaystyle\leq$ $\displaystyle g_{K}+\sum_{j=1}^{n}e_{j}+a_{n+1},$ (107)
$\displaystyle\sum_{j=K}^{n+1}R_{j}-T_{K}$ $\displaystyle\leq$
$\displaystyle\sum_{j=K}^{n}e_{j}+a_{n+1},$ (108)
where $l=2,3,\cdots,n$.
(b) Inequalities due to (101):
$\displaystyle R_{n+1}+T_{n+2}$ $\displaystyle\leq$ $\displaystyle
e_{n}+e_{n+1},$ (109) $\displaystyle T_{n+2}$ $\displaystyle\leq$
$\displaystyle e_{n+1},$ (110) $\displaystyle\sum_{j=l}^{n+1}R_{j}+T_{n+2}$
$\displaystyle\leq$
$\displaystyle\min\left\\{\sum_{j=l-1}^{n+1}e_{j},g_{l}+\sum_{j=l+1}^{n+1}e_{j}\right\\},$
$\displaystyle\sum_{j=1}^{n+1}R_{j}+T_{n+2}$ $\displaystyle\leq$
$\displaystyle g_{1}+\sum_{j=2}^{n+1}e_{j},$ (112)
$\displaystyle\sum_{j=K}^{n+1}R_{j}+T_{n+2}$ $\displaystyle\leq$
$\displaystyle g_{K}+\sum_{j=1}^{n+1}e_{j},$ (113)
$\displaystyle\sum_{j=K}^{n+1}R_{j}+T_{n+2}-T_{K}$ $\displaystyle\leq$
$\displaystyle\sum_{j=K}^{n+1}e_{j},$ (114)
where $l=2,3,\cdots,n$.
(c) Inequalities due to (102):
$\displaystyle 0$ $\displaystyle\leq$ $\displaystyle e_{n},$ (115)
$\displaystyle-R_{n+1}$ $\displaystyle\leq$ $\displaystyle 0,$ (116)
$\displaystyle\sum_{j=l}^{n}R_{j}$ $\displaystyle\leq$
$\displaystyle\min\left\\{\sum_{j=l-1}^{n}e_{j},g_{l}+\sum_{j=l+1}^{n}e_{j}\right\\},$
(117) $\displaystyle\sum_{j=1}^{n}R_{j}$ $\displaystyle\leq$ $\displaystyle
g_{1}+\sum_{j=2}^{n}e_{j},$ (118) $\displaystyle\sum_{j=K}^{n}R_{j}$
$\displaystyle\leq$ $\displaystyle g_{K}+\sum_{j=1}^{n}e_{j},$ (119)
$\displaystyle\sum_{j=K}^{n}R_{j}-T_{K}$ $\displaystyle\leq$
$\displaystyle\sum_{j=K}^{n}e_{j},$ (120)
where $l=2,3,\cdots,n$.
Inspecting the above three groups of inequalities, we can see that (104) and
(115) are obviously redundant. Also, (117) is redundant due to (-A), (118) is
redundant due to (87), (119) is redundant due to (88), and (120) is redundant
due to (92). Now, with these six redundant inequalities removed, the above
three groups of inequalities in (103)-(116) together with (82)-(93) form the
set of inequalities at the end of step $n+1$. It can be verified that this new
set of inequalities is exactly (82)-(102) with $n$ replaced by $n+1$. This
completes the induction part.
Now, we proceed with the $(K-1)$th step. At the end of this step,
$T_{1},T_{2},\cdots,T_{K-1}$ would all be removed and only $T_{K}$ would
remain. Because of the cyclic nature of the channel, the set of inequalities
(82)-(102) needs to be modified for this $n=K-1$ case. It can be verified that
at the end of the $(K-1)$th step of Fourier-Motzkin algorithm, we obtain the
following set of inequalities:
(a) Inequalities not including $T_{K}$: (82)-(88) with $n$ replaced by $K-1$
and
$\displaystyle\sum_{j=1}^{K}R_{j}\leq\sum_{j=1}^{K}e_{j}.$ (121)
(b) Inequalities including $T_{K}$ with a plus sign: (94)-(98) with $n$
replace by $K-1$. Note that, (99) becomes (121) when $n=K-1$.
(c) Inequalities including $T_{K}$ with a minus sign:
$\displaystyle R_{K}-T_{K}$ $\displaystyle\leq$ $\displaystyle a_{K},$ (122)
$\displaystyle\sum_{j=K}^{l}R_{j}-T_{K}$ $\displaystyle\leq$
$\displaystyle\sum_{j=K}^{l-1}e_{j}+a_{l},$ (123) $\displaystyle-T_{K}$
$\displaystyle\leq$ $\displaystyle 0,$ (124)
where $l=1,2,\cdots,K-1$.
In the $K$th step (final step) of the Fourier-Motzkin algorithm, $T_{K}$ is
eliminated by adding each of the inequalities involving $T_{K}$ with a plus
sign and each of the inequalities involving $T_{K}$ with a minus sign to
obtain new inequalities not involving $T_{K}$. (This is quite similar to the
procedure of obtaining (103)-(120).) Finally, after removing all the redundant
inequalities, we obtain the set of inequalities in Theorem 1.
### -B Proof of Theorem 2
We will prove the outer bounds from (20) to (23) one by one.
First, (20) is simply the cut-set upper bound for user $i$.
Second, (21) is the bound on the sum-rate of $l$ adjacent users starting from
$m$. According to Fano’s inequality, for a block of length $n$, we have
$\displaystyle n\left(\sum_{j=m}^{m+l-1}R_{j}-\epsilon_{n}\right)$ (125)
$\displaystyle\leq$ $\displaystyle\sum_{j=m}^{m+l-1}I(x_{j}^{n};y_{j}^{n})$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{\leq}}$ $\displaystyle
h(y_{m}^{n})-h(y_{m}^{n}|x_{m}^{n})+\sum_{j=m+1}^{m+l-2}I(x_{j}^{n};y_{j}^{n}s_{j}^{n})$
$\displaystyle+I(x_{m+l-1}^{n};y_{m+l-1}^{n}|x_{m+l}^{n})$ $\displaystyle=$
$\displaystyle h(y_{m}^{n})-h(s_{m+1}^{n})$
$\displaystyle+\sum_{j=m+1}^{m+l-2}\left[h(s_{j}^{n})-h(z_{j-1}^{n})+h(y_{j}^{n}|s_{j}^{n})-h(s_{j+1}^{n})\right]$
$\displaystyle+h(h_{m+l-1,m+l-1}x_{m+l-1}^{n}+z_{m+l-1}^{n})-h(z_{m+l-1}^{n})$
$\displaystyle=$ $\displaystyle
h(y_{m}^{n})-h(z_{m+l-1}^{n})+\sum_{j=m+1}^{m+l-2}\left[h(y_{j}^{n}|s_{j}^{n})-h(z_{j-1}^{n})\right]$
$\displaystyle+h(h_{m+l-1,m+l-1}x_{m+l-1}^{n}+z_{m+l-1}^{n})$
$\displaystyle-h(h_{m+l-1,m+l-2}x_{m+l-1}^{n}+z_{m+l-2}^{n})$
$\displaystyle\stackrel{{\scriptstyle(b)}}{{\leq}}$ $\displaystyle
n\left(\gamma_{m}+\sum_{j=m+1}^{m+l-2}\alpha_{j}+\beta_{m+l-1}\right),$
where in (a) we give genie $s_{j}^{n}$ to $y_{j}^{n}$ for $m+1\leq j\leq
m+l-2$ and $x_{m+l}^{n}$ to $y_{m+l-1}^{n}$ (genies $s_{j}^{n}$ are as defined
in [25, Theorem 2]), and (b) comes from the fact [8] that Gaussian inputs
maximize 1) entropy $h(y_{m}^{n})$, 2) conditional entropy
$h(y_{j}^{n}|s_{j}^{n})$ for any $j$, and 3) entropy difference
$h(h_{m+l-1,m+l-1}x_{m+l-1}^{n}+z_{m+l-1}^{n})-h(h_{m+l-1,m+l-2}x_{m+l-1}^{n}+z_{m+l-2}^{n})$.
This proves the first bound in (21).
Similarly, the second upper bound of (21) can be obtained by giving genie
$s_{j}^{n}$ to $y_{j}^{n}$ for $m\leq j\leq m+l-2$ and $x_{m+l}^{n}$ to
$y_{m+l-1}^{n}$:
$\displaystyle n\left(\sum_{j=m}^{m+l-1}R_{j}-\epsilon_{n}\right)$ (126)
$\displaystyle\leq$ $\displaystyle\sum_{j=m}^{m+l-1}I(x_{j}^{n};y_{j}^{n})$
$\displaystyle\leq$
$\displaystyle\sum_{j=m}^{m+l-2}I(x_{j}^{n};y_{j}^{n}s_{j}^{n})+I(x_{m+l-1}^{n};y_{m+l-1}^{n}|x_{m+1}^{n})$
$\displaystyle=$
$\displaystyle\sum_{j=m}^{m+l-2}\left[h(s_{j}^{n})-h(z_{j-1}^{n})+h(y_{j}^{n}|s_{j}^{n})-h(s_{j+1}^{n})\right]$
$\displaystyle+h(h_{m+l-1,m+l-1}x_{m+l-1}^{n}+z_{m+l-1}^{n})-h(z_{m+l-1}^{n})$
$\displaystyle=$ $\displaystyle
h(s_{m}^{n})-h(z_{m+l-1}^{n})+\sum_{j=m}^{m+l-2}\left[h(y_{j}^{n}|s_{j}^{n})-h(z_{j-1}^{n})\right]$
$\displaystyle+h(h_{m+l-1,m+l-1}x_{m+l-1}^{n}+z_{m+l-1}^{n})$
$\displaystyle-h(h_{m+l-1,m+l-2}x_{m+l-1}^{n}+z_{m+l-2}^{n})$
$\displaystyle\leq$ $\displaystyle
n\left(\mu_{m}+\sum_{j=m}^{m+l-2}\alpha_{j}+\beta_{m+l-1}\right).$
Combining (125) and (126) gives the upper bound in (21).
Third, the first upper bound in (22) is in fact the non-symmetric version of
[25, Theorem 2], from which we have
$\displaystyle R_{sum}-n\epsilon_{n}$ $\displaystyle\leq$
$\displaystyle\sum_{k=1}^{K}\\{h(y_{ki}|s_{ki})-h(z_{ki})\\}$ (127)
$\displaystyle\leq$ $\displaystyle n\sum_{j=1}^{K}\alpha_{j}.$
The other sum-rate upper bounds (i.e., $\rho_{l}$) can be derived by giving
genies $x_{l}^{n}$ to $y^{n}_{l-1}$ and $s_{j}^{n}$ to $y_{j}^{n}$ for
$j=1,2,\cdots,K,j\neq l,l-1$:
$\displaystyle n(R_{sum}-\epsilon_{n})$ (128) $\displaystyle\leq$
$\displaystyle
I(x_{1}^{n};y_{1}^{n})+I(x_{2}^{n};y_{2}^{n})+\cdots+I(x_{K}^{n};y_{K}^{n})$
$\displaystyle=$ $\displaystyle
I(x_{l-1}^{n};y_{l-1}^{n}|x_{l}^{n})+I(x_{l}^{n};y_{l}^{n})+\sum_{j=1,j\neq
l,l-1}^{K}I(x_{j}^{n};y_{j}^{n}s_{j}^{n})$ $\displaystyle=$ $\displaystyle
h(h_{l-1,l-1}x_{l-1}^{n}+z_{l-1}^{n})-h(z_{l-1}^{n})+h(y_{l}^{n})-h(s_{l+1}^{n})$
$\displaystyle+\sum_{j=1,j\neq
l,l-1}^{K}\left[h(s_{j}^{n})-h(z_{j-1}^{n})+h(y_{j}^{n}|s_{j}^{n})-h(s_{j+1}^{n})\right]$
$\displaystyle=$ $\displaystyle
h(y_{l}^{n})-h(z_{l-1}^{n})+h(h_{l-1,l-1}x_{l-1}^{n}+z_{l-1}^{n})$
$\displaystyle-h(h_{l-1,l-2}x_{l-1}^{n}+z_{l-2}^{n})$
$\displaystyle+\sum_{j=1,j\neq
l,l-1}^{K}\left[h(y_{j}^{n}|s_{j}^{n})-h(z_{j-1}^{n})\right]$
$\displaystyle\leq$ $\displaystyle
n\left(\beta_{l-1}+\gamma_{l}+\sum_{j=1,j\neq l,l-1}^{K}\alpha_{j}\right)$
$\displaystyle=$ $\displaystyle n\rho_{l}$
where $l=1,2,\cdots,K$.
Fourth, for the bound in (23), from Fano’s inequality, we have
$\displaystyle n(R_{sum}+R_{i}-\epsilon_{n})$ (129) $\displaystyle\leq$
$\displaystyle\sum_{j=1}^{K}I(x_{j}^{n};y_{j}^{n})+I(x_{i}^{n};y_{i}^{n})$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{\leq}}$ $\displaystyle
I(x_{i}^{n};y_{i}^{n})+I(x_{i}^{n};y_{i}^{n}|x_{i+1}^{n})+\sum_{j=1,j\neq
i}^{K}I(x_{j}^{n};y_{j}^{n}s_{j}^{n})$ $\displaystyle=$ $\displaystyle
h(y_{i}^{n})-h(s_{i+1}^{n})+h(h_{i,i}x_{i}^{n}+z_{i}^{n})-h(z_{i}^{n})$
$\displaystyle+\sum_{j=1,j\neq
i}^{K}\left[h(s_{j}^{n})-h(z_{j-1}^{n})+h(y_{j}^{n}|s_{j}^{n})-h(s_{j+1}^{n})\right]$
$\displaystyle=$ $\displaystyle
h(y_{i}^{n})-h(z_{i}^{n})+h(h_{i,i}x_{i}^{n}+z_{i}^{n})-h(h_{i,i-1}x_{i}^{n}+z_{i}^{n})$
$\displaystyle+\sum_{j=1,j\neq
i}^{K}\left[h(y_{j}^{n}|s_{j}^{n})-h(z_{j-1}^{n})\right]$ $\displaystyle\leq$
$\displaystyle n\left(\beta_{i}+\gamma_{i}+\sum_{j=1,j\neq
i}^{K}\alpha_{j}\right)$
where in (a) we give genie $x_{i+1}^{n}$ to $y_{i}^{n}$ and $s_{j}^{n}$ to
$y_{j}^{n}$ for $j=1,2,\cdots,K,j\neq i$.
### -C Proof of $\mathcal{R}_{\mathrm{HK-
TS}}^{(3)}\subseteq\mathcal{R}_{\mathrm{HK}}^{(3)}$
For a fixed $P_{3}\subseteq\mathcal{P}_{3}$, define
$P_{3}^{*}=\sum_{w_{1}}P_{3},\quad P_{3}^{**}=\sum_{w_{2}}P_{3},\quad
P_{3}^{***}=\sum_{w_{3}}P_{3}.$ (130)
We will show that
$\displaystyle\mathcal{R}_{\mathrm{HK-TS}}^{(3)}(P_{3})$
$\displaystyle\subseteq\mathcal{R}_{\mathrm{HK}}^{(3)}(P_{3})\cup\mathcal{R}_{\mathrm{HK}}^{(3)}(P_{3}^{*})\cup\mathcal{R}_{\mathrm{HK}}^{(3)}(P_{3}^{**})\cup\mathcal{R}_{\mathrm{HK}}^{(3)}(P_{3}^{***}).$
Suppose that rate triple $(R_{1},R_{2},R_{3})$ is in $\mathcal{R}_{\textrm{HK-
TS}}^{(3)}(P_{3})$ but not in $\mathcal{R}_{\textrm{HK}}^{(3)}(P_{3})$. Then
at least one of the following inequalities hold:
$\displaystyle a_{1}+e_{3}\leq R_{1}\leq d_{1},$ (132) $\displaystyle
a_{2}+e_{1}\leq R_{2}\leq d_{2},$ (133) $\displaystyle a_{3}+e_{2}\leq
R_{3}\leq d_{3},$ (134)
Without loss of generality, assume that (132) holds.
Substituting $W_{1}=\emptyset$ into $\mathcal{R}_{\mathrm{HK}}^{(3)}(P_{3})$,
we obtain $\mathcal{R}_{\mathrm{HK}}^{(3)}(P_{3}^{*})$ as follows:
$\displaystyle R_{1}$ $\displaystyle\leq$ $\displaystyle d_{1},$ (135)
$\displaystyle R_{2}$ $\displaystyle\leq$
$\displaystyle\min\\{d_{2},a_{2}+g_{1}\\},$ (136) $\displaystyle R_{3}$
$\displaystyle\leq$ $\displaystyle\min\\{I(Y_{3};X_{3}|Q),$ (137)
$\displaystyle e_{2}+I(Y_{3};X_{3}|W_{3},Q)\\},$ $\displaystyle R_{1}+R_{2}$
$\displaystyle\leq$ $\displaystyle a_{2}+g_{1},$ (138) $\displaystyle
R_{2}+R_{3}$ $\displaystyle\leq$
$\displaystyle\min\\{g_{2}+I(Y_{3};X_{3}|W_{3},Q),$ (139) $\displaystyle
g_{1}+e_{2}+I(Y_{3};X_{3}|W_{3},Q)\\},$ $\displaystyle R_{3}+R_{1}$
$\displaystyle\leq$ $\displaystyle\min\\{d_{1}+I(Y_{3};X_{3}|Q),$ (140)
$\displaystyle d_{1}+e_{2}+I(Y_{3};X_{3}|W_{3},Q)\\},$ $\displaystyle
R_{1}+R_{2}+R_{3}$ $\displaystyle\leq$ $\displaystyle
g_{1}+e_{2}+I(Y_{3};X_{3}|W_{3},Q).$ (141)
We will show that whenever $(\ref{R1_violated})$ is true, we have
$\mathcal{R}_{\mathrm{HK-
TS}}^{(3)}(P_{3})\subseteq\mathcal{R}_{\mathrm{HK}}^{(3)}(P_{3}^{*})$. To this
end, inspect $\mathcal{R}_{\mathrm{HK-TS}}^{(3)}(P_{3})$ in (47)-(54). From
(47), we have
$R_{1}\leq d_{1},$ (142)
and from (47) and (132) and (48), we have
$\displaystyle R_{2}$ $\displaystyle\leq$
$\displaystyle\min\\{d_{2},a_{2}+e_{1}-a_{1}\\}$ (143) $\displaystyle\leq$
$\displaystyle\min\\{d_{2},a_{2}+g_{1}\\},$
and from (132) and (50), we have
$\displaystyle R_{3}$ $\displaystyle\leq$
$\displaystyle\min\\{g_{3}-e_{3},e_{2}\\}$ (144) $\displaystyle\leq$
$\displaystyle\min\\{I(Y_{3};X_{3}|Q),e_{2}+I(Y_{3};X_{3}|W_{3},Q)\\},$
and from (48), we have
$R_{1}+R_{2}\leq a_{2}+g_{1},$ (145)
and from (132) and (51), we have
$\displaystyle R_{2}+R_{3}$ $\displaystyle\leq$
$\displaystyle\min\\{g_{2},e_{1}+e_{2}-a_{1}\\}$ (146) $\displaystyle\leq$
$\displaystyle\min\\{g_{2}+I(Y_{3};X_{3}|W_{3},Q),$ $\displaystyle
g_{1}+e_{2}+I(Y_{3};X_{3}|W_{3},Q)\\},$
and from (132) and (50), we have
$\displaystyle R_{3}+R_{1}$ $\displaystyle\leq$
$\displaystyle\min\\{d_{1}+g_{3}-a_{3},e_{2}+d_{1}\\}$ (147)
$\displaystyle\leq$ $\displaystyle\min\\{d_{1}+I(Y_{3};X_{3}|Q),$
$\displaystyle d_{1}+e_{2}+I(Y_{3};X_{3}|W_{3},Q)\\},$
and from (132) and (52), we have
$\displaystyle R_{1}+R_{2}+R_{3}$ $\displaystyle\leq$ $\displaystyle
g_{1}+e_{2}$ (148) $\displaystyle\leq$ $\displaystyle
g_{1}+e_{2}+I(Y_{3};X_{3}|W_{3},Q).$
It is easy to see that $(R_{1},R_{2},R_{3})$ satisfying the above constrains
(142)-(148) is within the rate region
$\mathcal{R}_{\mathrm{HK}}^{(3)}(P_{3}^{*})$. In the same way, we can prove
the cases for when (133) holds and when (134) holds.
Therefore, (-C) is true, and it immediately follows that
$\mathcal{R}_{\mathrm{HK-TS}}^{(3)}\subseteq\mathcal{R}_{\mathrm{HK}}^{(3)}.$
(149)
### -D Useful Inequalities
Keep in mind that, with the ETW’s power splitting strategy, i.e.,
$\mathsf{SNR}_{ip}=\min\\{\mathsf{SNR}_{i},\frac{\mathsf{SNR}_{i}}{\mathsf{INR}_{i}}\\}$,
we always have
$\mathsf{SNR}_{ip}>\frac{\mathsf{SNR}_{i}}{1+\mathsf{INR}_{i}}$. This appendix
presents several useful inequalities as follows. For all $i=1,2,\cdots,K$,
* •
$\lambda_{i}-d_{i}<1$, because
$\displaystyle\lambda_{i}-d_{i}$ $\displaystyle=$
$\displaystyle\log(1+\mathsf{SNR}_{i})-\log(2+\mathsf{SNR}_{i})+1$ (150)
$\displaystyle=$ $\displaystyle
1-\log\left(\frac{2+\mathsf{SNR}_{i}}{1+\mathsf{SNR}_{i}}\right)$
$\displaystyle\leq$ $\displaystyle 1$
* •
$\lambda_{i}-(a_{i}+e_{i-1})<2$, because
$\displaystyle\lambda_{i}-(a_{i}+e_{i-1})$ (151) $\displaystyle=$
$\displaystyle\log(1+\mathsf{SNR}_{i})-\log\left(2+\mathsf{SNR}_{ip}\right)+1$
$\displaystyle-\log\left(1+\mathsf{INR}_{i}+\mathsf{SNR}_{i-1,p}\right)+1$
$\displaystyle<$ $\displaystyle
2+\log(1+\mathsf{SNR}_{i})-\log\left(1+\frac{\mathsf{SNR}_{i}}{1+\mathsf{INR}_{i}}\right)$
$\displaystyle-\log\left(1+\mathsf{INR}_{i}\right)$ $\displaystyle=$
$\displaystyle
2-\log\left(1+\frac{\mathsf{INR}_{i}}{1+\mathsf{SNR}_{i}}\right)$
$\displaystyle\leq$ $\displaystyle 2$
* •
$\beta_{i}-a_{i}<1$, because
$\displaystyle\beta_{i}-a_{i}$ (152) $\displaystyle=$
$\displaystyle\log\left(\frac{1+\mathsf{SNR}_{i}}{1+\mathsf{INR}_{i}}\right)-\log\left(2+\mathsf{SNR}_{ip}\right)+1$
$\displaystyle<$
$\displaystyle\log\left(\frac{1+\mathsf{SNR}_{i}}{1+\mathsf{INR}_{i}}\right)-\log\left(1+\frac{\mathsf{SNR}_{i}}{1+\mathsf{INR}_{i}}\right)+1$
$\displaystyle=$ $\displaystyle
1-\log\left(1+\frac{\mathsf{INR}_{i}}{1+\mathsf{SNR}_{i}}\right)$
$\displaystyle\leq$ $\displaystyle 1$
* •
$\alpha_{i}-e_{i}<1$, because
$\displaystyle\alpha_{i}-e_{i}$ $\displaystyle=$
$\displaystyle\log\left(1+\mathsf{INR}_{i+1}+\frac{\mathsf{SNR}_{i}}{1+\mathsf{INR}_{i}}\right)$
(153)
$\displaystyle-\log\left(1+\mathsf{INR}_{i+1}+\mathsf{SNR}_{ip}\right)+1$
$\displaystyle\leq$ $\displaystyle 1$
* •
$\gamma_{i}-g_{i}=1$, because
$\displaystyle\gamma_{i}-g_{i}$ $\displaystyle=$
$\displaystyle\log\left(1+\mathsf{INR}_{i+1}+\mathsf{SNR}_{i}\right)$ (154)
$\displaystyle-\log\left(1+\mathsf{INR}_{i+1}+\mathsf{SNR}_{i}\right)+1$
$\displaystyle=$ $\displaystyle 1$
* •
$\mu_{i}-e_{i-1}<1$, because
$\displaystyle\mu_{i}-e_{i-1}$ $\displaystyle=$
$\displaystyle\log(1+\mathsf{INR}_{i})$ (155)
$\displaystyle-\log\left(1+\mathsf{INR}_{i}+\mathsf{SNR}_{i-1,p}\right)+1$
$\displaystyle\leq$ $\displaystyle 1$
## References
* [1] T. S. Han and K. Kobayashi, “A new achievable rate region for the interference channel,” _IEEE Trans. Inf. Theory_ , vol. 27, no. 1, pp. 49–60, Jan. 1981.
* [2] H. Chong, M. Motani, H. Garg, and H. El Gamal, “On the Han-Kobayashi region for the interference channel,” _IEEE Trans. Inf. Theory_ , vol. 54, no. 7, pp. 3188–3195, Jul. 2008.
* [3] A. B. Carleial, “Interference channels,” _IEEE Trans. Inf. Theory_ , vol. 24, no. 1, pp. 60–70, Jan. 1978.
* [4] H. Sato, “The capacity of the Gaussian interference channel under strong interference,” _IEEE Trans. Inf. Theory_ , vol. 27, no. 6, pp. 786–788, Nov. 1981.
* [5] V. S. Annapureddy and V. Veeravalli, “Gaussian interference networks: Sum capacity in the low interference regime and new outer bounds on the capacity region,” _IEEE Trans. Inf. Theory_ , vol. 55, no. 7, pp. 3032–3035, Jul. 2009.
* [6] A. S. Motahari and A. K. Khandani, “Capacity bounds for the Gaussian interference channel,” _IEEE Trans. Inf. Theory_ , vol. 55, no. 2, pp. 620–643, Feb. 2009.
* [7] X. Shang, G. Kramer, and B. Chen, “A new outer bound and the noisy-interference sum-rate capacity for Gaussian interference channels,” _IEEE Trans. Inf. Theory_ , vol. 55, no. 2, pp. 689–699, Feb. 2009.
* [8] R. Etkin, D. N. C. Tse, and H. Wang, “Gaussian interference channel capacity to within one bit,” _IEEE Trans. Inf. Theory_ , vol. 54, no. 12, pp. 5534–5562, Dec. 2008.
* [9] V. R. Cadambe and S. A. Jafar, “Interference alignment and the degrees of freedom for the K user interference channel,” _IEEE Trans. Inf. Theory_ , vol. 54, no. 8, pp. 3425–3441, Aug. 2008.
* [10] G. Bresler, A. Parekh, and D. N. C. Tse, “The approximate capacity of the many-to-one and one-to-many Gaussian interference channels,” _IEEE Trans. Inf. Theory_ , vol. 56, no. 9, pp. 4566 –4592, Sep. 2010.
* [11] X. Shang, G. Kramer, and B. Chen, “New outer bounds on the capacity region of Gaussian interference channels,” in _Proc. IEEE Int. Symp. Inf. Theory (ISIT)_ , Jul. 2008, pp. 245–249.
* [12] D. Tuninetti, “A new sum-rate outer bound for interference channels with three source-destination pairs,” in _Proc. Inf. Theory and App. (ITA)_ , Feb. 2011, pp. 1–8.
* [13] A. Chaaban and A. Sezgin, “The capacity region of the 3-user Gaussian interference channel with mixed strong-very strong interference,” in _Proc. Int. ITG Workshop on Smart Antennas (WSA)_ , Feb. 2011, pp. 1–5.
* [14] S. A. Jafar and S. Vishwanath, “Generalized degrees of freedom of the symmetric Gaussian K user interference channel,” _IEEE Trans. Inf. Theory_ , vol. 56, no. 7, pp. 3297–3303, July. 2010.
* [15] V. R. Cadambe and S. A. Jafar, “Interference alignment and a noisy interference regime for many-to-one interference channels,” _Submitted to IEEE Trans. Inf. Theory_ , Dec. 2009. [Online]. Available: http://arxiv.org/pdf/0912.3029
* [16] O. Ordentlich, U. Erez, and B. Nazer, “The approximate sum capacity of the symmetric Gaussian k-user interference channel,” _Submitted to IEEE Trans. Inf. Theory_ , Jun. 2012. [Online]. Available: http://arxiv.org/abs/1206.0197
* [17] O. Somekh, B. M. Zaidel, and S. Shamai, “Sum rate characterization of joint multiple cell-site processing,” _IEEE Trans. Inf. Theory_ , vol. 53, no. 12, pp. 4473–4497, Dec. 2007.
* [18] A. D. Wyner, “Shannon-theoretic approach to a Gaussian cellular multiple-access channel,” _IEEE Trans. Inf. Theory_ , vol. 40, no. 6, pp. 1713–1727, Nov. 1994.
* [19] Y. Liang and A. Goldsmith, “Symmetric rate capacity of cellular systems with cooperative base stations,” in _Proc. Global Telecommun. Conf. (Globecom)_ , Nov. 2006, pp. 1–5.
* [20] J. Sheng, D. N. C. Tse, J. Hou, J. B. Soriaga, and R. Padovani, “Multi-cell downlink capacity with coordinated processing,” in _Proc. Inf. Theory and App. (ITA)_ , Jan. 2007, pp. 1–5.
* [21] E. Sasoglu, “Successive cancellation for cyclic interference channels,” in _Proc. IEEE Inf. Theory Workshop (ITW)_ , May 2008, pp. 36–40.
* [22] K. Kobayashi and T. S. Han, “A further consideration on the HK and the CMG regions for the interference channel,” in _Proc. Inf. Theory and App. (ITA)_ , Jan. 2007.
* [23] A. Chaaban and A. Sezgin, “On the capacity of a class of multi-user interference channels,” in _Proc. Int. ITG Workshop on Smart Antennas (WSA)_ , Feb. 2011, pp. 1–5.
* [24] A. Raja, V. M. Prabhakaran, and P. Viswanath, “The two-user compound interference channel,” _IEEE Trans. Inf. Theory_ , vol. 55, no. 11, pp. 5100–5120, Nov. 2009.
* [25] L. Zhou and W. Yu, “On the symmetric capacity of the k-user symmetric cyclic Gaussian interference channel,” in _Proc. IEEE Conf. Inf. Sciences and Systems (CISS)_ , Mar. 2010, pp. 1–6.
Lei Zhou (S’05) received the B.E. degree in electronics engineering from
Tsinghua University, Beijing, China, in 2003 and M.A.Sc. degree in electrical
and computer engineering from the University of Toronto, ON, Canada, in 2008.
During 2008-2009, he was with Nortel Networks, Ottawa, ON, Canada. He is
currently pursuing the Ph.D. degree with the Department of Electrical and
Computer Engineering, University of Toronto, Canada. His research interests
include multiterminal information theory, wireless communications, and signal
processing. He is a recipient of the Shahid U.H. Qureshi Memorial Scholarship
in 2011, the Alexander Graham Bell Canada Graduate Scholarship for 2011-2013,
and the Chinese government award for outstanding self-financed students abroad
in 2012.
---
Wei Yu (S’97-M’02-SM’08) received the B.A.Sc. degree in Computer Engineering
and Mathematics from the University of Waterloo, Waterloo, Ontario, Canada in
1997 and M.S. and Ph.D. degrees in Electrical Engineering from Stanford
University, Stanford, CA, in 1998 and 2002, respectively. Since 2002, he has
been with the Electrical and Computer Engineering Department at the University
of Toronto, Toronto, Ontario, Canada, where he is now Professor and holds a
Canada Research Chair in Information Theory and Digital Communications. His
main research interests include multiuser information theory, optimization,
wireless communications and broadband access networks. Prof. Wei Yu currently
serves as an Associate Editor for IEEE Transactions on Information Theory. He
was an Editor for IEEE Transactions on Communications (2009-2011), an Editor
for IEEE Transactions on Wireless Communications (2004-2007), and a Guest
Editor for a number of special issues for the IEEE Journal on Selected Areas
in Communications and the EURASIP Journal on Applied Signal Processing. He is
member of the Signal Processing for Communications and Networking Technical
Committee of the IEEE Signal Processing Society. He received the IEEE Signal
Processing Society Best Paper Award in 2008, the McCharles Prize for Early
Career Research Distinction in 2008, the Early Career Teaching Award from the
Faculty of Applied Science and Engineering, University of Toronto in 2007, and
the Early Researcher Award from Ontario in 2006.
---
|
arxiv-papers
| 2010-10-06T00:54:02 |
2024-09-04T02:49:13.498180
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Lei Zhou and Wei Yu",
"submitter": "Lei Zhou",
"url": "https://arxiv.org/abs/1010.1044"
}
|
1010.1166
|
# Critical behavior of Binder ratios and ratios of higher order cumulants of
conserved charges in QCD deconfinement phase transition
Chen Lizhu Institute of Particle Physics, Hua-Zhong Normal University, Wuhan
430079, China Brookhaven National Laboratory, Upton, NY 11973, USA Pan Xue
Institute of Particle Physics, Hua-Zhong Normal University, Wuhan 430079,
China X. S. Chen Institute of Theoretial Physics, Chinese Academy of
Sciences, Beijing 100190, China Wu Yuanfang Institute of Particle Physics,
Hua-Zhong Normal University, Wuhan 430079, China Brookhaven National
Laboratory, Upton, NY 11973, USA Key Laboratory of Quark $\&$ Lepton Physics
(Huazhong Normal University), Ministry of Education, China
###### Abstract
Binder liked ratios of baryon number are firstly suggested in relativistic
heavy ion collisions. Using 3D-Ising model, the critical behavior of Binder
ratios and ratios of higher order cumulants of order parameter are fully
presented. Binder ratio is shown to be a step function of temperature. The
critical point is the intersection of the ratios of different system sizes
between two platforms. From low to high temperature through the critical
point, the ratios of third order cumulants change their values from negative
to positive in a valley shape, and ratios of fourth order cumulants oscillate
around zero. The normalized ratios, like the Skewness and Kurtosis, do not
diverge with correlation length, in contrary with corresponding cumulants.
Applications of these characters in search critical point in relativistic
heavy ion collisions are discussed.
###### pacs:
25.75.Gz,25.75.Ld
One of the main goals of current relativistic heavy ion experiments is to
locate the critical point of QCD deconfinement phase transition. The critical
character is that the correlation length $\xi$ goes to infinite larger at
infinite system. For finite system, like the formed one in relativistic heavy
ion collisions, the correlation length should be a finite maximum. Therefore,
the various correlation length related observables are suggested in
relativistic heavy ion collisions corr-fluc .
It has been recently shown that near the critical point, the density-density
correlator of baryon-number follow the same power law behavior as the
correlator of the sigma field, which is associated with the chiral order
parameter stephanov ; antoniou . Therefore, the baryon number is considered as
an equivalent order parameter of formed system in nuclear collisions kapusta .
From statistic physics, it also shows that the susceptibilities of order
parameter is directly related to the fluctuations of conserved charges, i.e.,
$\langle\delta N^{i}\rangle=VT\chi_{i}.$ (1)
$\chi_{i}$ is the $i$th order susceptibility. $\langle\delta
N^{i}\rangle=\langle(N-\bar{N})^{i}\rangle$ is the $i$th order cumulants of
the conserved charge number $N$. For three flavor QCD, the conserved charges
are baryon-number, strangeness, and electric charge koch .
The third and forth order cumulants of conserved charges are defined
respectively as,
$K_{3}=\langle\delta N^{3}\rangle,\ \ K_{4}=\langle\delta
N^{4}\rangle-3\langle\delta N^{2}\rangle^{2}.$ (2)
In the vicinity of critical point, they are argued to be proportional to the
higher power of correlation length, i.e., $\xi^{4.5}$ and $\xi^{7}$ stephanov
PRL ; rajargopal , respectively. So they are more sensitive to the correlation
length, and highly recommended.
In experiments star-prl , properly normalized cumulants, i.e., Skewness and
Kurtosis,
$K_{3}/K_{2}^{3/2}=\frac{\langle\delta N^{3}\rangle}{\langle\delta
N^{2}\rangle^{3/2}},\ \ K_{4}/K_{2}^{2}=\frac{\langle\delta
N^{4}\rangle}{\langle\delta N^{2}\rangle^{2}}-3,$ (3)
are actually presented. As the second cumulant is also proportional to a
certain power of correlation length non-monotonic , if such normalized
Skewness and Kurtosis diverge with correlation length is not clear from the
theoretical point of view.
From theoretical side, the ratios of high order cumulants to the second one,
e.g.,
$R_{3,2}=\frac{\langle\delta N^{3}\rangle}{\langle\delta N^{2}\rangle},\
R_{4,2}=\frac{\langle\delta N^{4}\rangle}{\langle\delta
N^{2}\rangle}-3\langle\delta N^{2}\rangle.$ (4)
are estimated karsch-prd ; plb09 ; Liuyx ; Skokov ; Wuyl . The Lattice QCD
with two light quark degrees of freedom shows that these ratios of the baryon
number, strangeness, and electric charge have pronounced peaks from low to
high temperature in the transition region of chiral symmetry break karsch-prd
. The effective models in the mean-field approximation also shows that there
are peak, valley, and oscillating structures near the deconfinement and chiral
phase transitions plb09 ; Liuyx ; Wuyl . However, all these are obtained under
some approximations due to the difficulties in Lattice QCD calculations Gupta
and model estimations Skokov .
Although the concrete form of interactions varies from one system to another,
according to the theory of university, the critical exponent of equivalent
measurement is identical in the same university class. This allows us to study
the critical behavior of complex system by known simple one.
It is known that the QCD deconfinement phase transition corresponds to the
restoration of O(1) symmetry, which is the same university class of 3D-Ising
model s-university . Therefore, the critical behavior of all above mentioned
cumulants of baryon number can be easily obtained from the corresponding
cumulants of order parameter in 3D-Ising model.
Moreover, it it known in statistical physics that the Binder ratio of order
parameter is a direct location of critical point binder . Generally, the
Binder liked ratios are normalized raw moments of order parameter. The third
and fourth Binder liked cumulant ratios can be simply defined as,
$B_{3}=\frac{\langle M^{3}\rangle}{\langle M^{2}\rangle^{3/2}},\ \
B_{4}=\frac{\langle M^{4}\rangle}{\langle M^{2}\rangle^{2}}.$ (5)
Here we take 3D-Ising model as an example. The order parameter in the model is
the magnet $M=\sum_{i=1}^{N_{L}}\vec{s_{i}}/N_{L}$ of spin $\vec{s}$ in all
lattice sites $N_{L}$.
Equivalently, the order parameter in relativistic heavy ion collisions is the
baryon number. The temperature, or the controlling parameter, is the incident
energy. The size of the formed system is mainly determined by the overlapped
area, i.e., centralities. So if we pass through the region of critical
incident energies in relativistic heavy ion collisions, the Binder ratios of
baryon number can be served as a good location of critical point of QCD
deconfinement phase transition.
In this paper, we firstly present the critical behavior of Binder ratios in
3D-Ising model, and demonstrate why they are helpful, in particular, in
locating the critical point in relativistic heavy ion collisions. Then, the
critical behavior of Skewness, Kurtosis, $R_{3,2}$, and $R_{4,2}$ are
presented and discussed, respectively. Meanwhile, from finite-size scaling of
the susceptibilities, the critical behavior of those ratios are estimated
model independently. Finally, the conclusions are drawn.
The critical behavior of Binder ratios, $B_{3}$ and $B_{4}$, in 3D-Ising for 4
different lattice sizes are presented in Fig. 1(a) and (b), respectively.
Where the simulation of 3D-Ising model is based on the wolff algorithm MCbook
. We can see that both $B_{3}$ and $B_{4}$ show a step jump in the vicinity of
critical temperature. The physical meaning of this jump is clear.
When the temperature is much lower than the critical one, the system is almost
order and the fluctuation of order parameter is very small, i.e.,
$\langle M^{n}\rangle\sim\langle M\rangle^{n}\;\;\;\;\;\;\;({\rm for}\
n=2,3,4\cdots).$ (6)
So it results the lower platform, which is 1 for all orders of Binder ratios
at all system sizes, as shown in Fig. 1. When the temperature approaches to
critical one, the correlation length starts to increase with temperature and
the fluctuations become larger and larger. Their critical behavior is system
size dependent and described by finite-size scaling. Only at critical
temperature, all size curves intersect to the fixed point, where they are
system size independent fs1 , as shown in Fig. 1. When the temperature is much
higher than the critical one, the system is totally disordered. It approaches
again to a constant. This forms the platform at high temperature. It is 1.6
and 3 times larger than the lower platforms for the third and fourth order
Binder ratios, respectively. So the higher the order of Binder ratio, the
larger the gap of the step function.
Figure 1: (Color online) The temperature dependence of Binder ratios in Eq.
(5) in the vicinity of critical temperature in 3D-Ising model for 4 different
lattice sizes.
This step function liked behavior can be served as a very good probe of
critical point in relativistic heavy ion collisions, where critical incident
energy is difficult to assign precisely in priori. So if we scan incident
energies, and observe two platforms at low and high energy regions,
respectively, then the critical one is most probably between them. We can
finely tune the incident energy in the region and precisely determine the
critical energy and exponents.
The Skewness and Kurtosis of order parameter in 3D-Ising model for 4 different
lattice sizes are presented in Fig. 2(a) and (b), respectively. We can see
from the figure that they change sharply in the vicinity of the critical
temperature. The Skewness first drops down and then goes up, and Kurtosis
oscillates with temperature. Their values are system size dependent. Their
signs change respectively near the critical point. Former in Fig. 2(a) changes
from negative to positive when the temperature is increased through the
critical point, while the later in Fig. 2(b) becomes negative only when the
temperature is close to the critical point. The sign change in Skewness, or
third order cumulants, is expected in effective models Asakawa ; Liuyx ; Wuyl
.
Figure 2: (Color online) The temperature dependence of Kurtosis (a) and
Skewness (b) in Eq. (3) in the vicinity of critical temperature in 3D Ising
model for 4 different lattice sizes.
As we know that the Skewness and Kurtosis measure the symmetry and sharpness
of the distribution, respectively. The distributions of order parameter $M$
near the critical point at system size $L=8$ are shown in Fig. 3. Where we can
clearly see that the long tail of the distributions changes from the left to
the right side when the temperature is increased through the critical point,
and the peak of the distribution vary from sharp to flat when temperature is
approached the critical point. The same trend has been observed in percolation
model, in studying clusterization phenomena in nuclear multi-fragmentation
percolation .
Figure 3: (Color online) The distributions of order parameter near critical
temperatures in 3D-Ising model at system size $L=8$.
This character can also be served as a signal associated with the appearance
of critical point in relativistic heavy ion collisions. If we observe sign
change of Skewness (Kurtosis) of baryon number at a certain incident energy
region, it most probably predicts the appearance of critical point in the
nearby incident energy region.
The Skewness and Kurtosis also converge to two constants when the temperature
is away from critical point, as shown in Fig. 2(a) and (b). But the constants
at low and high temperatures are close to zero and 1, respectively. The gap
between them are small and does not change very much with the order of
cumulants, unlike the Binder ratio.
Moreover, all size curves of Skewness (Kurtosis) intersect at critical point.
This can be easily understood from finite-size scaling of susceptibilities,
i.e.,
$\displaystyle\chi_{i}(t,L)=L^{\gamma_{i}/\nu}P_{\chi_{i}}(tL^{1/\nu}).$ (7)
Where the $\gamma_{i}$ is the critical exponents of $i$th order
susceptibility, and $\nu$ is the critical exponent of correlation length
$\xi_{\infty}=t^{-\nu}$ at infinite system. $t=\frac{T-T_{\rm c}}{T_{\rm c}}$
is reduced temperature, and $T_{\rm c}$ is critical temperature. In the
vicinity of critical point, the correlation length at finite system is
approximately the same order of the system size, i.e., $\xi\sim L=V^{1/3}$.
For $\chi_{3}$ and $\chi_{4}$, $\gamma_{\rm 3}/\nu=4.5$, $\gamma_{\rm
4}/\nu=7$, respectively stephanov PRL . So the critical behavior of the
Skewness and Kurtosis in Eq. (3) are,
$\displaystyle K_{3}/K_{2}^{3/2}$ $\displaystyle=$
$\displaystyle\frac{VT\chi_{3}}{(VT)^{3/2}\chi_{2}^{3/2}}\sim\frac{L^{3}L^{4.5}P_{\chi_{3}}(tL^{1/\nu})}{L^{4.5}L^{3}T^{1/2}P_{\chi_{2}}^{3/2}(tL^{1/\nu})}$
$\displaystyle=$ $\displaystyle T^{-1/2}F_{S}(tL^{1/\nu}),$ $\displaystyle
K_{4}/K_{2}^{2}$ $\displaystyle=$
$\displaystyle\frac{VT\chi_{4}}{(VT)^{2}\chi_{2}^{2}}\sim\frac{L^{3}L^{7}P_{\chi_{4}}(tL^{1/\nu})}{L^{6}L^{4}TP_{\chi_{2}}^{2}}$
(8) $\displaystyle=$ $\displaystyle T^{-1}F_{K}(tL^{1/\nu}).$
They no long diverge with correlation length, or system size. At the critical
temperature $t=0$, the scaling function, i.e., $F_{S}(0)$ or $F_{K}(0)$, is
system size independent constant. All size curves intersect to the constant,
i.e., the fixed point fs1 .
From this simple estimation and Fig. 2, we can see that normalized high order
cumulants, i.e., Skewness and Kurtosis, do not directly diverge with
correlation length any more, different from corresponding cumulants, $K_{3}$
and $K_{4}$, which are proportional to $\xi^{4.5}$ and $\xi^{7}$, respectively
stephanov PRL ; rajargopal .
The $R_{3,2}$, and $R_{4,2}$ of order parameter in 3D-Ising model for 4
different lattice sizes are presented in Fig. 4(a) and (b), respectively. We
can see again from Fig. 4(a) that $R_{3,2}$ changes its value sharply from
negative to positive when temperature is increased through the critical point.
$R_{4,2}$ in Fig. 4(b) oscillates greatly with temperature near the critical
point. These qualitative features, i.e., sign change in third moment, and
oscillating structure in forth cumulants, are consistent with estimations of
effective models Asakawa ; Liuyx ; Wuyl .
Figure 4: (Color online) The temperature dependence of $R_{3,2}$ (a), and
$R_{4,2}$ (b) in the vicinity of critical temperature in 3D-Ising model for 4
different lattice sizes.
$R_{3,2}$ and $R_{4,2}$ are very sensitive to the system size, or correlation
length. Their values become very large when system size increases. The
critical exponent of $R_{3,2}$, and $R_{4,2}$ can be roughly estimated from
finite-size scaling of susceptibilities, i.e.,
$\displaystyle R_{3,2}$ $\displaystyle=$
$\displaystyle\frac{VT\chi_{3}}{VT\chi_{2}}=\frac{L^{3}\xi^{4.5}P_{\chi_{3}}(tL^{1/\nu})}{L^{3}\xi^{2}P_{\chi_{2}}(tL^{1/\nu})}$
$\displaystyle=$ $\displaystyle\xi^{2.5}F_{R_{3,2}}(tL^{1/\nu})$
$\displaystyle R_{4,2}$ $\displaystyle=$
$\displaystyle\frac{VT\chi_{4}}{VT\chi_{2}}=\frac{L^{3}\xi^{7}P_{\chi_{4}}(tL^{1/\nu})}{L^{3}\xi^{2}P_{\chi_{2}}(tL^{1/\nu})}$
(9) $\displaystyle=$ $\displaystyle\xi^{5}F_{R_{4,2}}(tL^{1/\nu}).$
So $R_{3,2}$ and $R_{4,2}$ diverge with correlation length as $\xi^{2.5}$ and
$\xi^{5}$, respectively.
In this paper, the measurements of Binder liked ratios of conserved charges
are firstly suggested in relativistic heavy ion collisions. Using 3D-Ising
model, it is shown that near the critical temperature, Binder ratios is a step
function of temperature. The gap of the step function is 1.6 and 3 times wider
for the third and forth order Binder ratios, respectively. This can be served
as a good identification of critical behavior in relativistic heavy ion
collisions, where the critical incident energy is unknown in prior. The
critical point is the intersection of Binder ratios of different size systems
between two platforms.
The critical behavior of Skewness, Kurtosis, $R_{3,2}$ and $R_{4,2}$ at
various system sizes are also studied by 3D-Ising model, and estimated by
finite size scaling. When the temperature is increased through the critical
point, the ratios of the third order cumulants change their values from
negative to positive in a valley shape, and ratios of fourth order cumulants
oscillate around zero. All size curves of Skewness (Kurtosis) intersect at the
critical point. The normalized ratios, like the Skewness and Kurtosis, do not
diverge with correlation length. While, un-normalized ratios, $R_{3,2}$ and
$R_{4,2}$, are divergent with correlation length. They are proportional to
$\xi^{2.5}$ and $\xi^{5}$, respectively, and very sensitive to the system size
near the critical temperature.
These critical characters may show up at the energy dependence of
corresponding ratios of conserved charges. Their behavior at coming
relativistic heavy ion experiments at RHIC, SPS, and FAIR are called for.
We are grateful for stimulated discussions with Dr. Nu Xu. The first and last
authors are grateful for the hospitality of BNL STAR group. This work is
supported in part by the NSFC of China with project No. 10835005 and MOE of
China with project No. IRT0624 and No. B08033.
## References
* (1) M. A. Stephanov, K. Rajagopal, and E. Shuyak, Phys. Rev. Lett. 81, 4816(1998); S. Jeon and V. Koch, Phys. Rev. Lett. 85, 2076 (2000); M. Asakawa, U. Heinz and B. Müller, Phys. Rev. Lett. 85, 2072 (2000); H. Heiselberg, Phys. Rept. 351, 161(2001).
* (2) Y. Hatta and M. A. Stephanov, Phys. Rev. Lett. 91, 102003 (2003); Y. Hatta and T. Ikeda, Phys. Rev. D 67, 014028 (2003).
* (3) N. G. Antoniou, F. K. Diakonos,* and A. S. Kapoyannis, K. S. Kousouris, Phys. Rev. Lett., 97, 032002 (2006); D. Bower and S. Gavin, Phys. Rev. C 64, 051902(2001); N. G. Antoniou, Nucl. Phys. B, Proc. Suppl. 92, 26 (2001).
* (4) J. Kapusta, arXiv:1005.0860.
* (5) V. Koch, arXiv:0810.2520
* (6) M. A. Stephanov, Phys. Rev. Lett. 102, 032301(2009).
* (7) C. Athanasiou, K. Rajagopal, and M. Stephanov, arXiv:1006.4636; C. Athanasiou, K. Rajagopal, and M. Stephanov, arXiv:1008.3385.
* (8) M. M. Aggarwal et al., Phys. Rev. Lett. 105, 022302(2010).
* (9) M. A. Stephanov, hep-ph/0402115, Int. J. Mod. Phys. A 20 (2005) 4387.
* (10) C. R. Allton, M. Döring, S. Ejiri, S. J. Hands, O. Kaczmarek, F. Karsch, E. Laermann and K. Redlich. Phys. Rev. D 71, 054508(2005).
* (11) B. Stokić, B. Friman, K. Redlich, Phys. Lett. B 673, 192(2009).
* (12) Wei-jie Fu, Yu-xin Liu, Yue-Liang Wu, Phys. Rev. D 81, 014028 (2010)
* (13) V. Skokov,, B. Stokić, B. Friman, and K. Redlich, Phys. Rev. D 82, 034029(2010).
* (14) Wei-jie Fu, and Yue-liang Wu, arXiv: 1008.3684;
* (15) S. Gupta, arXiv:0909.4630.
* (16) J. Garcá, J. A. Gonzalo, Physica A 326 (2003) 464\.
* (17) K. Binder, Z. Phys. B 43, 119(1981). K. Binder, Rep. Prog. Phys. 60, 487(1997).
* (18) U. Wolff, Phys. Rev. Lett. 62, 361(1989).
* (19) Wu Yuanfang, Chen Lizhu, and X. S. Chen, PoS (CPOD, 2009)036; Chen Lizhu, X. S. Chen, and Wu Yuanfang, arXiv: 0904.1040; ibid.,1002:4139.
* (20) M. Asakawa, S. Ejiri, M. Kitazawa, Phys. Rev. Lett. 103, 262301(2009).
* (21) J. Brzychczyk, Phys. Rev. C 73, 024601 (2006).
|
arxiv-papers
| 2010-10-06T14:32:07 |
2024-09-04T02:49:13.530640
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Lizhu Chen, Xue Pan, Xiaosong Chen, and Yuanfang Wu",
"submitter": "Yuanfang Wu",
"url": "https://arxiv.org/abs/1010.1166"
}
|
1010.1189
|
# Development of nonlinear two fluid interfacial structures by combined action
of Rayleigh-Taylor, Kelvin-Helmholtz and Richtmyer-Meshkov
instabilities:Oblique shock
M. R. Gupta, Labakanta Mandal, Sourav Roy , Rahul Banerjee,Manoranjan Khan
Dept. of Instrumentation Science & Centre for Plasma Studies
Jadavpur University, Kolkata-700032, India
e-mail: mrgupta-cps@yahoo.co.ine-mail: laba.kanta@yahoo.come-mail:
phy.sou82@gmail.come-mail: rbanerjee.math@gmail.come-mail: mkhan-ju@yahoo.com
###### Abstract
The nonlinear evolution of two fluid interfacial structures like bubbles and
spikes arising due to the combined action of Rayleigh-Taylor and Kelvin-
Helmholtz instability or due to that of Richtmyer-Meshkov and Kelvin-Helmholtz
instability resulting from oblique shock is investigated. Using Layzer’s model
analytic expressions for the asymptotic value of the combined growth rate are
obtained in both cases for spikes and bubbles. However, if the overlying fluid
is of lower density the interface perturbation behaves in different ways.
Depending on the magnitude of the velocity shear associated with Kelvin-
Helmholtz instability both the bubble and spike amplitude may simultaneously
grow monotonically (instability) or oscillate with time or it may so happen
that while this spike steepens the bubble tends to undulate. In case of an
oblique shock which causes combined action of Richtmyer-Meshkov instability
arising due to the normal component of the shock and Kelvin Helmholtz
instability through creation of velocity shear at the two fluid interface due
to its parallel component, the instability growth rate-instead of behaving as
$1/t$ as $t\rightarrow\infty$ for normal shock, tends asymptotically to a
spike peak height growth velocity
$\sim\sqrt{\frac{5(1+A_{T})}{16(1-A_{T})}(\Delta v)^{2}}$ where $\Delta v$ is
the velocity shear and $A_{T}$ is the Atwood number. Implication of such
result in connection with generation of spiky fluid jets in astrophysical
context is discussed.
## I. INTRODUCTION
Rayleigh-Taylor (RTI) and Kelvin-Helmholtz (KHI) instabilities are associated
with the perturbation of the interface of two fluids of different densities
subject to the action of continuously acting acceleration (with respect to
time) and under the action of velocity shear,respectively. The perturbation
and the consequent instability may also be induced by a shock generated
impulsive acceleration known as Richtmyer-Meshkov (RMI) instability. Such
interfacial hydrodynamic instabilities occur in a wide range of physical
phenomenon from those associated with problems on wave generation by wind
blowing over water surface to problems related to Inertial Confinement Fusion
(ICF) or astrophysical problems like that of supernova explosion remnant which
belong to the domain of high energy density (HED) physics
${}^{\cite[cite]{[\@@bibref{}{rd06}{}{}]}}$. In ICF experiment HED plasmas may
be created due to multi kilo Joule laser with a pressure $\sim$ Mbar. In
ICF,in addition to RT and RM instabilities nonspherical implosion generate
shear flows; the later is also formed when shocks pass through irregular fluid
interfaces. The KHI and shear flow effects in general are also of practical
importance in a number of HED system. They should be considered in a multi
shock implosion schemes for direct drive capsule for ICF, since KHI may
accelerate the growth of turbulent mixing layer at the interface between the
ablator and solid deuterium-tritium nuclear fuel. In HED and astrophysical
system, it has been seen that structures driven by shear flow appear on the
high density spikes produced by R-T and R-M
instabilities${}^{\cite[cite]{[\@@bibref{}{kk03}{}{}]}}$. They may develop in
course of evolution of these instabilities
${}^{\cite[cite]{[\@@bibref{}{br06}{}{}]}}-^{\cite[cite]{[\@@bibref{}{db07}{}{}]}}$
and cover enormous range of spatial scales from $10^{17}$cm for jets from
young stellar objects to $10^{24}$cm for jets from quasars or active galactic
nuclei${}^{\cite[cite]{[\@@bibref{}{br06}{}{}]}}$. Examples are suggested to
be provided by pillars (”elephant trunk”) of Eagle Nebula which are identified
with spikes of a heavy fluid penetrating a light
fluid${}^{\cite[cite]{[\@@bibref{}{ls54}{}{}]}},^{\cite[cite]{[\@@bibref{}{ef54}{}{}]}}$.
Another example in astrophysics is the Herbig-Haro (HH) object like HH34,
where jets are observed with knots. Buhrke et al
${}^{\cite[cite]{[\@@bibref{}{tb88}{}{}]}}$ explained that Kelvin-Helmholtz
instability is the reason for knots in the jets. The jet must be $\sim 10$
times denser than its surrounding medium having velocity $\sim$300 km/sec and
Mach no. 30\. Steady isolated jets may form structure through the growth of
K-H modes. The stability properties of super magnetosonic astrophysical jets
are subject of current interest.
The linear theory of the combined effects of RT,KH and RM instabilities have
been investigated earlier ${}^{\cite[cite]{[\@@bibref{}{m94}{}{}]}}$. Weakly
nonlinear theoretical results of Kelvin-Helmholtz and Rayleigh-Taylor
instability growth rates together with different aspects of density and shear
velocity gradients have also been
discussed${}^{\cite[cite]{[\@@bibref{}{lw09}{}{}]}}-^{\cite[cite]{[\@@bibref{}{lw10}{}{}]}}$.
In case of the temporal evolution of these instabilities nonlinear structures
develop at the two fluid interface. The structure is called a bubble if the
lighter fluid pushes across the unperturbed surface into the heavier fluid and
a spike if the opposite takes place. The dynamics of such RTI and RMI
generated nonlinear structures have been studied
${}^{\cite[cite]{[\@@bibref{}{jh94}{}{}]}}-^{\cite[cite]{[\@@bibref{}{ps03}{}{}]}}$
under different physical situation using an expression near the tip of the
bubble or spike up to second order in the transverse coordinate to unperturbed
surface following Layzer’s
${}^{\cite[cite]{[\@@bibref{}{mr09}{}{}]}}$approach.
In the present paper, we investigate the combined effect of Rayleigh-
Taylor,Richtmyer-Meshkov and Kelvin Helmholtz instabilities by extending the
above method so as to include the effect of velocity shear induced
contribution to the growth rate of the tip of the nonlinear mushroom like
structures generated by shock wave (normal or oblique) incident on the
unperturbed interface.
In the event of excitation of RM instability due to normal incidence of shock
in absence of velocity shear of the growth rate of the height of the finger
like structures decay as
$(1/t)$${}^{\cite[cite]{[\@@bibref{}{vn02}{}{}]}},^{\cite[cite]{[\@@bibref{}{ps03}{}{}]}}$.
It is however interesting to note that if the shock incidence is oblique (or
if it passes across an irregular surface) the growth rate of the tip of the
spiky structure does not decrease as $(1/t)$ but attains a saturation value
proportional to $\sqrt{k^{2}(\Delta v)^{2}/(1-A_{T})}$ where $\Delta
v$=difference is the tangential velocity of the fluids at the interface and
$A_{T}$ is the Atwood number. Thus the growth rate may be quite large if
$A_{T}\rightarrow 1$ which may be likely in astrophysical situation and thus
play an important role in formation of
jets${}^{\cite[cite]{[\@@bibref{}{br06}{}{}]}},^{\cite[cite]{[\@@bibref{}{dd02}{}{}]}}$.
The paper is organized in the following manner. In section II is developed the
basic equations describing the dynamics of nonlinear structures which evolve
in consequence of the combined effects of these different types of
hydrodynamical instabilities. In section III it is shown that the classical
results${}^{\cite[cite]{[\@@bibref{}{dl55}{}{}]}}$ follow on linearization of
the evolution equation describing the bubbles and spikes. Numerical as well as
some analytical results regarding the saturation growth rates are presented in
section IV. Finally section V presents a brief summary of this results.
## II. BASIC EQUATIONS OF EVOLUTION OF THE HYDRODYNAMIC INSTABILITIES
Let the $y=0$ plane denote the unperturbed surface of separation of two fluids
(the line $y=0$ in the two dimensional form of this problem). The fluid with
density $\rho_{a}$ is assumed to overlie the fluid with density $\rho_{b}$.
The gravity $\overrightarrow{g}$ is assumed to act along the negative y- axis.
Any perturbation of the horizontal interface or a shock driven impulse gives
rise to Rayleigh-Taylor instability($\rho_{a}>\rho_{b}$) or Richtmyer -Meshkov
instability which in course of temporal evolution gives rise to nonlinear
interfacial structures.
The two fluids separated by the horizontal boundary are further assumed to be
in relative horizontal motion and thus subjected to Kelvin-Helmholtz
instability arising due to horizontal velocity shear. Thus we are faced with
the problem of the combined action of Rayleigh-Taylor and Kelvin-Helmholtz
instabilities.We shall see the same formulation will be applicable to
Richtmyer-Meshkov instability associated with an oblique shock incident on the
two fluid interface.
After perturbation the finger shaped interface is assumed to take up a
parabolic shape given by
$\displaystyle y(x,t)=\eta(x,t)=\eta_{0}(t)+\eta_{2}(t)(x-\eta_{1}(t))^{2}$
(1)
For a bubble (here the lower fluid is pushing across the interface into the
upper fluid with density $\rho_{a}>$ density $\rho_{b}$) we have,
$\displaystyle\mbox{}\qquad\qquad\eta_{0}>0\quad\mbox{and}\quad\eta_{2}<0$ (2)
and for spike:
$\displaystyle\qquad\qquad\eta_{0}<0\quad\mbox{and}\quad\eta_{2}>0$ (3)
The height of the vertex of the parabola i.e, the height of the peak of the
bubble (or spike) above the x-axis is $|\eta_{0}(t)|$. The position of the
peak at time t is at $x=\eta_{1}(t)$ and because of the relative streaming
motion of the two fluids the peak moves parallel to the x-axis with velocity
$\dot{\eta}_{1}(t)$. The densities of both fluids are uniform and fluid motion
is supposed to be single mode potential flow.
For the upper fluid with density $\rho_{a}$ we take the velocity potential
$\displaystyle\phi_{a}(x,y,t)=\left[\alpha_{a}(t)\cos{(k(x-\eta_{1}(t)))}+\beta_{a}(t)\sin{(k(x-\eta_{1}(t)))}\right]e^{-k(y-\eta_{0}(t))}-xu_{a}(t);\quad
y>0$ (4)
and for the lower fluid (density $\rho_{b}$) the velocity potential
$\displaystyle\phi_{b}(x,y,t)=\left[\alpha_{b}(t)\cos{(k(x-\eta_{1}(t)))}+\beta_{b}(t)\sin{(k(x-\eta_{1}(t)))}\right]e^{k(y-\eta_{0}(t))}-xu_{b}(t)+yb_{0}(t);\quad
y<0$ (5)
Before proceeding with the analysis of the kinematic and boundary conditions
using the two fluid interface perturbation $y=\eta(x,t)$ we forward the
following justification for restricting the expansion to terms
O$(x-\eta_{1}(t))^{2}$. We are concerned only motion very close to the tip of
the bubble or spike i.e., only in the region $k\mid x-\eta_{1}(t)\mid<<1.$
Consequently one is justified in neglecting terms O $(\mid
x-\eta_{1}(t)\mid)^{4}$ unless the coefficients of such terms are sufficiently
large. Further it has been shown ${}^{\cite[cite]{[\@@bibref{}{mr10}{}{}]}}$
that even it terms
$\sim\eta_{4}(t)(x-\eta_{1}(t))^{4}+\eta_{6}(t)(x-\eta_{1}(t))^{6}$ are
retained the contribution from coefficients
$\mid\eta_{4}\mid,\mid\eta_{6}\mid<<$ that from $\mid\eta_{2}\mid$ at least in
the asymptotic state $\tau\rightarrow\infty$.
The kinematical boundary conditions satisfied at the interfacial surface
$y=\eta(x,t)$are
$\displaystyle\frac{\partial\eta}{\partial t}-\frac{\partial\phi_{a}}{\partial
x}\frac{\partial\eta}{\partial x}=-\frac{\partial\phi_{a}}{\partial y}$ (6)
$\displaystyle-\frac{\partial\phi_{a}}{\partial x}\frac{\partial\eta}{\partial
x}+\frac{\partial\phi_{a}}{\partial y}=-\frac{\partial\phi_{b}}{\partial
x}\frac{\partial\eta}{\partial x}+\frac{\partial\phi_{b}}{\partial y}$ (7)
The dynamical boundary conditions are next obtained from Bernoulli’s equation
for the two fluids
$\displaystyle-\frac{\partial\phi_{a}}{\partial
t}+\frac{1}{2}(\vec{\nabla}\phi_{a})^{2}+gy\rho_{a}=-p_{a}+f_{a}(t)$ (8)
$\displaystyle-\frac{\partial\phi_{b}}{\partial
t}+\frac{1}{2}(\vec{\nabla}\phi_{b})^{2}+gy\rho_{b}=-p_{b}+f_{b}(t)$ (9)
by using the surface pressure equality
$\displaystyle p_{a}=p_{b}$ (10)
leading to
$\displaystyle-(\frac{\partial\phi_{a}}{\partial
t}-\frac{\partial\phi_{b}}{\partial
t})+\frac{1}{2}(\vec{\nabla}\phi_{a})^{2}-\frac{1}{2}(\vec{\nabla}\phi_{b})^{2}+g(\rho_{a}-\rho_{b})y$
$\displaystyle=f_{a}(t)-f_{b}(t)$ (11)
satisfied at the interface $y=\eta(x,t)$ Now from Eq.(1)
$\displaystyle\frac{\partial\eta}{\partial
t}=\dot{\eta}_{0}(t)-2\dot{\eta}_{1}(t)\eta_{2}(t)(x-\eta_{1}(t))+\dot{\eta}_{2}(t)(x-\eta_{1}(t))^{2}$
(12)
Also utilizing the property that close to the tip of the bubble or spike,
$k|x-\eta_{1}(t)|<<1$, we express the velocity components in the following
form
$\displaystyle v_{ax}=-\frac{\partial\phi_{a}}{\partial
x}=(u_{a}-k\beta_{a})+k^{2}\alpha_{a}(x-\eta_{1})+\beta_{a}k^{2}(\eta_{2}+k/2)(x-\eta_{1})^{2}$
(13) $\displaystyle v_{ay}=-\frac{\partial\phi_{a}}{\partial
y}=k\alpha_{a}+k^{2}\beta_{a}(x-\eta_{1})-k^{2}\alpha_{a}(\eta_{2}+k/2)(x-\eta_{1})^{2}$
(14)
and similar expressions for $v_{bx}$ and $v_{by}$.
Following Layzer’s${}^{\cite[cite]{[\@@bibref{}{dl55}{}{}]}}$ model we
substitute for $\eta_{t},\eta_{x},(v_{a{(b)}})_{x},(v_{a{(b)}})_{y}$in the
kinematic and boundary conditions represented by Eqs.(6),(7)and (11)and equate
coefficients of $(x-\eta_{1}(t))^{i};(i=0,1,2)$ and neglect terms
$O((x-\eta_{1}(t))^{i});(i\geq 3)$.This yields the following three algebraic
equations for the three unknown $b_{0},\alpha_{b},\beta_{b}$ :
$\displaystyle b_{0}=-\frac{6\eta_{2}}{(3\eta_{2}-k/2)}k\alpha_{a}$ (15)
$\displaystyle\alpha_{b}=\frac{(3\eta_{2}+k/2)}{(3\xi_{2}-k/2)}\alpha_{a}$
(16)
$\displaystyle\beta_{b}=\frac{(\eta_{2}+k/2)k\beta_{a}-{\eta_{2}(u_{a}-u_{b})}}{k(\eta_{2}-k/2)}$
(17)
and regarding the five other unknowns,viz
$\eta_{0}(t),\eta_{1}(t),\eta_{2}(t),\alpha_{a}(t),\beta_{b}(t)$ the following
five nonlinear ODE’s [Eqs.(18)-(22)].
$\displaystyle\frac{d\xi_{1}}{d\tau}=\xi_{3}$ (18)
$\displaystyle\frac{d\xi_{2}}{d\tau}=-\frac{1}{2}(6\xi_{2}+1)\xi_{3}$ (19)
$\displaystyle\frac{d\xi_{3}}{d\tau}=\frac{N_{1}(\xi_{2},r)}{D_{1}(\xi_{2},r)}\frac{\xi_{3}^{2}}{(6\xi_{2}-1)}+\frac{2(1-r)\xi_{2}(6\xi_{2}-1)}{D_{1}(\xi_{2},r)}+\frac{N_{2}(\xi_{2},r)}{D_{1}(\xi_{2},r)}\frac{(6\xi_{2}-1)\xi_{4}^{2}}{2\xi_{2}(2\xi_{2}-1)^{2}}\hskip
70.0pt$
$\displaystyle+2\frac{(4\xi_{2}-1)(6\xi_{2}-1)}{D_{1}(\xi_{2},r)(2\xi_{2}-1)^{2}}\left[(V_{a}-V_{b})^{2}\xi_{2}-(V_{a}-V_{b})(2\xi_{2}+1)\xi_{4}\right]$
(20)
$\displaystyle\frac{d\xi_{4}}{d\tau}=\frac{(2\xi_{2}-1)}{D_{2}(\xi_{2},r)}\left[(f_{b}-rf_{a})-r\frac{\xi_{3}\xi_{4}}{2\xi_{2}}\right]+\frac{2(f_{a}-f_{b})}{D_{2}(\xi_{2},r)}\xi_{2}\hskip
200.0pt$
$\displaystyle+\frac{(6\xi_{2}+1)\xi_{3}}{2D_{2}(\xi_{2},r)(6\xi_{2}-1)(2\xi_{2}-1)}\left[4(V_{a}-V_{b})(4\xi_{2}-1)-\frac{\xi_{4}}{\xi_{2}}(28\xi_{2}^{2}-4\xi_{2}-1)\right]$
(21)
$\displaystyle\frac{d\xi_{5}}{d\tau}=V_{a}-\frac{\xi_{4}(2\xi_{2}+1)}{2\xi_{2}}$
(22)
where
$\displaystyle\xi_{1}=k\eta_{0};\xi_{2}=\eta_{2}/k;\xi_{5}=k\eta_{1}$ (23)
$\displaystyle\xi_{3}=k^{2}\alpha_{a}/\sqrt{kg};\xi_{4}=k^{2}\beta_{a}/\sqrt{kg},\tau=t\sqrt{(}kg)$
(24) $\displaystyle
V_{a}=u_{a}\sqrt{(}k/g);V_{b}=u_{b}\sqrt{(}k/g);f_{a}=\frac{dV_{a}}{d\tau};f_{b}=\frac{dV_{b}}{d\tau}.$
(25)
The functions $N_{1,2}(\xi_{2},r),D_{1,2}(\xi_{2},r)$ where
$r=\frac{\rho_{a}}{\rho_{b}}$ is the density ratio are given by
$\displaystyle N_{1}(\xi_{2},r)=36(1-r)\xi_{2}^{2}+12(4+r)\xi_{2}+(7-r)$ (26)
$\displaystyle D_{1}(\xi_{2},r)=12(r-1)\xi_{2}^{2}+4(r-1)\xi_{2}-(r+1)$ (27)
$\displaystyle N_{2}(\xi_{2},r)=16(1-r)\xi_{2}^{3}+12(1+r)\xi_{2}^{2}-(1+r)$
(28) $\displaystyle D_{2}(\xi_{2},r)=2(1-r)\xi_{2}+(1+r)$ (29)
The above set of five Eqs. (18)-(22) together with Eqs. (23)-(29)which define
the different variables and functions describe the combined effect of RT and
KH instabilities.
On the other hand the impingement of an oblique shock on the two fluid
interface causes the joint effect of Richtmyer-Meshkov and Kelvin-Helmholtz
instability. The impact gives rise to an instantaneous acceleration which will
change the normal velocity (y-component) by an amount $\Delta
v=v_{after}-v_{before}$ and transverse velocity (x-component) by $\Delta
u_{a(b)}=(u_{a(b)})_{after}-(u_{a(b)})_{before}$. Taking nonzero values only
for the post shock velocities we replace the acceleration by their impulsive
values. We set:
$\displaystyle\frac{du_{a(b)}}{dt}=u_{a}\delta(t)\rightarrow\Delta v(t)$ (30)
and replace $g\rightarrow\Delta v\delta(t)$ The dynamical variables are non
dimensionalized using normalization in terms of $(k\Delta v)$instead of
$\sqrt{kg}$.
The combined effect of RM-KH instability resulting from oblique incidence of
shock on the two fluid interface is then described by the same set of
equations as Eqs.(18)-(22) together with the following replacements:
$(i)$The second term on the RHS of Eq.(20)drops out.
(ii) $\xi_{3},\xi_{4}$ and $\tau$ to be replaced by
$\overline{\xi_{3}}=\alpha_{a}k^{2}/(k\Delta
v),\overline{\xi_{4}}=\beta_{a}k^{2}/(k\Delta v)$ and $\tau=t(k\Delta v)$
respectively.
(iii) $V_{a}$ and $V_{b}$ by $\overline{V}_{a}=u_{a}/\Delta
v,\overline{V}_{b}=u_{b}/\Delta v$.
(iv) $f_{a}$ by
$\displaystyle\overline{f}_{a}=\frac{d\overline{v}_{a}}{d\overline{\tau}}=\frac{u_{a}}{\Delta
v}\Delta(\overline{\tau})\quad\mbox{and}\quad
f_{b}\quad\mbox{by}\quad\overline{f_{b}}=\frac{d\overline{v}_{b}}{d\overline{\tau}}=\frac{u_{b}}{\Delta
v}\Delta(\overline{\tau})$ (31)
## III. LINEAR APPROXIMATION
We now show that the usual combined RT and KH instability growth rates
${}^{\cite[cite]{[\@@bibref{}{sc81}{}{}]}}$ are recovered on linearization of
Eqs. (18)-(22). Let us put
$\displaystyle\frac{d(k\eta_{1})}{d\tau}=\frac{d\xi_{5}}{d\tau}=\alpha_{a}V_{a}+\alpha_{b}V_{b}\,;\,\,\,\,\left(\alpha_{a,(b)}=\frac{\rho_{a,(b)}}{\rho_{a}+\rho_{b}}\right)$
in Eq.(22)giving
$\displaystyle\xi_{4}=2\alpha_{b}(V_{a}-V_{b})\frac{\xi_{2}}{2\xi_{2}+1}\approx
2\alpha_{b}(V_{a}-V_{b})\xi_{2}$ (32)
on linearization . In absence of velocity shear $V_{a}-V_{b}=0$,we get
$\xi_{4}=0$. Thus the problem reduces to that of RT instability alone with no
contribution from KH instability. Linearizing Eqs. (19),(20)and (21) we get
$\displaystyle\frac{d\xi_{2}}{d\tau}=-\frac{1}{2}\xi_{3}$ (33)
$\displaystyle\frac{d\xi_{3}}{d\tau}=-2\left[A_{T}+\alpha_{a}\alpha_{b}(V_{a}-V_{b})^{2}\right]\xi_{2}$
(34) $\displaystyle\frac{d\xi_{4}}{d\tau}=-\rho_{a}(V_{a}-V_{b})\xi_{3}$ (35)
$A_{T}=\frac{\rho_{a}-\rho_{b}}{\rho_{a}+\rho_{b}}$ is the Atwood number.
Eq.(32) connecting $\xi_{2}$ and $\xi_{4}$ provides the consistency condition.
The exponential growth rate due to combined effect of RT and KH instability
coincides with the classical linear theory result
${}^{\cite[cite]{[\@@bibref{}{sc81}{}{}]}}$
$\displaystyle\gamma(k)=\sqrt{kg\left[A_{T}+\alpha_{a}\alpha_{b}(V_{a}-V_{b})^{2}\right]}$
(36)
## IV. RESULTS AND DISCUSSIONS
(A) Combined effect of RT and KH instability:
The growth rate of the RT instability induced nonlinear interfacial structures
is further enhanced due to KH instability. Setting $\frac{du_{a}}{dt}=0$ and
$\frac{du_{a}}{dt}=0$ the growth rate of the peak height of the bubbles and
spikes are obtained by numerical integration of Eqs. (18)-(22) and the results
are shown in Fig.1. The dependence of the growth rate on $V_{a}$ and $V_{b}$
keeping $(V_{a}-V_{b})$ unchanged are also indicated in the same diagrams.It
is found that for $V_{b}>V_{a}$ the growth rate is greater than that for
$V_{a}>V_{b}(\mid V_{a}-V_{b}\mid$ is the same for both cases); the asymptotic
values is the two cases are however identical. Moreover for
$\frac{\rho_{a}}{\rho_{b}}>1$ Eqs.(18)-(22) show that as
$\tau\rightarrow\infty$ there occurs growth rate saturation given by
$\displaystyle(\xi_{3})_{bubble}^{asym}=\sqrt{\frac{2A_{T}}{3(1+A_{T})}+\frac{5(1-A_{T})}{16(1+A_{T})}(V_{a}-V_{b})^{2}}$
(37)
and
$\displaystyle(\xi_{3})_{spike}^{asym}=\sqrt{\frac{2A_{T}}{3(1-A_{T})}+\frac{5(1+A_{T})}{16(1-A_{T})}(V_{a}-V_{b})^{2}}$
(38)
while
$\displaystyle(\xi_{4})^{asym}=0$
for both bubble and spike respectively.
Thus both saturation growth rate are enhanced for due to further
destabilization caused by the velocity shear.
On the other hand if
$r=\rho_{a}/\rho_{b}<1(A_{T}=\frac{\rho_{a}-\rho_{b}}{\rho_{a}+\rho_{b}}<0)$
there is no RT instability but it follows from Eqs.(37)and (38) that
instability due to velocity shear (Kelvin -Helmholtz instability) persists on
both the wind ward side and leeward side (i.e; both for bubbles and spikes) if
(see Fig.2)
$\displaystyle\frac{32|A_{T}|}{15(1+A_{T})}=\frac{16(1-r)}{15r}<(V_{a}-V_{b})^{2}$
(39)
and stabilized on both sides if (see Fig.3 which shows oscillation of
$\xi_{1}$and $\xi_{3}$ with respect to $\tau$)
$\displaystyle(V_{a}-V_{b})^{2}<\frac{16}{15}(1-r)$ (40)
If however $(V_{a}-V_{b})^{2}$ lies in the interval specified by the above
inequalities,i.e;
$\displaystyle\frac{16}{15}(1-r)<(V_{a}-V_{b})^{2}<\frac{16(1-r)}{15r};\mbox{(}r<1)$
(41)
it follows from the same two Eqs.(37)-(38) that the peak of the spike
continues to steeper (instability) with $\tau$ as the heavier fluid (density
$\rho_{b}$)pushes across the interface into the lighter fluid (density
$\rho_{a}$) while the bubble height will execute low finite amplitude
undulations. The above observation is shown to be suppressed in Fig.4. At time
t, the peak height which of the spike or the bubble occurs at
$x=\eta_{1}(\tau)$ and thus moves to the right (x-increases) as
$\eta_{1}(\tau)$ increases with $\tau$. The spike peak height increase
monotonically with t while that of the bubble undulates with low amplitude.
The three dimension representation of the steepening of the peak of the spike
as it moves along x-direction with time is shown in Fig.5.In this respect
there exists approximate qualitative agreement exists with the results of the
weakly nonlinear analysis${}^{\cite[cite]{[\@@bibref{}{lw10}{}{}]}}$.
(B) Combined effect of Richtmyer-Meshkov and Kelvin-Helmholtz instability:
oblique shock
The time evolution of the two fluid interfacial structure resulting from the
combined effect of Richtmyer-Meshkov and Kelvin-Helmholtz instabilities
consequent to impingement of an shock is described by the set of
Eqs.(18)-(22),(26)-(29) with modifications as shown in the set of Eq.(31). If
the shock incidence is oblique then the normal component generates velocity
shear and causes KH instability.${}^{\cite[cite]{[\@@bibref{}{m94}{}{}]}}$ The
shock generated initial values of $\overline{\xi}_{3}$ and
$\overline{\xi}_{4}$ are obtained from the impulsive accelerations represented
by the $\delta-$ function terms in Eq.(30) giving
$\displaystyle(\overline{\xi}_{3})_{\tau=0}=\left[\frac{2(1-r)\xi_{2}(6\xi_{2}-1)}{D_{1}(\xi_{2},r)}\right]_{{(\xi_{2})}{\tau=0}}$
(42)
$\displaystyle(\overline{\xi}_{4})_{\tau=0}=\frac{1}{D_{2}(\xi_{2},r)}\left[\frac{(2\xi_{2}-1)(u_{b}-ru_{a})+2\xi_{2}(u_{a}-u_{b})}{\Delta
v}\right]_{(\xi_{2})_{\tau=0}}$ (43)
Results obtained from numerical solution of Eqs.(18)-(22) with modifications
given by Eq.(31) subject to initial conditions (42) and (43) are presented in
Fig.6. The growth rate contributed in absence of velocity shear,i.e; by
normally incident shock induced Richtmyer-Meshkov instability varies as
$t\rightarrow\infty$. However in presence of velocity shear the growth rate
due to combined influence of RM and KH instability the growth rate approaches
finite saturation value asymptotically.
For RM-KH instability induced spikes it is given by the following closed
expression
$\displaystyle(\overline{\xi}_{3})_{t\rightarrow\infty}^{spike}=(\frac{d\xi_{1}}{dt})_{t\rightarrow\infty}^{spike}=\sqrt{\frac{5(1+A_{T})}{16(1-A_{T})}(u_{a}-u_{b})^{2}/(\Delta
v)^{2}}$ (44)
which becomes large as the Atwood number $A_{T}\rightarrow$1
(equivalently$\rho_{a}/\rho_{b}>>1)$
The following discussions suggest a higher plausibility of the effectiveness
of the joint influence of RM and KH instability in the explanation of certain
astrophysical phenomena.
Corresponding to parameter values for Eagle
Nebula($\rho_{a}/\rho_{b}=0.5\times 10^{2}$ and $|u_{a}-u_{b}|=2\times 10^{6}$
cm
sec-1)${}^{\cite[cite]{[\@@bibref{}{br06}{}{}]}},^{\cite[cite]{[\@@bibref{}{dd02}{}{}]}}$
the velocity of rise of the spike peak height hspike(the height of the pillar)
according to Eq.(44)is $(\frac{dh}{dt})_{t\rightarrow\infty}^{spike}\approx
0.79\times 10^{7}$cm sec-1. Modification through inclusion of Rayleigh-Taylor
instability effect (see Eq.(38)) can only slightly increases this value to
$\approx 10^{7}$cm sec-1.This gives the time to reach the observed pillar
height of $3\times 10^{19}$ cm of the Eagle Nebula $\approx 10^{4}$ years.
There are different time scales involved in the problem of development of the
pillar of the Eagle Nebula. As pointed out by Pound,
${}^{\cite[cite]{[\@@bibref{}{mw98}{}{}]}}$ there is a characteristic time
scale for hydrodynamic motion $\tau_{dyn}\approx(\Delta v)^{-1}$ where $\Delta
v$ is the velocity shear inside the cloud. Corresponding to data given in
ref.(3) this turns out to be $\tau_{hydrodanmic}\approx 10^{5}$yrs which is
the upper time limit for development of the Eagle Nebula pillar(”elephant
trunk”). But it is at least two orders of magnitude greater than the time
scale $\tau_{cool}\sim 10^{2}-10^{3}$ yrs imposed due to radiative cooling of
the cloud
${}^{\cite[cite]{[\@@bibref{}{br06}{}{}]}},^{\cite[cite]{[\@@bibref{}{dd02}{}{}]}}$.
In comparison the time scale of the development of the pillar is found here
$\approx 10^{4}$yrs. Thus consequent to the hydrodynamic model based on the
combined influence of Richtmyer-Meshkov and Kelvin-Helmholtz instability the
gap between the two time scales $\tau_{cool}$ and $\tau_{hydrodynamic}$is
reduced by one order of magnitude.
A high Mach number, radiatevily cooled jet of astrophysical interest has been
produced in laboratory using intense laser irradiation of a gold
cone${}^{\cite[cite]{[\@@bibref{}{df99}{}{}]}}$. The evolution of the jet was
imaged in emission and radiography.
K-H instability growth rate has recently been observed in HED plasma
experiment using Omega laser ($\lambda$)=0.351$\mu m$ delivering 4.3 $\pm
0.1$kJ to the target overlapping 10 drive beams on to the ablator
${}^{\cite[cite]{[\@@bibref{}{eh09}{}{}]}}$.Incompressible K-H growth rate
peak to valley at Foam-Plastic interface has been compared with several
analytical modes.
## V. Summary
Finally we summarize the results:
(a)If the heavier fluid overlies the lighter fluid the growth rate of both the
bubble and spike peak heights due to RT instability are enhanced due to
concurrent presence of velocity shear, i.e, K.H instability Fig.1. The
asymptotic growth rates are given by Eqs. (37) and (38).
(b) In the opposite case,i.e, if the overlying fluid is lighter and lower one
is heavier ($r=\rho_{a}/\rho_{b}<1$) both the spike and bubble peak
displacement increases continuously with time if
$\frac{16(1-r)}{15r}<(V_{a}-V_{b})^{2}$,i.e, instability persists (Fig.3)
while stabilization occurs if $(V_{a}-V_{b})^{2}<\frac{(16(1-r)}{15})$ (Fig. 4
shows oscillation of peak heights of bubbles and spikes).
(c) For $\frac{16(1-r)}{15}<(V_{a}-V_{b})^{2}<\frac{16(1-r)}{15r}$ for r$<1$
the spike steepens with time (the peak height continuously increases with time
as indicated in Fig.5. gives a three dimensional graph of displacement y
against x and $\tau$). But the peak displacement of the bubble undulates
within a small range Fig.4.
(d)If the two fluid interface is subjected to an oblique shock Kelvin-
Helmholtz instability due to generation of velocity shear occurs
simultaneously with Richtmyer-Meshkov instability.The growth rates of bubbles
and spikes due to this joint action are shown in Fig.6. respectively. It is
important to note that the growth rate of the combined action tends
asymptotically to a saturation value given by Eq.(44); this is in contrast to
that due to generation of RM instability due to normal shock incidence for
which the growth rate behaves as $\frac{1}{t}$ as t$\rightarrow\infty$.
Moreover this growth rate as shown by Eq.(44) the rate of growth of this spike
height has sufficiently large magnitude if the Atwood number
$A_{T}\rightarrow$1($\rho_{a}<<\rho_{b}$).This may have interesting
implication in the hydrodynamic explanation of formation of sufficiently long
spiky jets in astrophysical situation, e.g, in case of the Eagle Nebula.
## ACKNOWLEDGEMENTS
This work is supported by the Department of Science & Technology, Government
of India under grant no. SR/S2/HEP-007/2008.
## References
* [1] R.P.Drake, High Energy Density Physics, Spinger, (2006).
* [2] K.Kifonidis,T.Plewa,H.T.Janka,E.Muller, Astron. and Astrophys. 408,621 (2003).
* [3] B.A.Remington,R.P.Drake and D.D.Ryutov, Rev. Mod. Phys. 78, 755 (2006).
* [4] D.D.Ryutov and B.A Remington, Plasma Phys. Control Fusion 44, B407 (2002).
* [5] B.A. Remington,R.P.Drake,H.Takabe and D.Arnett, Phys. Plasmas 7, 1641 (2000).
* [6] L.Spitzer,Jr., Astrophys. J. 120, 1 (1954).
* [7] E.A.Frieman, Astrophys. J. 120, 18 (1954).
* [8] T.Buhrke,R.Mundt and T.P.Ray Astron. and Astrophys. 200,99 (1988).
* [9] K.O.Mikaelian, Phys. Fluids 6, 1943 (1994).
* [10] L.F.Wang,W.H.Ye,Z.F.Fan,Y.J.Li,X.T.He and M.Y.Yu, Europhys. Lett. 86, 15002 (2009).
* [11] L.F.Wang,W.H.Ye,Y.J.Li, Europhys. Lett. 87,54005 (2009).
* [12] L.F.Wang,C.Xue,W.H.Ye and Y.J.Li, Phys. Plasmas 16, 112104 (2009).
* [13] L.F.Wang,W.H.Ye and Y.J.Li, Phys. Plasmas 17,052305 (2010).
* [14] J.Hecht, U.Alon and D.Shvarts, Phys. Fluids 6,4019 (1994).
* [15] A.L.Velikovich and G.Dimonte, Phys.Rev.Lett.76,3112 (1996).
* [16] G.Hazak, Phys.Rev.Lett. 76,4167 (1996).
* [17] Q.Zhang, Phys.Rev.Lett. 81,3391 (1998).
* [18] V.N.Goncharov, Phys.Rev.Lett. 88,134502 (2002).
* [19] S.I.Sohn, Phys.Rev. E 67,026301 (2003).
* [20] M.R.Gupta,S.Roy,M.Khan,H.C.Pant,S.Sarkar and M.K.Srivastava, Phys.Plasma. 16,032303 (2009).
* [21] D.Layzer, Astrophys. J. 122,1 (1955).
* [22] S.Chandrasekhar, Hydrodynamics and Hydrodynamic Stability, Dover,New York 1981.
* [23] M.R.Gupta,L.K.Mandal,S.Roy and M.Khan, Phys.Plasma. 17,012306 (2010).
* [24] M.W.Pound, Astrophys. J. 493,L113 (1998).
* [25] D.R.Farely,K.G.Estabrook,S.G.Glendinning,S.H.Glenzer,B.A.Remington,K.Shigemori,
J.M.Stone,R.J.Wallace,G.B.Zimmerman and J.A.Harte, Phy.Rev.Lett. 83, 1982
(1999).
* [26] E.C.Harding,J.F.Hansen,O.A.Hurricane,R.P.Drake,H.F.Robey,C.C.Kuranz,B.A.Remington,
M.J.Bono,M.J.Grosskopf and R.S.Gillespie, Phy.Rev.Lett. 103, 045005 (2009).
> Figure 1: Initial values
> $r=\frac{\rho_{a}}{\rho_{b}}=1.5,\xi_{1}=-\xi_{2}=\xi_{3}=\xi_{4}=\xi_{5}=0.1$
> for bubble;$-\xi_{1}=\xi_{2}=-\xi_{3}=\xi_{4}=\xi_{5}=0.1$ for spike.Plot
> showing variation of $\xi_{1},\xi_{2}$,growth rate $\xi_{3},\xi_{4}$ and
> transverse displacement $\xi_{5}$ of bubble and spike with $V_{a}=V_{b}=0.0$
> for solid black line-spike and broken black line for
> bubble.$V_{a}=0.1,V_{b}=0.5$ for broken blue line-bubble and solid blue for
> spike,$V_{a}=0.5,V_{b}=0.1$,broken red line for bubble and solid red line-
> spike.
> and for following relation$\frac{16(1-r)}{15r}<(V_{a}-V_{b})^{2}.$
Figure 2: r=$\frac{\rho_{a}}{\rho_{b}}=0.4$;Lower fluid denser.Dashed line for
spike (heavier fluid pushes into lighter fluid) and unbroken line for
bubble.$V_{a}=0.8,V_{b}=-0.6$.Initial condition as in Fig.1.
> Figure 3: r=0.4,$V_{a}=0.0,V_{b}=0.2.(V_{a}-V_{b})^{2}<\frac{16(1-r)}{15}$
> .Initial condition as in Fig.1. Unbroken line for bubble and dashed line for
> spike.
> Figure 4:
> r=0.4,$V_{a}=0.6,V_{b}=-0.4;\frac{16(1-r)}{15}<(V_{a}-V_{b})^{2}<\frac{16(1-r)}{15r}$.Initial
> condition as before (Fig.3.).Unbroken line for bubble and dashed line for
> spike;height of spike peak increases monotonically with time
> (steepening);bubble depth undulates.
> Figure 5: 3 dimensional plot of spike(Interface
> $Y=\eta_{0}(\tau)+\eta_{2}(\tau)(x-\eta_{1}(\tau))^{2}$)belonging to the
> plot given in fig.4.
> Figure 6: Oblique shock: RM and KH instability for spike (dashed line) and
> bubble (unbroken line).Initial values as in fig.1. and
> $V_{a}=0.1,V_{b}=0.5.$
|
arxiv-papers
| 2010-10-06T16:14:02 |
2024-09-04T02:49:13.538250
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "M. R. Gupta, Labakanta Mandal, Sourav Roy, Rahul Banerjee, Manoranjan\n Khan",
"submitter": "Labakanta Mandal",
"url": "https://arxiv.org/abs/1010.1189"
}
|
1010.1242
|
# Variations in the axisymmetric transport of magnetic elements on the Sun:
1996-2010
David H. Hathaway NASA Marshall Space Flight Center, Huntsville, AL 35812 USA
david.hathaway@nasa.gov Lisa Rightmire Department of Physics, The University
of Alabama in Huntsville, Huntsville, AL 35899 USA lar0009@uah.edu
###### Abstract
We measure the axisymmetric transport of magnetic flux on the Sun by cross-
correlating narrow strips of data from line-of-sight magnetograms obtained at
a 96-minute cadence by the MDI instrument on the ESA/NASA SOHO spacecraft and
then averaging the flow measurements over each synodic rotation of the Sun.
Our measurements indicate that the axisymmetric flows vary systematically over
the solar cycle. The differential rotation is weaker at maximum than at
minimum. The meridional flow is faster at minimum and slower at maximum. The
meridional flow speed on the approach to the Cycle 23/24 minimum was
substantially faster than it was at the Cycle 22/23 minimum. The average
latitudinal profile is largely a simple sinusoid that extends to the poles and
peaks at about $35\arcdeg$ latitude. As the cycle progresses a pattern of in-
flows toward the sunspot zones develops and moves equatorward in step with the
sunspot zones. These in-flows are accompanied by the torsional oscillations.
This association is consistent with the effects of the Coriolis force acting
on the in-flows. The equatorward motions associated with these in-flows are
identified as the source of the decrease in net poleward flow at cycle maxima.
We also find polar counter-cells (equatorward flow at high latitudes) in the
south from 1996 to 2000 and in the north from 2002 to 2010. We show that these
measurements of the flows are not affected by the non-axisymmetric diffusive
motions produced by supergranulation.
Sun: rotation, Sun: surface magnetism, Sun: dynamo
## 1 INTRODUCTION
The structure and evolution of the magnetic field in the Sun’s photosphere is
believed to be produced by dynamo processes within the Sun (Charbonneau,
2005). This structure and evolution must be faithfully reproduced in any
viable dynamo model. Flux Transport Dynamo (FTD) models have recently been
used to predict the strength of the next solar cycle (Dikpati et al., 2006;
Choudhuri et al., 2007). In these FTD models the Sun’s axisymmetric flows
(differential rotaton and meridional flow) play key roles. The meridional
circulation transports magnetic flux at the surface to the poles, builds up
the polar fields, and sets the 11-year length of the solar cycle by its
presumed slow equatorward return at the base of the convection zone. The
differential rotation shears the poloidal magnetic field to produce strong
toroidal fields that erupt through the photosphere in sunspots and active
regions.
The structure and evolution of the photospheric magnetic field also serves as
the inner boundary condition for all of space weather – conditions on the Sun
and in the space environment that can influence the performance and
reliability of space-borne and ground-based technological systems. Surface
Flux Transport (SFT) models have been used since 1984 (DeVore et al., 1984) to
evolve the surface field using the flux that erupts in active regions as a
source term. This active region magnetic flux is then transported across the
surface by meridional flow, differential rotation, and diffusion by
supergranules – nonaxisymmetric, cellular flows that evolve on a time scale of
about 1-day. The magnetic field structure produced in SFT models has been used
to model solar wind structures (wind speed and interplanetary magnetic field)
for space weather forecasts (Arge & Pizzo, 2000) and to estimate the Sun’s
total irradiance since 1713 (Wang et al., 2005) for Sun-Climate studies.
The strength, structure, and evolution of the meridional flow in particular is
critically important in both FTD and SFT models. Unfortunately, the meridional
flow is difficult to measure due to its weakness. Supergranules have typical
flow speeds of about 300 $\rm m\ s^{-1}$ and differential rotation has a
typical velocity range of $\sim 200\ \rm m\ s^{-1}$. Yet, the axisymmetric
meridional flow has a top speed of only 10-20 $\rm m\ s^{-1}$.
The axisymmetric flows have been measured using a variety of techniques.
Feature tracking is amongst the simplest and oldest but gives different
results depending on the nature of the features themselves. Direct Doppler
measurements can give the plasma flow velocity in the photosphere but these
measurements are subject to systematic errors introduced by other solar
processes and only provide the line-of-sight velocity – which, for the
meridional flow, vanishes near the equator and limb. Global helioseismology
provides measurements of the differential rotation as a function of latitude,
radius, and time. Local helioseismology can provide measurements of the
meridional flow as a function of latitude, depth, and time using the methods
of ring diagram analysis or time-distance analysis.
Sunspots and sunspot groups were amongst the earliest features used to measure
the axisymmetric flows. Carrington (1859) measured the positions of sunspots
on consecutive days and noted the presence of an equatorial prograde current
and higher latitude retrograde flow. Newton & Nunn (1951) measured the
locations of recurrent sunspots groups on successive rotations as well as
individual sunspots on consecutive days and found slightly different rotation
profiles. Howard et al. (1986) made detailed measurements of individual
sunspot positions recorded on photographic plates at Mount Wilson Observatory
from 1921 to 1982. They found differential rotation with
$\omega=14.52-2.84\sin^{2}\lambda$ deg day-1 (where $\lambda$ is the
heliographic latitude) but noted that sunspot groups rotate more slowly than
individual sunspots and large sunspots rotate more slowly than small sunspots.
Sunspots and sunspot groups can also be used to measure the meridional flow.
Tuominen (1942) used the latitudinal positions of recurrent sunspot groups and
found equatorward flow of $\sim 1\ \rm m\ s^{-1}$ below $\sim 20\arcdeg$
latitude and poleward flow of similar strength at higher latitudes. Ward
(1973) used daily sunspot group positions to argue that there was no
meridional flow at the $1\ \rm m\ s^{-1}$ level. However, Howard & Gilman
(1986) measured the latitudinal drift of individual sunspots and found an
equatorward flow of about $3\ \rm m\ s^{-1}$ equatorward of $\sim 25\arcdeg$
with an even weaker poleward flow at higher latitudes. An obvious drawback to
tracking sunspots to measure the axisymmetric flows is the limited latitudinal
coverage (latitudes $<\sim 30\arcdeg$) and the complete lack of coverage at
times near sunspot cycle minima.
Smaller magnetic features, although often concentrated in the active
latitudes, do cover the entire solar surface and are present even at sunspot
cycle minima. Komm et al. (1993A) masked out the active regions in high-
resolution magnetograms ($2048\times 2048$ pixel full-disk arrays) and cross-
correlated the remaining magnetic features with those seen the next day from
1975 to 1991 for several hundred magnetogram pairs. They found differential
rotation with $\omega=14.43-1.77\sin^{2}\lambda-2.58\sin^{4}\lambda$ deg day-1
and noted that latitudinal profile was flatter at sunspot cycle maximum than
at minimum. Komm et al. (1993B) used the same technique to measure the
meridional flow and found a poleward flow that varied with sinusoidally
latitude, reaching a peak velocity of $\sim 13\ \rm m\ s^{-1}$ at $39\arcdeg$
latitude. Furthermore, they found that the flow speed was slower at the
sunspot cycle maximum than at minimum. Meunuer (1999) employed this technique
(without masking the active regions) using magnetogram pairs from the MDI
instrument (Scherrer et al., 1995) on the ESA/NASA SOHO mission over the
rising phase of sunspot cycle 23 from 1996 to 1998. She found that the
poleward meridional flow slowed in the presence of active regions. In a recent
paper Hathaway & Rightmire (2010) did a similar analysis (with masking of the
active regions) of MDI magnetograms over the time period from 1996 to 2009.
They obtained measurements from over 60,000 image pairs separated by 8-hours.
They also found that the meridional flow was poleward (with a peak velocity of
$\sim 11\ \rm m\ s^{-1}$ at $\sim 45\arcdeg$ latitude) and was fast at cycle
minimum but slow at cycle maximum. In addition they noted that the speed of
the meridional flow was substantially faster at the Cycle 23/24 minimum than
at the Cycle 22/23 minimum.
Larger magnetic features, and associated structures, yield substantially
different results for the meridional flow. Snodgrass & Dailey (1996) cross-
correlated Mt. Wilson coarse array magnetograms ($34\times 34$ pixel full-disk
arrays) obtained 24-38 days (a solar rotation) apart and found poleward flow
from $10\arcdeg$ to $60\arcdeg$ but equatorward flow at lower latitudes. Their
measurements extended from 1968 to 1992 – covering three sunspot cycle maxima
and two minima. They also found a systematic dependence of the meridional flow
pattern on the phase of the solar cycle. Out-flows from the sunspot zones were
observed to move toward the equator in step with the equatorward movement of
the sunspot zones themselves. Latushko (1994) used the same low resolution
data (after it was processed to construct synoptic maps for each solar
rotation) and also found out-flows from the sunspot zones. Švanda et al.
(2007) used a magnetic butterfly diagram constructed from synoptic maps of the
magnetic field averaged over longitude for 180 equispaced zones in sine-
latitude. They measured the slope – change in latitude vs. change in time – of
the magnetic features and found a meridional flow with peak velocities of
about $20\ \rm m\ s^{-1}$ at the poleward limit ($\sim 45\arcdeg$) of their
measurements.
Here we measure the axisymmetric motions of the the small magnetic elements
using the SOHO MDI data in which these elements are well resolved. These
magnetic elements are presisely those whose transport is modeled in SFT models
and in the surface transport of the FTD models.
## 2 DATA PREPARATION
High resolution full-disk images of the line-of-sight magnetic field have been
obtained at a 96-minute cadence since May 1996 by the SOHO MDI instrument.
These images were used in Hathaway & Rightmire (2010) to find the variation in
meridional flow strength over solar cycle 23. They noted in that paper that
the MDI imaging system appears to be rotated by $\sim 0.21\arcdeg$
counterclockwise with respect to the accepted position angle of the Sun’s
rotation axis. Furthermore, they found that the accepted position of the Sun’s
rotation axis is in error by $\sim 0.08\arcdeg$ as was noted previously by
Howard et al. (1986) and by Beck & Giles (2005). This small error introduced
annual variations in the apparent cross-equatorial meridional flow. Here we
account for those positional errors in mapping the full-disk magnetograms to
heliographic coordinates by using modified values for the position angle and
tilt of the Sun’s rotation axis. In addition, while reprocessing the data we
found a significant reduction in the scatter of the measurements if we took
the MDI image origin to be at the bottom left corner of the bottom left pixel
– not the center of the pixel as indicated in the MDI documentation. Here we
repeat the analyses in Hathaway & Rightmire (2010) using these corrected
magnetic maps and examine the variations in both the strength and structure of
the axisymmetric flows.
Figure 1: MDI magnetogram from 2001 June 5 04:48 UT mapped to heliographic
coordinates. Positive magnetic polarities are yellow, negative magnetic
polarities are blue, and masked areas are red. Tickmarks around the border are
at $15\arcdeg$ intervals in latitude and in longitude from the central
meridian.
Each full-disk magnetogram is mapped onto heliographic coordinates using bi-
cubic interpolation onto a grid with 2048 by 1024 equispaced points in
longitude and latitude for the entire surface of the Sun. This mapping gives a
close match to the spatial resolution of the MDI instrument and makes
longitudinal and latitudinal velocities linear functions of the displacements
in the mapped coordinates. The line-of-sight magnetic field is assumed to be
largely radial so we divide the magnetic field strength at each image pixel by
the cosine of the heliographic angle from disk center to minimize the apparent
variations in field strength with longitude from the central meridian. The
magnetic fields in sunspots are intense enough to produce magnetic pressures
similar to the plasma pressure (plasma $\beta\sim 1$). These intense magnetic
field elements resist the near-surface plasma flow and have their own peculiar
motions in longitude and latitude which vary depending on the size of the
sunspot and age of the active region (Howard et al., 1986). For this reason
sunspots and their immediate surroundings are masked out. We found that this
could be done quite effectively by identifying all mapped pixels with field
strengths $\left|B\right|>500$ G and all pixels within 5 mapped pixels of
those points with $\left|B\right|>100$ G as masked pixels. An example of one
of these mapped and masked magnetograms is shown in Fig. 1.
## 3 ANALYSIS PROCEDURES
The axisymmetric motions – differential rotation and meridional flow – of the
magnetic elements were determined by cross-correlating strips of pixels from
pairs of mapped images separated by 8 hours and finding the shift in longitude
and latitude that gave the strongest correlation. (Results obtained with image
pairs separated by 4.8 hours were substantially the same.) Each strip was 11
pixels ($\sim 2\arcdeg$) high in latitude and 600 pixels ($\sim 105\arcdeg$)
long in longitude. The shift in longitude and latitude producing the strongest
correlation was calculated to a fraction of a pixel by fitting parabolas in
longitude and latitude through the correlation coefficient peaks. This process
was performed at 860 latitude positions from $75\arcdeg$S to $75\arcdeg$N for
typically about 400 image pairs over each 27-day rotation of the Sun. In all
we obtained measurements from over 60,000 magnetogram pairs.
The average and the standard deviation of the differential rotation and
meridional flow velocities were calculated at each latitude for each solar
rotation of 27.25 days. The differential rotation and meridional flow profiles
for each rotation were fit with fourth order polynomials in $\sin\lambda$,
where $\lambda$ is the heliographic latitude. Errors in the fit coefficients
were estimated using a Monte Carlo method with random variations at each
latitude characterized by the standard deviations from the measurements. These
polynomial coefficients were also recast in terms of associated Legendre
polynomials of the first order. The Legendre polynomial coefficients are
better suited for studies of time variations based on the orthogonality of the
polynomials themselves (Snodgrass, 1984).
The latitudinal profiles of differential rotation and meridional flow as
measured with these data and this method represent the actual axisymmetric
motions of the magnetic elements. Since the magnetic elements are fully
resolved in these data the effects of supergranule diffusion are seen as
random motions of the magnetic elements and these random motions do not
introduce any systematic errors in our measurements as will be shown in
Section 7. Profiles were obtained for 178 rotations of the Sun from June 1996
to September 2010 with a gap from June 1998 to February 1999 when radio
contact with SOHO was lost and not fully recovered.
## 4 AVERAGE FLOW PROFILES
The average differential rotation profile from the entire dataset is shown in
Fig. 2. The velocities are taken relative to the Carrington frame of reference
which has a sidereal rotation rate of $14.184\rm{\ deg\ day}^{-1}$. The
average differential rotation profile is well represented by just the three
terms with symmetry across the equator –
$v_{\phi}(\lambda)=(a+b\sin^{2}\lambda+c\sin^{4}\lambda)\cos\lambda$ (1)
with
$a=35.6\pm 0.1\rm{\ m\ s}^{-1}$ (2)
$b=-208.6\pm 1.1\rm{\ m\ s}^{-1}$ (3)
$c=-420.6\pm 1.6\rm{\ m\ s}^{-1}$ (4)
This gives an angular rotation rate profile with
$\omega(\lambda)=A+B\sin^{2}\lambda+C\sin^{4}\lambda$ (5)
with
$A=14.437\pm 0.001\rm{\ deg\ day}^{-1}$ (6)
$B=-1.48\pm 0.01\rm{\ deg\ day}^{-1}$ (7)
$C=-2.99\pm 0.01\rm{\ deg\ day}^{-1}$ (8)
where coefficient $A$ includes the Carrington rotation rate.
This angular rotation rate is nearly identical to that found by Komm et al.
(1993A) for the time interval 1975-1991 using similar data and methods. We do
find a slight north-south asymmetry as seen in Fig. 2 by the deviation of the
measured profile from the symmetric profile given by the dashed line. The
differential rotation was slightly weaker in the south than in the north. We
also note the flattening of the profile at the equator with a slight ($\sim
1\rm{\ m\ s}^{-1}$) but significant dip from $\pm 5\arcdeg$ to the equator. A
similar “dimple” at the equator was seen previously in direct Doppler data by
Howard et al. (1980) and in magnetic element motions by Snodgrass (1983).
Figure 2: The average differential rotation profile with the $2\sigma$ error
range for the time interval 1996-2010. The symmetric profile given by Eqns.
1-4 is shown with the dashed line.
The average meridional flow profile for the entire dataset is shown in Fig. 3.
Although the average meridional flow profile does display substantial north-
south asymmetry, the profile is well represented with just the two anti-
symmetric terms –
$v_{\lambda}(\lambda)=(d\sin\lambda+e\sin^{3}\lambda)\cos\lambda$ (9)
with
$d=29.7\pm 0.3\rm{\ m\ s}^{-1}$ (10)
$e=-17.7\pm 0.7\rm{\ m\ s}^{-1}$ (11)
This gives a peak poleward meridional flow velocity of $11.2\rm{\ m\ s}^{-1}$
at a latitude of $35.2\arcdeg$. This is somewhat slower than the meridional
flow found by Komm et al. (1993B) for the time interval 1975 to 1991 but with
a peak at nearly the same latitude. Our average meridional flow profile shows
substantially different flows in the north and in the south. The flow velocity
is faster in the south and peaks at a higher latitude than in the north. The
flow in the north appears to nearly vanish at the extreme northern limit
($75\arcdeg$) of our measurements while the flow in the south is still
poleward with a speed of about $5\rm{\ m\ s}^{-1}$ at the southern limit.
Figure 3: The average meridional flow profile with $2\sigma$ error range for
the time interval 1996-2010. The anti-symmetric profile given by Eqns. 9-11 is
shown with the dashed line. This profile shows substantially different flow in
the north and south.
## 5 VARIATIONS IN FLOW SPEED
Variations in the amplitudes of the axisymmetric flow components were examined
by plotting the rotation-by-rotation histories of the Legendre polynomial
coefficients. The Legendre polynomials were normalized so that their maximum
values were either 1.0 or -1.0. The coefficients that multiply them then give
the peak velocity for that component. The normalized polynomials we used are
$P_{1}^{1}(\lambda)=\cos\lambda$ (12)
$P_{2}^{1}(\lambda)=2\sin\lambda\cos\lambda$ (13)
$P_{3}^{1}(\lambda)=\sqrt{135\over 256}(5\sin^{2}\lambda-1)\cos\lambda$ (14)
$P_{4}^{1}(\lambda)=0.947(7\sin^{3}\lambda-3\sin\lambda)\cos\lambda$ (15)
$P_{5}^{1}(\lambda)=0.583(21\sin^{4}\lambda-14\sin^{2}\lambda+1)\cos\lambda$
(16)
Figure 4: The differential rotation associated Legendre polynomial
coefficients (with $2\sigma$ error bars) for the time interval 1996-2010. The
coefficient T0 multiplies $P_{1}^{1}$, the polynomial of zeroth order in
$\sin\lambda$. The coefficient T2 multiplies $P_{3}^{1}$, the polynomial of
second order in $\sin\lambda$. The coefficient T4 multiplies $P_{5}^{1}$, the
polynomial of fourth order in $\sin\lambda$. The smoothed sunspot number
divided by 4 is shown in red for reference. The differential rotation is
slightly weaker (flatter) at sunspot cycle maximum.
The Legendre coefficient histories for the differential rotation are shown in
Fig. 4 along with the smoothed sunspot number for reference to the phase of
the sunspot cycle. The three symmetric components ($P_{1}^{1}$, $P_{3}^{1}$,
and $P_{5}^{1}$) dominate so we only show the three associated coefficient
histories. These three coefficients show only a slight variation over the
sunspot cycle with the amplitudes being smaller (less negative – weaker
differential rotation) at sunspot cycle maximum ($\sim 2002$). This “more
rigid” differential rotation at sunspot cycle maximum was previously noted by
Komm et al. (1993A).
Figure 5: The meridional flow Legendre polynomial coefficients (with
$2\sigma$ error bars) for the time interval 1996-2010. The coefficient S1
multiplies $P_{2}^{1}$, the polynomial of first order in $\sin\lambda$. The
coefficient S3 multiplies $P_{4}^{1}$, the polynomial of third order in
$\sin\lambda$. The smoothed sunspot number divided by 20 is shown in red. The
meridional flow is slower at sunspot cycle maximum but was even faster at
Cycle 23/24 minimum in 2008 than at Cycle 22/23 minimum in 1996.
The Legendre coefficient histories for the meridional flow are shown in Fig. 5
along with the smoothed sunspot number. The two anti-symmetric components
($P_{2}^{1}$, and $P_{4}^{1}$) dominate so we only show the two associated
coefficient histories. These two coefficients show substantial variations over
the sunspot cycle with the amplitudes being smaller at sunspot cycle maximum.
Komm et al. (1993B) found similar behavior for the time period 1978-1990.
In addition to this systematic trend over the sunspot cycle (fast at minimum
and slow at maximum) we find a secular variation in which the meridional flow
speed was substantially ($\sim 20\%$) faster at the Cycle 23/24 minimum in
2008 than at the Cycle 22/23 minimum in 1996. As in Hathaway & Rightmire
(2010) we note that the meridional flow speed was faster for the entire
interval from 2004 on, than it was at the cycle minimum in 1996. This increase
in meridional flow speed would explain the weak polar fields that were
produced during that time period in the SFT models of Schrijver & Liu (2008)
and Wang et al. (2009).
## 6 VARIATIONS IN STRUCTURE
The variations in flow speed shown in the last section are produced by and
accompanied by variations in flow structure. Our analyses produce latitudinal
profiles of the differential rotation and the meridional flow for each
individual solar rotation from June 1996 to September 2010. These profiles
were obtained at 860 latitude positions between $\pm 75\arcdeg$. For further
analysis we smoothed these profiles with a tapered Gaussian having a FWHM of 6
latitude points ($\sim 1\arcdeg$), resampled at intervals of $1\arcdeg$ in
latitude, and produced images of these latitudinally smoothed profiles and of
the differences between each such profile and the average symmetrized
profiles. Little, if any, variation can be seen in the full differential
rotation profile history. However, the meridional flow profile history shows
substantial variation as shown in Fig. 6.
Figure 6: The meridional flow profiles for individual solar rotations from
1996-2010. Poleward flow is indicated by shades of yellow. Equatorward flow is
indicated by shades of blue. The latitudinal centroid of the sunspot area in
each hemisphere for each rotation is shown in red. The weakening of the
meridional flow in the active latitudes near sunspot cycle maximum is evident
as are polar counter-cells (equatorward flow) in the south from 1996 to 2000
and in the north from 2002 to 2010.
The structure of the meridional flow changes substantially over the time
period represented in Fig. 6. The weakening of the poleward meridional flow at
sunspot cycle maximum (1999-2003) is evident in the muted colors surrounding
the sunspot zones. The strengthing of the meridional flow on the approach to
Cycle 23/24 minimum in late 2008 is evident in the intensified colors at most
latitudes after 2004.
Fig. 6 also reveals the existence of counter-cells (equatorward flow). One is
found in the south extending equatorward to about $60\arcdeg$S at the start of
the dataset in May of 1996 but that boundary moves poleward of our $75\arcdeg$
limit by mid-2000. A similar counter-cell is seen forming in the north in 2002
as it dips below $75\arcdeg$N and remains in evidence to the end of the
dataset in 2010. This long-lasting northern counter-cell is clearly the
primary source of the north-south asymmetry seen in the average meridional
flow profile (Fig. 3) and may be associated with the asymmetry in the
differential rotation (Fig. 2). The fact that it maintains its existence for
more than half of the time available in this dataset leaves its imprint on the
average meridional flow profile in the form of the rapid drop in poleward flow
in the north to near zero at $75\arcdeg$N latitude.
Figure 7: The differences between the meridional flow profiles for individual
solar rotations and the average, symmetric profile from 1996-2010. Poleward
flow (relative to the average profile) is indicated by shades of yellow.
Equatorward flow is indicated by shades of blue. The latitudinal centroid of
the sunspot area in each hemisphere for each rotation is shown in red. The
system of in-flows toward the sunspot zones is evident as poleward flow on the
equatorward sides of the sunspot zones and equatorward flow on the poleward
sides.
Additional details concerning the structural changes in the axisymmetric flows
are seen when the average symmetric flow profiles are subtracted from the
profiles for each individual rotation. These differences from the average for
the meridional flow are shown in Fig. 7. The two counter-cells are more
obvious here. In addition, these difference profiles show a system of in-flows
(relative to the average meridional flow) toward the sunspot zones with
poleward (yellow) flows on the equatorward sides and equatorward (blue) flows
on the poleward sides. This suggests that the slowdown in the poleward
meridional flow seen at sunspot cycle maxima is produced by the growing
strength and latitudinal extent of these in-flows.
The presence of these in-flows was nonetheless somewhat surprising. Snodgrass
& Dailey (1996) found _out-flows_ from the active latitudes with their low-
resolution magnetic data. Chou & Dai (2001) and Beck et al. (2002) also found
out-flows from the active latitudes using time-distance helioseismology.
However, González Hernández et al. (2010) found clear evidence for in-flows
much like what we see in Fig. 7 using ring-diagram helioseismology and the
structural changes seen in the magnetic element motions by Meunuer (1999) also
support the presence of these in-flows.
Figure 8: The differences between the differential rotation profiles for
individual solar rotations and the average, symmetric profile from 1996-2010.
Faster (prograde relative to the average profile) flow is indicated by shades
of yellow. Slower (retrograde) flow is indicated by shades of blue. The
latitudinal centroid of the sunspot area in each hemisphere for each rotation
is shown in red. The torsional oscillations are evident as faster flow on the
equatorward sides of the sunspot zones and slower flow on the poleward sides.
The in-flows toward the sunspot zones are accompanied by the torsional
oscillations – variations in the differential rotation seen as faster rotation
on the equatorward sides of the sunspot zones and slower rotation on the
poleward sides (Howard & LaBonte, 1980). This is shown in Fig. 8 by the
differences in the differential rotation profiles from the average symmetrized
differential rotation profile. (Note that there are instrumental artifacts at
the highest latitudes as evident by the annual variations in flow speed with
faster flow near the poles in the hemisphere tilted toward the observer. These
artifacts may be due to an elliptical distortion of the MDI image as reported
by Korzennik et al. (2004). However, our efforts to include this distortion
with either the angle they reported or the angle given in the MDI
documentation did not improve the results.)
These variations in the differential rotation are consistent with the effect
of the Coriolis force on the in-flows and the counter-cells. Material moving
equatorward from the higher latitudes will spin-down and give slower flows on
the poleward sides of the sunspot zones while material moving poleward from
the equator will spin-up and give faster flows on the equatorward sides. This
scenerio was suggested by Spruit (2003) as a response to cooling in the
sunspot zones by excess thermal emission from faculae. Earlier, Snodgrass
(1987) had suggested that in-flows and the torsional oscillations were part of
a system of azimuthal convection-rolls which migrate equatorward during each
sunspot cycle. These convection-rolls should have out-flows at some
undetermined depth below the surface – a possible source of the out-flows seen
in some of the helioseismology studies. The Coroilis force acting on the long-
lasting northern counter-cell should slowdown the rotation at the affected
latitudes. This may be the source of the north-south asymmetry in the average
differential rotation profile (Fig. 2).
## 7 EFFECTS OF DIFFUSION ON FLOW MEASUREMENTS
The magnetic elements under study here are also subject to a diffusion-like
random walk by the nonaxisymmetric cellular flows – supergranules in
particular (Leighton, 1964). This random walk transports the weak magnetic
elements in both longitude and latitude and leads to the formation of large
unipolar areas from the preceding and following magnetic flux in active
regions (Smithson, 1973). This random walk might contribute to the meridional
flow we measure due to resultant changes in the magnetic pattern. In SFT
models (DeVore et al., 1984; van Ballegooijen et al., 1998; Wang et al., 2002,
2005, 2009; Schrijver & Liu, 2008) this process is represented by a
diffusivity coupled with the Laplacian of the magnetic field. We would expect
that this might produce a meridional flow signal in the form of out-flows from
the sunspot zones where the magnetic field is concentrated. Although what we
observe is actually in-flows toward the sunspot zones, the effects of
diffusion might nonetheless alter the structure and evolution of the
meridional flow we measure. Given this caveat, we undertook an investigation
of the effects of supergranule diffusion on our measurements.
Hathaway et al. (2010) have recently produced a model of the photospheric
flows which includes the cellular flows, supergranules in particular, observed
with the SOHO MDI instrument. The cellular flows in this model have velocity
spectra, lifetimes, and motions that match those seen in the MDI data itself.
We have taken the vector velocities from this model and used them to transport
magnetic elements whose initial spatial distribution was taken from an MDI
synoptic magnetic map. We then used our analysis procedures to measure the
axisymmetric flows. We isolated the effects of diffusion by only including the
evolving cellular flows. We do not include the axisymmetric meridional flow or
differential rotation and the cellular flow pattern itself does not
participate in any axisymmetric meridional flow or differential rotation.
The cellular flow simulation produced vector velocities on a heliographic grid
with 4096 by 1500 equispaced points in longitude and latitude from an evolving
velocity spectrum that extended to spherical wavenumbers of 1500
(supergranules have spherical wavenumbers of $\sim 100$). The initial magnetic
field distribution was taken from an MDI synoptic magnetic chart for
Carrington rotation 2000 (mid-2003 – just after the peak of the sunspot
cycle). Our magnetic flux transport simulation was calculated on a grid the
same size as our mapped magnetograms. At each pixel in our simulated magnetic
map we introduced a number of 1000 G magnetic elements with filling factors of
5% until the average field strength in that pixel equaled the observed field
strength (a single element in a pixel would produce a field strength of 50 G).
This process required some 120,000 magnetic elements. These elements were then
transported explicitly by the velocity field from the cellular flow simulation
in 15-minute time steps for 10 days.
Figure 9: Simulated magnetic map regions at 1-day intervals. These regions
were extracted from the full simulated magnetic maps at the start of days 1-5
from an area bordered by the equator, $60\arcdeg$N, and longitudes
$109\arcdeg$ and $126\arcdeg$. The evolving magnetic network is evident in the
changing magnetic structures.
Examples from the simulated magnetic maps are shown in Fig. 9. The magnetic
elements are transported to the borders of the cells and then continue to move
as the cells themselves evolve. (This was shown in previous simulations by
Simon et al. (2001).) The magnetic elements retain their identities throughout
the simulation and do not interact with each other. If opposite polarities
occupy a pixel they do cancel each other in terms of the mapped magnetic field
strength but they continue to retain their identities and move with the
simulated flow.
These magnetic maps were processed with the same analysis procedures used with
the MDI magnetic maps by selecting a “central meridian” longitude and
correlating strips of pixels with those from a map 8-hours later. This was
done for a series of cental meridians at 1-hour intervals over the 10
simulated days. This resulted in 559 measurements of the axisymmetric flows
covering the full range of longitudes and the full 10 days. Fig. 10 shows the
meridional flow measured from these magnetic maps. The results have similar
noise levels to single rotation averages from MDI but show no evidence of any
systematic meridional flow.
Figure 10: Meridional flow profile measured from magnetic features subjected
to random walk by non-axisymmetric cellular flows. Our meridional flow
measurements do not include any systematic errors due to these random (and
spatially resolved) motions.
## 8 CONCLUSIONS
We have measured the axisymmetic motions of magnetic elements on the Sun by
cross-correlating strips of data from magnetic maps acquired at 96-minute
cadence by the MDI instrument on SOHO. Our measurements cover each rotation of
the Sun from June 1996 to September 2010 with the exception 8 rotations when
the data were unavailable. Although we exclude the magnetic elements in
sunspots themselves, the magnetic elements we track are in fact those whose
poleward motions produce the Sun’s polar fields in SFT models (DeVore et al.,
1984; van Ballegooijen et al., 1998; Wang et al., 2002, 2005, 2009; Schrijver
& Liu, 2008) and in FTD models (Dikpati et al., 2006; Choudhuri et al., 2007).
With these data these magnetic elements are well resolved and the random
motions due to supergranules appear as just that – random motions that do not
alter our measurements of the axisymmetric flows.
The differential rotation we measure agrees well with previous measurements
using similar data and methods (Komm et al., 1993A). Although the average
differential rotation profile is slightly asymmetric this asymmetry may be
specific to the time period and the presence of the meridional flow counter-
cell in the north. The torsional oscillation signal (Fig. 8) compares well
with the near surface pattern from helioseismology (Howe et al., 2009) and
does not require averaging the two hemispheres together.
The meridional flow we measure also agrees well with previous measurements
using similar data and methods (Komm et al., 1993B; Meunuer, 1999) but with
interesting differences and more detail. The average meridional flow speed we
found from 1996 to 2010 was somewhat slower than found by Komm et al. (1993B)
from 1978 to 1991. We both find that the flow is faster at cycle minima and
slower at maxima. Here we find that this slow-down can be attributed to a
system of in-flows toward the sunspot zones which, when superimposed on the
average meridional flow profile, lowers the peak flow velocity at cycle maxima
(Meunuer, 1999). Our slower average meridional flow speed is somewhat
surprising since our data included two (fast) minima and one maximum while the
Komm et al. (1993B) data included two (slow) maxima and one minimum.
An important difference for understanding the long, drawn-out, and low Cycle
23/24 minimum is the faster meridional flow after 2004 compared to the flow at
the Cycle 22/23 minimum in 1996. This faster meridional flow produces weaker
polar fields in the SFT models of Schrijver & Liu (2008) and Wang et al.
(2009). Weaker polar fields produce weak following cycles which typically have
long, low minima (Hathaway, 2010).
In spite of this agreement, our average meridional flow profile is problematic
for the SFT models. All of the SFT modeling groups use meridional flow
profiles which peak at low latitudes or do not extend poleward of $75\arcdeg$.
Comparisons between our symmetrized profile and those used in three SFT
calculations (van Ballegooijen et al., 1998; Wang et al., 2009; Schrijver &
Title, 2001) are shown in Fig 11.
Figure 11: Symmetrized meridional flow profile from this paper (solid line)
plotted with meridional flow profiles use in the Surface Flux Transport models
of Wang et al. (2009) (dashed line) van Ballegooijen et al. (1998) (dotted
line) and Schrijver & Title (2001) (dot-dashed line).
All three SFT profiles fall below our measured profile at the higher latitudes
– above $30\arcdeg$ for Wang et al. (2009), $45\arcdeg$ for Schrijver & Title
(2001), and $60\arcdeg$ for van Ballegooijen et al. (1998). Using our average
meridional flow profile in these models without compensating processes leads
to polar fields substantially stronger than those observed. Compensating
processes might include the counter-cells along with the north-south asymmetry
or neglected physical processes – for example radial diffusion suggested by
Baumann et al. (2006).
The nearly 20% change in meridional flow speed from Cycle 22/23 minimum in
1996 to Cycle 23/24 minimum in 2008 is problematic for the FTD models. Dikpati
& Charbonneau (1999) showed that with their FTD model increasing the surface
meridional flow speed from $2\rm{\ m\ s}^{-1}$ to $20\rm{\ m\ s}^{-1}$ changed
the surface polar field strength from 130G to 350G while changing the cycle
period from 77 years to 11 years. The faster meridional flow in this model
should have produced a shorter cycle with stronger polar fields. Yet,
observations reveal a very long cycle with much weaker polar fields.
We have shown that our data, with its high spatial resolution and rapid
cadence, fully resolve the magnetic element motions produced by supergranule
“diffusion” and thus yield measurements of the meridional flow without any
systematic errors due to that diffusion. Komm et al. (1993B) used data with
similar spatial resolution but lower cadence (daily rather than hourly) and
found similar results. However, Snodgrass & Dailey (1996) and Latushko (1994)
used data with much lower spatial resolution and much longer time-lags
(monthly) and found significant differences. These low spatial resolution data
do not resolve the individual magnetic elements. They image the emsemble
magnetic patches whose motions _do_ include the effects of diffusion. We
suspect that the magnetic pattern diffusion gave the equatorial flows at low
latitudes measured by Snodgrass & Dailey (1996) and the out-flows from the
sunspot zones seen by Snodgrass & Dailey (1996) and Latushko (1994), and more
rapid high-latitude flow seen by Švanda et al. (2007).
Comparisons of our measurements with those from other data types (direct
Doppler velocities, sunspot motions, and helioseismology) are subject to
problems associated with the characteristic depth of the measurements. The Sun
has a surface shear layer produced largely by the granule and supergranule
flows which tend to conserve angular momentum (Foukal & Jokipii, 1975) –
slowing down the rotation of the surface layers and speeding up the rotation
down to depths of about 35 Mm. This inward increase in rotation rate should be
accompanied by an inward decrease in the meridional flow speed (Hathaway,
1982) – a feature noted by Hathaway et al. (2010) in the meridional motion of
supergranules. This is consistent with the slower rotation rate and faster
meridional flow seen in direct Doppler measurements representative of the
photosphere (Ulrich et al., 1988; Ulrich, 2010) assuming that the magnetic
elements are rooted in somewhat deeper layers. Sunspots should be rooted even
deeper yet and sunspots show rotation rates which are even more rapid (Ward,
1966; Howard et al., 1986) and meridional motions that are vanishingly small
(Ward, 1973) or equatorward (Tuominen, 1942; Howard & Gilman, 1986). While
helioseismology studies indicate both out-flows (Chou & Dai, 2001; Beck et
al., 2002) and in-flows (González Hernández et al., 2010), this may be due to
differences in both the methods used and the associated depths of the
measurements. Helioseismology does provide supporting evidence for the
variations in meridional flow speed over the sunspot cycle (Basu & Antia,
2003; González Hernández et al., 2010).
Our observations of in-flows toward the sunspot zones may help us understand
the origins of the torsional oscillations. The strength and structure of these
in-flows are good matches to the flows predicted in the model of Spruit
(2003). However, helioseismology indicates that the torsional oscillations may
originate well below the surface at high latitudes (Basu & Antia, 2003) and
thus may not be forced by the effects of localized surface cooling.
Finally, we reitterate our point that the magnetic elements whose motions we
study are precisely those elements whose transport is modeled in SFT models
and at the surface in FTD models. Both SFT and FTD models must employ the
measured axisymmetric transport of those magnetic elements to conform with
observations.
DH would like to thank NASA for its support of this research through a grant
from the Heliophysics Causes and Consequences of the Minimum of Solar Cycle
23/24 Program to NASA Marshall Space Flight Center. LR would like to thank
NASA for its support through an EPSCoR grant to Dr. Gary P. Zank through The
University of Alabama in Huntsville. SOHO, is a project of international
cooperation between ESA and NASA.
## References
* Arge & Pizzo (2000) Arge, C. N., & Pizzo, V. J. 2000, J. Geophys. Res. 105, 10,465
* Basu & Antia (2003) Basu, S., & Antia, H. M. 2003, ApJ 585, 553
* Basu et al. (1999) Basu, S., Antia, H. M., & Tripathy, S. C. 1999, ApJ 512, 458
* Baumann et al. (2006) Baumann, I., Schmitt, D., & Schüssler, M. A&A 446, 307
* Beck & Giles (2005) Beck, J. G., & Giles, P. 2005, ApJ 621, L153
* Beck et al. (2002) Beck, J. G., Gizon, L., & Duvall, T. L., Jr. 2002, ApJ 575, L47
* Carrington (1859) Carrington, R. C. 1859, MNRAS 19, 81
* Carrington (1863) Carrington, R. C. 1863, Observations of the spots on the Sun from November 9, 1853, to March 24, 1861, made at Redhill (London: Williams & Norgate)
* Charbonneau (2005) Charbonneau, P. 2005, Living Rev. Solar Phys. 2, 2
* Chou & Dai (2001) Chou, D.-Y., & Dai, D.-C, 2001, ApJ 559, L175
* Choudhuri et al. (2007) Choudhuri, A. R., Chatterjee, P., & Jiang, J. 2007, Phys. Rev. Lett. 98, 131103-1
* DeVore et al. (1984) DeVore, C. R., Sheeley, N. R., Jr., & Boris, J. P. 1984, Sol. Phys. 92, 1
* Dikpati & Charbonneau (1999) Dikpati, M., & Charbonneau, P. 1999, ApJ 518, 508
* Dikpati et al. (2006) Dikpati, M., de Toma, G., & Gilman, P. A. 2006, Geophys. Res. Lett. 33, L05102
* Foukal & Jokipii (1975) Foukal, P., & Jokipii, R. 1975, ApJ 199, L71
* Giles et al. (1997) Giles, P. M., Duvall, T. L., Jr., Scherrer, P. H., & Bogart, R. S. 1997, Nature 390, 52
* Gizon et al. (2003) Gizon, L., Duvall, T. L., Jr., & Schou, J. 2003, Nature 421, 43
* González Hernández et al. (2010) González Hernández, I., Howe, R., Komm, R., & Hill, F. 2010, ApJ 713, L16
* Hathaway (1982) Hathaway, D. H. 1982, Sol. Phys. 77, 341
* Hathaway (2010) Hathaway, D. H. 2010, Living Rev. Solar Phys. 7, 1
* Hathaway & Rightmire (2010) Hathaway, D. H. & Rightmire, L. 2010, Science 327, 1350
* Hathaway et al. (2010) Hathaway, D. H., Williams, P. E., Dela Rosa, K., & Cuntz, M. 2010, ApJ 725, 1082
* Howard et al. (1980) Howard, R. F., Boyden, J. E. & LaBonte, B. J. 1980, Sol. Phys. 66, 167
* Howard & LaBonte (1980) Howard, R. F. & LaBonte, B. J. 1980, ApJ 239, L33
* Howard & Gilman (1986) Howard, R. F., & Gilman, P. A. 1986, ApJ 307, 389
* Howard et al. (1986) Howard, R. F., Gilman, P. A., & Gilman, P. I. 1984, ApJ 283, 373
* Howe et al. (2009) Howe, R., Christensen-Dalsgaard, J., Hill, F., Komm, R., Schou, J., & Thompson, M. J. 2009, ApJ 701, L87
* Komm et al. (1993A) Komm, R. W., Howard, R. F., & Harvey, J. W. 1993, Sol. Phys. 145, 1
* Komm et al. (1993B) Komm, R. W., Howard, R. F., & Harvey, J. W. 1993, Sol. Phys. 147, 207
* Korzennik et al. (2004) Korzennik, S. G., Rebello-Soares, M. C., & Schou, J. 2004, ApJ 602, 481
* Latushko (1994) Latushko, S. 1994, Sol. Phys. 149, 231
* Leighton (1964) Leighton, R. B. 1964, ApJ 140, 1547
* Meunuer (1999) Meunier, N., 1999, ApJ 527, 967
* Newton & Nunn (1951) Newton, H. W., & Nunn, M. L. 1951, MNRAS 111, 413
* Scherrer et al. (1995) Scherrer, P. H., Bogart, R. S., Bush, R. I., Hoeksema, J. T., Kosovichev, A. G., Schou, J., Rosenberg, W., Springer, L., Tarbell, T. D., Title, A., Wolfson, C. J., Zayer, I., and the MDI Engineering Team 1995, Sol. Phys. 162, 129
* Schou (2003) Schou, J. 2003, ApJ 596, L259
* Schrijver & Liu (2008) Schrijver, C. J. & Liu, Y. 2008, Sol. Phys. 252, 19
* Schrijver & Title (2001) Schrijver, C. J. & Title A. M. 2001, ApJ 551, 1099
* Simon et al. (2001) Simon, G. W., Title, A. M.,& Weiss, N. O. 2001, ApJ 561, 427
* Spruit (2003) Spruit, H. C. 2003, Sol. Phys. 213, 1
* Smithson (1973) Smithson, R. C. 1973, Sol. Phys. 29, 365
* Snodgrass (1983) Snodgrass, H. B. 1983, ApJ 270, 288
* Snodgrass (1984) Snodgrass, H. B. 1984, Sol. Phys. 94, 13
* Snodgrass (1987) Snodgrass, H. B. 1987, ApJ 316, L91
* Snodgrass & Dailey (1996) Snodgrass, H. B. & Dailey, S. B. 1996, Sol. Phys. 163, 21
* Švanda et al. (2007) Švanda, M. Kosovichev, A. G.& Zhao, J. 2007, ApJ 670, L69
* Tuominen (1942) Tuominen, J. 1942, ZAp 21, 96
* Ulrich (2010) Ulrich, R. K., ApJ 725, 658
* Ulrich et al. (1988) Ulrich, R. K., Boyden, J. E., Webster, L., Snodgrass, H. B., Padilla, S. P., Gilman, P., and Shieber, T. 1988, Sol. Phys. 117, 291
* van Ballegooijen et al. (1998) van Ballegooijen, A. A., Cartledge, N. P., & Priest, E. R. 1998, ApJ 501, 866
* Wang et al. (2002) Wang, Y.-M., Sheeley, N. R., Jr. & Lean, J. 2002, ApJ 580, 1188
* Wang et al. (2005) Wang, Y.-M., Lean, J. & Sheeley, N. R., Jr. 2005, ApJ 625, 522
* Wang et al. (2009) Wang, Y.-M., Robbrecht, E. & Sheeley, N. R., Jr. 2009, ApJ 707, 1372
* Ward (1966) Ward, F. 1966, ApJ 145, 416
* Ward (1973) Ward, F. 1973, Sol. Phys. 30, 527
|
arxiv-papers
| 2010-10-06T19:40:30 |
2024-09-04T02:49:13.547944
|
{
"license": "Public Domain",
"authors": "David H. Hathaway and Lisa Rightmire",
"submitter": "David Hathaway",
"url": "https://arxiv.org/abs/1010.1242"
}
|
1010.1307
|
# Constraints on the Dark Side of the Universe
and Observational Hubble Parameter Data
Tong-Jie Zhang tjzhang@bnu.edu.cn Department of Astronomy, Beijing Normal
University, Beijing 100875, P. R. China Center for High Energy Physics,
Peking University, Beijing 100871, P. R. China Cong Ma Department of
Astronomy, Beijing Normal University, Beijing 100875, P. R. China Tian Lan
Department of Astronomy, Beijing Normal University, Beijing 100875, P. R.
China
###### Abstract
This paper is a review on the observational Hubble parameter data that have
gained increasing attention in recent years for their illuminating power on
the dark side of the universe — the dark matter, dark energy, and the dark
age. Currently, there are two major methods of independent observational
$H(z)$ measurement, which we summarize as the “differential age method” and
the “radial BAO size method”. Starting with fundamental cosmological notions
such as the spacetime coordinates in an expanding universe, we present the
basic principles behind the two methods. We further review the two methods in
greater detail, including the source of errors. We show how the observational
$H(z)$ data presents itself as a useful tool in the study of cosmological
models and parameter constraint, and we also discuss several issues associated
with their applications. Finally, we point the reader to a future prospect of
upcoming observation programs that will lead to some major improvements in the
quality of observational $H(z)$ data.
###### pacs:
98.80.Es, 95.36.+x, 95.35.+d, 98.62.Ai, 98.62.Py, 98.65.Dx
## I Introduction
The expansion of our universe has been one of the greatest attractions of
scientific talents since the seminal work of Edwin Powell Hubble (Hubble,
1929) in 1929. Hubble’s compilation of observational distance-redshift
(expressed in terms of radial velocity) data suggested a linear pattern of
“extra-Galactic nebulae” (an archaic term for galaxies) receding from each
other:
$\dot{\boldsymbol{x}}=H\boldsymbol{x},$ (1)
where $H$ is the proportional constant now bearing his name, and $x$ is the
positional coordinates of a galaxy measured with our Galaxy as the origin.
The discovery of Hubble’s Law marked the commencement of the era of
quantitative cosmology in which theories of the universe can be subjected to
observational test. Since the days of Hubble, advances in technology have
enabled astronomers to measure the light from increasingly deeper space and
more ancient time, and our ideas of the entire history of the expanding
universe have been gradually converging into a unified picture of Big
Bang–Cold Dark Matter universe. In this picture, the dominating form of energy
density transited from radiation to dark matter, and relics of primordial
perturbation were imprinted on today’s observable CMB anisotropy and large-
scale structures (LSS). This picture is obtained from its two ends: the CMB
last-scattering surface at $z\approx 1000$ and the LSS around us at $z\approx
0$. The vast spacetime extent between both ends, in particular the era before
reionization, remains mostly hidden from our view. In addition, the past two
decades’ cosmological observations, especially those of type Ia supernovae
(SNIa), indicated that the recent history of universal expansion is an
acceleration, possibly driven by an unknown “dark energy” (Riess _et al._ ,
1998; Perlmutter _et al._ , 1999) whose physical nature has not been
identified.
Therefore it appears to us that our understanding of the universe is currently
under the shade of three dark clouds — the mysterious dark energy that drives
late-time accelerated expansion, the nature of dark matter that is vital to
the formation of structures, and the unfathomable dark age that has not yet
revealed itself to observations. This is the “3-D universe” in which possible
answers to some of the most profound questions of physics are hidden.
In the face of these vast unknown sectors of the universe, any observational
probe into its past history is invaluable. Recently, the direct measurement of
the expansion rate, expressed in terms of the Hubble parameter $H(z)$, is
gaining increasing attention. As a cosmological test, it can help with the
determination of important parameters that affects the evolution of the
universe, and reconstruct the history around key events such as the turning
point from deceleration to acceleration. As an observable, it manifests itself
in various forms in different eras, especially in the baryon acoustic
oscillation (BAO) features in the LSS that may be detectable in the dark age.
This paper is a review on the current status of observational Hubble parameter
data and its application in cosmology. In Section II we briefly review the
cosmological background of an expanding universe. In Section III we present
two important observational methods of $H(z)$ observation, their principles
and implementations. Next, we review the important role of the observational
$H(z)$ data in the study of cosmological models in Section IV. We will also
discuss some issues associated with their application. Finally, in Section V,
we briefly discuss some ongoing efforts that promise possible improvements
over the current status of $H(z)$ measurements.
## II Background
In this section, we will review some basic ideas and definitions in cosmology
that must be kept in mind in order to understand and interpret the
observational $H(z)$ data and their implications.
### II.1 Spacetime, Metric, and Coordinates
The spacetime structure of the homogeneous, isotropic, and spatially flat
universe is characterized by the Friedmann-Robertson-Walker (FRW) metric
$\displaystyle g_{\mu\nu}=\left(\begin{array}[]{cccc}-1&&&\\\ &a^{2}(t)&&\\\
&&a^{2}(t)&\\\ &&&a^{2}(t)\end{array}\right).$ (6)
The presence of the scale factor $a(t)$ means that the spacetime is not
necessarily static. In reality, we know that the universe is expanding, and
$a(t)$ increases with time.
Using the metric (6), the infinitesimal spacetime interval scalar
$\mathrm{d}s^{2}=\mathrm{d}x_{\nu}\mathrm{d}x^{\nu}=g_{\mu\nu}\mathrm{d}x^{\mu}\mathrm{d}x^{\nu}$
is obviously
$\mathrm{d}s^{2}=-c^{2}\mathrm{d}t^{2}+a^{2}(t)\mathrm{d}x^{i}\mathrm{d}x^{i}.$
(7)
Here we have used the four-coordinate vector
$x^{\alpha}=(ct,\boldsymbol{x}^{i})$ that has the dimension of length.
It is often useful to express the spatial components of the four-coordinate
vector, i.e. the “comoving position”, in dimensionless spherical coordinates
$\boldsymbol{x}^{i}=(r,\theta,\phi)$ in order to extend the metric to non-flat
situations, and give the scale factor the dimension of length. Under this
convention, the spacetime interval (7) can be re-written as
$\mathrm{d}s^{2}=-c^{2}\mathrm{d}t^{2}+a^{2}(t)\left(\frac{\mathrm{d}r^{2}}{1-kr^{2}}+r^{2}\mathrm{d}\theta^{2}+r^{2}\sin^{2}\theta\mathrm{d}\phi^{2}\right)$
(8)
where $k$ is one of $\left\\{-1,0,1\right\\}$. The parameter $k$ is the sign
of the spatial curvature, and $k=0$ if the universe is spatially flat.
We can further transform equation (8) by introducing the coordinate
$\chi=\int^{r}_{0}\frac{\mathrm{d}r^{\prime}}{\sqrt{1-kr^{\prime
2}}}=\operatorname{sinn}^{-1}r$ (9)
where the $\operatorname{sinn}$ function is a shorthand notation:
$\displaystyle\operatorname{sinn}x=\begin{cases}\sin x&\text{for }k=1,\\\
x&\text{for }k=0,\\\ \sinh x&\text{for }k=-1.\end{cases}$ (10)
Switching to the spatial coordinates $(\chi,\theta,\phi)$, the interval
$\mathrm{d}s^{2}$ can be written as
$\mathrm{d}s^{2}=-c^{2}\mathrm{d}t^{2}+a^{2}(t)\left[\mathrm{d}\chi^{2}+\operatorname{sinn}^{2}\chi\left(\mathrm{d}\theta^{2}+\sin^{2}\theta\mathrm{d}\phi^{2}\right)\right].$
(11)
The physical interpretation of $\chi$ can bee seen by placing ourselves at the
origin $r=0$ and consider a distant, comoving photon emitter in our line-of-
sight direction with the coordinate $r=r_{e}$. Rotate the coordinates so that
the direction of the emitter has $\theta=0,\phi=0$, we find
$\mathrm{d}s^{2}=-c^{2}\mathrm{d}t^{2}+a^{2}(t)\frac{\mathrm{d}r^{2}}{1-kr^{2}}$
(12)
along the line-of-sight. Let $t_{e}$ be the time of photon emission and
$t_{0}$ that of its reception. Since light-like worldlines have
$\mathrm{d}s^{2}=0$, we find, for the photon:
$\int^{t_{0}}_{t_{e}}\frac{c\mathrm{d}t}{a(t)}=\int^{r_{e}}_{0}\frac{\mathrm{d}r}{\sqrt{1-kr^{2}}}=\chi(r_{e}).$
(13)
Consider the integrand in the left-hand side of equation (13). The line
element $\mathrm{d}x=c\mathrm{d}t$ is the physical distance the photon has
traveled during the time interval $\mathrm{d}t$. But by dividing the physical
distance by $a(t)$ we get the comoving distance, therefore $\chi$ can be
interpreted as the total, integrated comoving distance between the emitter and
us. If the space is flat, this comoving distance is just the difference in the
radial coordinate $\Delta r=r_{e}-0=r_{e}$.
Sometimes it is convenient to introduce the conformal time, or the comoving
horizon $\eta$ as the time component of the four-coordinate. The conformal
time is defined as
$\eta(t)=\int^{t}_{0}\frac{\mathrm{d}t^{\prime}}{a(t^{\prime})}$ (14)
where we integrate from the “beginning of time”. Using $c\eta$ as the time
component, the comoving four-coordinate can be written as a dimensionless
vector $x^{\alpha}=(c\eta,\chi,\theta,\phi)$ and the FRW metric takes the form
$\displaystyle g_{\mu\nu}=a^{2}(\eta)\left(\begin{array}[]{cccc}-1&&&\\\
&1&&\\\ &&\operatorname{sinn}^{2}\chi&\\\
&&&\operatorname{sinn}^{2}\chi\sin^{2}\theta\end{array}\right).$ (19)
### II.2 Expansion, Redshift, and the Hubble parameter
In the introduction we mentioned Hubble’s Law discovered in 1929. Hubble’s
original paper had profound impact upon the history of astrophysics and, to a
greater extent, mankind’s perception of the universe, but here we only take
some time to appreciate two of his timeless insights.
At the end of his paper Hubble briefly discussed the possible mechanisms for
“displacements of the spectra” (i.e. redshift, in modern terms) in the de
Sitter cosmology model in which the expansion of the universe is dominated by
a vacuum energy. He pointed out the two sources of the redshift: the first
being “an apparent slowing down of atomic vibrations” and the other attributed
to “a general tendency of material particles to scatter”. In today’s words,
the first is the special-relativistic effect of Doppler shift caused by the
peculiar motion of galaxies, and the latter the general-relativistic,
cosmological redshift which is linked to the expansion of the comoving grid
itself. In the rest of this article we will see how these two effects arise in
modern cosmology and end up in our observational figures.
Hubble also noted that his proportional law might be “a first approximation
representing a restricted range in distance”, therefore deviating from the
pure de Sitter model in which the Hubble constant $H$ should indeed be
constant everywhere and throughout the history. This is exactly how we see it
now. In the contemporary context, we usually define the Hubble parameter $H$
to be the relative expansion rate of the universe:
$H=\frac{\dot{a}}{a},$ (20)
and its value is usually expressed in the unit of $\mathrm{km\ s^{-1}\
Mpc^{-1}}$. The Hubble constant, $H_{0}$, now officially refers to the current
value of the Hubble parameter.
However, it is not apparent how this definition is related to observable
quantities. Therefore we have to relate equation (20) to physical observables
such as the length, the time, and the redshift.
First, we note that the cosmological redshift $z$ at any time $t$ is related
to the scale factor $a$. Let $t_{e}$ be the time of a photon’s emission by a
distant source and $t_{0}$ the time of its reception by an observer “here and
now.” The observed redshift $z$ of the source satisfies
$1+z=\frac{a(t_{0})}{a(t_{e})}.$ (21)
Consider an observer who surveys various sources with different redshifts. The
ideal survey is assumed to complete instantly — all the observations are done
at exactly the same time instance $t_{0}$. Of course this is not strictly
true, but we do not expect the scale factor $a(t_{0})$ to change “too fast”,
and we expect the redshift not to change too much during the temporal scale of
our interest (i.e. typical lifetime of humans or observation programs). If we
do allow $t_{0}$ to change however, we are led to the Sandage-Loeb test
(Sandage, 1962; Loeb, 1998) that observes the drifting of redshift during a
long period of time. Recently, the variation in the apparent magnitude of
stable sources over $t_{0}$ has also been proposed as a possible cosmological
test (Qi and Lu, 2010). To our best knowledge, no data have been produced
using these methods by now, and the proposed observation plans usually require
$\sim$10 years to yield meaningful results (Corasaniti _et al._ , 2007; Zhang
_et al._ , 2010) (however, we note that the idea of “real-time cosmology” is
gaining interest recently, as reviewed by Quercellini _et al._ (2010)). In
this paper we will not focus on these methods, and we therefore neglect the
passing of $t_{0}$.
We therefore differentiate equation (21) with respect to $t_{e}$, setting
$t_{0}$ as a constant:
$\frac{\mathrm{d}a(t_{e})}{\mathrm{d}t_{e}}=-\frac{a(t_{0})}{(1+z)^{2}}\frac{\mathrm{d}z}{\mathrm{d}t_{e}}=-\frac{a(t_{e})}{1+z}\frac{\mathrm{d}z}{\mathrm{d}t_{e}}.$
(22)
Dividing both sides by $a(t_{e})$ we immediately find
$H(z)=-\frac{1}{1+z}\frac{\mathrm{d}z}{\mathrm{d}t_{e}}.$ (23)
In Section III.1, we will see how equation (23) is useful in measuring $H(z)$
by observing passively evolving galaxies.
Another way to relate $H(z)$ to observable quantities is to use the notion of
the comoving distance $\chi$ introduced in equation (9). Take the time
derivative of equation (13), we find
$\frac{\mathrm{d}\chi}{\mathrm{d}t_{e}}=-\frac{c}{a(t_{e})}.$ (24)
On the other hand, equation (22) tells us about another derivative
$\mathrm{d}t_{e}/\mathrm{d}z$. Therefore we can find the derivative of $\chi$
with respect to the redshift:
$\displaystyle\frac{\mathrm{d}\chi}{\mathrm{d}z}=\frac{\mathrm{d}\chi}{\mathrm{d}t_{e}}\frac{\mathrm{d}t_{e}}{\mathrm{d}z}$
$\displaystyle=$
$\displaystyle\frac{c}{a(t_{e})}\frac{a(t_{e})}{(1+z)}\frac{\mathrm{d}t_{e}}{\mathrm{d}a(t_{e})}$
(25) $\displaystyle=$
$\displaystyle\frac{ca(t_{e})}{a(t_{0})}\frac{\mathrm{d}t_{e}}{\mathrm{d}a(t_{e})}$
$\displaystyle=$ $\displaystyle\frac{c}{a(t_{0})H},$
that is,
$\frac{\mathrm{d}\left[a(t_{0})\chi\right]}{\mathrm{d}z}=\frac{c}{H(z)}$ (26)
(also see, for example (Seo and Eisenstein, 2003; Bernstein, 2006), but beware
of different notation conventions). If an observable object spans the length
$a(t_{0})\Delta\chi$ along the line-of-sight in some redshift slice $\Delta
z$, we can estimate $H(z)$. But how do we find such objects, i.e. “standard
rods”? The idea is not to use the length of a concrete object. Instead, we
explore the spatial distribution of matter in the universe and focus on its
statistical features, such as the BAO peaks in the two-point correlation
function of the density field. This is another method for extracting $H(z)$
data from observations. (The quantity $a(t_{0})\chi$ can be seen as a distance
measure. It is closely related to the “structure distance” $d_{S}=a(t_{0})r$
defined by Weinberg (2008, Chapter 8) that naturally arises in calculating the
power spectrum of LSS. From equation (9) we can see that the structure
distance is equivalent to $a(t_{0})\chi$ if the space is flat, or if the
object is not too far away.)
We remark that the derivation of $H(z)$ expressed in terms of the standard
rod, equation (26), is only part of the story, for we have only considered a
standard rod placed in the line-of-sight direction. The transversely aligned
test body is related to another important cosmological measure, namely the
angular diameter distance $D_{A}(z)=a(z)r(z)$. In an expanding universe, the
angle $\Delta\theta$ subtended by a distant source is
$\Delta\theta=\frac{a(z)}{D_{A}(z)}\Delta
r_{\bot}=\frac{a(t_{0})}{(1+z)D_{A}(z)}\Delta r_{\bot},$ (27)
where $\Delta r_{\bot}$ is the transverse spatial span of the source measured
in the difference of comoving coordinate $r$ (Weinberg, 1972; Hogg, 1999).
Naturally, once the physical scale of BAO is known and the BAO signal
measured, the corresponding angular diameter distance can also be used as a
cosmological test.
A classical cosmological test is the Alcock-Paczyński (AP) test (Alcock and
Paczyński, 1979) that can be expressed as another combination of $H(z)$ and
$D_{A}(z)$. The observable of the AP test is the quantity $A(z)=\Delta
z/(z\Delta\theta)$ of some extended, spherically symmetric sources, where
$\Delta z$ is the difference in redshft between the near and far ends of the
object, and $\Delta\theta$ the angular diameter. By our equations (26) and
(27) it can be expressed as
$A(z)=\frac{\Delta
z}{z\Delta\theta}=\frac{1+z}{z}D_{A}(z)H(z)\frac{\Delta\chi}{\Delta
r_{\bot}}.$ (28)
A well-localized object placed in a region not too far away from us (so the
non-trivial spatial geometry can be neglected) will have
$\Delta\chi\approx\Delta r_{\shortparallel}$, the difference in the comoving
coordinate along the line-of-sight. Furthermore, for a nearly spherical object
the approximation $\Delta r_{\shortparallel}\approx\Delta r_{\bot}$ holds, and
$A(z)$ is reduced to
$A(z)=\frac{1+z}{z}D_{A}(z)H(z).$ (29)
Clearly it cannot constrain $H(z)$ or $D_{A}(z)$ separately, but a combination
of both. The AP test, in more modern context, is usually understood as a
geometrical effect on the statistical distribution of objects instead of
concrete celestial bodies (see (Matsubara and Suto, 1996; Ballinger _et al._
, 1996; Matsubara and Szalay, 2003), and also (Seo and Eisenstein, 2003;
Matsubara, 2004) where the BAO effects were explicitly treated in the
analysis).
Another combination of $H(z)$ and $D_{A}(z)$ naturally arises in the
application of BAO scales measured in the spherically averaged galaxy
distribution, namely the distance measure $D_{V}$ (Percival _et al._ , 2007)
defined by
$D_{V}(z)=\left[\frac{cz(1+z)^{2}D_{A}^{2}(z)}{H(z)}\right]^{1/3}.$
To break the degeneracy between $H(z)$ and $D_{A}(z)$ in $D_{V}(z)$, the full
2-dimensional galaxy distribution must be used, with the correlation function
conveniently decomposed into the line-of-sight and transverse components (see
section III.2, but also see (Padmanabhan and White, 2008) for another
decomposition scheme).
## III Hubble Parameter from Observations
Equations (23) and (26) are the bare-bone descriptions of two established
methods for $H(z)$ determination: the differential age method and the radial
BAO size method respectively. Either has been made possibly only by virtue of
state-of-the-art redshift surveys such as the Sloan Digital Sky Survey (SDSS)
111http://www.sdss.org/. In this section, we will review both methods and the
data they produced.
### III.1 The Differential Age Method
As equation(23) suggests, to apply age-dating to the expansion history, we
look for the variation of ages, $\Delta t$, in a redshift bin $\Delta z$
(Jimenez and Loeb, 2002). The aging of stars serves as an observable indicator
of the aging of the universe, because the evolution of stars is a well-studied
subject, and stars’ spectra can be taken and analysed to reveal information
about their ages. However, at cosmological distance scales it is not practical
to observe the stars one by one: we can only take the spectra of galaxies that
are ensembles of stars, possibly of different populations. Since different
star populations are formed at drastically different epochs, it is important
for us to identify galaxies that comprises relatively uniform star
populations, and to look for more realistic models of star formation.
The identification of such “clock” galaxies and the observation of their
spectra have been carried out for archival data (Jimenez _et al._ , 2003),
and surveys such as the Gemini Deep Deep Survey (GDDS) (McCarthy _et al._ ,
2004), VIMOS-VLT Deep Survey (VVDS) and the SDSS (Stern _et al._ , 2010a). In
addition, high-quality spectroscopic data have been acquired from the Keck I
telescope for red galaxies in galaxy clusters (Stern _et al._ , 2010b). Among
the galaxies being observed, special notices should be paid to the luminous
red galaxies (LRGs). LRGs are massive galaxies whose constituent star
populations are fairly homogeneous. They make up a fair proportion in the SDSS
sample and, beyond serving as “clocks”, also trace the underlying distribution
of matter in the universe (albeit with bias). Therefore, they reveal BAO
signature in the density autocorrelation function that is used as the
“standard rod” in the size method.
The identification and spectroscopic observations of these galaxies have led
to direct determinations of $H(z)$ in low and intermediate redshift ranges.
Jimenez _et al._ (2003) first obtained a determination of $H(z)=69\pm 12\
\mathrm{km\ s^{-1}\ Mpc^{-1}}$ at an effective redshift $z\approx 0.09$ by the
differential age method. The work was later expanded by Simon _et al._ (2005)
who extended the determination of $H(z)$ to 8 more redshift bins up to
$z\approx 1.8$. This dataset was brought up-to-date by Stern _et al._ (2010a,
Table 2). Recently, new age-redshift datasets for different galaxy velocity
dispersion groups have been made available (Carson and Nichol, 2010) from SDSS
data release (DR) 7 LRG samples. We will see how these data are used in the
study of cosmology models in Section IV.
One may wonder why we take the effort to calculate the age differences in
redshift bins when the age (or lookback time) data themselves can also be used
to test cosmological models. Indeed, the absolute age has been very useful in
the estimation of cosmological parameters (Stockton _et al._ , 1995; Dunlop
_et al._ , 1996; Spinrad _et al._ , 1997; Alcaniz and Lima, 2001).
Nevertheless, precise age-dating with low systematic biases can be only
carried out on a narrow selection of sources. On the other hand, by taking the
difference of the ages in narrow redshift bins, the systematic bias in the
absolute ages can hopefully cancel each other (Jimenez _et al._ , 2004). Of
course, we are not gaining anything for nothing even if the systematics
perfectly cancel, for the binning of data lowers the total amount of
measurements we can have.
A further approximation is that the majority of stars in the galaxies are
formed almost instantaneously, in a single “burst”(Jimenez _et al._ , 1999),
therefore the intrinsic spread of the measured age arising from a
heterogeneous star formation history can be expected to be small when fitting
the observed spectra to stellar population models (specifically the single-
stellar population (SSP) model used in (Simon _et al._ , 2005) and (Stern
_et al._ , 2010a)). However, recent developments in the study of the formation
history of galaxies and their stellar populations have led us to re-consider
the assumptions made in previous works. For example, using galaxy samples
selected from numerical simulations, Crawford _et al._ (2010) have shown that
the SSP assumption may contribute to the systematic bias that varies across
redshift ranges (hence failing to cancel, and propagating into the
differential ages), while models that take the extended star formation history
into account can be used to reduce the errors on $H(z)$.
In addition to the complexities in the stellar populations in each galaxy, the
heterogeneity of galaxies in the sample also contributes to the errors in
$H(z)$ measurements. In (Crawford _et al._ , 2010), new sample selection
criteria have been proposed that could help with obtaining more homogeneous
galaxy samples for future analyses.
### III.2 The Radial BAO Size Method
In Section II.2, we mentioned that the “standard rod” we seek in the sky is
not an actual object but a statistical feature. Indeed, the physical sizes of
distant celestial objects are usually poorly known. Worse still, even the
apparent, i.e. angular, sizes of galaxies are ambiguous because galaxies do
not show sharp edges, and they appear fuzzy in images. It can be imagined that
size measurements along the line-of-sight could only lead to more problems,
because even the angular sizes cannot help us much in this case. Therefore,
identifying a statistical “standard rod” becomes a necessity.
In the study of LSS, correlation functions are a simple and convenient measure
of the statistical features in the spatial distribution of matter in the
universe. (For an early yet important treatment of the topic in the context of
galaxy surveys, see (Peebles, 1973). For an example of other statistics in the
context of BAO, see (Xu _et al._ , 2010).) The two-point autocorrelation
(i.e. the correlation of a density field with itself) function
$\xi(\boldsymbol{r}_{1},\boldsymbol{r}_{2})$ is one of the most used member in
the correlation function family. It measures the relatedness of position pairs
in the same density field: the joint probability of finding two galaxies in
volume elements $\mathrm{d}V_{1}$ and $\mathrm{d}V_{2}$ located in the
neighborhood of spatial positions $\boldsymbol{r}_{1}$ and
$\boldsymbol{r}_{2}$ respectively is
$\mathrm{d}P_{12}=n^{2}\left[1+\xi(\boldsymbol{r}_{1},\boldsymbol{r}_{2})\right]\mathrm{d}V_{1}\mathrm{d}V_{2}$
(30)
where $n$ is the mean number density. If we believe that our universe is
homogeneous in a statistical sense (i.e. that the probabilistic distribution,
or ensemble, from which the densities anywhere in our particular instance of
the universe is drawn, does not vary from one area of the universe to
another), the autocorrelation function becomes a function of
$\boldsymbol{r}=\boldsymbol{r}_{1}-\boldsymbol{r}_{2}$ only. If we further
assumes the (statistical) isotropy of the universe, the direction of
$\boldsymbol{r}$ becomes unimportant, and the autocorrelation is dependent on
the magnitude of $\boldsymbol{r}$ only (that is, $\xi=\xi(r)$). Actually, our
assumption of homogeneity is unnecessarily strong if we only work with two-
point statistics, and all we need is the homogeneity in the first two moments
of the underlying ensemble. Such an ensemble is known as a wide-sense
stationary (WSS) one.
For a WSS ensemble, the famous Wiener-Khinchin theorem says that the
autocorrelation and the power spectrum $P(\boldsymbol{k})$ form a Fourier
transform pair:
$\displaystyle P(k)=P(\boldsymbol{k})$ $\displaystyle=$
$\displaystyle\int\xi(\boldsymbol{r})e^{i\boldsymbol{k}\cdot\boldsymbol{r}}\mathrm{d}^{3}r,$
$\displaystyle\xi(r)=\xi(\boldsymbol{r})$ $\displaystyle=$
$\displaystyle\frac{1}{(2\pi)^{3}}\int
P(\boldsymbol{k})e^{-i\boldsymbol{k}\cdot\boldsymbol{r}}\mathrm{d}^{3}k.$ (31)
(Here we write the power spectrum as $P(k)$, independent of the direction of
the wave vector $\boldsymbol{k}$, under the same assumption of statistical
isotropy mentioned above, but see discussion about redshift distortion below.)
Therefore, either the power spectrum or the autocorrelation can serve as a
statistical tool to reveal the information contained in the LSS. Methods of
estimating $P(k)$ has been developed and the importance of the power spectrum
emphasized (Feldman _et al._ , 1994; Percival _et al._ , 2004). On the other
hand, for BAO surveys the autocorrelation function is probably a more
straightforward way of presenting the results and testing their significance,
because the BAO scales manifest themselves as protruding features (“peaks” or
“bulges”) in $\xi(r)$. Actually, an estimator to the autocorrelation, along
with its variance, can also be conveniently constructed from survey data using
pair counts between the survey and random fields (Landy and Szalay, 1993).
Needless to say, the “true” autocorrelation of the ensemble can never be fully
known, because we have only one realization of the random field which is the
universe we live in. However, estimating the autocorrelation still makes sense
because for today’s large and well-sampled surveys the assumption of
ergodicity is valid, under which the statistics can be performed to infer
knowledges about the underlying ensemble (Weinberg, 2008, Chapter 8 and
Appendix D).
Thus, if a random process induces some features in the spatial distribution of
matter, the autocorrelation can be numerically computed to reveal such
features that are otherwise hidden in the seemingly stochastic distribution.
Furthermore, if the mechanism and properties of this process is well
understood and quantitatively modelled, parameter estimation using these
features becomes a possibility.
One of such possibility is provided by the BAO signatures in the LSS. The
mechanism of BAO effects must be traced back to the early universe before
recombination, when the Compton scattering rate was much higher than the
cosmic expansion rate. Under this extreme limit, the tightly coupled photons
and baryons can be treated as a fluid in which the perturbations drive sound
waves. The BAO effect in the cosmic microwave background (CMB) radiation has
been subjected to extensive theoretical studies (see the early work of Peebles
and Yu (1970), a powerful analytical treatment by Hu and Sugiyama (1995) in
Fourier space, another by Bashinsky and Bertschinger (2002) in position space,
and a review by Hu and Dodelson (2002)). It has been confirmed and measured by
CMB observations such as the Wilkinson Microwave Anisotropy Probe (WMAP)
(Hinshaw _et al._ , 2003; Page _et al._ , 2003; Hinshaw _et al._ , 2007;
Nolta _et al._ , 2009; Larson _et al._ , 2010). We will not discuss CMB in
detail, and mainly concern ourselves with the aftereffect of BAO, namely its
imprints on the large-scale distribution of matter.
The imprints of BAO in the observable distribution of galaxies today was
predicted in theory (see (Goldberg and Strauss, 1998; Meiksin _et al._ ,
1999), and note that these papers were mainly written in the language of
$P(k)$ rather than $\xi(r)$). They were first detected in SDSS data by
Eisenstein _et al._ (2005). In (Percival _et al._ , 2007), BAO measurements
were made for SDSS and 2dF survey data using the power spectrum, and the
results were presented as a general test of cosmological models. The usage of
BAO signatures in the LSS as a probe of $H(z)$ was discussed in (Seo and
Eisenstein, 2003) (see also (Blake and Glazebrook, 2003; Seo and Eisenstein,
2005, 2007)).
The idea of using BAO scales may appear to be simple and straightforward by
our description so far, but in reality the autocorrelation function is
subjected to various distortion effects that must be accounted for.
First, galaxies are not comoving objects. Their apparent redshifts are
inevitably a combined effect of the cosmological redshift and peculiar
velocities (which was once contemplated by E .P. Hubble, see Section II.2).
Peculiar motion distorts the apparent correlation pattern in the redshift
space and makes it anisotropic (see (Davis and Peebles, 1983; Kaiser, 1987)).
Therefore, the isotropic autocorrelation function $\xi(r)$ fails to be a good
measure. In the literature the autocorrelation is usually expressed as a
function of scales in the radial (line-of-sight) direction $\pi$ and
transverse direction $\sigma$: $\xi=\xi(\sigma,\pi)$ with
$r=\sqrt{\sigma^{2}+\pi^{2}}$. The observed $\xi(\sigma,\pi)$ will be a
convolution between $\xi(r)$ and the peculiar velocity field.
Second, geometry of the spacetime also distorts the correlation pattern as the
observation goes into deeper distances, where the spacetime geometry becomes
non-trivial (Magira _et al._ , 2000). This is not a major concern for the
analyses we will review in the rest of this section, because the survey data
were from our local section of the universe ($z\approx 0$), and for $H(z)$
measurements only some thin slices in the redshift space were used. However,
future work that deals with deep survey data must take the geometrical
distortions into analysis.
There is also the more delicate issue of biasing, meaning that the correlation
pattern of the observed “indicators” does not necessarily reflect that of the
underlying matter distribution (Kaiser, 1984). Among the effects contributing
to the bias, the magnification effect by weak lensing is worthy of notice for
our discussion, because it has a large effect on the radial autocorrelation
function (Hui _et al._ , 2007, 2008).
Using SDSS LRG samples in the redshift range $0.16\leq z\leq 0.47$, BAO
signature was detected in $\xi(\sigma,\pi)$ by Okumura _et al._ (2008). In
their work the magnification bias by weak lensing was neglected, but in the
redshift range it contributes little to the spherically averaged
autocorrelation $\xi_{0}$ (Hui _et al._ , 2007), also known as the monopole:
$\xi_{0}(r)=\frac{1}{2}\int_{-1}^{1}\xi(\sigma,\pi)\mathrm{d}\mu,$ (32)
where $r=\sqrt{\sigma^{2}+\pi^{2}}$, and $\mu=\pi/r$. In (Okumura _et al._ ,
2008) the BAO peak was detected in the monopole significantly, while the
ridge-like BAO feature was weak in the anisotropic $\xi(\sigma,\pi)$.
Using improved LRG samples from SDSS DRs 6 and 7, and by modelling the weak
lensing magnification bias, radial BAO detection and $H(z)$ measurements were
made in redshift slices $z=0.15\sim 0.30$ and $z=0.40\sim 0.47$ by Gaztañaga
_et al._ (2009a) (see Figure 1 for a presentation of the BAO detection).
Because these redshift slices were well separated, the two measurements were
independent from each other. (In previous works such as (Percival _et al._ ,
2007) the samples overlapped and the results at different $z$’s were
correlated.)
Figure 1: Detection of radial ($\pi$-direction) BAO by Gaztañaga _et al._
(2009a, Figure 13) in the full LRG sample. This is the correlation pattern
along the $\pi$-direction, and should not be confused with the monopole
pattern in Figure 3 of (Okumura _et al._ , 2008). The effect of weak lensing
magnification can bee seen by comparing the solid and short dashed curves,
which shows that the magnification systematically moves the peak location
towards the higher scales. The dash-dotted (blue) curve shows the $1\sigma$
range by allowing the fiducial distance-redshift relation used in the analysis
to vary in a parameterized way, accounting for the systematic error introduced
by the mere using of a fiducial model.
These $H(z)$ measurements were the first implementation of the radial BAO
method. Due to the distortion effects, confirming the significance of the
baryon ridge detection becomes a demanding process, since each distortion
effect has to be carefully modelled. However, exact modelling of all the
distortion effects on all scales is difficult, and when such modelling cannot
be done exactly, these effects introduces systematic errors in the measurement
of the BAO ridge’s scale.
Despite these, the radial BAO size method still surpasses the age method in
precision. In fact, the combined statistical and systematic uncertainties
presented an precision of $\sim$4$\%$ in $H(z)$ (Gaztañaga _et al._ , 2009a,
Table 3). This is intuitively perceptible. As we have seen in Section III.1,
the age method is affected by the (possibly very large) systematic errors in
age determination. Since we can measure spatial quantities of galaxies, i.e.
the distribution of their positions, with much greater accuracy than we can do
with temporal quantities related to some vaguely defined event (namely the
time duration from star formation in the red galaxies to now), one may
intuitively expect lower uncertainties from the radial size method than the
differential age method.
A subtle issue of possible circular logic in the analysis also contributes to
the systematic errors in this method. In (Gaztañaga _et al._ , 2009a), a
fiducial flat $\mathrm{\Lambda CDM}$ model and parameters were used to convert
redshifts into distances, and to gauge the comoving BAO scales in the selected
redshift slice, $r_{\mathrm{BAO}}$ to that of the CMB measured by 5-year WMAP,
$r_{\mathrm{WMAP}}=153.3\pm 2.0\mathrm{Mpc}$ (see (Komatsu _et al._ , 2009))
to yield the estimation $H_{\mathrm{BAO}}(z)$:
$\frac{H_{\mathrm{BAO}}(z)}{r_{\mathrm{BAO}}}=\frac{H_{\mathrm{fid}}(z)}{r_{\mathrm{WMAP}}},$
(33)
where
$H_{\mathrm{fid}}(z)=H_{0}\sqrt{\Omega_{\mathrm{m}}(1+z)^{3}+(1-\Omega_{\mathrm{m}})}$
and $\Omega_{\mathrm{m}}=0.25$ 222Another way to present the measurement
results for use in cosmological parameter constraint $\Delta
z_{\mathrm{BAO}}=r_{\mathrm{BAO}}H(z)/c$. Schematically, this is done by
approximating the derivative in equation (26) with a ratio of differences, and
identifying the interval $a(t_{0})\Delta\chi$ with the measured comoving BAO
scale. In Section IV we briefly discuss its usage.. The use of a fiducial
model introduces bias in all measurements, which is hard to model exactly, but
an analysis of this effect was performed using Monte Carlo simulations so that
its contribution to the systematic uncertainties could be assessed. The
authors of (Gaztañaga _et al._ , 2009a) hence argued that the measurement
results are model-independent, therefore is useful as a general cosmological
test. The reader may also consult (Percival _et al._ , 2007) for a different
approach to this issue, using cubit spline fit of the distance-redshift
relation so that the result could be applied to a large class of models
without having to re-analyze the power spectra for each model to be tested.
#### A Word on the Dispute over the Radial BAO Detection.
Currently there is some dispute over the claimed detection of radial BAO and
measurement of $H(z)$ in (Gaztañaga _et al._ , 2009a). Miralda-Escudé (2009)
argued against the methods in (Gaztañaga _et al._ , 2009a) and the
statistical significance of the claimed BAO detection. Kazin _et al._ (2010)
analyzed the SDSS DR7 sample of LRGs and obtained similar results to
(Gaztañaga _et al._ , 2009a), but offered another interpretation using the
$\chi^{2}/(\text{degree of freedom})$ statistic and the Bayesian evidence
(Liddle, 2009) that disfavors a statistically significant detection. On the
other hand, the recent research of Tian _et al._ (2010) claims that the
radial BAO feature is not a fluke, albeit certain assumptions made this re-
assessment somewhat optimistic. The authors of (Gaztañaga _et al._ , 2009a)
also defended their work in (Cabré and Gaztañaga, 2010). We refer to these
variety of arguments and opinions to remind the reader of these ongoing
investigations. Nevertheless, we believe that the general method of measuring
$H(z)$ using radial BAO is well-motivated and promising regardless of its
current implementation, as it is expected to give more definitive results of
radial BAO and $H(z)$ measurement with upcoming redshift survey projects
(Kazin _et al._ , 2010).
## IV Observational Hubble Parameter as a Cosmological Test
The efforts in obtaining observational $H(z)$ data was certainly done with the
goal of testing cosmological models in mind. In (Jimenez _et al._ , 2003) the
observation $H(z)$ at $z\approx 0.09$ was used to constrain the equation of
state parameter of dark energy. In (Simon _et al._ , 2005) the redshift-
variability of a slow-roll scalar field dark energy potential was constrained
by the differential age $H(z)$ data. The same dataset was also utilized in the
study of the $\mathrm{\Lambda CDM}$ universe, especially the summed neutrino
masses $m_{\nu}$, the effective number of relativistic neutrino species
$N_{\mathrm{rel}}$, the spatial curvature $\Omega_{\mathrm{k}}$, and the dark
energy equation of state parameter $\omega$ (Figueroa _et al._ , 2008). The
updated $H(z)$ data presented in (Stern _et al._ , 2010a) was used by their
authors to improve the results obtained in earlier papers.
In particular, the combination of CMB and $H(z)$ observation is a very
effective way to constrain $N_{\mathrm{rel}}$ (Reid _et al._ , 2010, see the
reproduced Figure 2 in this paper). In this paper we will not go further into
the topic of cosmic neutrinos, which is intrinsically related to fundamental
physics. However, we should point out a remarkable result, that the $H(z)$
data, when used jointly with CMB and other late-era cosmological tests, offer
valuable insight into the neutrino properties related to the much earlier
universe, independent of Big-Bang neucleosynthesis (BBN) (Izotov _et al._ ,
2007; Iocco _et al._ , 2009) tests. Moreover, the BBN constraints are
obtained using Helium abundance measurements that are subjected to the
systematic biasing effects arising from late-time neucleosysthesis. Therefore,
$H(z)$ data is an important consistency check measure in the presence of this
systematic uncertainty (Reid _et al._ , 2010).
Figure 3 shows that adding $H(z)$ data helps with breaking the degeneracy
between spatial curvature and dark energy equation of state. In the
$\mathrm{\Lambda CDM}$ universe, both the dark energy and spatial curvature
becomes dominant in recent epochs. Therefore, separating their respective
effects on the expansion of the universe becomes important, as well as
challenging (Clarkson _et al._ , 2007; Vardanyan _et al._ , 2009). While
other tests using the combination of weak lensing and BAO are likely to
measure the curvature distinctively in the future (Bernstein, 2006; Zhan _et
al._ , 2009), our current knowledge of $H(z)$ is still a valuable complement
to other tests in the sense of DE-curvature degeneracy breaking (Figueroa _et
al._ , 2008).
The data produced by the BAO size method in (Gaztañaga _et al._ , 2009a) is
scarcer in quantity but of higher precision. In (Gaztañaga _et al._ , 2009a)
they were extrapolated to $z=0$ to offer an independent estimation of the
Hubble constant $H_{0}$, and were used to test the accelerated expansion of
the universe. It has been demonstrated that the radial $\Delta
z_{\mathrm{BAO}}$ measurements is able to put stringent constraints over the
dark energy parameters (Gaztañaga _et al._ , 2009b).
In the papers cited above, the parameter constraints obtained from
observational $H(z)$ data were shown to be consistent with other cosmological
tests, such as the CMB anisotropy. In this way, the observational $H(z)$ data
presents themselves as a useful, independent cosmological test. In particular,
it serves as a powerful tool to break the degeneracy between the curvature and
dark energy parameters.
Figure 2: Constraint on the effective number of relativistic neutrino species,
$N_{\mathrm{rel}}$, by Stern _et al._ (2010a) using their $H(z)$ measurements
by the differential age method. Dotted line plots the 5-year WMAP (Dunkley
_et al._ , 2009) likelihood, dashed line plots the likelihood with WMAP and
$H_{0}$ determined by Riess _et al._ (2009), and the solid like the
likelihood with WMAP, $H_{0}$ and $H(z)$ data. Adding $H(z)$ data helped
refining the constraint to $N_{\mathrm{rel}}=4\pm 0.5$ at 1-$\sigma$ level.
The improvement in the constraint by adding $H(z)$ data is evident. Note that
this figure displays the deviation of the $\chi^{2}$ statistic from its
minimum inverted ($\Delta\chi^{2}=\chi^{2}_{\mathrm{min}}-\chi^{2}$). The
intersections of the $\Delta\chi^{2}$ plots with the constant
$\Delta\chi^{2}=4$ line correspond to 2-$\sigma$ constraints. Figure 3: Joint
constraint on the energy density corresponding to the spatial curvature,
$\Omega_{\mathrm{k}}$, and the dark energy equation of state parameter, $w$,
by Stern _et al._ (2010a). The large, irregular regions bounded by dark
contours were from 5-year WMAP alone. The blue contours were obtained by
adding $H_{0}$ constraints. Filled regions were obtained by further adding
$H(z)$ data. The application of $H(z)$ data helps with breaking the degeneracy
between $\Omega_{\mathrm{k}}$ and $w$.
These up-to-date data are summarized in Table 1. In Figure 4 we plot the
$H(z)$ data versus the redshift. To help visualizing the data, we also plot a
spatially flat $\mathrm{\Lambda CDM}$ model with
$\Omega_{\mathrm{m}}=0.25,\Omega_{\mathrm{\Lambda}}=0.75,\text{and }H_{0}=72\
\mathrm{km\ s^{-1}\ Mpc^{-1}}$.
Table 1: The set of available observational $H(z)$ data $z$ | $H(z)\,\pm\,1\sigma\text{ error}$11footnotemark: 1 | References | Remarks
---|---|---|---
$0.09$ | $69\,\pm\,12$ | (Jimenez _et al._ , 2003; Stern _et al._ , 2010a) |
$0.17$ | $83\,\pm\,8$ | (Stern _et al._ , 2010a) |
$0.24$ | $79.69\,\pm\,2.65$222H(z) figures are in the unit of km s^-1 Mpc^-1. | (Gaztañaga _et al._ , 2009a) | In the redshift slice $0.15\sim 0.30$
$0.27$ | $77\,\pm\,14$ | (Stern _et al._ , 2010a) |
$0.4$ | $95\,\pm\,17$ | (Stern _et al._ , 2010a) |
$0.43$ | $86.45\,\pm\,3.68$222Including both statistical and systematic uncertainties: σ= σ^2_sta + σ^2_sys. | (Gaztañaga _et al._ , 2009a) | In the redshift slice $0.40\sim 0.47$
$0.48$ | $97\,\pm\,62$ | (Stern _et al._ , 2010a) |
$0.88$ | $90\,\pm\,40$ | (Stern _et al._ , 2010a) |
$0.9$ | $117\,\pm\,23$ | (Stern _et al._ , 2010a) |
$1.3$ | $168\,\pm\,17$ | (Stern _et al._ , 2010a) |
$1.43$ | $177\,\pm\,18$ | (Stern _et al._ , 2010a) |
$1.53$ | $140\,\pm\,14$ | (Stern _et al._ , 2010a) |
$1.75$ | $202\,\pm\,40$ | (Stern _et al._ , 2010a) |
Figure 4: Top panel — the available $H(z)$ data from both differential age
method and radial BAO size method (see Table 1 and references therein). The
solid curve plots the theoretical Hubble parameter $H_{\mathrm{fid}}$ as a
function of $z$ from the spatially flat $\mathrm{\Lambda CDM}$ model with
$\Omega_{\mathrm{m}}=0.25,\Omega_{\mathrm{\Lambda}}=0.75,\text{and }H_{0}=72\
\mathrm{km\ s^{-1}\ Mpc^{-1}}$. Bottom panel — the same data, but the
residuals with respect to the theoretical model $H_{\mathrm{fid}}$ are
plotted. In both panels, the $z$ error bars on the measurements from the
radial BAO method are used to mark the extents of the two independent redshift
slices in which the BAO peaks were measured.
In addition to the above authors, the observational $H(z)$ datasets have been
widely used to put various cosmological models under test. The first adopters
included Yi and Zhang (2007) and Samushia and Ratra (2006) who made use of the
$H(z)$ results of (Simon _et al._ , 2005) in the study of dark energy. In (Yi
and Zhang, 2007) the $H(z)$ data alone were used to constrain the parameters
of the holographic dark energy model, especially th $c$ parameter that
determines the dynamical history of the expanding universe (see Figure 5). The
same dataset has also been used to study modified gravity theory such as
$f(R)$ gravity in the context of cosmology (Carvalho _et al._ , 2008). The
updated data in (Stern _et al._ , 2010a) and (Gaztañaga _et al._ , 2009a)
have been adopted to constrain the parameters in more exotic dark energy
models, e.g. (Xu and Wang, 2010; Durán _et al._ , 2010).
Figure 5: Parameter constraints for the holographic dark energy model in the
$\Omega_{\mathrm{m}}$-$c$ plane, by Yi and Zhang (2007). The constraints were
obtained using age-determined $H(z)$ data in (Simon _et al._ , 2005) alone.
The cross in the lower-left marks the best-fit value. The dash-dotted, solid,
and dotted contours marks the $68.3\%$, $95.4\%$, and $99.7\%$ confidence
regions respectively. Although some degeneracy exists, it is evident that the
data favor models with $c<1$.
Beyond parameter constraints, the observational $H(z)$ data are also
applicable in non-parametric, model-independent cosmological tests. For
example, the $Om$ statistic by Sahni _et al._ (2008), defined by
$Om(z)=\frac{h^{2}(z)-1}{(1+z)^{3}-1},$ (34)
where $h$ is the dimensionless Hubble parameter, $h=H(z)/H_{0}$. This
statistic is useful as a null test of dark energy being a cosmological
constant $\mathrm{\Lambda}$, and is more robust than parameterizations of the
dark energy equation of state. Another result for testing $\mathrm{\Lambda}$
that incorporates $H(z)$ data (the $\mathcal{L}_{\mathrm{gen}}$ test) is given
by Zunckel and Clarkson (2008), with the addition of distance information. In
either paper however, the Hubble parameter data used were not the independent
observational measurements discussed in this review, but the ones
reconstructed using SNIa luminosity distances. In a similar fashion, it has
been shown that $H(z)$ and distance measurements can further test the spatial
flatness of the universe, or even the Copernican Principle of large-scale
homogeneity and isotropy that is behind the mathematical form of the FRW
metric (6) by a model-independent approach (Clarkson _et al._ , 2008;
Shafieloo and Clarkson, 2010). In (Shafieloo and Clarkson, 2010) the use of
$H(z)$ in some of these tests was demonstrated with real-world observational
data reviewed here.
Despite the wide application of the $H(z)$ datasets in the literature, we
would like to point out some issues associated with their usage.
First, in some papers (Xu and Wang, 2010; Durán _et al._ , 2010; Pan _et
al._ , 2010) that made use of $H(z)$ data derived from radial BAO by Gaztañaga
_et al._ (2009a) in $\chi^{2}$ analyses, the measurement at a middle redshift
$z=0.34$ was used in conjunction with those from the two independent redshift
slices near $z=0.24$ and $0.43$, under the tacit assumption of being
independent from each other. However, this is not true, because the
determination at the middle redshift was not made from a separate, non-
overlapping redshift slice, but from the whole sample of galaxies, including
the lower and upper redshift ranges. If the data is to be used in quantitative
works, this interdependency should not be ignored and must be explicitly
analysed. A related issue is combining the $H(z)$ data determined from radial
BAO peaks with the $\Delta z_{\mathrm{BAO}}$ data derived using the same
method under the assumption of their independence (this practice can be found,
for example, in (Pan _et al._ , 2010)). To be rigorous (or pedantic,
depending on your point of view), we do not believe that this is the best way
to use the data, and we insist on an analysis involving the (non-diagonal)
covariance between these datasets. On the other hand, the combination of
$\Delta z_{\mathrm{BAO}}$ data and age-dated $H(z)$ is mostly free from this
interdependence problem, and they actually complement each other well (Zhai
_et al._ , 2010, in particular Figures 1 and 2). We also note that in
qualitative explorations one may choose to relax this restriction to some
reasonable extent, for example in the discussion of accelerate expansion in
(Gaztañaga _et al._ , 2009a, Section 4.4).
Another topic that cold be worthy of future discussions is the possible
tension between the $H(z)$ datasets and other observational data. As noted by
Figueroa _et al._ (2008), datasets of different physical natures and
systematic effects can be safely combined only if they agree with each other
well (see also (Verde, 2010)). In this regard, we note that there is possibly
some tension between $H(z)$ and type Ia supernova (SNIa) luminosity distances
as shown in (Zhai _et al._ , 2010) (see Figure 6). However, this apparent
tension could be statistical in nature and may simply be a consequence of not
having enough independent measurements of $H(z)$. We hope that future expanded
$H(z)$ datasets would allow us to check its consistency with other data in a
quantitative manner.
Figure 6: Possible tension between $H(z)$ and type Ia supernovae data depicted
in the $\chi^{2}$ fitting results for the spatially flat XCDM model (similar
to $\mathrm{\Lambda CDM}$, except that the dark energy equation of state
parameter $\omega$ is set free instead of being fixed at $\omega=-1$). The SN
data favor a phantom dark energy with $\omega<-1$ while other data, including
observational $H(z)$ (OHD), are consistent with $\mathrm{\Lambda CDM}$. The
OHD used in this figure were the measurements by (Simon _et al._ , 2005)
using the differential age method, and the SN data were from (Riess _et al._
, 2004). The RBAO contours were found using the $\Delta z_{\mathrm{BAO}}$ data
in (Gaztañaga _et al._ , 2009a). Confidence regions are $68.3\%$, $95.4\%$,
and $99.7\%$ respectively. This figure first appeared in (Zhai _et al._ ,
2010, Fig. 4).
## V Future Directions
The available $H(z)$ data have so far proven to be a useful tool in the
pursuit of understanding the expansion history of the universe and the
possible nature of dark energy. However, these datasets do not have very good
redshift coverage. The current measurements have gone as deep as $z=1.75$, and
this redshift range is only sparsely covered. There is also another issue of
the large error bars associated with the $H(z)$ figures from the differential
age method. On the other hand, the collection of more and higher quality
$H(z)$ data will not only help us constrain the parameters, but will also
allow us to understand the possible tension between $H(z)$ and other
cosmological tests. The latter is important, because tension is usually an
indicator of systematic errors in the data. By understanding the tension, we
may finally conquer the systematic effects that have not yet been modelled
well enough.
In this section, we will describe a few directions of future cosmological
observations and their implications in the measurements of the Hubble
parameter.
### V.1 Future Improvements in the Differential Age Method
The relatively large uncertainties in the differential age method could be
partially compensated if future datasets could offer better coverage in the
redshift range accessible by this method. Using mock data, we recently
estimated that future $H(z)$ datasets would offer similar or even higher
parameter-constraining power compared with current SNIa datasets if it could
add as many as $\sim$60 independent measurements to cover the redshift range
$0\leq z\leq 2$ (Ma and Zhang, 2010). To achieve this level of data coverage,
future surveys must be able to offer a large sample of LRGs to be used in age-
dating. According to (Simon _et al._ , 2005), the Atacama Cosmology Telescope
(ACT) 333http://www.physics.princeton.edu/act/index.html can be utilized in
the future to identify passively evolving, red galaxies by their Sunyaev-
Zel’dovich effect. These galaxies can in turn be spectroscopically measured
and age-dated, and it has been estimated that they could yield $\sim$1000
$H(z)$ measurements. This means the quality of current differential age $H(z)$
data can be expected to increase significantly.
The error model used in the analysis of differential age $H(z)$ data in (Ma
and Zhang, 2010) was empirical, which may have underestimated possible future
improvements. In (Crawford _et al._ , 2010) it has been estimated that $H(z)$
may be measured within $3\%$ relative error at $z\approx 0.42$ in realistic
observations if the star formation systematics could be properly accounted
for. This level of precision is on par with the current status of the radial
BAO method, and we hope it could be achieved in the near future.
### V.2 Future Improvements in the Radial BAO Size Method
The radial BAO size method has already been demonstrated to provide highly
accurate $H(z)$ measurements. However, this accuracy came at a cost, for
spectroscopic data must be taken for the great number of galaxies under survey
to find their redshifts, which is time-consuming. Fortunately it turns out
that for low redshift ranges, photometric redshift surveys can be a sufficient
and promising approach (Benítez _et al._ , 2009; Arnalte-Mur _et al._ ,
2009; Roig _et al._ , 2009) to the detection and measurements of radial BAO
features in the autocorrelation function. Photometry has several advantages
over spectroscopy – it is cheaper, faster, and able to reach fainter sources.
Shortly before this review is written, the WiggleZ redshift survey
444http://wigglez.swin.edu.au/ of emission-line galaxies produced its first
data release (Drinkwater _et al._ , 2010). As the data is being released, it
is expected that the radial BAO signal can be put to further scrutiny (Kazin
_et al._ , 2010).
The BAO method is unique in that it allows us to reconstruct the cosmic
expansion through a vast range of eras. Unlike the differential age method in
which the observable indicators of time are located within a limited redshift
range, BAO signal detection is possible as long as the distribution of matter,
regardless of its form, can be traced. Even if the current implementation of
the radial BAO method is mainly confined in the redshift range of $z\approx
0$, future redshift surveys such as the planned SDSS III project
555http://www.sdss3.org/ are designed to reach into deeper universe and
measure $H(z)$ at redshifts up to $z\approx 2.5$ by observing the
Lyman-$\alpha$ forest absorption spectra of high-redshift quasars (see
(McDonald and Eisenstein, 2007) for a discussion of high-$z$ measurement of
radial BAO and $H(z)$ and its implication for dark energy, and (Norman _et
al._ , 2009; White _et al._ , 2010) for numerical simulation studies).
Recently, in the wake of the proposed Euclid satellite project (Cimatti _et
al._ , 2009; Laureijs, 2009), the enormous potential of space-based redshift
surveys in the determination of $H(z)$ and other parameters has been studied
in (Wang _et al._ , 2010). Finally, the proposed observational programs of
the 21 cm background may further extend our knowledge of $H(z)$ into even
deeper redshift ranges before or near the reionization era (Barkana and Loeb,
2005; Mao and Wu, 2008; Seo _et al._ , 2010), the “dark ages” that have not
been extensively explored by current observations yet.
It is also worth noting that the previous works on the analysis and
measurement of $H(z)$ from the clustering of LSS have mostly concentrated on
the BAO features alone. However, Shoji _et al._ (2009) shows that accurate
estimates of $H(z)$ and $D_{A}(z)$ could be made using the full galaxy power
spectrum in the extraction of cosmological information instead of BAO features
alone, provided that the non-linear clustering effects are well controlled. We
hope that the future redshift surveys observations, as well as advances in
better understanding of nonlinear-regime redshift-space distortions, could
lead to successful realization of their method.
## VI Summary
In this paper, we reviewed the current status of observationally measured
Hubble parameter data. We presented the principle ideas behind the two
important and independent methods of $H(z)$ measurement, namely the
differential age method and the radial BAO size method. Both methods have been
successfully implemented over the years to yield $H(z)$ data that are of
varying precision and redshift coverage, and the up-to-date results have been
summarized in Table 1. These data are valuable for the study of the expanding
universe. They have seen wide application by cosmologists to put various
cosmological models under test, and to constrain important cosmological
parameters either independently or in conjunction with data of different
physical natures. However, we also pointed out several issues in the usage of
observational $H(z)$ data. Finally, despite some current shortcomings, we find
the $H(z)$ data of great potential, as future observational programs can be
expected to improve significantly the quality of $H(z)$ data that may lead us
into unexplored realms of the universe.
###### Acknowledgements.
We gratefully acknowledge Chris Clarkson, Eyal A. Kazin, and Varun Sahni for
their helpful suggestions. We would like to thank the anonymous referee for
critically reviewing the manuscript and providing insightful comments that
helped us improve this paper greatly. CM thanks Zhongfu Yu for his help in
preparing some of the materials in the bibliography list. This work was
supported by the National Science Foundation of China (Grants No. 10473002),
the Ministry of Science and Technology National Basic Science program (project
973) under grant No. 2009CB24901, the Fundamental Research Funds for the
Central Universities.
## References
* Hubble (1929) E. P. Hubble, “A relation between distance and radial velocity among extra-galactic nebulae,” Proc. Natl. Acad. Sci. USA 15, 168–173 (1929).
* Riess _et al._ (1998) A. G. Riess, A. V. Filippenko, P. Challis, A. Clocchiatti, A. Diercks, P. M. Garnavich, R. L. Gilliland, C. J. Hogan, S. Jha, R. P. Kirshner, B. Leibundgut, M. M. Phillips, D. Reiss, B. P. Schmidt, R. A. Schommer, R. C. Smith, J. Spyromilio, C. Stubbs, N. B. Suntzeff, and J. Tonry, “Observational evidence from supernovae for an accelerating universe and a cosmological constant,” AJ 116, 1009–1038 (1998), arXiv:astro-ph/9805201 .
* Perlmutter _et al._ (1999) S. Perlmutter, G. Aldering, G. Goldhaber, R. A. Knop, P. Nugent, P. G. Castro, S. Deustua, S. Fabbro, A. Goobar, D. E. Groom, I. M. Hook, A. G. Kim, M. Y. Kim, J. C. Lee, N. J. Nunes, R. Pain, C. R. Pennypacker, R. Quimby, C. Lidman, R. S. Ellis, M. Irwin, R. G. McMahon, P. Ruiz-Lapuente, N. Walton, B. Schaefer, B. J. Boyle, A. V. Filippenko, T. Matheson, A. S. Fruchter, N. Panagia, H. J. M. Newberg, W. J. Couch, and The Supernova Cosmology Project, “Measurements of $\Omega$ and $\Lambda$ from 42 high-redshift supernovae,” ApJ 517, 565–586 (1999), arXiv:astro-ph/9812133 .
* Sandage (1962) A. Sandage, “The change of redshift and apparent luminosity of galaxies due to the deceleration of selected expanding universes,” ApJ 136, 319–333 (1962).
* Loeb (1998) A. Loeb, “Direct measurement of cosmological parameters from the cosmic deceleration of extragalactic objects,” ApJ 499, L111–L114 (1998), arXiv:astro-ph/9802112 .
* Qi and Lu (2010) S. Qi and T. Lu, “Possible direct measurement of the expansion rate of the universe,” preprint (2010), arXiv:1001.3975 [astro-ph.CO] .
* Corasaniti _et al._ (2007) P.-S. Corasaniti, D. Huterer, and A. Melchiorri, “Exploring the dark energy redshift desert with the Sandage-Loeb test,” Phys. Rev. D 75, 062001 (2007), arXiv:astro-ph/0701433 .
* Zhang _et al._ (2010) J. Zhang, L. Zhang, and X. Zhang, “Sandage-Loeb test for the new agegraphic and Ricci dark energy models,” Phys. Lett. B 691, 11–17 (2010), arXiv:1006.1738 [astro-ph.CO] .
* Quercellini _et al._ (2010) C. Quercellini, L. Amendola, A. Balbi, P. Cabella, and M. Quartin, “Real-time cosmology,” Phys. Rep., submitted (2010), arXiv:1011.2646 [astro-ph.CO] .
* Seo and Eisenstein (2003) H.-J. Seo and D. J. Eisenstein, “Probing dark energy with baryonic acoustic oscillations from future large galaxy redshift surveys,” ApJ 598, 720–740 (2003), arXiv:astro-ph/0307460 .
* Bernstein (2006) G. Bernstein, “Metric tests for curvature from weak lensing and baryon acoustic oscillations,” ApJ 637, 598–607 (2006), arXiv:astro-ph/0503276 .
* Weinberg (2008) S. Weinberg, _Cosmology_ (Oxford Univ. Press Inc., New York, 2008).
* Weinberg (1972) S. Weinberg, _Gravitation and cosmology: Principles and applications of the General Theory of Relativity_ (John Wiley & Sons, Inc., New York, 1972) Chapter 14 of the book is dedicated to the topic of cosmography – the geometrical description and measurement of the universe.
* Hogg (1999) D. W. Hogg, “Distance measures in cosmology,” preprint (1999), arXiv:astro-ph/9905116 .
* Alcock and Paczyński (1979) C. Alcock and B. Paczyński, “An evolution free test for non-zero cosmological constant,” Nature 281, 358–359 (1979).
* Matsubara and Suto (1996) T. Matsubara and Y. Suto, “Cosmological redshift distortion of correlation functions as a probe of the density parameter and the cosmological constant,” ApJ 470, L1–L5 (1996), arXiv:astro-ph/9604142 .
* Ballinger _et al._ (1996) W. E. Ballinger, J. A. Peacock, and A. F. Heavens, “Measuring the cosmological constant with redshift surveys,” MNRAS 282, 877–888 (1996), arXiv:astro-ph/9605017 .
* Matsubara and Szalay (2003) T. Matsubara and A. S. Szalay, “Apparent clustering of intermediate-redshift galaxies as a probe of dark energy,” Phys. Rev. Lett. 90, 021302 (2003), arXiv:astro-ph/0208087 .
* Matsubara (2004) T. Matsubara, “Correlation function in deep redshift space as a cosmological probe,” ApJ 615, 573–585 (2004), arXiv:astro-ph/0408349 .
* Percival _et al._ (2007) W. J. Percival, S. Cole, D. J. Eisenstein, R. C. Nichol, J. A. Peacock, A. C. Pope, and A. S. Szalay, “Measuring the baryon acoustic oscillation scale using the Sloan Digital Sky Survey and 2dF Galaxy Redshift Survey,” MNRAS 381, 1053–1066 (2007), arXiv:0705.3323 .
* Padmanabhan and White (2008) N. Padmanabhan and M. White, “Constraining anisotropic baryon oscillations,” Phys. Rev. D 77, 123540 (2008), arXiv:0804.0799 .
* Note (1) http://www.sdss.org/.
* Jimenez and Loeb (2002) R. Jimenez and A. Loeb, “Constraining cosmological parameters based on relative galaxy ages,” ApJ 573, 37–42 (2002), arXiv:astro-ph/0106145 .
* Jimenez _et al._ (2003) R. Jimenez, L. Verde, T. Treu, and D. Stern, “Constraints on the equation of state of dark energy and the Hubble constant from stellar ages and the cosmic microwave background,” ApJ 593, 622–629 (2003), arXiv:astro-ph/0302560 .
* McCarthy _et al._ (2004) P. J. McCarthy, D. Le Borgne, D. Crampton, H.-W. Chen, R. G. Abraham, K. Glazebrook, S. Savaglio, R. G. Carlberg, R. O. Marzke, K. Roth, I. Jørgensen, I. Hook, R. Murowinski, and S. Juneau, “Evolved galaxies at $z>1.5$ from the Gemini Deep Deep Survey: The formation epoch of massive stellar systems,” ApJ 614, L9–L12 (2004), arXiv:astro-ph/0408367 .
* Stern _et al._ (2010a) D. Stern, R. Jimenez, L. Verde, M. Kamionkowski, and S. A. Stanford, “Cosmic chronometers: constraining the equation of state of dark energy. I: ${H}(z)$ measurements,” J. Cosmology Astropart. Phys 2, 8 (2010a), arXiv:0907.3149 [astro-ph.CO] .
* Stern _et al._ (2010b) D. Stern, R. Jimenez, L. Verde, S. A. Stanford, and M. Kamionkowski, “Cosmic chronometers: Constraining the equation of state of dark energy. II: A spectroscopic catalog of red galaxies in galaxy clusters,” ApJS 188, 280–289 (2010b), arXiv:0907.3152 [astro-ph.CO] .
* Simon _et al._ (2005) J. Simon, L. Verde, and R. Jimenez, “Constraints on the redshift dependence of the dark energy potential,” Phys. Rev. D 71, 123001 (2005), arXiv:astro-ph/0412269 .
* Carson and Nichol (2010) D. P. Carson and R. C. Nichol, “The age-redshift relation for luminous red galaxies in the Sloan Digital Sky Survey,” MNRAS 408, 213–233 (2010), arXiv:1006.2830 [astro-ph.CO] .
* Stockton _et al._ (1995) A. Stockton, M. Kellogg, and S. E. Ridgway, “The nature of the stellar continuum in the radio galaxy 3C 65,” ApJ 443, L69–L72 (1995).
* Dunlop _et al._ (1996) J. Dunlop, J. Peacock, H. Spinrad, A. Dey, R. Jimenez, D. Stern, and R. Windhorst, “A 3.5-Gyr-old galaxy at redshift 1.55,” Nature 381, 581–584 (1996).
* Spinrad _et al._ (1997) H. Spinrad, A. Dey, D. Stern, J. Dunlop, J. Peacock, R. Jimenez, and R. Windhorst, “LBDS 53W091: an old, red galaxy at $z=1.552$,” ApJ 484, 581–601 (1997), arXiv:astro-ph/9702233 .
* Alcaniz and Lima (2001) J. S. Alcaniz and J. A. S. Lima, “Dark energy and the epoch of galaxy formation,” ApJ 550, L133–L136 (2001), arXiv:astro-ph/0101544 .
* Jimenez _et al._ (2004) R. Jimenez, J. MacDonald, J. S. Dunlop, P. Padoan, and J. A. Peacock, “Synthetic stellar populations: single stellar populations, stellar interior models and primordial protogalaxies,” MNRAS 349, 240–254 (2004), arXiv:astro-ph/0402271 .
* Jimenez _et al._ (1999) R. Jimenez, A. C. S. Friaca, J. S. Dunlop, R. J. Terlevich, J. A. Peacock, and L. A. Nolan, “Premature dismissal of high-redshift elliptical galaxies,” MNRAS 305, L16–L20 (1999), arXiv:astro-ph/9812222 .
* Crawford _et al._ (2010) S. M. Crawford, A. L. Ratsimbazafy, C. M. Cress, E. A. Olivier, S.-L. Blyth, and K. J. van der Heyden, “Luminous red galaxies in simulations: cosmic chronometers?” MNRAS 406, 2569–2577 (2010), arXiv:1004.2378 [astro-ph.CO] .
* Peebles (1973) P. J. E. Peebles, “Statistical analysis of catalogs of extragalactic objects. I. Theory,” ApJ 185, 413–440 (1973), The term covariance function used in this article is also known as the correlation function which occurs more frequently in the contemporary literature.
* Xu _et al._ (2010) X. Xu, M. White, N. Padmanabhan, D. J. Eisenstein, J. Eckel, K. Mehta, M. Metchnik, P. Pinto, and H.-J. Seo, “A new statistic for analyzing baryon acoustic oscillations,” ApJ 718, 1224–1234 (2010), arXiv:1001.2324 [astro-ph.CO] .
* Feldman _et al._ (1994) H. A. Feldman, N. Kaiser, and J. A. Peacock, “Power-spectrum analysis of three-dimensional redshift surveys,” ApJ 426, 23–37 (1994), arXiv:astro-ph/9304022 .
* Percival _et al._ (2004) W. J. Percival, L. Verde, and J. A. Peacock, “Fourier analysis of luminosity-dependent galaxy clustering,” MNRAS 347, 645–653 (2004), arXiv:astro-ph/0306511 .
* Landy and Szalay (1993) S. D. Landy and A. S. Szalay, “Bias and variance of angular correlation functions,” ApJ 412, 64–71 (1993).
* Peebles and Yu (1970) P. J. E. Peebles and J. T. Yu, “Primeval adiabatic perturbation in an expanding universe,” ApJ 162, 815–836 (1970).
* Hu and Sugiyama (1995) W. Hu and N. Sugiyama, “Anisotropies in the cosmic microwave background: an analytic approach,” ApJ 444, 489–506 (1995), arXiv:astro-ph/9407093 .
* Bashinsky and Bertschinger (2002) S. Bashinsky and E. Bertschinger, “Dynamics of cosmological perturbations in position space,” Phys. Rev. D 65, 123008 (2002), arXiv:astro-ph/0202215 .
* Hu and Dodelson (2002) W. Hu and S. Dodelson, “Cosmic microwave background anisotropies,” ARA&A 40, 171–216 (2002), arXiv:astro-ph/0110414 .
* Hinshaw _et al._ (2003) G. Hinshaw, D. N. Spergel, L. Verde, R. S. Hill, S. S. Meyer, C. Barnes, C. L. Bennett, M. Halpern, N. Jarosik, A. Kogut, E. Komatsu, M. Limon, L. Page, G. S. Tucker, J. L. Weiland, E. Wollack, and E. L. Wright, “First-year Wilkinson Microwave Anisotropy Probe (WMAP) observations: The angular power spectrum,” ApJS 148, 135–159 (2003), arXiv:astro-ph/0302217 .
* Page _et al._ (2003) L. Page, M. R. Nolta, C. Barnes, C. L. Bennett, M. Halpern, G. Hinshaw, N. Jarosik, A. Kogut, M. Limon, S. S. Meyer, H. V. Peiris, D. N. Spergel, G. S. Tucker, E. Wollack, and E. L. Wright, “First-year Wilkinson Microwave Anisotropy Probe (WMAP) observations: Interpretation of the TT and TE angular power spectrum peaks,” ApJS 148, 233–241 (2003), arXiv:astro-ph/0302220 .
* Hinshaw _et al._ (2007) G. Hinshaw, M. R. Nolta, C. L. Bennett, R. Bean, O. Doré, M. R. Greason, M. Halpern, R. S. Hill, N. Jarosik, A. Kogut, E. Komatsu, M. Limon, N. Odegard, S. S. Meyer, L. Page, H. V. Peiris, D. N. Spergel, G. S. Tucker, L. Verde, J. L. Weiland, E. Wollack, and E. L. Wright, “Three-year Wilkinson Microwave Anisotropy Probe (WMAP) observations: Temperature analysis,” ApJS 170, 288–334 (2007), arXiv:astro-ph/0603451 .
* Nolta _et al._ (2009) M. R. Nolta, J. Dunkley, R. S. Hill, G. Hinshaw, E. Komatsu, D. Larson, L. Page, D. N. Spergel, C. L. Bennett, B. Gold, N. Jarosik, N. Odegard, J. L. Weiland, E. Wollack, M. Halpern, A. Kogut, M. Limon, S. S. Meyer, G. S. Tucker, and E. L. Wright, “Five-year Wilkinson Microwave Anisotropy Probe observations: Angular power spectra,” ApJS 180, 296–305 (2009), arXiv:0803.0593 .
* Larson _et al._ (2010) D. Larson, J. Dunkley, G. Hinshaw, E. Komatsu, M. R. Nolta, C. L. Bennett, B. Gold, M. Halpern, R. S. Hill, N. Jarosik, A. Kogut, M. Limon, S. S. Meyer, N. Odegard, L. Page, K. M. Smith, D. N. Spergel, G. S. Tucker, J. L. Weiland, E. Wollack, and E. L. Wright, “Seven-year Wilkinson Microwave Anisotropy Probe (WMAP) observations: Power spectra and WMAP-derived parameters,” ApJS, in press (2010), arXiv:1001.4635 [astro-ph.CO] .
* Goldberg and Strauss (1998) D. M. Goldberg and M. A. Strauss, “Determination of the baryon density from large-scale galaxy redshift surveys,” ApJ 495, 29–43 (1998), arXiv:astro-ph/9707209 .
* Meiksin _et al._ (1999) A. Meiksin, M. White, and J. A. Peacock, “Baryonic signatures in large-scale structure,” MNRAS 304, 851–864 (1999), arXiv:astro-ph/9812214 .
* Eisenstein _et al._ (2005) D. J. Eisenstein, I. Zehavi, D. W. Hogg, R. Scoccimarro, M. R. Blanton, R. C. Nichol, R. Scranton, H.-J. Seo, M. Tegmark, Z. Zheng, S. F. Anderson, J. Annis, N. Bahcall, J. Brinkmann, S. Burles, F. J. Castander, A. Connolly, I. Csabai, M. Doi, M. Fukugita, J. A. Frieman, K. Glazebrook, J. E. Gunn, J. S. Hendry, G. Hennessy, Z. Ivezić, S. Kent, G. R. Knapp, H. Lin, Y.-S. Loh, R. H. Lupton, B. Margon, T. A. McKay, A. Meiksin, J. A. Munn, A. Pope, M. W. Richmond, D. Schlegel, D. P. Schneider, K. Shimasaku, C. Stoughton, M. A. Strauss, M. SubbaRao, A. S. Szalay, I. Szapudi, D. L. Tucker, B. Yanny, and D. G. York, “Detection of the baryon acoustic peak in the large-scale correlation function of SDSS luminous red galaxies,” ApJ 633, 560–574 (2005), arXiv:astro-ph/0501171 .
* Blake and Glazebrook (2003) C. Blake and K. Glazebrook, “Probing dark energy using baryonic oscillations in the galaxy power spectrum as a cosmological ruler,” ApJ 594, 665–673 (2003), arXiv:astro-ph/0301632 .
* Seo and Eisenstein (2005) H.-J. Seo and D. J. Eisenstein, “Baryonic acoustic oscillations in simulated galaxy redshift surveys,” ApJ 633, 575–588 (2005), arXiv:astro-ph/0507338 .
* Seo and Eisenstein (2007) H.-J. Seo and D. J. Eisenstein, “Improved forecasts for the baryon acoustic oscillations and cosmological distance scale,” ApJ 665, 14–24 (2007), arXiv:astro-ph/0701079 .
* Davis and Peebles (1983) M. Davis and P. J. E. Peebles, “A survey of galaxy redshifts. V - The two-point position and velocity correlations,” ApJ 267, 465–482 (1983).
* Kaiser (1987) N. Kaiser, “Clustering in real space and in redshift space,” MNRAS 227, 1–21 (1987).
* Magira _et al._ (2000) H. Magira, Y. P. Jing, and Y. Suto, “Cosmological redshift-space distortion on clustering of high-redshift objects: Correction for nonlinear effects in the power spectrum and tests with $N$-body simulations,” ApJ 528, 30–50 (2000), arXiv:astro-ph/9907438 .
* Kaiser (1984) N. Kaiser, “On the spatial correlations of Abell clusters,” ApJ 284, L9–L12 (1984).
* Hui _et al._ (2007) L. Hui, E. Gaztañaga, and M. Loverde, “Anisotropic magnification distortion of the 3D galaxy correlation. I. Real space,” Phys. Rev. D 76, 103502 (2007), arXiv:0706.1071 .
* Hui _et al._ (2008) L. Hui, E. Gaztañaga, and M. Loverde, “Anisotropic magnification distortion of the 3D galaxy correlation. II. Fourier and redshift space,” Phys. Rev. D 77, 063526 (2008), arXiv:0710.4191 .
* Okumura _et al._ (2008) T. Okumura, T. Matsubara, D. J. Eisenstein, I. Kayo, C. Hikage, A. S. Szalay, and D. P. Schneider, “Large-scale anisotropic correlation function of SDSS luminous red galaxies,” ApJ 676, 889–898 (2008), arXiv:0711.3640 .
* Gaztañaga _et al._ (2009a) E. Gaztañaga, A. Cabré, and L. Hui, “Clustering of luminous red galaxies - IV. Baryon acoustic peak in the line-of-sight direction and a direct measurement of $H(z)$,” MNRAS 399, 1663–1680 (2009a), arXiv:0807.3551 .
* Komatsu _et al._ (2009) E. Komatsu, J. Dunkley, M. R. Nolta, C. L. Bennett, B. Gold, G. Hinshaw, N. Jarosik, D. Larson, M. Limon, L. Page, D. N. Spergel, M. Halpern, R. S. Hill, A. Kogut, S. S. Meyer, G. S. Tucker, J. L. Weiland, E. Wollack, and E. L. Wright, “Five-year Wilkinson Microwave Anisotropy Probe Observations: Cosmological interpretation,” ApJS 180, 330–376 (2009), arXiv:0803.0547 .
* Note (2) Another way to present the measurement results for use in cosmological parameter constraint $\Delta z_{\mathrm{BAO}}=r_{\mathrm{BAO}}H(z)/c$. Schematically, this is done by approximating the derivative in equation (26) with a ratio of differences, and identifying the interval $a(t_{0})\Delta\chi$ with the measured comoving BAO scale. In Section IV we briefly discuss its usage.
* Miralda-Escudé (2009) J. Miralda-Escudé, “Comment on the claimed radial BAO detection by Gaztañaga et al.” preprint (2009), arXiv:0901.1219 [astro-ph.CO] .
* Kazin _et al._ (2010) E. A. Kazin, M. R. Blanton, R. Scoccimarro, C. K. McBride, and A. A. Berlind, “Regarding the line-of-sight baryonic acoustic feature in the Sloan Digital Sky Survey and Baryon Oscillation Spectroscopic Survey luminous red galaxy samples,” ApJ 719, 1032–1044 (2010), arXiv:1004.2244 [astro-ph.CO] .
* Liddle (2009) A. R. Liddle, “Statistical methods for cosmological parameter selection and estimation,” Annu. Rev. Nucl. Part. Sci. 59, 95–114 (2009), arXiv:0903.4210 [hep-th] .
* Tian _et al._ (2010) H. J. Tian, M. C. Neyrinck, T. Budavári, and A. S. Szalay, “Redshift-space enhancement of line-of-sight baryon acoustic oscillations in the SDSS main-galaxy sample,” ApJ, submitted (2010), arXiv:1011.2481 [astro-ph.CO] .
* Cabré and Gaztañaga (2010) A. Cabré and E. Gaztañaga, “Have Baryonic Acoustic Oscillations in the galaxy distribution really been measured?” MNRAS, submitted (2010), arXiv:1011.2729 [astro-ph.CO] .
* Figueroa _et al._ (2008) D. G. Figueroa, L. Verde, and R. Jimenez, “Improved cosmological parameter constraints from CMB and $H(z)$ data,” J. Cosmology Astropart. Phys 10, 38 (2008), arXiv:0807.0039 .
* Reid _et al._ (2010) B. A. Reid, L. Verde, R. Jimenez, and O. Mena, “Robust neutrino constraints by combining low redshift observations with the CMB,” J. Cosmology Astropart. Phys 1, 3 (2010), arXiv:0910.0008 [astro-ph.CO] .
* Izotov _et al._ (2007) Y. I. Izotov, T. X. Thuan, and G. Stasińska, “The primordial abundance of 4He: A self-consistent empirical analysis of systematic effects in a large sample of low-metallicity H II regions,” ApJ 662, 15–38 (2007), arXiv:astro-ph/0702072 .
* Iocco _et al._ (2009) F. Iocco, G. Mangano, G. Miele, O. Pisanti, and P. D. Serpico, “Primordial nucleosynthesis: From precision cosmology to fundamental physics,” Phys. Rep. 472, 1–76 (2009), arXiv:0809.0631 .
* Clarkson _et al._ (2007) C. Clarkson, M. Cortês, and B. Bassett, “Dynamical dark energy or simply cosmic curvature?” J. Cosmology Astropart. Phys 8, 11 (2007), arXiv:astro-ph/0702670 .
* Vardanyan _et al._ (2009) M. Vardanyan, R. Trotta, and J. Silk, “How flat can you get? a model comparison perspective on the curvature of the Universe,” MNRAS 397, 431–444 (2009), arXiv:0901.3354 [astro-ph.CO] .
* Zhan _et al._ (2009) H. Zhan, L. Knox, and J. A. Tyson, “Distance, growth factor, and dark energy constraints from photometric baryon acoustic oscillation and weak lensing measurements,” ApJ 690, 923–936 (2009), arXiv:0806.0937 .
* Gaztañaga _et al._ (2009b) E. Gaztañaga, R. Miquel, and E. Sánchez, “First cosmological constraints on dark energy from the radial baryon acoustic scale,” Phys. Rev. Lett. 103, 091302 (2009b), arXiv:0808.1921 .
* Dunkley _et al._ (2009) J. Dunkley, E. Komatsu, M. R. Nolta, D. N. Spergel, D. Larson, G. Hinshaw, L. Page, C. L. Bennett, B. Gold, N. Jarosik, J. L. Weiland, M. Halpern, R. S. Hill, A. Kogut, M. Limon, S. S. Meyer, G. S. Tucker, E. Wollack, and E. L. Wright, “Five-year Wilkinson Microwave Anisotropy Probe observations: Likelihoods and parameters from the WMAP data,” ApJS 180, 306–329 (2009), arXiv:0803.0586 .
* Riess _et al._ (2009) A. G. Riess, L. Macri, S. Casertano, M. Sosey, H. Lampeitl, H. C. Ferguson, A. V. Filippenko, S. W. Jha, W. Li, R. Chornock, and D. Sarkar, “A redetermination of the Hubble constant with the Hubble Space Telescope from a differential distance ladder,” ApJ 699, 539–563 (2009), arXiv:0905.0695 .
* Yi and Zhang (2007) Z.-L. Yi and T.-J. Zhang, “Constraints on holographic dark energy models using the differential ages of passively evolving galaxies,” Mod. Phys. Lett. A 22, 41–53 (2007), arXiv:astro-ph/0605596 .
* Samushia and Ratra (2006) L. Samushia and B. Ratra, “Cosmological constraints from Hubble parameter versus redshift data,” ApJ 650, L5–L8 (2006), arXiv:astro-ph/0607301 .
* Carvalho _et al._ (2008) F. C. Carvalho, E. M. Santos, J. S. Alcaniz, and J. Santos, “Cosmological constraints from the Hubble parameter on $f(R)$ cosmologies,” J. Cosmology Astropart. Phys 9, 8 (2008), arXiv:0804.2878 .
* Xu and Wang (2010) L. Xu and Y. Wang, “Observational constraints to Ricci dark energy model by using: SN, BAO, OHD, fgas data sets,” J. Cosmology Astropart. Phys 6, 2 (2010), arXiv:1006.0296 [astro-ph.CO] .
* Durán _et al._ (2010) I. Durán, D. Pavón, and W. Zimdahl, “Observational constraints on a holographic, interacting dark energy model,” J. Cosmology Astropart. Phys 7, 18 (2010), arXiv:1007.0390 [astro-ph.CO] .
* Sahni _et al._ (2008) V. Sahni, A. Shafieloo, and A. A. Starobinsky, “Two new diagnostics of dark energy,” Phys. Rev. D 78, 103502 (2008), arXiv:0807.3548 .
* Zunckel and Clarkson (2008) C. Zunckel and C. Clarkson, “Consistency tests for the cosmological constant,” Phys. Rev. Lett. 101, 181301 (2008), arXiv:0807.4304 .
* Clarkson _et al._ (2008) C. Clarkson, B. Bassett, and T. H.-C. Lu, “A general test of the Copernican Principle,” Phys. Rev. Lett. 101, 011301 (2008), arXiv:0712.3457 .
* Shafieloo and Clarkson (2010) A. Shafieloo and C. Clarkson, “Model independent tests of the standard cosmological model,” Phys. Rev. D 81, 083537 (2010), arXiv:0911.4858 [astro-ph.CO] .
* Pan _et al._ (2010) N. Pan, Y. Gong, Y. Chen, and Z.-H. Zhu, “Improved cosmological constraints on the curvature and equation of state of dark energy,” Class. Quantum Grav. 27, 155015 (2010), arXiv:1005.4249 [astro-ph.CO] .
* Zhai _et al._ (2010) Z.-X. Zhai, H.-Y. Wan, and T.-J. Zhang, “Cosmological constraints from radial baryon acoustic oscillation measurements and observational Hubble data,” Phys. Lett. B 689, 8–13 (2010), arXiv:1004.2599 [astro-ph.CO] .
* Verde (2010) L. Verde, “Statistical methods in cosmology,” in _Lectures on Cosmology_, Lecture Notes in Physics, Vol. 800, edited by G. Wolschin (Springer, Berlin, 2010) pp. 147–177, arXiv:0911.3105 [astro-ph.CO] .
* Riess _et al._ (2004) A. G. Riess, L.-G. Strolger, J. Tonry, S. Casertano, H. C. Ferguson, B. Mobasher, P. Challis, A. V. Filippenko, S. Jha, W. Li, R. Chornock, R. P. Kirshner, B. Leibundgut, M. Dickinson, M. Livio, M. Giavalisco, C. C. Steidel, T. Benítez, and Z. Tsvetanov, “Type Ia supernova discoveries at $z>1$ from the Hubble Space Telescope: Evidence for past deceleration and constraints on dark energy evolution,” ApJ 607, 665–687 (2004), arXiv:astro-ph/0402512 .
* Ma and Zhang (2010) C. Ma and T.-J. Zhang, “Power of observational Hubble parameter data: a figure of merit exploration,” preprint (2010), arXiv:1007.3787 [astro-ph.CO] .
* Note (3) http://www.physics.princeton.edu/act/index.html.
* Benítez _et al._ (2009) N. Benítez, E. Gaztañaga, R. Miquel, F. Castander, M. Moles, M. Crocce, A. Fernández-Soto, P. Fosalba, F. Ballesteros, J. Campa, L. Cardiel-Sas, J. Castilla, D. Cristóbal-Hornillos, M. Delfino, E. Fernández, C. Fernández-Sopuerta, J. García-Bellido, J. A. Lobo, V. J. Martínez, A. Ortiz, A. Pacheco, S. Paredes, M. J. Pons-Bordería, E. Sánchez, S. F. Sánchez, J. Varela, and J. F. de Vicente, “Measuring baryon acoustic oscillations along the line of sight with photometric redshifts: The PAU survey,” ApJ 691, 241–260 (2009), arXiv:0807.0535 .
* Arnalte-Mur _et al._ (2009) P. Arnalte-Mur, A. Fernández-Soto, V. J. Martínez, E. Saar, P. Heinämäki, and I. Suhhonenko, “Recovering the real-space correlation function from photometric redshift surveys,” MNRAS 394, 1631–1639 (2009), arXiv:0812.4226 .
* Roig _et al._ (2009) D. Roig, L. Verde, J. Miralda-Escudé, R. Jimenez, and C. Peña-Garay, “Photo-$z$ optimization for measurements of the BAO radial scale,” J. Cosmology Astropart. Phys 4, 8 (2009), arXiv:0812.3414 .
* Note (4) http://wigglez.swin.edu.au/.
* Drinkwater _et al._ (2010) M. J. Drinkwater, R. J. Jurek, C. Blake, D. Woods, K. A. Pimbblet, K. Glazebrook, R. Sharp, M. B. Pracy, S. Brough, M. Colless, W. J. Couch, S. M. Croom, T. M. Davis, D. Forbes, K. Forster, D. G. Gilbank, M. Gladders, B. Jelliffe, N. Jones, I.-H. Li, B. Madore, D. C. Martin, G. B. Poole, T. Small, E. Wisnioski, T. Wyder, and H. K. C. Yee, “The WiggleZ Dark Energy Survey: survey design and first data release,” MNRAS 401, 1429–1452 (2010), arXiv:0911.4246 [astro-ph.CO] .
* Note (5) http://www.sdss3.org/.
* McDonald and Eisenstein (2007) P. McDonald and D. J. Eisenstein, “Dark energy and curvature from a future baryonic acoustic oscillation survey using the Lyman-$\alpha$ forest,” Phys. Rev. D 76, 063009 (2007), arXiv:astro-ph/0607122 .
* Norman _et al._ (2009) M. L. Norman, P. Paschos, and R. Harkness, “Baryon acoustic oscillations in the Lyman alpha forest,” Journal of Physics Conference Series 180, 012021 (2009), arXiv:0908.0964 [astro-ph.CO] .
* White _et al._ (2010) M. White, A. Pope, J. Carlson, K. Heitmann, S. Habib, P. Fasel, D. Daniel, and Z. Lukic, “Particle mesh simulations of the Ly$\alpha$ forest and the signature of baryon acoustic oscillations in the intergalactic medium,” ApJ 713, 383–393 (2010), arXiv:0911.5341 [astro-ph.CO] .
* Cimatti _et al._ (2009) A. Cimatti, M. Robberto, C. Baugh, S. V. W. Beckwith, R. Content, E. Daddi, G. De Lucia, B. Garilli, L. Guzzo, G. Kauffmann, M. Lehnert, D. Maccagni, A. Martínez-Sansigre, F. Pasian, I. N. Reid, P. Rosati, R. Salvaterra, M. Stiavelli, Y. Wang, M. Zapatero Osorio, M. Balcells, M. Bersanelli, F. Bertoldi, J. Blaizot, D. Bottini, R. Bower, A. Bulgarelli, A. Burgasser, C. Burigana, R. C. Butler, S. Casertano, B. Ciardi, M. Cirasuolo, M. Clampin, S. Cole, A. Comastri, S. Cristiani, J.-G. Cuby, F. Cuttaia, A. de Rosa, A. D. Sanchez, M. di Capua, J. Dunlop, X. Fan, A. Ferrara, F. Finelli, A. Franceschini, M. Franx, P. Franzetti, C. Frenk, J. P. Gardner, F. Gianotti, R. Grange, C. Gruppioni, A. Gruppuso, F. Hammer, L. Hillenbrand, A. Jacobsen, M. Jarvis, R. Kennicutt, R. Kimble, M. Kriek, J. Kurk, J.-P. Kneib, O. Le Fevre, D. Macchetto, J. MacKenty, P. Madau, M. Magliocchetti, D. Maino, N. Mandolesi, N. Masetti, R. McLure, A. Mennella, M. Meyer, M. Mignoli, B. Mobasher, E. Molinari, G. Morgante, S. Morris, L. Nicastro, E. Oliva, P. Padovani, E. Palazzi, F. Paresce, A. Perez Garrido, E. Pian, L. Popa, M. Postman, L. Pozzetti, J. Rayner, R. Rebolo, A. Renzini, H. Röttgering, E. Schinnerer, M. Scodeggio, M. Saisse, T. Shanks, A. Shapley, R. Sharples, H. Shea, J. Silk, I. Smail, P. Spanó, J. Steinacker, L. Stringhetti, A. Szalay, L. Tresse, M. Trifoglio, M. Urry, L. Valenziano, F. Villa, I. Villo Perez, F. Walter, M. Ward, R. White, S. White, E. Wright, R. Wyse, G. Zamorani, A. Zacchei, W. W. Zeilinger, and F. Zerbi, “SPACE: the spectroscopic all-sky cosmic explorer,” Exp. Astron. 23, 39–66 (2009), arXiv:0804.4433 .
* Laureijs (2009) R. Laureijs, “Euclid assessment study report for the ESA Cosmic Visions,” preprint (2009), arXiv:0912.0914 [astro-ph.CO] .
* Wang _et al._ (2010) Y. Wang, W. Percival, A. Cimatti, P. Mukherjee, L. Guzzo, C. M. Baugh, C. Carbone, P. Franzetti, B. Garilli, J. E. Geach, C. G. Lacey, E. Majerotto, A. Orsi, P. Rosati, L. Samushia, and G. Zamorani, “Designing a space-based galaxy redshift survey to probe dark energy,” MNRAS 409, 737–749 (2010), arXiv:1006.3517 [astro-ph.CO] .
* Barkana and Loeb (2005) R. Barkana and A. Loeb, “Probing the epoch of early baryonic infall through 21-cm fluctuations,” MNRAS 363, L36–L40 (2005), arXiv:astro-ph/0502083 .
* Mao and Wu (2008) X.-C. Mao and X.-P. Wu, “Signatures of the baryon acoustic oscillations on 21 cm emission background,” ApJ 673, L107–L110 (2008), arXiv:0709.3871 .
* Seo _et al._ (2010) H.-J. Seo, S. Dodelson, J. Marriner, D. Mcginnis, A. Stebbins, C. Stoughton, and A. Vallinotto, “A ground-based 21 cm baryon acoustic oscillation survey,” ApJ 721, 164–173 (2010), arXiv:0910.5007 [astro-ph.CO] .
* Shoji _et al._ (2009) M. Shoji, D. Jeong, and E. Komatsu, “Extracting angular diameter distance and expansion rate of the universe from two-dimensional galaxy power spectrum at high redshifts: Baryon acoustic oscillation fitting versus full modeling,” ApJ 693, 1404–1416 (2009), arXiv:0805.4238 .
|
arxiv-papers
| 2010-10-07T00:37:12 |
2024-09-04T02:49:13.560452
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Tong-Jie Zhang (1 and 2), Cong Ma (1), Tian Lan (1) ((1) Department of\n Astronomy, Beijing Normal University, (2) Center for High Energy Physics,\n Peking University)",
"submitter": "Cong Ma",
"url": "https://arxiv.org/abs/1010.1307"
}
|
1010.1526
|
∎
11institutetext: Zoltán Prekopcsák 22institutetext: Budapest University of
Technology and Economics, Hungary
Magyar tudósok körútja 2, Budapest, H-1117 Hungary
22email: prekopcsak@tmit.bme.hu, Phone: +36 1 4633119, Fax: +36 1 4633107
33institutetext: Daniel Lemire 44institutetext: LICEF, Université du Québec à
Montréal (UQAM)
100 Sherbrooke West, Montreal, QC, H2X 3P2 Canada
# Time Series Classification by Class-Based Mahalanobis Distances
Zoltán Prekopcsák Daniel Lemire
(Received: date / Accepted: date)
###### Abstract
To classify time series by nearest neighbors, we need to specify or learn one
or several distances. We consider variations of the Mahalanobis distances
which rely on the inverse covariance matrix of the data. Unfortunately – for
time series data – the covariance matrix has often low rank. To alleviate this
problem we can either use a pseudoinverse, covariance shrinking or limit the
matrix to its diagonal. We review these alternatives and benchmark them
against competitive methods such as the related Large Margin Nearest Neighbor
Classification (LMNN) and the Dynamic Time Warping (DTW) distance. As we
expected, we find that the DTW is superior, but the Mahalanobis distances are
computationally inexpensive in comparison. To get best results with
Mahalanobis distances, we recommend learning one distance per class using
either covariance shrinking or the diagonal approach.
###### Keywords:
Time-series classification Distance learning Nearest Neighbor Mahalanobis
distance
###### MSC:
62-07 62H30
††journal: Advances in Data Analysis and Classification
## 1 Introduction
Time series are sequences of values measured over time. Examples include
financial data, such as stock prices, or medical data, such as blood sugar
levels. Classifying time series is an important class of problems which is
applicable to music classification (Weihs et al, 2007), medical diagnostic
(Sternickel, 2002) or bioinformatics (Legrand et al, 2008).
Nearest Neighbor (NN) methods classify time series efficiently and accurately
(Ding et al, 2008). The 1-NN method is especially simple: we merely have to
find the nearest labeled instance.
We need to specify a distance: the Euclidean and Dynamic Time Warping (Sakoe
and Chiba, 1978a) distances are popular choices. However, we can also learn a
distance based on some training data (Yang and Jin, 2006; Weinberger and Saul,
2009). Given the training data set made of classes of time series instances,
we can either learn a single (global) distance function, or learn one distance
function per class (Csatári and Prekopcsák, 2010; Paredes and Vidal, 2000,
2006). That is, to compute the distance between a test element and an instance
of class $j$, we use a distance function specific to class $j$.
Because the Euclidean distance is popular for NN classification, it is
tempting to consider generalized ellipsoid distances (Ishikawa et al, 1998),
that is, distances of the form
$\displaystyle D(x,y)=(x-y)^{\top}M(x-y).$
When $M$ is a positive semi-definite matrix, the square root of this distance
is a pseudometric: it is symmetric, non-negative and it satisfies the triangle
inequality ($\sqrt{D(x,y)}+\sqrt{D(y,z)}\leq\sqrt{D(x,z)}$). Further, when $M$
is a positive definite matrix, then the square root becomes a metric because
$D(x,x)=0\Rightarrow x=0$. When the matrix $M$ is the identity matrix, we
recover the (squared) Euclidean distance. We get the Mahalanobis distance when
solving for the matrix $M$ minimizing the sum of distances $\sum_{x,y}D(x,y)$
(see § 3). We can require $M$ to be diagonal, thus defining the diagonal
Mahalanobis distance (Paredes and Vidal, 2006; Ishikawa et al, 1998). In the
more general case, solving for $M$ can be difficult, as it often involves
inverting a low-rank matrix. Perhaps partly due to these mathematical
difficulties, there has been no attempt to use the general Mahalanobis
distance to classify time series (to our knowledge). Thus, for the first time,
we apply the full-matrix Mahalanobis distance for time series classification.
To solve the mathematical difficulties, we use both an approach based on a
pseudoinverse and on covariance shrinkage. With both the diagonal and the
covariance shrinkage approaches, the square root of the distance $D$ is a
metric (under mild assumptions) whereas our approach based on a pseudoinverse
merely generates a pseudo-metric.
While we find that the full-matrix Mahalanobis distance is not competitive
when relying on a pseudoinverse, we get good results with covariance shrinkage
or using the diagonal Mahalanobis distance. Moreover, we find that the class-
based Mahalanobis distance is preferable to the global Mahalanobis distance.
## 2 Related Works
Several distance functions are used for time series classification, such as
* •
Dynamic Time Warping (DTW) (Ratanamahatana and Keogh, 2004),
* •
DISSIM (Frentzos et al, 2007),
* •
Threshold Queries (Aßfalg et al, 2006),
* •
Edit distances (Chen and Ng, 2004; Chen et al, 2005),
* •
Longest Common Subsequences (LCSS) (Vlachos et al, 2002),
* •
Swale (Morse and Patel, 2007),
* •
SpADe (Chen et al, 2007),
* •
and Cluster, Then Classify (CTC) (Keogh and Pazzani, 1998).
Ding et al (2008) presented an extensive comparison of these distance
functions and concluded that DTW is among the best measures and that the
accuracy of the Euclidean distance converges to DTW as the size of the
training set increases.
In a general Machine Learning setting, Paredes and Vidal (2000, 2006) compared
Euclidean distance with the conventional and class-based Mahalanobis
distances. One of our contribution is to validate these generic results on
time series: instead of tens of features, we have hundreds or even thousands
of values which makes the problem mathematically more challenging: the rank of
our covariance matrices are often tiny compared to their sizes.
More generally, distance metric learning has an extensive literature
(Wettschereck et al, 1997; Hastie and Tibshirani, 1996; Chai et al, 2010;
Short and Fukunaga, 1980). We refer the reader to Weinberger and Saul (2009)
for a review.
A conventional distance-learning approach is to find an optimal generalized
ellipsoid distance with respect to a specific loss function. The LMNN
algorithm proposed by Weinberger and Saul (2009) takes a different approach.
It seeks to force nearest neighbors to belong to the same class and it
separates instances from different classes by a large margin. LMNN can be
formulated as a semi-definite programming problem. They also propose a
modification which they call multiple metrics LMNN as it learns different
distances for each class.
There are many extensions and alternatives to NN classification. For example,
Jahromi et al (2009) use instance weights to improve classification.
Meanwhile, Zhan et al (2009) learn a distance per instance.
### 2.1 Dynamic Time Warping (DTW)
As already stated, The DTW is one of the most accurate distance function for
time-series classification. The DTW was invented to recognize spoken words
(Sakoe and Chiba, 1978b), but it is also used for various problems such as
handwriting recognition (Bahlmann, 2004; Niels and Vuurpijl, 2005), chromosome
classification (Legrand et al, 2007), networking (Ang et al, 2010) or shape
retrieval (Bartolini et al, 2005; Marzal et al, 2006).
For simplicity, consider two time series $x$ and $y$ of equal length ($n$).
Recall that the $l_{p}$ distance between two time series is
$\displaystyle\sqrt[p]{\sum_{i=1}^{n}|x_{i}-y_{i}|^{p}}$
if $p$ is a positive integer or $\max_{i=1,\ldots,n}|x_{i}-y_{i}|$ if
$p=\infty$. The $l_{1}$ distance is also called the Manhattan distance whereas
the $l_{2}$ distance is the Euclidean distance.
The intuition behind the DTW is that one could speak the same sentence by
speeding up and slowing down without changing the meaning of the sounds. The
DTW is a generalization of the $l_{p}$ distance which allows the data to be
realigned. To compute the DTW between $x$ and $y$, you must find a many-to-
many matching between the data points in $x$ and the data points in $y$. That
is each data point from one series must be matched with at least one data
point with the other series. One such matching is the trivial one, which maps
the first data point from $x$ to the first data point in $y$, the second data
point in $x$ to the second data point in $y$, and so on. Write the set matches
$(i,j)$ as $\Gamma$ so that the trivial matching is just
$\Gamma=\\{(1,1),(2,2),\ldots,(n,n)\\}$. The $l_{p}$ cost corresponding to a
matching is defined as
$\displaystyle\sqrt[p]{\sum_{(i,j)\in\Gamma}|x_{i}-y_{j}|^{p}}$
if $p$ is in an integer or $\max_{(i,j)\in\Gamma}|x_{i}-y_{j}|$ if $p=\infty$.
Typically, $p$ is either 1 or 2: for our purposes we assume $p=2$. For a given
$p$, the DTW is defined as the minimal cost over all allowed matchings
$\Gamma$. Typically, we require matchings to be monotonic: if both $(i,j)$ and
$(i+1,j^{\prime})$ are in $\Gamma$ then $j^{\prime}\geq j$, that is, we cannot
warp back in time. Moreover, some matches might be forbidden, maybe because
the data points are too far apart (Itakura, 1975; Sakoe and Chiba, 1978b). Yu
et al (2011) has proposed learning this warping constraint from the data.
Except when $p=\infty$, the DTW fails to satisfy the triangle inequality: the
DTW is not a metric distance.
The computational cost of the DTW is sometimes a challenge (Salvador and Chan,
2007). To alleviate this problem, several strategies have been proposed
including lower bounds and R*-tree indexes (Ratanamahatana and Keogh, 2005;
Lemire, 2009; Ouyang and Zhang, 2010).
Gaudin and Nicoloyannis (2006) proposed a weighted version of the DTW called
Adaptable Time Warping. Instead of computing
$\sum_{(i,j)\in\Gamma}|x_{i}-y_{j}|^{p}$, it computes
$\sum_{(i,j)\in\Gamma}M_{i,j}|x_{i}-y_{j}|^{p}$ where $M$ is some matrix.
Unfortunately, finding the optimal matrix $M$ can be a challenge. Jeong et al
(2010) investigated another form of weighted DTW where you seek the minimize
$\displaystyle\sqrt[p]{\sum_{(i,j)\in\Gamma}w_{|i-j|}|x_{i}-y_{j}|^{p}}$
where $w$ is some weight vector. Many other variations on the DTW distance
have been proposed, e.g., Chouakria and Nagabhushan (2007).
## 3 Mahalanobis distance
For completeness, we derive the Mahalanobis distance (Mahalanobis, 1936) as an
optimal form of generalized ellipsoid distance. We seek $M$ minimizing
$\displaystyle\sum_{x,y\in S}{(x-y)^{\top}M(x-y)}=\sum_{x,y\in
S}{\left(\sum_{k=1}^{n}\sum_{l=1}^{n}{(x_{k}-y_{k})m_{kl}(x_{l}-y_{l})}\right)}$
where $S$ is some class of time series. We add a regularization constraint on
the determinant ($\det(M)=1$).
We solve the minimization problem by the Lagrange’s multiplier method with the
Lagrangian
$\displaystyle L(M,\lambda)$ $\displaystyle=$ $\displaystyle\sum_{x,y\in
S}\left(\sum_{k=1}^{n}\sum_{l=1}^{n}{(x_{k}-y_{k})m_{k,l}(x_{l}-y_{l})}\right)-\lambda(\det
M-1).$
We want to compute the derivative of the Lagrangian, and of $\det(M)$, with
respect to $m_{k,l}$. By Laplace expansion, we have for all $l$ that
$\displaystyle\det(M)=\sum_{k=1}^{n}{(-1)^{k+l}m_{k,l}\det(M_{k,l})}=1$
where $M_{k,l}$ is the $(k,l)$ minor of $M$: an $(n-1)\times(n-1)$ matrix
obtained by deleting $k$-th row and $l$-th column of $M$. Thus, we have
${\partial\det(M)}/{\partial m_{k,l}}=(-1)^{k+l}\det(M_{k,l})$. Setting the
derivatives to zero, we get
$\displaystyle\frac{\partial L(M,\lambda)}{\partial m_{k,l}}=\sum_{x,y\in
S}{(x_{k}-y_{k})(x_{l}-y_{l})}-\lambda(-1)^{k+l}\det(M_{k,l})=0$
and therefore
$\displaystyle\det(M_{k,l})=\frac{\sum_{x,y\in
S}{(x_{k}-y_{k})(x_{l}-y_{l})}}{\lambda(-1)^{k+l}}.$
Because $\det(M)=1$ and using Cramer’s rule, the inverse matrix $M^{-1}$ can
be represented as
$\displaystyle
m_{k,l}^{-1}=\frac{(-1)^{k+l}\det(M_{k,l})}{\det(M)}=(-1)^{k+l}\det(M_{k,l}).$
Hence, we have
$\displaystyle m_{k,l}^{-1}=\frac{\sum_{x,y\in
S}{(x_{k}-y_{k})(x_{l}-y_{l})}}{\lambda}.$
Thus, we have that $M^{-1}\propto C$ where $C$ is the covariance matrix.
Because we require $M$ to be positive definite and to satisfy $\det(M)=1$, we
set $M=(\det(C))^{\frac{1}{n}}C^{-1}$ which produces the Mahalanobis
distance111The original Mahalanobis distance is defined with $M=C^{-1}$.
Our derivation assumes that the covariance matrix is (numerically) invertible.
This fails in practice. In § 4, we review some solutions.
## 4 Computing Mahalanobis distances for time series
As a rule of thumb, the covariance matrix becomes singular when the number of
instances is smaller or about the same as the number of attributes. This is a
common problem with time series: whereas individual time series might have
thousands of samples, there may only be a few labeled time series in each
class.
The most straight-forward solution is to limit the matrix $M$ to its
diagonal–thus producing a weighted Euclidean distance. Revisiting the
derivation of § 3 where we require $m_{k,l}=0$ for $k\neq l$, we get that the
inverse of the Mahalanobis matrix $M$ must be equal to the inverse of the
diagonal of the covariance matrix: $M^{-1}\propto\mathrm{diag}(C)$. As long as
the variance of each attribute in our training sets is different from zero – a
condition satisfied in practice in our experiments, the problem is well posed
and the result is a positive-definite matrix. In such a diagonal case, the
number of parameters to learn grows only linearly with the number of
attributes in the time series. In contrast, the number of elements in the full
covariance matrix grows quadratically. The speed of the computation of the
distances also depends on the number of non-zero elements in the Mahalanobis
matrix $M$.
Alas, the diagonal Mahalanobis distance fails to take into account the
information off the diagonal in the covariance matrix. See Figure 1 for the
covariance matrix of a class of time series. It is clear from the figure that
the covariance matrix has significant values off the diagonal. There are even
block-like patterns in the matrix corresponding to specific time intervals.
(a) Sample of time series (b) Sample covariance
Figure 1: The Cylinder class from the CBF data set and its sample covariance.
Higher absolute values in the matrix are presented using darker colors.
Could it be that non-diagonal Mahalanobis distance could be superior or at
least competitive with the diagonal Mahalanobis distance? It is tempting to
use banded matrices, but the restriction of a positive definite matrix to a
band may fail to be positive definite. Block-diagonal matrices (Matton et al,
2010) can preserve positive definiteness, but learning which blocks to use in
the context of time series might be difficult. Instead, we propose two
approaches: one is based on the widely used Moore-Penrose pseudoinverse, and
the other is covariance shrinkage. See Figure 2 for the three different
covariance estimates of the same class.
(a) Sample covariance (b) Shrinked covariance (c) Diagonal covariance
Figure 2: The covariance estimates of the Funnel class in the CBF data set.
Large absolute values are in darker colors. Both the shrinked and diagonal
covariances are positive definite whereas the sample covariance matrix is
singular.
The approach based on the pseudoinverse is based on the singular value
decomposition (SVD). We write the SVD as $C=U\Sigma V^{*}$ where $\Sigma$ is a
diagonal matrix with eigenvalues $\lambda_{1},\lambda_{2},\ldots$ and $U$ and
$V$ are unitary matrices. The Moore-Penrose pseudoinverse is given by
$V\Sigma^{+}U^{*}$ where $\Sigma^{+}$ is the diagonal matrix made of the
eigenvalues $1/\lambda_{1},1/\lambda_{2},\ldots$ with the convention that
$1/0=0$. The pseudo-determinant is the product of the non-zero eigenvalues of
$\Sigma$. We set $M$ equal to the pseudoinverse of the covariance
matrix—normalized so that it has a pseudo-determinant of one. This solution is
equivalent to projecting the time series data on the subspace corresponding to
the non-zero eigenvalues of $\Sigma$. That is, the matrix $M$ will be
singular.
Covariance shrinkage is an estimation method for problems with small number of
instances and large number of attributes (Stein, 1956). It has better
theoretical and practical properties for such data sets as the estimated
covariance matrix is guaranteed to be non-singular. Let
$x_{1},x_{2},\ldots,x_{n}$ be a set of time series. We write
$x_{i}=(x_{i1},x_{i2},\ldots,x_{ip})$ where $x_{ik}$ is the $k^{\mathrm{th}}$
value (attribute) of the time series $x_{i}$. One element of the (sample)
covariance matrix $S$ is
$\displaystyle
s_{ij}=\frac{1}{n-1}\sum_{k=1}^{n}(x_{ki}-\bar{x}_{i})(x_{kj}-\bar{x}_{j})$
where $\bar{x}_{i}=\frac{1}{n}\sum_{k=1}^{n}x_{ki}$. It is positive semi-
definite but can be singular. To fix the problem that $S$ is singular, we can
replace it with an estimation of the form
$\displaystyle C^{*}=\lambda T+(1-\lambda)S$
for some suitably chosen target $T$. If $T$ is a positive definite matrix and
$\lambda\in(0,1]$, we have that $\lambda T+(1-\lambda)S$ must be positive
definite. Moreover, the smallest eigenvalue of $\lambda T+(1-\lambda)S$ must
be at least as large as $\lambda$ times the smallest eigenvalue of $T$. We
have used the target recommended by Schäfer and Strimmer (2005) which is the
diagonal of the unrestricted covariance estimate, $T=\mathrm{diag}(C)$. It is
positive definite in our examples. For $\lambda$, we use the estimation
proposed by Schäfer and Strimmer (2005): it is computationally inexpensive.
Thus, finally, we consider six types of Mahalanobis distances: two localities
(global or class-based) and three estimators (pseudoinverse, shrinkage, or
diagonal).
## 5 Experiments
The main goal of our experiments is to evaluate Mahalanobis distances and the
class-based approach on time series. A secondary goal is to evaluate the LMNN
method.
We begin all tests with training data set made of classes of instances. When
applicable, distances are learned from this data set. We then attempt to
classify some test data using 1-NN and we report the classification error.
The code for the experiments is available online (Prekopcsák, 2011) with
instructions on how the results can be reproduced. For LMNN, we use the source
code provided by Weinberger and Saul (2008) for the experiments with default
parameters.
### 5.1 Data sets
We use the UCR time series classification benchmark (Keogh et al, 2006) for
our experiments as it includes diverse time series data sets from many
domains. It has predefined training-test splits for the experiments, so the
results can be compared across different papers. We removed the two data that
are not z-normalized by default (Beef and Coffee). Indeed, z-normalization
improves substantially the classification accuracy—irrespective of the chosen
distance. Thus, for fair results, we should z-normalize them, but this may
create confusion with previously reported numbers. We also removed the Wafer
data set as all distances classify it nearly perfectly. The remaining 17 data
sets were used for the comparison of different methods.
### 5.2 Best Mahalanobis distance for 1-NN accuracy
We compare the various Mahalanobis distances in Table 1. We have left out the
Moore-Penrose pseudoinverse, because its error rates were twice as high on
average compared to the other variants. What is immediately apparent is that
the class-based metrics give better classification results.
The diagonal Mahalanobis is somewhat better and they are also considerably
faster computationally, but the shrinkage estimate yields significantly better
results for several data sets (e.g. Adiac and Face (four)). Thus, out of the
six variations, we recommend the class-based shrinkage estimate and the class-
based diagonal Mahalanobis distance.
Table 1: Classification error for the various Mahalanobis distances. Data set | Shrinkage | Diagonal
---|---|---
global | class-based | global | class-based
50 words | 0.36 | 0.71 | 0.34 | 0.32
Adiac | 0.33 | 0.28 | 0.37 | 0.36
CBF | 0.52 | 0.04 | 0.16 | 0.05
ECG | 0.13 | 0.09 | 0.10 | 0.08
Fish | 0.33 | 0.15 | 0.19 | 0.18
Face (all) | 0.31 | 0.27 | 0.32 | 0.25
Face (four) | 0.45 | 0.10 | 0.16 | 0.17
Gun-Point | 0.06 | 0.10 | 0.10 | 0.11
Lighting-2 | 0.49 | 0.31 | 0.25 | 0.25
Lighting-7 | 0.59 | 0.32 | 0.36 | 0.23
OSU Leaf | 0.69 | 0.69 | 0.46 | 0.46
OliveOil | 0.17 | 0.17 | 0.17 | 0.13
Swedish Leaf | 0.25 | 0.14 | 0.21 | 0.18
Trace | 0.27 | 0.09 | 0.21 | 0.07
Two Patterns | 0.10 | 0.10 | 0.12 | 0.12
Synthetic Control | 0.23 | 0.10 | 0.13 | 0.09
Yoga | 0.24 | 0.21 | 0.17 | 0.17
# of best errors | 2 | 6 | 3 | 10
### 5.3 Comparing competitive distances
How do the class-based Mahalanobis distances fare compared to competitive
distances? Computationally, the diagonal Mahalanobis is inexpensive compared
to schemes such as the DTW or LMNN. Regarding the 1-NN classification error
rate, we give the results in Table 2. As expected (Ding et al, 2008), no
distance is better on all data sets. However, because the diagonal Mahalanobis
distance is closely related to the Euclidean distance, we compare their
classification accuracy. In two data sets, the Euclidean distance outperformed
the class-based Mahalanobis distance and only by small differences (0.09
versus 0.10-0.12). Meanwhile, the class-based diagonal Mahalanobis
outperformed the Euclidean distance 12 times, and sometimes by large margins
(0.07 versus 0.24 and 0.05 versus 0.15).
The LMNN is also competitive: its classification error is sometimes half that
of the Euclidean distance. However, the class-based LMNN gets the best result
among all methods only twice as opposed to five times for the global LMNN.
Moreover, the global LMNN significantly outperforms the class-based LMNN on
the Two Patterns data set (0.05 versus 0.24). For time series data sets, the
class-based LMNN is not an improvement over the global LMNN.
We have to note that DTW has the lowest error rates and provides best results
for half of the data sets, but it is much slower than Mahalanobis distances.
Table 2: Classification errors for some competitive schemes. We use class-based Mahalanobis distances. For the 50 words data set, the LMNN computation fails because it has a class with only one instance. Data set | Euclidean | DTW | C.-b. Mahalanobis | LMNN |
---|---|---|---|---|---
| | | shrink. | diag. | | c.-b. |
50 words | 0.37 | 0.31 | 0.71 | 0.32 | — | — |
Adiac | 0.39 | 0.40 | 0.28 | 0.36 | 0.23 | 0.32 |
CBF | 0.15 | 0.00 | 0.04 | 0.05 | 0.15 | 0.15 |
ECG | 0.12 | 0.23 | 0.09 | 0.08 | 0.10 | 0.07 |
Fish | 0.22 | 0.17 | 0.15 | 0.18 | 0.13 | 0.14 |
Face (all) | 0.29 | 0.19 | 0.27 | 0.25 | 0.16 | 0.20 |
Face (four) | 0.22 | 0.17 | 0.10 | 0.17 | 0.16 | 0.16 |
Gun-Point | 0.09 | 0.09 | 0.10 | 0.11 | 0.05 | 0.09 |
Lighting-2 | 0.25 | 0.13 | 0.31 | 0.25 | 0.41 | 0.34 |
Lighting-7 | 0.42 | 0.27 | 0.32 | 0.23 | 0.51 | 0.48 |
OSU Leaf | 0.48 | 0.41 | 0.69 | 0.46 | 0.57 | 0.54 |
OliveOil | 0.13 | 0.13 | 0.17 | 0.13 | 0.13 | 0.13 |
Swedish Leaf | 0.21 | 0.21 | 0.14 | 0.18 | 0.21 | 0.19 |
Trace | 0.24 | 0.00 | 0.09 | 0.07 | 0.20 | 0.20 |
Two Patterns | 0.09 | 0.00 | 0.10 | 0.12 | 0.05 | 0.24 |
Synthetic Control | 0.12 | 0.01 | 0.10 | 0.09 | 0.03 | 0.09 |
Yoga | 0.17 | 0.16 | 0.21 | 0.17 | 0.18 | 0.18 |
# of best errors | 1 | 9 | 2 | 2 | 5 | 2 |
### 5.4 Effect of the number of instances per class
Whereas Table 2 shows that the Mahalanobis distances are far superior to the
Euclidean distance on some data sets, this result is linked to the number of
instances per class. For example, on the Wafer data set (which we removed),
there are many instances per class (500), and correspondingly, all distances
give a negligible classification error.
Thus, we considered three different synthetic time-series data sets with
varying numbers of instances per class: Cylinder-Bell-Funnel (CBF) (Saito,
1994), Control Charts (CC) (Pham and Chan, 1998) and Waveform (Breiman, 1998).
Test sets have 1 000 instances per class whereas training sets have between 10
to 1 000 instances. We repeated each test ten times, with different training
sets. Fig. 3 shows that whereas the class-based diagonal Mahalanobis is
superior to the Euclidean distance when there are few instances, this benefit
is less significant as the number of instances increases. Indeed, the
classification accuracy of the Euclidean distance grows closer to perfection
and it becomes more difficult for alternatives to be far superior.
Figure 3: Ratios of the 1-NN classification accuracies using the class-based
diagonal Mahalanobis and Euclidean distances
## 6 Conclusion
The Mahalanobis distances have received little attention for time series
classification and we are not surprised given their poor performance as a 1-NN
classifier when used in a straight-forward manner. However, by learning one
Mahalanobis distance per class we get a competitive classifier when using
either covariance shrinkage or a diagonal approach. Moreover, the diagonal
Mahalanobis distance is particularly appealing computationally: we only need
to compute the variances of the components. Meanwhile, we get good results
with the LMNN on time series data, though it is more expensive. The DTW is
superior, but computationally expensive.
###### Acknowledgements.
This work is supported by NSERC grant 261437.
## References
* Ang et al (2010) Ang H, Gopalkrishnan V, Ng W, Hoi S (2010) On classifying drifting concepts in P2P networks. In: Balcázar J, Bonchi F, Gionis A, Sebag M (eds) Machine Learning and Knowledge Discovery in Databases, Lecture Notes in Computer Science, vol 6321, Springer Berlin / Heidelberg, pp 24–39
* Aßfalg et al (2006) Aßfalg J, Kriegel H, Kröger P, Kunath P, Pryakhin A, Renz M (2006) Similarity search on time series based on threshold queries. EDBT 2006 pp 276–294
* Bahlmann (2004) Bahlmann C (2004) The writer independent online handwriting recognition system frog on hand and cluster generative statistical Dynamic Time Warping. Writer 26(3):299–310
* Bartolini et al (2005) Bartolini I, Ciaccia P, Patella M (2005) WARP: Accurate retrieval of shapes using phase of fourier descriptors and time warping distance. IEEE Transactions on Pattern Analysis and Machine Intelligence 27(1):142–147, DOI http://doi.ieeecomputersociety.org/10.1109/TPAMI.2005.21
* Breiman (1998) Breiman L (1998) Classification and Regression Trees. Chapman & Hall/CRC
* Chai et al (2010) Chai J, Liu H, Chen B, Bao Z (2010) Large margin nearest local mean classifier. Signal Process 90(1):236–248
* Chen and Ng (2004) Chen L, Ng R (2004) On the marriage of $l_{p}$-norms and edit distance. In: VLDB’04, pp 1040–1049
* Chen et al (2005) Chen L, Özsu M, Oria V (2005) Robust and fast similarity search for moving object trajectories. In: SIGMOD’05, ACM, pp 491–502
* Chen et al (2007) Chen Y, Nascimento M, Ooi B, Tung A (2007) SpaDe: On shape-based pattern detection in streaming time series. In: ICDE’07, pp 786–795
* Chouakria and Nagabhushan (2007) Chouakria A, Nagabhushan P (2007) Adaptive dissimilarity index for measuring time series proximity. Advances in Data Analysis and Classification 1:5–21
* Csatári and Prekopcsák (2010) Csatári B, Prekopcsák Z (2010) Class-based attribute weighting for time series classification. In: POSTER 2010: Proceedings of the 14th International Student Conference on Electrical Engineering
* Ding et al (2008) Ding H, Trajcevski G, Scheuermann P, Wang X, Keogh E (2008) Querying and mining of time series data: experimental comparison of representations and distance measures. In: VLDB’08, pp 1542–1552
* Frentzos et al (2007) Frentzos E, Gratsias K, Theodoridis Y (2007) Index-based most similar trajectory search. In: ICDE 2007, pp 816–825
* Gaudin and Nicoloyannis (2006) Gaudin R, Nicoloyannis N (2006) An adaptable time warping distance for time series learning. In: Proceedings of the 5th International Conference on Machine Learning and Applications, IEEE Computer Society, Washington, DC, USA, pp 213–218, DOI 10.1109/ICMLA.2006.12
* Hastie and Tibshirani (1996) Hastie T, Tibshirani R (1996) Discriminant adaptive nearest neighbor classification. IEEE Transactions on Pattern Analysis and Machine Intelligence 18(6):607–616
* Ishikawa et al (1998) Ishikawa Y, Subramanya R, Faloutsos C (1998) Mindreader: Querying databases through multiple examples. In: VLDB’98, pp 218–227
* Itakura (1975) Itakura F (1975) Minimum prediction residual principle applied to speech recognition. IEEE Transactions on Acoustics, Speech, and Signal Processing 23(1):67–72
* Jahromi et al (2009) Jahromi MZ, Parvinnia E, John R (2009) A method of learning weighted similarity function to improve the performance of nearest neighbor. Inf Sci 179(17):2964–2973
* Jeong et al (2010) Jeong YS, Jeong MK, Omitaomu OA (2010) Weighted dynamic time warping for time series classification. Pattern Recognition In Press, Corrected Proof:–, DOI DOI:10.1016/j.patcog.2010.09.022
* Keogh and Pazzani (1998) Keogh E, Pazzani M (1998) An enhanced representation of time series which allows fast and accurate classification, clustering and relevance feedback. In: Proceedings of the 4th International Conference of Knowledge Discovery and Data Mining, pp 239–241
* Keogh et al (2006) Keogh E, Xi X, Wei L, Ratanamahatana CA (2006) The UCR time series classification/clustering homepage. http://www.cs.ucr.edu/~eamonn/time_series_data/ [last checked on 9/10/2010]
* Legrand et al (2007) Legrand B, Chang CS, Ong SH, Neo SY, Palanisamy N (2007) Chromosome classification using dynamic time warping. Pattern Recognition Letters 29(3):215–222
* Legrand et al (2008) Legrand B, Chang C, Ong S, Neo SY, Palanisamy N (2008) Chromosome classification using dynamic time warping. Pattern Recognition Letters 29(3):215 – 222
* Lemire (2009) Lemire D (2009) Faster retrieval with a two-pass dynamic-time-warping lower bound. Pattern Recognition 42:2169–2180
* Mahalanobis (1936) Mahalanobis PC (1936) On the generalised distance in statistics. In: Proceedings of the National Institute of Sciences of India, pp 49–55
* Marzal et al (2006) Marzal A, Palazon V, Peris G (2006) Contour-based shape retrieval using Dynamic Time Warping. Lecture notes in Computer Science 4177:190
* Matton et al (2010) Matton M, Compernolle DV, Cools R (2010) Minimum classification error training in example based speech and pattern recognition using sparse weight matrices. Journal of Computational and Applied Mathematics 234(4):1303–1311
* Morse and Patel (2007) Morse MD, Patel JM (2007) An efficient and accurate method for evaluating time series similarity. In: SIGMOD ’07, pp 569–580
* Niels and Vuurpijl (2005) Niels R, Vuurpijl L (2005) Using Dynamic Time Warping for intuitive handwriting recognition. In: IGS2005, pp 217–221
* Ouyang and Zhang (2010) Ouyang Y, Zhang F (2010) Histogram distance for similarity search in large time series database. In: Proceedings of the 11th international conference on Intelligent data engineering and automated learning, Springer-Verlag, Berlin, Heidelberg, IDEAL’10, pp 170–177
* Paredes and Vidal (2000) Paredes R, Vidal E (2000) A class-dependent weighted dissimilarity measure for nearest neighbor classification problems. Pattern Recognition Letters 21(12):1027–1036
* Paredes and Vidal (2006) Paredes R, Vidal E (2006) Learning prototypes and distances: A prototype reduction technique based on nearest neighbor error minimization. Pattern Recognition 39(2):180–188
* Pham and Chan (1998) Pham DT, Chan AB (1998) Control chart pattern recognition using a new type of self-organizing neural network. Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering 212(2):115–127
* Prekopcsák (2011) Prekopcsák Z (2011) Matlab code for the experiments. http://github.com/Preko/Time-series-classification
* Ratanamahatana and Keogh (2004) Ratanamahatana CA, Keogh E (2004) Making time-series classification more accurate using learned constraints. In: SDM’04, pp 11–22
* Ratanamahatana and Keogh (2005) Ratanamahatana CA, Keogh E (2005) Three myths about Dynamic Time Warping data mining. In: SDM’05
* Saito (1994) Saito N (1994) Local feature extraction and its applications using a library of bases. PhD thesis, Yale University, New Haven, CT, USA
* Sakoe and Chiba (1978a) Sakoe H, Chiba S (1978a) Dynamic programming algorithm optimization for spoken word recognition. Acoustics, Speech and Signal Processing, IEEE Transactions on 26(1):43 – 49, DOI 10.1109/TASSP.1978.1163055
* Sakoe and Chiba (1978b) Sakoe H, Chiba S (1978b) Dynamic programming algorithm optimization for spoken word recognition. IEEE Transactions on Acoustics, Speech, and Signal Processing 26(1):43–49
* Salvador and Chan (2007) Salvador S, Chan P (2007) FastDTW: Toward accurate dynamic time warping in linear time and space. Intelligent Data Analysis 11(5):561–580
* Schäfer and Strimmer (2005) Schäfer J, Strimmer K (2005) A shrinkage approach to large-scale covariance matrix estimation and implications for functional genomics. Statistical applications in genetics and molecular biology 4(1):32
* Short and Fukunaga (1980) Short R, Fukunaga K (1980) A new nearest neighbor distance measure. In: Proceedings of the Fifth International Conference on Pattern Recognition, pp 81–86
* Stein (1956) Stein C (1956) Inadmissibility of the usual estimator for the mean of a multivariate normal distribution. In: Proceedings of the Third Berkeley Symposium on Mathematical and Statistical Probability, pp 197––206
* Sternickel (2002) Sternickel K (2002) Automatic pattern recognition in ECG time series. Computer Methods and Programs in Biomedicine 68(2):109 – 115
* Vlachos et al (2002) Vlachos M, Gunopoulos D, Kollios G (2002) Discovering similar multidimensional trajectories. In: ICDE ’02, p 673
* Weihs et al (2007) Weihs C, Ligges U, Mörchen F, Müllensiefen D (2007) Classification in music research. Advances in Data Analysis and Classification 1:255–291
* Weinberger and Saul (2008) Weinberger K, Saul L (2008) Large margin nearest neighbor – matlab code. http://www.cse.wustl.edu/~kilian/Downloads/LMNN.html
* Weinberger and Saul (2009) Weinberger K, Saul L (2009) Distance metric learning for large margin nearest neighbor classification. The Journal of Machine Learning Research 10:207–244
* Wettschereck et al (1997) Wettschereck D, Aha DW, Mohri T (1997) A review and empirical evaluation of feature weighting methods for a class of lazy learning algorithms. Artif Intell Rev 11(1-5):273–314
* Yang and Jin (2006) Yang L, Jin R (2006) Distance metric learning: A comprehensive survey. Tech. rep., Michigan State University
* Yu et al (2011) Yu D, Yu X, Hu Q, Liu J, Wu A (2011) Dynamic time warping constraint learning for large margin nearest neighbor classification. Information Sciences 181(13):2787 – 2796, including Special Section on Databases and Software Engineering
* Zhan et al (2009) Zhan DC, Li M, Li YF, Zhou ZH (2009) Learning instance specific distances using metric propagation. In: ICML ’09, pp 1225–1232
|
arxiv-papers
| 2010-10-07T19:48:23 |
2024-09-04T02:49:13.587203
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Zolt\\'an Prekopcs\\'ak and Daniel Lemire",
"submitter": "Zoltan Prekopcsak",
"url": "https://arxiv.org/abs/1010.1526"
}
|
1010.1597
|
# Additive energy and the Falconer distance problem in finite fields
Doowon Koh and Chun-Yen Shen Department of Mathematics
Michigan State University
East Lansing, MI 48824, USA koh@math.msu.edu Department of Applied
Mathematics
National Chiao Tung University
Hsinchu 300, Taiwan chunyshen@gmail.com
###### Abstract.
We study the number of the vectors determined by two sets in $d$-dimensional
vector spaces over finite fields. We observe that the lower bound of
cardinality for the set of vectors can be given in view of an additive energy
or the decay of the Fourier transform on given sets. As an application of our
observation, we find sufficient conditions on sets where the Falconer distance
conjecture for finite fields holds in two dimension. Moreover, we give an
alternative proof of the theorem, due to Iosevich and Rudnev, that any Salem
set satisfies the Falconer distance conjecture for finite fields.
###### 2000 Mathematics Subject Classification:
52C10
Key words and phrases: additive energy , distances, the Falconer distance
conjecture
###### Contents
1. 1 Introduction
1. 1.1 Purpose of this paper
2. 2 Cardinality of difference sets
3. 3 Sets in $\mathbb{F}_{q}^{2}$ satisfying the Falconer distance conjecture
1. 3.1 Examples of the Falconer conjecture sets in two dimension
4. 4 Salem sets and difference sets
## 1\. Introduction
Let $\mathbb{F}_{q}^{d},d\geq 1,$ be $d$-dimensional vector space over the
finite field $\mathbb{F}_{q}$ with $q$ elements. Given
$A,B\subset\mathbb{F}_{q}^{d},$ one may ask what is the cardinality of the set
$A-B,$ where the difference set $A-B$ is defined by
$A-B=\\{x-y\in\mathbb{F}_{q}^{d}:x\in A,y\in B\\}.$
It is clear that $|A-B|\geq\max\\{|A|,|B|\\},$ here, and throughout the paper,
we denote by $|E|$ the cardinality of the set $E.$ However, taking
$A=B=\mathbb{F}_{q}^{s},1\leq s\leq d,$ shows that the trivial estimate for
$|A-B|$ is sharp in general, because $|A-B|=|\mathbb{F}_{q}^{s}|=q^{s}.$
Moreover, if $s=d-1$, then the size of $A-B$ is much smaller than that of
$\mathbb{F}_{q}^{d},$ although $|A||B|=q^{2d-2}$ is somewhat big. Therefore,
it may be interesting to find some conditions on the sets
$A,B\subset\mathbb{F}_{q}^{d}$ such that the cardinality of $A-B$ is much
bigger than the trivial lower bound, $\max\\{|A|,|B|\\},$ of $|A-B|,$ or the
difference set $A-B$ contains a positive proportion of all vectors in
$\mathbb{F}_{q}^{d},$ that is $|A-B|\gtrsim|\mathbb{F}_{q}^{d}|=q^{d}.$ Here,
we recall that for $l,m>0,$ the expression $l\gtrsim m$ or $m\lesssim l$ means
that there exists a constant $c>0$ independent of $q,$ the size of the
underlying finite field $\mathbb{F}_{q},$ such that $cl\geq m.$ The problem to
consider the size of difference sets is strongly motivated by the Falconer
distance problem for finite fields, which was introduced by Iosevich and
Rudnev [9]. In this paper, we shall make an effort to find the connection
between the size of the difference set $A-B$ and the cardinality of the
distance set determined by $A,B\subset\mathbb{F}_{q}^{d}.$ As one of the main
results, we shall give some examples for sets satisfying the Falconer distance
conjecture for finite fields.
First, let us review the Falconer distance problem for the Euclidean case and
the finite field case. In the Euclidean setting, the Falconer distance problem
is to determine the Hausdorff dimensions of compact sets
$E,F\subset\mathbb{R}^{d},d\geq 2,$ such that the Lebesgue measure of the
distance set
$\Delta(E,F):=\\{|x-y|:x\in E,y\in F\\}$
is positive. In the case when $E=F$, Falconer [4] first addressed this problem
and showed that if the Hausdorff dimension of the compact set $E$ is greater
than $(d+1)/2$, then the Lebesgue measure of $\Delta(E,E)$ is positive. He
also conjectured that every compact set with the Hausdorff dimension $>d/2$
yields a distance set with a positive Lebesgue measure. This problem is called
as the Falconer distance conjecture which has not been solved in all
dimensions. The best known result for this problem is due to Wolff [16] in two
dimension and Erdog̃an [3] in all other dimensions. They proved that if the
Hausdorff dimension of any compact set $E\subset\mathbb{R}^{d}$ is greater
than $d/2+1/3$, then the Lebesgue measure of $\Delta(E,E)$ is positive. These
results are a culmination of efforts going back to Falconer [4] in 1985 and
Mattila [13] a few years later. The Falconer distance problem on generalized
distances was also studied in [1], [6], [7], [8], and [10].
In the Finite field setting, one can also study the Falconer distance problem.
Given $A,B\subset\mathbb{F}_{q}^{d},d\geq 2,$ the distance set $\Delta(A,B)$
is given by
$\Delta(A,B)=\\{\|x-y\|\in\mathbb{F}_{q}:x\in A,y\in B\\},$
where $\|\alpha\|=\alpha_{1}^{2}+\dots+\alpha_{d}^{2}$ for
$\alpha=(\alpha_{1},\dots,\alpha_{d})\in\mathbb{F}_{q}^{d}.$ It is clear that
$|\Delta(A,B)|\leq q,$ because the distance set is a subset of the finite
field with $q$ elements. In this setting, the Falconer distance problem is to
determine the minimum value of $|A||B|$ such that $|\Delta(A,B)|\gtrsim q.$ In
the case when $A=B$, this problem was introduced by Iosevich and Rudnev [9]
and they proved that if $A=B$ and $|A|\gtrsim q^{(d+1)/2},$ then
$|\Delta(A,B)|\gtrsim q.$ It turned out in [5] that if the dimension $d$ is
odd, then the theorem due to Alex and Rudnev gives the best possible result on
the Falconer distance problem for finite fields. However, if the dimension $d$
is even, then it has been believed that the aforementioned authors’ result may
be improved to the following conjecture.
###### Conjecture 1.1 (Iosevich and Rudnev [9]).
Let $K\subset\mathbb{F}_{q}^{d}$ with $d\geq 2$ even. If $|K|\geq
Cq^{\frac{d}{2}},$ with $C>0$ sufficiently large, then
$|\Delta(K,K)|\gtrsim q.$
This conjecture has not been solved in all dimensions. The exponent $(d+1)/2$
obtained by Iosevich and Rudnev is currently the best known result for all
dimensions except two dimension. In two dimension, this exponent was improved
by $4/3$ (see [2] or [11]). We may consider the following general version of
Conjecture 1.1:
###### Conjecture 1.2.
Let $A,B\subset\mathbb{F}_{q}^{d}$ with $d\geq 2$ even. If $|A||B|\geq
Cq^{d},$ with $C>0$ large enough, then
$|\Delta(A,B)|\gtrsim q.$
Theorem 2.1 in [14] due to Shparlinski implies that if
$A,B\subset\mathbb{F}_{q}^{d},d\geq 2,$ and $|A||B|\gtrsim q^{d+1},$ then
$|\Delta(A,B)|\gtrsim q.$ This was improved by authors [11] who showed that if
$|A||B|\gtrsim q^{8/3}$ for $A,B\subset\mathbb{F}_{q}^{2},$ then
$|\Delta(A,B)|\gtrsim q.$ For a variant of the Falconer distance problem for
finite fields, see [15] and [12].
### 1.1. Purpose of this paper
The goal of this paper is to find some sets
$A,B\subset\mathbb{F}_{q}^{d},d\geq 2,$ for which Conjecture 1.2 holds. In
general, it may not be easy to construct such examples, supporting the claim
that Conjecture 1.2 holds. A well-known example is due to Iosevich and Rudnev
[9] who showed that if $K\subset\mathbb{F}_{q}^{d},d\geq 2,$ is a Salem set
and $|K|\gtrsim q^{d/2},$ then $|\Delta(K,K)|\gtrsim q.$ Here, we recall that
we say that $E\subset\mathbb{F}_{q}^{d}$ is a Salem set if for every
$m\in\mathbb{F}_{q}^{d}\setminus\\{(0,\dots,0)\\}$,
$|\widehat{E}(m)|:=|q^{-d}\sum_{x\in E}\chi(-x\cdot
m)|\lesssim\frac{\sqrt{E}}{q^{d}}.$
They obtained this example by showing that the formula of $|\Delta(K,K)|$ is
closely related to the decay of the Fourier transform on the set $K.$ In this
paper, we take a new approach to find such examples. First, we shall show that
if $A,B\subset\mathbb{F}_{q}^{d},d\geq 2$ and $|A-B|\gtrsim q^{d}$, then
$|\Delta(A,B)|\gtrsim q.$ Second, we find some conditions on the set
$A,B\subset\mathbb{F}_{q}^{d}$ such that $|A-B|\sim q^{d}.$ Thus, estimating
the size of the difference set $A-B$ makes an important role. For example,
using our approach we can recover the example by Iosevich and Rudnev.
Moreover, we can find a stronger result that if one of
$A,B\subset\mathbb{F}_{q}^{d}$ is a Salem set and $|A||B|\gtrsim q^{d},$ then
$A-B$ contains a positive proportion of all elements in $\mathbb{F}_{q}^{d}.$
In particular, our method yields that if one of $A,B\subset\mathbb{F}_{q}^{2}$
intersects with $\sim q$ points in an algebraic curve which does not contain
any line, and $|A||B|\gtrsim q^{2},$ then the sets $A,B$ satisfies Conjecture
1.2 in two dimension.
## 2\. Cardinality of difference sets
In this section we introduce the formulas for the lower bound of difference
sets. Such formulas are closely related to the additive energy
$\Lambda(A,B)=|\\{(x,y,z,w)\in A\times A\times B\times B:x-y+z-w=0\\}|.$
In fact, applying the Cauchy-Schwarz inequality shows that if
$A,B\subset\mathbb{F}_{q}^{d},d\geq 2,$ then
$|A|^{2}|B|^{2}=\left(\sum_{c\in
A-B}|A\cap(B+c)|\right)^{2}\leq|A-B|\sum_{c\in\mathbb{F}_{q}^{d}}|A\cap(B+c)|^{2}.$
Observing that $\sum_{c\in\mathbb{F}_{q}^{d}}|A\cap(B+c)|^{2}=\Lambda(A,B),$
it follows that
(2.1) $|A-B|\geq\frac{|A|^{2}|B|^{2}}{\Lambda(A,B)}.$
Since $\Lambda(A,B)\leq\min\\{|A|^{2}|B|,|A||B|^{2}\\},$ it is clear that
$|A-B|\geq\max\\{|A|,|B|\\},$
which is in fact a trivial bound of $|A-B|.$ However, if we take a subspace as
$A,B$ with $A=B,$ then the trivial bound is the best bound. In this case, the
difference set $A-B$ has much smaller cardinality than $|A||B|$. It therefore
is natural to guess that if $A$ and $B$ do not contain a big subspace, then
$|A-B|$ can be large. In this paper, we shall deal with this issue.
The lower bound of $|A-B|$ can be written in terms of the Fourier transforms
on $A,B.$ To see this, using the definition of the Fourier transform and the
orthogonality relation of the nontrivial additive character of
$\mathbb{F}_{q},$ observe that
$\Lambda(A,B)=q^{3d}\sum_{m\in\mathbb{F}_{q}^{d}}|\widehat{A}(m)|^{2}|\widehat{B}(m)|^{2},$
Here, we recall that the Fourier transform on the set
$E\subset\mathbb{F}_{q}^{d}$ is defined by
$\widehat{E}(m)=\frac{1}{q^{d}}\sum_{x\in E}\chi(-x\cdot
m)\quad\mbox{for}~{}~{}m\in\mathbb{F}_{q}^{d},$
where $\chi$ denotes a nontrivial additive character of $\mathbb{F}_{q}.$
Therefore, the formula (2.1) can be replaced by
(2.2)
$|A-B|\geq\frac{|A|^{2}|B|^{2}}{q^{3d}\sum_{m\in\mathbb{F}_{q}^{d}}|\widehat{A}(m)|^{2}|\widehat{B}(m)|^{2}}.$
This formula indicates that if the Fourier decay on $A$ or $B$ is good, then
several kinds of vectors are contained in the difference set $A-B.$ For
example, if $A$ or $B$ takes a Salem set such as the paraboloid or the sphere,
then $|A-B|$ is big and so a lot of distances can be determined by $A,B.$
## 3\. Sets in $\mathbb{F}_{q}^{2}$ satisfying the Falconer distance
conjecture
In view of the sizes of difference sets, we shall find some sets
$A,B\subset\mathbb{F}_{q}^{2}$ where the Falconer distance conjecture
(Conjecture 1.2) holds. Simple but core idea is due to the following fact.
###### Lemma 3.1.
Let $E\subset\mathbb{F}_{q}^{2}.$ If $|E|\geq cq^{2}$ for some $0<c\leq 1,$
then we have
$|\\{\|x\|\in\mathbb{F}_{q}:x\in E\\}|\geq\frac{cq}{2}.$
###### Proof.
For each $a\in\mathbb{F}_{q},$ consider a vertical line
$L_{a}=\\{(a,t)\in\mathbb{F}_{q}^{2}:t\in\mathbb{F}_{q}\\}.$ Since $|E|\geq
cq^{2},$ it is clear from the pigeonhole principle that there exists a line
$L_{b}$ for some $b\in\mathbb{F}_{q}$ with $|E\cap L_{b}|\geq cq.$ Thus, Lemma
follows from the following observation that for the fixed
$b\in\mathbb{F}_{q},$
$|\\{b^{2}+t^{2}\in\mathbb{F}_{q}:(b,t)\in E\cap L_{b}\\}|\geq\frac{cq}{2}.$
∎
If $|A-B|\gtrsim|A||B|\gtrsim q^{2}$, then Lemma 3.1 implies that
$A,B\subset\mathbb{F}_{q}^{2}$ are the sets to satisfy the Falconer
conjecture. Thus, the main task is to fine sets $A,B$ such that $|A-B|$ is
extremely large. The following lemma tells us some properties of sets $A,B$
where the size of $A-B$ can be large.
###### Lemma 3.2.
Let $B\subset\mathbb{F}_{q}^{2}.$ Suppose that there exists a set
$W\subset\mathbb{F}_{q}^{2}$ with $|W|\sim 1$ such that
(3.1) $|B\cap(B+c)|\lesssim 1\quad\mbox{for
all}~{}~{}c\in\mathbb{F}_{q}^{2}\setminus W.$
Then, for any $A\subset\mathbb{F}_{q}^{2},$ we have
$|A-B|\gtrsim\min(|A||B|,|B|^{2}).$
###### Proof.
From (2.1), it suffices to show that
$\Lambda(A,B)=|\\{(x,y,z,w)\in A\times A\times B\times
B:x-y+z-w=0\\}|\lesssim|A||B|+|A|^{2}.$
It follows that
$\Lambda(A,B)=\sum_{x,y\in A}\left(\sum_{w,z\in
B:z-w=y-x}1\right)=\sum_{x,y\in A}|B\cap(B+y-x)|$ $=\sum_{x,y\in A:y-x\notin
W}|B\cap(B+y-x)|+\sum_{x,y\in A:y-x\in W}|B\cap(B+y-x)|$
$=\mbox{I}+\mbox{II}.$
From the assumption (3.1), it is clear that $|\mbox{I}|\lesssim|A|^{2}.$ On
the other hand, the value II can be estimated as follows.
$\mbox{II}=\sum_{\beta\in W}\sum_{x,y\in
A:y-x=\beta}|B\cap(B+\beta)|\leq\sum_{\beta\in W}\sum_{x,y\in
A:y-x=\beta}|B|.$
Whenever we fix $x\in A$ and $\beta\in W,$ there is at most one $y\in A$ such
that $y-x=\beta.$ We therefore see
$\mbox{II}\leq|W||A||B|\sim|A||B|.$
Thus, we complete the proof.∎
### 3.1. Examples of the Falconer conjecture sets in two dimension
First recall that the Bezout’s theorem says that two algebraic curves of
degrees $d_{1}$ and $d_{2}$ intersect in $d_{1}\cdot d_{2}$ points and cannot
meet in more than $d_{1}\cdot d_{2}$ points unless they have a component in
common. As a direct application of the Bezout’s theorem, it can be shown that
subsets of certain algebraic curves in two dimension satisfy the condition in
(3.1). This observation yields the following theorem.
###### Theorem 3.3.
Let $P(x)\in\mathbb{F}_{q}[x_{1},x_{2}]$ be an polynomial which does not have
any liner factor. Define an algebraic variety
$V=\\{x\in\mathbb{F}_{q}^{2}:P(x)=0\\}.$ If $B\subset V$, then for any
$A\subset\mathbb{F}_{q}^{2},$ we have
$|A-B|\gtrsim\min(|A||B|,|B|^{2}).$
###### Proof.
First recall that we always assume that the degree of the polynomial is $\sim
1.$ Thus, if $B\subset V$, then the pigeonhole principle implies that we can
choose a subvariety $V^{\prime}$ of $V$ and a set $B^{\prime}\subset
V^{\prime}$ with $|B^{\prime}|\sim|B|.$ Therefore, we may assume that $V$ is a
variety generated by an irreducible polynomial with degree $k\geq 2.$ Applying
the Bezout’s theorem shows that for any
$c\in\mathbb{F}_{q}^{2}\setminus\\{(0,0)\\},$
$|V\cap(V+c)|\leq k^{2}\lesssim 1.$
Therefore, the proof is complete from Lemma 3.2. ∎
The following corollary follows immediately from Lemma 3.2 and Lemma 3.1.
###### Corollary 3.4.
Let $B\subset\mathbb{F}_{q}^{2}$ with $|B|\gtrsim q.$ Suppose that
$W\subset\mathbb{F}_{q}^{2}$ with $|W|\sim 1,$ and $|B\cap(B+c)|\lesssim 1$
for any $c\in\mathbb{F}_{q}^{2}\setminus W.$ Then, for any
$A\subset\mathbb{F}_{q}^{2}$ with $|A|\gtrsim q,$ we have
$|\Delta(A,B)|=|\\{||x-y||\in\mathbb{F}_{q}:x\in A,y\in B\\}|\gtrsim q.$
Notice that such sets $A,B$ as in this corollary satisfy the Falconer distance
conjecture. Moreover, the difference set $A-B$ contains a positive proportion
of all elements in $\mathbb{F}_{q}^{2}.$ As a consequence of Theorem 3.3 and
corollary 3.4, more concrete examples for the Falconer distance conjecture
sets can be found.
###### Example 3.5.
First,we choose a polynomial $P\in\mathbb{F}_{q}[x_{1},x_{2}]$ which does not
contain any linear factor. Second, consider a variety
$V=\\{x\in\mathbb{F}_{q}:P(x)=0\\}.$ If we can check that $|V|\gtrsim q$, then
choose a subset $B\subset V$ with $|B|\sim q.$ Finally, choose any subset $A$
of $\mathbb{F}_{q}^{2},$ whose cardinality is $\sim q.$ Then, the difference
set $A-B$ contains the positive proportion of all elements in
$\mathbb{F}_{q}^{2}$ and so $|\Delta(A,B)|\sim q.$ Since $|A||B|\sim q^{2},$
the sets $A,B$ are of the Falconer distance conjecture sets.
Observe that if both $A$ and $B$ contain many points in some lines
$L_{1},L_{2}$ respectively, then we can not proceed such steps as in above
example. If sets $A,B$ possess the structures like product sets, then it seems
that two sets $A,B$ determine the distance set $\Delta(A,B)$ with a small
cardinality.
## 4\. Salem sets and difference sets
If the decay of the Fourier transform on $A,B\subset\mathbb{F}_{q}^{d}$ is
known, then the formula (2.2) can be very useful to measure the lower bound of
$|A-B|.$ Here, we shall show that if one of $A$ and $B$ is a Salem set, then
$|A-B|$ is so big that $A,B$ satisfy the Falconer distance conjecture. We need
the following lemma which shows the relation between the Fourier decay of sets
and the size of difference sets.
###### Lemma 4.1.
Let $A,B\subset\mathbb{F}_{q}^{d}.$ Suppose that for every
$m\in\mathbb{F}_{q}^{d}\setminus\\{(0,\dots,0)\\},$
(4.1) $|\widehat{B}(m)|\lesssim q^{\beta}\quad\mbox{for
some}~{}~{}\beta\in\mathbb{R}.$
Then, we have
$|A-B|\gtrsim\min\left(q^{d},\frac{|A||B|^{2}}{q^{2d+2\beta}}\right).$
###### Proof.
The proof is based on the formula (2.2) and discrete Fourier analysis. It
follows that
$q^{3d}\sum_{m\in\mathbb{F}_{q}^{d}}|\widehat{A}(m)|^{2}|\widehat{B}(m)|^{2}$
$\leq
q^{3d}|\widehat{A}(0,\dots,0)|^{2}|\widehat{B}(0,\dots,0)|^{2}+q^{3d}\left(\max_{m\in\mathbb{F}_{q}^{d}\setminus(0,\dots,0)}|\widehat{B}(m)|^{2}\right)\sum_{m\in\mathbb{F}_{q}^{d}}|\widehat{A}(m)|^{2}$
$=\mbox{I}+\mbox{II}.$
By the definition of the Fourier transform, it is clear that
$\mbox{I}=q^{-d}|A|^{2}|B|^{2}.$ On the other hand, using the assumption (4.1)
and the Plancherel theorem, we obtain that $\mbox{II}\lesssim
q^{2d+2\beta}|A|.$ Thus, we have
$q^{3d}\sum_{m\in\mathbb{F}_{q}^{d}}|\widehat{A}(m)|^{2}|\widehat{B}(m)|^{2}\lesssim
q^{-d}|A|^{2}|B|^{2}+q^{2d+2\beta}|A|.$
Thus, Lemma 2.2 can be used to obtain that
$|A-B|\gtrsim\frac{|A|^{2}|B|^{2}}{q^{-d}|A|^{2}|B|^{2}+q^{2d+2\beta}|A|}\gtrsim\min\left(q^{d},\frac{|A||B|^{2}}{q^{2d+2\beta}}\right),$
which completes the proof. ∎
As mentioned in introduction, it is known that if $B\subset\mathbb{F}_{q}^{d}$
with $|B|\gtrsim q^{d/2}$ is a Salem set, then $|\Delta(B,B)\gtrsim q.$
Namely, the Salem set $B$ is of the Falconer distance conjecture sets. In this
case, we can state a strong fact that $B-B$ contains a positive proportion of
all elements in $\mathbb{F}_{q}^{d}.$ More precisely, we have the following
theorem.
###### Theorem 4.2.
If $B\subset\mathbb{F}_{q}^{d}$ is a Salem set, then for any
$A\subset\mathbb{F}_{q}^{d}$ with $|A||B|\gtrsim q^{d},$ we have
$|A-B|\gtrsim q^{d}.$
###### Proof.
Since $B\subset\mathbb{F}_{q}^{d}$ is a Salem set, taking
$q^{\beta}=q^{-d}\sqrt{|B|}$ from Lemma 4.1 shows that
(4.2) $|A-B|\gtrsim\min\\{q^{d},|A||B|\\}.$
Since $|A||B|\gtrsim q^{d},$ the proof is complete. ∎
The following corollary follows immediately from above theorem and Lemma 3.1.
###### Corollary 4.3.
Let $A\subset\mathbb{F}_{q}^{d}$ is a Salem set. Then, for any
$B\subset\mathbb{F}_{q}^{d}$ with $|A||B|\gtrsim q^{d},$ we have
$|\Delta(A,B)|\gtrsim q.$
In other words, the sets $A,B$ satisfy the Falconer distance conjecture.
## References
* [1] G. Arutyunyants and A. Iosevich, _Falconer conjecture, spherical averages and discrete analogs_ , In Towards a theory of geometric graphs, 15-24, Contemp. Math. 342, Amer. Math. Soc., Providence, (2004).
* [2] J. Chapman, M. Erdog̃an, D. Hart, A. Iosevich, and D. Koh, _Pinned distance sets, Wolff’s exponent in finite fields and sum-product estimates_ , arXiv:0903.4218v2, (2009).
* [3] M. Erdog̃an, _A bilinear Fourier extension theorem and applications to the distance set problem,_ Internat. Math. Res. Notices 23 (2005), 1411-1425.
* [4] K. Falconer, _On the Hausdorff dimensions of distance sets,_ Mathematika, 32 (1985), 206–212.
* [5] D. Hart, A. Iosevich, D. Koh and M. Rudnev, _Averages over hyperplanes, sum-product theory in vector spaces over finite fields and the Erdös-Falconer distance conjecture_ , Trans. Amer. Math. Soc. (2010) To appear.
* [6] S. Hofmann and A. Iosevich, _Circular averages and Falconer/Erdo”s distance conjecture in the plane for random metrics_ , Proc. Amer. Math. Soc. 133 (2005) 133-143.
* [7] A. Iosevich and I. Laba, _$K$ -distance, Falconer conjecture, and discrete analogs_, Integers, Electronic Journal of Combinatorial Number Theory, Proceedings of the Integers Conference in honor of Tom Brown, (2005) 95–106.
* [8] A. Iosevich and M. Rudnev, _On distance measures for well-distributed sets_ , Journal of Discrete and Computational Geometry, 38, (2007).
* [9] A. Iosevich and M. Rudnev, _Erdös distance problem in vector spaces over finite fields_ , Trans. Amer. Math. Soc. 359 (2007), 6127-6142.
* [10] A. Iosevich and M. Rudnev, _Freiman’s theorem, Fourier transform, and additive structure of measures_ , Journal of the Australian Mathematical Society, 86, (2009), 97–109.
* [11] D. Koh and C. Shen, _Sharp extension theorems and Falconer distance problems for algebraic curves in two dimensional vector spaces over finite fields_ , preprint (2010), arxiv.org.
* [12] D. Koh and C. Shen, _The generalized Erdös-Falconer distance problems in vector spaces over finite fields_ , preprint, arxiv.org.
* [13] P. Mattila, _Spherical averages of Fourier transforms of measures with finite energy: dimension of intersections and distance sets,_ Mathematika 34(1987), 207–228.
* [14] I. Shparlinski, _On the set of distance between two sets over finite fields_ , International Journal of Mathematics and Mathematical Sciences Volume 2006, Article ID 59482, Pages 1–5.
* [15] V. Vu, _Sum-product estimates via directed expanders_ , Math. Res. Lett. 15 (2008), 375–388.
* [16] T. Wolff, _Decay of circular means of Fourier transforms of measures,_ Internat. Math. Res. Notices 1999, 547–567.
|
arxiv-papers
| 2010-10-08T05:56:21 |
2024-09-04T02:49:13.607600
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Doowon Koh and Chun-Yen Shen",
"submitter": "Doowon Koh",
"url": "https://arxiv.org/abs/1010.1597"
}
|
1010.1623
|
# Statistical Properties of Ideal Ensemble of Disordered 1$D$ Steric Spin-
Chains
A. S. Gevorkyan g˙ashot@sci.am Institute for Informatics and Automation
Problems, NAS of Armenia H. G. Abajyan habajyan@ipia.sci.am Institute for
Informatics and Automation Problems, NAS of Armenia H. S. Sukiasyan
haikarin@netsys.am Institute of Mathematics, NAS of Armenia
###### Abstract
The statistical properties of ensemble of disordered 1$D$ steric spin-chains
(SSC) of various length are investigated. Using 1$D$ spin-glass type classical
Hamiltonian, the recurrent trigonometrical equations for stationary points and
corresponding conditions for the construction of stable 1$D$ SSCs are found.
The ideal ensemble of spin-chains is analyzed and the latent interconnections
between random angles and interaction constants for each set of three nearest-
neighboring spins are found. It is analytically proved and by numerical
calculation is shown that the interaction constant satisfies Lev́y’s alpha-
stable distribution law. Energy distribution in ensemble is calculated
depending on different conditions of possible polarization of spin-chains. It
is specifically shown that the dimensional effects in the form of set of local
maximums in the energy distribution arise when the number of spin-chains
$M<<N^{2}_{x}$ (where $N_{x}$ is number of spins in a chain) while in the case
when $M\propto N^{2}_{x}$ energy distribution has one global maximum and
ensemble of spin-chains satisfies Birkhoff’s ergodic theorem. Effective
algorithm for parallel simulation of problem which includes calculation of
different statistic parameters of 1$D$ SSCs ensemble is elaborated.
neural networks, spin glass Hamiltonian, ergodic hypothesis, statistic
distributions, parallel simulation.
###### pacs:
71.45.-d, 75.10.Hk, 75.10.Nr, 81.5Kf
## I Formulation of the problem
Let us consider classical ensemble of disordered 1$D$ steric spin-chains
(SSC), where it is supposed that interactions between spin-chains are absent
(later it will be called an ideal ensemble) and that there are $N_{x}$ spins
in an each chain. Despite some ideality of the model it can be interesting
enough and rather convenient for investigation of a number of important and
difficult applied problems of physics, chemistry, material science, biology,
evolution, organization dynamics, hard-optimization, environmental and social
structures, human logic systems, financial mathematics etc (see for example
Young ; Bov ; Fisch ; Tu ; Chary ; Baake ). As was shown by authors spin-glass
model can be used for investigation of media’s properties on scales of space-
time periods of an external fields at conditions far from a usual equilibrium
of media gev .
Mathematically mentioned type of ideal ensemble can be generated by 1$D$
Heisenberg spin-glass Hamiltonian without external field Bind ; Mezard ; Young
:
Figure 1: A stable $1D$ steric spin-chain with random interactions and the
length of $L_{x}$. The spherical angles $\varphi$ and $\psi$ describe the
spatial orientation of $\textbf{\emph{S}}_{0}$ spin, the pair of angles
$(\varphi_{i},\psi_{i})$ correspondingly defines the spatial orientation of
the spin $\textbf{\emph{S}}_{i}$, the distance between two neighboring spins
in $1D$ lattice is $d_{0}$.
$\displaystyle
H_{0}(N_{x})=-\sum_{i=0}^{N_{x}-1}J_{i\,i+1}{\textbf{\emph{S}}}_{i}{\textbf{\emph{S}}}_{i+1}.$
(1)
where ${\textbf{\emph{S}}}_{i}$ describes the $i$-th spin which is a unit
length vector and has a random orientation. In the expression (1) $J_{i\,i+1}$
characterizes a random interaction constant between $i$ and $i+1$ spins, which
can have positive and negative values as well EdwAnd .
In other words we consider the mathematical model of spin-chains ensemble
where every spin-chain is like a regular 1$D$ lattice with the length
$L_{x}=d_{0}N_{x}$, where spins are put on nodes of lattice and interactions
between them are random (see FIG 1).
The distribution of spin-spin interaction constant $W(J)$ is chosen from
considerations of convenience and as a rule it is a Gauss-Edwards-Anderson
model EdwAnd (see also Bind ):
$\displaystyle
W(J)=\frac{1}{\sqrt{2\pi(\Delta{J})^{2}}}\exp\biggl{\\{}-\frac{\bigl{(}J-J_{0}\bigr{)}^{2}}{2(\Delta{J})^{2}}\biggr{\\}},$
(2)
where $J_{0}=\bigl{<}J\bigr{>}_{av}$ and
$\bigl{(}\Delta{J}\bigr{)}^{2}=\bigl{<}J^{2}\bigr{>}_{av}-\bigl{<}J\bigr{>}_{av}^{2}$.
Let us recall that $J_{0}$ and $\Delta{J}$ for this model are independent from
the distance and scaled with the spin number $N_{x}$ as:
$\displaystyle\bigl{<}J\bigr{>}_{av}=J_{0}\propto{N_{x}^{-1}},\qquad\Delta{J}\propto{N_{x}^{-1/2}},$
(3)
in order to ensure a sensible thermodynamic limit. $\bigl{<}...\bigr{>}_{av}$
in Eqs. (2) and (3) describes the averaging procedure. Below we will
investigate the issue of how much lawful the choice of this model is.
For further investigations it is useful to rewrite the Hamiltonian (1) in
spherical coordinates (see FIG 1):
$\displaystyle
H_{0}(N_{x})=-\sum_{i=0}^{N_{x}-1}J_{i\,i+1}\bigl{[}\cos\psi_{i}\cos\psi_{i+1}\cos(\varphi_{i}-\varphi_{i+1})+\sin\psi_{i}\sin\psi_{i+1}\bigr{]}.$
(4)
A stationary point of the Hamiltonian is given by the system of
trigonometrical equations:
$\frac{\partial{H_{0}}}{\,\,\partial\psi_{i}}=0,\qquad\qquad\frac{\partial{H_{0}}}{\,\,\partial\varphi_{i}}=0,$
(5)
where ${\Theta}_{i}=(\psi_{i},\varphi_{i})$ are angles of $i$-th spin in the
spherical coordinates system ($\psi_{i}$ is a polar and $\varphi_{i}$ is an
azimuthal angles),
$\mathbf{\Theta}=({\Theta_{1}},{\Theta_{2}}....{\Theta_{N_{x}}})$ respectively
describe the angular part of a spin-chain configuration.
Now using expression (4) and equations (5) it is easy to find the following
system of trigonometrical equations:
$\displaystyle\sum_{\nu=i-1;\,\,\nu\neq i}^{i+1}J_{\nu
i}\bigl{[}\sin\psi_{\nu}-\tan\psi_{i}\cos\psi_{\nu}\cos(\varphi_{i}-\varphi_{\nu})\bigr{]}=0,$
$\displaystyle\sum_{\nu=i-1;\,\,\nu\neq i}^{i+1}J_{\nu
i}\,\cos\psi_{\nu}\sin(\varphi_{i}-\varphi_{\nu})=0,\,\,\,\qquad\,J_{\nu
i}\equiv J_{i\nu}.$ (6)
In case when all the interaction constants between $i$-th spin with its
nearest-neighboring spins $J_{i-1\,i}$, $J_{i\,i+1}$ and angle configurations
$\bigl{(}\psi_{i-1},\varphi_{i-1}\bigr{)}$,
$\bigl{(}\psi_{i},\varphi_{i}\bigr{)}$ are known, it is possible to explicitly
calculate the pair of angles
${\Theta_{i+1}}=\bigl{(}\psi_{i+1},\varphi_{i+1}\bigr{)}$. Correspondingly,
the $i$-th spin will be in the ground state (in the state of minimum energy)
if in the stationary point
${\Theta_{i}^{0}}=\bigl{(}\psi_{i}^{0},\varphi_{i}^{0}\bigr{)}$ the following
conditions are satisfied:
$A_{\psi_{i}\psi_{i}}({\Theta_{i}^{0}})>0,\qquad
A_{\psi_{i}\psi_{i}}({\Theta_{i}^{0}})\,A_{\varphi_{i}\varphi_{i}}({\Theta_{i}^{0}})-A_{\psi_{i}\phi_{i}}^{2}({\Theta_{i}^{0}})>0,$
(7)
where
$A_{\alpha_{i}\alpha_{i}}({\Theta_{i}^{0}})={\partial^{2}{H_{0}}}/{\partial\alpha_{i}^{2}},\quad
A_{\alpha_{i}\beta_{i}}({\Theta_{i}^{0}})=A_{\beta_{i}\alpha_{i}}({\Theta_{i}^{0}})={\partial^{2}{H_{0}}}/{\partial\alpha_{i}\partial\beta_{i}}$,
in addition:
$A_{\psi_{i}\psi_{i}}({\Theta_{i}^{0}})\,=\,\biggl{\\{}\,\sum_{\nu=i-1;\,\,\nu\neq
i}^{i+1}J_{\nu
i}\bigl{[}\cos\psi_{\nu}\cos(\varphi_{\nu}-\varphi_{i}^{0})+\tan\psi_{i}^{0}\sin\psi_{\nu}\bigr{]}\biggr{\\}}\cos\psi_{i}^{0},\,\,\,\,$
$\displaystyle
A_{\varphi_{i}\varphi_{i}}({\Theta_{i}^{0}})=\biggl{\\{}\,\sum_{\nu=i-1;\,\,\nu\neq
i}^{i+1}J_{\nu
i}\cos\psi_{\nu}\cos(\varphi_{\nu}-\varphi_{i}^{0})\biggr{\\}}\cos\psi_{i}^{0},\qquad\qquad\qquad$
$\displaystyle
A_{\psi_{i}\phi_{i}}({\Theta_{i}^{0}})=\biggl{\\{}\,\sum_{\nu=i-1;\,\,\nu\neq
i}^{i+1}J_{\nu
i}\cos\psi_{\nu}\sin(\varphi_{\nu}-\varphi_{i}^{0})\,\biggr{\\}}\sin\psi_{i}^{0}.\qquad\qquad\qquad$
(8)
Taking into account the second equation in (6) we can reduce condition (7) to
the following kind:
$A_{\psi_{i}\psi_{i}}({\Theta_{i}^{0}})>0,\qquad\qquad
A_{\varphi_{i}\varphi_{i}}({\Theta_{i}^{0}})>0.$ (9)
So, with the help of Eq.s (6) and conditions (9) huge number of stable $1D$
SSCs may be calculated and on its basis it is possible to further construct
the statistical properties of 1$D$ SSCs ensemble. It is important to note that
the average polarization of 1$D$ SSCs ensemble is supposed to be equal to
zero.
Now we can construct the distribution function of energy in 1$D$ SCCs
ensemble. To this effect it is useful to divide the nondimensional energy axis
$\varepsilon=\epsilon/\delta\epsilon$ into regions
$0>\varepsilon_{0}>...>\varepsilon_{n}$, where $n>>1$ and $\epsilon$ is a real
energy axis. The number of stable 1$D$ SSC configurations with length of
$L_{x}$ in the range of energy
$[\varepsilon-\delta\varepsilon,\varepsilon+\delta\varepsilon]$ will be
denoted by $M_{L_{x}}(\varepsilon)$ while the number of all stable 1$D$ SSC
configurations - correspondingly by symbol
$M_{L_{x}}^{full}=\sum_{j=1}^{n}M_{L_{x}}(\varepsilon_{j})$. Accordingly, the
energy distribution function into the 1$D$ SSCs ensemble may be defined by
expressions:
$F_{L_{x}}(\varepsilon;d_{0}(T))=M_{L_{x}}(\varepsilon)/M_{L_{x}}^{full},$
(10)
where distribution function is normalized to unit:
$\lim_{n\to\infty}\sum^{n}_{j=1}F_{L_{x}}(\varepsilon_{j};d_{0}(T))\delta\varepsilon_{j}=\int^{\,0}_{-\infty}F_{L_{x}}(\varepsilon;d_{0}(T))d\varepsilon=1.$
By similar way we can define also distributions for polarization and for a
spin-spin interaction constant.
## II Algorithm of 1$D$ SSCs Ideal Ensemble Simulation
Now our aim is elaboration of algorithm for parallel simulation of ideal
ensemble of $1D$ SSCs.
Using equations (6) for stationary points of Hamiltonian $H_{0}(N_{x})$ we can
find the following equations system:
$\displaystyle
J_{i-1\,i}\bigl{[}\sin\psi_{i-1}-\tan\psi_{i}\cos\psi_{i-1}\cos(\varphi_{i}-\varphi_{i-1})\bigr{]}+J_{i\,i+1}\bigl{[}\sin\psi_{i+1}$
$\displaystyle-\tan\psi_{i}\cos\psi_{i+1}\cos(\varphi_{i}-\varphi_{i+1})\bigr{]}=0,$
$\displaystyle
J_{i-1\,i}\,\cos\psi_{i-1}\sin(\varphi_{i}-\varphi_{i-1})\,+J_{i+1\,i}\,\cos\psi_{i+1}\sin(\varphi_{i}-\varphi_{i+1})=0.$
(11)
After designations:
$x=\cos\psi_{i+1},\qquad y=\sin(\varphi_{i}-\varphi_{i+1}),$ (12)
the system (11) may be transformed to the following form:
$\displaystyle
C_{1}+J_{i\,i+1}\bigl{[}\sqrt{1-x^{2}}-\tan\psi_{i}\,x\sqrt{1-y^{2}}\bigr{]}=0,\qquad
C_{2}+J_{i\,i+1}\,x\,y=0,$ (13)
where parameters $C_{1}$ and $C_{2}$ are defined by expressions:
$\displaystyle
C_{1}=J_{i-1\,i}\bigl{[}\sin\psi_{i-1}-\tan\psi_{i}\cos\psi_{i-1}\cos(\varphi_{i}-\varphi_{i-1})\bigr{]},$
$\displaystyle
C_{2}=J_{i-1\,i}\cos\psi_{i-1}\sin(\varphi_{i}-\varphi_{i-1}).\qquad\qquad\qquad\qquad\,\,$
(14)
From the system of equations (13) we can find the equation for the unknown
variable $y$:
$C_{1}y+C_{2}\sqrt{1-y^{2}}\tan\psi_{i}+\sqrt{J_{i\,i+1}^{2}y^{2}-C_{2}^{2}}=0.$
(15)
We can transform the equation (15) to the following equation of fourth order:
$\bigl{[}A^{2}+4C_{1}^{2}C_{2}^{2}\sin\psi_{i}\bigl{]}y^{4}-2\bigl{[}AC_{2}^{2}+2C_{1}C_{2}^{2}\sin^{2}\psi_{i}\bigr{]}y^{2}+C_{2}^{4}=0,$
(16)
where
$A=J^{2}_{i\,i+1}\cos^{2}\psi_{i}-C_{1}^{2}+C_{2}^{2}\sin^{2}\psi_{i}.$ (17)
Discriminant of equation (16) is equal to:
$D=C_{2}^{4}\bigl{(}A+2C_{1}\sin^{2}\psi_{i}\bigr{)}^{2}-C_{2}^{4}\bigl{(}A^{2}+4C_{1}^{2}C_{2}^{2}\sin^{2}\psi_{i}\bigr{)}$
$=4C^{4}_{2}C^{2}_{1}\sin^{2}\psi_{i}\bigl{(}A+C^{2}_{1}\sin^{2}\psi_{i}-C^{2}_{2}).\qquad\qquad$
From the condition of nonnegativity of discriminant $D\geq 0$ we can find the
following condition:
$A+C^{2}_{1}\sin^{2}\psi_{i}-C^{2}_{2}\geq 0.$ (18)
Further substituting the value of $A$ from (17) into (18) we can find the new
condition to which the interaction constant between two successive spins
should satisfy:
$J_{i\,i+1}^{2}\geq C^{2}_{1}+C^{2}_{2}.$ (19)
Now we can write the following expressions for unknown variables $x$ and $y$:
$\displaystyle x^{2}$ $\displaystyle=\frac{C_{2}^{2}}{J_{i\,i+1}^{2}y^{2}},$
$\displaystyle y^{2}$
$\displaystyle=C_{2}^{2}\,\frac{\cos^{2}\psi_{i}J_{i\,i+1}^{2}\pm
2C_{1}\sin\psi_{i}\cos\psi_{i}\sqrt{J_{i\,i+1}^{2}-C_{1}^{2}-C_{2}^{2}}+C_{3}+2C_{1}^{2}\sin^{2}\psi_{i}}{\cos^{4}\psi_{i}J_{i\,i+1}^{4}+2C_{3}\cos^{2}\psi_{i}J_{i\,i+1}^{2}+(C_{1}^{2}+\sin^{2}\psi_{i}C_{2}^{2})^{2}},$
(20)
where $C_{3}=-C_{1}^{2}+C_{2}^{2}\,\sin^{2}\psi_{i}.$
Finally taking into account designations (12) we can find new conditions of
restriction of the calculated angles
$\bigl{(}\varphi_{i+1},\psi_{i+1}\bigr{)}$:
$0\leq x^{2}\leq 1,\qquad 0\leq y^{2}\leq 1.$ (21)
These conditions are very important for elaborating correct and effective
algorithm for numerical simulations.
### II.1 Algorithm description
This is parallel algorithm for simulation of 1$D$ SSCs ensemble, which
consists of separate iterative calculations of nodes in 1$D$ SSC. The first
and second nodes are initialized randomly, then $i$-th node is obtained from
$(i-2)$-th and $(i-1)$-th layers nodes. Every node contains the following
information:
$\varphi$-polar angle,
$\psi$-azimuthal angle,
$J$-interaction coefficient,
The following parameters are initializes in the following way:
$\varphi_{0}$ and $\varphi_{1}$ \- rand()${}^{\ast}2^{\ast}\pi^{\ast}R$;
$\psi_{0}$ and $\psi_{1}$ \- acos (rand());
$J_{0\,1}$ \- rand();
where rand() function generates uniformly distributed random numbers on the
interval $(0,1)$.
The algorithm pseudo-code is following:
// generate $n$ separate independent sets of problem in parallel
for $i=1:N_{x}$
for $j=1:R$ // regenerate $J_{i}$ maximum $R$ times if needed
for $k=1:L_{i}$ // go through all elements in the $i$-th layer if conditions
// (9) are satisfied
begin
// calculate energy on $i$-th layer,
// calculate polarization on $x,y$ and $z$-axis
// calculate $x_{i+1}$ and $y_{i+1},$
// save $J_{i}$ value
. . . .
end
endfor
endfor
endfor
if ($i==N_{x}$) // reached the $N_{x}$-th layer
begin
// save energy, polarizations values
end
endif
// construct distribution functions of energy $\varepsilon$, polarization $p$
and
// interaction constant $J$
// calculate the mean value of energy $\bar{\varepsilon}$, polarization
$\bar{p}$, interaction constant $\bar{J}$ and
// its variance $\bar{J^{2}}$.
## III Numerical Simulation
We will consider an ideal ensemble of 1$D$ SSCs which consists of $M$ number
of spin-chains each of them with the length 25$d_{0}$. For realization of
parallel simulation we will use algorithm A (see FIG 2).
The parallel algorithm works in the following way. Randomly $M$ sets of
initial parameters are generated and parallel calculations of equations (20)
for unknown variables $x$ and $y$ transact with taking into account conditions
(21). However only specifying of initial conditions is not enough for solution
of these equations. Evidently these equations can be solved after definition
of the constant $J_{0\,1}$, which is also randomly generated. In the case when
solutions are found then conditions of stability of spin in node (9) are
checked. The solution proceeds for the following spin if the specified
conditions (9) are satisfied. If conditions are not satisfied, a new constant
$J_{0\,1}$ is randomly generated and correspondingly new solutions are found
which are checked later on conditions (9). This cycle on each spin repeats
until the solutions do not satisfy to conditions of the minimum spin energy in
the node.
Figure 2: The algorithm of $1D$ SSCs of ideal ensemble parallel simulation of
statistical parameters.
At first we have conducted numerical simulation for definition of different
statistical parameters of the ensemble which consists of $10^{2}$ spin-chains.
Let us recall that the number of simulation of spin-chains define the number
of spin-chains in the ensemble. As the simulation shows (see the left picture
in FIG 3) the energy distribution function has a set of local maximums
($\varepsilon^{(0)},...,\varepsilon^{(m)})$. Obviously they are dimensional
effects and are similar to the first-order phase transitions which often
happen in spin-glass systems Bind ).
Figure 3: The energy distribution where there are apparently many local
minimum of energy for ensemble of 1$D$ SSCs with the length of
$L_{x}=25d_{0}$, which consists of $10^{2}$ spin-chains (the left picture). On
the right picture polarization distributions of ensemble on coordinates $x,y$
and $z$ are shown. Figure 4: In the left picture is shown the energy
distribution in the ensemble of 1$D$ SSCs with the length of $L_{x}=25d_{0}$,
which consists of 2$\cdot 10^{3}$ spin-chains. Apparently, the number of local
minimums of energy is promptly reduced comparing with the increase of spin-
chains. On the right picture polarization distributions of ensemble on
coordinates $x,y$ and $z$ are shown.
Let us note that during simulation we suppose that spin-chains can be
polarized up to 20 percent i.e. the total value of spins sum in each chain can
be in an interval of $-5\leq p\leq 5$, where $p$ designates the polarization
of spin-chain. In other words each spin-chain is a vector of certain length
which is directed to coordinate $x$. As calculations show, in the ensemble
consisting of a small number of spin-chains, for example, of the order
$10^{2}$, the self-averaging of spin-chains does not occur in full measure
i.e. the total polarization of an ensemble differs from zero:
$p_{x}=-0.33099,\,p_{y}=-0.035191,\,p_{z}=-0.024543$ where
$p=\int_{-\infty}^{+\infty}F(p)dp$, where it is supposed that
$p=(p_{x},\,p_{y},\,p_{z})$. In this case the average energy of an ensemble is
equal to $\bar{\varepsilon}=-14.121$, where
$\bar{\varepsilon}=\int_{-\infty}^{0}F(\varepsilon)\varepsilon d\varepsilon$.
For the ensemble which consists of $2.10^{3}$ spin-chains (see FIG 4), the
dimensional effects practically disappear. The summary polarization of
ensemble in this case is very small:
$p_{x}=-0.020538,\,p_{y}=-0.047634,\,p_{z}=-0.12687$ and correspondingly the
average energy of $1D$ SSC is equal to $\bar{\varepsilon}=-13.603$.
Ensemble which consists of $10^{4}$ spin-chains has an energy distribution
$F(\varepsilon)$ with one global maximum (see Fig 5). As to polarization
distributions, $F(p_{x})$ $F(p_{y}),$ and $F(p_{z})$, in the considered case
are obviously very symmetric in comparison with similar distributions of
previous ensembles (see FIG 3 and Fig 4). The average values of polarizations
on coordinates for this ensemble are much smaller
$p_{x}=-0.0072863,\,p_{y}=-0.014242,\,p_{z}=-0.018387$, correspondingly the
average energy is equal to $\bar{\varepsilon}=-13.634$. Thus in the case when
ensemble consists of a big number of spin-chains, the self-averaging of spin-
chains system occurs with high accuracy. Whereas the summation procedure on
the number of spins in chain or spin-chains ensemble is similar to the
procedure of averaging by the natural parameter or ”timing” in the dynamical
system, it is possible to introduce the concept of ergodicity for the both
separate spin-chains and ensemble as a whole.
Figure 5: The energy distribution and its fitted curve (left picture) in
ensemble of 1$D$ SSCs with the length of $L_{x}=25d_{0}$, which consists of
$10^{4}$ spin-chains. Evidently there is only one global maximum for energy
distribution. In the right picture polarization distributions are shown
correspondingly on coordinates $x,y$ and $z$.
Thus as calculations show Birkhoff ergodic hypothesis Birkhoff may be used
for ensembles which consist of $M\sim N_{x}^{2}$ spin-chains in order to
change the summation of spin-chains on the integration by the energy
distribution of the ensemble. The energy distribution of ensemble does not
depend on the length of the spin-chain in the limit of ergodicity and it can
be fitted very precisely with Eckart function Eckart (see FIG 5, the smooth
$F(\varepsilon)=C(a,b,c,\gamma)\biggl{\\{}\frac{a}{b+e^{-2\gamma\varepsilon}}+\frac{c\gamma^{2}}{(e^{-\gamma\varepsilon}+e^{\gamma\varepsilon})^{2}}\biggr{\\}},$
(22)
where $a,b,c$ and $\gamma$ some constants, in addition $C$ is a normalization
constant and can be found from the condition:
$\int_{-\infty}^{0}F(\varepsilon)d\varepsilon=1.$ (23)
By placing (22) into (23) we can find:
$C^{-1}(a,b,c,\gamma)=\frac{a}{2b\gamma}\ln(1+b)+\frac{c\gamma}{4}.$ (24)
After fitting the energy distribution by means of analytical function (22) we
find values of constants by entering into the function:
$a=131.4,\,b=3138.2,\,\,c=-1.20344$ and $\gamma=0.162174.$
Figure 6: The energy distributions for ensembles consisting of 1$D$ SSCs of
the length $L_{x}=25d_{0}$, with spin-chains polarization correspondingly up
to $20,\,40$ and $100$ percents (left picture). Note that all the ensembles
consist of $10^{4}$ spin-chains and their distributions practically do not
differ. On the right picture the distribution of the spin-spin interaction
constant is shown which differs essentially from Gauss-Edwards-Anderson
distribution model (2).
We have also calculated 1$D$ SSCs ensemble with the length of spin-chains
25$d_{0}$ and correspondingly with polarizations of spin-chains up to 20, 40,
and 100 percents (see Fig 6, the left picture). In particular, as it follows
from the picture the energy distribution does not depend on the degree of
spin-chains polarization. We also have conducted simulation of ensembles which
consist of spin-chains with lengths $100d_{0}$ and $1000d_{0}$
correspondingly. As the numerical modeling shows, statistical properties of
ensembles are similar. In the considered cases distributions of energy
concentrate correspondingly on scales $100d_{0}$ and $1000d_{0}$. Limits of
ergodicities of ensembles are also investigated and it is shown that in these
cases too it is of an order $N_{x}^{2}$.
Finally it is important to note that the distribution of spin-spin interaction
constant is not defined apriori with the help of expression (2) but with the
mass calculations of equations (6). On the basis of the obtained numerical
data, the distribution of interaction constant $W(J)\equiv F(J)$ is
constructed (see Fig 6, the right picture) from which it follows, that it
essentially differs from the Gauss-Edwards-Anderson distribution model (2).
The obtained distribution relatively is well fitted by the normalized to the
unit of nonsymmetric Cauchy function Spiegle :
$F(J)=\frac{g+\beta J}{\pi\bigl{[}g^{2}+(J-a_{0})^{2}\bigr{]}}.$ (25)
where $g,\,\beta$ and $a_{0}$ are some adjusting parameters which are found
from the condition of a good approximation of the data visualization curve. In
the considered case they are correspondingly equal to:
$g=0.27862,\,\beta=0.009$ and $a_{0}=0.083236$. Nevertheless, as the detailed
analysis of curve of numerical data visualization shows (in particular its
asymptotes) the distribution of interaction constant can be approximated
precisely by Lev́y skew alpha-stable distribution function. Let us recall that
Lev́y skew alpha-stable distribution is a continuous probability and a limit
of certain random process $X(\alpha,\beta,\gamma,\delta;k)$ where parameters
describe correspondingly: an index of stability or characteristic exponent
$\alpha\in(0;2]$, a skewness parameter $\beta\in[-1;1]$, a scale parameter
$\gamma>0$, a location parameter $\delta\in\mathbb{R}$ and an integer $k$
shows the certain parametrization (see in more detailed references Ibragimov ;
Nolan ). Let us note, that the mean of distribution and its variance are
infinite. However, taking into account that spin-spin interaction constant has
limited value in real physical systems, it is possible to calculate
distribution mean and its variance. In particular if $J\in[-5,+5]$ then
$\overline{J}=0.50113$ and $\overline{J^{2}}=2,1052$.
## IV Conclusion
The investigation of statistical properties of classical spin-glass system of
various sizes is very important for understanding possibilities of effective
influence and control over parameters of medium with the help of weak external
fields. Evidently, when we put the spin-glass in external field the space-time
periods define scales on which probably an essential changes in medium occur.
For simplicity we suppose that the spin-glass system is an ensemble which
consists of disordered 1$D$ steric spin-chains of $L_{x}$ lengths, between
which interaction is absent (ideal ensemble). This type of classical ensemble
is described by Heisenberg Hamiltonian (2). We have researched conditions of
arising of stable spin-chains Eqs. (11) and nonequalities (9) and found a
latent connection between random variables (see expression (19)), which shows
that the distribution for spin-spin interaction constant can not be described
by Guss-Edwards-Anderson model. In the result of equations of stationary
points analysis (11) we have found system of recurrent equations (20) and new
conditions (21). On the basis of obtained mathematical formulas the effective
parallel algorithm for numerical simulation is developed which was realized on
the example of the ensemble which consists of 1$D$ SSCs with length 25$d_{0}$.
Similar to the dynamical systems, we have introduced the idea of Birkhoff
ergodic hypothesis Birkhoff for the statical spin-glass systems. In this case
the number of spin-chains of ensemble plays a role of the natural or ”timing”
parameter of the system. Numerical simulations show that the ergodic
hypothesis may be used for the case when ensemble consists of $M\propto
N_{x}^{2}$ spin-chains in order to change the summation of spin-chains on the
integration by the energy (polarization, etc.) distribution of the ensemble.
In particular, we have made numerical experiments for ensembles which include
$10^{2}$, $2\cdot 10^{3}$ and $10^{4}$ spin-chains. As it was shown by
simulations in the case when $M\ll N_{x}^{2}$ for an ensemble, they are
characteristic dimensional effects in energy distribution (the left picture on
FIG 3). When the number of spin-chains is $M$ of order $2\cdot 10^{3}$ or more
$10^{4}$, dimensional effects disappear and correspondingly energy
distribution functions have one global maximum (see left pictures on FIG 4 and
FIG 5). As it was shown, when increasing spin-chains number, the total and
partial polarizations of the ensemble disappear. Let us note, that at
modelling by algorithm (see scheme on FIG 2) condition (19) specifies the
region of localization of random interaction constant $J_{i\,i+1}$ which
depends on angular configurations $(i-1)$-th and $i$-th spins and interaction
constant $J_{i-1\,i}$ between them. As a result, it allows to accelerate
calculations of each spin-chain and hence the speed of parallel calculations
of ensemble is increased essentially.
Finally it is important to note that it is proved, that the spin-spin
interaction constant $J_{i\,i+1}$ has a form of Lev́y skew alpha-stable
distribution (see the right picture on FIG 6). The considered scheme of
solution of 1$D$ steric spin-glass problem can be used in different applied
fields (see e.g. Helmut ). It can also be useful for analyzing 3$D$ spin-glass
problem and creation of an effective parallel simulation algorithm of the
spin-glass system with large dimensionality.
## References
* (1) K. Binder and A. P. Young, Spin glasses: Experimental facts, theoretical concepts, and open questions. Rev. Mod. Physics, 58(4), 801-976 (1986).
* (2) M. Mézard, G. Parisi, M. A. Virasoro, Spin Glass Theory and Beyond (World Scientific, Singapore, 1987)
* (3) A. P. Young (ed.), Spin Glasses and Random Fields (World Scientific, Singapore, 1998)
* (4) R. Fisch and A. B. Harris, Spin-glass model in continuous dimensionality, Phys. Rev. Let., 47, 620 (1981).
* (5) A. Bovier, Statistical Mechanics of Disordered Systems: A Mathematical Perspective, Cambridge Series in Statistical and Probabilistic Mathematics, p 308 (2006).
* (6) Y. Tu, J. Tersoff and G. Grinstein, Structure and Energetic of the $Si$ and $SiO_{2}$ Interface, Phys. Rev. Lett., 81, 4899 (1998).
* (7) K. V. R. Chary, G. Govil, NMR in Biological Systems: From Molecules to Human (Focus on Structural Biology 6), Springer, p 511, (2008).
* (8) E. Baake, M. Baake and H. Wagner, Ising Quantum Chain is a Equivalent to a Model of Biological Evolution, Phys. Rev. Let., 78(3), 559-562 (1997.)
* (9) A S Gevorkyan et al., New Mathematical Conception and Computation Algorithm for Study of Quantum 3D Disordered Spin System Under the Influence of External Field, Trans. On Comput. Sci., VII, LNCS 132-153, Spinger-Verlage, 10.1007/978-3-642-11389-58
* (10) S. F. Edwards and P. W. Anderson, Theory of spin glasses, J. Phys. F 9, 965 (1975).
* (11) J. von Neuman, Physical Applications of the Ergodic Hypothesis, Proc. Nat. Acad. Sci. USA, 18(3): 263-266 (1932).
G. D. Birkhoff, What is ergodic theorem? American Mathematical Monthly, 49(4):
222-226 (1931).
* (12) S. Flügge, Practical Quantum Mechanics I, (Springer-Verlag, Berlin-Heidelberg- New York 1971).
* (13) M. R. Spiegle, Theory and Problems of Probability and Stochastics, (New-York, McGraw-Hill, pp 114-115, 1992).
* (14) I. Ibragimov and Yu. Linnik, Independent and Stationary Sequences of Random Variebles, (Wolters-Noordhoff Publishing Groningen, The Netherlands 1971).
* (15) J. P. Nolan, Stable Distributions: Models for Heavy Tailed Data (2009-02-21). $en.wikipedia.org/Stable_{/}distribution$.
* (16) H. G. Katzgraber, A. K. Hartmann and A. P. Young, New Insights from One-Dimensional Spin Glasses, (2008) ArXiv:0803.3417v1 [cond-mat.dis-nn].
|
arxiv-papers
| 2010-10-08T08:01:45 |
2024-09-04T02:49:13.615339
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ashot Gevorkyan, Hakob Abajyan and Haik Sukiasyan",
"submitter": "Ashot Gevorkyan S",
"url": "https://arxiv.org/abs/1010.1623"
}
|
1010.1845
|
# Navigation in non-uniform density social networks
Yanqing Hu, Yong Li, Zengru Di, Ying Fan111yfan@bnu.edu.cn Department of
Systems Science, School of Management and Center for Complexity Research,
Beijing Normal University, Beijing 100875, China
###### Abstract
Recent empirical investigations suggest a universal scaling law for the
spatial structure of social networks. It is found that the probability density
distribution of an individual to have a friend at distance $d$ scales as
$P(d)\propto d^{-1}$. Since population density is non-uniform in real social
networks, a scale invariant friendship network(SIFN) based on the above
empirical law is introduced to capture this phenomenon. We prove the time
complexity of navigation in 2-dimensional SIFN is at most $O(\log^{4}n)$. In
the real searching experiment, individuals often resort to extra information
besides geography location. Thus, real-world searching process may be seen as
a projection of navigation in a $k$-dimensional SIFN($k>2$). Therefore, we
also discuss the relationship between high and low dimensional SIFN.
Particularly, we prove a 2-dimensional SIFN is the projection of a
3-dimensional SIFN. As a matter of fact, this result can also be generated to
any $k$-dimensional SIFN.
Navigation, Non-uniform Population Density, Spatial Structure
## I Introduction
To understand the structure of the social networks in which we live is a very
interesting problem. As part of the recent surge of interest in networks,
there have been active research about social networksKleinberg-covergence ;
SWNature ; BA model ; Human travel ; Human travel commen ; Mobile travel
patten . Besides some well known common properties such as small-world and
community structurefull model navigation ; Givens-Newman ; Newman-review ,
much attention has been dedicated to navigation in real social networks.
In the 1960s, Milgram and his co-workers conducted the first small-world
experiment the oldest experiment . Randomly chosen individuals in the United
States were asked to send a letter to a particular recipient using only
friends or acquaintances. The results of the experiment reveal that the
average number of intermediate steps in a successful chain is about six. Since
then, “six degrees of separation” has became the subject of both experimental
and theoretical researchNews ; Play . Recently, Dodds et al carried out an
experiment study in a global social network consisting about 60,000 email
users recent experiment . They estimated that social navigation can reach
their targets in a median of five to seven steps, which is similar to the
results of Milgram’s experiment.
The first theoretical navigation model was proposed by Kleinbergnavigation
brief nature ; navigation full . He introduced an $n\times n$ lattice to model
social networks. In addition to the links between nearest neighbors, each node
$u$ is connected to a random node $v$ with a probability proportional to
$d(u,v)^{-r}$, where $d(u,v)$ denotes the lattice distance between $u$ and
$v$. Kleinberg has proved that the optimal navigation can be obtained when the
power-law exponent $r$ equals to $d$, where $d$ is the dimensionality of the
lattice, and the time complexity of navigation in that case is at most
$O(\log^{2}n)$. Since then, much attention has been dedicated to Kleinberg’s
navigation modelAnalyzing kleinberg ; Oskar licentiate thesis ; efficient
routing . Roberson et al. studied the navigation problem in fractal networks,
where they proved that $r=d$ was also the optimal power-law exponent in the
fractal caseRob06 . Carmi, Cartozo and their cooperators have provided exact
solutions respectively for the asymptotic behavior of Kleinberg’s navigation
modelAsymptotic behavior ; Extended Navigability . More recently, the
navigation probolem with a total cost restriction has also been discussed,
where the cost denotes the length of the long-range connectionsTotal cost ;
Total cost1 .
Meanwhile, recent empirical investigations suggest a universal spatial scaling
law on social networks. Liben-Nowell et al explored the role of geography
alone in routing messages within the LiveJournal social networkUse Kleinberg
search . They found that the probability density function (PDF) of geographic
distance $d$ between friendship was about $P(d)\propto d^{-1}$. Adamic and Ada
also observed the $P(d)\propto d^{-1}$ law when investigating the Hewlett-
Packard Labs email networkpower-law networks Kleinberg search . Lambiotte et
al analyzed the statistical properties of a communication network constructed
from the records of a mobile phone company Renaud Lambiotte . Their empirical
results showed that the probability that two people $u$ and $v$ living at a
geographic distance $d(u,v)$ were connected by a link was proportional to
$d(u,v)^{-2}$. Because the number of nodes having distance $d$ to any given
node is proportional to $d$ in 2-dimensional world, so the probability for an
individual to have a friend at distance $d$ should be $P(d)\propto d\cdot
d^{-2}=d^{-1}$. More recently, Goldenberg et al studied the effect of IT
revolution on social interactionsdistance . Through analyzing an extensive
data set of the Facebook online social network, they pointed out that social
communication decrease inversely with the distance $d$ following the scaling
law $P(d)\propto d^{-1}$ as well.
Such as in the LiveJournal social network, population density is non-uniform
in real social networksUse Kleinberg search . To deal with the navigation
problem with non-uniform population density, a scale invariant friendship
network (SIFN for short) model based on the above spatial scaling law
$P(d)\propto d^{-1}$ of social networks is proposed in this paper. We prove
the time complexity of navigation in a 2-dimensional SIFN is at most
$O(\log^{4}n)$, which indicates social networks is navigable. Dodds et al have
pointed out that individuals often resort to extra information such as
education and professional information besides geography location in the real
searching experimentrecent experiment . Considering this phenomenon,
navigation process in real world may be seen as the projection of navigation
in a higher dimensional SIFN. Therefore, we further discuss the relationship
between high and low dimensional SIFN. Particularly, we prove that a
2-dimensional SIFN can be seen as the projection of any $k$-dimensional
SIFN($k>2$) through theoretical analysis.
## II Navigation In Non-uniform Density Social Networks
To deal with the non-uniform population density in real social networks, we
divide the whole population into small areas and give the following two
assumptions. First, the population density is uniform in each small area.
Second, the minimum population density among the areas is $m$, while the
maximum is $M$. We set $m>0$ to guarantee that a searching algorithm can
always make some progress toward any target at every step of the chain.
Like Kleinberg’s network (KN for short) and Liben-Nowell’s rank-based
friendship network (RFN for short), we employ an $n\times n$ lattice to
construct SIFN. Without loss of generality, we assume each node $u$ has $q$
directed long-range connections, where $q$ is a constantnavigation full . To
generate a long-range connection of node $u$, we first randomly choose a
distance $d$ according to the observed scaling law $P(d)\propto d^{-1}$ in
social networks. Then randomly choose a node $v$ from the node set, whose
elements have the same lattice distance $d$ to node $u$, and create a directed
long-range connection from $u$ to $v$. The lattice is assumed to be large
enough that the long-range connections will not overlap.
For simplicity, we set $q=1$. Let $S$ denote the set of all nodes, then the
probability that $u$ chooses $v$ as its long-rang connection in SIFN can be
given by eq.(1).
$Pr_{\text{SIFN}}(u,v)=\frac{1}{c(u,v)}\frac{d(u,v)^{-1}}{\sum_{d=1}^{n}d^{-1}}$
(1)
where $c(u,v)=|\\{x|d(u,x)=d(u,v),x\in S\\}|$ and $d(u,v)$ denotes the lattice
distance between nodes $u$ and $v$. Likewise, the probability that $u$ chooses
$v$ as its long-rang connection in KN and RFN are given respectively by eq.(2)
and eq.(3).
$Pr_{\text{KN}}(u,v,r)=\frac{d(u,v)^{-r}}{\sum_{w\neq u}d(u,w)^{-r}}$ (2)
$Pr_{\text{RFN}}(u,v)=\frac{rank_{u}(v)^{-1}}{\sum_{w\neq u}rank_{u}(w)^{-1}}$
(3)
where $rank_{u}(v)=|\\{w|d(u,w)<d(u,v),x\in S\\}|$ denotes the number of nodes
within distance $d(u,v)$ to node $u$ in RFNnavigation full ; Use Kleinberg
search . Notice that, the number of nodes with a distance $d(u,v)$ in a
$k$-dimensional($k>1$) lattice is proportional to $d{(u,v)^{k-1}}$. Thus, a
node $u$ connects to node $v$ with probability proportional to $d(u,v)^{-a}$
does not mean $P(d)\propto d^{-a}$ but $P(d)\propto d^{-a+k-1}$ instead.
Therefore, ${Pr_{\text{KN}}(u,v,k)}$, ${Pr_{\text{SIFN}}(u,v)}$ and
$Pr_{\text{RFN}}(u,v)$ are exactly the same for any $k$-dimensional lattice
based network when population density is uniform. However, SIFN always
satisfies the empirical results $P(d)\propto d^{-1}$ in social networks
compared with KN and RFN. Further,
${Pr_{KN}(u,v,k)}$,${Pr_{\text{SIFN}}(u,v)}$ and $Pr_{\text{RFN}}(u,v)$ can be
quite different when the population density is non-uniform.
Figure 1: Two strategies of sending message in a 2-dimensional SIFN. Strategy
$\mathcal{A}$, send the message directly to target $t$ from the current
message holder using Kleinberg’s greedy routing strategy. At each step, the
message is sent to one of its neighbors who is most close to the target in the
sense of lattice distance. Strategy $\mathcal{B}$, the message is first sent
to a given node $j$ using Kleinberg’s greedy strategy and then to the target
node $t$ using the same strategy. Suppose we start from a source node $s$,
after one step, the message reaches nodes $A_{1}$ and $B_{1}$ respectively
with strategy $\mathcal{A}$ and $\mathcal{B}$. Consider $B_{1}$ as the new
source node, then we should get $A_{2}$ and $B_{2}$ respectively with
strategies $\mathcal{A}$ and $\mathcal{B}$ in the next step.
Since our 2-dimensional SIFN captures the non-uniform population density
property in the real social networks, we purposefully divide the navigation
process into two stages for simplicity. First send messages inside a small
area and then among the areas. To analyze the time complexity of navigation in
a 2-dimensional SIFN, we first compare the following two searching strategies
as shown in FIG.1. Strategy $\mathcal{A}$, send the message directly to target
$t$ from the current message holder using Kleinberg’s greedy routing strategy.
At each step, the message is sent to one of its neighbors who is most close to
the target in the sense of lattice distance. Strategy $\mathcal{B}$, the
message is first sent to a given node $j$ using Kleinberg’s greedy strategy
and then to the target node $t$ using the same strategy. It can be proved that
strategy $\mathcal{A}$ performs better than strategy $\mathcal{B}$ on average.
Suppose we start sending message from the source node $s$, the message reaches
nodes $A_{1}$ and $B_{1}$ respectively with strategy $\mathcal{A}$ and
$\mathcal{B}$ after one step. It is always correct that lattice distance
$d(A_{1},t)$ $\leq$ $d(B_{1},t)$, because greedy routing strategy always
choose the node most close to target $t$ from its neighbors. According to the
results ofAsymptotic behavior ; efficient routing , the longer the distance
between a source and a given target, the more is the expected steps. Thus we
should have $T(A_{1}\rightarrow t)\leq T(B_{1}\rightarrow t)$, where
$T(A_{1}\rightarrow t)$ and $T(B_{1}\rightarrow t)$ denote the expected
delivery time to target $t$ from $A_{1}$ and $B_{1}$ respectively.
Let $T(s\rightarrow j\rightarrow t)$ denote the expected delivery time from
$s$ to $t$ via a transport node $j$, then we have $T(s\rightarrow t)\leq
T(s\rightarrow B_{1}\rightarrow t)$. Consider $B_{1}$ as a new source node,
then message will reach $A_{2}$ and $B_{2}$ with strategies $\mathcal{A}$ and
$\mathcal{B}$ respectively in the next step. Following the same deduction, we
have $T(B_{1}\rightarrow A_{2}\rightarrow t)\leq T(B_{1}\rightarrow
B_{2}\rightarrow t)$. Repeat this process until the message reaches the given
node $j$ with strategy $\mathcal{B}$, then we should have a monotone
increasing sequence of expected delivery time {$T(s\rightarrow
B_{1}\rightarrow t)$, $T(s\rightarrow B_{2}\rightarrow t)$ , $\cdots$,
$T(s\rightarrow j\rightarrow t)$ }. Therefore, we can obtain $T(s\rightarrow
t)$ $\leq$ $T(s\rightarrow j\rightarrow t)$, which means strategy
$\mathcal{A}$ is better than strategy $\mathcal{B}$. This analysis can be
extended to any $k$-dimensional SIFN.
Based on the first assumption and the fact that SIFN is identical to KN when
population density is uniform, the expected steps spent in each small area
using Kleinberg greedy algorithm is at most $O(\log^{2}n)$. Consider each
small area as a node, we will get a new 2-dimensional weighted lattice. The
weight (population) of the nodes is between $m$ and $M$ based on the second
assumption. Thus we have
$c\frac{m}{M}d^{-1}\leq Pr_{\text{SIFN}}^{{}^{\prime}}(u,v)\leq
c\frac{M}{m}d^{-1}$ (4)
where $c$ is a constant and $Pr_{\text{SIFN}}^{{}^{\prime}}(u,v)$ represents
the probability that area $u$ is connected to area $v$ in the new weighted
lattice.
We say that the execution of greedy algorithm is in phase $j$ ($j>0$) when the
lattice distance from the current node to target $t$ is greater than $2^{j}$
and at most $2^{j+1}$. Obviously, we have
$\sum\limits_{d=1}^{n}{d^{-1}}\leq 1+\int\limits_{1}^{n}{x^{-1}dx=1+\log
n<2\log n}.$ (5)
Further, we define $B_{j}$ as the node set whose elements are within lattice
distance $2^{j}+2^{j+1}<2^{j+2}$ to $u$. Let $|B_{j}|$ denote the number of
nodes in set $B_{j}$, we should have
$|B_{j}|>1+\sum\limits_{i=1}^{2^{j}}{i>2^{2j-1}}.$ (6)
Suppose that the message holder is currently in phase $j$, then the
probability that the node is connected by a long-range link to a node in phase
$j-1$ is at least $(Mm^{-1}2\log n\cdot 4\cdot 2^{2j+4})^{-1}$. The
probability $\psi(x)$ to reach the next phase $j-1$ in more than $x$ steps can
be given by
$\psi(x)={(1-{(M{m^{-1}}2\log n\cdot 4\cdot{2^{2j+4}})^{-1}})^{x}}$ (7)
and the average number of steps required to reach phase $j-1$ is
$<x>=\sum\limits_{i=1}^{\infty}{{{(1-\frac{m}{{256M\log
n}})}^{i-1}}}=\frac{{256M\log n}}{m}.$ (8)
Since the initial value of $j$ is at most $\log n$, then the expected total
number of steps required to reach the target is at most
$O(\frac{M}{m}\log^{2}n)$.
As a matter of fact, it means that we are using strategy $\mathcal{B}$ to send
message in 2-dimensional SIFN when the navigation process is divided into the
above 2 stages. Thus, the time complexity of navigation in SIFN with strategy
$\mathcal{B}$ is at most $O(\frac{M}{m}\log^{4}n)$. However, actual navigation
process in real world should be carried out regardless of the above two
assumptions, which indicates individuals should use strategy $\mathcal{A}$.
Based on the above analysis, strategy $\mathcal{A}$ performs better than
strategy $\mathcal{B}$ on average. Therefore, the time complexity of
navigation in 2-dimensional SIFN is at most $O(\log^{4}n)$ with non-uniform
population density.
## III Relationship between high and low dimensional SIFN
The empirical results show individuals always resort to extra information such
as profession and education information besides the target’s geography
location when routing messagesrecent experiment . Then, real navigation
process in social networks may be modeled with a higher dimensional SIFN. In
the following, we will discuss the relationship between the high and low
dimensional SIFN and prove that a 2-dimensional SIFN can be obtained by any
$k$-dimensional SIFN ($k>2$). Particularly, we will provide the theoretic
analysis for the case where $k=3$. The analysis can be generated to any $k$
dimensional cases.
We employe a random variable $D_{3}$ to denote the friendship distance in a
3-dimensional SIFN. For simplicity, a continuous expressions is used. Since,
the long-range connections in 3-dimensional SIFN satisfies the above empirical
law, the PDF of $D_{3}$ can be expressed by
$P(D_{3}=d)=\frac{1}{\ln d_{M}-\ln d_{m}}\frac{1}{d},d_{m}\leq d\leq d_{M}$
(9)
where $d_{m}$ and $d_{M}$ denote the minimum and maximum distance respectively
in the 3-dimensional SIFN.
We can obtain a 2-dimensional network model if we project a 3-dimensional SIFN
to a 2-dimensional world. Similarly, a random variable $D_{2}$ is used to
denote the friendship distance in the new 2-dimensional network model. It is
not difficult to understand that the condition for a 2-dimensional SIFN should
be the PDF of $D_{2}$ satisfies $P(d)\propto d^{-1}$. Since $D_{2}$ is the
projection of $D_{3}$, then $D_{2}$ can be seen as the product of $D_{3}$ and
$X$. Here random variable $X$ is independent on $D_{3}$ and its PDF can be
given by eq.(10).
$P(X=x)=\frac{1}{\lambda},0\leq x\leq\lambda$ (10)
where $0\leq\lambda\leq 1$. Finally, the PDF of $D_{2}$ can be written as
$P(D_{2}=d)=\begin{cases}0,&\text{$d\leq 0$}\\\
\frac{d_{M}-d_{m}}{d_{M}d_{m}\lambda(\ln d_{M}-\ln d_{m})},&\text{$0<d\leq
d_{m}\lambda$}\\\ \frac{\frac{1}{d}-\frac{1}{d_{M}\lambda}}{\ln d_{M}-\ln
d_{m}},&\text{$d_{m}\lambda<d\leq d_{M}\lambda$}\\\
0,&\text{$d>d_{M}\lambda$}\end{cases}$ (11)
When taking account of real social networks, $d_{M}$ is large enough that the
term $\frac{1}{d_{M}\lambda}$ will approach its limit of 0. Meanwhile, the
term $d_{m}\lambda$ can be neglected when compared with $d_{M}\lambda$,
because $\lambda\leq 1$ and $d_{m}$ is relatively small. Thus the PDF of
$D_{2}$ can be simplified into $P(d)\propto d^{-1}$, which is identical to
that of $D_{3}$ in a 3-dimensional SIFN.
Through theoretical analysis, we have proved a 2-dimensional SIFN can be seen
as the projection of a 3-dimensional SIFN. Likewise, we can get a
2-dimensional SIFN from any $k$-dimensional($k>2$) SIFN. Notice that
individuals are always restricted on the 2-dimensional geography world even
they possess extra information from other dimensions. Thus, real-world
searching process may be seen as the projection of navigation in a high
dimensional SIFN. Our analysis indicate that SIFN model may explain the
navigability of real social networks even take account of the fact that
individuals always resort to extra information in real searching experiments.
## IV Conclusion
Recent investigations suggest that the probability distribution of having a
friend at distance $d$ scales as $P(d)\propto d^{-1}$. We propose an SIFN
model based on this spatial property to deal with navigation problem with non-
uniform population density. It has been proved that the time complexity of
navigation in 2-dimensional SIFN is at most $O(\log^{4}n)$, which corresponds
to the upper bond of navigation in real social networks. Given the fact that
individuals are always restricted on the 2-dimensional geography world even
they possess information of the higher dimensions, actual searching process
can be seen as a projection of navigation in a higher $k$-dimensional SIFN.
Through theoretical analysis, we prove that the projection of a higher
$k$-dimensional SIFN results in a 2-dimensional SIFN. Therefore, SIFN model
may explain the navigability of real social networks even take account of the
information from higher dimensions other than geography dimensions.
## V Acknowledgement
We thank Prof. Shlomo Havlin for some useful discussions. This work is
partially supported by the Fundamental Research Funds for the Central
Universities and NSFC under Grants No.70771011 and No. 60974084 and
NCET-09-0228. Yanqing Hu is supported by Scientific Research Foundation and
Excellent Ph.D Project of Beijing Normal University.
## References
* (1) J. Kleinberg, The Convergence of Social and Technological Networks, Communications of the ACM. november 51, 66-72 (2008).
* (2) D. Brockmann, L. Hufnagel, and T. Geisel, The scaling laws of human travel, Nature(London) 439 462-465 (2006).
* (3) M. F. Shlesinger, Random walks: Follow the money, Nature Physics 2, 69-70 (2006).
* (4) M. C. Gonzalez, C. A. Hidalgo, and A.-L. Barabasi, Understanding individual human mobility patterns, Nature(London) 453, 779-782 (2008).
* (5) S. H. Strogatz, Exploring complex networks, Nature(London) 410, 268-276 (2001).
* (6) R. Albert and A.-L Barabasi, Statistical mechanics of complex networks, Reviews of Modern Physics 74, 47-97 (2002).
* (7) M. Girvan and M. E. J. Newman, Community structure in social and biological networks, Proceedings of the National Academy of Sciences 99, 7821-7826 (2004).
* (8) M. E. J. Newman, The structure and function of complex networks, SIAM Rev. 45, 167-256 (2003).
* (9) D. J. Watts, P. S. Dodds, and M. E. J. Newman, Identity and search in social networks, Science 296, 1302-1305 (2002).
* (10) S. Milgram, The small world problem, Psychology today 2, 60-67 (1967).
* (11) J. Guare, Six degrees of separation: A play, (Vintage Books, New York,1990)
* (12) F. Macrae, Microsoft proves you ARE just six degrees of separation from anyone in the world, Mailonline, Science and Tech, 04th August,(2008).
* (13) P. S. Dodds, R. Muhamad and D.J. Watts, An experimental study of search in global social networks, Science 301, 827 - 829 (2003).
* (14) J. Kleinberg, Navigation in a small world, Nature(London) 406, 845-845 (2000).
* (15) J. Kleinberg, The Small-World Phenomenon: An Algorithmic Perspective, Proceedings of the thirty-second annual ACM symposium on Theory of computing, 163-170 (2000).
* (16) C. Martel, Analyzing Kleinberg’s (and other) small-world models, Proceedings of the twenty-third annual ACM symposium on Principles of distributed computing, 179-188 (2004).
* (17) O. Sandberg, Searching in a Small World, Licentiate thesis, Chalmers University, (2005).
* (18) L. Barri re, P. Fraigniaud, E. Kranakis, and D. Krizanc, Efficient routing in networks with long range contacts (Springer,New York, 2001)
* (19) M. R. Roberson and D. Ben-Avraham, Kleinberg navigation in fractal small-world networks, Physical Review E 74, 17101 (2006).
* (20) C. Caretta Cartozo and P. De Los Rios, Extended navigability of small world networks: exact results and new insights, Physical Review Letters 102, 238703 (2009).
* (21) S. Carmi, S. Carter, J. Sun, and D. Ben-Avraham , Asymptotic behavior of the Kleinberg model, Physical Review Letters 102, 238702 (2009).
* (22) G. Li, S. D. S. Reis2 A. A. Moreira, S. Havlin, H. E. Stanley1 and J. S. Andrade, Jr., Towards design principles for optimal transport networks, Physical Review Letters 104, 018701 (2010).
* (23) Y. Li, D. Zhou, Y. Hu, J. Zhang and Z. Di, Exact solution for optimal navigation with total cost restriction, EPL 92, 58002 (2010).
* (24) D. Liben-Nowell, J. Novak, R. Kumar, P. Raghavan, and A. Tomkins, Geographic routing in social networks, Proceedings of the National Academy of Sciences 102, 11623-11628 (2005).
* (25) L. Adamic and E. Adar, How to search a social network, Social Networks 27, 187-203 (2005).
* (26) R. Lambiotte, V. D. Blondel, C. de Kerchove ,E. Huens , C. Prieur , Z. Smoreda, and P. V. Dooren, Geographical dispersal of mobile communication networks, Physica A 387, 5317 C5325 (2008).
* (27) J. Goldenberg and M. Levy, Distance is not dead: Social interaction and geographical distance in the internet era, arXiv:0906.3202v2.
|
arxiv-papers
| 2010-10-09T13:18:02 |
2024-09-04T02:49:13.628290
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yanqing Hu, Yong Li, Zengru Di, Ying Fan",
"submitter": "Li Yong",
"url": "https://arxiv.org/abs/1010.1845"
}
|
1010.1912
|
# $\bar{B}\to X_{s}\gamma$ constraints on the top quark anomalous $t\to
c\gamma$ coupling
Xingbo Yuan1, Yang Hao1 and Ya-Dong Yang1,2
1Institute of Particle Physics, Huazhong Normal University, Wuhan, Hubei
430079, P. R. China
2Key Laboratory of Quark & Lepton Physics, Ministry of Education, Huazhong
Normal University,
Wuhan, Hubei, 430079, P. R. China
###### Abstract
Observation of top quark flavor changing neutral process $t\to c+\gamma$ at
the LHC would be the signal of physics beyond the Standard Model. If anomalous
$t\to c\gamma$ coupling exists, it will affect the precisely measured
$\mathcal{B}(\bar{B}\to X_{s}\gamma)$. In this paper, we study the effects of
a dimension 5 anomalous $tc\gamma$ operator in $\bar{B}\to X_{s}\gamma$ decay
to derive constraints on its possible strength. It is found that, for real
anomalous $t\to c\gamma$ coupling $\kappa_{\rm{tcR}}^{\gamma}$, the
constraints correspond to the upper bounds $\mathcal{B}(t\to
c+\gamma)<6.54\times 10^{-5}$ (for $\kappa_{\rm{tcR}}^{\gamma}>0$) and
$\mathcal{B}(t\to c+\gamma)<8.52\times 10^{-5}$ (for
$\kappa_{\rm{tcR}}^{\gamma}<0$), respectively, which are about the same order
as the $5\sigma$ discovery potential of ATLAS ($9.4\times 10^{-5}$) and
slightly lower than that of CMS ($4.1\times 10^{-4}$) with $10\ \rm{fb}^{-1}$
integrated luminosity operating at $\sqrt{s}=14$ TeV.
## I Introduction
In the Standard Model (SM), top quark lifetime is dominated by the $t\to
bW^{+}$ process, and its flavor changing neutral current (FCNC) processes
$t\to qV(q=u,c;V=\gamma,Z,g)$ are extremely suppressed by GIM mechanism. It is
known that the SM predicts very tiny top FCNC branching ratio
$\mathcal{B}(t\to qV)$, less than $\mathcal{O}(10^{-10})$ gadi , which would
be inaccessible at the CERN Large Hadron Collider(LHC). In the literature
gadicp ; Beneke , however, a number of interesting questions have been
intrigued by the large top quark mass which is close to the scale of
electroweak symmetry breaking. For example, one may raise the question whether
new physics (NP) beyond the SM could manifest itself in nonstandard couplings
of top quark which would show up as anomalies in the top quark productions and
decays.
At present, the direct constraints on $\mathcal{B}(t\to qV)$ are still very
weak. For its radiative decay, the available experimental bounds are
$\mathcal{B}(t\to u\gamma)<0.75\%$ from ZEUS ZEUS and $\mathcal{B}(t\to
q\gamma)<3.2\%$ from CDF CDF at $95\%$ C.L., respectively. These constraints
will be improved greatly by the large top quark sample to be available at the
LHC, which is expected to produce $8\times 10^{6}$ top quark pairs and another
few million single top quarks per year at low luminosity ($10\
\rm{fb}^{-1}$/year). Both ATLAS ATLAS and CMS CMS have got analyses ready
for hunting out top quark FCNC processes as powerful probes for NP. With $10\
\rm{fb}^{-1}$ data, it is expected that both ATLAS and CMS could observe $t\to
q\gamma$ decays if their branching ratios are enhanced to
$\mathcal{O}(10^{-4})$ by anomalous top quark couplings ATLAS ; CMS . However,
if the top quark anomalous couplings present, they will affect some precisely
measured qualities with virtual top quark contribution. Inversely, these
qualities can also restrict the possible number of top quark FCNC decay
signals at the LHC. The precisely measured inclusive decay $B\to X_{s}\gamma$
is one of the well known sensitive probes for extensions of the SM, especially
the NPs which alter the strength of FCNCs top . Thus, when performing the
study of the possible strength of $t\to c\gamma$ decays at the LHC, one should
take into account the constraints from $B\to X_{s}\gamma$ wtb ; Fox .
In this paper, we will study the contribution of anomalous $t\gamma c$
operators to the $\bar{B}\to X_{s}\gamma$ branching ratio and derive
constraints on its strength. In the next section, after a brief discussion of
a set of model-independent dimension 5 effective operators relevant to $t\to
c\gamma$ decay, we calculate the effects of operator
$\bar{c}_{L}\sigma^{\mu\nu}t_{R}F_{\mu\nu}$ in $B\to X_{s}\gamma$ decay, which
result in a modification to $C_{7\gamma}$. In Sec. III we present our
numerical results of the constraints on its strength and the corresponding
upper limits on branching ratio of $t\to c\gamma$ decays. Finally, conclusions
are made in Sec. IV. Calculation details are presented in Appendix A, and
input parameters are collected in Appendix B.
## II Top quark anomalous couplings and their effects in $\bar{B}\to
X_{s}\gamma$ decay
Without resorting to the detailed flavor structure of a specific NP model, the
Lagrangian describing the top quark anomalous couplings can be written in a
model independent way with dimension 5 operators lag1
$\displaystyle{\mathcal{L}}_{5}=$ $\displaystyle-
g_{s}\sum_{q=u,c,t}\frac{\kappa^{g}_{tqL}}{\Lambda}\bar{q}_{R}\sigma^{\mu\nu}T^{a}t_{L}G^{a}_{\mu\nu}-\frac{g}{\sqrt{2}}\sum_{q=d,s,b}\frac{\kappa^{W}_{tqL}}{\Lambda}\bar{q}_{R}\sigma^{\mu\nu}t_{L}W^{-}_{\mu\nu}-e\sum_{q=u,c,t}\frac{\kappa^{\gamma}_{tqL}}{\Lambda}\bar{q}_{R}\sigma^{\mu\nu}t_{L}F_{\mu\nu}$
$\displaystyle-\frac{g}{2\cos\theta_{W}}\sum_{q=u,c,t}\frac{\kappa^{Z}_{tqL}}{\Lambda}\bar{q}_{R}\sigma^{\mu\nu}t_{L}Z_{\mu\nu}+(R\leftrightarrow
L)+h.c.,$ (1)
where $\kappa$ is the complex coupling of its corresponding operator,
$\theta_{W}$ is the weak angle, and $T^{a}$ is the Gell-Mann matrix. $\Lambda$
is the possible new physics scale, which is unknown but may be much larger
than the electroweak scale. There are also Lagrangian describing the top quark
anomalous interactions with dimension 4 and 6 operators, and the dimension 4
and 5 terms can be traced back to dimension 6 operators wyler ; list . In fact
top quark anomalous interactions can be generally described by the gauge-
invariant effective Lagrangian with dimension 6 operators in a form without
redundant operators and parameters Fox ; Saavedra . A recent full list of
dimension 6 operators could be found in Ref. SM6 . But for on-shell gauge
bosons, the Lagrangian in Eq. (1) works and is commonly employed in high
energy phenomenology analysis Beneke ; ATLAS ; Li .
The operators in Eq. (1) relevant to $t\to q\gamma$ decays read
$\displaystyle{\mathcal{L}}_{\gamma}=-e\sum_{q=u,c}\frac{\kappa^{\gamma}_{tqL}}{\Lambda}\bar{q}_{R}\sigma^{\mu\nu}t_{L}F_{\mu\nu}-e\sum_{q=u,c}\frac{\kappa^{\gamma}_{tqR}}{\Lambda}\bar{q}_{L}\sigma^{\mu\nu}t_{R}F_{\mu\nu}+h.c..$
(2)
It is understood that the Dirac matrix $\sigma_{\mu\nu}$ connects left-handed
fields to right-handed fields, the $t\to c\gamma$ transition will involve two
independent operators $m_{q}\bar{q}_{R}\sigma^{\mu\nu}t_{L}F_{\mu\nu}$ and
$m_{t}\bar{q}_{L}\sigma^{\mu\nu}t_{R}F_{\mu\nu}$, where the mass factors must
appear whenever a chirality flip $L\to R$ or $R\to L$ occurs. Due to the mass
hierarchy $m_{t}\gg m_{c}$, the effect of
$m_{q}\bar{q}_{R}\sigma^{\mu\nu}t_{L}F_{\mu\nu}$ can be neglected unless
$\kappa^{\gamma}_{tqL}$ is enhanced to be comparable to
$\tfrac{m_{t}}{m_{c}}\kappa^{\gamma}_{tqR}$ by unknown mechanism.
The anomalous $t\gamma q$ coupling affects $b\to s\gamma$ decays through the
two Feynman diagrams depicted in Figs. 1 and 1. It is interesting to note that
the CKM factors in Fig. 1 and Fig. 1 are $V_{tb}V_{qs}^{*}$ and
$V_{qb}V_{ts}^{*}$, respectively. Since
$|V_{tb}V_{qs}^{*}|\gg|V_{qb}V_{ts}^{*}|$ for $q=u,c$, the contribution of
Fig. 1 would be much stronger than that of Fig. 1. Furthermore, given the
strengths of $t\to u\gamma$ and $t\to c\gamma$ comparable, the contribution of
Fig. 1 to $b\to s\gamma$ is still dominated by $t\to c\gamma$ because of
$|V_{cs}|\gg|V_{us}|$. Hence we will only consider Fig. 1 with anomalous
$tc\gamma$ coupling. From the Feynman diagram of Fig. 1, it is easy to observe
that the large CKM factor $V_{tb}V_{cs}\approx 1$ makes $b\to s\gamma$ very
sensitive to the strength of anomalous $tc\gamma$ coupling.
The calculation of Fig. 1 can be carried out straightforwardly. The
calculation details are presented in Appendix A, and the final result reads
$\displaystyle i\mathcal{M}(b\to s\gamma)$ $\displaystyle=$
$\displaystyle\bar{s}[e\Gamma^{\nu}(k)]b\epsilon_{\nu}(k),$ $\displaystyle
e\Gamma^{\nu}(p,k)$ $\displaystyle=$ $\displaystyle
ie\frac{G_{F}}{4\sqrt{2}\pi^{2}}V^{*}_{cs}V_{tb}\left[i\sigma^{\nu\mu}k_{\mu}(m_{s}f_{L}(x)L+m_{b}f_{R}(x)R)\right].$
(3)
Usually $m_{s}$ term can be neglected, and the function $f_{\rm{R}}(x)$ is
calculated to be
$f_{\rm{R}}(x)=\frac{\kappa^{\gamma}_{\rm{tcR}}}{\Lambda}2m_{t}\left[-\frac{1}{(x_{c}-1)(x_{t}-1)}-\frac{x_{c}^{2}}{(x_{c}-1)^{2}(x_{c}-x_{t})}\ln
x_{c}+\frac{x_{t}^{2}}{(x_{t}-1)^{2}(x_{c}-x_{t})}\ln x_{t}\right],$ (4)
with $x_{q}=m_{q}^{2}/m_{W}^{2}$. Now we are ready to incorporate the NP
contribution into its SM counterpart for ${\bar{B}}\to X_{s}\gamma$ decay.
Figure 1: Feynman diagrams for $b\to s\gamma$. (a) and (b) are the penguin
diagrams with the anomalous $tq\gamma$ couplings. (c) Sample LO penguin
diagram in the SM.
In the SM, it is known that ${\bar{B}}\to X_{s}\gamma$ decay is governed by
the effective Hamiltonian at scale $\mu=\mathcal{O}(m_{b})$ Buras1
$\displaystyle\mathcal{H}_{\rm{eff}}(b\to
s\gamma)=-\frac{4G_{F}}{\sqrt{2}}V_{ts}^{*}V_{tb}\left[\sum_{i=1}^{6}C_{i}(\mu)Q_{i}(\mu)+C_{7\gamma}(\mu)O_{7\gamma}(\mu)+C_{8g}(\mu)O_{8g}(\mu)\right],$
(5)
where $C_{i}(\mu)$ are the Wilsion coefficients, $O_{i=1-6}$ are the effective
four quark operators and
$\displaystyle
O_{7\gamma}=\frac{e}{16\pi^{2}}m_{b}(\bar{s}_{L}\sigma^{\mu\nu}b_{R})F_{\mu\nu},~{}~{}~{}~{}O_{8g}=\frac{g}{16\pi^{2}}m_{b}(\bar{s}_{L}\sigma^{\mu\nu}T^{a}b_{R})G_{\mu\nu}^{a}.$
(6)
For calculating $\mathcal{B}(\bar{B}\to X_{s}\gamma)$, instead of the original
Wision coefficients $C_{i}$, it is convenient to use the so called “effective
coefficients” Buras94
$\displaystyle
C_{7\gamma}^{(0)\rm{eff}}(m_{b})=\eta^{\frac{16}{23}}C_{7\gamma}^{(0)\rm{SM}}(M_{W})+\frac{8}{3}(\eta^{\frac{14}{23}}-\eta^{\frac{16}{23}})C_{8g}^{(0)\rm{SM}}(M_{W})+C_{2}^{(0)\rm{SM}}(M_{W})\sum_{i=1}^{8}h_{i}\eta^{a_{i}},$
(7)
where $\eta=\alpha_{s}(\mu_{W})/\alpha_{s}(\mu_{b})$ and
$\displaystyle h_{i}$ $\displaystyle=\bigl{(}$
$\displaystyle\tfrac{626126}{272277}$ $\displaystyle-\tfrac{56281}{51730}$
$\displaystyle-\tfrac{3}{7}$ $\displaystyle-\tfrac{1}{14}$ $\displaystyle-$
$\displaystyle 0.6494$ $\displaystyle-0.0380$ $\displaystyle-0.0185$
$\displaystyle-0.0057$ $\displaystyle\bigr{)},$ (8) $\displaystyle a_{i}$
$\displaystyle=\bigl{(}$ $\displaystyle\tfrac{14}{23}$
$\displaystyle\tfrac{16}{23}$ $\displaystyle\tfrac{6}{23}$
$\displaystyle-\tfrac{12}{23}$ $\displaystyle 0.4086$ $\displaystyle-0.4230$
$\displaystyle-0.8994$ $\displaystyle 0.1456$ $\displaystyle\bigr{)}.$ (9)
To the leading order approximation, the $\mathcal{B}(\bar{B}\to X_{s}\gamma)$
is proportional to $|C_{7\gamma}^{(0)\rm{eff}}(m_{b})|^{2}$ Buras .
In terms of the operator basis in Eq. (5), the contribution of the anomalous
$t\to c\gamma$ couplings in Eq. (3) would result in the deviation of
$C_{7\gamma}(M_{W})\to
C^{\prime}_{7\gamma}(M_{W})=C_{7\gamma}(M_{W})+C^{\rm{NP}}_{7\gamma}(M_{W})$
(10)
and $C^{\rm{NP}}_{7\gamma}(M_{W})$ can be read from Eq. (3) as
$C_{7\gamma}^{\rm{NP}}(M_{W})=\frac{\kappa_{\rm{tcR}}^{\gamma}}{\Lambda}\frac{V_{cs}^{*}}{V_{ts}^{*}}m_{t}\left[\frac{1}{(x_{c}-1)(x_{t}-1)}+\frac{x_{c}^{2}}{(x_{c}-1)^{2}(x_{c}-x_{t})}\log
x_{c}-\frac{x_{t}^{2}}{(x_{t}-1)^{2}(x_{c}-x_{t})}\ln x_{t}\right].$ (11)
From this equation, one can see that the NP contribution is suppressed by a
factor of $m_{t}/\Lambda$ but enhanced by $V_{cs}/V_{ts}$.
Since NP contribution does not bring about any new operator, the
renormalization group evolution of $C_{7\gamma}^{\rm eff}$ from $M_{W}$ to
$m_{b}$ scale is just the same as the SM one in Eq. (7). For $m_{t}=172$ GeV,
$m_{b}=4.67$ GeV, $\alpha_{s}(M_{Z})=0.118$ and $\Lambda=1$ TeV, we have
$\displaystyle C_{7\gamma}^{\prime\rm{eff}}(m_{b})$ $\displaystyle=$
$\displaystyle\eta^{\frac{16}{23}}\left[C_{7\gamma}^{(0)\rm{SM}}(M_{W})+C_{7\gamma}^{(0)\rm{NP}}(M_{W})\right]+\frac{8}{3}(\eta^{\frac{14}{23}}-\eta^{\frac{16}{23}})C_{8g}^{(0)\rm{SM}}(M_{W})+C_{2}^{(0)\rm{SM}}(M_{W})\sum_{i=1}^{8}h_{i}\eta^{a_{i}}$
(12) $\displaystyle=$ $\displaystyle
0.665\left[C_{7\gamma}^{(0)\rm{SM}}(M_{W})+C_{7\gamma}^{(0)\rm{NP}}(M_{W})\right]+0.093\
C_{8g}^{(0)\rm{SM}}(M_{W})-0.158\ C_{2}^{(0)\rm{SM}}(M_{W})$ $\displaystyle=$
$\displaystyle
0.665\left[-0.189+\kappa_{\rm{tcR}}^{\gamma}(-1.092)\right]+0.093\
(-0.095)-0.158.$
In principle, $C_{7\gamma}^{\prime\rm{eff}}(m_{b})$ will receive corrections
from anomalous $t\to cg$ couplings in Eq. (1) which will cause a deviation to
$C_{8g}^{(0)\rm{SM}}(M_{W})$. However, as shown by Eq. (12), the coefficient
$\eta^{\frac{16}{23}}$ of $C_{7\gamma}^{(0)}(M_{W})$ is about one order larger
than $\frac{8}{3}(\eta^{\frac{14}{23}}-\eta^{\frac{16}{23}})$ of
$C_{8g}^{(0)\rm{NP}}(M_{W})$. Given the relative strength of
$C_{8g}^{(0)\rm{NP}}(M_{W})$ to $C_{8g}^{(0)\rm{SM}}(M_{W})$ at $10\%$ level,
$C_{7\gamma}^{\prime\rm{eff}}(m_{b})$ will be shifted by only few percentage.
For simplifying the numerical analysis, we would neglect the contribution of
the anomalous $t\to cg$ couplings. We also find that the operator
$\bar{q}_{R}\sigma^{\mu\nu}t_{L}F_{\mu\nu}$ contributes to $\bar{B}\to
X_{s}\gamma$ only through the term
$m_{s}\bar{s}\sigma_{\mu\nu}(1-\gamma_{5})b$ as shown by Eq. (3) and Eq. (7).
Combined with the previous remarks on this operator, the effects of
$\bar{q}_{R}\sigma^{\mu\nu}t_{L}F_{\mu\nu}$ could be safely neglected.
## III Numerical results and discussions
The current average of experimental results of $\mathcal{B}(\bar{B}\to
X_{s}\gamma)$ by Heavy Flavor Average Group is HFAG
$\mathcal{B}^{\rm exp}(\bar{B}\to X_{s}\gamma)=(3.55\pm 0.24\pm 0.09)\times
10^{-4}.$ (13)
On the theoretical side, the NLO calculation has been completed Misiak ; Buras
, and gives
$\mathcal{B}(\bar{B}\to X_{s}\gamma)=(3.57\pm 0.30)\times 10^{-4}.$ (14)
The recent estimation at NNLO Misiak1 gives $\mathcal{B}(\bar{B}\to
X_{s}\gamma)=(3.15\pm 0.23)\times 10^{-4}$, which is about $1\sigma$ lower
than the experimental average in Eq. (13). Thus the experimental measurement
of $\mathcal{B}(\bar{B}\to X_{s}\gamma)$ is in good agreement with the SM
predictions with roughly $10\%$ errors on each side. The agreement would
provide strong constraints on the top quark anomalous interactions beyond the
SM wtb ; Fox .
The decay amplitude of $t\to c\gamma$ has been calculated up to NLO Li . For a
consistent treatment of the constraints from $t\to c\gamma$ and $b\to s\gamma$
decays, we use the NLO formulas in Ref. Misiak to calculate
$\mathcal{B}(\bar{B}\to X_{s}\gamma)$. The experimental inputs and main
formulas are collected in Appendix B. For numerical analysis, we will use the
notation $\kappa_{\rm tcR}^{\gamma}=|\kappa_{\rm tcR}^{\gamma}|e^{i\theta_{\rm
tcR}^{\gamma}}$ and set $\Lambda=1$ TeV.
Figure 2: The contour-plot describes the dependence of $\mathcal{B}(\bar{B}\to
X_{s}\gamma)(\times 10^{-4})$ on $|\kappa_{\rm{tcR}}^{\gamma}/\Lambda|$ and
$\theta_{\rm{tcR}}^{\gamma}$. The dashed lines correspond to the experimental
center value of $\mathcal{B}(\bar{B}\to X_{s}\gamma)$.
At first, we analyze the dependence of $\mathcal{B}^{\rm SM+NP}(\bar{B}\to
X_{s}\gamma)$ on the new physics parameters
$|\kappa_{\rm{tcR}}^{\gamma}/\Lambda|$ and $\theta_{\rm{tcR}}^{\gamma}$, which
is shown in Fig. 2. From the figure, one can find that the contribution of
anomalous $t\to c\gamma$ coupling is constructive to the SM one for
$\theta_{\rm{tcR}}^{\gamma}\in[-50^{\circ},50^{\circ}]$, thus
$\mathcal{B}(\bar{B}\to X_{s}\gamma)$ is very sensitive to
$|\kappa_{\rm{tcR}}^{\gamma}|$. However, when
$|\theta_{\rm{tcR}}^{\gamma}|\in[80^{\circ},130^{\circ}]$, the sensitivity of
$\mathcal{B}(\bar{B}\to X_{s}\gamma)$ to $|\kappa_{\rm{tcR}}^{\gamma}|$
becomes weak. For $|\theta_{\rm{tcR}}^{\gamma}|\sim 180^{\circ}$, the
contribution of anomalous $t\to c\gamma$ coupling is destructive to the SM one
and there are two separated possible strengths for
$|\kappa_{\rm{tcR}}^{\gamma}/\Lambda|$.
Figure 3: The $95\%$ C.L. upper bounds on anomalous coupling
$|\kappa_{\rm{tcR}}^{\gamma}/\Lambda|$ as a function of
$\theta_{\rm{tcR}}^{\gamma}$. The shadowed region is allowed by
$\mathcal{B}^{\rm exp}(\bar{B}\to X_{s}\gamma)$ and the dash-line is the CDF
CDF upper limit. Figure 4: $\mathcal{B}(t\to c\gamma)$ as a function of
$\theta_{\rm{tcR}}^{\gamma}$. The shadowed region is allowed by the combined
constraints of $\mathcal{B}(\bar{B}\to X_{s}\gamma)$ and CDF searching at 95%
C.L.
The allowed region for the parameters $|\kappa_{\rm{tcR}}^{\gamma}/\Lambda|$
and $\theta_{\rm{tcR}}^{\gamma}$ under the constraints from
$\mathcal{B}(\bar{B}\to X_{s}\gamma)$ at $95\%$ C.L. is shown in Fig. 3. The
corresponding $95\%$ C.L. upper bound on $\mathcal{B}(t\to c\gamma)$ is shown
in Fig. 4.
Now we turn to discuss the our numerical results. From Eq. (12), the explicit
relation between the SM and the $t\to c\gamma$ coupling contributions is
$C_{7\gamma}^{\prime\rm{eff}}(m_{b})=-0.293-0.726~{}\kappa_{\rm{tcR}}^{\gamma}.$
(15)
Obviously, when $\rm{Re}~{}\kappa_{\rm{tcR}}^{\gamma}>0$, the interference
between them is constructive, and it turns to be destructive when
$\theta_{\rm{tcR}}^{\gamma}>90^{\circ}$. Thus the features of these
constraints shown in Figs. 3 and 4 for different $\theta_{\rm{tcR}}^{\gamma}$
are
1. (i)
the bound on $|\kappa_{\rm{tcR}}^{\gamma}/\Lambda|$ is very strong for
$\theta_{\rm{tcR}}^{\gamma}\in[-50^{\circ},50^{\circ}]$. For
$\theta_{\rm{tcR}}^{\gamma}\approx 0^{\circ}$, as shown in Fig. 3, we obtain
the most restrictive upper bound
$|\kappa_{\rm{tcR}}^{\gamma}/\Lambda|<4.9\times 10^{-5}\ \rm{GeV}^{-1}$, which
implies $\mathcal{B}(t\to c\gamma)<6.54\times 10^{-5}$;
2. (ii)
the bound on $|\kappa_{\rm{tcR}}^{\gamma}/\Lambda|$ is rather weak for
$\theta_{\rm{tcR}}^{\gamma}$ around $110^{\circ}$. For such a case,
$\rm{Re}~{}\kappa_{\rm{tcR}}^{\gamma})$ is destructive to the SM contribution
as shown by Eq. (15), so, the allowed strength for the anomalous coupling is
much larger than the one for real $\kappa_{\rm{tcR}}^{\gamma}$. When
$|\theta_{\rm{tcR}}^{\gamma}|\approx 135^{\circ}$ and
$|\kappa_{\rm{tcR}}^{\gamma}|\approx 0.571$,
$C_{7\gamma}^{\prime\rm{eff}}(m_{b})$ is almost imaginary since ${\rm
Re}~{}C_{7\gamma}^{\prime\rm{eff}}(m_{b})\approx 0$. Then the restriction on
$|\kappa_{\rm{tcR}}^{\gamma}/\Lambda|$ is provided by the CDF search for
$\mathcal{B}(t\to c\gamma)$ CDF ;
3. (iii)
as shown in Fig. 3, when $\theta_{\rm{tcR}}^{\gamma}\sim\pm 180^{\circ}$,
there are two solutions for $|\kappa_{\rm{tcR}}^{\gamma}/\Lambda|$. The larger
one $|\kappa^{\gamma}_{\rm{tcR}}/\Lambda|\sim 1.4\times
10^{-3}~{}\rm{GeV}^{-1}$(S2 column in Table 1) corresponds to the situation
that the sign of $C^{\rm{eff}}_{7\gamma}$ is flipped. However, it has been
excluded by the CDF upper bound of $\mathcal{B}(t\to c\gamma)<0.032$ CDF . The
another solution (S1 column in Table 1)
$|\kappa_{\rm{tcR}}^{\gamma}/\Lambda|<5.6\times 10^{-5}\ \rm{GeV}^{-1}$ will
result in the upper limit $\mathcal{B}(t\to c\gamma)<8.52\times 10^{-5}$.
Taking $\theta^{\gamma}_{\rm{tcR}}=0^{\circ},~{}\pm 180^{\circ}$ and $\pm
110^{\circ}$ as benchmarks, we summarize our numerical constraints on
$\kappa^{\gamma}_{\rm{tcR}}$ and their corresponding upper limits on
$\mathcal{B}(t\to c\gamma)$ in Table 1. From the table, we can find that our
indirect bound on real $\kappa^{\gamma}_{\rm{tcR}}$ is much stronger than the
CDF direct bound. The corresponding upper limits on $\mathcal{B}(t\to
c\gamma)$ are about the same order as the ATLAS sensitivity $\mathcal{B}(t\to
c\gamma)>9.4\times 10^{-5}$ ATLAS and CMS sensitivity $\mathcal{B}(t\to
c\gamma)>4.1\times 10^{-4}$ CMS with an integrated luminosity of $10\
\rm{fb}^{-1}$ of the LHC operating at ${\sqrt{s}}=14$ TeV ATLAS .
Table 1: The 95% C.L. constraints on the anomalous $t\to c\gamma$ coupling by $\mathcal{B}(\bar{B}\to X_{s}\gamma)$ and $\mathcal{B}(t\to c\gamma)$ for some specific $\theta_{\rm{tcR}}^{\gamma}$ values. | $\theta_{\rm{tcR}}^{\gamma}=0^{\circ}$ | $\theta_{\rm{tcR}}^{\gamma}=\pm 180^{\circ}$ S1 | $\theta_{\rm{tcR}}^{\gamma}=\pm 180^{\circ}$ S2 | $\theta_{\rm{tcR}}^{\gamma}=\pm 110^{\circ}$
---|---|---|---|---
$\mathcal{B}(\bar{B}\to X_{s}\gamma)$ | $|\kappa_{\rm{tcR}}^{\gamma}|<0.049$ | $|\kappa_{\rm{tcR}}^{\gamma}|<0.056$ | $1.35<|\kappa_{\rm{tcR}}^{\gamma}|<1.45$ | $|\kappa_{\rm{tcR}}^{\gamma}|<0.55$
$\mathcal{B}(t\to c\gamma)$ CDF boundsCDF | $|\kappa_{\rm{tcR}}^{\gamma}|<1.09\enskip$ | $|\kappa_{\rm{tcR}}^{\gamma}|<1.09\enskip$ | $|\kappa_{\rm{tcR}}^{\gamma}|<1.09$ | $|\kappa_{\rm{tcR}}^{\gamma}|<1.09$
Combined bounds | $|\kappa_{\rm{tcR}}^{\gamma}|<0.049$ | $|\kappa_{\rm{tcR}}^{\gamma}|<0.056$ | $-$ | $|\kappa_{\rm{tcR}}^{\gamma}|<0.55$
$\mathcal{B}(t\to c\gamma)$ | $<6.54\times 10^{-5}$ | $<8.52\times 10^{-5}$ | $-$ | $<8.17\times 10^{-3}$
## IV Conclusions
In this paper, starting with model independent dimension five anomalous
$tc\gamma$ operators, we have studied their contributions to
$\mathcal{B}(\bar{B}\to X_{s}\gamma)$. It is noted that the $t\to c\gamma$
transition will involve two independent operators
$\kappa^{\gamma}_{\rm{tcR}}\bar{c}_{L}\sigma^{\mu\nu}t_{R}F_{\mu\nu}$ and
$\kappa^{\gamma}_{\rm{tcL}}\bar{c}_{R}\sigma^{\mu\nu}t_{L}F_{\mu\nu}$. The
first operator will produce a left-handed photon in $t\to c\gamma$ decay,
while the second one will produce a right-handed photon. It is found that
$\bar{B}\to X_{s}\gamma$ is sensitive to the first operator, but not to the
second one.
For real $\kappa^{\gamma}_{\rm{tcR}}$, the constraint on the presence of
$\kappa^{\gamma}_{\rm{tcR}}\bar{c}_{L}\sigma^{\mu\nu}t_{R}F_{\mu\nu}$ is very
strong, which corresponds to the indirect upper limits $\mathcal{B}(t\to
c\gamma)<6.54\times 10^{-5}$ (for positive $\kappa^{\gamma}_{\rm{tcR}}$) and
$\mathcal{B}(t\to c\gamma)<8.52\times 10^{-5}$ (for negative
$\kappa^{\gamma}_{\rm{tcR}}$), respectively. These upper limits for
$\mathcal{B}(t\to c\gamma)$ are close to the $5\sigma$ discovery sensitivities
of ATLAS ATLAS and slightly lower than that of CMS CMS with $10\
\rm{fb}^{-1}$ integrated luminosity operating at $\sqrt{s}=14$ TeV. For nearly
imaginary $\kappa^{\gamma}_{\rm{tcR}}$, the constraints are rather weak since
$C_{7\gamma}$ in the SM is a real number. If $\mathcal{B}(t\to c\gamma)$ were
found to be of the order of $\mathcal{O}(10^{-3})$ at the LHC in the future,
it would imply the weak phase of $\kappa^{\gamma}_{\rm{tcR}}$ to be around
$\pm 100^{\circ}$. However, such a coupling might be ruled out by the other
observable in B meson decays xqli .
In summary, we have studied the interesting interplay between the precise
measurement of $b\to s\gamma$ decay at B factories and the possible $t\to
c\gamma$ decay at the LHC. For real anomalous coupling, it is shown that
$\mathcal{B}(t\to c\gamma)$ has been restricted to be blow $10^{-4}$ at $95\%$
C.L. by $\bar{B}\to X_{s}\gamma$ decay, which is already two order lower than
the direct upper bound from CDF CDF . The result also implies that one may
need data sample much larger than $10\ \rm{fb}^{-1}$ to hunt out $t\to
c\gamma$ signals at the LHC.
## ACKNOWLEDGMENTS
The work is supported by National Natural Science Foundation under contract
Nos.11075059 and 10735080. We thank Xinqiang Li for many helpful discussions
and cross-checking calculations.
Figure 5: (a) the Feynman rules of $t\gamma c$ interactions in the Lagrangian
of Eq. 1. (b) penguin diagram contribution to $b\to s\gamma$ with top quark
anomalous interactions.
## Appendix A The calculation of $C_{7\gamma}^{\rm NP}(\mu_{W})$
Using the Feynman rules in Fig 5, the amplitude of penguin diagram in Fig 5
can be written as,
$\displaystyle i\mathcal{M}$
$\displaystyle=\bar{u}_{s}(p^{\prime})[e\Gamma^{\nu}(p,k)]u_{b}(p)\epsilon_{\nu}(k),$
(1) $\displaystyle\Gamma^{\nu}(p,k)$
$\displaystyle=-\frac{ig^{2}}{\Lambda}V^{*}_{cs}V_{tb}\int\frac{d^{4}q}{(2\pi)^{4}}\frac{N}{[(p\prime-q)^{2}-m_{c}^{2}+i\epsilon][(p-q)^{2}-m_{t}^{2}+i\epsilon][q^{2}-m_{W}^{2}+i\epsilon]},$
(2) $\displaystyle N$
$\displaystyle=\gamma_{\alpha}L(\displaystyle{\not}p\prime-\displaystyle{\not}q+m_{q})\sigma^{\mu\nu}(\kappa^{\gamma}_{\rm{tcR}}R+\kappa^{\gamma}_{\rm{tcL}}L)(\displaystyle{\not}p-\displaystyle{\not}q+m_{t})\gamma_{\beta}Lg^{\alpha\beta}k_{\mu},$
(3)
with $R=(1+\gamma^{5})/2$ and $L=(1-\gamma^{5})/2$. By Dirac algebra
$\displaystyle\gamma_{\alpha}L\displaystyle{\not}q\sigma^{\mu\nu}(\kappa^{\gamma}_{\rm{tcR}}R+\kappa^{\gamma}_{\rm{tcL}}L)\displaystyle{\not}q\gamma_{\beta}L=0,$
(4)
the terms with $q^{2}$ in $N$ vanishes and N becomes
$\displaystyle N$
$\displaystyle=m_{c}\kappa^{\gamma}_{\rm{tcL}}[2(\displaystyle{\not}p-\displaystyle{\not}q)\sigma^{\mu\nu}+(4-D)\sigma^{\mu\nu}(\displaystyle{\not}p-\displaystyle{\not}q)]Lk_{\mu}$
$\displaystyle\;+m_{t}\kappa^{\gamma}_{\rm{tcR}}[2\sigma^{\mu\nu}(\displaystyle{\not}p\prime-\displaystyle{\not}q)+(4-D)(\displaystyle{\not}p\prime-\displaystyle{\not}q)\sigma^{\mu\nu}]Rk_{\mu}.$
(5)
Thus, there is no divergence in $\Gamma^{\nu}(p,k)$. After integrating out $q$
in the $\Gamma^{\nu}(p,k)$ and using on-shell condition, $\Gamma^{\nu}(p,k)$
can be written in the following form,
$\displaystyle
e\Gamma^{\nu}(p,k)=ie\frac{G_{F}}{4\sqrt{2}\pi^{2}}V^{*}_{cs}V_{tb}\left[i\sigma^{\nu\mu}k_{\mu}(m_{s}f_{\rm{L}}(x)L+m_{b}f_{\rm{R}}(x)R)\right],$
(6)
where
$\displaystyle
f_{\rm{L}}(x)=\frac{\kappa^{\gamma}_{\rm{tcL}}}{\Lambda}2m_{c}\left[-\frac{1}{(x_{c}-1)(x_{t}-1)}-\frac{x_{c}^{2}}{(x_{c}-1)^{2}(x_{c}-x_{t})}\ln
x_{c}+\frac{x_{t}^{2}}{(x_{t}-1)^{2}(x_{c}-x_{t})}\ln x_{t}\right],$ (7)
$\displaystyle
f_{\rm{R}}(x)=\frac{\kappa^{\gamma}_{\rm{tcR}}}{\Lambda}2m_{t}\left[-\frac{1}{(x_{c}-1)(x_{t}-1)}-\frac{x_{c}^{2}}{(x_{c}-1)^{2}(x_{c}-x_{t})}\ln
x_{c}+\frac{x_{t}^{2}}{(x_{t}-1)^{2}(x_{c}-x_{t})}\ln x_{t}\right],$ (8)
Using the convention of Ref. Buras , we have
$\displaystyle C_{7\gamma}^{(0)\rm NP}(M_{W})$
$\displaystyle=-\frac{1}{2}\frac{V_{cs}^{*}}{V_{ts}^{*}}f_{\rm{R}}(x)$
$\displaystyle=\frac{V_{cs}^{*}}{V_{ts}^{*}}m_{t}\frac{\kappa^{\gamma}_{\rm{tcR}}}{{\Lambda}}\left[\frac{1}{(x_{c}-1)(x_{t}-1)}+\frac{x_{c}^{2}}{(x_{c}-1)^{2}(x_{c}-x_{t})}\ln
x_{c}-\frac{x_{t}^{2}}{(x_{t}-1)^{2}(x_{c}-x_{t})}\ln x_{t}\right].$ (9)
## Appendix B Main formulas and inputs
Following the notation in Ref. Misiak , the branching ratio of $\bar{B}\to
X_{s}\gamma$ can be expressed as
$\mathcal{B}[\bar{B}\to X_{s}\gamma]_{E_{\gamma}>E_{0}}=\mathcal{B}^{\rm
exp}[\bar{B}\to
X_{c}e\bar{\nu}]\left|\frac{V^{*}_{ts}V_{tb}}{V_{cb}}\right|^{2}\frac{6\alpha_{\rm
em}}{\pi\;C}\left[P(E_{0})+N(E_{0})\right],$ (1)
where $P(E_{0})$ is the perturbative ratio
$\frac{\Gamma[\bar{B}\to
X_{s}\gamma]_{E_{\gamma}>E_{0}}}{|V_{cb}/V_{ub}|^{2}\;\Gamma[\bar{B}\to
X_{u}e\bar{\nu}]}=\left|\frac{V^{*}_{ts}V_{tb}}{V_{cb}}\right|^{2}\frac{6\alpha_{\rm
em}}{\pi}P(E_{0}),$ (2)
which includes the Wilson coefficients of Eq. 7. $N(E_{0})$ denotes the non-
perturbative corrections. The semileptonic phase space factor
$C=\left|\frac{V_{ub}}{V_{cb}}\right|^{2}\frac{\Gamma[\bar{B}\to
X_{c}e\bar{\nu}]}{\Gamma[\bar{B}\to X_{u}e\bar{\nu}]}$ (3)
can be obtained from a fit of the experimental spectrum of the $\bar{B}\to
X_{c}l\bar{\nu}$ C .
For calculating $\mathcal{B}(t\to c\gamma)$, we use the NLO formulas in Ref.
Li and Li1 . Because $t\to bW$ is the dominant top quark decay mode, the
branching ratio of $t\to c\gamma$ is defined as
$\mathcal{B}(t\to c\gamma)=\frac{\Gamma(t\to c\gamma)}{\Gamma(t\to bW)}.$ (4)
The partial width $\Gamma(t\to c\gamma)$ at the NLO can be found in Ref. Li ,
namely,
$\Gamma_{\rm{NLO}}(t\to c\gamma)=\frac{2\alpha_{s}}{9\pi}\Gamma_{0}(t\to
c\gamma)\left[-3\log\left(\frac{\mu^{2}}{m_{t}^{2}}\right)-2\pi^{2}+8\right],$
(5)
where $\Gamma_{0}(t\to c\gamma)=\alpha
m_{t}^{3}\left(\kappa^{\gamma}_{\rm{tcR}}/\Lambda\right)^{2}$ is the LO
partial decay width.
The partial width of $t\to bW$ has been calculated in Ref. Li1 at the NLO,
which reads
$\displaystyle\Gamma_{\rm{NLO}}(t\to bW)=\Gamma_{0}(t\to
bW)\biggl{\\{}1+\frac{2\alpha_{s}}{3\pi}\biggl{[}2\left(\frac{(1-\beta_{W}^{2})(2\beta_{W}^{2}-1)(\beta_{W}^{2}-2)}{\beta_{W}^{4}(3-2\beta_{W}^{2})}\right)\ln(1-\beta_{W}^{2})$
$\displaystyle-\frac{9-4\beta_{W}^{2}}{3-2\beta_{W}^{2}}\ln\beta_{W}^{2}+2\mathrm{Li}_{2}(\beta_{W}^{2})-2\mathrm{Li}_{2}(1-\beta_{W}^{2})-\frac{6\beta_{W}^{4}-3\beta_{W}^{2}-8}{2\beta_{W}^{2}(3-2\beta_{W}^{2})}-\pi^{2}\biggr{]}\biggr{\\}}$
(6)
with $\Gamma_{0}(t\to
bW)=\frac{G_{F}m_{t}^{3}}{8\sqrt{2}\pi}|V_{tb}|^{2}\beta_{W}^{4}(3-2\beta_{W}^{2})$
and $\beta_{W}\equiv(1-m_{W}^{2}/m_{t}^{2})^{1/2}$.
Table 2: Experimental inputs for calculating the branching ratio of
$\bar{B}\to X_{s}\gamma$ and $t\to c\gamma$. Experimental Inputs
---
$\alpha_{em}=1/137.036$ PDG | $M_{Z}=91.1876\pm 0.0021\ \rm GeV$ PDG
$\alpha_{s}(M_{Z})=0.1184\pm 0.0007$ PDG | $M_{W}=80.399\pm 0.023\ \rm GeV$ PDG
$G_{\rm F}=1.16637\times 10^{-5}\ \rm GeV^{-2}$ PDG | $m_{b}^{\rm 1S}=4.67_{-0.06}^{+0.18}\ \rm GeV$ PDG
$A=0.812^{+0.013}_{-0.027}$ CKMfitter | $m_{c}(m_{c})=(1.224\pm 0.017\pm 0.054)\ \rm GeV$ Manohar
$\lambda=0.22543\pm 0.00077$ CKMfitter | $m_{t,pole}=172.0\pm 0.9\pm 1.3\ \rm GeV$ PDG
$\bar{\rho}=0.144\pm 0.025$ CKMfitter | $\mathcal{B}^{\rm exp}[\bar{B}\to X_{c}e\bar{\nu}]=(10.64\pm 0.17\pm 0.06)\%$ BABAR
$\bar{\eta}=0.342^{+0.016}_{-0.015}$ CKMfitter | $C=0.580\pm 0.016$ C
$\left|V^{*}_{ts}V_{tb}/V_{cb}\right|^{2}=0.9625$ | $\epsilon_{\rm ew}=0.0071$ Misiak ; Gambino
$(V_{us}^{*}V_{ub})/(V_{ts}^{*}V_{tb})=-0.007+0.018\rm i$ | $N(E_{0})=0.0036\pm 0.0006$ Misiak
$V_{cs}^{*}/V_{ts}^{*}=-24.023-0.432\rm i$ | $E_{0}=1.6\ \rm GeV$
The experimental inputs are collected in Table. 2, in which the CKM factors
are derived from the Wolfenstein parameters A, $\lambda$, $\bar{\rho}$ and
$\bar{\eta}$.
## References
* (1) G. Eilam, J.L. Hewett and A.Soni, Phys. Rev. D44, 1473(1991), Erratum-ibid D59, 039901(1999).
* (2) D. Atwood, S. Bar-Shalom, G. Eilam and A. Soni, Phys. Rept. 347, 1(2001), arXiv: hep-ph/0006032.
* (3) There are many papers on top quark rare decays. For a review, we refer to M. Beneke et al., arXiv:hep-ph/0003033 and references therein.
* (4) S. Chekanov et al. [ZEUS Collaboration], Phys. Lett. B 559, 153 (2003) [arXiv:hep-ex/0302010].
* (5) F. Abe et al. [CDF Collaboration], Phys. Rev. Lett. 80, 2525 (1998).
* (6) J. Carvalho et al. [ATLAS Collaboration], Eur. Phys. J. C 52, 999 (2007) [arXiv:0712.1127 [hep-ex]]. F. Veloso, et al. CERN-THESIS-2008-106.
* (7) L. Benucci and A. Kyriakis, Nucl. Phys. Proc. Suppl. 177-178, 258 (2008).
* (8) M. Antonelli et al., Phys. Rept.494, 197(2010), arXiv:0907.5386 [hep-ph].
* (9) B. Grzadkowski and M. Misiak, Phys. Rev. D 78, 077501(2008).
* (10) P. J. Fox, Z. Ligeti, M. Papucci, G. Perez, and M. D. Schwartz, Phys. Rev. D 78, 054008(2008).
* (11) W. Hollik, J. I. Illana, S. Rigolin, C. Schappacher and D. Stockinger, Nucl. Phys. B 551, 3 (1999) [Erratum-ibid. B 557, 407 (1999)] [arXiv:hep-ph/9812298].
* (12) W. Buchmuller and D. Wyler, Nucl. Phys. B 268, 621 (1986).
* (13) K. Hagiwara, S. Ishihara, R. Szalapski and D. Zeppenfeld, Phys. Rev. D 48, 2182 (1993). G. J. Gounaris, F. M. Renard and C. Verzegnassi, Phys. Rev. D 52, 451 (1995) [arXiv:hep-ph/9501362]. G. J. Gounaris, F. M. Renard and N. D. Vlachos, Nucl. Phys. B 459, 51 (1996) [arXiv:hep-ph/9509316]. K. Whisnant, J. M. Yang, B. L. Young and X. Zhang, Phys. Rev. D 56, 467 (1997) [arXiv:hep-ph/9702305]. J. M. Yang and B. L. Young, Phys. Rev. D 56, 5907 (1997) [arXiv:hep-ph/9703463].
* (14) J. A. Aguilar-Saavedra, Nucl. Phys. B 812, 181 (2009) [arXiv:0811.3842 [hep-ph]].
* (15) B. Grzadkowski, M. Iskrzyński, M. Misiak, and J. Rosiek, arXiv: 1008.4884 [hep-ph].
* (16) J. J. Zhang, C. S. Li, J. Gao, H. Zhang, Z. Li, C. P. Yuan and T. C. Yuan, Phys. Rev. Lett. 102, 072001 (2009) [arXiv:0810.3889 [hep-ph]]. J. Drobnak, S. Fajfer, and Jernej F. Kamenik, Phys.Rev.Lett. 104, 252001(2010) [arXiv:1004.0620 [hep-ph]]
* (17) A. J. Buras, arXiv:hep-ph/9806471. G. Buchalla, A. J. Buras. and M.E. Lautenbacher, Rev. Mod. Phys. 68, 1125(1996).
* (18) A.J. Buras, M. Misiak, M. Münz, and S. Pokorski, Nucl. Phys. 424, 374(1994)
* (19) E. Barberio et al. [Heavy Flavor Averaging Group], arXiv:0808.1297 [hep-ex].
* (20) P. Gambino and M. Misiak, Nucl. Phys. B 611, 338 (2001) [arXiv:hep-ph/0104034].
* (21) A. J. Buras, A. Czarnecki, M. Misiak and J. Urban, Nucl. Phys. B 631, 219 (2002) [arXiv:hep-ph/0203135].
* (22) M. Misiak et al., Phys. Rev. Lett. 98, 022002 (2007) [arXiv:hep-ph/0609232].
* (23) C. W. Bauer, Z. Ligeti, M. Luke, A. V. Manohar and M. Trott, Phys. Rev. D 70, 094017 (2004) [arXiv:hep-ph/0408002].
* (24) C. S. Li, R. J. Oakes and T. C. Yuan, Phys. Rev. D 43, 3759 (1991).
* (25) K. Nakamura et al. (Particle Data Group), J. Phys. G 37, 075021 (2010) Cut-off date for this update was January 15, 2010.
* (26) J. Charles et al. [CKMfitter Group], Eur. Phys. J. C 41, 1 (2005) [arXiv:hep-ph/0406184].
* (27) A. H. Hoang and A. V. Manohar, Phys. Lett. B 633, 526 (2006) [arXiv:hep-ph/0509195].
* (28) B. Aubert et al. [BABAR Collaboration], Phys. Rev. D 81, 032003 (2010) [arXiv:0908.0415 [hep-ex]].
* (29) P. Gambino and U. Haisch, JHEP 0110, 020 (2001) [arXiv:hep-ph/0109058].
* (30) X.Q. Li, work in preparation.
|
arxiv-papers
| 2010-10-10T11:40:44 |
2024-09-04T02:49:13.637214
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xing-Bo Yuan, Yang Hao, Ya-Dong Yang",
"submitter": "Yadong Yang",
"url": "https://arxiv.org/abs/1010.1912"
}
|
1010.2014
|
# Recurrence and Pólya number of general one-dimensional random walks
Xiao-Kun Zhang, Jing Wan, Jing-Ju Lu and Xin-Ping Xu xuxinping@suda.edu.cn
School of Physical Science and Technology, Soochow University, Suzhou 215006,
China
###### Abstract
The recurrence properties of random walks can be characterized by Pólya
number, i.e., the probability that the walker has returned to the origin at
least once. In this paper, we consider recurrence properties for a general 1D
random walk on a line, in which at each time step the walker can move to the
left or right with probabilities $l$ and $r$, or remain at the same position
with probability $o$ ($l+r+o=1$). We calculate Pólya number $P$ of this model
and find a simple expression for $P$ as, $P=1-\Delta$, where $\Delta$ is the
absolute difference of $l$ and $r$ ($\Delta=|l-r|$). We prove this rigorous
expression by the method of creative telescoping, and our result suggests that
the walk is recurrent if and only if the left-moving probability $l$ equals to
the right-moving probability $r$.
###### pacs:
05.40.Fb, 05.60.Cd, 05.40.Jc
Random walk is related to the diffusion models and is a fundamental topic in
discussions of Markov processes. Several properties of (classical) random
walks, including dispersal distributions, first-passage times and encounter
rates, have been extensively studied. The theory of random walk has been
applied to computer science, physics, ecology, economics, and a number of
other fields as a fundamental model for random processes in time rn1 ; rn2 ;
rn3 ; rn4 .
An interesting question for random walks is whether the walker eventually
returns to the starting point, which can be characterized by Pólya number,
i.e., the probability that the walker has returned to the origin at least once
during the time evolution. The concept of Pólya number was proposed by George
Pólya, who is a mathematician and first discussed the recurrence property in
classical random walks on infinite lattices in 1921 rn5 ; rn6 . Pólya pointed
out if the number equals one, then the walk is called recurrent, otherwise the
walk is transient because the walker has a nonzero probability to escape rn7 .
As a consequence, Pólya showed that for one and two dimensional infinite
lattices the walks are recurrent, while for three dimension or higher
dimensions the walks are transient and a unique Pólya number is calculated for
them rn8 . Recently, M. Štefaňák et al. extend the concept of Pólya number to
characterize the recurrence properties of quantum walks rn9 ; rn10 ; rn11 .
They point out that the recurrence behavior of quantum walks is not solely
determined by the dimensionality of the structure, but also depend on the
topology of the walk, choice of coin operators, and the initial coin state,
etc rn9 ; rn10 ; rn11 . This suggests the Pólya number of random walks or
quantum walks may depends on a variety of ingredients including the structural
dimensionality and model parameters.
In this paper, we consider recurrence properties for a general one-dimensional
random walk. The walk starts at $x=0$ on a line and at each time step the
walker moves one unit towards the left or right with probabilities $l$ and
$r$, or remain at the same position with probability $o$ ($l+r+o=1$). This
general random walk model has some useful application in physical or chemical
problems, and some of its dynamical properties requires a further study.
Previous studies of one-dimensional random walk focus on the simple symmetric
case where the walker moves to left and right with equal probability
($l=r=1/2$) rn10 . For instance, Pólya showed that the symmetric random walk
is recurrent and its Pólya number equals to 1 rn12 ; rn13 . However,
recurrence properties of this general random walk defined here are still
unknown. As a consequence, we will calculate the Pólya number for this general
random model and discuss its recurrence properties. We will try to derive an
explicit expression for Pólya number, and reveal its dependence on the model
parameters $l$, $r$ and $o$.
Pólya number of random walks can be expressed in terms of the return
probability $p_{0}(t)$ rn12 ; rn10 , i.e., the probability for the walker
returns to its original position $x=0$ at step $t$,
$P=1-\frac{1}{\sum_{t=0}^{\infty}p_{0}(t)}.$ (1)
Hence, the recurrence behavior of random walk is determined solely by the
infinite summation of return probabilities. It is evident that if the
summation of return probabilities diverges the walk is recurrent ($P=1$), and
if the summation converges the walk is transient ($P<1$). To calculate the
Pólya number, it is crucial to obtain the return probabilities. In the
following, we will calculate the return probabilities for our general random
walk model.
The return probability $p_{0}(t)$ can be obtained using the trinomial
coefficients of $(l+o+r)^{t}$. Considering an ensemble of random walks after
$t$ steps, in which the walker has $L$ steps moving left, $R$ steps moving
right and $O$ steps remaining at the same position, then the probability for
such random walks is $\frac{t!}{O!L!R!}o^{O}l^{L}r^{R}$ ($l+o+r=1$,
$L+O+R=t$). Since the walker’s position $x$ is only dependant on the
difference of right-moving steps $R$ and left-moving steps $L$, $x=R-L$,
returning to the original position $x=0$ requires $R=L$. Therefore, the
ensemble of random walks returning to $x=0$ involves sum over all possible $O$
subject to the constraints $R=L$ and $R+L+O=t$. Because $R+L$ is an even
number, $t$ and $O$ must have the same parity. Here, we suppose $t=2n$, $O=2i$
for even $t$ and $O$, and $t=2n+1$, $O=2i+1$ for odd $t$ and $O$ ($i$ and $n$
are nonnegative integers, and $i\leq n$). We calculate the return probability
for even $t$ and odd $t$ separately. For even $t$, the return probability is
given by,
$p_{0}(t)|_{t=2n}=\sum_{i=0}^{n}\frac{(2n)!}{(2i)!(n-i)!(n-i)!}o^{2i}l^{n-i}r^{n-i},$
(2)
where $t=2n$, $O=2i$, $R=L=(t-O)/2=n-i$ are used in the above equation.
Analogously, for odd $t$, the return probability is given by,
$p_{0}(t)|_{t=2n+1}=\sum_{i=0}^{n}\frac{(2n+1)!}{(2i+1)!(n-i)!(n-i)!}o^{2i+1}l^{n-i}r^{n-i}.$
(3)
The infinite summation of return probabilities $S$ can be determined by the
sum of $p_{0}(2n)$ and $p_{0}(2n+1)$,
$S=\sum_{t=0}^{\infty}p_{0}(t)=\sum_{n=0}^{\infty}\Big{(}p_{0}(t)|_{t=2n}+p_{0}(t)|_{t=2n+1}\Big{)}.$
(4)
In order to get a simple expression for $S$, we define $\Delta=|r-l|$, thus
$lr=\big{(}(1-o)^{2}-\Delta^{2}\big{)}/4$. Substituting this relation into Eq.
(4), we get
$\begin{array}[]{ll}S&=\displaystyle{\sum_{n=0}^{\infty}}\Big{(}p_{0}(t)|_{t=2n}+p_{0}(t)|_{t=2n+1}\Big{)}\\\
&=\displaystyle{\sum_{n=0}^{\infty}}\Big{(}\sum_{i=0}^{n}\frac{(2n)!}{(2i)!(n-i)!(n-i)!}o^{2i}\big{(}\frac{(1-o)^{2}-\Delta^{2}}{4}\big{)}^{n-i}\\\
&\
+\displaystyle{\sum_{i=0}^{n}}\frac{(2n+1)!}{(2i+1)!(n-i)!(n-i)!}o^{2i+1}\big{(}\frac{(1-o)^{2}-\Delta^{2}}{4}\big{)}^{n-i}\Big{)}\\\
&=\displaystyle{\sum_{n=0}^{\infty}}\frac{(2n)!}{(n!)^{2}}\big{(}\frac{(1-o)^{2}-\Delta^{2}}{4}\big{)}^{n}\\\
&\times\Big{(}\ _{2}F_{1}(-n,-n,1/2,\frac{o^{2}}{(1-o)^{2}-\Delta^{2}})+\\\
&(2n+1)o\
_{2}F_{1}(-n,-n,3/2,\frac{o^{2}}{(1-o)^{2}-\Delta^{2}})\Big{)},\end{array}$
(5)
where ${}_{2}F_{1}(a,b,c,z)$ is the Gauss Hypergeometric function. $S$ can be
further simplified, for the sake of clarity, we first consider the case $o=0$.
When $o=0$ the Hypergeometric function equals to 1, $S$ can be simplified as,
$S=\sum_{n=0}^{\infty}\frac{(2n)!}{(n!)^{2}}\Big{(}\frac{1-\Delta^{2}}{4}\Big{)}^{n}=\frac{1}{\Delta}.$
(6)
The last equality follows from the Taylor series expansion at $z=0$ for the
function $1/\sqrt{1-4z}$.
For $o>0$, we find that $S$ also equals to $1/\Delta$. This result is
surprising because $S$ does not depend on the remaining unmoving probability
$o$. This suggests that, for all $o$ and $\Delta$, Eq. (5) can be simplified
as,
$S=\frac{1}{\Delta},\ \ \ \ \ \forall\ \ 0<o,\Delta\leq 1,o+\Delta\leq 1.$ (7)
It is difficult to simplify Eq. (5) or prove Eq. (7) using the usual
mathematical methods. Here, in the appendix, we prove this rigorous expression
(7) by the method of creative telescoping. The method of creative telescoping
rn14 ; rn15 ; rn16 is an algorithm to compute hypergeometric summation,
definite integration, and prove combinatorial identity. Using this method, we
transfer $S$ to the solution of a partial differential equation (See the proof
in the appendix).
The Pólya number in Eq. (1) can be written as,
$P=1-\frac{1}{S}=1-\Delta.$ (8)
Consequently, we find a simple explicit expression for Pólya number, which is
solely determined by the absolute difference of $l$ and $r$, $\Delta=|l-r|$.
According to Eq. (8), Pólya number $P$ equals to 1 for $\Delta=0$. This
suggests that the walk is recurrent if and only if the left-moving probability
$l$ equals to the right-moving probability $r$. Our result is consistent with
previous conclusion that one-dimensional symmetric random walk ($l=r=1/2$) is
recurrent. Our result also indicates that the infinite summation of return
probabilities $S$ diverges for $\Delta=0$ and converges for $\Delta\neq 0$. To
verify this point, we plot the return probability $p_{0}(t)$ as a function of
step $t$ in Fig. 1. We find that $p_{0}(t)$ is a power-law decay as
$p_{0}(t)\sim t^{-0.5}$ for $\Delta=0$ (See Fig. 1 (a) in the log-log plot)
and exponential decay for $\Delta\neq 0$ (See Fig. 1 (b), (c) in the log-
linear plot). Since $p_{0}(t)$ for $\Delta=0$ decays slower than $t^{-1}$ and
decays faster than $t^{-1}$ for $\Delta\neq 0$, the infinite summation $S$
diverges for $\Delta=0$ and converges otherwise. Particularly, by means of
Stirling’s approximation $n!\approx\sqrt{2\pi n}(n/e)^{n}$ for $o=0$, we find
an asymptotic form for the return probability in Eq. (6):
$p_{0}(t)\approx\sqrt{\frac{2}{\pi t}}(1-\Delta^{2})^{t/2}$ for even $t$ and
$p_{0}(t)=0$ for odd $t$. For a certain value of $\Delta>0$, the decay
behavior of $p_{0}(t)$ seems different for different values of $o$ (See Fig. 1
(b), (c)). However, the summations of $p_{0}(t)$ for different $o$ are
identical and equal to $1/\Delta$. This result is some what unexpected and we
provide a strict proof in the appendix.
Figure 1: (Color online) Return probability $p_{0}(t)$ as a function of step
$t$ for $\Delta=0$ (a), $\Delta=0.2$ (b) and $\Delta=0.4$ (c). For each value
of $\Delta$, we plot $p_{0}(t)$ vs $t$ for $o=0$ (black squares), $o=0.2$ (red
dots) and $o=0.4$ (blue triangles). The critical decay for convergence
$p_{0}(t)\sim t^{-1}$ are also plotted in the figure. $p_{0}(t)$ shows a
power-law decay $t^{-0.5}$ for $\Delta=0$ (See (a)), and $p_{0}(t)$ exhibits
exponential decay for $\Delta>0$ (See (b) and (c)). It should be pointed out
that for the case $o>0$, $p_{0}(t)$ is nonzero at all values of $t$, while
$p_{0}(t)$ is zero at odd $t$ for $o=0$.
In summary, we have studied recurrence properties for a general 1D random walk
on a line, in which at each time step the walker can move to the left or right
with probabilities $l$ and $r$, or remain at the same position with
probability $o$ ($l+r+o=1$). We calculate Pólya number $P$ of this model for
the first time, and find a simple explicit expression for $P$ as,
$P=1-\Delta$, where $\Delta$ is the absolute difference of $l$ and $r$
($\Delta=|l-r|$). We prove this rigorous relation by the method of creative
telescoping, and our result suggests that the walk is recurrent if and only if
the left-moving probability $l$ equals to the right-moving probability $r$.
We thank Armin Straub and Dr. Koutschan for useful discussions. This work is
supported by National Natural Science Foundation of China under project
10975057, the new Teacher Foundation of Soochow University under contracts
Q3108908, Q4108910, and the extracurricular research foundation of
undergraduates under project KY2010056A.
## Appendix A The method of creative telescoping (MCT)
The method of creative telescoping, also known as Zeilberger’s algorithm rn14
; rn15 ; rn16 , is a powerful tool for solving problem involving definite
integration and summation of hypergeometric function. Suppose we are given a
certain holonomic function of two variables $F(z,n)$ ($n\in Integers$, $z\in
Reals$), and it is required to prove that the summation of $F(z,n)$ over $n$
equals to $f(z)$,
$\sum_{n}F(z,n)=f(z).$ (9)
The basic idea of creative telescoping algorithm is to find a linear
recurrence equation for the summands $F(z,n)$. This could be done by
constructing a differential operator $\hat{L}$ with coefficients being
polynomials in $z$, and a new function $G(z,n)$ satisfying,
$\hat{L}(z)F(z,n)=G(z,n+1)-G(z,n).$ (10)
Thus $\hat{L}(z)$ operating on the summation $\sum_{n}F(z,n)$ is determined by
the difference of upper bound and lower bound
$G_{0}(z)=G(z,n_{max})-G(z,n_{min})$. Then we just need to check both sides of
Eq. (9) satisfy recurrence equations: $\hat{L}(z)\sum_{n}F(z,n)=G_{0}(z)$,
$\hat{L}(z)f(z)=G_{0}(z)$, and check Eq. (9) holds for some initial
conditions.
Several algorithms for computing creative telescoping relations have been
developed in the past rn17 . The main programs are Zeilberger’s Maple program
and Mathematica program written by Peter Paule and Markus Schorn rn17 ; rn18 ;
rn19 . Here, we use the mathematical program to compute the creative
telescoping relation for our problem.
## Appendix B Proof of $S=\frac{1}{\Delta}$ using MCT
In this section, we prove $S=\frac{1}{\Delta}$ using the method of creative
telescoping (MCT). We use the Mathematica package Holonomic Functions rn17 ;
rn20 ; rn21 to create a recurrence relation for the summands
$s_{n}(o,\Delta)$ in Eq. (5),
$\Big{(}2oD_{o}+\Delta
D_{\Delta}+1+(S_{n}-1)\frac{1}{\Delta}D_{\Delta}\Big{)}s_{n}(o,\Delta)=0,$
(11)
where $D_{o}$, $D_{\Delta}$ are the partial differential operator
($D_{o}\equiv\partial/\partial o$, $D_{\Delta}\equiv\partial/\partial\Delta$),
$S_{n}$ is the shift operator satisfying $S_{n}f(n)=f(n+1)$.
Summing over $n$ leads to,
$\Big{(}2oD_{o}+\Delta
D_{\Delta}+1\Big{)}S+\displaystyle{\sum_{n=0}^{\infty}}(S_{n}-1)\frac{1}{\Delta}D_{\Delta}s_{n}(o,\Delta)=0.$
(12)
The second term in the above equation is a telescoping series, the central
terms are cancelled and only leave the last term and first term. Noting that
$\frac{1}{\Delta}D_{\Delta}s_{n}(o,\Delta)$ are zero for $n=0$ and
$n\rightarrow\infty$, the second term in Eq. (12) equals to $0$. Hence the
infinite summation of return probabilities $S$ satisfies,
$\Big{(}2oD_{o}+\Delta D_{\Delta}+1\Big{)}S=0.$ (13)
It is easy to check $\frac{1}{\Delta}$ also satisfies the above partial
differential equation. Combining with the initial condition
$S=\frac{1}{\Delta}$ for $o=0$ (See Eq. (6)), $S=\frac{1}{\Delta}$ holds for
all $o$ and $\Delta$.
## References
* (1) N. Guillotin-Plantard and R. Schott, _Dynamic Random Walks: Theory and Application_ (Elsevier, Amsterdam, 2006).
* (2) W. Woess, _Random Walks on Infinite Graphs and Groups_ (Cambridge: Cambridge University Press, 2000).
* (3) F Spitzer, _Principles of random walk_(Springer, Berlin, 2000).
* (4) G. H. Weiss, _Aspects and applications of the random walk_ , (North-Holland, New york, 1994).
* (5) G. Pólya, _How to Solve It_ , (Princeton University Press, 1945).
* (6) G. L. Alexanderson, _The Random Walks of George Pólya_ , (Mathematical Association of America, 2000).
* (7) W. E. Weisstein, _Pólya’s Random Walk Constants_ , From MathWorld-A Wolfram Web Resource.
http://mathworld.wolfram.com/PolyasRandomWalkConstants.html
* (8) S. R. Finch, _Pólya’s Random Walk Constant_ , in §5.9 Mathematical Constants (Cambridge University Press, pp. 322-331, 2003).
* (9) M. Štefaňák, I. Jex and T. Kiss, Phys. Rev. Lett 100, 020501 (2008).
* (10) M. Štefaňák, T. Kiss and I. Jex, Phys. Rev. A 78, 032306 (2008).
* (11) Z. Darázs and T. Kiss, Phys. Rev. A 81, 062319 (2010).
* (12) C. Domb, _On Multiple Returns in the Random-Walk Problem_ , Proc. Cambridge Philos. Soc. 50, 586-591 (1954).
* (13) E. W. Montroll, _Random Walks in Multidimensional Spaces, Especially on Periodic Lattices_ , J. SIAM 4, 241-260 (1956).
* (14) D. Zeilberger, _The Method of Creative Telescoping_ , J. Symbolic Computation 11, 195-204 (1991).
* (15) D. Zeilberger, _A Holonomic Systems Approach to Special Function Identities_ , J. Comput. Appl. Math. 32, 321-368 (1990).
* (16) D. Zeilberger, _A Fast Algorithm for Proving Terminating Hypergeometric Series Identities_ , Discrete Math. 80, 207-211 (1990).
* (17) M. Petkovšek, H. S. Wilf and D. Zeilberger, _A=B_ , (AK Peters, Ltd. 1996)).
* (18) http://www.math.temple.edu/`~`zeilberg/programs.html.
* (19) P. Paule and M. Schorn, _A Mathematica Version of Zeilberger’s Algorithm for Proving Binomial Coefficient Identities_ , J. Symbolic Comput. 20, pp. 673-698 (1995).
* (20) C. Koutschan, _A Fast Approach to Creative Telescoping_ , arxiv:1004.3314
* (21) The Holonomic package can be downloaded at
http://www.risc.uni-linz.ac.at/research/combinat/software/
|
arxiv-papers
| 2010-10-11T06:49:51 |
2024-09-04T02:49:13.651294
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xiao-Kun Zhang, Jing Wan, Jing-Ju Lu, Xin-Ping Xu",
"submitter": "Xin-Ping Xu",
"url": "https://arxiv.org/abs/1010.2014"
}
|
1010.2020
|
# Symmetry and special relativity in Finsler spacetime with constant curvature
Xin Li1,3 lixin@itp.ac.cn Zhe Chang2,3 changz@ihep.ac.cn 1Institute of
Theoretical Physics, Chinese Academy of Sciences, 100190 Beijing, China
2Institute of High Energy Physics, Chinese Academy of Sciences, 100049
Beijing, China
3Theoretical Physics Center for Science Facilities, Chinese Academy of
Sciences
###### Abstract
Within the framework of projective geometry, we investigate kinematics and
symmetry in $(\alpha,\beta)$ spacetime-one special types of Finsler spacetime.
The projectively flat $(\alpha,\beta)$ spacetime with constant flag curvature
is divided into four types. The symmetry in type A-Riemann spacetime with
constant sectional curvature is just the one in de Sitter special relativity.
The symmetry in type B-locally Minkowski spacetime is just the one in very
special relativity. It is found that type C-Funk spacetime and type D-scaled
Berwald’s metric spacetime both possess the Lorentz group as its isometric
group. The geodesic equation, algebra and dispersion relation in the
$(\alpha,\beta)$ spacetime are given. The corresponding invariant special
relativity in the four types of $(\alpha,\beta)$ spacetime contain two
parameters-the speed of light and a geometrical parameter which may relate to
the new physical scale. They all reduce to Einstein’s special relativity while
the geometrical parameter vanishes.
special relativity; Finsler spacetime; projectively flat
###### pacs:
03.30.+p,02.40.Dr,11.30.-j
## I Introduction
Lorentz Invariance (LI) is one of the foundations of the Standard model of
particle physics. Of course, it is very interesting to test the fate of the LI
both on experiments and theories. The theoretical approach of investigating
the LI violation is studying the possible spacetime symmetry, and erecting
some counterparts of special relativity. Recently, there are a few
counterparts of special relativity. The first one is doubly special relativity
(DSR)Amelino1 ; Amelino2 ; Amelino3 . In DSR, the Planck-scale effects have
been taken into account by introducing an invariant Planckin parameter
$\kappa$. Together with the speed of light $c$, DSR has two invariant
parameters. The second one is very special relativity (VSR) Coleman1 ;
Coleman2 . Coleman and Glashow have set up a perturbative framework for
investigating possible departures of local quantum field theory from LI. The
symmetry group of VSR is some certain subgroups of Poincare group, which
contains the spacetime translations and proper subgroups of Lorentz
transformations. The last is the de Sitter(dS)/anti de Sitter(AdS) invariant
special relativity (dSSR) Look ; Look1 . The dSSR suggests that the principle
of relativity should be generalized to constant curvature spacetime with
radius $R$ in Riemannian manifold.
In fact, the three kinds of modified special relativity share common ground.
Historically, Snyder proposed a quantized spacetime model Snyder . In his
model, the spacetime coordinates were defined as translation generators of dS-
algebra $\mathfrak{so}(1,4)$ and become noncommutative. It has already been
pointed out in Ref. Guo that there is a dual one-to-one correspondence
between Snyder’s quantized spacetime model as a DSR and the dSSR. Actually,
the Plackin parameter $\kappa$ in DSR is related to the parameter $a$ in
Snyder’s model in addition to $c$. Furthermore, the dSSR can be regarded as a
spacetime counterpart of Snyder’s model. VSR can be realized on a
noncommutative Moyal plane with light-like noncommutativity Sheikh . Thus, the
three kinds of modified special relativity all have noncommutative
realization.
On the other hand, these counterparts of special relativity have connections
with Finsler geometry Bao , which is a natural generalization of Riemannian
geometry. The noncommutativity effects may be regarded as the deviation of
Finsler spacetime from Riemann spacetime. Ref.Ghosh gave a canonical
description of DSR and showed that the DSR admits a modified dispersion
relation (MDR) as well as noncommutative $\kappa$-Minkowskian phase space.
Furthermore, Girelli et al.Girelli showed that the MDR in DSR could be
incorporated into the framework of Finsler geometry. As for VSR, Gibbons et
al. have pointed out that general VSR is Finsler Geometry Gibbons .
Therefore, It is reasonable to assume that these counterparts of special
relativity may have a corporate origin in Finsler geommetry. In order to
investigate the counterpart of special relativity in a systematic way, first,
we should erect the inertial frames in Finsler spacetime. Second, we should
investigate the symmetry in Finsler spacetime. The way of describing spacetime
symmetry in a covariant language (the symmetry should not depend on any
particular choice of coordinate system) involves the concept of isometric
transformations. In fact, the symmetry of spacetime is described by the so
called isometric group. The generators of isometric group is directly
connected with the Killing vectorsKilling . Actually, the symmetry of deformed
relativity has been studied by investigating the Killing vectorsAlvarez . It
is well known that the isometric group is a Lie group in Riemannian manifold.
This fact also holds in Finslerian manifoldDeng . The counterparts of Poincare
algebra in Finsler spacetime could be studied. At last, we should give the
kinematic and dispersion law in Finsler spacetime.
This paper is organized as follows. In Sec.2, we present basic notations of
Finsler geometry and discuss inertial frames in Finsler spacetime. In Sec.3,
we use the isometric group to investigate the symmetry of Finsler spacetime.
In Sec.4, we discuss the kinematics in projectively flat $(\alpha,\beta)$
spacetime with constant flag curvature. The isometric groups and the
corresponded Lie algebras for different types of $(\alpha,\beta)$ spacetime
are given. At last, we give the concluding remarks. The counterpart of
sectional curvature in Riemann geometry-flag curvature is introduced in
appendix.
## II Finsler spacetime
The inertial frame means a particle in it continue at rest or in uniform
straight motion. In an inertial system, the inertial motion is described by
$x^{i}=v^{i}(t-t_{0})+x^{i}_{0},~{}~{}~{}v^{i}\equiv\frac{dx^{i}}{dt}=consts.$
(1)
It should be notice that such definition for inertial motion (1) does not
involve any specific requirements on the metric of spacetime. In fact,
Einstein just assumed that the spacetime should be Euclidean which inherited
from NewtonEinstein . If we loose the requirement that the spacetime should be
Euclidean and require that the spacetime should be Riemannian, there exists
three classes of inertial frames. Historically, de Sitter first used the
projective coordinates (or Beltrami coordinates) to erect a spacetime with
constant sectional curvature-the de Sitter spacetime. De Sitter used his dS
spacetime to debate with Einstein on ‘relative inertial’. Actually, the dS
spacetime is one kinds of locally projectively flat spacetime.
A spacetime is said to be locally projectively flat if at every point, the
geodesics are straight lines
$x^{\mu}(\tau)=f(\tau)m^{\mu}+n^{\mu},$ (2)
where $\tau$ is the parameter of the curve, $f(\tau)$ is a function which
depends on the metric of spacetime and $m^{\mu},n^{\mu}$ are constants.
Clearly, the definition of projectively flat spacetime (2) implies the
inertial motion. If $x^{0}$ denotes time, one could obtain the formula (1)
from (2). In Riemannian manifold, Beltrami’s theorem tells us that a
Riemannian metric is locally projectively flat if and only if it is of
constant sectional curvature. It is well known that there are three kinds of
spacetime with constant sectional curvature. They are Minkowski (Mink)
spacetime and dS/AdS spacetime. That is why there only exists three classes of
inertial frames in Riemannian spacetime. The three classes of inertial frames
are the basis of the dSSR.
If we further loose the requirement for spacetime, just require that the
spacetime should be Finslerian, various inertial frames could be obtained,
including the inertial frames for VSR and DSR.
Instead of defining an inner product structure over the tangent bundle in
Riemann geometry, Finsler geometry is based on the so called Finsler structure
$F$ with the property $F(x,\lambda y)=\lambda F(x,y)$ for all $\lambda>0$,
where $x\in M$ represents position and $y\equiv\frac{dx}{d\tau}$ represents
velocity. The Finsler metric is given asBook by Bao
$g_{\mu\nu}\equiv\frac{\partial}{\partial y^{\mu}}\frac{\partial}{\partial
y^{\nu}}\left(\frac{1}{2}F^{2}\right).$ (3)
Finsler geometry has its genesis in integrals of the form
$\int^{r}_{s}F(x^{1},\cdots,x^{n};\frac{dx^{1}}{d\tau},\cdots,\frac{dx^{n}}{d\tau})d\tau~{}.$
(4)
The Finsler structure represents the length element of Finsler space. Two
types of Finsler space should be noticed. One is the Riemann space. A Finsler
metric is said to be Riemannian, if $F^{2}$ is quadratic in $y$. Another is
locally Minkowski space. A Finsler metric is said to be locally Minkowskian if
at every point, there is a local coordinate system, such that $F=F(y)$ is
independent of the position $x$ Book by Bao .
The geodesic equation for Finsler manifold is given asBook by Bao
$\frac{d^{2}x^{\mu}}{d\tau^{2}}+2G^{\mu}=0,$ (5)
where
$G^{\mu}=\frac{1}{4}g^{\mu\nu}\left(\frac{\partial^{2}F^{2}}{\partial
x^{\lambda}\partial y^{\nu}}y^{\lambda}-\frac{\partial F^{2}}{\partial
x^{\nu}}\right)$ (6)
is called geodesic spray coefficient. Obviously, if $F$ is Riemannian metric,
then
$G^{\mu}=\frac{1}{2}\gamma^{\mu}_{\nu\lambda}y^{\nu}y^{\lambda},$ (7)
where $\gamma^{\mu}_{\nu\lambda}$ is the Riemannian Christoffel symbol. By
making use of the geodesic equation (5), one could find that a Finsler metric
is locally projectively flat if and only if $G^{\mu}$ satisfies
$G^{\mu}=P(x,y)y^{\mu},$ (8)
where $P(x,y)$ is a function of $x$ and $y$. It is equivalent to the following
equation that was proposed by Hamel Hamel
$\frac{\partial^{2}F}{\partial x^{\lambda}\partial
y^{\nu}}y^{\lambda}=\frac{\partial F}{\partial x^{\nu}}.$ (9)
By making use of the Hamel equation (9), we get
$G^{\mu}=\left(\frac{\partial F}{\partial x^{\nu}}y^{\nu}/2F\right)y^{\mu}.$
(10)
It means that $P=\frac{\partial F}{\partial x^{\nu}}y^{\nu}/2F$. One should
notice that
$\displaystyle\frac{dF}{d\tau}$ $\displaystyle=$ $\displaystyle\frac{\partial
F}{\partial x^{\mu}}\frac{dx^{\mu}}{d\tau}+\frac{\partial F}{\partial
y^{\mu}}\frac{dy^{\mu}}{d\tau}$ (11) $\displaystyle=$ $\displaystyle
2PF-2PF=0,$
where we has already used the formula for $P$ and the geodesic equation (5) to
deduce the second equation.
## III Symmetry in Finsler space
To investigate the Killing vectors, we should construct the isometric
transformations of Finsler structure. It is convenient to discuss the
isometric transformations under an infinitesimal coordinate transformation for
$x$
$\bar{x}^{\mu}=x^{\mu}+\epsilon V^{\mu},$ (12)
together with a corresponding transformation for $y$
$\bar{y}^{\mu}=y^{\mu}+\epsilon\frac{\partial V^{\mu}}{\partial
x^{\nu}}y^{\nu},$ (13)
where $|\epsilon|\ll 1$. Under the coordinate transformation (12) and (13), to
first order in $|\epsilon|$, we obtain the expansion of the Finsler structure,
$\bar{F}(\bar{x},\bar{y})=\bar{F}(x,y)+\epsilon V^{\mu}\frac{\partial
F}{\partial x^{\mu}}+\epsilon y^{\nu}\frac{\partial V^{\mu}}{\partial
x^{\nu}}\frac{\partial F}{\partial y^{\mu}},$ (14)
where $\bar{F}(\bar{x},\bar{y})$ should equal to $F(x,y)$. Under the
transformation (12) and (13), a Finsler structure is called isometry if and
only if
$F(x,y)=\bar{F}(x,y).$ (15)
Deducing from the (14), we obtain the Killing equation $K_{V}(F)$ in Finsler
space
$K_{V}(F)\equiv V^{\mu}\frac{\partial F}{\partial
x^{\mu}}+y^{\nu}\frac{\partial V^{\mu}}{\partial x^{\nu}}\frac{\partial
F}{\partial y^{\mu}}=0.$ (16)
Searching the Killing vectors for general Finsler manifold is a difficult
task. Here, we give the Killing vectors for a class of Finsler
space-$(\alpha,\beta)$ spaceShen with metric defining as
$\displaystyle F=\alpha\phi(s),~{}~{}~{}s=\frac{\beta}{\alpha},$ (17)
$\displaystyle\alpha=\sqrt{a_{\mu\nu}y^{\mu}y^{\nu}}~{}~{}{\rm
and}~{}~{}\beta=b_{\mu}(x)y^{\mu},$ (18)
where $\phi(s)$ is a smooth function, $\alpha$ is a Riemannian metric and
$\beta$ is a one form. Then, the Killing equation (16) in $(\alpha,\beta)$
space reads
$\displaystyle 0$ $\displaystyle=$ $\displaystyle K_{V}(\alpha)\phi(s)+\alpha
K_{V}(\phi(s))$ (19) $\displaystyle=$
$\displaystyle\left(\phi(s)-s\frac{\partial\phi(s)}{\partial
s}\right)K_{V}(\alpha)+\frac{\partial\phi(s)}{\partial s}K_{V}(\beta).$
And by making use of the Killing equation (16), we obtain
$\displaystyle K_{V}(\alpha)$ $\displaystyle=$
$\displaystyle\frac{1}{2\alpha}(V_{\mu|\nu}+V_{\nu|\mu})y^{\mu}y^{\nu},$ (20)
$\displaystyle K_{V}(\beta)$ $\displaystyle=$
$\displaystyle\left(V^{\mu}\frac{\partial b_{\nu}}{\partial
x^{\mu}}+b_{\mu}\frac{\partial V^{\mu}}{\partial x^{\nu}}\right)y^{\nu},$ (21)
where $``|"$ denotes the covariant derivative with respect to the Riemannian
metric $\alpha$. The solutions of the Killing equation (19) have three viable
scenarios. The first one is
$\phi(s)-s\frac{\partial\phi(s)}{\partial s}=0~{}~{}{\rm
and}~{}~{}K_{V}(\beta)=0,$ (22)
which implies $F=\lambda\beta$ for all $\lambda\in\mathbb{R}$. The second one
is
$\frac{\partial\phi(s)}{\partial s}=0~{}~{}{\rm and}~{}~{}K_{V}(\alpha)=0,$
(23)
which implies $F=\lambda\alpha$ for all $\lambda\in\mathbb{R}$. The above two
scenarios are just trivial space. Here we focus on the case of
$\phi(s)-s\frac{\partial\phi(s)}{\partial s}\neq 0$ and
$\frac{\partial\phi(s)}{\partial s}\neq 0$. This will induce the last
scenario.
Apparently, in the last scenario we have such solutions
$\displaystyle V_{\mu|\nu}+V_{\nu|\mu}$ $\displaystyle=$ $\displaystyle 0,$
(24) $\displaystyle V^{\mu}\frac{\partial b_{\nu}}{\partial
x^{\mu}}+b_{\mu}\frac{\partial V^{\mu}}{\partial x^{\nu}}$ $\displaystyle=$
$\displaystyle 0.$ (25)
The equation (24) is no other than the Riemannian Killing equation. The
equation (25) can be regarded as the constraint on the Killing vectors that
satisfy the Killing equation (24). Here, we must point out that additional
solutions of Killing equation (19) for $(\alpha,\beta)$ space exist, besides
the solutions (24) and (25). It will be discussed in next section.
However, the Killing equation for one type of $(\alpha,\beta)$ space-Randers
spaceRanders only have solutions (24) and (25). In Randers space, the
$\phi(s)$ is set as $\phi(s)=1+s$. Then, the Killing equation (19) reduces to
$K_{V}(\alpha)+K_{V}(\beta)=0.$ (26)
The $K_{V}(\alpha)$ contains irrational term of $y^{\mu}$ and $K_{V}(\beta)$
only contains rational term of $y^{\mu}$, therefore the equation (26)
satisfies if and only if $K_{V}(\alpha)=0$ and $K_{V}(\beta)=0$.
## IV Lie algebra and kinematics in projectively flat $(\alpha,\beta)$
spacetime
An $n$ $(n>3)$ dimensional $(\alpha,\beta)$ space is projectively flat with
constant flag curvature if and only if one of the following holdsBLi
* A.
it is Riemann spacetime with constant sectional curvature;
* B.
it is locally Minkowski spacetime;
* C.
it is locally isometric to a generalized Funk spacetimeFunk ;
* D.
it is locally isometric to Berwald’s metric spacetimeBerwald .
We will discuss the four types of projectively flat space respectively.
Throughout this section the $\cdot$ denotes the inner product of Minkowski
space $x\cdot x=\eta_{\mu\nu}x^{\mu}x^{\nu}$, where $\eta_{\mu\nu}={\rm
diag}(1,-1,-1,-1)$.
### IV.1 Symmetry in type A $(\alpha,\beta)$ spacetime and dSSR
The metric of Riemann spacetime with constant sectional curvature can be given
by the projective coordinate system
$F_{R}=\frac{\sqrt{(y\cdot y)(1-\mu(x\cdot x))+\mu(x\cdot
y)^{2}}}{1-\mu(x\cdot x)},$ (27)
where the sectional curvature $\mu$ of metric (27) is constant. Clearly, the
signature $+,0,-$ of $\mu$ corresponds to the dS spacetime, Mink spacetime and
AdS spacetime, respectively. Such a metric (27) is invariant under the
fractional linear transformations (FLT), and it is
$ISO(1,3)/SO(1,4)/SO(2,3)$\- invariant Mink/dS/AdS-spacetimeGuo .
By making use of the formula (6), we know that the geodesic spray coefficient
$G^{\mu}$ for metric (27) is given as
$G^{\mu}_{R}=\frac{\mu(x\cdot y)}{1-\mu(x\cdot x)}y^{\mu}.$ (28)
Thus, the geodesic equation for metric (27) is of the form
$\frac{d^{2}x^{\mu}}{d\tau^{2}}+\frac{2\mu(x\cdot\frac{dx}{d\tau})}{1-\mu(x\cdot
x)}\frac{dx^{\mu}}{d\tau}=0.$ (29)
In fact, the geodesic equation is equivalent to
$\frac{dp^{\mu}}{d\tau}=0,~{}~{}p^{\mu}\equiv\frac{m_{R}}{F_{R}}\frac{1}{1-\mu(x\cdot
x)}\frac{dx^{\mu}}{d\tau},$ (30)
where $m_{R}$ is the mass of the particle. Thus, $p^{\mu}$ is a constant along
the geodesic. It could be regarded as the counterpart of momentum. From
$F^{2}_{R}=g_{\mu\nu}y^{\mu}y^{\nu}$, we get
$g_{\mu\nu}p^{\mu}p^{\nu}=\frac{1}{(1-\mu(x\cdot x))^{2}}m^{2}_{R}.$ (31)
It is obvious that if $\mu=0$, the above relation returns to the dispersion
relation in Minkowski spacetime. The counterpart of angular momentum tensor
could be defined as
$L^{\mu\nu}\equiv x^{\mu}p^{\nu}-x^{\nu}p^{\mu}.$ (32)
It is also a conserved quantities along the geodesic, for
$\frac{dL^{\mu\nu}}{d\tau}=0$. The dispersion law in dSSR Guo is given as
$p\cdot p-\frac{|\mu|}{2}L\cdot L=m_{R}^{2}.$ (33)
By making use of the Killing equation (16), we obtain the Killing vectors for
Riemmannian metric (27)
$V^{\mu}=Q^{\mu}_{~{}\nu}x^{\nu}+C^{\mu}-\mu(x\cdot C)x^{\mu},$ (34)
where $Q_{\mu\nu}=\eta_{\rho\mu}Q^{\rho}_{~{}\nu}$ is an arbitrary constant
skew-symmetric matrix and $C_{\mu}=\eta_{\rho\mu}C^{\rho}$ is an arbitrary
constant vector. The isometric group of a Finsler space is a Lie group Deng .
One should notice that translation-like generators are induced by $C^{\mu}$
and Lorentz generators are induced by $Q_{\mu\nu}$. The generators of
isometric group in Riemannian space (27) read
$\displaystyle\eta_{\mu\nu}\hat{p}^{\nu}=\hat{p}_{\mu}=i(\partial_{\mu}-\mu
x_{\mu}(x\cdot\partial)),$ (35)
$\displaystyle\hat{L}_{\mu\nu}=x_{\mu}\hat{p}_{\nu}-x_{\nu}\hat{p}_{\mu}=i(x_{\mu}\partial_{\nu}-x_{\nu}\partial_{\mu}).$
(36)
The non-trivial Lie algebra corresponded to the Killing vectors (34) is given
as
$\displaystyle~{}[\hat{p}_{\mu},\hat{p}_{\nu}]$ $\displaystyle=$
$\displaystyle i\mu\hat{L}_{\mu\nu},$
$\displaystyle~{}[\hat{L}_{\mu\nu},\hat{p}_{\rho}]$ $\displaystyle=$
$\displaystyle i\eta_{\nu\rho}\hat{p}_{\mu}-i\eta_{\mu\rho}\hat{p}_{\nu},$
(37) $\displaystyle~{}[\hat{L}_{\mu\nu},\hat{L}_{\rho\lambda}]$
$\displaystyle=$ $\displaystyle
i\eta_{\mu\lambda}\hat{L}_{\nu\rho}-i\eta_{\mu\rho}\hat{L}_{\nu\lambda}+i\eta_{\nu\rho}\hat{L}_{\mu\lambda}-i\eta_{\nu\lambda}\hat{L}_{\mu\rho}.$
While sectional curvature of Riemannian spacetime (27) $\mu$ vanishes, the
dS/AdS spacetime reduce to Mink spacetime, the momentum tensors and angular
momentum tensors reduce to the one in Mink spacetime, and the Lie algebra (37)
in dSSR reduces to Poincare algebra.
The sectional curvature $\mu$ is linked with Guo the cosmological constant
$\Lambda$ Copeland , and the Newton-Hooke constant $\nu$ Huang
$\mu\simeq\frac{\Lambda}{3},~{}~{}\nu\equiv c\sqrt{\mu}\sim
10^{-35}s^{-2}~{}.$ (38)
### IV.2 Symmetry in type B $(\alpha,\beta)$ spacetime and VSR
A Finsler metric is said to be locally Minkowskian if at every point, there is
a local coordinate system, such that $F=F(y)$ is independent of the position
$x$. It is clear from the definition (6) that the geodesic spray coefficient
$G^{\mu}$ vanishes in locally Minkowski space. Thus, the geodesic equation of
locally Minkowshi space is of simply form
$\frac{d^{2}x^{\mu}}{d\tau^{2}}=0.$ (39)
The momentum tensor $p^{\mu}=\frac{m_{V}}{F_{V}}\frac{dx^{\mu}}{d\tau}$ and
angular momentum tensor $L^{\mu\nu}\equiv x^{\mu}p^{\nu}-x^{\nu}p^{\mu}$ are
conserved quantities along the geodesic, for
$\frac{dp^{\mu}}{d\tau}=0,~{}~{}\frac{dL^{\mu\nu}}{d\tau}=0.$ (40)
Besides the Minkowski space, locally Minkowski space still involve a various
types of metric space. But not all of them has physical implication. Here, we
just focus on the locally Minkowski space which is invariant under the VSR
symmetric group.
The VSR preserves the law of energy-momentum conservationGlashow . It implies
that the translation invariance should be contained in the symmetries of the
VSR. The left symmetries of the VSR include four possible subgroups of Lorentz
group. We introduce the notation $T_{1}=(K_{x}+J_{y})/\sqrt{2}$ and
$T_{2}=(K_{y}-J_{x})/\sqrt{2}$ (the index $x,y,z$ denote the space
coordinate), where $J$ and $K$ are the generators of rotations and boosts,
respectively. The four subgroups of Lorentz group are given asSheikh :
i)$T(2)$, the Abelian subgroup of the Lorentz group, generated by $T_{1}$ and
$T_{2}$, with the structure:
$[T_{1},T_{2}]=0;$ (41)
ii)$E(2)$, the group of two-dimensional Euclidean motion, generated by
$T_{1}$, $T_{2}$ and $J_{z}$, with the structure:
$[T_{1},T_{2}]=0,~{}[J_{z},T_{1}]=-iT_{2},~{}[J_{z},T_{2}]=iT_{1};$ (42)
iii)$HOM(2)$, the group of orientation-preserving similarity transformations,
generated by $T_{1}$, $T_{2}$ and $K_{z}$, with the structure:
$[T_{1},T_{2}]=0,~{}[T_{1},K_{z}]=iT_{1},~{}[T_{2},K_{z}]=iT_{2};$ (43)
iv)$SIM(2)$, the group isomorphic to the four-parametric similitude group,
generated by $T_{1}$, $T_{2}$, $J_{z}$ and $K_{z}$, with the structure:
$\displaystyle~{}[T_{1},T_{2}]=0,$ $\displaystyle~{}[T_{1},K_{z}]=iT_{1}$
$\displaystyle,~{}[T_{2},K_{z}]=iT_{2},$ $\displaystyle~{}[J_{z},K_{z}]=0,$
$\displaystyle~{}[J_{z},T_{1}]=-iT_{2}$
$\displaystyle,~{}[J_{z},T_{2}]=iT_{1}.$ (44)
We will show that there is a relation between the isometric group of the
Finsler structureGibbons
$F_{V}=(\eta_{\mu\nu}y^{\mu}y^{\nu})^{(1-n)/2}(b_{\rho}y^{\rho})^{n}$ (45)
and symmetries of the VSR. Here $n$ is an arbitrary constant, $\eta_{\mu\nu}$
is Minkowskian metric and $b_{\rho}=\eta_{\mu\rho}b^{\mu}$ is a constant
vector. It is referred as the VSR metric. By making use of the Killing
equation (16), we obtain Killing equation for the VSR metric
$y^{\nu}\frac{\partial V^{\mu}}{\partial
x^{\nu}}\left(\frac{(1-n)y_{\mu}(b_{\rho}y^{\rho})^{n}+n(\eta_{\alpha\beta}y^{\alpha}y^{\beta})^{1/2}b_{\mu}(b_{\rho}y^{\rho})^{n-1}}{(\eta_{\alpha\beta}y^{\alpha}y^{\beta})^{(1+n)/2}}\right)=0.$
(46)
The Eq. (46) has solutions
$\displaystyle V^{\mu}$ $\displaystyle=$ $\displaystyle
Q^{\mu}_{~{}\nu}x^{\nu}+C^{\mu},$ (47) $\displaystyle b_{\mu}Q^{\mu}_{~{}\nu}$
$\displaystyle=$ $\displaystyle 0,$ (48)
where $Q_{\mu\nu}=\eta_{\rho\mu}Q^{\rho}_{~{}\nu}$ is an arbitrary constant
skew-symmetric matrix and $C_{\mu}=\eta_{\rho\mu}C^{\rho}$ is an arbitrary
constant vector. If one requires that the transformation group for the vectors
no other than the Lorentz one or subgroup of Lorentz one, formula (47)
togethers with the constraint (48) is the only solution of Killing equation
(16) for the VSR metric.
Taking the light cone coordinate Kogut
$\eta_{\alpha\beta}y^{\alpha}y^{\beta}=2y^{+}y^{-}-y^{i}y^{i}$ (with $i$
ranging over the values 1 and 2) and supposing
$b_{\mu}=\\{0,0,0,b_{-}\\}$($b_{-}=1$), we know that in general
$Q^{-}_{~{}\mu}\neq 0$. It means that the Killing vectors of the VSR metric
(45) do not have non-trivial components $Q_{+-}$ and $Q_{+i}$. The isometric
group of a Finsler space is a Lie group Deng . The non-trivial Lie algebra
corresponded to the Killing vectors (47), which satisfies the constraint (48),
is given as
$\displaystyle~{}[J_{z},T^{i}]=i\epsilon_{ij}T^{j},$
$\displaystyle[J_{z},P^{i}]=i\epsilon_{ij}P^{j},$
$\displaystyle~{}[T_{i},P^{-}]=-iP_{i},$
$\displaystyle[T_{i},P^{j}]=-i\delta_{ij}P^{+},$ (49)
where $\epsilon_{12}=-\epsilon_{12}=1,\epsilon_{11}=\epsilon_{22}=0$ and
$P^{\pm}=(P_{0}\pm P_{z})/\sqrt{2}$. It is obvious that the generators of the
isometric group of the VSR metric include generators of $E(2)$ and four
spacetime translation generators. This result induces the $E(2)$ scenario of
VSR from the VSR metric (45). The $HOM(2)$ scenario of VSR could be induced in
the same approach.
The above investigations are under the premise that the direction of spacetime
is arbitrary or the transformation group for the vectors no other than the
Lorentz group or subgroups of Lorentz group. It means that no preferred
direction exists in spacetime. If the spacetime does have a special direction,
the Killing equation (16) for the VSR metric will have a special solution. The
VSR metric was first suggested by Bogoslovsky Bogoslovsky . He assumed that
the spacetime has a preferred direction. Following the assumption and taking
the null direction to be the preferred direction, we obtain the solution of
Killing equation (46)
$V^{\mu}=(Q^{\mu}_{~{}\nu}+\delta^{\mu}_{~{}\nu})x^{\nu}+C^{\mu},$ (50)
where $Q^{\mu}_{~{}\nu}$ is an antisymmetrical matrix and satisfies the
requirement
$Q_{+-}n^{-}=-n^{-}.$ (51)
Here $n^{-}$ is a null direction. One can check that the Killing vectors (50)
does not have non-trivial components $Q_{+i}$. It implies that the null
direction is invariant under the transformation
$\Lambda^{-}_{~{}-}n^{-}\equiv\left(\delta^{-}_{~{}-}+\epsilon(n\delta^{-}_{~{}-}+Q^{-}_{~{}-})\right)n^{-}=\left(1+\epsilon(n-1)\right)n^{-}.$
(52)
Here, $\Lambda^{\mu}_{~{}\nu}$ denotes the counterpart of Lorentz
transformation. Therefore, if the spacetime has a preferred direction in null
direction, the symmetry corresponded to $Q_{+-}$ is restored. One can see that
the Killing vectors (50) have a non-trivial component
$\delta^{\mu}_{~{}\nu}x^{\nu}$. It represents the dilations. Thus, we know
that the transformation group for the VSR metric (45) contains dilations,
while the null direction is a preferred direction. One could obtain the Lie
algebra for such transformation group. In fact, the non-trivial Lie algebra is
just the algebra of $DISIM(2)$ group proposed by Gibbons et al.Gibbons
$\displaystyle~{}[K_{z},P^{\pm}]=-i(n\pm 1)P^{\pm},$
$\displaystyle[K_{z},P^{i}]=-inP^{i},$
$\displaystyle~{}[K_{z},T_{i}]=-iT_{i},$
$\displaystyle[J_{z},T^{i}]=i\epsilon_{ij}T^{j},$
$\displaystyle~{}[J_{z},P^{i}]=i\epsilon_{ij}P^{j},$
$\displaystyle[T_{i},P^{-}]=-iP_{i},$ (53)
$\displaystyle[T_{i},P^{j}]=-i\delta_{ij}P^{+}.$
The $DISIM(2)$ group is a subgroup of Weyl group, it contains a subgroup
$E(2)$ together with a combination of a boost in the $+-$ direction and a
dilation. It should be noticed that the deformed generator $K_{z}$ acts not
only as a boost but also a dilation. The transformation acts by $K_{z}$ is
given as
$\bar{x}^{\pm}=\left(\exp(\phi)\right)^{\pm
1+n}x^{\pm},~{}~{}\bar{x}^{i}=(\exp(\phi))^{n}x^{i},$ (54)
where $\exp(\phi)=\sqrt{\frac{1+v/c}{1-v/c}}$. The transformations act by
other generators of $DISIM(2)$ group are same with Lorentz one.
If $b_{\mu}$ in the VSR metric (45) has the form
$b_{\mu}=\\{0,b_{x},0,b_{-}\\}$($b_{x}=b_{-}=1$), solutions of Killing
equation (46) show that the Killing vectors just have non-trivial components
$Q_{-y}$ and $C^{\mu}$. However, the corresponded Lie algebra does not exist.
For the generators corresponded to $Q_{-y}$ together with the generators of
translations can not form a subalgebra of the Poincare algebra. Consequently,
we show that the investigation of Killing equation for VSR metric (45) could
account for the $E(2)$, $HOM(2)$ and $SIM(2)$($DISIM(2)$) scenarios of the
VSR.
The Lagrangian for VSR metric is given as
$\mathcal{L}=m_{V}F_{V}=m_{V}(\eta_{\mu\nu}y^{\mu}y^{\nu})^{(1-n)/2}(b_{\rho}y^{\rho})^{n}.$
(55)
The corresponding dispersion relation is of the form
$\eta^{\mu\nu}p_{\mu}p_{\nu}=m_{V}^{2}(1-n^{2})\left(\frac{n^{\rho}p_{\rho}}{m(1-n)}\right)^{2n/(1+n)}.$
(56)
The dispersion relation (56) is not Lorentz-invariant, but invariant under the
transformations of $DISIM(2)$ group. Ref. Bogoslovsky showed that the ether-
drift experiments gives a constraint $|n|<10^{-10}$ for the parameter $n$ of
the VSR metric (45).
### IV.3 Symmetry in type C $(\alpha,\beta)$ spacetime
The generalized Funk metric Funk has two geometrical parameters. For physical
consideration and simplicity, as DSR, VSR and dSSR, only one geometrical
parameter is needed. Therefore, we just investigate the Funk metric of this
form
$F_{F}=\frac{\sqrt{(y\cdot y)(1-\kappa^{2}(x\cdot x))+\kappa^{2}(x\cdot
y)^{2}}-\kappa(x\cdot y)}{1-\kappa^{2}(x\cdot x)}.$ (57)
Apparently, the Funk metric (57) is of Randers type,
$\displaystyle F_{F}=\alpha_{F}+\beta_{F},~{}~{}\alpha_{F}=\frac{\sqrt{(y\cdot
y)(1-\kappa^{2}(x\cdot x))+\kappa^{2}(x\cdot y)^{2}}}{1-\kappa^{2}(x\cdot
x)},~{}~{}\beta_{F}=\frac{-\kappa(x\cdot y)}{1-\kappa^{2}(x\cdot x)}.$ (58)
As discussed in Sec.3, the Killing vectors of Funk metric of Randers type must
satisfy both $K_{V}(\alpha)=0$ and $K_{V}(\beta)=0$, and it is the only
solutions of the Killing equation (16). The solution of equation
$K_{V}(\alpha)=0$ gives
$V^{\mu}=Q^{\mu}_{~{}\nu}x^{\nu}+C^{\mu}-\kappa^{2}(x\cdot C)x^{\mu},$ (59)
where $Q_{\mu\nu}=\eta_{\rho\mu}Q^{\rho}_{~{}\nu}$ is an arbitrary constant
skew-symmetric matrix and $C_{\mu}=\eta_{\rho\mu}C^{\rho}$ is an arbitrary
constant vector. And the solution of equation $K_{V}(\beta)=0$ gives
$\kappa C^{\nu}=0.$ (60)
The solutions (59) and (60) imply that the Killing vectors of Funk metric (57)
is of the form
$V^{\mu}=Q^{\mu}_{~{}\nu}x^{\nu},$ (61)
if $\kappa\neq 0$. While $\kappa=0$, the Funk metric (57) reduces to Minkowski
metric, the solutions (59) and (60) reduce to
$V^{\mu}=Q^{\mu}_{~{}\nu}x^{\nu}+C^{\mu},$ (62)
as expected. The non-trivial Lie algebra of non-trivial Funk spacetime (57)
($\kappa\neq 0$) corresponded to the Killing vectors (61) is given as
$[\hat{L}_{\mu\nu},\hat{L}_{\rho\lambda}]=i\eta_{\mu\lambda}\hat{L}_{\nu\rho}-i\eta_{\mu\rho}\hat{L}_{\nu\lambda}+i\eta_{\nu\rho}\hat{L}_{\mu\lambda}-i\eta_{\nu\lambda}\hat{L}_{\mu\rho},$
(63)
where $\hat{L}_{\mu\nu}=i(x_{\mu}\partial_{\nu}-x_{\nu}\partial_{\mu})$. It
means that the non-trivial Funk metric (57) is invariant just under the
Lorentz group.
By making use of the formula (6), the geodesic spray coefficient $G^{\mu}$ for
metric (57) is given as
$G^{\mu}_{F}=-\kappa\frac{F_{F}}{2}y^{\mu}.$ (64)
Thus, the geodesic equation for metric (57) is given as
$\frac{d^{2}x^{\mu}}{d\tau^{2}}-\kappa F_{F}\frac{dx^{\mu}}{d\tau}=0.$ (65)
Actually, the geodesic equation (65) is related to the scaled Berwald’s metric
$F_{B}$, which will be discussed in the next subsection. And the geometrical
parameter in $F_{B}$ is set as $\kappa$. The derivative of $F_{B}$ with
respect to the curve parameter $\tau$ in Funk metric (57) reads
$\displaystyle\frac{dF_{B}}{d\tau}$ $\displaystyle=$
$\displaystyle\frac{\partial F_{B}}{\partial
x^{\mu}}\frac{dx^{\mu}}{d\tau}+\frac{\partial F_{B}}{\partial
y^{\mu}}\frac{dy^{\mu}}{d\tau}$ (66) $\displaystyle=$ $\displaystyle-2\kappa
F_{B}F_{F}+\kappa F_{B}F_{F}$ $\displaystyle=$ $\displaystyle-\kappa
F_{B}F_{F}.$
Therefore, the geodesic equation (65) is equivalent to
$\frac{dp^{\mu}}{d\tau}=0,~{}~{}p^{\mu}\equiv\frac{m_{F}F_{B}}{F_{F}^{2}}\frac{dx^{\mu}}{d\tau},$
(67)
where $m_{F}$ is the mass of the particle in Funk spacetime.
The dispersion relation in Funk spacetime (57) is given as
$F^{2}_{F}(x,p)=g_{\mu\nu}p^{\mu}p^{\nu}=m_{F}^{2}\frac{F_{F}^{2}(x,y)}{F_{B}^{2}(x,y)}.$
(68)
The flag curvature of Funk spacetime is $K_{F}=\frac{1}{4}\kappa^{2}$. As the
discussion about dSSR, the constant flag curvature may relate to new physical
scale (like cosmological constant), and it is very tiny. Therefore, such
counterpart of special relativity-Funk special relativity also cannot be
excluded by the experiments. To first order in $\kappa$, we obtain the
expansion of the dispersion relation (68)
$p\cdot p-2\kappa(x\cdot p)\sqrt{p\cdot p}=m^{2}_{F}.$ (69)
Such dispersion relation (69) could be regarded as one type of modified
dispersion law in DSR.
### IV.4 Symmetry in type D $(\alpha,\beta)$ spacetime
The metric constructed by Berwald Berwald is of the form
$F=\frac{\left(\sqrt{(y\cdot y)(1-x\cdot x)+(x\cdot y)^{2}}+(x\cdot
y)\right)^{2}}{(1-(x\cdot x))^{2}\sqrt{(y\cdot y)(1-x\cdot x)+(x\cdot
y)^{2}}}.$ (70)
It is projectively flat with constant flag curvature $K_{B}=0$. One important
property of projective geometry shows that a projectively flat space is still
projectively flat after a scaling on $x$. It can be proved by using the Hamel
equation (9). Thus, the scaled Berwald’s metric is given as
$F_{B}=\frac{\left(\sqrt{(y\cdot y)(1-\lambda^{2}(x\cdot
x))+\lambda^{2}(x\cdot y)^{2}}-\lambda(x\cdot
y)\right)^{2}}{(1-\lambda^{2}(x\cdot x))^{2}\sqrt{(y\cdot
y)(1-\lambda^{2}(x\cdot x))+\lambda^{2}(x\cdot y)^{2}}},$ (71)
where $\lambda$ is a constant. The flag curvature of scaled Berwald’s metric
(71) is $K_{B}=0$.
Defining
$\alpha_{B}=\frac{\sqrt{(y\cdot y)(1-\lambda^{2}(x\cdot x))+\lambda^{2}(x\cdot
y)^{2}}}{1-\lambda^{2}(x\cdot x)},~{}~{}\beta_{B}=\frac{-\lambda(x\cdot
y)}{1-\lambda^{2}(x\cdot x)},$ (72)
we have
$F_{B}=\frac{(\alpha_{B}+\alpha_{B})^{2}}{\alpha_{B}(1-x\cdot x)}.$ (73)
Substituting the metric (73) into the Killing equation (16), we get
$K_{V}(F_{B})=\frac{\alpha_{B}+\beta_{B}}{\alpha_{B}(1-\lambda^{2}(x\cdot
x))}\left((1-\beta_{B}/\alpha_{B})K_{V}(\alpha_{B})+2K_{V}(\beta_{B})+2\lambda^{2}(\alpha_{B}+\beta_{B})\frac{x_{\mu}V^{\mu}}{1-\lambda^{2}(x\cdot
x)}\right)=0.$ (74)
The equations $K_{V}(\alpha_{B})=0$ and $K_{V}(\beta_{B})=0$ imply
$V^{\mu}=Q^{\mu}_{~{}\nu}x^{\nu},$ (75)
if $\lambda\neq 0$, where $Q_{\mu\nu}=\eta_{\rho\mu}Q^{\rho}_{~{}\nu}$ is an
arbitrary constant skew-symmetric matrix. Furthermore, it is obvious that
$x_{\mu}Q^{\mu}_{~{}\nu}x^{\nu}=0$. Therefore, the Killing vectors of the form
(75) is a solution of the Killing equation (74). The Killing vector of the
form (75) means that the scaled Berwald’s metric spacetime (71) is isotropic
about a given point. Therefore, the Killing vectors which implies such
symmetry (isotropic about a given point) reach its maximal numbers. And
additional solutions of Killing equations (74) must have the form
$V^{\mu}=f^{\mu}(x,C),$ (76)
where $C_{\mu}=\eta_{\rho\mu}C^{\rho}$ is an arbitrary constant vector. If
$V^{\mu}=\\{f(x,c),0,0,0\\}$ is a solution of Killing equation (74), it is
clear that $V^{\mu}=\\{f(x,c),f(x,c),f(x,c),f(x,c)\\}$ is also a solution of
(74). Therefore, the maximal dimension of isometric group of 4 dimensional
scaled Berwald’s spacetime equals either $6$ or $10$. It is known Egorov that
the maximal dimension of isometric group in an n dimensional non Riemannian
Finslerian space is $\frac{n(n-1)}{2}+2$. The scaled Berwald’s metric
spacetime is non Riemannian. We conclude that the solution of Killing equation
(74) only have solutions of the form (75).
The Lie algebra of non-trivial scaled Berwald’s metric spacetime (71)
($\lambda\neq 0$) corresponded to the Killing vectors (75) is given as
$[\hat{L}_{\mu\nu},\hat{L}_{\rho\lambda}]=i\eta_{\mu\lambda}\hat{L}_{\nu\rho}-i\eta_{\mu\rho}\hat{L}_{\nu\lambda}+i\eta_{\nu\rho}\hat{L}_{\mu\lambda}-i\eta_{\nu\lambda}\hat{L}_{\mu\rho},$
(77)
where $\hat{L}_{\mu\nu}=i(x_{\mu}\partial_{\nu}-x_{\nu}\partial_{\mu})$.
By making use of the formula (6), we obtain the geodesic spray coefficient
$G^{\mu}$ for metric (71)
$G^{\mu}_{B}=-\lambda\frac{\sqrt{(y\cdot y)(1-\lambda^{2}(x\cdot
x))+\lambda^{2}(x\cdot y)^{2}}-\lambda(x\cdot y)}{1-\lambda^{2}(x\cdot
x)}y^{\mu}=-\lambda F_{F}y^{\mu},$ (78)
where $F_{F}$ is the Funk metric, and the parameter in $F_{F}$ is set as
$\lambda$. Thus, the geodesic equation for metric (71) is given as
$\frac{d^{2}x^{\mu}}{d\tau^{2}}-2\lambda F_{F}\frac{dx^{\mu}}{d\tau}=0.$ (79)
One should notice that the derivatives of $F_{F}$ with respect to the curve
parameter $\tau$ in scaled Berwald’s metric (71) reads
$\frac{dF_{F}}{d\tau}=\frac{\partial F_{F}}{\partial
x^{\mu}}\frac{dx^{\mu}}{d\tau}+\frac{\partial F_{F}}{\partial
y^{\mu}}\frac{dy^{\mu}}{d\tau}=-\lambda F_{F}^{2}+2\lambda F_{F}^{2}=-\lambda
F_{F}^{2}.$ (80)
Therefore, the geodesic equation (79) is equivalent to
$\frac{dp^{\mu}}{d\tau}=0,~{}~{}p^{\mu}\equiv\frac{m_{B}F_{F}^{2}}{F_{B}^{3}}\frac{dx^{\mu}}{d\tau},$
(81)
where $m_{B}$ is the mass of the particle in scaled Berwald’s metric
spacetime. The dispersion relation in scaled Berwald’s metric spacetime (71)
is given as
$F^{2}_{B}(x,p)=g_{\mu\nu}p^{\mu}p^{\nu}=m_{B}^{2}\frac{F_{F}^{4}(x,y)}{F_{B}^{4}(x,y)}.$
(82)
The parameter $\lambda$ in scaled Berwald’s metric spacetime (71) may relate
to new physical scale and it is very tiny. To first order in $\lambda$, we
obtain the expansion of the dispersion law (82)
$p\cdot p-4\lambda(x\cdot p)\sqrt{p\cdot p}=m^{2}_{B}.$ (83)
Here, we find that Funk spacetime (57) and scaled Berwald’s metric spacetime
(71) have same isometric group. And the geodesic equations in Funk spacetime
and scaled Berwald’s metric spacetime are alike, if they both take the same
geometrical parameter. Also, to first order in geometrical parameter, the
dispersion relation are almost the same.
## V Conclusion
In this paper, we have extended the concept of inertial motion in the
framework of the projective geometry. The inertial frames in projectively flat
Finsler spacetime are investigated. We have studied the inertial motion in a
special Finsler spacetime-the projectively flat $(\alpha,\beta)$ spacetime
with constant flag curvature (the counterpart of sectional curvature). The
projectively flat $(\alpha,\beta)$ spacetime with constant flag curvature can
be divided into four types. We have showed that the inertial motion and
symmetry in Type A and Type B spacetime are just the one in dSSR and VSR,
respectively. And the dispersion law in Type C and Type D could be regarded as
one types of modified dispersion law in DSR. The four types of
$(\alpha,\beta)$ spacetime involve two parameters-the speed of light and a
geometrical parameter which may relate to new physical scale. While the
geometrical parameter vanishes, the four types of spacetime reduce to
Minkowski spacetime, the momentum tensors and angular momentum tensors reduce
to the one in Minkowski spacetime, the corresponded Lie algebra reduces to
Poincare algebra, and the inertial motions reduce to the one in special
relativity. In the following table, we list basic features of the kinematics
and symmetry in the four types spacetime.
Table 1: the projectively flat $(\alpha,\beta)$ space with constant flag curvature Type | parameter | geodesic equation | momentum | isometric group
---|---|---|---|---
A | $\mu$ | $\frac{d^{2}x^{\mu}}{d\tau^{2}}+\frac{2\mu(x\cdot\frac{dx}{d\tau})}{1-\mu(x\cdot x)}\frac{dx^{\mu}}{d\tau}=0$ | $p^{\mu}_{R}\equiv m_{R}\frac{1}{F_{R}}\frac{1}{1-\mu(x\cdot x)}\frac{dx^{\mu}}{d\tau}$ | dS/AdS group
B | $n$ | $\frac{d^{2}x^{\mu}}{d\tau^{2}}=0$ | $p^{\mu}_{V}\equiv m_{V}\frac{1}{F_{V}}\frac{dx^{\mu}}{d\tau}$ | DISIM(2) group
C | $\kappa$ | $\frac{d^{2}x^{\mu}}{d\tau^{2}}-\kappa F_{F}(\kappa)\frac{dx^{\mu}}{d\tau}=0$ | $p^{\mu}_{F}\equiv m_{F}\frac{F_{B}(\kappa)}{F_{F}^{2}(\kappa)}\frac{dx^{\mu}}{d\tau}$ | Lorentz group
D | $\lambda$ | $\frac{d^{2}x^{\mu}}{d\tau^{2}}-2\lambda F_{F}(\lambda)\frac{dx^{\mu}}{d\tau}=0$ | $p^{\mu}_{B}\equiv m_{B}\frac{F_{F}^{2}(\lambda)}{F_{B}^{3}(\lambda)}\frac{dx^{\mu}}{d\tau}$ | Lorentz group
###### Acknowledgements.
We would like to thank Prof. C. J. Zhu, C. G. Huang and Z. Shen for useful
discussions. The work was supported by the NSF of China under Grant No.
10525522, 10875129 and 11075166. *
## Appendix A Flag curvature
The flag curvature Book by Bao ; Shen1 in Finsler geometry is the counterpart
of the sectional curvature in Riemannian geometry. It is a geometrical
invariant. Furthermore, the same flag curvature is obtained for any connection
chosen in Finsler space. The curvature tensor $R^{\mu}_{~{}\nu}$ is defined as
$R^{\mu}_{~{}\nu}(x,y)\equiv-\left(\frac{\partial G^{\mu}}{\partial
x^{\nu}}-y^{\lambda}\frac{\partial^{2}G^{\mu}}{\partial x^{\lambda}\partial
y^{\nu}}+2G^{\lambda}\frac{\partial^{2}G^{\mu}}{\partial y^{\lambda}\partial
y^{\nu}}-\frac{\partial G^{\mu}}{\partial x^{\lambda}}\frac{\partial
G^{\lambda}}{\partial x^{\nu}}\right),$ (84)
where $G^{\mu}$ is geodesic spray coefficient. For a tangent plane $\Pi\subset
T_{x}M$ and a non-zero vector $y\in T_{x}M$, the flag curvature is defined as
$K(\Pi,y)\equiv\frac{g_{\lambda\mu}R^{\mu}_{~{}\nu}u^{\nu}u^{\lambda}}{F^{2}g_{\rho\theta}u^{\rho}u^{\theta}-(g_{\sigma\kappa}y^{\sigma}u^{\kappa})^{2}},$
(85)
where $u\in\Pi$. If $F$ is projectively flat, substituting
$G^{\mu}=P(x,y)y^{\mu}$ into the definition of flag curvature (85), and by
making use of formula (84), we obtain that
$K=-\frac{P^{2}-\frac{\partial P}{\partial x^{\mu}}y^{\mu}}{F^{2}}.$ (86)
The curvature tensor $R^{\mu}_{~{}\nu}$ defined above is presented as
$-\bar{R}^{\mu}_{~{}\nu}$ in Ref.Shen1 . The notation we used here keeps the
sectional curvature of dS spacetime to be positive and of AdS spacetime to be
negative. By making use of the formula for the flag curvature of projectively
flat Finsler spacetime (86), we get the flag curvature for dS/AdS spacetime
(27), Funk spacetime (57) and scaled Berwald’s metric spacetime (71),
respectively,
$K_{R}=\mu,~{}~{}~{}K_{F}=\frac{1}{4}\kappa^{2},~{}~{}~{}K_{B}=0$ (87)
And the flag curvature of locally Minkowski spacetime equals zero.
## References
* (1) G. Amelino-Camelia, Phys. Lett. B510, 255 (2001).
* (2) G. Amelino-Camelia, Int. J. Mod. Phys. D11, 35 (2002).
* (3) G. Amelino-Camelia, Nature 418, 34 (2002).
* (4) S.R. Coleman and S.L. Glashow, Phys. Lett. B405, 249 (1997).
* (5) S.R. Coleman and S.L. Glashow, Phys. Rev. D59, 116008 (1999).
* (6) A.G. Cohen and S.L. Glashow, Phys. Rev. Lett. 97 021601 (2006).
* (7) H. Y. Guo, C. G. Huang, Z. Xu, et al., Mod. Phys. Lett. A, 19, 1701 (2004); Phys. Lett. A 331, 1 (2004).
* (8) K. H. Look, C. L. Tsou(Z. L. Zou) and H. Y. Kuo(H. Y Guo), Acta Phys. Sin. 23, 225 (1974); H.Y. Guo, Nucl. Phys. B (Proc. Suppl.) 6, 381 (1989).
* (9) H. S. Snyder, Phys. Rev. 71, 38 (1947).
* (10) H. Y. Guo, et al., Class. Quantum Grav. 24, 4009 (2007).
* (11) M. M. Sheikh-Jabbari, A. Tureanu, Phys. Rev. Lett. 101, 261601 (2008).
* (12) D. Bao, S. S. Chern and Z. Shen, An Introduction to Riemann-Finsler Geometry, Graduate Texts in Mathematics 200, Springer, New York, 2000.
* (13) S. Ghosh and P. Pal, Phys. Rev. D 75, 105021 (2007).
* (14) F. Girelli, S. Liberati and L. Sindoni, Phys. Rev. D75, 064015 (2007).
* (15) G. W. Gibbons, J. Gomis and C. N. Pope, Phys. Rev. D 76, 081701 (2007).
* (16) W. Killing, J. f. d. reine u. angew. Math. (Crelle), 109, 121 (1892).
* (17) E.Alvarez, R.Vidal, arXiv: 0803,1949V1 (2008); F.Cardone et al., EJTP6, No 20, 59 (2009).
* (18) S. Deng and Z. Hou, Pac. J. Math. 207, 149 (2002).
* (19) A. Einstein, Ann. de Phys. 17, 891 (1905).
* (20) D. Bao, S. S. Chern and Z. Shen, An Introduction to Riemann–Finsler Geometry, Graduate Texts in Mathematics 200, Springer, New York, 2000.
* (21) G. Hamel, Über die Geometrieen in denen die Geraden die Kürzesten sind, Math. Ann. 57, 231(1903).
* (22) Z. Shen, Some perspectives in Finsler geometry, MSRI Publication Series. Cambridge: Cambridge university press, 2004.
* (23) G. Randers, Phys. Rev. 59, 195 (1941).
* (24) B. Li and Z. Shen, International Journal of Mathematics, 18, 1 (2007).
* (25) P. Funk, Math. Ann. 101, 226 (1929); Z. Shen, Differential geometry of spray and Finsler spaces, Dordrecht: Kluwer, 2001.
* (26) L. Berwald, Math. Z. 30, 449 (1929).
* (27) E. J. Copeland, M. Sami and S. Tsujikawa, Int. J. Mod. Phys. D 15, 1753 (2006).
* (28) C. G. Huang, H. Y. Guo, Y. Tian, Z. Xu and B. Zhou, Int. J. Mod. Phys. A 22, 2535 (2004); Y. Tian, H. Y. Guo, C. G. Huang, Z. Xu and B. Zhou, Phys. Rev. D 71, 044030 (2005).
* (29) J. B. Kogut and D. E. Soper, Phys. Rev. D 1, 2901 (1970).
* (30) G. Bogoslovsky, arXiv:gr-qc/0706.2621.
* (31) A. I. Egorov, Math. USSR Sb. 44, 279 (1983).
* (32) Z. Shen, Trans. of Amer. Math. Soc. 355, 1713 (2003).
|
arxiv-papers
| 2010-10-11T07:37:11 |
2024-09-04T02:49:13.659190
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xin Li and Zhe Chang",
"submitter": "Xin Li",
"url": "https://arxiv.org/abs/1010.2020"
}
|
1010.2058
|
# Observational Constraints on Exponential Gravity
Louis Yang louis.lineage@msa.hinet.net Chung-Chi Lee g9522545@oz.nthu.edu.tw
Ling-Wei Luo d9622508@oz.nthu.edu.tw Chao-Qiang Geng geng@phys.nthu.edu.tw
Department of Physics, National Tsing Hua University, Hsinchu 300, Taiwan
###### Abstract
We study the observational constraints on the exponential gravity model of
$f(R)=-\beta R_{s}(1-e^{-R/R_{s}})$. We use the latest observational data
including Supernova Cosmology Project (SCP) Union2 compilation, Two-Degree
Field Galaxy Redshift Survey (2dFGRS), Sloan Digital Sky Survey Data Release 7
(SDSS DR7) and Seven-Year Wilkinson Microwave Anisotropy Probe (WMAP7) in our
analysis. From these observations, we obtain a lower bound on the model
parameter $\beta$ at 1.27 (95% CL) but no appreciable upper bound. The
constraint on the present matter density parameter is
$0.245<\Omega_{m}^{0}<0.311$ (95% CL). We also find out the best-fit value of
model parameters on several cases.
###### pacs:
98.80.-k, 04.50.Kd, 95.36.-x
## I Introduction
Cosmic observations from type Ia supernovae (SNe Ia) (Riess1998a, ;
Perlmutter1999a, ), large scale structure (LSS) (Tegmark2004a, ; Seljak2005a,
), baryon acoustic oscillations (BAO) (Eisenstein2005, ) and cosmic microwave
background (CMB) (Spergel2003, ; Spergel2007, ) indicate that our universe is
undergoing an accelerating expansion. The reason for this acceleration, the
so-called dark energy problem, remains a fascinating question today. The
simplest model to explain this problem is the $\Lambda$CDM model, in which a
time independent energy density is added to the universe. However, the
$\Lambda$CDM model suffers from both fine-tuning and coincidence problems
(Weinberg1989, ; Sahni2000, ; Carroll2001, ; Peebles2003, ; Padmanabhan2003b,
; Copeland2006, ). In general, the ways to understand the cosmic acceleration
can be separated into two branches. One is to modify the matter by introducing
some kind of “dark energy”. The other one is to modify Einstein’s general
relativity – the modification of gravity.
In modified gravity, one of the popular approaches is to promote the Ricci
scalar $R$ in the Einstein-Hibert action to a function, $f(R)$. Although there
are several viable $f(R)$ models, many of them are restricted to the regimes
to be effectively identical to the $\Lambda$CDM by the observational
constraints. Recently, Linder (Linder2009, ) has explored an $f(R)$ theory
named “exponential gravity”, which has also been discussed in Refs.
(Zhang2006, ; Zhang2007, ; Cognola2008, ). The exponential gravity has the
feature that it allows the relaxation of fine-tuning and it has only one more
parameter than the $\Lambda$CDM model. In addition, the exponential gravity
satisfies all conditions for the viability (Bamba2010, ) such as the local
gravity constraint, stability of the late-time de Sitter point, constraints
from the violation of the equivalence principle, stability of cosmological
perturbations, positivity of the effective gravitational coupling, and
asymptotic behavior to the $\Lambda$CDM model in the high curvature regime. In
this paper, we will study the constraints given by latest observational data,
reexamine the alleviation of the fine-tuning problem, and find the possibility
of the derivation from $\Lambda$CDM. We use units of
$k_{\mathrm{B}}=c=\hbar=1$ and the gravitational constant is given by
$G=M_{\mathrm{Pl}}^{-2}$ with the Planck mass of $M_{\mathrm{Pl}}=1.2\times
10^{19}$ GeV.
The paper is organized as follows. In Sec. II, we review equations of motion
and the asymptotic behavior at the high redshift regime in the exponential
gravity model. In Sec. III, we discuss the observations and methods. We show
our results in Sec. IV. Finally, conclusions are given in Sec. V.
## II Exponential Gravity
The action of $f(R)$ gravity with matter is given by
$S=\frac{1}{2\kappa^{2}}\int d^{4}x\sqrt{-g}\left[R+f(R)\right]+S_{m},$ (1)
where $\kappa^{2}\equiv 8\pi G$ and $f(R)$ is a function of the Ricci scalar
curvature $R$. In this paper, we focus on the exponential gravity model
(Linder2009, ), given by
$f(R)=-\beta R_{s}(1-e^{-R/R_{s}}),$ (2)
where $R_{s}$ is related to the characteristic curvature modification scale.
Since the product of $\beta$ and $R_{s}$ can be determined by the present
matter density $\Omega_{m}^{0}$ (Linder2009, ), we can choose $\beta$ and
$\Omega_{m}^{0}$ as the free parameters in the model.
We use the standard metric formalism. From the action (1), the modified
Friedmann equation of motion becomes (Song2007, )
$H^{2}=\frac{\kappa^{2}\rho_{M}}{3}+\frac{1}{6}(f_{R}R-f)-H^{2}\left(f_{R}+f_{RR}R^{\prime}\right),$
(3)
where $H\equiv\dot{a}/a$ is the Hubble parameter, a subscript R denotes the
derivative with respect to R, a prime represents $d/d\ln a$, and
$\rho_{M}=\rho_{m}+\rho_{r}$ is the energy density of all perfect fluids of
generic matter including (non-relativistic) matter, denoted by $m$, and
relativistic particles, denoted by $r$. Here, we only consider the matter
density. Since the modification by the exponential gravity only happens at the
low redshift, the contributions from relativistic particles are negligible. In
a flat spacetime, the Ricci scalar is given by
$R=12H^{2}+6HH^{\prime}.$
Following Hu and Sawicki’s parameterization (Hu2007, ), we define
$y_{H}\equiv\frac{\rho_{DE}}{\rho_{m}^{0}}=\frac{H^{2}}{m^{2}}-a^{-3},\quad
y_{R}\equiv\frac{R}{m^{2}}-3a^{-3},$ (4)
where $m^{2}\equiv\kappa^{2}\rho_{m}^{0}/3$, $\rho_{DE}$ is the effective dark
energy density, and $\rho_{m}^{0}$ is the present matter density. Then, Eqs.
(3) and (II) can be rewritten as two coupled differential equations,
$y_{H}^{\prime}=\frac{y_{R}}{3}-4y_{H}$ (5)
and
$y_{R}^{\prime}=9a^{-3}-\frac{1}{H^{2}f_{RR}}\left[y_{H}+f_{R}\left(\frac{H^{2}}{m^{2}}-\frac{R}{6m^{2}}\right)+\frac{f}{6m^{2}}\right],$
(6)
where $R$ and $H^{2}$ can be further replaced by $y_{R}$ and $y_{H}$ from
equations in (4). Combining Eqs. (5) and (6), we obtain a second order
differential equation of $y_{H}$,
$y_{H}^{\prime\prime}+J_{1}y_{H}^{\prime}+J_{2}y_{H}+J_{3}=0,$ (7)
where
$\displaystyle J_{1}$ $\displaystyle=$ $\displaystyle
4-\frac{1}{y_{H}+a^{-3}}\frac{f_{R}}{6m^{2}f_{RR}},$ $\displaystyle J_{2}$
$\displaystyle=$
$\displaystyle-\frac{1}{y_{H}+a^{-3}}\frac{f_{R}-1}{3m^{2}f_{RR}},$
$\displaystyle J_{3}$ $\displaystyle=$
$\displaystyle-3a^{-3}+\frac{f_{R}a^{-3}+f/3m^{2}}{y_{H}+a^{-3}}\frac{1}{6m^{2}f_{RR}},$
(8)
with
$\displaystyle
R=m^{2}\left[3\left(y_{H}^{\prime}+4y_{H}\right)+3a^{-3}\right].$ (9)
Solving Eq. (7) numerically, we can get the evolution of the Hubble parameter
in the low redshift regime ($z=0\sim 4$). The effective dark energy equation
of state $w_{DE}$ is given by
$\displaystyle w_{DE}=-1-\frac{y_{H}^{\prime}}{3y_{H}}.$ (10)
In the high redshift regime ($z\gtrsim 4$), the exponential factor
$e^{-R/R_{S}}$ of $f(R)$ in Eq. (2) becomes negligible
($e^{-R/R_{S}}<10^{-5}$). The exponential gravity model behaves essentially
like a cosmological constant model with the dark energy density parameter
$\Omega_{\Lambda}=\beta R_{S}/6H_{0}^{2}\cong\Omega_{m}^{0}y_{H}(z_{high})$.
Thus, the Hubble parameter as a function of $z$ in this regime can be
expressed as
$\displaystyle H(z)$ $\displaystyle=$ $\displaystyle
H_{0}\sqrt{\Omega_{m}^{0}\left(1+z\right)^{3}+\Omega_{r}^{0}\left(1+z\right)^{4}+\frac{\beta
R_{S}}{6H_{0}^{2}}},$ (11)
where $\Omega_{r}^{0}$ is the density parameter of relativistic particles
including photons and
neutrinos111$\Omega_{r}^{0}=\Omega_{\gamma}^{0}\left(1+0.2271N_{eff}\right)$,
where $\Omega_{\gamma}^{0}$ is the present fractional photon energy density
and $N_{eff}=3.04$ is the effective number of neutrino species (Komatsu2010,
).. The equation (11) will be used in the data fitting of CMB and the high
redshift part of BAO in section III.
## III Observational Constraints
To constrain the free parameters of $\beta$ and $\Omega_{m}^{0}$ in the
exponential gravity model, we use three kinds of the observational data
including SNe Ia, BAO and CMB. The SNe Ia and CMB data lead to constraints at
the low and high redshift regimes, respectively, while the BAO data provide
constraints at the both regimes.
### III.1 Type Ia Supernovae (SNe Ia)
The observations of SNe Ia, known as “standard candles”, give us the
information about the luminosity distance $D_{L}$ as a function of the
redshift $z$. The distance modulus $\mu$ is defined as
$\displaystyle\mu_{th}(z_{i})\equiv 5\log_{10}D_{L}(z_{i})+\mu_{0},$ (12)
where $\mu_{0}\equiv 42.38-5\log_{10}h$ with $H_{0}=h\cdot 100km/s/Mpc$ is the
present value of the Hubble parameter. The Hubble-free luminosity distance for
the flat universe is
$\displaystyle D_{L}(z)=(1+z)\int_{0}^{z}\frac{dz^{\prime}}{E(z^{\prime})},$
(13)
where $E(z)=H(z)/H_{0}$. The $\chi^{2}$ of the SNe Ia data is
$\displaystyle\chi_{SN}^{2}=\sum_{i}\frac{\left[\mu_{obs}(z_{i})-\mu_{th}(z_{i})\right]^{2}}{\sigma_{i}^{2}},$
(14)
where $\mu_{obs}$ is the observed value of the distance modulus. Since the
absolute magnitude of SNe Ia is unknown, we should minimize $\chi_{SN}^{2}$
with respect to $\mu_{0}$, which relates to the absolute magnitude, and expand
it to be (Nesseris2005, ; Perivolaropoulos2005, )
$\displaystyle\chi_{SN}^{2}=A-2\mu_{0}B+\mu_{0}^{2}C,$ (15)
where
$\displaystyle A$ $\displaystyle=$
$\displaystyle\sum_{i}\frac{\left[\mu_{obs}(z_{i})-\mu_{th}(z_{i};\mu_{0}=0)\right]^{2}}{\sigma_{i}^{2}},$
$\displaystyle B$ $\displaystyle=$
$\displaystyle\sum_{i}\frac{\mu_{obs}(z_{i})-\mu_{th}(z_{i};\mu_{0}=0)}{\sigma_{i}^{2}},\quad
C=\sum_{i}\frac{1}{\sigma_{i}^{2}}.$ (16)
The minimum of $\chi_{SN}^{2}$ with respect to $\mu_{0}$ is
$\displaystyle\tilde{\chi}_{SN}^{2}=A-\frac{B^{2}}{C}.$ (17)
We adopt this $\tilde{\chi}_{SN}^{2}$ for our later $\chi^{2}$ minimization.
We will use the data from the Supernova Cosmology Project (SCP) Union2
compilation, which contains 557 supernovae (Amanullah2010, ), ranging from
$z=0.015$ to $z=1.4$.
### III.2 Baryon Acoustic Oscillations (BAO)
The observation of BAO measures the distance ratios of $d_{z}\equiv
r_{s}(z_{d})/D_{V}(z)$, where $D_{V}$ is the volume-averaged distance, $r_{s}$
is the comoving sound horizon and $z_{d}$ is the redshift at the drag epoch
(Percival2010, ). The volume-averaged distance $D_{V}(z)$ is defined as
(Eisenstein2005, )
$\displaystyle
D_{V}(z)\equiv\left[(1+z)^{2}D_{A}^{2}(z)\frac{z}{H(z)}\right]^{1/3},$ (18)
where $D_{A}(z)$ is the proper angular diameter distance:
$\displaystyle
D_{A}(z)=\frac{1}{1+z}\int_{0}^{z}\frac{dz^{\prime}}{H(z^{\prime})},\quad\textrm{(for
flat universe)}.$ (19)
The comoving sound horizon $r_{s}(z)$ is given by
$\displaystyle
r_{s}(z)=\frac{1}{\sqrt{3}}\int_{0}^{1/(1+z)}\frac{da}{a^{2}H({\scriptstyle
z^{\prime}=\frac{1}{a}-1})\sqrt{1+(3\Omega_{b}^{0}/4\Omega_{\gamma}^{0})a}},$
(20)
where $\Omega_{b}^{0}$ and $\Omega_{\gamma}^{0}$ are the present values of
baryon and photon density parameters, respectively. We use
$\Omega_{b}^{0}=0.022765h^{-2}$ and $\Omega_{\gamma}^{0}=2.469\times
10^{-5}h^{-2}$ (Komatsu2010, ). The fitting formula for $z_{d}$ is given by
(Eisenstein1998, )
$\displaystyle
z_{d}=\frac{1291(\Omega_{m}^{0}h^{2})^{0.251}}{1+0.659(\Omega_{m}^{0}h^{2})^{0.828}}\left[1+b_{1}(\Omega_{b}^{0}h^{2})^{b2}\right],$
(21)
where
$\displaystyle b_{1}$ $\displaystyle=$ $\displaystyle
0.313(\Omega_{m}^{0}h^{2})^{-0.419}\left[1+0.607(\Omega_{m}^{0}h^{2})^{0.674}\right],$
$\displaystyle b_{2}$ $\displaystyle=$ $\displaystyle
0.238(\Omega_{m}^{0}h^{2})^{0.223}.$ (22)
The typical value of $z_{d}$ is about 1021 with $\Omega_{m}^{0}=0.276$ and
$h=0.705$. Since $z_{d}$ is in the high redshift regime, we use Eq. (11) to
calculate $r_{s}(z_{d})$. On the other hand, $D_{V}(z)$ is evaluated by the
numerical result of Eq. (7) as it is in the low redshift regime.
The BAO data from the Two-Degree Field Galaxy Redshift Survey (2dFGRS) and the
Sloan Digital Sky Survey Data Release 7 (SDSS DR7) (Percival2010, ) measured
the distance ratio $d_{z}$ at two redshifts $z=0.2$ and $z=0.35$ to be
$d_{z=0.2}^{obs}=0.1905\pm 0.0061$ and $d_{z=0.35}^{obs}=0.1097\pm 0.0036$
with the inverse covariance matrix:
$\displaystyle C_{BAO}^{-1}=\left(\begin{array}[]{cc}30124&-17227\\\
-17227&86977\end{array}\right).$ (25)
The $\chi^{2}$ for the BAO data is
$\displaystyle\chi_{BAO}^{2}=(x_{i,BAO}^{th}-x_{i,BAO}^{obs})(C_{BAO}^{-1})_{ij}(x_{j,BAO}^{th}-x_{j,BAO}^{obs}),$
(26)
where $x_{i,BAO}\equiv\left(d_{0.2},d_{0.35}\right)$.
### III.3 Cosmic Microwave Background (CMB)
The CMB is sensitive to the distance to the decoupling epoch $z_{*}$
(Komatsu2009, ). It can give constraints on the model in the high redshift
regime ($z\sim 1000$). The CMB data are taken from Wilkinson Microwave
Anisotropy Probe (WMAP) observations (Komatsu2010, ). To use the WMAP data, we
compare three quantities: (i) the acoustic scale $l_{A}$,
$\displaystyle l_{A}(z_{*})\equiv(1+z_{*})\frac{\pi
D_{A}(z_{*})}{r_{S}(z_{*})},$ (27)
(ii) the shift parameter $R$ (Bond1997, ),
$\displaystyle R(z_{*})\equiv\sqrt{\Omega_{m}^{0}}H_{0}(1+z_{*})D_{A}(z_{*}),$
(28)
and (iii) the redshift of the decoupling epoch $z_{*}$. The fitting function
of $z_{*}$ is given by (Hu1996, )
$\displaystyle
z_{*}=1048\left[1+0.00124(\Omega_{b}^{0}h^{2})^{-0.738}\right]\left[1+g_{1}(\Omega_{m}^{0}h^{2})^{g2}\right],$
(29)
where
$\displaystyle
g_{1}=\frac{0.0783(\Omega_{b}^{0}h^{2})^{-0.238}}{1+39.5(\Omega_{b}^{0}h^{2})^{0.763}},\quad
g_{2}=\frac{0.560}{1+21.1(\Omega_{b}^{0}h^{2})^{1.81}}.$ (30)
The $\chi^{2}$ of the CMB data is
$\displaystyle\chi_{CMB}^{2}=(x_{i,CMB}^{th}-x_{i,CMB}^{obs})(C_{CMB}^{-1})_{ij}(x_{j,CMB}^{th}-x_{j,CMB}^{obs}),$
(31)
where $x_{i,CMB}\equiv\left(l_{A}(z_{*}),R(z_{*}),z_{*}\right)$ and
$C_{CMB}^{-1}$ is the inverse covariance matrix. The data from Seven-Year
Wilkinson Microwave Anisotropy Probe (WMAP7) observations (Komatsu2010, ) lead
to $l_{A}(z_{*})=302.09$, $R(z_{*})=1.725$ and $z_{*}=1091.3$ with the inverse
covariance matrix:
$\displaystyle C_{CMB}^{-1}=\left(\begin{array}[]{ccc}2.305&29.698&-1.333\\\
29.698&6825.27&-113.180\\\ -1.333&-113.180&3.414\end{array}\right).$ (35)
Finally, the $\chi^{2}$ of all the observational data is
$\displaystyle\chi^{2}=\tilde{\chi}_{SN}^{2}+\chi_{BAO}^{2}+\chi_{CMB}^{2}.$
(36)
In our fitting process, we did not use the Markov chain Monte Carlo (MCMC)
approach because the numerical calculation for each solution of $f(R)$ theory
is very time-consuming, and the necessary change to the code like CosmoMC
Lewis2002 is very extensive with no obvious benefit in our study of the
exponential gravity. Therefore, we take the simple $\chi^{2}$ method as our
main fitting procedure. The $\Lambda$CDM result obtained from SNe Ia, BAO and
CMB constraints with this $\chi^{2}$ method is
$\Omega_{m}^{0}=0.276_{-0.013}^{+0.014}$, while that with the MCMC method is
$\Omega_{m}^{0}=0.272_{-0.011}^{+0.013}$ Gong2010 . We note that the fitting
in Ref. Gong2010 has also included the observational constraints from the
radial BAO and Hubble parameter H(z).
Figure 1: The 68.3%, 95.4% and 99.7% confidence intervals for the exponential gravity model, constrained by the SNe Ia, BAO, and CMB data. The best-fit point in this parameter region is marked with a plus sign. Table 1: The best-fit values of the matter density parameter $\Omega_{m}^{0}$ (68% CL) and $\chi^{2}$ for the exponential gravity model with $\beta=2,3,4$ and the $\Lambda$CDM model. Note that the error for $\Omega_{m}^{0}$ is obtained when $\beta$ is fixed. Model | | $\Omega_{m}^{0}$ | $\chi^{2}$
---|---|---|---
| $\beta=2$ | $0.274_{-0.013}^{+0.014}$ | 546.7136
Exponential Gravity | $\beta=3$ | $0.276_{-0.013}^{+0.014}$ | 545.3836
| $\beta=4$ | $0.276_{-0.013}^{+0.014}$ | 545.1721
$\Lambda$CDM | | $0.276_{-0.013}^{+0.014}$ | 545.1522
## IV Results
Based on the methods described in Sec. III, we now examine the parameter space
of the exponential gravity model. In Fig. 1, we present likelihood contour
plots at 68.3, 95.4 and 99.7% confidence levels obtained from the SNe Ia, BAO
and CMB constraints. The results show that the observational data give no
upper bound on the model parameter $\beta$, making it a free parameter. Hence,
there is no fine-tuning problem. However, a larger value of $\beta$, which is
closer to the $\Lambda$CDM model, is slightly preferred by the observational
data as expected. The lower bound on $\beta$ is $\beta>1.27$ (95% CL). The
present matter density parameter $\Omega_{m}^{0}$ is constrained to
$0.245<\Omega_{m}^{0}<0.311$ (95% CL), which agrees with the current
observations. The best-fit value (smallest $\chi^{2}$) in the parameter space
between $\beta=1$ and 4222We only concentrate on the region of $1<\beta<4$.
For $\beta>4$, it is almost the $\Lambda$CDM model. For $\beta<1$, it is ruled
out by the local gravity constraints and the stability of the de-Sitter phase.
is $\chi^{2}=545.1721$ with $\beta=4$ and $\Omega_{m}^{0}=0.276$. The
comparison of the best-fit $\Omega_{m}^{0}$ and $\chi^{2}$ for the model with
$\beta=2,3,4$ and $\Lambda$CDM is shown in Table 1.
In Fig. 2, we illustrate the evolution of the effective dark energy equation
of state $w_{DE}$ for $\beta=2,3,4$ with their best-fit $\Omega_{m}^{0}$,
which is given in Table 1. We can see that, for every value of $\beta$, the
effective dark energy equation of state $w_{DE}$ starts at the phase of a
cosmological constant $w_{DE}=-1$ and evolves from the phantom phase
($w_{DE}<-1$) to the non-phantom phase ($w_{DE}>-1$). And, for larger value of
$\beta$, the deviation from cosmological constant phase ($w_{DE}=-1$) become
smaller. For $\beta=2$, there is still another small oscillation after the
main phantom phase crossing. Negative $z$ means the future evolution. It is
clear that the exponential gravity model has the feature of crossing the
phantom phase in the past as well as the future (BGL2, ).
In Fig. 3, we depict the effective dark energy density $\Omega_{DE}$ and non-
relativistic matter density $\Omega_{m}$ vs. the redshift $z$.
Figure 2: Evolution of the effective dark energy equation of state $w_{DE}$
corresponding to $\beta=2,3,4$ with their best-fit $\Omega_{m}^{0}$ given in
Table 1. Figure 3: The evolutions of the effective dark energy density
parameter $\Omega_{DE}$ and non-relativistic matter density parameter
$\Omega_{m}$ as functions of $z$, where the solid lines indicate the
exponential gravity model with $\beta=1.27$ and the best-fit
$\Omega_{m}^{0}=0.270$ and the dashed lines represent the $\Lambda$CDM model
with $\Omega_{m}^{0}=0.276$. For a higher value of $\beta$, the evolution
becomes closer to that in $\Lambda$CDM.
## V Conclusion
We have studied the exponential gravity model. In the low redshift regime, we
follow Hu and Sawicki’s parameterization to form the differential equation for
the exponential gravity and solve it numerically. In the high redshift regime,
we take advantage of the asymptotic behavior of the exponential gravity toward
an effective cosmological constant. The analytical form of the Hubble
parameter $H$ as a function of the redshift $z$ can be expressed in the high
redshift limit. We have constrained the parameter space of the model by the
SNe Ia, BAO and CMB data. We have found that there is a lower bound on the
model parameter $\beta$ at 1.27 but no upper limit, and $\Omega_{m}^{0}$ is
constrained to the concordance value. This means that the exponential gravity
model shows no need of fine-tuning. Nevertheless, the $\Lambda$CDM model is
still included by the observational constraints since $\beta\rightarrow\infty$
corresponds to the model. Current observational data still lack the ability to
distinguish between the $\Lambda$CDM and exponential gravity models.
Finally, we remark that as seen from Fig. 3, the noticeable difference between
the exponential gravity and $\Lambda$CDM models lies in the regime $0.2<z<1$,
and is maximized at $z=0.5$ if we compare their expected distance modulus. An
improvement on the BAO observation may give a stronger constraint on this
redshift regime or higher. The ongoing and future dark energy survey projects
which will observe BAO include WiggleZ Glazebrook2007 , BOSS (Baryon
Oscillation Spectroscopic Survay) BOSS , HETDEX (Hobby-Eberly Dark Energy
Experiment) HETDEX , EUCLID Euclid , JDEM (Joint Dark Energy Mission)/Omega
with Wide Field Infrared Survey Telescope (WFIRST) JDEM , BigBOSS (Big Baryon
Oscillation Spec-troscopic Survay) BigBOSS , SKA (Square Kilometer Array) SKA
, LSST (Large Synoptic Survey Telescope) Tyson2002 and DES (Dark Energy
Survey) DES . In addition, it is known that the measurement on the growth rate
of $f_{g}(z)=d\ln\delta_{m}/d\ln a$ has the potential to distinguish the
models with the same expansion history but different physics. In the
exponential gravity case, the growth index is $\gamma=0.540$ for $\beta=2$. It
is clear that if those surveys such as WiggleZ, EUCLID, BigBOSS and JDEM/Omega
can measure the growth rate with a high accuracy, they will be able to
discriminate the exponential gravity from the $\Lambda$CDM model.
###### Acknowledgements.
We thank Dr. K. Bamba for many helpful discussions and suggestions. The work
is supported in part by the National Science Council of R.O.C. under: Grant #:
NSC-98-2112-M-007-008-MY3 and National Tsing Hua University under the Boost
Program #: 97N2309F1.
## References
* (1) A. G. Riess et al. [SNST Collaboration], Astron. J. 116, 1009 (1998).
* (2) S. Perlmutter et al. [SNCP Collaboration], Astrophys. J. 517, 565 (1999).
* (3) M. Tegmark et al., [SDSS Collaboration], Phys. Rev. D 69, 103501 (2004).
* (4) U. Seljak et al. [SDSS Collaboration], Phys. Rev. D 71, 103515 (2005).
* (5) D. J. Eisenstein et al. [SDSS Collaboration], Astrophys. J. 633, 560 (2005).
* (6) D. N. Spergel et al. [WMAP Collaboration], Astrophys. J. Suppl. 148, 175 (2003).
* (7) D. N. Spergel et al. [WMAP Collaboration], Astrophys. J. Suppl. 170, 377 (2007).
* (8) S. Weinberg, Rev. Mod. Phys. 61, 1 (1989).
* (9) V. Sahni and A. A. Starobinsky, Int. J. Mod. Phys. D 9, 373 (2000).
* (10) S. M. Carroll, Living Reviews in Relativity 4 (2001).
* (11) P. J. E. Peebles and B. Ratra, Rev. Mod. Phys. 75, 559 (2003).
* (12) T. Padmanabhan, Phys. Rept. 380, 235 (2003).
* (13) E. J. Copeland, M. Sami and S. Tsujikawa, Int. J. Mod. Phys. D 15, 1753 (2006).
* (14) E. V. Linder, Phys. Rev. D 80, 123528 (2009).
* (15) P. Zhang, Phys. Rev. D 73, 123504 (2006).
* (16) P. Zhang, M. Liguori, R. Bean, and S. Dodelson, Phys. Rev. Lett. 99, 141302 (2007).
* (17) G. Cognola, E. Elizalde, S. Nojiri, S. D. Odintsov, L. Sebastiani, and S. Zerbini, Phys. Rev. D 77, 046009 (2008).
* (18) K. Bamba, C. Q. Geng and C. C. Lee, JCAP 1008, 021 (2010) [arXiv:1005.4574 [astro-ph.CO]].
* (19) Y. S. Song, W. Hu, and I. Sawicki, Phys. Rev. D 75, 044004 (2007).
* (20) W. Hu and I. Sawicki, Phys. Rev. D 76, 064004 (2007).
* (21) E. Komatsu et al. [WMAP Collaboration], arXiv:1001.4538 [astro-ph.CO].
* (22) S. Nesseris and L. Perivolaropoulos, Phys. Rev. D 72, 123519 (2005).
* (23) L. Perivolaropoulos, Phys. Rev. D 71, 063503 (2005).
* (24) R. Amanullah et al., Astrophys. J. 716, 712 (2010), arXiv:1004.1711 [astro-ph.CO].
* (25) W. J. Percival et al., Mon. Not. Roy. Astron. Soc. 401, 2148 (2010), arXiv:0907.1660 [astro-ph.CO].
* (26) D. J. Eisenstein and W. Hu, Astrophys. J. 496, 605 (1998), arXiv:astro-ph/9709112.
* (27) E. Komatsu et al. [WMAP Collaboration], Astrophys. J. Suppl. 180, 330 (2009), arXiv:0803.0547 [astro-ph].
* (28) J. R. Bond, G. Efstathiou, and M. Tegmark, Mon. Not. Roy. Astron. Soc. 291, L33 (1997), arXiv:astro-ph/9702100.
* (29) W. Hu and N. Sugiyama, Astrophys. J. 471, 542 (1996), arXiv:astro-ph/9510117.
* (30) A. Lewis and S. Bridle, Phys. Rev. D 66, 103511 (2002), arXiv:astro-ph/0205436.
* (31) Y. Gong, X. ming Zhu, and Z.-H. Zhu, arXiv:1008.5010 [astro-ph.CO].
* (32) K. Bamba, C. Q. Geng and C. C. Lee, arXiv:1007.0482 [astro-ph.CO].
* (33) K. Glazebrook et al., ASP conference series 379, 72 (2007), arXiv:astro-ph/0701876.
* (34) D. Schlegel, M. White, and D. Eisenstein [with input from the SDSS-III], arXiv:0902.4680 [astro-ph.CO].
* (35) G. J. Hill et al., ASP conference series 399, 115 (2008), arXiv:0806.0183 [astro-ph].
* (36) European Space Agency Euclid Mission, http://sci.esa.int/euclid/.
* (37) N. Gehrels, arXiv:1008.4936 [astro-ph.CO]. See also http://jdem.gsfc.nasa.gov/.
* (38) D. J. Schlegel et al., arXiv:0904.0468 [astro-ph.CO].
* (39) The Square Kilometre Array, http://www.skatelescope.org/.
* (40) J. A. Tyson, D. M. Wittman, J. F. Hennawi, and D. N. Spergel, Nucl. Phys. Proc. Suppl. 124, 21 (2002), arXiv:astro-ph/0209632; J. A. Tyson [LSST Collaboration], Proc. SPIE Int. Soc. Opt. Eng. 4836, 10 (2002), arXiv:astro-ph/0302102; AIP Conf. Proc. 870, 44 (2006), arXiv:astro-ph/0609516; see also http://www.lsst.org/.
* (41) The Dark Energy Survey, http://www.darkenergysurvey.org/.
|
arxiv-papers
| 2010-10-11T10:27:33 |
2024-09-04T02:49:13.669154
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Louis Yang, Chung-Chi Lee, Ling-Wei Luo, and Chao-Qiang Geng",
"submitter": "Chung-Chi Lee",
"url": "https://arxiv.org/abs/1010.2058"
}
|
1010.2133
|
Constraint structure and Hamiltonian treatment of Nappi-Witten model
M. Dehghani111mdehghani@ph.iut.ac.ir A. Shirzad222shirzad@ipm.ir
Department of Physics, Isfahan University of Technology
P.O.Box 84156-83111, Isfahan, IRAN,
School of Physics, Institute for Research in Fundamental Sciences (IPM)
P.O.Box 19395-5531, Tehran, IRAN.
###### Abstract
We investigate the Hamiltonian analysis of Nappi-Witten model (WZW action
based on non semi simple gauge group) and find a time dependent non-
commutativity by canonical quantization. Our procedure is based on constraint
analysis of the model in two parts. A first class analysis is used for gauge
fixing the original model following by a second class analysis in which the
boundary condition are treated as Dirac constraints. We find the reduced phase
space by imposing our second class constraints on the variables in an extended
Fourier space.
Keywords:Noncommutativity, constraint analysis
## 1 Introduction
Treating boundary conditions as Dirac constrains has been considered in the
recent years by so many authors [1, 2, 3, 4]. This approach has been used
first in studying the Polyakov string coupled to a B-field. The common feature
of all works is non commutativity of the coordinate fields on the boundaries
which may lie on some brains, as first predicted by [5]. However, there are
different approaches in defining the constraints and investigating their
consistency in time. We have reviewed the whole subject in our previous work
[6] and showed if we impose the set of constraints on the Fourier expansions
of the fields, the redundant modes will be omitted in a natural way.
For simple physical models obeying linear equations of motion, the ordinary
Fourier expansion gives appropriate coordinates to reach the reduced phase
space. In other words, the infinite set of second class constraints emerging
as the result of boundary conditions, forces us to omit a number of Fourier
modes. However, ordinary Fourier transformation is not essential for
quantization; it is just one tool that acts well for most physical models at
hand. In the general case one should search for ”appropriate coordinates”, in
which imposing the set of second class constraints is equivalent to omitting
some canonical pairs from the theory.
In this paper we study the constraint structure of the Nappi-Witten model in
the Hamiltonian formalism. This model acquires complicated boundary conditions
so that the ordinary Fourier expansion seems inadequate to impose the whole
set of constraints which emerge from the boundary conditions. Nevertheless,
the Nappi-Witten model, on its own grands, is an attractive one since it
describes a non semi simple gauge group as well as giving time dependent non
commutativity in some gauges [7]. Our next interest is to emphasize that
solving the equations of motion is not necessarily needed for quantizing a
theory; the only necessity is finding the dynamics of the constraints and
construct their algebra with the Hamiltonian such that they remain consistent
with time on the constraint surface.
We give a precise Hamiltonian treatment of the model in which the constraint
structure is followed step by step from the initial action to the final
reduced phase space. In section 2 we introduce the model and find primary and
secondary constraints of the system. Section 3 is devoted to fixing the gauge
by introducing appropriate gauge fixing conditions. In section 4 we follow our
strategy of treating the boundary conditions as primary Dirac constraints and
follow their consistencies. The boundary conditions which come from the
original action, in fact, make the system more complicated. So, it is not
possible to write down the solutions in a closed form similar to a simple
Fourier expansion (see reference [8]). We try to find a basis which is
appropriate for imposing the infinite set of constraints in section 5. In
section 6 we will give our concluding remarks and will compare our results
with parallel approaches.
## 2 Hamiltonian structure of the model
The Nappi-Witten model describes a 4-component bosonic string
$X_{a}=(a_{1},a_{2},u,v)$ living in the background metric $G_{ab}(X)$ and
coupled to a $B$-field. The action is given as:
$S=\int
d^{2}\sigma\bigg{[}\sqrt{-g}g^{ij}G_{ab}\partial_{i}X^{a}\partial_{j}X^{b}+B_{ab}\epsilon^{ij}\partial_{i}X^{a}\partial_{j}X^{b}\bigg{]},$
(1)
where
$G(X)=\left(\begin{array}[]{llll}1&0&\frac{a_{2}}{2}&0\\\
0&1&-\frac{a_{1}}{2}&0\\\ \frac{a_{2}}{2}&-\frac{a_{1}}{2}&b&1\\\
0&0&1&0\end{array}\right),B(X)=\left(\begin{array}[]{llll}0&u&0&0\\\
-u&0&0&0\\\ 0&0&0&0\\\ 0&0&0&0\end{array}\right).$ (2)
The special form of $G(X)$ and $B(X)$ are chosen so that the gauge group of
the model is non semi-simple [8]. The metric field can be written in terms of
the following variables:
$\begin{array}[]{lll}N_{1}=\frac{1}{g^{00}\sqrt{-g}},&N_{2}=-\frac{g^{01}}{g^{00}},&N_{3}=\sqrt{-g}=\frac{1}{\sqrt{(g^{01})^{2}-g^{00}g^{11}}}.\\\
\end{array}$ (3)
In terms of the variables $X_{a}$ and $N_{\alpha}$ the action becomes:
$S=\int
d^{2}\sigma\bigg{[}\frac{1}{N_{1}}G_{ab}(X)(\dot{X}^{a}\dot{X}^{b}-2N_{2}\dot{X}^{a}X^{\prime
b}+(N_{2}^{2}-N_{1}^{2})X^{\prime a}X^{\prime b})+2B_{ab}\dot{X}^{a}X^{\prime
b}\bigg{]},$ (4)
where dot and prime means temporal and spatial derivatives, respectively. The
canonical momenta $\pi^{\alpha}$ and $p_{a}$ conjugate to $N_{\alpha}$ and
$X^{a}$ are:
$\begin{array}[]{l}\pi^{\alpha}=0\ \ \ \ \ \ \ \ \ \ \ \alpha=1,2,3\\\
p_{i}=\frac{1}{N_{1}}(2\dot{a}_{i}+\dot{u}\epsilon_{ij}a_{j})-\frac{N_{2}}{N_{1}}(2a^{\prime}_{i}+u^{\prime}\epsilon_{ij}a_{j})+2u\epsilon_{ij}a^{\prime}_{j}\\\
p_{u}=\frac{1}{N_{1}}(2b\dot{u}+2\dot{v}+\epsilon_{ij}\dot{a}_{i}a_{j})-\frac{N_{2}}{N_{1}}(2bu^{\prime}+2v^{\prime}+\epsilon_{ij}a^{\prime}_{i}a_{j})\\\
p_{v}=\frac{2\dot{u}}{N_{1}}-\frac{N_{2}}{N_{1}}2u^{\prime}.\\\ \end{array}$
(5)
The Canonical Hamiltonian reads:
$H=\int
d^{2}\sigma\frac{1}{N_{1}}G_{ab}(F^{a}F^{b}-(N_{2}^{2}-N_{1}^{2})X^{\prime
a}X^{\prime b}),$ (6)
where
$F^{a}=\dot{X}^{a}=N_{1}(G^{-1})^{ab}(p_{b}-B_{bc}X^{\prime
c})+N_{2}B_{ab}X^{\prime b}$ (7)
In terms of component fields $a_{i}$, $u$ and $v$ we have
$H=\int d^{2}\sigma(N_{1}\Psi^{1}+N_{2}\Psi^{2})$ (8)
where
$\begin{array}[]{l}\Psi^{1}={1\over 4}p_{i}^{2}+{1\over
4}\epsilon_{ij}p_{v}a_{i}p_{j}+{1\over 2}p_{u}p_{v}-{1\over
4}bp_{v}^{2}+{1\over 16}a_{i}^{2}p_{v}^{2}\\\ \hskip
22.76219pt+u^{\prime}\epsilon_{ij}a^{\prime}_{i}a_{j}+\epsilon_{ij}ua^{\prime}_{i}p_{j}+\frac{1}{2}up_{v}a^{\prime}_{i}a_{i}+(1+u^{2})a^{\prime
2}_{i}+bu^{\prime 2}+2u^{\prime}v^{\prime}\\\
\Psi^{2}=a^{\prime}_{i}p_{i}+u^{\prime}p_{u}+v^{\prime}p_{v},\\\ \end{array}$
(9)
As can be seen from Eqs. (5) the momenta $\pi^{\alpha}$ are primary
constraints. The dynamics of the system is achieved by the total Hamiltonian:
$H_{T}=H+\int d\sigma\lambda_{\alpha}\pi^{\alpha}(\sigma,\tau),$ (10)
in which $\lambda_{\alpha}$ are Lagrange multipliers. As usual we should
impose the consistency conditions on the constraints so that they remain valid
during the time. For this reason we demand $\dot{\pi}^{\alpha}\approx 0$,
where $\approx$ means weak equality i.e. equality on the constraint surface.
Using Eqs. (10) and (6) we have:
$\begin{array}[]{l}\dot{\pi}^{1}=\\{\pi^{1},H_{T}\\}=-\Psi^{1}\\\
\dot{\pi}^{2}=\\{\pi^{2},H_{T}\\}=-\Psi^{2}\\\
\dot{\pi}^{3}=\\{\pi^{3},H_{T}\\}=0,\\\ \end{array}$ (11)
Therefore, the consistency of three primary constraints $\pi^{\alpha}$ gives
two second level constraints $\Psi^{1}$ and $\Psi^{2}$. In this way we have so
far two levels of constraints as
$\begin{array}[]{lll}\pi^{1}&\pi^{2}&\pi^{3}\\\ \Psi^{1}&\Psi^{2}&\end{array}\
.$ (12)
In order to investigate the consistency of second level constraints, we need
to calculate the Poisson brackets of $\Psi^{1}(\sigma,\tau)$ and
$\Psi^{2}(\sigma,\tau)$ at different points. Direct calculation, using the
fundamental Poisson brackets among the four conjugate pairs $(u,p_{u})$,
$(v,p_{v})$ and $(a_{i},p_{i})$ gives:
$\begin{array}[]{l}\\{\Psi^{1}(\sigma,\tau),\Psi^{1}(\sigma^{\prime},\tau)\\}=\frac{1}{2}(\Psi^{2}(\sigma,\tau)\partial_{\sigma}-\Psi^{2}(\sigma^{\prime},\tau)\partial_{\sigma^{\prime}})\delta(\sigma-\sigma^{\prime})\\\
\\{\Psi^{1}(\sigma,\tau),\Psi^{2}(\sigma^{\prime},\tau)\\}=\Psi^{1}(\sigma,\tau)\partial_{\sigma}\delta(\sigma-\sigma^{\prime})\\\
\\{\Psi^{2}(\sigma,\tau),\Psi^{2}(\sigma^{\prime},\tau)\\}=\frac{1}{2}(\Psi^{2}(\sigma,\tau)\partial_{\sigma}-\Psi^{2}(\sigma^{\prime},\tau)\partial_{\sigma^{\prime}})\delta(\sigma-\sigma^{\prime}),\\\
\end{array}$ (13)
where
$\delta^{\prime}(\sigma-\sigma^{\prime})\equiv\frac{\partial}{\partial\sigma}\delta(\sigma-\sigma^{\prime})$.
It should be noted that each of the above Poisson brackets leads to a set of
terms at different points $\sigma$ and $\sigma^{\prime}$ multiplied by
$\frac{\partial}{\partial\sigma}\delta(\sigma-\sigma^{\prime})$ or
$\frac{\partial}{\partial\sigma^{\prime}}\delta(\sigma-\sigma^{\prime})$ which
equals to $-\frac{\partial}{\partial\sigma}\delta(\sigma-\sigma^{\prime})$.
However, since these terms have only non vanishing value when
$\sigma^{\prime}$ approaches to $\sigma$, one can consider all of them at the
same point. Then they add up to give the above results. The algebra (13) shows
that $\Psi^{1}(\sigma,\tau)$ and $\Psi^{2}(\sigma,\tau)$ are first class
constraints. Moreover, from (8) we see that:
$\begin{array}[]{l}\\{\Psi^{1},H\\}={N_{2}}^{\prime}\Psi^{1}+{N_{1}}^{\prime}\Psi^{2}+\frac{1}{2}N_{1}{\Psi^{2^{\prime}}}\approx
0\\\
\\{\Psi^{2},H\\}={N_{1}}^{\prime}\Psi^{1}+{N_{2}}^{\prime}\Psi^{2}+{N_{1}}\Psi^{1}+\frac{1}{2}N_{2}{\Psi^{2^{\prime}}}\approx
0\end{array}$ (14)
This shows that the consistency of $\Psi^{1}(\sigma,\tau)$ and
$\Psi^{2}(\sigma,\tau)$ does not give any new constraint, and we are left with
the five first class constraints given in (12).
In this way we have derived three constraint chains
$\left(\begin{array}[]{l}\pi^{1}\\\ \Psi^{1}\end{array}\right)$ ,
$\left(\begin{array}[]{l}\pi^{2}\\\ \Psi^{2}\end{array}\right)$ and
$\left(\begin{array}[]{l}\pi^{3}\end{array}\right)$ in the terminology of
reference [9]. In fact, the chain relation
$\\{\pi^{\alpha},H\\}=\Psi^{\alpha}$ holds for all of the chains. However the
first two chains are correlated, since the Poisson bracket of the last element
of each chain with the Hamiltonian contains the other constraint. This means
that it is not possible to construct closed algebra within each chain. The
last chain contains just one element and is not correlated to other chains,
since it commutes with all of them as well as with Hamiltonian.
As in ordinary Polyakov string one can show that $\pi^{3}$ generates the Weyl
symmetry of the model which affects only the components of the world-sheet
metric. In terms of the variables $N_{\alpha}$ we have $N_{3}\rightarrow
N_{3}+\epsilon$ under Weyl transformation. On the other hand the constraint
chains $\left(\begin{array}[]{l}\pi^{1}\\\ \Psi^{1}\end{array}\right)$ ,
$\left(\begin{array}[]{l}\pi^{2}\\\ \Psi^{2}\end{array}\right)$ can be shown
that generate the effect of reparametrization invariance on the metric
variables $N_{1}$ and $N_{2}$ as well as the variables $X_{a}$.
## 3 Gauge fixing
We began the theory with 14 field variables in the phase space, i.e. $X^{a}$,
$N_{\alpha}$ and their corresponding momentum fields $p_{a}$ and
$\pi^{\alpha}$. Then we derived 5 first class constraints given in (12). As is
well known from Dirac theory the first class constraints are generators of
gauge transformations [10]. One needs to consider additional conditions to fix
the gauges. These ”gauge fixing conditions” are functions of phase space
variables which should vanish to fix the gauges. The gauge fixing conditions
should fulfill two conditions. First, they should constitute a system of
second class constraints when added to the original first class constraints of
the system. This condition is necessary to fix the values of variables which
vary under the action of gauge generators [12]. Second, they should have a
closed algebra under the consistency conditions, i.e. under the successive
Poisson brackets with the Hamiltonian.
For a ”complete gauge fixing” the number of independent gauge fixing
conditions should be equal to the number of first class constraints [13]. In
this way, we should suggest 5 gauge fixing conditions to fix the gauges
generated by the constraints given in (12), and reach a ”reduced phase space”
of 4 field variables. Since the momenta $\pi^{\alpha}$ are generators of
transformations in $N_{\alpha}$, we fix the corresponding gauge by choosing
the values of $N_{\alpha}$ as $N_{1}\approx 1,\ N_{2}\approx 0$ and
$N_{3}\approx 1$. These values are chosen such that $g_{ij}=\eta_{ij}$. In
this way we have so far introduced three gauge fixing conditions
$\begin{array}[]{l}\Omega_{1}\equiv N_{1}-1,\\\ \Omega_{2}\equiv N_{2},\\\
\Omega_{3}\equiv N_{3}-1.\end{array}$ (15)
It can easily seen that the system of 6 constraints $\pi^{\alpha}$ and
$\Omega_{\alpha}$ are second class. The consistency of $\Omega_{\alpha}$’s by
the use of total Hamiltonian (10) determines the lagrange multipliers
$\lambda_{\alpha}$ to be zero and does not give any new constraint. This makes
us sure that the two criterions of a good gauge mentioned above are satisfied.
In fact, by the above gauge fixing three degrees of freedom $N_{\alpha}$ are
removed completely from the theory. This gauge has fixed the Weyl symmetry as
well as the effect of the reparametrization on the metric variables $N_{1}$
and $N_{2}$. On the other hand, we are still left with the remaining gauges
generated by $\Psi^{1}$ and $\Psi^{2}$ which generate the effect of
reparametrization on the variables $X_{a}$. In fact, since we have fixed the
gauge from the middle of the constraint chains, the gauge is fixed partially
in the language of reference [13]. In partial gauge fixing the Lagrange
multipliers are determined while the variations generated by some of the gauge
generators are not fixed.
To fix the effect of the parametrization of the world-sheet on $X_{a}$’s, as
in so many models in string theory we need to determine some definite
combinations of fields as the time variable in target space. Taking a look on
the form of the constraints $\Psi^{1}$ and $\Psi^{2}$ in (9) shows that the
choice $u=\mu\tau$ is more economical in the sense that simplifies the
constraints better. Here $\mu$ is a parameter with dimension of
$(\mbox{length})^{-1}$. We recall that all of the dynamical variables in the
action are dimensionless. Hence, we consider the gauge fixing condition
$\Omega_{4}=u-\mu\tau.$ (16)
To fulfill the second criterion of a good gauge we choose the last gauge
fixing condition as
$\displaystyle\Omega_{5}$ $\displaystyle\equiv$
$\displaystyle\dot{\Omega}_{4}$ $\displaystyle=$
$\displaystyle\\{\Omega_{1},H_{T}\\}+\frac{\partial\Omega_{1}}{\partial\tau}$
$\displaystyle\approx$ $\displaystyle p_{v}-2\mu$
This new constraint should also be valid during the time. Since
$\dot{\Omega}_{5}=2\mu(-\frac{N_{2}}{N_{1}}+N^{\prime}_{2})\approx 0,$ (18)
the chosen gauges are consistent and make a closed algebra with the
Hamiltonian. It is also clear that $\Omega_{4}$ and $\Omega_{5}$ make a second
class system with $\Psi^{1}$ and $\Psi^{2}$. Imposing strongly the constraints
(16) and (3) on the system, simplifies the constraints $\Psi_{1}$ and
$\Psi_{2}$ as
$\begin{array}[]{l}\Psi^{1}\rightarrow\bar{\Psi}^{1}=\frac{1}{4}p_{i}^{2}+\frac{1}{2}\epsilon_{ij}\mu
a_{i}p_{j}+\epsilon_{ij}\mu\tau
a^{\prime}_{i}p_{j}+(1+\mu^{2}\tau^{2})a^{\prime 2}_{i}+\mu
p_{u}-b\mu^{2}+\frac{1}{2}\mu^{2}a_{i}^{2}+\mu^{2}\tau a_{i}a^{\prime}_{i},\\\
\Psi^{2}\rightarrow\bar{\Psi}^{2}=a^{\prime}_{i}p_{i}+2\mu
v^{\prime},\end{array}$ (19)
This shows that $p_{u}$ and $v$ can be derived on the constraint surface, i.e.
from identities $\bar{\Psi}_{1}=0$ and $\bar{\Psi}_{2}=0$, in terms of the
physical variables $a_{i}$ and $p_{i}$. In this way the reduced phase space is
just the four dimensional space of $(a_{i},p_{i})$ whose original Poisson
brackets serve as the Dirac brackets in the remaining physical space. The
terms $\mu p_{u}$ and $\mu^{2}b$ in the expressions of $\bar{\Psi}_{1}$ have
nothing to do with the dynamics of $(a_{i},p_{i})$ and can be dropped. The
parameter $b$ has in fact no important role in the theory and only shifts the
spectrum of the energy with a constant value.
As in reference [8] we consider the dimensionless quantity $\mu l$ as a small
parameter which should be considered only in the first order. Therefore, in
all of the foregoing calculations we keep only linear terms with respect to
$\mu$, assuming that $l$ is finite. Therefore, the Hamiltonian (8) in the
reduced phase space can be written in terms of the Hamiltonian density:
$\mathcal{H}_{GF}=\frac{1}{4}p_{i}^{2}+\frac{1}{2}\epsilon_{ij}\mu
a_{i}p_{j}+\epsilon_{ij}\mu\tau a^{\prime}_{i}p_{j}+a^{\prime 2}_{i}.$ (20)
Since $B(X)$ in (2) is linear with respect to $u$ one may think of $\mu$ as
the order of magnitude of the $B$-field. This assumption is equivalent to
considering the effect of the $B$-field only up to the first order.
## 4 Boundary conditions as constraints
From now on we forget about the original theory and suppose we are given a
theory with two degrees of freedom $a_{i}$ and the corresponding momenta
$p_{i}$ whose dynamics is given by the final Hamiltonian (20). We make a
change of variables from $(a_{i},p_{i})$ to
$(A_{i}=\epsilon_{ij}a_{j},P_{i}=p_{i})$. Then the the fundamental Poisson
brackets which is the same as the final Dirac bracket of the original theory
read
$\begin{array}[]{l}\\{A_{i}(\sigma,\tau),P_{j}(\sigma^{\prime},\tau)\\}=\epsilon_{ij}\delta(\sigma-\sigma^{\prime}),\\\
\\{A_{i}(\sigma,\tau),A_{j}(\sigma^{\prime},\tau)\\}=\\{P_{i}(\sigma,\tau),P_{j}(\sigma^{\prime},\tau)\\}=0\end{array}$
(21)
The Hamiltonian equation of motion for the remaining fields, can be written as
$\begin{array}[]{l}\dot{A}_{i}={1\over 2}\epsilon_{ij}(P_{j}-2\mu\tau
A^{\prime}_{j}-\mu A_{j})\\\ \dot{P}_{i}=-\epsilon_{ij}({1\over 2}\mu
P_{j}-\mu\tau P^{\prime}_{j}+2A^{\prime\prime}_{j})\\\ \end{array}$ (22)
The only things that should be brought from the original theory are the
boundary conditions. Using the original action (4) the boundary condition
after gauge fixing emerge in terms of phase space variables as:
$\Phi_{i}^{(1)}=\mu\tau P_{i}-2A^{\prime}_{i}=0\;\;\;\;\;\ \mbox{at
$\sigma=0,l$}$ (23)
We have shown in the appendix that the boundary condition (23) can also be
derived from the parallel approach as the equations of motion of the end
points in the discretized version.
As mentioned in the introduction we do not want to find the general solution
of the dynamical equations of motion. On the other hand, we are interested to
follow the dynamics of the boundary conditions which means investigating the
consistency of primary constraints $\Phi^{(1)}_{i}(\sigma)|_{\sigma=0}$ and
$\Phi^{(1)}_{i}(\sigma)|_{\sigma=l}$. Using the gauge fixed Hamiltonian of the
previous section (20) the total Hamiltonian at this stage is
$\overline{H}_{T}=\int_{0}^{l}d\sigma[\frac{1}{4}P_{i}P_{i}-\frac{1}{2}\mu
A_{i}P_{i}-\mu\tau
A^{\prime}_{i}P_{i}+A^{\prime}_{i}A^{\prime}_{i}]+\Lambda_{1}^{i}\Phi^{(1)}_{i}(\sigma)|_{\sigma=0}+\Lambda_{2}^{i}\Phi^{(1)}_{i}(\sigma)|_{\sigma=l}.$
(24)
The consistency of primary constraints for instance at $\sigma=0$ gives
$0=\left[\mu
P_{i}-\epsilon_{ij}P^{\prime}_{j}+\mu\epsilon_{ij}A^{\prime}_{j}\right]_{\sigma=0}+\Lambda_{1}^{j}\
\left\\{\Phi^{(1)}_{i}|_{\sigma=0}\ ,\Phi^{(1)}_{j}|_{\sigma=0}\right\\}$ (25)
Similar equations should be written at the end-point $\sigma=l$. As discussed
in details in [14] the first term in the LHS of Eq. (25) has not the same
order as the coefficient of $\Lambda^{i}_{1}$ (and $\Lambda^{i}_{2}$) in the
second term when regularizing the Dirac delta function. Therefore this
condition can be fulfilled identically only if $\Lambda^{i}_{1,2}$ as well as
the first term vanish simultaneously. In this way we have used the consistency
conditions of the constraints for simultaneously determining the undetermined
Lagrange multiplier and finding the next level of constraints as
$\Phi_{i}^{(2)}(0)$ and $\Phi_{i}^{(2)}(l)$ where
$\Phi_{i}^{(2)}(\sigma)=P_{i}-\epsilon_{ij}P^{\prime}_{j}+\mu\epsilon_{ij}A^{\prime}_{j}.$
(26)
Then we should consider the consistency of second level constraints by using
the Hamiltonian
$\overline{H}=\int_{0}^{l}d\sigma[\frac{1}{4}P_{i}P_{i}-\frac{1}{2}\mu
A_{i}P_{i}-\mu\tau A^{\prime}_{i}P_{i}+A^{\prime}_{i}A^{\prime}_{i}]$ (27)
which is the same as the total Hamiltonian (24) after imposing
$\Lambda^{i}_{1,2}=0$. This gives the third level of constraints. Subsequent
levels of constraints can be derived in the same way. Using the relations:
$\begin{array}[]{l}\\{A_{i}^{(n)},\overline{H}\\}=\frac{1}{2}\epsilon_{ij}(P^{(n)}_{j}-\mu
A_{j}^{(n)}-2\mu\tau A^{(n+1)}_{j})+{\cal O}(\mu^{2})\\\
\\{P_{i}^{(n)},\overline{H}\\}=-\epsilon_{ij}(\frac{1}{2}\mu
P^{(n)}_{j}-\mu\tau P^{(n+1)}_{j}+2A^{(n+2)}_{j})+{\cal O}(\mu^{2}),\\\
\end{array}$ (28)
where $A_{i}^{(n)}=\partial_{\sigma}^{n}A_{i}$ and
$P_{i}^{(n)}=\partial_{\sigma}^{n}P_{i}$ one can inductively show that the
full set of constraints are $\Phi_{i}^{(N)}(0)\approx 0$ and
$\Phi_{i}^{(N)}(l)\approx 0$ where
$\begin{array}[]{ll}\Phi_{i}^{(2n+1)}=-n\mu P_{i}^{(2n-1)}+\mu\tau
P_{i}^{(2n)}-2n\mu\epsilon_{ij}A^{(2n)}_{j}-2A^{(2n+1)}_{i}+{\cal
O}(\mu^{2}),&\\\ \Phi_{i}^{(2n+2)}=(n+1)\mu
P^{(2n)}_{i}-\epsilon_{ij}P^{(2n+1)}_{j}+(2n+1)\mu\epsilon_{ij}A_{j}^{(2n+1)}+{\cal
O}(\mu^{2})&n=0,1,2,\cdots\\\ \end{array}$ (29)
For practical calculations we write the constraints as ordinary functions in
the bulk of the string and then integrate them with the use of
$\delta(\sigma)$ and $\delta(\sigma-l)$ respectively.
Now we want to investigate whether the constraints are first or second class.
For this reason one should calculate the Poisson brackets of the constraints.
Since the constraints contain different orders of derivatives of
$A_{i}(\sigma,\tau)$ and $P_{i}(\sigma,\tau)$, the Poisson brackets
$C_{ij}^{k,k^{\prime}}\equiv\\{\Phi_{i}^{k},\Phi_{j}^{k^{\prime}}\\}$ contain
derivatives of orders $k+k^{\prime}$, $k+k^{\prime}-1$, etc, of the Dirac
delta function, which are highly divergent and independent of each other. One
way of treating the matrix of Poisson brackets is regularizing the delta
functions as gaussian functions of width $\varepsilon$ and let
$\varepsilon\rightarrow 0$ after all. A tedious calculation gives
$\begin{array}[]{l}C^{2m+1,2n+1}_{ij}=\frac{-2\mu\epsilon_{ij}}{\sqrt{\pi}}\varepsilon^{-2(m+n+1)}(\varepsilon(m+n)H_{2m+2n}(0)-2\tau
H_{2m+2n+1}(0))+\mathcal{O}(\mu^{2})\\\
C^{2m+2,2n+1}_{ij}=\frac{-2}{\sqrt{\pi}}\varepsilon^{-2(m+n+1)-1}(n\mu\varepsilon\epsilon_{ij}H_{2m+2n+1}(0)+\delta_{ij}H_{2m+2n+2}(0))+\mathcal{O}(\mu^{2}),\\\
C^{2m+2,2n+2}_{ij}=\frac{2\mu\epsilon_{ij}}{\sqrt{\pi}}\varepsilon^{-2(m+n+1)-1}H_{2m+2n+2}(0)+\mathcal{O}(\mu^{2})\end{array}$
(30)
where $H_{n}(x)$ are Hermite polynomials. Similar expressions should be
considered with $H_{n}(1)$ at the end-point $\sigma=l$. The non vanishing
elements on each row are located such that no vanishing linear combination of
rows may be found. This means that the infinite dimensional matrix
$C_{ij}^{k,k^{\prime}}$ is not singular and can in principle be inverted.
Therefore, all of the constraints are second class. However, it is not
practically possible to find the inverse of $C_{ij}^{k,k^{\prime}}$. The
problem is how we can find the Dirac brackets of the fields which need to have
$C^{-1}$.
## 5 Reduced phase space
As stated before, we seek for appropriate coordinates in which imposing the
constraints (29) lead to omitting a set of canonical pairs. Here we have a
difficult problem in which the ordinary Fourier expansion does not do this
job. However, in the limit $\mu\rightarrow 0$ the boundary condition (23) is
the ordinary Neumann one and the Hamiltonian (27) has a simple quadratic form
in terms of coordinates and momenta. Hence, we need to write extended Fourier
transformations for the fields $A_{i}$ and $P_{i}$ that include at most linear
corrections with respect to the parameter $\mu$ and go to the ordinary Fourier
transformation in the limit $\mu\rightarrow 0$. Since $\mu\tau$ and
$\mu\sigma$ are the only dimensionless quantities that can be used for this
correction, what can we do is correcting the Fourier coefficients by
correction terms linear in $\tau$ or $\sigma$. The linear term in $\tau$,
however, is not needed at this stage, since it can be considered as part of
the solution of the equations of motion. Adding all these points up together
we suggest the following extended Fourier transformations for the fields
$A_{i}(\sigma,\tau)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}dk\left[\left(A_{i}(k,\tau)+\mu\sigma\alpha_{i}(k,\tau)\right)\cos
k\sigma+\left(B_{i}(k,\tau)+\mu\sigma\beta_{i}(k,\tau)\right)\sin
k\sigma\right],$ (31)
$P_{i}(\sigma,\tau)=\frac{-\epsilon_{ij}}{\sqrt{2\pi}}\int_{-\infty}^{\infty}dk\left[\left(C_{j}(k,\tau)+\mu\sigma\gamma_{j}(k,\tau)\right)\cos
k\sigma+\left(D_{j}(k,\tau)+\mu\sigma\delta_{j}(k,\tau)\right)\sin
k\sigma\right].$ (32)
In ordinary Fourier expansions the coefficients $A_{i}(k,\tau)$,
$B_{i}(k,\tau)$, $C_{i}(k,\tau)$ and $D_{i}(k,\tau)$ contain the same amount
of data as the original fields $A_{i}(\sigma,\tau)$ and $P_{i}(\sigma,\tau)$.
Comparing the expansions (31) and (32) with ordinary Fourier expansions shows
that we have used a duplicated basis including $\sin$’s, $\cos$’s, $\sigma$
times $\sin$’s and $\sigma$ times $\cos$’s for expanding our fields. This
basis is complete but its elements are not independent. Mathematically it is
allowed to use a basis which is ”larger than necessary”. However, the
essential point is that one should assume appropriate Poisson brackets among
the extended Fourier modes such that the desired fundamental Poisson brackets
(21) remain valid. In other words, we should tune their brackets in such a way
that our physical phase space variables, which are half of the extended phase
space variables, do obey the right Poisson brackets. Direct calculation shows
that the following Poisson brackets lead to the standard Poisson algebra (21)
for the physical fields,
$\begin{array}[]{l}\\{A_{i}(k,\tau),C_{j}(k^{\prime},\tau)\\}=\\{B_{i}(k,\tau),D_{j}(k^{\prime},\tau)\\}=\delta_{ij}\delta(k-k^{\prime}),\\\
\\{\alpha_{i}(k,\tau),D_{j}(k^{\prime},\tau)\\}=\\{\gamma_{i}(k,\tau),B_{j}(k^{\prime},\tau)\\}=\delta_{ij}\partial_{k^{\prime}}\delta(k-k^{\prime}).\end{array}$
(33)
All other Poisson brackets are assumed to vanish. Specially the modes
$\beta_{i}$ and $\delta_{i}$ have vanishing Poisson brackets with all other
variables in the extended Fourier space and so decouple from the theory. This
means that we can put them away and write down the expansions only with linear
terms in the cosine modes. We will see on the other hand that omitting the
modes $\beta_{i}$ and $\delta_{i}$ does not disturb our analysis of imposing
the boundary conditions. We have, up to this point, 6 sets of real variables
in the extended Fourier space which depend on real, continues and positive
variable $k$.
Now we want to impose the full set of constraints (29) on the fields. Using
the expansions (31) and (32) the constraints at the end-point $\sigma=0$ lead
to
$\begin{array}[]{l}\int_{-\infty}^{\infty}dkk^{2n}\left[\mu\tau\epsilon_{ij}C_{j}+2n\epsilon_{ij}A_{j}+(4n+2)\alpha_{i}+2k\tilde{B}_{i}\right]+{\cal
O}(\mu^{2})=0\\\
\int_{-\infty}^{\infty}dkk^{2n-1}\left[(n+1)\epsilon_{ij}C_{j}+(2n+1)\gamma_{i}+k\tilde{D}_{i}\right]+{\cal
O}(\mu^{2})=0\end{array}$ (34)
where $B_{i}=\mu\tilde{B}_{i}$ and $D_{i}=\mu\tilde{D}_{i}$. Since these
conditions should be satisfied for arbitrary values of $n$ we have
$\begin{array}[]{l}\mu\tau\epsilon_{ij}C_{j}+2n\epsilon_{ij}A_{j}+(4n+2)\alpha_{i}+2k\tilde{B}_{i}=0,\\\
(n+1)\epsilon_{ij}C_{j}+(2n+1)\mu\gamma_{i}+k\tilde{D}_{i}=0.\end{array}$ (35)
The difficulty arises here since the integer $n$, which shows the level of
constraints, has appeared in the form of relations among the Fourier modes.
This means that it is not possible to satisfy the constraints of all levels
just by considering simple linear relations among the Fourier modes of a given
$k$ as can be done in ordinary Dirichlet, Neumann, or even mixed boundary
conditions [6]. In fact, this phenomenon is the reason which makes the
ordinary Fourier expansion inadequate for realizing the constraints. However,
we have the opportunity of existence of extra variables in the extended phase
space, which provides us additional tools for satisfying the constraints. In
this way we are allowed to assume that the coefficients of $n$ besides the
terms independent of $n$ in (35) vanish. This gives
$\begin{array}[]{ll}\alpha_{i}=-\frac{1}{2}\epsilon_{ij}A_{j}+{\cal
O}(\mu^{2})&\tilde{B}_{i}=\frac{1}{2k}\epsilon_{ij}(A_{j}-\tau C_{j})+{\cal
O}(\mu^{2})\\\ \gamma_{i}=-\frac{1}{2}\epsilon_{ij}C_{j}+{\cal
O}(\mu^{2})&\tilde{D}_{i}=-\frac{1}{2k}\epsilon_{ij}C_{j}+{\cal
O}(\mu^{2})\end{array}$ (36)
Hence the main fields $A_{i}(\sigma,\tau)$ and $P_{i}(\sigma,\tau)$ can be
written in terms of two remaining sets of Fourier modes $A_{i}(k,\tau)$ and
$C_{i}(k,\tau)$ as
$A_{i}(\sigma,\tau)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}dk\left[(\delta_{ij}-\frac{1}{2}\mu\sigma\epsilon_{ij})A_{j}\cos
k\sigma+\frac{\mu}{2k}\epsilon_{ij}(A_{j}-\tau C_{j})\sin k\sigma\right],$
(37)
$P_{i}(\sigma,\tau)=\frac{-1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}dk\left[(\epsilon_{ij}+\frac{1}{2}\mu\sigma\delta_{ij})C_{j}\cos
k\sigma+\frac{\mu}{2k}C_{i}\sin k\sigma\right].$ (38)
As expected, the zeroth order (with respect to $\mu$) of the Eqs. (37) and
(38) is the expansion of a simple bosonic string with Neumann boundary
condition at the end point $\sigma=0$. The linear term with respect to
$\sigma$ in cosine modes as well as the sin term itself are appeared as the
first order corrections.
Next we should impose the constraints (29) at the end-point $\sigma=l$ on the
fields derived recently in Eqs. (37) and (38). Hence we find
$\begin{array}[]{l}\int_{-\infty}^{\infty}dkk^{2n-1}(-1)^{n}[n\mu\epsilon_{ij}C_{j}+2k^{2}(A_{i}-\frac{1}{2}\mu\sigma
A_{j})]\sin(kl)+{\cal O}(\mu^{2})=0,\\\ \\\
\int_{-\infty}^{\infty}dkk^{2n+1}(-1)^{n}[(\delta_{ij}-\frac{1}{2}\mu\sigma\epsilon_{ij})C_{j}-(2n+1)\mu\epsilon_{ij}A_{j}]\sin(kl)+{\cal
O}(\mu^{2})=0.\end{array}$ (39)
The above constraints are satisfied identically for $kl=m\pi$. However, for
$k\neq\frac{m\pi}{l}$ there is no way for satisfying the constraints for
arbitrary $n$ except assuming that
$A_{i}(k,\tau)=C_{i}(k,\tau)=0\hskip 28.45274pt\mbox{for}\ \ \
k\neq\frac{m\pi}{l}$ (40)
This leads to descritizing the Fourier modes.
Before writing the final form of the fields in terms of the set of enumerable
Fourier modes, care is needed to write the zero modes. The contributions due
to cosine modes come out automatically by letting $k=0$. However,
contributions to zero modes originating from sine terms should be derived by
taking the following limits:
$\lim_{k\rightarrow 0}\tilde{B}_{i}\sin
k\sigma=\frac{1}{2}\sigma\epsilon_{ij}(A_{j}(0,\tau)-\tau
C_{j}(0,\tau)),\hskip 28.45274pt\lim_{k\rightarrow 0}\tilde{D}_{i}\sin
k\sigma=-\frac{1}{2}\sigma\epsilon_{ij}C_{j}(0,\tau),$ (41)
which follow from Eqs.(36). Adding these two contributions the zero mode part
of the fields are so far as follows
$\begin{array}[]{l}A^{0}_{i}(\sigma,\tau)=A_{i}^{0}(\tau)-\frac{1}{2}\mu\sigma\tau\epsilon_{ij}C^{0}_{j}(\tau)\\\
P^{0}_{i}(\sigma,\tau)=-(\epsilon_{ij}+\mu\sigma\delta_{ij})C^{0}_{j}(\tau)\end{array}$
(42)
At this point we want to notice the reader to a global symmetry of the gauged
fixed Lagrangian. If we turn off the B-field we would have an ordinary bosonic
string in which only the derivatives of the A-fields are present in the
Lagrangian. This allows one to shift the fields by a constant amount without
any change in the Lagrangian. When the B-field is on, Eq. (20) shows that the
A-field itself is present in the gauged fixed Hamiltonian. However, the
relevant term, i.e. the second term in Eq. (20), is proportional to $\mu$.
This shows that the theory is symmetric, up to second order terms with respect
to $\mu$, under the following transformation
$A_{i}(\sigma,\tau)\rightarrow A_{i}(\sigma,\tau)+\mu f(\tau)$ (43)
where $f(\tau)$ is an arbitrary function of time. This symmetry leads to an
ambiguity in the zero mode of the A-field. Hence we should correct the first
row of Eq. (42) in the most general case as follows
$A^{0}_{i}(\sigma,\tau)=A_{i}^{0}(\tau)-\frac{1}{2}\mu\sigma\tau\epsilon_{ij}C^{0}_{j}(\tau)+\mu
l[(a_{ij}A^{0}_{j}(\tau)+b_{ij}C^{0}_{j}(\tau)]$ (44)
Note that $\mu l$ is the only relevant dimensionless quantity which is first
order in $\mu$. The unknown coefficients $a_{ij}$ and $b_{ij}$ should be
determined upon suitable assumptions about the algebra of the fields. The best
assumption seems to be keeping the standard algebra (21) in the bulk of the
string and letting all changes in the algebra of the fields lay on the
boundaries. If we make this choice the final form of the physical fields in
terms of the set of discrete Fourier modes $A_{i}^{m}(\tau)\equiv
A_{i}(\frac{m\pi}{l},\tau)$ and $C_{i}^{m}(\tau)\equiv
C_{i}(\frac{m\pi}{l},\tau)$ are as follows
$\begin{array}[]{lll}A_{i}(\sigma,\tau)&=&\frac{1}{\sqrt{l}}\bigg{[}A_{i}^{0}(\tau)-\frac{1}{2}\mu\tau(\sigma-\frac{l}{2})\epsilon_{ij}C_{j}^{0}(\tau)-\frac{1}{2}\mu
l\epsilon_{ij}A_{j}^{0}(\tau)\bigg{]}\\\
{}&+&\sqrt{\frac{2}{l}}\sum_{m=1}^{\infty}\bigg{[}(A_{i}^{m}(\tau)-\frac{1}{2}\mu\sigma\epsilon_{ij}A_{j}^{m}(\tau))\cos\frac{m\pi\sigma}{l}+\frac{\mu
l}{2m\pi}\epsilon_{ij}(A_{j}^{m}(\tau)-\tau
C_{j}^{m}(\tau))\sin\frac{m\pi\sigma}{l}\bigg{]}\end{array}$ (45)
$\begin{array}[]{lll}P_{i}(\sigma,\tau)&=&-\frac{1}{\sqrt{l}}\bigg{[}\epsilon_{ij}C_{j}^{0}(\tau)+\mu\sigma
C_{i}^{0}(\tau)\bigg{]}\\\
{}&-&\sqrt{\frac{2}{l}}\sum_{m=1}^{\infty}\bigg{[}(\epsilon_{ij}C_{j}^{m}(\tau)+\frac{1}{2}\mu\sigma
C_{i}^{m}(\tau))\cos\frac{m\pi\sigma}{l}+\frac{\mu
l}{2n\pi}C_{i}^{m}(\tau)\sin\frac{m\pi\sigma}{l}\bigg{]}\end{array}$ (46)
The normalization factor $\frac{1}{\sqrt{2\pi}}$ is replaced by
$\sqrt{\frac{2}{l}}$ for oscillatory modes and $\frac{1}{\sqrt{l}}$ for zero
mode upon going from the continues parameter $k$ to the discrete parameter
$m$. 333Since another length scale, i.e. $\mu^{-1}$, exists in the model, one
may suppose that the normalization factors should differ from the ordinary
Fourier series. However, it can be shown that such corrections only changes
the observables by amounts of ${\cal O}(\mu^{2})$ which is not important With
this normalization the brackets of the discrete modes should also be given in
terms of Kronecker delta as
$\\{A_{i}^{m},C_{j}^{m^{\prime}}\\}=\delta_{ij}\delta_{mm^{\prime}},$ (47)
$\\{A_{i}^{m},A_{j}^{m^{\prime}}\\}=\\{C_{i}^{m},C_{j}^{m^{\prime}}\\}=0.$
(48)
In fact, the remaining canonical pairs $A_{i}^{m}$ and $C_{i}^{m}$ as a small
part of the original phase space are natural coordinates of the reduced phase
space. On the other hand, a great part of the initial phase space variables
are omitted due to the constraints.
Remember that if one is able to omit the redundant variables due to all kinds
of constraints and write down the relevant fields in terms of final canonical
coordinates of the reduced phase space, then there is no need to find the
Dirac brackets. In other words, we pay the expense of using the Dirac brackets
whenever it is not possible to find a canonical basis to describe the reduced
phase space. Hence, we will find the Dirac brackets of the original fields
$A_{i}(\sigma,\tau)$ and $P_{i}(\sigma,\tau)$ if we calculate their brackets
by using the brackets (47) and (48).
Eq. (46) shows that the momentum-fields $P_{i}(\sigma,\tau)$ just include the
variables $C_{i}^{m}$ and have vanishing brackets:
$\\{P_{i}(\sigma,\tau),P_{j}(\sigma^{\prime},\tau)\\}=0.$ (49)
Straightforward calculations gives the brackets of coordinate and momentum
fields as
$\\{A_{i}(\sigma,\tau),P_{j}(\sigma^{\prime},\tau)\\}=\epsilon_{ij}\delta_{N}(\sigma,\sigma^{\prime}),$
(50)
where
$\delta_{N}(\sigma,\sigma^{\prime})\equiv\delta(\sigma-\sigma^{\prime})+\delta(\sigma+\sigma^{\prime}).$
Since both $\sigma$ and $\sigma^{\prime}$ lie in the interval $[0,l]$ their
sum never vanishes. So the second delta function does not have any role and
Eq. (50) reduces to the usual form of Eq. (21). However, since in the
expansion of $A$-fields both variables $A_{i}^{m}$ and $C_{j}^{m}$ are
present, the interesting phenomenon appears in the bracket of coordinate
fields at different points. Direct calculation gives
$\\{A_{i}(\sigma,\tau),A_{j}(\sigma^{\prime},\tau)\\}=\frac{1}{2}\mu\tau\epsilon_{ij}\left(\frac{\sigma+\sigma^{\prime}}{l}-1+\frac{2}{\pi}\sum_{n=1}^{\infty}\frac{1}{n}\sin\frac{n\pi}{l}(\sigma+\sigma^{\prime})\right).$
(51)
This result is similar to what derived in [6] for a string coupled to constant
background B-field. The right hand side of Eq. (51) vanishes in the bulk of
the string, i.e. when $\sigma$ or $\sigma^{\prime}$ does not lie on the end
points. It gives (-2) when $\sigma=\sigma^{\prime}=0$ and (+2) when
$\sigma=\sigma^{\prime}=l$. However, as the B-field itself, the amount of non
commutativity grows linearly with time. Our result here defers from reference
[7] with a term proportional to $\mu\tau^{2}$ which is the same on both
boundaries as well as in the bulk of the string. If, however, we add a term
$-\frac{1}{2}\mu\tau^{2}\epsilon_{ij}C_{j}^{0}(\tau)$ to the zero mode part of
the field $A_{i}(\sigma,\tau)$ in Eq. (45), our result will coincide with
reference [7]. This correction is allowed according to the global symmetry of
Eq. (43). This means that we have forgiven our previous assumption that the
components of the A-field commute in the bulk of the string. With this
assumption the resulted brackets can be summarized as follows
$\begin{array}[]{l}\\{A_{i}(\sigma,\tau),P_{j}(\sigma^{\prime},\tau)\\}=\epsilon_{ij}\delta\sigma,\sigma^{\prime}),\\\
\\{P_{i}(\sigma,\tau),P_{j}(\sigma^{\prime},\tau)\\}=0\\\
\\{A_{i}(\sigma,\tau),A_{j}(\sigma^{\prime},\tau)\\}=\left\\{\begin{array}[]{ll}\frac{\mu\tau^{2}\epsilon_{ij}}{2l}&\sigma\neq
0,l\ \ \mbox{or}\ \ \sigma^{\prime}\neq 0,l\\\
\mu\tau\epsilon_{ij}(1+\frac{\tau}{2l})&\sigma=\sigma^{\prime}=0\\\
\mu\tau\epsilon_{ij}(-1+\frac{\tau}{2l})&\sigma=\sigma^{\prime}=l\end{array}\right.\end{array}$
(52)
This shows that the fundamental characters of the $A$-fields and $P$-fields as
coordinate and momentum fields are remained almost as before and the time
dependent B-field leads to a time dependent non commutativity in the
coordinate fields all over the string.
## 6 Concluding remarks
In this paper we gave a complete Hamiltonian treatment of the Nappi-Witten
model (WZW model based on non semi simple gauge group) as an interesting and
non trivial system in which complicated boundary conditions make the physical
subset of variables far from reaching. The initial dynamical variables in this
model are 4 components of a bosonic string, $X_{a}=(a_{1},a_{2},u,v)$, and the
components of world-sheet metric. We used appropriate variables to find 3
primary and 2 secondary first class constraints. It can be shown that these
constraints are generators of reparametrizations as well as Weyl
transformations. Then we fixed the gauge such that the world-sheet metric is
flat and $u=\mu\tau$ where the small parameter $\mu$ is proportional to the
strength of the B-field. In this way the components of the world-sheet metric
and the variables $u$ and $v$ disappeared as the result of constraints and
gauge fixing conditions. Hence, we derived a smaller theory with two
coordinate fields $a_{1}$ and $a_{2}$ and their corresponding momentum fields.
The most important part of the problem seems to be the boundary conditions
which should be brought from the original theory. Considering the boundary
condition as Dirac constraints and following their consistency, we found two
infinite chains of second class constraints at the end-points which restricted
the space of physical variables to a much smaller set. Due to complicated form
of the boundary conditions, it is not an easy task to impose them on the space
of the physical variables. In fact, with an ordinary Fourier expansion the
constraints do not lead simply to omitting some Fourier modes as in Dirichlet
or Neumann boundary conditions.
To overcome this difficulty we extended the phase space to a larger one which
is given by an extended Fourier expansion in which the Fourier modes are
replaced by linear functions of the variables. In this basis the infinite set
of constraints can be imposed more easily by using the arbitrariness due to
extra variables. This results to disappearing of so many canonical pairs among
the used extended Fourier basis and finally a set of discrete modes remain
which act as the canonical coordinates of the reduced phase space. Then all
physical objects including the original coordinate and momentum fields can be
expanded in terms of these modes.
Using these expansions we found that the commutation relations of the
coordinate and momentum fields are almost as usual, except that the coordinate
fields do not commute at the boundaries, with an amount proportional to time
and/or B-field but with opposite signs at two boundaries. We showed that it is
allowed to insert a term which gives non commutativity proportional to
$\tau^{2}$ throughout the string. This correction may make our results
consistent with those of reference [7] in which the authors have given
iterative solutions for the equations of motion.
We think that our method here has two main advantages in two different areas.
First, we do not solve the equation of motion. Therefore, in our final result
the time dependence of remaining modes are not specified. However, this time
dependence is not essential for quantization of the model. If needed, one can
use the Hamiltonian written in terms of the final modes and then derive their
time dependence. In fact, our main objective is that for quantizing a theory,
i.e. investigating the algebraic structure of the observables, it is not
needed to follow the full dynamics of the system; it is just enough to study
the dynamics of constraints. As a matter of fact, for simple models it may
seem more simple and economic to solve the equations of motion and then
quantize the theory, since this procedure contains the dynamics of the
constraints within itself. But this may not be the case for a complicated
model such as the model considered in this paper.
The next advantage is in the context of constraint systems. As we see in the
literature [1, 14] the main difficulty in considering the infinite set of
constraints due to boundary conditions is deriving the Dirac brackets. In this
paper, as in our previous work [6] we showed that if one is able to find a set
of canonical variables describing the reduced phase space, then there is
naturally no need to calculate the Dirac brackets. In fact, this was the main
brilliant idea of Dirac [11], who gave his famous formula of Dirac brackets in
such a way that it is equivalent to calculating the Poisson brackets only in
the space of canonical variables describing the reduced phase space.
## References
* [1] C.S Chu, P.M Ho, Nucl. Phys. B 550 (1999) 151.
* [2] F. Ardalan, H. Arfaei, M.M. Sheikh-Jabbari, Nucl. Phys. B 576 (2000) 578.
* [3] T. Lee, Phys. Rev. D 62 (2000) 024022.
* [4] R. Banerjee, B. Chakraborty, K. Kumar, Nucl. Phys. B 668 (2003) 179.
* [5] N. Seiberg, E. Witten, JHEP 09 (1999) 032.
* [6] M. Dehghani and A. Shirzad, Eur. Phys. J. C48 (2006) 315.
* [7] L. Dolan and C. R. Nappi, Phys. Lett. B 551 (2003) 369.
* [8] C.R. Nappi , E. Witten, Phys. Rev. Lett. 71 (1993) 3751.
* [9] F. Loran and A. Shirzad, Int. J. Mod. Phys. A 17 (2002) 625.
* [10] M. Henneaux, C. Teitelboim, J. Zanelli, Nucl. Phys. B 332 (1990) 169.
* [11] P.A.M. Dirac, Lecture Notes on Quantum Mechanics, Yeshiva University New York, 1964. Also see P.A.M. Dirac, Proc. Roy. Soc. London. ser. A, 246, 326, 1950\.
* [12] A. Shirzad, F. Loran, Int. J. Mod. Phys. A 17 (2002) 4801.
* [13] A. Shirzad, J. Math. Phys 48 (2007) 082303.
* [14] M.M. Sheikh Jabbari, A. Shirzad, Eur. Phys. J. C 19 (2001) 383.
|
arxiv-papers
| 2010-10-11T15:22:41 |
2024-09-04T02:49:13.678883
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Mehdi Dehghani and Ahmad Shirzad",
"submitter": "Mehdi Dehghani",
"url": "https://arxiv.org/abs/1010.2133"
}
|
1010.2174
|
# Near-IR H2 Emission of Protostars: Probing Circumstellar Environments111The
data presented herein were obtained at the W.M. Keck Observatory from
telescope time allocated to the National Aeronautics and Space Administration
through the agency’s scientific partnership with the California Institute of
Technology and the University of California. The Observatory was made possible
by the generous financial support of the W.M. Keck Foundation.
Thomas P. Greene NASA Ames Research Center, M.S. 245-6, Moffett Field, CA
94035 tom.greene@nasa.gov Mary Barsony22affiliation: Space Science Institute,
4750 Walnut Street, Suite 205, Boulder, CO 80301 Department of Physics and
Astronomy, San Francisco State University, 1600 Holloway Drive, San Francisco,
CA 94132 mbarsony@SpaceScience.org David A. Weintraub Department of Physics
and Astronomy, Vanderbilt University, Nashville, TN 37235
david.a.weintraub@vanderbilt.edu
###### Abstract
We present new observations of near-infrared molecular hydrogen (H2) line
emission in a sample of 18 Class I and flat-spectrum low mass protostars,
primarily in the Tau-Aur and $\rho$ Oph dark clouds. The line emission is
extended by up to several arcseconds (several hundred AU) for most objects,
and there is little night-to-night variation in line strength coincident with
the continuum point source. Flux ratios of H2 $v=2-1$ $S(1)$ and $v=1-0$
$S(1)$ lines are consistent with this emission arising in jets or winds in
many objects. However, most objects have only small offsets (under 10 km s-1)
between their H2 and photospheric radial velocities. No objects have line
ratios which are clearly caused solely by UV excitation, but the H2 emission
of several objects may be caused by UV or X-ray excitation in the presence of
circumstellar dust. There are several objects in the sample whose observed
velocities and line fluxes suggest quiescent, non-mechanical origins for their
molecular hydrogen emissions. Overall we find the H2 emission properties of
these protostars to be similar to the T Tauri stars studied in previous
surveys.
ISM: jets and outflows — stars: pre-main-sequence, formation — infrared: stars
— techniques: spectroscopic
††slugcomment: Accepted by ApJ on 8 October 2010
## 1 Introduction
Embedded low-mass protostars have been identified from their infrared (IR)
energy distributions for over two decades, but their high extinctions and
relatively small sizes make it very difficult to observe how radiation from
the central protostars interacts with their inner, pre-planetary circumstellar
disks. Recent high sensitivity, high resolution spectroscopic surveys have
revealed the detailed stellar properties of significant numbers of these
objects (Doppmann et al., 2005; White & Hillenbrand, 2004; Greene & Lada,
2002), but less is known about the physical conditions in their disks,
especially of the gaseous component, or about their inner disks where planets
might form and migrate. Here, we explore the gaseous component of the
innermost regions of self-embedded (Class I and flat-spectrum; hereafter FS)
protostars via high-resolution near-infrared (NIR) spectroscopy of the most
abundant molecule in proto-planetary disks, molecular hydrogen (H2). Near-
infrared H2 transitions are used, since the high extinctions to these sources
preclude UV spectroscopy.
The NIR ro-vibrational lines of H2 are good tracers of physical conditions in
inner circumstellar disks and winds close to protostars. There are several
different mechanisms that might be responsible for the production of NIR H2
emission lines in late-stage protostars. These include: i) shock heating of
the ambient medium by winds or jets, ii) X-ray heating, or iii) UV-heating,
level pumping and fluorescence.
Protostars must accrete mass at mean rates of $\sim 10^{-6}-10^{-5}M_{\sun}$
yr-1 to assemble themselves on time scales of several $10^{5}$ yr, and these
high rates produce significant UV flux when the accreting matter impacts the
stellar surface (e.g., Gullbring et al., 2000). Protostars and T Tauri stars
are also known to be strong and variable sources of X-ray emission (e.g.,
Imanishi et al., 2001, 2003; Güdel et al., 2007; Flaccomio et al., 2009). Both
UV and X-rays can excite vibrational states of H2, producing NIR line emission
(e.g., Gredel & Dalgarno, 1995; Nomura et al., 2007). Many protostars and T
Tauri stars also shed mass in jets that drive molecular outflows, and these
jets also frequently excite NIR vibrational H2 line emission (Zinnecker et
al., 1998).
The most commonly encountered excitation mechanism for NIR H2 emission
associated with protostars to date has been shock-excitation222See
http://www.jach.hawaii.edu/UKIRT/MHCat/ for an up-to-date listing of
“Molecular Hydrogen Emission-Line Objects (MHOs) in Outflows from Young
Stars.” (e.g., Davis et al., 2010), consistent with many self-embedded
protostars being associated with large-scale molecular outflows (e.g.,
Moriarty-Schieven et al., 1992). These outflows are understood to be driven by
powerful stellar winds (Masson & Chernin, 1993). Such winds are often detected
as blue-shifted absorption components in forbidden emission lines or in the
HeI 1.0830 $\mu$m line associated with protostars and young T Tauri stars
(e.g., Edwards et al., 1993; Kwan et al., 2007). Winds are also believed to
cause the shocked IR H2 emission seen in jets from heavily extinguished young
stellar objects. Such stellar winds are inferred to be driven by mass
accretion onto young stars. Jets detected in NIR H2 lines frequently are
displaced in space and in radial velocity from photospheric absorption
features and usually have relatively large linewidths of a couple dozen km s-1
or more. For example, in a recent VLT/ISAAC pilot study of the H2 1$-$0 S(1)
emission from embedded Class I sources, mean intensity-weighted velocities
were blue-shifted by -90 to -10 km s-1 and velocity widths of the lines varied
from $\sim$ 45 km s-1 to $\sim$ 80 km s-1 (Chrysostomou et al., 2008).
X-rays and UV may also excite molecular hydrogen emission in the inner
circumstellar environments of protostars. Both of these processes offer a
radiative alternative to the mechanical excitation excitation of the NIR H2
emission. If dominant, these excitation mechanisms may produce emission lines
with lower velocity widths or offsets than if the H2 were mechanically excited
in shocks or jets. We now consider the mechanisms, observational evidence, and
implications for these processes in protostellar environments.
It is now well established that Class I/FS protostars are copious X-ray
emitters (10${}^{29}\leq L_{x}\leq 10^{31}$ erg s-1, $\sim$100$-$1000 times
more X-ray luminous than main-sequence stars), and emit higher energy X-rays
(4-6 keV vs. 1-2 keV) than their older, T Tauri cousins (e.g., Casanova et
al., 1995; Grosso et al., 1997; Imanishi et al., 2001; Feigelson et al.,
2007). One would, therefore, expect significant X-ray heating of the
disk/inner envelope material surrounding these central objects (e.g.,
Meijerink et al., 2009, 2007). X-rays from young stellar objects (YSOs) can
penetrate disk atmospheres to fairly large surface densities and can ionize
circumstellar gas at a level greater than Galactic cosmic rays out to large
distances ($\sim$104 AU; Glassgold et al., 2005).
The mechanism of H2 excitation by X-rays requires impacts from energetic
electrons. X-rays impact hydrogen molecules, ejecting electrons. These high
energy electrons subsequently collide with and ionize or dissociate ambient
gas, losing kinetic energy in the process. Some of electrons will eventually
have energies appropriate to excite ambient H2 molecules into excited states
instead of dissociating them completely (e.g., Gredel & Dalgarno, 1995; Tine
et al., 1997; Maloney et al., 1996). Direct evidence for X-ray heating of gas
other than H2 in disks has been found for several Class I sources in which the
6.4 keV line from neutral iron has been detected (Giardino et al., 2007, and
references therein).
Glassgold and co-workers especially have emphasized the importance of X-ray
heating of disk atmospheres out to relatively large distances from the central
source (Glassgold et al., 2007, 2004). Such heating can extend to large
distances because of disk flaring, first proposed to account for the spectral
energy distributions (SEDs) produced by dust (Kenyon & Hartmann, 1987; Chiang
& Goldreich, 1997). More recent disk modeling, especially studies involving
disk chemistry, routinely use vertically stratified models, with different gas
and dust scale heights (d’Alessio et al., 1999; Aikawa et al., 2002; Gorti &
Hollenbach, 2008; Lacy et al., 2010). In X-ray heated disk models, there is a
low column density (N${}_{H}\sim$1020 cm-2) surface layer of hot (T$\sim$4000
K) gas that extends to $\gtrsim$10 AU radius (Najita et al., 2009).
In the context of disk heating by X-rays, it must be noted that current disk
models assume irradiation by a central source with X-ray spectra typical of T
Tauri stars: with plasma having $kT_{X}=$1 keV and a low-energy cut-off of 100
eV (e.g., Glassgold et al., 2007). However, Class I and FS sources are known
to have harder X-ray spectra, with 4 keV $\leq kT_{x}\leq$6 keV being typical
(e.g., Imanishi et al., 2001). Unsurprisingly, the column densities of
hydrogen gas inferred towards Class I/FS sources from X-ray observations are
$\sim\ N_{H}\ =\ 1-5\times 10^{22}$ cm-2, 1-2 orders of magnitude greater than
inferred for T Tauri stars. Finally, the quantity, Lx/Lbol, is systematically
smaller for Class I/FS objects relative to T Tauri stars, consistent with the
interpretation of higher accretion rates in these systems. The higher
accretion rates create higher optical depths in the X-ray absorbing gas,
obscuring the lower energy X-rays and producing relatively lower Lx/Lbol
ratios. Higher accretion rates would also lead to higher heating rates of the
disk gas in self-embedded protostars than in classical T Tauri star (CTTS)
disks (e.g., d’Alessio et al., 2004).
The possibility of UV excitation of the NIR H2 lines also exists for Class
I/FS protostars, given their consistently higher accretion rates relative to T
Tauri stars. The excess UV continuum emission observed in CTTSs has been
modeled as being produced by the impact of accretion columns onto the pre-
main-sequence stellar surface (e.g., Gullbring et al., 2000). This UV
continuum excess could potentially excite molecular hydrogen, producing
emission lines in the near-infrared. Strong Lyman-$\alpha$ emission from the
central object can irradiate the disk’s surface, and, if H2 is present at
$\sim$ 2000K, can excite the H2 into electronic states which produce a rich UV
emission-line spectrum as observed in some T Tauri stars (Herczeg et al.,
2006). Finally, if the stellar EUV flux is sufficiently strong, it can ionize
hydrogen, which produces high temperatures (T$\approx$ 104K) and small mean
molecular weights at the disk surface. Outside some critical radius, the gas
becomes unbound, and a slow, v$\approx$10 km s-1, photoevaporative wind forms
(Alexander et al., 2006a, b; Gorti et al., 2009; Woitke et al., 2009).
Like molecular hydrogen, [NeII] emission can also arise from high energy
photons in dense circumstellar disks or outflows. [NeII] line emission at
12.81 $\mu$m and [NeIII] emission at 15.5 $\mu$m was predicted to be
detectable in the case of X-ray heated disk gas. The 12.81 $\mu$m [NeII] line
has subsequently been detected in a number of young stars with the Spitzer
Space Telescope (Glassgold et al., 2007; Pascucci et al., 2007; Lahuis et al.,
2007; Flaccomio et al., 2009; Najita et al., 2010) and has been studied at
high spectral resolution from the ground in three objects (Herczeg et al.,
2007; Najita et al., 2009). Although a disk origin has been postulated or
confirmed for the [NeII] emission observed in the objects studied at high
spectral resolution, there are also sources in which [NeII] is detected in
outflows close to the sources (e.g., Neufeld et al., 2006; van Boekel et al.,
2009).
In general, gas motions in the inner circumstellar environments (within $\sim$
100 AU) of protostars have not been studied well except for a very few cases
of velocity resolved NIR CO observations, and this has limited our
understanding of how they accrete matter, form winds, shed angular momentum,
and disperse their natal circumstellar envelopes. We seek to understand these
processes as well as the UV and X-ray radiation environments of their
circumstellar disks by embarking on a new study of protostars’ NIR H2 line
strengths, morphologies, and velocities.
H2 NIR ro-vibrational line ratios yield excitation temperatures and also
provide clues to excitation mechanisms. The intensity ratio of $S(1)$ lines in
the $v\ =\ 2\rightarrow 1$ and $v\ =\ 1\rightarrow 0$ transitions is often
used as a diagnostic. This ratio has a value of 0.13 in shocked gas at 2000 K
and a value of 0.54 for UV pumping. For X-ray excitation, this ratio is
predicted to be 0.06 in gas of low fractional ionization (10-4) and 0.54 in
gas of high fractional ionization (10-2) (Gredel & Dalgarno, 1995). Any
observed variability in these emission lines can also provide clues to the
nature of their excitation and the location of the excited and emitting gas.
Several recent studies have made good progress in diagnosing the nature of H2
excitation in the inner circumstellar disks of CTTSs using observations of
their NIR ro-vibrational emission lines. A growing body of work shows that the
NIR H2 line radial velocities and linewidths of many CTTS and Herbig AeBe
stars are consistent with UV or X-ray excitation in a circumstellar disk
(Weintraub et al., 2000; Bary et al., 2002, 2003; Carmona et al., 2007; Bary
et al., 2008). By contrast, in a recently completed NIR adaptive optics (AO)
integral field spectroscopic study of several CTTS, the H2 emissions were most
consistent with shocks arising in winds (Beck et al., 2008).
Despite this progress in understanding H2 emission in CTTSs, little has been
done to date in probing the nature of such emission in more embedded Class I
and flat–spectrum protostars. Greene & Lada (1996) reported that these
protostars were significantly more likely to have NIR H2 emission than CTTSs,
and Doppmann et al. (2005) found that 23 of 52 observed protostars showed NIR
H2 emission. The most embedded protostars also have significant envelopes that
may have enough column density to generate observable H2 emission if the
hydrogen molecules there receive sufficient radiative or mechanical energy. If
this envelope source existed, it would be a new source of emission not present
in CTTSs. Unfortunately, earlier studies did not have adequate spectral
resolution or spectral range to observe multiple NIR ro–vibrational lines
simultaneously with high spectral resolution and good signal-to-noise: all
necessary ingredients for measuring line velocities and line ratios in order
to diagnose excitation.
An early motivation for a new observational study was the identification of
outflow drivers amongst late-stage protostars via detection of NIR H2 emission
at high spectral resolution observed directly towards the putative powering
source (Barsony, 2005). However, careful examination of the H2 $v=1-0$ $S(0)$
line profiles (at 2.2235 $\mu$m) observed at R$\sim$17,000 in the work of
Doppmann et al. (2005), showed that a substantial number of sources exhibited
relatively narrow linewidths ($\Delta v\leq$ 15 km s-1). Furthermore, in many
cases, the H2 emission line centers were not significantly displaced from the
central object’s radial velocity. Taken together, these two observations call
into question an outflow origin for the observed NIR H2 emission in some
sources.
We have conducted a new study of Class I and flat-spectrum protostars with
data sufficient for diagnosing the natures of their H2 emission in their inner
circumstellar disks and envelopes. We report on the sample and observations in
§2 and present our analysis of these data in §3. We discuss our results in §4
and summarize our work in §5.
## 2 Observations and Data Reduction
High resolution NIR spectra of 18 Class I and flat-spectrum protostars
previously searched for H2 $v=1-0$ $S(0)$ emission by Doppmann et al. (2005)
were re-observed with the Keck II telescope on Mauna Kea, Hawaii using its
NIRSPEC multi-order cryogenic echelle facility spectrograph (McLean et al.,
1998). This included 17 objects found to exhibit H2 $v=1-0$ $S(0)$ emission in
this previous study, and one (03260+311A) without. All new spectra were
acquired on 2007 June 24 and 25 UT ($\rho$ Oph and Ser objects) and 2008
January 24 and 25 (Tau-Aur and Per objects). The late type (K1 – M2.5) dwarfs
HD 20165, HD 28343, and HD 285968 with precision velocities measured by
Nidever et al. (2002) were observed on 2008 January 24 UT to serve as radial
velocity references.
Spectra were acquired with a 0$\farcs$58 (4-pixel) wide slit, providing
spectroscopic resolution $R\equiv\lambda/\delta\lambda$ = 18,000 (16.7 km
s-1). The plate scale was 0$\farcs$20 pixel-1 along the 12$\arcsec$ slit
length, and the seeing was typically 0$\farcs$5–0$\farcs$6\. The NIRSPEC
gratings were oriented to observe the 2.1218 $\mu$m H2 $v=1-0$ $S(1)$, 2.2477
$\mu$m H2 $v=2-1$ $S(1)$, and 2.386 $\mu$m H2 $v=3-2$ $S(1)$ lines on the
instrument’s 1024 $\times$ 1024 pixel InSb detector array in a single
exposure. The 2.2233 $\mu$m H2 $v=1-0$ $S(0)$ line previously observed by
Doppmann et al. (2005) was also captured in this grating setting. The
NIRSPEC-7 blocking filter was used to image these orders on the detector.
NIRSPEC was configured to acquire simultaneously multiple cross-dispersed
echelle orders 32–37 (2.05–2.40 $\mu$m, non-continuous) for all objects. Each
order had an observed spectral range $\Delta\lambda\simeq\lambda/67$ ($\Delta
v\simeq$ 4450 km s-1).
The slit was held physically stationary during the exposures and thus rotated
on the sky as the non-equatorially-mounted telescope tracked when observing.
Data were acquired in pairs of exposures of durations from 180–600 s each,
with the telescope nodded 3$\arcsec$ or 6$\arcsec$ along the slit between
frames so that object spectra were acquired in all exposures. Most of the
targets were observed twice on consecutive nights. The observation dates,
total integration times, slit angles and coordinates of all Class I and flat-
spectrum objects are given in Table 1. The early-type (B9–A0) dwarfs HD 28354,
HR 6070, HR 5993, and HD 168966 were observed for telluric correction of the
target spectra. The telescope was automatically guided with frequent images
from the NIRSPEC internal “SCAM” IR camera during all exposures of more than
several seconds duration. Spectra of the internal NIRSPEC continuum lamp were
taken for flat fields, and exposures of the Ar, Ne, Kr, and Xe lamps were used
for wavelength calibrations.
All data were reduced with IRAF. First, object and sky frames were differenced
and then divided by normalized flat fields. Next, bad pixels were fixed via
interpolation, and spectra were extracted with the APALL task. Spectra were
wavelength calibrated using low-order fits to lines in the arc lamp exposures,
and spectra at each slit position of each object were co-added. Instrumental
and atmospheric features were removed by dividing wavelength-calibrated object
spectra by spectra of early-type stars observed at similar airmass at each
slit position. Final spectra were produced for each night by combining the
spectra of both slit positions for each object and then multiplying them by
spectra of 10,000 K blackbodies to rectify the spectral shapes induced when
dividing by the telluric stars with that effective temperature. Spectra
acquired on different nights were not combined in any way.
## 3 Analysis and Results
The spectra of all objects in Table 1 were analyzed by measuring their H2
$v=1-0$ $S(1)$, $v=2-1$ $S(1)$, and $v=1-0$ $S(0)$ line fluxes, H2 line radial
velocities, and the radial velocities of photospheric absorption lines. The
2.386 $\mu$m H2 $v=3-2$ $S(1)$ line is in a region of poor atmospheric
transmission and was not significantly detected in any object. The values of
the H2 line properties and their night-to-night variations were analyzed for
physical insights into their excitation as described in this section.
Spectra of the H2 $v=1-0$ $S(1)$ region are shown for all objects in Figure 1;
the first epoch (2007 Jun 24 and 2008 Jan 24) is shown in the left panel and
the second (2007 Jun 25 and 2008 Jan 25) in the right. Of the 18 objects, 14
were observed in both epochs. When present, the $v=1-0$ $S(1)$ line was the
strongest H2 feature observed in all objects. All emission spectra shown in
the figures and all derived strength and velocity values presented in Table 2
are for the H2 emission that was spatially coincident with each object’s
continuum source, typically limited by the $0\farcs 6$ seeing, corresponding
to a spatial extent of $\sim 100$ AU for most sources.
### 3.1 H2 Emission Morphologies
The spatial extent of the H2 emission along the spectrograph slit was measured
for each object by examining the differenced and flat-fielded spectral images
that were created prior to spectral extraction. The non-rotating spectrograph
slit was projected onto the sky at different position angles each night, so
the two observational epochs sample the spatial extent of any extended
emission differently (see position angles in Table 1).
The angular extent of the H2 emission along the length of the slit is reported
in Table 2) for each observation. Ten of the 18 protostars showed H2 emission
that was over $1\arcsec$ in spatial extent in at least one observation (at
least one position angle). Of the 14 objects observed in two epochs, 6 had H2
$v=1-0$ $S(1)$ emission extended by $\sim 1\arcsec$ or less on both nights.
Therefore the molecular hydrogen emission of these 6 objects is contained
within about 70 AU ($\sim 0\farcs 5$) of their central stars. The highest H2
surface brightness was generally found to be coincident with each object’s
unresolved point source, but in some cases the extended emission may have more
integrated flux and luminosity than the point source component. Beck et al.
(2008) found this to be true in their AO integral field study of several CTTSs
that sampled their immediate circumstellar environments completely instead of
evaluating just two slit position angles as done in this study.
Objects with significantly different spatial extents between epochs (position
angles) may have their H2 emission confined to asymmetrical structures like
jets. 03260+3111B, 04158+2805, and 04264+2433 are examples of objects with
these potentially jet-like morphologies. 03260+3111A shows significantly
extended emission, but this is spatially displaced from the stellar continuum
which has no coincident H2 emission.
### 3.2 Variability
We found H2 emission in all of the 17 objects also found to have H2 emission
at earlier epochs by Doppmann et al. (2005). The one source without emission
(03260+3111A) in that earlier study also did not show H2 emission in our new
observations. This indicates that these these protostars do not terminate or
initiate their H2 emissions on time scales as short as 5 – 10 years,
suggesting that their H2 lines are emitted over physically large regions or
else are excited by processes with relatively stable fluxes.
Fourteen of the 18 objects were observed twice, with the two observations
spaced approximately 24 hr apart. The position angles of both observations
were sometimes similar (within $\sim 10\deg$), but often they varied by
$60\deg$ or more (Table 1). Therefore the different slit position angles must
be considered when comparing H2 line fluxes measured on the different dates to
determine whether any differences are due to true temporal variation or the
rotation of different spatial features (i.e., jets) onto or off of the slit.
However, the measured equivalent H2 emission widths and line velocities were
similar for both observations of each object (see Table 2), so there appears
to be little temporal variation on $\sim 1$ day time scales for the emission
that is spatially confined to be coincident with the protostar continuum
source confined within the slit width.
### 3.3 Radial Velocities and Line Widths
Stellar photospheric radial velocities were computed from the continuum
absorption lines of all observed objects. First, the object spectra were
cross-correlated (using fxcor in IRAF) with spectra of the radial velocity
standards, always HD 28343 (K7 V) and sometimes HD 20165 (K1 V) or HD 285968
(M2.5 V), using up to 4 spectral orders that contained photospheric lines but
no emission lines. Heliocentric $V_{LSR}$ radial velocities were computed for
each object by summing the radial velocity shift measured with the cross-
correlations, the mean measured radial velocities of the 3 standards (Nidever
et al., 2002), and the $V_{LSR}$ corrections for the objects and standards at
their time of observation (from the IRAF rvcorrect task). The $V_{LSR}$ radial
velocities computed in this way generally agreed well with those of the 10
objects also reported by Covey et al. (2006). We observed these 10 objects a
total of 18 times, and we measured the offset from the Covey et al. (2006)
velocities to be $2.2\pm 3.5$ km s-1. This offset is consistent with zero, and
the standard deviation is not surprising given the 17 km s-1 spectral
resolution and the generally heavily veiled spectra.
Radial velocities of H2 lines were computed by measuring the central
wavelengths of Gaussian profiles fit to the lines using the IRAF splot task
and then converting these values to velocities. These observed radial
velocities were converted to $V_{LSR}$ values by adding the $V_{LSR}$
correction offset computed for each object as described above. The velocities
of the H2 lines relative to photospheric lines of each object were computed by
differencing these two $V_{LSR}$ radial velocities, and the results are shown
in Table 2 and Figure 2. FWHM velocity widths of the $v=1-0$ $S(1)$ lines were
computed by subtracting the instrumental line width of 17 km s-1 in quadrature
from the FWHM values of the Gaussian fits. These resultant line widths are
also reported in Table 2, and their histogram is shown in Figure 3. Objects
observed twice on successive nights had similar radial and FWHM velocities
(Table 2), so these values were averaged to reduce noise in Figures 2 and 3.
The other H2 lines generally had similar FWHM values, but the $v=1-0$ $S(1)$
lines had the highest signal-to-noise, so only those values are presented. We
estimate that all reported velocities have uncertainties of a few km s-1.
### 3.4 H2 Line Fluxes and Ratios
Line luminosities were estimated by scaling the relative fluxes measured in
each H2 line to the spatially coincident 2.2 $\mu$m continuum and multiplying
this by the absolute 2.2 $\mu$m continuum flux estimated from each object’s
2MASS K-band magnitude after correcting for extinction. Extinctions were
calculated by de-reddening each object’s $JHK$ 2MASS magnitudes to the CTTS
locus (Meyer et al., 1997). Extinctions were estimated at each H2 line
wavelength using $A_{v}=9.09[(J-H)-(J-H)_{0}]$, $A_{k}=0.09A_{v}$, and
$A_{\lambda}\propto\lambda^{-1.9}$. These values were computed by and derived
from Cohen et al. (1981) for the CIT photometric system, which is essentially
identical to that of 2MASS (Carpenter, 2001).
Distances were assumed to be 140 pc for Tau-Aur (Kenyon et al., 1994), 140 pc
for $\rho$ Oph (Mamajek, 2008), 260 pc for SVS 2 in Serpens (Straižys et al.,
1996), and 320 pc for the 03260+3111 objects in Perseus (Herbig, 1998). H2
line equivalent widths and luminosities are presented in Table 2, and a
histogram of luminosities of the H2 $v=1-0$ $S(1)$ line is shown in Figure 4.
A histogram of H2 $1-0/2-1$ $S(1)$ line ratios is presented in Figure 5. As
done in previous figures, values derived from observations of objects acquired
on successive nights are averaged in these figures as well. Note that the H2
line ratio value plotted in Figure 5 is the inverse of the values presented in
Table 2 (column 8).
### 3.5 Correlations
We examined whether correlations exist between measured H2 line properties and
other protostellar activity indicators in an attempt to isolate the origins of
the H2 line emissions.
First we computed the correlations between the H2 line properties measured in
this new survey. The FWHM velocity widths and the $1-0/2-1$ $S(1)$ line ratios
in Table 2 have a correlation coefficient of 0.61, which improves to 0.71 when
both observations of a single object are averaged into single points. This
value drops to only 0.17 (individual or mean values) if the observations of
03260+3111B are excluded; its high FWHM velocity and relatively high $1-0/2-1$
$S(1)$ line ratio drives this correlation. Thus the object sample as a whole
does not show a good correlation between its H2 FWHM velocities and $1-0/2-1$
$S(1)$ line ratios. We then examined correlations between H2 FWHM velocities
and velocity offsets between H2 and photospheric lines. These values had
correlation coefficients of -0.43 and -0.47 for individual and averaged
values, respectively. The square of the correlation coefficient is below 0.25
in both cases, indicating that less than 25% of the variance of the two
quantities are in common for this sample of protostars, a poor correlation.
Next we correlated the protostars’ H2 line properties with other physical
characteristics measured in other studies. Twelve of the 18 objects have
measured X-ray luminosities or upper limits (Güdel et al., 2007; Flaccomio et
al., 2009), and the correlation coefficient between the logarithms of their H2
$1-0$ $S(1)$ line luminosities and the logarithms of their X-ray luminosities
is -0.20, suggesting a very weak or nonexistent inverse correlation between
these properties. Finally, we re-analyzed the spectra of Doppmann et al.
(2005) and used their equivalent width measurements of H2 $1-0$ $S(0)$ and HI
Br $\gamma$ emission lines to evaluate the correlation of these properties in
that somewhat earlier epoch. We found that these values had a correlation
coefficient of 0.20, indicating another poor correlation. In summary, we find
little correlation among the NIR H2 line emission properties or between these
properties and other young stellar activity indicators.
## 4 Discussion
The different radiative and collisional excitation mechanisms of H2 are well
matched to the radiative and mass flux processes in the environments of
protostars and T Tauri stars. We now interpret the results of the preceding
analysis in terms of several of these possible processes in order to constrain
the H2 excitation mechanisms of the sample and to understand better the
circumstellar environments of these protostars.
### 4.1 Emission Morphologies, Variability, and Velocities
In addition to their compact H2 line emissions, all but 4 of the 18 objects
also showed $v=1-0$ $S(1)$ molecular hydrogen emission extended by $\sim
1\arcsec$ or more along the slit (see Table 2), corresponding to 70 AU ($\sim
0\farcs 5$) or more projected radial distance. It is unlikely that UV
radiation could travel that far from the central protostars without
significant attenuation by gaseous and dusty envelopes, so it is likely that
this extended emission is excited by either stellar winds or high energy
X-rays. The expected small size of the UV emission region on the protostellar
photosphere also suggests that UV may not be a good candidate for exciting the
observed steady and long-lived H2 line emissions. Two objects, GY 21 and IRS
43, have extended $1-0$ $S(1)$ emission with broad line widths, FWHM $\gtrsim$
40 km s-1, about twice that of the compact emission spatially coincident with
their stellar continua. The extended emissions of these 2 objects are good
candidates for excitation in shocks caused by stellar winds.
The relatively stable values of the point source H2 emission over 1 day and
several-year time scales (see §3.2 and Table 1) also provide clues to the
nature of these emissions. Numerous Class I and FS protostars have been
observed to undergo rapid (several hr), large amplitude X-ray emission
variations (e.g., Imanishi et al., 2001, 2003; Güdel et al., 2007; Flaccomio
et al., 2009). This X-ray flaring of several protostars (e.g., IRS 43) has
also been observed to appear or disappear in data taken $\sim$ 5 – 10 years
apart. If these X-ray flares were exciting H2 close to the stars, then it is
likely that we would see significant night-to-night or year-to year variations
in their point-source near-IR line fluxes, but this is not seen in our data.
The very stable observed molecular hydrogen emission is more consistent with
mechanical (wind or jet) excitation as well as excitation by steady, non-
flaring X-ray emission from the protostars.
The preceding analysis of the velocity widths and velocity shifts of the H2
line emission in §3.3 also provides clues to the nature of its excitation.
Collisional excitations in jets or winds are likely to result in H2 emission
line radial velocities displaced from photospheric absorption lines by over 10
km s-1, exhibition of broad line wings, or large full-width half maximum
velocities of several 10 km s-1 or more (Maloney et al., 1996; Montmerle et
al., 2000; Nomura et al., 2007; Beck et al., 2008). Jets are also often
significantly collimated and spatially extended, making them easy to identify
in one or two dimensional spectral images (e.g., Schwartz & Greene, 2003).
The significant spatial extension of the H2 line emission of many objects is
consistent with collisional excitation in winds or by jets. However, only 5 of
13 protostars show radial velocity offsets between their H2 lines and
photospheric absorption lines with absolute value of greater than 4 km s-1
(see Fig. 2). This value is similar to the 3.5 km s-1 uncertainty we measured
for radial velocity standards (see §3.3), so we do not consider velocity
offsets less than 4 km s-1 to be significant. Only 3 protostars have velocity
offsets of at least 10 km s-1, significant at about the 3-$\sigma$ confidence
level or greater. This evidence suggests that collisional excitation in jets
is unlikely to be the molecular hydrogen excitation mechanism in most objects
(except the 3 with significant velocity differences).
However, the on-source H2 $v=1-0$ $S(1)$ line FWHM line widths of all 17
protostars with this on-source feature are broader than 10 km s-1 (see Fig.
3). Six of the emitting objects (35%) exhibit FWHM greater that 20 km s-1. All
6 of these objects also show spatially extended H2 emission many tens of AU
from their central stars; this combination of factors makes them good
candidates for collisional excitation in jets or winds. Seven of the emitting
objects (41%) have FWHM line widths below 16 km s-1. This is similar to the 9
to 14 km s-1 line widths found by Bary et al. (2003) and Bary et al. (2008)
for 8 of 10 T Tauri stars found to have H2 emission, which they interpreted as
evidence for quiescent emission in circumstellar disks. We conclude that the
molecular hydrogen emission line velocities and FWHM values of at least 3 to 6
of the 17 emitting objects are consistent with collisional excitation in jets,
and at least 7 or 8 objects have H2 velocity parameters consistent with
quiescent (non-collisional) excitation.
### 4.2 Emission Line Strengths and Ratios
NIR vibrational H2 line ratios are also sensitive to the gas excitation levels
and excitation mechanisms. Gredel & Dalgarno (1995) show that the ratios of
the H2 $v=1-0$ $S(1)$ to $v=2-1$ $S(1)$ lines are relatively sensitive to
excitation mechanisms. They compute the ratios of these 2 lines to be 1.9 for
UV excitation, 7.7 for shocked gas at $T=2000$ K, and 16.7 for X-ray
excitation of low ionization H2. Black & van Dishoeck (1987) also found
similar differences between UV and shock excitation of H2. However, there are
limits to the usefulness of these ratios as diagnostics of excitation in
circumstellar disks. In practice it is difficult to distinguish between
shocked and X-ray excited H2 emission from examining only a few NIR lines.
Collisions will thermalize the excitation levels of H2 in a sufficiently dense
gas, so $v=1-0$ $S(1)$ to $v=2-1S(1)$ line ratios indicative of cold-to-warm
gas in equilibrium are not always useful for distinguishing between excitation
mechanisms (Gredel & Dalgarno, 1995; Maloney et al., 1996; Tine et al., 1997;
Nomura et al., 2007; Beck et al., 2008).
However, these line ratios may be useful for diagnosing excitation processes
in extreme cases. Black & van Dishoeck (1987) note that fluorescent UV
excitation of H2 produces significant population of vibrational levels $v\geq
2$ and therefore strong emission in the $v=2-1$ $S(1)$ line when not
thermalized in a high density environment. This level population can be
characterized by a temperature of $T_{\rm vib}\approx 6000-9000$ K, much
larger than the $T_{\rm vib}=T\simeq 2000$ K characteristic of shock
excitation. Therefore any objects with observed ratios of H2 $v=1-0$ $S(1)$ to
$v=2-1$ $S(1)\simeq 2$ ($2-1/1-0\simeq 0.5$) may be exhibiting UV-excited H2
emission. However, there are no objects in our sample with H2 $v=2-1/1-0$
$S(1)>0.25$, so we do not have any good candidates for purely UV excitation in
a dust-free environment as modeled by Black & van Dishoeck (1987).
The presence of dust grains can significantly alter these ratios and
complicate their interpretation. Nomura et al. (2007) have computed the
expected NIR H2 line fluxes for conditions in circumstellar disks around young
stars, modeling X-ray and UV heating in the presence of both gas and dust and
accounting for thermalization at high gas densities. They find that the H2
$v=2-1$ $S(1)$ to $v=1-0$ $S(1)$ line ratio is greatly impacted by the
presence of dust grains of different sizes (see their Figure 17). For UV
excited emission, Nomura et al. (2007) find that this line ratio is $\simeq
0.025$ for a power law dust grain distribution with a maximum size of 10
$\mu$m - 1 mm, roughly consistent with that expected for a protostar’s
circumstellar disk. They find that the grains must be much larger ($\sim 10$
cm or more) for this ratio to approach the dust-free value of 0.5 computed by
Gredel & Dalgarno (1995). This line ratio is much less sensitive to grain size
in the case of X-ray excitation; Nomura et al. (2007) find that this value is
close to the Gredel & Dalgarno (1995) value of 0.06 for a maximum grain size
of 10 $\mu$m - 10 cm. Thus they find that the H2 $v=2-1$ $S(1)$ to $v=1-0$
$S(1)$ line ratio differs by only about a factor of 2 for a circumstellar disk
with a power law grain size distribution with a maximum size 10 $\mu$m - 1 mm.
Nomura et al. (2007) do not consider collisional excitation by winds or jets,
but collisional excitation may produce a fairly wide range of gas temperatures
and H2 line ratios as discussed previously.
If excited by collisions in shocks, ratios of the $v=1-0$ $S(1)$ to $v=1-0$
$S(0)$ emission lines can be used to estimate ortho:para ratios of molecular
hydrogen and to assess the thermal history of the emitting gas. The values of
ortho:para ratios were modeled for C- and J-type shocks by Wilgenbus et al.
(2000); see also their summary of previous work. Kristensen et al. (2007) and
Harrison et al. (1998) showed that the $v=1-0$ $S(1)$ to $v=1-0$ $S(0)$
emission line ratio directly yields the molecular hydrogen ortho:para ratio
with little sensitivity to H2 rotation temperature. Using Eq. 5 of Kristensen
et al. (2007) and assuming an H2 rotation temperature of 3500 K, we find that
the mean ortho:para ratio for our object sample is $<o/p>=3.2\pm 0.8$.
Hydrogen atom exchanges in shocks set the high temperature limit to o/p $\leq
3$ (e.g., see Wilgenbus et al., 2000). Thus it appears that not all objects in
our sample have molecular hydrogen emission consistent with production in
shocks since a number of objects have o/p $>3$. Unfortunately we were unable
to use line ratios to diagnose the nature of the spatially extended H2
emissions seen in many objects (see Table 2 and §4.1). This emission was
generally much weaker than the point source emission, and it was not detected
significantly in any NIR H2 line except $v=1-0$ $S(1)$ for any object.
### 4.3 Molecular Hydrogen Excitation Mechanisms in Observed Protostars
Our sample has 5 objects with H2 $v=2-0$ $S(1)$ to $v=1-0$ $S(1)$ line ratios
in the 0.025 - 0.06 range, consistent with UV or X-ray excitation in the
presence of dust. Of these, 04264+2433, 04295+2251, 04365+2535, and IRS 43
have relatively low mean H2 $v=1-0$ $S(1)$ line widths, FWHM $\lesssim 15$ km
s-1. All but 04264+2433 have been detected in X-rays, so X-ray or UV
excitation may be possible for these protostars. WL 12 is the other protostar
with a line ratio in this range, and it has a broad FWHM $\simeq 30$ km s-1
and is associated with a molecular outflow (Bontemps et al., 1996). Therefore
its H2 may be collisionally excited. However, WL 12 has ortho:para ratios of
about 4.5, larger than the o/p $\leq 3$ limit that can be produced in C-shocks
or J-shocks. The objects 04158+2805, 04181+2654, and WL 6 also have estimated
o/p $\geq 3.5$, indicating non-shock excitation even if their H2 rotation
temperature is somewhat higher than the assumed 3500 K. Therefore the NIR H2
emission of these objects may not be excited in jets. The latter 3 objects
also have H2 radial velocity offset by less than 5 km s-1 from their
photospheric velocities. However, 04158+2805 and 04181+2654 have H2 FWHM line
widths $\geq$ 20 km s-1, clouding a non-mechanical interpretation of their
molecular hydrogen excitation.
Interestingly, there are several objects in the sample whose observed
velocities and line fluxes suggest quiescent, non-mechanical origins for their
molecular hydrogen emissions. 04361+2547, WL 6, and IRS 67 all have small H2
FWHM line widths, small H2 velocity offsets from photospheric velocities, and
small H2 emission spatial extents (See Table 2). Interestingly, at least one
measurement of the $v=1-0$ $S(1)$ to $v=2-1$ $S(1)$ line ratios is $\sim$0.07
for each object, similar to the value of 0.06 computed by Gredel & Dalgarno
(1995) and Nomura et al. (2007) for X-ray excitation of H2. Thus these objects
appear to be the best candidates for non-mechanical excitation of their
molecular hydrogen emissions.
We conclude this discussion by noting that several of the protostars have NIR
ro-vibrational emission properties consistent with collisional excitation, and
some others appear to be good candidates for X-ray and / or UV excitation.
This is similar to the results found by Beck et al. (2008) and Bary et al.
(2008) in their surveys of T Tauri stars. We also find no individual or set of
spectral features that are inconsistent with previous observations of NIR H2
emission in CTTSs. It appears that there is no single NIR line diagnostic that
can clearly identify the excitation mechanisms of H2 in the circumstellar
disks and environments of protostars, and correlations between diagnostics are
not strong (§3.5). However, the measures of emission line morphologies,
velocity widths, velocity shifts, and line ratios can constrain the various
emission mechanisms when interpreted within an appropriate theoretical model.
## 5 Summary
We present new observations of near-infrared H2 line emission in a sample of
18 Class I and flat-spectrum low mass protostars, primarily in the Tau-Aur and
$\rho$ Oph dark clouds. We reach the following conclusions from analyzing
these data:
1\. All 17 objects found to have NIR H2 ro-vibrational line emission spatially
conincident with their continuum sources in an earlier epoch were also found
to have this emission in this new study, 5 – 10 years later. There appears to
be little temporal variation of this emission on $\sim 1$ day time scales. Ten
of the 18 protostars showed H2 $v=1-0$ $S(1)$ line emission that was over
$1\arcsec$ in spatial extent in at least one observation (at least one
position angle).
2\. Nearly all of the protostars have H2 $v=1-0$ $S(1)$ line emission radial
velocities within 10 km s-1 of their stellar photospheric line velocities;
only 3 objects have H2 velocity offsets greater than or equal to 10 km s-1.
This evidence suggests that collisional excitation in jets is unlikely to be
the molecular hydrogen excitation mechanism in many objects.
3\. The H2 $v=1-0$ $S(1)$ line FWHM line widths of all 17 protostars with this
feature on-source are broader than 10 km s-1 (Fig. 3). Six of the emitting
objects (35%) exhibit FWHM greater that 20 km s-1, and these are good
candidates for collisional excitation in jets or winds. Seven of the emitting
objects (41%) have point source FWHM line widths below 16 km s-1. The
spatially extended H2 $v=1-0$ $S(1)$ line emission of two objects had line
widths FWHM $\gtrsim$ 40 km s-1, about twice that of their central point
source emission. This is consistent with collisional excitation by jets or
winds.
4\. The molecular hydrogen emission line velocities and FWHM values of at
least 3 to 6 of the 17 objects with on-source emission are consistent with
collisional excitation in jets. At least 7 or 8 objects have H2 velocity
parameters consistent with quiescent (non-collisional) excitation. There are
several objects whose small emission line widths, small H2 – photospheric
radial velocity differences, and small spatial extents are more consistent
with quiescent molecular hydrogen emission and not collisional excitation.
5\. Several of the protostars have H2 $v=2-0$ $S(1)$ to $v=1-0$ $S(1)$ line
ratios indicative of X-ray or UV excitation (in the presence of dust) and are
known X-ray emitters. 04361+2547, WL 6, and IRS 67 are the best examples of
such protostars. However, we see no rapid variation in the H2 $\Delta v=1$
$S(1)$ line fluxes on $\sim$24 hr time scales as might be expected from
excitation by X-ray flaring events.
6\. We find that the mean ortho:para ratio for our object sample is
$<o/p>=3.2\pm 0.8$. Hydrogen atom exchanges in shocks set the high temperature
limit to o/p $\leq 3$ (e.g., see Wilgenbus et al., 2000). Thus it appears that
not all objects in our sample have molecular hydrogen emission consistent with
production in shocks since a number of objects have o/p $>3$. 04158+2805,
04181+2654, WL 6, and WL 12 are all estimated to have ortho:para ratios
significantly higher than this value. However, WL 6 is the only protostar with
H2 line ratios and velocities also indicative of non-mechanical excitation.
We thank D. Hollenbach and U. Gorti for helpful discussions of our data and
its interpretation via theoretical models. We also thank G. Herczeg for
discussing pre-publication data and thank the anonymous referee for thoughtful
suggestions that improved this paper. The Keck Observatory Observing
Assistants H. Hershley and C. Parker are thankfully acknowledged for
assistance with the observations. The authors wish to recognize and
acknowledge the very significant cultural role and reverence that the summit
of Mauna Kea has always had within the indigenous Hawaiian community. We are
most fortunate to have the opportunity to conduct observations from this
mountain. TPG acknowledges support from NASA’s Origins of Solar Systems
program via WBS 811073.02.07.01.89. MB and TPG would like to acknowledge NASA
support via NExScI for travel expenses to the W.M. Keck Observatory for
acquiring the observations for this project. Facilities: Keck (NIRSpec), IRAF,
2MASS
## References
* Aikawa et al. (2002) Aikawa, Y., van Zadelhoff, G.J., van Dishoeck, E.F., & Herbst, E. 2002, A&A, 386, 622
* Alexander et al. (2006a) Alexander, R.D., Clarke, C.J., & Pringle, J.E. 2006a, MNRAS, 369, 216
* Alexander et al. (2006b) Alexander, R.D., Clarke, C.J., & Pringle, J.E. 2006b, MNRAS, 369, 229
* Bary et al. (2002) Bary, J. S., Weintraub, D. A., & Kastner, J. H. 2002, ApJ, 576, L73
* Bary et al. (2003) Bary, J. S., Weintraub, D. A., & Kastner, J. H. 2003, ApJ, 586, 1136
* Bary et al. (2008) Bary, J. S., Weintraub, D. A., Shukla, S., Leisenring, J.M., & Kastner, J. H. 2008, ApJ, 678, 1088
* Barsony (2005) Barsony, M. 2005, in “High-Resolution Infrared Spectroscopy in Astronomy,” Proc. of an ESO Workshop held at Garching, Germany, 18-21 November 2003, eds. Käufl, H.U., Siebenmorgen, R., & A. Moorwood, pp.125-130
* Beck et al. (2008) Beck, T. L., McGregor, P. J., Takami, M., & a, T.-S. 2008, ApJ, 676, 472
* Black & van Dishoeck (1987) Black, J. H., & van Dishoeck, E. F. 1987, ApJ, 322, 412
* Bontemps et al. (1996) Bontemps, S., Andre, P., Terebey, S., & Cabrit, S. 1996, A&A, 311, 858
* Burton et al. (1989) Burton, M. G., Brand, P. W. J. L., a, T. R., & Webster, A. S. 1989, MNRAS, 236, 409
* Carmona et al. (2007) Carmona, A., van den Ancker, M.E., Henning, Th., Goto, M., Fedele, D., & Stecklum, B. 2007, A&A, 476, 853
* Carpenter (2001) Carpenter, J. M. 2001, AJ, 121, 2851
* Casanova et al. (1995) Casanova, S., Montmerle, T., a, E. D., & Andre, P. 1995, ApJ, 439, 752
* Chiang & Goldreich (1997) Chiang, E.I. & Goldreich, P. 1997, ApJ, 490, 368
* Chrysostomou et al. (2008) Chrysostomou, A., Bacciotti, F., Nisini, B., Ray, T.P., Eislöffel, J., Davis, C.J., & Takami, M. 2008, A&A, 482, 575
* Cohen et al. (1981) Cohen, J. G., a, S. E., Elias, J. H., & aa, J. A. 1981, ApJ, 249, 481
* Covey et al. (2006) Covey, K. R., Greene, T. P., Doppmann, G. W., & a, C. J. 2006, AJ, 131, 512
* d’Alessio et al. (1999) d’Alessio, P., Calvet, N., Hartmann, L., Lizano, S., & Cantó, J. 1999, ApJ, 527, 893
* d’Alessio et al. (2004) d’Alessio, P., Calvet, N., Hartmann, L., Muzerolle, J., & Sitko, M. 2004, in Star Formation at High Angular Resolution, IAU Symp. 221, 403, eds. M.G. Burton, R. Jayawardhana, & T.L. Bourke
* Davis et al. (2010) Davis, C.J., Gell, R., Khanzadyan, T., Smith, M.D., & Jenness, T. 2010, A&A, 511, 24
* Doppmann et al. (2005) Doppmann, G. W., Greene, T. P., Covey, K. R., & Lada, C. J. 2005, AJ, 130, 1145
* Edwards et al. (1993) Edwards, S., Ray, T., & Mundt, R. 1993, Protostars and Planets III, 567
* Feigelson et al. (2007) Feigelson, E., Townsley, L., Güdel, M., & Stassun, K. 2007, Protostars and Planets V, 313
* Flaccomio et al. (2009) Flaccomio, E., Stelzer, B., Sciortino, S., Micela, G., Pillitteri, I., & Testi, L. 2009, A&A, 505, 695
* Giardino et al. (2007) Giardino, G., Favata, F., Pillitteri, I., Flaccomio, F., Micela, G., & Sciortino, S. 2007, A&A, 475, 891
* Glassgold et al. (2005) Glassgold, A.E., Feigelson, E.D., Montmerle, T., & Wolk, S. 2005, ASP Conf. Ser. 341, p.165
* Glassgold et al. (2007) Glassgold, A.E., Najita, J.R., Igea, J. 2007, ApJ, 656, 515
* Glassgold et al. (2004) Glassgold, A.E., Najita, J.R., Igea, J. 2004, ApJ, 615, 972
* Gorti & Hollenbach (2008) Gorti, U. & Hollenbach, D. 2008, ApJ, 683, 287
* Gorti et al. (2009) Gorti, U., Dullemond, C.P., & Hollenbach, D. 2009, ApJ, 705, 1237
* Gredel & Dalgarno (1995) Gredel, R., & a, A. 1995, ApJ, 446, 852
* Greene & Lada (1996) Greene, T. P., & Lada, C. J. 1996, AJ, 112, 2184
* Greene & Lada (2002) Greene, T. P., & Lada, C. J. 2002, AJ, 124, 2185
* Grosso et al. (1997) Grosso, N., Montmerle, T., Feigelson, E. D., André, P., Casanova, S., & Gregorio-Hetem, J. 1997, Nature, 387, 56
* Güdel et al. (2007) Güdel, M., et al. 2007, A&A, 468, 353
* Gullbring et al. (2000) Gullbring, E., Calvet, N., Muzerolle, J., & Hartmann, L. 2000, ApJ, 544, 927
* Harrison et al. (1998) Harrison, A., Puxley, P., Russell, A., & Brand, P. 1998, MNRAS, 297, 624
* Herbig (1998) Herbig, G. H. 1998, ApJ, 497, 736
* Herczeg et al. (2007) Herczeg, G. J., Najita, J. R., Hillenbrand, L. A., & Pascucci, I. 2007, ApJ, 670, 509
* Herczeg et al. (2006) Herczeg, G. J., Linsky, J. L., Walter, F. M., Gahm, G. F., & Johns-Krull, C. M. 2006, ApJS, 165, 256
* Imanishi et al. (2001) Imanishi, K., Koyama, K., & Tsuboi, Y. 2001, ApJ, 557, 747
* Imanishi et al. (2003) Imanishi, K., Nakajima, H., Tsujimoto, M., Koyama, K., & Tsuboi, Y. 2003, PASJ, 55, 653
* Kenyon et al. (1994) Kenyon, S. J., Dobrzycka, D., & Hartmann, L. 1994, AJ, 108, 1872
* Kenyon & Hartmann (1987) Kenyon, S. J., & Hartmann, L. 1987, ApJ, 323 714
* Kristensen et al. (2007) Kristensen, L. E., Ravkilde, T. L., Field, D., Lemaire, J. L., & Pineau Des Forêts, G. 2007, A&A, 469, 561
* Kwan et al. (2007) Kwan, J., Edwards, S., & Fischer, W. 2007, ApJ, 657, 897
* Lacy et al. (2010) Lacy, J.H., Harrold, S.T., Watson, D.M. 2010, BAAS, 41, 346
* Lahuis et al. (2007) Lahuis, F., van Dishoeck, E.F., Blake, G.A., Evans, N.J. II, Kessler-Silacci, J.E., & Pontopiddan, K.M. 2007, ApJ, 665, 492L
* Maloney et al. (1996) Maloney, P. R., Hollenbach, D. J., & Tielens, A. G. G. M. 1996, ApJ, 466, 561
* Mamajek (2008) Mamajek, E. E. 2008, Astronomische Nachrichten, 329, 10
* Masson & Chernin (1993) Masson, C. R., & Chernin, L. M. 1993, ApJ, 414, 230
* McLean et al. (1998) McLean, I. S. et al. 1998, Proc. SPIE, 3354, 566
* Meyer et al. (1997) Meyer, M. R., Calvet, N., & Hillenbrand, L. A. 1997, AJ, 114, 288
* Montmerle et al. (2000) Montmerle, T., Grosso, N., Tsuboi, Y., & Koyama, K. 2000, ApJ, 532, 261
* Moriarty-Schieven et al. (1992) Moriarty-Schieven, G. H., Wannier, P. G., Tamura, M., & Keene, J. 1992, ApJ, 400, 260
* Meijerink et al. (2009) Meijerink, R., Poelman, D.R., Glassgold, A.E., Najita, J.R., Tielens, A.G.G.M., & Spaans, M. 2009, “The Evolving ISM in the Milky Way and Nearby Galaxies,” Proc. of the 4th Spitzer Science Center Conference, eds. K. Sheth, A. Noriega-Crespo, J. Ingalls, & R. Palladini, on-line at http://ssc.spitzer.caltech.edu/mtgs/ismevol/
* Meijerink et al. (2007) Meijerink, R., Glassgold, A., & Najita, J. 2007, in “The Spirit of Bernard Lyot: The Direct Detection of Planets and Circumstellar Disks in the 21st Century,” Proc. of Conf. held 4-8 June 2007 at U.C. Berkeley, ed. P. Kalas
* Najita et al. (2010) Najita, J.R., Carr, J.S., Strom, S.E., Watson, D.M., Pascucci, I., Hollenbach, D., Gorti, U., & L. Keller, 2010, ApJ, 712, 274
* Najita et al. (2009) Najita, J.R., Doppmann, G.W., Bitner, M.A., Richter, M.J., Lacy, J.H., Jaffe, D.T., Carr, J.S., Meijerink, R., Blake, G.A., Herczeg, G.J.., & Glassgold, A.E. 2009, ApJ, 697, 957
* Neufeld et al. (2006) Neufeld, D., et al. 2006, ApJ, 649, 816
* Nidever et al. (2002) Nidever, D. L., Marcy, G. W., Butler, R. P., Fischer, D. A., & Vogt, S. S. 2002, ApJS, 141, 503
* Nomura et al. (2007) Nomura, H., Aikawa, Y., Tsujimoto, M., Nakagawa, Y., & Millar, T. J. 2007, ApJ, 661, 334
* Pascucci et al. (2007) Pascucci, I., Hollenbach, D., Najita, J., Muzerolle, J., Gorti, U., Herczeg, G.J., Hillenbrand, L.A., Kim, J.S., Carpenter, J.M., Meyer, M.R., Mamajek, E.E., & Bouwman, J. 2007, ApJ, 663, 383
* Rieke & Lebofsky (1985) Rieke, G. H., & Lebofsky, M. J. 1985, ApJ, 288, 618
* Schwartz & Greene (2003) Schwartz, R. D., & Greene, T. P. 2003, AJ, 126, 339
* Straižys et al. (1996) Straižys, V., Černis, K., & Bartašiūtė, S. 1996, Baltic Astronomy, 5, 125
* Tine et al. (1997) Tine, S., Lepp, S., Gredel, R., & Dalgarno, A. 1997, ApJ, 481, 282
* van Boekel et al. (2009) van Boekel, R., Güdel, M., Henning, Th., Lahuis, F., & Pantin, E. 2009, A&A, 497, 137
* Weintraub et al. (2000) Weintraub, D. A., Kastner, J. H., & Bary, J. S. 2000, ApJ, 541, 767
* White & Hillenbrand (2004) White, R. J., & Hillenbrand, L. A. 2004, ApJ, 616, 998
* Wilgenbus et al. (2000) Wilgenbus, D., Cabrit, S., Pineau des Forêts, G., & Flower, D. R. 2000, A&A, 356, 1010
* Woitke et al. (2009) Woitke, P. et al. 2009, in Cool Stars, Stellar Systems, and the Sun: Proc. of the 15th Cambridge Workshop, AIP Conf. Proc., 1094, 225
* Zinnecker et al. (1998) Zinnecker, H., McCaughrean, M. J., & Rayner, J. T. 1998, Nature, 394, 862
Figure 1: H2 $v=1-0$ $S(1)$ line spectra. Spectra acquired on 2007 Jun 24 and
2008 Jan 24 (first epoch) appear in the left panel, and spectra acquired on
2007 Jun 25 and 2008 Jan 25 (second epoch) appear in the right panel. These
spectra were extracted only in the region containing the mostly point source
continuum emission; any extended H2 emission is not included.
Figure 2: Histogram of mean H2 $v=1-0$ $S(1)$ line velocities minus
photospheric velocities for all observations.
Figure 3: Histogram of H2 $v=1-0$ $S(1)$ line FWHM velocities. The
instrumental line width of 17 km s-1 has been subtracted in quadrature from
each value before binning.
Figure 4: Histogram of H2 $v=1-0$ $S(1)$ line luminosities.
Figure 5: Histogram of H2 $v=1-0/2-1$ $S(1)$ line ratios. Values of 1.9, 7.7,
and 17 are indicated, nominally correspond to UV, shock, and X-ray secondary
electron impact excitation respectively in the presence of no dust (Gredel &
Dalgarno, 1995). These values are the inverse of those presented in column 8
of Table 2.
Table 1: Journal of Observations Object | Region | $\alpha$(J2000) | $\delta$(J2000) | UT Date | Int. Time | Slit PA
---|---|---|---|---|---|---
| | (hh mm ss.s) | ($\arcdeg$ $\arcmin$ $\arcsec$) | | (minutes) | ($\arcdeg$E of N)
03260+3111B | Per | 03 29 07.7 | 31 21 58 | 2008 Jan 24 | 20.0 | 103
| | | | 2009 Jan 25 | 20.0 | 149
03260+3111A | Per | 03 29 10.7 | 31 21 59 | 2008 Jan 24 | 20.0 | 118
| | | | 2008 Jan 25 | 12.0 | 180
04108+2803B | Tau-Aur | 04 13 54.9 | 28 11 31 | 2008 Jan 24 | 20.0 | -123
| | | | 2008 Jan 25 | 25.0 | 121
04158+2805 | Tau-Aur | 04 18 58.2 | 28 12 24 | 2008 Jan 24 | 20.0 | 73
| | | | 2008 Jan 25 | 20.0 | 63
04181+2654AB | Tau-Aur | 04 21 11.5 | 27 01 09 | 2008 Jan 24 | 10.0 | 48
| | | | 2008 Jan 25 | 30.0 | 172
DG Tau | Tau-Aur | 04 27 04.8 | 26 06 17 | 2008 Jan 25 | 10.0 | 57
04264+2433 | Tau-Aur | 04 29 30.0 | 24 39 56 | 2009 Jan 24 | 12.0 | 57
| | | | 2008 Jan 25 | 12.0 | -106
04295+2251 | Tau-Aur | 04 32 32.1 | 22 57 27 | 2009 Jan 24 | 40.0 | -106
| | | | 2008 Jan 25 | 30.0 | 75
04361+2547 | Tau-Aur | 04 39 13.5 | 25 53 20 | 2008 Jan 24 | 12.0 | 68
| | | | 2008 Jan 25 | 12.0 | -110
04365+2535 | Tau-Aur | 04 39 35.2 | 25 41 45 | 2008 Jan 24 | 44.0 | 97
| | | | 2008 Jan 25 | 33.3 | 99
GSS 30 | Oph | 16 26 21.4 | -24 23 06 | 2007 Jun 25 | 14.0 | -124
GY 21 | Oph | 16 26 23.6 | -24 24 38 | 2007 Jun 24 | 20.0 | -56
| | | | 2007 Jun 25 | 20.0 | -100
WL 12 | Oph | 16 26 44.1 | -24 34 48 | 2007 Jun 24 | 8.0 | -47
| | | | 2007 Jun 25 | 10.0 | -114
WL 6 | Oph | 16 27 21.6 | -24 29 51 | 2007 Jun 24 | 20.0 | -72
| | | | 2007 Jun 25 | 30.0 | -89
IRS 43 | Oph | 16 27 27.0 | -24 40 50 | 2007 Jun 24 | 45.0 | -78
| | | | 2007 Jun 25 | 60.0 | -66
YLW 16A | Oph | 16 27 27.8 | -24 39 32 | 2007 Jun 24 | 12.0 | -85
| | | | 2007 Jun 25 | 12.0 | -133
IRS 67 | Oph | 16 32 01.1 | -24 56 45 | 2007 Jun 24 | 60.0 | -59
SVS 2 | Ser | 18 29 56.8 | 01 14 46 | 2007 Jun 25 | 28.0 | -85
Table 2: Protostar H2 Line Analysis Source | UT Date | 1–0 S(0) | 1–0 S(1) | 2–1 S(1) | AvbbV magnitude extinction was computed using each objects 2MASS $JHK$ colors, an extinction law, and estimated intrinsic CTTS locus colors as explained in the text in §3.4 | 1–0 S(1) | 2–1/1–0 | FWHMcc The mean FWHM velocity of the 1–0 S(1) H2 line, where the intrinsic instrumental line width of 17 km s-1 has been removed in quadrature. | V(H2 \- *)dd The radial velocity of the stellar photosphere subtracted from the mean radial velocity of the 1–0 S(0), 1–0 S(1), and 2–1 S(1) H2 emission lines (uncertain lines not used). No data indicate that the star lacked either H2 or photospheric lines. | H2 1–0 S(1)
---|---|---|---|---|---|---|---|---|---|---
| | EW (Å)aaPositive equivalent widths indicate emission. Uncertainties are $\sim 0.02\AA$ for 2007 data and $\sim 0.05\AA$ for 2008 data. | EW (Å)aaPositive equivalent widths indicate emission. Uncertainties are $\sim 0.02\AA$ for 2007 data and $\sim 0.05\AA$ for 2008 data. | EW (Å)aaPositive equivalent widths indicate emission. Uncertainties are $\sim 0.02\AA$ for 2007 data and $\sim 0.05\AA$ for 2008 data. | (mag) | Log L(W) | S(1) | (km s-1) | (km s-1) | extent (″)
03260+3111B | 2008 Jan 24ee Spectrum displays $\Delta v=2-0$ CO emission. | 0.50 | 2.12 | 0.49 | 3.5 | 22.0 | 0.25 | 43 | … | $\gtrsim 10$
| 2008 Jan 25 | 0.51 | 2.69 | 0.39 | 3.5 | 22.1 | 0.14 | 44 | … | $\lesssim 1$
03260+3111AffExtended H2 emission is spatially displaced from the star. No H2 emission is coincident with the stellar continuum (as also observed by Doppmann et al. 2005). | 2008 Jan 24 | $<0.1$ | $<0.1$ | $<0.1$ | … | … | … | … | … | $\sim 7$
| 2008 Jan 25 | $<0.1$ | $<0.1$ | $<0.1$ | … | … | … | … | … | $\gtrsim 10$
04108+2803B | 2008 Jan 24 | 0.92 | 4.29 | 0.39 | 19 | 22.0 | 0.08 | 18 | -1 | $\sim 1$
| 2008 Jan 25 | 0.69 | 3.43 | 0.29 | 19 | 21.9 | 0.07 | 18 | -1 | $\sim 1$
04158+2805 | 2008 Jan 24 | 0.49 | 2.57 | 0.26: | 3.1 | 21.1 | 0.08: | 20 | -4 | $<1$
| 2008 Jan 25 | 0.35 | 2.70 | 0.22: | 3.1 | 21.2 | 0.06: | 20 | -4 | $\sim 2$
04181+2654AB | 2008 Jan 24 | 0.23 | 1.46 | 0.04:: | 23 | 22.0 | … | 24 | -1 | $\sim 1$
| 2008 Jan 25 | 0.39 | 1.37 | 0.00:: | 23 | 21.9 | … | 25 | 8 | $<1$
DG Tau | 2008 Jan 25 | 0.14 | 0.54 | 0.06:: | 0 | 22.0 | 0.08:: | 14 | -9 | $<1$
04264+2433 | 2008 Jan 24 | 1.56 | 5.78 | 0.53 | 6.1 | 21.7 | 0.06 | 10 | 4 | $\sim 2$
| 2008 Jan 25 | 1.44 | 6.09 | 0.52 | 6.1 | 21.7 | 0.06 | 13 | -4 | 6
04295+2251 | 2008 Jan 24 | 0.59 | 2.42 | 0.14: | 17 | 22.0 | 0.04: | 16 | … | $\sim 2$
| 2008 Jan 25 | 0.59 | 2.43 | 0.16: | 17 | 22.0 | 0.05: | 16 | … | $\sim 2$
04361+2547 | 2008 Jan 24 | 1.06 | 4.11 | 0.45 | 22 | 22.2 | 0.08 | 14 | … | 0
| 2008 Jan 25 | 0.66 | 2.10 | 0.21: | 22 | 21.9 | 0.07: | 13 | … | 0
04365+2535 | 2008 Jan 24 | 0.57 | 2.58 | 0.18: | 19 | 21.9 | 0.06: | 19 | … | 0
| 2008 Jan 25 | 0.35 | 1.59 | 0.09: | 19 | 21.6 | 0.05: | 16 | … | 0
GSS 30 | 2007 Jun 25 | 0.53 | 2.35 | 0.20 | 20 | 22.6 | 0.07 | 25 | -11 | 2
GY 21ggThe H2 emission slightly spatially displaced from the stellar continuum has velocity FWHM $\Delta v\sim 60$ km s-1, about twice that of the H2 emission spatially coincident with the stellar continuum. | 2007 Jun 24 | 0.37 | 1.80 | 0.20 | 16 | 21.8 | 0.07 | 28 | -11 | $\sim 1$
| 2007 Jun 25 | 0.52 | 2.21 | 0.20 | 16 | 21.9 | 0.07 | 29 | -10: | $\sim 1$
WL 12 | 2007 Jun 24 | 1.43 | 12.4 | 0.79 | 18 | 22.1 | 0.06 | 32 | -2 | $\sim 2$
| 2007 Jun 25 | 2.06 | 11.0 | 0.73 | 18 | 22.1 | 0.06 | 29 | -12 | $\sim 1$
WL 6 | 2007 Jun 24 | 0.06: | 0.34 | 0.06: | 37 | 21.6 | 0.15: | 15 | 0 | 0
| 2007 Jun 25 | 0.09 | 0.37 | 0.03: | 37 | 21.6 | 0.07: | 16 | -3 | 0
IRS 43hhExtended H2 emission shows velocity structure with a maximum FWHM $\Delta v\sim 40$ km s-1, about twice that of the H2 emission spatially coincident with the stellar continuum. | 2007 Jun 24 | 0.09 | 0.28 | 0.02:: | 33 | 21.8 | 0.05:: | 9 | 0 | $\sim 3$
| 2007 Jun 25 | 0.17 | 0.64 | 0.04: | 33 | 22.1 | 0.05:: | 22 | -4 | $\sim 3$
YLW 16A | 2007 Jun 24 | 0.32 | 1.41 | 0.20 | 17 | 21.6 | 0.12 | 25 | 0 | $\sim 2$
| 2007 Jun 25 | 0.31 | 1.25 | 0.13 | 17 | 21.6 | 0.09 | 28 | -4 | $\sim 3$
IRS 67 | 2007 Jun 24 | 0.13 | 0.37 | 0.03:: | 22 | 21.1 | 0.07:: | 15 | 0 | $\sim 1$
SVS 2iiThe extended H2 emission is separated from the stellar continuum and its line emission by $\sim 2\arcsec$. The extended emission has FWHM $\Delta v\sim 24$ km s-1, about twice that of the H2 emission spatially coincident with the stellar continuum. H2 line strength ratio was not calculated due to the very low value and high uncertainty ($\sim 50$%) of the 2–1 S(1) line measurement. | 2007 Jun 25 | 0.09 | 0.39 | 0.01::: | 0 | 21.5 | … | 13 | 19 | $\sim 3$
|
arxiv-papers
| 2010-10-11T17:51:24 |
2024-09-04T02:49:13.690457
|
{
"license": "Public Domain",
"authors": "Thomas P. Greene, Mary Barsony, and David A. Weintraub",
"submitter": "Tom Greene",
"url": "https://arxiv.org/abs/1010.2174"
}
|
1010.2296
|
# Rainbow Connection Number and
Connected Dominating Sets
L. Sunil Chandran Department of Computer Science and Automation,
Indian Institute of Science,
Bangalore -560012, India.
{sunil, anita, deepakr}@csa.iisc.ernet.in Anita Das Department of Computer
Science and Automation,
Indian Institute of Science,
Bangalore -560012, India.
{sunil, anita, deepakr}@csa.iisc.ernet.in Deepak Rajendraprasad Department
of Computer Science and Automation,
Indian Institute of Science,
Bangalore -560012, India.
{sunil, anita, deepakr}@csa.iisc.ernet.in Nithin M. Varma Department of
Computer Science and Engineering,
National Institute of Technology,
Calicut - 673 601, India.
nithvarma@gmail.com
###### Abstract
Rainbow connection number $rc(G)$ of a connected graph $G$ is the minimum
number of colours needed to colour the edges of $G$, so that every pair of
vertices is connected by at least one path in which no two edges are coloured
the same. In this paper we show that for every connected graph $G$, with
minimum degree at least $2$, the rainbow connection number is upper bounded by
$\gamma_{c}(G)+2$, where $\gamma_{c}(G)$ is the connected domination number of
$G$. Bounds of the form $diameter(G)\leq rc(G)\leq diameter(G)+c$, $1\leq
c\leq 4$, for many special graph classes follow as easy corollaries from this
result. This includes interval graphs, AT-free graphs, circular arc graphs,
threshold graphs, and chain graphs all with minimum degree at least $2$ and
connected. We also show that every bridge-less chordal graph $G$ has
$rc(G)\leq 3.radius(G)$. In most of these cases, we also demonstrate the
tightness of the bounds.
An extension of this idea to two-step dominating sets is used to show that for
every connected graph on $n$ vertices with minimum degree $\delta$, the
rainbow connection number is upper bounded by $3n/(\delta+1)+3$. This solves
an open problem from [Schiermeyer, 2009], improving the previously best known
bound of $20n/\delta$ [Krivelevich and Yuster, 2010]. Moreover, this bound is
seen to be tight up to additive factors by a construction mentioned in [Caro
et al., 2008].
Keywords: rainbow connectivity, rainbow colouring, connected dominating set,
connected two-step dominating set, radius, minimum degree.
## 1 Introduction
Edge colouring of a graph is a function from its edge set to the set of
natural numbers. A path in an edge coloured graph with no two edges sharing
the same colour is called a rainbow path. An edge coloured graph is said to be
rainbow connected if every pair of vertices is connected by at least one
rainbow path. Such a colouring is called a rainbow colouring of the graph. The
minimum number of colours required to rainbow colour a connected graph is
called its rainbow connection number, denoted by $rc(G)$. For example, the
rainbow connection number of a complete graph is $1$, that of a path is its
length, that of an even cycle is its diameter, that of an odd cycle is one
more than its diameter, and that of a star is its number of leaves. Note that
disconnected graphs cannot be rainbow coloured and hence the rainbow
connection number for them is left undefined. For a basic introduction to the
topic, see Chapter $11$ in [Chartrand and Zhang, 2008].
The concept of rainbow colouring was introduced in [Chartrand et al., 2008].
Precise values of rainbow connection number for many special graphs like
complete multi-partite graphs, Peterson graph and wheel graphs were also
determined there. It was shown in [Chakraborty et al., 2009] that computing
the rainbow connection number of an arbitrary graph is NP-Hard. To rainbow
colour a graph, it is enough to ensure that every edge of some spanning tree
in the graph gets a distinct colour. Hence order of the graph minus one is an
upper bound for rainbow connection number. There have been attempts to find
better upper bounds for the same in terms of other graph parameters like
connectivity, minimum degree etc.
In the search towards good upper bounds for rainbow connection number, an idea
that turned out to be successful more than once is a “strengthened” notion of
connected $k$-step dominating set (Definition 2 in Section 1.1): a
strengthening so that a rainbow colouring of the induced graph on such a set
can be extended to the whole graph using a constant number of additional
colours. Theorem $1.4$ in [Caro et al., 2008] was proved using a strengthened
connected $1$-step dominating set and Theorem $1.1$ in [Krivelevich and
Yuster, 2010] was proved using a strengthened connected $2$-step dominating
set. A closer examination revealed to us that the additional requirements
imposed on the connected dominating sets in both those cases were far more
restrictive than what was essential. This led us to the investigation of what
is the weakest possible strengthening of a connected dominating set which can
achieve the same. Since every edge incident on a pendant vertex will need a
different colour, it is easy to see that such a dominating set should
necessarily include all the pendant vertices in the graph. Quite surprisingly,
it turns out that this obvious necessary condition is also sufficient!
(Theorem 1 in Section 2). For rainbow connection number of many special graph
classes, the above result gives tight upper bounds which were otherwise
difficult to obtain (Theorem 4 in Section 2). The farthest we could get with
the idea was a curious theorem about chordal graphs (Theorem 5 in Section 2).
A similar inquiry for the weakest strengthening a connected two-step
dominating set (Theorem 7 in Section 2) led us to the solution of an important
open problem in this area regarding the optimal upper bound of rainbow
connection number in terms of minimum degree. (See Theorem 10 in Section 2 and
the remarks therein). As an intermediate step in solving the above problem, we
also discovered a tight upper bound on the size of a minimum connected two-
step dominating set of a graph in terms of its minimum degree (Theorem 8 in
Section 2). To the best of our knowledge, this bound is not yet reported in
literature. It may have applications beyond the realm of rainbow colouring.
For instance, Theorem 8 immediately gives an upper bound on radius of every
graph in terms of its minimum degree (Corollary 9 in Section 2) which
marginally improves the one reported in [Erdős et al., 1989].
### 1.1 Preliminaries
See Table 1 for the notations employed throughout the paper.
Table 1: Notations. $G$ is a graph, $v$ a vertex in $G$ and $S$ a subset of vertices in $G$. $k$ is a non-negative integer. $V(G)$ | Vertex set of $G$.
---|---
$E(G)$ | Edge set of $G$.
$|G|$ | Number of vertices in $G$ or order of $G$.
$\delta(G)$ | Minimum degree of $G$
$pen(G)$ | Number of pendant vertices in $G$
$rc(G)$ | Rainbow connection number of $G$
$d(u,v)$ | Distance between vertices $u$ and $v$
$ecc(v)$ | Eccentricity of $v$
$diam(G)$ | Diameter of $G$
$rad(G)$ | Radius of $G$
$\gamma_{c}^{k}(G)$ | Connected $k$-step domination number of $G$
$\gamma_{c}(G)$ | $\gamma^{1}_{c}(G)$, Connected domination number of $G$
$N^{k}(S)$ | Set of all vertices at distance exactly $k$ from set $S$
$N^{k}(v)$ | $N^{k}(\\{v\\})$
$N(S)$ | $N^{1}(S)$, Neighbourhood of $S$
$N(v)$ | $N^{1}(\\{v\\})$, Neighbourhood of $v$
$G[S]$ | Induced subgraph of $G$ on $S$
All graphs considered in this article are finite, simple and undirected. The
length of a path is its number of edges. An edge in a connected graph is
called a bridge, if its removal disconnects the graph. A graph with no bridges
is called a bridge-less graph.
###### Definition 1.
Let $G$ be a connected graph. The distance between two vertices $u$ and $v$ in
$G$, denoted by $d(u,v)$ is the length of a shortest path between them in $G$.
The eccentricity of a vertex $v$ is $ecc(v):=\max_{x\in V(G)}{d(v,x)}$. The
diameter of $G$ is $diam(G):=\max_{x\in V(G)}{ecc(x)}$. The radius of $G$ is
$rad(G):=\min_{x\in V(G)}{ecc(x)}$. Distance between a vertex $v$ and a set
$S\subseteq V(G)$ is $d(v,S):=\min_{x\in S}{d(v,x)}$. The $k$-step open
neighbourhood of a set $S\subseteq V(G)$ is $N^{k}(S):=\\{x\in
V(G)|d(x,S)=k\\}$, $k\in\\{0,1,2,\cdots\\}$. The degree of a vertex $v$ is
$degree(v):=|N^{1}(\\{v\\})|$. The minimum degree of $G$ is
$\delta(G):=\min_{x\in V(G)}{degree(x)}$. A vertex is called pendant if its
degree is $1$ and isolated if its degree is $0$.
###### Definition 2.
Given a graph $G$, a set $D\subseteq V(G)$ is called a $k$-step dominating set
of $G$, if every vertex in $G$ is at a distance at most $k$ from $D$. Further,
if $D$ induces a connected sub-graph of $G$, it is called a connected $k$-step
dominating set of $G$. The cardinality of a minimum connected $k$-step
dominating set in $G$ is called its connected $k$-step domination number
$\gamma^{k}_{c}(G)$. When $k=1$, we may omit the qualifier “$1$-step” in the
above names and the superscript $1$ in the notation.
Note that connected $k$-step dominating sets exist only for connected graphs.
Connected $k$-step domination number is left undefined otherwise.
###### Definition 3.
An intersection graph of a family of sets $\mathcal{F}$, is a graph whose
vertices can be mapped to sets in $\mathcal{F}$ such that there is an edge
between two vertices in the graph if and only if the corresponding two sets in
$\mathcal{F}$ have a non-empty intersection. An interval graph is an
intersection graph of intervals on the real line. A unit interval graph is an
intersection graph of unit length intervals on the real line. A circular arc
graph is an intersection graph of arcs on a circle.
###### Definition 4.
An independent triple of vertices $x$, $y$, $z$ in a graph $G$ is an
asteroidal triple $($AT$)$, if between every pair of vertices in the triple,
there is a path that does not contain any neighbour of the third. A graph
without asteroidal triples is called an AT-free graph.
###### Definition 5.
A graph $G$ is a threshold graph, if there exists a weight function
$w:V(G)\rightarrow\mathbb{R}$ and a real constant $t$ such that two vertices
$u,v\in V(G)$ are adjacent if and only if $w(u)+w(v)\geq t$.
###### Definition 6.
A bipartite graph $G(A,B)$ is called a chain graph if the vertices of $A$ can
be ordered as $A=(a_{1},a_{2},\cdots,a_{k})$ such that $N(a_{1})\subseteq
N(a_{2})\subseteq\cdots\subseteq N(a_{k})$ [Yannakakis, 1982].
###### Definition 7.
A graph $G$ is called chordal, if there is no induced cycle of length greater
than $3$.
## 2 Our Results
The main ideas in this paper are captured in Theorem 1, Theorem 7 and Theorem
8. The other results are consequences of them. Among the results, Theorem 10
demands a special mention due to the prominence of the question it answers in
the area of rainbow colouring. To state Theorem 1 in its full generality, we
need to make one new definition.
###### Definition 8 (Two-way dominating set).
A dominating set $D$ in a graph $G$ is called a two-way dominating set if
every pendant vertex of $G$ is included in $D$. In addition, if $G[D]$ is
connected, we call $D$ a connected two-way dominating set.
###### Remark 1.
If $\delta(G)\geq 2$, then every (connected) dominating set in $G$ is a
(connected) two-way dominating set. We use the name “two-way domination” since
the definition implies that every vertex in $V(G)\backslash D$ has at least
two edge disjoint paths to $D$.
###### Theorem 1.
If $D$ is a connected two-way dominating set in a graph $G$, then
$rc(G)\leq rc(G[D])+3.$
Proof is given in Section 3.1
###### Remark 2.
The reader may wonder why the pendant vertices had to be included in the
dominating set $D$. Our strategy is to colour $G[D]$ first and then colour all
the edges outside using a constant number (in this case $3$) of additional
colours ensuring rainbow connectivity. Pendent vertices are always a
bottleneck for rainbow colouring since no two pendant edges (edges incident on
pendant vertices) can share the same colour. Hence the restriction.
###### Corollary 2.
For every connected graph $G$, with $\delta(G)\geq 2$,
$rc(G)\leq\gamma_{c}(G)+2.$
###### Proof.
This follows from Theorem 1 since (i) in this case, every connected dominating
set in $G$ is a connected two-way dominating set and (ii)
$rc(G[D])\leq|D|-1=\gamma_{c}(G)-1$ for a minimum connected dominating set $D$
in $G$. ∎
###### Corollary 3.
For every connected graph $G$,
$rc(G)\leq\gamma_{c}(G)+pen(G)+2.$
###### Proof.
This follows from Theorem 1 since adding all the pendant vertices to a minimum
connected dominating set gives a connected two-way dominating set of size at
most $\gamma_{c}(G)+pen(G)$. ∎
Diameter of a graph is a trivial lower bound for its rainbow connection
number. Theorem 1 gives upper bounds which are only a small additive constant
above the diameter for many special graph classes.
###### Theorem 4.
Let $G$ be a connected graph with $\delta(G)\geq 2$. Then,
1. $(i)$
if $G$ is an interval graph, $diam(G)\leq rc(G)\leq diam(G)+1$,
2. $(ii)$
if $G$ is AT-free, $diam(G)\leq rc(G)\leq diam(G)+3$,
3. $(iii)$
if $G$ is a threshold graph, $diam(G)\leq rc(G)\leq 3$,
4. $(iv)$
if $G$ is a chain graph, $diam(G)\leq rc(G)\leq 4$,
5. $(v)$
if $G$ is a circular arc graph, $diam(G)\leq rc(G)\leq diam(G)+4$.
Moreover, there exist interval graphs, threshold graphs and chain graphs with
minimum degree at least $2$ and rainbow connection number equal to the
corresponding upper bound above. There exists an AT-free graph $G$ with
minimum degree at least $2$ and $rc(G)=diam(G)+2$, which is $1$ less than the
upper bound above.
###### Remark 3.
The upper bounds follow from Theorem 1 since (i) every interval graph $G$
which is not isomorphic to a complete graph has a dominating path of length at
most $diam(G)-2$, (ii) every AT-free graph $G$ has a dominating path of length
at most $diam(G)$, (iii) a maximum weight vertex in a connected threshold
graph $G$ is a dominating vertex, (iv) every connected chain graph $G$ has a
dominating edge, and (v) every circular arc graph $G$, which is not an
interval graph, has a dominating cycle of diameter at most $diam(G)$.
Tight examples and proofs for non-trivial claims made in the above remark are
given in Section 3.4. We could not find tight examples for AT-free and
circular arc graphs. It may be interesting to see whether those two upper
bounds can be improved.
Another interesting application of Theorem 1 is the following result on
chordal graphs. It is curious since chordal graphs, unlike interval graphs or
AT-free graphs, can grow in more than two directions and hence they need not
contain dominating paths in general.
###### Theorem 5.
If $G$ is a bridge-less chordal graph, then $rc(G)\leq 3.rad(G)$. Moreover,
there exists a bridge-less chordal graph with $rc(G)=3.rad(G)$.
Proof is given in Section 3.6. The main idea is that we can induct on the
radius of the graph and use Theorem 1 to prove the induction step.
Theorem 4$(i)$ gives $rc(G)\leq diam(G)+1$ for every unit interval graph $G$.
We have a stronger result, using a different approach.
###### Theorem 6.
If $G$ is a unit interval graph such that $\delta(G)\geq 2$, then
$rc(G)=diam(G)$.
Proof is given in Section 3.6
The extension of the idea of two-way domination to two-way two-step domination
is what gives the remaining results. We need to make one more definition to
state the next major theorem (Theorem 7) in its full generality.
###### Definition 9 (Two-way two-step dominating set).
A (connected) two-step dominating set $D$ of vertices in a graph $G$ is called
a $($connected$)$ two-way two-step dominating set if (i) every pendant vertex
of $G$ is included in $D$ and (ii) every vertex in $N^{2}(D)$ has at least two
neighbours in $N^{1}(D)$.
###### Remark 4.
As in the two-way ($1$-step) dominating set, here too every vertex $v\in
V(G)\backslash D$ has two edge disjoint paths into $D$. Hence the adjective
“two-way”. It may be noted that, just like pendant edges, no two bridges in a
graph can be coloured the same in any rainbow colouring. Hence the restriction
of two-way domination is in some sense necessary to obtain colouring
strategies which use only a constant number of extra colours outside the
dominating set.
###### Theorem 7.
If $D$ is a connected two-way two-step dominating set in a graph $G$, then
$rc(G)\leq rc(G[D])+6.$
Proof is given in Section 3.2
###### Theorem 8.
(i) Every connected graph $G$ of order $n$ and minimum degree $\delta$ has a
connected two-step dominating set $D$ of size at most
$\frac{3(n-|N^{2}(D)|)}{\delta+1}-2$. (ii) Every connected graph $G$ of order
$n\geq 4$ and minimum degree $\delta$ has a connected two-way two-step
dominating set $D^{\prime}$ of size at most $\frac{3n}{\delta+1}-2$. Moreover,
for every $\delta\geq 2$, there exist infinitely many connected graphs $G$
such that $\gamma^{2}_{c}(G)\geq\frac{3(n-2)}{\delta+1}-4$.
Proof is given in Section 3.3
It is easy to see that the radius of any connected graph is at most $k$ more
than the radius of its $k$-step connected dominating set. Moreover, the radius
of any graph $H$ is at most $|H|/2$. Hence the following corollary is also
immediate.
###### Corollary 9.
For every connected graph $G$ of order $n$ and minimum degree $\delta$,
$rad(G)\leq\frac{3n}{2(\delta+1)}+1.$
This bound marginally improves the one reported in [Erdős et al., 1989],
namely $\frac{3(n-3)}{2(\delta+1)}+5$ and the proof is shorter. Note that we
can similarly upper bound the diameter of $G$ by $\frac{3n}{\delta+1}+1$. But
the corresponding bound reported in [Erdős et al., 1989] is better, namely
$\frac{3n}{\delta+1}-1$.
###### Theorem 10.
For every connected graph $G$ of order $n$ and minimum degree $\delta$,
$rc(G)\leq\frac{3n}{\delta+1}+3.$
Moreover, for every $\delta\geq 2$, there exist infinitely many graphs $G$
such that $rc(G)\geq\frac{3(n-2)}{\delta+1}-1$.
###### Proof.
Observe that the connected (two-way two-step dominating) set $D$ can be
rainbow coloured using $|D|-1$ colours by ensuring that every edge of some
spanning tree gets distinct colours. So the upper bound follows immediately
from Theorems 7 and 8(ii). The family of tight examples is demonstrated in
[Caro et al., 2008]. ∎
###### Remark 5.
Theorem 10 nearly settles the investigation for an optimal upper bound of
rainbow connection number in terms of minimum degree which was initiated in
[Caro et al., 2008]. There it was shown that, if $\delta(G)\geq 3$, then
$rc(G)<5n/6$. For general $\delta$, they had given two upper bounds viz.,
$(1+o_{\delta}(1))n\ln(\delta)/\delta$ and $(4\ln(\delta)+3)n/\delta$. They
had also shown a construction for a family of graphs with
$diam(G)=\frac{3(n-2)}{\delta+1}-1$, leaving a gap of $\ln(\delta)$ factor
between the bound and the construction. They remarked that the problem of
finding an optimum bound for $rc(G)$ in terms of $\delta$ is an intriguing
problem and conjectured that for $\delta(G)\geq 3$, $rc(G)<3n/4$. Schiermeyer
proved the above conjecture and raised the question whether $rc(G)\leq
3n/(\delta+1)$ for all values of $\delta$ [Schiermeyer, 2009]. If the answer
is yes, then for graphs with linear minimum degree $\epsilon n$, the rainbow
connection number is bounded by a constant. This was indeed shown to be the
case in [Chakraborty et al., 2009]. But their proof employed Szemerédi’s
Regularity Lemma and hence the bound was a tower function in $1/\epsilon$.
This was considerably improved in [Krivelevich and Yuster, 2010], where it was
shown that $rc(G)\leq 20n/\delta$ for any connected graph. This is the best
known bound for the problem till date. Theorem 10 improves it and answers the
question from [Schiermeyer, 2009] in affirmative but up to an additive
constant of $3$. Moreover, this bound is seen to be tight up to additive
factors by the construction mentioned in [Caro et al., 2008].
## 3 Proofs
### 3.1 Proof of Theorem 1
Statement. If $D$ is a connected two-way dominating set in a graph $G$, then
$rc(G)\leq rc(G[D])+3$.
###### Proof.
We prove the theorem by demonstrating a rainbow colouring that uses at most
$rc(G[D])+3$ colours. For $x\in N^{1}(D)$, its neighbours in $D$ will be
called foots of $x$, and the corresponding edges will be called legs of $x$.
Any rainbow path whose edge colours are contained in $\\{1,2,\cdots,k\\}$ will
be called a $k$-rainbow path.
Rainbow colour $G[D]$ using colours $\\{1,2,\cdots,k:=rc(G[D])\\}$. Let
$H:=G[V(G)\backslash D]$. Partition $V(H)$ into sets $X$, $Y$ and $Z$ as
follows. $Z$ is the set of all isolated vertices of $H$. In every non-
singleton connected component of $H$, choose a spanning tree. This gives a
spanning forest on $V(H)\backslash Z$ with no isolated vertices. Choose $X$
and $Y$ as any one of the bipartitions defined by this forest. Colour every
$X\mbox{--}D$ edge with $k+1$, every $Y\mbox{--}D$ edge with $k+2$ and every
edge in $H$ with $k+3$. Since $D$ is a two-way dominating set, there are no
pendant vertices outside $D$. Therefore, every vertex in $Z$ will have at
least two legs. Colour one of them with $k+1$ and all the others with $k+2$.
We show that the above colouring is a rainbow colouring of $G$. For pairs in
$D\times D$, there is already a $k$-rainbow path connecting them in $G[D]$.
For a pair $(x,y)\in N^{1}(D)\times D$, join any leg of $x$ (coloured $k+1$ or
$k+2$) with the $k$-rainbow path from the corresponding foot to $y$ in $G[D]$.
For a pair $(x,y)\in(X\cup Z)\times(Y\cup Z)$ join a $k+1$ leg of $x$ and a
$k+2$ leg of $y$ with a $k$-rainbow path between the corresponding foots in
$G[D]$. For a pair in $(x,x^{\prime})\in X\times X$, $x$ has a neighbour
$y(x)\in Y$ from the spanning forest. Join the corresponding $x\mbox{--}y(x)$
edge (coloured $k+3$) with the $y(x)\mbox{--}x^{\prime}$ $(k+2)$-rainbow path
mentioned earlier. Similarly every pair $(y,y^{\prime})\in Y\times Y$ is also
rainbow connected. ∎
### 3.2 Proof of Theorem 7
Statement. If $D$ is a connected two-way two-step dominating set in a graph
$G$, then $rc(G)\leq rc(G[D])+6$.
###### Proof.
We prove the theorem by demonstrating a rainbow colouring that uses at most
$rc(G[D])+6$ colours. For $x\in N^{k}(D)$, its neighbours in $N^{k-1}(D)$,
$k=1,2$ will be called foots of $x$ and the corresponding edges will be called
legs. Any rainbow path whose edge colours are contained in
$\\{1,2,\cdots,k\\}$ will be called a $k$-rainbow path.
Rainbow colour $G[D]$ using colours $\\{1,2,\cdots,k:=rc(G[D])\\}$. Construct
a new graph $H$ on $N^{1}(D)$ with the edge set
$\displaystyle E(H)$ $\displaystyle=$ $\displaystyle\\{\\{x,y\\}|x,y\in
N^{1}(D),\\{x,y\\}\in E(G)\textnormal{ or }$ $\displaystyle\exists z\in
N^{2}(D)\textnormal{ such that }\\{x,z\\},\\{y,z\\}\in E(G)\\}.$
Recall that, in a two-way two-step dominating set $D$, there are no pendant
vertices outside $D$ and every vertex in $N^{2}(D)$ has at least two
neighbours in $N^{1}(D)$. Hence in the above graph $H$, the isolated vertices
are only those which have all their neighbours (at least $2$) in $D$. Call
their collection $Z$. Choose a spanning tree in every non-singleton connected
component of $H$. This gives a spanning forest of $V(H)\backslash Z$ with no
isolated vertices. Let $X$ and $Y$ be any bipartition defined by this forest.
Colour every $X\mbox{--}D$ edge with $(k+1)$ and every $Y\mbox{--}D$ edge with
$(k+2)$. For every vertex in $Z$, colour one of its legs with $(k+1)$ and the
remaining with $(k+2)$. Colour every edge of $G$ within $N^{1}(D)$ by $k+3$.
Partition the vertices of $N^{2}(D)$ into $A$ and $B$ as follows. $A=\\{x\in
N^{2}(D)|N(x)\cap X\neq\emptyset\textnormal{ and }N(x)\cap Y\neq\emptyset\\}$
and $B=N^{2}(D)\backslash A$. Colour every $A\mbox{--}X$ edge with $(k+3)$ and
every $A\mbox{--}Y$ edge with $(k+4)$. First we claim that
$G^{\prime}:=G[D\cup N^{1}(D)\cup A]$ is rainbow connected.
By following the same arguments as in proof of Theorem 1, it can be easily
seen that every pair in $D\times D$, is connected by a $k$-rainbow path and
every pair in $N^{1}(D)\times D$ and $(X\cup Z)\times(Y\cup Z)$ is connected
by a $(k+2)$-rainbow path. Notice that for every vertex $x\in X$, there exists
$y(x)\in Y$ such that $x\mbox{--}y(x)$ is an edge in the spanning forest.
Vertices $x$ and $y(x)$ are connected either by a single $(k+3)$ edge or a
$(k+3,k+4)$ path. Hence between any pair $(x,x^{\prime})\in X\times X$, we can
find a rainbow path by joining the $x\mbox{--}y(x)$ path with the
$y(x)\mbox{--}x^{\prime}$ $(k+2)$-rainbow path. Similarly any pair in $Y\times
Y$ is also rainbow connected. Any pair $(a,a^{\prime})\in A\times A$ can be
rainbow connected by joining the $(k+3)$ leg of $a$ whose foot will be in $X$
and $(k+4)$ leg of $a^{\prime}$ whose foot will be in $Y$ with the
$(k+2)$-rainbow path between the two foots. Similarly we can connect any
vertex $a\in A$ with any vertex in $x\in D\cup N^{1}(D)$ by using the $(k+3)$
leg of $a$ if $x\in Y$ and the $(k+4)$ leg of $a$ otherwise. Hence
$G^{\prime}$ is rainbow coloured using colours $1$ to $k+4$.
Now only the vertices of $B$ remain. All of them have at least two neighbours
in $G^{\prime}$. Colour one edge to $G^{\prime}$ with $k+5$ and all the other
edges with $k+6$. It is easily seen that we now have a rainbow colouring of
entire $G$. ∎
### 3.3 Proof of Theorem 8
Statement. (i) Every connected graph $G$ of order $n$ and minimum degree
$\delta$ has a connected two-step dominating set $D$ of size at most
$\frac{3(n-|N^{2}(D)|)}{\delta+1}-2$. (ii) Every connected graph $G$ of order
$n\geq 4$ and minimum degree $\delta$ has a connected two-way two-step
dominating set $D^{\prime}$ of size at most $\frac{3n}{\delta+1}-2$. Moreover,
for every $\delta\geq 2$, there exist infinitely many connected graphs $G$
such that $\gamma^{2}_{c}(G)\geq\frac{3(n-2)}{\delta+1}-4$.
###### Proof.
The case when $\delta\leq 1$ can be checked easily. So we assume $\delta\geq
2$ and execute the following two stage procedure.
1. Stage $1$.
$D=\\{u\\}$, for some $u\in V(G)$.
While $N^{3}(D)\neq\emptyset$,
{
Pick any $v\in N^{3}(D)$. Let $(v,x_{2},x_{1},x_{0})$, $x_{0}\in D$ be a
shortest $v\mbox{--}D$ path.
$D=D\cup\\{x_{1},x_{2},v\\}$.
}
2. Stage $2$.
$D^{\prime}=D$.
While $\exists v\in N^{2}(D^{\prime})$ such that $|N(v)\cap
N^{2}(D^{\prime})|\geq\delta-1$,
{
$D^{\prime}=D^{\prime}\cup\\{x_{1},v\\}$ where $(v,x_{1},x_{0})$, $x_{0}\in
D^{\prime}$ is a shortest $v\mbox{--}D^{\prime}$ path.
}
Clearly $D$ remains connected after every iteration in Stage $1$. Since Stage
$1$ ends only when $N^{3}(D)=\emptyset$, the final $D$ is a two step
dominating set. Let $k_{1}$ be the number of iterations executed in Stage $1$.
$|D\cup N^{1}(D)|\geq\delta+1$ when Stage $1$ starts. Since a new vertex from
$N^{3}(D)$ is added to $D$, $|D\cup N^{1}(D)|$ increases by at least
$\delta+1$ in every iteration. Therefore, when Stage $1$ ends,
$k_{1}+1\leq\frac{|D\cup N^{1}(D)|}{\delta+1}=\frac{n-|N^{2}(D)|}{\delta+1}$.
Since $D$ starts as a singleton set and each iteration adds $3$ more vertices,
$|D|=3k_{1}+1\leq\frac{3(n-|N^{2}(D)|)}{\delta+1}-2$. This proves Part (i) of
the theorem.
$D^{\prime}$ remains a connected two-step dominating set throughout Stage $2$.
Stage $2$ ends only when every vertex $v\in N^{2}(D^{\prime})$ has at most
$\delta-2$ neighbours in $N^{2}(D^{\prime})$. Hence at least two neighbours of
$v$ are in $N^{1}(D^{\prime})$. Moreover, since $\delta\geq 2$, there are no
pendant vertices in $G$. So the final $D^{\prime}$ is a connected two-way two-
step dominating set. Let $k_{2}$ be the number of iterations executed in Stage
$2$. Since we add to $D^{\prime}$ a vertex who has at least $\delta-1$
neighbours in $N^{2}(D^{\prime})$, $|N^{2}(D^{\prime})|$ reduces by at least
$\delta$ in every iteration. Since we started with $|N^{2}(D)|$ vertices,
$k_{2}\leq\frac{|N^{2}(D)|}{\delta}$. Since we add $2$ vertices to
$D^{\prime}$ in each iteration,
$|D^{\prime}|=|D|+2k_{2}\leq\frac{3(n-|N^{2}(D)|)}{\delta+1}-2+\frac{2|N^{2}(D)|}{\delta}\leq\frac{3n}{\delta+1}-2$
for $\delta\geq 2$. This proves Part (ii).
For every $\delta>2$, construction for infinitely many graphs $G$ with
$diam(G)=\frac{3(n-2)}{\delta+1}-1$ is reported in [Erdős et al., 1989] and
[Caro et al., 2008]. It is easy to see that for every graph $G$,
$diam(G)\leq\gamma^{2}_{c}(G)+3$. Hence
$\gamma^{2}_{c}(G)\geq\frac{3(n-2)}{\delta+1}-4$ for every graph in that
family. ∎
### 3.4 Proof of Theorem 4
Among the remarks made below Theorem 4, only (i), (ii) and (v) are non-
trivial. Proof of (ii) can be found in [Corneil et al., 1997]. We give proofs
of (i) and (v) below.
Statement (i). Every interval graph $G$ which is not isomorphic to a complete
graph has a dominating path of length at most $diam(G)-2$.
###### Proof.
Consider an interval representation of $G=(V,E)$. For $u\in V$ let $l(u)$ and
$r(u)$ represent the left end point and the right end point of $u$,
respectively. Let $A=\min_{u\in V}r(u)$ and $B=\max_{u\in V}l(u)$. Let $a$ and
$b$ be vertices such that $r(a)=A$ and $l(b)=B$. Let $S_{1}=\\{u\in V|l(u)\leq
A\\}$ and $S_{2}=\\{u\in V|r(u)\geq b\\}$. Clearly $S_{1}$ and $S_{2}$ induce
cliques in $G$ and thus we can assume that for each $u\in S_{1}$, $l(u)=A$ and
for each $u\in S_{2}$, $r(u)=B$. Thus the intervals corresponding to $a$ and
$b$ are point intervals. Since $G$ is not a complete graph, $A\neq B$.
Moreover, since $G$ is connected, there exists $u\in S_{1}$ and $v\in S_{2}$
such that $u,v$ are not point intervals. Let $x\in S_{1}$ and $y\in S_{2}$ be
two not necessarily distinct vertices such that the distance from $x$ to $y$
is minimum among all such pairs. Clearly $x\neq a$, $y\neq b$ and the shortest
path $P$ between $x$ and $y$ is a dominating path in $G$. Moreover, since $a$
and $b$ are point intervals, $d(a,b)\geq d(x,y)+2$. Hence length of $P$ is at
most $d(a,b)-2\leq diam(G)-2$ as required. ∎
Statement (v). Every circular arc graph $G$, which is not an interval graph,
has a dominating cycle of diameter at most $diam(G)$.
###### Proof.
Let $\mathcal{C}$ denote the circle in the circular arc representation of $G$.
We will use the same symbol to denote a vertex of $G$ and its corresponding
arc if there is no chance of confusion. Let $C_{k}=(c_{1},c_{2},\cdots,c_{k})$
be a minimum collection of arcs that cover $\mathcal{C}$. It is easy to see
that $C_{k}$ is a dominating cycle of $G$. We claim that $diam(C_{k})\leq
diam(G)$.
For contradiction, let us assume $diam(C_{k})>diam(G)$. Hence there exists
$c_{i},c_{j}\in V(C_{k}),\;i<j$, such that their distance in $G$ is less than
their distance in $C_{k}$. Let $\mathcal{S}_{1}$ and $\mathcal{S}_{2}$ denote
the two disjoint segments of $\mathcal{C}\backslash(c_{i}\cup c_{j})$. Let
$(c_{i}=x_{0},x_{1},\cdots,x_{l},x_{l+1}=c_{j})$ be a shortest
$c_{i}\mbox{--}c_{j}$ path in $G$. The set of arcs
$X=(x_{1},x_{2},\cdots,x_{l})$ will surely cover at least one of
$\mathcal{S}_{1}$ or $\mathcal{S}_{2}$. Let $R:=(c_{i+1},c_{i+2},\cdots
c_{j-1})$ and $L:=C_{k}\mbox{--}(R\cup\\{c_{i},c_{j}\\})$. Since arcs
corresponding to $C_{k}$ cover the circle, the arcs corresponding one of them,
say $L$ will cover the $\mathcal{S}_{i}$ not covered by $X$. By assumption
$|L|,|R|>|X|=l$. So we can get a smaller collection of arcs covering
$\mathcal{C}$ by replacing $R$ with $X$ in $C_{k}$ contradicting the
minimality of $C_{k}$. ∎
#### Tight Examples
We give examples to show that the upper bounds in $(i)$, $(iii)$ and $(iv)$ of
Theorem 4 are tight. We also give an example to show that the upper bound in
$(ii)$ is nearly tight.
$x_{1}$$x_{2}$$x_{3}$$x_{d-3}$$x_{d-2}$$x_{d-1}$$y_{1}$$y_{2}$$y_{3}$$y_{4}$$z_{1}$$z_{2}$$z_{3}$$z_{4}$
Figure 1: Example of an interval graph with minimum degree $2$, diameter $d$
and rainbow connection number $d+1$.
###### Example 1 (An interval graph $G$ with $\delta(G)\geq 2$ and $rc(G)\geq
d+1$ for any given diameter $d$).
Consider the graph in Figure 1. It is an interval graph with minimum degree
$2$ and diameter $d$. We claim that it cannot be rainbow coloured using $d$
colours.
Let $Y=\\{y_{1},y_{2},y_{3},y_{4}\\}$ and $Z=\\{z_{1},z_{2},z_{3},z_{4}\\}$.
Every pair $(y,z)\in Y\times Z$ is at a distance $d$ apart and they have only
one $d$-length path between them. Hence every shortest $Y\mbox{--}Z$ path
should be rainbow coloured. So in any rainbow colouring which used only $d$
colours, every $Y\mbox{--}x_{1}$ edge is forced to share the same colour.
Hence there is no rainbow path between $y_{1}$ and $y_{3}$.
###### Example 2 (An AT-free graph $G$ with $\delta(G)\geq 2$ and
$rc(G)=diam(G)+2$).
$K_{2,n}$, the complete bipartite graph with $2$ vertices in one part and $n$
in the other, is an AT-free graph with minimum degree $2$ and diameter $2$.
For $n\geq 10$, its rainbow connection number is known to be $4$ (Theorem
$2.6$ in [Chartrand et al., 2008]).
$e$$y_{1}$$y_{2}$$x_{1}$$x_{2}$$x_{3}$$x_{n-3}$$x_{n-2}$$x_{n-1}$ Figure 2:
Example of a threshold graph with minimum degree $2$ and rainbow connection
number $3$. Also an example of a bridge-less chordal graph with
$rc(G)=3.rad(G)$.
###### Example 3 (A threshold graph $G$ with $\delta(G)\geq 2$ and $rc(G)\geq
3$).
Consider the graph $G$ in Figure 2 which can be obtained by adding an edge $e$
between the two vertices in the smaller part of $K_{2,n-1}$, $n\geq 10$. It is
easily seen to be a threshold graph (Two dominating vertices, $y_{1}$ and
$y_{2}$, can be given a weight $1$, others a weight $0$ and threshold being
$1$). For contradiction let us assume that $G$ can be coloured using $2$
colours. Subdividing $e$ gives a $K_{2,n}$. It is easy to see that by
retaining the colour of $e$ to one of the two new edges and giving a third
colour to the other is a rainbow colouring of $K_{2,n}$. This is a
contradiction to the fact that $rc(K_{2,n})=4$ for $n\geq 10$. (Theorem $2.6$
in [Chartrand et al., 2008]).
###### Example 4 (A chain graph $G$ with $\delta(G)\geq 2$ and $rc(G)\geq
4$).
$K_{2,n}$ is a chain graph with minimum degree $2$ and diameter $2$. For
$n\geq 10$, it is known to have a rainbow connection number of $4$ (Theorem
$2.6$ in [Chartrand et al., 2008]).
### 3.5 Proof of Theorem 5
###### Lemma 11.
If $v$ is a vertex of eccentricity $r$ in a bridge-less chordal graph $G$,
then $G[\bigcup_{i=0}^{l}{N^{i}(v)}]$ is a bridge-less chordal graph for all
$l\in\\{0,1,\cdots,r\\}$.
###### Proof.
It is enough to show that $G^{\prime}=G[\bigcup_{i=0}^{r-1}{N^{i}(v)}]$ is a
bridge-less chordal graph. The general result will follow by repeated
application of the above. Every induced subgraph of a chordal graph is also
chordal. Hence it suffices to show that $G^{\prime}$ is bridge-less.
For contradiction, let us assume that $b=(x,y)\in E(G^{\prime})$ is a bridge
of $G^{\prime}$. Consider a BFS tree $T^{\prime}$ of $G^{\prime}$ rooted on
$v$. Since $b$ is a bridge, $b\in E(T^{\prime})$ (else $G^{\prime}\backslash
b$ will be connected). Without loss of generality let $x\in N^{i-1}(v)$ and
$y\in N^{i}(v)$, $i\leq r-1$. Since $G$ is bridge-less by assumption, there
exists a path from $x$ to $y$ in $G\backslash b$. Consider a shortest such
path $P$. Since $P$ is a shortest path, $P\cup b$ is an induced cycle in $G$.
Since $G^{\prime}\backslash b$ is disconnected, $P$ has to contain at least
one vertex $z$ from $N^{r}(v)$. Further, since $x\in N^{i-1}(v),\;i-1\leq r-2$
cannot be adjacent to $z$, $P$ should contain at least one more vertex $w$
from $G^{\prime}$. Hence $P\cup\\{b\\}$ is an induced cycle of length at least
$4$ in $G$ which contradicts the assumption that $G$ is chordal. ∎
With the above lemma, now we can easily give the proof of Theorem 5
Statement. If $G$ is a bridge-less chordal graph, then $rc(G)\leq 3.rad(G)$.
Moreover, there exists a bridge-less chordal graph with $rc(G)=3.rad(G)$.
###### Proof.
We will prove the statement by an induction on radius. Any graph with radius
zero is a singleton vertex which can be rainbow coloured using zero colours.
Hence the statement is true for radius zero. Let the statement be true up till
a radius of $r-1$.
Now, let $G$ be any bridge-less chordal graph with radius $r$. Let $v$ be a
central vertex of $G$, i.e., a vertex with eccentricity $r$. By Lemma 11,
$G^{\prime}=G[\bigcup_{i=0}^{r-1}{N^{i}(v)}]$ is also a bridge-less chordal
graph and its radius is at most $r-1$. Hence by induction hypothesis
$rc(G^{\prime})\leq 3(r-1)$. Since minimum degree is at least two for any
bridge-less graph, $V(G^{\prime})$ is a connected two-way dominating set for
$G$. Hence by Theorem 1, $rc(G)\leq rc(G^{\prime})+3\leq 3r$. Thus the
statement is true for all values of radius.
Consider the graph $G$ in Figure 2. It is a bridge-less chordal graph with
radius $1$ and rainbow connection number is $3$. (See the argument under
Example 3 in Section 3.4.)
∎
### 3.6 Proof of Theorem 6
Statement. If $G$ is a unit interval graph such that $\delta(G)\geq 2$, then
$rc(G)=diam(G)$.
###### Proof.
Let $G$ be a unit interval graph with $\delta(G)\geq 2$. Consider a unit
interval representation of $G$. For $u\in V(G)$, let $l(u)$ and $r(u)$
represent the left and right end points of $u$ respectively. Let $x$ and $y$
be the vertices corresponding to the intervals with leftmost left end and
rightmost right end respectively. Consider a shortest path $P$ between $x$ and
$y$, say $P=(x=x_{1},x_{2},\ldots,x_{k}=y)$. Clearly $k\leq diam(G)+1$ and $P$
is a dominating path in $G$. Let $S_{i}=\\{u\in V(G)|l(u)\leq r(x_{i})\leq
r(u)\\}$, $i=1,2,\cdots,k-1$. It is easily seen that each $S_{i}$ induces a
clique in $G$. Let $H$ be a subgraph of $G$ with
$V(H)=\bigcup_{i=1}^{k-1}{S_{i}}$ and $E(H)=\bigcup_{i=1}^{k-1}E(G[S_{i}])$.
Since $G$ is a unit interval graph, $H$ contains all edges incident on $P$
(including those in $P$). Hence $H$ is a spanning subgraph of $G$. Colour
every edge in $E(G[S_{i}])\backslash\bigcup_{j=1}^{i-1}{E(G[S_{j}])}$ with
colour $i$ for $i=1,2,\cdots,k-1$. This colours every edge of $H$ using at
most $k-1$ colours. Colour the remaining edges of $G$ using colour $1$. We
claim that this is a rainbow colouring of $G$.
For any pair of vertices, $u$ and $v$ in $V(G)$, consider any shortest path
$R$ between them in $H$. Clearly $R$ does not contain more than one edge from
a single clique. In the above colouring, two edges of $H$ will get the same
colour only if they belong to the same clique. Hence $R$ is a rainbow path. So
$rc(G)\leq k-1\leq diam(G)$. Since diameter is a lower bound for rainbow
connection number, $rc(G)=diam(G)$. ∎
## References
* [Caro et al., 2008] Caro, Y., Lev, A., Roditty, Y., Tuza, Z., and Yuster, R. (2008). On rainbow connection. the electronic journal of combinatorics, 15(R57):1.
* [Chakraborty et al., 2009] Chakraborty, S., Fischer, E., Matsliah, A., and Yuster, R. (2009). Hardness and algorithms for rainbow connection. Journal of Combinatorial Optimization, pages 1–18.
* [Chartrand et al., 2008] Chartrand, G., Johns, G., McKeon, K., and Zhang, P. (2008). Rainbow connection in graphs. Math. Bohem, 133(1):85–98.
* [Chartrand and Zhang, 2008] Chartrand, G. and Zhang, P. (2008). Chromatic graph theory. Chapman & Hall.
* [Corneil et al., 1997] Corneil, D., Olariu, S., and Stewart, L. (1997). Asteroidal triple-free graphs. SIAM journal on discrete mathematics, 10(3):399–430.
* [Erdős et al., 1989] Erdős, P., Pach, J., Pollack, R., and Tuza, Z. (1989). Radius, diameter, and minimum degree. Journal of Combinatorial Theory, Series B, 47(1):73–79.
* [Krivelevich and Yuster, 2010] Krivelevich, M. and Yuster, R. (2010). The rainbow connection of a graph is (at most) reciprocal to its minimum degree. Journal of Graph Theory, 63(3):185–191.
* [Schiermeyer, 2009] Schiermeyer, I. (2009). Rainbow connection in graphs with minimum degree three. In Fiala, J., Kratochvíl, J., and Miller, M., editors, Combinatorial Algorithms, volume 5874 of Lecture Notes in Computer Science, pages 432–437. Springer Berlin / Heidelberg.
* [Yannakakis, 1982] Yannakakis, M. (1982). The complexity of the partial order dimension problem. SIAM Journal on Algebraic and Discrete Methods, 3(3):351–358.
|
arxiv-papers
| 2010-10-12T04:44:21 |
2024-09-04T02:49:13.704550
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "L. Sunil Chandran, Anita Das, Deepak Rajendraprasad and Nithin M.\n Varma",
"submitter": "Deepak Rajendraprasad",
"url": "https://arxiv.org/abs/1010.2296"
}
|
1010.2301
|
# Rapidity distribution as a probe for elliptical flow at intermediate
energies
Sanjeev Kumar Varinderjit Kaur Suneel Kumar suneel.kumar@thapar.edu School
of Physics and Materials Science, Thapar University, Patiala-147004, Punjab
(India)
###### Abstract
Interplay between the spectator and participant matter in heavy-ion collisions
is investigated within isospin dependent quantum molecular dynamics (IQMD)
model in term of rapidity distribution of light charged particles. The effect
of different type and size rapidity distributions is studied in elliptical
flow. The elliptical flow patterns show important role of the nearby spectator
matter on the participant zone. This role is further explained on the basis of
passing time of the spectator and expansion time of the participant zone. The
transition from the in-plane to out-of-plane is observed only when the mid-
rapidity region is included in the rapidity bin, otherwise no transition
occurs. The transition energy is found to be highly sensitive towards the size
of the rapidity bin, while weakly on the type of the rapidity distribution.
The theoretical results are also compared with the experimental findings and
are found in good agreement.
###### pacs:
25.70.-z, 25.75.Ld
††preprint: APS/123-QED
## I Introduction
Since last many years, investigation about the nuclear equation of state
(NEOS) at the extreme conditions of density and temperature has been one of
the primary driving forces in heavy ion studies at intermediate energies. The
interest in low energies, however, are for isospin effects in fusion process
Aich91 ; Kuma105 . These investigations has been performed with the help of
rare phenomena such as multifragmentation, collective flow, particle
production as well as nuclear stopping Kuma105 ; Stoc86 ; West935 ; Luka055 ;
Andr055 ; Luka045 ; Sood065 ; Chen065 ; Sing005 ; Saks105 ; Zhan065 ; Huan93 ;
Gyul835 . The relation between the nuclear EOS and flow phenomena has been
explored extensively in the simulations.
Recently the analysis of transverse-momentum dependence of elliptical flow has
also been put forwarded Luka055 ; Andr055 ; Dani00 . The elliptical flow is
shaped by the interplay between the geometry and mean field and, when gated by
the transverse momentum, reveals the momentum dependence of the mean field at
supra-normal densities. The parameter of the elliptic flow is quantified by
the second-order Fourier coefficient Volo965
$v_{2}~{}=~{}<cos2\phi>~{}=~{}\langle\frac{p_{x}^{2}-p_{y}^{2}}{p_{x}^{2}+p_{y}^{2}}\rangle,$
(1)
from the azimuthal distribution of detected particles at mid rapidity as
$\frac{dN}{d\phi}=p_{0}(1+2v_{1}cos\phi+2v_{2}cos2\phi+.....),$ (2)
where $p_{x}$ and $p_{y}$ are the ${\it x}$ and ${\it y}$ components of
momentum. The $p_{x}$ is in the reaction plane, while, $p_{y}$ is
perpendicular to the reaction plane and $\phi$ is the azimuthal angle of
emitted particles momentum relative to the x-axis. The positive values of
$<cos2\phi>$ reflect preferential in-plane emission, while negative values
reflect preferential out-of-plane emission. The pulsating of sign observed
recently at intermediate energies has received particular attention as it
reflects the increased pressure buildup in the non isotropic collision zone
Adle03 .
After the pioneering measurements at Saturne Demo90 and Bevalac Gutb89 , a
wealth of experimental results have been obtained at Bevalac and SIS Luka055 ;
Andr055 ; Luka045 ; Wang96 as well as at AGS Pink995 , SPS Adle03 and RHIC
Acke01 . In recent years, the FOPI, INDRA, and PLASTIC BALL Collaborations
Luka055 ; Andr055 ; Luka045 are actively involved in measuring the excitation
function of elliptical flow from Fermi energies to relativistic one. In most
of these studies, collisions of ${}_{79}Au^{197}~{}+~{}_{79}Au^{197}$ is
undertaken. Interestingly, elliptical flow was reported to change from
positive (in-plane) taken. negative (out-of-plane) values around 100
MeV/nucleon and maximum squeeze-out is observed around 400 MeV/nucleon. These
observations were reported recently by us and others theoretically Luka055 ;
Andr055 ; Zhan065 ; Dani00 . A careful analysis reveals that elliptic flow is
very sensitive towards the choice of rapidity cut which makes the difference
between spectator and participant matter.
The elliptical flow pattern of participant matter is affected by the presence
of cold spectators Dani00 . When nucleons are decelerated in the participant
region, the longitudinal kinetic energy associated with the initial colliding
nuclei is converted into the thermal and potential compression energy. In a
subsequent rapid expansion(or explosion), the collective transverse energy
develops Dani00 and many particles from the participant region get emitted
into the transverse directions. The particles emitted towards the reaction
plane can encounter the cold spectator pieces and, hence, get redirected. In
contrast, the particles emitted essentially perpendicular to the reaction
plane are largely unhindered by the spectators. Thus, for the beam energies
leading to rapid expansion in the vicinity of the spectators, elliptic flow
directed out of the reaction plane (squeeze-out) is expected. This squeeze-out
is related with the pace at which expansion develops, and is, therefore,
related to the EOS. This contribution of the participant and spectator matter
Dani00 in the intermediate energy heavy-ion collisions motivated us to
perform a detailed analysis of the excitation function of elliptical flow over
different regions of participant and spectator matter. If excitation function
is found to be affected by the different contributions it will definitely,
also affect the in-plane to out-of-plane emission i.e. transition energy.
The rapidity distribution is an important parameter to study the participant-
spectator contribution in the intermediate energy heavy-ion collisions Sood09
. In this paper, we will study the effect of participant and spectator matters
in term of different rapidity distributions on the excitation function of
elliptical flow. Attempts shall also be made to parameterize the transition
energy in term of different rapidity bins.
The entire work is carried out in the framework of isospin-dependent quantum
molecular dynamics (IQMD) Hart98 . The IQMD model is discussed in detail in
the following section.
## II ISOSPIN-dependent QUANTUM MOLECULAR DYNAMICS (IQMD) MODEL
The IQMD model Saks105 ; Hart98 , which is an improved version of QMD model
Aich91 developed by J. Aichelin and coworkers, then has been used
successfully to various phenomena such as collective flow, disappearance of
flow, fragmentation & elliptical flow
Stoc86 ; Sood065 ; Sood09 ; Acke01 . The isospin degree of freedom enters into
the calculations via symmetry potential, cross-sections and Coulomb
interaction Saks105 ; Hart98 . The details about the elastic and inelastic
cross-sections for proton-proton and neutron-neutron collisions can be found
in Ref. Hart98 .
In IQMD model, the nucleons of target and projectile interact via two and
three-body Skyrme forces, Yukawa potential and Coulomb interactions. In
addition to the use of explicit charge states of all baryons and mesons, a
symmetry potential between protons and neutrons corresponding to the Bethe-
Weizsacker mass formula has been included.
The hadrons propagate using classical Hamilton equations of motion:
$\frac{d\vec{r_{i}}}{dt}~{}=~{}\frac{d\it{\langle~{}H~{}\rangle}}{d\vec{p_{i}}}~{}~{};~{}~{}\frac{d\vec{p_{i}}}{dt}~{}=~{}-\frac{d\it{\langle~{}H~{}\rangle}}{d\vec{r_{i}}},$
(3)
with
$\displaystyle\langle~{}H~{}\rangle$ $\displaystyle=$
$\displaystyle\langle~{}T~{}\rangle+\langle~{}V~{}\rangle$ (4)
$\displaystyle=$
$\displaystyle\sum_{i}\frac{p_{i}^{2}}{2m_{i}}+\sum_{i}\sum_{j>i}\int
f_{i}(\vec{r},\vec{p},t)V^{\it ij}({\vec{r}^{\prime},\vec{r}})$
$\displaystyle\times
f_{j}(\vec{r}^{\prime},\vec{p}^{\prime},t)d\vec{r}d\vec{r}^{\prime}d\vec{p}d\vec{p}^{\prime}.$
The baryon-baryon potential $V^{ij}$, in the above relation, reads as:
$\displaystyle V^{ij}(\vec{r}^{\prime}-\vec{r})$ $\displaystyle=$
$\displaystyle V^{ij}_{Skyrme}+V^{ij}_{Yukawa}+V^{ij}_{Coul}+V^{ij}_{sym}$ (5)
$\displaystyle=$
$\displaystyle\left(t_{1}\delta(\vec{r}^{\prime}-\vec{r})+t_{2}\delta(\vec{r}^{\prime}-\vec{r})\rho^{\gamma-1}\left(\frac{\vec{r}^{\prime}+\vec{r}}{2}\right)\right)$
$\displaystyle+~{}t_{3}\frac{exp(|\vec{r}^{\prime}-\vec{r}|/\mu)}{(|\vec{r}^{\prime}-\vec{r}|/\mu)}~{}+~{}\frac{Z_{i}Z_{j}e^{2}}{|\vec{r}^{\prime}-\vec{r}|}$
$\displaystyle+t_{6}\frac{1}{\varrho_{0}}T_{3}^{i}T_{3}^{j}\delta(\vec{r_{i}}^{\prime}-\vec{r_{j}}).$
Here $Z_{i}$ and $Z_{j}$ denote the charges of $i^{th}$ and $j^{th}$ baryon,
and $T_{3}^{i}$, $T_{3}^{j}$ are their respective $T_{3}$ components (i.e. 1/2
for protons and -1/2 for neutrons). The parameters $\mu$ and
$t_{1},.....,t_{6}$ are adjusted to the real part of the nucleonic optical
potential. For the density dependence of nucleon optical potential, standard
Skyrme-type parameterization is employed. The potential part resulting from
the convolution of the distribution function with the Skyrme interactions
$V_{\it Skyrme}$ reads as :
${\it
V}_{Skyrme}~{}=~{}\alpha\left(\frac{\rho_{int}}{\rho_{0}}\right)+\beta\left(\frac{\rho_{int}}{\rho_{0}}\right)^{\gamma}~{}~{}\cdot$
(6)
The two of the three parameters of equation of state are determined by
demanding that at normal nuclear matter density, the binding energy should be
equal to 16 MeV. The third parameter $\gamma$ is usually treated as a free
parameter. Its value is given in term of the compressibility:
$\kappa~{}=~{}9\rho^{2}\frac{\partial^{2}}{\partial\rho^{2}}\left(\frac{E}{A}\right)~{}~{}\cdot$
(7)
The different values of compressibility give rise to Soft and Hard equations
of state. It is worth mentioning that Skyrme forces are very successful in
analysis of low energy phenomena like fusion, fission and cluster-
radioactivity.
As noted Zhan02 , elliptical flow is weakly affected by the choice of equation
of state. On the other hand, in the refs. Sood065 ; Sood09 ; Mage00 , the hard
equation of state is used to study the directed as well as elliptical flow.
For the present analysis, a hard (H) equation of state has been employed along
with standard energy dependent cross-section.
## III Results and Discussion
For the present analysis, simulations are carried out for thousand of events
for the reaction of ${}_{79}Au^{197}~{}+~{}_{79}Au^{197}$ at semi-central
geometry using a hard equation of state. The whole of the analysis is
performed for light charged particles (LCP’s)[1$\leq$ A $\leq$ 4]. The
reaction conditions and fragments are chosen on the basis of availability of
experimental data Luka055 ; Andr055 ; Luka045 . As noted in Ref. Lehm93 , the
relativistic effects do not play role at these incident energies and the
intensity of sub-threshold particle production is very small. The phase space
generated by the IQMD model has been analyzed using the minimum spanning tree
(MST) Sood065 method. The MST method binds two nucleons in a fragment if
their distance is less than 4 fm. This is the one of the widely used method in
the intermediate energy heavy-ion collisions. However, some improvements like
momentum and binding energy are also discussed in the literature Sood065 . In
recent years, more sophisticated and complicated algorithms are also available
in the literature saca . The entire calculations are performed at $t=200$
fm/c. This time is chosen by keeping in view the saturation of the collective
flow Sood065 .
As our purpose of present study is to understand the effect of participant-
spectator matter on the excitation function of elliptical flow, one will
concentrate on the rapidity distribution only Sood09 . The important concept
of spectators and participants in collisions was first introduced by Bowman et
al. Bowm73 and later employed for the description of a wide-angle energetic
particle emission by Westfall et al. West76 . The two nuclei slamming against
one other can be viewed as producing cylindrical cuts through each other. The
swept-out nucleons or participants (from projectile and target) undergo a
violent collision process. In the meantime, the remnants of the projectile and
target continue with largely undisturbed velocities, and are much less
affected by the collision process than the participant nucleons. On one hand,
this picture is supported by features of the data Andr055 and, on the other,
by dynamic simulations Gait00 . During the violent stage of a reaction, the
spectators can influence the behavior of participant matter. Following the
same, we are dividing the rapidity distribution in the different cuts in term
of $Y_{c.m.}/Y_{beam}$ parameter, which is given as:
$Y(i)=\frac{1}{2}~{}ln\frac{E(i)+P_{z}(i)}{E(i)-P_{z}(i)}$ (8)
where E(i) and $P_{z}(i)$ are respectively, the total energy and longitudinal
momentum of $i^{th}$ particle.
As we are interested in studying the effect of rapidity distribution on the
incident energy dependence of elliptical flow, one has to understand the
$dN/dY$ as a function of $Y_{c.m.}/Y_{beam}~{}=~{}Y^{red}$ at different
energies. For this, in Fig.1, we display the rapidity distribution for the
light charged fragments (LCP’s) at different incident energies. The rapidity
distribution is found to vary drastically throughout the range from
$|Y^{red}|~{}\leq~{}1.75$.
This is indicating the compressed or participant zone around $0$ value and
decay into spectator zone towards both side of the $0$ value. However around
$0$ value, the region between $-0.1$ to $0.1$ is specified as mid-rapidity
region in the Literature Andr055 ; Zhan065 . The decay from $-0.1$ towards
negative side approach to target like spectator, while, other side is known as
projectile like spectator. Interestingly, these regions are found to affect
drastically with incident energy. With increase in incident energy, number of
particles are increasing in the region from $-1.25$ to $1.25$, while
decreasing away from this region on both side. The inset in the figure is
showing the clear view of change in behavior around $-1.25$, which is also
true for other side also. The rapidity distribution is also used as
thermalization source in the literature many times Kuma105 . On the basis of
this, we have divided the rapidity distribution in five different zones out of
which three are with including $|Y^{red}|~{}<~{}0.1$(participant zone), while
other two are with excluding this region (target and projectile
spectator).They are as: (i) From -0.1 $\leq$ $Y^{red}$ $\leq$ 0.1 with
increment of 0.2 on both side upto -1.5 $\leq$ $Y^{red}$ $\leq$ 1.5. (ii) From
0 $\leq$ $Y^{red}$ $\leq$ 0.1 with increment of 0.2 on latter side upto 0
$\leq$ $Y^{red}$ $\leq$ 1.5. (iii) From -0.1 $\leq$ $Y^{red}$ $\leq$ 0 with
decrement of 0.2 the on former side upto -1.5 $\leq$ $Y^{red}$ $\leq$ 0\. (iv)
From $Y^{red}$ $\geq$ 0.1 with increment of 0.2 upto $Y^{red}$ $\geq$ 1.5. and
(v)From $Y^{red}$ $<$ -0.1 with decrement of 0.2 upto $Y^{red}$ $<$ -1.5.
Figure 1: The rapidity distribution dN/dY as a function of $Y_{c.m.}/Y_{beam}$
at different incident energies ranging from 50 to 1000 MeV/nucleon for light
charged particles (LCP’s)[1$\leq$ A $\leq$ 4]. The inset is showing only one
side of the rapidity distribution. Figure 2: The incident energy dependence of
elliptical flow for LCP’s collectively for projectile as well as target matter
including mid-rapidity region. The different lines are at different size of
the rapidity bin, which includes the participant as well as spectator matter.
Let us now understand the effect of these rapidity cuts on the excitation
function of elliptical flow. In Fig.2, we present the excitation function of
elliptical flow for the first condition. This condition is the mixture of the
participant as well as spectator zone from projectile as well as target
matter. There are two points to discuss here. One is the change in the
elliptical flow with incident energy and other is to see the effect of bin
size on elliptical flow. The elliptical flow evolves from a positive value,
rotational-like, emission to an negative value, collective expansion, with
increase in the incident energy. In other words, transition from the in-plane
to out-of-plane takes place. The energy at which this transition takes place
is known as transition energy. This transition is due to the competition
between the mean field at low incident energy and NN collisions at high
incident energies. The incident energy dependence of directed flow also show
such type of transitions from negative to positive value, which is know as
balance energy Sood065 . After this transition, the strength of collective
expansion overcomes the rotational like motion Wang96 . This leads to increase
of out-of-plane emission towards a maximum around 400 MeV/nucleon. This maxima
is further supported by the nuclear stopping at 400 MeV/nucleon Reis04 .
Beyond this energy, elliptical flow decreases indicating a transition to in-
plane preferentially emission Pink995 . This rise and fall behavior of the
elliptical flow in the expansion region is due to the variation in the passing
time $t_{pass}$ of the spectator and expansion time of the participant zone
Andr055 . In a simple participant spectator model,
$t_{pass}~{}=~{}2R/(\gamma_{s}v_{s})$, where R is the radius of the nucleus at
rest, $v_{s}$ is the spectator velocity in c.m. and $\gamma_{s}$ the
corresponding Lorentz factor. Due to the comparable value of the passing time
and expansion time in the collective expansion region up to 400 MeV/nucleon,
the elliptical flow results an interplay between fireball expansion and
spectator shadowing. In other words, due to the comparable size of two
times(the fireball expansion and spectator shadowing), the participant
particles which will come in the way of spectator shadowing are deflected
towards the out-of-plane and hence more squeeze out. However, in the energy
range from 400 MeV/nucleon to 1.49 GeV/nucleon, $t_{pass}$ decreases from 30
to 16 fm/c (not shown here), implying that overall the expansion gets about
two times faster in this energy region Andr055 . This is supported by the
average expansion velocities extracted from the particle spectra Wang96 . In
this case, corresponding to much shorter passing time compared to the
expansion time, the participant zone is affected by the shadowing of the
spectator at very early times, however no shadowing effect is observed at
later time where expansion of the participant matter is still happening. In
other words, the participant particles at later times are not blocked by the
shadowing of spectator and hence decrease is observed in the out-of-plane
emission after 400 MeV/nucleon.
On the other hand, with increase in the rapidity region, the transition energy
is found to affect to a great extent. In literature, the balance energy using
directed flow is also calculated, but was over the entire rapidity
Distribution Sood065 . The rapidity distribution affects many phenomena in
intermediate energy region like particle production, nuclear stopping Kuma105
and now the elliptical flow. The transition energy increases with rapidity
region between $|Y^{red}|~{}\leq~{}0.1$ and $|Y^{red}|~{}\leq~{}1.5$. With the
increase in the rapidity region, dominance of the spectator matter from
projectile as well as target takes place that will further result in the
dominance of the mean field up to higher energies. After the transition
energy, the collective expansion is found to have less squeeze out with an
increase in the rapidity region. In this region, the spectator zone
contribution along with participant zone starts to come in play. As we know,
the passing time for the spectator is very less compared to expansion time of
the participant zone, leading to the decreasing effect of the spectator
shadowing on the participant zone. The chances of the participant to move in-
plane increases with increase in the rapidity bin and hence less squeeze out
is observed with increase in the size of the rapidity bin. If one see
carefully, no transition is observed after $|Y^{red}|~{}\leq~{}1.1$. This is
due to negligible effect of the participant zone compared to the spectator
zone. The inset in the figure shows interesting results: the intersection of
all the rapidity bins takes place at a particular incident energy. Below and
above this energy, incident energy dependence of elliptical flow is changing
the behavior with rapidity distribution. One can have a good study of
elliptical flow below and above this particular incident energy with variation
in the rapidity distribution.
Figure 3: Same as in fig.2, but the contribution for fragments is from the
projectile like matter only. All the bins includes mid-rapidity region. Figure
4: Same as in fig.2, but the contribution for fragments is from the target
like matter only. All the bins includes mid-rapidity region.
In Fig.2, we had displayed the effect on the participant as well as spectator
matter from the projectile as well as target collectively including the
distribution at $|Y^{red}|~{}<~{}0.1$. It becomes quite interesting to study
the participant and spectator matter contribution on the incident energy
dependence of elliptical flow for projectile as well as target matter
separately. For this, in Figs.3 and 4, the incident energy dependence of the
elliptical flow are displayed from the mid-rapidity to spectator zone for
projectile and target matter. Both of the figures are following the universal
behavior of the in-plane to out-of-plane emission with increase in incident
energy, as is displayed in Fig.2. On the other hand, the effect of rapidity
bins is also quite similar as seen in of Fig.2.
If one observes the maximum squeeze out values in Figs. 2-4 around 400
MeV/nucleon, it is found that less squeeze out is observed for projectile
matter and spectator matter, separately as compared to projectile and target
matter, collectively. However, it is almost same for the projectile or target
matter, separately. This is obvious as when rapidity bin is from 0 to 0.1 or
-0.1 to 0, then participant zone is less compared to the region -0.1 to 0.1.
This can be further clarify from the Fig.1, where rapidity distribution for
different bins is displayed. This is true for all the bins under
investigation. The effect of rapidity distribution on the transition energy
will be discussed later under different conditions.
Figure 5: Same as in fig.2, but the contribution for fragments is from the
projectile like matter only. All the bins excludes the mid-rapidity region. In
this case, we move from participant+spectator contribution towards spectator
contribution. Figure 6: Same as in fig.2, but the contribution for fragments
is from the target like matter only. All the bins excludes the mid-rapidity
region. In this case, we move from participant+spectator contribution towards
spectator contribution.
One notices from the above figures that major contribution for elliptical flow
comes from the mid rapidity region. One is further interested to know the fall
of elliptical flow if the mid-rapidity region is excluded. For this, by
excluding the $|Y^{red}|~{}<~{}0.1$ region, we have displayed the incident
energy dependence for projectile and target matter from participant to purely
projectile or target spectator matter in Fig.5 and Fig.6, respectively. The
noted behavior is entirely different compared to previous three figures. The
differences are: (a) The behavior of rise and fall in the elliptical flow at a
particular incident energy is obtained as were in the previous figures, but,
no transition is observed from in-plane to out-of-plane emission. Over all the
incident energies, the value of elliptical flow remains positive, however,
some competition is observed at low incident energies around 150 MeV/nucleon.
This is due to the dominance of attractive mean field from the spectator
matter, which restricts the effects of participant zone.
(b) A higher value of in-plane flow is observed at high incident energies
compared to low incident energies upto $Y^{red}~{}\geq~{}0.7$ in Fig.5 for
projectile matter and $Y^{red}~{}\leq~{}-0.7$ for target matter, which is in
contrast to the previous figures. This is due to the effect that at low
incident energies, the passing time of spectator and expansion time of the
participant and comparable, while, with increase in the incident energy, the
passing time is found to be decrease as compared to expansion time. This will
reduce the shadowing effect at higher incident energies on the particles of
the participant zone, results in the enhanced in plane flow. Moreover, more
higher value of in-plane flow at higher incident energies as compared to lower
one is due to the absence of most compressible region from the region
$|Y^{red}|~{}\leq~{}-0.1$, which competes with the mean field of the spectator
to a great extent.
(c) With increase in the rapidity region from participant and spectator
($Y^{red}~{}\geq~{}0.1$ )to purely spectator matter ($Y^{red}~{}\geq~{}1.5$ ),
the dominance of in-plane flow takes place upto $Y^{red}~{}\geq~{}0.7$ and
after that suddenly fall in the in-plane flow takes place upto
$Y^{red}~{}\geq~{}1.5$ in Fig.5, which is also true for the Fig.6. The fall in
the in-plane flow after $Y^{red}~{}\geq~{}0.7$ can be explained with the help
of the Fig.1. After this region, the contribution of the participant is almost
negligible and size of the spectator also decreases. Due to the decreasing
size of the spectator matter, the violence of the incident energy as well as
short passing time of the spectator, they start to fly out-of-plane, which
were enjoying in-plane earlier due to their heavier size. Hence, minimum in-
plane flow is observed for $Y^{red}~{}\geq~{}1.5$ where only 5-7 particles
exist as is clear from the Fig.1. The same behavior is observed in the Fig.6.
Figure 7: (Color online) Comparison of the results of incident energy
dependence of elliptical flow including different rapidity bins with the
experimental results of different collaborations Luka055 ; Andr055 . The left
hand side is for Z $\leq$ 2 particles, while, right hand side is for the
protons. Figure 8: The dependence of rapidity distributions on the transition
energy. The transition energies are extracted from Fig.2, 3and 4 for top,
middle and bottom panels, respectively. The numbering from 1to 7 is
representing the different rapidity bin from the respective figures mentioned
above. All the panels are parameterized with the straight line interpolation
$Y=mX~{}+~{}C$, where m is the slope, displayed in respective panels.
Going through all the aspects of rapidity distribution, it is important to
include the mid-rapidity region $|Y^{red}|~{}<~{}0.1$ in the study of
elliptical flow in heavy-ion collisions. In order to compare the findings with
the experimental one, one must have the transition from in-plane to out-of-
plane, which are observed in Figs.2-4. Moreover, more negative values are
observed in Fig.2 compared to other two one. From this discussion, it is
fruitful to compare the findings of Fig.2 with experimental findings of
different collaborations. This is displayed in Fig.7 for LCP’s (Z $\leq$ 2)
and protons. The curves have the similar trend as displayed in Fig.2.
Interestingly, it is observed that there is a competition in the rapidity bin
$|Y^{red}|\leq 0.1$ and $|Y^{red}|\leq 0.3$. After that systematic deviation
is observed in the elliptical flow values from the data values with increase
in the rapidity bin. There are some deviation in the middle region in case of
LCP’s between theory and data, while data is well explained for the protons.
From here, it is concluded that one can vary the mid rapidity region from the
$|Y^{red}|\leq 0.1$ to $|Y^{red}|\leq 0.3$ to get the better agreement with
the experimental one. However, the discrepancy with the data for LCP’s can be
reduced by varying the isospin-dependent cross sections.
Last, but not least, the rapidity distribution dependence of the transition
energy is displayed in the Fig.8. The top, middle and bottom panels are
representing the transition energies extracted from Fig.2-4, respectively. The
numbering from 1 to 7 in the figure is representing the bin size from the
respective figures. All the curves are fitted with straight line equation
$Y~{}=~{}mX~{}+~{}C$, where m is slope of line and C is a constant. The
transition energy is found to be sensitive towards the different bins of
rapidity distributions. It is observed that transition energy is found to
increase with the size of the rapidity bin for light charged particles. The
necessary condition for the transition energy is that the mid-rapidity region
must be included in the rapidity distribution bin. It is also observed that no
transition energy is obtained when the rapidity distribution region extends
away from $|Y_{red}|~{}\leq~{}1.1$. This is due to the dominance of the
spectator matter, which enjoys in-plane as compared to out-of-plane. The
transition energy is found to be weakly sensitive towards the choice of the
different type of rapidity distributions (projectile-target or projectile or
target) while we have included the mid-rapidity region. However, transition
energy is found to disappear when mid-rapidity region is excluded. In other
studies, system size dependence of the transition energy is also studied by us
and others for different kind of Fragments Zhan065 . Recently, we have studied
the fragment size dependence of transition energy under the influence of
different equations of state, nucleon-nucleon cross sections etc Sood065 .
These studies were made at a fixed rapidity bin. The detailed analysis of the
transition energy with rapidity distributions has revealed many interesting
aspects for the first time.
For more interest, in the near future, we are performing the comparative study
of different QMD and IQMD simulations. From the preliminary results, it is
observed that isospin content of the colliding partners $(N/Z~{}=~{}1.49)$ is
supposed to playing the appreciable role in the analysis of elliptical flow in
intermediate energy heavy-ion collisions in term of (i) closeness of the
results with experimental findings of different collaborations and (ii) effect
on the elliptical flow with change in the rapidity bins. The detailed analysis
of these findings will be presented in the near future.
## IV Conclusion
Within the semi classical transport simulations of energetic semi central
collisions of ${}_{79}Au^{197}~{}+~{}_{79}Au^{197}$ reaction, we have carried
out a new investigation of the interplay between the participant and spectator
regions in term of rapidity distributions. The maxima and minima in the
incident energy dependence of elliptical flow is produced due to the different
contributions of passing time of the spectator and expansion time of the
participant. The shadowing of the spectator matter plays important role up to
later times due to the comparable magnitude of the passing and expansion time
up to energy 400 MeV/nucleon, however at high energies, shadowing effect is
dominant only at earlier times due to the shorter passing time as compared to
expansion time. The transition from in-plane to out-of-plane is observed only
when the mid-rapidity region is included in the rapidity bin, otherwise, no
transition is observed. The transition energy is found to be strongly
dependent on the size of the rapidity bin, while, weakly dependent on the type
of the rapidity distributions. The transition energy is parameterized with a
straight line interpolation. Comparison with experimental bin, reveals the
competition is observed between the rapidity bin of $|Y^{red}|~{}\leq~{}0.1$
and $|Y^{red}|~{}\leq~{}0.3$. To remove this discrepancy in the middle region
for LCP’s, one has to reduce the strength of nucleon-nucleon cross-section.
###### Acknowledgements.
This work has been supported by the Grant no. 03(1062)06/ EMR-II, from the
Council of Scientific and Industrial Research (CSIR) New Delhi, Govt. of
India.
## References
* (1) J. Aichelin, Phys. Rep. 202, 233(1991);G. Peilert, H. Stocker, W. Greiner, A. Rosenhauer, A. Bohnet and J. Aichelen, Phys. Rev. C 39, 1402(1989).
* (2) S. Kumar, S. Kumar and R. K. Puri, Phys. Rev. C 81, 014601 (2010); G. Lehaut et al., Phys. Rev. Lett. 104, 232701 (2010); S. Kumar and S. Kumar, Chin. Phys. Lett. 27, 062504 (2010). R. K. Puri, S. S. Malik and R. K. Gupta, Eur. Phys. Lett. 9, 767(1989); R. K. Puri and R. K. Gupta, J. Phys. G: Nucl. Part. Phys. 18, 903(1992); R. K. Gupta, M. Balasubramanium, R. K. Puri and W. Schied, J. Phys. G: Nucl. Part. Phys. 26, L23(2000). R. K. Puri and R. K. Gupta, Phys. Rev. C 45, 1837(1992); R. K. Puri, P. Chattopadhyay and R.K. Gupta, Phys. Rev. C 43, 315(1991).
* (3) A. Le Fevre and J. Aichelin, Phys. Rev. Lett. 100, 042701 (2008); A. Le Fevre et al,Phys. Rev. C 80, 044615(2009); Y. K. Vermani, J. K. Dhawan, S. Goyal, R. K. Puri and J. Aichelin, J. Phys. G: Nucl. Part. Phys. 37, 015105(2010).
* (4) G. D. Westfall et al., Phys. Rev. Lett. 71, 1986 (1993); M. B. Tsang et al., Phys. Rev. C 53, 1959 (1996); Y. M. Zheng, C. M. Ko, B. A. Li, and B. Zhang, Phys. Rev. Lett. 83, 2534 (1999); A. B. Larionov, W. Cassing, C. Greiner, and U. Mosel, Phys. Rev. C 62, 064611 (2000); B. A. Li, A. T. Sustich, and B. Zhang, ibid. 64, 054604 (2001); C. Alt et al., ibid.68, 034903 (2003).
* (5) J. Lukasik, G. Auger, and M. L. Begemann-Blaich et al., Phys. Lett. B 608, 223 (2005).
* (6) A. Andronic et al., Nucl. Phys. A 679, 765 (2001); Phys. Lett. B 612, 173 (2005).
* (7) J. Lukasik, et.al., INDRA Collaborations, Int. Workshop on Multifragmentation and related topics (IWM 2003) Caen, France (2003).
* (8) A. D. Sood and R. K. Puri, Phys. Rev. C 69, 054612 (2004); Eur. Phys. A 30, 571 (2006); A. D. Sood and R. K. Puri, Phys. Rev. C 70, 034611 (2004); S. Kumar, M. K. Sharma, R. K. Puri, K. P. Singh, and I. M. Govil, ibid. 58, 3494 (1998), A. D. Sood, R. K. Puri, and J. Aichelin, Phys. Lett. B 594, 260 (2004).S. Kumar and R. K. Puri, Phys. Rev. C 60, 054607 (1999); S. Kumar, S. Kumar, and R. K. Puri, Phys. Rev. C 78, 064602 (2008); Y. K. Vermani, S. Goyal, and R. K. Puri, ibid.79, 064613 (2009); R. Chugh and R. K. Puri, Phys. Rev. C 82, 014603 (2010).
* (9) L. W. Chen and C. M. Ko, Phys. Lett. B 634, 205 (2006); Phys. Rev. C 73, 014906 (2006).
* (10) J. Singh, S. Kumar, and R. K. Puri, Phys. Rev. C 62, 044617 (2000); S. Kumar and R. K. Puri, ibid.58, 1618 (1998).
* (11) S. Gautam et al., J. of Phys. G 37, 085102 (2010).
* (12) Y. Zhang and Z. Li, Phys. Rev. C 74, 014602 (2006).
* (13) S. W. Huang, A. Faesseler, G. Q. Li, R. K. Puri, E. Lehmann, D. T. Khoa, and M. A. Matin, Phys. Lett. B 298, 41 (1993); G. Batko et al., J. Phys. G: Nucl. Part. Phys. 20, 461(1994).
* (14) M. Gyulassy, K. A. Frankel, and H. Stöcker, Phys. Lett. B 110, 185 (1982).
* (15) P. Danielewicz, Nucl. Phys. A673, 375 (2000). T. Z. Yan et al., Chin. Phys. lett. 26, 112501 (2009).
* (16) S. Voloshin and Y. Zhang, Z. Phys. C 70, 665 (1996).
* (17) C. Adler et al., Phys. Rev. Lett. 90, 032301 (2003).
* (18) M. Demoulins et al., Phys. Lett. B 241, 476 (1990).
* (19) H. H. Gutbrod et al., Phys. Lett. B 216, 267 (1989).
* (20) S. Wang et al., Phys. Rev. Lett. 76, 3911 (1996); D. Brill et al., Z. Phys. A 355, 61 (1996); M.B. Tsang et al., Phys. Rev. C 53, 1959 (1996); N. Bastid et al., Nucl. Phys. A 622, 573 (1997); P. Crochet et al., Nucl. Phys. A 624, 755 (1997); G. Stoicea et al., Phys. Rev. Lett. 92, 072303 (2004).
* (21) C. Pinkenburg et al., Phys. Rev. Lett. 83, 1295 (1999); A. Andronic et al., Eur. Phys. J A 30, 31 (2006).
* (22) K.H. Ackermann et al., Phys. Rev. Lett. 86, 402 (2001).
* (23) A. D. Sood and R. K. Puri, Phys. Rev. C 79, 064618 (2009).
* (24) C. Hartnack et al., Eur. Phys. J. A1, 151 (1998).
* (25) H. Y. Zhang et al., J. Phys. G: Nucl. and Part. 28, 2397 (2002).
* (26) D. J. Magestro, W. Bauer, and G. D. Westfall, Phys. Rev. C 62, 041603(R) (2000); J. Singh and R. K. Puri, Phys. Rev. C 62, 054602 (2000); A. D. Sood and R. K. Puri, Phys. Rev. C 73, 067602 (2006).
* (27) E. Lehmann et.al., Prog. Part. Nucl. Phys. 30, 219 (1993); Phys. Rev. C 51, 2113 (1995).
* (28) R. K. Puri, Ch. Hartnack and J. Aichelin, Phys. Rev. C 54, R28 (1996). P. B. Gossiaux, R. K. Puri, Ch. Hartnack and J. Aichelin, Nucl. Phys. A 619, 379(1997). R. K. Puri and J. Aichelin, J. Comp. Phys. 162, 245(2000).
* (29) J. D. Bowman, W. J. Swiatecki and C. F. Tsang, Lawrence Berkeley Laboratory Report No. LBL-2908, 1973 (unpublished).
* (30) G. Westfall et al., Phys. Rev. Lett. 18, 1202 (1976).
* (31) T. Gaitanos, H. H. Wolter, C. Fuchs, Phys. Lett. B 478, 79 (2000); J. Gosset, H. H. Gotbrod, W. G. Meyer, A. M. Poskanzer, A. Sandoval, R. Stock, and G. D. Westfall, Phys. Rev. C 16, 629 (1977).
* (32) W. Reisdorf et al., Phys. Rev. Lett. 92, 232301 (2004).
|
arxiv-papers
| 2010-10-12T05:12:14 |
2024-09-04T02:49:13.714150
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Sanjeev Kumar, Varinderjit Kaur, and Suneel Kumar",
"submitter": "Sanjeev Kumar",
"url": "https://arxiv.org/abs/1010.2301"
}
|
1010.2306
|
11footnotetext: Department of Mathematics, Huzhou University, Huzhou,
313000,China, Email: 071018034@fudan.edu.cn.22footnotetext: School of
Sciences, China University of Mining and Technology, Xuzhou, 221008, China,
Email: yzsung@gmail.com
# Non-zero Sum Stochastic Differential Games of Fully Coupled Forward-
Backward Stochastic Systems
Maoning Tang1 Qingxin Meng1 Yongzheng Sun2
###### Abstract
In this paper, an open-loop two-person non-zero sum stochastic differential
game is considered for forward-backward stochastic systems. More precisely,
the controlled systems are described by a fully coupled nonlinear multi-
dimensional forward-backward stochastic differential equation driven by a
multi-dimensional Brownian motion. one sufficient (a verification theorem) and
one necessary conditions for the existence of open-loop Nash equilibrium
points for the corresponding two-person non-zero sum stochastic differential
game are proved. The control domain need to be convex and the admissible
controls for both players are allowed to appear in both the drift and
diffusion of the state equations.
Keywords: N-person differential games, forward-backward stochastic
differential equation, Nash equilibrium point.
## 1 Introduction
Differential Game theory had been an active area of research and a useful tool
in many applications, particularly in biology and economic. The so called
differential games are the ones in which the position, being controlled by the
players, evolves continuously. On the one hand, since the study of
differential games was initiated by Isaacs [18], many papers (see [4, 3, 5, 6,
7, 13, 14, 15]) have appeared which developed the foundations for two-person
zero sum differential games. For this case, there a single performance
criterion which one player tries to minimize and the other tries to maximize.
On the other hand, many authors (see [16, 8, 9, 11, 19, 22, 23, 25, 27, 26,
30]discussed N-person non-zero sum differential games. For this case, there
may be more than two players and each player tries to minimize his individual
performance criterion, and the sum of all player’s criteria is not zero or is
it constant.
All the above mentioned paper are restricted deterministic system. On the
differential games of stochastic systems, we can refer to[2, 17, 29]. In 2008,
Tang and Li [28] established the minimax principle for N-person differential
games governed by forward stochastic systems with the control appearing in the
diffusion term. In 2010, wang and Yu [31] studied the Non-zero sum
differential games of backward stochastic systems, and they established a
necessary condition and a sufficient condition in the form of stochastic
maximum principle for open-loop Nash equilibrium.
Forward-Backward stochastic systems are not only used in mathematical
economics (see Antonelli [1], Duffie and Epstein [10], for example), but also
used in mathematical finance(see El Karoui, Peng and Quenez [12]). It now
becomes more clear that certain important problems in mathematical economics
and mathematical finance, especially in the optimization problem, can be
formulated to be Forward-backward stochastic system. So the optimal control
problem for Forward-backward stochastic system and the corresponding
stochastic maximum principle are extensively studied in this literature. We
refer to [33, 32, 24] and references therein. They established the necessary
maximum principle in the case the control domain is convex or the forward
diffusion coefficients can not contain a control variable. In 2010, Yong[34]
proved necessary conditions for the optimal control of forward-backward
stochastic systems where the control domain is not assumed to be convex and
the control appears in the diffusion coefficient of the forward equation.
In this paper we will discuss non-zero sum stochastic differential games for
forward-backward stochastic systems. To our best knowledge, very little work
has been published on this subject. In section 2, we state the problem and our
main assumptions. In section 3, we state and prove our main results: a
sufficient condition for the existence of open-loop Nash equilibrium point
which can check whether the candidate equilibrium points are optimal or not.
Section 4 is devoted to present a necessary condition for the existence of
open-loop Nash equilibrium point by the stochastic maximum principle for the
optimal control of the optimal control problem of forward-backward stochastic
systems established in [32].
Moreover, we refer to [21, 20] on the existence and uniqueness of solutions to
the fully coupled forward-backward stochastic differential equations.
## 2 Problem formulation and main assumptions
Let $(\Omega,{\mathcal{F}},\\{{\mathcal{F}}_{t}\\}_{t\geq 0},P)$ be a complete
probability space, on which a $d$-dimensional standard Brownian motion
$B(\cdot)$ is defined with $\\{{\mathcal{F}}_{t}\\}_{t\geq 0}$ being its
natural filtration, augmented by all $P$-null sets in ${\mathcal{F}}.$ Let
$T>0$ be a fixed time horizon. Let E be a Euclidean space. The inner product
in E is denoted by $\langle\cdot,\cdot\rangle$, and the norm in E is denoted
by $|\cdot|.$ We further introduce some other spaces that will be used in the
paper. Denote by $L^{2}(\Omega,{\mathcal{F}}_{T},P;E)$ the the set of all
$E$-valued ${\mathcal{F}}_{T}$-measurable random variable $\eta$ such that
$E|\eta|^{2}<\infty.$ Denote by $M^{2}(0,T;E)$ the set of all $E$-valued
$\mathcal{F}_{t}$-adapted stochastic processes $\\{\varphi(t):t\in[0,T]\\}$
which satisfy $E\int_{0}^{T}|\varphi(t)|^{2}dt<\infty.$ Denote by
$\mathcal{S}^{2}(0,T;E)$ the set of all $E$-valued $\mathcal{F}_{t}$-adapted
continuous stochastic processes $\\{\varphi(t):t\in[0,T]\\}$ which satisfy
$E\sup_{0\leq t\leq T}|\varphi(t)|^{2}dt<\infty.$
In this paper, we consider the system which is given by a controlled fully
coupled nonlinear forward-backward stochastic differential equations (abbr.
FBSDEs) of the form
$\displaystyle\left\\{\begin{array}[]{lll}dx(t)&=&b(t,x(t),y(t),z(t),u_{1}(t),u_{2}(t))dt\\\
&{}{}&+\sigma(t,x(t),y(t),z(t),u_{1}(t),u_{2}(t))dB(t),\\\ \displaystyle
dy(t)&=&-f(t,x(t),y(t),z(t),u_{1}(t),u_{2}(t))dt\\\ &{}{}&+z(t)dB(t),\\\
\displaystyle x(0)&=&a,\\\ \displaystyle y(T)&=&\xi.\par\end{array}\right.$
(2.1)
Here $\displaystyle b:[0,T]\times R^{n}\times R^{m}\times R^{m\times
d}\times{U}_{1}\times{U}_{2}\rightarrow R^{n},\displaystyle\sigma:[0,T]\times
R^{n}\times R^{m}\times R^{m\times d}\times{U}_{1}\times{U}_{2}\rightarrow
R^{n\times d},\displaystyle f:[0,T]\times R^{n}\times R^{m}\times R^{m\times
d}\times{U}_{1}\times{U}_{2}\rightarrow R^{m}$ are given mapping, $a$ and
$T>0$ are given constants, and$\xi\in
L^{2}(\Omega,{\mathcal{F}}_{T},P;R^{m})$. The processes $u_{1}(\cdot)$ and
$u_{2}(\cdot)$ in the system (2.1) are the open-loop control processes which
present the controls of the two players, required to have values in two given
nonempty convex sets ${U}_{1}\subset R^{k_{1}}$ and ${U}_{2}\subset R^{k_{2}}$
respectively. The admissible control process $(u_{1}(\cdot),u_{2}(\cdot))$ is
defined as a ${\mathcal{F}}_{t}$-adapted process with values in $U_{1}\times
U_{2}$ such that
$E\displaystyle\int_{0}^{T}(|u_{1}(t)|^{2}+|u_{2}(t)|^{2})dt<+\infty.$ The set
of all admissible control processes is denoted by
${\mathcal{A}}_{1}\times{\mathcal{A}}_{2}.$
For each one of the two player, there is a cost functional
$\begin{array}[]{ll}&J_{i}(u_{1}(\cdot),u_{2}(\cdot))\\\
=&E\bigg{[}\displaystyle\int_{0}^{T}l_{i}(t,x(t),y(t),z(t),u_{1}(t),u_{2}(t))dt\\\
&+\phi_{i}(x(T))+h_{i}(y(0))\bigg{]},\end{array}$ (2.2)
where $l_{i}:[0,T]\times R^{n}\times R^{m}\times R^{m\times
d}\times{\mathcal{U}}_{1}\times{\mathcal{U}}_{2}\rightarrow
R,\displaystyle\phi_{i}:R^{n}\rightarrow R,\displaystyle
h_{i}:R^{m}\rightarrow R$ are given mapping $(i=1,2)$.
Now we make the main assumptions throughout the paper.
###### Assumption 2.1.
$f,g,\sigma$ are continuously differentiable with respect to
$(x,y,z,u_{1},u_{2})$. The derivatives of $f,g,\sigma$ are bounded. For any
admissible control $(u_{1}(\cdot),u_{2}(\cdot)),$ the forward-backward
stochastic system satisfies the assumptions (H2.1) and (H2.2) in Wu[32].
###### Assumption 2.2.
$l_{i},\phi_{i}$ and $h_{i}$ are continuously differentiable with respect to
$(x,y,z,u_{1},u_{2}),x$ and $y,(i=1,2).$ And $l_{i}$ is bounded by
$C(1+|x|^{2}+|y|^{2}+|z|^{2}+|u_{1}|^{2}+|u_{2}|^{2}).$ And the derivatives of
$l_{i}$ are bounded by $C(1+|x|+|y|+|z|+|u_{1}|+|u_{2}|).$ And $\phi_{i}$ and
$h_{i}$ are bounded by $C(1+|x|^{2})$ and $C(1+|y|^{2})$ respectively. And the
derivatives of $\phi_{i}$ and $h_{i}$ with respect to $x$ and $y$ are bounded
by $C(1+|x|)$ and $C(1+|y|)$ respectively. $(i=1,2)$.
Under Assumption 2.1, from Theorem 2.1 in Wu [32], we see that for any given
admissible control $(u_{1}(\cdot),u_{2}(\cdot)$, the system (2.1) admits a
unique solution
$(x(\cdot),y(\cdot),z(\cdot))\in S_{\mathcal{F}}^{2}(0,T;R^{n})\times\in
S_{\mathcal{F}}^{2}(0,T;R^{m})\times\in M_{\mathcal{F}}^{2}(0,T;R^{m\times
d}).$
Then we call $(x(\cdot),y(\cdot),z(\cdot))$ the state process corresponding to
the control process $(u_{1}(\cdot),u_{2}(\cdot)$ and
$((u_{1}(\cdot),u_{2}(\cdot);y(\cdot),q(\cdot),z(\cdot))$ the admissible pair.
Furthermore, from Assumption 2.2, it is easy to check
that$|J_{i}(u_{1}(\cdot),u_{2}(\cdot))|<\infty.$$(i=1,2).$
Then we can pose the following two-person non-zero sum stochastic differential
game problem
###### Problem 2.1.
Find an open-loop admissible control
$(\bar{u}_{1}(\cdot),\bar{u}_{2}(\cdot))\in\mathcal{A}_{1}\times\mathcal{A}_{2}$
such that
$J_{1}(\bar{u}_{1}(\cdot),\bar{u}_{2}(\cdot))=\displaystyle\inf_{u_{1}(\cdot)\in{\mathcal{A}}_{1}}J_{1}({u}_{1}(\cdot),\bar{u}_{2}(\cdot))$
(2.3)
and
$J_{2}(\bar{u}_{1}(\cdot),\bar{u}_{2}(\cdot))=\displaystyle\inf_{u_{2}(\cdot)\in{\mathcal{A}}_{2}}J_{2}(\bar{u}_{1}(\cdot),u_{2}(\cdot)).$
(2.4)
Any
$(\bar{u}_{1}(\cdot),\bar{u}_{2}(\cdot))\in\mathcal{A}_{1}\times\mathcal{A}_{2}$
satisfying the above is called a open-loop Nash equilibrium point of Problem
2.1. Such an admissible control allows two players to play individual optimal
control strategies simultaneously.
## 3 A Verification Theorem
In this section we state and prove a verification theorem for the Nash
equilibrium points of Problem 2.1.
For any given admissible pair
$(u_{1}(\cdot),u_{2}(\cdot);x(\cdot),y(\cdot),z(\cdot)),$ We can introduce the
following adjoint forward-backward stochastic differential equations of the
system (2.1)
$\left\\{\begin{array}[]{ll}\displaystyle
dk^{i}(t)=&-\bigg{[}b_{y}^{*}(t,x(t),y(t),z(t),u_{1}(t),u_{2}(t))p^{i}(t)\\\
&+\sigma_{y}^{*}(t,x(t),y(t),z(t),u_{1}(t),u_{2}(t))q^{i}(t)\\\
&-f_{y}^{*}(t,x(t),y(t),z(t),u_{1}(t),u_{2}(t))k^{i}(t)\\\
&\displaystyle+l_{iy}(t,x(t),y(t),z(t),u_{1}(t),u_{2}(t))\bigg{]}dt\\\
{}{}&-\bigg{[}b_{z}^{*}(t,x(t),y(t),z(t),u_{1}(t),u_{2}(t))p^{i}(t)\\\
&\displaystyle+\sigma_{z}^{*}(t,x(t),y(t),z(t),u_{1}(t),u_{2}(t))q^{i}(t)\\\
{}{}&-f_{z}^{*}(t,x(t),y(t),z(t),u_{1}(t),u_{2}(t))k^{i}(t)\\\
&+l_{iz}(t,x(t),y(t),z(t),u_{1}(t),u_{2}(t))\bigg{]}dB(t)\\\ \displaystyle
dp^{i}(t)=&-\bigg{[}b_{x}^{*}(t,x(t),y(t),z(t),u_{1}(t),u_{2}(t))p^{i}(t)\\\
&+\sigma_{x}^{*}(t,x(t),y(t),z(t),u_{1}(t),u_{2}(t))q^{i}(t)\\\
{}{}&-f_{x}(t,x(t),y(t),z(t),u_{1}(t),u_{2}(t))k^{i}(t)\\\
&\displaystyle+l_{ix}(t,x(t),y(t),z(t),u_{1}(t),u_{2}(t))\bigg{]}dt\\\
&+q^{i}(t)dB(t)\\\ \displaystyle
k^{i}(0)=&-h_{iy}(y_{0}),~{}~{}~{}p^{i}(T)=\phi_{ix}(x(T)),\\\ &~{}~{}0\leq
t\leq T,(i=1.2).\end{array}\right.$ (3.1)
Under Assumptions2.1-2.2, according to Theorem 2.2 in [32], the above adjoint
equation has a unique solution
$(k^{i}(\cdot),p^{i}(\cdot),q^{i}(\cdot))\in{\mathcal{S}}_{\mathcal{F}}(0,T;R^{m})\times\in{\mathcal{S}}_{\mathcal{F}}^{2}(0,T;R^{n})\times\in
M_{\mathcal{F}}(0,T;R^{n\times d}),(i=1.2).$
We define the Hamiltonian functions $H_{i}:[0,T]\times R^{n}\times R^{m}\times
R^{m\times d}\times{\mathcal{U}}_{1}\times{\mathcal{U}}_{2}\times R^{n}\times
R^{n\times d}\times R^{m}\rightarrow R$ by
$\begin{array}[]{ll}&\displaystyle H_{i}(t,x,y,z,u_{1},u_{2},p,q,k)=\langle
k,-f(t,x,y,z,u_{1},u_{2}\rangle\\\ &+\langle
p,b(t,x,y,z,u_{1},u_{2})\rangle+l_{i}(t,x,y,z,u_{1},u_{2})\\\
\displaystyle&+\langle
q,\sigma(t,x,y,z,u_{1},u_{2})\rangle,(i=1,2).\end{array}$ (3.2)
Then we can rewrite the equations (3.1) in Hamiltonian system’s form:
$\left\\{\begin{array}[]{ll}\displaystyle
dk^{i}(t)&=-H_{iy}(t,x(t),y(t),z(t),u_{1}(t),u_{2}(t),p^{i}(t),q^{i}(t),k^{i}(t))dt\\\
&-H_{iz}(t,x(t),y(t),z(t),u_{1}(t),u_{2}(t),p^{i}(t),q^{i}(t),k^{i}(t))dB(t)\\\
\displaystyle
dp^{i}(t)&=-H_{ix}(t,x(t),y(t),z(t),u_{1}(t),u_{2}(t),p^{i}(t),q^{i}(t),k^{i}(t))dt\\\
&+q^{i}(t)dB(t)\\\
k^{i}(0)=&-h_{iy}(y_{0}),~{}~{}~{}p^{i}(T)=\phi_{ix}(x(T)),(i=1,2).\end{array}\right.$
(3.3)
We are now coming to a verification theorem for an Nash equilibrium point of
Problem 2.1.
###### Theorem 3.1.
Under Assumptions 2.1-2.2, let
$(\bar{u}_{1}(\cdot),\bar{u}_{2}(\cdot);\bar{x}(\cdot),\bar{y}(\cdot),\bar{z}(\cdot))$
be an admissible pair. Let
$({\bar{p}^{i}}(\cdot),\bar{q}^{i}(\cdot),\bar{k}^{i}(\cdot))$$(i=1,2)$ be the
unique solution of the corresponding adjoint equation (3.1). Suppose that for
almost all $(t,\omega)\in[0,T]\times\Omega$ , $(x,y,z,u_{1})\mapsto
H_{1}(t,x,y,z,{u}_{1},\bar{u}_{2}(t),\\\
\bar{p}^{1}(t),\bar{q}^{1}(t),\bar{k}^{1}(t))$ is convex with respect to
$(x,y,z,u_{1})$, $(x,y,z,u_{2})\mapsto
H_{2}(t,x,y,z,\bar{u}_{1}(t),{u}_{2},\bar{p}^{2}(t),\\\
\bar{q}^{2}(t),\bar{k}^{2}(t))$ is convex with respect to $(x,y,z,u_{2})$,
$x\mapsto h_{i}(x)$ is convex with respect with to $x$, and
$y\mapsto\phi_{i}(y)$ is convex with respect to $y$ (i=1,2), and the following
optimality condition holds
$\begin{array}[]{ll}&\displaystyle\max_{u_{1}\in{\mathcal{U}}_{1}}H_{1}(t,\bar{x}(t),\bar{y}(t),\bar{z}(t),u_{1},\bar{u}_{2}(t),\bar{p}^{1}(t),\bar{q}^{1}(t),\bar{k}^{1}(t))\\\
&~{}~{}~{}~{}=H_{1}(t,\bar{x}(t),\bar{y}(t),\bar{z}(t),\bar{u}_{1}(t),\bar{u}_{2}(t),\bar{p}^{1}(t),\bar{q}^{1}(t),\bar{k}^{1}(t)),\end{array}$
(3.4)
and
$\begin{array}[]{ll}&\displaystyle\max_{u_{2}\in{\mathcal{U}}_{2}}H_{2}(t,\bar{x}(t),\bar{y}(t),\bar{z}(t),\bar{u}_{1}(t),u_{2},\bar{p}^{2}(t),\bar{q}^{2}(t),\bar{k}^{2}(t))\\\
&~{}~{}~{}~{}=H_{2}(t,\bar{x}(t),\bar{y}(t),\bar{z}(t),\bar{u}_{1}(t),\bar{u}_{2}(t),\bar{p}^{2}(t),\bar{q}^{2}(t),\bar{k}^{2}(t)).\end{array}$
(3.5)
Then $(\bar{u}_{1}(\cdot),\bar{u}_{2}(\cdot))$ is Nash equilibrium point of
Problem 2.1
###### Proof.
(i) we consider an stochastic optimal control problem. The system is the
following controlled forward-backward stochastic differential equation
$\displaystyle\left\\{\begin{array}[]{lll}dx(t)&=&b(t,x(t),y(t),z(t),u_{1}(t),\bar{u}_{2}(t))dt\\\
&&+\sigma(t,x(t),y(t),z(t),u_{1}(t),\bar{u}_{2}(t))dB(t),\\\ \displaystyle
dy(t)&=&-f(t,x(t),y(t),z(t),u_{1}(t),\bar{u}_{2}(t))dt\\\ &&+z(t)dB(t),\\\
\displaystyle x(0)&=&a\\\ \displaystyle y(T)&=&\xi,\end{array}\right.$ (3.6)
where $u_{1}(\cdot)$ is any given admissible control in $\mathcal{A}_{1}.$ The
cost function is defined as
$\begin{array}[]{ll}&J_{1}(u_{1}(\cdot),\bar{u}_{2}(\cdot))\\\
=&E\bigg{[}\displaystyle\int_{0}^{T}l_{1}(t,x(t),y(t),z(t),u_{1}(t),\bar{u}_{2}(t))dt\\\
&+\phi_{1}(x(T))+h_{1}(y_{0})\bigg{]},\end{array}$ (3.7)
where $(x(\cdot),y(\cdot),z(\cdot))$ is the solution to the forward-backward
stochastic system (3.6) corresponding to the control
$u_{1}(\cdot)\in\mathcal{A}_{1}.$
The optimal control problem is minimize $J(u_{1}(\cdot),\bar{u}_{2}(\cdot))$
over $u_{1}(\cdot)\in{\mathcal{A}}_{1}$. Now will show the admissible control
$\bar{u}_{1}(\cdot)$ is an optimal control of the problem, i.e,
$J_{1}(\bar{u}_{1}(\cdot),\bar{u}_{2}(\cdot))=\displaystyle\min_{u_{1}(\cdot)\in{\mathcal{A}}_{1}}J_{1}(u_{1}(\cdot),\bar{u}_{2}(\cdot)).$
(3.8)
In fact, Let $u_{1}(\cdot)$ be any admissible control in ${\mathcal{A}}_{1},$
$(x(\cdot),y(\cdot),z(\cdot))$ be the corresponding state process of the
system (3.6). It is easy to check that for the control $\bar{u}_{1}(\cdot)$,
the corresponding state process of the system (3.6) is indeed
$(\bar{x}(\cdot),\bar{y}(\cdot),\bar{z}(\cdot)).$
From (3.7), we have
$\begin{array}[]{ll}&J_{1}(u_{1}(\cdot),\bar{u}_{2}(\cdot))-J_{1}(\bar{u}_{1}(\cdot),\bar{u}_{2}(\cdot))\\\
&~{}~{}~{}~{}~{}~{}=E\displaystyle\int_{0}^{T}\bigg{[}l_{1}(t,x(t),y(t),z(t),u_{1}(t),\bar{u}_{2}(t))\\\
&~{}~{}~{}~{}~{}~{}~{}~{}~{}-l_{1}(t,\bar{x}(t),\bar{y}(t),\bar{z}(t),\bar{u}_{1}(t),\bar{u}_{2}(t))\bigg{]}dt\\\
&~{}~{}~{}~{}~{}~{}~{}~{}~{}+E\bigg{[}\phi_{1}(x(T))-\phi_{1}(\bar{x}(T))\displaystyle\bigg{]}\\\
&~{}~{}~{}~{}~{}~{}~{}~{}~{}+E\bigg{[}h_{1}(y(0))-h_{1}(\bar{y}(0))\bigg{]}\\\
&~{}~{}~{}~{}~{}~{}=I_{1}+I_{2},\end{array}$ (3.9)
where
$\displaystyle\begin{split}I_{1}&=E\displaystyle\int_{0}^{T}\bigg{[}l_{1}(t,x(t),y(t),z(t),u_{1}(t),\bar{u}_{2}(t))\\\
&~{}~{}~{}~{}-l_{1}(t,\bar{x}(t),\bar{y}(t),\bar{z}(t),\bar{u}_{1}(t),\bar{u}_{2}(t))\bigg{]}dt,\end{split}$
(3.10)
$\begin{array}[]{ll}\displaystyle
I_{2}=E\bigg{[}\phi_{1}(x(T))-\phi_{1}(\bar{x}(t))\displaystyle\bigg{]}+E\bigg{[}h_{1}(y(0))-h_{1}(\bar{y}(0))\bigg{]}.\end{array}$
(3.11)
Using Convexity of $\phi_{1}$ and $h_{1}$, and Itô formula to
$\langle\bar{p}^{1}(t),x(t)-\bar{x}(t)\rangle+\langle\bar{k}^{1}(t),y(t)-\bar{y}(t)\rangle,$
we get
$\begin{array}[]{ll}I_{2}&=E\big{[}\phi_{1}(x(T))-\phi_{1}(\bar{x}(T))\big{]}+E\big{[}h_{1}(y(0)-h_{1}(\bar{y}(0))\big{]}\\\
&\geq E\langle\phi_{1x}(\bar{x}(T)),x(T)-\bar{x}(T)\rangle+E\langle
h_{1y}(\bar{y}_{0}),y_{0}-\bar{y}_{0}\rangle\\\
&=E\langle\bar{p}^{1}(T)),x(T)-\bar{x}(T)\rangle+E\langle\bar{k}^{1}(0),y_{0}-\bar{y}_{0}\rangle\\\
&=-E\displaystyle\int_{0}^{T}\langle
H_{1x}(t,\bar{x}(t),\bar{y}(t),\bar{z}(t),\bar{u}_{1}(t),\bar{u}_{2}(t),\bar{p}^{1}(t),\bar{q}^{1}(t),\bar{k}^{1}(t)),x(t)-\bar{x}(t)\rangle
dt\\\ &~{}~{}-E\displaystyle\int_{0}^{T}\langle
H_{1y}(t,\bar{x}(t),\bar{y}(t),\bar{z}(t),\bar{u}_{1}(t),\bar{u}_{2}(t),\bar{p}^{1}(t),\bar{q}^{1}(t),\bar{k}^{1}(t)),y(t)-\bar{y}(t)\rangle
dt\\\ &~{}~{}-E\displaystyle\int_{0}^{T}\langle
H_{1z}(t,\bar{x}(t),\bar{y}(t),\bar{z}(t),\bar{u}_{1}(t),\bar{u}_{2}(t),\bar{p}^{1}(t),\bar{q}^{1}(t),\bar{k}^{1}(t)),z(t)-\bar{z}(t)\rangle
dt\\\
&~{}~{}+E\displaystyle\int_{0}^{T}\langle\bar{p}^{1}(t),b(t,x(t),y(t),z(t),u_{1}(t),\bar{u}_{2}(t))-b(t,\bar{x}(t),\bar{y}(t),\bar{z}(t),\bar{u}_{1}(t),\bar{u}_{2}(t))\rangle
dt\\\
&~{}~{}+E\displaystyle\int_{0}^{T}\langle\bar{q}^{1}(t),\sigma(t,x(t),y(t),z(t),u_{1}(t),\bar{u}_{2}(t))-\sigma(t,\bar{x}(t),\bar{y}(t),\bar{z}(t),\bar{u}_{1}(t),\bar{u}_{2}(t))\rangle
dt\\\
&~{}~{}+E\displaystyle\int_{0}^{T}\langle\bar{k}^{1}(t),-(f(t,x(t),y(t),z(t),u_{1}(t),\bar{u}_{2}(t))-f(t,\bar{x}(t),\bar{y}(t),\bar{z}(t),\bar{u}_{1}(t),\bar{u}_{2}(t)))\rangle
dt\\\ &=-J_{1}+J_{2},\end{array}$ (3.12)
where
$\begin{array}[]{ll}J_{1}&=E\displaystyle\int_{0}^{T}\langle
H_{1x}(t,\bar{x}(t),\bar{y}(t),\bar{z}(t),\bar{u}_{1}(t),\bar{u}_{2}(t),\bar{p}^{1}(t),\bar{q}^{1}(t),\bar{k}^{1}(t)),x(t)-\bar{x}(t)\rangle
dt\\\ &~{}~{}+E\displaystyle\int_{0}^{T}\langle
H_{1y}(t,\bar{x}(t),\bar{y}(t),\bar{z}(t),\bar{u}_{1}(t),\bar{u}_{2}(t),\bar{p}^{1}(t),\bar{q}^{1}(t),\bar{k}^{1}(t)),y(t)-\bar{y}(t)\rangle
dt\\\ &~{}~{}+E\displaystyle\int_{0}^{T}\langle
H_{1z}(t,\bar{x}(t),\bar{y}(t),\bar{z}(t),\bar{u}_{1}(t),\bar{u}_{2}(t),\bar{p}^{1}(t),\bar{q}^{1}(t),\bar{k}^{1}(t)),z(t)-\bar{z}(t)\rangle
dt,\\\
J_{2}=&E\displaystyle\int_{0}^{T}\langle\bar{p}^{1}(t),b(t,x(t),y(t),z(t),u_{1}(t),\bar{u}_{2}(t))-b(t,\bar{x}(t),\bar{y}(t),\bar{z}(t),\bar{u}_{1}(t),\bar{u}_{2}(t))\rangle
dt\\\
&~{}~{}+E\displaystyle\int_{0}^{T}\langle\bar{q}^{1}(t),\sigma(t,x(t),y(t),z(t),u_{1}(t),\bar{u}_{2}(t))-\sigma(t,\bar{x}(t),\bar{y}(t),\bar{z}(t),\bar{u}_{1}(t),\bar{u}_{2}(t))\rangle
dt\\\
&~{}~{}+E\displaystyle\int_{0}^{T}\langle\bar{k}^{1}(t),-(f(t,x(t),y(t),z(t),u_{1}(t),\bar{u}_{2}(t))-f(t,\bar{x}(t),\bar{y}(t),\bar{z}(t),\bar{u}_{1}(t),\bar{u}_{2}(t)))\rangle
dt\end{array}$
and we have used the fact that
$y(T)-\bar{y}(T)=\xi-\xi=0,x(0)-\bar{x}(0)=a-a=0.$
On the other hand, in view of the definition of Hamilton function $H_{1}$ (see
(3.2)), the integration $I_{1}$ can be rewritten as
$\begin{array}[]{ll}I_{1}=&E\displaystyle\int_{0}^{T}\bigg{[}l_{1}(t,x(t),y(t),z(t),u_{1}(t),\bar{u}_{2}(t))\\\
&~{}~{}~{}~{}-l_{1}(t,\bar{x}(t),\bar{y}(t),\bar{z}(t),\bar{u}_{1}(t),\bar{u}_{2}(t))\bigg{]}dt\\\
~{}~{}~{}=&E\displaystyle\int_{0}^{T}\bigg{[}H_{1}(t,x(t),y(t),z(t),u_{1}(t),\bar{u}_{2}(t),\bar{p}^{1}(t),\bar{q}^{1}(t),\bar{k}^{1}(t))\\\
&-H_{1}(t,\bar{x}(t),\bar{y}(t),\bar{z}(t),\bar{u}_{1}(t),\bar{u}_{2}(t),\bar{p}^{1}(t),\bar{q}^{1}(t),\bar{k}^{1}(t))\bigg{]}dt\\\
&-E\displaystyle\int_{0}^{T}\langle\bar{p}^{1}(t),b(t,x(t),y(t),z(t),u_{1}(t),\bar{u}_{2}(t))-b(t,\bar{x}(t),\bar{y}(t),\bar{z}(t),\bar{u}_{1}(t),\bar{u}_{2}(t))\rangle
dt\\\
&-E\displaystyle\int_{0}^{T}\langle\bar{q}^{1}(t),\sigma(t,x(t),y(t),z(t),u_{1}(t),\bar{u}_{2}(t))-\sigma(t,\bar{x}(t),\bar{y}(t),\bar{z}(t),\bar{u}_{1}(t),\bar{u}_{2}(t))\rangle
dt\\\
&-E\displaystyle\int_{0}^{T}\langle\bar{k}^{1}(t),-(f(t,x(t),y(t),z(t),u_{1}(t),\bar{u}_{2}(t))-f(t,\bar{x}(t),\bar{y}(t),\bar{z}(t),\bar{u}_{1}(t),\bar{u}_{2}(t)))\rangle
dt\\\ ~{}~{}~{}=&J_{3}-J_{2},\end{array}$ (3.13)
where
$\displaystyle\begin{split}J_{3}=&E\displaystyle\int_{0}^{T}\bigg{[}H_{1}(t,x(t),y(t),z(t),u_{1}(t),\bar{u}_{2}(t),\bar{p}^{1}(t),\bar{q}^{1}(t),\bar{k}^{1}(t))\\\
&-H_{1}(t,\bar{x}(t),\bar{y}(t),\bar{z}(t),\bar{u}_{1}(t),\bar{u}_{2}(t),\bar{p}^{1}(t),\bar{q}^{1}(t),\bar{k}^{1}(t))\bigg{]}dt\end{split}$
(3.14)
From the optimality condition (3.4), we have
$\begin{array}[]{ll}&\bigg{\langle}H_{1u_{1}}(t,\bar{x}(t),\bar{y}(t),\bar{z}(t),\bar{u}_{1}(t),\bar{u}_{2}(t),\bar{p}^{1}(t),\bar{q}^{1}(t),\bar{k}^{1}(t)),u_{1}(t)-\bar{u}_{1}(t)\bigg{\rangle}\geq
0,a.s.a.e..\end{array}$ (3.15)
Using convexity of
$H_{1}(t,x,y,z,u_{1},\bar{u}_{2}(t),\bar{p}^{1}(t),\bar{q}^{1}(t),\bar{k}^{1}(t))$
with respect to $(x,y,z,u_{1})$, and noting (3.14) and (3.15), we have
$\begin{array}[]{ll}J_{3}&\geq E\displaystyle\int_{0}^{T}\langle
H_{1x}(t,\bar{x}(t),\bar{y}(t),\bar{z}(t),\bar{u}_{1}(t),\bar{u}_{2}(t),\bar{p}^{1}(t),\bar{q}^{1}(t),\bar{k}^{1}(t)),x(t)-\bar{x}(t)\rangle
dt\\\ &~{}~{}+E\displaystyle\int_{0}^{T}\langle
H_{1y}(t,\bar{x}(t),\bar{y}(t),\bar{z}(t),\bar{u}_{1}(t),\bar{u}_{2}(t),\bar{p}^{1}(t),\bar{q}^{1}(t),\bar{k}^{1}(t)),y(t)-\bar{y}(t)\rangle
dt\\\ &~{}~{}+E\displaystyle\int_{0}^{T}\langle
H_{1z}(t,\bar{x}(t),\bar{y}(t),\bar{z}(t),\bar{u}_{1}(t),\bar{u}_{2}(t),\bar{p}^{1}(t),\bar{q}^{1}(t),\bar{k}^{1}(t)),z(t)-\bar{z}(t)\rangle
dt\\\ &=J_{1}.\end{array}$ (3.16)
Therefore, it follows from (3.9), (3.12),(3.13) and (3.16) that
$\begin{array}[]{ll}J(u_{1}(\cdot),\bar{u}_{2}(\cdot))-J(\bar{u}_{1}(\cdot),\bar{u}_{2}(\cdot))&=I_{1}+I_{2}=(J_{3}-J_{2})+I_{2}\\\
&\geq(J_{1}-J_{2})+(-J_{1}+J_{2})=0.\end{array}$
Since $u_{1}(\cdot)\in{\mathcal{A}}_{1}$ is arbitrary, we conclude that
$\displaystyle\begin{split}J_{1}(\bar{u}_{1}(\cdot),\bar{u}_{2}(\cdot))=\displaystyle\min_{u_{1}(\cdot)\in{\mathcal{A}}_{1}}J_{1}(u_{1}(\cdot),\bar{u}_{2}(\cdot)).\end{split}$
(3.17)
(ii) Now we consider another stochastic optimal control problem. The system is
the following controlled forward-backward stochastic differential equation
$\displaystyle\left\\{\begin{array}[]{lll}dx(t)&=&b(t,x(t),y(t),z(t),\bar{u}_{1}(t),{u}_{2}(t))dt\\\
&&+\sigma(t,x(t),y(t),z(t),\bar{u}_{1}(t),{u}_{2}(t))dB(t)\\\ \displaystyle
dy(t)&=&-f(t,x(t),y(t),z(t),\bar{u}_{1}(t),{u}_{2}(t))dt\\\ &&+z(t)dB(t)\\\
\displaystyle x(0)&=&a\\\ \displaystyle y(T)&=&\xi,\end{array}\right.$ (3.18)
where $u_{2}(\cdot)$ is any given admissible control in $\mathcal{A}_{2}.$ The
cost function is defined as
$\begin{array}[]{ll}&J_{2}(\bar{u}_{1}(\cdot),{u}_{2}(\cdot))\\\
=&E\bigg{[}\displaystyle\int_{0}^{T}l_{2}(t,x(t),y(t),z(t),\bar{u}_{1}(t),{u}_{2}(t))dt\\\
&+\phi_{2}(x(T))+h_{2}(y_{0})\bigg{]},\end{array}$ (3.19)
where $(x(\cdot),y(\cdot),z(\cdot))$ is the solution to the system (3.18)
corresponding to the control $u_{2}(\cdot)\in\mathcal{A}_{2}.$
The optimal control problem is minimize $J(\bar{u}_{1}(\cdot),{u}_{2}(\cdot))$
over $u_{2}(\cdot)\in{\mathcal{A}}_{2}$. As in (i), we can similarly show the
admissible control $\bar{u}_{2}(\cdot)$ is an optimal control of the problem,
i.e,
$J_{1}(\bar{u}_{1}(\cdot),\bar{u}_{2}(\cdot))=\displaystyle\min_{u_{2}(\cdot)\in{\mathcal{A}}_{2}}J_{1}(\bar{u}_{1}(\cdot),{u}_{2}(\cdot)).$
(3.20)
So from (3.17) and (3.20), we can conclude that
$(\bar{u}_{1}(\cdot),\bar{u}_{2}(\cdot))$ is an equilibrium point of Problem
2.1. The proof is complete.
∎
## 4 Necessary optimality conditions
###### Theorem 4.1.
Under Assumptions 2.1-2.2, let $(\bar{u}_{1}(\cdot),\bar{u}_{2}(\cdot))$ be a
Nash equilibrium point of Problem 2.1. Suppose that
$(\bar{x}(\cdot),\bar{y}(\cdot),\bar{z}(\cdot))$ is the state process of the
system (2.1) corresponding to the admissible control
$(\bar{u}_{1}(\cdot),\bar{u}_{2}(\cdot)).$ Let
$({\bar{p}^{i}}(\cdot),\bar{q}^{i}(\cdot),\bar{k}^{i}(\cdot))$$(i=1,2)$ be the
unique solution of the adjoint equation (3.1) corresponding
$(\bar{u}_{1}(\cdot),\bar{u}_{2}(\cdot);\bar{x}(\cdot),\bar{y}(\cdot),\bar{z}(\cdot))$.
Then we have
$\begin{array}[]{ll}\big{\langle}H_{1u_{1}}(t,\bar{x}(t),\bar{y}(t),\bar{z}(t),\bar{u}_{1}(t),\bar{u}_{2}(t),\bar{p}^{1}(t),\bar{q}^{1}(t),\bar{k}^{1}(t)),u_{1}-\bar{u}_{1}(t)\big{\rangle}\geq
0,\forall u_{1}\in U_{1}a.s.a.e.,\end{array}$ (4.1)
$\begin{array}[]{ll}\big{\langle}H_{1u_{2}}(t,\bar{x}(t),\bar{y}(t),\bar{z}(t),\bar{u}_{1}(t),\bar{u}_{2}(t),\bar{p}^{1}(t),\bar{q}^{2}(t),\bar{k}^{2}(t)),u_{2}-\bar{u}_{2}(t)\big{\rangle}\geq
0,\forall u_{2}\in U_{2},a.s.a.e..\end{array}$ (4.2)
###### Proof.
Since $(\bar{u}_{1}(\cdot),\bar{u}_{2}(\cdot))$ be an equilibrium point, then
$J_{1}(\bar{u}_{1}(\cdot),\bar{u}_{2}(\cdot))=\displaystyle\min_{u_{1}(\cdot)\in{\mathcal{A}}_{1}}J_{1}(u_{1}(\cdot),\bar{u}_{2}(\cdot)).$
(4.3)
and
$J_{2}(\bar{u}_{1}(\cdot),\bar{u}_{2}(\cdot))=\displaystyle\min_{u_{2}(\cdot)\in{\mathcal{A}}_{2}}J_{2}(\bar{u}_{1}(\cdot),{u}_{2}(\cdot)).$
(4.4)
By (4.3), $\bar{u}_{1}(\cdot)$ can be regarded as an optimal control of the
optimal control problem where the controlled system is (3.6) and the cost
functional is (3.7). For this case, it is easy to see that the Hamilton
function is $H_{1}$ (see (3.2)) and the correspond adjoint equation is
$\eqref{eq:2.1}$ for $i=1,$ and
$(\bar{x}(\cdot),\bar{y}(\cdot),\bar{z}(\cdot))$ is the corresponding optimal
state process. Thus applying the stochastic maximum principle for the optimal
control of the forward-backward stochastic system (see Theorem 3.3 in [32]),
we can obtain (4.1). Similarly, from (4.4), we can obtain (4.2). The proof is
complete.
## 5 Conclution
In this paper, we have discussed two-person non-zero sum differential game
governed by a fully coupled forward-backward stochastic system with the
control process $u(\cdot)$ appearing in the forward diffusion term. The
verification theory is obtained as a sufficient condition for the existence of
open-loop Nash equilibrium point. On the other hand, applying the stochastic
maximum principle for the optimal control problem of the forward-backward
stochastic system, we derive the the stochastic maximum principle in a local
formulation as a necessary condition for the existence of open-loop Nash
equilibrium point.
∎
## References
* [1] F. Antonelli. Backward-forward stochastic differential equations. Ann. Appl. Probab., 3:777–793, 1993.
* [2] T. basar and S. Li. In Proceedings of IFAC 10th Word Congress, pages 231–236.
* [3] L. D. Berkovitz. A differential games without pure strategies solutions on an open set, advances in game theory. In Annuals of Mathematica Studies, number 52, pages 174–194. Princeton University Press, Providence, New Jersey, 1964.
* [4] L. D. Berkovitz. A variational approach to differential games, advances in game theory. In Annuals of Mathematica Studies, number 52, pages 127–174. Princeton University Press, Providence, New Jersey, 1964.
* [5] L. D. Berkovitz. Necessary conditions for optimal strategies in a class of differential games and control problems. SIAM J. Control Optim., 5:1–24, 1967.
* [6] L. D. Berkovitz. A Survey of Differential Games. Mathematical Theory of Control. Academic Press, 1967.
* [7] L. D. Berkovitz. The existence of value and saddle point in games of fixed duration. SIAM J. Control Optim., 23:173–196, 1985.
* [8] J. H. Case. Toward a theory of many-player differential games. SIAM J. Control Optim., 7:179–197, 1969.
* [9] G. Chen and Q. Zheng. N-person differential games, part 1-3. Mathematial Research reports, Pennsylvania State University, 1984\.
* [10] D. Duffie and L. Epstein. Asset pricing with stochastic differential utilities. Rev. Financial Stud., 5:411–436, 1992.
* [11] T. Eisele. Nonexistence and nonuniqueness of open-loop equilibrium in linear-quadratic differential games. Journal of Optimization Theory and Applications, 37:443–468, 1982\.
* [12] N. El Karoui, S. Peng, and M. C. Quenez. Backward stochastic differential equations in finance. Math. Finance, 7:1–71, 1997.
* [13] R. J. Elliott and N. J. Kalton. The existence of value in differential games. Memories of the Amercian mathematical Society, 126:1–67, 1972.
* [14] W. H. Flemming. the convergence prblem for differential games. Journal of Mathematical Analysis and Applications, 3:102–116, 1961\.
* [15] W. H. Flemming. The convergence problem for differential games, part 2, advances in games theory. In Annuals of Mathematica Studies, number 52, pages 195–210. Princeton University Press, Providence, New Jersey, 1964.
* [16] A. Friedman. Linear-quadratic differential games with nonzero sum and n-players. Journal of Rational Mechanics and Analysis, 34:165–178, 1969.
* [17] A. Friedman. Stochastic differential games. J. Differential Equations, 11:79–108, 1972.
* [18] R. Isaacs. Differential games, parts 1-4. The RAND Corporation, Research Memorandums, pages Nos. RM–1391, RM–1399, RM–1411, RM–1486,, 1954-55.
* [19] D. L. Lukes and D. L. Russell. A global theory for linear-quadratic differential games. Journal of Mathematical Analysis and Applications, 33:96–123, 1971\.
* [20] J. Ma and J. Yong. On linear, degenerate backward stochastic partial differential equations. Probab. Theory Relat. Fields, 113:135–170, 1999.
* [21] S. Peng and Z. Wu. Fully coupled forward-backward stochastic differential equations and applications to optimal control. SIAM J. Control Optim., 37:825–843, 1999.
* [22] I. G. Sarma and U. R. Prasad. A note on the necessary conditions for optimal strategies in a class of noncoperative n-person differential games. SIAM J. Control. Optim., 9:441–445, 1971.
* [23] I. G. Sarma, R. K. Ragade, and U. R. Prasad. Necessary conditions for optimal strategies in a class of noncoperative n-person differential games. SIAM J. Control. Optim., 7:637–644, 1969.
* [24] J. Shi and Z. Wu. The maximum principle for fully coupled forward backward stochastic control system. Acta Automat Sinica, 32:375–380, 2006.
* [25] A. W. Starr. Nonzero-sum differential games: Concepts and models,. Harvard University, Engineering and Applied Physics, Technical Report, (590), 1969.
* [26] A. W. Starr and Y. C. Ho. Further properties of nonzero-sum differential games. Journal of Optimization Theory and Applications, 3:207–219, 1969\.
* [27] A. W. Starr and Y. C. Ho. Nonzero-sum differential games. Journal of Optimization Theory and Application, 3:184–206, 1969\.
* [28] S. Tang and X. Li. Differential games of n players in stochastic systems with controlled diffusion terms. Automation and Remote Control, 69:874–890, 2008.
* [29] K. Uchida. On existence of a nash equilibrium point in n-person nonzero sum stochastic differential games. SIAM J. Control Optim., 16, 1978.
* [30] P. Varaiya. N-person nonzero-sum differential games with linear dynamics. SIAM J. Control Optim., 8:441–449, 1970.
* [31] G. Wang and Z. Yu. A pontryagin’s maximum principle for non-zero sum differential games of bsdes with applications. IEEE Transactions on Automatic Control, 55:1742–1747, 2010.
* [32] Z. Wu. Maximum principle for optimal control problem of fully coupled forward-backward stochastic systems. Acta Math. Appl. Sinica (English Ser.), 11:249–259, 1998.
* [33] W. Xu. Stochastic maximum principle for optimal control problem of forward and backward system. J. Aust. Math., 37:172–185, 1995.
* [34] J. Yong. Optimality variational principle for controlled forward-backward stochastic differential equations with mixed initial-terminal conditions. SIAM J. Control Optim., 48:4119–4156, 2010.
|
arxiv-papers
| 2010-10-12T06:31:49 |
2024-09-04T02:49:13.723280
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Maoning Tang and Qingxin Meng and Yongzheng Sun",
"submitter": "Meng Qingxin",
"url": "https://arxiv.org/abs/1010.2306"
}
|
1010.2330
|
arxiv-papers
| 2010-10-12T09:38:41 |
2024-09-04T02:49:13.730850
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "M.H. Seymour (Manchester U., CERN)",
"submitter": "Scientific Information Service Cern",
"url": "https://arxiv.org/abs/1010.2330"
}
|
|
1010.2391
|
In this paper we prove that periodic boundary-value
problems (BVPs) for delay differential equations are locally
equivalent to finite-dimensional algebraic systems of equations.
We rely only on regularity assumptions that follow those of the
review by Hartung et al. (2006). Thus, the equivalence
result can be applied to differential equations with state-dependent
delays (SD-DDEs), transferring many results of bifurcation theory
for periodic orbits to this class of systems. We demonstrate this by
using the equivalence to give an elementary proof of the Hopf
bifurcation theorem for differential equations with state-dependent
delays. This is an alternative and extension to the original Hopf
bifurcation theorem for SD-DDEs by Eichmann (2006).
§ INTRODUCTION
If a dynamical system is described by a differential equation where
the derivative at the current time may depend on states in the past
one speaks of delay differential or, more generally, functional
differential equations (FDEs). A reasonably general formulation of an
autonomous dynamical system of this type looks like this:
\begin{equation}\label{eq:ivp}
\dot x(t)=f(x_t,\mu)
\end{equation}
where $\tau>0$ is an upper bound for the delay. On the right-hand side
$f$ is a functional, mapping $C^0([-\tau,0];\R^n)$ (the space of
continuous functions on the interval $[-\tau,0]$ with values in
$\R^n$) into $\R^n$. The dependent variable $x$ is a function on
$[-\tau,T_{\max})$ for some $T_{\max}>0$, and $x_t$ is the current
function segment: $x_t(s)=x(t+s)$ for $s\in[-\tau,0]$ such that
$x_t\in C^0([-\tau,0];\R^n)$. The second argument $\mu\in\R^\nu$ is a
system parameter. For a system of the form (<ref>) one would
have to prescribe a continuous function $x$ on the interval
$[-\tau,0]$ as the initial value and then extend $x$ toward time
$T_{\max}$ (see textbooks on functional differential equations such as
[4, 11, 22]).
A long-standing problem with certain types of FDEs is that they do
not fit well into the general framework of smooth infinite-dimensional
dynamical system theory. The problem occurs whenever the functional
$f$ invokes the evaluation operation in a non-trivial way, that is,
for example, if one has a state-dependent delay. A prototypical caricature
example would be the functional
\begin{align}
\label{eq:example}
f:&\ U\times\R\mapsto \R\mbox{,}& f(x,\mu)&=\mu-x(-x(0))\mbox{,}
\intertext{where $U=\{x\in C^0([-\tau,0];\R): 0<x(0)<\tau\}$ is an
open set in $C^0([-\tau,0];\R)$. The corresponding FDE is} \dot
\end{align}
Here, $f$ evaluates its first argument $x$ at a point that itself
depends on $x$. We restrict ourselves to solutions $x$ of
(<ref>) with $x(t)\in(0,\tau)$ for $t\geq0$ to avoid
problems with causality and to limit the maximal delay to $\tau$
(always keeping $x_t$ in $U$).
The difficulty with (<ref>) stems from the fact that $f$
as a map is only as smooth as its argument $x$. Specifically, the
derivative of $f$ with respect to its first argument in this example
exists only for $x\in C^1([-\tau,0];\R)$ (the space of all
continuously differentiable functions on $[-\tau,0]$):
\begin{equation}\label{eq:exdf}
\begin{split}
\partial^1 f&:C^1([-\tau,0];\R)\times\R\times
\partial^1f&(x,\mu,y)=x'(-x(0))\,y(0)-y(-x(0))\mbox{.}
\end{split}
\end{equation}
So, if we choose $U$ as the phase space for initial-value problems
(IVPs) in example (<ref>) then the functional $f$ is not
differentiable for all elements of $U$. In fact, it is not even
locally Lipschitz continuous in $U$. Indeed, Winston [25] gave
an example of an initial condition in $U$ for (<ref>)
(with $\mu=0$ and $\tau>1$), for which the IVP did not have a unique
solution. This counterexample is not surprising since the right-hand
side $f$ does not fit into the framework that the textbooks
[4, 11, 22] assume to be present. A result of Walther
[24] rescues IVPs with state-dependent delays (such as
(<ref>)) by restricting the phase space in general to the
closed submanifold $C_c$ of $C^1([-\tau,0];\R^n)$:
C_c={x∈C^1([-τ,0];^n): x'(0)=f(x)}
Walther [24] could prove the existence of a semiflow inside this
manifold that is continuously differentiable with respect to its
initial conditions. However, this result is restricted to a single
degree of differentiability. Results about higher degrees of
smoothness are lacking for the semiflow [12].
A typical task one wants to perform for problems of type
(<ref>), or example (<ref>), is bifurcation
analysis of equilibria and periodic orbits. Equilibria
are solutions $x$ of (<ref>) that are constant in time, and
periodic orbits are solutions $x\in C^1(\R;\R^n)$ of (<ref>)
that satisfy $x(t+T)=x(t)$ for some $T>0$ and all $t\in\R$. Equilibria
of the general FDE (<ref>) can be determined by finding the
solutions $(p,\mu)\in\R^n\times\R^\nu$ of the algebraic system of
\begin{equation}
\label{eq:eqsys}
\end{equation}
where $E_0$ is the trivial embedding
We observe that, even though the FDE (<ref>) is an
infinite-dimensional system, its equilibria can be found as roots of
the finite-dimensional system (<ref>) of algebraic
equations. Moreover, the regularity problems of the semiflow do not
affect (<ref>): in the example (<ref>), the
algebraic equation (<ref>) reads $0=\mu-p$, which is smooth
to arbitrary degree, and can be solved even for negative $\mu$ (near
equilibria with $\mu=p<0$ the semiflow does not exist).
In this paper we establish a system similar to (<ref>), but
for periodic orbits: we find a finite-dimensional algebraic system of
equations that does not suffer from the regularity problems affecting
the semiflow, and an equivalence between solutions of this algebraic
system and periodic orbits of (<ref>). In comparison, for
ordinary differential equations (ODEs) of the form $\dot
x(t)=f(x(t),\mu)$ with a smooth $f:\R^n\times\R^\nu\mapsto\R^n$, the
fact that the problem of finding periodic orbits can be reduced to
algebraic root-finding is well known [9]. For example, in ODEs
one can use the algebraic system $0=X(T;p,\mu)-p$ where $t\mapsto
X(t;p,\mu)$ is the trajectory defined by the IVP starting from
$p\in\R^n$ and using parameter $\mu\in\R^\nu$.
A central notion in the construction of the equivalent algebraic
system for periodic orbits of FDEs are periodic boundary-value
problems (BVPs) for FDEs on the interval $[-\pi,\pi]$ with periodic
boundary conditions (which we identify with the unit circle
$\T$). Periodic orbits of (<ref>) can then be found as
solutions of periodic BVPs. If one wants to make the equivalence
result useful in practical applications, one has to find a regularity
(smoothness) condition on the right-hand side $f$ that includes the
class of state-dependent delay equations reviewed in [12],
while still ensuring that it is possible to prove the existence of an
equivalent algebraic system. We use exactly the same condition as used
by Walther in [24] to prove the existence of a continuously
differentiable semiflow, the so-called extendable continuous
differentiability (originally introduced as “almost Frechét
differentiability” in [21]), which implies a restricted form
of local Lipschitz continuity. We generalize restricted continuous
differentiability to higher degrees of restricted smoothness (which we
call $EC^k$ smoothness) in a similar fashion as Krisztin [19]
did for the proof of the existence and smoothness of local unstable
manifolds of equilibria. Our definition of $EC^k$ smoothness is
comparatively simple to state and check, and lends itself easily to
inductive proofs.
After introducing the notation for periodic BVPs and $EC^k$ smoothness
we state the main result, an equivalence theorem between periodic BVPs
and algebraic systems of equations in Section <ref>. The
equivalence theorem reduces statements about existence and smooth
dependence of periodic orbits of FDEs to root-finding problems of
smooth algebraic equations. The result is weaker than the
corresponding results for equilibria of FDEs and for periodic orbits
of ODEs because the equivalence is only valid locally. For any given
periodic function $x_0$ with Lipschitz continuous time derivative we
construct an algebraic system that is equivalent to the periodic BVP
in a sufficiently small open neighborhood of $x_0$. However, the
result is still useful, as we then demonstrate in
Section <ref>. We apply the equivalence theorem in the vicinity
of equilibria for which the linearization of (<ref>) has
eigenvalues on the imaginary axis (for example, near $x_0=\mu=\pi/2$
in example (<ref>)) to prove the Hopf Bifurcation
Theorem. The equivalence theorem reduces the
proof of the Hopf Bifurcation Theorem to an application of the
Algebraic Branching Lemma [1]. This provides a complete proof
for the Hopf Bifurcation Theorem for FDEs with state-dependent delays,
including the regularity of the emerging periodic orbits. We discuss
differences to the first version of the proof by Eichmann [5]
and the approach of Hu and Wu [13] in Section <ref>.
The equivalence is applicable in other scenarios where one would
expect branching of periodic solutions. Examples are period doublings,
the branching from periodic orbits with resonant Floquet multipliers
on the unit circle in Arnol'd tongues, and branching scenarios in FDEs
with symmetries. We give a tentative list of straightforward
applications and generalizations of the equivalence theorem in the
conclusion (Section <ref>).
We note that the theorem stated in Section <ref> differs
from statements about numerical approximations. As part of the theorem
we also provide a map $X$ that maps the root of the algebraic system
back into a function space to give the exact solution of the
periodic BVP, and a projection $P$ that maps functions to
finite-dimensional vectors (and, hence, periodic orbits to roots of
the algebraic system). In numerical methods one typically has to
increase the dimension of the algebraic system in order to get more
and more accurate approximations of the true solution whereas
the dimension of the algebraic system constructed in
Section <ref> is finite.
§ THE EQUIVALENCE THEOREM
This section states the assumptions and conclusions of the main result
of the paper, the Equivalence Theorem stated in
Theorem <ref>. Before doing so, we introduce some basic
notation (function spaces on intervals with periodic boundary
conditions and projections onto the leading Fourier modes).
§.§.§ Periodic BVPs
We first state precisely what we mean by periodic BVP and introduce
the usual hierarchy of continuous, continuously differentiable and
Lipschitz continuous functions on the compact interval $[-\pi,\pi]$
with periodic boundary conditions. For $j\geq0$ we will use the
notation $C^j(\T;\R^n)$ for the spaces of all functions $x$ on the
interval $[-\pi,\pi]$ with continuous derivatives up to order $j$
(including order $0$ and $j$) satisfying the periodic boundary conditions
$x^{(l)}(-\pi)=x^{(l)}(\pi)$ for $l=0\ldots j$. Elements of
$C^0(\T;\R^n)$ are continuous and satisfy $x(-\pi)=x(\pi)$. For
derivatives of order $j>0$, $x^{(j)}(-\pi)$ is the right-sided $j$th
derivative of $x$ in $-\pi$, and $x^{(j)}(\pi)$ is the left-sided $j$th
derivative of $x$ in $\pi$. The norm in $C^j(\T;\R^n)$ is
We can extend any function $x$ in $C^j(\T;\R^n)$ to arguments in $\R$
by defining $x(t)=x(t-2k\pi)$ where $k$ is an integer chosen such that
$-\pi\leq t-2k\pi<\pi$ (we will write $t_{\mod[-\pi,\pi)}$
later). Thus, every element of $C^j(\T;\R^n)$ is also an element of
$BC^j(\R;\R^n)$, the space of functions with bounded continuous
derivatives up to order $j$ on the real line. We use the notation
$t\in\T$ for arguments $t$ of $x$, and also call $\T$ the unit
circle. This make sense because the parametrization of the unit circle
by angle provides a cover, identifying $\T$ with $\R$ where we use
$[-\pi,\pi)$ as the fundamental interval.
Additional useful function spaces are the space of Lipschitz
continuous functions and, correspondingly, spaces with Lipschitz
continuous derivatives, denoted by $C^{j,1}(\T;\R^n)$, which are
equipped with the norm
\begin{equation}\label{eq:cj1def}
\|x\|_{j,1}=\max\left\{\|x\|_j,
\sup_{
\begin{subarray}{c}
t\neq s
\end{subarray}
}\frac{|x^{(j)}(s)-x^{(j)}(t)|}{|s-t|} \right\}
\end{equation}
($x^{(0)}(t)$ refers to $x(t)$). Note that we used the notation
$t,s\in\R$ in the index of the supremum, as we can apply arbitrary
arguments in $\R$ to a function $x\in C^0(\T;\R^n)$ by considering it
as an element of $BC^0(\R;\R^n)$, as explained above. We use the same
notation ($C^j(J;\R^n)$ and $C^{j,1}(J;\R^n)$) also for functions on
an arbitrary compact interval $J\subset\R$ without periodic boundary
conditions (and one-sided derivatives at the boundaries). As any
function $x\in C^j(\T;\R^n)$ is also an element of $BC^j(\R;\R^n)$, it
is also an element of $C^j(J;\R^n)$ for any compact interval $J$ (and
the norm of the embedding operator equals unity). On the function
spaces $C^j(\T;\R^n)$ we define the time shift operator
The operator $\Delta_t$ is linear and has norm $1$ in all spaces
$C^j(\T;\R^n)$. Similarly, $\Delta_t$ maps also
$C^{j,1}(\T;\R^n)\mapsto C^{j,1}(\T;\R^n)$, and has unit norm there as
Let $f$ be a continuous functional on the space of continuous periodic
functions, that is,
The right-hand side $f$, together with the shift $\Delta_t$, creates
an operator in $C^0(\T;\R^n)$, defined as
\begin{align}
\label{eq:fdef}
F&:C^0(\T;\R^n)\mapsto C^0(\T;\R^n) &
\end{align}
The operator $F$ is invariant with respect to time shift by construction:
$F(\Delta_tx)=\Delta_tF(x)$. We consider autonomous periodic
boundary-value problems for differential equations where $f$ is the
right-hand side:
\begin{equation}
\label{eq:perbvp}
\begin{split}
\dot x(t)&=f(\Delta_tx)=F(x)(t)\mbox{.}
\end{split}
\end{equation}
A function $x\in \C^1(\T;\R^n)$ is a solution of (<ref>) if
$x$ satisfies equation (<ref>) for all $t\in\T$ (for each
$t\in\T$ equation (<ref>) is an equation in $\R^n$). In
contrast to the introduction we do not expressly include a parameter
$\mu$ as an argument of $f$. This does not reduce generality as we
will explain in Section <ref>. The main result, the Equivalence
Theorem <ref>, will be concerned with equivalence of the
periodic BVP (<ref>) to an algebraic system of
equations. The notion of the shift $\Delta_t$ on the unit circle and
the operator $F$, combining $f$ with the shift, is specific to
periodic BVPs such that the BVP (<ref>) looks different from
the IVP (<ref>) in the introduction. Several results stating
how regularity of $f$ transfers to regularity of $F$ are collected in
Appendix <ref>.
§.§.§ Definition of $EC^k$ smoothness and local (restricted)
$EC$ Lipschitz continuity
Continuity of the functional $f$ is not strong enough as a condition
to prove the Equivalence Theorem. Rather, we need a notion of
smoothness for $f$. However, as explained in the introduction, we
cannot assume that $f$ is continuously differentiable with degree
$k\geq1$, if we want to include examples such as $f(x)=-x(-x(0))$ (see
FDE (<ref>) for $\mu=0$) into the class under
The review by Hartung et al. [12] observed the
following typical property of functionals $f$ appearing in equations
of type (<ref>): the derivative $\tpartial^1f(x)$ of $f$ in
$x$ as a linear map from $C^1(\T;\R^n)$ into $\R^n$ can be extended to
a bounded linear map from $C^0(\T;\R^n)$ into $\R^n$, and the mapping
^1f: C^1(;^n)×C^0(;^n) ↦^n
(x,y)↦^1 f(x,y)
is continuous as a function of both arguments. In other words, the
derivative of $f$ may depend on $x'$ but not on $y'$. For the example
$f(x)=-x(-x(0))$ this is true (see (<ref>)). Most of the
fundamental results establishing basic dynamical systems properties
for FDEs with state-dependent delay in [12] rest on this
extendability of $\partial^1f$.
We also rely strongly on this notion of extendable
continuous differentiability. The precise definition is given below in
Definition <ref>. In this definition we permit the
argument range $J$ to be any compact interval or $\T$. We use the
notation of a subspace of higher-order continuous differentiability
not only for $C^j(J;\R^n)$ but also for products of such spaces in a
natural way. Say, if
\begin{equation}\label{eq:Dspacedef}
D=C^{k_1}(J;\R^{m_1})\times\ldots\times C^{k_\ell}(J;\R^{m_\ell})
\mbox{,}
\end{equation}
where $\ell\geq 1$, and $k_j\geq0$ and $m_j\geq1$ are integers, and
denoting the natural maximum norm on the product $D$ by
then for integers $r\geq0$ the space $D^r$ is defined in the natural
way as
\begin{align*}
D^r&=C^{k_1+r}(J;\R^{m_1})\times\ldots\times C^{k_\ell+r}(J;\R^{m_\ell})
\mbox{,\quad with \ }\\
\|x\|_{D,r}&=\max_{
\begin{subarray}{c}
0\leq j\leq r\\[0.2ex]
1\leq i\leq \ell
\end{subarray}
} \|x_i^{(j)}\|_{k_i}\mbox{.}
\end{align*}
For the simplest example, $D=C^0(J;\R^n)$, $D^k$ is $C^k(J;\R^n)$. If
$J=\T$ then the time shift $\Delta_t$ extends naturally to products of spaces:
Let $D$ be a product space of the type (<ref>), and let
$f:D\mapsto\R^n$ be continuous. We say that $f$ has an extendable
continuous derivative if there exists a map $\partial^1f$
that is continuous in both arguments $(u,v)\in D^1\times D$ and
linear in its second argument $v\in D$, such that for all $u\in D^1$
\begin{equation}\label{eq:ass:contdiff:j}
\lim_{
\begin{subarray}{c}
v\in D^1\\[0.2ex]
\|v\|_{D,1}\to 0
\end{subarray}
} \frac{|f(u+v)-f(u)-\partial_1f(u,v)|}{\|v\|_{D,1}}=0\mbox{.}
\end{equation}
We say that $f$ is $k$ times continuously differentiable in this
extendable sense if the map $\partial^kf$, recursively defined as
$\partial^kf=\partial^1[\partial^{k-1}f]$, exists and satisfies the
limit condition (<ref>) for $\partial^{k-1}f$. We
abbreviate this notion by saying that $f$ is $EC^k$ smooth in $D$.
The limit in (<ref>) is a limit in $\R$. For $k=1$
the definition is identical to property (S) in the review
[12], one of the central assumptions for fundamental results
on the semiflow. Extendable continuous differentiability requires the
derivative to exist only in points in $D^1$ and with respect to
deviations in $D^1$, but it demands that the derivative must extend in
its second argument to $D$ ($\partial^1f$ is linear in its second
argument). This is the motivation for calling this property
extendable continuous differentiability.
The definition of $EC^k$ smoothness for $k>1$ uses the notation that a
functional (say, $\partial^1f$) of two arguments (say, $u\in D^1$ and
$v\in D$) for which one would write $\partial^1f(u,v)$, is also a
functional of a single argument $w=(u,v)\in D^1\times D$, such that
one can also write $\partial^1f(w)$. When using this notation we observe that
the space $D^1\times D$ is again a product of type
(<ref>) such that $\partial^1f$ is again a functional of
the same structure as $f$. For example, let us consider the functional
$f:x\mapsto -x(-x(0))$ from example (<ref>) (setting
$\mu=0$). The functional is well defined and continuous also on
$D=C^0(\T;\R)$. Moreover, $f$ is $EC^k$ smooth in $D$ to arbitrary
degree $k$. Its first two derivatives are:
\begin{align}
&\partial^1f: C^1(\T;\R)\times C^0(\T;\R)\mbox{,}\nonumber\\
&\partial^1f(u,v)=u'(-u(0))\,v(0)-v(-u(0))\mbox{, and}\label{eq:exampledf1}\\
&\partial^2f: \left[C^2(\T;\R)\times C^1(\T;\R)\right]\times
\left[C^1(\T;\R)\times C^0(\T;\R)\right]\mbox{,}\nonumber\\
\begin{aligned}
\partial_2f(u,v,w,x)=&-u''(-u(0))\,w(0)\,v(0)+u'(-u(0))\,x(0)\\
&+w'(-u(0))\,v(0)- v'(-u(0))\,w(0)-x(-u(0))\mbox{.}
\end{aligned}\label{eq:exampledf2}
\end{align}
As one can see, the first derivative $\partial^1f$ has the same
structure as $f$ itself if we replace $D=C^0(\T;\R^n)$ by $D^1\times
D$. So, it is natural to apply the definition again to $\partial^1f$
on the space $D^1\times D$.
Assuming that $f$ is $EC^1$ smooth on $C^0(J;\R^n)$ implies classical
continuous differentiability of $f$ as a map from $C^1(J;\R^n)$ into
$\R^n$ and is, thus, strictly stronger than assuming that $f$ is
continuously differentiable on $C^1(\T;\R^n)$.
Since every element of $C^j(\T;\R^n)$ is also an element of
$C^j(J;\R^n)$ for any compact interval $J$ (and the embedding operator
has unit norm), any $EC^k$ smooth functional $f:C^0(J;\R^m)\mapsto\R^n$
is also a $EC^k$ smooth functional from $C^0(\T;\R^m)$ into $\R^n$.
It is worth comparing Definition <ref> with the definition
for higher degree of regularity used by Krisztin in [19]. With
Definition <ref> the $k$th derivative has $2^k$
arguments. In contrast to this, the $k$th derivative as defined in
[19] has only $k+1$ arguments (the first argument is the base
point, and the derivative is a $k$-linear form in the other $k$
arguments). The origin of this difference can be understood by looking
at the example $f(x)=-x(-x(0))$ and its derivatives in
(<ref>)–(<ref>). Krisztin's definition
applied to the second derivative does not include the derivative of
$\partial^1f$ with respect to the linear second argument $v$ (as is
often convention, because it is the identity). One would obtain the
second derivative according to Krisztin's definition by setting $x=0$
in (<ref>). Indeed, the terms containing the argument
$x$ in (<ref>) are simply $\partial^1f(u,x)$, as one
expects when differentiating $\partial^1f(u,v)$ with respect to $v$,
calling the deviation $x$. While in practical examples it is often
more economical to use the compact notation with $k$-forms, inductive
proofs of higher-order differentiability using the full derivative
only require the notion of at most bi-linear forms, making them less
If $f$ is $EC^1$ smooth then it automatically
satisfies a restricted form of local Lipschitz continuity [12],
which we call local $EC$ Lipschitz continuity:
We say that $f:C^0(\T;\R^n)\mapsto \R^n$ is locally $EC$ Lipschitz
continuous if for every $x_0\in C^1(\T;\R^n)$ there exists a
neighborhood $U(x_0)\subset C^1(\T;\R^n)$ and a constant $K$ such
\begin{equation}\label{eq:loclip}
|f(y)-f(z)|\leq K\|y-z\|_0
\end{equation}
holds for all $y$ and $z$ in $U(x_0)$.
That $EC^1$ smoothness implies local $EC$ Lipschitz continuity has
been shown, for example, in [24] (but see also
Lemma <ref> in Appendix <ref>). Note that the
estimate (<ref>) uses the $\|\cdot\|_0$-norm for the upper
bound. This is a sharper estimate than one would obtain using the
expected $\|\cdot\|_1$-norm. The constant $K$ may depend on the
derivatives of the elements in $U(x_0)$ though. For example, for
$f(x)=-x(-x(0))$ as in (<ref>) with $\mu=0$, one would have
the estimate
This means that in this example, the neighborhood $U(x_0)$ can be
chosen arbitrarily large as long as it is bounded in
The following lemma states that we can extend the neighborhood
$U(x)$ in Definition <ref> into the space of Lipschitz
continuous functions ($C^{0,1}$ instead of $C^1$) and include time
shifts (which possibly increases the bound $K$).
Let $f$ be locally $EC$ Lipschitz continuous, and let $x_0$ be in
$C^{0,1}(\T;\R^n)$. Then there exists a bounded neighborhood
$U(x_0)\subset C^{0,1}(\T;\R^n)$ and a constant $K$ such that
holds for all $y$ and $z$ in $U(x_0)$, and for all $t\in\T$. Thus,
for all $y$ and $z$ in $U(x_0)$.
Recall that $F(x)(t)=f(\Delta_tx)$. See Lemma <ref> and
Lemma <ref> in Appendix <ref> for the
proof of Lemma <ref>.
A consequence of Lemma <ref> is that the time
derivative of a solution $x_0$ of the periodic BVP is also Lipschitz
continuous (in time): if $\dot x_0(t)=f(\Delta_tx_0)$ then there
exists a constant $K$ such that
\begin{equation}\label{eq:lipx0t}
\|x_0'(t)-x_0'(s)\|_0\leq K|t-s|
\end{equation}
Thus, $x_0\in C^{1,1}(\T;\R^n)$. This follows from
Lemma <ref> by inserting $\Delta_tx_0$ and
$\Delta_sx_0$ for $y$ and $z$ and using that $x_0'(t)=f(\Delta_tx_0)$
(it is enough to show (<ref>) for $|t-s|$ small).
§.§.§ Projections onto subspaces spanned by Fourier modes
The variables of the algebraic system in the Equivalence
Theorem will be the coefficients of the first $N$
Fourier modes (where $N$ will be determined as sufficiently large
later) of elements of $C^{0,1}(\T;\R^n)$ (the space of Lipschitz
continuous functions on $\T$). Consider the functions on $\T$
for $k=1,\ldots,\infty$ (which is the classical Fourier basis of
$\Lint^2(\T;\R)$). For any $m\geq1$ we define the projectors and maps
\begin{equation}\label{eq:proj}
\begin{aligned}
P_N&:C^j(\T;\R^m)\mapsto C^j(\T;\R^m)\mbox{,}& [P_Nx](t)_i&=\sum_{k=-N}^N
\left[\frac{1}{\pi}\int_{-\pi}^\pi b_k(s)x_i(s)\d s\right]\,b_k(t)\mbox{,}
\allowdisplaybreaks\\
E_N&:\R^{m\times (2N+1)}\mapsto C^j(\T;\R^m)\mbox{,}&
R_N&:C^j(\T;\R^m)\mapsto \R^{m\times(2N+1)}\mbox{,} &
[R_Nx]_{i,k}\ \ &=\frac{1}{\pi}\int_{-\pi}^\pi b_k(s)x_i(s)\d s\mbox{,}
\allowdisplaybreaks\\
L_{\phantom{N}}&:C^j(\T;\R^m)\mapsto C^j(\T;\R^m)\mbox{,}& [Lx](t)\ \ \,&=\int_0^t x(s)-
R_0x\,\d s=\int_0^t Q_0[x](s)\d s\mbox{.}
\end{aligned}
\end{equation}
The projector $P_N$ projects a periodic function onto the subspace
spanned by the first $2N+1$ Fourier modes, and $Q_N$ is its
complement. The map $E_N$ maps a vector $p$ of $2N+1$ Fourier
coefficients (which are each vectors of length $n$ themselves) to the
periodic function that has these Fourier coefficients. The map $R_N$
extracts the first $2N+1$ Fourier coefficients from a function. The
simple relation $P_N=E_NR_N$ holds. The vector $R_0x$ is the average
of a function $x$, and $Q_0$ subtracts the average from a periodic
function. The operator $L$ takes the anti-derivative of a periodic
function after subtracting its average (to ensure that $L$ maps back
into the space of periodic functions). In all of the definitions the
degree $j$ of smoothness of the vector space $C^j$ can be any
non-negative integer. The operator $L$ not only maps $C^j$ back into
itself, but it maps $C^j(\T;\R^m)$ into $C^{j+1}(\T;\R^m)$.
We do not attach an index $m$ to the various maps to indicate how many
dimensions the argument and, hence, the value has because there is no
room for confusion: for example, if $x\in C^0(\T;\R^2)$ then $P_Nx\in
C^0(\T;\R^2)$ such that we use the same notation $P_Nx$ for
$x:\T\mapsto\R^m$ with arbitrary $m$. Similarly, we apply all maps
also on product spaces $D$ of the type
$C^{k_1}(\T;\R^{m_1})\times\ldots\times C^{k_\ell}(\T;\R^{m_\ell})$
introduced in Equation (<ref>) by applying the maps
element-wise. For example,
\begin{align*}
P_Nx&=(P_Nx_1,\ldots,P_Nx_\ell) &&\mbox{for
$x=(x_1,\ldots,x_\ell)\in D$,}\\
E_Np&=(E_Np_1,\ldots,E_Np_\ell) &&\mbox{for
\R^{m_1\times(2N+1)}\times\ldots\times\R^{m_\ell\times(2N+1)}$.}
\end{align*}
§.§.§ Equivalent integral equation
We note the fact that a function $x\in C^1(\T;\R^n)$ solves the
periodic BVP $\dot x(t)=f(\Delta_tx)=F(x)(t)$ if and only if it
satisfies the equivalent integral equation
\begin{equation}\label{eq:inteq}
x(t)=x(0)+\int_0^tF(x)(s)\d s\mbox{\quad for all $t\in\T$.}
\end{equation}
For each $t\in\T$, Equation (<ref>) is an equation in
$\R^n$. In particular, the term $x(0)$ is in $\R^n$. Thus, the
integral equation (<ref>) is very similar to the
corresponding integral equation used in the proof of the
Picard-Lindelöf Theorem for ODEs [3]. This is in contrast
to the abstract integral equations used by Diekmann et al.
[4] to construct unique solutions to IVPs, in which equality
at every point in time is an equality in function spaces. It is the
similarity of (<ref>) to its ODE equivalent that makes the
reduction of periodic BVPs to finite dimensional algebraic equations
possible. One minor problem is that the Picard iteration for
(<ref>) cannot be expected to converge. In fact, the integral
term $\int_0^tF(x)(s)\d s$ does not even map back into the space
$C^0(\T;\R^n)$ of periodic functions, even if $x$ is in
$C^0(\T;\R^n)$. However, a simple algebraic manipulation using the
newly introduced maps $L$, $P_N$, $Q_N$, $E_N$ and $R_N$ removes this
problem (remember that $F(x)(t)=f(\Delta_tx)$):
Let $N\geq0$ be an arbitrary integer. A function $x\in C^0(\T;\R^n)$
and a vector $p\in\R^{n\times(2\,N+1)}$ satisfy
\begin{align}
\dot x(t)&=f(\Delta_tx)\mbox{\quad and\quad}
\intertext{if and only if they satisfy the system}
\label{eq:lowmodes:intro}\mbox{.}
\end{align}
Note that the map $R_N$ extracts the lowest $2N+1$ Fourier
coefficients from a periodic function. Equation (<ref>)
can be viewed as a fixed-point equation in $C^{0,1}(\T;\R^n)$,
parametrized by $p$. We will apply the Picard iteration to this
fixed-point equation instead of
(<ref>). Equation (<ref>) is an equation in
$\R^{n\times(2\,N+1)}$. If the Picard iteration converges then the
fixed-point equation (<ref>) can be used to construct
(for sufficiently large $N$) a map
$X:U\subset\R^{n\times(2\,N+1)}\mapsto C^{0,1}(\T;\R^n)$, which maps
the parameter $p$ to its corresponding fixed point $x$. Inserting this
fixed point $x=X(p)$ into (<ref>) turns
(<ref>) into a system of $n\times(2\,N+1)$ algebraic
equations for the $n\times(2\,N+1)$-dimensional variable $p$, making
the periodic BVP for $x$ equivalent to an algebraic system for its
first $2N+1$ Fourier coefficients, $p$. The proof of
Lemma <ref> is simple algebra, see
Section <ref>.
§.§.§ Statement of the Equivalence Theorem
Using the Splitting Lemma <ref> we can now state the central
result of the paper. The intention to treat (<ref>) as a
fixed-point equation motivates the introduction of the map
\begin{align*}
M_N&:C^{0,1}(\T;\R^n)\times \R^{n\times(2\,N+1)}\mapsto C^{0,1}(\T;\R^n)
\mbox{\quad given by}\\
\end{align*}
This means that we will look for fixed points of the map
$M_N(\cdot,p)$ for given $p$ and sufficiently large $N$. We will do
this in small closed balls in $C^{0,1}(\T;\R^n)$ (the space of
Lipschitz continuous functions) such that it is useful to introduce
the notation
for $\delta>0$ and $x_0\in C^{0,1}(\T;\R^n)$. That is,
$B_\delta^{0,1}(x_0)$ is the closed ball of radius $\delta$ around
$x_0\in C^{0,1}(\T;\R^n)$ in the $\|\cdot\|_{0,1}$-norm (the Lipschitz
norm on $\T$).
Let $f$ be $EC^{j_{\max}}$ smooth, and let $x_0$ have a Lipschitz continuous
derivative, that is, $x_0\in C^{1,1}(\T;\R^n)$. Then there exist a
$\delta>0$ and a positive integer $N$ such that the map
$M_N(\cdot,p)$ has a unique fixed point in $B_{6\delta}^{0,1}(x_0)$ for
all $p$ in the neighborhood $U$ of $R_Nx_0$ given by
U={p∈^n×(2 N+1):
The maps
\begin{align*}
X&:U\mapsto C^0(\T;\R^n)\mbox{,}&
X(p)&=\mbox{\ fixed point of $M_N(\cdot,p)$ in $B_{6\delta}^{0,1}(x_0)$,}\\
g&:U\mapsto\R^{n\times(2\,N+1)}\mbox{,} &
\end{align*}
are $j_{\max}$ times continuously differentiable with respect to
their argument $p$, and $X(p)$ is an element of
$C^{j_{\max}+1}(\T;\R^n)$. Moreover, for all $x\in
B_{\delta}^{0,1}(x_0)$ the following equivalence holds:
\begin{align*}
\dot x(t)&=f(\Delta_tx)\\
\intertext{if and only if $p=R_Nx$ is in $U$ and satisfies}
g(p)&=0\mbox{\quad and\quad} x=X(p)\mbox{.}
\end{align*}
Theorem <ref> is the central result of the paper. It implies
that, for any $x_0\in C^{1,1}(\T;\R^n)$ all solutions of the periodic
BVP in a sufficiently small neighborhood of $x_0$ lie in the graph
$X(U)$ of a finite-dimensional manifold. Moreover, these solutions can
be determined by finding the roots of $g$ in $U\subset
\R^{n\times(2\,N+1)}$. We note that Theorem <ref> is
different from statements about numerical approximations. Even though
the integer $N$ is finite, solving the algebraic system $g(p)=0$ and
then mapping the solutions with the map $X$ into the function space
$C^0(\T;\R^n)$ gives an exact solution $x=X(p)$ of the periodic BVP
$\dot x(t)=f(\Delta_tx)$.
The size of the radius $\delta$ of the ball in which the equivalence
holds depends on how large one can choose $\delta$ such that a local
$EC$ Lipschitz constant $K$ for $F$ exists for
$B_{6\delta}^{0,1}(x_0)$ (such neighborhoods exist according to
Lemma <ref>). In many applications (in particular, in the
example (<ref>)) this can be any closed ball in which the
right-hand side $f$ is well defined (at the expense of increasing $K$
for larger balls). Once the local $EC$ Lipschitz constant $K$ is
determined, one can find a uniform upper bound $R$ for the norm
$\|F(x)\|_{0,1}$ for all $x\in B_{6\delta}^{0,1}(x_0)$ (see
Lemma <ref>). The integer $N$, which determines the
dimension of the algebraic system, is then chosen depending on $R$,
$K$ and $\|x_0'\|_{0,1}$.
Section <ref> contains the complete proof of
Theorem <ref>. The first step of the proof of Equivalence
Theorem <ref> is the existence of the fixed point of $M_N$ in
$B_{6\delta}^{0,1}(x_0)$ for $p\in U$. This is achieved by applying
Banach's contraction mapping principle to the map $M_N(\cdot,p)$ in
the closed ball $B_{6\delta}^{0,1}(x_0)$. The only peculiarity in this
step is that we apply the principle to $B_{6\delta}^{0,1}(x_0)$, which
is a closed bounded set of Lipschitz continuous functions, using the
(weaker) maximum norm ($\|\cdot\|_0$). This is possible because closed
balls in $C^{0,1}(\T;\R^n)$ are complete also with respect to the norm
$\|\cdot\|_0$. With respect to the maximum norm the map
$M_N(\cdot,p):x\mapsto E_Np+Q_NLF(x)$ becomes a contraction for
sufficiently large $N$ (because the norm of the operator $Q_NL$ is
bounded by $C\log(N)/N$, and $F$ has a Lipschitz constant $K$ with
respect to $\|\cdot\|_0$ in $B_{6\delta}^{0,1}(x_0)$).
After the existence of the fixed point of $M_N(\cdot,p)$ is
established in Section <ref> the equivalence between the
algebraic system $g(p)=0$ and the periodic BVP $\dot
x(t)=f(\Delta_tx)$ in the smaller ball $B_\delta^{0,1}(x_0)$ follows
from the Splitting Lemma <ref>.
The smoothness (in the classical sense) of the maps $X$ and $g$
follows, colloquially speaking, from implicit differentiation of the
fixed-point problem $x=E_Np+Q_NLF(x)$ with respect to $p$.
Section <ref> checks the uniform convergence of the
difference quotient in detail, Section <ref> uses the
higher degrees of $EC^{j_{\max}}$ smoothness of $f$ to prove higher
degrees of smoothness for $X$ and $g$. For proving higher-order
smoothness one has to check only if the spectral radius of a linear
operator is less than unity, but the inductive argument requires more
elaborate notation than the first-order continuous differentiability.
§ APPLICATION TO PERIODIC ORBITS OF AUTONOMOUS FDES
— HOPF BIFURCATION THEOREM
Let us come back to the original problem, the parameter-dependent FDE
(<ref>) $\dot x = f(x_t,\mu)$, where $\mu\in\R^\nu$ is a system
parameter and the functional $f:C^0(J;\R^n)\times\R^\nu\mapsto\R^n$ is
defined for first arguments that exist on an arbitrary compact
interval $J$. Periodic orbits are solutions $x$ of $\dot x =
f(x_t,\mu)$ that are defined on $\R$ and satisfy $x(t)=x(t+T)$ for
some $T>0$ and all $t\in\R$.
Let $x$ be a periodic function of period $T=2\pi/\omega$. Then the
function $y(s)=x(s/\omega)$ is a function of period $2\pi$
($s\in\T$). This makes it useful to define the map
\begin{align*}
S:&BC^0(\R;\R^n)\times\R\mapsto BC^0(\R;\R^n) &
[S(y,\omega)](s)=y(\omega s)\mbox{,}
\end{align*}
such that $S(y,\omega)(t)=x(t)$ for all $t\in\R$ (remember that
$BC^0(\R;\R^n)$ is the space of bounded continuous functions on the
real line). Then $x\in C^1(\R;\R^n)$ satisfies the differential
\begin{equation}
\label{eq:ft}
\dot x(t)=f(x_t,\mu)
\end{equation}
on the real line and has period $2\pi/\omega$ if and only if
$y=S(x,1/\omega)\in C^1(\T;\R^n)$ satisfies the differential equation
Let us define an extended differential equation
\begin{align}
\label{eq:ftext}
\dot x_\mathrm{ext}(s)&=f_\mathrm{ext}(\Delta_sx_\mathrm{ext})\mbox{,}
\end{align}
where $f_\mathrm{ext}$ maps $C^0(\T;\R^{n+1+\nu})$ into
$\R^{n+1+\nu}$ and is defined by
\begin{align*}
\begin{pmatrix}
y \\ \omega\\ \mu
\end{pmatrix}&=
\begin{bmatrix}
f\left(S(y,R_0\omega),R_0\mu\right)/\cut(R_0\omega)\\ 0\\ 0
\end{bmatrix}\mbox{,\quad where}\\
\cut(\omega)&=
\begin{cases}
\omega &\mbox{if $\omega>\omega_\mathrm{cutoff}>0$}\\
\mbox{smooth, uniformly non-negative extension} & \mbox{for
\end{cases}
\end{align*}
for $y\in C^0(\T;\R^n)$, $\omega\in C^0(\T;\R)$ and $\mu\in
C^0(\T;\R^\nu)$ (recall that $R_0$ takes the average of a function on
$\T$). We have used in our definition that any functional $f$ defined
for $x\in C^0(J;\R^n)$ is also a functional on $C^0(\T;\R^n)$
(periodic functions have a natural extension
$x(t)=x(t_{\mod[-\pi,\pi)})$ if $t\in \R$ is arbitrary). The extended
system has introduced the unknown $\omega$ and the system parameter
$\mu$ as functions of time, and the additional differential equations
$\dot \omega=0$, $\dot\mu=0$, which force the new functions to be
constant for solutions of (<ref>). We have also introduced a
cut-off for $\omega$ close to zero to keep $f_\mathrm{ext}$ globally
defined. The extended BVP (<ref>) is in the form of periodic
BVPs covered by the Equivalence Theorem <ref>. Thus, if
$f_\mathrm{ext}$ is $EC^{j_{\max}}$ smooth then BVP (<ref>)
satisfies the assumptions of Theorem <ref> in the vicinity of
every periodic function $x_{0,\mathrm{ext}}\in
C^{1,1}(\T;\R^{n+\nu+1})$. Any solution
$x_\mathrm{ext}=(y,\omega,\mu)$ that we find for (<ref>)
corresponds to a periodic solution $t\mapsto y(\omega t)$ of period
$2\pi/R_0\omega$ at parameter $R_0\mu$ for (<ref>) and vice
versa, as long as $R_0\omega>\omega_\mathrm{cutoff}$. The condition of
$EC^{j_{\max}}$ smoothness has to be checked only for the first $n$
components of the function $f_\mathrm{ext}$ since its final $\nu+1$
components are zero.
Application of the Equivalence Theorem <ref> results in a
system of algebraic equations that has $(n+\nu+1)(2N+1)$ variables and
equations, where $N$ is the positive integer proven to exist in
Theorem <ref>. Let us denote as $F=(F_y,F_\omega,F_\mu)$ the
components of the right-hand side $F_\mathrm{ext}$ (given by
of which $F_\mu$ and $F_\omega$ are identically zero. Let
$p=(p_y,p_\omega,p_\mu)$ be the $2N+1$ leading Fourier coefficients of
$y$, $\omega$ and $\mu$, respectively (these are the variables of the
algebraic system constructed via Theorem <ref>), and
$X(p)=(X_y(p),X_\omega(p),X_\mu(p))$ be the map from
$R^{(n+\nu+1)(2N+1)}$ into $C^{j_{\max}}(\T;\R^{n+\nu+1})$. Then
several of the components of $p$ can be eliminated as variables, and
the equations for $p$ resulting from Theorem <ref>
correspondingly simplified. Since $F$ is identically zero in its last
$\nu+1$ components we have
\begin{align*}
X_\omega(p)&=E_Np_\omega\mbox{,} &
\end{align*}
Hence, the right-hand side
defined in Theorem <ref>, has $\nu+1$ components that are
identical to zero (since $P_0F(X(p))=0$ for the equations
$\dot\omega=0$ and $\dot\mu=0$). Furthermore, $g(p)=0$ contains the
equations $R_NQ_0E_Np_\omega=0$ and $R_NQ_0E_Np_\mu=0$, which require
that all Fourier coefficients (except the averages $R_0\omega$ and
$R_0\mu$) of $\mu$ and $\omega$ are equal to zero. This means
(unsurprisingly) that the algebraic system forces $\omega$ and $\mu$
to be constant. Thus, we can eliminate $R_NQ_0E_Np_\omega$ and
$R_NQ_oE_Np_\mu$ (which are $2N(\nu+1)$ variables), replacing them by
zero, and drop the corresponding equations. Since $\omega$ and $\mu$
must be constant, we can replace the arguments $p_\omega$ and $p_\mu$
of $X$ by the scalar $R_0E_Np_\omega$ (which we re-name back to
$\omega$) and the vector $R_0E_Np_\mu\in\R^\nu$ (which we re-name back
to $\mu$). This leaves the first $n(2N+1)$ algebraic equations
\begin{align}\label{eq:lowmodes:par}
\end{align}
which depend smoothly (with degree $j_{\max}$) on the $n(2N+1)$
variables $p_y$ and the parameters $\omega\in\R$ and
$\mu\in\R^\nu$. Overall, (<ref>) is a system of
$n\times(2\,N+1)$ equations.
§.§.§ Rotational Invariance
The original nonlinearity $F$, defined by $[F(x)](t)=f(\Delta_tx)$ is
equivariant with respect to time shift: $\Delta_tF(x)=F(\Delta_tx)$ for
all $t\in\T$ and $x\in C^0(\T;\R^n)$. Furthermore, $\Delta_t$ commutes
with the following operations:
\begin{align*}
\Delta_tQ_NL&=Q_NL\Delta_t \mbox{\quad (if $N\geq0$) and }
\end{align*}
This property gets passed on to the algebraic equation in the
following sense: let us define the operation $\Delta_t$ for a vector
$p$ in $\R^{n(2N+1)}$, which we consider as a vector of Fourier
coefficients of the function $E_Np\in C^0(\T;\R^n)$, by
With this definition $\Delta_t$ commutes with $R_N$ and $E_N$. It is a
group of rotation matrices: $\Delta_t$ is regular for all $t$, and
$\Delta_{2k\pi}$ is the identity for all integers $k$. The definition
of $X(p)$ as a fixed point of $x\mapsto E_Np+Q_NLF(x)$ implies that
$\Delta_tX(p)=X(\Delta_tp)$. From this it follows that the algebraic
system of equations is also equivariant with respect to $\Delta_t$. If
we denote the right-hand-side of the overall system
(<ref>) by $G(p_y,\omega,\mu)$ then $G$ satisfies
\begin{align*}
\Delta_t G(p_y,\omega,\mu)=G(\Delta_tp_y,\omega,\mu)\mbox{\quad for
all $t\in\T$, $p_y\in\R^{n(2N+1)}$, $\omega>0$ and
\end{align*}
§.§.§ Application to Hopf bifurcation
One useful aspect of the Equivalence Theorem is that it provides an
alternative approach to proving the Hopf Bifurcation Theorem for
equations with state-dependent delays. The first proof that the Hopf
bifurcation occurs as expected is due to Eichmann [5]. The
reduction of periodic boundary-value problems to smooth algebraic
equations reduces the Hopf bifurcation problem to an equivariant
algebraic pitchfork bifurcation.
Let us consider the equation
\begin{equation}
\label{eq:dynsys}
\end{equation}
where $f:C^0(J;\R^n)\times\R\mapsto\R^n$, $\mu\in\R$, $J$ is a compact
interval, $x_0\in\R^n$, and the operator $E_0$ (as defined in
(<ref>) in Section <ref>) extends a constant to a
function on $\T$ (and thus, on $J$). This means that
(<ref>) is a system of $n$ algebraic equations for the $n+1$
variables $(x_0,\mu)$. The definition of $EC^k$ smoothness does not
cover functionals that depend on parameters. We avoid the introduction
of a separate definition of $EC^k$ smoothness for parameter-dependent
functionals that distinguishes between parameters and functional
arguments. We rather extend Definition <ref>:
Let $J=[a,b]$ be a compact interval (or $J=\T$),
and $D$ be a product space of the form
$D=C^{k_1}(J;\R^{m_1})\times\ldots\times C^{k_\ell}(J;\R^{m_\ell})$
where $\ell\geq 1$, and $k_j\geq0$ and $m_j\geq1$ are integers. We
say that $f:D\times\R^\nu\mapsto \R^n$ is $EC^k$ smooth if the
\begin{equation}\label{eq:ass:pareck}
(x,y)\in D\times C^0(J;\R^\nu)\mapsto f(x,y(a))\in\R^n
\end{equation}
is $EC^k$ smooth (if $J=\T$ we use $a=-\pi$).
Requiring $EC^k$-smoothness of the parameter-dependent functional $f$
in this sense, implies that the algebraic system $0=f(E_0x_0,\mu)$ is
$k$ times continuously differentiable. Let us assume that the
algebraic system $0=f(E_0x_0,\mu)$ has a regular solution
$x_0(\mu)\in\R^n$ for $\mu$ close to $0$. Without loss of generality
we can assume that $x_0(\mu)=0$, otherwise, we introduce the new
variable $x_\mathrm{new}=x_\mathrm{old}-E_0x_0(\mu)\in
C^0(J;\R^n)$. Hence, $f(0,\mu)=0$ for all $\mu$ close to $0$.
The $EC^1$ derivative of $f$ in $(0,\mu)$ is a linear functional,
mapping $C^0(J;\R^{n+\nu})$ into $\R^n$. Let us denote its first $n$
components (the derivative with respect to the first argument $x$ of
$f$) by $a(\mu)$. The linear operator $a(\mu)$ can easily be
complexified by defining $a(\mu)[x+iy]=a(\mu)[x]+ia(\mu)[y]$ for
$x+iy\in C^0([-\tau,0];\C^n)$. If $f$ is $EC^k$ smooth with $k\geq2$
then the $n\times n$-matrix $K(\lambda,\mu)$ (called the
characteristic matrix), defined by
\begin{equation}\label{eq:charmdef}
K(\lambda,\mu)\,v=\lambda v-a(\mu)[v\exp(\lambda t)]
\end{equation}
is analytic in its complex argument $\lambda$ and $k-1$ times
differentiable in its real argument $\mu$ (since the functions
$t\mapsto v\exp(\lambda t)$ to which $a(\mu)$ is applied are all
elements of $C^k(J;\R^n)$). The Hopf Bifurcation Theorem states the
Assume that $f$ is $EC^k$ smooth ($k\geq 2$) in the sense of
Definition <ref> and that the characteristic matrix
$K(\lambda,\mu)$ satisfies the following conditions:
* (Imaginary eigenvalue) there
exists an $\omega_0>0$ such that $\det K(i\omega_0,0)=0$ and
$i\omega_0$ is an isolated root of $\lambda\mapsto \det
K(\lambda,0)$. We denote the corresponding null vector by
$v_1=v_r+iv_i\in\C^n$ (scaling it such that $|v_r|^2+|v_i|^2=1$).
* (Non-resonance) $\det
K(ik\omega,0)\neq 0$ for all integers $k\neq \pm1$.
* (Transversal crossing) The
local root curve $\mu\mapsto \lambda(\mu)$ of $\det
K(\lambda,\mu)$ that corresponds to the isolated root $i\omega_0$
at $\mu=0$ (that is, $\lambda(0)=i\omega_0$) has a non-vanishing
derivative of its real part:
Then there exists a $k-1$ times differentiable curve
such that for sufficiently small $\epsilon>0$ the following holds:
* $x(\omega\cdot)$ (or $S(x,\omega)$)
is a periodic orbit of $\dot x(t)=f(x_t,\mu)$ of period
$2\pi/\omega$, that is, $x\in C^1(\T;\R^n)$ and
\begin{equation}
\dot x(t)=
\frac{1}{\omega}f(S(\Delta_tx,\omega),\mu)\mbox{,}\label{eq:hopf:po}
\end{equation}
* the first Fourier coefficients of $x$
are equal to $(0,\beta)$, that is,
\begin{equation}
\begin{split}
\Re\left[v_1\exp(it)\right]^T x(t)\d t\mbox{,\quad and}\\
\beta&=-\frac{1}{\pi}
\int_{-\pi}^\pi\Im\left[v_1\exp(it)\right]^T x(t)\d t\mbox{,}
\end{split}
\label{eq:hopfphase}
\end{equation}
* $x\vert_{\beta=0}=0$,
$\mu\vert_{\beta=0}=0$ and $\omega\vert_{\beta=0}=\omega_0$, that
is, the solution $x$, the system parameter $\mu$ and the frequency
$\omega$ of $x$, which are differentiable functions of the
amplitude $\beta$, are equal to $x=0$, $\mu=0$, $\omega=\omega_0$
for $\beta=0$.
The statement is identical to the classical Hopf Bifurcation Theorem
for ODEs in its assumptions and conclusions apart from the regularity
assumption on $f$ specific to FDEs. Note that the existence of the
one-parameter family (parametrized in $\beta$) automatically implies
the existence of a two-parameter family for $\beta\neq0$ due to the
rotational invariance: if $x$ is a solution of (<ref>) then
$\Delta_sx$ is also a solution of (<ref>) for every fixed
$s\in\T$. Condition (<ref>) fixes the time shift $s$ of
$x$ such that $x$ is orthogonal to $\Re[v_1\exp(it)]$ using the
$\Lint^2$ scalar product on $\T$.
The proof of the Hopf Bifurcation Theorem is a simple fact-checking
exercise. We have to translate the assumptions on the derivative of
$f:C^0(J;\R^n)\mapsto\R^n$ into properties of the right-hand side of
the nonlinear algebraic system (<ref>) near
$(x,\omega,\mu)=(0,\omega_0,0)$, and then apply algebraic bifurcation
theory to the algebraic system. The only element of the proof that is
specific to functional differential equations comes in at the linear
level: the fact that the eigenvalue $i\omega_0$ is simple implies that
the right nullvector $v_1\in\C^n$ and any non-trivial left nullvector
$w_1$ satisfy
\begin{align*}
\lambda}K(\lambda,0)\right\vert_{
\textstyle\lambda=i\omega_0}\right]
\end{align*}
This is the generalization of the orthogonality condition
$w_1^Hv_1\neq 0$, known from ordinary matrix eigenvalue problems, to
exponential matrix eigenvalue problems of the type
$K(\lambda,\mu)\,v=0$. The proof of Theorem <ref> is
entirely based on the standard calculus arguments for branching of
solutions to algebraic systems as can be found in textbooks
[1]. The details of the proof can be found in
Section <ref>.
The statements in Theorem <ref> should be compared to two
previous works considering the same situation: the branching of
periodic orbits from an equilibrium losing its stability in FDEs with
state-dependent delays. The PhD thesis of Eichmann (2006, [5])
proves the existence of the curve
$\beta\mapsto(x(\beta),\mu(\beta),\omega(\beta))$ and that it is once
continuously differentiable (assuming only $EC^2$ smoothness of
$f$). Since $\mu'(0)=0$ due to rotational symmetry (see proof in
Section <ref>) this is not enough to determine if the
non-trivial periodic solutions exist for $\mu>0$ or for $\mu<0$ for
small $\beta$ (the so-called criticality of the Hopf
bifurcation, which is of interest in applications [15, 16]).
Moreover, the non-resonance condition in [5] is slightly too
strong, requiring that $i\omega_0$ is the only purely imaginary root
of $\det K(\lambda,0)$ (this assumption is different in the summary
given in the review by Hartung et al. [12]; note that
the publicly available version of [5] has a typo in the
corresponding assumption L1), and only the pure-delay case (where the
time interval $J$ equals $[-\tau,0]$) was considered. However, the
techniques employed in [5], based on the Fredholm alternative,
are likely to yield exactly the same result as stated in
Theorem <ref> if one assumes general $EC^k$ smoothness with
$k\geq2$ (the formulation of $EC^2$ smoothness is already rather
convoluted in [5]).
Hu and Wu [13] use $S^1$-degree theory [7, 18] to
prove the existence of a branch of non-trivial periodic solutions near
$(x,\mu,\omega)=(0,0,\omega_0)$. This type of topological methods
gives generally weaker results concerning the local uniqueness of
branches of periodic orbits or their regularity, but they require only
weaker assumptions ([13] still needs to assume $EC^2$
smoothness, though). Degree methods also give global existence results
by placing restrictions on the number of branches that can occur.
§ CONCLUSION, APPLICATIONS AND GENERALIZATIONS
Periodic boundary-value problems for functional differential equations
(FDEs) are equivalent to systems of smooth algebraic equations if the
functional $f$ defining the right-hand side of the boundary-value
problem satisfies natural smoothness assumptions. These assumptions
are identical to those imposed in the review by Hartung et al.
[12] and do not exclude FDEs with state-dependent
delay. There are several immediate extensions of the results presented
in this paper. The list below indicates some of them.
§.§.§ Further potential applications of the Equivalence
Theorem <ref>
Theorem <ref> on the Hopf bifurcation is not the central
result of the paper, even though it is a moderate extension of the
theorem proved in [5]. Rather, it is a demonstration of the use
of the Equivalence Theorem <ref>. The main strength of the
equivalence result stated in Theorem <ref> is that it permits
the straightforward application of continuation and Lyapunov-Schmidt
reduction techniques to FDE problems involving periodic orbits of
finite period, regardless if the delay is state dependent, or if the
equation is of so-called mixed type (that is, positive and negative
delays are present). A source of complexity, for example, in
[5, 7, 10, 13, 18, 26], is that techniques such as
Lyapunov-Schmidt reduction or $S^1$-degree theory had to be applied in
Banach spaces. Theorem <ref> removes the need for this,
reducing the analysis of periodic orbits to root-finding in
$\R^{n\times(2\,N+1)}$. For example, Humphries et al.
[14] study periodic orbits in FDEs with two state-dependent
delays numerically using DDE-Biftool [6], alluding to
theoretical results about bifurcations of periodic orbits that have
been proven only for constant delay. Fist of all, Humphries et
al. [14] continue branches of periodic orbits.
Theorem <ref> makes clear when these branches as curves of
points $(x,\omega,\mu)$ in the extended space
$C^0(\T;\R^n)\times\R\times\R$ are smooth to arbitrary degree: the
Jacobian of (<ref>) with respect to $(p_y,\omega,\mu)$
along the curve has to have full rank. Along these branches Humphries
et al. [14] encounter degeneracies of the linearization
and conjecture the existence of the corresponding bifurcations (backed
up with numerical evidence) such as: fold bifurcations,
period-doubling bifurcations, or the branching off of resonance
surfaces (Arnol'd tongues) when resonant Floquet multipliers of the
linearized equation cross the unit circle [20]. The Equivalence
Theorem <ref> provides a straightforward route to proofs
that these scenarios occur as expected. Similarly,
Theorem <ref> will likely not only simplify the proofs about
bifurcations of symmetric periodic orbits such as those of Wu
[26], but also extend them to the case of FDEs with
state-dependent delays. As long as one considers branching of periodic
orbits with finite periods, the problem can be reduced locally (and
often in every ball of finite size) to a finite-dimensional
root-finding problem. This transfers also a list of results of
symmetric bifurcation theory found in textbooks [8] to FDEs
with state-dependent delays.
§.§.§ Globally valid algebraic system
The main result was
formulated locally in the neighborhood of a given $x_0\in
C^{1,1}(\T;\R^n)$ and required only local Lipschitz continuity. The
proof makes obvious that the domain of definition for the map $X$,
which maps between the function space and the finite-dimensional space
is limited by the size of the neighborhood of $x_0$ for which one can
find a uniform ($EC$) Lipschitz constant of the right-hand side
$F$. In problems with delay the right-hand side is typically a
combination of Nemytskii operators generated by smooth functions and
the evaluation operator $\ev:C^0(\T;\R^n)\times\T\mapsto\R^n$, given by
$\ev(x,t)=x(t)$. These typically satisfy a (semi-)global Lipschitz
condition (see also condition (Lb) in [12]): for all $R$
there exists a constant $K$ such that
for all $x$ and $y$ satisfying $\|x\|_1\leq R$ and $\|y\|_1\leq R$.
Under this condition one can choose for any bounded ball an algebraic
system that is equivalent to the periodic boundary-value problem in
this bounded ball. For periodic orbits of autonomous systems this
means that one can find an algebraic system such that all periodic
orbits of amplitude less than $R$ and of period and frequency at most
$R$ are given by the roots of the algebraic system.
§.§.§ Implicitly given delays
In practical applications the delay is sometimes given implicitly, for
example, the position control problem considered in [23] and the
cutting problem in [15, 16] contain a separate algebraic
equation, which defines the delay implicitly. In simple cases these
problems can be reduced to explicit differential equations using the
standard Baumgarte reduction [2] for index-1 differential
algebraic equations. For example, in the cutting problem the delay
$\tau$ depends on the current position $x$ via the implicit linear
\begin{equation}\label{eq:cutdelay}
\tau(t)=a-bx(t)-bx(t-\tau(t))\mbox{,}
\end{equation}
which can be transformed into a differential equation by
differentiation with respect to time:
\begin{equation}\label{eq:cutdelaydiff}
\dot\tau(t)=
\frac{-bv(t)-\left[\tau(t)-a+bx(t)+bx(t-\tau(t))\right]}{1+bv(t-\tau(t))}
\end{equation}
($v(t)=\dot x(t)$ is explicitly present as a variable in the cutting
model, which is a second-order differential equation). The original
model accompanied with the differential equation
(<ref>) instead of the algebraic equation
(<ref>) fits into the conditions of the Equivalence
Theorem <ref>. The regularity statement of the Equivalence
Theorem guarantees that the resulting periodic solutions have
Lipschitz continuous derivatives with respect to time. This implies
that the defect $d=\tau-(a-bx-bx(t-\tau))$ occurring in the algebraic
condition (<ref>) satisfies $\dot d(t)=-d(t)$ along
solutions. Since the solutions are periodic the defect $d$ is
periodic, too, and, hence, $d$ is identically zero. The denominator
appearing in Equation (<ref>) becomes zero exactly in
those points in which the implicit condition (<ref>)
cannot be solved for the delay $\tau$ with a regular derivative.
The same argument can be applied to the position control problem as
long as the object, at position $x$, does not hit the base at position
$-w$ (the model contains the term $|x+w|$ in the right-hand side).
§.§.§ Neutral equations
The index reduction works only if the delay
$\tau$, which is itself a function of time, is not evaluated at
different time points. For example, changing $bx(t-\tau(t))$ to
$bx(t-\tau(t-1))$ on the right-hand-side of (<ref>) would
make the index reduction impossible. However, certain simple neutral
equations permit a similar reduction directly on the function space
level. Consider
\begin{equation}
\label{eq:neutral}
\frac{\d}{\d t} \left[\Delta_t(x+g(x))\right]=f(\Delta_t x)
\end{equation}
where the functional $f$ satisfies the local $EC$ Lipschitz condition,
defined in Definition <ref>, in a neighborhood $U$ of a
point $x_0\in\C^{1,1}(\T;\R^n)$, and $g:C^0(\T;\R^n)\mapsto\R^n$ has a
global (classical) Lipschitz constant less than unity (this excludes
state-dependent delays in the essential part of the neutral
equation). Then one can define the map $X_g(y)$ as the unique solution
$x$ of the fixed point problem
which reduces (<ref>) to the equation
\begin{equation}\label{eq:neutralred}
\dot y(t)=f(\Delta_tX_g(y))=f(X_g(\Delta_ty))\mbox{.}
\end{equation}
Equation (<ref>) satisfies the conditions of the
Equivalence Theorem <ref>. One implication of this reduction
is that periodic solutions of (<ref>) are $k$ times
continuously differentiable if the functional $f$ is $EC^k$ smooth in
the sense of Definition <ref> and $g$ is $k$ times
continuously differentiable as a map from $C^0(\T;\R^n)$ into $\R^n$.
§ PROOF OF THE EQUIVALENCE THEOREM <REF>
Theorem <ref> is proved in three steps. First, we establish
the existence of a locally unique fixed point of the map
$M_N(\cdot,p)$ using Banach's contraction mapping principle. This step
requires only local $EC$ Lipschitz continuity in the sense of
Definition <ref>. In the second step we prove continuous
differentiability of the map $X$ and the right-hand side $g$ of the
algebraic system assuming that $f$ is $EC^1$ smooth. In the final step
we prove higher-order differentiability, assuming that $f$ is $EC^k$
smooth for degrees $k$ up to $j_{\max}$.
§.§ Decay of Fourier coefficients for integrals and smooth functions
The following preparatory lemma states the well-known fact that,
colloquially speaking, integrating a function makes its high-frequency
Fourier coefficients smaller. In the fixed-point
equation (<ref>) of Theorem <ref> the term
$Q_NL$ occurs, and we need this term to be small for large $N$. Recall
that $Q_N$ removes the first $N$ Fourier modes from a periodic
function and $Lx$ is the anti-derivative of $x$ (after subtracting the
average of $x$), see Equation (<ref>) for the precise
The norm of the linear operator $Q_NL$, mapping the space
$C^j(\T;\R^n)$ back into itself, is bounded by
where $C$ is a constant. The same holds in the Lipschitz norm (with
the same constant $C$):
We find the norm $\|Q_NL\|_0$ first, and start out with the well-known
estimate for interpolating trigonometric polynomials for continuous
functions on $\T$. Let $x$ be a continuous function on $\T$ with
modulus of continuity $\omega$. Then (see [17])
where $C_0$ is a constant that does not depend on $x$ or $N$. A
function $\omega:[0,\infty)\mapsto[0,\infty)$ is called a modulus of
continuity of a continuous function $x$ if
holds for all $s$ and $t\in\T$. For a function $x\in C^0(\T;\R^n)$ the
[Lx](t)=∫_0^tx(s)-R_0x ṣ
has the Lipschitz constant $\|x\|_0=\max\{|x(t)|:t\in\T\}$ such that a
modulus of continuity for $Lx$ is $\omega(h)=\|x\|_0
h$. Consequently,
\begin{equation}\label{eq:qnlf0}
\|Q_NLx\|_0\leq C_0\frac{2\pi\|x\|_0}{N}\log N\mbox{,}
\end{equation}
where $C_0$ does not depend on $x$ or $N$. This proves the claim of the
lemma for $j=0$. For $x\in C^j(\T;\R^n)$ we notice that all
derivatives of $x$ up to order $j$ are continuous. Applying estimate
(<ref>) to each of the derivatives of $x$ we get
Q_NLx^(l)_0≤2πC_0/NlogN x^(l)_0
Consequently, the maximum of the left-hand sides over all
$l\in\{0,\ldots,j\}$ must be less than the maximum of the right-hand
Q_NLx_j=max_l=0,…,jQ_NLx^(l)_0≤2πC_0/NlogN max_l=0,…,jx^(l)_0=
which implies the desired estimate for $\|Q_NL\|_j$ using the constant
$C=2\pi C_0$.
The estimate of $Q_NL$ in the Lipschitz norm is a continuity
argument. The operator $Q_NL$ is bounded (and, thus, continuous) on
$C^{0,1}(\T;\R^n)$. For every element $y$ of $C^1(\T;\R^n)$ (which is
a dense subspace of $C^{0,1}(\T;\R^n)$) the Lipschitz constant is
identical to $\|y'\|_0=\max_{t\in\T}|y'(t)|$, and, thus,
$\|y\|_1=\|y\|_{0,1}$. Let $x_n\in C^1(\T;\R^n)$ be a sequence of
continuously differentiable functions that converges to $x\in
C^{0,1}(\T;\R^n)$ in the $\|\cdot\|_{0,1}$-norm: $\|x_n-x\|_{0,1}\to0$
for $n\to\infty$. Then
On both sides of the inequality the limit for $n\to\infty$ exists,
resulting in the desired estimate for $\|Q_NL\|_{0,1}$.
A direct consequence of Lemma <ref> is that the Lipschitz norm
of $Q_Nx$, $\|Q_Nx\|_{0,1}$, goes to zero for $N\to\infty$ for
elements of $C^{1,1}(\T;\R^n)$, so, for example, for a solution $x$ of
a periodic BVP:
\begin{equation}\label{eq:qn}
\|Q_Nx\|_{0,1}=\|Q_NLx'\|_{0,1}\leq
C\frac{\log N}{N}\|x'\|_{0,1}\leq C\frac{\log N}{N}\|x\|_{1,1}\mbox{.}
\end{equation}
§.§ Proof of Splitting Lemma <ref>
For any given integer $N\geq0$ we have to show that the pair $(x,p)\in
C^0(\T;\R^n)\times\R^{n\times(2\,N+1)}$ satisfies
\begin{align}\label{eq:inteq:app}
x(t)&=x(0)+\int_0^tF(x)(s)\d s\mbox{\quad for all $t\in\T$, and}\\
p&=R_Nx\mbox{,\quad (or, equivalently, $E_Np=P_Nx$)}\label{eq:peqpnx:app}\\
\intertext{ if and only if it satisfies the system}
\label{eq:lowmodes:app}\mbox{,}
\end{align}
“$\Rightarrow$”: Assume that $x\in C^0(\T;\R^n)$ satisfies
(<ref>), and let $p=R_Nx$. Subtracting
equation (<ref>) for $t=-\pi$ from (<ref>)
for $t=\pi$ implies that the average of $F(x)$ is zero. Thus,
$R_0F(x)=0$ and $P_0F(x)=0$. Since $Ly=\int_0^ty(s)-R_0y\d s$, the
identity (<ref>) implies (in combination with
\begin{equation}\label{eq:intident:app}
\end{equation}
Applying projection $Q_N$ to this identity we obtain
$Q_Nx=Q_NLF(x)$. Adding (<ref>) to this we obtain
equation (<ref>). Applying projection $P_NQ_0$ (which is
the same as $Q_0P_N$) to (<ref>) we obtain
$Q_0P_Nx=Q_0P_NLF(x)$. Inserting $E_Np$ for $P_Nx$ into this identity
leads to $Q_0[E_Np-P_NLF(x)]=0$. Since $P_0F(x)=0$, this in turn
implies (<ref>).
“$\Leftarrow$”: Applying $P_N$ to (<ref>) implies
$P_Nx=E_Np$ (and $p=R_Nx$) immediately. The expression inside the
parentheses of $R_N$ in equation (<ref>) is a sum of
two parts that each have to be zero (since they are both in the image
of $P_N$ on which $R_N$ in injective). The projection $Q_0$ subtracts
the average from its argument. Hence, $(x,p)\in
C^0(\T;\R^n)\times\R^{n\times(2\,N+1)}$ satisfies
(<ref>)–(<ref>) if and only if there
exists a constant $c\in\R^n$ such that the triple $(c,x,p)$ satisfies
the system of equations consisting of (<ref>) and
\begin{align}
\end{align}
Note that $E_0$ maps the constant $c\in\R^n$ to a function that equals
this constant for all $t\in\T$. In this system, (<ref>)
ensures that the average of $F(x)$ is
zero. Equation (<ref>) is an equation in the
finite-dimensional space $\rg P_N$. Subtracting
(<ref>) from (<ref>) gives
This equals (<ref>), keeping in mind that
$[Ly](t)=\int_0^ty(s)-R_0y\d s$ ($[Ly](0)=0$ for all $y\in
C^0(\T;\R^n)$, hence, $x(0)=c$), and using $R_0F(x)=0$ (see
equation (<ref>)).
§.§ Unique solvability of the fixed point problem (<ref>)
Let $x_0$ be an element of $C^{1,1}(\T;\R^n)$, for example, a solution
of the periodic boundary value problem $\dot
x(t)=f(\Delta_tx)=F(x)(t)$. Consider a closed ball
$B_\delta^{0,1}(x_0)$ of radius $\delta$ around $x_0$ in the Lipschitz
The superscript “$0,1$” indicates which norm is used to measure the
distance from $x_0$ and that only elements of $C^{0,1}(\T;\R^n)$ are
Lemma <ref> implies that $F$ is Lipschitz continuous with
respect to the $\|\cdot\|_0$-norm in $B_\delta^{0,1}(x_0)$ if we
choose $\delta$ sufficiently small (thus, $F$ is also called $EC$
Lipschitz continuous in $B_\delta^{0,1}(x_0)$):
\begin{equation}\label{eq:flip0}
\|F(x)-F(y)\|_0\leq K\|x-y\|_0
\end{equation}
for all $x$ and $y$ in $B_\delta^{0,1}(x_0)$ and a fixed $K>0$. In any
ball $B_\delta^{0,1}$, in which $F$ is $EC$ Lipschitz continuous, $F$
is also bounded in the Lipschitz norm:
\begin{equation}\label{eq:Fboundlip}
\|F(x)\|_{0,1}\leq R\mbox{\quad for all $x\in B_\delta^{0,1}(x_0)$.}
\end{equation}
See Lemma <ref> in Appendix <ref> for the
We can now formulate a lemma about the unique solvability of the fixed
point problem
This unique solvability and the Splitting Lemma <ref> allow
us to reduce the periodic BVP $\dot x(t)=f(\Delta_tx)$ to a system of
algebraic equations. Remember that $E_Np$ takes a vector $p$ of $2N+1$
Fourier coefficients and maps it to the periodic function having these
Fourier coefficients, $R_Nx$ extracts the first $2N+1$ Fourier
coefficients from a periodic function $x$, $P_Nx$ projects the
periodic function $x$ onto the space spanned by the basis
$b_{-N},\ldots,b_N$ and $Q_N=\id-P_N$ sets the first Fourier modes of
a function to zero. ($P_N$ and $Q_N$ are projections in the function
space, and $R_N$ and $E_N$ map between the finite-dimensional subspace
$\rg P_N$ and $\R^{n\times(2N+1)}$.)
Let $x_0$ be in $C^{1,1}(\T;\R^n)$, and let $\delta>0$ be such that
\begin{equation}\label{eq:fbounds}
\|F(x)\|_{0,1}\leq R\mbox{\quad and\quad} \|F(x)-F(y)\|_0\leq K\|x-y\|_0
\end{equation}
for all $x$ and $y\in B_{6\delta}^{0,1}(x_0)$ and for some constants
$K>0$ and $R>0$ depending on $\delta$. Then for any sufficiently
large $N$ the fixed point problem
\begin{equation}\label{eq:fixp}
\end{equation}
has a unique solution $x\in B_{6\delta}^{0,1}(x_0)$ for all vectors
$p\in\R^{n\times(2N+1)}$ in the neighborhood $U$ of $R_Nx_0$ given
\begin{equation}
\label{eq:pclose}
\left\|E_N\left[p-R_Nx_0\right]\right\|_{0,1}<2\delta\right\}\mbox{.}
\end{equation}
Moreover, if $x\in B_\delta^{0,1}(x_0)$ is continuously
differentiable and satisfies $x'=F(x)$ then its projection $p=R_Nx$ is
in the neighborhood $U$, and $x$ and $p$ satisfy (<ref>).
Note that $U$ is an open set of $\R^{n\times(2N+1)}$ since $E_N$ is an
isomorphism between $\rg P_N$, equipped with the
$\|\cdot\|_{0,1}$-norm, and $\R^{n\times(2N+1)}$. We have to prove the
unique solvability of the fixed-point problem in a slightly larger
ball (radius $6\delta$) and for a slightly larger range of parameters
$p$ (note the $2\delta$ in (<ref>)) in order to establish
one-to-one correspondence in the ball of radius $\delta$.
The idea is, of course, that the function
\begin{align*}
M_N(\cdot,p): x\mapsto E_Np+Q_NLF(x)
\end{align*}
maps the closed ball $B_{6\delta}^{0,1}(x_0)$ back into itself and is
uniformly contracting for suitably large $N$ and vectors $p\in U$.
First, any closed ball $B_r^{0,1}(x_0)$ is closed (and, thus, forms a
complete metric space) with respect to the $\|\cdot\|_0$-norm. This
completeness is a simple continuity argument: let $y_n=x_0+z_n$ be a
Cauchy sequence in $B_r^{0,1}(x_0)$ with respect to the
$\|\cdot\|_0$-norm. Then $z_n$ converges to a continuous function $z$,
and, since $\|z_n\|_0\leq\|z_n\|_{0,1}\leq r$, for all $n$, the
maximum norm of $z$ is also bounded by $r$: $\|z\|_0\leq r$. We only
have to show that the Lipschitz constant of $z$ is bounded by
$r$, too. Let $\epsilon>0$ be arbitrary and let $t\neq s$ be
arbitrary in $\T$. We select some $n$ such that $\|z-z_n\|_0<
\epsilon|t-s|/2$. Then
\begin{align*}
&< \epsilon|t-s|+r|t-s|\leq (r+\epsilon)|t-s|\mbox{.}
\end{align*}
Thus, the Lipschitz constant of $z$ is less than $r+\epsilon$ for
arbitrary $\epsilon>0$. Hence, $\|z\|_{0,1}\leq r$, completing the
argument for completeness of $B_r^{0,1}(x_0)$ with respect to the
$\|\cdot\|_0$-norm. This completeness implies that we can apply
Banach's contraction mapping principle in a ball $B_r^{0,1}(x_0)$, a
ball of Lipschitz continuous functions, using the weaker maximum norm
in the following.
We choose the radius $r$ of the ball equal to $6\delta$ ($\delta$ was
chosen in the lemma such that the estimates (<ref>) are
true for the constants $K$ and $R$), Thus, $B_{6\delta}^{0,1}(x_0)$ is
the set to which we want to apply Banach's contraction mapping
principle. To ensure that the map $M_N(\cdot,p)$ maps into
$B_{6\delta}^{0,1}(x_0)$ for $p\in U$, and that $M_N(\cdot,p)$ is a
contraction we pick $N$ large enough. Specifically, we pick $N$ such
\begin{equation}\label{eq:nchoice}
\begin{aligned}
\|Q_Nx_0\|_{0,1}&\leq 2\delta\mbox{,}&
\|Q_NL\|_{0,1}&\leq \frac{2\delta}{R}\mbox{,}\\
\|Q_NL\|_0&\leq \frac{1}{2K}\mbox{,}& C\frac{\log
N}{N}&< 1/\max\left\{1,\left(R+\|x_0\|_{1,1}\right)/\delta\right\}\mbox{,}
\end{aligned}
\end{equation}
where $R$ and $K$ are the bounds on $F$ given in the conditions of the
lemma, in Equation (<ref>). We know that these bounds exist
due to Lemma <ref> (see Equation (<ref>)) and
Lemma <ref> (see Equation (<ref>)). We
know that choosing $N$ according to (<ref>) is possible
from Lemma <ref> and estimate (<ref>) following
Lemma <ref>.
Let us check first that $x\mapsto E_Np+Q_NLF(x)$ maps the closed ball
$B_{6\delta}^{0,1}(x_0)$ back into itself:
\begin{align*}
\lefteqn{\left\|E_Np+Q_NLF(x)-x_0\right\|_{0,1}\leq}&\\
&\leq \left\|E_N\left[p-R_Nx_0\right]\right\|_{0,1}+
\left\|Q_Nx_0\right\|_{0,1}+
\left\|Q_NL\right\|_{0,1}\left\|F(x)\right\|_{0,1}\\
\end{align*}
Here we used the bounds (<ref>) implied by our choice of
$N$ and the definition (<ref>) of the set $U$ of permitted
$p$, and the bound on $\|F(x)\|_{0,1}$, which is determined in
(<ref>) by our choice of $\delta$.
Second, let us check that $x\mapsto E_Np+Q_NLF(x)$ is a uniform
contraction in $B_{6\delta}^{0,1}$ with respect to the
\begin{align*}
\left\|Q_NL\left[F(x)-F(y)\right]\right\|_0\leq
\left\|Q_NL\right\|_0\left\|F(x)-F(y)\right\|_0\leq
\frac{1}{2K}K\|x-y\|_0 \leq \frac{1}{2}\|x-y\|_0\mbox{.}
\end{align*}
Again, we exploited the bounds (<ref>), implied by our
choice of $N$, and the Lipschitz constant $K$ of $F$ determined in
(<ref>) by our choice of $\delta$.
Since $B_{6\delta}^{0,1}(x_0)$ is complete with respect to the
$\|\cdot\|$-norm Banach's contraction mapping principle implies that
the fixed point problem (<ref>) has a unique solution $x\in
B_\delta^{0,1}(x_0)$ for $p\in U$.
Finally, let us check that for $x\in B_\delta^{0,1}(x_0)\cap
C^1(\T;\R^n)$ satisfying the periodic BVP $x'=F(x)$ the projection
$p=R_Nx$ is in $U$. For this we have to prove that if
$\|x-x_0\|_{0,1}\leq \delta$ and $x'=F(x)$ then
$\|P_N(x-x_0)\|_{0,1}< 2\delta$. We can estimate $\|P_N(x-x_0)\|_{0,1}$ via
\begin{align}
\|P_N(x-x_0)\|_{0,1} &\leq \|(I-Q_N)(x-x_0)\|_{0,1}
\leq \|x-x_0\|_{0,1}+\|Q_N(x-x_0)\|_{0,1}
\label{eq:pnxx0:tri}\\
&\leq \delta +C\frac{\log N}{N}\|x-x_0\|_{1,1}
\label{eq:pnxx0:estqnl}\\
&\leq\delta +C\frac{\log N}{N}\max\{|x-x_0\|_{0,1},\|x'-x_0'\|_{0,1}\}
\label{eq:pnxx0:defn11}\\
&\leq \delta +C\frac{\log N}{N}\max\{\delta,\|x'\|_{0,1}+\|x_0'\|_{0,1}\}
\label{eq:pnxx0:balltri}\\
&= \delta +C\frac{\log N}{N}\max\{\delta,\|F(x)\|_{0,1}+\|x_0\|_{1,1}\}
\label{eq:pnxx0:xpFx}\\
&\leq \delta+C\frac{\log N}{N}\max\{\delta,R+\|x_0\|_{1,1}\}<2\delta\mbox{.}
\label{eq:pnxx0:fbound}
\end{align}
The inequality (<ref>) follows from the definition of
$P_N$ and $Q_N$ and the triangular inequality for the
$\|\cdot\|_{0,1}$-norm. The step from (<ref>) to
(<ref>) uses the estimate (<ref>) for the norm
$\|Q_Ny\|_{0,1}$ for elements $y$ of $C^{1,1}(\T;\R^n)$. It also
bounds $\|x-x_0\|_{0,1}$ by the radius $\delta$ of the ball. Step
(<ref>) splits up the $\|\cdot\|_{1,1}$ norm into its
two parts which are estimated separately in the following steps. One
part, $\|x-x_0\|_{0,1}$ is bounded by $\delta$ (the radius of the
ball), the difference of the derivatives is bounded by a triangular
inequality for its parts, $\|x'\|_{0,1}$ and $\|x_0'\|_{0,1}$ in
(<ref>). To get to (<ref>) we use that
$x$ satisfies the BVP $x'=F(x)$. We also bound the norm of $x_0'$ by
$\|x_0\|_{1,1}$. Finally, in (<ref>) we estimate the
Lipschitz norm of $F(x)$, $\|F(x)\|_{0,1}$ by the bound $R$ determined
in (<ref>) by our choice of $\delta$. The right-hand side
of (<ref>) is (strictly) less than $2\delta$ by our
choice of $N$, see (<ref>).
§.§ Lipschitz continuity of the algebraic system
The Splitting Lemma <ref> guarantees in combination with the
unique existence of the fixed point of $M_N(\cdot,p)$, proven in
Lemma <ref>, the equivalence between the periodic BVP $\dot
x(t)=f(\Delta_tx)$ and the algebraic equation $g(p)=0$ for $x$ inside
the ball $B_\delta^{0,1}(x_0)$, where $g$ is given in
(<ref>) by
\begin{align}
\label{eq:lowmodes}
g&:p\in U\mapsto
\in\R^{n\times(2\,N+1)}\mbox{, where}\\
\label{eq:xpdef}
X&:p\in U\mapsto C^0(\T;\R^n) \mbox{,\quad and $X(p)$ is the fixed point
of $M_N(\cdot,p)$ in $B_{6\delta}^{0,1}(x_0)$}\mbox{.}
\end{align}
The relation between $p\in U$ and $x\in B_\delta^{0,1}(x_0)$ is given
via $p=R_Nx$ and $x=X(p)$: if $x$ satisfies the periodic BVP then
$p=R_Nx$ satisfies $g(p)=0$, and, vice versa, if $p\in U$ satisfies
$g(p)=0$ then $x=X(p)$ satisfies the periodic BVP. The domain of
definition, $D(X)=U$ is an open set, however the map $X$ (and, thus,
$g$) can be extended continuously to the boundary of $U$:
$M_N(\cdot,p)$ maps into the closed ball $B_{6\delta}^{0,1}$ back into
itself also for $p$ on the boundary of $U$ and it still has
contraction rate $1/2$ with respect to the $\|\cdot\|_0$-norm.
The remainder of the section addresses the remaining open claim of the
Equivalence Theorem <ref>, namely the regularity of the maps
$X$ and $g$. Using only local $EC$ Lipschitz continuity
(Definition <ref>) we can prove the Lipschitz continuity of
$g$ and $X$:
* For all $p$ in the neighborhood $U=D(X)$,
defined in (<ref>), the image $X(p)$ is in
$C^{1,1}(\T;\R^n)$ (that is, $X(p)\in C^1(\T;\R^n)$ and its time
derivative is Lipschitz continuous),
* $X$ is Lipschitz continuous with respect to
the $\|\cdot\|_1$-norm for its images: there exists
a constant $C_N$ such that
* the map $p\in U\mapsto
\left[R_0F(X(p)),P_NLF(X(p))\right]\in\R^n\times\R^{n\times(2\,N+1)}$
is Lipschitz continuous in $U$.
Proof For a function $y\in\rg P_N$, differentiation is a
bounded operator: $y'=D_Ny$. The vector $R_Ny$ of the first $2N+1$
Fourier coefficients of a function $y$ and the vector $R_N[y']$
satisfy $R_N[y']=\tilde D_NR_Ny$ where $\tilde D_N$ is a matrix
(independent of $y$). Hence, $y'=E_N\tilde D_NR_Ny$ for all $y\in\rg
P_N$ such that we can define $D_N=E_N\tilde D_NR_N$. Denote $X(p)$ as
$x$. By definition of the map $X$, $x=E_Np+Q_NLF(x)$. The right-hand
side of this fixed-point equation is differentiable with respect to
time, giving
\begin{equation}\label{eq:xpform}
\end{equation}
This guarantees that $x\in
C^1(\T;\R^n)$. Equation (<ref>) ensures that
$\|F(x)\|_{0,1}\leq R$, which implies that the right-hand side of
(<ref>) is Lipschitz continuous in time. This in turn
implies that $x'$ is Lipschitz continuous in time (thus, $x\in
C^{1,1}(\T;\R^n)$), and
Representation (<ref>) also implies point <ref>:
let $x=X(p)$ and $y=X(q)$ be two functions in the image of $X$:
\begin{equation}\label{eq:xlip1norm}
\|x'-y'\|_0\leq\|D_NE_N\|_0|p-q|+(\|Q_0\|_0+\|D_NP_NL\|_0)K\|x-y\|_0\mbox{,}
\end{equation}
where $K$ was the $EC$ Lipschitz constant of $F$ in
$B_{6\delta}^{0,1}(x_0)$. The difference $x-y$ in the
$\|\cdot\|_0$-norm is bounded due to the contractivity of the
right-hand side in fixed point problem (<ref>) defining $X$
(the $\|\cdot\|_0$-norm was the metric used to apply the contraction
mapping principle):
\begin{align*}
\|x-y\|_0&\leq \|E_N\|_0|p-q|+\|Q_NL[F(x)-F(y)\|_0\leq
\|E_N\|_0|p-q|+\frac{1}{2}\|x-y\|_0\mbox{.}
\intertext{Thus,}
\|x-y\|_0&\leq 2\|E_N\|_0|p-q|\mbox{,}
\end{align*}
which, combined with (<ref>), gives Lipschitz continuity of
$X$ as a map from $U$ into $C^1(\T;\R^n)$:
\begin{equation}\label{eq:xlip10norm}
\|x'-y'\|_0\leq\left[\|D_NE_N\|_0+(\|Q_0\|_0+\|D_NP_NL\|_0)2K\|E_N\|_0\right]
\end{equation}
Point <ref> is a direct consequence of the Lipschitz
continuity of $F$ with respect to $\|\cdot\|_0$-norm in
$B_{6\delta}^{0,1}(x_0)$, the Lipschitz continuity of $X$ on $U$ in
the $\|\cdot\|_0$-norm, and the fact that $X$ maps into
§.§ First-order differentiability
of the algebraic system
Until now we have only used the $EC$ Lipschitz continuity (in the
sense of Definition <ref>) of the right-hand side $F$ in
the ball $B_{6\delta}^{0,1}(x_0)$ with respect to the
$\|\cdot\|_0$-norm. We can expect that the right-hand side $g$ of the
algebraic system, defined in (<ref>), is smooth only if we
require more smoothness of the right-hand side $f$ (which enters $F$
in the algebraic system).
We first discuss first-order differentiability of the map $X$ and the
right-hand side $g$, defined in (<ref>) and
(<ref>). For this we assume $EC^1$ smoothness of $f$ as
defined in Definition <ref>. For $x\in C^1(\T;\R^n)\cap
B_{6\delta}^{0,1}(x_0)$ the norm of $\partial^1f(x,\cdot)$ as an
element of $L(C^0(\T;\R^n);\R^n)$ (the space of linear functionals
mapping $C^0(\T;\R^n)$ into $\R^n$) is less than or equal to $K$, the
$EC$ Lipschitz constant of $F$ (and, hence, $f$) in
$B_{6\delta}^{0,1}(x_0)$ assumed to exist in the conditions of
Lemma <ref>.
Let us define the map
\begin{align*}
\partial^1F:&C^1(\T;\R^n)\times C^0(\T;\R^n)\mapsto
C^0(\T;\R^n)\mbox{,} &
\left[\partial^1F(v,w)\right](t)&= \partial^1f(\Delta_t
\end{align*}
If $v\in C^1(\T;\R^n)$ and $w\in C^{0,1}(\T;\R^n)$ then the map
$\partial^1F$ defined above is indeed the derivative of $F$ in
$v$ with respect to the deviation $w$ (see Lemma <ref> in
Appendix <ref>):
\begin{equation}\label{eq:Fdiff1}
\lim_{
\begin{subarray}{c}
w\in C^{0,1}(\T;\R^n)\\[0.2ex]
\|w\|_{0,1}\to0
\end{subarray}
\frac{\|F(v+w)-F(v)-\partial^1F(v,w)\|_0}{\|w\|_{0,1}}=0\mbox{.}
\end{equation}
Part of the definition of $EC^1$ smoothness for $f$ is that the map
$\partial^1f$ is continuous in both arguments, $v\in C^1(\T;\R^n)$ and
$w\in C^0(\T;\R^n)$. One can then apply Lemma <ref> to
$\partial^1f$ to conclude that the map $\partial^1F$ (a composition of
$\Delta_t$ and $\partial^1f$) is continuous with respect to the
$\|\cdot\|_0$-norm in its image space as a map of both arguments (in
their respective norm),
For $v\in B_{6\delta}^{0,1}(x_0)$ the norm of the linear map
$\partial^1F(v,\cdot)$ as an element of
$L(C^0(\T;\R^n);C^0(\T;\R^n))$, the space of continuous linear
functionals from $C^0(\T;\R^n)$ back to itself, is bounded by the $EC$
Lipschitz constant $K$ of $F$ in $B_{6\delta}^{0,1}(x_0)$.
The additional regularity assumption on $f$ and its implications for
$F$ permit us to improve our statements about regularity of $X$ and
the algebraic system:
Assume that the right-hand side $f$ is $EC^1$ smooth in the
sense of Definition <ref>. Then the regularity
statements about the map $X$, defined in (<ref>), and the
right-hand side of the algebraic system, defined in
(<ref>), can be extended:
* $X(p)$ is in $C^2(\T;\R^n)$ for all $p\in U=D(X)$, the domain
of definition of $X$, and $p\mapsto X(p)$ is continuous with
respect to the $\|\cdot\|_2$-norm for its images.
* The map $X$, which maps $U$ into $C^1(\T;\R^n)$ according to
Lemma <ref>, is continuously differentiable with
respect to its argument $p$ using the $\|\cdot\|_1$-norm for its images.
* The map $p\in U\mapsto
\left[R_0F(X(p)),P_NLF(X(p))\right]\in\R^n\times\R^{n\times(2\,N+1)}$
is continuously differentiable with respect to $p$.
Proof Let $p\in U=D(X)\subset \R^{n\times(2\,N+1)}$, where
$U$ is defined in (<ref>), and let us denote $X(p)$ by
$x$. Lemma <ref> ensures already that $x$ is in
$C^{1,1}(\T;\R^n)$. Lemma <ref> in
Appendix <ref> proves that $F(x)\in C^1(\T;\R^n)$ for
$x\in C^1(\T;\R^n)$ (choosing $D=C^0(\T;\R^n)$ and $k=0$ in the
assumptions of Lemma <ref>). This implies the first
statement, that $X(p)\in C^2(\T;\R^n)$: since
\begin{equation}\label{eq:Xdiff:proof:xpc2}
\end{equation}
and $X(p)\in C^{1,1}(\T;\R^n)$ (see Lemma <ref>), $F(X(p))$
is in $C^1(\T;\R^n)$, and, thus, $LF(X(p))$ is in
$C^2(\T;\R^n)$. Hence, $X(p)$ is an element of $C^2(\T;\R^n)$,
too. Furthermore, Lemma <ref> states that $F$ is continuous
as a map from $C^1(\T;\R^n)$ into $C^1(\T;\R^n)$. Since $X$ is
continuous as a map from $U$ into $C^1(\T;\R^n)$ (in fact, it is
Lipschitz continuous, see Lemma <ref>), the right-hand side
of (<ref>) in $p$ is continuous with respect to the
$\|\cdot\|_1$-norm. This proves the first point.
Concerning the second statement: again, let $p_0$ be in $U=D(X)$, and
choose a small open neighborhood $U(p_0)$ which has a positive
distance to the boundary of $U$. We will prove point two for all $p\in
U(p_0)$. Let us choose an initial $\epsilon_0$ sufficiently small such
that $p+hq$ is still in $U$ for $h\in(-\epsilon_0,\epsilon_0)$, all
$p\in U(p_0)$, and all $q$ with $|q|\leq 1$. Let us introduce the
difference quotient for $h\in(-\epsilon_0,\epsilon_0)\setminus\{0\}$:
\begin{equation}\label{eq:Xdiffproof:zdef}
z(h,q,p)=\frac{1}{h}\left[X(p+hq)-X(p)\right] \mbox{.}
\end{equation}
The maps $z$ maps
$\left[(-\epsilon_0,\epsilon_0)\setminus\{0\}\right]\times B_1(0)\times
U(p_0)\subset \R\times \R^{n\times(2\,N+1)}\times
\R^{n\times(2\,N+1)}$ into $C^1(\T;\R^n)$. We first prove that $z$
has a limit for $h\to0$ in $C^1(\T;\R^n)$, and that this limit is
achieved uniformly for all $p\in U(p_0)$ and $|q|\leq 1$. By
definition of $X$, $z$ satisfies the fixed point equation (dropping
all arguments from $z$)
\begin{equation}\label{eq:gfixpd0}
\end{equation}
for $h\in(-\epsilon_0,\epsilon_0)\setminus\{0\}$. Let us introduce
\begin{equation}\label{eq:Xdiffproof:tA1def}
\tilde A_1(p,z,h)=
\begin{cases}
\frac{1}{h}\left[F(X(p)+hz)-F(X(p))\right] & \mbox{if $h\neq0$}\\
\partial^1F(X(p),z) & \mbox{if $h=0$,}
\end{cases}
\end{equation}
which maps $U(p_0)\times C^{0,1}(\T;\R^n)\times\R$ into
$C^0(\T;\R^n)$. The limit (<ref>) implies that $\tilde A_1$
is continuous in all arguments (insert $v=x$, $w=hz$ into
(<ref>)). Using $\tilde A_1$ we extend the fixed point
problem (<ref>) to $h=0$:
\begin{equation}\label{eq:gfixpd1}
z=E_Nq+Q_NL\tilde A_1(X(p),z,h)\mbox{.}
\end{equation}
The following intermediate lemma proves that the fixed point problem
(<ref>) has a unique solution:
There exists an $\epsilon>0$ and constants $C_0>0$ and $C_1>0$ such
that the map
which depends on the additional parameters $p$, $q$ and $h$, has a
unique fixed point $z_*$ in
for all $h\in(-\epsilon,\epsilon)$, all $p\in U(p_0)\subset
U=D(X)\subset\R^{n\times(2\,N+1)}$ and all $q\in\R^{n(2N+1)}$ with
$|q|<1$. The fixed point $z_*$ is an element of $C^1(\T;\R^n)$ and
depends continuously on $h$, $p$ and $q$ with respect to the
Note that the $\epsilon$ we have to choose in Lemma <ref>
is smaller than the initial $\epsilon_0$ for which the difference
quotient $z$ is defined.
Proof of Lemma <ref> First of all, since $\tilde
A_1$ is continuous in all arguments, the map $\gamma$ is
continuous. Moreover, since $x'=(X(p))'$ and $x=X(p)$ depend
continuously on $p$ (see Lemma <ref> and expression
(<ref>)), the map $\gamma$ also depends continuously on the
parameters $p$, $q$ and $h$ (that is, the expression $E_nq+Q_NL\tilde
A_1(X(p),z,h)$, defining $\gamma$, depends continuously on $z$, $p$, $q$
and $h$ with respect to the $\|\cdot\|_{0,1}$-norm). We choose the
constants $C_0>0$ and $C_1>0$ such that
\begin{align}
\label{eq:c0choice}
\label{eq:c1choice}
C_1&\geq \|D_NE_N\|_0+
\left(\|Q_0\|_0+\|D_NP_NL\|_0\right)KC_0\mbox{,}
\end{align}
where $K$ is the Lipschitz constant of $F$ with respect to the
$\|\cdot\|_0$-norm in $B_{6\delta}^{0,1}$.
We choose $\epsilon\leq\epsilon_0$ such that for all $z$ satisfying
$\|z\|_{0,1}\leq C_1$ and all $p\in U(p_0)$ the function $X(p)+hz$ is
in $B_{6\delta}^{0,1}(x_0)$ for all $h\in(-\epsilon,\epsilon)$. This
implies that for any $z_1$ and $z_2$ satisfying $\|z_1\|_{0,1}\leq
C_1$ and $\|z_2\|_{0,1}\leq C_1$ we have
\begin{equation}
\begin{split}
\frac{1}{h}\|F(X(p)+hz_1)-F(X(p)+hz_2)\|_0&\leq K\|z_1-z_2\|_0\mbox{,}\\
\|\partial^1F(X(p),z_1-z_2)\|_0&\leq K\|z_1-z_2\|_0\mbox{,}
\end{split}\label{eq:g0lip}
\end{equation}
where $K$ was the Lipschitz constant for $F$ in
$B_{6\delta}^{0,1}(x_0)$, and, thus,
\begin{align}
\|\gamma(z_1)-\gamma(z_2)\|_0&\leq \frac{1}{2}\|z_1-z_2\|_0\label{eq:g0contract}
\end{align}
for all $h\in(-\epsilon,\epsilon)$ by choice of $N$ ($N$ was such that
$\|Q_NL\|_0\leq (2K)^{-1}$). This estimate for $\gamma$ implies
\begin{equation}\label{eq:g0bound}
\|\gamma(z)\|_0\leq \|E_N\|_0+\frac{1}{2}\|z\|_0
\mbox{\qquad if $\|z\|_{0,1}\leq C_1$,}
\end{equation}
since $\gamma(0)=E_Nq$ and $|q|\leq1$. Moreover, the two
inequalities (<ref>) imply that for
$h\in(-\epsilon,\epsilon)$, $\|z\|_{0,1}\leq C_1$ and $p\in U(p_0)$
the maximum norm of $\tilde A_1(p,z,h)$ is bounded by $K\|z\|_0$:
\begin{equation}
\label{eq:tA1bound}
\|\tilde A_1(p,z,h)\|_0\leq K\|z\|_0
\end{equation}
The time derivative of $\gamma(z)$ exists and its $\|\cdot\|_0$-norm can be
estimated by differentiating the expression $E_nq+Q_NL\tilde
A_1(X(p),z,h)$, defining $\gamma$, with respect to time in the same manner
as we obtained (<ref>) (we insert (<ref>) to
bound $\|\tilde A_1(p,z,h)\|_0$):
\begin{equation}\label{eq:g1bound}
\left\|\frac{\d}{\d t}\gamma(z)\right\|_0\leq
\|D_NE_N\|_0+(\|Q_0\|_0+\|D_NP_NL\|_0)K\|z\|_0\mbox{.}
\end{equation}
The combination of the bounds (<ref>) and
(<ref>) and the definition of the constants $C_0$ and $C_1$
guarantee that $\gamma(z)$ maps the set
back into itself. The contraction estimate (<ref>) for
the $\|\cdot\|_0$-norm and the completeness of $B$ with respect to the
$\|\cdot\|_0$-norm make the contraction mapping principle applicable
with a uniform contraction rate for all $p\in U(p_0)$, all $|q|\leq1$
and $h\in(-\epsilon,\epsilon)$. This ensures that the fixed point
$z_*$ depends continuously on $p$, $q\in\R^{n(2N+1)}$ and
$h\in(-\epsilon,\epsilon)$ with respect to the $\|\cdot\|_0$-norm
(since $\gamma$ is continuous with respect to $z$, $h$, $q$ and $p$).
The time derivative $z_*'$ of $z_*$ also exists and is continuous in
$p$, $q$ and $h$: we differentiate the fixed point equation
(<ref>) for $z_*$ with respect to time (in the same way as
done in (<ref>)) to get
\begin{align}\label{eq:zpform}
z_*'=&D_NE_Nq+Q_0\tilde A_1(X(p),z_*,h)- D_NP_NL\tilde
\end{align}
which is a continuous function in $p$, $q$ and $h$ with respect to the
$\|\cdot\|_0$-norm (note that $z_*$ depends on $p\in U(p_0)$, $q$ and
$h$). Thus, the fixed point $z_*$ is in $C^1(\T;\R^n)$ and depends
continuously on $p$, $q$ and $h$ with respect to the
Proof of Lemma <ref> continued As a consequence of
Lemma <ref> we may write the fixed point $z_*$ of $\gamma$ as a
function of $h$, $q$ and $p$: $z_*(h,q,p)$ maps
$h\in(-\epsilon,\epsilon)$, $q$ in the unit ball of
$\R^{n\times(2\,N+1)}$ and $p\in U(p_0)$ continuously into
$C^1(\T;\R^n)$. It is also identical to $z(h,q,p)$, defined in
(<ref>) as the directional difference quotient of
$X$. Thus, the directional difference quotient $z(h,q,p)$ has a limit
for $h\to0$ in the $\|\cdot\|_1$-norm, and this limit equals
$z_*(0,q,p)$. Moreover, this limit $z_*(0,q,p)$ depends continuously
on $p$ and $q$ in the $\|\cdot\|_1$-norm (as proved in
Lemma <ref>), and it is linear in $q$ (since $\tilde
A_1(p,z,0)$ is linear in $z$). Thus, $z_*(0,q,p)$ is the Frechét
\begin{equation}\label{eq:Xdiff1proof:Frechet0}
\lim_{q\to0}\frac{\|X(p+q)-X(p)-z_*(0,q,p)\|_1}{|q|}=0\mbox{.}
\end{equation}
Consequently, the map
$(p,q)\mapsto z_*(0,q,p)=\partial^1X(p)\,q$ is continuous in the
$\|\cdot\|_1$-norm as claimed in the lemma.
The third statement of Lemma <ref> is a consequence of the
second statement and the fact that the difference quotient of $F$ has
a limit in the $\|\cdot\|_0$-norm if it is taken between arguments in
$C^1(\T;\R^n)$ (see (<ref>)). We split the difference
quotients into two parts:
\begin{align}\label{eq:Fd1expr1}
\frac{F(X(p+hq))-F(X(p))}{h}=&\ \frac{F(X(p)+h\partial^1
X(p)q)-F(X(p))}{h} +\\
&\ +\frac{F(X(p+hq))-F(X(p)+h\partial^1 X(p)q)}{h}\label{eq:Fd1expr2}
\end{align}
The right-hand side in (<ref>) converges in the
$\|\cdot\|_0$-norm to $\partial^1F(X(p),\partial^1X(p)q)$ for $h\to0$,
since $X(p)$ and $\partial^1X(p)q$ are in $C^1(\T;\R^n)$ because $F$
is $EC^1$ continuous (see the second point of the lemma for the
regularity of $\partial^1X(p)q$ and Lemma <ref> for the
regularity of $X(p)$). For the term in (<ref>) we can
apply the local $EC$ Lipschitz continuity (all arguments are in
$B_{6\delta}^{0,1}(x_0)$ for $p\in U(p_0)$, $|q|\leq 1$ and
$h\in(\epsilon,\epsilon)$) such that we get
which converges to $0$ for $h\to0$ due to the second statement of the
lemma ($K$ is the $EC$ Lipschitz constant of $F$ in
$B_{6\delta}^{0,1}(x_0)$). Consequently, we obtain from the limit of
(<ref>) for $h\to0$ that the directional derivative of
$F(X(p))$ in direction $q$ is equal to $\partial^1F(X(p),\partial^1
X(p)q)$, which is continuous with respect to $p$ and $q$ and linear in
$q$. Thus,
\begin{equation}\label{eq:Fd1expr}
\left[\frac{\partial}{\partial p}F(X(p))\right]q
=\partial^1F(X(p),\partial X(p)q)\mbox{,}
\end{equation}
and $p\mapsto F(X(p))$ is continuously differentiable with respect to
$p$ in the $\|\cdot\|_0$-norm. Note that we use the notation not
enclosing $q$ in the bracket in (<ref>) to highlight that
this is a classical derivative with respect to a finite-dimensional
variable. The linear operators $R_0$ and $P_NL$ preserve the
continuity (and the linearity in $q$) of (<ref>).
For $x=X(p)$ (where $p\in U=D(X)$) consider
the linear map
The spectral radius of $M$ as a map from $C^0(\T;\R^n)$ back into
itself, or as a map from $C^1(\T;\R^n)$ back into itself, is less or equal $1/2$.
Proof Since $M$ is compact as an element of
$L(C^k(\T;\R^n);C^k(\T;\R^n))$ (the space of linear functionals from
$C^k(\T;\R^n)$ back to itself) for $k=0$ and $k=1$, the spectral radius
is identical to the modulus of the maximal (in modulus) eigenvalue,
which is of finite algebraic multiplicity if it is different from
zero. An eigenvector $z$ corresponding to this maximal eigenvalue is
an element of $C^1(\T;\R^n)$ such that the spectral radius of $M$ is
the same for $k=0$ and $k=1$.
Since $x$ and $z$ are both in $C^1(\T;\R^n)$ we have that
\begin{equation}\label{eq:sradproof:d1f}
\partial^1F(x)z=\lim_{h\to0}\frac{1}{h}\left[F(x+hz)-F(x)\right]
\end{equation}
For $x=X(p)$ where $p\in U$, and $h$ sufficiently small the arguments
of $F$, $x+hz$ and $x$, both lie inside $B_{6\delta}^{0,1}$ such that
the $EC$ Lipschitz constant $K$ applies to the difference:
Since $\|Q_NL\|_0\leq 1/(2K)$,
(<ref>) and (<ref>) combine to
As $z$ is an eigenvector corresponding to the largest eigenvalue, the
spectral radius of $M$ is less or equal $1/2$.
Thus, the derivative $z=\tpartial X(p)\,q$ of $X$ in $p$ is the unique
solution of the contractive linear fixed point problem in
\begin{equation}\label{eq:dxfixp}
\end{equation}
§.§ Higher degrees of smoothness
We observe that $(x,y)=(X(p),\partial^1X(p)\,q)$ satisfies the
system of equations
\begin{equation}\label{eq:fixp:ext}
\begin{split}
\end{split}
\end{equation}
This has a similar structure to the original fixed point problem
(<ref>) but in dimension $n_1=2n$ with the variables $(x,y)$
and parameters $(p,q)$. Thus, we aim to apply a linear version of the
arguments of Section <ref> recursively, assuming that $f$
is $EC^k$ smooth as recursively defined in
Definition <ref>. Throughout this section we assume that
$f$ is $EC^k$ smooth for all degrees up to order $j_{\max}$.
For higher-order derivatives, we introduce the spaces $D_j$ and the
operators $\partial^jF$ for $j\geq0$ recursively:
\begin{align*}
D_0&=C^0(\T;\R^n) & D_j&=D_{j-1}^1\times D_{j-1}\\
\partial^jF:&D_j\mapsto C^0(\T;\R^n)\mbox{,} &
\end{align*}
The spaces $D_j$ are products of the type (<ref>), and
the argument $x$ of $\partial^jF$ and $\partial^jf$ is in $D_j$, a
product of $2^j$ spaces. We also recall that the notion of subspaces
$D_j^k$ of higher-oder ($k\geq0$) differentiability for product spaces
such as $D_j$ was introduced in Section <ref>.
For example,
\begin{align*}
D_1^k&=D_0^{k+1}\times D_0^k=C^{k+1}(\T;\R^n)\times C^k(\T;\R^n)\mbox{,}\\
D_2^k&=D_1^{k+1}\times D_1^k=C^{k+2}(\T;\R^n)\times
C^{k+1}(\T;\R^n)\times C^{k+1}(\T;\R^n)\times C^k(\T;\R^n)\mbox{, etc.,}
\end{align*}
all with their natural maximum norms. The maps $\partial^jF$ are all
continuous and map indeed into $C^0(\T;\R^n)$ due to the continuity of
$\partial^jf$ and $\Delta_t$ (applying Lemma <ref> to $D_j$,
$\partial^jF$ and $\partial^jf$). It is also clear from the definition
that $\partial^{j+k}F=\partial^j[\partial^kF]$ if $j+k\leq
j_{\max}$. We will also use the notation $L(D_j^k;D_i^\ell)$ for the
space of linear bounded functionals mapping from $D_k^k$ into
The following lemma is a consequence of the $EC^k$ smoothness of
For $j+k\leq j_{\max}$ the operator $\partial^jF$ is a continuous map from
$D_j^k$ into $C^k(\T;\R^n)$.
Proof of Lemma We have to apply Lemma <ref> from
Appendix <ref> inductively over the order of
differentiability ($k$). To start the induction for $k=0$ we can apply
Lemma <ref> to $D_j$, $\partial^jF$ and $\partial^jf$. For
the inductive step let us assume that for $k$ we know that
$\partial^jF:D_j^k\mapsto C^k(\T;\R^n)$ is continuous for all $j\leq
j_{\max}-k$. Let us fix a $j\leq j_{\max}-k-1$. We have to show that
$\partial^jF$ maps $D_j^{k+1}$ continuously into
$C^{k+1}(\T;\R^n)$. We know (by inductive assumption) that
$\partial^jF$ maps $D_j^k$ continuously into $C^k(\T;\R^n)$ and that
$\partial^{j+1}F$ maps $D_{j+1}^k=D_j^{k+1}\times D_j^k$ continuously
into $C^k(\T;\R^n)$. Thus, we can apply Lemma <ref> to
$\partial^jF$ (this takes the place of the operator $F$ in
Lemma <ref>) and $D=D_j^k$, obtaining that
$\partial^jF:D_j^{k+1}\mapsto C^{k+1}(\T;\R^n)$ is continuous.
An immediate consequence of Lemma <ref> is that $X(p)$
and $\partial X(p)\,q$, as constructed in Section <ref>,
are as smooth as the right-hand-side:
Let $f$ be $EC^{j_{\max}}$ smooth. For every $p\in U=D(X)$ and every
$q\in R^{n(2N+1)}$ the functions $X(p)$ and $\partial X(p)\,q$
satisfy $X(p)\in C^{j_{\max}+1}(\T;\R^n)$ and $\partial X(p)\,q\in
C^{j_{\max}}(\T;\R^n)$. Moreover, the maps
\begin{align*}
p&\mapsto X(p)\in C^{j_{\max}+1}(\T;\R^n)\mbox{\quad and\quad}
[p,q]\mapsto \partial X(p)\,q\in C^{j_{\max}}(\T;\R^n)
\end{align*}
are continuous.
Proof The function $x=X(p)$ satisfies
$x=E_Np+Q_NLF(x)$. Since $F$ maps $D_0^k=C^k(\T;\R^n)$ back into
itself continuously for all $k\leq j_{\max}$, $Q_NL$ maps $D_0^k$ into
$D_0^{k+1}$ continuously for all $k$, and $E_Np\in C^\infty(\T;\R^n)$,
the fixed point equation implies the following: if $x\in D_0^k$ then
$F(x)\in D_0^k$, thus, $x=E_Np+Q_NLF(x)\in D_0^{k+1}$(for all $k\leq
j_{\max}$). Similarly, $z=E_Nq+Q_NL\partial^1F(x)\,z$, and
$\partial^1F$ maps $D_1^k$ into $D_0^k$ for all $k\leq
j_{\max}-1$. Thus, the fixed point equation implies: if $z\in D_0^k$
and $x\in D_0^{k+1}$ then $(x,z)\in D_1^k$, thus, $\partial^1F(x,z)\in
D_0^k$, thus, $z=E_Nq+Q_NL\partial^1F(x,z)\in D_0^{k+1}$ for all
$k\leq j_{\max}-1$. All of the above dependencies are continuous such
that the continuous dependence on $p$ and $q$ in the norms of
$D_0^{j_{\max}+1}$ and $D_0^{j_{\max}}$, respectively, follows.
We plan to find the derivatives of the map $X$ inductively through
fixed point equations of the form (<ref>). In order to set
up the recursion we define inductively the operators $F_j$ by
\begin{align}
F_0(x)&=F(x) &&\mbox{for $x\in D_0$}\label{eq:F0def}\\
\begin{pmatrix}
x\\ y
\end{pmatrix}
\begin{bmatrix}
\partial^1F_{j-1}(x,y)
\end{bmatrix}\mbox{,} &&\mbox{for\ }
\begin{bmatrix}
x\\ y
\end{bmatrix}\in D_j=D_{j-1}^1\times D_j\mbox{.}\label{eq:Fjdef}
\end{align}
Note that $F_j$ is always linear in its second argument, $y$, since
$\partial^1F_{j-1}$ is linear in its second argument. The operators
$F_j$ are combinations of derivatives of $F$. The plan is to study
fixed-point problems of the type $x=E_Np+Q_NLF_j(x)$ (with $j=1$ we
obtain (<ref>)). Before doing so, we establish which
spaces the operators $F_j$ map into:
For $j+l+k\leq j_{\max}$ the operator $\partial^lF_j$ maps
$D_{j+l}^k$ continuously into $D_j^k$. In particular, $F_j$ maps
$D_j$ continuously back into itself.
Proof The statement of the lemma follows inductively from the
definition of $F_j$ and $D_j^k$. We apply Lemma <ref>
to start our induction over $j$ (for $j=0$ the statement is identical
to Lemma <ref>). For the inductive step let us assume
that we know that $\partial^lF_{j-1}$ maps $D_{j+l-1}^k$ continuously
into $D_{j-1}^k$ for all $k$ and $l$ satisfying $l+k\leq
j_{\max}-j+1$. By definition (<ref>) of $F_j$ the derivative
$\partial^lF_j$ for $l\leq j_{\max}-j$ is
\begin{align*}
\partial^lF_j
\begin{pmatrix}
x\\ y
\end{pmatrix}=
\begin{bmatrix}
\partial^lF_{j-1}(x)\\
\partial^{l+1}F_{j-1}(x,y)
\end{bmatrix}
&&\mbox{for\ }
\begin{bmatrix}
x\\ y
\end{bmatrix}\in D_{l+j}=D_{l+j-1}^1\times D_{l+j-1}\mbox{.}
\end{align*}
The first component, $\partial^lF_{j-1}$ maps $D_{l+j-1}^{k+1}$
continuously into $D_{j-1}^{k+1}$ for all $k$ from $0$ to
$j_{\max}-l-j$ (this is the assumption of the inductive step when one
shifts the index $k$ down by $1$). Similarly, $\partial^{l+1}F_{j-1}$
maps $D_{j+l-1}^{1+k}\times D_{j+l-1}^k=D_{j+l}^k$ continuously into
$D_{j-1}^k$ for all $k$ from $0$ to $j_{\max}-l-j$, again due to the
assumption of the inductive step. Consequently, $\partial^lF_j$ maps
$D_{j+l-1}^{k+1}\times D_{j+l-1}^k=D_{j+l}^k$ continuously into
$D_{j}^k$ for all $k$ from $0$ to $j_{\max}-l-j$, which is the
statement we had to prove for the inductive step.
Even though the map $x\in D_j^1\mapsto \partial^1F_j(x,\cdot)\in
L(D_j;D_j)$ is in general not continuous, the following map is:
For $j<j_{\max}$ the map $x\in D_j^1\mapsto
Q_NL\partial^1F_j(x,\cdot)\in L(D_j^1;D_j^1)$ is continuous with
respect to $x\in D_j^1$.
Proof of Lemma <ref> The $EC^k$ smoothness of
$f$ (for $k\leq j_{\max}$) implies that $F_j$ is continuously
differentiable (in the classical sense) as a map from $D_j^1$ into
$D_j$. Thus, the map $x\mapsto\partial^1F_j(x,\cdot)$ as a map from
$D_j^1$ into $L(D_j^1;D_j)$ is continuous. Recall that the operator
$L$ involves taking the anti-derivative of its argument such that
$L:D_j\mapsto D_j^1$. Since $Q_NL$ maps $D_j$ continuously into
$D_j^1$, the map $x\mapsto Q_NL\partial^1F_j(x,\cdot)$ is continuous
as a map from $D_j^1$ into $L(D_j^1;D_j^1)$.
The following theorem provides continuous differentiability of order
$j_{\max}$ for $X$ and the map $p\mapsto F(X(p))$ if the right-hand
side is $EC^k$ smooth in the sense of Definition <ref> for
$k\leq j_{\max}$:
Define $n_0=n(2N+1)$ and $n_j=2^jn_0$, and the maps
\begin{align*}
X_0&: p\in U=D(X)\subseteq \R^{n_0}\mapsto X(p)\in D_0\mbox{\ and}\\
Y_0&:p\in U=D(X)\subseteq \R^{n_0}\mapsto F(X(p))\in D_0\mbox{,}
\end{align*}
and assume that $f:D_0=C^0(\T;\R^n)\mapsto\R^n$ is $EC^{j_{\max}}$
smooth. Then the following maps exist and are continuous for all $j$
up to $j_{\max}$:
\begin{align*}
X_j&:[p,q]\in D(X_j):=D(X_{j-1})\times\R^{n_{j-1}}\subseteq\R^{n_j}
\mapsto [X_{j-1}(p),\partial X_{j-1}(p)\,q]\in D_j\mbox{,}\\
Y_j&:[p,q]\in D(X_j)
\phantom{\ :=D(X_{j-1})\times\R^{n_{j-1}}\subseteq\R^{n_j}}
\mapsto [Y_{j-1}(p),\partial Y_{j-1}(p)\,q]\in D_j\mbox{.}
\end{align*}
The proof of Theorem <ref> does not require the application
of the contraction mapping principle for nonlinear maps. It uses only
Lemma <ref>, Lemma <ref> and
Lemma <ref> inductively.
Proof of Theorem <ref> The main work is the proof
of the existence and continuity of $X_j$, which we will do
The assumption of the inductive step is
comprised of the following two statements. We assume for $j$:
* The map $(p_1,p_2)\in D(X_{j-1})\times
\R^{n_{j-1}}\mapsto X_j(p_1,p_2)\in D_j$ exists and is
continuous. Moreover, the pair $(x_1,x_2)=X_j(p_1,p_2)$ satisfies
\begin{align}
\end{align}
* The linear map $z\mapsto
Q_NL\partial^1F_{j-1}(x_1,z)$ maps $D_{j-1}^1$ back into itself and
has spectral radius less or equal $1/2$.
Both statements of the assumption of the inductive step have been
proven for $j=1$ in Lemma <ref> and
Lemma <ref> . Let $j$ be smaller than $j_{\max}$.
§.§.§ Regularity of $X_j(p)$
Let us first establish that the map
∂^1X_j-1(p_1) p_2
does not only map continuously into $D_j$ but into $D_j^k$ for all
$k\leq j_{\max}-j+1$.
The argument is the same as in the proof of
Lemma <ref>: the map $F_j$ maps $D_j^k$ continuously
back into $D_j^k$ for all $k\leq j_{\max}-j$. If $x\in D_j^k$ then
$F_j(x)\in D_j^k$, thus, $x=E_Np+Q_NLF(x)\in D_j^{k+1}$ for all $k\leq
j_{\max}-j$ (and the dependence on $p$ is continuous because all
dependencies are continuous).
§.§.§ Proof of existence and continuity of $\partial^1X_j(p)\,q$
Let us use the notation $p=(p_1,p_2)$ and $x=(x_1,x_2)=X_j(p)$. Let
$p_0\in D(X_j)$ be arbitrary. We first show that $\partial^1X_j(p)\,q$
exists for all $p$ in a neighborhood $U(p_0)$ with positive distance
to the boundary of $D(X_j)$. We can choose $\epsilon>0$ sufficiently
small such that $p+hq\in D(X_j)$ for all $h\in(-\epsilon,\epsilon)$,
all $q=(q_1,q_2)$ with $|q|<1$ and all $p\in U(p_0)$. Consider the
difference quotient
∂^1X_j-1(p_1+hq_1) [p_2+hq_2]-∂^1X_j-1(p_1) p_2
By assumption of the inductive step, $X_{j-1}$ is continuously
differentiable such that the first row of this difference quotient
has the form
\begin{equation}\label{eq:xjm1p1q1}
\int_0^1\partial^1X_{j-1}(p_1+hsq_1)\,q_1\d s
\end{equation}
for $h\neq0$. As established above
$(p_1,q_1)\mapsto \partial^1X_{j-1}(p_1)\,q_1 \in D_{j-1}^k$ is
continuous for all $k\leq j_{\max}-j+1$ such that
z_1(h,p_1,q_1)∈D_j-1^j_max-j+1 ⊆D^2_j-1
($j_{\max}-j+1\geq2$ since $j<j_{\max}$), and $z(h,p_1,q_1)$ depends
continuously on its arguments, also when $h=0$. Let us use the
\begin{align*}
x_1(p_1)&=X_{j-1}(p_1)\mbox{,} \\
\int_0^1\partial^1X_{j-1}(p_1+hsq_1)\,q_1\d s\\
-\partial^1X_{j-1}(p_1)\,p_2\right]\mbox{\quad for $h\neq0$.}
\end{align*}
With these notations we have $X_{j-1}(p_1+hq_1)=x_1+hz_1$ and, for
non-zero $h$, $\partial^1X_{j-1}(p_1+hq_1)[p_2+hq_2]=x_2+hz_2$. The
fixed-point equations (<ref>) and (<ref>) imply a
fixed-point equation for the difference quotient $z_2$ for non-zero
\begin{align}
\partial^1F_{j-1}(x_1,x_2)\right]\nonumber\\
\mbox{\quad where}\label{eq:z2fixp}\\
\tilde z(p_1,p_2,q_1,h)&=Q_NL\frac{\partial^1F_{j-1}(x_1+hz_1,x_2)-
\partial^1F_{j-1}(x_1,x_2)}{h}\nonumber
\end{align}
The regularity of $x_1$, $x_2$ and $z_1$ is:
\begin{equation}\label{eq:x1x2z1reg}
\begin{split}
x_1&\in D_{j-1}^{j_{\max}-j+2}\subseteq D_{j-1}^3\mbox{,} \\
x_2&\in D_{j-1}^{j_{\max}-j+1}\subseteq D_{j-1}^2\mbox{\quad and} \\
z_1&\in D_{j-1}^{j_{\max}-j+1}\subseteq D_{j-1}^2\mbox{.}
\end{split}
\end{equation}
We can apply the mean value theorem to the difference quotient
appearing in $\tilde z$ since $x_1$ and $z_1$ are at least in $D_{j-1}^2$
and $x_2$ is at least in $D_{j-1}^1$ (see
Lemma <ref>, and Lemma <ref> and
Lemma <ref> in Appendix <ref>):
\begin{align*}
\tilde z(p_1,p_2,q_1,h)&=
Q_NL\int_0^1\partial^2F_{j-1}(x_1+shz_1,x_2,z_1,0)\d s\mbox{.}
\end{align*}
The map
$(x_1,x_2,z_1,h)\mapsto \partial^2F_{j-1}(x_1+shz_1,x_2,z_1,0)$ maps
$x_1$, $x_2$, $z_1$ and $h$ continuously into the space
$D_{j-1}^{j_{\max}-j-1}$ (we see this by applying
Lemma <ref> to $\partial^2F_{j-1}$, setting $k$ in
Lemma <ref> to $j_{\max}-j-1$). Thus, the quantity
$\tilde z(p_1,p_2,q_1,h)$ is in $D_{j-1}^{j_{\max}-j}\subseteq
D_{j-1}^1$ (since $j\leq j_{\max}-1$). It depends continuously on
$p_1$, $p_2$, $q_1$ and $h$ in this space, and can be extended to
$h=0$ continuously (such that $\tilde z(p_1,p_2,q_1,0)\in
D_{j-1}^{j_{\max}-j}$, too).
Hence, (<ref>) is a linear fixed-point problem for $z_2$
where the inhomogeneity is in $D_{j-1}^{j_{\max}-j}$ and depends
continuously on $(p,q,h)$. The linear map $M(h):z_2\mapsto
Q_NL\partial^1F_{j-1}(x_1+hz_1)\,z_2$ in front of $z_2$ on the
right-hand side of (<ref>) depends continuously on $h$ as an
element of $L(D_{j-1}^1;D_{j-1}^1)$ (see Lemma <ref> and
note that $x_1$ and $z_1$ are in $D_{j-1}^1$). Since the spectral
radius of the map $M(0)$ (for $h=0$) is less or equal than $1/2$ by
assumption of our inductive step, the spectral radius of $M(h)$ is
less than unity if we choose $h$ sufficiently small. Thus, for all
$p\in D(X_j)$ and $q\in \R^{n_j}$ and sufficiently small $h$, $z_2$
satisfies a contractive linear fixed point equation with an
inhomogeneity in $D_{j-1}^1$ and a contractive linear map that maps
into $D_{j-1}^1$ where all coefficients depend continuously on
$(h,p,q)$. Consequently, $z_2$ has a limit in $D_{j-1}^1$ for $h\to0$
that depends continuously on $(p,q)$. For $h=0$ the fixed point
equation for $(z_1,z_2)$ simplifies to
\begin{equation}
\label{eq:isz}
\begin{split}
\partial^1F_{j-1}(x_1,z_2)\right]\mbox{.}
\end{split}
\end{equation}
Both equations are linear in $q$ and $z=(z_1,z_2)$. Consequently,
$z(0,p,q)$, which is by definition the directional derivative of $X_j$
in $p$ in direction $q$, depends linearly on $q$ and continuously on
$p$ and $q$. Consequently,
is the Frechét derivative of $X_j$.
§.§.§ Collection to finish proof of statement 1 of inductive step
The functions $x=X_j(p)$ and $z=\partial^1X_j(p)q$ satisfy
\begin{equation}\label{eq:ivfjp1}
\begin{aligned}
x=&E_Np+Q_NLF_j(x) &&\mbox{by inductive assumption
\eqref{eq:ivf}--\eqref{eq:ivderiv},}\\
z=&E_Nq+Q_NL\partial^1F_j(x,z) &&\mbox{by \eqref{eq:isz} and
definition of $F_j$.}
\end{aligned}
\end{equation}
The variable $x=X_j(p)$ depends continuously on $p$ with respect to
the norm of $D_j^1$ by the assumption of the inductive step and the
step “Regularity of $X_j(p)$”. The variable $z=\partial^1X(p)\,q$
depends continuously on $p$ and $q$ as shown in the previous step,
“Existence and continuity of $\partial^1X(p)\,q$”. Thus
$(x,z)=(X_j(p),\partial^1X(p)\,q)=X_{j+1}(p,q)\in D_j^1\times
D_j=D_{j+1}$ depends continuously on $(p,q)$, and satisfies
(<ref>)–(<ref>) for $j+1$ (which is identical to
system (<ref>)). This completes the proof of
statement <ref> of the inductive assumption for $j+1$.
§.§.§ Spectral radius of map $z\mapsto Q_NL\partial^1F_j(x,z)$
The map $\partial^1F_j$ maps $D_{j+1}$ continuously into $D_j$ (by
Lemma <ref>). Thus, for fixed $x$ the linear map
$z\mapsto \partial^1F_j(x,z)$ maps $D_j$ continuously into $D_j$, and,
hence, the map $M_j(x): z\mapsto Q_NL\partial^1F_j(x,z)$ maps $D_j$
continuously into $D_j^1$, making $M_j(x)$ a compact linear
operator. Thus, the spectral radius of $M_j(x)$ is determined by its
largest eigenvalue (which has finite modulus and algebraic
multiplicity if it is non-zero). Splitting $M_j(x)$ into its two
components we get
Q_NL[ ∂^1F_j-1(x_1,z_1)
If $(\lambda,(z_1,z_2))$ is an eigenpair of $M_j(x)$ then the first
row of the definition of $M_j(x)$ implies that, either $(\lambda,z_1)$
is an eigenpair of $z_1\mapsto Q_NL\partial^1F_{j-1}(x_1,z_1)$, or
$z_1=0$. If $(\lambda,z_1)$ is an eigenpair of $z_1\mapsto
Q_NL\partial^1F_{j-1}(x_1,z_1)$ then, by inductive assumption,
$|\lambda|\leq 1/2$. If $z_1=0$ then the term
$\partial^2F_{j-1}(x_1,x_2,z_1,0)$ vanishes in the second row, such
that $(\lambda,z_2)$ is an eigenpair of $z_2\mapsto
Q_NL\partial^1F_{j-1}(x_1,z_2)$. Thus, by inductive assumption,
$|\lambda|\leq 1/2$ in this case, too. Consequently, the spectral
radius of $M_j(x)$ is also less or equal to $1/2$, which proves
statement <ref> of the inductive assumption for
Existence of $Y_j$
We show inductively that
$Y_j(p)=F_j(X_j(p))$. For $j=1$ this statement was proven in
Lemma <ref>. Let $j<j_{\max}$ and assume that
$Y_j=F_j(X_j(p))$ for $p\in D(X_j)$. Since
and $F_j$ maps $D_j^1$ into $D_j^1$, $X_j$ is an element of
$D_j^1$. Let $q\in \R^{n_j}$ be arbitrary, and let us denote
$(x_1,x_2)=(X_j(p),\partial^1X_j(p)\,q)=X_{j+1}(p,q)$. The component $x_2$ satisfies
such that $x_2$ is in $D_j^1$, too. Consequently,
\begin{align}
\frac{Y_j(p+hq)-Y_j(p)}{h}&=
\frac{F_j(X_j(p+hq))-F_j(X_j(p))}{h}\nonumber\\
\frac{F_j(X_j(p+hq))-F_j(x_1+hx_2)}{h}\mbox{.}\label{eq:Yjsplit}
\end{align}
Since $F_j$ is continuously differentiable for $x_1\in D_j^1$ and
deviations $hx_2\in D_j^1$ the first quotient in the
expression (<ref>) converges to
$\partial^1F_j(x_1,x_2)$. Since $F_j$ as a map from $D_j^1$ into $D_j$
is locally Lipschitz continuous the second term in (<ref>)
can be bounded by
\begin{eqnarray*}
\lefteqn{\left\| \frac{F_j(X_j(p+hq))-
\end{eqnarray*}
with some constant $K_1$, for sufficiently small $h$, which converges
to zero for $h\to0$ because $X_j$ is differentiable. Consequently, the
directional derivative of $Y_j$ in $p$ in direction $q$ is
$\partial^1F_j(X_j(p))[\partial X_j(p)\,q]$, which is continuous in
$p$ and $q$ and linear in $q$. Therefore, the Frechét derivative
of $Y_j$ exists and
∂^1F_j(X_j(p),∂X_j(p) q)
which implies by definition of $F_j$ and $X_j$ that $Y_{j+1}=F_{j+1}(X_{j+1}(p,q))$.
We can refine the statement of Theorem <ref> slightly by
noting that $X_j:D(X_j)\mapsto D_j^1$ is continuous for all $j\leq
j_{\max}$ (instead of $X_j:D(X_j)\mapsto D_j$). This follows from
the continuity of $Y_j=F_j(X_j(p))$ as a map into $D_j$ and the
Theorem <ref> completes the proof of the Equivalence
Theorem <ref>. The refinement (that $X_j$ maps into $D_j^1$)
ensures that the image $X(p)$ is in $C^{j_{\max}+1}(\T;\R^n)$, as
claimed in Theorem <ref>
§ PROOF OF HOPF BIFURCATION THEOREM
First, we note that $x\mapsto S(x,\omega)^{-1}=x(\omega^{-1}\cdot)$
maps $C^k(\T;\R^n)$ into a closed subspace of $C^k([-\tau,0];\R^n)$,
if we extend functions $x$ on $\T$ to the whole real line by setting
$x(t)=x(t_{\mod[-\pi,\pi)})$. This implies that, if the functional
$f:C^0([-\tau,0];\R^n)\times\R\mapsto\R^n$ is $EC^k$ smooth then the
is $EC^k$ smooth, too, such that we can reduce the problem of finding
periodic orbits of frequency $\omega$ to the algebraic system
(<ref>). The right-hand side $F_y$ in
(<ref>) is defined by
Let us choose the periodic orbit $x_0=(x,\omega,\mu)$ with $x=0$,
$\omega=\omega_0$, $\mu=0$ as the solution in the neighborhood of
which we construct the equivalent algebraic system. We choose the
number $N$ of Fourier modes and the size $\delta$ of the neighborhood
$B_\delta^{0,1}(x_0)$ in $C^{0,1}(\T;\R^{n+2})$ such that the
conditions of Theorem <ref> are satisfied in
$B_\delta^{0,1}(x_0)$. The full algebraic system
(<ref>) then reads (after multiplication with $\omega$
and mapping it onto the space $\rg P_N$ from $\R^{n(2N+1)}$ by
applying $R_N^{-1}$)
\begin{equation}
\begin{aligned}
\omega Q_0P_NE_Np- Q_0P_NL F_y(X_y(p,\omega,\mu),\omega,\mu)
\end{aligned}
\label{eq:hopf:modes}
\end{equation}
The variables are $p\in\R^{n(2N+1)}$ (which was called $p_y$ in
(<ref>)), $\mu$ and $\omega$. We know from
Theorem <ref> that
\begin{align*}
\mapsto F(X_y(p,\omega,\mu),\omega),\mu)\in C^0(\T;\R^n)\mbox{,}\\
X_y:&(p,\mu,\omega)\in\R^{n(2N+1)}\times\R\times\R\mapsto X_y(p,\omega,\mu)
\in C^0(\T;\R^n)
\end{align*}
are $k$ times differentiable, and note that
\begin{equation}\label{eq:hopf:fzero}
\end{equation}
for all $\omega\approx\omega_0$ and $\mu\approx0$ (because
$x_0=(0,\omega,\mu)$ is a solution). The derivative of the right-hand
side $F_y$ in $x=0$, $\omega\approx\omega_0$ and $\mu\approx0$ with
respect to $x$ is $A(\omega,\mu)x$, defined by
\begin{align*}
\left[A(\omega,\mu)x\right](t)=a(\mu)\left[x(t+\omega\cdot)\right]\mbox{,}
\end{align*}
where $a(\mu)$ is the same linear functional as used in the definition
of the characteristic matrix $K(\lambda,\mu)$ in (<ref>)
(the derivatives of $F$ with respect to $\omega$ and $\mu$ are zero
due to (<ref>)). We observe that $A(\omega,\mu)$
commutes with $P_j$ and $Q_j$ for all $j\geq0$.
Let us now determine the linearization of $X_y(p,\omega,\mu)$ in
$(p,\omega,\mu)=(0,\omega,\mu)$. Due to (<ref>)
$X_y(0,\omega,\mu)$ is equal to zero for all $\omega\approx\omega_0$
and $\mu\approx0$: since $0$ is a solution to the periodic BVP and
$P_N0=0$, the zero solution must also be equal to
$X_y(0,\omega,\mu)$. Thus, we have
\begin{align*}
0&=\left.\frac{\partial}{\partial\omega} X_y(p,\omega,\mu)
\right\vert_{\textstyle p=0}\mbox{\quad and} &
X_y(p,\omega,\mu)\right\vert_{\textstyle p=0}\mbox{.}
\end{align*}
Moreover, the fixed point equation (<ref>) defining
$z=[\partial X_y/\partial p] (p,\omega,\mu)\,q$, evaluated in
$(p,\omega,\mu)=(0,\omega,\mu)$ reads
\begin{equation}\label{eq:hopf:qnz}
\end{equation}
exploiting that $Q_NL=Q_NLQ_N$ and
$Q_NA(\omega,\mu)=A(\omega,\mu)\,Q_N$. In the neighborhood
$B_\delta^{0,1}(x_0)$ the spectral radius of $Q_NLA(\mu,\omega)$ is
less than unity (see Lemma <ref>). Application of $Q_N$
to (<ref>) gives $Q_Nz=Q_NLA(\mu,\omega)Q_Nz$. Since the
spectral radius of $Q_NLA(\mu,\omega)$ is less than unity this implies
that $Q_Nz=0$, and, thus
X_y(p,ω,μ)|_p=0]q= E_Nq
Consequently, the linearization of the algebraic system
(<ref>) in $(0,\omega,\mu)$ with respect
to the first variable is
\begin{equation}\label{eq:hopf:lin}
0=P_0A(\omega,\mu)E_Np+\omega Q_0P_NE_Np-Q_0P_NLA(\omega,\mu)E_Np
\end{equation}
for all $\omega\approx \omega_0$ and $\mu\approx 0$ (also using $p$
for the argument of the linearization in (<ref>)). We
observe that the linear system (<ref>) decouples into
equations for
\begin{align*}
y_0&=P_oE_Np=E_0p=p_0 &&\mbox{(the average of $E_Np$),}\\
y_1&=Q_0E_1p=p_{-1}\sin t+p_1\cos t &&\mbox{(the first Fourier component of $E_Np$),}\\
y_j&=Q_{j-1}E_jp=p_{-j}\sin(jt)+p_j\cos(jt) &&\mbox{(the $j$-th
Fourier component of $E_Np$,}\\
&&&\mbox{$2\leq j\leq N$),}
\end{align*}
where we denote the components of $p$ by $p_j\in \R^n$ ($j=-N\ldots N$).
This decoupling is achieved by pre-multiplication of
(<ref>) with $P_0$ and $Q_{j-1}P_j$ for $j=1\ldots N$:
\begin{align}
\label{eq:hopf:lin:dec0}\\
0&=\omega y_1-Q_0LA(\omega,\mu)\,y_1
\label{eq:hopf:lin:dec1}\\
0&=\omega y_j-Q_0LA(\omega,\mu)\,y_j &&
\mbox{\ for $j=2\ldots N$.}
\label{eq:hopf:lin:decj}
\end{align}
Inserting the definition of $y_j$ into the equations (<ref>)
and (<ref>) gives for $j\geq 1$
\begin{align*}
Q_0\int_0^ta(\mu)[p_j\sin(js+j\omega\cdot)+p_j\cos(js+j\omega\cdot)]\d s\\
\frac{1}{j}\sin(jt)a(\mu)[p_{-j}\sin(j\omega\cdot)+p_j\cos(j\omega\cdot)]\\
\end{align*}
These equations are satisfied if and only if the coefficients in front
of $\sin(jt)$ and $\cos(jt)$ are zero. The resulting system of
equations reads in complex notation (splitting up again into the cases
$j=1$ and $j>1$)
\begin{align}
\label{eq:hopf:red1}
i\omega u_1-a(\mu)\left[u_1\exp(i\omega s)\right]&=K(i\omega,\mu)\,u_1=0\mbox{,}\\
\label{eq:hopf:redj}
ij\omega u_j-a(\mu)\left[u_j\exp(ij\omega s)\right]&=K(ij\omega,\mu)\,u_j=0
\mbox{\quad ($2\leq j\leq N$),}
\end{align}
that is, $u_j=p_{-j}+ip_j\in\C^n$ is a solution of
(<ref>) (or (<ref>), respectively) if and
only if $y_j=p_{-j}\sin(jt)+p_j\cos(jt)$ is a solution of
(<ref>) (or (<ref>), respectively).
The non-resonance assumption of the theorem guarantees that equation
(<ref>) is a regular linear system for $p_0$, and
that (<ref>) is a regular linear algebraic system for
$p_{-j}$ and $p_j$ ($j\geq 2$) at $\mu=0$ and $\omega=\omega_0$ (and,
hence, for all $\omega$ and $\mu$ near-by). The condition on the
simplicity of the eigenvalue $i\omega_0$ of $K$ ensures that equation
(<ref>) (and, thus, (<ref>)) has a
one-dimensional (in complex notation) subspace of solutions for
$\omega=\omega_0$ and $\mu=0$, spanned by the nullvector $v_1$ of
$K(i\omega,0)$. Let us denote the adjoint nullvector of
$K(i\omega_0,0)$ by $w_1\in\C^n$ (again, using complex notation,
$w_1^HK(i\omega_0,0)=0$). Since $i\omega_0$ is simple, the
w_1^H∂K/∂λ(iω,0) v_1≠0
holds which implies that we can choose $w_1\in\C^n$ without
loss of generality such that
w_1^H∂K/∂λ(iω,0) v_1=1
With this convention we observe that
\begin{align}
w_1^H\frac{\partial K}{\partial\mu}(i\omega,0)\,v_1&=
=:c_\mu\in\C\mbox{, and}&
\end{align}
where $\Re c_\mu\neq0$ by the transversal crossing assumption of the
theorem. In complex notation any scalar multiple of the nullvector
$v_1=v_r+iv_i$ is also a nullvector. Thus, the complex scalar factor
$\alpha+i\beta$ in front of $v_1$ makes up two components of the
variable $p$ (in real notation): in short, $p$ solves the linearized
algebraic system (<ref>) if and only if all $p_j$ with
$|j|\neq1$ are zero and $p_{-1}\sin t+p_1\cos
t=\Re\left[(\alpha+i\beta)v_1\exp(it)\right]$ for some
$\alpha,\beta\in\R$, that is,
\begin{equation}\label{eq:hopf:abintro}
\begin{bmatrix}
p_{-1}\\ p_{1\phantom{-}}
\end{bmatrix}=
\alpha
\begin{bmatrix}
-v_i\\ \phantom{-}v_r
\end{bmatrix}+\beta
\begin{bmatrix}
-v_r\\ -v_i
\end{bmatrix}=:\alpha b_r+\beta b_i\mbox{.}
\end{equation}
Let us collect the statements so far and introduce coordinates. We
collect all components $p_j$ with $|j|\neq1$ and the orthogonal
complement in $\R^{2n}$ of the space spanned by $\{b_1,b_2\}$ into a
single variable $q$ (of real dimension $n_q=n(2N-1)+2(n-1)$). Then a
set of coordinates for $p$ are the variables
\begin{align*}
(\alpha,\beta)&=:r\in\R^2\mbox{,\quad and\quad}
\end{align*}
We split up the full algebraic system of equations
(<ref>) in the same way as we did for the linearized
problem, by pre-multiplication with $P_0$ and $Q_{j-1}P_j$ for
$j=1\ldots N$:
\begin{align}
\label{eq:hopf:nlin0}\\
0&=\omega Q_0E_1p-Q_0P_1LF(X_y(p,\omega,\mu),\omega,\mu)
\label{eq:hopf:nlin1}\\
Q_{j-1}P_j\cdot\mbox{\eqref{eq:hopf:modes}}:&& 0&=\omega
\label{eq:hopf:nlinj}
\end{align}
We split equation (<ref>) further using $w_1^H$ and its
orthogonal complement, the projection $w_1^\perp=\id-w_1w_1^H/(w_1^Hw_1)$. This gives
rise to a splitting into two real equations
($w_1^H\cdot$(<ref>)) and $2(n-1)$ real equations
($w_1^\perp\cdot$(<ref>)). Collecting
$w_1^\perp\cdot$(<ref>) and the equations
(<ref>) and (<ref>) into a subsystem of
$n(2N-1)+2(n-1)=n_q$ equations the full algebraic system
(<ref>) in the coordinates $(r,q)$ has the form
\begin{equation}\label{eq:hopf:nlin:matrixform}
M_{rr}(r,q,\omega,\mu) & M_{rq}(r,q,\omega,\mu)\\
M_{qr}(r,q,\omega,\mu) & M_{qq}(r,q,\omega,\mu)
\end{bmatrix}
\begin{bmatrix}
r\\ q
\end{bmatrix}\mbox{.}
\end{equation}
By our choice of coordinates the matrices $M_{rr}\in\R^{2\times2}$,
$M_{rq}\in\R^{2\times n_q}$ and $M_{qr}\in\R^{n_q\times2}$ are
identically zero in $r=0$, $q=0$, $\mu=0$, $\omega=i\omega_0$ such
that the system matrix has the form
0 0
0 0
0 … 0
0 … 0
0 0
⋮ ⋮
0 0
$(r,q,\mu,\omega)=(0,0,0,\omega_0)$. Thus, we can perform a
Lyapunov-Schmidt reduction: we eliminate $q$ by solving the $n_q$
lower equations for $q$, obtaining a graph $q(r,\omega,\mu)\,r$
locally in a neighborhood of $(r,q,\mu,\omega)=(0,0,0,\omega_0)$. This
graph respects rotational invariance:
Note that the application of $\Delta_s$ to $r=(\alpha,\beta)$
corresponds to the rotation of $r$ by angle $s$ (the same as the
multiplication $\exp(is)(\alpha+i\beta)$). The Lyapunov-Schmidt
reduction of (<ref>), replacing $q$ by the
graph $q(r,\omega,\mu)\,r$, then reads
\begin{equation}
\label{eq:hopf:nlin:reduced}
\end{equation}
where $M_r$ is still rotationally symmetric in $r$:
Equation (<ref>) in real notation implies that
\begin{align*}
\frac{\partial M_r}{\partial\omega}(0,\omega_0,0)&=
\frac{\partial M_{rr}}{\partial\omega}(0,0,\omega_0,0)=
\begin{bmatrix}
0&{-1}\\ 1 &\phantom{-}0
\end{bmatrix}\mbox{,} \\
\frac{\partial M_r}{\partial\mu}(0,\omega_0,0)&=
\frac{\partial M_{rr}}{\partial\mu}(0,0,\omega_0,0)=
\begin{bmatrix}
\Re c_\mu&-\Im c_\mu\\ \Im c_\mu &\phantom{-}\Re c_\mu
\end{bmatrix}\mbox{.}
\end{align*}
Equation (<ref>) is a system of two equations
with four unknowns ($r=(\alpha,\beta)$, $\omega$ and $\mu$). We now
fix one of the unknowns setting
such that we can expect one-parametric families of solutions
Introducing $M_\beta$ as the second column of $M_r$ and dropping the
dependence on $\alpha$ (which is zero), the first derivative of
$M_\beta(\beta,\omega,\mu)$ in $(0,\omega_0,0)$ with respect to the
pair $\omega$ and $\mu$ is:
\begin{align*}
\begin{bmatrix}
{\displaystyle\frac{\partial M_\beta}{\partial \omega}}&
{\displaystyle\frac{\partial M_\beta}{\partial \mu}}
\end{bmatrix}
\begin{bmatrix}
-1 &-\Im c_\mu\\ \phantom{-}0& \phantom{-}\Re c_\mu
\end{bmatrix}\mbox{,}
\end{align*}
which is regular (as established in (<ref>), since $\Re
c_\mu\neq0$ due to the assumption that the eigenvalue crosses the
imaginary axis transversally). Note that $M_\beta$ itself is a
projection of the first derivative of the original right-hand side of
the full algebraic system (<ref>). Thus,
$M_\beta$ is $k-1$ times continuously differentiable, and we end up
with a system of two equations for three scalar variables
\begin{align*}
\end{align*}
Hence, either $\beta=0$, which corresponds to the trivial solution or
(after division by $\beta$)
\begin{align}
\end{align}
where $M_\beta(0,\omega_0,0)=(0,0)$ and the derivative with respect to
the pair $(\omega,\mu)$ is regular in $(0,\omega_0,0)$. Thus, we can
apply the Implicit Function Theorem to (<ref>) to
obtain a unique graph $(\omega(\beta),\mu(\beta))$ solving
(<ref>). The graph satisfies
$(\omega(0),\mu(0))=(\omega_0,0)$, and, thus, branches off from the
trivial solution (which has $\beta=0$ and $\omega$ and $\mu$
arbitrary). The rotational symmetry of $M_r$ implies reflection
symmetry of $M_\beta$ in $\beta$ such that
$M_\beta(-\beta,\omega,\mu)=M_\beta(\beta,\omega,\mu)$ for all
$\beta$, $\omega$ and $\mu$. Hence, the solution graph is reflection
symmetric, too: $\omega(-\beta)=\omega(\beta)$ and
$\mu(-\beta)=\mu(\beta)$. Thus, for small $\beta$ there is a unique
non-trivial solution of the full algebraic system of the form
$r=(0,\beta)$, $q=q(r,\omega(\beta),\mu(\beta))\,r$. As
Equation (<ref>) shows, we can extract the coordinates
$\alpha$ (which is zero) and $\beta$ from the full solution $x\in
C^k(\T;\R^n)$ by applying the projections
\begin{align*}
\frac{1}{\pi}\int_{-\pi}^\pi\cos(t)v_r^Tx(t)-\sin(t)v_i^Tx(t)\d t&=
\frac{1}{\pi}\int_{-\pi}^\pi\Re\left[v_1\exp(it)\right]^Tx(t)\d t=\alpha\mbox{,}\\
\frac{1}{\pi}\int_{-\pi}^\pi\sin(t)v_r^Tx(t)+\cos(t)v_i^Tx(t)\d t&=
\frac{1}{\pi}\int_{-\pi}^\pi\Im\left[v_1\exp(it)\right]^Tx(t)\d t=-\beta\mbox{,}
\end{align*}
which determines the First Fourier coefficients of
$x$ as claimed in (<ref>) in
Theorem <ref>. (Recall that the vector $v_1=v_r+v_i$ was
scaled to have unit length and that the decomposition was
[1]
E. Allgower and M. K. Georgi.
Introduction to Numerical Continuation Methods.
SIAM, 1979.
[2]
J. Baumgarte.
Stabilization of constraints and integrals of motion in dynamical
Computer Methods in Applied Mechanics and Engineering,
1(1):1–16, 1972.
[3]
E.A. Coddington and N. Levinson.
Theory of ordinary differential equations.
International series in pure and applied mathematics. McGraw-Hill,
New York, 1955.
[4]
O. Diekmann, S.A. van Gils, S.M. Verduyn Lunel, and H.-O. Walther.
Delay equations, volume 110 of Applied Mathematical
Springer-Verlag, New York, 1995.
[5]
Markus Eichmann.
A local Hopf Bifurcation Theorem for differential equations
with state-dependent delays.
PhD thesis, University of Giessen, 2006.
[6]
K. Engelborghs, T. Luzyanina, and D. Roose.
Numerical bifurcation analysis of delay differential equations using
ACM Transactions on Mathematical Software, 28(1):1–21, 2002.
[7]
L. H. Erbe, W. Krawcewicz, and K. Geba.
${S}^1$-degree and global bifurcation theory of
functional-differential equations.
J. Differ. Eq., 98:277–298, 1992.
[8]
M. Golubitsky, D.G. Schaeffer, and I. Stewart.
Singularities and groups in bifurcation theory, volume 69 of
Applied Mathematical Sciences.
Springer, New York, 1988.
[9]
J. Guckenheimer and P. Holmes.
Nonlinear oscillations, dynamical systems, and bifurcations of
vector fields, volume 42 of Applied Mathematical Sciences.
Springer-Verlag, New York, 1990.
[10]
S. Guo.
Equivariant Hopf bifurcation for functional differential equations
of mixed type.
Applied Mathmeatics Letters, 24:724–730, 2011.
[11]
J.K. Hale and S.M. Verduyn Lunel.
Introduction to functional-differential equations, volume 99
of Applied Mathematical Sciences.
Springer-Verlag, New York, 1993.
[12]
F. Hartung, T. Krisztin, H.-O. Walther, and J. Wu.
Functional differential equations with state-dependent delays:
theory and applications.
In P. Drábek, A. Cañada, and A. Fonda, editors, Handbook
of Differential Equations: Ordinary Differential Equations, volume 3,
chapter 5, pages 435–545. North-Holland, 2006.
[13]
Q. Hu and J. Wu.
Global Hopf bifurcation for differential equations with
state-dependent delay.
Journal of Differential Equations, 248(10):2801–2840, 2010.
[14]
A. R. Humphries, O. DeMasi, F. M. Magpantay, and F. Upham.
Dynamics of a delay differential equation with multiple state
dependent delays.
Discrete and Continuous Dynamical Systems A, to appear.
[15]
T. Insperger, D. A. W. Barton, and G. Stépán.
Criticality of Hopf bifurcation in state-dependent delay model of
turning processes.
International Journal of Non-Linear Mechanics, 43(2):140 –
149, 2008.
[16]
T. Insperger, G. Stépán, and J. Turi.
State-dependent delay model for regenerative cutting processes.
In Proceedings of the Fifth EUROMECH Nonlinear Dynamics
Conference, Eindhoven, Netherlands, 2005.
[17]
D. Jackson.
The Theory of Approximation, volume XI.
AMS Colloquium Publication, New York, 1930.
[18]
W. Krawcewicz and J. Wu.
Theory of degrees with applications to bifurcations and
differential equations.
Canadian Mathematical Society Series of Monographs and Advanced
Texts. John Wiley & Sons Inc., New York, 1997.
[19]
T. Krisztin.
A local unstable manifold for differential equations with
state-dependent delay.
Discrete Contin. Dynam. Systems, 9(4):993–1028, 2003.
[20]
Y. A Kuznetsov.
Elements of Applied Bifurcation Theory, volume 112 of Applied Mathematical Sciences.
Springer-Verlag, New York, third edition, 2004.
[21]
J. Mallet-Paret, R. D. Nussbaum, and P. Paraskevopoulos.
Periodic solutions for functional differential equations with
multiple state-dependent delays.
Topological Methods in Nonlinear Analysis, 3(2):101–162, 1994.
[22]
G Stépán.
Retarded Dynamical Systems: Stability and Characteristic
Longman Scientific and Technical, Harlow, Essex, 1989.
[23]
H.-O. Walther.
Stable periodic motion of a system with state-dependent delay.
Differential and Integral Equations, 15:923–944, 2002.
[24]
H.-O. Walther.
Smoothness properties of semiflows for differential equations with
state-dependent delays.
Journal of Mathematical Sciences, 124(5):5193–5207, 2004.
[25]
E. Winston.
Uniqueness of the zero solution for differential equations with
state-dependent delay.
J. Diff. Eqs., 7:395–405, 1970.
[26]
J. Wu.
Symmetric functional differential equations and neural networks with
Transactions of the AMS, 350(12):4799–4838, 1998.
§ BASIC DIFFERENTIABILITY PROPERTIES OF THE RIGHT-HAND SIDE
Let $J$ be a compact interval or $\T$. Let $(D,\|\cdot\|_D)$ be a
Banach space of the form
where $\ell\geq 1$, the integers $k_j$ are non-negative and the integers
$m_j$ are positive. We use the natural maximum norm on the product $D$:
and use the notation
\begin{align*}
D^k&=C^{k_1+k}(J;\R^{m_1})\times\ldots\times C^{k_\ell+k}(J;\R^{m_\ell})\mbox{,}
& \|x\|_{D,k}&=\max_{0\leq j\leq
k} \|x^{(j)}\|_D\mbox{,}\\
D^{0,1}&=\left\{x\in D: L(x)<\infty\right\}\mbox{, with the norm}&
% \mbox{with \ }
\|x\|_{D,L}&=\max\left\{\|x\|_D,L(x)\right\}\mbox{,}
\intertext{where $x^{(j)}$ is the component-wise $j$th derivative
and the Lipschitz constant $L(x)$ is defined as} L(x)&=\sup_{
\begin{subarray}{c}
t\neq s
\end{subarray}
}\ \max_{j=1\ldots,\ell}\ \frac{|x_j^{(k_j)}(t)-x_j^{(k_j)}(s)|}{|t-s|}
\mbox{,}
\end{align*}
where $t$ and $s$ in the index of $\sup$ are taken from $J$, if $J$ is
a compact interval, and from $\R$ if $J=\T$. Balls that are closed and
bounded in $D^{0,1}$ are complete with respect to the norm of $D$.
§.§ Basic properties of $f$
This section proves three properties that $EC^1$ smooth
functionals $f$ have: first that the derivative limit
(<ref>) exists also for Lipschitz continuous
deviations, second a weaker form of the mean value theorem, and third,
local $EC$ Lipschitz continuity.
Let $f:D\mapsto\R^n$ be $EC^1$ smooth in the sense of
Definition <ref>. Then the limit required to exist in
Definition<ref> exists also in the
$\|\cdot\|_{D,L}$-norm: for all $x\in D^1$
\begin{align}\allowdisplaybreaks
\label{eq:ass:contdifflip}
\lim_{
\begin{subarray}{c}
y\in D^{0,1}\\[0.2ex]
\|y\|_{D,L}\to0
\end{subarray}
\frac{|f(x+y)-f(x)-\partial^1f(x,y)|}{\|y\|_{D,L}}=0\mbox{.}
\end{align}
Note that in (<ref>) the norm in which $y$ goes to
zero is $\|\cdot\|_{D,L}$ instead of $\|\cdot\|_{D,1}$.
Proof This is a simple continuity argument. Let $\epsilon>0$
be arbitrary. We pick $\delta>0$ such that
\begin{equation}\label{eq:ydsmall}
|f(x+\tilde y)-f(x)-\partial^1f(x,\tilde y)|<\epsilon\|\tilde y\|_{D,1}
\end{equation}
for all $\tilde y\in D^1$ satisfying $\|\tilde y\|_{D,1}<\delta$. Let
$y\in D^{0,1}$ be such that $\|y\|_{D,L}<\delta$. We can choose a
$\tilde y\in D^1$ that satisfies
\begin{align}
\|\tilde y\|_{D,1}&<\min\{\delta,2\|y\|_{D,L}\}\label{eq:yydsmall}\\
|f(x+y)-f(x+\tilde y)|&<\epsilon\|y\|_{D,L}\label{eq:fyydsmall}\\
|\partial^1f(x,y-\tilde y)|&<\epsilon\|y\|_{D,L}\label{eq:ayydsmall}\mbox{.}
\end{align}
Condition (<ref>) can be achieved because $D^1$ is a dense
subspace in $D^{0,1}$, and for every element $\tilde y$ of $D^1$ the
$\|\cdot\|_{D,1}$-norm is not larger than the $\|\cdot\|_{D,L}$-norm: $\|\tilde y\|_{D,1}\leq\|\tilde
y\|_{D,L}$. (<ref>) follows from the continuity of $f$
and the density of $D^{0,1}$ in $D^1$, and (<ref>)
follows from the continuity of $\partial^1f$ as a map on $D^1\times
D$, and the density of $D^{0,1}$ in $D^1$. Combining estimate
(<ref>) with (<ref>)–(<ref>) we
There exists a continuous function
which is linear in its third argument and satisfies for all $x,y\in
\begin{equation}\label{eq:meanval}
f(x+y)-f(x)=\tilde a(x,y,y)\mbox{.}
\end{equation}
Moreover, $\tilde a(x,0,y)=\partial^1f(x,y)$ for all $x\in D^1$ and
$y\in D$.
The argument for the existence of a mean value follows exactly the
proof of the general mean value theorem [12]: the candidate
for $\tilde a(u,v,w)$ is
\begin{equation}\label{eq:meandiff}
\tilde a(u,v,w)=\int_0^1\partial^1f(u+sv,w)\d s\mbox{.}
\end{equation}
Since $\partial^1f$ is assumed to be continuous in its arguments the
integral is well defined and continuous in its arguments $u\in D^1$,
$v\in D^1$, $w\in D$. All one has to show is that the $\tilde a$
defined in (<ref>) satisfies (<ref>): let $x,y
\in D^1$ and $\epsilon>0$ be arbitrary, and choose $m$ such that
uniformly for all $s\in[0,1]$
\begin{align*}
\left|\int_0^1\partial^1f(x+sy,y)\d s- \frac{1}{m}\sum_{k=0}^{m-1}
\partial^1f\left(x+\frac{k}{m}y,y\right)\right|&<\epsilon\mbox{,}\\
\left|f\left(x+sy+\frac{y}{m}\right)
\end{align*}
Then it follows that
Since $\epsilon>0$ was arbitrary the left-hand side must be zero.
For all $x\in D^{0,1}$ there exists a neighborhood $U(x)\subseteq
D^{0,1}$ and a constant $K_x>0$ such that for all $y_1$ and $y_2\in
U(x)$ the following Lipschitz estimate holds:
Note that the upper bound depends only on the $\|\cdot\|_D$-norm, not
on the $\|\cdot\|_{D,L}$-norm, which would be a weaker statement.
We prove the Lipschitz continuity first for $y_1$ and $y_2$ from a
sufficiently small neighborhood $U(x)\cap D^1\subseteq D^1$ of $x\in
Let $x$ be an element of $D^1$. Since the mean value $\tilde a$ is
continuous in $(x,0,0)$, and $\tilde a(x,0,0)=0$, we have a $\delta>0$
such that for all $u,v\in D^1$ and $w\in D$ satisfying
$\|u\|_{D,1}<\delta$, $\|v\|_{D,1}<\delta$ and $\|w\|_D<\delta$
This implies that $|\tilde a(x+u,v,w)|<[\epsilon/\delta]\|w\|_D$ for
$u$ and $v$ with $\max\{\|u\|_{D,1},\|v\|_{D,1}\}<\delta$ and $w\in D$
(since $\tilde a$ is linear in its third argument). Thus, $\|\tilde
a(x+u,v,\cdot)\|_D\leq\epsilon/\delta$ for $\tilde
a(x+u,v,\cdot)$ as an element of $L(D;D)$ in the operator norm
corresponding to $D$. Consequently, if $\|y_1-x\|_{D,1}<\delta/2$ and
such that we can choose $K_x=\epsilon/\delta$. The extension of the
statement to $D^{0,1}$ follows from the continuity of $f$ in $D$:
$U(x_0)\cap D^1$ is dense in $U(x_0)\subset D^{0,1}$ using the
$\|\cdot\|_{D,L}$-norm. Pick two sequences $y_n$ and $z_n$ in
$U(x_0)\cap D^1$ that converge to $y$ and $z$ in $U(x_0)$ in the
Lipschitz norm. Then $f(y_n)\to f(y)$ and $f(z_n)\to f(z)$ since $f$
is continuous in $D$. Moreover, $\|y_n-z_n\|_D\to\|y-z\|_D$ for
$n\to\infty$. Since
\begin{equation} |f(y_n)-f(z_n)|\leq
\end{equation}
for all $n$ the inequality also holds for the limit for $n\to\infty$.
§.§ Basic properties of $F$
In this section we restrict ourselves to the periodic case:
$J=\T$. Let $F:D\mapsto C^0(\T;\R^n)$ be defined as
Let $f:D\mapsto\R^n$ be continuous. Then $F:D\mapsto C^0(\T;\R^n)$
is also continuous.
Proof This is a simple consequence of the continuity of $f$,
the continuity of $(t,x)\mapsto \Delta_tx$ with respect to both
arguments ($t$ and $x$) in the $\|\cdot\|_0$-norm, and the
compactness of $\T$. Let $\epsilon>0$ and $x\in D$ be arbitrary. We
want to prove continuity of $F$ in $x$. So, we have to find a
$\delta>0$ such that
\begin{equation}\label{eq:Fcont:epsdelta}
\left|f(\Delta_sx+h)-f(\Delta_sx)\right|<\epsilon
\mbox{\quad for all $s\in\T$ and $h\in D$, satisfying $\|h\|_D<\delta$.}
\end{equation}
(Since $\|\Delta_sh\|_D=\|h\|_D$ we can replace $\Delta_sh$ by $h$.)
The continuity of $f$ implies that for every $r>0$ and every $t\in\T$
we find a $\delta_x(t,r)$ such that
\begin{equation}
\mbox{\quad whenever $\|h\|_D<\delta_x(t,r)$.}\label{eq:Fcont:fdelta}
\end{equation}
For every $t\in\T$ there exists an
open neighborhood $U(t)\subset \T$ such that
because the function $t\in\T\mapsto \Delta_tx$ is continuous in $t$.
These neighborhoods $U(t)$ are an open cover of the compact set $\T$,
so there exist finitely many $t_1,\ldots,t_m\in\T$ such that the union
of the neighborhoods $U(t_j)$ contains all points $s\in\T$. We choose
which is a positive quantity. Let $s\in\T$ be arbitrary and let $h\in
D$ satisfy $\|h\|_D<\delta$. We have to check the inequality
(<ref>). The point $s$ is in one of the
neighborhoods $U(t_j)$, say without loss of generality, $s\in
U(t_1)$. Thus,
$\|\Delta_sx-\Delta_{t_1}x\|_D<\delta_x(t_1,\epsilon/2)/2$, and,
$\|\Delta_sx-\Delta_{t_1}x+h\|_D<\delta_x(t_1,\epsilon/2)$ (because
also $\|h\|_D<\delta\leq \delta_x(t_1,/\epsilon/2)/2$). Therefore, we
can split up the difference $|f(\Delta_sx+h)-f(\Delta_sx)|$:
\begin{align*}
\left|\left[f\left(\Delta_{t_1}x+(\Delta_sx-\Delta_{t_1}x+h)\right)
&\ +\left|\left[f\left(\Delta_{t_1}x+(\Delta_sx-\Delta_{t_1}x)\right)
<&\ \epsilon/2+\epsilon/2=\epsilon
\end{align*}
Note that the deviations from $\Delta_{t_1}x$ in the arguments of $f$
in both terms of the sum are less than or equal to
$\delta_x(t_1,\epsilon/2)$ such that we can apply
(<ref>) for $t=t_1$, $r=\epsilon/2$.
The following lemma lists properties that $F$ has if $f$ satisfies
local $EC$ Lipschitz continuity in the sense of
Definition <ref>. That is, we do not assume that $f$
is $EC^1$ smooth in the sense of Definition <ref> for
Lemma <ref>. Since Lemma <ref> was proved using
only the assumption of $EC^1$ smoothness of $f$, local $EC$ Lipschitz
continuity is a weaker condition.
Assume that $f:D\mapsto\R^n$ is locally $EC$ Lipschitz continuous in the sense of
Definition <ref>. Then $F$ has the following properties:
* for all $x\in D^{0,1}$ there exists a neighborhood
$U(x)\subseteq D^{0,1}$ and a constant $K_x>0$ such that for all
$y_1$ and $y_2\in U(x)$
* $F$ maps elements of $D^{0,1}$ into
$C^{0,1}(\T;\R^n)$. Moreover, for every $x\in D^{0,1}$, any
bounded neighborhood $U(x)\subseteq D^{0,1}$ for which the
Lipschitz constant $K_x$ exists has a bounded image under $F$:
there exists a bound $R>0$ such that $\|F(y)\|_{0,1}\leq R$ for
all $y\in U(x)$ ($R$ depends on $U(x)$).
Statement <ref> is a consequence of the local $EC$
Lipschitz continuity of $f$ and the compactness of $\T$ (which allows
one to choose a uniform Lipschitz bound $K_x$ for all $t\in\T$).
Concerning statement <ref>: let $x\in D^{0,1}$ be
arbitrary, and let the neighborhood $U(x)$ be bounded (say,
$\|y-x\|_{D,L}\leq \delta$) such that $F$ has a Lipschitz constant
$K_x$ in $U(x)$. Then we have for all $y,z\in U(x)$ and $t,s\in\T$
the estimate
Initially setting $z=x$ and $s=t$ we get a bound on $\|F(y)\|_0$:
$\|F(y)\|_0\leq \|F(x)\|_0+K_x\delta=:R_0$ for all $y\in U(x)$. It
remains to be shown that the Lipschitz constant of $F(y)$ is bounded
for $y\in U(x)$:
\begin{align*}
\leq K_x\|\Delta_ty-\Delta_sy\|_D
\leq K_x\|y\|_{D,L}|t-s|\mbox{.}
\end{align*}
Since $\|y-x\|_{D,L}\leq \delta$ for $y\in U(x)$,
ensures that $\|F(y)\|_{0,1}\leq R$.
Define the maps
\begin{align*}
\partial^1F(u,v)(t)&=\partial^1f(\Delta_tu,\Delta_tv)
&&\mbox{for $u\in D^1$, $v\in D$,}\\
\tilde A(u,v,w)(t)&=\tilde a(\Delta_tu,\Delta_tv,\Delta_tw)
&&\mbox{for $u\in D^1$, $v\in D^1$, $w\in D$.}
\end{align*}
The following Lemma <ref>, and Lemma <ref>
assume that $f$ is $EC^1$ smooth in $D$ in the sense of
Definition <ref>.
Let $f:D\mapsto\R^n$ be $EC^1$ smooth. Then $F$, $\partial^1F$ and
$\tilde A$ have the following properties.
* The map $(u,v)\mapsto\partial^1F(u,v)$ is
continuous in both arguments (and linear in its second argument)
as a map from $D^1\times D$ into $C^0(\T;\R^n)$. It satisfies for
all $x\in D^1$
\begin{equation}\label{eq:Fcontdiff}
\lim_{
\begin{subarray}{c}
y\in D^{0,1}\\[0.2ex]
\|y\|_{D,L}\to0
\end{subarray}
\partial^1F(x,y)\|_0}{\|y\|_{D,L}}=0\mbox{.}
\end{equation}
* The map $\tilde A(u,v,w)$ is continuous in
all three arguments (and linear in its third argument) as a map
from $D^1\times D^1\times D$ into $C^0(\T;\R^n)$. It satisfies for
all $x,y\in D^1$
Moreover, $\tilde A(x,0,y)=\partial^1F(x,y)$ for all $x\in D^1$
and $y\in D$.
Note that in the limit (<ref>) we allow for deviations
$y\in D^{0,1}$.
Proof The continuity of $\partial^1F$ follows from the
continuity of $\partial^1f$ by applying Lemma <ref> to
$\partial^1f:D^1\times D\mapsto\R^n$ instead of $f$. The linearity of
$\partial^1F$ in its second argument follows from the linearity of
$\partial^1f$ in its second argument.
The limit (<ref>) also follows from the corresponding
limit (<ref>): let $x\in D^1$ and $\epsilon>0$ be
arbitrary. For every fixed $t$ there exists a $\delta(t)>0$ such that
\begin{equation}\label{eq:Fcontdiffproof:ineq}
\frac{|f(\Delta_tx+\Delta_ty)-f(\Delta_tx)-
\partial^1f(\Delta_tx,\Delta_ty)|}{\|y\|_{D,L}}<\epsilon
\end{equation}
for all $y$ with $\|y\|_{D,L}<\delta(t)$. As $f$ and $\partial^1f$
are continuous in their arguments $x\in D^1$ and $y\in D^{0,1}$, the
inequality also holds for all $s$ in a sufficiently small open
neighborhood of $t$, $U(t)$. The set of neighborhoods $U(t)$ for all
$t\in\T$ are a cover of the compact set $\T$ by open sets. Choosing a
finite subcover from this cover, and labeling the times
$t_1,\ldots,t_m$, we can choose
such that (<ref>) holds for all uniformly
$t\in\T$. This proves statement <ref> of the lemma.
Concerning statement <ref>: for the continuity of $\tilde
A$ we invoke again Lemma <ref>, this time for $\tilde a$ on
$D^1\times D^1\times D$. The linearity of $\tilde A$ in its third
argument follows from the linearity of $\tilde a$ in its third
argument. The relations $F(x+y)(t)-F(x)(t)=\tilde A(x,y,y)(t)$ and
$\tilde A(x,0,y)(t)=\partial^1F(x,y)(t)$ in every $t\in\T$
follow from the corresponding relations for $f$ and $\tilde a$, as
stated in Lemma <ref>.
Let $f:D\mapsto\R^n$ be $EC^1$ smooth and let $k\geq0$ be some
integer. We assume that $F:D\mapsto C^k(\T;\R^n)$ and
$\partial^1F:D^1\times D\mapsto C^k(\T;\R^n)$ are continuous maps.
Then $F$ maps elements of $D^1$ into $C^{k+1}(\T;\R^n)$, and $F$ is
continuous as a map from $D^1$ to $C^{k+1}(\T;\R^n)$.
Let $x$ be in $D^1$, that is, $x'\in D$. If $\partial^1F:D^1\times
D\mapsto C^k(\T;\R^n)$ is continuous then $\tilde A:D^1\times
D^1\times D\mapsto C^k(\T;\R^n)$, which is given by $\tilde
A(u,v,w)=\int_0^1\partial^1F(u+sv,w)\d s$, is continuous, too. Using
statement <ref> of Lemma <ref> we have
\begin{align}
\frac{F(\Delta_hx)-F(x)}{h}&=\tilde
\label{eq:Fimage:meanval}
\end{align}
On the right side $\|\Delta_hx-x\|_{D,1}$ converges to $0$ for
$h\to0$. Also,
because $x\in D^1$. Since $\tilde A$ is continuous in its arguments
the right side converges to $\tilde A(x,0,x')=\partial^1F(x,x')\in
C^k(\T;\R^n)$ for $h\to0$. Thus, the limit of the left-hand side in
(<ref>) for $h\to0$ exists, too, such that $F(x)\in
C^{k+1}(\T;\R^n)$ and the time derivative $(F(x))'$ equals
$\partial^1F(x,x')$. Since $(v,w)\in D^1\times
D\mapsto \partial^1F(v,w)\in C^k(\T;\R^n)$ is continuous in $(u,v)$,
the time derivative of $F(x)$, $(F(x))'=\partial^1F(x,x')$ is also
continuous in $x$ if we use the norm $\|\cdot\|_{D,1}$ for the
argument and $\|\cdot\|_k$ for the image.
|
arxiv-papers
| 2010-10-12T13:49:56 |
2024-09-04T02:49:13.740308
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Jan Sieber",
"submitter": "Jan Sieber",
"url": "https://arxiv.org/abs/1010.2391"
}
|
1010.2392
|
# Saturation of interband absorption in graphene
F.T. Vasko ftvasko@yahoo.com Institute of Semiconductor Physics, NAS of
Ukraine, Pr. Nauky 41, Kiev, 03028, Ukraine
###### Abstract
The transient response of an intrinsic graphene, which is caused by the
ultrafast interband transitions, is studied theoretically for the range of
pumping correspondent to the saturated absorption regime. Spectral and
temporal dependencies of the photoexcited concentration as well as the
transmission and relitive absotption coefficients are considered for mid-IR
and visible (or near-IR) spectral regions at different durations of pulse and
broadening energies. The characteristic intencities of saturation are
calculated and the results are compared with the experimental data measured
for the near-IR lasers with a saturable absorber. The negative absorption of a
probe radiation during cascade emission of optical phonons is obtained.
###### pacs:
78.47.jb, 78.67.Wj, 42.65.-k
## I Introduction
The character of nonlinear response under the ultrafast interband excitation
of an intrinsic graphene is determined by a several physical processes which
are dependent on conditions of pumping. Under a low excitation level, when the
one-photon transitions take place, the energy relaxation and recombination of
photoexcited carriers were studied with the use of the time-resolved pump-
probe measurements, see experimental data and theoretical discussions in Refs.
1 and 2, respectively. Under an extremely high pumping, the multi-quantum
transitions, which cause the harmonics generation and the hybridization of
electron-hole states, take place. This regime is not investigated completely
for graphene, see general consideration for bulk materials or quantum wells in
Sects. 10 and 56 of Ref. 3. The nonlinear response is also possible within the
single-photon approach because the Rabi oscillations of coherent response
should take place in graphene if a pulse duration is less 100 fs. 4 Beside of
this, the nonlinear regime of energy relaxation and recombination due to the
Pauli blocking effect takes place if the photogeneration rate is comparable to
the energy relaxation or recombination rates. A saturation of transient
absorption, which have been investigated recently, 5 ; 6 is the most
important manifistation of such a regime. It is because this phenomenon was
exploited for realization of an ultrafast laser with a graphene saturable
absorber in the telecommunication spectral region. To the best of our
knowledge, a complete theoretical treatment of the saturation mechanism is not
performed yet and an investigation of this phenomena is timely now.
In this paper, we consider the temporal nonlinear evolution of carriers under
photoexcitation by the ultrafast pulses in mid-IR and visible (or near-IR)
spectral regions. For the mid-IR pump, one can neglect the relaxation and
recombination processes (the quasielastic relaxation due to acoustic phonons
remains uneffective up to nanoseconds) and the transient distribution of
nonequilibrium carriers is determined by the broadening of interband
transitions due to an elastic scattering and by the parameters of excitation.
For the excitation energies, $\hbar\Omega$, above the optical phonon energy,
$\hbar\omega_{\eta}$ ($\eta=\Gamma,~{}K$ labels the phonon modes correspondent
to the intra- and intervalley transitions), when a cascade emission of the
optical phonons dominates in relaxation, the transient distribution transforms
into a set of peaks. The effective electron-hole recombination takes place if
the lowest peak is placed around the half-energy of optical phonon,
$\hbar\omega_{\eta}/2$. So that, the character of response modifies
essentially if the frequency $\Omega$ varies over the $\omega_{\eta}$-range.
The saturation process is described within the framework of the temporally
local approach, when the decoherentization time (which determines the
broadening of the interband transitions) is shorter in comparision with the
duration of pumping. Spectral and temporal dependencies of the photoexcited
concentration and the response on the probe radiation (transmission and
absorption coefficients) are presented. The thresholds for saturation of
response are estimated to be about 0.2 MW/cm2, 60 MW/cm2, and 0.6 GW/cm2 for
$\hbar\Omega\sim$0.12, 0.8, and 1.5 eV, respectively (mid-IR, near-IR, and
visible spectral regions). These results are dependent on the
decoherentization and relaxation mechanisms and the electrodynamics
conditions. Their are discussed in comparison with the experimental data for
the near-IR pumping case 5 ; 6 . Conditions for the transient negative
absorption of a probe radiation during cascade emission of optical phonons are
also analyzed (this phenomenon under a steady-state pumping was considered
recently 7 in connection with a possibility of the THz lasing effect).
The consideration below is organized as follows. The temporally local approach
for description of the response of photoexcited carriers is developed in Sec.
II. Spectral and temporal dependencies of the response are described in Sects.
III and IV for the cases of excitation in mid-IR and visible (or near-IR)
spectral regions, respectively. A discussion of experimental data, the list of
assumptions used, and concluding remarks are given in the last section. In
Appendix we consider the mechanism of saturation caused by the collisionless
Rabi oscillations.
## II Temporally local approach
Since the symmetry of the energy spectrum and scattering processes for
electrons and holes in an intrinsic graphene, we describe the phenomena under
consideration by the same distribution functions for the both types of
carriers, $f_{pt}$. According to Refs. 2b and 4, the kinetic equation for
$f_{pt}$ takes form:
$\frac{df_{pt}}{dt}=\nu_{pt}(1-2f_{pt})+J(f_{t}|p)$ (1)
and it should be solved with the initial condition $f_{pt\to\infty}=0$. Here
$J(f_{t}|p)$ is the collision integral, which is described the relaxation and
recombination processes, and $\nu_{pt}$ is the interband generation rate due
to the in-plane electric field ${\bf E}w_{t}\exp(-i\Omega t)+c.c.$, where
${\bf E}$ is the field strength, $\Omega$ is the pumping frequency, and
$w_{t}$ is the envelope form-factor of pulse with duration 2$\tau_{p}$
centered at $t=0$. Supposing that $\tau_{p}$ exceeds the dephasing time, we
have used in Eq. (1) the temporally local approach with the separated filling
factor, $(1-2f_{pt})$, and with the rate of photoexcitaion:
$\nu_{pt}=\nu_{R}w_{t}^{2}\Delta\left(\frac{2\upsilon
p-\hbar\Omega}{\gamma}\right),~{}~{}~{}~{}\nu_{R}=\frac{\pi(eE\upsilon/\Omega)^{2}}{\hbar\gamma}.$
(2)
Here $\upsilon=10^{8}$ cm/s is the velocity of neutrinolike quasiparticles,
$2\gamma$ is the broadening of the interband excitation described by the
phenomenological factor $\Delta(z)$. Below we consider the Lorentzian
lineshape of photoexcitation, when $\Delta(z)=\left[\pi(1+z^{2})\right]^{-1}$
and the Gaussian temporal envelope
$w_{\tau}=\sqrt[4]{2/\pi}\exp[-(t/\tau_{p})^{2}]$.
The solution of Eq. (1) determines both the photoinduced concentration, which
given by the standard formula
$n_{t}=\frac{2}{\pi\hbar^{2}}\int\limits_{0}^{\infty}{dpp}f_{pt},$ (3)
and the transient response on a probe radiation of frequency $\omega$
($\propto\exp(-i\omega t)$, which is described by the dynamic conductivity
$\sigma_{\omega t}$. For the collisionless case $\hbar\omega/\gamma\gg 1$,
when the parametric dependency on time takes place, 9 the real part of
$\sigma_{\omega t}$ is written as follows:
${\rm Re}\sigma_{\omega t}=\frac{e^{2}}{4\hbar}(1-2f_{p_{\omega},t}),$ (4)
where $p_{\omega}=\hbar\omega/\upsilon$. The imaginary part of $\sigma_{\omega
t}$ is determined through ${\rm Re}\sigma_{\omega t}$ with the use of the
dispersion relation and one can check that the carrier-induced contribution to
${\rm Im}\sigma_{\omega t}$ appears to be weak in comparison with (4) for the
peak-like distributions of carriers considered below. Thus, the only filling
factor in ${\rm Re}\sigma_{\omega t}$ is responsible for the nonlinear
behavior of the response under consideration.
We restrict ourselves by the the geometry of normal propagation of radiation.
The relative absorption of graphene sheet, $\xi_{\omega t}$, as well as the
reflection and transmission coefficients, $R_{\omega t}$ and $T_{\omega t}$,
are determined through $\sigma_{\omega t}$. Since the energy conservation
requirement, 8
$R_{\omega t}+T_{\omega t}+\xi_{\omega t}=1,$ (5)
we consider below only the absorption and transmission coefficients:
$\displaystyle\xi_{\omega t}\simeq\frac{16\pi}{\sqrt{\varepsilon}c}\frac{{\rm
Re}\sigma_{\omega t}}{|1+\sqrt{\epsilon}+4\pi\sigma_{\omega
t}/c|^{2}}\approx\xi_{m}(1-2f_{p_{\omega}t}),$ $\displaystyle T_{\omega
t}\simeq\frac{4\sqrt{\epsilon}}{\left|1+\sqrt{\epsilon}+4\pi\sigma_{\omega
t}/c\right|^{2}}\approx\frac{T_{m}}{(1-af_{p_{\omega}t})^{2}}.$ (6)
Here $\sqrt{\epsilon}$ is the refraction index of a thick substrate (for SiO2
substrate $\sqrt{\epsilon}\simeq$1.45 and dispersion of $\epsilon$ can be
neglected) and we approximately separated the carrier contributions using the
coefficients $\xi_{m}\approx
4\pi\alpha/\left[\sqrt{\epsilon}(1+\sqrt{\epsilon})\right]$, $T_{m}\approx
4\sqrt{\epsilon}/(1+\sqrt{\epsilon}+\pi\alpha)^{2}$, and $a\approx
2\pi\alpha/(1+\sqrt{\epsilon}+\pi\alpha)$ with $\alpha=e^{2}/\hbar c$. Notice,
that the negative absorption regime $\xi_{\omega t}<0$ takes place if
$f_{p_{\omega},t}>1/2$, under the population inversion condition (see
discussion in Sec. IV).
At $\omega=\Omega$ these relations describe the propagation of pumping pulse
with the time-dependent intensity $Sw_{t}^{2}$, where $S$ is the maximal
intensity. Performing the averaging of (6) over the pulse duration one obtains
$\left|\begin{array}[]{*{20}c}\xi_{S}\\\
T_{S}\end{array}\right|=\int\limits_{-\infty}^{\infty}\frac{dt}{\tau_{p}}w_{t}^{2}\left|\begin{array}[]{*{20}c}\xi_{\Omega
t}\\\ T_{\Omega t}\end{array}\right|,$ (7)
where we used $\int\limits_{-\infty}^{\infty}dtw_{t}^{2}/\tau_{p}=1$. Below we
solve Eq. (1) and analyze the responses (6) and (7) for different parameters
of pump and probe radiations.
## III Mid-IR excitation
We consider here the mid-IR pumping case when the energy relaxation of
carriers is ineffective and $J(f_{t}|p)$ in Eq. (1) can be neglected. As a
result, the solution of the problem (1) takes form:
$f_{pt}=\int\limits_{-\infty}^{t}dt^{\prime}\nu_{pt^{\prime}}\exp\left(-2\int\limits_{t^{\prime}}^{t}d\tau\nu_{p\tau}\right).$
(8)
Evolution of such a distribution from zero value at $t\ll-\tau_{p}$ to the
saturated peak with the maximal value $f_{max}=1/2$ is shown in Fig. 1a versus
dimensionless time and energy at the pumping intensity $S=$1 MW/cm2. Temporal
dependencies of $f_{p_{\Omega}t}$ at different $S$ are shown in Fig. 1b. These
calculations were performed for $\hbar\Omega\simeq$120 meV (pumping by
CO2-laser), the pulse duration $2\tau_{p}\simeq$1 ps, and the broadening
energy $\gamma\simeq$6 meV which is in agreement with the mobility data for
the case of elastic scattering. 9 The temporally-dependent photoinduced
concentration $n_{t}$ is plotted Fig. 1c for the same parameters. The
saturated concentration versus intensity, which is attained at $t>\tau_{p}$,
is presented for $\gamma=$6 and 12 meV in Fig. 1d. These dependencies can be
fitted as
$n_{S}\approx\frac{bS}{1+S/S_{n}},$ (9)
where $b\simeq$6 or 12.2 MW-1 [$n_{S}$ is measured in 1011 cm-2] and
$S_{n}\simeq$1.76 or 10 MW/cm2 for and $\gamma=$6 or 12 meV, respectively.
Figure 1: (Color online) (a) Photoexcited distribution $f_{pt}$ versus energy
$\upsilon p$ and dimensionless time, $t/\tau_{p}$ at mid-IR pumping level
$S=$1 MW/cm2. (b) Temporal evolution of $f_{\Omega t}\equiv f_{p_{\Omega}t}$
at $S=$0.1, 0.3, 1, 3, and 6 MW/cm2 (marked as 1-5). (c) Potoinduced
concentration versus $t/\tau_{p}$ for the same conditions as in panel (b). (d)
Concentration $n_{S}$ at $t/\tau_{p}\to\infty$ versus $S$ for the different
broadening energies $\gamma$. Dotted curves are correspondent to the fit (9).
The relative absorption and transmission coefficients of a probe radiation of
frequency $\omega$ are determined through $f_{p_{\omega}t}$ according to Eqs.
(6). Spectral and temporal dependencies of $\xi_{\omega t}$ are shown in Fig.
2a for the conditions used in Fig. 1a. Since $af_{p_{\omega}t}\ll 1$, the peak
of relative transmission $T_{\omega t}/T_{m}$ resembles $f_{p_{\Omega}t}$
presented in Fig. 1a. Here $T_{m}=\simeq$0.95 is the transmission coefficient
without for non-doped graphene. The temporally-dependent relative absorption
and transmission at the pumping frequency $\Omega$ and at different $S$ are
presented in Figs. 2b and 2c, respectively. The saturated values of
$\xi_{S}/\xi_{m}$ and $T_{S}$ versus intensity are plotted in the upper and
lower panels of Fig. 2d for the parameters used in Fig. 1d ($\xi_{S}$ and
$T_{S}$ have only a weak dependency on $\gamma$). These curves can be fitted
as
$\xi_{S}\approx\frac{\xi_{m}}{1+S/\overline{S}},~{}~{}~{}~{}T_{S}\approx
T_{m}+\frac{hS}{1+S/\overline{S}},$ (10)
where $\overline{S}\approx$0.2 MW/cm2 and $h\approx$0.09 cm2/MW. The
saturation of $\xi_{S}$ and $T_{S}$ takes place at lower threshold intensities
in comparision to $n_{S}$, c. f. Figs. 1c and 2d, 2e. Thus, for the pumping
range $\geq$1 MW/cm2 the one-photon absorption is suppressed and a damage of
graphene by mid-IR radiation with $\tau_{p}\lesssim$1 ps is not possible.
Figure 2: (Color online) (a) Spectral and temporal dependencies of relative
absorption, $\xi_{\omega t}$ at $S=$1 MW/cm2. (b) Temporal evolution of
$\xi_{\Omega t}$ at $S$ used in Fig. 1b (marked). (c) The same as in panel (b)
for transmission, $T_{\Omega t}$. (d) Avaraged over pulse absorption and
transmission coefficients (upper and lower panels, respectively) versus $S$
[dashed curves are correspondent to Eq. (10)].
## IV Cascade emission effect
In this section we consider the photoexcitation by visible and near-IR
radiation, when the cascade emission of optical phonons should be taken into
account in Eq. (1). For the temperatures below the optical phonon energies,
the spontaneous emission processes are only essential and the collision
integral is given by the finite-difference form (see evaluation in Refs. 2b
and 10)
$\displaystyle
J\left(f_{t}|p\right)=\sum_{\eta}\left[\nu_{p+p_{\eta}}\left(1-f_{pt}\right)f_{p+p_{\eta}t}\right.$
(11)
$\displaystyle\left.-\nu_{p-p_{\eta}}\left(1-f_{p-p_{\eta}t}\right)f_{pt}-\widetilde{\nu}_{p_{\eta}-p}f_{p_{\eta}-pt}f_{pt}\right].$
Here $\eta=\Gamma,~{}K$ is correspondent to the intra- and intervalley
transitions with the energy transfer, $\hbar\omega_{\eta}$, and the momentum
transfer, $p_{\eta}=\hbar\omega_{\eta}/\upsilon$. The last contribution of Eq.
(11) is responsible for the recombination process while the first and second
terms describe the interband cascade relaxation of carriers. The relaxation
rates $\nu_{p}$ and $\widetilde{\nu}_{p}$ are proportional to the density of
states,
$\nu_{p}\approx\widetilde{\nu}_{p}\approx\theta(p)\upsilon_{\eta}p/\hbar$,
where the characteristic velocities $\upsilon_{\Gamma,K}$ can be estimated
crudely as $\upsilon_{K}\approx 2\times 10^{6}$ cm/s and
$\upsilon_{\Gamma}\approx 10^{6}$ cm/s. 10 Thus, the $K$-mode emission gives
a dominant contribution to the relaxation process; moreover, the only
interband recombination is possible in the passive region, $0<\upsilon
p<\hbar\omega_{K}=$170 meV. Below we neglect other relaxation processes, so
that a peak-like transient distribution of carriers takes place due to the
negligible phonon dispersion and a narrow distribution of photoexcited
carriers, under the condition $\gamma\ll\hbar\omega_{K}$. For the sake of
simplicity, the cases of effective or suppressed recombination, when the lower
peak in the passive region is placed around or outside the energy
$\hbar\omega_{K}$ are considered. It is convenient to analyze calculations for
the near-IR and visible pumping cases separately.
Figure 3: (Color online) (a) Contour plots of photoexcited distributions
$f_{pt}$ versus energy and time for pulse duration $2\tau_{p}=$0.6 ps at
pumping level 200 MW/cm2 and $\hbar\Omega=$850 meV. (b) Transient evolution of
concentration $n_{t}$ and and populations of peaks around $\sim$43, 213, and
383 meV (marked as $n_{1}$, $n_{2}$, and $n_{3}$, respectively), for the same
conditions as in panel (a). (c) Evolution of $n_{t}$ for pumping levels
$S=$100, 200, and 300 MW/cm2 (marked as 1, 2, and 3). Solid and dashed curves
are plotted for $\hbar\Omega=$ 850 meV and 765 meV.
### IV.1 Near-IR pumping
First, we consider the three-step cascade processes under the near-IR pumping
with wavelengths around $\sim 1.5~{}\mu$m and the pulse duration determined by
$\tau_{p}=$ 0.3 ps. We consider the regimes of the enhanced or suppressed
recombination supposing $\hbar\Omega=$ 850 meV or 765 meV. For this energy
region, the broadening of photoexcited peak is taken as $\gamma\simeq$18 meV,
so that $\hbar/\gamma\ll\tau_{p}$. The numerical solution of Eq. (1) with the
collision integral (11) is performed with the use of the temporal iterations
11 at different $S$. Figure 2a shows the contour plot of the three-peak
distribution function $f_{pt}$ for the case of efficient recombination
($\hbar\Omega=$ 850 meV) at $S=$200 MW/cm2. The carrier concentrations over
the peaks 1-3 and the total consentration $n_{t}$ given by Eq. (3) are shown
in Fig. 3b for the same parameters as in Fig. 1a. Since the relaxation rate in
Eq. (11) is proportional to the density of states, $\nu_{p}\propto p$, the
bottleneck effect takes place under the transition between the second and
third peaks and $n_{2}$ exceeds $n_{1,3}$. The transient evolution of
concentration for different $S$ is shown in Fig. 3c where the maximal
concentration exceeds 1012 cm-2 at $t\sim\tau_{p}$ and $S\geq$0.3 GW/cm2.
During the further evolution, $n_{t}$ decays due to the recombination process.
The case of the suppressed recombination ($\hbar\Omega=$765 meV) is different
because of, first, the peaks are shifted below (about 43 meV) and, second, the
decreasing of $f_{pt}$ and $n_{t}$ due to recombination is absent. The
saturated concentrations (dashed curves in Fig. 3c) exceed the peak
concentrations (solid curves in Fig. 3c) by factor $\sim$1.3. Note, that
$n_{t}$ decreases with increasing of $\gamma$ at fixed $S$ (not shown in Fig.
3).
Figure 4: (Color online) (a) Transient evolution of relative absorption for
$\hbar\Omega=$ 850 and 765 meV (solid and dotted curves, respectively) at
$S=$50, 100 and 200 MW/cm2 (marked as 1-3). (b) The same as in panel (a) for
transmission coefficient. (c) Avaraged over pulse absorption versus $S$
[dashed curve is correspondent to Eq. (9)]. (d) The same as in panel (c) for
transmission coefficient.
Transient evolutions of the absorption and transmission coefficients given by
Eq. (6) are shown in Figs. 4a and 4b at the different pumping frequencies
$\Omega$ (solid and dashed curves) and at different $S$. For
$t/\tau_{p}\leq$0.5, the temporal evolution of $\xi_{\Omega t}$ and $T_{\Omega
t}$ do not dependent on the character of recombination. For
$t/\tau_{p}\lesssim$1.5 this evolution is completely different: a quenching of
photoresponse or a steady-state contribution take place for the effective or
suppressed recombination cases. At $S\geq$300 MW/cm2 and $t/\tau_{p}\sim$0 one
obtains the saturated absorption around $\xi_{\Omega t}\sim$0.1. The negative
absorption takes place for a probe radiation with $\hbar\omega$ around the
first and second peaks. It is because $f_{p_{\omega}t}>1/2$, see Eq. (4) and
the contour plot in Fig. 3, where the regions of negative absorption are
separated by the thick (red) curves. Thus, the negative absorption (and a
possible stimulated emission of mid-IR radiation) is realized at $S\geq$100
MW/cm2 during time intervals $t\lesssim 5\tau_{p}$.
The absorption and transition coefficients averaged over pulse duration
according to Eq. (7) are shown in Figs. 4c and 4d. Since the transient
response at $|t|\lesssim\tau_{p}$ does not depend on the recombination
mechanism (see Figs. 3c, 4a, and 4b), the variation of $\xi_{S}$ and $T_{S}$
with $\hbar\Omega$ is less than 5%. These dependencies can be fitted by Eq.
(10) with the characteristic intensity $\overline{S}\approx$60 MW/cm2 and the
coefficient $h\approx$0.3 cm2/GW.
Figure 5: (Color online) (a) Transient evolution of concentration $n_{t}$ and
and populations of peaks around $\sim$0.09, 0.26, 0.43, 0.6, and 0.77 eV
(marked as $n_{1-5}$, respectively) for pulse duration $2\tau_{p}=$0.4 ps at
pumping level 0.4 GW/cm2 and $\hbar\Omega=$1.53 eV. (b) The same as in panel
(a) at $\hbar\Omega=$1.615 eV for peak’s positions $\sim$0.13, 0.3, 0.47,
0.64, and 0.81 eV marked as $n_{1-5}$. (c) Evolution of $n_{t}$ for pumping
levels $S=$0.2, 0.3, 0.4, 0.6 and 0.8 GW/cm2 (marked as 1-5, respectively) for
$\hbar\Omega=$1.53 eV. (d) The same as in panel (c) for $\hbar\Omega=$1.615
eV.
### IV.2 Visible pumping
Next, we consider the visible light pumping, with wavelengths around $\sim
0.75~{}\mu$m, using the pulse duration $2\tau_{p}=$0.4 ps and the broadening
$\gamma\approx$34 meV (so that $\hbar/\gamma\ll\tau_{p}$). Supposing
$\hbar\Omega=$ 1.53 and 1.615 eV for the enhanced and suppressed recombination
regimes one arrive to the distribution function formed during the five-step
cascade process. Transient evolutions of the concentrations over the peaks 1-5
and of the total concentration $n_{t}$ at $S=$0.4 GW/cm2 are shown in Figs. 5a
and 5b for the cases of enhanced and suppressed recombination, respectively.
Similarly to the near-IR pumping case, the upper peak concentrations decrease
fast at $t>\tau_{p}$ and a maximal population of the second peak takes place
due to the bottleneck effect. Once again, at $t<\tau_{p}$ the shapes of
$n_{t}$ are the same for the both cases. At $t>\tau_{p}$ a quenching of
$n_{t}$ due to recombination takes place in Fig. 5a while there is no a
decreasing of $n_{t}$ in Fig. 5b. The temporal dependencies of concentration
for different $S$ are shown in Figs. 5c and 5d for the two recombination
regimes under consideration. The maximal concentration range up to 1013 cm-2
at $t\sim\tau_{p}$ and $S\approx$1 GW/cm2.
Figure 6: (Color online) (a) Transient evolution of relative absorption for
$\hbar\Omega=$1.53 and 1.615 eV, (solid and dotted curves respectively) at
$S=$0.2, 0.4, 0.6, 0.8 and 1.2 GW/cm2 (marked as 1-5). (b) The same as in
panel (a) for transmission coefficient. (c) Avaraged over pulse absorption
versus $S$ [dashed curve is correspondent to Eq. (9)]. (d) The same as in
panel (c) for transmission coefficient.
The temporal evolution of $\xi_{\Omega t}$ and $T_{\Omega t}$ at frequency
$\Omega$ (solid and dotted curves are correspondent to the two recombination
cases under consideration) are plotted in Figs. 5a and 5b. By analogy with
Sect. IVA, the negative absorption regime takes place at $S\geq$0.3 GW/cm2.
Beside of this, the conditions $\xi_{\omega t}<0$ take place around the peak
positions at $\omega<\Omega$ (not plotted, see a similar behavior in Fig. 3a);
for the first and second peaks the negative absorption regime is realized up
to $t\sim 5\tau_{p}$ at $S\geq$0.1 GW/cm2. In addition, at $t<0.5\tau_{p}$ the
response does not dependent on recombination and at $t>1.5\tau_{p}$ a damping
or time-independent response is realized for the effective or suppressed
recombination.
The averaged according to Eq. (7) absorption and transmission coefficients,
which do not depend on the recombination mechanism, are plotted in Figs. 6c
and 6d. Once again, $\xi_{S}$ and $T_{S}$ can be fitted by Eqs. (10) with the
characteristic intensity $\overline{S}\approx$0.56 GW/cm2 and the coefficient
$h\approx$0.03 cm2/GW. Since the departure rate from the photoexcited peak
increases if $\hbar\Omega$ grows, the characteristic intensity $\overline{S}$
is also increased in the visible spectral region in comparison with the near-
IR pumping case.
## V Discussion and conclusions
To summarize, we have developed the nonlinear theory of transient response of
an intrinsic graphene under the ultrafast interband excitation. Within the
local time approach, the conditions of saturation of absorption were found in
the mid-IR, near-IR, and visible spectral regions. In addition, we have
demonstrated a possibility for the stimulated mid-IR radiation due to the
bottleneck effect during the cascade emission of optical phonons. Our
consideration is based on the set of assumptions about relaxation mechanisms.
First of all, the phenomenological model for the broadening with the
characteristic energy $\gamma$ is used for description of the intersubband
transitions. In Sects. III and IV we estimated $\gamma$ from the experimental
data for the departure relaxation rates. 1 ; 2 ; 9 Secondary, a simplified
description of energy relaxation is employed. We neglect the Coulomb
scattering which is not a dominant relaxation channel at $t\lesssim\tau_{p}$,
so that the results for $\xi_{S}$ and $T_{S}$ should not be modified
essentially. But a transient distribution at $t\gg\tau_{p}$ and a condition
for the negative absorption of a probe radiation in the mid-IR region can be
modified. Also, a possible contribution of the substrate vibration 12 is not
taken into account. These points require a special consideration but, anyway,
our calculation gives a lower bound of $S$. The other assumptions (parameters
for the electron-phonon coupling, conditions for the temporally-local
approach, and description of the interband response) are rather standard for
the calculations of the optical properties and the relaxation phenomena in
graphene. In addition, an inhomogenity of pumping, which causes the lateral
diffusion of carriers, 13 and a heating of phonons 14 may be essential;
these phenomena requre a special treatment, both experimental and theoretical.
We turn now to discussion of the experimental data available for the near-IR
spectral region. 5 ; 6 Numerical estimates for the saturation thresholds and
for the concentrations of the photoexcited carriers are in a qualitative
agreement with the consideration performed. But an accurate comparison with
the results presented is not possible for the two reasons. First, the graphene
structure was embedded into the laser cavity in 5 ; 6 so that the
electrodynamical conditions (for a propagated, reflected, and absorbed
radiation) were different from the simple geometry considered here. Second,
the multi-layer graphene or the graphene flakes were used, while a single-
layer graphene case was not under a detailed treatment. Thus, a special
measurements with the use of the simplest geometry of a well-characterized
sample placed over a semi-infinite substrate are necessary.
In closing, we have analyzed theoretically the conditions for realization of
an efficient graphene-based saturable absorber and have performed a comparison
with the experimental data. More extended treatment of this phenomena under
near-IR pumping, including an above-mentioned special measurements, in order
to improve an efficiency of the graphene based saturable absorber in the
lasers for telecommunications. An additional study in the mid-IR and visible
spectral regions should be useful for verification of different relaxation
mechanisms.
The author would like to thank E. I. Karp for insightful comments.
*
## Appendix A Rabi oscillations regime
Below we describe the saturation of the averaged absorption and transmission
coefficients (7) under an ultrafast pumping for the case when the Rabi
oscillations conditions are satisfied. 4 The collisionless regime of response
is described by the $S$-dependent contribution to the distribution function
$1-2f_{pt}=\cos\left(\sqrt{\frac{S}{S_{R}}}\int\limits_{-\infty}^{t}\frac{dt^{\prime}}{\tau_{p}}w_{t^{\prime}}\right).$
(12)
Here the characteristic intensity is given by
$S_{R}=\frac{\sqrt{\epsilon}c}{4\pi}\left(\frac{\hbar\Omega}{e\tau_{p}\upsilon}\right)^{2}$
(13)
and $S_{R}\simeq$0.6 MW/cm2 for CO2 pumping with $\tau_{p}\simeq$0.1 ps. For
the near-IR or visible pumping with $\tau_{p}\simeq$30 fs, one obtains
$S_{R}\simeq$0.3 or 1 GW/cm2. Notice, that $S_{R}\propto(\Omega/\tau_{p})^{2}$
and (A.1) is not dependent on any other parameter if $\tau_{p}$ is shorter
than the dephasing relaxation time.
Figure 7: (Color online) Normalized absorption coefficient given by (A.3)
versus dimensionless intensity $S/S_{R}$. Dotted curve presents a monotonic
fit.
Substituting the distribution (A.1) into Eqs. (6) and (7) one obtains the
following analytical expression for the relative absorption
$\frac{\xi_{S}}{\xi_{m}}=\int\limits_{-\infty}^{\infty}\frac{dt}{\tau_{p}}w_{t}^{2}\cos\left(\sqrt{\frac{S}{S_{R}}}\int\limits_{-\infty}^{t}\frac{dt^{\prime}}{\tau_{p}}w_{t^{\prime}}\right)$
(14)
while the transmission coefficient is given by $T_{S}\approx
T_{m}(1-\widetilde{a}\xi_{S})$ with
$\widetilde{a}=\sqrt{\epsilon}(1+\sqrt{\epsilon})/2$. In Fig. 7 we plot the
function $\xi_{S}/\xi_{m}$ versus dimensionless intensity $S/S_{R}$ and the
oscillating character of response at $S/S_{R}\geq$4\. The oscillations appears
due to the dynamic inversion of transient population, see Ref. 4. The fit of
(A.3) at $S/S_{R}\leq 1$ is given by Eq. (10) with the characteristic
intensity $\overline{S}=2S_{R}$.
## References
* (1) J. M. Dawlaty, S. Shivaraman, M. Chandrashekhar, F. Rana, and M. G. Spencer Appl. Phys. Lett. 92, 042116 (2008); D. Sun, Z.-K. Wu, C. Divin, X. Li, C. Berger, W. A. de Heer, P. N.First, and T. B. Norris, Phys. Rev. Lett. 101, 157402 (2008); R. W. Newson, J. Dean, B. Schmidt, and H. M. van Driel, Opt. Exp. 17, 2326 (2009).
* (2) F. Rana, P. A. George, J. H. Strait, J. Dawlaty, S. Shivaraman, Mvs Chandrashekhar, and M. G. Spencer, Phys. Rev. B 79, 115447 (2009); P. N. Romanets and F.T. Vasko, Phys. Rev. B 81, 085421 (2010).
* (3) F.T. Vasko and O.E. Raichev, Quantum Kinetic Theory and Applications (Springer, N.Y., 2005).
* (4) P. N. Romanets and F.T. Vasko, Phys. Rev. B 81, 241411 (2010).
* (5) Z. Sun, T. Hasan, F. Torrisi, D. Popa, G.Privitera, F. Wang, F. Bonaccorso, D. M. Basko, and A. C. Ferrari, ACS Nano 4, 803 (2010); Z. Sun, D. Popa, T. Hasan, F. Torrisi, F. Wang, E. J. R. Kelleher, J. C. Travers, and A. C. Ferrari, Nano Research 3, 653 (2010).
* (6) H. Zhang, D.Y. Tang, L.M. Zhao, Q. Bao, K.P. Loh, Opt. Express, 17 17630 (2009); H. Zhang, D. Tang, R.J. Knize, L. Zhao, Q. Bao, K.P. Loh, Appl. Phys. Lett., 96, 111112 (2010).
* (7) V. Ryzhii, M. Ryzhii, and T. Otsuji, J. Appl. Phys. 101, 083114 (2007); A. Satou, F. T. Vasko, and V. Ryzhii, Phys. Rev. B 78, 115431 (2008).
* (8) L.A. Falkovsky, Phys. Usp. 51 887 (2008); M. V. Strikha and F.T. Vasko, Phys. Rev. B 81, 115413 (2010).
* (9) N.M.R. Peres, Rev. Mod. Phys. 82, 2673 (2010); F. T. Vasko and V. Ryzhii, Phys. Rev. B 76, 233404 (2007); X. Hong, K. Zou, and J. Zhu, Phys. Rev. B 80, 241415 (2009).
* (10) H. Suzuura and T. Ando, J. Phys. Soc. Japan, 77, 044703 (2008); S. Piscanec, M. Lazzeri, F. Mauri, A. C. Ferrari, and J. Robertson, Phys. Rev. Lett. 93, 185503 (2004).
* (11) D. Potter, Computational Physics (J. Wiley, London, 1973).
* (12) J. H. Chen, C. Jang, S. Xiao, M. Ishigami, and M. S. Fuhrer, Nature Nanotech. 3, 206 (2008); S. Fratini and F. Guinea, Phys. Rev. B 77, 195415 (2008).
* (13) B. A. Ruzicka, S. Wang, L. K. Werake, B. Weintrub, K. P. Loh, and H. Zhao, arXiv:1005.3850.
* (14) C. H. Lui, K. F. Mak, J. Shan, and T. F. Heinz, Phys. Rev. Lett. 105, 127404 (2010); H. Wang, J. H. Strait, P. A. George, S. Shivaraman, V. B. Shields, Mvs Chandrashekhar, J. Hwang, F. Rana, M. G. Spencer, C. S. Ruiz-Vargas, and J. Park, Appl. Phys. Lett. 96, 081917 (2010).
|
arxiv-papers
| 2010-10-12T13:54:25 |
2024-09-04T02:49:13.758898
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "F.T. Vasko",
"submitter": "Fedir Vasko T",
"url": "https://arxiv.org/abs/1010.2392"
}
|
1010.2511
|
# The use of machine learning with signal- and NLP processing of source code
to fingerprint, detect, and classify vulnerabilities and weaknesses with
MARFCAT
Serguei A. Mokhov
Concordia University Montreal, QC, Canada mokhov@cse.concordia.ca
###### Abstract
We present a machine learning approach to static code analysis and
findgerprinting for weaknesses related to security, software engineering, and
others using the open-source MARF framework and the MARFCAT application based
on it for the NIST’s SATE 2010 static analysis tool exposition workshop.
###### Contents
1. 0.1 Introduction
2. 0.2 Related Work
3. 0.3 Methodology
1. 0.3.1 Core principles
2. 0.3.2 CVEs – the “Knowledge Base”
3. 0.3.3 Categories for Machine Learning
4. 0.3.4 Basic Methodology
5. 0.3.5 Line Numbers
4. 0.4 Results
1. 0.4.1 Preliminary Results Summary
2. 0.4.2 Version SATE.4
3. 0.4.3 Version SATE.5
4. 0.4.4 Version SATE.6
5. 0.4.5 Version SATE.7
5. 0.5 Conclusion
1. 0.5.1 Shortcomings
2. 0.5.2 Advantages
3. 0.5.3 Practical Implications
4. 0.5.4 Future Work
6. .6 Classification Result Tables
###### List of Tables
1. 1 CVE Stats for Wireshark 1.2.0, Quick Enriched, version SATE.4
2. 2 CVE NLP Stats for Wireshark 1.2.0, Quick Enriched, version SATE.4
3. 3 CVE NLP Stats for Wireshark 1.2.0, Quick Enriched, version SATE.4
4. 4 CVE Stats for Chrome 5.0.375.54, Quick Enriched, (clean CVEs) version SATE.4
5. 5 CWE Stats for Chrome 5.0.375.54, (clean CVEs) version SATE.5
6. 6 CVE Stats for Tomcat 5.5.13, version SATE.5
7. 7 CWE Stats for Tomcat 5.5.13, version SATE.5
8. 8 CVE NLP Stats for Tomcat 5.5.13, version SATE.5
9. 9 CWE NLP Stats for Tomcat 5.5.13, version SATE.5
10. 10 CVE NLP Stats for Chrome 5.0.375.54, version SATE.7
11. 11 CWE NLP Stats for Chrome 5.0.375.54, version SATE.7
## 0.1 Introduction
This paper elaborates on the details of the methodology and the corresponding
results of application of the machine learning techniques along with signal
processing and NLP alike to static source code analysis in search for
weaknesses and vulnerabilities in such a code. This work resulted in a proof-
of-concept tool, code-named MARFCAT, a MARF-based Code Analysis Tool [Mok11],
presented at the Static Analysis Tool Exposition (SATE) workshop 2010 [ODBN10]
collocated with the Software Assurance Forum on October 1, 2010.
This paper is a “rolling draft” with several updates expected to be made
before it reaches more complete final-like version as well as combined with
the open-source release of the MARFCAT tool itself [Mok11]. As-is it may
contain inaccuracies and incomplete information.
At the core of the workshop there were C/C++-language and Java language tracks
comprising CVE-selected cases as well as stand-alone cases. The CVE-selected
cases had a vulnerable version of a software in question with a list of CVEs
attached to it, as well as the most know fixed version within the minor
revision number. One of the goals for the CVE-based cases is to detect the
known weaknesses outlined in CVEs using static code analysis and also to
verify if they were really fixed in the “fixed version” [ODBN10].
The test cases at the time included CVE-selected:
* •
C: Wireshark 1.2.0 (vulnerable) and Wireshark 1.2.9 (fixed)
* •
C++: Chrome 5.0.375.54 (vulnerable) and Chrome 5.0.375.70 (fixed)
* •
Java: Tomcat 5.5.13 (vulnerable) and Tomcat 5.5.29 (fixed)
and non-CVE selected:
* •
C: Dovecot 2.0-beta6
* •
Java: Pebble 2.5-M2
We develop MARFCAT to machine-learn from the CVE-based vulnerable cases and
verify the fixed versions as well as non-CVE based cases from similar
programming languages.
### Organization
We develop this “running” article gradually. The related work, some of the
present methodology is based on, is referenced in Section 0.2. The methodology
summary is in Section 0.3. We present the results, most of which were reported
at SATE2010, in Section 0.4. We then describe the machine learning aspects as
well as mathematical estimates of functions of how to determine line numbers
of unknown potentially weak code fragments in Section 0.3.5. (The latter is
necessary since during the representation of the code a wave form (i.e.
signal) with current processing techniques the line information is lost (e.g.
filtered out as noise) making reports less informative, so we either machine-
learn the line numbers or provide a mathematical estimate and that section
describes the proposed methodology to do so, some of which was implemented.)
Then we present a brief summary, description of the limitations of the current
realization of the approach and concluding remarks in Section 0.5.
## 0.2 Related Work
Related work (to various degree of relevance) can be found below (this list is
not exhaustive):
* •
Taxonomy of Linux kernel vulnerability solutions in terms of patches and
source code as well as categories for both are found in [MLB07].
* •
The core ideas and principles behind the MARF’s pipeline and testing
methodology for various algorithms in the pipeline adapted to this case are
found in [Mok08]. There also one can find the core options used to set the
configuration for the pipeline in terms of algorithms used.
* •
A binary analysis using machine learning approach for quick scans for files of
known types in a large collection of files is described in [MD08].
* •
The primary approach here is similar in a way that was done for DEFT2010
[Mok10b, Mok10a] with the corresponding DEFT2010App and its predecessor
WriterIdentApp [MSS09].
* •
Tlili’s 2009 PhD thesis covers topics on automatic detection of safety and
security vulnerabilities in open source software [Tli09].
* •
Statistical analysis, ranking, approximation, dealing with uncertainty, and
specification inference in static code analysis are found in the works of
Engler’s team [KTB+06, KAYE04, KE03].
* •
Kong et al. further advance static analysis (using parsing, etc.) and
specifications to eliminate human specification from the static code analysis
in [KZL10].
* •
Spectral techniques are used for pattern scanning in malware detection by Eto
et al. in [ESI+09].
* •
Researchers propose a general data mining system for incident analysis with
data mining engines in [IYE+09].
* •
Hanna et al. describe a synergy between static and dynamic analysis for the
detection of software security vulnerabilities in [HLYD09] paving the way to
unify the two analysis methods.
* •
The researchers propose a MEDUSA system for metamorphic malware dynamic
analysis using API signatures in [NJG+10].
## 0.3 Methodology
Here we briefly outline the methodology of our approach to static source code
analysis in its core principles in Section 0.3.1, the knowledge base in
Section 0.3.2, machine learning categories in Section 0.3.3, and the high-
level step-wise description in Section 0.3.4.
### 0.3.1 Core principles
The core methodology principles include:
* •
Machine learning
* •
Spectral and NLP techniques
We use signal processing techniques, i.e. presently we do not parse or
otherwise work at the syntax and semantics levels. We treat the source code as
a “signal”, equivalent to binary, where each $n$-gram ($n=2$ presently, i.e.
two consecutive characters or, more generally, bytes) are used to construct a
sample amplitude value in the signal.
We show the system examples of files with weaknesses and MARFCAT learns them
by computing spectral signatures using signal processing techniques from CVE-
selected test cases. When some of the mentioned techniques are applied (e.g.
filters, silence/noise removal, other preprocessing and feature extraction
techniques), the line number information is lost as a part of this process.
When we test, we compute how similar or distant each file is from the known
trained-on weakness-laden files. In part, the methodology can approximately be
seen as some signature-based antivirus or IDS software systems detect bad
signature, except that with a large number of machine learning and signal
processing algorithms, we test to find out which combination gives the highest
precision and best run-time.
At the present, however, we are looking at the files overall instead of
parsing the fine-grained details of patches and weak code fragments, which
lowers the precision, but is fast to scan all the files.
### 0.3.2 CVEs – the “Knowledge Base”
The CVE-selected test cases serve as a source of the knowledge base to gather
information of how known weak code “looks like” in the signal form, which we
store as spectral signatures clustered per CVE or CWE. Thus, we:
* •
Teach the system from the CVE-based cases
* •
Test on the CVE-based cases
* •
Test on the non-CVE-based cases
### 0.3.3 Categories for Machine Learning
The tow primary groups of classes we train and test on include:
* •
CVEs [NIS11a, NIS11b]
* •
CWEs [VM10] and/or our custom-made, e.g. per our classification methodology in
[MLB07]
The advantages of CVEs is the precision and the associated meta knowledge from
[NIS11a, NIS11b] can be all aggregated and used to scan successive versions of
the the same software or derived products. CVEs are also generally uniquely
mapped to CWEs. The CWEs as a primary class, however, offer broader
categories, of kinds of weaknesses there may be, but are not yet well assigned
and associated with CVEs, so we observe the loss of precision.
Since we do not parse, we generally cannot deduce weakness types or even
simple-looking aspects like line numbers where the weak code may be. So we
resort to the secondary categories, that are usually tied into the first two,
which we also machine-learn along, shown below:
* •
Types (sink, path, fix)
* •
Line numbers
### 0.3.4 Basic Methodology
Algorithmically-speaking, MARFCAT performs the following steps to do its
learning analysis:
1. 1.
Compile meta-XML files from the CVE reports (line numbers, CVE, CWE, fragment
size, etc.). Partly done by a Perl script and partly manually. This becomes an
index mapping CVEs to files and locations within files.
2. 2.
Train the system based on the meta files to build the knowledge base (learn).
Presently in these experiments we use simple mean clusters of feature vectors
per default MARF specification ([Mok08, The11]).
3. 3.
Test on the training data for the same case (e.g. Tomcat 5.5.13 on Tomcat
5.5.13) with the same annotations to make sure the results make sense by being
high and deduce the best algorithm combinations for the task.
4. 4.
Test on the testing data for the same case (e.g. Tomcat 5.5.13 on Tomcat
5.5.13) without the annotations as a sanity check.
5. 5.
Test on the testing data for the fixed case of the same software (e.g. Tomcat
5.5.13 on Tomcat 5.5.29).
6. 6.
Test on the testing data for the general non-CVE case (e.g. Tomcat 5.5.13 on
Pebble).
### 0.3.5 Line Numbers
As was earlier mentioned, line number reporting with MARFCAT is an issue
because the source text is essentially lost without line information preserved
(filtered out as noise or silence or mixed in with another signal sample).
Therefore, some conceptual ideas were put forward to either derive a
heuristic, a function of a line number based on typical file attributes as
described below, or learn the line numbers as a part of the machine learning
process. While the methodology of the line numbers discussed more complete
scenarios and examples, only and approximation subset was actually implemented
in MARFCAT.
#### Line Number Estimation Methodology
Line number is a function of the file’s dimensions in terms of line numbers,
size in bytes, and words. The meaning of $W$ may vary. The implementations of
$f$ may vary and can be purely mathematical or relativistic and with side
effects. These dimensions were recorded in the meta XML files along with the
other indexing information. This gives as the basic Equation 1.
$l=f(L_{T},B,W)$ (1)
where
* •
$L_{T}$ – number of lines of text in a file
* •
$B$ – the size of the file in bytes
* •
$W$ – number of words per wc [Fre09], but can be any blank delimited printable
character sequence; can also be an $n$-gram of $n$ characters.
The function should be additive to allow certain components to be zero if the
information is not available or not needed, in particular $f(B)$ and $f(W)$
may fall into this category. The ceiling $\lceil\ldots\rceil$ is required when
functions return fractions, as shown in Equation 2.
$f(L_{T},B,W)=\lceil f(L_{T})+f(B)+f(W)\rceil$ (2)
Constraints on parameters:
* •
$l\in[1,\ldots,L_{T}]$ – the line number must be somewhere within the lines of
text.
* •
$f(L_{T})>0$ – the component dependent on the the lines of text $L_{T}$ should
never be zero or less.
* •
$EOL=\\{\mathtt{\n},\mathtt{\r},\mathtt{\r\n},\mathtt{EOF}\\}$. The inclusion
of EOF accounts for the last line of text missing the traditional line
endings, but is non-zero.
* •
$L_{T}>0\implies B>0$
* •
$B>0\implies L_{T}>0$ under the above definition of EOL; if EOF is excluded
this implication would not be true
* •
$B=0\implies L_{T}=0,W=0$
Affine combination is in Equation 3:
$f(L_{T},B,W)=\lceil k_{L}\cdot f(L_{T})+k_{B}\cdot f(B)+k_{W}\cdot
f(W)\rceil$ (3)
* •
$k_{L}+k_{B}+k_{W}<1\implies$ the line is within the triangle
Affine combination with context is in Equation 4:
$f(L_{T},B,W)=\lceil k_{L}\cdot f(L_{T})+k_{B}\cdot f(B)+k_{W}\cdot
f(W)\rceil\pm\Delta c$ (4)
where $\pm\Delta c$ is the amount of context surrounding the line, like in
diff [MES02]; with $c=0$ we are back to the original affine combination.
##### Learning approach with matrices and probabilities from examples.
This case of the line number determination must follow the preliminary
positive test with some certainty that a give source code file contains
weaknesses and vulnerabilities. This methodology in itself would be next to
useless if this preliminary step is not performed.
In a simple case a line number is a cell in the 3D matrix $M$ given the file
dimensions alone, as in Equation 5. The matrix is sparse and unknown entries
are 0 by default. Non-zero entries are learned from the examples of files with
weaknesses. This matrix is capable of encoding a single line location per file
of the same dimensions. As such it can’t handle multiple locations per file or
two or more distinct unrelated files with different line numbers for a single
location. However, it serves as a starting point to develop a further and
better model.
$l=f(L_{T},B,W)=M[L_{T},B,W]$ (5)
To allow multiple locations per file we either replace the $W$ dimension with
the locations dimension $N$ if $W$ is not needed, as e.g. in Equation 6, or
make the matrix 4D by adding $N$ to it, as in Equation 7. This will take care
of the multiple locations issue mentioned earlier. $N$ is not known at the
classification stage, but the coordinates $L_{T},B,W$ will give a value in the
3D matrix, which is a vector of locations $\vec{n}$. At the reporting stage we
simply report all of the elements in $\vec{n}$.
$\vec{l}=f(L_{T},B,W)=M[L_{T},B,N]$ (6)
$\vec{l}=f(L_{T},B,W)=M[L_{T},B,W,N]$ (7)
In the above matrices $M$, the returned values are either a line number $l$ or
a collection of line numbers $\vec{l}$ that were learned from examples for the
files of those dimensions. However, if we discovered a file tested positive to
contain a weakness, but we have never seen its dimensions (even taking into
the account we can sometimes ignore $W$), we’ll get a zero. This zero presents
a problem: we can either (a) rely on one of the math functions described
earlier to fill in that zero with a non-zero line number or (b) use
probability values, and convert $M$ to $M_{p}$, as shown in Equation 8.
The $M_{p}$ matrix would contain a vector value $\vec{n_{p}}$ of probabilities
a given line number is a line number of a weakness.
$\vec{l_{p}}=f(L_{T},B,W)=M_{p}[L_{T},B,W,N]$ (8)
We then select the most probable ones from the list with the highest
probabilities. The index $i$ within $\vec{l_{p}}$ represents the line number
and the value at that index is the probability $p=\vec{l_{p}}[i]$.
Needless to say this 4D matrix is quite sparse and takes a while to learn. The
learning is performed by counting occurrences of line numbers of weaknesses in
the training data over total of entries. To be better usable for the unseen
cases the matrix needs to be smoothed using any of the statistical estimators
available, e.g. from NLP, such as add-delta, ELE, MLE, Good-Turing, etc. by
spreading the probabilities over to the zero-value cells from the non-zero
ones. This is promising to be the slowest but the most accurate method.
In MARF, $M$ is implemented using marf.util.Matrix, a free-form matrix that
grows upon the need lazily and allows querying beyond physical dimensions when
needed.
#### Classes of Functions
Define is the meaning of:
* •
$k_{?}=\frac{L_{T}}{B}$
* •
$k_{?}=\frac{W}{B}$
Non-learning:
1. 1.
* •
$k_{*}=1$
* •
$f(L_{T})=L_{T}/2$
* •
$f(B)=0$
* •
$f(W)=0$
2. 2.
* •
$k_{L}=\frac{W}{B}$
* •
$f(L_{T})=L_{T}/2$
* •
$f(B)=0$
* •
$f(W)=0$
3. 3.
* •
$k_{L}=\frac{L_{T}}{B}$
* •
$f(L_{T})=L_{T}/2$
* •
$f(B)=0$
* •
$f(W)=0$
4. 4.
* •
$k_{*}=1$
* •
$f(L_{T})=random(L_{T})$
* •
$f(B)=0$
* •
$f(W)=0$
## 0.4 Results
The preliminary results of application of our methodology are outlined in this
section. We summarize the top precisions per test case using either signal-
processing or NLP-processing of the CVE-based cases and their application to
the general cases. Subsequent sections detail some of the findings and issues
of MARFCAT’s result releases with different versions.
The results currently are being gradually released in the iterative manner
that were obtained through the corresponding versions of MARFCAT as it was
being designed and developed.
### 0.4.1 Preliminary Results Summary
Current top precision at the SATE2010 timeframe:
* •
Wireshark:
* –
CVEs (signal): 92.68%, CWEs (signal): 86.11%,
* –
CVEs (NLP): 83.33%, CWEs (NLP): 58.33%
* •
Tomcat:
* –
CVEs (signal): 83.72%, CWEs (signal): 81.82%,
* –
CVEs (NLP): 87.88%, CWEs (NLP): 39.39%
* •
Chrome:
* –
CVEs (signal): 90.91%, CWEs (signal): 100.00%,
* –
CVEs (NLP): 100.00%, CWEs (NLP): 88.89%
* •
Dovecot:
* –
14 warnings; but it appears all quality or false positive
* –
(very hard to follow the code, severely undocumented)
* •
Pebble:
* –
none found during quick testing
What follows are some select statistical measurements of the precision in
recognizing CVEs and CWEs under different configurations using the signal
processing and NLP processing techniques.
“Second guess” statistics provided to see if the hypothesis that if our first
estimate of a CVE/CWE is incorrect, the next one in line is probably the
correct one. Both are counted if the first guess is correct.
### 0.4.2 Version SATE.4
#### Wireshark 1.2.0
Typical quick run on the enriched Wireshark 1.2.0 on CVEs is in Table 1. All
22 CVEs are reported. Pretty good precision for options -diff and -cheb (Diff
and Chebyshev distance classifiers, respectively [Mok08]). In Unigram, Add-
Delta NLP results on Wireshark 1.2.0’s training file for CVEs, the precision
seems to be overall degraded compared to the classical signal processing
pipeline. Only 20 out of 22 CVEs are reported, as shown in Table 2. CWE-based
testing on Wireshark 1.2.0 (also with some basic line heuristics that does not
impact the precision) is in Table 3.
The following select reports are about Wireshark 1.2.0 using a small subset of
algorithms. There are line numbers that were machine-learned from the
_train.xml file. The two XML report files are the best ones we have chosen
among several of them. Their precision rate using machine learning techniques
is 92.68% after several bug corrections done. All CVEs are reported making
recall 100%. The stats-*.txt files are there summarizing the evaluation
precision. The results are as good as the training data given; if there are
mistakes in the data selection and annotation XML files, then the results will
also have mistakes accordingly.
The best reports are:
report-noprepreprawfftcheb-wireshark-1.2.0-train.xml
report-noprepreprawfftdiff-wireshark-1.2.0-train.xml
The first one validates with both sate2010 schemas, but the latter has
problems with the exponential -E notation.
##### Files.
The corresponding *.log files are there for references, but contain a lot of
debug information from the tool. The tool is using thresholding to reduce the
amount of noise going into the reports.
marfcat-nopreprep-raw-fft-cheb.log
marfcat-nopreprep-raw-fft-diff.log
marfcat--super-fast.log (primaily training log)
report-noprepreprawfftcheb-wireshark-1.2.0-train.xml
report-noprepreprawfftdiff-wireshark-1.2.0-train.xml
stats--super-fast.txt
wireshark-1.2.0_train.xml
#### Wireshark 1.2.9
The following analysis reports are about Wireshark 1.2.9 using a small subset
of MARF’s algorithms. The system correctly does not report the fixed CVEs
(currently, the primary class), so most of the reports come up empty (no
noise). All example reports (one per configuration) validate with the schemas
sate_2010.xsd and sate_2010.pathcheck.xsd.
The best (empty) reports are:
report-noprepreprawfftcheb-wireshark-1.2.9-test.xml
report-noprepreprawfftdiff-wireshark-1.2.9-test.xml
report-noprepreprawffteucl-wireshark-1.2.9-test.xml
report-noprepreprawffthamming-wireshark-1.2.9-test.xml
The below particular report shows the Minkowski distance classifier (-mink)
was not perhaps the best choice, as it mistakingly reported a known CVE that
was in fact fixed, this is an example of machine learning “red herring”:
report-noprepreprawfftmink-wireshark-1.2.9-test.xml
##### Files.
All the corresponding tool-specific *.log files are there for reference.
marfcat-nopreprep-raw-fft-cheb.log
marfcat-nopreprep-raw-fft-diff.log
marfcat-nopreprep-raw-fft-eucl.log
marfcat-nopreprep-raw-fft-hamming.log
marfcat-nopreprep-raw-fft-mink.log
marfcat--super-fast-wireshark.log (training log)
report-noprepreprawfftcheb-wireshark-1.2.9-test.xml
report-noprepreprawfftdiff-wireshark-1.2.9-test.xml
report-noprepreprawffteucl-wireshark-1.2.9-test.xml
report-noprepreprawffthamming-wireshark-1.2.9-test.xml
report-noprepreprawfftmink-wireshark-1.2.9-test.xml
#### Chrome 5.0.375.54
This version’s CVE testing result of Chrome 5.0.375.54 (after updates and
removal unrelated CVEs per SATE organizers) is in Table 4. The corresponding
select reports produced below are about Chrome 5.0.375.54 using a small subset
of algorithms. There are line numbers that were machine-learned from the
*_train.xml file. The two report-*.xml files are ones of the best ones we have
picked. Their precision rate using machine learning techniques is 90.91% after
all the corrections done. The stats-*.txt file is there summarizing the
evaluation precision in the end of that file. Again, the results are as good
as the training data given; if there are mistakes in the data selection and
annotation XML files, then the results will also have mistakes accordingly.
The best reports are:
report-noprepreprawfftcheb-chrome-5.0.375.54-train.xml
report-noprepreprawfftdiff-chrome-5.0.375.54-train.xml
Both validate with both sate2010 schemas.
##### Files.
The corresponding *.log files are there for references, but contain A LOT of
debug info from the tool. The tool is using thresholding to reduce the amount
of noise going into the reports, but if you are curious to examine the logs,
they are included.
chrome-5.0.375.54_train.xml
marfcat-nopreprep-raw-fft-cheb.log
marfcat-nopreprep-raw-fft-diff.log
marfcat--super-fast-chrome.log
README.txt
report-noprepreprawfftcheb-chrome-5.0.375.54-train.xml
report-noprepreprawfftdiff-chrome-5.0.375.54-train.xml
stats--super-fast.txt
#### Chrome 5.0.375.70
The following reports are about Chrome 5.0.375.70 using a small subset of
algorithms. The system correctly does not report the fixed CVEs, so most of
the reports come up empty (no noise) as they are expected to be for known CVE-
selected weaknesses. All example reports (one per configuration) validate with
the schema sate_2010.xsd and sate_2010.pathcheck.xsd.
The best (empty) reports are:
report-noprepreprawfftcheb-chrome-5.0.375.70-test.xml
report-noprepreprawfftdiff-chrome-5.0.375.70-test.xml
report-noprepreprawffteucl-chrome-5.0.375.70-test.xml
report-noprepreprawffthamming-chrome-5.0.375.70-test.xml
report-noprepreprawfftmink-chrome-5.0.375.70-test.xml
##### Files.
All the corresponding tool-specific *.log files are there for reference.
chrome-5.0.375.70_test.xml
marfcat-nopreprep-raw-fft-cheb.log
marfcat-nopreprep-raw-fft-diff.log
marfcat-nopreprep-raw-fft-eucl.log
marfcat-nopreprep-raw-fft-hamming.log
marfcat-nopreprep-raw-fft-mink.log
marfcat--super-fast-chrome.log
report-noprepreprawfftcheb-chrome-5.0.375.70-test.xml
report-noprepreprawfftdiff-chrome-5.0.375.70-test.xml
report-noprepreprawffteucl-chrome-5.0.375.70-test.xml
report-noprepreprawffthamming-chrome-5.0.375.70-test.xml
report-noprepreprawfftmink-chrome-5.0.375.70-test.xml
### 0.4.3 Version SATE.5
#### Chrome 5.0.375.54
Here we complete the CVE results from the MARFCAT SATE.5 version by using
Chrome 5.0.375.54 training on Chrome 5.0.375.54 with classical CWEs as opposed
to CVEs. The result summary is in Table 5.
#### Tomcat 5.5.13
With this MARFCAT version we did first CVE-based testing on training for
Tomcat 5.5.13. Classifiers corresponding to -cheb (Chebyshev distance) and
-diff (Diff distance) continue to dominate as in the other test cases. An
observation: for some reason, -cos (cosine similarity classifier) with the
same settings as for the C/C++ projects (Wireshark and Chrome) actually
preforms well and *_report.xml is not as noisy; in fact comparable to -cheb
and -diff. These CVE-based results are summarized in Table 6. Further, we
perform quick CWE-based testing on Tomcat 5.5.13. Reports are quite larger for
-cheb, -diff, and -cos, but not for other classifiers. The precision results
are illustrated in Table 7. Then, in SATE.5, quick Tomcat 5.5.13 CVE NLP
testing shows higher precision of 87.88%, but the recall is poor, 25/31 – 6
CVEs are missing out (see Table 8). Subsequent, quick Tomcat 5.5.13 CWE NLP
testing was surprisingly poor topping at 39.39% (see Table 9). The resulting
select reports about this Apache Tomcat 5.5.13 test case using a small subset
of algorithms are mentioned below with some commentary.
##### CVE-based training and reporting:
As before, there are line numbers that were machine-learned from the
_train.xml file as well as the types of locations and descriptions provided by
the SATE organizers and incorporated into the reports via machine learning.
This includes the types of locations, such as “fix”, “sink”, or “path” learned
from the ogranizers-provided XML/spreadsheet as well as the source code files.
Two of all the produced XML reports are the best ones. The macro precision
rate in there using machine learning techniques is 83.72%. The stats-*.txt
files are there summarizing the evaluation precision.
The best reports are:
report-noprepreprawfftcheb-apache-tomcat-5.5.13-train-cve.xml
report-noprepreprawfftdiff-apache-tomcat-5.5.13-train-cve.xml
(does not validate three tool-specific lines)
Other reports are, to a various degree of detail and noise:
report-noprepreprawfftcos-apache-tomcat-5.5.13-train-cve.xml
(does not validate two lines)
report-noprepreprawffteucl-apache-tomcat-5.5.13-train-cve.xml
(does not validate three tool-specific lines)
report-noprepreprawffthamming-apache-tomcat-5.5.13-train-cve.xml
report-noprepreprawfftmink-apache-tomcat-5.5.13-train-cve.xml
report-nopreprepcharunigramadddelta-apache-tomcat-5.5.13-train-cve-nlp.xml
The \--nlp version reports use the NLP techniques with the machine learning
instead of signal processing techniques. Those reports are largely comparable,
but have smaller recall, i.e. some CVEs are completely missing out from the
reports in this version. Some reports have problems with tool-specific ranks
like: $4.199735736674989E-4$, which we will have to see how to reduce these.
##### CWE-based training and reporting:
The CWE-based reports use the CWE as a primary class instead of CVE for
training and reporting, and as such currently do not report on CVEs directly
(i.e. no direct mapping from CWE to CVE exists unlike in the opposite
direction); however, their recognition rates are not very low either in the
same spots, types, etc. In the future version of MARFCAT the plan is to
combine the two machine learning pipeline runs of CVE and CWE together to
improve mutual classification, but right now it is not available. The CWE-
based training is also used on the testing files say of Pebble to see if there
are any similar weaknesses to that of Tomcat found, again e.g. in Pebble.
CWEs, unlike CVEs for most projects, represent better cross-project classes as
they are largely project-independent. Both CVE-based and CWE-base methods use
the same data for training. CWEs are recognized correctly 81.82% for Tomcat.
NLP-based CWE testing is not included as its precision was quite low ($\approx
39\%$).
The best reports are:
report-cweidnoprepreprawfftcheb-apache-tomcat-5.5.13-train-cwe.xml
(does not validate)
report-cweidnoprepreprawfftdiff-apache-tomcat-5.5.13-train-cwe.xml
(does not validate)
Other reports are, to a various degree of detail and noise:
report-cweidnoprepreprawfftcos-apache-tomcat-5.5.13-train-cwe.xml
report-cweidnoprepreprawffteucl-apache-tomcat-5.5.13-train-cwe.xml
(does not validate)
report-cweidnoprepreprawffthamming-apache-tomcat-5.5.13-train-cwe.xml
report-cweidnoprepreprawfftmink-apache-tomcat-5.5.13-train-cwe.xml
##### Files.
The corresponding *.log files are there for references, but contain A LOT of
debug info from the tool. The tool is using thresholding to reduce the amount
of noise going into the reports, but if you are curious to examine the logs,
they are included.
apache-tomcat-5.5.13-src_train.xml (meta training file)
marfcat-cweid-nopreprep-raw-fft-cheb.log
marfcat-cweid-nopreprep-raw-fft-cos.log
marfcat-cweid-nopreprep-raw-fft-diff.log
marfcat-cweid-nopreprep-raw-fft-eucl.log
marfcat-cweid-nopreprep-raw-fft-hamming.log
marfcat-cweid-nopreprep-raw-fft-mink.log
marfcat-nopreprep-char-unigram-add-delta.log
marfcat-nopreprep-raw-fft-cheb.log
marfcat-nopreprep-raw-fft-cos.log
marfcat-nopreprep-raw-fft-diff.log
marfcat-nopreprep-raw-fft-eucl.log
marfcat-nopreprep-raw-fft-hamming.log
marfcat-nopreprep-raw-fft-mink.log
marfcat--super-fast-tomcat-train-cve.log
marfcat--super-fast-tomcat-train-cve-nlp.log
marfcat--super-fast-tomcat-train-cwe.log
report-cweidnoprepreprawfftcheb-apache-tomcat-5.5.13-train-cwe.xml
report-cweidnoprepreprawfftcos-apache-tomcat-5.5.13-train-cwe.xml
report-cweidnoprepreprawfftdiff-apache-tomcat-5.5.13-train-cwe.xml
report-cweidnoprepreprawffteucl-apache-tomcat-5.5.13-train-cwe.xml
report-cweidnoprepreprawffthamming-apache-tomcat-5.5.13-train-cwe.xml
report-cweidnoprepreprawfftmink-apache-tomcat-5.5.13-train-cwe.xml
report-nopreprepcharunigramadddelta-apache-tomcat-5.5.13-train-cve-nlp.xml
report-noprepreprawfftcheb-apache-tomcat-5.5.13-train-cve.xml
report-noprepreprawfftcos-apache-tomcat-5.5.13-train-cve.xml
report-noprepreprawfftdiff-apache-tomcat-5.5.13-train-cve.xml
report-noprepreprawffteucl-apache-tomcat-5.5.13-train-cve.xml
report-noprepreprawffthamming-apache-tomcat-5.5.13-train-cve.xml
report-noprepreprawfftmink-apache-tomcat-5.5.13-train-cve.xml
stats-per-cve-nlp.txt
stats-per-cve.txt
stats-per-cwe.txt
#### Pebble 2.5-M2
Using the machine learning approach of MARF by using the Tomcat 5.5.13 as a
source of training on a Java project with known weaknesses, we used that
(rather small) “knowledge base” to test if anything weak similar to the
weaknesses in Tomcat are also present in the supplied version of Pebble
2.5-M2. The current result is that under the version of MARFCAT SATE.5 all
reports come up empty under the current thresholding rules meaning the tool
was not able to identify similar weaknesses in files in Pebble. The
corresponding tool-specific log files are also provided if of interest, but
the volume of data in them is typically large. It is planned to lower the
thresholds after reviewing logs in detail to see if anything interesting comes
up that we missed otherwise.
##### Files.
marfcat--super-fast-tomcat13-pebble-cwe.log
marfcat-cweid-nopreprep-raw-fft-cheb.log
marfcat-cweid-nopreprep-raw-fft-cos.log
marfcat-cweid-nopreprep-raw-fft-diff.log
marfcat-cweid-nopreprep-raw-fft-eucl.log
marfcat-cweid-nopreprep-raw-fft-hamming.log
marfcat-cweid-nopreprep-raw-fft-mink.log
report-cweidnoprepreprawfftcheb-pebble-test-cwe.xml
report-cweidnoprepreprawfftcos-pebble-test-cwe.xml
report-cweidnoprepreprawfftdiff-pebble-test-cwe.xml
report-cweidnoprepreprawffteucl-pebble-test-cwe.xml
report-cweidnoprepreprawffthamming-pebble-test-cwe.xml
report-cweidnoprepreprawfftmink-pebble-test-cwe.xml
#### Tomcat and Pebble Testing Results Summary
* •
Tomcat 5.5.13 on Tomcat 5.5.29 classical CVE testing produced only report with
-cos with 10 weaknesses, some correspond to the files in training. However,
the line numbers reported are midline, so next to meaningless.
* •
Tomcat 5.5.13 on Tomcat 5.5.29 classical CWE testing also report with -cos
with 2 weaknesses.
* •
Tomcat 5.5.13 on Tomcat 5.5.29 NLP CVE testing single report (quick testing
only does add-delta, unigram) came up empty.
* •
Tomcat 5.5.13 on Tomcat 5.5.29 NLP CWE testing, also with a single report
(quick testing only does add-delta, unigram) came up empty.
* •
Tomcat 5.5.13 on Pebble classical CVE reports are empty.
* •
Tomcat 5.5.13 on Pebble NLP CVE report is not empty, but reports wrongly on
blank.html (empty HTML file) on multiple CVEs. The probability $P=0.0$ for all
in this case CVEs, not sure why it is at all reported. A red herring.
* •
Tomcat 5.5.13 on Pebble classical CWE reports are empty.
* •
Tomcat 5.5.13 on Pebble NLP CWE is similar to the Pebble NLP CVE report on
blank.html entries, but fewer of them. All the other symptoms are the same.
### 0.4.4 Version SATE.6
#### Dovecot 2.0.beta6
This is a quick test and a report for Dovecot 2.0.beta6, with line numbers and
other information. The report is ‘raw’, without our manual evaluation and
generated as-is at this point.
The report of interest:
report-cweidnoprepreprawfftcos-dovecot-2.0.beta6-wireshark-test-cwe.xml
It appears though from the first glance most of the are warnings are ‘bogus’
or ‘buggy’, but could indicate potential presence of weaknesses in the flagged
files. One thing is for sure the Dovecode’s source code’s main weakness is a
near chronic lack of comments, which is also a weakness of a kind. Other
reports came up empty. The source for learning was Wireshark 1.2.0.
##### Files.
dovecot-2.0.beta6_test.xml
marfcat--super-fast-dovecot-wireshark-test-cwe.log
marfcat-cweid-nopreprep-raw-fft-cheb.log
marfcat-cweid-nopreprep-raw-fft-cos.log
marfcat-cweid-nopreprep-raw-fft-diff.log
marfcat-cweid-nopreprep-raw-fft-eucl.log
marfcat-cweid-nopreprep-raw-fft-hamming.log
marfcat-cweid-nopreprep-raw-fft-mink.log
report-cweidnoprepreprawfftcheb-dovecot-2.0.beta6-wireshark-test-cwe.xml
report-cweidnoprepreprawfftcheb-wireshark-1.2.0_train.xml.xml
report-cweidnoprepreprawfftcos-dovecot-2.0.beta6-wireshark-test-cwe.xml
report-cweidnoprepreprawfftdiff-dovecot-2.0.beta6-wireshark-test-cwe.xml
report-cweidnoprepreprawffteucl-dovecot-2.0.beta6-wireshark-test-cwe.xml
report-cweidnoprepreprawffthamming-dovecot-2.0.beta6-wireshark-test-cwe.xml
report-cweidnoprepreprawfftmink-dovecot-2.0.beta6-wireshark-test-cwe.xml
#### Tomcat 5.5.29
This is another quick CVE-based evaluation of Tomcat 5.5.29, with line
numbers, etc. They are ’raw’, without our manual evaluation and generated as-
is.
The reports of interest:
report-noprepreprawfftcos-apache-tomcat-5.5.29-test-cve.xml
report-cweidnoprepreprawfftcos-apache-tomcat-5.5.29-test-cwe.xml
As for the Dovecot case, it appears though from the first glance most of the
warnings are either ‘bogus’ or ‘buggy’, but could indicate potential presence
of weaknesses in the flagged files or fixed as such. Need more manual
inspection to be sure. Other XML reports came up empty. The source for
learning was Tomcat 5.5.13.
##### Files.
marfcat--super-fast-tomcat13-tomcat29-cve.log
marfcat--super-fast-tomcat13-tomcat29-cwe.log
marfcat-cweid-nopreprep-raw-fft-cheb.log
marfcat-cweid-nopreprep-raw-fft-cos.log
marfcat-cweid-nopreprep-raw-fft-diff.log
marfcat-cweid-nopreprep-raw-fft-eucl.log
marfcat-cweid-nopreprep-raw-fft-hamming.log
marfcat-cweid-nopreprep-raw-fft-mink.log
marfcat-nopreprep-raw-fft-cheb.log
marfcat-nopreprep-raw-fft-cos.log
marfcat-nopreprep-raw-fft-diff.log
marfcat-nopreprep-raw-fft-eucl.log
marfcat-nopreprep-raw-fft-hamming.log
marfcat-nopreprep-raw-fft-mink.log
report-cweidnoprepreprawfftcheb-apache-tomcat-5.5.29-test-cwe.xml
report-cweidnoprepreprawfftcos-apache-tomcat-5.5.29-test-cwe.xml
report-cweidnoprepreprawfftdiff-apache-tomcat-5.5.29-test-cwe.xml
report-cweidnoprepreprawffteucl-apache-tomcat-5.5.29-test-cwe.xml
report-cweidnoprepreprawffthamming-apache-tomcat-5.5.29-test-cwe.xml
report-cweidnoprepreprawfftmink-apache-tomcat-5.5.29-test-cwe.xml
report-noprepreprawfftcheb-apache-tomcat-5.5.29-test-cve.xml
report-noprepreprawfftcos-apache-tomcat-5.5.29-test-cve.xml
report-noprepreprawfftdiff-apache-tomcat-5.5.29-test-cve.xml
report-noprepreprawffteucl-apache-tomcat-5.5.29-test-cve.xml
report-noprepreprawffthamming-apache-tomcat-5.5.29-test-cve.xml
report-noprepreprawfftmink-apache-tomcat-5.5.29-test-cve.xml
### 0.4.5 Version SATE.7
Up until this version NLP processing of Chrome was not successful. Errors
related to the number of file descriptors opened and “mark invalid” for NLP
processing of Chrome 5.0.375.54 for both CVEs and CWEs have been corrected, so
we have produced the results for these cases. CVEs are reported in Table 10.
CWEs are further reported in Table 11.
## 0.5 Conclusion
We review the current results of this experimental work, its current
shortcomings, advantages, and practical implications. We also release MARFCAT
Alpha version as open-source that can be found at [Mok11]. This is following
the open-source philosophy of greater good (MARF itself has been open-source
from the very beginning [The11]).
### 0.5.1 Shortcomings
The below is a list of most prominent issues with the presented approach. Some
of them are more “permanent”, while others are solvable and intended to be
addressed in the future work. Specifically:
* •
Looking at a signal is less intuitive visually for code analysis by humans.
* •
Line numbers are a problem (easily “filtered out” as high-frequency “noise”,
etc.). A whole “relativistic” and machine learning methodology developed for
the line numbers in Section 0.3.5 to compensate for that. Generally, when CVEs
is the primary class, by accurately identifying the CVE number one can get all
the other pertinent details from the CVE database, including patches and line
numbers.
* •
Accuracy depends on the quality of the knowledge base (see Section 0.3.2)
collected. “Garbage in – garbage out.”
* •
To detect CVE or CWE signatures in non-CVE cases requires large knowledge
bases (human-intensive to collect).
* •
No path tracing (since no parsing is present); no slicing, semantic
annotations, context, locality of reference, etc. The “sink”, “path”, and
“fix” results in the reports also have to be machine-learned.
* •
A lot of algorithms and their combinations to try (currently $\approx 1800$
permutations) to get the best top N. This is, however, also an advantage of
the approach as the underlying framework can quickly allow for such testing.
* •
File-level training vs. fragment-level training – presently the classes are
trained based on the entire file where weaknesses are found instead of the
known fragments from CVE-reported patches. The latter would be more fine-
grained and precise than whole-file classification, but slower. However,
overall the file-level processing is a man-hour limitation than a
technological one.
* •
No nice GUI. Presently the application is script/command-line based.
### 0.5.2 Advantages
There are some key advantages of the approach presented. Some of them follow:
* •
Relatively fast (e.g. Wireshark’s $\approx 2400$ files train and test in about
3 minutes) on a now-commodity desktop.
* •
Language-independent (no parsing) – given enough examples can apply to any
language, i.e. methodology is the same no matter C, C++, Java or any other
source or binary languages (PHP, C#, VB, Perl, bytecode, assembly, etc.).
* •
Can automatically learn a large knowledge base to test on known and unknown
cases.
* •
Can be used to quickly pre-scan projects for further analysis by humans and
other tools that do in-depth semantic analysis.
* •
Can learn from other SATE’10 reports.
* •
Can learn from SATE’09 and SATE’08 reports.
* •
High precision in CVEs and CWE detection.
* •
Lots of algorithms and their combinations to select the best for a particular
task or class (see Section 0.3.3).
### 0.5.3 Practical Implications
Most practical implications of all static code analyzers are obvious – to
detect and report source code weaknesses and report them appropriately to the
developers. We outline additional implications this approach brings to the
arsenal below:
* •
The approach can be used on any target language without modifications to the
methodology or knowing the syntax of the language. Thus, it scales to any
popular and new language analysis with a very small amount of effort.
* •
The approach can nearly identically be transposed onto the compiled binaries
and bytecode, detecting vulnerable deployments and installations – sort of
like virus scanning of binaries, but instead scanning for infected binaries,
one would scan for security-weak binaries on site deployments to alert system
administrators to upgrade their packages.
* •
Can learn from binary signatures from other tools like Snort [Sou10].
### 0.5.4 Future Work
There is a great number of possibilities in the future work. This includes
improvements to the code base of MARFCAT as well as resolving unfinished
scenarios and results, addressing shortcomings in Section 0.5.1, testing more
algorithms and combinations from the related work, and moving onto other
programming languages (e.g. PHP, ASP, C#). Furthermore, plan to conceive
collaboration with vendors such as VeraCode, Coverity, and others who have
vast data sets to test the full potential of the approach with the others and
a community as a whole. Then move on to dynamic code analysis as well applying
similar techniques there.
## References
* [ESI+09] Masashi Eto, Kotaro Sonoda, Daisuke Inoue, Katsunari Yoshioka, and Koji Nakao. A proposal of malware distinction method based on scan patterns using spectrum analysis. In Proceedings of the 16th International Conference on Neural Information Processing: Part II, ICONIP’09, pages 565–572, Berlin, Heidelberg, 2009. Springer-Verlag.
* [Fre09] Free Software Foundation, Inc. wc – print newline, word, and byte counts for each file. GNU coreutils 6.10, 2009. man 1 wc.
* [HLYD09] Aiman Hanna, Hai Zhou Ling, Xiaochun Yang, and Mourad Debbabi. A synergy between static and dynamic analysis for the detection of software security vulnerabilities. In Robert Meersman, Tharam S. Dillon, and Pilar Herrero, editors, OTM Conferences (2), volume 5871 of Lecture Notes in Computer Science, pages 815–832. Springer, 2009.
* [IYE+09] Daisuke Inoue, Katsunari Yoshioka, Masashi Eto, Masaya Yamagata, Eisuke Nishino, Jun’ichi Takeuchi, Kazuya Ohkouchi, and Koji Nakao. An incident analysis system NICTER and its analysis engines based on data mining techniques. In Proceedings of the 15th International Conference on Advances in Neuro-Information Processing – Volume Part I, ICONIP’08, pages 579–586, Berlin, Heidelberg, 2009. Springer-Verlag.
* [KAYE04] Ted Kremenek, Ken Ashcraft, Junfeng Yang, and Dawson Engler. Correlation exploitation in error ranking. In Foundations of Software Engineering (FSE), 2004.
* [KE03] Ted Kremenek and Dawson Engler. Z-ranking: Using statistical analysis to counter the impact of static analysis approximations. In SAS 2003, 2003.
* [KTB+06] Ted Kremenek, Paul Twohey, Godmar Back, Andrew Ng, and Dawson Engler. From uncertainty to belief: Inferring the specification within. In Proceedings of the 7th Symposium on Operating System Design and Implementation, 2006.
* [KZL10] Ying Kong, Yuqing Zhang, and Qixu Liu. Eliminating human specification in static analysis. In Proceedings of the 13th international conference on Recent advances in intrusion detection, RAID’10, pages 494–495, Berlin, Heidelberg, 2010. Springer-Verlag.
* [MD08] Serguei A. Mokhov and Mourad Debbabi. File type analysis using signal processing techniques and machine learning vs. file unix utility for forensic analysis. In Oliver Goebel, Sandra Frings, Detlef Guenther, Jens Nedon, and Dirk Schadt, editors, Proceedings of the IT Incident Management and IT Forensics (IMF’08), LNI140, pages 73–85. GI, September 2008.
* [MES02] D. Mackenzie, P. Eggert, and R. Stallman. Comparing and merging files. [online], 2002. http://www.gnu.org/software/diffutils/manual/ps/diff.ps.gz.
* [MLB07] Serguei A. Mokhov, Marc-André Laverdière, and Djamel Benredjem. Taxonomy of linux kernel vulnerability solutions. In Innovative Techniques in Instruction Technology, E-learning, E-assessment, and Education, pages 485–493, University of Bridgeport, U.S.A., 2007. Proceedings of CISSE/SCSS’07.
* [Mok07] Serguei A. Mokhov. Introducing MARF: a modular audio recognition framework and its applications for scientific and software engineering research. In Advances in Computer and Information Sciences and Engineering, pages 473–478, University of Bridgeport, U.S.A., December 2007\. Springer Netherlands. Proceedings of CISSE/SCSS’07.
* [Mok08] Serguei A. Mokhov. Study of best algorithm combinations for speech processing tasks in machine learning using median vs. mean clusters in MARF. In Bipin C. Desai, editor, Proceedings of C3S2E’08, pages 29–43, Montreal, Quebec, Canada, May 2008. ACM.
* [Mok10a] Serguei A. Mokhov. Complete complimentary results report of the MARF’s NLP approach to the DEFT 2010 competition. [online], June 2010. http://arxiv.org/abs/1006.3787.
* [Mok10b] Serguei A. Mokhov. L’approche MARF à DEFT 2010: A MARF approach to DEFT 2010\. In Proceedings of TALN’10, July 2010. To appear in DEFT 2010 System competition at TALN 2010.
* [Mok11] Serguei A. Mokhov. MARFCAT – MARF-based Code Analysis Tool. Published electronically within the MARF project, http://sourceforge.net/projects/marf/files/Applications/MARFCAT/, 2010–2011. Last viewed February 2011.
* [MSS09] Serguei A. Mokhov, Miao Song, and Ching Y. Suen. Writer identification using inexpensive signal processing techniques. In Tarek Sobh and Khaled Elleithy, editors, Innovations in Computing Sciences and Software Engineering; Proceedings of CISSE’09, pages 437–441. Springer, December 2009. ISBN: 978-90-481-9111-6, online at: http://arxiv.org/abs/0912.5502.
* [NIS11a] NIST. National Vulnerability Database. [online], 2005–2011. http://nvd.nist.gov/.
* [NIS11b] NIST. National Vulnerability Database statistics. [online], 2005–2011. http://web.nvd.nist.gov/view/vuln/statistics.
* [NJG+10] Vinod P. Nair, Harshit Jain, Yashwant K. Golecha, Manoj Singh Gaur, and Vijay Laxmi. MEDUSA: MEtamorphic malware dynamic analysis using signature from API. In Proceedings of the 3rd International Conference on Security of Information and Networks, SIN’10, pages 263–269, New York, NY, USA, 2010\. ACM.
* [ODBN10] Vadim Okun, Aurelien Delaitre, Paul E. Black, and NIST SAMATE. Static Analysis Tool Exposition (SATE) 2010. [online], 2010. See http://samate.nist.gov/SATE.html and http://samate.nist.gov/SATE2010Workshop.html.
* [Sou10] Sourcefire. Snort: Open-source network intrusion prevention and detection system (IDS/IPS). [online], 2010. http://www.snort.org/.
* [The11] The MARF Research and Development Group. The Modular Audio Recognition Framework and its Applications. [online], 2002–2011. http://marf.sf.net and http://arxiv.org/abs/0905.1235, last viewed April 2010.
* [Tli09] Syrine Tlili. Automatic detection of safety and security vulnerabilities in open source software. PhD thesis, Concordia Institute for Information Systems Engineering, Concordia University, Montreal, Canada, 2009. ISBN: 9780494634165.
* [VM10] Various contributors and MITRE. Common Weakness Enumeration (CWE) – a community-developed dictionary of software weakness types. [online], 2010. See http://cwe.mitre.org.
## .6 Classification Result Tables
What follows are result tables with top classification results ranked from
most precise at the top. This include the configuration settings for MARF by
the means of options (the algorithm implementations are at their defaults
[Mok07]).
Table 1: CVE Stats for Wireshark 1.2.0, Quick Enriched, version SATE.4 guess | run | algorithms | good | bad | %
---|---|---|---|---|---
1st | 1 | -nopreprep -raw -fft -diff | 38 | 3 | 92.68
1st | 2 | -nopreprep -raw -fft -cheb | 38 | 3 | 92.68
1st | 3 | -nopreprep -raw -fft -eucl | 29 | 12 | 70.73
1st | 4 | -nopreprep -raw -fft -hamming | 26 | 15 | 63.41
1st | 5 | -nopreprep -raw -fft -mink | 23 | 18 | 56.10
1st | 6 | -nopreprep -raw -fft -cos | 37 | 51 | 42.05
2nd | 1 | -nopreprep -raw -fft -diff | 39 | 2 | 95.12
2nd | 2 | -nopreprep -raw -fft -cheb | 39 | 2 | 95.12
2nd | 3 | -nopreprep -raw -fft -eucl | 34 | 7 | 82.93
2nd | 4 | -nopreprep -raw -fft -hamming | 28 | 13 | 68.29
2nd | 5 | -nopreprep -raw -fft -mink | 31 | 10 | 75.61
2nd | 6 | -nopreprep -raw -fft -cos | 38 | 50 | 43.18
guess | run | class | good | bad | %
1st | 1 | CVE-2009-3829 | 6 | 0 | 100.00
1st | 2 | CVE-2009-2563 | 6 | 0 | 100.00
1st | 3 | CVE-2009-2562 | 6 | 0 | 100.00
1st | 4 | CVE-2009-4378 | 6 | 0 | 100.00
1st | 5 | CVE-2009-4376 | 6 | 0 | 100.00
1st | 6 | CVE-2010-0304 | 6 | 0 | 100.00
1st | 7 | CVE-2010-2286 | 6 | 0 | 100.00
1st | 8 | CVE-2010-2283 | 6 | 0 | 100.00
1st | 9 | CVE-2009-3551 | 6 | 0 | 100.00
1st | 10 | CVE-2009-3550 | 6 | 0 | 100.00
1st | 11 | CVE-2009-3549 | 6 | 0 | 100.00
1st | 12 | CVE-2009-3241 | 16 | 8 | 66.67
1st | 13 | CVE-2010-1455 | 34 | 20 | 62.96
1st | 14 | CVE-2009-3243 | 18 | 11 | 62.07
1st | 15 | CVE-2009-2560 | 8 | 6 | 57.14
1st | 16 | CVE-2009-2561 | 6 | 5 | 54.55
1st | 17 | CVE-2010-2285 | 6 | 5 | 54.55
1st | 18 | CVE-2009-2559 | 6 | 5 | 54.55
1st | 19 | CVE-2010-2287 | 6 | 6 | 50.00
1st | 20 | CVE-2009-4377 | 12 | 15 | 44.44
1st | 21 | CVE-2010-2284 | 6 | 9 | 40.00
1st | 22 | CVE-2009-3242 | 7 | 12 | 36.84
2nd | 1 | CVE-2009-3829 | 6 | 0 | 100.00
2nd | 2 | CVE-2009-2563 | 6 | 0 | 100.00
2nd | 3 | CVE-2009-2562 | 6 | 0 | 100.00
2nd | 4 | CVE-2009-4378 | 6 | 0 | 100.00
2nd | 5 | CVE-2009-4376 | 6 | 0 | 100.00
2nd | 6 | CVE-2010-0304 | 6 | 0 | 100.00
2nd | 7 | CVE-2010-2286 | 6 | 0 | 100.00
2nd | 8 | CVE-2010-2283 | 6 | 0 | 100.00
2nd | 9 | CVE-2009-3551 | 6 | 0 | 100.00
2nd | 10 | CVE-2009-3550 | 6 | 0 | 100.00
2nd | 11 | CVE-2009-3549 | 6 | 0 | 100.00
2nd | 12 | CVE-2009-3241 | 17 | 7 | 70.83
2nd | 13 | CVE-2010-1455 | 44 | 10 | 81.48
2nd | 14 | CVE-2009-3243 | 18 | 11 | 62.07
2nd | 15 | CVE-2009-2560 | 9 | 5 | 64.29
2nd | 16 | CVE-2009-2561 | 6 | 5 | 54.55
2nd | 17 | CVE-2010-2285 | 6 | 5 | 54.55
2nd | 18 | CVE-2009-2559 | 6 | 5 | 54.55
2nd | 19 | CVE-2010-2287 | 12 | 0 | 100.00
2nd | 20 | CVE-2009-4377 | 12 | 15 | 44.44
2nd | 21 | CVE-2010-2284 | 6 | 9 | 40.00
2nd | 22 | CVE-2009-3242 | 7 | 12 | 36.84
Table 2: CVE NLP Stats for Wireshark 1.2.0, Quick Enriched, version SATE.4 guess | run | algorithms | good | bad | %
---|---|---|---|---|---
1st | 1 | -nopreprep -char -unigram -add-delta | 30 | 6 | 83.33
2nd | 1 | -nopreprep -char -unigram -add-delta | 31 | 5 | 86.11
guess | run | class | good | bad | %
1st | 1 | CVE-2009-3829 | 1 | 0 | 100.00
1st | 2 | CVE-2009-2563 | 1 | 0 | 100.00
1st | 3 | CVE-2009-2562 | 1 | 0 | 100.00
1st | 4 | CVE-2009-4378 | 1 | 0 | 100.00
1st | 5 | CVE-2009-2561 | 1 | 0 | 100.00
1st | 6 | CVE-2009-4377 | 1 | 0 | 100.00
1st | 7 | CVE-2009-4376 | 1 | 0 | 100.00
1st | 8 | CVE-2010-2286 | 1 | 0 | 100.00
1st | 9 | CVE-2010-0304 | 1 | 0 | 100.00
1st | 10 | CVE-2010-2285 | 1 | 0 | 100.00
1st | 11 | CVE-2010-2284 | 1 | 0 | 100.00
1st | 12 | CVE-2010-2283 | 1 | 0 | 100.00
1st | 13 | CVE-2009-2559 | 1 | 0 | 100.00
1st | 14 | CVE-2009-3550 | 1 | 0 | 100.00
1st | 15 | CVE-2009-3549 | 1 | 0 | 100.00
1st | 16 | CVE-2010-1455 | 8 | 1 | 88.89
1st | 17 | CVE-2009-3243 | 3 | 1 | 75.00
1st | 18 | CVE-2009-3241 | 2 | 2 | 50.00
1st | 19 | CVE-2009-2560 | 1 | 1 | 50.00
1st | 20 | CVE-2009-3242 | 1 | 1 | 50.00
2nd | 1 | CVE-2009-3829 | 1 | 0 | 100.00
2nd | 2 | CVE-2009-2563 | 1 | 0 | 100.00
2nd | 3 | CVE-2009-2562 | 1 | 0 | 100.00
2nd | 4 | CVE-2009-4378 | 1 | 0 | 100.00
2nd | 5 | CVE-2009-2561 | 1 | 0 | 100.00
2nd | 6 | CVE-2009-4377 | 1 | 0 | 100.00
2nd | 7 | CVE-2009-4376 | 1 | 0 | 100.00
2nd | 8 | CVE-2010-2286 | 1 | 0 | 100.00
2nd | 9 | CVE-2010-0304 | 1 | 0 | 100.00
2nd | 10 | CVE-2010-2285 | 1 | 0 | 100.00
2nd | 11 | CVE-2010-2284 | 1 | 0 | 100.00
2nd | 12 | CVE-2010-2283 | 1 | 0 | 100.00
2nd | 13 | CVE-2009-2559 | 1 | 0 | 100.00
2nd | 14 | CVE-2009-3550 | 1 | 0 | 100.00
2nd | 15 | CVE-2009-3549 | 1 | 0 | 100.00
2nd | 16 | CVE-2010-1455 | 8 | 1 | 88.89
2nd | 17 | CVE-2009-3243 | 3 | 1 | 75.00
2nd | 18 | CVE-2009-3241 | 3 | 1 | 75.00
2nd | 19 | CVE-2009-2560 | 1 | 1 | 50.00
2nd | 20 | CVE-2009-3242 | 1 | 1 | 50.00
Table 3: CVE NLP Stats for Wireshark 1.2.0, Quick Enriched, version SATE.4 guess | run | algorithms | good | bad | %
---|---|---|---|---|---
1st | 1 | -cweid -nopreprep -raw -fft -cheb | 31 | 5 | 86.11
1st | 2 | -cweid -nopreprep -raw -fft -diff | 31 | 5 | 86.11
1st | 3 | -cweid -nopreprep -raw -fft -eucl | 29 | 7 | 80.56
1st | 4 | -cweid -nopreprep -raw -fft -hamming | 22 | 14 | 61.11
1st | 5 | -cweid -nopreprep -raw -fft -cos | 33 | 25 | 56.90
1st | 6 | -cweid -nopreprep -raw -fft -mink | 20 | 16 | 55.56
2nd | 1 | -cweid -nopreprep -raw -fft -cheb | 33 | 3 | 91.67
2nd | 2 | -cweid -nopreprep -raw -fft -diff | 33 | 3 | 91.67
2nd | 3 | -cweid -nopreprep -raw -fft -eucl | 33 | 3 | 91.67
2nd | 4 | -cweid -nopreprep -raw -fft -hamming | 27 | 9 | 75.00
2nd | 5 | -cweid -nopreprep -raw -fft -cos | 41 | 17 | 70.69
2nd | 6 | -cweid -nopreprep -raw -fft -mink | 22 | 14 | 61.11
guess | run | class | good | bad | %
1st | 1 | CWE-399 | 6 | 0 | 100.00
1st | 2 | NVD-CWE-Other | 17 | 3 | 85.00
1st | 3 | CWE-20 | 50 | 10 | 83.33
1st | 4 | CWE-189 | 8 | 2 | 80.00
1st | 5 | NVD-CWE-noinfo | 72 | 40 | 64.29
1st | 6 | CWE-119 | 13 | 17 | 43.33
2nd | 1 | CWE-399 | 6 | 0 | 100.00
2nd | 2 | NVD-CWE-Other | 17 | 3 | 85.00
2nd | 3 | CWE-20 | 52 | 8 | 86.67
2nd | 4 | CWE-189 | 8 | 2 | 80.00
2nd | 5 | NVD-CWE-noinfo | 83 | 29 | 74.11
2nd | 6 | CWE-119 | 23 | 7 | 76.67
Table 4: CVE Stats for Chrome 5.0.375.54, Quick Enriched, (clean CVEs) version SATE.4 guess | run | algorithms | good | bad | %
---|---|---|---|---|---
1st | 1 | -nopreprep -raw -fft -eucl | 10 | 1 | 90.91
1st | 2 | -nopreprep -raw -fft -cos | 10 | 1 | 90.91
1st | 3 | -nopreprep -raw -fft -diff | 10 | 1 | 90.91
1st | 4 | -nopreprep -raw -fft -cheb | 10 | 1 | 90.91
1st | 5 | -nopreprep -raw -fft -mink | 9 | 2 | 81.82
1st | 6 | -nopreprep -raw -fft -hamming | 9 | 2 | 81.82
2nd | 1 | -nopreprep -raw -fft -eucl | 11 | 0 | 100.00
2nd | 2 | -nopreprep -raw -fft -cos | 11 | 0 | 100.00
2nd | 3 | -nopreprep -raw -fft -diff | 11 | 0 | 100.00
2nd | 4 | -nopreprep -raw -fft -cheb | 11 | 0 | 100.00
2nd | 5 | -nopreprep -raw -fft -mink | 10 | 1 | 90.91
2nd | 6 | -nopreprep -raw -fft -hamming | 10 | 1 | 90.91
guess | run | class | good | bad | %
1st | 1 | CVE-2010-2301 | 6 | 0 | 100.00
1st | 2 | CVE-2010-2300 | 6 | 0 | 100.00
1st | 3 | CVE-2010-2299 | 6 | 0 | 100.00
1st | 4 | CVE-2010-2298 | 6 | 0 | 100.00
1st | 5 | CVE-2010-2297 | 6 | 0 | 100.00
1st | 6 | CVE-2010-2304 | 6 | 0 | 100.00
1st | 7 | CVE-2010-2303 | 6 | 0 | 100.00
1st | 8 | CVE-2010-2295 | 10 | 2 | 83.33
1st | 9 | CVE-2010-2302 | 6 | 6 | 50.00
2nd | 1 | CVE-2010-2301 | 6 | 0 | 100.00
2nd | 2 | CVE-2010-2300 | 6 | 0 | 100.00
2nd | 3 | CVE-2010-2299 | 6 | 0 | 100.00
2nd | 4 | CVE-2010-2298 | 6 | 0 | 100.00
2nd | 5 | CVE-2010-2297 | 6 | 0 | 100.00
2nd | 6 | CVE-2010-2304 | 6 | 0 | 100.00
2nd | 7 | CVE-2010-2303 | 6 | 0 | 100.00
2nd | 8 | CVE-2010-2295 | 10 | 2 | 83.33
2nd | 9 | CVE-2010-2302 | 12 | 0 | 100.00
Table 5: CWE Stats for Chrome 5.0.375.54, (clean CVEs) version SATE.5 guess | run | algorithms | good | bad | %
---|---|---|---|---|---
1st | 1 | -cweid -nopreprep -raw -fft -cheb | 9 | 0 | 100.00
1st | 2 | -cweid -nopreprep -raw -fft -cos | 9 | 0 | 100.00
1st | 3 | -cweid -nopreprep -raw -fft -diff | 9 | 0 | 100.00
1st | 4 | -cweid -nopreprep -raw -fft -eucl | 8 | 1 | 88.89
1st | 5 | -cweid -nopreprep -raw -fft -hamming | 8 | 1 | 88.89
1st | 6 | -cweid -nopreprep -raw -fft -mink | 6 | 3 | 66.67
2nd | 1 | -cweid -nopreprep -raw -fft -cheb | 9 | 0 | 100.00
2nd | 2 | -cweid -nopreprep -raw -fft -cos | 9 | 0 | 100.00
2nd | 3 | -cweid -nopreprep -raw -fft -diff | 9 | 0 | 100.00
2nd | 4 | -cweid -nopreprep -raw -fft -eucl | 8 | 1 | 88.89
2nd | 5 | -cweid -nopreprep -raw -fft -hamming | 8 | 1 | 88.89
2nd | 6 | -cweid -nopreprep -raw -fft -mink | 8 | 1 | 88.89
guess | run | class | good | bad | %
1st | 1 | CWE-79 | 6 | 0 | 100.00
1st | 2 | NVD-CWE-noinfo | 6 | 0 | 100.00
1st | 3 | CWE-399 | 6 | 0 | 100.00
1st | 4 | CWE-119 | 6 | 0 | 100.00
1st | 5 | CWE-20 | 6 | 0 | 100.00
1st | 6 | NVD-CWE-Other | 10 | 2 | 83.33
1st | 7 | CWE-94 | 9 | 3 | 75.00
2nd | 1 | CWE-79 | 6 | 0 | 100.00
2nd | 2 | NVD-CWE-noinfo | 6 | 0 | 100.00
2nd | 3 | CWE-399 | 6 | 0 | 100.00
2nd | 4 | CWE-119 | 6 | 0 | 100.00
2nd | 5 | CWE-20 | 6 | 0 | 100.00
2nd | 6 | NVD-CWE-Other | 11 | 1 | 91.67
2nd | 7 | CWE-94 | 10 | 2 | 83.33
Table 6: CVE Stats for Tomcat 5.5.13, version SATE.5 1st | 1 | -nopreprep -raw -fft -diff | 36 | 7 | 83.72
---|---|---|---|---|---
1st | 2 | -nopreprep -raw -fft -cheb | 36 | 7 | 83.72
1st | 3 | -nopreprep -raw -fft -cos | 37 | 9 | 80.43
1st | 4 | -nopreprep -raw -fft -eucl | 34 | 9 | 79.07
1st | 5 | -nopreprep -raw -fft -mink | 28 | 15 | 65.12
1st | 6 | -nopreprep -raw -fft -hamming | 26 | 17 | 60.47
2nd | 1 | -nopreprep -raw -fft -diff | 40 | 3 | 93.02
2nd | 2 | -nopreprep -raw -fft -cheb | 40 | 3 | 93.02
2nd | 3 | -nopreprep -raw -fft -cos | 40 | 6 | 86.96
2nd | 4 | -nopreprep -raw -fft -eucl | 36 | 7 | 83.72
2nd | 5 | -nopreprep -raw -fft -mink | 31 | 12 | 72.09
2nd | 6 | -nopreprep -raw -fft -hamming | 29 | 14 | 67.44
guess | run | algorithms | good | bad | %
1st | 1 | CVE-2006-7197 | 6 | 0 | 100.00
1st | 2 | CVE-2006-7196 | 6 | 0 | 100.00
1st | 3 | CVE-2006-7195 | 6 | 0 | 100.00
1st | 4 | CVE-2009-0033 | 6 | 0 | 100.00
1st | 5 | CVE-2007-3386 | 6 | 0 | 100.00
1st | 6 | CVE-2009-2901 | 3 | 0 | 100.00
1st | 7 | CVE-2007-3385 | 6 | 0 | 100.00
1st | 8 | CVE-2008-2938 | 6 | 0 | 100.00
1st | 9 | CVE-2007-3382 | 6 | 0 | 100.00
1st | 10 | CVE-2007-5461 | 6 | 0 | 100.00
1st | 11 | CVE-2007-6286 | 6 | 0 | 100.00
1st | 12 | CVE-2007-1858 | 6 | 0 | 100.00
1st | 13 | CVE-2008-0128 | 6 | 0 | 100.00
1st | 14 | CVE-2007-2450 | 6 | 0 | 100.00
1st | 15 | CVE-2009-3548 | 6 | 0 | 100.00
1st | 16 | CVE-2009-0580 | 6 | 0 | 100.00
1st | 17 | CVE-2007-1355 | 6 | 0 | 100.00
1st | 18 | CVE-2008-2370 | 6 | 0 | 100.00
1st | 19 | CVE-2008-4308 | 6 | 0 | 100.00
1st | 20 | CVE-2007-5342 | 6 | 0 | 100.00
1st | 21 | CVE-2008-5515 | 19 | 5 | 79.17
1st | 22 | CVE-2009-0783 | 11 | 4 | 73.33
1st | 23 | CVE-2008-1232 | 13 | 5 | 72.22
1st | 24 | CVE-2008-5519 | 6 | 6 | 50.00
1st | 25 | CVE-2007-5333 | 6 | 6 | 50.00
1st | 26 | CVE-2008-1947 | 6 | 6 | 50.00
1st | 27 | CVE-2009-0781 | 6 | 6 | 50.00
1st | 28 | CVE-2007-0450 | 5 | 7 | 41.67
1st | 29 | CVE-2007-2449 | 6 | 12 | 33.33
1st | 30 | CVE-2009-2693 | 2 | 6 | 25.00
1st | 31 | CVE-2009-2902 | 0 | 1 | 0.00
2nd | 1 | CVE-2006-7197 | 6 | 0 | 100.00
2nd | 2 | CVE-2006-7196 | 6 | 0 | 100.00
2nd | 3 | CVE-2006-7195 | 6 | 0 | 100.00
2nd | 4 | CVE-2009-0033 | 6 | 0 | 100.00
2nd | 5 | CVE-2007-3386 | 6 | 0 | 100.00
2nd | 6 | CVE-2009-2901 | 3 | 0 | 100.00
2nd | 7 | CVE-2007-3385 | 6 | 0 | 100.00
2nd | 8 | CVE-2008-2938 | 6 | 0 | 100.00
2nd | 9 | CVE-2007-3382 | 6 | 0 | 100.00
2nd | 10 | CVE-2007-5461 | 6 | 0 | 100.00
2nd | 11 | CVE-2007-6286 | 6 | 0 | 100.00
2nd | 12 | CVE-2007-1858 | 6 | 0 | 100.00
2nd | 13 | CVE-2008-0128 | 6 | 0 | 100.00
2nd | 14 | CVE-2007-2450 | 6 | 0 | 100.00
2nd | 15 | CVE-2009-3548 | 6 | 0 | 100.00
2nd | 16 | CVE-2009-0580 | 6 | 0 | 100.00
2nd | 17 | CVE-2007-1355 | 6 | 0 | 100.00
2nd | 18 | CVE-2008-2370 | 6 | 0 | 100.00
2nd | 19 | CVE-2008-4308 | 6 | 0 | 100.00
2nd | 20 | CVE-2007-5342 | 6 | 0 | 100.00
2nd | 21 | CVE-2008-5515 | 19 | 5 | 79.17
2nd | 22 | CVE-2009-0783 | 12 | 3 | 80.00
2nd | 23 | CVE-2008-1232 | 13 | 5 | 72.22
2nd | 24 | CVE-2008-5519 | 12 | 0 | 100.00
2nd | 25 | CVE-2007-5333 | 6 | 6 | 50.00
2nd | 26 | CVE-2008-1947 | 6 | 6 | 50.00
2nd | 27 | CVE-2009-0781 | 12 | 0 | 100.00
2nd | 28 | CVE-2007-0450 | 7 | 5 | 58.33
2nd | 29 | CVE-2007-2449 | 8 | 10 | 44.44
2nd | 30 | CVE-2009-2693 | 4 | 4 | 50.00
2nd | 31 | CVE-2009-2902 | 0 | 1 | 0.00
Table 7: CWE Stats for Tomcat 5.5.13, version SATE.5 guess | run | algorithms | good | bad | %
---|---|---|---|---|---
1st | 1 | -cweid -nopreprep -raw -fft -cheb | 27 | 6 | 81.82
1st | 2 | -cweid -nopreprep -raw -fft -diff | 27 | 6 | 81.82
1st | 3 | -cweid -nopreprep -raw -fft -cos | 24 | 9 | 72.73
1st | 4 | -cweid -nopreprep -raw -fft -eucl | 13 | 20 | 39.39
1st | 5 | -cweid -nopreprep -raw -fft -hamming | 12 | 21 | 36.36
1st | 6 | -cweid -nopreprep -raw -fft -mink | 9 | 24 | 27.27
2nd | 1 | -cweid -nopreprep -raw -fft -cheb | 32 | 1 | 96.97
2nd | 2 | -cweid -nopreprep -raw -fft -diff | 32 | 1 | 96.97
2nd | 3 | -cweid -nopreprep -raw -fft -cos | 29 | 4 | 87.88
2nd | 4 | -cweid -nopreprep -raw -fft -eucl | 17 | 16 | 51.52
2nd | 5 | -cweid -nopreprep -raw -fft -hamming | 18 | 15 | 54.55
2nd | 6 | -cweid -nopreprep -raw -fft -mink | 13 | 20 | 39.39
guess | run | class | good | bad | %
1st | 1 | CWE-264 | 7 | 0 | 100.00
1st | 2 | CWE-255 | 6 | 0 | 100.00
1st | 3 | CWE-16 | 6 | 0 | 100.00
1st | 4 | CWE-119 | 6 | 0 | 100.00
1st | 5 | CWE-20 | 6 | 0 | 100.00
1st | 6 | CWE-200 | 22 | 4 | 84.62
1st | 7 | CWE-79 | 24 | 21 | 53.33
1st | 8 | CWE-22 | 35 | 61 | 36.46
2nd | 1 | CWE-264 | 7 | 0 | 100.00
2nd | 2 | CWE-255 | 6 | 0 | 100.00
2nd | 3 | CWE-16 | 6 | 0 | 100.00
2nd | 4 | CWE-119 | 6 | 0 | 100.00
2nd | 5 | CWE-20 | 6 | 0 | 100.00
2nd | 6 | CWE-200 | 23 | 3 | 88.46
2nd | 7 | CWE-79 | 30 | 15 | 66.67
2nd | 8 | CWE-22 | 57 | 39 | 59.38
Table 8: CVE NLP Stats for Tomcat 5.5.13, version SATE.5 guess | run | algorithms | good | bad | %
---|---|---|---|---|---
1st | 1 | -nopreprep -char -unigram -add-delta | 29 | 4 | 87.88
2nd | 1 | -nopreprep -char -unigram -add-delta | 29 | 4 | 87.88
guess | run | class | good | bad | %
1st | 1 | CVE-2006-7197 | 1 | 0 | 100.00
1st | 2 | CVE-2006-7196 | 1 | 0 | 100.00
1st | 3 | CVE-2009-2901 | 1 | 0 | 100.00
1st | 4 | CVE-2006-7195 | 1 | 0 | 100.00
1st | 5 | CVE-2009-0033 | 1 | 0 | 100.00
1st | 6 | CVE-2007-1355 | 1 | 0 | 100.00
1st | 7 | CVE-2007-5342 | 1 | 0 | 100.00
1st | 8 | CVE-2009-2693 | 1 | 0 | 100.00
1st | 9 | CVE-2009-0783 | 1 | 0 | 100.00
1st | 10 | CVE-2008-2370 | 1 | 0 | 100.00
1st | 11 | CVE-2007-2450 | 1 | 0 | 100.00
1st | 12 | CVE-2008-2938 | 1 | 0 | 100.00
1st | 13 | CVE-2007-2449 | 3 | 0 | 100.00
1st | 14 | CVE-2007-1858 | 1 | 0 | 100.00
1st | 15 | CVE-2008-4308 | 1 | 0 | 100.00
1st | 16 | CVE-2008-0128 | 1 | 0 | 100.00
1st | 17 | CVE-2009-3548 | 1 | 0 | 100.00
1st | 18 | CVE-2007-5461 | 1 | 0 | 100.00
1st | 19 | CVE-2007-3382 | 1 | 0 | 100.00
1st | 20 | CVE-2007-0450 | 2 | 0 | 100.00
1st | 21 | CVE-2009-0580 | 1 | 0 | 100.00
1st | 22 | CVE-2007-6286 | 1 | 0 | 100.00
1st | 23 | CVE-2008-5515 | 3 | 1 | 75.00
1st | 24 | CVE-2008-1232 | 1 | 2 | 33.33
1st | 25 | CVE-2009-2902 | 0 | 1 | 0.00
2nd | 1 | CVE-2006-7197 | 1 | 0 | 100.00
2nd | 2 | CVE-2006-7196 | 1 | 0 | 100.00
2nd | 3 | CVE-2009-2901 | 1 | 0 | 100.00
2nd | 4 | CVE-2006-7195 | 1 | 0 | 100.00
2nd | 5 | CVE-2009-0033 | 1 | 0 | 100.00
2nd | 6 | CVE-2007-1355 | 1 | 0 | 100.00
2nd | 7 | CVE-2007-5342 | 1 | 0 | 100.00
2nd | 8 | CVE-2009-2693 | 1 | 0 | 100.00
2nd | 9 | CVE-2009-0783 | 1 | 0 | 100.00
2nd | 10 | CVE-2008-2370 | 1 | 0 | 100.00
2nd | 11 | CVE-2007-2450 | 1 | 0 | 100.00
2nd | 12 | CVE-2008-2938 | 1 | 0 | 100.00
2nd | 13 | CVE-2007-2449 | 3 | 0 | 100.00
2nd | 14 | CVE-2007-1858 | 1 | 0 | 100.00
2nd | 15 | CVE-2008-4308 | 1 | 0 | 100.00
2nd | 16 | CVE-2008-0128 | 1 | 0 | 100.00
2nd | 17 | CVE-2009-3548 | 1 | 0 | 100.00
2nd | 18 | CVE-2007-5461 | 1 | 0 | 100.00
2nd | 19 | CVE-2007-3382 | 1 | 0 | 100.00
2nd | 20 | CVE-2007-0450 | 2 | 0 | 100.00
2nd | 21 | CVE-2009-0580 | 1 | 0 | 100.00
2nd | 22 | CVE-2007-6286 | 1 | 0 | 100.00
2nd | 23 | CVE-2008-5515 | 3 | 1 | 75.00
2nd | 24 | CVE-2008-1232 | 1 | 2 | 33.33
2nd | 25 | CVE-2009-2902 | 0 | 1 | 0.00
Table 9: CWE NLP Stats for Tomcat 5.5.13, version SATE.5 guess | run | algorithms | good | bad | %
---|---|---|---|---|---
1st | 1 | -cweid -nopreprep -char -unigram -add-delta | 13 | 20 | 39.39
2nd | 1 | -cweid -nopreprep -char -unigram -add-delta | 17 | 16 | 51.52
guess | run | class | good | bad | %
1st | 1 | CWE-16 | 1 | 0 | 100.00
1st | 2 | CWE-255 | 1 | 0 | 100.00
1st | 3 | CWE-264 | 2 | 0 | 100.00
1st | 4 | CWE-119 | 1 | 0 | 100.00
1st | 5 | CWE-20 | 1 | 0 | 100.00
1st | 6 | CWE-200 | 3 | 1 | 75.00
1st | 7 | CWE-22 | 3 | 13 | 18.75
1st | 8 | CWE-79 | 1 | 6 | 14.29
2nd | 1 | CWE-16 | 1 | 0 | 100.00
2nd | 2 | CWE-255 | 1 | 0 | 100.00
2nd | 3 | CWE-264 | 2 | 0 | 100.00
2nd | 4 | CWE-119 | 1 | 0 | 100.00
2nd | 5 | CWE-20 | 1 | 0 | 100.00
2nd | 6 | CWE-200 | 4 | 0 | 100.00
2nd | 7 | CWE-22 | 5 | 11 | 31.25
2nd | 8 | CWE-79 | 2 | 5 | 28.57
Table 10: CVE NLP Stats for Chrome 5.0.375.54, version SATE.7 guess | run | algorithms | good | bad | %
---|---|---|---|---|---
1st | 1 | -nopreprep -char -unigram -add-delta | 9 | 0 | 100.00
2nd | 1 | -nopreprep -char -unigram -add-delta | 9 | 0 | 100.00
guess | run | class | good | bad | %
1st | 1 | CVE-2010-2304 | 1 | 0 | 100.00
1st | 2 | CVE-2010-2298 | 1 | 0 | 100.00
1st | 3 | CVE-2010-2301 | 1 | 0 | 100.00
1st | 4 | CVE-2010-2295 | 2 | 0 | 100.00
1st | 5 | CVE-2010-2300 | 1 | 0 | 100.00
1st | 6 | CVE-2010-2303 | 1 | 0 | 100.00
1st | 7 | CVE-2010-2297 | 1 | 0 | 100.00
1st | 8 | CVE-2010-2299 | 1 | 0 | 100.00
2nd | 1 | CVE-2010-2304 | 1 | 0 | 100.00
2nd | 2 | CVE-2010-2298 | 1 | 0 | 100.00
2nd | 3 | CVE-2010-2301 | 1 | 0 | 100.00
2nd | 4 | CVE-2010-2295 | 2 | 0 | 100.00
2nd | 5 | CVE-2010-2300 | 1 | 0 | 100.00
2nd | 6 | CVE-2010-2303 | 1 | 0 | 100.00
2nd | 7 | CVE-2010-2297 | 1 | 0 | 100.00
2nd | 8 | CVE-2010-2299 | 1 | 0 | 100.00
Table 11: CWE NLP Stats for Chrome 5.0.375.54, version SATE.7 guess | run | algorithms | good | bad | %
---|---|---|---|---|---
1st | 1 | -cweid -nopreprep -char -unigram -add-delta | 8 | 1 | 88.89
2nd | 1 | -cweid -nopreprep -char -unigram -add-delta | 8 | 1 | 88.89
guess | run | class | good | bad | %
1st | 1 | CWE-399 | 1 | 0 | 100.00
1st | 2 | NVD-CWE-noinfo | 1 | 0 | 100.00
1st | 3 | CWE-79 | 1 | 0 | 100.00
1st | 4 | NVD-CWE-Other | 2 | 0 | 100.00
1st | 5 | CWE-119 | 1 | 0 | 100.00
1st | 6 | CWE-20 | 1 | 0 | 100.00
1st | 7 | CWE-94 | 1 | 1 | 50.00
2nd | 1 | CWE-399 | 1 | 0 | 100.00
2nd | 2 | NVD-CWE-noinfo | 1 | 0 | 100.00
2nd | 3 | CWE-79 | 1 | 0 | 100.00
2nd | 4 | NVD-CWE-Other | 2 | 0 | 100.00
2nd | 5 | CWE-119 | 1 | 0 | 100.00
2nd | 6 | CWE-20 | 1 | 0 | 100.00
2nd | 7 | CWE-94 | 1 | 1 | 50.00
## Index
* API
* DEFT2010App 4th item
* marf.util.Matrix §0.3.5
* WriterIdentApp 4th item
* C 1st item, 1st item, §0.1, §0.4.3, 2nd item
* C++ 2nd item, §0.1, §0.4.3, 2nd item
* Chrome
* 5.0.375.54 Table 10, Table 11, Table 4, Table 5, 2nd item, §0.4.2, §0.4.2, §0.4.3, §0.4.3, §0.4.5
* 5.0.375.70 2nd item, §0.4.2, §0.4.2
* CVE
* CVE-2006-7195 Table 6, Table 6, Table 8, Table 8
* CVE-2006-7196 Table 6, Table 6, Table 8, Table 8
* CVE-2006-7197 Table 6, Table 6, Table 8, Table 8
* CVE-2007-0450 Table 6, Table 6, Table 8, Table 8
* CVE-2007-1355 Table 6, Table 6, Table 8, Table 8
* CVE-2007-1858 Table 6, Table 6, Table 8, Table 8
* CVE-2007-2449 Table 6, Table 6, Table 8, Table 8
* CVE-2007-2450 Table 6, Table 6, Table 8, Table 8
* CVE-2007-3382 Table 6, Table 6, Table 8, Table 8
* CVE-2007-3385 Table 6, Table 6
* CVE-2007-3386 Table 6, Table 6
* CVE-2007-5333 Table 6, Table 6
* CVE-2007-5342 Table 6, Table 6, Table 8, Table 8
* CVE-2007-5461 Table 6, Table 6, Table 8, Table 8
* CVE-2007-6286 Table 6, Table 6, Table 8, Table 8
* CVE-2008-0128 Table 6, Table 6, Table 8, Table 8
* CVE-2008-1232 Table 6, Table 6, Table 8, Table 8
* CVE-2008-1947 Table 6, Table 6
* CVE-2008-2370 Table 6, Table 6, Table 8, Table 8
* CVE-2008-2938 Table 6, Table 6, Table 8, Table 8
* CVE-2008-4308 Table 6, Table 6, Table 8, Table 8
* CVE-2008-5515 Table 6, Table 6, Table 8, Table 8
* CVE-2008-5519 Table 6, Table 6
* CVE-2009-0033 Table 6, Table 6, Table 8, Table 8
* CVE-2009-0580 Table 6, Table 6, Table 8, Table 8
* CVE-2009-0781 Table 6, Table 6
* CVE-2009-0783 Table 6, Table 6, Table 8, Table 8
* CVE-2009-2559 Table 1, Table 1, Table 2, Table 2
* CVE-2009-2560 Table 1, Table 1, Table 2, Table 2
* CVE-2009-2561 Table 1, Table 1, Table 2, Table 2
* CVE-2009-2562 Table 1, Table 1, Table 2, Table 2
* CVE-2009-2563 Table 1, Table 1, Table 2, Table 2
* CVE-2009-2693 Table 6, Table 6, Table 8, Table 8
* CVE-2009-2901 Table 6, Table 6, Table 8, Table 8
* CVE-2009-2902 Table 6, Table 6, Table 8, Table 8
* CVE-2009-3241 Table 1, Table 1, Table 2, Table 2
* CVE-2009-3242 Table 1, Table 1, Table 2, Table 2
* CVE-2009-3243 Table 1, Table 1, Table 2, Table 2
* CVE-2009-3548 Table 6, Table 6, Table 8, Table 8
* CVE-2009-3549 Table 1, Table 1, Table 2, Table 2
* CVE-2009-3550 Table 1, Table 1, Table 2, Table 2
* CVE-2009-3551 Table 1, Table 1
* CVE-2009-3829 Table 1, Table 1, Table 2, Table 2
* CVE-2009-4376 Table 1, Table 1, Table 2, Table 2
* CVE-2009-4377 Table 1, Table 1, Table 2, Table 2
* CVE-2009-4378 Table 1, Table 1, Table 2, Table 2
* CVE-2010-0304 Table 1, Table 1, Table 2, Table 2
* CVE-2010-1455 Table 1, Table 1, Table 2, Table 2
* CVE-2010-2283 Table 1, Table 1, Table 2, Table 2
* CVE-2010-2284 Table 1, Table 1, Table 2, Table 2
* CVE-2010-2285 Table 1, Table 1, Table 2, Table 2
* CVE-2010-2286 Table 1, Table 1, Table 2, Table 2
* CVE-2010-2287 Table 1, Table 1
* CVE-2010-2295 Table 10, Table 10, Table 4, Table 4
* CVE-2010-2297 Table 10, Table 10, Table 4, Table 4
* CVE-2010-2298 Table 10, Table 10, Table 4, Table 4
* CVE-2010-2299 Table 10, Table 10, Table 4, Table 4
* CVE-2010-2300 Table 10, Table 10, Table 4, Table 4
* CVE-2010-2301 Table 10, Table 10, Table 4, Table 4
* CVE-2010-2302 Table 4, Table 4
* CVE-2010-2303 Table 10, Table 10, Table 4, Table 4
* CVE-2010-2304 Table 10, Table 10, Table 4, Table 4
* CWE
* CWE-119 Table 11, Table 11, Table 3, Table 3, Table 5, Table 5, Table 7, Table 7, Table 9, Table 9
* CWE-16 Table 7, Table 7, Table 9, Table 9
* CWE-189 Table 3, Table 3
* CWE-20 Table 11, Table 11, Table 3, Table 3, Table 5, Table 5, Table 7, Table 7, Table 9, Table 9
* CWE-200 Table 7, Table 7, Table 9, Table 9
* CWE-22 Table 7, Table 7, Table 9, Table 9
* CWE-255 Table 7, Table 7, Table 9, Table 9
* CWE-264 Table 7, Table 7, Table 9, Table 9
* CWE-399 Table 11, Table 11, Table 3, Table 3, Table 5, Table 5
* CWE-79 Table 11, Table 11, Table 5, Table 5, Table 7, Table 7, Table 9, Table 9
* CWE-94 Table 11, Table 11, Table 5, Table 5
* NVD-CWE-noinfo Table 11, Table 11, Table 3, Table 3, Table 5, Table 5
* NVD-CWE-Other Table 11, Table 11, Table 3, Table 3, Table 5, Table 5
* Dovecot 1st item, §0.4.4, §0.4.4
* Files
* report-cweidnoprepreprawfftcheb-apache-tomcat-5.5.13-train-cwe.xml §0.4.3
* report-cweidnoprepreprawfftcos-apache-tomcat-5.5.13-train-cwe.xml §0.4.3
* report-cweidnoprepreprawfftcos-apache-tomcat-5.5.29-test-cwe.xml §0.4.4
* report-cweidnoprepreprawfftcos-dovecot-2.0.beta6-wireshark-test-cwe.xml §0.4.4
* report-cweidnoprepreprawfftdiff-apache-tomcat-5.5.13-train-cwe.xml §0.4.3
* report-cweidnoprepreprawffteucl-apache-tomcat-5.5.13-train-cwe.xml §0.4.3
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* sate_2010.pathcheck.xsd §0.4.2, §0.4.2
* sate_2010.xsd §0.4.2, §0.4.2
* Frameworks
* MARF 6.§, §0.1, 2nd item, item 2, §0.3.5, §0.4.2, §0.4.3, §0.5, The use of machine learning with signal- and NLP processing of source code to fingerprint, detect, and classify vulnerabilities and weaknesses with MARFCAT
* Java 3rd item, 2nd item, §0.1, §0.4.3, 2nd item
* Libraries
* MARF 6.§, §0.1, 2nd item, item 2, §0.3.5, §0.4.2, §0.4.3, §0.5, The use of machine learning with signal- and NLP processing of source code to fingerprint, detect, and classify vulnerabilities and weaknesses with MARFCAT
* MARF 6.§, §0.1, 2nd item, item 2, §0.3.5, §0.4.2, §0.4.3, §0.5, The use of machine learning with signal- and NLP processing of source code to fingerprint, detect, and classify vulnerabilities and weaknesses with MARFCAT
* Applications
* MARFCAT §0.1, §0.1, §0.3.1, §0.3.4, §0.3.5, §0.4, §0.4, §0.4.3, §0.4.3, §0.4.3, §0.4.3, §0.5, §0.5.4, The use of machine learning with signal- and NLP processing of source code to fingerprint, detect, and classify vulnerabilities and weaknesses with MARFCAT
* MARFCAT §0.1, §0.1, §0.3.1, §0.3.4, §0.3.5, §0.4, §0.4, §0.4.3, §0.4.3, §0.4.3, §0.4.3, §0.5, §0.5.4, The use of machine learning with signal- and NLP processing of source code to fingerprint, detect, and classify vulnerabilities and weaknesses with MARFCAT
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* Tomcat
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* 5.5.29 3rd item, item 5, 1st item, 2nd item, 3rd item, 4th item, §0.4.4, §0.4.4
* Tools
* diff §0.3.5
* wc 3rd item
* Wireshark
* 1.2.0 Table 1, Table 2, Table 3, 1st item, §0.4.2, §0.4.2, §0.4.2, §0.4.4
* 1.2.9 1st item, §0.4.2, §0.4.2
|
arxiv-papers
| 2010-10-12T20:37:06 |
2024-09-04T02:49:13.774517
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Serguei A. Mokhov",
"submitter": "Serguei Mokhov",
"url": "https://arxiv.org/abs/1010.2511"
}
|
1010.2642
|
11institutetext: CERN, Geneva, Switzerland
# Particle cosmology
A. Riotto
###### Abstract
In these lectures the present status of the so-called standard cosmological
model, based on the hot Big Bang theory and the inflationary paradigm is
reviewed. Special emphasis is given to the origin of the cosmological
perturbations we see today under the form of the cosmic microwave background
anisotropies and the large scale structure and to the dark matter and dark
energy puzzles.
## 0.1 Introduction
The evolution of the universe is determined to a large extent by the same
microphysics laws of physics that govern high-energy physics phenomena. Hence,
any progress in particle physics has a large impact on the cosmological
model(s) and, conversely, any new step taken towards the understanding of the
past, present and future of our universe might provide a hint of high-energy
physics beyond the one we currently know. This is the reason why these
lectures are entitled Particle Cosmology. If the reader takes only one lesson
home from them it is that particle physics and cosmology are nowadays
intimately connected.
There are fundamental questions we are on the edge of answering: what is the
origin of our universe? Why is the universe so homogeneous and isotropic on
large scales? What are the origins of dark matter and dark energy? What is the
fate of our universe? While these lectures will certainly not be able to give
definite answers to them, we shall try to provide the students with some tools
they might find useful in order to solve these overwhelming mysteries
themselves.
These lectures will contain a short review of the standard Big Bang model; a
rather long discussion of the inflation paradigm with particular emphasis on
the possibility that the cosmological seeds originated from a period of
primordial acceleration; the physics of the Cosmic Microwave Background (CMB)
anisotropies, and a discussion of the dark matter and dark energy puzzles.
Since these lectures were delivered at a school, we shall not provide an
exhaustive list of references to original material, but refer to several basic
cosmology books and reviews where students can find the references to the
original material [1, 2, 3, 4, 5, 6, 7, 8].
## 0.2 Basics of the Big Bang model
We know two basic facts about our local universe (the universe we may
observe). First, it is homogeneous and isotropic on sufficiently large
cosmological scales [2]. Once this experimental evidence is accepted, one can
promote it to a principle, dubbed “the cosmological principle”. Secondly, it
expands. The next question would then be: how can we describe such a universe?
The standard cosmology is based upon the maximally spatially symmetric
Friedmann–Robertson–Walker (FRW) line element
$ds^{2}=-dt^{2}+a(t)^{2}\left[{dr^{2}\over
1-kr^{2}}+r^{2}(d\theta^{2}+\sin^{2}\theta\,d\phi^{2})\right]\,;$ (1)
where $a(t)$ is the cosmic-scale factor, $R_{\rm curv}\equiv a(t)|k|^{-1/2}$
is the curvature radius, and $k=-1,0,1$ is the curvature signature. All three
models are without boundary: the positively curved model is finite and curves
back on itself; the negatively curved and flat models are infinite in extent.
The Robertson–Walker metric embodies the observed isotropy and homogeneity of
the universe. It is interesting to note that this form of the line element was
originally introduced for the sake of mathematical simplicity; we now know
that it is well justified at early times or today on large scales ($\gg
10\,{\rm Mpc}$), at least within our visible patch.
The coordinates, $r$, $\theta$, and $\phi$, are referred to as co-moving
coordinates: A particle at rest in these coordinates remains at rest, i.e.,
constant $r$, $\theta$, and $\phi$. A freely moving particle eventually comes
to rest in these coordinates, as its momentum is redshifted by the expansion,
$p\propto a^{-1}$. Motion with respect to the co-moving coordinates (or cosmic
rest frame) is referred to as peculiar velocity; unless supported by the
inhomogeneous distribution of matter, peculiar velocities decay away as
$a^{-1}$. Thus the measurement of peculiar velocities, which is not easy as it
requires independent measures of both the distance and velocity of an object,
can be used to probe the distribution of mass in the universe.
Physical separations between freely moving particles scale as $a(t)$; or said
another way the physical separation between two points is simply $a(t)$ times
the coordinate separation. The momenta of freely propagating particles
decrease, or redshift, as $a(t)^{-1}$, and thus the wavelength of a photon
stretches as $a(t)$, which is the origin of the cosmological redshift. The
redshift suffered by a photon emitted from a distant galaxy $1+z=a_{0}/a(t)$;
that is, a galaxy whose light is redshifted by $1+z$, emitted that light when
the universe was a factor of $(1+z)^{-1}$ smaller. When the light from the
most distant quasar yet seen ($z=4.9$) was emitted, the universe was a factor
of almost six smaller; when CMB photons last scattered, the universe was about
$1100$ times smaller.
### 0.2.1 Friedmann equations
The evolution of the scale factor $a(t)$ is governed by Einstein equations
$R_{\mu\nu}-\frac{1}{2}\,R\,g_{\mu\nu}\equiv G_{\mu\nu}=8\pi G\,,T_{\mu\nu}$
(2)
where $R_{\mu\nu}$ $(\mu,\nu=0,\cdots 3)$ is the Riemann tensor and $R$ is the
Ricci scalar constructed via the metric (1) [2], and $T_{\mu\nu}$ is the
energy-momentum tensor. $G=m_{\rm Pl}^{-2}$ is the Newton constant. Under the
hypothesis of homogeneity and isotropy, we can always write the energy-
momentum tensor under the form $T_{\mu\nu}={\rm diag}\left(\rho,P,P,P\right)$
where $\rho$ is the energy density of the system and $P$ its pressure. They
are functions of time.
The evolution of the cosmic-scale factor is governed by the Friedmann equation
$H^{2}\equiv\left({\dot{a}\over a}\right)^{2}={8\pi G\rho\over 3}-{k\over
a^{2}}\,,$ (3)
where $\rho$ is the total energy density of the universe, matter, radiation,
vacuum energy, and so on.
Differentiating wrt to time both members of Eq. (3) and using the the mass
conservation equation
$\dot{\rho}+3H(\rho+P)=0\,,$ (4)
we find the equation for the acceleration of the scale factor
$\frac{\ddot{a}}{a}=-\frac{4\pi G}{3}(\rho+3P).$ (5)
Combining Eqs. (3) and (5) we find
$\dot{H}=-4\pi G\left(\rho+P\right).$ (6)
The evolution of the energy density of the universe is governed by
$d(\rho a^{3})=-Pd\left(a^{3}\right);$ (7)
which is the first law of thermodynamics for a fluid in the expanding
universe. (In the case that the stress energy of the universe is comprised of
several, non-interacting components, this relation applies to each separately;
e.g., to the matter and radiation separately today.) For $P=\rho/3$, ultra-
relativistic matter, $\rho\propto a^{-4}$ and $a\sim t^{\frac{1}{2}}$; for
$P=0$, very nonrelativistic matter, $\rho\propto a^{-3}$ and $a\sim
t^{\frac{2}{3}}$; and for $P=-\rho$, vacuum energy, $\rho=\,$const. If the rhs
of the Friedmann equation is dominated by a fluid with equation of state
$P=w\rho$, it follows that $\rho\propto a^{-3(1+w)}$ and $a\propto
t^{2/3(1+w)}$.
We can use the Friedmann equation to relate the curvature of the universe to
the energy density and expansion rate:
$\Omega-1={k\over a^{2}H^{2}}\,;\qquad\Omega={\rho\over\rho_{\rm crit}}\,;$
(8)
and the critical density today $\rho_{\rm crit}=3H^{2}/8\pi G=1.88h^{2}{\,{\rm
g\,cm^{-3}}}\simeq 1.05\times 10^{4}{\,\rm eV}{\,{\rm cm}^{-3}}$. There is a
one-to-one correspondence between $\Omega$ and the spatial curvature of the
universe: positively curved, $\Omega_{0}>1$; negatively curved,
$\Omega_{0}<1$; and flat ($\Omega_{0}=1$). Further, the fate of the universe
is determined by the curvature: model universes with $k\leq 0$ expand forever,
while those with $k>0$ necessarily recollapse. The curvature radius of the
universe is related to the Hubble radius and $\Omega$ by
$R_{\rm curv}={H^{-1}\over|\Omega-1|^{1/2}}\,.$ (9)
In physical terms, the curvature radius sets the scale for the size of spatial
separations where the effects of curved space become pronounced. And in the
case of the positively curved model it is just the radius of the 3-sphere.
The energy content of the universe consists of matter and radiation (today,
photons and neutrinos). Since the photon temperature is accurately known,
$T_{0}=2.73\pm 0.01\,$K, the fraction of critical density contributed by
radiation is also accurately known: $\Omega_{R}h^{2}=4.2\times 10^{-5}$, where
$h=0.72\pm 0.07$ is the present Hubble rate in units of $100$ km ${\rm
s}^{-1}$ ${\,{\rm Mpc}}^{-1}$ [9]. The remaining content of the universe is
another matter. Rapid progress has been made recently toward the measurement
of cosmological parameters [10]. Over the past years the basic features of our
universe have been determined. The universe is spatially flat; accelerating;
comprised of one third dark matter and two thirds a new form of dark energy.
The measurements of the cosmic microwave background anisotropies at different
angular scales performed by the WMAP Collaboration [9] have recently
significantly increased the case for accelerated expansion in the early
universe (the inflationary paradigm) and at the current epoch (dark energy
dominance), especially when combined with data on high-redshift supernovae
(SN1) and large-scale structure (LSS) [10]. The CMB$+$LSS$+$SN1 data give [9]
$\Omega_{0}=1.00^{+0.07}_{-0.03}\,,$
meaning that the present universe is spatially flat (or at least very close to
being flat). Restricting to $\Omega_{0}=1$, the dark matter density is given
by [9]
$\Omega_{\rm DM}h^{2}=0.11^{+0.0034}_{-0.059}\,,$
and a baryon density
$\Omega_{B}=0.045\pm 0.0015,$
while the Big Bang nucleosynthesis estimate is $\Omega_{B}h^{2}=0.019\pm
0.002.$ Substantial dark (unclustered) energy is inferred:
$\Omega_{\rm DE}\approx 0.72\pm 0.015\,.$
What is most relevant for us is that this universe was apparently born from a
burst of rapid expansion, inflation, during which quantum noise was stretched
to astrophysical size seeding cosmic structure. This is exactly the phenomenon
we want to address in part of these lectures.
### 0.2.2 The early, radiation-dominated universe
In any case, at present, matter outweighs radiation by a wide margin. However,
since the energy density in matter decreases as $a^{-3}$, and that in
radiation as $a^{-4}$ (the extra factor due to the redshifting of the energy
of relativistic particles), at early times the universe was radiation
dominated—indeed the calculations of primordial nucleosynthesis provide
excellent evidence for this. Denoting the epoch of matter and radiation
equality by subscript ‘EQ,’ and using $T_{0}=2.73\,$K, it follows that
$a_{\rm EQ}=4.18\times 10^{-5}\,(\Omega_{0}h^{2})^{-1}\,;\qquad T_{\rm
EQ}=5.62(\Omega_{0}h^{2}){\,\rm eV}\,;$ (10) $t_{\rm EQ}=4.17\times
10^{10}(\Omega_{0}h^{2})^{-2}{\rm s}\,.$ (11)
At early times the expansion rate and age of the universe were determined by
the temperature of the universe and the number of relativistic degrees of
freedom:
$\rho_{\rm rad}=g_{*}(T){\pi^{2}T^{4}\over 30};\qquad H\simeq
1.67g_{*}^{1/2}T^{2}/{m_{\rm Pl}};$ (12) $\Rightarrow a\propto t^{1/2};\qquad
t\simeq 2.42\times 10^{-6}g_{*}^{-1/2}(T/\,{\rm GeV})^{-2}\,{\rm s}\,;$ (13)
where $g_{*}(T)$ counts the number of ultra-relativistic degrees of freedom
($\approx$ the sum of the internal degrees of freedom of particle species much
less massive than the temperature) and ${m_{\rm Pl}}\equiv G^{-1/2}=1.22\times
10^{19}\,{\rm GeV}$ is the Planck mass. For example, at the epoch of
nucleosynthesis, $g_{*}=10.75$ assuming three, light ($\ll\,{\rm MeV}$)
neutrino species; taking into account all the species in the Standard Model,
$g_{*}=106.75$ at temperatures much greater than $300\,{\rm GeV}$.
A quantity of importance related to $g_{*}$ is the entropy density in
relativistic particles,
$s={\rho+p\over T}={2\pi^{2}\over 45}g_{*}T^{3},$
and the entropy per co-moving volume,
$S\ \ \propto\ \ a^{3}s\ \ \propto\ \ g_{*}a^{3}T^{3}.$
By a wide margin most of the entropy in the universe exists in the radiation
bath. The entropy density is proportional to the number density of
relativistic particles. At present, the relativistic particle species are the
photons and neutrinos, and the entropy density is a factor of 7.04 times the
photon-number density: $n_{\gamma}=413{\,{\rm cm}^{-3}}$ and $s=2905{\,{\rm
cm}^{-3}}$.
In thermal equilibrium—which provides a good description of most of the
history of the universe—the entropy per co-moving volume $S$ remains constant.
This fact is very useful. First, it implies that the temperature and scale
factor are related by
$T\propto g_{*}^{-1/3}a^{-1},$ (14)
which for $g_{*}=\,$const leads to the familiar $T\propto a^{-1}$.
Second, it provides a way of quantifying the net baryon number (or any other
particle number) per co-moving volume:
$N_{B}\equiv R^{3}n_{B}={n_{B}\over s}\simeq(4-7)\times 10^{-11}.$ (15)
The baryon number of the universe tells us two things: (1) the entropy per
particle in the universe is extremely high, about $10^{10}$ or so compared to
about $10^{-2}$ in the Sun and a few in the core of a newly formed neutron
star. (2) The asymmetry between matter and antimatter is very small, about
$10^{-10}$, since at early times quarks and antiquarks were roughly as
abundant as photons. One of the great successes of particle cosmology is
baryogenesis, the idea that $B$, $C$, and $CP$ violating interactions
occurring out-of-equilibrium early on allow the universe to develop a net
baryon number of this magnitude. Finally, the constancy of the entropy per co-
moving volume allows us to characterize the size of co-moving volume
corresponding to our present Hubble volume in a very physical way: by the
entropy it contains,
$S_{U}={4\pi\over 3}H_{0}^{-3}s\simeq 10^{90}.$ (16)
The standard cosmology is tested back to times as early as about 0.01 s; it is
only natural to ask how far back one can sensibly extrapolate. Since the
fundamental particles of Nature are point-like quarks and leptons whose
interactions are perturbatively weak at energies much greater than $1\,{\rm
GeV}$, one can imagine extrapolating as far back as the epoch where general
relativity becomes suspect, i.e., where quantum gravitational effects are
likely to be important: the Planck epoch, $t\sim 10^{-43}{\rm s}$ and $T\sim
10^{19}\,{\rm GeV}$. Of course, at present, our firm understanding of the
elementary particles and their interactions only extends to energies of the
order of $100\,{\rm GeV}$, which corresponds to a time of the order of
$10^{-11}{\rm s}$ or so. We can be relatively certain that at a temperature of
100–200 MeV ($t\sim 10^{-5}{\rm s}$) there was a transition (likely a second-
order phase transition) from quark/gluon plasma to very hot hadronic matter,
and that some kind of phase transition associated with the symmetry breakdown
of the electroweak theory took place at a temperature of the order of
$300\,{\rm GeV}$ ($t\sim 10^{-11}{\rm s}$).
### 0.2.3 The concept of particle horizon
In spite of the fact that the universe was vanishingly small at early times,
the rapid expansion precluded causal contact from being established
throughout. Photons travel on null paths characterized by $dr=dt/a(t)$; the
physical distance that a photon could have travelled since the bang until time
$t$, the distance to the particle horizon, is
$\displaystyle R_{H}(t)$ $\displaystyle=$ $\displaystyle
a(t)\int_{0}^{t}{dt^{\prime}\over a(t^{\prime})}$ (17) $\displaystyle=$
$\displaystyle\frac{t}{(1-n)}=n\,\frac{H^{-1}}{(1-n)}\sim H^{-1}\qquad{\rm
for}\ a(t)\propto t^{n},\ \ n<1.$
Using the conformal time $d\tau=dt/a$, the particle horizon becomes
$R_{H}(t)=a(\tau)\int_{\tau_{0}}^{\tau}\,d\tau,$ (18)
where $\tau_{0}$ indicates the conformal time corresponding to $t=0$. Note, in
the standard cosmology the distance to the horizon is finite, and up to
numerical factors, equal to the age of the universe or the Hubble radius,
$H^{-1}$. For this reason, we shall use horizon and Hubble radius
interchangeably111As we shall see, in inflationary models the horizon and
Hubble radius are not roughly equal as the horizon distance grows
exponentially relative to the Hubble radius; in fact, at the end of inflation
they differ by $e^{N}$, where $N$ is the number of e-folds of inflation.
However, we shall slip and use “horizon” and “Hubble radius” interchangeably,
though we shall always mean Hubble radius..
Note also that a physical length scale $\lambda$ is within the horizon if
$\lambda<R_{H}\sim H^{-1}$. Since we can identify the length scale $\lambda$
with its wavenumber $k$, $\lambda=2\pi a/k$, we shall have the following rule
$\displaystyle\frac{k}{aH}$ $\displaystyle\ll$ $\displaystyle
1\Longrightarrow{\rm SCALE}\leavevmode\nobreak\ \leavevmode\nobreak\
\lambda\leavevmode\nobreak\ \leavevmode\nobreak\ {\rm
OUTSIDE}\leavevmode\nobreak\ \leavevmode\nobreak\ {\rm
THE}\leavevmode\nobreak\ \leavevmode\nobreak\ {\rm HORIZON}$
$\displaystyle\frac{k}{aH}$ $\displaystyle\gg$ $\displaystyle
1\Longrightarrow{\rm SCALE}\leavevmode\nobreak\ \leavevmode\nobreak\
\lambda\leavevmode\nobreak\ \leavevmode\nobreak\ {\rm
WITHIN}\leavevmode\nobreak\ \leavevmode\nobreak\ {\rm THE}\leavevmode\nobreak\
\leavevmode\nobreak\ {\rm HORIZON}$
---
## 0.3 The shortcomings of the standard Big Bang theory
By now the shortcomings of standard cosmology are well appreciated: the
horizon or large-scale smoothness problem; the small-scale inhomogeneity
problem (origin of density perturbations); and the flatness or oldness
problem. we shall briefly review only the horizon problem here here.
### 0.3.1 The horizon problem
According to standard cosmology, photons decoupled from the rest of the
components (electrons and baryons) at a temperature of the order of 0.3 eV.
This corresponds to the so-called surface of ‘last scattering’ at a redshift
of about $1100$ and an age of about $180000\,(\Omega_{0}h^{2})^{-1/2}{\rm
yr}$. From the epoch of last scattering onwards, photons free-stream and reach
us basically untouched. Detecting primordial photons is therefore equivalent
to take a picture of the universe when the latter was about 300 000 years old.
The spectrum of the cosmic background radiation (CBR) is consistent with that
of a black body at temperature 2.73 K over more than three decades in
wavelength.
The most accurate measurement of the temperature and spectrum is that by the
WMAP5 instrument on the COBE satellite which determined its temperature to be
$2.726\pm 0.01\,$K [9]. The length corresponding to our present Hubble radius
(which is approximately the radius of our observable universe) at the time of
last scatteringwas
$\lambda_{H}(t_{\rm LS})=R_{H}(t_{0})\left(\frac{a_{\rm
LS}}{a_{0}}\right)=R_{H}(t_{0})\left(\frac{T_{0}}{T_{\rm LS}}\right).$
On the other hand, during the matter-dominated period, the Hubble length
decreased with a different law
$H^{2}\propto\rho_{M}\propto a^{-3}\propto T^{3}.$
At last-scattering
$H_{LS}^{-1}=R_{H}(t_{0})\left(\frac{T_{LS}}{T_{0}}\right)^{-3/2}\ll
R_{H}(t_{0}).$
The length corresponding to our present Hubble radius was much larger that the
horizon at that time. This can be by shown comparing the volumes corresponding
to these two scales
$\frac{\lambda^{3}_{H}(T_{LS})}{H_{LS}^{-3}}=\left(\frac{T_{0}}{T_{LS}}\right)^{-\frac{3}{2}}\approx
10^{6}.$ (19)
Figure 1: The horizon scale (solid line) and a physical scale $\lambda$
(dashed line) as function of the scale factor $a$
There were $\sim 10^{6}$ casually disconnected regions within the volume that
now corresponds to our horizon! It is difficult to come up with a process
other than an early hot and dense phase in the history of the universe that
would lead to a precise black body for a bath of photons which were causally
disconnected the last time they interacted with the surrounding plasma.
The horizon problem is well represented by Fig. 1 where the solid line
indicates the horizon scale and the dashed line any generic physical length
scale $\lambda$. Suppose, indeed, that $\lambda$ indicates the distance
between two photons we detect today. From Eq. (19) we discover that at the
time of emission (last-scattering) the two photons could not talk to each
other, the dashed line is above the solid line. There is another aspect of the
horizon problem which is related to the problem of initial conditions for the
cosmological perturbations. We have every indication that the universe at
early times, say $t\ll 300\,000{\rm\leavevmode\nobreak\ yr}$, was very
homogeneous; however, today inhomogeneity (or structure) is ubiquitous: stars
($\delta\rho/\rho\sim 10^{30}$), galaxies ($\delta\rho/\rho\sim 10^{5}$),
clusters of galaxies ($\delta\rho/\rho\sim 10$—$10^{3}$), superclusters, or
“clusters of clusters” ($\delta\rho/\rho\sim 1$), voids
($\delta\rho/\rho\sim-1$), great walls, and so on. For some twenty-five years
standard cosmology has provided a general framework for understanding this
picture. Once the universe becomes matter dominated (around 1000 yr after the
bang) primeval density inhomogeneities ($\delta\rho/\rho\sim 10^{-5}$) are
amplified by gravity and grow into the structure we see today [2]. The
existence of density inhomogeneities has another important consequence:
fluctuations in the temperature of the CMB radiation of a similar amplitude.
The temperature difference measured between two points separated by a large
angle ($\mathrel{\mathchoice{\lower
3.6pt\vbox{\halign{$\mathsurround=0pt\displaystyle\hfil#\hfil$\cr>\crcr\sim\crcr}}}{\lower
3.6pt\vbox{\halign{$\mathsurround=0pt\textstyle\hfil#\hfil$\cr>\crcr\sim\crcr}}}{\lower
3.6pt\vbox{\halign{$\mathsurround=0pt\scriptstyle\hfil#\hfil$\cr>\crcr\sim\crcr}}}{\lower
3.6pt\vbox{\halign{$\mathsurround=0pt\scriptscriptstyle\hfil#\hfil$\cr>\crcr\sim\crcr}}}}1^{\circ}$)
arises due to a very simple physical effect: the difference in the
gravitational potential between the two points on the last scatteringsurface,
which in turn is related to the density perturbation, determines the
temperature anisotropy on the angular scale subtended by that length scale,
$\left({\delta T\over
T}\right)_{\theta}\approx\left({\delta\rho\over\rho}\right)_{\lambda},$ (20)
where the scale $\lambda\sim 100h^{-1}\,{\rm Mpc}(\theta/{\rm deg})$ subtends
an angle $\theta$ on the last-scattering surface. This is known as the
Sachs–Wolfe effect [11, 12]. We shall come back to this piece of physics.
Figure 2: The CMBR anisotropy as function of $\ell$ (from Ref. [9])
The temperature anisotropy is commonly expanded in spherical harmonics
$\frac{\Delta T}{T}(x_{0},\tau_{0},{\bf n})=\sum_{\ell
m}a_{\ell,m}(x_{0})Y_{\ell m}({\bf n}),$ (21)
where $x_{0}$ and $\tau_{0}$ are our position and the preset time,
respectively, ${\bf n}$ is the direction of observation, $\ell^{\prime}$s are
the different multipoles and222An alternative definition is
$C_{\ell}=\langle\left|a_{\ell
m}\right|^{2}\rangle=\frac{1}{2\ell+1}\sum_{m=-\ell}^{\ell}\left|a_{\ell
m}\right|^{2}$.
$\langle a_{\ell
m}a^{*}_{\ell^{\prime}m^{\prime}}\rangle=\delta_{\ell,\ell^{\prime}}\delta_{m,m^{\prime}}C_{\ell},$
(22)
where the deltas are due to the fact that the process that created the
anisotropy is statistically isotropic. The $C_{\ell}$’s are the so-called CMB
power spectrum. For homogeneity and isotropy, the $C_{\ell}$’s are neither a
function of $x_{0}$, nor of $m$. The two-point correlation function is related
to the $C_{l}$’s in the following way
$\displaystyle\Big{<}\frac{\delta T({\bf n})}{T}\frac{\delta T({\bf
n}^{\prime})}{T}\Big{>}$ $\displaystyle=$
$\displaystyle\sum_{\ell\ell^{\prime}mm^{\prime}}\langle a_{\ell
m}a^{*}_{\ell^{\prime}m^{\prime}}\rangle Y_{\ell m}({\bf
n})Y^{*}_{\ell^{\prime}m^{\prime}}({\bf n}^{\prime})$ (23) $\displaystyle=$
$\displaystyle\sum_{\ell}C_{\ell}\sum_{m}Y_{\ell m}({\bf n})Y^{*}_{\ell
m}({\bf
n}^{\prime})=\frac{1}{4\pi}\sum_{\ell}(2\ell+1)C_{\ell}P_{\ell}(\mu={\bf
n}\cdot{\bf n}^{\prime})$
where we have used the addition theorem for the spherical harmonics, and
$P_{\ell}$ is the Legendre polynom of order $\ell$. In expression (23) the
expectation value is an ensemble average. It can be regarded as an average
over the possible observer positions, but not in general as an average over
the single sky we observe, because of the cosmic variance333The usual
hypothesis is that we observe a typical realization of the ensemble. This
means that we expect the difference between the observed values $|a_{\ell
m}|^{2}$ and the ensemble averages $C_{\ell}$ to be of the order of the mean-
square deviation of $|a_{\ell m}|^{2}$ from $C_{\ell}$. The latter is called
cosmic variance and, because we are dealing with a Gaussian distribution, it
is equal to $2C_{\ell}$ for each multipole $\ell$. For a single $\ell$,
averaging over the $(2\ell+1)$ values of $m$ reduces the cosmic variance by a
factor $(2\ell+1)$, but it remains a serious limitation for low multipoles..
WMAP5 data are given in Fig. 2.
Let us now consider the last scatteringsurface. In co-moving coordinates the
latter is ‘far’ from us a distance equal to
$\int_{t_{\rm LS}}^{t_{0}}\,\frac{dt}{a}=\int_{\tau_{\rm
LS}}^{\tau_{0}}\,d\tau=\left(\tau_{0}-\tau_{\rm LS}\right).$ (24)
A given co-moving scale $\lambda$ is therefore projected on the last
scatteringsurface sky on an angular scale
$\theta\simeq\frac{\lambda}{\left(\tau_{0}-\tau_{\rm LS}\right)},$ (25)
where we have neglected tiny curvature effects. Consider now that the scale
$\lambda$ is of the order of the co-moving sound horizon at the time of last-
scattering, $\lambda\sim c_{s}\tau_{\rm LS}$, where $c_{s}\simeq 1/\sqrt{3}$
is the sound velocity at which photons propagate in the plasma at the last-
scattering. This corresponds to an angle
$\theta\simeq c_{s}\frac{\tau_{\rm LS}}{\left(\tau_{0}-\tau_{\rm
LS}\right)}\simeq c_{s}\frac{\tau_{\rm LS}}{\tau_{0}},$ (26)
where the last passage has been performed knowing that $\tau_{0}\gg\tau_{\rm
LS}$. Since the universe is matter-dominated from the time of last
scatteringonwards, the scale factor has the following behaviour: $a\sim
T^{-1}\sim t^{2/3}\sim\tau^{2}$. The angle $\theta_{\rm HOR}$ subtended by the
sound horizon on the last-scattering surface then becomes
$\theta_{\rm HOR}\simeq c_{s}\left(\frac{T_{0}}{T_{\rm LS}}\right)^{1/2}\sim
1^{\circ},$ (27)
where we have used $T_{\rm LS}\simeq 0.3$ eV and $T_{0}\sim 10^{-13}$ GeV.
This corresponds to a multipole $\ell_{\rm HOR}$
$\ell_{\rm HOR}=\frac{\pi}{\theta_{\rm HOR}}\simeq 200\,.$ (28)
From these estimates we conclude that two photons which on the last
scatteringsurface were separated by an angle larger than $\theta_{\rm HOR}$,
corresponding to multipoles smaller than $\ell_{\rm HOR}\sim 200$, were not in
causal contact. On the other hand, from Fig. 2 it is clear that small
anisotropies, of the same order of magnitude $\delta T/T\sim 10^{-5}$ are
present at $\ell\ll 200$. We conclude that one of the striking features of the
CMB fluctuations is that they appear to be non-causal. Photons at the last
scatteringsurface which were causally disconnected have the same small
anisotropies! The existence of particle horizons in the standard cosmology
precludes explaining the smoothness as a result of microphysical events: the
horizon at decoupling, the last time one could imagine temperature
fluctuations being smoothed by particle interactions, corresponds to an
angular scale on the sky of about $1^{\circ}$, which precludes temperature
variations on larger scales from being erased. To account for the small-scale
lumpiness of the universe today, density perturbations with horizon-crossing
amplitudes of $10^{-5}$ on scales of $1\,{\rm Mpc}$ to $10^{4}\,{\rm Mpc}$ or
so are required.
As can be seen in Fig. 1, in the standard cosmology the physical size of a
perturbation, which grows as the scale factor, begins larger than the horizon
and, relatively late in the history of the universe, crosses inside the
horizon. This precludes a causal microphysical explanation for the origin of
the required density perturbations.
From the considerations made so far, it appears that solving the horizon
problem of the standard Big Bang theory requires that the universe go through
a primordial period during which the physical scales $\lambda$ evolve faster
than the horizon scale $H^{-1}$.
Figure 3: The behaviour of a generic scale $\lambda$ and the horizon scale
$H^{-1}$ in the standard inflationary model
If there is period during which physical length scales grow faster than
$H^{-1}$, length scales $\lambda$ which are within the horizon today,
$\lambda<H^{-1}$ (such as the distance between two detected photons) and were
outside the horizon for some period, $\lambda>H^{-1}$ (for instance at the
time of last scatteringwhen the two photons were emitted), had a chance to be
within the horizon at some primordial epoch, $\lambda<H^{-1}$ again, see Fig.
3. If this happens, the homogeneity and the isotropy of the CMB can easily be
explained: photons that we receive today and were emitted from the last
scattering surface from causally disconnected regions have the same
temperature because they had a chance to ‘talk’ to each other at some
primordial stage of the evolution of the universe.
The second condition can easily be expressed as a condition on the scale
factor $a$. Since a given scale $\lambda$ scales like $\lambda\sim a$ and
$H^{-1}=a/\dot{a}$, we need to impose that there is a period during which
$\left(\frac{\lambda}{H^{-1}}\right)^{\cdot}=\ddot{a}>0\,.$
We can therefore introduce the following rigorous definition: an inflationary
stage is a period of the universe during which the latter accelerates
INFLATION ⟺ ¨a>0.
---
Comment: Let us stress that during such an accelerating phase the universe
expands adiabatically. This means that during inflation one can exploit the
usual FRW equations (3) and (5). It must be clear therefore that the non-
adiabaticity condition is satisfied not during inflation, but during the phase
transition between the end of inflation and the beginning of the radiation-
dominated phase. At this transition phase a large entropy is generated under
the form of relativistic degrees of freedom: the Big Bang has taken place.
## 0.4 The standard inflationary universe
From the previous section we have learned that an accelerating stage during
the primordial phases of the evolution of the universe might be able to solve
the horizon problem. From Eq. (5) we learn that
$\ddot{a}>0\Longleftrightarrow(\rho+3P)<0\,.$
An accelerating period is obtainable only if the overall pressure $p$ of the
universe is negative: $p<-\rho/3$. Neither a radiation-dominated phase nor a
matter-dominated phase (for which $p=\rho/3$ and $p=0$, respectively) satisfy
such a condition. Let us postpone for the time being the problem of finding a
‘candidate’ able to provide the condition $P<-\rho/3$. For sure, inflation is
a phase of the history of the universe occurring before the era of
nucleosynthesis ($t\approx 1$ s, $T\approx 1$ MeV) during which the light
elements abundances were formed. This is because nucleosynthesis is the
earliest epoch from which we have experimental data and they are in agreement
with the predictions of the standard Big Bang theory. However, the thermal
history of the universe before the epoch of nucleosynthesis is unknown.
In order to study the properties of the period of inflation, we assume the
extreme condition $p=-\rho$ which considerably simplifies the analysis. A
period of the universe during which $P=-\rho$ is called the de Sitter stage.
By inspecting Eqs. (3) and (4), we learn that during the de Sitter phase
$\displaystyle\rho$ $\displaystyle=$ $\displaystyle\leavevmode\nobreak\
\leavevmode\nobreak\ {\rm constant}\,,$ $\displaystyle H_{I}$ $\displaystyle=$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ {\rm constant}\,,$
where we have indicated by $H_{I}$ the value of the Hubble rate during
inflation. Correspondingly, solving Eq. (3) gives
$a=a_{i}\,e^{H_{I}(t-t_{i})},$ (29)
where $t_{i}$ denotes the time at which inflation starts. Let us now see how
such a period of exponential expansion takes care of the shortcomings of the
standard Big Bang Theory444 Despite the fact that the growth of the scale
factor is exponential and the expansion is superluminal, this is not in
contradiction with what is dictated by relativity. Indeed, it is the spacetime
itself which is progating so fast and not a light signal in it..
### 0.4.1 Inflation and the horizon problem
During the inflationary (de Sitter) epoch the horizon scale $H^{-1}$ is
constant. If inflation lasts long enough, all the physical scales that have
left the horizon during the radiation-dominated or matter-dominated phase can
re-enter the horizon in the past: this is because such scales are
exponentially reduced. As we have seen in the previous section, this explains
both the problem of the homogeneity of CMB and the initial condition problem
of small cosmological perturbations. Once the physical length is within the
horizon, microphysics can act, the universe can be made approximately
homogeneous and the primeval inhomogeneities can be created.
Let us see how long inflation must be sustained in order to solve the horizon
problem. Let $t_{i}$ and $t_{f}$ be, respectively, the time of beginning and
end of inflation. We can define the corresponding number of e-foldings $N$
$N={\rm ln}\left[H_{I}(t_{e}-t_{i})\right].$ (30)
A necessary condition to solve the horizon problem is that the largest scale
we observe today, the present horizon $H_{0}^{-1}$, was reduced during
inflation to a value $\lambda_{H_{0}}(t_{i})$ smaller than the value of
horizon length $H_{I}^{-1}$ during inflation. This gives
$\lambda_{H_{0}}(t_{i})=H^{-1}_{0}\left(\frac{a_{t_{f}}}{a_{t_{0}}}\right)\left(\frac{a_{t_{i}}}{a_{t_{f}}}\right)=H_{0}^{-1}\left(\frac{T_{0}}{T_{f}}\right)e^{-N}\mathrel{\mathchoice{\lower
3.6pt\vbox{\halign{$\mathsurround=0pt\displaystyle\hfil#\hfil$\cr<\crcr\sim\crcr}}}{\lower
3.6pt\vbox{\halign{$\mathsurround=0pt\textstyle\hfil#\hfil$\cr<\crcr\sim\crcr}}}{\lower
3.6pt\vbox{\halign{$\mathsurround=0pt\scriptstyle\hfil#\hfil$\cr<\crcr\sim\crcr}}}{\lower
3.6pt\vbox{\halign{$\mathsurround=0pt\scriptscriptstyle\hfil#\hfil$\cr<\crcr\sim\crcr}}}}H_{I}^{-1},$
where we have neglected for simplicity the short period of matter-domination
and we have called $T_{f}$ the temperature at the end of inflation (to be
indentified with the reheating temperature $T_{RH}$ at the beginning of the
radiation-dominated phase after inflation, see later). We get
$N\mathrel{\mathchoice{\lower
3.6pt\vbox{\halign{$\mathsurround=0pt\displaystyle\hfil#\hfil$\cr>\crcr\sim\crcr}}}{\lower
3.6pt\vbox{\halign{$\mathsurround=0pt\textstyle\hfil#\hfil$\cr>\crcr\sim\crcr}}}{\lower
3.6pt\vbox{\halign{$\mathsurround=0pt\scriptstyle\hfil#\hfil$\cr>\crcr\sim\crcr}}}{\lower
3.6pt\vbox{\halign{$\mathsurround=0pt\scriptscriptstyle\hfil#\hfil$\cr>\crcr\sim\crcr}}}}\ln\left(\frac{T_{0}}{H_{0}}\right)-\ln\left(\frac{T_{f}}{H_{I}}\right)\approx
67+\ln\left(\frac{T_{f}}{H_{I}}\right).$
Apart from the logarithmic dependence, we obtain $N\mathrel{\mathchoice{\lower
3.6pt\vbox{\halign{$\mathsurround=0pt\displaystyle\hfil#\hfil$\cr>\crcr\sim\crcr}}}{\lower
3.6pt\vbox{\halign{$\mathsurround=0pt\textstyle\hfil#\hfil$\cr>\crcr\sim\crcr}}}{\lower
3.6pt\vbox{\halign{$\mathsurround=0pt\scriptstyle\hfil#\hfil$\cr>\crcr\sim\crcr}}}{\lower
3.6pt\vbox{\halign{$\mathsurround=0pt\scriptscriptstyle\hfil#\hfil$\cr>\crcr\sim\crcr}}}}70$.
### 0.4.2 A prediction of inflation
Since during inflation the Hubble rate is constant
$\Omega-1=\frac{k}{a^{2}H^{2}}\propto\frac{1}{a^{2}}\ .$
On the other hand it is easy to show that to reproduce a value of
$(\Omega_{0}-1)$ of order of unity today, the initial value of $(\Omega-1)$ at
the beginning of the radiation-dominated phase must be
$\left|\Omega-1\right|\sim 10^{-60}$. Since we identify the beginning of the
radiation-dominated phase with the beginning of inflation, we require
$\left|\Omega-1\right|_{t=t_{f}}\sim 10^{-60}.$
During inflation
$\frac{\left|\Omega-1\right|_{t=t_{f}}}{\left|\Omega-1\right|_{t=t_{i}}}=\left(\frac{a_{i}}{a_{f}}\right)^{2}=e^{-2N}.$
(31)
Taking $\left|\Omega-1\right|_{t=t_{i}}$ of order unity, it is enough to
require that $N\approx 70$. However, IF the period of inflation lasts longer
than 70 e-foldings the present-day value of $\Omega_{0}$ will be equal to
unity with great precision. One can say that a generic prediction of inflation
is that
INFLATION ⟹ Ω_0=1.
---
This statement, however, must be taken cum grano salis and properly specified.
Inflation does not change the global geometric properties of the space-time.
If the universe is open or closed, it will always remain flat or closed,
independently from inflation. What inflation does is to magnify the radius of
curvature $R_{\rm curv}$ defined in Eq. (9) so that locally the universe is
flat with a great precision. As we shall see, the current data on the CMB
anisotropies confirm this prediction.
### 0.4.3 Inflation and the inflaton
In the previous subsections we have described the various advantages of having
a period of accelerating phase. The latter required $P<-\rho/3$. Now, we would
like to show that this condition can be attained by means of a simple scalar
field. We shall call this field the inflaton $\phi$.
The action of the inflaton field reads
$S=\int
d^{4}x\,\sqrt{-g}\,\mathcal{L}=\int\,d^{4}x\,\sqrt{-g}\,\left[\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi+V(\phi)\right],$
(32)
where $\sqrt{-g}=a^{3}$ for the FRW metric (1). From the Euler–Lagrange
equations
$\partial^{\mu}\frac{\delta(\sqrt{-g}\mathcal{L})}{\delta\,\partial^{\mu}\phi}-\frac{\delta(\sqrt{-g}\mathcal{L})}{\delta\phi}=0\,,$
(33)
we obtain
$\ddot{\phi}+3H\dot{\phi}-\frac{\nabla^{2}\phi}{a^{2}}+V^{\prime}(\phi)=0\,,$
(34)
where $V^{\prime}(\phi)=\left(dV(\phi)/d\phi\right)$. Note, in particular, the
appearance of the friction term $3H\dot{\phi}$: a scalar field rolling down
its potential suffers a friction due to the expansion of the universe.
We can write the energy momentum tensor of the scalar field
$T_{\mu\nu}=\partial_{\mu}\phi\partial_{\nu}\phi-g_{\mu\nu}\,\mathcal{L}\,.$
The corresponding energy density $\rho_{\phi}$ and pressure density $P_{\phi}$
are
$\displaystyle
T_{00}=\rho_{\phi}=\frac{\dot{\phi}^{2}}{2}+V(\phi)+\frac{(\nabla\phi)^{2}}{2a^{2}},$
(35) $\displaystyle
T_{ii}=P_{\phi}=\frac{\dot{\phi}^{2}}{2}-V(\phi)-\frac{(\nabla\phi)^{2}}{6a^{2}}\,.$
(36)
Note that, if the gradient term were dominant, we would obtain
$P_{\phi}=-\frac{\rho_{\phi}}{3}$, not enough to drive inflation. We can now
split the inflaton field in
$\phi(t)=\phi_{0}(t)+\delta\phi({\bf x},t)\,,$
where $\phi_{0}$ is the ‘classical’ (infinite wavelength) field, that is the
expectation value of the inflaton field on the initial isotropic and
homogeneous state, while $\delta\phi({\bf x},t)$ represents the quantum
fluctuations around $\phi_{0}$. In this section, we shall be concerned only
with the evolution of the classical field $\phi_{0}$. The next section will be
devoted to the crucial issue of the evolution of quantum perturbations during
inflation. This separation is justified by the fact that quantum fluctuations
are much smaller than the classical value and therefore negligible when
looking at the classical evolution. Not to be overwhelmed by the notation, we
shall indicate the classical value of the inflaton field by $\phi$ from now
on. The energy momentum tensor becomes
$\displaystyle T_{00}=\rho_{\phi}=\frac{\dot{\phi}^{2}}{2}+V(\phi)$ (37)
$\displaystyle T_{ii}=P_{\phi}=\frac{\dot{\phi}^{2}}{2}-V(\phi).$ (38)
If
$V(\phi)\gg\dot{\phi}^{2}$
we obtain the following condition
$P_{\phi}\simeq-\rho_{\phi}\,.$
From this simple calculation, we realize that a scalar field whose energy is
dominant in the universe and whose potential energy dominates over the kinetic
term gives inflation. Inflation is driven by the vacuum energy of the inflaton
field.
### 0.4.4 Slow-roll conditions
Let us now quantify better under which circumstances a scalar field may give
rise to a period of inflation. The equation of motion of the field is
$\ddot{\phi}+3H\dot{\phi}+V^{\prime}(\phi)=0\,.$ (39)
If we require that $\dot{\phi}^{2}\ll V(\phi)$, the scalar field is slowly
rolling down its potential. This is the reason why such a period is called
slow-roll. We may also expect that since the potential is flat, $\ddot{\phi}$
is negligible as well. We shall assume that this is true and we shall quantify
this condition soon. The FRW equation (3) becomes
$H^{2}\simeq\frac{8\pi G}{3}\,V(\phi),$ (40)
where we have assumed that the inflaton field dominates the energy density of
the universe. The new equation of motion becomes
$3H\dot{\phi}=-V^{\prime}(\phi)$ (41)
which gives $\dot{\phi}$ as a function of $V^{\prime}(\phi)$. Using Eq. (41)
slow-roll conditions then require
$\dot{\phi}^{2}\ll V(\phi)\\\ \Longrightarrow\\\ \frac{(V^{\prime})^{2}}{V}\ll
H^{2}$
and
$\ddot{\phi}\ll 3H\dot{\phi}\\\ \Longrightarrow\\\ V^{\prime\prime}\ll H^{2}.$
It is now useful to define the slow-roll parameters $\epsilon$ and $\eta$ in
the following way
$\displaystyle\epsilon$ $\displaystyle=$
$\displaystyle-\frac{\dot{H}}{H^{2}}=4\pi
G\frac{\dot{\phi}^{2}}{H^{2}}=\frac{1}{16\pi
G}\left(\frac{V^{\prime}}{V}\right)^{2},$ $\displaystyle\eta$ $\displaystyle=$
$\displaystyle\frac{1}{8\pi
G}\left(\frac{V^{\prime\prime}}{V}\right)=\frac{1}{3}\frac{V^{\prime\prime}}{H^{2}},$
$\displaystyle\delta$ $\displaystyle=$
$\displaystyle\eta-\epsilon=-\frac{\ddot{\phi}}{H\dot{\phi}}\,.$
---
It might be useful to have the same parameters expressed in terms of conformal
time
$\displaystyle\epsilon$ $\displaystyle=$ $\displaystyle 1-\frac{\,{\cal
H}^{\prime}}{\,{\cal H}^{2}}=4\pi G\frac{\phi{{}^{\prime}}^{2}}{\,{\cal
H}^{2}}$ $\displaystyle\delta$ $\displaystyle=$
$\displaystyle\eta-\epsilon=1-\frac{\phi^{\prime\prime}}{\,{\cal
H}\phi^{\prime}}\,.$
---
The parameter $\epsilon$ quantifies how much the Hubble rate $H$ changes with
time during inflation. Notice that, since
$\frac{\ddot{a}}{a}=\dot{H}+H^{2}=\left(1-\epsilon\right)H^{2},$
inflation can be attained only if $\epsilon<1$:
INFLATION ⟺ ϵ<1.
---
As soon as this condition fails, inflation ends. In general, slow-roll
inflation is attained if $\epsilon\ll 1$ and $|\eta|\ll 1$. During inflation
the slow-roll parameters $\epsilon$ and $\eta$ can be considered to be
approximately constant since the potential $V(\phi)$ is very flat.
Comment: In the following, we shall work at first-order perturbation in the
slow-roll parameters, that is we shall take only the first power of them.
Since, using their definition, it is easy to see that
$\dot{\epsilon},\dot{\eta}={\cal O}\left(\epsilon^{2},\eta^{2}\right)$, this
amounts to saying that we shall treat the slow-roll parameters as constant in
time.
Within these approximations, it is easy to compute the number of e-foldings
between the beginning and the end of inflation. If we indicate by $\phi_{i}$
and $\phi_{f}$ the values of the inflaton field at the beginning and at the
end of inflation, respectively, we find that the total number of e-foldings is
$\displaystyle N$ $\displaystyle\equiv$
$\displaystyle\int_{t_{i}}^{t_{f}}\,H\,dt$ (42) $\displaystyle\simeq$
$\displaystyle H\int^{\phi_{f}}_{\phi_{i}}\frac{d\phi}{\dot{\phi}}$
$\displaystyle\simeq$
$\displaystyle-3H^{2}\int^{\phi_{f}}_{\phi_{i}}\frac{d\phi}{V^{\prime}}$
$\displaystyle\simeq$ $\displaystyle-8\pi
G\int^{\phi_{f}}_{\phi_{i}}\frac{V}{V^{\prime}}\,d\phi\,.$
We may also compute the number of e-foldings $\Delta N$ which are left to go
to the end of inflation
$\Delta N\simeq 8\pi G\int^{\phi_{\Delta
N}}_{\phi_{f}}\frac{V}{V^{\prime}}\,d\phi,$ (43)
where $\phi_{\Delta N}$ is the value of the inflaton field when there are
$\Delta N$ e-foldings to the end of inflation.
1\. Comment: According to the criterion given in Subsection 2.4, a given scale
length $\lambda=a/k$ leaves the horizon when $k=aH_{k}$ where $H_{k}$ is the
the value of the Hubble rate at that time. One can easily compute the rate of
change of $H^{2}_{k}$ as a function of $k$
$\frac{d{\rm ln}\,H_{k}^{2}}{d{\rm ln}\,k}=\left(\frac{d{\rm
ln}\,H_{k}^{2}}{dt}\right)\left(\frac{dt}{d{\rm
ln}\,a}\right)\left(\frac{d{\rm ln}\,a}{d{\rm
ln}\,k}\right)=2\frac{\dot{H}}{H}\times\frac{1}{H}\times
1=2\frac{\dot{H}}{H^{2}}=-2\epsilon.$ (44)
2\. Comment: Take a given physical scale $\lambda$ today which crossed the
horizon scale during inflation. This happened when
$\lambda\left(\frac{a_{f}}{a_{0}}\right)e^{-\Delta
N_{\lambda}}=\lambda\left(\frac{T_{0}}{T_{f}}\right)e^{-\Delta
N_{\lambda}}=H_{I}^{-1}$
where $\Delta N_{\lambda}$ indicates the number of e-foldings from the time
the scale crossed the horizon during inflation and the end of inflation. This
relation gives a way to determine the number of e-foldings to the end of
inflation corresponding to a given scale
$\Delta N_{\lambda}\simeq 65+{\rm ln}\left(\frac{\lambda}{3000\,\,{\rm
Mpc}}\right)+2\,{\rm ln}\left(\frac{V^{1/4}}{10^{14}\,\,{\rm GeV}}\right)-{\rm
ln}\left(\frac{T_{f}}{10^{10}\,\,{\rm GeV}}\right).$
Scales relevant for the CMB anisotropies correspond to $\Delta N\sim$60.
Inflation ended when the potential energy associated with the inflaton field
became smaller than the kinetic energy of the field. By that time, any pre-
inflation entropy in the universe had been inflated away, and the energy of
the universe was entirely in the form of coherent oscillations of the inflaton
condensate around the minimum of its potential. The universe may be said to be
frozen after the end of inflation. We know that somehow the low-entropy cold
universe dominated by the energy of coherent motion of the $\phi$ field must
be transformed into a high-entropy hot universe dominated by radiation. The
process by which the energy of the inflaton field is transferred from the
inflaton field to radiation has been dubbed reheating. In the theory of
reheating, the simplest way to envisage this process is if the co-moving
energy density in the zero mode of the inflaton decays into normal particles,
which then scatter and thermalize to form a thermal background. It is usually
assumed that the decay width of this process is the same as the decay width of
a free inflaton field.
Of particular interest is a quantity usually known as the reheat temperature,
denoted as $T_{RH}$555So far, we have indicated it by $T_{f}$.. The reheat
temperature is calculated by assuming an instantaneous conversion of the
energy density in the inflaton field into radiation when the decay width of
the inflaton energy, $\Gamma_{\phi}$, is equal to $H$, the expansion rate of
the universe.
The reheat temperature is calculated quite easily. After inflation the
inflaton field executes coherent oscillations about the minimum of the
potential. Averaged over several oscillations, the coherent oscillation energy
density redshifts as matter: $\rho_{\phi}\propto a^{-3}$, where $a$ is the
Robertson–Walker scale factor. If we denote as $\rho_{I}$ and $a_{I}$ the
total inflaton energy density and the scale factor at the initiation of
coherent oscillations, then the Hubble expansion rate as a function of $a$ is
$H^{2}(a)=\frac{8\pi}{3}\frac{\rho_{I}}{{m_{\rm
Pl}}^{2}}\left(\frac{a_{I}}{a}\right)^{3}.$ (45)
Equating $H(a)$ and $\Gamma_{\phi}$ leads to an expression for $a_{I}/a$. Now
if we assume that all available coherent energy density is instantaneously
converted into radiation at this value of $a_{I}/a$, we can find the reheat
temperature by setting the coherent energy density,
$\rho_{\phi}=\rho_{I}(a_{I}/a)^{3}$, equal to the radiation energy density,
$\rho_{R}=(\pi^{2}/30)g_{*}T_{RH}^{4}$, where $g_{*}$ is the effective number
of relativistic degrees of freedom at temperature $T_{RH}$. The result is
$T_{RH}=\left(\frac{90}{8\pi^{3}g_{*}}\right)^{1/4}\sqrt{\Gamma_{\phi}{m_{\rm
Pl}}}\ =0.2\left(\frac{200}{g_{*}}\right)^{1/4}\sqrt{\Gamma_{\phi}{m_{\rm
Pl}}}\ .$ (46)
## 0.5 Inflation and the cosmological perturbations
As we have seen in the previous section, the early universe was made very
nearly uniform by a primordial inflationary stage. However, the important
caveat in that statement is the word ‘nearly’. Our current understanding of
the origin of structure in the universe is that it originated from small
‘seed’ perturbations, which over time grew to become all of the structure we
observe. Once the universe becomes matter dominated (around 1000 yrs after the
bang) primeval density inhomogeneities ($\delta\rho/\rho\sim 10^{-5}$) are
amplified by gravity and grow into the structure we see today [4]. The fact
that a fluid of self-gravitating particles is unstable to the growth of small
inhomogeneities was first pointed out by Jeans and is known as the Jeans
instability. Furthermore, the existence of these inhomogeneities was confirmed
by the COBE discovery of CMB anisotropies; the temperature anisotropies
detected almost certainly owe their existence to primeval density
inhomogeneities, since, as we have seen, causality precludes microphysical
processes from producing anisotropies on angular scales larger than about
$1^{\circ}$, the angular size of the horizon at last-scattering.
The growth of small matter inhomogeneities of wavelength smaller than the
Hubble scale ($\lambda\mathrel{\mathchoice{\lower
3.6pt\vbox{\halign{$\mathsurround=0pt\displaystyle\hfil#\hfil$\cr<\crcr\sim\crcr}}}{\lower
3.6pt\vbox{\halign{$\mathsurround=0pt\textstyle\hfil#\hfil$\cr<\crcr\sim\crcr}}}{\lower
3.6pt\vbox{\halign{$\mathsurround=0pt\scriptstyle\hfil#\hfil$\cr<\crcr\sim\crcr}}}{\lower
3.6pt\vbox{\halign{$\mathsurround=0pt\scriptscriptstyle\hfil#\hfil$\cr<\crcr\sim\crcr}}}}H^{-1}$)
is governed by a Newtonian equation:
${\ddot{\delta}}_{\bf k}+2H{\dot{\delta}}_{\bf
k}+v_{s}^{2}\frac{k^{2}}{a^{2}}\delta_{\bf k}=4\pi G\rho_{M}\delta_{\bf k},$
(47)
where $v_{s}^{2}=\partial P/\partial\rho_{M}$ is the square of the speed of
sound and we have expanded the perturbation to the matter density in plane
waves
${\delta\rho_{m}({\bf x},t)\over\rho_{m}}={1\over(2\pi)^{3}}\int
d^{3}k\,\delta_{\bf k}(t)e^{-i{\bf k}\cdot{\bf x}}.$ (48)
Competition between the pressure term and the gravity term on the rhs of Eq.
(47) determines whether or not pressure can counteract gravity: perturbations
with wavenumber larger than the Jeans wavenumber, $k_{J}^{2}=4\pi
Ga^{2}\rho_{m}/v_{s}^{2}$, are Jeans stable and just oscillate; perturbations
with smaller wavenumber are Jeans unstable and can grow.
Let us discuss solutions to this equation under different circumstances.
First, consider the Jeans problem, evolution of perturbations in a static
fluid, i.e., $H=0$. In this case Jeans unstable perturbations grow
exponentially, $\delta_{\bf k}\propto\exp(t/\tau)$ where
$\tau=1/\sqrt{4G\pi\rho_{M}}$. Next, consider the growth of Jeans unstable
perturbations in a matter-dominated universe, i.e., $H^{2}=8\pi G\rho_{M}/3$
and $a\propto t^{2/3}$. Because the expansion tends to pull particles away
from one another, the growth is only power law, $\delta_{\bf k}\propto
t^{2/3}$; i.e., at the same rate as the scale factor. Finally, consider a
radiation-dominated universe. In this case, the expansion is so rapid that
matter perturbations grow very slowly, as $\ln a$ in a radiation-dominated
epoch. Therefore, perturbations may grow only in a matter-dominated period.
Once a perturbation reaches an overdensity of order unity or larger it
separates from the expansion, i.e., it becomes its own self-gravitating system
and ceases to expand any further. In the process of virial relaxation, its
size decreases by a factor of two—density increases by a factor of 8;
thereafter, its density contrast grows as $a^{3}$ since the average matter
density is decreasing as $a^{-3}$, though smaller scales could become Jeans
unstable and collapse further to form smaller objects of higher density.
In order for structure formation to occur via gravitational instability, there
must have been small pre-existing fluctuations on physical length scales when
they crossed the Hubble radius in the radiation-dominated and matter-dominated
eras. In the standard Big Bang model these small perturbations have to be put
in by hand, because it is impossible to produce fluctuations on any length
scale while it is larger than the horizon. Since the goal of cosmology is to
understand the universe on the basis of physical laws, this appeal to initial
conditions is unsatisfactory. The challenge is therefore to give an
explanation to the small seed perturbations which allow the gravitational
growth of the matter perturbations.
Our best guess for the origin of these perturbations is quantum fluctuations
during an inflationary era in the early universe. Although originally
introduced as a possible solution to the cosmological conundrums such as the
horizon, flatness and entropy problems, by far the most useful property of
inflation is that it generates spectra of both density perturbations and
gravitational waves. These perturbations extend from extremely short scales to
scales considerably in excess of the size of the observable universe.
During inflation the scale factor grows quasi-exponentially, while the Hubble
radius remains almost constant. Consequently the wavelength of a quantum
fluctuation— either in the scalar field whose potential energy drives
inflation or in the graviton field—soon exceeds the Hubble radius. The
amplitude of the fluctuation therefore becomes ‘frozen in’. This is quantum
mechanics in action at macroscopic scales.
According to quantum field theory, empty space is not entirely empty. It is
filled with quantum fluctuations of all types of physical fields. The
fluctuations can be regarded as waves of physical fields with all possible
wavelenghts, moving in all possible directions. If the values of these fields,
averaged over some macroscopically large time, vanish then the space filled
with these fields seems to us empty and can be called the vacuum.
In the exponentially expanding universe the vacuum structure is much more
complicated. The wavelenghts of all vacuum fluctuations of the inflaton field
$\phi$ grow exponentially in the expanding universe. When the wavelength of
any particular fluctuation becomes greater than $H^{-1}$, this fluctuation
stops propagating, and its amplitude freezes at some non-zero value
$\delta\phi$ because of the large friction term $3H\dot{\phi}$ the equation of
motion of the field $\phi$. The amplitude of this fluctuation then remains
almost unchanged for a very long time, whereas its wavelength grows
exponentially. Therefore, the appearance of such frozen fluctuation is
equivalent to the appearance of a classical field $\delta\phi$ that does not
vanish after having averaged over some macroscopic interval of time. Because
the vacuum contains fluctuations of all possible wavelengths, inflation leads
to the creation of more and more new perturbations of the classical field with
wavelength larger than the horizon scale.
Once inflation has ended, however, the Hubble radius increases faster than the
scale factor, so the fluctuations eventually re-enter the Hubble radius during
the radiation- or matter-dominated eras. The fluctuations that exit around 60
$e$-foldings or so before reheating re-enter with physical wavelengths in the
range accessible to cosmological observations. These spectra provide a
distinctive signature of inflation. They can be measured in a variety of
different ways including the analysis of microwave background anisotropies.
Quantum fluctuations of the inflaton field are generated during inflation.
Since gravity talks to any component of the universe, small fluctuations of
the inflaton field are intimately related to fluctuations of the space-time
metric, giving rise to perturbations of the curvature ${\cal R}$ (which will
be defined in the following; the reader may loosely think of it as a
gravitational potential). The wavelengths $\lambda$ of these perturbations
grow exponentially and leave the horizon soon when $\lambda>R_{H}$. On
superhorizon scales, curvature fluctuations are frozen in and may be
considered as classical. Finally, when the wavelength of these fluctuations
re-enters the horizon, at some radiation- or matter-dominated epoch, the
curvature (gravitational potential) perturbations of the space-time give rise
to matter (and temperature) perturbations $\delta\rho$ via the Poisson
equation. These fluctuations will then start growing, giving rise to the
structures we observe today.
In summary, these are the key ingredients for understanding the observed
structures in the universe within the inflationary scenario:
* •
Quantum fluctuations of the inflaton field are excited during inflation and
stretched to cosmological scales. At the same time, being the inflaton
fluctuations connected to the metric perturbations through Einstein’s
equations, ripples on the metric are also excited and stretched to
cosmological scales.
* •
Gravity acts a messenger since it communicates the small seed perturbations to
photons and baryons once a given wavelength becomes smaller than the horizon
scale after inflation.
Let us now see how quantum fluctuations are generated during inflation. we
shall proceed by steps. First, we shall consider the simplest problem of
studying the quantum fluctuations of a generic scalar field during inflation:
we shall learn how perturbations evolve as a function of time and compute
their spectrum. Then—since a satisfactory description of the generation of
quantum fluctuations has to take both the inflaton and the metric
perturbations into account— we shall study the system composed by quantum
fluctuations of the inflaton field and quantum fluctuations of the metric.
## 0.6 Quantum fluctuations of a generic massless scalar field during
inflation
Let us first see how the fluctuations of a generic scalar field $\chi$, which
is not the inflaton field, behave during inflation. To warm up we first
consider a de Sitter epoch during which the Hubble rate is constant.
### 0.6.1 Quantum fluctuations of a generic massless scalar field during a de
Sitter stage
We assume this field to be massless. The massive case will be analysed in the
next subsection.
Expanding the scalar field $\chi$ in Fourier modes
$\delta\chi({\bf x},t)=\int\,\frac{d^{3}{\bf k}}{(2\pi)^{3/2}}\,e^{i{\bf
k}\cdot{\bf x}}\,\delta\chi_{{\bf k}}(t),$
we can write the equation for the fluctuations as
$\delta\ddot{\chi}_{\bf k}+3H\,\delta\dot{\chi}_{\bf
k}+\frac{k^{2}}{a^{2}}\,\delta\chi_{\bf k}=0\,.$ (49)
Let us study the qualitative behaviour of the solution to Eq. (49).
* •
For wavelengths within the horizon, $\lambda\ll H^{-1}$, the corresponding
wave-number satisfies the relation $k\gg a\,H$. In this regime, we can neglect
the friction term $3H\,\delta\dot{\chi}_{\bf k}$ and Eq. (49) reduces to
$\delta\ddot{\chi}_{\bf k}+\frac{k^{2}}{a^{2}}\,\delta\chi_{\bf k}=0,$ (50)
which is basically the equation of motion of an harmonic oscillator. Of
course, the frequency term $k^{2}/a^{2}$ depends upon time because the scale
factor $a$ grows exponentially. On the qualitative level, however, one expects
that when the wavelength of the fluctuation is within the horizon, the
fluctuation oscillates.
* •
For wavelengths above the horizon, $\lambda\gg H^{-1}$, the corresponding
wave-number satisfies the relation $k\ll aH$ and the term $k^{2}/a^{2}$ can be
safely neglected. Equation (49) reduces to
$\delta\ddot{\chi}_{\bf k}+3H\,\delta\dot{\chi}_{\bf k}=0,$ (51)
which tells us that on superhorizon scales $\delta\chi_{\bf k}$ remains
constant.
We have therefore the following picture: take a given fluctuation whose
initial wavelength $\lambda\sim a/k$ is within the horizon. The fluctuations
oscillate till the wavelength becomes of the order of the horizon scale. When
the wavelength crosses the horizon, the fluctuation ceases to oscillate and
gets frozen in.
Let us now study the evolution of the fluctuation in a more quantitative way.
To do so, we perform the following redefinition
$\delta\chi_{\bf k}=\frac{\delta\sigma_{\bf k}}{a}$
and we work in conformal time $d\tau=dt/a$. For the time being, we solve the
problem for a pure de Sitter expansion and we take the scale factor
exponentially growing as $a\sim e^{Ht}$; the corresponding conformal factor
reads (after choosing properly the integration constants)
$a(\tau)=-\frac{1}{H\tau}\,\,\,\,(\tau<0).$
In the following we shall also solve the problem in the case of quasi de
Sitter expansion. The beginning of inflation coincides with some initial time
$\tau_{i}\ll 0$. We find that Eq. (49) becomes
$\delta\sigma^{\prime\prime}_{\bf
k}+\left(k^{2}-\frac{a^{\prime\prime}}{a}\right)\delta\sigma_{\bf k}=0.$ (52)
We obtain an equation which is very ‘close’ to the equation for a Klein–Gordon
scalar field in flat space-time, the only difference being a negative time-
dependent mass term $-a^{\prime\prime}/a=-2/\tau^{2}$. Equation (52) can be
obtained from an action of the type
$\delta S_{\bf k}=\int\,d\tau\,\left[\frac{1}{2}\delta\sigma^{\prime 2}_{\bf
k}-\frac{1}{2}\left(k^{2}-\frac{a^{\prime\prime}}{a}\right)\delta\sigma^{2}_{\bf
k}\right],$ (53)
which is the canonical action for a simple harmonic oscillator with canonical
commutation relations $\delta\sigma^{*}_{\bf k}\delta\sigma^{\prime}_{\bf
k}-\delta\sigma_{\bf k}\delta\sigma^{*\prime}_{\bf k}=-i$.
Let us study the behaviour of this equation on subhorizon and superhorizon
scales. Since
$\frac{k}{aH}=-k\,\tau\,,$
on subhorizon scales $k^{2}\gg a^{\prime\prime}/a$ Equation (52) reduces to
$\delta\sigma^{\prime\prime}_{\bf k}+k^{2}\,\delta\sigma_{\bf k}=0\,,$
whose solution is a plane wave
$\delta\sigma_{\bf k}=\frac{e^{-ik\tau}}{\sqrt{2k}}\,\,\,\,(k\gg aH)\,.$ (54)
We find again that fluctuations with wavelength within the horizon oscillate
exactly like in flat space-time. This does not come as a surprise. In the
ultraviolet regime, that is for wavelengths much smaller than the horizon
scale, one expects that approximating the space-time as flat is a good
approximation.
On superhorizon scales, $k^{2}\ll a^{\prime\prime}/a$ Equation (52) reduces to
$\delta\sigma^{\prime\prime}_{\bf
k}-\frac{a^{\prime\prime}}{a}\delta\sigma_{\bf k}=0,$
which is satisfied by
$\delta\sigma_{\bf k}=B(k)\,a\,\,\,\,(k\ll aH)\,$ (55)
where $B(k)$ is a constant of integration. Roughly matching the (absolute
values of the) solutions $(\ref{q1})$ and $(\ref{x2})$ at $k=aH$ ($-k\tau=1$),
we can determine the (absolute value of the) constant $B(k)$
$\left|B(k)\right|a=\frac{1}{\sqrt{2k}}\Longrightarrow\left|B(k)\right|=\frac{1}{a\sqrt{2k}}=\frac{H}{\sqrt{2k^{3}}}.$
Going back to the original variable $\delta\chi_{\bf k}$, we obtain that the
quantum fluctuation of the $\chi$ field on superhorizon scales is constant and
approximately equal to
|δχ_k|≃H2k3 (ON SUPERHORIZON SCALES)
---
In fact we can do much better, since Eq. (52) has an exact solution:
$\delta\sigma_{\bf
k}=\frac{e^{-ik\tau}}{\sqrt{2k}}\left(1+\frac{i}{k\tau}\right).$ (56)
This solution reproduces all that we have found by qualitative arguments in
the two extreme regimes $k\ll aH$ and $k\gg aH$. We have performed the
matching procedure to show that the latter can be very useful to determine the
behaviour of the solution on superhorizon scales when the exact solution is
not known.
### 0.6.2 The power spectrum
Let us define now the power spectrum, a useful quantity to characterize the
properties of the perturbations. For a generic quantity $g({\bf x},t)$, which
can expanded in Fourier space as
$g({\bf x},t)=\int\,\frac{d^{3}{\bf k}}{(2\pi)^{3/2}}\,e^{i{\bf k}\cdot{\bf
x}}\,g_{{\bf k}}(t),$
the power spectrum can be defined as
$\langle 0|g^{*}_{{\bf k}_{1}}g_{{\bf
k}_{2}}|0\rangle\equiv\delta^{(3)}\left({\bf k}_{1}-{\bf
k}_{2}\right)\,\frac{2\pi^{2}}{k^{3}}\,{\cal P}_{g}(k),$ (57)
where $\left|0\right.\rangle$ is the vacuum quantum state of the system. This
definition leads to the usual relation
$\langle 0|g^{2}({\bf x},t)|0\rangle=\int\,\frac{dk}{k}\,{\cal P}_{g}(k).$
(58)
### 0.6.3 Quantum fluctuations of a generic scalar field in a quasi de Sitter
stage
So far, we have computed the time evolution and the spectrum of the quantum
fluctuations of a generic scalar field $\chi$ supposing that the scale factor
evolves like in a pure de Sitter expansion, $a(\tau)=-1/(H\tau)$. However,
during inflation the Hubble rate is not exactly constant, but changes with
time as $\dot{H}=-\epsilon\,H^{2}$ (quasi de Sitter expansion). In this
subsection, we shall solve for the perturbations in a quasi de Sitter
expansion. Using the definition of the conformal time, one can show that the
scale factor for small values of $\epsilon$ becomes
$a(\tau)=-\frac{1}{H}\frac{1}{\tau(1-\epsilon)}.$
The fluctuation mass-squared mass term is
$M^{2}(\tau)=m_{\chi}^{2}a^{2}-\frac{a^{\prime\prime}}{a},$
where
$\displaystyle\frac{a^{\prime\prime}}{a}$ $\displaystyle=$ $\displaystyle
a^{2}\left(\frac{\ddot{a}}{a}+H^{2}\right)=a^{2}\left(\dot{H}+2\,H^{2}\right)$
(59) $\displaystyle=$ $\displaystyle
a^{2}\left(2-\epsilon\right)H^{2}=\frac{\left(2-\epsilon\right)}{\tau^{2}\left(1-\epsilon\right)^{2}}$
$\displaystyle\simeq$
$\displaystyle\frac{1}{\tau^{2}}\left(2+3\epsilon\right).$
Armed with these results, we may compute the variance of the perturbations of
the generic $\chi$ field
$\displaystyle\langle 0|\left(\delta\chi({\bf x},t)\right)^{2}|0\rangle$
$\displaystyle=$
$\displaystyle\int\,\frac{d^{3}k}{(2\pi)^{3}}\,\left|\delta\chi_{\bf
k}\right|^{2}$ (60) $\displaystyle=$
$\displaystyle\int\,\frac{dk}{k}\,\frac{k^{3}}{2\pi^{2}}\,\left|\delta\chi_{\bf
k}\right|^{2}$ $\displaystyle=$ $\displaystyle\int\,\frac{dk}{k}\,{\cal
P}_{\delta\chi}(k),$
which defines the power spectrum of the fluctuations of the scalar field
$\chi$
${\cal P}_{\delta\chi}(k)\equiv\frac{k^{3}}{2\pi^{2}}\,\left|\delta\chi_{\bf
k}\right|^{2}.$ (61)
Since we have seen that fluctuations are (nearly) frozen in on superhorizon
scales, a way of characterizing the perturbations is to compute the spectrum
on scales larger than the horizon. For a massive scalar field, we obtain
${\cal
P}_{\delta\chi}(k)=\left(\frac{H}{2\pi}\right)^{2}\left(\frac{k}{aH}\right)^{3-2\nu_{\chi}},$
(62)
where, taking $m_{\chi}^{2}/H^{2}=3\eta_{\chi}$ and expanding for small values
of $\epsilon$ and $\eta$,
$\nu_{\chi}\simeq\frac{3}{2}+\epsilon-\eta_{\chi}.$ (63)
We may also define the spectral index $n_{\delta\chi}$ of the fluctuations as
n_δχ-1= d ln Pδϕd ln k=3-2ν_χ= 2η_χ-2ϵ.
---
The power spectrum of fluctuations of the scalar field $\chi$ is therefore
nearly flat, that is is nearly independent of the wavelength $\lambda=\pi/k$:
the amplitude of the fluctuation on superhorizon scales does almost not depend
upon the time at which the fluctuation crosses the horizon and becomes frozen
in. The small tilt of the power spectrum arises from the fact that the scalar
field $\chi$ is massive and because during inflation the Hubble rate is not
exactly constant, but nearly constant, where ‘nearly’ is quantified by the
slow-roll parameters $\epsilon$. Adopting the traditional terminology, we may
say that the spectrum of perturbations is blue if $n_{\delta\chi}>1$ (more
power in the ultraviolet) and red if $n_{\delta\chi}<1$ (more power in the
infrared). The power spectrum of the perturbations of a generic scalar field
$\chi$ generated during a period of slow-roll inflation may be either blue or
red. This depends upon the relative magnitude between $\eta_{\chi}$ and
$\epsilon$.
Comment: We might have computed the spectral index of the spectrum ${\cal
P}_{\delta\chi}(k)$ by first solving the equation for the perturbations of the
field $\chi$ in a di Sitter stage, with $H=$ constant and therefore
$\epsilon=0$, and then taking into account the time evolution of the Hubble
rate introducing the subscript in $H_{k}$ whose time variation is determined
by Eq. (44). Correspondingly, $H_{k}$ is the value of the Hubble rate when a
given wavelength $\sim k^{-1}$ crosses the horizon (from that point on the
fluctuation remains frozen in). The power spectrum in such an approach would
read
${\cal
P}_{\delta\chi}(k)=\left(\frac{H_{k}}{2\pi}\right)^{2}\left(\frac{k}{aH}\right)^{3-2\nu_{\chi}}$
(64)
with $3-2\nu_{\chi}\simeq\eta_{\chi}$. Using Eq. (44), one finds
$n_{\delta\chi}-1=\frac{d{\rm ln}\,{\cal P}_{\delta\phi}}{d{\rm
ln}\,k}=\frac{d{\rm ln}\,H_{k}^{2}}{d{\rm
ln}\,k}+3-2\nu_{\chi}=2\eta_{\chi}-2\epsilon$
which reproduces our previous findings.
Comment: Since on superhorizon scales
$\delta\chi_{\bf
k}\simeq\frac{H}{\sqrt{2k^{3}}}\left(\frac{k}{aH}\right)^{\eta_{\chi}-\epsilon}\simeq\frac{H}{\sqrt{2k^{3}}}\left[1+\left(\eta_{\chi}-\epsilon\right){\rm
ln}\,\left(\frac{k}{aH}\right)\right],$
we discover that
$\left|\delta\dot{\chi}_{\bf
k}\right|\simeq\left|H\left(\eta_{\chi}-\epsilon\right)\,\delta\chi_{\bf
k}\right|\ll\left|H\,\delta\chi_{\bf k}\right|,$ (65)
that is, on superhorizon scales the time variation of the perturbations can be
safely neglected.
## 0.7 Quantum fluctuations during inflation
As we have mentioned in the previous section, the linear theory of the
cosmological perturbations represents a cornerstone of modern cosmology and is
used to describe the formation and evolution of structures in the universe as
well as the anisotropies of the CMB. The seeds for these inhomogeneities were
generated during inflation and stretched over astronomical scales because of
the rapid superluminal expansion of the universe during the (quasi) de Sitter
epoch.
In the previous section we have already seen that pertubations of a generic
scalar field $\chi$ are generated during a (quasi) de Sitter expansion. The
inflaton field is a scalar field and, as such, we conclude that inflaton
fluctuations will be generated as well. However, the inflaton is special from
the point of view of perturbations. The reason is very simple. By assumption,
the inflaton field dominates the energy density of the universe during
inflation. Any perturbation in the inflaton field means a perturbation of the
stress energy momentum tensor
$\delta\phi\Longrightarrow\delta T_{\mu\nu}.$
A perturbation in the stress energy momentum tensor implies, through
Einstein’s equations of motion, a perturbation of the metric
$\delta T_{\mu\nu}\Longrightarrow\left[\delta
R_{\mu\nu}-\frac{1}{2}\delta\left(g_{\mu\nu}R\right)\right]=8\pi G\delta
T_{\mu\nu}\Longrightarrow\delta g_{\mu\nu}.$
On the other hand, a pertubation of the metric induces a back-reaction on the
evolution of the inflaton perturbation through the perturbed Klein–Gordon
equation of the inflaton field
$\delta
g_{\mu\nu}\Longrightarrow\delta\left(\partial_{\mu}\partial^{\mu}\phi+\frac{\partial
V}{\partial\phi}\right)=0\Longrightarrow\delta\phi.$
This logic chain makes us conclude that the perturbations of the inflaton
field and of the metric are tightly coupled to each other and have to be
studied together
δϕ⟺δg_μν .
---
As we shall see shortly, this relation is stronger than one might think
because of the issue of gauge invariance.
Before launching ourselves into the problem of finding the evolution of the
quantum perturbations of the inflaton field when they are coupled to gravity,
let us give a heuristic explanation of why we expect that during inflation
such fluctuations are indeed present.
If we take Eq. (34) and split the inflaton field as its classical value
$\phi_{0}$ plus the quantum flucutation $\delta\phi$, $\phi({\bf
x},t)=\phi_{0}(t)+\delta\phi({\bf x},t)$, the quantum perturbation
$\delta\phi$ satisfies the equation of motion
$\delta\ddot{\phi}+3H\,\delta{\dot{\phi}}-\frac{\nabla^{2}\delta\phi}{a^{2}}+V^{\prime\prime}\,\delta\phi=0.$
(66)
Differentiating Eq. (39) wrt time and taking $H$ constant (de Sitter
expansion) we find
$({\phi}_{0})^{\cdot\cdot\cdot}+3H\ddot{\phi}_{0}+V^{\prime\prime}\,\dot{\phi}_{0}=0.$
(67)
Let us consider for simplicity the limit ${\bf k}^{2}/a^{2}\ll 1$ and let us
disregard the gradient term. Under this condition we see that $\dot{\phi}_{0}$
and $\delta\phi$ solve the same equation. The solutions have therefore to be
related to each other by a constant of proportionality which depends upon time
$\delta\phi=-\dot{\phi}_{0}\,\delta t({\bf x}).$ (68)
This tells us that $\phi({\bf x},t)$ will have the form
$\phi({\bf x},t)=\phi_{0}\left({\bf x},t-\delta t({\bf x})\right).$
This equation indicates that the inflaton field does not acquire the same
value at a given time $t$ in all the space. On the contrary, when the inflaton
field is rolling down its potential, it acquires different values from one
spatial point ${\bf x}$ to the next. The inflaton field is not homogeneous and
fluctuations are present. These fluctuations, in turn, will induce
fluctuations in the metric.
### 0.7.1 The metric fluctuations
The mathematical tool to describe the linear evolution of the cosmological
perturbations is obtained by perturbing at the first order the FRW metric
$g^{(0)}_{\mu\nu}$, see Eq. (1)
$g_{\mu\nu}\quad=\quad
g^{(0)}_{\mu\nu}(t)\,+\,g_{\mu\nu}(\mathbf{x},t)\,;\qquad
g_{\mu\nu}\,\ll\,g^{(0)}_{\mu\nu}\,.$ (69)
The metric perturbations can be decomposed according to their spin with
respect to a local rotation of the spatial coordinates on hypersurfaces of
constant time. This leads to
* •
scalar perturbations
* •
vector perturbations
* •
tensor perturbations
Tensor perturbations or gravitational waves have spin 2 and are the true
degrees of freedom of the gravitational fields in the sense that they can
exist even in the vacuum. Vector perturbations are spin 1 modes arising from
rotational velocity fields and are also called vorticity modes. Finally,
scalar perturbations have spin 0.
Let us do a simple exercise to count how many scalar degrees of freedom are
present. Take a space-time of dimensions $D=n+1$, of which $n$ coordinates are
spatial coordinates. The symmetric metric tensor $g_{\mu\nu}$ has
$\frac{1}{2}(n+2)(n+1)$ degrees of freedom. We can perform $(n+1)$ coordinate
transformations in order to eliminate $(n+1)$ degrees of freedom, this leaves
us with $\frac{1}{2}n(n+1)$ degrees of freedom. These $\frac{1}{2}n(n+1)$
degrees of freedom contain scalar, vector and tensor modes. According to
Helmholtz’s theorem we can always decompose a vector $u_{i}$ $(i=1,\cdots,n)$
as $u_{i}=\partial_{i}v+v_{i}$, where $v$ is a scalar (usually called
potential flow) which is curl-free, $v_{[i,j]}=0$, and $v_{i}$ is a real
vector (usually called vorticity) which is divergence-free, $\nabla\cdot v=0$.
This means that the real vector (vorticity) modes are $(n-1)$. Furthermore, a
generic traceless tensor $\Pi_{ij}$ can always be decomposed as
$\Pi_{ij}=\Pi^{S}_{ij}+\Pi_{ij}^{V}+\Pi_{ij}^{T}$, where
$\Pi^{S}_{ij}=\left(-\frac{k_{i}k_{j}}{k^{2}}+\frac{1}{3}\delta_{ij}\right)\Pi$,
$\Pi^{V}_{ij}=(-i/2k)\left(k_{i}\Pi_{j}+k_{j}\Pi_{i}\right)$
$(k_{i}\Pi_{i}=0)$ and $k_{i}\Pi^{T}_{ij}=0$. This means that the true
symmetric, traceless and transverse tensor degreees of freedom are
$\frac{1}{2}(n-2)(n+1)$.
The number of scalar degrees of freedom is therefore
$\frac{1}{2}n(n+1)-(n-1)-\frac{1}{2}(n-2)(n+1)=2,$
while the degrees of freedom of true vector modes are $(n-1)$ and the number
of degrees of freedom of true tensor modes (gravitational waves) is
$\frac{1}{2}(n-2)(n+1)$. In four dimensions $n=3$, meaning that one expects 2
scalar degrees of freedom, 2 vector degrees of freedom and 2 tensor degrees of
freedom. As we shall see, to the 2 scalar degrees of freedom from the metric,
one has to add another one, the inflaton field perturbation $\delta\phi$.
However, since Einstein’s equations will tell us that the two scalar degrees
of freedom from the metric are equal during inflation, we expect a total
number of scalar degrees of freedom equal to 2.
At the linear order, the scalar, vector, and tensor perturbations evolve
independently (they decouple) and it is therefore possible to analyse them
separately. Vector perturbations are not excited during inflation because
there are no rotational velocity fields during the inflationary stage. we
shall analyse the generation of tensor modes (gravitational waves) in the
following. For the time being we want to focus on the scalar degrees of
freedom of the metric.
Considering only the scalar degrees of freedom of the perturbed metric, the
most generic perturbed metric reads
$g_{\mu\nu}\,=\,a^{2}\left(\begin{array}[]{c c}-1\,-\,2\,\Phi&\partial_{i}B\\\
\partial_{i}B&\left(1\,-\,2\,\psi\right)\delta_{ij}\,+\,D_{ij}E\\\
\end{array}\right),$ (70)
while the line-element can be written as
$ds^{2}\,=\,a^{2}\big{(}(-1-2\,\Phi)d\tau^{2}\,+\,2\,\partial_{i}B\,d\tau\,dx^{i}\,+\,\left((1-2\,\psi)\delta_{ij}\,+\,D_{ij}E\right)\,dx^{i}\,dx^{j}\big{)}.$
(71)
Here
$D_{ij}\,=\left(\partial_{i}\partial_{j}\,-\,\frac{1}{3}\,\delta_{ij}\,\nabla^{2}\right)$.
### 0.7.2 The issue of gauge invariance
When studying the cosmological density perturbations, what we are interested
in is following the evolution of a space-time which is neither homogeneous nor
isotropic. This is done by following the evolution of the differences between
the actual space-time and a well understood reference space-time. So we shall
consider small perturbations away from the homogeneous, isotropic space-time.
The reference system in our case is the spatially flat
Friedmann–Robertson–Walker (FRW) space-time, with line element
$ds^{2}=a^{2}(\tau)\left\\{d\tau^{2}-\delta_{ij}dx^{i}dx^{j}\right\\}$. Now,
the key issue is that general relativity is a gauge theory where the gauge
transformations are the generic coordinate transformations from one local
reference frame to another.
When we compute the perturbation of a given quantity, this is defined to be
the difference between the value that this quantity assumes on the real
physical space-time and the value it assumes on the unperturbed background.
Nonetheless, to perform a comparison between these two values, it is necessary
to compute them at the same space-time point. Since the two values live on two
different geometries, it is necessary to specify a map which allows one to
link univocally the same point on the two different space-times. This
correspondence is called a gauge choice and changing the map means performing
a gauge transformation.
Fixing a gauge in general relativity implies choosing a coordinate system. A
choice of coordinates defines a threading of space-time into lines
(corresponding to fixed spatial coordinates ${\bf x}$) and a slicing into
hypersurfaces (corresponding to fixed time $\tau$). A choice of coordinates is
called a gauge and there is no unique preferred gauge
GAUGE CHOICE ⟺ SLICING AND THREADING
---
From a more formal point of view, operating an infinitesimal gauge
transformation on the coordinates
$\widetilde{x^{\mu}}\,=\,x^{\mu}\,+\,\delta x^{\mu}$ (72)
implies on a generic quantity $Q$ a transformation on its perturbation
$\widetilde{{\delta Q}}\,=\,{\delta Q}\,+\,\pounds_{\delta x}\,Q_{0}\,$ (73)
where $Q_{0}$ is the value assumed by the quantity $Q$ on the background and
$\pounds_{\delta x}$ is the Lie-derivative of $Q$ along the vector $\delta
x^{\mu}$.
Decomposing in the usual manner the vector $\delta x^{\mu}$
$\displaystyle\delta x^{0}\,$ $\displaystyle=$
$\displaystyle\,\xi^{0}(x^{\mu})\,;$ $\displaystyle\delta x^{i}\,$
$\displaystyle=$
$\displaystyle\,\partial^{i}\beta(x^{\mu})\,+\,v^{i}(x^{\mu})\,;\qquad\partial_{i}v^{i}\,=\,0\,,$
(74)
we can easily deduce the transformation law of a scalar quantity $f$ (like the
inflaton scalar field $\phi$ and energy density $\rho$). Instead of applying
the formal definition (73), we find the transformation law in an alternative
(and more pedagogical) way. We first write $\delta f(x)=f(x)-f_{0}(x)$, where
$f_{0}(x)$ is the background value. Under a gauge transformation we have
$\widetilde{\delta
f}(\widetilde{x^{\mu}})=\widetilde{f}(\widetilde{x^{\mu}})-\widetilde{f}_{0}(\widetilde{x^{\mu}})$.
Since $f$ is a scalar we can write $f(\widetilde{x^{\mu}})=f(x^{\mu})$ (the
value of the scalar function in a given physical point is the same in all the
coordinate system). On the other side, on the unperturbed background
hypersurface $\widetilde{f}_{0}=f_{0}$. We have therefore
$\displaystyle\widetilde{\delta f}(\widetilde{x^{\mu}})$ $\displaystyle=$
$\displaystyle\widetilde{f}(\widetilde{x^{\mu}})-\widetilde{f}_{0}(\widetilde{x^{\mu}})$
$\displaystyle=$ $\displaystyle f(x^{\mu})-f_{0}(\widetilde{x^{\mu}})$
$\displaystyle=$ $\displaystyle
f\left(\widetilde{x^{\mu}}\right)-f_{0}(\widetilde{x^{\mu}})$ $\displaystyle=$
$\displaystyle f(\widetilde{x^{\mu}})-\delta x^{\mu}\,\frac{\partial
f}{\partial x^{\mu}}(\widetilde{x})-f_{0}(\widetilde{x^{\mu}}),$
from which we finally deduce, being $f_{0}=f_{0}(x^{0})$,
~δf=δf-f^′ ξ^0
---
For the spin-zero perturbations of the metric, we can proceed analogously. We
use the following trick. Upon a coordinate transformation
$x^{\mu}\rightarrow\widetilde{x^{\mu}}=x^{\mu}+\delta x^{\mu}$, the line
element is left invariant, $ds^{2}=\widetilde{ds^{2}}$. This implies, for
instance, that
$a^{2}(\widetilde{x^{0}})\left(1+\widetilde{\Phi}\right)\left(d\widetilde{x^{0}}\right)^{2}=a^{2}(x^{0})\left(1+\Phi\right)(dx^{0})^{2}$.
Since $a^{2}(\widetilde{x^{0}})\simeq a^{2}(x^{0})+2a\,a^{\prime}\,\xi^{0}$
and $d\widetilde{x^{0}}=\left(1+\xi^{0\prime}\right)dx^{0}+\frac{\partial
x^{0}}{\partial x^{i}}\,dx^{i}$, we obtain
$1+2\Phi=1+2\widetilde{\Phi}+2\,{\cal H}\xi^{0}+2\xi^{0\prime}$. We now may
introduce in detail some gauge-invariant quantities which play a major role in
the computation of the density perturbations. In the following we shall be
interested only in the coordinate transformations on constant time
hypersurfaces and therefore gauge invariance will be equivalent to
independence of the slicing.
### 0.7.3 The co-moving curvature perturbation
The intrinsic spatial curvature on hypersurfaces on constant conformal time
$\tau$ and for a flat universe is given by
${}^{(3)}R=\frac{4}{a^{2}}\nabla^{2}\,\psi.$
The quantity $\psi$ is usually referred to as the curvature perturbation. We
have seen, however, that the curvature potential $\psi$ is not gauge
invariant, but is defined only on a given slicing. Under a transformation on
constant time hypersurfaces $t\rightarrow t+\delta\tau$ (change of the
slicing)
$\psi\rightarrow\psi+\,{\cal H}\,\delta\tau.$
We now consider the co-moving slicing which is defined to be the slicing
orthogonal to the worldlines of co-moving observers. The latter are are free-
falling and the expansion defined by them is isotropic. In practice, what this
means is that there is no flux of energy measured by these observers, that is
$T_{0i}=0$. During inflation this means that these observers measure
$\delta\phi_{\rm com}=0$ since $T_{0i}$ goes like $\partial_{i}\delta\phi({\bf
x},\tau)\phi^{\prime}(\tau)$.
Since $\delta\phi\rightarrow\delta\phi-\phi^{\prime}\delta\tau$ for a
transformation on constant time hypersurfaces, this means that
$\delta\phi\rightarrow\delta\phi_{\rm
com}=\delta\phi-\phi^{\prime}\,\delta\tau=0\Longrightarrow\delta\tau=\frac{\delta\phi}{\phi^{\prime}},$
that is $\delta\tau=\frac{\delta\phi}{\phi^{\prime}}$ is the time-displacement
needed to go from a generic slicing with generic $\delta\phi$ to the co-moving
slicing where $\delta\phi_{\rm com}=0$. At the same time the curvature
perturbation $\psi$ transforms into
$\psi\rightarrow\psi_{\rm com}=\psi+\,{\cal H}\,\delta\tau=\psi+\,{\cal
H}\frac{\delta\phi}{\phi^{\prime}}.$
The quantity
R=ψ\+ Hδϕϕ′=ψ+Hδϕ˙ϕ
---
is the co-moving curvature perturbation. This quantity is gauge invariant by
construction and is related to the gauge-dependent curvature perturbation
$\psi$ on a generic slicing to the inflaton perturbation $\delta\phi$ in that
gauge. By construction, the meaning of ${\cal R}$ is that it represents the
gravitational potential on co-moving hypersurfaces where $\delta\phi=0$ or the
inflaton fluctuation hypersurfaces where $\psi=0$:
${\cal
R}=\left.\psi\right|_{\delta\phi=0}=\left.H\frac{\delta\phi}{\dot{\phi}}\right|_{\psi=0}.$
The power spectrum of the curvature perturbation may then be easily computed
${\cal R}_{\bf k}=H\,\frac{\delta\phi_{\bf k}}{\dot{\phi}}.$ (76)
We may now compute the power spectrum of the co-moving curvature perturbation
on superhorizon scales
P_R(k)=12mPl2ϵ(H2π)^2 (kaH)^n_R-1≡A^2_R (kaH)^n_R-1
---
where we have defined the spectral index $n_{{\cal R}}$ of the co-moving
curvature perturbation as
n_R-1= d ln PRd ln k=3-2ν= 2η-6ϵ.
---
We conclude that inflation is responsible for the generation of
adiabatic/curvature perturbations with an almost scale-independent spectrum.
To compute the spectral index of the spectrum ${\cal P}_{{\cal R}}(k)$ we have
proceeded as follows: first solve the equation for the perturbation
$\delta\phi_{\bf k}$ in a de Sitter stage, with $H=$ constant
($\epsilon=\eta=0$), whose solution is Eq. (56) and then taking into account
the time-evolution of the Hubble rate and of $\phi$ introducing the subscript
in $H_{k}$ and $\dot{\phi}_{k}$. The time variation of the latter is
determined by
$\frac{d{\rm ln}\,\dot{\phi}_{k}}{d{\rm ln}\,k}=\left(\frac{d{\rm
ln}\,\dot{\phi}_{k}}{dt}\right)\left(\frac{dt}{d{\rm
ln}\,a}\right)\left(\frac{d{\rm ln}\,a}{d{\rm
ln}\,k}\right)=\frac{\ddot{\phi}_{k}}{\dot{\phi}_{k}}\times\frac{1}{H}\times
1=-\delta=\epsilon-\eta.$ (77)
Correspondingly, $\dot{\phi}_{k}$ is the value of the time derivative of the
inflaton field when a given wavelength $\sim k^{-1}$ crosses the horizon (from
that point on the fluctuations remains frozen in). The curvature perturbation
in such an approach would read
${\cal R}_{\bf k}\simeq\frac{H_{k}}{\dot{\phi}_{k}}\,\delta\phi_{\bf
k}\simeq\frac{1}{2\pi}\left(\frac{H_{k}^{2}}{\dot{\phi}_{k}}\right).$
Correspondingly
$n_{{\cal R}}-1=\frac{d{\rm ln}\,{\cal P}_{{\cal R}}}{d{\rm
ln}\,k}=\frac{d{\rm ln}\,H_{k}^{4}}{d{\rm ln}\,k}-\frac{d{\rm
ln}\,\dot{\phi}_{k}^{2}}{d{\rm
ln}\,k}=-4\epsilon+(2\eta-2\epsilon)=2\eta-6\epsilon.$
During inflation the curvature perturbation is generated on superhorizon
scales with a spectrum which is nearly scale invariant [13], that is, is
nearly independent of the wavelength $\lambda=\pi/k$: the amplitude of the
fluctuation on superhorizon scales does not (almost) depend upon the time at
which the fluctuation crosses the horizon and becomes frozen in. The small
tilt of the power spectrum arises from the fact that the inflaton field is
massive, giving rise to a non-vanishing $\eta$ and because during inflation
the Hubble rate is not exactly constant, but nearly constant, where ‘nearly’
is quantified by the slow-roll parameters $\epsilon$.
Comment: From what we have found so far, we may conclude that on superhorizon
scales the co-moving curvature perturbation ${\cal R}$ and the uniform-density
gauge curvature $\zeta$ satisfy on superhorizon scales the relation
$\dot{\cal R}_{\bf k}\simeq 0.$
### 0.7.4 Gravitational waves
Quantum fluctuations in the gravitational fields are generated in a similar
fashion to that of the scalar perturbations discussed so far. A gravitational
wave may be viewed as a ripple of space-time in the FRW background metric (1)
and in general the linear tensor perturbations may be written as
$g_{\mu\nu}=a^{2}(\tau)\left[-d\tau^{2}+\left(\delta_{ij}+h_{ij}\right)dx^{i}dx^{j}\right],$
where $\left|h_{ij}\right|\ll 1$. The tensor $h_{ij}$ has six degrees of
freedom, but, as we studied in Subsection 7.1, the tensor perturbations are
traceless, $\delta^{ij}h_{ij}=0$, and transverse $\partial^{i}h_{ij}=0$
$(i=1,2,3)$. With these four constraints, there remain two physical degrees of
freedom, or polarizations, which are usually indicated $\lambda=+,\times$.
More precisely, we can write
$h_{ij}=h_{+}\,e_{ij}^{+}+h_{\times}\,e_{ij}^{\times},$
where $e^{+}$ and $e^{\times}$ are the polarization tensors which have the
following properties
$e_{ij}=e_{ji},\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
k^{i}e_{ij}=0,\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
,e_{ii}=0,$ $e_{ij}(-{\bf k},\lambda)=e^{*}_{ij}({\bf
k},\lambda),\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\sum_{\lambda}\,e^{*}_{ij}({\bf k},\lambda)e^{ij}({\bf k},\lambda)=4.$
Notice also that the tensors $h_{ij}$ are gauge-invariant and therefore
represent physical degrees of freedom.
If the stress-energy momentum tensor is diagonal, as the one provided by the
inflaton potential $T_{\mu\nu}=\partial_{\mu}\phi\partial_{\nu}\phi-
g_{\mu\nu}{\cal L}$, the tensor modes do not have any source in their equation
and their action can be written as
$\frac{{m_{\rm
Pl}}^{2}}{2}\,\int\,d^{4}x\,\sqrt{-g}\,\frac{1}{2}\partial_{\sigma}h_{ij}\,\partial^{\sigma}h_{ij},$
that is the action of four independent massless scalar fields. The gauge-
invariant tensor amplitude
$v_{\bf k}=a{m_{\rm Pl}}\frac{1}{\sqrt{2}}\,h_{\bf k},$
satisfies therefore the equation
$v_{\bf k}^{\prime\prime}+\left(k^{2}-\frac{a^{\prime\prime}}{a}\right)v_{\bf
k}=0,$
which is the equation of motion of a massless scalar field in a quasi-de
Sitter epoch. We can therefore make use of the results present in Subsection
6.5 and Eq. (63) to conclude that on superhorizon scales the tensor modes
scale like
$\left|v_{\bf
k}\right|=\left(\frac{H}{2\pi}\right)\left(\frac{k}{aH}\right)^{\frac{3}{2}-\nu_{T}},$
where
$\nu_{T}\simeq\frac{3}{2}-\epsilon.$
Since fluctuations are (nearly) frozen in on superhorizon scales, a way of
characterizing the tensor perturbations is to compute the spectrum on scales
larger than the horizon
${\cal P}_{T}(k)=\frac{k^{3}}{2\pi^{2}}\sum_{\lambda}\left|h_{\bf
k}\right|^{2}=4\times 2\frac{k^{3}}{2\pi^{2}}\left|v_{\bf k}\right|^{2}.$ (78)
This gives the power spectrum on superhorizon scales
P_T(k)=8mPl2(H2π)^2 (kaH)^n_T≡A^2_T (kaH)^n_T
---
where we have defined the spectral index $n_{T}$ of the tensor perturbations
as
n_T= d ln PTd ln k=3-2ν_T= -2ϵ.
---
The tensor perturbation is almost scale-invariant. Notice that the amplitude
of the tensor modes depends only on the value of the Hubble rate during
inflation. This amounts to saying that it depends only on the energy scale
$V^{1/4}$ associated to the inflaton potential. A detection of gravitational
waves from inflation will therefore be a direct measurement of the energy
scale associated to inflation.
### 0.7.5 The consistency relation
The results obtained so far for the scalar and tensor perturbations allow one
to predict a consistency relation which holds for the models of inflation
addressed in these lectures, i.e., the models of inflation driven by one-
single field $\phi$. We define the tensor-to-scalar amplitude ratio to be
$r=\frac{\frac{1}{100}A_{T}^{2}}{\frac{4}{25}A_{\cal
R}^{2}}=\frac{\frac{1}{100}8\left(\frac{H}{2\,\pi\,{m_{\rm
Pl}}}\right)^{2}}{\frac{4}{25}(2\epsilon)^{-1}\left(\frac{H}{2\,\pi\,{m_{\rm
Pl}}}\right)^{2}}=\epsilon.$
This means that
r=-nT2
---
One-single models of inflation predict that during inflation driven by a
single scalar field, the ratio between the amplitude of the tensor modes and
that of the curvature perturbations is equal to minus one-half of the tilt of
the spectrum of tensor modes. If this relation turns out to be falsified by
the future measurements of the CMB anisotropies, this does not mean that
inflation is wrong, but only that inflation has not been driven by only one
field.
### 0.7.6 From the inflationary seeds to the matter power spectrum
As the curvature perturbations enter the causal horizon during radiation- or
matter-domination, they create density fluctuations $\delta\rho_{\bf k}$ via
gravitational attractions of the potential wells. The density contrast
$\delta_{\bf k}=\frac{\delta\rho_{\bf k}}{\overline{\rho}}$ can be deduced
from the Poisson equation
$\frac{k^{2}\Phi_{\bf k}}{a^{2}}=-4\pi G\,\delta\rho_{\bf k}=-4\pi
G\,\frac{\delta\rho_{\bf
k}}{\overline{\rho}}\,\overline{\rho}=\frac{3}{2}\,H^{2}\,\frac{\delta\rho_{\bf
k}}{\overline{\rho}}$
where $\overline{\rho}$ is the background average energy density. This means
that
$\delta_{\bf k}=\frac{2}{3}\,\left(\frac{k}{aH}\right)^{2}\,\Phi_{\bf k}.$
From this expression we can compute the power spectrum of matter density
perturbations induced by inflation when they re-enter the horizon during
matter-domination:
${\cal P}_{\delta\rho}=\langle\left|\delta_{\bf
k}\right|^{2}\rangle=A\,\left(\frac{k}{aH}\right)^{n}=\frac{2\pi^{2}}{k^{3}}\left(\frac{2}{5}\right)^{2}A^{2}_{\cal
R}\left(\frac{k}{aH}\right)^{4}\,\left(\frac{k}{aH}\right)^{n_{{\cal R}}-1}$
from which we deduce that matter perturbations scale linearly with the wave-
number and have a scalar tilt
$n=n_{{\cal R}}=1+2\eta-6\epsilon.$
The primordial spectrum ${\cal P}_{\delta\rho}$ is of course reprocessed by
gravitational instabilities after the universe becomes matter-dominated.
Indeed, as we have seen in Section 6, perturbations evolve after entering the
horizon and the power spectrum will not remain constant. To see how the
density contrast is reprocessed we have first to analyse how it evolves on
superhorizon scales before horizon-crossing. We use the following trick.
Consider a flat universe with average energy density $\overline{\rho}$. The
corresponding Hubble rate is
$H^{2}=\frac{8\pi G}{3}\,\overline{\rho}.$
A small positive fluctuation $\delta\rho$ will cause the universe to be
closed:
$H^{2}=\frac{8\pi
G}{3}\left(\overline{\rho}+\delta\rho\right)-\frac{k}{a^{2}}.$
Substracting the two equations we find
$\frac{\delta\rho}{\rho}=\frac{3}{8\pi
G}\frac{k}{a^{2}\rho}\sim\left\\{\begin{array}[]{cc}a^{2}&{\rm RD}\\\ a&{\rm
MD}\end{array}\right.$
Notice that $\Phi_{\bf k}\sim\delta\rho a^{2}/k^{2}\sim(\delta\rho/\rho)\rho
a^{2}/k^{2}=$ constant for both RD and MD which confirms our previous
findings.
When the matter densities enter the horizon, they do not increase appreciably
before matter-domination because before matter-domination pressure is too
large and does not allow the matter inhomogeneities to grow. On the other
hand, the suppression of growth due to radiation is restricted to scales
smaller than the horizon, while large-scale perturbations remain unaffected.
This is why the horizon size at equality sets an important scale for structure
growth:
$k_{\rm EQ}=H^{-1}\left(a_{\rm EQ}\right)\simeq 0.08\,h\,{\rm Mpc}^{-1}.$
Therefore, perturbations with $k\gg k_{\rm EQ}$ are perturbations which have
entered the horizon before matter-domination and have remained nearly constant
till equality. This means that they are suppressed with respect to those
perturbations having $k\ll k_{\rm EQ}$ by a factor $(a_{\rm ENT}/a_{\rm
EQ})^{2}=(k_{\rm EQ}/k)^{2}$. If we define the transfer function $T(k)$ by the
relation ${\cal R}_{\rm final}=T(k)\,{\cal R}_{\rm initial}$ we find therefore
that roughly speaking
$T(k)=\left\\{\begin{array}[]{cc}1&k\ll k_{\rm EQ},\\\ (k_{\rm EQ}/k)^{2}&k\gg
k_{\rm EQ}.\end{array}\right.$
The corresponding power spectrum will be
${\cal
P}_{\delta\rho}(k)\sim\left\\{\begin{array}[]{cc}\left(\frac{k}{aH}\right)&k\ll
k_{\rm EQ},\\\ \left(\frac{k}{aH}\right)^{-3}&k\gg k_{\rm
EQ}.\end{array}\right.$
Of course, a more careful computation needs to include many other effects such
as neutrino free-streaming, photon diffusion and the diffusion of baryons
along with photons. It is encouraging, however, that this rough estimate turns
out to be confirmed by present data on large-scale structures [4].
The next step would be to investigate how the primordial perturbations
generated by inflation flow into the CMB to produce their anisotropies.
## 0.8 From inflation to large-angle CMB anisotropy
As we have already mentioned, the high temperature of the early universe
maintained a low equilibrium fraction of neutral atoms, and a correspondingly
high number density of free electrons. Coulomb scattering between the ions and
electrons kept them in local kinetic equilibrium, and Thomson scattering of
photons tended to maintain the isotropy of the CMB in the baryon rest frame.
As the universe expanded and cooled, the dominant element hydrogen started to
recombine when the temperature fell below $\sim$ 4000 K. This is a factor of
40 lower than might be anticipated from the 13.6 eV ionization potential of
hydrogen, and is due to the large ratio of the number of photons to baryons.
Through recombination, the mean-free path for Thomson scattering grew to the
horizon size and CMB photons “decoupled” from matter. More precisely, the
probability density that photons last scattered at some time defines the
visibility function. This is now known to peak 380 kyr after the Big Bang with
a width $\sim 120$ kyr. Since then, CMB photons have propagated relatively
unimpeded for $13.7\leavevmode\nobreak\ \mathrm{Gyr}$, covering a co-moving
distance $\sim 14.1\,\mathrm{Gpc}$. The distribution of their energies carries
the imprint of fluctuations in the radiation temperature, the gravitational
potentials, and the peculiar velocity of the radiation where they last
scattered, as the temperature anisotropies that we observe today.
Temperature fluctuations in the CMB arise due to various distinct physical
effects: our peculiar velocity with respect to the cosmic rest frame;
fluctuations in the gravitational potential on the last scattering surface;
fluctuations intrinsic to the radiation field itself on the last
scatteringsurface; the peculiar velocity of the last scatteringsurface and
damping of anisotropies if the universe should be re-ionized after decoupling.
The first effect gives rise to the dipole anisotropy. Finally, there is the
contribution from the evolution of the anisotropies from the last scattering
surface till today (which we shall neglect from now on).
The second effect, known as the Sachs–Wolfe effect is the dominat contribution
to the anisotropy on large-angular scales, $\theta\gg\theta_{\rm HOR}\sim
1^{\circ}$. The last three effects provide the dominant contributions to the
anisotropy on small-angular scales, $\theta\ll 1^{\circ}$.
### 0.8.1 Sachs–Wolfe plateau
We consider first the temperature fluctuations on large-angular scales that
arise due to the Sachs–Wolfe effect. These anisotropies probe length scales
that were superhorizon-sized at photon decoupling and therefore insensitive to
microphysical processes. On the contrary, they provide a probe of the original
spectrum of primeval fluctuations produced during inflation.
To proceed, we consider the CMB anisotropy measured at positions other than
our own and at earlier times. This is called the brightness function
$\Theta(t,{\bf x},{\bf n})\equiv\delta T(t,{\bf x},{\bf n})/T(t)$. The photons
with momentum ${\bf p}$ in a given range $d^{3}p$ have intensity $I$
proportional to $T^{4}(t,{\bf x},{\bf n})$ and therefore $\delta I/I=4\Theta$.
The brightness function depends upon the direction ${\bf n}$ of the photon
momentum or, equivalently, on the direction of observation ${\bf e}=-{\bf n}$.
Because the CMB travels freely from the last-scattering, we can write
$\frac{\delta T}{T}=\Theta\left(t_{\rm LS},{\bf x}_{\rm LS},{\bf
n}\right)+\left(\frac{\delta T}{T}\right)_{*},$
where ${\bf x}_{\rm LS}=-x_{\rm LS}{\bf n}$ is the point of the origin of the
photon coming from the direction ${\bf e}$. The co-moving distance of the last
scatteringdistance is $x_{\rm LS}=2/H_{0}$. The first term corresponds to the
anisotropy already present at last scattering and the second term is the
additional anisotropy acquired during the travel towards us, equal to minus
the fractional pertubation in the redshift of the radiation. Notice that the
separation between each term depends on the slicing, but the sum does not.
Consider the redshift perturbation on co-moving slicing. We imagine the
universe populated by co-moving observers along the line of sight. The
relative velocity of adjacent co-moving observers is equal to their distance
times the velocity gradient measured along ${\bf n}$ of the photon. In the
unperturbed universe, we have ${\bf u}=H{\bf r}$, leading to the velocity
gradient $u_{ij}=\partial u_{i}/\partial r_{j}=u_{ij}=H(t)\delta_{ij}$ with
zero vorticity and shear. Including a peculiar velocity field as perturbation,
${\bf u}=H{\bf r}+{\bf v}$ and
$u_{ij}=H(t)\delta_{ij}+\frac{1}{a}\frac{\partial v_{i}}{\partial v_{j}}$. The
corresponding Doppler shift is
$\frac{d\lambda}{\lambda}=\frac{da}{a}+n_{i}n_{j}\frac{\partial
v_{i}}{\partial x_{j}}dx.$
The perturbed FRW equation is
$\delta H=\frac{1}{3}\nabla\cdot{\bf v},$
while
$(\delta\rho)^{\cdot}=-3\rho\delta H-3H\delta\rho.$
Instead of $\delta\rho$, let us work with the density contrast
$\delta=\delta\rho/\rho$. Remembering that $\rho\sim a^{-3}$, we find that
$\dot{\delta}=-3\delta H$, which gives
$\nabla\cdot{\bf v}=-\dot{\delta}_{\bf k}.$
From the Euler equation $\dot{\bf u}=-\rho^{-1}\nabla p-\nabla\Phi$, we deduce
$\dot{\bf v}+H{\bf v}=-\nabla\Phi-\rho^{-1}\nabla p$. Therefore, for $a\sim
t^{2/3}$ and negligible pressure gradient, since the gravitational potential
is constant, we find
${\bf v}=-t\nabla\Phi$
leading to
$\left(\frac{\delta T}{T}\right)_{*}=\int_{0}^{x_{\rm
LS}}\,\frac{t}{a}\frac{d^{2}\Phi}{dx^{2}}\,dx.$ (79)
The photon trajectory is $ad{\bf x}/dt={\bf n}$. Using $a\sim t^{2/3}$ gives
$x(t)=\int_{t}^{t_{0}}\frac{dt^{\prime}}{a}=3\left(\frac{a_{0}}{t_{0}}-\frac{t}{a}\right).$
Integrating by parts Eq. (79), we finally find
$\left(\frac{\delta T}{T}\right)_{*}=\frac{1}{3}\left[\Phi({\bf x}_{\rm
LS})-\Phi(0)\right]+{\bf e}\cdot\left[{\bf v}(0,t_{0})-{\bf v}({\bf x}_{\rm
LS},t_{\rm LS})\right].$
The potential at our position contributes only to the unobservable monopole
and can be dropped. On scales outside the horizon, ${\bf v}=-t\nabla\Phi\sim
0$. The remaining term is the Sachs–Wolfe effect
$\frac{\delta T({\bf e})}{T}=\frac{1}{3}\Phi({\bf x}_{\rm
LS})=\frac{1}{5}{\cal R}({\bf x}_{\rm LS}).$
This relation has been obtained as follows. The co-moving curvature
perturbation is given during the radiation phase by ${\cal
R}=\psi+H\delta\rho/\dot{\rho}=\psi-1/3\delta\rho_{\gamma}/\rho_{\gamma}$.
Einstein equations set $\psi=\Phi$ and
$\delta\rho_{\gamma}/\rho_{\gamma}=-2\Phi$ on super-horizon scales. Therefore
${\cal R}=5/3\Phi$ beyond the horizon.
At large angular scales, the theory of cosmological perturbations predicts a
remarkably simple formula relating the CMB anisotropy to the curvature
perturbation generated during inflation.
We have seen previously that the temperature anisotropy is commonly expanded
in spherical harmonics $\frac{\Delta T}{T}(x_{0},\tau_{0},{\bf n})=\sum_{\ell
m}a_{\ell,m}(x_{0})Y_{\ell m}({\bf n}),$ where $x_{0}$ and $\tau_{0}$ are our
position and the preset time, respectively, ${\bf n}$ is the direction of
observation, $\ell^{\prime}$s are the different multipoles, and $\langle
a_{\ell
m}a^{*}_{\ell^{\prime}m^{\prime}}\rangle=\delta_{\ell,\ell^{\prime}}\delta_{m,m^{\prime}}C_{\ell}$,
where the deltas are due to the fact that the process that created the
anisotropy is statistically isotropic. The $C_{\ell}$’s are the so-called CMB
power spectrum. For homogeneity and isotropy, the $C_{\ell}$’s are neither a
function of $x_{0}$, nor of $m$. The two-point-correlation function is related
to the $C_{l}$’s according to Eq. (23).
For adiabatic perturbations we have seen that on large scales, larger than the
horizon on the last scatteringsurface (corresponding to angles larger than
$\theta_{\rm HOR}\sim 1^{\circ}$) $\delta T/T=\frac{1}{3}\Phi({\bf x}_{\rm
LS})$. In Fourier transform
$\frac{\delta T({\bf k},\tau_{0},{\bf n})}{T}=\frac{1}{3}\Phi_{{\bf
k}}\,e^{i\,{\bf k}\cdot{\bf n}(\tau_{0}-\tau_{{\rm LS}})}.$ (80)
Using the decomposition
$\exp(i\,{\bf k}\cdot{\bf n}(\tau_{0}-\tau_{{\rm
LS}}))=\sum_{\ell=0}^{\infty}(2\ell+1)i^{\ell}j_{\ell}(k(\tau_{0}-\tau_{{\rm
LS}}))P_{\ell}({\bf k}\cdot{\bf n})$ (81)
where $j_{\ell}$ is the spherical Bessel function of order $\ell$ and
substituting, we get
$\displaystyle\Big{<}\frac{\delta T(x_{0},\tau_{0},{\bf n})}{T}\frac{\delta
T(x_{0},\tau_{0},{\bf n^{\prime}})}{T}\Big{>}=$ (82)
$\displaystyle=\frac{1}{V}\int d^{3}x\Big{<}\frac{\delta T(x_{0},\tau_{0},{\bf
n})}{T}\frac{\delta T(x_{0},\tau_{0},{\bf n}^{\prime})}{T}\Big{>}=$
$\displaystyle=\frac{1}{(2\pi)^{3}}\int d^{3}k\Big{<}\frac{\delta T({\bf
k},\tau_{0},{\bf n})}{T}\left(\frac{\delta T({\bf k},\tau_{0},{\bf
n}^{\prime})}{T}\right)^{*}\Big{>}=$ $\displaystyle=\frac{1}{(2\pi)^{3}}\int
d^{3}k\Big{(}\Big{<}\frac{1}{3}|\Phi|^{2}\Big{>}\sum_{\ell,\ell^{\prime}=0}^{\infty}(2\ell+1)(2\ell^{\prime}+1)j_{\ell}(k(\tau_{0}-\tau_{\rm
LS}))$ $\displaystyle j_{\ell^{\prime}}(k(\tau_{0}-\tau_{{\rm
LS}}))P_{\ell}({\bf k}\cdot{\bf n})P_{\ell^{\prime}}({\bf k}^{\prime}\cdot{\bf
n}^{\prime})\Big{)}$ (83)
Inserting $P_{\ell}({\bf k}\cdot{\bf
n})=\frac{4\pi}{2\ell+1}\sum_{m}Y^{*}_{lm}({\bf k})Y_{\ell m}({\bf n})$ and
analogously for $P_{\ell}({\bf k}^{\prime}\cdot{\bf n}^{\prime})$, integrating
over the directions $d\Omega_{k}$ generates
$\delta_{\ell\ell^{\prime}}\delta_{mm^{\prime}}\sum_{m}Y^{*}_{\ell m}({\bf
n})Y_{\ell m}({\bf n}^{\prime})$. Using as well $\sum_{m}Y^{*}_{\ell m}({\bf
n})Y_{\ell m}({\bf n}^{\prime})=\frac{2\ell+1}{4\pi}P_{\ell}({\bf n}\cdot{\bf
n}^{\prime})$, we get
$\displaystyle\Big{<}\frac{\delta T(x_{0},\tau_{0},{\bf n})}{T}\frac{\delta
T(x_{0},\tau_{0},{\bf n}^{\prime})}{T}\Big{>}$ (84)
$\displaystyle=\Sigma_{\ell}\frac{2\ell+1}{4\pi}P_{\ell}({\bf n}\cdot{\bf
n}^{\prime})\frac{2}{\pi}\int\frac{dk}{k}\Big{<}\frac{1}{9}|\Phi|^{2}\Big{>}k^{3}j^{2}_{\ell}(k(\tau_{0}-\tau_{\rm
LS})).$
Comparing this expression with that for the $C_{\ell}$, we get the expression
for the $C^{{\rm AD}}_{\ell}$, where the suffix “AD” stands for adiabatic:
$C^{\rm
AD}_{\ell}=\frac{2}{\pi}\int\frac{dk}{k}\Big{<}\frac{1}{9}\left|\Phi\right|^{2}\Big{>}k^{3}j^{2}_{\ell}(k(\tau_{0}-\tau_{\rm
LS}))$ (85)
which is valid for $2\leq\ell\ll(\tau_{0}-\tau_{\rm LS})/\tau_{\rm LS}\sim
100$.
If we generically indicate by $\langle|\Phi_{\bf k}|^{2}\rangle
k^{3}=A^{2}\,(k\tau_{0})^{n-1}$, we can perform the integration and get
$\frac{\ell(\ell+1)C^{\rm
AD}_{\ell}}{2\pi}=\left[\frac{\sqrt{\pi}}{2}\ell(\ell+1)\frac{\Gamma(\frac{3-n}{2})\Gamma(\ell+\frac{n-)}{2})}{\Gamma\left(\frac{4-n}{2}\right)\Gamma\left(\ell+\frac{5-n}{2}\right)}\right]\frac{A^{2}}{9}\left(\frac{H_{0}}{2}\right)^{n-1}.$
(86)
For $n\simeq 1$ and $\ell\gg 1$, we can approximate this expression to
$\frac{\ell(\ell+1)C^{\rm AD}_{l}}{2\pi}=\frac{A^{2}}{9}.$ (87)
This result shows that inflation predicts a very flat spectrum for low $\ell$.
Furthermore, since inflation predicts $\Phi_{\bf k}=\frac{3}{5}{\cal R}_{\bf
k}$, we find that
$\pi\,\ell(\ell+1)C^{\rm AD}_{l}=\frac{A_{\cal
R}^{2}}{25}=\frac{1}{25}\frac{1}{2\,{m_{\rm
Pl}}^{2}\,\epsilon}\left(\frac{H}{2\pi}\right)^{2}.$ (88)
WMAP5 data imply that $\frac{\ell(\ell+1)C^{\rm AD}_{l}}{2\pi}\simeq 10^{-10}$
or
(Vϵ)^1/4≃6.7×10^16 GeV
---
### 0.8.2 Acoustic peaks
To be able to calculate the power spectrum of the anisotropies even on angular
scales larger than $1^{\circ}$, we need to consider the evolution of the
photon anistropies. As we already mentioned, before recombination Thomson
scattering was very efficient. As a result it is a good approximation to treat
photons and baryons as a single fluid. This treatment is called the tight-
coupling approximation and will allow us to evolve the perturbations until
recombination.
The equation for the photon density perturbations for one Fourier mode of
wave-number $k$ is that of a forced and damped harmonic oscillator
$\displaystyle\ddot{\delta}_{\gamma}+{\dot{R}\over(1+R)}\dot{\delta}_{\gamma}+k^{2}c^{2}_{s}\delta_{\gamma}=F,$
$\displaystyle F=4[\ddot{\psi}+{\dot{R}\over(1+R)}\dot{\psi}-{1\over
3}k^{2}\Phi],$ $\displaystyle\dot{\delta}_{\gamma}=-{4\over
3}kv_{\gamma}+4\dot{\psi}.$ (89)
The photon–baryon fluid can sustain acoustic oscillations. The inertia is
provided by the baryons, while the pressure is provided by the photons. The
sound speed is $c_{s}^{2}=1/3(1+R)$, with $R=3\rho_{b}/4\rho_{\gamma}=31.5\
(\Omega_{b}h^{2})(T/2.7)^{-4}[(1+z)/10^{3}]^{-1}$. As the baryon fraction goes
down, the sound speed approaches $c_{s}^{2}\rightarrow 1/3$. The third
equation above is the continuity equation.
As a toy problem, we shall solve Eq. (0.8.2) under some simplifying
assumptions. If we consider a matter-dominated universe, the driving force
becomes a constant, $F=-4/3k^{2}\Phi$, because the gravitational potential
remains constant in time. We neglect anisotropic stresses so that $\psi=\Phi$,
and, furthermore, we neglect the time dependence of $R$. Equation (0.8.2)
becomes that of a harmonic oscillator that can be trivially solved. This is a
very simplified picture, but it captures most of the relevant physics we want
to discuss.
To obtain the final solution we need again to specify the initial conditions.
we shall restrict ourselves to adiabatic initial conditions, the most natural
outcome of inflation. In our context this means that initially
$\Phi=\psi=\Phi_{0}$, $\delta_{\gamma}=-8/3\Phi_{0}$, and $v_{\gamma}=0$. We
have denoted $\Phi_{0}$ the initial amplitude of the potential fluctuations.
We shall take $\Phi_{0}$ to be a Gaussian random variable with power spectrum
$P_{\Phi_{0}}$.
We have made enough approximations that the evaluation of the sources in the
integral solution has become trivial. The solution for the density and
velocity of the photon fluid at recombination is
$\displaystyle\left({\delta_{\gamma}\over 4}+\Phi\right)|_{\rm LS}$
$\displaystyle=$ $\displaystyle{\Phi_{0}\over 3}(1+3R)\cos(kc_{s}\tau_{\rm
LS})-\Phi_{0}R,$ $\displaystyle v_{\gamma}|_{\tau_{\rm LS}}$ $\displaystyle=$
$\displaystyle-\Phi_{0}(1+3R)c_{s}\sin(kc_{s}\tau_{\rm LS}).$ (90)
Equation (0.8.2) is the solution for a single Fourier mode. All quantities
have an additional spatial dependence ($e^{i\bf k\cdot\bf x}$), which we have
not included in order to make the notation more compact. With that additional
term the solution we have is
$\displaystyle\frac{\delta T}{T}({\bf n})$ $\displaystyle=$ $\displaystyle
e^{ikD_{\rm LS}\cos\theta}S$ $\displaystyle S$ $\displaystyle=$
$\displaystyle\Phi_{0}{(1+3R)\over 3}[\cos(kc_{s}\tau_{\rm
LS})-{3R\over(1+3R)},$ (91) $\displaystyle-i\sqrt{3\over
1+R}\cos\theta\sin(kc_{s}\tau_{\rm LS})],$
where we have neglected the $\Phi$ on the left-hand side because it is a
constant. We have introduced $\cos\theta$, the cosine of the angle between the
direction of observation and the wavevector ${\bf k}$; for example, ${\bf
k}\cdot{\bf x}=kD_{\rm LS}\cos\theta$ . The term proportional to $\cos\theta$
is the Doppler contribution.
Once the temperature perturbation produced by one Fourier mode has been
calculated, we need to expand it into spherical harmonics. The power spectrum
of temperature anisotropies is expressed in terms of the $a_{lm}$ coefficients
as $C_{T\ell}=\sum_{m}|a_{\ell m}|^{2}$. The contribution to $C_{Tl}$ from
each Fourier mode is weighted by the amplitude of primordial fluctuations in
this mode, characterized by the power spectrum of $\Phi_{0}=3/5{\cal R}$,
$P_{\Phi_{0}}=Ak^{-3}$ as dictated by inflation. In practice, fluctuations on
angular scale $\ell$ receive most of their contributions from wavevectors
around $k_{*}=\ell/D_{\rm LS}$, so roughly the amplitude of the power spectrum
at multipole $\ell$ is given by the value of the sources in Eq. (0.8.2) at
$k_{*}$.
After summing the contributions from all modes, the power spectrum is roughly
given by
$\displaystyle\ell(\ell+1)C_{Tl}$ $\displaystyle\approx$ $\displaystyle
A\\{[{(1+3R)\over 3}\cos(k_{*}c_{s}\tau_{\rm LS})-R]^{2}+$ (92)
$\displaystyle{(1+3R)^{2}\over 3}c_{s}^{2}\sin^{2}(k_{*}c_{s}\tau_{\rm
LS})\\}.$
Equation (92) can be used to understand the basic features in the CMB power
spectra. The baryon drag on the photon–baryon fluid reduces its sound speed
below $1/3$ and makes the monopole contribution dominant (the one proportional
to $\cos(k_{*}c_{s}\tau_{\rm LS}$). Thus, the $C_{Tl}$ spectrum peaks where
the monopole term peaks, $k_{*}c_{s}\tau_{\rm LS}=\pi,2\pi,3\pi,\cdots$, which
correspond to $\ell_{\rm peak}=n\pi D_{\rm LS}/c_{S}\tau_{\rm LS}$.
It is very important to understand the origin of the acoustic peaks. In this
model the universe is filled with standing waves; all modes of wave-number $k$
are in phase, which leads to the oscillatory terms. The sine and cosine in Eq.
(92) originate in the time dependence of the modes. Each mode $\ell$ receives
contributions preferentially from Fourier modes of a particular wavelength
$k_{*}$ (but pointing in all directions), so to obtain peaks in $C_{\ell}$, it
is crucial that all modes of a given $k$ be in phase. If this is not the case,
the features in the $C_{T\ell}$ spectra will be blurred and can even
disappear. This is what happens when one considers the spectra produced by
topological defects. The phase coherence of all modes of a given wave-number
can be traced to the fact that perturbations were produced very early on and
had wavelengths larger than the horizon during many expansion times.
There are additional physical effects we have neglected. The universe was
radiation dominated early on, and modes of wavelength smaller and bigger than
the horizon at matter-radiation equality behave differently. During the
radiation era the perturbations in the photon–baryon fluid are the main source
for the gravitational potentials which decay once a mode enters into the
horizon. The gravitational potential decay acts as a driving force for the
oscillator in Eq. (0.8.2), so a feedback loop is established. As a result, the
acoustic oscillations for modes that entered the horizon before matter-
radiation equality have a higher amplitude. In the $C_{T\ell}$ spectrum the
separation between modes that experience this feedback and those that do not
occurs at $\ell\sim D_{\rm LS}/\tau_{\rm LS}$. Larger $\ell$ values receive
their contributions from modes that entered the horizon before matter-
radiation equality. Finally, when a mode is inside the horizon during the
radiation era the gravitational potentials decay.
There is a competing effect, Silk damping, that reduces the amplitude of the
large-$l$ modes. The photon–baryon fluid is not a perfect fluid. Photons have
a finite mean free path and thus can random-walk away from the peaks and
valleys of the standing waves. Thus perturbations of wavelength comparable to
or smaller than the distance the photons can random-walk get damped. This
effect can be modelled by multiplying Eq. 91 by $\exp(-k^{2}/k_{s}^{2})$, with
$k^{-1}_{s}\propto\tau_{\rm LS}^{1/2}(\Omega_{b}h^{2})^{-1/2}$. Silk damping
is important for multipoles of order $\ell_{\rm Silk}\sim k_{s}D_{\rm LS}$.
Finally, the last scatteringsurface has a finite width. Perturbations with
wavelength comparable to this width get smeared out due to cancellations along
the line of sight. This effect introduces an additional damping with a
characteristic scale $k^{-1}_{w}\propto\delta\tau_{\rm LS}$.
The location of the first peak is by itself a measurement of the geometry of
the universe. In fact, photons propagating on geodesics from the last
scattering surface to us feel the spatial geometry, whose properties we
learned are dictated by $\Omega_{0}$. In fact, the location of the first peak
is given by $\ell_{1}\simeq 220/\sqrt{\Omega_{0}}$. WMAP5 gives
$\Omega_{0}=1.00^{+0.07}_{-0.03}$. This tells us that the spatial (local)
geometry of the universe is flat. This is precisely what inflation predicts.
### 0.8.3 The polarization of the CMB anisotropies
The anisotropy field is characterized by a $2\times 2$ intensity tensor
$I_{ij}$. For convenience, we normalize this tensor so that it represents the
fluctuations in units of the mean intensity ($I_{ij}=\delta I/I_{0}$). The
intensity tensor is a function of direction on the sky, ${\bf n}$, and two
directions perpendicular to ${\bf n}$ that are used to define its components
(${\bf e}_{1}$,${\bf e}_{2}$). The Stokes parameters $Q$ and $U$ are defined
as $Q=(I_{11}-I_{22})/4$ and $U=I_{12}/2$, while the temperature anisotropy is
given by $T=(I_{11}+I_{22})/4$ (the factor of $4$ relates fluctuations in the
intensity with those in the temperature, $I\propto T^{4}$). When representing
polarization using “rods” in a map, the magnitude is given by
$P=\sqrt{Q^{2}+U^{2}}$, and the orientation makes an angle $\alpha={1\over
2}\arctan({U/Q})$ with ${\bf e}_{1}$. In principle the fourth Stokes parameter
$V$ that describes circular polarization is needed, but we ignore it because
it cannot be generated through Thomson scattering, so the CMB is not expected
to be circularly polarized. While the temperature is invariant under a right-
handed rotation in the plane perpendicular to direction ${\bf n}$, $Q$ and $U$
transform under rotation by an angle $\psi$ as
$(Q\pm iU)^{\prime}({\bf n})=e^{\mp 2i\psi}(Q\pm iU)({\bf n}),$ (93)
where ${\bf e}_{1}^{\prime}=\cos\psi\ {\bf e}_{1}+\sin\psi\ {\bf e}_{2}$ and
${\bf e}_{2}^{\prime}=-\sin\psi\ {\bf e}_{1}+\cos\psi\ {\bf e}_{2}$. The
quantities $Q\pm iU$ are said to be spin 2.
We already mentioned that the statistical properties of the radiation field
are usually described in terms of the spherical harmonic decomposition of the
maps. This basis, basically the Fourier basis, is very natural because the
statistical properties of anisotropies are rotationally invariant. The
standard spherical harmonics are not the appropriate basis for $Q\pm iU$
because they are spin-2 variables, but generalizations (called ${}_{\pm
2}Y_{lm}$) exist. We can expand
$\displaystyle(Q\pm iU)({\bf n})$ $\displaystyle=$ $\displaystyle\sum_{\ell
m}a_{\pm 2,\ell m}\;{}_{\pm 2}Y_{\ell m}({\bf n}).$ (94)
Here $Q$ and $U$ are defined at each direction $\hat{{\bf n}}$ with respect to
the spherical coordinate system $({\bf e}_{\theta},{\bf e}_{\phi})$. To ensure
that $Q$ and $U$ are real, the expansion coefficients must satisfy $a_{-2,\ell
m}^{*}=a_{2,\ell-m}$. The equivalent relation for the temperature coefficients
is $a_{T,\ell m}^{*}=a_{T,\ell-m}$. Instead of $a_{\pm 2,\ell m}$, it is
convenient to introduce their linear combinations $a_{E,\ell m}=-(a_{2,\ell
m}+a_{-2,\ell m})/2$ and $a_{B,\ell m}=i(a_{2,\ell m}-a_{-2,\ell m})/2$. We
define two quantities in real space, $E({\bf n})=\sum_{\ell,m}a_{E,\ell
m}Y_{\ell m}({\bf n})$ and $B({\bf n})=\sum_{\ell,m}a_{B,\ell m}Y_{\ell
m}({\bf n})$. Here $E$ and $B$ completely specify the linear polarization
field.
The temperature is a scalar quantity under a rotation of the coordinate
system, $T^{\prime}({\bf n}^{\prime}={\bf\cal R}{\bf n})=T({\bf n})$, where
$\bf{\cal R}$ is the rotation matrix. We denote with a prime the quantities in
the transformed coordinate system. While $Q\pm iU$ are spin 2, $E({\bf n})$
and $B({\bf n})$ are invariant under rotations. Under parity, however, $E$ and
$B$ behave differently, $E$ remains unchanged, while $B$ changes sign.
Figure 4: Examples of $E$\- and $B$-mode patterns of polarization
To characterize the statistics of the CMB perturbations, only four power
spectra are needed, those for $T$, $E$, $B$ and the cross correlation between
$T$ and $E$. The cross correlation between $B$ and $E$ or $B$ and $T$ vanishes
if there are no parity-violating interactions because $B$ has the opposite
parity to $T$ or $E$. The power spectra are defined as the rotationally
invariant quantities $C_{T\ell}={1\over 2\ell+1}\sum_{m}\langle a_{T,\ell
m}^{*}a_{T,\ell m}\rangle$, $C_{E\ell}={1\over 2\ell+1}\sum_{m}\langle
a_{E,\ell m}^{*}a_{E,\ell m}\rangle$, $C_{B\ell}={1\over
2\ell+1}\sum_{m}\langle a_{B,\ell m}^{*}a_{B,\ell m}\rangle$, and
$C_{C\ell}={1\over 2\ell+1}\sum_{m}\langle a_{T,\ell m}^{*}a_{E,\ell
m}\rangle$. The brackets $\langle\cdots\rangle$ denote ensemble averages.
Polarization is generated by Thomson scattering between photons and electrons,
which means that polarization cannot be generated after recombination (except
for re-ionization, which we shall discuss later). But Thomson scattering is
not enough. The radiation incident on the electrons must also be anisotropic.
In fact, its intensity needs to have a quadrupole moment. This requirement of
having both Thomson scattering and anisotropies is what makes polarization
relatively small. After recombination, anisotropies grow by free streaming,
but there is no scattering to generate polarization. Before recombination
there were so many scatterings that they erased any anisotropy present in the
photon–baryon fluid.
Figure 5: Thomson scattering of radiation where quadrupole anisotropy
generates linear polarization
In the context of anisotropies induced by density perturbations, velocity
gradients in the photon–baryon fluid are responsible for the quadrupole that
generates polarization. Let us consider a scattering occurring at position
$\hbox{\boldmath{$x$}}_{0}$: the scattered photons came from a distance of
order the mean free path ($\lambda_{T}$) away from this point. If we are
considering photons traveling in direction $\hat{n}$, they roughly come from
$\hbox{\boldmath{$x$}}=\hbox{\boldmath{$x$}}_{0}+\lambda_{T}\hbox{\boldmath{$\hat{n}$}}$.
The photon–baryon fluid at that point was moving at velocity
$\hbox{\boldmath{$v$}}(\hbox{\boldmath{$x$}})\approx\hbox{\boldmath{$v$}}(\hbox{\boldmath{$x$}}_{0})+\lambda_{T}{\hbox{\boldmath{$\hat{n}$}}}_{i}\partial_{i}{\hbox{\boldmath{$v$}}}(\hbox{\boldmath{$x$}}_{0})$.
Due to the Doppler effect the temperature seen by the scatterer at
$\hbox{\boldmath{$x$}}_{0}$ is $\delta
T(\hbox{\boldmath{$x$}}_{0},\hbox{\boldmath{$\hat{n}$}})=\hbox{\boldmath{$\hat{n}$}}\cdot[\hbox{\boldmath{$v$}}(\hbox{\boldmath{$x$}})-\hbox{\boldmath{$v$}}(\hbox{\boldmath{$x$}}_{0})]\approx\lambda_{T}{\hbox{\boldmath{$\hat{n}$}}}_{i}{\hbox{\boldmath{$\hat{n}$}}}_{j}\partial_{i}{\hbox{\boldmath{$v$}}}_{j}(\hbox{\boldmath{$x$}}_{0})$,
which is quadratic in $\hat{n}$ (i.e., it has a quadrupole). Velocity
gradients in the photon–baryon fluid lead to a quadrupole component of the
intensity distribution, which, through Thomson scattering, is converted into
polarization.
The polarization of the scattered radiation field, expressed in terms of the
Stokes parameters $Q$ and $U$, is given by $(Q+iU)\propto\sigma_{T}\int
d\Omega^{\prime}({\bf m}\cdot\hat{{\bf n}}^{\prime})^{2}T(\hat{{\bf
n}}^{\prime})$ $\propto\lambda_{p}{\bf m}^{i}{\bf
m}^{j}\partial_{i}v_{j}|_{\tau_{\rm LS}}$, where $\sigma_{T}$ is the Thomson
scattering cross-section and we have written the scattering matrix as $P({\bf
m},\hat{{\bf n}}^{\prime})=-3/4\sigma_{T}({\bf m}\cdot\hat{{\bf
n}}^{\prime})^{2}$, with ${\bf m}=\hat{{\bf e}}_{1}+i\hat{{\bf e}}_{2}$ . In
the last step, we integrated over all directions of the incident photons
$\hat{{\bf n}}^{\prime}$. As photons decouple from the baryons, their mean
free path grows very rapidly, so a more careful analysis is needed to obtain
the final polarization:
$\displaystyle(Q+iU)(\hat{{\bf n}})\approx\epsilon\delta\tau_{\rm LS}{\bf
m}^{i}{\bf m}^{j}\partial_{i}v_{j}|_{\tau_{\rm LS}},$ (95)
where $\delta\tau_{\rm LS}$ is the width of the last scattering surface and
gives a measure of the distance that photons travel between their last two
scatterings, and $\epsilon$ is a numerical constant that depends on the shape
of the visibility function. The appearance of ${\bf m}^{i}{\bf m}^{j}$ in Eq.
(95) ensures that $(Q+iU)$ transforms correctly under rotations of $(\hat{{\bf
e}}_{1},\hat{{\bf e}}_{2})$.
If we evaluate Eq. (95) for each Fourier mode and combine them to obtain the
total power, we get the equivalent of Eq. (92),
$\displaystyle\ell(\ell+1)C_{E\ell}\approx
A\epsilon^{2}(1+3R)^{2}(k_{*}\delta\tau_{\rm
LS})^{2}\sin^{2}(k_{*}c_{s}\tau_{\rm LS}),$ (96)
where we are assuming $n=1$ and that $\ell$ is large enough that factors like
$(\ell+2)!/(\ell-2)!\approx\ell^{4}$. The extra $k_{*}$ in Eq. (96) originates
in the gradient in Eq. (95). The large-angular scale polarization is greatly
suppressed by the $k\delta\tau_{\rm LS}$ factor. Correlations over large
angles can only be created by the long-wavelength perturbations, but these
cannot produce a large polarization signal because of the tight coupling
between photons and electrons prior to recombination. Multiple scatterings
make the plasma very homogeneous; only wavelengths that are small enough to
produce anisotropies over the mean free path of the photons will give rise to
a significant quadrupole in the temperature distribution, and thus to
polarization. Wavelengths much smaller than the mean free path decay due to
photon diffusion (Silk damping) and so are unable to create a large quadrupole
and polarization. As a result polarization peaks at the scale of the mean free
path.
On sub-degree angular scales, temperature, polarization, and the cross-
correlation power spectra show acoustic oscillations. In the polarization and
cross-correlation spectra the peaks are much sharper. The polarization is
produced by velocity gradients of the photon—baryon fluid at the last
scatteringsurface. The temperature receives contributions from density and
velocity perturbations, and the oscillations in each partially cancel one
another, making the features in the temperature spectrum less sharp. The
dominant contribution to the temperature comes from the oscillations in the
density [Eq. (0.8.2)], which are out of phase with the velocity. This explains
the difference in location between the temperature and polarization peaks. The
extra gradient in the polarization signal, Eq. (95), explains why its overall
amplitude peaks at a smaller angular scale.
Now, as photons travel in the metric perturbed by a GW [$ds^{2}=a^{2}(\tau)$
$[-d\tau^{2}$ $+(\delta_{ij}+h^{T}_{ij})dx^{i}dx^{j}]$], they get redshifted
or blueshifted depending on their direction of propagation relative to the
direction of propagation of the GW and the polarization of the GW. For
example, for a GW travelling along the $z$ axis, the frequency shift is given
by
${1\over\nu}{d\nu\over d\tau}={1\over 2}\
\hat{n}^{i}\hat{n}^{j}{\dot{h}}^{T(\pm)}_{ij}={1\over 2}\
(1-\cos^{2}\theta)e^{\pm i2\phi}\ \ \dot{h}_{t}\ \exp(i\bf k\cdot\bf x),$ (97)
where $(\theta,\phi)$ describe the direction of propagation of the photon, the
$\pm$ correspond to the different polarizations of the GW, and $h_{t}$ gives
the time-dependent amplitude of the GW. During the matter-dominated era, for
example, $h_{t}=3j_{1}(k\tau)/k\tau$: time changes in the metric lead to
frequency shifts (or equivalently shifts in the temperature of the black body
spectrum). Notice that the angular dependence of this frequency shift is
quadrupolar in nature. As a result, the temperature fluctuations induced by
this effect as photons travel between successive scatterings before
recombination produce a quadrupole intensity distribution, which, through
Thomson scattering, lead to polarization. Both $E$ and $B$ power spectra are
generated by GW. The current push to improve polarization measurements follows
from the fact that density perturbations, to linear order in perturbation
theory, cannot create any $B$-type polarization. As a rough rule of thumb, the
amplitude of the peak in the $B$-mode power spectrum for GW is
[ℓ(ℓ+1)C_Bl/ 2π]^1/2=0.024 (V^1/4/ 10^16GeV)^2 μK
---
where
$V^{1/4}\simeq 6.7\,r^{1/4}\,\times 10^{16}\,{\rm GeV}$ (98)
is the energy scale of inflation. A future experiment like CMBPol [14] can
probe values of $r$ as small as $10^{-2}$, corresponding to an inflation
energy scale of about $2\times\times 10^{16}$ GeV. Furthermore, using the
consistency relation $r=\epsilon$ valid in one-single field models of
inflation, one deduces that
$\frac{\Delta\phi}{m_{\rm Pl}}\simeq\left(\frac{r}{10^{-2}}\right)^{1/2},$
(99)
meaning that a future measurement of the $B$-mode of CMB polarization will
imply an inflaton excursus of Planckian values. Therefore, A future
measurement of the $B$-mode polarization of the CMB will allow a determination
of the value of the energy scale of inflation. This explains the utility of
CMB polarization measurements as probes of the physics of inflation. A
detection of primordial $B$-mode polarization would also demonstrate that
inflation occurred at a very high energy scale, and that the inflaton
traversed a super-Planckian distance in field space.
Figure 6: E- and B-mode power spectra for a tensor-to-scalar ratio saturating
the current bounds, $r=0.3$ and for $r=0.01$. Shown are the experimental
sensitivities of WMAP, Planck and two different realizations of CMBPol (EPIC-
LC and EPIC-2m)
## 0.9 The dark puzzles
Having explored the physics of the primordial epochs of the evolution of the
universe, such as inflation, and its impact on the present-day observables, we
now devote the remaining space to a short discussion of the dark puzzles of
the present-day universe: the dark energy and the dark matter puzzles.
### 0.9.1 A present-day accelerating universe
In 1998 the accelerated expansion of the universe was pointed out by two
groups from the observations of Type Ia Supernova (SN Ia) [15, 16]. Let us see
how this came about.
An important concept related to observational tools in an expanding background
is associated with the definition of a distance. A way of defining a distance
is through the luminosity of a stellar object. The distance $d_{L}$ known as
the luminosity distance, plays a very important role in astronomy including in
supernovae observations. It proves to be convenient to write the metric as
$\displaystyle
ds^{2}=-dt^{2}+a^{2}(t)\left[dr^{2}+f_{K}^{2}(r)(d\theta^{2}+\sin^{2}\theta
d\phi^{2})\right]\,,$ (100)
where
$\displaystyle f_{K}(r)=\left\\{\begin{array}[]{lll}{\rm sin}r\,,&K=+1\,,\\\
r\,,&K=0\,,\\\ {\rm sinh}r\,,&K=-1\,.\end{array}\right.$ (104)
In Minkowski space time the absolute luminosity $L_{s}$ of the source and the
energy flux ${\cal F}$ at a distance $d$ is related through ${\cal
F}=L_{s}/(4\pi d^{2})$. By generalizing this to an expanding universe, the
luminosity distance, $d_{L}$, is defined as
$d_{L}^{2}\equiv\frac{L_{s}}{4\pi{\cal F}}\,.$ (105)
Let us consider an object with absolute luminosity $L_{s}$ located at a co-
moving coordinate distance $r$ from an observer at $r=0$. The energy of light
emitted from the object with time interval $\Delta t_{e}$ is denoted as
$\Delta E_{e}$, whereas the energy which reaches at the sphere with radius $r$
is written as $\Delta E_{r}$. We note that $\Delta E_{e}$ and $\Delta E_{r}$
are proportional to the frequencies of light at $r$ and $r=0$, respectively,
i.e., $\Delta E_{e}\propto\nu_{e}$ and $\Delta E_{r}\propto\nu_{r}$. The
luminosities $L_{r}$ and $L_{e}$ are given by
$L_{r}=\frac{\Delta E_{e}}{\Delta t_{e}}\,,\quad L_{e}=\frac{\Delta
E_{e}}{\Delta t_{e}}.$ (106)
The speed of light is given by $c=\nu_{e}\lambda_{e}=\nu_{r}\lambda_{r}$,
where $\lambda_{e}$ and $\lambda_{r}$ are the wavelengths at $r$ and $r=0$.
Then, we find
$\frac{\lambda_{r}}{\lambda_{e}}=\frac{\nu_{e}}{\nu_{r}}=\frac{\Delta
t_{r}}{\Delta t_{e}}=\frac{\Delta E_{e}}{\Delta E_{r}}=1+z,$ (107)
where we have also used $\nu_{r}\Delta t_{r}=\nu_{e}\Delta t_{e}$. Combining
Eq. (106) with Eq. (107), we obtain
$L_{e}=L_{r}(1+z)^{2}.$ (108)
The light travelling along the $r$ direction satisfies the geodesic equation
$ds^{2}=-dt^{2}+a^{2}(t)dr^{2}=0$. We then obtain
$r=\int_{0}^{r}\,dr^{\prime}=\int_{t_{e}}^{t_{r}}\frac{dt}{a(t)}$ (109)
From the metric (100) we find that the area of the sphere at $t=t_{r}$ is
given by $S=4\pi(a_{r}f_{K}(r))^{2}$. Hence the observed energy flux is
${\cal F}=\frac{L_{r}}{4\pi(a_{r}f_{K}(r))^{2}}\,.$ (110)
Substituting Eqs. (109) and (110) for Eq. (105), we obtain the luminosity
distance in an expanding universe:
$d_{L}=a_{r}f_{K}(r)(1+z).$ (111)
In the flat FRW background with $f_{K}(r)=r$ we find
$d_{L}=\left(\frac{1+z}{H_{0}}\right)\int_{0}^{z}\,dz^{\prime}\frac{H_{0}}{H(z^{\prime})}.$
(112)
Then the Hubble rate $H(z)$ can be expressed in terms of $d_{L}(z)$:
$H(z)=\left\\{\frac{d}{dz}\left(\frac{d_{L}(z)}{1+z}\right)\right\\}^{-1}\,.$
(113)
If we measure the luminosity distance observationally, we can determine the
expansion rate of the universe.
The energy density $\rho$ on the right-hand side of Einstein equations
includes all components present in the universe, namely, non-relativistic
particles, relativistic particles, cosmological constant and so on
$\rho=\sum_{i}\rho^{(0)}_{i}(a/a_{0})^{-3(1+w_{i})}=\sum_{i}{\rho_{i}^{(0)}(1+z)^{3(1+w_{i})}}.$
(114)
Here $w_{i}$ and $\rho^{(0)}_{i}$ correspond to the equation of state and the
present energy density of each component, respectively. The Hubble parameter
takes the convenient form
$H^{2}=H^{2}_{0}\sum_{i}{\Omega^{(0)}_{i}(1+z)^{3(1+w_{i})}},$ (115)
where $\Omega^{(0)}_{i}\equiv 8\pi
G\rho_{i}^{(0)}/(3H_{0}^{2})=\rho_{i}^{(0)}/\rho_{c}^{(0)}$ is the density
parameter for an individual component at the present epoch. Hence the
luminosity distance in a flat geometry is given by
$d_{L}=\frac{(1+z)}{H_{0}}\int^{z}_{0}{\frac{dz^{\prime}}{\sqrt{\sum_{i}{\Omega_{i}^{(0)}(1+z^{\prime})^{3(1+w_{i})}}}}}\,.$
(116)
The direct evidence for the current acceleration of the universe is related to
the observation of luminosity distances of high redshift supernovae [15, 16].
The apparent magnitude $m$ of the source with an absolute magnitude $M$ is
related to the luminosity distance $d_{L}$ via the relation [17]
$m-M=5\log_{10}\left(\frac{d_{L}}{{\rm Mpc}}\right)+25\,.$ (117)
This comes from taking the logarithm of Eq. (105) by noting that $m$ and $M$
are related to the logarithms of ${\cal F}$ and $L_{s}$, respectively. The
numerical factors arise because of conventional definitions of $m$ and $M$ in
astronomy. Type Ia supernovae (SN Ia) can be observed when white dwarf stars
exceed the mass of the Chandrasekhar limit and explode. The belief is that SN
Ia are formed in the same way irrespective of where they are in the universe,
which means that they have a common absolute magnitude $M$ independent of the
redshift $z$. Thus they can be treated as an ideal standard candle. We can
measure the apparent magnitude $m$ and the redshift $z$ observationally, which
of course depends upon the objects we observe.
In order to get a feeling of the phenomenon let us consider two supernovae
1992P at low-redshift $z=0.026$ with $m=16.08$ and 1997ap at high-redshift
$z=0.83$ with $m=24.32$ [15]. As we have already mentioned, the luminosity
distance is approximately given by $d_{L}(z)\simeq z/H_{0}$ for $z\ll 1$.
Using the apparent magnitude $m=16.08$ of 1992P at $z=0.026$, we find that the
absolute magnitude is estimated by $M=-19.09$ from Eq. (117). Here we adopted
the value $H_{0}^{-1}=2998h^{-1}\,{\rm Mpc}$ with $h=0.72$. Then the
luminosity distance of 1997ap is obtained by substituting $m=24.32$ and
$M=-19.09$ for Eq. (117): $H_{0}d_{L}\simeq 1.16$ for $z=0.83$. From Eq. (116)
the theoretical estimate for the luminosity distance in a two-component flat
universe is $H_{0}d_{L}\simeq 0.95$ for $\Omega_{m}^{(0)}\simeq 1$ and
$H_{0}d_{L}\simeq 1.23$ for $\Omega_{m}^{(0)}\simeq 0.3,\leavevmode\nobreak\
\Omega_{\rm DE}\simeq 0.7$
Figure 7: The luminosity distance versus redshift for a flat cosmological
model. Black points are from the “Gold” data sets [18]; red points are from
recent data from HST
In 2004 Riess et al. [18] reported the measurement of 16 high-redshift SN Ia
with redshift $z>1.25$ with the Hubble Space Telescope (HST). By including 170
previously known SN Ia data points, they showed that the universe exhibited a
transition from deceleration to acceleration at $>99$% confidence level. A
best-fit value of $\Omega_{m}^{(0)}$ was found to be
$\Omega_{m}^{(0)}=0.29^{+0.05}_{-0.03}$ (the error bar is $1\sigma$). This
shows that a matter-dominated universe without a cosmological constant does
not fit the data.
We should emphasize that the accelerated expansion is by cosmological
standards really a late-time phenomenon, starting at a redshift $z\sim 1$.
From Eq. (115) the deceleration parameter, $q\equiv-a\ddot{a}/\dot{a}^{2}$, is
given by
q(z)=32 ∑iΩi(0)(1+wi)(1+z)3(1+wi)∑iΩi(0)(1+z)3(1+wi)-1.
---
For the two-component flat cosmology, the universe enters an accelerating
phase ($q<0$) for $z<z_{c}\equiv(2\Omega_{\rm DE}/\Omega_{\rm DM})^{1/3}-1$.
When $\Omega_{\rm DM}=0.3$ and $\Omega_{\rm DE}=0.7$, we have $z_{c}=0.67$.
The problem of why an accelerated expansion should occur now in the long
history of the universe is called the “coincidence problem”.
Figure 8: The dark energy (vacuum energy) and dark matter (mass density)
abundances from SN, CMB, and galaxy clustering observations
#### The origin of the acceleration
Once the idea of the accelerating universe is accepted, the next pressing
question is: Why? There are various explanations available that we may mention
briefly. The general trend is to accept that there is a form of Dark Energy
(DE) fluid dominating the energy density of the present day. Its pressure is
$P=w\rho$ and $w$ needs to be smaller than $-1/3$ for this fluid to cause the
acceleration. Having learned how to use scalar fields to accelerate the
universe at primordial epochs, the most natural way to explain DE whould be to
introduce a scalar field $\phi$ dubbed quintessence, with potential
${\cal V}(\phi)=V_{0}+V(\phi),$ (118)
Now, if $V_{0}\gg V(\phi)$ (at least at present epochs), the DE is in practice
a Cosmological Constant (CC). Its value must be extremely small,
$V_{0}^{1/4}\simeq(H_{0}m_{\rm Pl})^{1/2}\simeq 10^{-3}$ eV. Why it is so
small is a mystery that earned the name “the CC problem”. On the other hand,
if $V_{0}\ll V(\phi)$, then the dynamics of the quintessence field dominates.
However, another problem arises at this stage. Having learned from inflation
that the field must be slow-rolling to cause the acceleration of the universe,
we have to assume that $(m_{\rm Pl}^{2}V^{\prime\prime}/V)$ is smaller than
unity. This implies that $\phi$ is of order of the Planck scale and that its
mass squared is such that $V^{\prime\prime}\sim H_{0}^{2}\sim(10^{-33}\,{\rm
eV})^{2}$. The quintessence field has a Compton wavelegth as large as the
entire observed universe.
If the reader does not like all this fine-tuning, there are at least two other
explanations for the acceleration of the universe. The first one goes under
the name of modified gravity and is in fact rather intuitive. If gravity gets
weaker at large distances, objects far from us may recede at a velocity larger
than what they would do in the traditional Newtonian gravity case. For this to
work, we have to suppose that the gravitational force has a transient at some
critical (and cosmological) scale $r_{c}$, from the usual $1/r^{2}$ to, say
$1/r^{3}$. How to get this transition is unfortunately beyond the scope of
these lectures. Another alternative goes under the name of the “anthropic
principle” and is based on the following point. As we have seen, in a static
universe, overdense regions will increase their density at an exponential
rate. In an expanding universe, however, there is a competition between the
expansion and the gravitational collapse. More rapid expansion, as induced by
DE, retards the growth of structure. General relativity provides the following
useful relation in linear perturbation theory between the growth factor $g(z)$
and the expansion history of the universe
$\ddot{g}+2H\dot{g}=4\pi G\rho_{m}=\frac{3\Omega_{\rm DM}H_{0}^{2}}{2a^{3}}g.$
(119)
If the universe is always matter-dominated, then $g\sim a$; however, in a DE
dominated universe $g$ scales slower than the scale factor. Now, if the CC is
too large, structure does not have time to develop: the initial condition is
$\delta\rho_{m}/\rho_{m}\sim 10^{-5}$ at the last scattering surface ($z\sim
10^{3})$ and needs to becomes order unity by now. Now, if we impose that
structures might have been able to develop by now even in the presence of a
CC, one obtains a reassuring bound, the CC $V_{0}^{1/4}$ must be smaller than
about $10^{-1}$ eV. In other words, the CC may not be far from the value we
observe (if it is non-zero) because otherwise we would not be here to discuss
about it. A great deal of observational effort of the next decades will be
devoted to understand the cause of the acceleration of the universe [19]. Four
observational techniques are currently receiving much attention: 1) Baryoniuc
Acoustic Oscillations (BAO) are observed on large-scale surveys of the spatial
distribution of matter. They are caused by the same oscillations that left an
imprint in the CMB under the form of acoustic peaks. The BAO technique is
sensitive to the DE through its effect on the angular-diameter distance vs.
redshift relation and through its effect on the time evolution of the
expansion rate; 2) Galaxy Cluster (CL) surveys measure the spatial density and
distribution of galaxy clusters. The CL technique is sensitive to DE through
its effect in the angular-diameter distance vs. redshift relation and through
its effect on the time evolution of the expansion rate and the growth rate of
perturbations; 3) supernovae as standard candles to determine the luminosity
distance vs. redshift relation; 4) Weak Lensing (WL) surveys measure the
distortion of background images due to the bending of light as it passes by
galaxies or clusters of galaxies. The WL technique is sensitive to DE through
its effect on the angular-diameter distance vs. redshift relation and the
growth rate of perturbations. All these techniques will not only shed light on
the nature of DE, but will also help us to discriminate the various
possibilities to explain the present-day acceleration. For instance, the
modified gravity scenario predicts a growth function which is different from
the one predicted in a CC dominated universe. Future applications of the
techniques briefly summarized above should be able to determine which scenario
is more likely.
### 0.9.2 Dark matter
The evidence that 95% of the mass of galaxies and clusters is made of some
unknown component of Dark Matter (DM) comes from (i) rotation curves (out to
tens of kpc), (ii) gravitational lensing (out to 200 kpc), and (iii) hot gas
in clusters. They lead us to believe that DM makes up about 30% of the entire
energy of the universe. A nice review about DM can be found in Ref. [20].
In the 1970s, Ford and Rubin discovered that rotation curves of galaxies are
flat. The velocities of objects (stars or gas) orbiting the centres of
galaxies, rather than decreasing as a function of the distance from the
galactic centres as had been expected, remain constant out to very large
radii. Similar observations of flat rotation curves have now been found for
all galaxies studied, including our Milky Way. The simplest explanation is
that galaxies contain far more mass than can be explained by the bright
stellar objects residing in galactic disks. This mass provides the force to
speed up the orbits. To explain the data, galaxies must have enormous dark
haloes made of unknown matter. Indeed, more than 95% of the mass of galaxies
consists of dark matter. The baryonic matter which accounts for the gas and
disk cannot alone explain the galactic rotation curve. However, adding a DM
halo allows a good fit to data.
The limitations of rotation curves are that one can only look out as far as
there is light or neutral hydrogen (21 cm), namely to distances of tens of
kpc. Thus one can see the beginnings of DM haloes, but cannot trace where most
of the DM is. The lensing experiments discussed in the next section go beyond
these limitations.
Einstein’s theory of General Relativity predicts that mass bends, or lenses,
light. This effect can be used to gravitationally ascertain the existence of
mass even when it emits no light. Lensing measurements confirm the existence
of enormous quantities of DM both in galaxies and in clusters of galaxies.
Observations are made of distant bright objects such as galaxies or quasars.
As the result of intervening matter, the light from these distant objects is
bent towards the regions of large mass. Hence there may be multiple images of
the distant objects, or, if these images cannot be individually resolved, the
background object may appear brighter. Some of these images may be distorted
or sheared. The Sloan Digital Sky Survey used weak lensing (statistical
studies of lensed galaxies) to conclude that galaxies, including the Milky
Way, are even larger and more massive than previously thought, and require
even more DM out to great distances. Again, the predominance of DM in galaxies
is observed. The key success of the lensing of DM to date is the evidence that
DM is seen out to much larger distances than could be probed by rotation
curves: the DM is seen in galaxies out to 200 kpc from the centres of
galaxies, in agreement with N-body simulations. On even larger Mpc scales,
there is evidence for DM in filaments (the cosmic web). Another piece of
gravitational evidence for DM is the hot gas in clusters. The X-ray data
indicates the presence of hot gas. The existence of this gas in the cluster
can only be explained by a large DM component that provides the potential well
to hold on to the gas. In summary, the evidence is overwhelming for the
existence of an unknown component of DM that comprises 95% of the mass in
galaxies and clusters.
There is another basic reason why DM is necessary: to form structures as we
observe them. Let us assume that the matter content of the universe is
dominated by a pressureless and self-gravitating fluid. This approximation
holds if we are dealing with the evolution of the perturbations in the DM
component or in case we are dealing with structures whose size is much larger
than the typical Jeans scale length of baryons. Let us also define $\rm{\bf
x}$ to be the co-moving coordinate and ${\bf r}=a(t){\bf x}$ the proper
coordinate, $a(t)$ being the cosmic expansion factor. Furthermore, if ${\bf
v}=\dot{\bf r}$ is the physical velocity, then ${\bf v}=\dot{a}{\bf x}+{\bf
u}$, where the first term describes the Hubble flow, while the second term,
${\bf u}=a(t)\dot{\bf x}$, gives the peculiar velocity of a fluid element
which moves in an expanding background.
In this case the equations that regulate the Newtonian description of the
evolution of density perturbations are the continuity equation:
${\partial\delta\over\partial t}+\nabla\cdot[(1+\delta){\bf u}]=0\,,$ (120)
which gives the mass conservation, the Euler equation
${\partial{\bf u}\over\partial t}+2H(t){\bf u}+({\bf u}\cdot\nabla){\bf
u}=-{\nabla\phi\over a^{2}}\,,$ (121)
which gives the relation between the acceleration of the fluid element and the
gravitational force, and the Poisson equation
$\nabla^{2}\phi=4\pi G\bar{\rho}a^{2}\delta$ (122)
which specifies the Newtonian nature of the gravitational force. In the above
equations, $\nabla$ is the gradient computed with respect to the co-moving
coordinate ${\bf x}$, $\phi({\bf x})$ describes the fluctuations of the
gravitational potential, and $H(t)=\dot{a}/a$ is the Hubble parameter at the
time $t$. Its time-dependence is given by $H(t)=E(t)H_{0}$, where
$E(z)=[(1+z)^{3}\Omega_{m}+(1+z)^{2}(1-\Omega_{m}-\Omega_{DE})+(1+z)^{3(1+w)}\Omega_{DE}]^{1/2}.$
(123)
In the case of small perturbations, these equations can be linearized by
neglecting all the terms which are of second order in the fields $\delta$ and
${\bf u}$. In this case, using the Euler equation to eliminate the term
$\partial\rm{\bf u}/\partial t$, and using the Poisson equation to eliminate
$\nabla^{2}\phi$, one ends up with
${\partial^{2}\delta\over\partial t^{2}}+2H(t){\partial\delta\over\partial
t}-4\pi G\bar{\rho}\delta=0\,.$ (124)
This equation describes the Jeans instability of a pressureless fluid, with
the additional “Hubble drag” term $2H(t){\partial\delta/\partial t}$, which
describes the counter-action of the expanding background on the perturbation
growth. Its effect is to prevent the exponential growth of the gravitational
instability taking place in a non-expanding background. The solution of the
above equation can be cast in the form:
$\delta({\bf x},t)=\delta_{+}({\bf x},t_{i})D_{+}(t)+\delta_{-}({\bf
x},t_{i})D_{-}(t)\,,$ (125)
where $D_{+}$ and $D_{-}$ describe the growing and decaying modes of the
density perturbation, respectively. In the case of an Einstein–de-Sitter (EdS)
universe ($\Omega_{m}=1$, $\Omega_{DE}=0$), it is $H(t)=2/(3t)$, so that
$D_{+}(t)=(t/t_{i})^{2/3}$ and $D_{-}(t)=(t/t_{i})^{-1}$. The fact that
$D_{+}(t)\propto a(t)$ for an EdS universe should not be surprising. Indeed,
the dynamical time-scale for the collapse of a perturbation of uniform density
$\rho$ is $t_{\rm dyn}\propto(G\rho)^{-1//2}$, while the expansion time-scale
for the EdS model is $t_{\rm exp}\propto(G\bar{\rho})^{-1//2}$, where
$\bar{\rho}$ is the mean cosmic density. Since for a linear (small)
perturbation it is $\rho\simeq\bar{\rho}$, then $t_{\rm dyn}\sim t_{\rm exp}$,
thus showing that the cosmic expansion and the perturbation evolution take
place at the same pace. This argument also leads to understanding the
behaviour for a $\Omega_{m}<1$ model. In this case, the expansion time scale
becomes shorter than the above one at the redshift at which the universe
recognizes that $\Omega_{m}<1$. This happens at $1+z\simeq\Omega_{m}^{-1/3}$
or at $1+z\simeq\Omega_{m}^{-1}$ in the presence or absence of a cosmological
constant term, respectively. Therefore, after this redshift, cosmic expansion
takes place at a quicker pace than gravitational instability, with the result
that the perturbation growth is frozen.
The exact expression for the growing model of perturbations is given by
$D_{+}(z)\,=\,{5\over 2}\,\Omega_{m}E(z)\,\int_{z}^{\infty}{1+z^{\prime}\over
E(z^{\prime})^{3}}\,dz^{\prime}.$ (126)
The EdS has the faster evolution, while the slowing down of the perturbation
growth is more apparent for the open low-density model, the presence of a
cosmological constant providing an intermediate degree of evolution. The key
point is, however, that a pressureless fluid such as DM is needed for the
perturbations to grow to give rise to collapsed objects. Baryon perturbations,
being coupled to photons till the last-scattering epoch, feel a non-vanishing
pressure and therefore they may not grow. After the last-scattering stage, the
baryons fall into the gravitational potential generated by DM and the baryonic
perturbations may promptly catch up with those of DM.
#### Dark matter candidates
There is a plethora of dark matter candidates. MACHOs, or Massive Compact Halo
Objects, are made of ordinary matter in the form of faint stars or stellar
remnants; they could also be primordial black holes or mirror matter. However,
there are not enough of them to completely resolve the question. Of the non-
baryonic candidates, the most popular are the WIMPS (Weakly Interacting
Massive Particles) and the axions, as these particles have been proposed for
other reasons in particle physics. Ordinary massive neutrinos are too light to
be cosmologically significant, though sterile neutrinos remain a possibility.
Other candidates include primordial black holes, non-thermal WIMPzillas, and
Kaluza–Klein particles which arise in higher dimensional theories.
About axions, the good news is that cosmologists do not need to “invent” new
particles. Two candidates already exist in particle physics for other reasons:
axions and WIMPs. Axions with masses in the range $10^{-(3-6)}$ eV arise in
the Peccei–Quinn solution to the strong-CP problem in the theory of strong
interactions.
WIMPs are also natural dark matter candidates from particle physics. These
particles, if present in thermal abundances in the early universe, annihilate
with one another so that a predictable number of them remain today. The relic
density of these particles comes out to be the right value:
$\Omega_{\rm DM}h^{2}=(3\times 10^{-26}{\rm cm}^{3}/{\rm s})/\langle\sigma
v\rangle_{\rm A}$ (127)
where the annihilation cross-section $\langle\sigma v\rangle_{\rm A}$ of weak
interaction strength automatically gives the right answer. The reason why the
final abundance is inversely proportional to the annihilation cross-section is
rather clear: the larger the annihilation cross-section, the more WIMPs
annihilate and the fewer of them are left behind. Furthermore, annihilation is
not eternal: owing to the expansion of the universe, annihilation stops when
its rate becomes smaller than the expansion rate of the universe. When this
happens, the abundance is said to freeze-out.
Figure 9: The abundance of WIMPs of a given mass $m$ as a function of
temperature and for various annihilation cross-sections
This coincidence is known as ‘the WIMP miracle’ and is the reason why WIMPs
are taken so seriously as DM candidates. The best WIMP candidate is motivated
by Supersymmetry (SUSY): the lightest neutralino in the Minimal Supersymmetric
Standard Model. Supersymmetry in particle theory is designed to keep particle
masses at the right value. As a consequence, each particle we know has a
partner: the photino is the partner of the photon, the squark is the quark’s
partner, and the selectron is the partner of the electron. The lightest
superysmmetric partner is a good dark matter candidate.
There are several ways to search for dark WIMPs. SUSY particles may be
discovered at the LHC as missing energy in an event. In that case one knows
that the particles live long enough to escape the detector, but it will still
be unclear whether they are long-lived enough to be the dark matter. Thus
complementary astrophysical experiments are needed. In direct detection
experiments, the WIMP scatters off a nucleus in the detector, and a number of
experimental signatures of the interaction can be detected. In indirect
detection experiments, neutrinos that arise as annihilation products of
captured WIMPs exit from the Sun and can be detected on Earth. Another way to
detect WIMPs is to look for anomalous cosmic rays from the Galactic Halo:
WIMPs in the Halo can annihilate with one another to give rise to antiprotons,
positrons, or neutrinos. In addition, neutrinos, gamma rays, and radio waves
may be detected as WIMP annihilation products from the Galactic Centre. For
lack of time these issues were not discussed extensively in the lectures. The
interested reader may find more about these issues in Ref. [20].
## 0.10 Conclusions
The period when we say that cosmology is entering a golden age has already
passed: cosmology is in the middle of its golden age. Present observational
data pose various puzzles whose solutions might either be around the corner or
decades far in the future. It will require some young and creative researcher
sitting in this room to solve them. This is why the cosmological puzzles are
dark, but the future is brighter.
## Acknowledgements
It is a great pleasure to thank all the organizers, N. Ellis, E. Lillistol, D.
Metral, and especially M. Losada and E. Nardi, for having created such a
stimulating atmosphere. All students are also acknowledged for their never-
ending enthusiasm.
## References
* [1] A. D. Linde, Particle Physics and Inflationary Cosmology (Harwood, Chur, Switzerland, 1990) and references therein.
* [2] E. W. Kolb and M. S. Turner, The Early Universe (Addison-Wesley, Redwood City, CA, 1990).
* [3] A. R. Liddle and D. H. Lyth, Phys. Rep. 231, 1 (1993) and references therein.
* [4] A. R. Liddle and D. H. Lyth, Cosmological Inflation and Large-Scale Structure (Cambridge University Press, 2000) and references therein.
* [5] D. H. Lyth and A. Riotto, Phys. Rep. 314, 1 (1999) and references therein.
* [6] A. Riotto, arXiv:hep-ph/0210162 and references therein.
* [7] For a review, see N. Bartolo, S. Matarrese, and A. Riotto, arXiv:astro-ph/0703496 and references therein.
* [8] E. J. Copeland, M. Sami, and S. Tsujikawa, Int. J. Mod. Phys. D 15, 1753 (2006) and references therein.
* [9] E. Komatsu et al. [WMAP Collaboration], Astrophys. J. Suppl. 180, 330 (2009) [arXiv:0803.0547 [astro-ph]].
* [10] For a review, see, for instance, N. A. Bahcall, J. P. Ostriker, S. Perlmutter, and P. J. Steinhardt, Science 284, 1481 (1999).
* [11] R. K. Sachs and A. M. Wolfe, Astrophys. J. 147, 73 (1967).
* [12] S. Dodelson, Modern Cosmology (Academic Press, New York, 2003).
* [13] V. F. Mukhanov, H. A. Feldman, and R. H. Brandenberger, Phys. Rep. 215, 203 (1992) and references therein.
* [14] S. Dodelson et al., CMBPol Science White Paper submitted to the US Astro2010 Decadal Survey, arXiv:0902.3796v1.
* [15] S. Perlmutter et al., Astrophys. J. 517, 565 (1999).
* [16] A. G. Riess et al., Astron. J. 116, 1009 (1998); Astron. J. 117, 707 (1999).
* [17] T. Padmanabhan, Phys. Rep. 380, 235 (2003); T. Padmanabhan, Current Science, 88, 1057 (2005) [arXiv:astro-ph/0510492].
* [18] A. G. Riess et al. [Supernova Search Team Collaboration], Astrophys. J. 607, 665 (2004).
* [19] A. J. Albrecht et al., arXiv:astro-ph/0609591.
* [20] G. Bertone, D. Hooper, and J. Silk, Phys. Rep. 405, 279 (2005).
|
arxiv-papers
| 2010-10-13T12:16:07 |
2024-09-04T02:49:13.792810
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "A. Riotto (CERN)",
"submitter": "Antonio Riotto",
"url": "https://arxiv.org/abs/1010.2642"
}
|
1010.2647
|
11institutetext: APC, Paris, France
# High-energy astroparticle physics
D. Semikoz
###### Abstract
In these three lectures I discuss the present status of high-energy
astroparticle physics including Ultra-High-Energy Cosmic Rays (UHECR), high-
energy gamma rays, and neutrinos. The first lecture is devoted to ultra-high-
energy cosmic rays. After a brief introduction to UHECR I discuss the
acceleration of charged particles to highest energies in the astrophysical
objects, their propagation in the intergalactic space, recent observational
results by the Auger and HiRes experiments, anisotropies of UHECR arrival
directions, and secondary gamma rays produced by UHECR. In the second lecture
I review recent results on TeV gamma rays. After a short introduction to
detection techniques, I discuss recent exciting results of the H.E.S.S.,
MAGIC, and Milagro experiments on the point-like and diffuse sources of TeV
gamma rays. A special section is devoted to the detection of extragalactic
magnetic fields with TeV gamma-ray measurements. Finally, in the third lecture
I discuss Ultra-High-Energy (UHE) neutrinos. I review three different UHE
neutrino detection techniques and show the present status of searches for
diffuse neutrino flux and point sources of neutrinos.
## 0.1 Ultra-high-energy cosmic rays
### 0.1.1 Introduction
Figure 1: Left: The cosmic ray spectrum $I(E)$ as function of kinetic energy
$E$, compiled using results from the LEAP, proton, Akeno, and HiRes
experiments [2, 3]. The energy region influenced by the Sun is marked in
yellow and an $1/E^{2.7}$ power-law is also shown. Right: The same spectrum at
high energies $E>10^{11}$ eV multiplied by $E^{3}$ [4]. Spectrum changes are
called the ‘knee’ at $10^{15}$ eV and the ‘ankle’ at $10^{19}$ eV.
Particles coming from space to the atmosphere of the Earth historically were
called cosmic rays. Most cosmic rays, however, are not ‘rays’ or photons, but
charged particles, protons and nuclei. Real high-energy gamma rays coming from
space to the Earth are only a small fraction of total flux, and they will be
discussed in Section 0.2. The measured spectrum of cosmic rays from 100 GeV to
highest energies $E>10^{20}$ eV is presented in Fig. 1 (left). The yellow
strip at low energies presents the contribution of the Sun. The remaining
spectrum can be fitted with a single power law $1/E^{2.7}$ up to highest
energies. The main contribution to it above 100 GeV gives galactic sources.
After multiplication of the spectrum on the energy cube, one can see changes
of power law in Fig. 1 (right). At $E>10^{15}$ eV the spectrum becomes
steeper. This change in the spectrum called the ‘knee’ and associated energy
$E=10^{15}$ eV is the maximum energy up to which galactic sources accelerate
cosmic rays. The next change of the spectrum is located at $E=3\cdot 10^{18}$
eV and has two possible interpretations. Either this is the place where
extragalactic sources start to dominate or it is the result of pair-
production energy loss by extragalactic protons (see Section 0.1.3). At the
end of the spectrum there is a cutoff, which was not seen in the old
experiments presented in Fig. 1 (right) due to small statistics, but it was
observed recently by the HiRes [3] and Auger [5] experiments.
In this lecture I briefly discuss the theory and observations of Ultra-High
Energy Cosmic Rays (UHECR), the highest-energy particles measured on Earth
with energy $E>10^{18}$ eV. Such particles, protons and nuclei, can be
accelerated in astrophysical objects, propagate through intergalactic space,
losing energy in the interactions with Cosmic Microwave Background (CMB).
UHECR are charged particles. Therefore they are also deflected in the Galactic
and intergalactic magnetic fields on the way from the source to the Earth. For
a more detailed introduction to UHECR I recommend recent lectures by M.
Kachelriess [6].
There are several important scales commonly used in astroparticle physics.
Distance is usually measured in parsecs, $\rm{1\leavevmode\nobreak\ pc}=3\cdot
10^{18}$ cm. Corresponding larger units are kiloparsec
$\rm{1\leavevmode\nobreak\ kpc}=10^{3}\rm{\leavevmode\nobreak\ pc}$ and
megaparsec $\rm{1\leavevmode\nobreak\ Mpc}=10^{6}\rm{\leavevmode\nobreak\
pc}$. Energy at highest energies is usually expressed in units of
$\rm{EeV}=10^{18}$ eV.
The plan of this lecture is as follows. In Section 0.1.2 I shall discuss
possible acceleration mechanisms of cosmic rays and astrophysical objects
which potentially can be their sources. In Section 0.1.3 I present the main
energy loss processes for UHECR particles and briefly discuss their deflection
in the magnetic fields. In Section 0.1.4 I sum up recent observational results
from the Pierre Auger Observatory and other experiments. In Section 0.1.5
results on anisotropy at highest energy are discussed. In Section 0.1.6 I
review expectations on secondary photons and neutrinos from UHECR protons.
Results are summed up in Section 0.1.7.
### 0.1.2 Acceleration
There are several possible acceleration mechanisms that can work in
astrophysical objects. These include first-order Fermi acceleration on the
shocks in plasma or acceleration in the potential difference, which we call
one-shot acceleration below. However, in any case, the Larmor radius of a
particle does not exceed the accelerator size, otherwise the particle escapes
from the accelerator and cannot gain energy further. This criterion is called
the Hillas condition [7] and sets the limit
${\cal E}\leq{\cal E}_{\rm H}=qBR$ (1)
for the energy ${\cal E}$ gained by a particle with charge $q$ in the region
of size $R$ with the magnetic field $B$.
Figure 2: The Hillas plot with constraints from geometry and radiation losses
for $10^{20}$ eV protons (left) and iron (right). The thick line represents
the lower boundary of the area allowed by the Hillas criterion, Eq. (1).
Shaded areas are allowed by the radiation-loss constraints as well: light grey
corresponds to one-shot acceleration in the curvature-dominated regime only;
grey allows also for one-shot acceleration in the synchrotron-dominated
regime; dark grey allows for both one-shot and diffusive (e.g., shock)
acceleration.
The maximum energy of the accelerated particle can be restricted even more
than required by Eq. (1) if one takes into account energy losses during
acceleration. Unavoidable losses come from particle emission in the external
magnetic field, which can be either synchrotron-dominated if the velocity of
the particle is not parallel to the magnetic field, or curvature-dominated in
the opposite case.
In Fig. 2 in the plane magnetic field versus acceleration region size, the
Hillas condition Eq. (1) is shown by a thick black line. The left figure is
for protons and the right one for iron nuclei. Possible acceleration in
different astrophysical objects is shown with thin solid figures. Notations
are the following: NS are neutron stars, GRB are gamma-ray bursts, BH are
black holes, AD are accretion disks, jets are jets in active galaxies, K and
HS are knots and hot spots in the jets, L are lobes of radio galaxies,
clusters are clusters of galaxies, starbursts are starburst galaxies, voids
are voids in large-scale structure. Additional notations in brackets are
subtypes of active galaxies: Sy for Seyfert galaxies, BL for BL Lac galaxies
and RG for radio galaxies. Only objects above the Hillas line have the
potential possibility to accelerate particles to $10^{20}$ eV. This is a
necessary condition, but not enough for a specific acceleration mechanism. As
seen from Fig. 2, for example, neutron stars cannot accelerate particles to
highest energies under any condition, while shock acceleration would work only
for objects presented in the dark grey corner of this plot.
### 0.1.3 Propagation
Owing to expansion of the Universe, particles which come from sources at
redshift $z$ lose their energy as
$E_{P}\rightarrow E^{\prime}_{P}=E_{P}/(1+z)\leavevmode\nobreak\ .$ (2)
A typical energy loss distance, i.e., distance at which particles lose a
significant part of their energy for this process, is of the order of $z\sim
1$ (50% of energy), i.e., $R\sim 3\leavevmode\nobreak\ \mbox{Gpc}=10^{28}$ cm.
As well as during propagation in the intergalactic space, protons lose energy
due to two other main processes of interactions with Cosmic Microwave
Background (CMB) photons. Those are electron–positron pair production and pion
production. In both processes massive particles have to be produced and they
have threshold energy. Since the typical energy of CMB photons is very small,
$\epsilon_{CMB}=6\times 10^{-4}$ eV, the threshold for those processes is very
high. Only at energies above $E_{th}=m_{e}^{2}/\epsilon_{CMB}\sim 10^{15}$ eV
does the electron–positron pair-production process become important:
$P+\gamma_{CMB}\rightarrow P+e^{+}+e^{-}.$ (3)
The typical energy loss distance for this process is
$R=\frac{M_{P}}{2m_{e}}\frac{1}{\sigma_{P{e^{+}e^{-}}}n_{CMB}}=600\leavevmode\nobreak\
\mbox{Mpc}=2\times 10^{27}\leavevmode\nobreak\ \mbox{cm}\leavevmode\nobreak\
,$ (4)
where $n_{CMB}=400/\mbox{cm}^{3}$ is the density of CMB photons,
$\sigma_{P{e^{+}e^{-}}}\approx 10^{-27}/\mbox{cm}^{2}$ is the proton-pair
production cross-section. The factor $M_{P}/2m_{e}$ comes here from the fact
that in every interaction a proton loses only a tiny fraction of its energy
proportional to the proton/electron mass ratio.
Figure 3: Left: Nucleon interaction length as function of energy from Ref.
[8]. Attenuation due to pion production Eq. (5) presented by the thick solid
line, the same for pair production Eq. (3) presented by the thin solid line.
Right: Horizon (maximal distance) from which protons with given or higher
energy can arrive. Lines for 10%, 30%, 50%, 70% and 90% of events are shown
[9].
At energies above threshold $E_{th}\approx
m_{\pi}M_{P}/\epsilon_{CMB}=10^{20}$ eV, the pion production process dominates
energy losses. This process for cosmic rays was first considered by Greizen,
Zatsepin, and Kuzmin in 1966 [10] and is now named the GZK process.
$P+\gamma_{CMB}\rightarrow\left\\{\begin{array}[]{c}P+\pi^{0}+\sum_{i}\pi_{i}\\\
N+\pi^{+}+\sum_{i}\pi_{i}\end{array}\right.$ (5)
The typical energy loss distance for this process is
$R=\frac{M_{P}}{m_{\pi}}\frac{1}{\sigma_{P\gamma}n_{CMB}}=100\leavevmode\nobreak\
\mbox{Mpc}=3\times 10^{26}\leavevmode\nobreak\ \mbox{cm}\leavevmode\nobreak\
,$ (6)
where $\sigma_{P\gamma}\approx 6\times 10^{-28}/\mbox{cm}^{2}$ is the proton
pion production cross-section. The factor $M_{P}/m_{\pi}$ comes here from the
fact that in every interaction the proton loses only 15–20% of its energy
proportional to the proton/pion mass ratio. Note that at higher energies the
dominating process in Eq. (5) is multi-pion production, in which the proton
loses 50% of its energy in every interaction, however, the cross-section for
this process $\sigma_{\sum\pi}=10^{-28}/\mbox{cm}^{2}$ is a factor 6 lower
than single pion production.
None of the above processes allows a proton with high energy to come from a
very large distance. The distance from which protons can come as a function of
energy is presented in Fig. 3 (left) [8]. The interaction length for pion
production [Eq. (5)] is shown by the dashed line. Attenuation due to pion
production [Eq. (6)] is presented by the thick solid line, the same for pair
production [Eq. (3)] is presented by the thin solid line. Figure 3 (left)
shows the average distance travelled by a single particle. However, for
searches of UHECR sources the important question is the maximum distance or
horizon from which UHECR can come to the detector. In Fig. 3 (right) we
present the horizon as a function of minimal proton energy. The lines 10%,
30%, 50%, 70% and 90% show the fraction of events which come from a given
distance. For example, 90% of events with $E>10^{20}$ eV should come from
distances $R<100$ Mpc. This distance is sometimes called the GZK distance,
because energy losses in this case are dominated by the GZK process of Eq.
(5).
The dominant loss process for nuclei of energy $E\raise
1.29167pt\hbox{$\;>$\kern-7.5pt\raise-4.73611pt\hbox{$\sim\;$}}10^{19}\,$eV is
photodisintegration $A+\gamma\to(A-1)+N$ in the CMB and the infrared
background due to the giant dipole resonance [11]. The threshold for this
reaction follows from the binding energy per nucleon, $\sim 10\>$MeV. Photo-
disintegration leads to a suppression of the flux of nuclei above an energy
that varies between $3\times 10^{19}\>$eV for He and $8\times 10^{19}\>$eV for
Fe.
Figure 4: Sky map of the UHECR proton deflections for energy $E=40$ EeV in
two different models of Galactic magnetic field from Ref. [12]. The colours
show deflections from 0 to 10 degrees.
Figure 5: Fraction of the sky in which the deflection in the extra-Galactic
magnetic field is bigger than the given value. Left: Constraint simulation of
K. Dolag et al. [13]. Right: Simulation of Sigl et al. [14].
Since UHECR are charged particles, they not only lose energy in the
interactions with background photons, but also when deflected by Galactic and
intergalactic magnetic fields.
The magnetic field of the Milky Way galaxy is conventionally modelled as a sum
of the regular and turbulent components of the field in the disk and halo of
the Galaxy. This means that the deflection in the Galactic field $\theta_{\rm
Gal}$ is a superposition of at least four terms:
$\theta_{\rm Gal}=\theta_{\rm Disk}^{\rm regular}+\theta_{\rm Disk}^{\rm
turbulent}+\theta_{\rm Halo}^{\rm regular}+\theta_{\rm Halo}^{\rm
turbulent}\leavevmode\nobreak\ .$ (7)
The deflection angle of UHECR in a regular magnetic field after propagation of
distance $D$ is given by:
$\theta^{\rm regular}\simeq\frac{ZeB_{\bot}D}{E_{\rm UHECR}}\simeq
5^{\circ}Z\left[\frac{E_{\rm UHECR}}{4\cdot 10^{19}\mbox{
eV}}\right]^{-1}\left[\frac{B_{\bot}}{2\cdot 10^{-6}\mbox{
G}}\right]\left[\frac{D}{2\mbox{ kpc}}\right]\leavevmode\nobreak\ ,$ (8)
where where $B_{\bot}$ is the magnetic field component orthogonal to the line
of sight, $E_{\rm UHECR}$ is the particle energy, and $Z$ is the atomic
charge. In the case of deflection by the turbulent field on the distance $D$
much larger than the correlation length of the field $\lambda_{B}$ and where
the deflection angle is small, the deflection is given by
$\displaystyle\theta^{\rm turb}$ $\displaystyle\simeq$
$\displaystyle\frac{1}{\sqrt{2}}\frac{ZeB_{\bot}\sqrt{D\lambda_{B}}}{E_{\rm
UHECR}}\simeq 1.2^{\circ}Z\left[\frac{E_{\rm UHECR}}{4\cdot 10^{19}\mbox{
eV}}\right]^{-1}\left[\frac{B_{\bot}}{4\cdot 10^{-6}\mbox{
G}}\right]\left[\frac{D}{2\mbox{
kpc}}\right]^{1/2}\left[\frac{\lambda_{B}}{50\mbox{ pc}}\right]^{1/2}$ (9)
The deflection angles by regular and turbulent components of Galactic Disk and
Halo, $\theta^{\rm Disk}_{\rm regular}$ and $\theta^{\rm Disk}_{\rm
turbulent}$ are given by Eqs. (8), (9). Contributions of the Halo fields
$\theta^{\rm Halo}_{\rm regular}$ and $\theta^{\rm Halo}_{\rm turbulent}$ are
less certain, but the result in deflections is usually assumed to be less than
the one for disk fields.
Deflections of UHECR by the regular field in the disk $\theta_{\rm Disk}^{\rm
regular}$ have been studied in many theoretical models. A sky map of the
deflections of UHECR with $E=40$ EeV in two different models is presented in
Fig. 4. Despite both models being consistent on average with the expectation
of Eq. (8), predictions in any given direction are strongly model-dependent.
Extragalactic magnetic fields are unknown except in the centres of galaxy
clusters. Therefore one has to use theoretical models for the evolution of the
magnetic fields. In such models magnetic fields follow the formation of large-
scale structures. Because the structure of extragalactic magnetic fields is
very non-trivial with very large fields near large-scale structures and tiny
fields in the voids, one cannot use Eq. (9) everywhere. Instead one can
introduce a fraction of the sky with deflections lower than a given value.
Unfortunately, modern models give a very broad range of predictions. In Fig. 5
we show calculations by two groups that show very different results. The group
of K. Dolag et al. made constraint simulations of local large-scale structures
within 100 Mpc around the Earth [13]. This means that all big structures such
as clusters of galaxies are located in exactly the same places as in the real
sky. Also the density of points in this simulation is adaptive with more
points at clusters and fewer on filaments. The results of this simulation are
shown in Fig. 5 (left). According to this simulation, at 100 Mpc from the
Earth only in 2% of the sky are deflections bigger than
$\theta^{EGMF}=1^{\circ}$. Those places are centres of galaxy clusters. In
contrast the simulation of G. Sigl et al. [14] uses a uniform grid with more
points on filaments and fewer on clusters. Unfortunately this simulation is
not constrained and thus cannot be directly compared to local large-scale
structures. In this simulation $\theta^{EGMF}>50^{\circ}$ in 60% of the sky.
Let us note here that propagation energy losses are extremely important for
the understanding of experimental results on spectrum and composition
discussed in the next section. Deflections in turn are a key issue in the
anisotropy studies in Section 0.1.5.
### 0.1.4 Observations
In this section we shall discuss the present status of UHECR observations.
Figure 6: Extensive air shower produced by a UHECR particle in the atmosphere
We start with the detection of UHECR in the atmosphere. The typical column
density of the atmosphere is 1000 g/cm2 and in 1g there are $N=10^{24}$
protons, while a strong cross-section is $\sigma_{PP}\sim
10^{-25}\leavevmode\nobreak\ \mbox{cm}^{2}$. Thus UHECR protons or nuclei
should interact within the atmosphere many times before they reach the Earth’s
surface. In these interactions it would produce extensive air showers. An
example of such a shower is illustrated in Fig. 6. After first interaction the
primary proton or nuclei would produce a large number of pions. Neutral pions
would start an electromagnetic cascade, while charge pions would produce
muons. At the maximum of shower development one expects $N=10^{9-10}$
particles distributed in an area with a radius of a few kilometres. At this
point the shower mostly consists of 10 MeV electrons and photons and only
5–10% of its energy is in muons. If the shower is not vertical, the column
density increases as $1/\cos(\theta_{zenith})$ and reaches 2000 g/cm2 for
$\theta_{zenith}=60^{\circ}$. At such a depth all electromagnetic components
of the shower disappear and the shower would consist of muons only. Another
way to detect the shower is to look for fluorescent UV light of nitrogen atoms
in the atmosphere. This method is called ‘calorimetric’ and gives a
3-dimensional image of the shower. The main problem with this method is that
detection is possible only on a moonless night, making the duty cycle of such
detectors possible only 10–15% of the time. Finally one can detect direct
Cherenkov light of the charged particles, but since this light is concentrated
only within the central kilometre of the shower, one cannot use this technique
at highest energies.
Figure 7: Pierre Auger Observatory detector with more than 1600 water tanks
and 4 fluorescence telescopes (see Ref. [15] for details) Figure 8: Energy
spectrum of UHECR as a function of energy measured by the Pierre Auger
Observatory and model predictions for iron nuclei (blue) and protons (red) [5]
The Pierre Auger Observatory (Auger) is the largest UHECR detector in the
world at the moment with an area of 3000 km2. Such a big area is required to
collect enough statistics with UHECR at highest energies $E>60$ EeV, because
the flux of such UHECR is tiny, one particle per 100 km2 per year. Auger is
located on the plateau at an altitude of 1000 metres in the Mendoza province
of Argentina. The ground detector consists of 1600 water tanks distributed 1.5
km from each other as presented in Fig. 7. Also there are four fluorescence
telescopes pointed at the atmosphere above the ground detectors as shown in
Fig. 7. Detection of 10% of showers both by fluorescence detectors (FD) and by
ground detectors guarantees a good quality of events and at the same time
allows one to calibrate the ground detectors by FD.
In Fig. 8 we show a recent energy spectrum which was measured by Auger before
31 March 2009 [5]. The steeply falling flux of UHECR is multiplied by $E^{3}$
in order to show details of the spectrum. The total systematic energy error is
22% and is shown in the top right corner of the figure. For energy bins with
$E<3\cdot 10^{19}$ eV statistical errors are not important, while at highest
energies $E>60$ EeV the shape of the spectrum is still uncertain and more
statistics are needed. On the other hand, the suppression of the spectrum is
statistically significant and is clearly seen in Fig. 8.
This is an important experimental result, since it is independent confirmation
of similar observations made by the HiRes experiment [3]. Thus cutoff in the
energy spectrum exists. However, there are several questions to be answered
before one can tell that this really is a GZK cutoff. First, is this cutoff
due to the maximum energy of sources, or to energy losses? In Section 0.1.2 we
have seen that indeed the maximum energy for many types of sources is close to
$10^{20}$ eV. The ultimate answer to this question would be the detection of
several sources at different distances with cutoff following expectations of
energy losses.
Figure 9: Left: Measurement of shower development by signals in the
fluorescence detectors as a function of depth in the atmosphere. The maximum
of shower development in this example is $X_{max}=753\leavevmode\nobreak\
g/\mbox{cm}^{2}$ [16]. Right: Average $X_{max}$ of showers measured by HiRes
and Auger and $RMS$ of $X_{max}$ measured by Auger in 2009 [17].
Second, is the chemical composition of UHECR proton-dominated at those
energies? Since in our Galaxy all elements up to iron are accelerated to
energies around the knee $E=10^{15}$ eV, the same situation can exist in
astrophysical objects which accelerate to highest energies. Experimental
answers to this question can be found in the future even by Auger, but already
current data show that the composition becomes heavy at high energies. Indeed,
recent Auger results from Refs. [16, 17] are shown in Fig. 9. In Fig. 9 (left)
we present the shape of the shower development in the atmosphere as seen by a
fluorescence telescope. The signal is proportional to the number of electrons
and positrons in the shower. Signals grow due to the development of
electromagnetic cascades. The maximum of the signal corresponds to the maximum
development of the cascade in the atmosphere. After that the shower loses its
energy due to dissipation effects. The depth of the atmosphere corresponding
to the maximum of the shower development is called $X_{max}$. For the example
presented, this maximum is at $X_{max}$ = 753 g/cm2 and the energy of the
event is $E=1.6\cdot 10^{19}$ eV [16]. At the same energy, protons on average
interact much deeper in the atmosphere than heavy nuclei.
Figure 10: The layout around the Interaction Point 1 (IP1) of the LHC. The
structure at the centre indicates the ATLAS detector surrounding the collision
point. The LHCf detectors are installed in the instrumentation slot of the
TANs located $\pm 140$ m from IP1. Two independent detectors, LHCf Arm1 and
LHCf Arm2 are installed at either side of IP1[18].
On the top panel of Fig. 9 (right) one can see the results of the most common
hadronic models presented with red lines for protons and with blue lines for
iron. The example event in Fig. 9 (left) is definitely proton-like. The
averaged $X_{max}$ values in each bin are presented in the same figure for
both the Auger and HiRes experiments. Both results are consistent with each
other showing a relatively light composition from $10^{18}$ eV to $10^{19}$
eV. However, Auger shows heavier composition at highest energies.
The main problem when measuring the composition with $X_{max}$ is its strong
model dependence, as seen on the top panel of Fig. 9 (right). There are two
complementary ways out. One is to use a composition-sensitive parameter that
weakly depends on the model choice. Such a parameter is
$RMS(X_{max})=\sqrt{\langle X_{max}^{2}\rangle-\langle X_{max}\rangle^{2}}$,
presented in the lower panel of Fig. 9 (right). One can see that according to
this measurement the composition becomes heavier at high energy. Another
important way is to test models and find the best one. For this purpose a
dedicated experiment LHC forward (LHCf) was constructed at CERN. The idea of
this experiment is to measure the neutral particles emitted in the very
forward region of LHC collisions at low luminosity. The configuration of this
experiment is presented in Fig. 10. Data required for testing the hadronic
models will be collected in the first scientific runs of the LHC [18]. Thus in
the near future we shall have better knowledge of hadronic models and more
understanding of the composition of UHECR at highest energy. At present the
Auger results indicate heavy composition; this was not confirmed by
independent measurements and the fraction of light nuclei in the data remains
uncertain.
This is a very important question for searches of UHECR sources, which we
shall discuss in the next section.
### 0.1.5 Anisotropy
Figure 11: Left: Sky map of arrival directions of UHECR with $E>40$ EeV in
old experiments. Right: Probability that this anisotropy is a function of
angular distance between events [19].
Since for every UHECR event the arrival direction is detected, for tens of
years many attempts were made to find sources of UHECR in the experimental
data. Unfortunately none of them has been confirmed so far. There are two ways
to look for the sources. One is to look for the data itself and try to find
anisotropy in autocorrelation factions. The second is to pick up a catalogue
of possible sources and look for the cross-correlations with this catalogue.
This second way always requires confirmation by an independent data set, since
completeness of the catalogue is a very complicated issue and it is difficult
to estimate the probability due to the parameter choice a posteriori.
Here we start with autocorrelations. In the left panel of Fig. 11 one can see
the sky map with the arrival direction of events with $E>40$ EeV in several
old experiments, including SUGAR, AGASA, HiRes, Yakutsk, Havera Park, Volcano
Ranch, and Fly’s Eye. On the right panel of the same figure one can see the
probability that autocorrelations between selected events are by chance within
a given angle. One can see that the probability is minimal at angles 20–25
degrees. After penalization on the choice of angle, the probability that this
happened by chance is $P=0.3$% [19]. This clustering of events on rather
moderate scales can be due to the location of the sources in the Large Scale
Structure.
Figure 12: Probability of autocorrelations as a function of energy and
angular distance between events, see Ref. [20]
The same probability in the first Auger data is presented in Fig. 12 as a
function of both energy and angle. One can see that for exactly the same
energy $E=40$ EeV and angle $\theta=20^{\circ}$–$25^{\circ}$ the probability
is $P\sim 10^{-2}$. However, recent results with larger statistics did not
show more significant anisotropy at such energies [21]. This makes the
situation with anisotropy in the data less clear.
Figure 13: Left: Sky map of arrival directions of 27 UHECR with $E>57$ EeV
measured by the Pierre Auger Observatory before August 2007 in galactic
coordinates (circles) and 472 nearby AGNs (red stars) [22]. Blue contours show
the Auger exposure. Right: Likelihood ratio for events after formulation of
the prescription. Period II is for data on the left panel. Period III is for
new data up to March 2009 [21].
Now let us discuss correlations with astrophysical objects. First Auger data
have shown strong correlations with nearby active galaxies called Active
Galactic Nuclei (AGN). Namely, 12 out of 14 events with E > 57 EeV were
correlated within $\theta<3.1^{\circ}$ from 472 AGNs from the Veron catalogue
with distances $R<75$ Mpc. This correlation was considered by the Auger
Collaboration as a formal way to study the deviation of cosmic rays from
isotropic distribution. Data from Period I was tested with the prescription
during Period II, where 13 new events were detected, out of which 9 obeyed
prescription parameters. The prescription was fulfilled, i.e., the observed
sky was considered anisotropic at the 99% confidence level [22]. Data used in
this publication and shown in Fig. 13 (left) correspond to Periods I (not
shown) and II shown in Fig. 13 (right) before the vertical line. Unfortunately
this correlation was not confirmed in the later data [Period III in Fig. 13
(right)].
Figure 14: Angular distribution of events around the Cen A galaxy in Auger
data compared to isotropic ones
It does not mean that all anisotropy signals in Auger have completely
disappeared. There is still a remaining excess of events around the Cen A
galaxy on scales of 20 degrees, see Fig. 14. This anisotropy has to be tested
by future data.
### 0.1.6 Secondary photons and neutrinos from UHECR
Figure 15: Left: Fluxes of protons and secondary photons as a function of
energy. Primary protons with spectrum $1/E^{2.6}$ and maximum energy
$E_{max}=10^{21}$ eV are shown by the thin red line. Secondary protons fit the
UHECR spectrum from $E>10^{18}$ eV (thick red line). Secondary photons from
all reactions are shown by the blue dashed line and from pion production only,
Eq. (5) by the magenta line. Right: Fluxes of UHECR and secondary photons in
the case of iron nuclei primaries with spectrum $1/E^{2.1}$ and maximum energy
$E_{max}=10^{21}$ eV. The remaining iron nuclei are shown by the green line.
Secondary protons by the magenta line. Secondary photons by the blue line.
Figure 16: Contribution of secondary photons from UHECR to the extragalactic
gamma-ray background as a function of energy and other possible sources which
contribute to the same background [23]
As was discussed in Section 0.1.3, protons lose their energy in pair
production and pair production reactions. Since secondary pions quickly decay,
secondary photons and neutrinos are produced. Neutrinos propagate to the Earth
without interactions on the way, but photons cannot. They start to interact
with background photons and produce pairs. Electrons and positrons in turn up-
scatter CMB photons or produce synchrotron radiation:
$\displaystyle\gamma+\gamma_{background}$ $\displaystyle\rightarrow$
$\displaystyle e^{+}+e^{-}$ $\displaystyle e^{\pm}+\gamma_{background}$
$\displaystyle\rightarrow$ $\displaystyle e^{\pm}+\gamma$ (10) $\displaystyle
e^{\pm}+B$ $\displaystyle\rightarrow$ $\displaystyle e^{\pm}+\gamma_{synch}$
The sequence of processes in Eq. (0.1.6) is called an electromagnetic cascade.
At energies above $10^{15}$ eV the cascade proceeds on the CMB background
(400/cm3), but at lower energies pair production on CMB is impossible. At such
energies the cascade continues on a much less abundant infrared background
(1/cm3) and at lower energies on optical background (0.01/cm3). Then it stops
at the multi-GeV energies of gamma rays.
In Fig. 15 we plot primary cosmic-ray and secondary photon fluxes from primary
protons (left) and iron (right) from Ref. [23]. Secondary protons after
interaction fit the UHECR spectrum from $E>10^{18}$ eV in Fig. 15 (left).
Secondary photons cascade down to the GeV region. Only a small fraction of
photons come from the pion production reaction (magenta dotted line). Most of
the photons generated are from the $e^{+}e^{-}$ production reaction with total
flux shown by the dash-dotted blue line. The number of secondary protons is
much lower in the case of iron primaries, as shown by the dotted magenta line
in Fig. 15 (right). As a result, the secondary photon flux in the GeV region
is much smaller in this case, on the level of 0.2% of the EGRET measurement.
Also very high energy photons are absent in this case due to low maximum
proton energy.
Figure 17: Left: Example of GZK photon flux from Ref. [24]. UHECR protons fit
the HiRes spectrum. Secondary neutrinos are shown by a green line. The
remaining secondary photons are in the range between the blue lines. Right:
Experimental upper limits on the photon fraction in the UHECR spectrum from
Ref. [21].
In Fig. 16 we compare the range of the electromagnetic cascade fluxes from
UHECR with other possible astrophysical contributions in the EGRET band. Note
that most of the uncertainty of the UHECR cascade flux comes from an unknown
source evolution. The scatter for a given class of sources is thus much
smaller, as seen from Fig. 16 for the case of AGNs.
In Fig. 17 (left) we plot the possible range of GZK gamma-ray fluxes for a
given proton flux which fit the UHECR spectrum. The range of fluxes comes from
the variation of possible values of the extragalactic magnetic field and the
range of the models for extragalactic radio background. Also on the same
figure the corresponding neutrino flux is shown by a green line. In Fig. 17
(right) we show the experimental upper limits on the fraction of photons in
the UHECR flux. The range of possible GZK photon fluxes corresponds to protons
with a range of power law injection spectra and source evolution fitting the
UHECR spectrum. One can note that the current best upper limits of Auger are
still above the range of expected theoretical values. On the other hand,
existing limits already exclude some exotic models.
### 0.1.7 Summary
In the first lecture we briefly discussed many aspects of UHECR physics.
Observed cosmic rays have energies up to $10^{20}$ eV. Acceleration in
astrophysical objects to such energies is a very non-trivial task and there
are no objects in our Galaxy which can do this job. There are very few classes
of exceptionally powerful objects in the Universe, some of which can be real
sources of UHECR. Accelerated particles lose their energy in interactions with
the CMB background and are also deflected by electromagnetic fields during
their propagation from sources to the Earth.
There are three important experimental challenges in UHECR physics: the
spectrum of cosmic rays, the chemical composition of cosmic rays, and the
search for anisotropies in the sky with the ultimate goal of finding UHECR
sources.
The cutoff in the energy spectrum at highest energies $E>6\cdot 10^{19}$ eV
has now been established by two independent experiments, HiRes and Auger.
The most striking result of 2009 was evidence of heavy composition, shown by
the Auger experiment at highest energies, Fig. 9. This result still needs
independent confirmation. Also the interpretation of composition measurements
is affected by uncertainty in the hadronic models. This question can be
clarified in the near future by the LHCf experiment.
Finally, most challenging is the search for UHECR sources. The last result in
this direction was made by Auger in 2007. They found that the sky is
anisotropic at the highest energies, at least at the 99% C.L., by looking at
the correlations with nearby AGNs. Unfortunately those correlations were not
confirmed in the new data, and the only anisotropy excess remaining in the
Auger data at the highest energies is an excess around the Cen A galaxy, see
Fig. 14.
During energy losses the UHECR protons produce secondary photons and
neutrinos. Most of the secondary photons cascade down to the GeV energies,
where this contributes to the diffuse extragalactic background. An
experimental search for the remaining gamma rays at highest energies
$E>10^{18}$ eV is challenging and existing upper limits are just above
theoretical predictions, see Fig. 17.
Thus there are many unsolved problems in UHECR physics. They require both
theoretical and experimental efforts in the near and more distant future.
## 0.2 High-energy gamma rays
### 0.2.1 Introduction
In this lecture I shall discuss the theory of TeV gamma rays and recent
observations made in this field. I shall give a brief introduction to the
experimental detection techniques and present some selected results on the
subject. For more detailed study I would like to recommend the recent review
by F. Aharonian, J. Buckley, T. Kifune, and G. Sinnis [25].
Relativistic particles can travel with a speed larger than the speed of light
in the medium $V>V_{M}=c/n$. Here $n>1$ is the refractive index of the medium.
This index in the air is $n_{\mbox{a}}=1.008$ and in water
$n_{\mbox{w}}=1.33$.
The charged particles polarize the molecules of the medium, which then return
rapidly to their ground state, emitting prompt radiation called Cherenkov
radiation. This radiation is emitted under a constant Cherenkov angle with the
particle trajectory, given by
$\cos\delta=\frac{V_{M}}{V}=\frac{c}{nV}=\frac{1}{\beta n}\leavevmode\nobreak\
.$ (11)
Figure 18: Detection of high-energy gamma rays by Cherenkov telescopes in air
(left) and in water (right)
Figure 19: Examples of gamma-ray experiments: Cherenkov telescope H.E.S.S.
(left) and water pool Milagro (right)
The minimal energy of a charged particle is
$\gamma_{min}=\frac{E_{min}}{M}=\frac{n}{\sqrt{n^{2}-1}}\leavevmode\nobreak\
.$ (12)
Particles with higher energy will produce a cone of Cherenkov light. This
effect is used by Cherenkov telescopes for air (H.E.S.S., MAGIC, Veritas, CTA)
and by ground experiments in water (Milagro, HAWK). Detection of the shower in
air and in water is illustrated in Fig. 18.
We present examples of such experiments in Fig. 19. On the left panel we show
a view of the H.E.S.S. experiment. This experiment made the most significant
contribution to the development of TeV gamma-ray astrophysics in recent years.
On the right panel we show the Milagro experiment, a pioneering experiment in
water Cherenkov techniques.
### 0.2.2 Point sources of TeV gamma rays
Figure 20: Sky in the TeV gamma rays with 3 sources in 1995 (top left), 32
sources in 2005 (top right), and 80 sources in 2009 (bottom)
TeV gamma-ray astrophysics is developing very quickly. One can see the number
of detected sources in the sky as a function of time in Fig. 20. From 3
sources in 1995 one has 32 sources in 2005 and 80 sources in 2008. In addition
not only does the number of observed sources grow, but also the number of
different populations of sources. This is a very important fact for future
experiments with better sensitivity like the Cherenkov Telescope Array (CTA).
They would have a very large potential for detecting many different classes of
sources.
In particular, in Fig. 20 on the bottom panel, red circles show extragalactic
sources which contain BL Lac objects, radio galaxies, and starburst galaxies.
Also in the galactic plane there are many different classes of objects, which
include supernova shells, pulsar wind nebulas, pulsars, binary systems and
dark objects. Dark objects mean they were detected in gamma rays, but there is
no corresponding source in other wavebands.
Figure 21: Sensitivity of gamma-ray detectors to point sources, from Ref.
[25] Figure 22: Redshift for gamma rays as a function of energy. Lines show
constant optical depth in two models of IR/O background. Figure 23: Redshift
for gamma rays as a function of energy. Lines show constant optical depth in
two models of IR/O background.
The sensitivity of gamma-ray detectors to point sources as a function of
energy is shown in Fig. 21. The sensitivity of air telescopes is shown for 50
hours of observation for one source. The sensitivity for ground experiments is
shown for 5 years, but they observe all the sky ($2\pi$sr). At low energies
$E<10$ GeV the sensitivity of the GLAST (Fermi) satellite is the best. One
year of observations are shown. At large energies $E>10$ TeV the ground air-
shower experiments (Tibet) have the best sensitivity. Future CTA projects will
be orders of magnitude better than present-day experiments from 10 GeV to 10
TeV energies.
Another important fact is that gamma rays cannot travel freely in the
intergalactic space. They interact with optical/infrared background photons
and disappear producing pairs of electrons and positrons. In Fig. 22 one can
see the main backgrounds for gamma-ray propagation. They are shown in units of
photon density per cm3. The largest contribution comes from the CMB background
with 400 photons per cm3. However, owing to the small energy of CMB photons,
this background is important only for $E>1000$ TeV. For the experimentally
interesting energy range $E<100$ TeV the main backgrounds are infrared and
optical. Since those backgrounds are created by galaxies and partly by dust
they are strongly model dependent both as a function of energy and as a
function of redshift.
Optical depth can be defined as
$\tau(E)=R\cdot\sigma_{\gamma\gamma}(E)\cdot
n_{back}(z,\epsilon)\leavevmode\nobreak\ ,$ (13)
where $R$ is the distance travelled by photons, $\sigma_{\gamma\gamma}(E)$ is
the pair-production cross section, and $n_{back}(z,\epsilon)$ is the density
of background photons. Distances on the cosmological scale are often expressed
in terms of redshift. One can express it through the Hubble law $R=z\cdot
c/H_{0}$, where $H_{0}=70$ km/s/Mpc is the Hubble constant. In Fig. 23
contours of constant optical depth $\tau(E)$ are shown on the plane redshift
versus energy for $\tau(E)=1,3,10$ in two different models of IR/O background.
Figure 24: Observation of Mkn 421 as a function of time
There is one important difference between air Cherenkov telescopes and water
Cherenkov detectors. In Fig. 24 we plot world-wide monitoring of the nearby BL
Lac object Mkn 421 as a function of time. One can see that air Cherenkov
telescopes can see a signal only on moonless nights, which restricts their
operation to the corresponding intervals of time. On the contrary, water
Cherenkov telescopes operate all the time they can see a source, which will
allow source activity to be detected all the time. On the other hand, the
problem of water Cherenkov experiments is poor sensitivity, which will prevent
them from detection of relatively low fluxes and very fast variations in time.
Thus both techniques are complementary to each other.
Figure 25: Central part of the Milky Way galaxy in infrared, optical, and in
TeV gamma rays. The TeV gamma-ray sky from H.E.S.S. observations with a large
number of sources. Figure 26: The Milky Way galaxy in TeV gamma rays from
galactic longitude 20∘ to 220∘ and galactic latitude from $-10^{\circ}$ to
$10^{\circ}$. The image is the culmination of a seven-year exposure by the
Milagro instrument.
In Fig. 25 one can see a view of the central part of the Milky Way galaxy in
three energy bands: optical, infrared, and TeV gamma rays. At least three
astronomical source populations: supernova remnants (SNRs), pulsar wind
nebulae (PWNe), and binary systems (BSs) are represented in this figure. In
addition, the H.E.S.S. observations of the central region of our Galaxy
revealed a diffuse TeV $\gamma$-ray emission component which is apparently
dominated by contributions from giant molecular clouds (GMCs). These massive
complexes of gas and dust most likely serve as effective targets for
interactions of relativistic particles from nearby active or recent
accelerators. Thus one may claim that four galactic source populations are
already firmly established as effective TeV $\gamma$-ray emitters. Meanwhile,
many sources discovered by H.E.S.S. in the galactic plane remain unidentified.
Although some of these sources might have direct or indirect links to SNRs,
PWNe, and GMCs, one cannot exclude that a fraction of the H.E.S.S.
unidentified sources are related to other source classes.
The Milagro telescope has made the first measurement of the diffuse TeV gamma-
ray flux from the Galactic Disk. Figure 26 shows the Galaxy (as visible from
the Northern Hemisphere) in TeV gamma rays. In addition to the individual
sources discussed above, the image (compiled from Milagro data) shows the
existence of a diffuse TeV gamma-ray flux between galactic longitudes of 30∘
and 90∘.
### 0.2.3 Extragalactic magnetic fields
Figure 27: Detection of EGMF through observation of secondary emissions
around a point source [26]
Another very important field which will benefit in the near future from TeV
gamma rays is Extragalactic Magnetic Fields.
Indeed, as discussed above, TeV gamma rays emitted by astrophysical sources
can be measured by detectors on Earth. Practically all TeV gamma rays from
galactic sources come directly to the detectors. However, this is not true for
extragalactic sources. As one can see from Fig. 23, even for nearby sources
like Mkn 501, gamma rays with $E>10$ TeV cannot come freely to the detector.
The pair production on Extragalactic Background Light (EBL) reduces the flux
of $\gamma$-rays from the source by
$F(E_{\gamma_{0}})=F_{0}(E^{\prime}_{\gamma_{0}}(z_{E}))e^{-\tau(E_{\gamma_{0}},z_{E})},$
(14)
where $F(E_{\gamma_{0}})$ is the detected spectrum,
$F_{0}(E^{\prime}_{\gamma_{0}})$ is the initial spectrum of the source, and
$\tau(E_{\gamma_{0}},z_{E})$ is the optical depth Eq. (13). The typical
distance which a primary gamma ray travels is
$D_{\gamma_{0}}=D_{\gamma}(E_{\gamma_{0}}^{\prime},z)=40\frac{\kappa}{(1+z)^{2}}\left[\frac{E_{\gamma_{0}}^{\prime}}{20\mbox{
TeV}}\right]^{-1}\mbox{ Mpc}\leavevmode\nobreak\ ,$ (15)
where a numerical factor $\kappa=\kappa(E_{\gamma_{0}},z)\sim 1$ accounts for
the model uncertainties.
The cascade electrons lose their energy via Inverse Compton (IC) scattering of
the CMB photons within the distance
$D_{e}=\frac{3m_{e}^{2}c^{3}}{4\sigma_{T}U_{\rm
CMB}^{\prime}E_{e}^{\prime}}\simeq
10^{23}(1+z_{\gamma\gamma})^{-4}\left[\frac{E_{e}^{\prime}}{10\mbox{
TeV}}\right]^{-1}\mbox{ cm}$ (16)
The deflection angle of the $e^{+}e^{-}$ pairs, accumulated over the cooling
distance, depends on the correlation length of the magnetic field,
$\lambda_{B}$. Note also that electrons and positrons travel much shorter
distances than primary photons.
The $e^{+}e^{-}$ pairs produced in interactions of multi-TeV $\gamma$-rays
with EBL photons produce secondary $\gamma$-rays via IC scattering of the
Cosmic Microwave Background (CMB) photons. Typical energies of the IC photons
reaching the Earth are
$E_{\gamma}=\frac{4}{3}(1+z_{\gamma\gamma})^{-1}\epsilon_{CMB}^{\prime}\frac{E_{e}^{\prime
2}}{m_{e}^{2}}\simeq 0.32\left[\frac{E^{\prime}_{\gamma_{0}}}{20\mbox{
TeV}}\right]^{2}\mbox{ TeV}$ (17)
where $\epsilon_{CMB}^{\prime}=6\times 10^{-4}(1+z_{\gamma\gamma})$ eV is the
typical energy of CMB photons. In the above equation we have assumed that the
energy of a primary $\gamma$-ray is $E^{\prime}_{\gamma_{0}}\simeq
2E_{e}^{\prime}$ with $E_{\gamma_{0}}^{\prime}$ being the energy of the
primary $\gamma$-rays at the redshift of the pair production. Upscattering of
the infrared/optical background photons gives a sub-dominant contribution to
the IC scattering spectrum because the energy density of CMB is much higher
than the density of the infrared/optical background.
Deflections of $e^{+}e^{-}$ pairs produced by the $\gamma$-rays, which were
initially emitted slightly away from the observer, could lead to ‘redirection’
of the secondary cascade photons toward the observer. This effect leads to the
appearance of two potentially observable effects: extended emission around an
initially point source of $\gamma$-rays [26, 27, 28] and delayed ‘echo’ of
$\gamma$-ray flares of extragalactic sources [29, 30].
Figure 28: Model predictions and estimates for the EGMF strength. Cyan shaded
region excluded by present day measurements. Black ellipses show measurements
of the field in the Galaxy and galaxy clusters. Left panel: left and right
hatched regions show theoretically allowed range of values of
($\lambda_{B}$,B) for non-helical and helical fields generated at the epoch of
electroweak phase transition during radiation-dominated era. Middle panel:
left and right hatched region show ranges of possible ($\lambda_{B}$,B) for
non-helical and helical magnetic fields produced during the QCD phase
transition. Right panel: hatched region is the range of possible
($\lambda_{B}$,B) for EGMF generated during recombination epoch. Dark grey
shaded region shows the range of ($\lambda_{B}$,B) parameter space accessible
for the $\gamma$-ray measurements via $\gamma$-ray observations [31].
The above processes are illustrated in Fig. 27. Electron deflection $\delta$
depends on the magnetic field in the region of deflection. Note, that, in
principle, EGMF depends on the redshift, $B^{\prime}=B^{\prime}(z)$. In the
simplest case, when the magnetic field strength changes only as a result of
expansion of the Universe, $B^{\prime}(z)\sim B_{0}(1+z)^{2}$, where $B_{0}$
is the present epoch EGMF strength. This gives
$\displaystyle\delta$ $\displaystyle=$ $\displaystyle\frac{D_{e}}{R_{L}}\simeq
3\times 10^{-6}(1+z_{\gamma\gamma})^{-4}\left[\frac{B^{\prime}}{10^{-18}\mbox{
G}}\right]\left[\frac{E_{e}^{\prime}}{10\mbox{ TeV}}\right]^{-2}$ (18)
$\displaystyle\simeq$ $\displaystyle 3\times
10^{-6}(1+z_{\gamma\gamma})^{-2}\left[\frac{B_{0}}{10^{-18}\mbox{
G}}\right]\left[\frac{E_{e}^{\prime}}{10\mbox{ TeV}}\right]^{-2}$
Knowing the deflection angle of electrons, one can readily find the angular
extension of the secondary IC emission from the $e^{+}e^{-}$ pairs
$\Theta_{\rm
ext}\simeq\left\\{\begin{array}[]{ll}0.5^{\circ}(1+z)^{-2}\left[\frac{\tau_{\theta}}{10}\right]^{-1}&\\\
\left[\frac{\displaystyle E_{\gamma}}{\displaystyle 0.1\mbox{
TeV}}\right]^{-1}\left[\displaystyle\frac{B_{0}}{\displaystyle 10^{-14}\mbox{
G}}\right],&\lambda_{B}^{\prime}\gg D_{e}\\\ &\\\
0.07^{\circ}(1+z)^{-1/2}\left[\frac{\tau_{\theta}}{10}\right]^{-1}&\\\
\left[\frac{\displaystyle E_{\gamma}}{\displaystyle 0.1\mbox{
TeV}}\right]^{-3/4}\left[\frac{\displaystyle B_{0}}{\displaystyle
10^{-14}\mbox{ G}}\right]\left[\frac{\displaystyle\lambda_{B0}}{\displaystyle
1\mbox{ kpc}}\right]^{1/2},&\lambda_{B}^{\prime}\ll D_{e}\end{array}\right.$
(19)
This is a key point for detection of the field, since extended emission
depends on energy in a well-defined way and can be reconstructed using
independent measurements at different energies.
The possible ranges of the ($\lambda_{B}$,B) parameter space are shown in Fig.
28 for the cases when magnetogenesis proceeds during electroweak or QCD phase
transitions or at the moment of recombination.
It is interesting to note that predictions for the strength and correlation
length of the primordial magnetic fields fall in a region of ($\lambda_{B}$,B)
parameter space which is not accessible for the existing measurement
techniques, such as Faraday rotation or Zeeman splitting methods. However, it
turns out that this region of ($\lambda_{B}$,B) parameter space is accessible
for the measurement techniques which exploit the potential of the newly opened
field of very-high-energy (VHE) $\gamma$-ray astronomy [31].
### 0.2.4 Summary
Gamma-ray astronomy works, hundreds of sources have been detected in the GeV
energy range and about one hundred in TeV energies.
There are several major questions to be answered in the near future:
* •
One needs to understand the hadronic component in a variety of astrophysical
sources.
* •
Extragalactic IR/O backgrounds have already been constrained by observations
of TeV sources to factor two uncertainty. The next step is precision
determination of those backgrounds using measurements of many sources at
different redshifts.
* •
For the first time one has a possibility to study primordial magnetic fields
through TeV gamma-ray measurements. We can test models of primordial magnetic
fields in the near future.
There are several other important issues which were not discussed in this
Lecture due to lack of time. The corresponding questions are:
* •
Good measurements of blazar flairs can help to understand gravity near black
holes.
* •
TeV gamma rays give one more constraint/signature on Dark Matter.
* •
Constraints on exotic physics (LIV, etc.) will be improved.
## 0.3 High-energy neutrinos
### 0.3.1 Introduction
In this lecture we discuss theoretical predictions and experimental efforts to
detect Ultra-High Energy neutrinos. In Section 0.3.2 we discuss possible ways
to detect UHE neutrinos and their corresponding experiments. In Section 0.3.3
we show theoretical predictions for UHE neutrino fluxes and present the status
of experimental searches for such fluxes. In Section 0.3.4 we discuss another
possibility to detect Galactic neutrino sources at multi-TeV energies. In
Section 0.3.5 we summarize all the results of this lecture.
### 0.3.2 High-energy neutrino experiments
There are three types of ultra-high energy (UHE) neutrino experiments.
First, neutrinos can be detected by UHECR experiments. There are two
possibilities for this. First, one can use the fact that the atmosphere
horizontally has depth 36 times the vertical depth. Relatively young
electromagnetic horizontal showers can be caused by neutrinos only. Hadronic
showers at such a depth consist of muons only. Second, one can look for events
penetrating the Earth in the tau-neutrino channel, i.e., look for upward-going
events. This was used by the Auger experiment (see Fig. 7). The resulting
limit on neutrino flux is shown in Fig. 31. Also less significant limits were
presented by previous UHECR experiments including Fly’s Eye, AGASA, and HiRes
(the HiRes limit is also shown in Fig. 31).
Figure 29: IceCube detector. Left: Configuration of the IceCube detector.
Eighty strings will be located at a depth of 1.5 km in the Antarctic ice
filling a volume of one cubic kilometre. The present construction stage is
also shown [32]. Right: Simulation of a high-energy neutrino event in the
IceCube detector [33].
Second, one can detect neutrinos in the water or in the ice by detecting
Cherenkov light created by corresponding leptons after neutrino interaction in
the medium. There are two important backgrounds for such measurement. First,
secondary leptons, mostly muons, should not be confused with secondary muons
from extensive air showers in the atmosphere. In order to reduce the
background of atmospheric muons one has to put the detector at a depth greater
than one kilometre from the surface. Second, there are atmospheric neutrinos
created by the same cosmic rays, which would produce isotropy in the space
energy-dependent background. In order to fight this background, one either has
to go to high energies $E>10^{15-16}$ eV, where it is small, or look for point
sources on top of this background.
Experiments that worked with these techniques in the past were Baikal and
ANTARES in water and AMANDA in ice. All those experiments had a volume $0.1$
km3 or less. The new-generation experiment IceCube with a volume of 1 km3 is
in the construction stage at the moment. In Fig. 29 in the left panel one can
see the configuration of this experiment, which consists of 80 strings,
filling a cubic kilometre volume in the Antarctic ice at a depth of 1.5 km
from the surface. Strings already implemented are shaded blue on top of the
picture (see Ref. [32] for more details). Also, as shown in the figure the top
of the detector is covered by an array of ice tanks (ice top). In the right
panel one can see a Monte Carlo simulation of a high-energy neutrino event,
detected by the IceCube experiment. First results of this experiment will be
discussed in Section 0.3.4.
Figure 30: ANtarctic Impulsive Transient Array (ANITA) radio balloon
experiment. Array of radio antennas flying in the ballon, as shown on the left
panel. It flies in circles over the Antarctic ice at a height of 37 km (see
right panel) and looks for radio signals which UHE neutrinos create in the
ice.
Finally, radio neutrino experiments exploit the Askaryan effect in which
strong coherent radio emission arises from electromagnetic showers in any
dielectric medium. High-energy neutrinos trigger a cascade of electromagnetic
particles in the medium, which has net charge and can emit an analogue of
Cherenkov light in the radio energy range. The main point of this effect is
that the length of the radio wave is macroscopic (tens of centimetres) and is
bigger than the size of the cascade itself. This in turn means that all
electrons in the cascade emit coherently. The effect was first observed in
2000 at SLAC. Recently the Askaryan effect has been clearly confirmed and
characterized for ice as the medium, as part of the pre-flight calibration of
the ANITA-1 payload. The Askaryan effect can be seen only at high energies
$E>10^{17-18}$ eV. Experiments using this effect benefit from the absence of
atmospheric neutrino flux at such high energies, but they also have to look
over a huge effective volume in order to see tiny neutrino fluxes at highest
energies.
Experiments that used this effect to search for UHE neutrinos are FORTE [34],
RICE [35], and ANITA [36, 37]. FORTE is a satellite experiment, which, in
particular, looked over the Greenland ice. Unfortunately, the threshold of
this experiment was very high, $E_{\nu}>10^{22}$ eV, so it could test only
exotic top-down models. RICE was an array of radio antennas located in the ice
at the South Pole at the same place as the AMANDA experiment. This experiment
presented its final results in 2006. Finally, the most advanced for the moment
of this kind of experiment is the ANtarctic Impulsive Transient Array (ANITA)
radio balloon experiment, see Fig. 30. In the left panel one can see an array
of radio antennas in the balloon. In the right panel one can see a schematic
map of flight over the Antarctic at a height of 37 km.
### 0.3.3 Search for cosmogenic neutrinos
Figure 31: Predictions of cosmogenic neutrino fluxes and theoretical bounds
on them [39, 40]
As discussed in Section 0.1.3 UHECR protons lose their energy in interactions
with CMB photons and produce pions at energies above threshold $E>6\cdot
10^{19}$ eV. This GZK threshold was found in 1966 [10]. As long ago as 1969
Berezinsky and Zatsepin suggested that one can try to observe secondary
neutrinos from pion decays and called them cosmogenic neutrinos [38]. Recently
the ANITA Collaboration proposed to call such neutrinos Berezinsky–Zatsepin
neutrinos, or BZ neutrinos [37]. Below we follow this suggestion.
One can calculate the flux of BZ neutrinos theoretically, after fitting the
corresponding proton spectrum to the experimental flux above some energy. The
absolute limit for neutrino flux comes from the fact that gamma rays
unavoidably produced from $\pi^{0}$ decays and from electrons $\pi^{\pm}$
decays cascade down to GeV energies and the maximum flux of such gamma rays
cannot overshoot the EGRET measuremen shown in Fig. 16. This bound on the BZ
neutrino flux is called “gamma-ray bound” in Fig. 31. Note that there are many
additional ways to create photons in the EGRET energy range, including
electron–positron pair production discussed in the previous section, so the
real BZ neutrino flux is always lower than this region.
Figure 32: Experimental limits on cosmogenic neutrino flux. Best up-to-date
ANITA-1 limits based on no surviving candidates for 18 days of live time shown
as ANITA-2008 [37]. Also limits from Auger [41], HiRes [42], FORTE [34], Anita
prototype ANITAlite [36], RICE [35], and AMANDA II [43] are shown.
Also in Fig. 31 we plot two theoretical limits derived under a set of
theoretical assumptions. One is called the Waxman–Bahcall (WB) bound and the
other the MPR bound. On the same figure we show several examples of
theoretical neutrino fluxes which violate both WB and MPR bounds, but all of
them are consistent with the experimental gamma-ray bound.
In Fig. 32 we show present-day experimental bounds confronting theoretical
predictions for BZ neutrinos from Ref. [37]. One can see that the best up-to-
date experimental bounds come from the ANITA experiment. ANITA-1 was able to
view a volume of ice of $\sim 1.6$ Mkm3 during 17.3 days, however, volumetric
acceptance to a diffuse neutrino flux, accounting for the small solid angle of
acceptance for any given volume element, is several hundred km3 water-
equivalent steradians at $E_{\nu}=10^{19}$ eV. This allowed them for the first
time a tough theoretically interesting region, excluding part of the parameter
space with highest neutrino fluxes.
On the same figure one can see existing limits on diffuse neutrino flux from
the Auger [41], HiRes [42], FORTE [34], Anita prototype ANITAlite [36], RICE
[35], and AMANDA II [43] experiments.
Let us note also that in Fig. 32 the composition is assumed to be proton-
dominated. If recent Auger results presented in Fig. 9 are confirmed,
theoretical expectations for neutrino flux in Fig. 32 will be strongly
reduced. This will make observations of the diffused flux of UHE neutrinos an
even more complicated issue. However, at lower energies one still can have a
hope of seeing point sources with neutrinos, as will be discussed in the next
section.
### 0.3.4 Point sources of UHE neutrinos
Figure 33: Neutrino–nucleon cross-section as a function of the neutrino
energy. Charge-current and neutral-current contributions to the cross-section
are shown with thin solid and dashed lines. The total cross-section is
presented by a thick solid line. See Ref. [44] for details.
At highest energies the neutrino flux is too low to detect one single source
of neutrinos, but at lower energies $E<1000$ TeV the flux from a single source
can be high enough to detect it. Indeed, in Fig. 33 the neutrino–nucleon
cross-section is shown as a function of energy. This cross-section is
proportional to $E$ at low energies $E<1$ TeV and to $E^{0.4}$ at high
energies $E>10^{6}$ GeV. Good candidates for neutrino sources in the Galaxy
are objects emitting TeV gamma rays. They can produce neutrinos in the
proton–proton collisions in objects in the case of binary systems and in the
interaction with molecular clouds in the Galaxy. In the 10 TeV energy range
$\sigma_{p\nu}(10\textrm{ TeV})=10^{-34}\leavevmode\nobreak\
\mbox{cm}^{2}\leavevmode\nobreak\ .$ (20)
In the IceCube detector only a small fraction of neutrinos will produce a
signal:
$\tau_{\nu}=\sigma_{p\nu}n_{ICE}R\sim\leavevmode\nobreak\
10^{-5}\leavevmode\nobreak\ ,$ (21)
where $n_{ICE}\sim 10^{24}/\mbox{cm}^{3}$ is the density of the ice and $R=1$
km is the height of the IceCube detector.
The expected flux of neutrinos produced in the proton–proton collisions in the
Galactic sources is
$F_{\nu}\sim F_{\gamma}=10^{-12}\frac{1}{\mbox{cm}^{2}\mbox{s}}\approx 3\cdot
10^{5}\frac{1}{\mbox{km}^{2}\mbox{yr}}\leavevmode\nobreak\ .$ (22)
Thus in the IceCube detector one can expect three events per year for a $10$
TeV neutrino flux.
Figure 34: Left: Simulated detection of Milagro TeV galactic sources by
IceCube. Right: Significance of Milagro hotspots after five years of
observation of IceCube.
In Fig. 34 one can see a simulation of Milagro sources from Fig. 26 after five
years of working of the IceCube detector.
Figure 35: Equatorial sky-map of events (points) and pre-trial significances
(p-value) of the all-sky point source search in the 22-string IceCube detector
[45]. The solid curve is the galactic plane. The most significant spot arrives
in a random sky with probability $P\sim 1\%$.
We now present recent results for point-source searches using data recorded
during 2007–08 with 22 strings of IceCube (1/4 of the detector). An all-sky
search within the declination range $-5^{\circ}$ to $+85^{\circ}$ found the
most significant deviation from the background at $153.4^{\circ}$ r.a.,
$11.4^{\circ}$ dec. Accounting for all trials in the point-source search, the
final p-value for this result is 1.34%, consistent with the null hypothesis of
background-only events at the 2.2$\sigma$ level. No obvious source candidates
are near this location, and an analysis of the timing of the events did not
find any evidence of a burst in time. The location can be added to the a
priori source candidate list for analysis using future IceCube data, in which
case a similar excess would be identified with much higher significance [45].
### 0.3.5 Summary
IceCube is half-complete. If it observes first sources, a new field of
astroparticle physics will be started: neutrino astrophysics. If not, much
bigger detectors are needed with a size of at least 10 km3. Secondary neutrino
flux from UHECR protons can be detected by future radio experiments, like
ANITA. Neutrinos from some bright galactic sources can be detected by IceCube.
Extragalactic sources can be observed during bright flair activity. In order
to detect continuous flux from sources like Cen A one needs detectors much
larger than 1 km3. Galactic SN can be detected with neutrinos at low and high
energies. Cubic-kilometre water detectors will be constructed if IceCube gives
positive results.
### Acknowledgements
I would like to thank the Organizing Committee of the 5th CERN Latin American
School for giving me the opportunity to present lectures there and for the
excellent organization of the School.
## References
* [1]
* [2] E. S. Seo et al., Measurement of cosmic-ray proton and helium spectra during the 1987 solar minimum, Astrophys. J. 378, 763 (1991); M. Nagano et al., Energy spectrum of primary cosmic rays between $10^{14.5}$ and $10^{18}\,$eV, J. Phys. G10, 1295 (1984); M. Nagano et al., Energy spectrum of primary cosmic rays above $10^{17}\,$eV determined from extensive air shower experiments at Akeno, J. Phys. G18, 423 (1992); N. L. Grigorov et al., Energy spectrum of primary cosmic rays in the $10^{11}-10^{15}\,$eV according to the data of Proton-4 measurements, Proceedings 12th ICRC, 1, 1760 (1971).
* [3] R. Abbasi et al. [HiRes Collaboration], Phys. Rev. Lett. 100, 101101 (2008) [arXiv:astro-ph/0703099].
* [4] M. Nagano and A. A. Watson, Rev. Mod. Phys. 72, 689 (2000).
* [5] J. Abraham et al. [The Pierre Auger Collaboration], The Cosmic Ray Energy Spectrum and Related Measurements with the Pierre Auger Observatory, arXiv:0906.2189.
* [6] M. Kachelriess, Lecture Notes on High Energy Cosmic Rays, 2008, arXiv:0801.4376 [astro-ph].
* [7] A. M. Hillas, Annu. Rev. Astron. Astrophys. 22, 425 (1984).
* [8] P. Bhattacharjee and G. Sigl, Phys. Rep. 327, 109 (2000) [arXiv:astro-ph/9811011].
* [9] M. Kachelriess, E. Parizot and D. V. Semikoz, JETP Lett. 88, 553 (2009) [arXiv:0711.3635 [astro-ph]].
* [10] K. Greisen, Phys. Rev. Lett. 16, 748 (1966). G. T. Zatsepin and V. A. Kuzmin, JETP Lett. 4, 78 [Pisma Zh. Eksp. Teor. Fiz. 4, 114 (1966)].
* [11] F. W. Stecker, Phys. Rev. 180, 1264 (1969).
* [12] M. Kachelriess, P. D. Serpico and M. Teshima, Astropart. Phys. 26, 378 (2006) [arXiv:astro-ph/0510444].
* [13] K. Dolag, D. Grasso, V. Springel and I. Tkachev, JCAP 0501, 009 (2005) [arXiv:astro-ph/0410419].
* [14] G. Sigl, F. Miniati and T. A. Ensslin, Phys. Rev. D 70, 043007 (2004) [arXiv:astro-ph/0401084].
* [15] J. Abraham et al. [The Pierre Auger Collaboration], Operations of and Future Plans for the Pierre Auger Observatory, arXiv:0906.2354.
* [16] J. Abraham et al. [The Pierre Auger Collaboration], Studies of Cosmic Ray Composition and Air Shower Structure with the Pierre Auger Observatory, arXiv:0906.2319.
* [17] M. Unger, Study of the Cosmic Ray Composition with the PAO, talk at conference Searching for the Origins of Cosmic Rays, Trondheim, Norway, 2009, http://web.phys.ntnu.no/$\sim$mika/unger2.pdf.
* [18] T. Sako et al., Current status and plan of the LHCf experiment, Proceedings of the 31st ICRC, Lodz, 2009.
* [19] M. Kachelrieß and D. V. Semikoz, Clustering of ultra-high energy cosmic ray arrival directions on medium scales, Astropart. Phys. 26, 10 (2006) [arXiv:astro-ph/0512498].
* [20] S. Mollerach and the Pierre Auger Collaboration, Nucl. Phys. Proc. Suppl. 190, 198 (2009) [arXiv:0901.4699 [astro-ph.HE]].
* [21] J. Abraham et al. [The Pierre Auger Collaboration], arXiv:0906.2347 [astro-ph.HE].
* [22] J. Abraham et al. [Pierre Auger Collaboration], Science 318, 938 (2007) [arXiv:0711.2256 [astro-ph]].
* [23] O. E. Kalashev, D. V. Semikoz and G. Sigl, Phys. Rev. D 79, 063005 (2009) [arXiv:0704.2463 [astro-ph]].
* [24] G. Gelmini, O. Kalashev and D. V. Semikoz, J. Exp. Theor. Phys. 106, 1061 (2008) [arXiv:astro-ph/0506128]. Astropart. Phys. 28, 390 (2007) [arXiv:astro-ph/0702464].
* [25] F. Aharonian, J. Buckley, T. Kifune and G. Sinnis, Rep. Prog. Phys. 71, 096901 (2008).
* [26] A. Neronov and D. V. Semikoz, JETP Lett. 85, 473 (2007) [arXiv:astro-ph/0604607].
* [27] A. Elyiv, A. Neronov and D. V. Semikoz, Phys. Rev. D 80, 023010 (2009) [arXiv:0903.3649 [astro-ph.CO]].
* [28] K. Dolag, M. Kachelriess, S. Ostapchenko and R. Tomas, Astrophys. J. 703, 1078 (2009) [arXiv:0903.2842 [astro-ph.HE]].
* [29] R. Plaga, Nature 374, 430 (1995).
* [30] K. Murase, K. Takahashi, S. Inoue, K. Ichiki and S. Nagataki, Astrophys. J. 686 L67 (2008) [ arXiv:0806.2829 [astro-ph]].
* [31] A. Neronov and D. Semikoz, Sensitivity of gamma-ray telescopes for detection of magnetic fields in arXiv:0910.1920 [astro-ph.CO]. To appear in Physical Review D.
* [32] T. DeYoung [for the IceCube Collaboration], Recent Results from IceCube and AMANDA, arXiv:0910.3644 [astro-ph.HE].
* [33] F. Halzen, A. Kappes and A. O’Murchadha, Phys. Rev. D 78, 063004 (2008) [arXiv:0803.0314 [astro-ph]].
* [34] N. G. Lehtinen, P. W. Gorham, A. R. Jacobson and R. A. Roussel-Dupre, Phys. Rev. D 69, 013008 (2004) [arXiv:astro-ph/0309656].
* [35] I. Kravchenko et al., Phys. Rev. D 73, 082002 (2006) [arXiv:astro-ph/0601148].
* [36] S. W. Barwick et al. [ANITA Collaboration], Phys. Rev. Lett. 96, 171101 (2006) [arXiv:astro-ph/0512265].
* [37] P. W. Gorham et al. [ANITA collaboration], Phys. Rev. Lett. 103, 051103 (2009) [arXiv:0812.2715 [astro-ph]].
* [38] V. S. Berezinsky and G. T. Zatsepin, Phys. Lett. B 28, 423 (1969).
* [39] O. E. Kalashev, V. A. Kuzmin, D. V. Semikoz and G. Sigl, Phys. Rev. D 66, 063004 (2002) [arXiv:hep-ph/0205050].
* [40] D. V. Semikoz and G. Sigl, JCAP 0404, 003 (2004) [arXiv:hep-ph/0309328].
* [41] J. Abraham et al. [The Pierre Auger Collaboration], Phys. Rev. Lett. 100, 211101 (2008) [arXiv:0712.1909 [astro-ph]].
* [42] R. U. Abbasi et al., An upper limit on the electron-neutrino flux from the HiRes detector, arXiv:0803.0554 [astro-ph].
* [43] M. Ackermann et al. [IceCube Collaboration], Astrophys. J. 675, 1014 (2008) [arXiv:0711.3022 [astro-ph]].
* [44] R. Gandhi, C. Quigg, M. H. Reno and I. Sarcevic, Astropart. Phys. 5, 81 (1996) [arXiv:hep-ph/9512364].
* [45] R. Abbasi et al. [IceCube Collaboration], Astrophys. J. 701, L47 (2009) [arXiv:0905.2253 [astro-ph.HE]].
|
arxiv-papers
| 2010-10-13T12:37:01 |
2024-09-04T02:49:13.814346
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "D. Semikoz (APC)",
"submitter": "Scientific Information Service Cern",
"url": "https://arxiv.org/abs/1010.2647"
}
|
1010.2666
|
Flavour physics and CP violation
Y. Nir
Weizmann Institute of Science, Rehovot, Israel
This is a written version of a series of lectures aimed at graduate
students in particle theory/string theory/particle experiment
familiar with the basics of the Standard Model. We explain the many
reasons for the interest in flavour physics. We describe flavour
physics and the related CP violation within the Standard Model, and
explain how the B-factories proved that the Kobayashi-Maskawa
mechanism dominates the CP violation that is observed in meson
decays. We explain the implications of flavour physics for new
physics. We emphasize the “new physics flavour puzzle”. As an
explicit example, we explain how the recent measurements of
$D^0-\overline{D}^0$ mixing constrain the supersymmetric flavour
structure. We explain how the ATLAS and CMS experiments can solve
the new physics flavour puzzle and perhaps shed light on the standard
model flavour puzzle. Finally, we describe various interpretations of
the neutrino flavour data and their impact on flavour models.
§ WHAT IS FLAVOUR?
The term `flavours' is used, in the jargon of particle
physics, to describe several copies of the same gauge representation,
namely several fields that are assigned the same quantum
charges. Within the Standard Model, when thinking of its unbroken
$SU(3)_\text{C}\times U(1)_\text{EM}$ gauge group, there are four
different types of particles, each coming in three flavours:
* Up-type quarks in the $(3)_{+2/3}$ representation: $u,c,t$.
* Down-type quarks in the $(3)_{-1/3}$ representation: $d,s,b$.
* Charged leptons in the $(1)_{-1}$ representation: $e,\mu,\tau$.
* Neutrinos in the $(1)_{0}$ representation: $\nu_1,\nu_2,\nu_3$.
The term `flavour physics' refers to interactions that
distinguish between flavours. By definition, gauge interactions,
namely interactions that are related to unbroken symmetries and
mediated therefore by massless gauge bosons, do not distinguish among
the flavours and do not constitute part of flavour physics. Within the
Standard Model, flavour physics refers to the weak and Yukawa
The term `flavour parameters' refers to parameters that carry
flavour indices. Within the Standard Model, these are the nine masses of
the charged fermions and the four `mixing parameters' (three angles
and one phase) that describe the interactions of the
charged weak-force carriers ($W^\pm$) with quark–antiquark pairs. If
one augments the Standard Model with Majorana mass terms for the
neutrinos, one should add to the list three neutrino masses and six
mixing parameters (three angles and three phases) for the $W^\pm$
interactions for lepton–antilepton pairs.
The term `flavour universal' refers to interactions with couplings
(or to flavour parameters) that are proportional to the unit matrix in
flavour space. Thus, the strong and electromagnetic interactions are
flavour universal[In the interaction basis, the weak interactions are also
flavour universal, and one can identify the source of all flavour
physics in the Yukawa interactions among the gauge-interaction
An alternative term for `flavour universal' is `flavour blind'.
The term `flavour diagonal' refers to interactions with couplings (or
to flavour parameters) that are diagonal, but not necessarily
universal, in the flavour space. Within the Standard Model, the Yukawa
interactions of the Higgs particle are flavour diagonal in the mass
The term `flavour changing' refers to processes where the
initial and final flavour-numbers (that is, the number of particles of a
certain flavour minus the number of antiparticles of the same flavour)
are different. In `flavour-changing charged current' processes, both
up-type and down-type flavours, and/or both charged lepton and neutrino
flavours are involved. Examples are (i) muon decay via $\mu\to
e\bar\nu_i\nu_j$, and (ii) $K^-\to\mu^-\bar\nu_j$ (which corresponds,
at the quark level, to $s\bar u\to\mu^-\bar\nu_j$). Within the
Standard Model, these processes are mediated by the $W$ bosons and
occur at tree level. In `flavour-changing neutral current' (FCNC)
processes, either up-type or down-type flavours but not both, and/or
either charged lepton or neutrino flavours but not both, are involved.
Examples are (i) muon decay via $\mu\to e\gamma$ and (ii)
$K_L\to\mu^+\mu^-$ (which corresponds, at the quark level, to $s\bar
d\to\mu^+\mu^-$). Within the Standard Model, these processes do not
occur at tree level, and are often highly suppressed.
Another useful term is `flavour violation'. We shall explain
it later in these lectures.
§ WHY IS FLAVOUR PHYSICS INTERESTING?
* Flavour physics can discover new physics or probe it before it
is directly observed in experiments. Here are some examples from
the past:
* The smallness of $\frac{\Gamma(K_L\to\mu^+\mu^-)}
{\Gamma(K^+\to\mu^+\nu)}$ led to the prediction of a fourth
(the charm) quark.
* The size of $\Delta m_K$ led to a successful prediction of the
charm mass.
* The size of $\Delta m_B$ led to a successful prediction of the
top mass.
* The measurement of $\varepsilon_K$ led to the prediction of
the third generation.
* CP violation is closely related to flavour physics. Within the
Standard Model, there is a single CP-violating parameter, the
Kobayashi–Maskawa phase $\delta_\text{KM}$
[1]. Baryogenesis tells us, however, that
there must exist new sources of CP violation. Measurements of CP
violation in flavour-changing processes might provide evidence
for such sources.
* The fine-tuning problem of the Higgs mass, and the puzzle of
dark matter imply that there exists new physics at, or below,
the scale. If such new physics had a generic flavour
structure, it would contribute to flavour-changing neutral
current (FCNC) processes orders of magnitude above the observed
rates. The question of why this does not happen constitutes the
new physics flavour puzzle.
* Most of the charged fermion flavour parameters are small and
hierarchical. The Standard Model does not provide any explanation of
these features. This is the Standard Model flavour puzzle. The
puzzle became even deeper after neutrino masses and mixings were
measured because, so far, neither smallness nor hierarchy in these
parameters have been established.
§ FLAVOUR IN THE STANDARD MODEL
A model of elementary particles and their interactions is defined by
the following ingredients: (i) The symmetries of the Lagrangian and
the pattern of spontaneous symmetry breaking; (ii) The representations
of fermions and scalars. The Standard Model (SM) is defined as
(i) The gauge symmetry is
\begin{equation}\label{smsym}
G_\text{SM}=SU(3)_\text{C}\times SU(2)_\text{L}\times U(1)_\text{Y}.
\end{equation}
It is spontaneously broken by the VEV of a single Higgs scalar,
$\phi(1,2)_{1/2}$ ($\langle\phi^0\rangle=v/\sqrt{2}$):
\begin{equation}\label{smssb}
G_\text{SM} \to SU(3)_\text{C}\times U(1)_\text{EM}.
\end{equation}
(ii) There are three fermion generations, each consisting of five
representations of $G_\text{SM}$:
\begin{equation}\label{ferrep}
Q_{Li}(3,2)_{+1/6},\ \ U_{Ri}(3,1)_{+2/3},\ \
D_{Ri}(3,1)_{-1/3},\ \ L_{Li}(1,2)_{-1/2},\ \ E_{Ri}(1,1)_{-1}.
\end{equation}
§.§ The interactions basis
The Standard Model Lagrangian, $\mathcal{L}_\text{SM}$, is the most
general renormalizable Lagrangian that is consistent with the gauge
symmetry (<ref>), the particle content (<ref>) and the
pattern of spontaneous symmetry breaking (<ref>). It can be
divided into three parts:
\begin{equation}\label{LagSM}
\mathcal{L}_\text{SM}=\mathcal{L}_\text{kinetic}+\mathcal{L}_\text{Higgs}
\end{equation}
For the kinetic terms, to maintain gauge invariance, one has to
replace the derivative with a covariant derivative:
\begin{equation}\label{SMDmu}
D^\mu=\partial^\mu+ig_s G^\mu_a L_a+ig W^\mu_b T_b+ig^\prime B^\mu Y.
\end{equation}
Here $G^\mu_a$ are the eight gluon fields, $W^\mu_b$ the three weak
interaction bosons, and $B^\mu$ the single hypercharge boson. The
$L_a$'s are $SU(3)_\text{C}$ generators (the $3\times3$ Gell-Mann
matrices $\frac{1}{2}\lambda_a$ for triplets, $0$ for singlets), the
$T_b$'s are $SU(2)_\text{L}$ generators (the $2\times2$ Pauli matrices
$\frac{1}{2}\tau_b$ for doublets, $0$ for singlets), and the $Y$'s are
the $U(1)_\text{Y}$ charges. For example, for the quark doublets
$Q_L$, we have
\begin{equation}\label{DmuQL}
\mathcal{L}_\text{kinetic}(Q_L)= i{\overline{Q_{Li}}}\gamma_\mu
\left(\partial^\mu+\frac{i}{2}g_s G^\mu_a\lambda_a
+\frac{i}{2}g W^\mu_b\tau_b+\frac{i}{6}g^\prime
\end{equation}
while for the lepton doublets $L_L^I$, we have
\begin{equation}\label{DmuLL}
\mathcal{L}_\text{kinetic}(L_L)= i{\overline{L_{Li}}}\gamma_\mu
\left(\partial^\mu+\frac{i}{2}g W^\mu_b\tau_b-\frac i2 g^\prime
\end{equation}
The unit matrix in flavour space, $\delta_{ij}$, signifies that
these parts of the interaction Lagrangian are flavour universal. In
addition, they conserve CP.
The Higgs potential, which describes the scalar self-interactions, is
given by
\begin{equation}\label{HiPo}
\mathcal{L}_\text{Higgs}=\mu^2\phi^\dagger\phi-\lambda(\phi^\dagger\phi)^2.
\end{equation}
For the Standard Model scalar sector, where there is a single doublet,
this part of the Lagrangian is also CP conserving.
The quark Yukawa interactions are given by
\begin{equation}\label{Hqint}
-\mathcal{L}_\text{Y}^{q}=Y^d_{ij}{\overline {Q_{Li}}}\phi D_{Rj}
+Y^u_{ij}{\overline {Q_{Li}}}\tilde\phi U_{Rj}+\text{h.c.},
\end{equation}
(where $\tilde\phi=i\tau_2\phi^\dagger$) while the lepton Yukawa
interactions are given by
\begin{equation}\label{Hlint}
-\mathcal{L}_\text{Y}^{\ell}=Y^e_{ij}{\overline {L_{Li}}}\phi E_{Rj}
\end{equation}
This part of the Lagrangian is, in general, flavour dependent (that is,
$Y^f\not\propto\mathbf{1}$) and CP violating.
§.§ Global symmetries
In the absence of the Yukawa matrices $Y^d$, $Y^u$ and $Y^e$, the SM
has a large $U(3)^5$ global symmetry:
\begin{equation}\label{gglobal}
G_\text{global}(Y^{u,d,e}=0)=SU(3)_q^3\times SU(3)_\ell^2\times U(1)^5,
\end{equation}
\begin{eqnarray}\label{susuu}
SU(3)_q^3&=&SU(3)_Q\times SU(3)_U\times SU(3)_D,\nonumber\\
SU(3)_\ell^2&=&SU(3)_L\times SU(3)_E,\nonumber\\
U(1)^5&=&U(1)_B\times U(1)_L\times U(1)_Y\times U(1)_\text{PQ}\times
\end{eqnarray}
Out of the five $U(1)$ charges, three can be identified with baryon
number ($B$), lepton number ($L$), and hypercharge ($Y$), which are
respected by the Yukawa interactions. The two remaining $U(1)$ groups
can be identified with the PQ symmetry whereby the Higgs and $D_R,E_R$
fields have opposite charges, and with a global rotation of $E_R$
The point that is important for our purposes is that
$\mathcal{L}_\text{kinetic}+\mathcal{L}_\text{Higgs}$ respect the
non-Abelian flavour symmetry $S(3)_q^3\times SU(3)_\ell^2$, under which
\begin{equation}\label{symkh}
Q_L\to V_QQ_L,\ \ \ U_R\to V_U U_R,\ \ \ D_R\to V_D D_R,\ \ L_L\to V_L
L_L,\ \ \ E_R\to V_E E_R,
\end{equation}
where the $V_i$ are unitary matrices.
The Yukawa interactions (<ref>) and (<ref>) break the
global symmetry,
\begin{equation}\label{globre}
G_\text{global}(Y^{u,d,e}\neq0)= U(1)_B\times U(1)_e\times
U(1)_\mu\times U(1)_\tau.
\end{equation}
(Of course, the gauged $U(1)_Y$ also remains a good symmetry.) Thus,
the transformations of symkh are not a symmetry of
$\mathcal{L}_\text{SM}$. Instead, they correspond to a change of the
interaction basis. These observations also offer an alternative way of
defining flavour physics: it refers to interactions that break the
$SU(3)^5$ symmetry (<ref>). Thus, the term `flavour
violation' is often used to describe processes or parameters that
break the symmetry.
One can think of the quark Yukawa couplings as spurions that break the
global $SU(3)_q^3$ symmetry (but are neutral under $U(1)_B$),
\begin{equation}\label{Gglobq}
Y^u\sim(3,\bar3,1)_{SU(3)_q^3},\ \ \
\end{equation}
and of the lepton Yukawa couplings as spurions that break the global
$SU(3)_\ell^2$ symmetry (but are neutral under $U(1)_e\times
U(1)_\mu\times U(1)_\tau$),
\begin{equation}\label{Gglobl}
\end{equation}
The spurion formalism is convenient for several purposes: parameter
counting (see below), identification of flavour suppression factors
(see sec:nppuzzle), and the idea of minimal flavour
violation (see sec:lhc).
§.§ Counting parameters
How many independent parameters are there in $\mathcal{L}_\text{Y}^q$?
The two Yukawa matrices, $Y^u$ and $Y^d$, are $3\times3$ and complex.
Consequently, there are 18 real and 18 imaginary parameters in these
matrices. Not all of them are, however, physical. The pattern of
$G_\text{global}$ breaking means that there is freedom to remove 9
real and 17 imaginary parameters (the number of parameters in three
$3\times3$ unitary matrices minus the phase related to $U(1)_B$). For
example, we can use the unitary transformations $Q_L\to V_QQ_L$,
$U_R\to V_U U_R$, and $D_R\to V_D D_R$ to lead to the following
interaction basis:
\begin{equation}\label{speint}
Y^d=\lambda_d,\ \ \ Y^u=V^\dagger\lambda_u,
\end{equation}
where $\lambda_{d,u}$ are diagonal,
\begin{equation}\label{deflamd}
\lambda_d=\text{diag}(y_d,y_s,y_b),\ \ \
\lambda_u=\text{diag}(y_u,y_c,y_t),
\end{equation}
while $V$ is a unitary matrix that depends on three real angles and
one complex phase. We conclude that there are 10 quark flavour
parameters: 9 real ones and a single phase. In the mass basis, we
shall identify the nine real parameters as six quark masses and three
mixing angles, while the single phase is $\delta_\text{KM}$.
How many independent parameters are there in $\mathcal{L}_\text{Y}^\ell$?
The Yukawa matrix $Y^e$ is $3\times3$ and complex. Consequently, there
are 9 real and 9 imaginary parameters in this matrix. There is,
however, freedom to remove 6 real and 9 imaginary parameters (the
number of parameters in two $3\times3$ unitary matrices minus the
phases related to $U(1)^3$). For example, we can use the unitary
transformations $L_L\to V_LL_L$ and $E_R\to V_E E_R$ to lead to the
following interaction basis:
\begin{equation}\label{speintl}
\end{equation}
We conclude that there are three real lepton flavour parameters. In
the mass basis, we shall identify these parameters as the three
charged lepton masses. We must, however, modify the model when we take
into account the evidence for neutrino masses.
§.§ The mass basis
Upon the replacement $\re{\phi^0}\to\frac{v+H^0}{\sqrt2}$, the Yukawa
interactions (<ref>) give rise to the mass matrices
\begin{equation}\label{YtoMq}
\end{equation}
The mass basis corresponds, by definition, to diagonal mass
matrices. We can always find unitary matrices $V_{qL}$ and $V_{qR}$
such that
\begin{equation}\label{diagMq}
V_{qL}M_q V_{qR}^\dagger=M_q^\text{diag}\equiv\frac{v}{\sqrt2}\lambda_q.
\end{equation}
The four matrices $V_{dL}$, $V_{dR}$, $V_{uL}$, and $V_{uR}$ are then
the ones required to transform to the mass basis. For example, if we
start from the special basis (<ref>), we have
$V_{dL}=V_{dR}=V_{uR}=\mathbf{1}$ and $V_{uL}=V$. The combination
$V_{uL}V_{dL}^\dagger$ is independent of the interaction basis from
which we start this procedure.
We denote the left-handed quark mass eigenstates as $U_L$ and $D_L$.
The charged-current interactions for quarks [that is the interactions of the
charged $SU(2)_\text{L}$ gauge bosons $W^\pm_\mu=\frac{1}{\sqrt{2}}
(W^1_\mu\mp iW_\mu^2)$], which in the interaction basis are described
by (<ref>), have a complicated form in the mass basis:
\begin{equation}\label{Wmasq}
-\mathcal{L}_{W^\pm}^q=\frac{g}{\sqrt{2}}{\overline {U_{Li}}}\gamma^\mu
V_{ij}D_{Lj} W_\mu^++\text{h.c.}\ ,
\end{equation}
where $V$ is the $3\times3$ unitary matrix ($VV^\dagger=V^\dagger
V=\mathbf{1}$) that appeared in speint. For a general
interaction basis,
\begin{equation}\label{VCKM}
\end{equation}
$V$ is the Cabibbo–Kobayashi–Maskawa (CKM) mixing matrix for
quarks [2, 1]. As a result of the fact
that $V$ is not diagonal, the $W^\pm$ gauge bosons couple to quark
mass eigenstates of different generations. Within the Standard
Model, this is the only source of flavour-changing quark
Exercise 1: Prove that, in the absence of neutrino masses, there
is no mixing in the lepton sector.
Exercise 2: Prove that there is no mixing in the $Z$
couplings. (In the jargon of physics, there are no flavour-changing
neutral currents at tree level.)
The detailed structure of the CKM matrix, its parametrization, and the
constraints on its elements are described in Appendix <ref>.
§ TESTING CKM
Measurements of rates, mixing, and CP asymmetries in $B$ decays in the
two B factories, BaBar and Belle, and in the two Tevatron detectors,
CDF and D0, signified a new era in our understanding of CP
violation. The progress is both qualitative and quantitative. Various
basic questions concerning CP and flavour violation have, for the
first time, received answers based on experimental information. These
questions include, for example,
* Is the Kobayashi–Maskawa mechanism at work (namely, is
* Does the KM phase dominate the observed CP violation?
As a first step, one may assume the SM and test the overall
consistency of the various measurements. However, the richness of data
from the B factories allows us to go a step further and answer these
questions model independently, namely allowing new physics to
contribute to the relevant processes. We here explain the way in which
this analysis proceeds.
§.§ $S_{\psi K_S}$
The CP asymmetry in $B\to\psi K_S$ decays plays a major role in
testing the KM mechanism. Before we explain the test itself, we should
understand why the theoretical interpretation of the asymmetry is
exceptionally clean, and what are the theoretical parameters on which
it depends, within and beyond the Standard Model.
The CP asymmetry in neutral meson decays into final CP eigenstates
$f_{\CP}$ is defined as follows:
\begin{equation}\label{asyfcpt}
\mathcal{A}_{f_{\CP}}(t)\equiv\frac{d\Gamma/dt[\Bzb_\text{phys}(t)\to f_{\CP}]-
d\Gamma/dt[\Bz_\text{phys}(t)\to f_{\CP}]}
{d\Gamma/dt[\Bzb_\text{phys}(t)\to f_{\CP}]+d\Gamma/dt[\Bz_\text{phys}(t)\to
f_{\CP}]}\; .
\end{equation}
A detailed evaluation of this asymmetry is given in Appendix
<ref>. It leads to the following form:
\begin{eqnarray}\label{asyfcpbt}
\mathcal{A}_{f_{\CP}}(t)&=&S_{f_{\CP}}\sin(\Delta
mt)-C_{f_{\CP}}\cos(\Delta mt),\nonumber\\
\ \ C_{f_{\CP}}\equiv\frac{1-|\lambda_{f_{\CP}}|^2}{1+|\lambda_{f_{\CP}}|^2}
\; ,
\end{eqnarray}
\begin{equation}\label{lamhad}
\lambda_{f_{\CP}}=e^{-i\phi_B}(\overline{A}_{f_{\CP}}/A_{f_{\CP}}) \; .
\end{equation}
Here $\phi_B$ refers to the phase of $M_{12}$ [see
defmgam]. Within the Standard Model, the corresponding
phase factor is given by
\begin{equation}\label{phimsm}
e^{-i\phi_B}=(V_{tb}^* V_{td}^{})/(V_{tb}^{}V_{td}^*) \;.
\end{equation}
The decay amplitudes $A_f$ and $\overline{A}_f$ are defined in
[]Feynman diagrams for (a) tree and (b) penguin amplitudes
contributing to $B^0\to f$ or $B_{s}\to f$ via a $\bar
b\to\bar q q\bar q^\prime$ quark-level process
The $B^0\to J/\psi K^0$ decay [3, 5]
proceeds via the quark transition $\bar b\to\bar c c\bar s$. There are
contributions from both tree ($t$) and penguin ($p^{q_u}$, where
$q_u=u,c,t$ is the quark in the loop) diagrams (see fig:diags)
which carry different weak phases:
\begin{equation}\label{ckmdec}
A_f = \left(V^\ast_{cb} V^{}_{cs}\right) t_f +
\sum_{q_u= u,c,t}\left(V^\ast_{q_u b} V^{}_{q_u s}\right) p^{q_u}_f \; .
\end{equation}
(The distinction between tree and penguin contributions is a heuristic
one, the separation by the operator that enters is more precise. For a
detailed discussion of the more complete operator product approach,
which also includes higher order QCD corrections, see, for example,
Buchalla:1995vs.) Using CKM unitarity, these decay
amplitudes can always be written in terms of just two CKM
\begin{equation}\label{btoccs}
A_{\psi K}=\left(V^\ast_{cb} V^{}_{cs}\right)T_{\psi
K}+\left(V^\ast_{ub} V^{}_{us}\right)P^u_{\psi K},
\end{equation}
where $T_{\psi K}=t_{\psi K}+p^c_{\psi K}-p^t_{\psi K}$ and $P^u_{\psi
K}=p^u_{\psi K}-p^t_{\psi K}$. A subtlety arises in this decay that is
related to the fact that ${B}^0\to J/\psi K^0$ and $\overline{B}^0\to
J/\psi\overline{K}{}^0$. A common final state, $J/\psi K_S$, can
be reached via $K^0$–$\overline{K}{}^0$ mixing. Consequently, the
phase factor corresponding to neutral $K$ mixing,
$e^{-i\phi_K}=(V^*_{cd}V^{}_{cs})/(V^{}_{cd}V^*_{cs})$, plays a role:
\begin{equation}\label{psikmix}
\frac{\overline{A}_{\psi K_S}}{A_{\psi K_S}}
=-\frac{\left(V^{}_{cb} V^\ast_{cs}\right)T_{\psi
K}+\left(V^{}_{ub} V^\ast_{us}\right)P^u_{\psi K}}
{\left(V^\ast_{cb} V^{}_{cs}\right)T_{\psi
K}+\left(V^\ast_{ub} V^{}_{us}\right)P^u_{\psi K}}\times
\frac{V_{cd}^\ast V_{cs}^{}}{V_{cd}^{}V_{cs}^\ast}.
\end{equation}
The crucial point is that, for $B\to J/\psi K_S$ and other $\bar
b\to\bar cc\bar s$ processes, we can neglect the $P^u$ contribution to
$A_{\psi K}$, in the SM, to an approximation that is better than one
per cent:
\begin{equation}\label{smapprox}
|P^u_{\psi K}/T_{\psi K}|\times|V_{ub}/V_{cb}|\times|
V_{us}/V_{cs}|\sim(\text{loop\ factor})\times0.1\times0.23\lesssim0.005.
\end{equation}
Thus, to an accuracy of better than one per cent,
\begin{equation}
\lambda_{\psi K_S}=\left(\frac{V_{tb}^*
\end{equation}
where $\beta$ is defined in abcangles, and consequently
\begin{equation}\label{btopsik}
S_{\psi K_S}=\sin2\beta,\ \ \ C_{\psi K_S}=0 \; .
\end{equation}
(Below the per cent level, several effects modify this equation
[7, 8, 9, 10].)
Exercise 3: Show that, if the $B\to\pi\pi$ decays were dominated
by tree diagrams, then $S_{\pi\pi}=\sin2\alpha$.
Exercise 4: Estimate the accuracy of the predictions
$S_{\phi K_S}=\sin2\beta$ and $C_{\phi K_S}=0$.
When we consider extensions of the SM, we still do not expect any
significant new contribution to the tree level decay, $b\to c\bar cs$,
beyond the SM $W$-mediated diagram. Thus the expression $\bar A_{\psi
K_S}/A_{\psi K_S}=(V_{cb}V_{cd}^*)/(V_{cb}^*V_{cd})$ remains valid,
though the approximation of neglecting sub-dominant phases can be
somewhat less accurate than smapprox. On the other hand,
$M_{12}$, the $B^0$–$\overline{B}^0$ mixing amplitude, can in
principle get large and even dominant contributions from new
physics. We can parametrize the modification to the SM in terms of two
parameters, $r_d^2$ signifying the change in magnitude, and
$2\theta_d$ signifying the change in phase:
\begin{equation}\label{derthed}
M_{12}=r_d^2\ e^{2i\theta_d}\ M_{12}^\text{SM}(\rho,\eta).
\end{equation}
This leads to the following generalization of btopsik:
\begin{equation}\label{btopsiknp}
S_{\psi K_S}=\sin(2\beta+2\theta_d),\ \ \ C_{\psi K_S}=0 \; .
\end{equation}
The experimental measurements give the following ranges [11]:
\begin{equation}\label{scpkexp}
S_{\psi K_S}=0.671\pm0.024,\ \ \ C_{\psi K_S}=0.005\pm0.019 \; .
\end{equation}
§.§ Self-consistency of the CKM assumption
The three-generation Standard Model has room for CP violation, through
the KM phase in the quark mixing matrix. Yet, one would like to make
sure that CP is indeed violated by the SM interactions, namely that
$\sin\delta_\text{KM}\neq0$. If we establish that this is the case, we
would further like to know whether the SM contributions to CP
violating observables are dominant. More quantitatively, we would like
to put an upper bound on the ratio between the new physics and the SM
As a first step, one can assume that flavour-changing processes are
fully described by the SM, and check the consistency of the various
measurements with this assumption. There are four relevant mixing
parameters, which can be taken to be the Wolfenstein parameters
$\lambda$, $A$, $\rho$, and $\eta$ defined in wolpar. The values
of $\lambda$ and $A$ are known rather accurately [12]
from, respectively, $K\to\pi\ell\nu$ and $b\to c\ell\nu$ decays:
\begin{equation}\label{lamaexp}
\lambda=0.2257\pm0.0010,\ \ \ A=0.814\pm0.022.
\end{equation}
Then, one can express all the relevant observables as a function of
the two remaining parameters, $\rho$ and $\eta$, and check whether
there is a range in the $\rho$–$\eta$ plane that is consistent with
all measurements. The list of observables includes the following:
* The rates of inclusive and exclusive charmless semileptonic $B$
decays depend on $|V_{ub}|^2\propto\rho^2+\eta^2$.
* The CP asymmetry in $B\to\psi K_S$, $S_{\psi
* The rates of various $B\to DK$ decays depend on the phase
$\gamma$, where $e^{i\gamma}=\frac{\rho+i\eta}{\rho^2+\eta^2}$.
* The rates of various $B\to\pi\pi,\rho\pi,\rho\rho$ decays depend
on the phase $\alpha=\pi-\beta-\gamma$.
* The ratio between the mass splittings in the neutral $B$ and
$B_s$ systems is sensitive to
* The CP violation in $K\to\pi\pi$ decays, $\epsilon_K$, depends
in a complicated way on $\rho$ and $\eta$.
The resulting constraints are shown in fg:UT.
[]Allowed region in the $\rho$–$\eta$ plane. Superimposed are
the individual constraints from charmless semileptonic
$B$ decays ($|V_{ub}/V_{cb}|$), mass differences in the
$B^0$ ($\Delta m_d$) and $B_s$ ($\Delta m_s$) neutral
meson systems, and CP violation in $K\to\pi\pi$
($\varepsilon_K$), $B\to\psi K$ ($\sin2\beta$),
$B\to\pi\pi,\rho\pi,\rho\rho$ ($\alpha$), and $B\to DK$
($\gamma$). Taken from ckmfitter.
The consistency of the various constraints is impressive. In
particular, the following ranges for $\rho$ and $\eta$ can account for
all the measurements [12]:
\begin{equation}
\rho=0.135^{+0.031}_{-0.016},\ \ \ \eta=0.349\pm0.017.
\end{equation}
One can then make the following statement [15]:
Very likely, CP violation in flavour-changing processes is
dominated by the Kobayashi–Maskawa phase.
In the next two subsections, we explain how we can remove the phrase
`very likely' from this statement, and how we can quantify the
KM dominance.
§.§ Is the Kobayashi–Maskawa mechanism at work?
In proving that the KM mechanism is at work, we assume that
charged-current tree-level processes are dominated by the $W$-mediated
SM diagrams (see, for example, Grossman:1997dd). This is a
very plausible assumption. I am not aware of any viable well-motivated
model where this assumption is not valid. Thus we can use all tree-level
processes and fit them to $\rho$ and $\eta$, as we did
before. The list of such processes includes the following:
* Charmless semileptonic $B$-decays, $b\to u\ell\nu$, measure
$R_u$ [see RbRt].
* $B\to DK$ decays, which go through the quark transitions $b\to
c\bar u s$ and $b\to u\bar cs$, measure the angle $\gamma$ [see
* $B\to\rho\rho$ decays (and, similarly, $B\to\pi\pi$ and
$B\to\rho\pi$ decays) go through the quark transition $b\to
u\bar ud$. With an isospin analysis, one can determine the
relative phase between the tree decay amplitude and the mixing
amplitude. By incorporating the measurement of $S_{\psi K_S}$,
one can subtract the phase from the mixing amplitude, finally
providing a measurement of the angle $\gamma$ [see
In addition, we can use loop processes, but then we must allow for new
physics contributions, in addition to the $(\rho,\eta)$-dependent SM
contributions. Of course, if each such measurement adds a separate
mode-dependent parameter, then we do not gain anything by using this
information. However, there are a number of observables where the only
relevant loop process is $B^0$–$\overline{B}{}^0$ mixing. The list
includes $S_{\psi K_S}$, $\Delta m_B$, and the CP asymmetry in
semileptonic $B$ decays:
\begin{align}\label{apksNP}
S_{\psi K_S} &=\sin(2\beta+2\theta_d),\nonumber\\
\Delta m_{B} &=r_d^2(\Delta m_B)^\text{SM},\nonumber\\
\mathcal{A}_\text{SL}&=-
\mathcal{R}e \left(\frac{\Gamma_{12}}{M_{12}}\right)^\text{SM}
\frac{\sin2\theta_d}{r_d^2}
\frac{\cos2\theta_d}{r_d^2}.
\end{align}
As explained above, such processes involve two new parameters [see
derthed]. Since there are three relevant observables, we can
further tighten the constraints in the $(\rho,\eta)$ plane. Similarly,
one can use measurements related to $B_s$–$\overline{B}_s$
mixing. One gains three new observables at the cost of two new
parameters (see, for example, Grossman:2006ce).
The results of such a fit, projected on the $\rho$–$\eta$ plane, can be
seen in fig:re_tree. It gives [14]
\begin{equation}
\eta=0.44^{+0.05}_{-0.23}\ \ (3\sigma).
\end{equation}
[A similar analysis in Bona:2007vi obtains the $3\sigma$
range $(0.31$–$0.46)$.] It is clear that $\eta\neq0$ is well
The Kobayashi–Maskawa mechanism of CP violation is at work.
[]The allowed region in the $\rho$–$\eta$ plane, assuming
that tree diagrams are dominated by the Standard Model
[14]
Another way to establish that CP is violated by the CKM matrix is to
find, within the same procedure, the allowed range for $\sin2\beta$
\begin{equation}\label{stbth}
\sin2\beta^\text{tree}=0.76\pm0.04.
\end{equation}
([b]ckmfitter finds $0.82^{+0.02}_{-0.13}$.) Thus,
$\beta\neq0$ is well established.
The consistency of the experimental results (<ref>) with the
SM predictions (<ref>,<ref>) means that the KM mechanism
of CP violation dominates the observed CP violation. In the next
subsection, we make this statement more quantitative.
§.§ How much can new physics contribute to
$B^0$–$\overline{B}{}^0$ mixing?
All that we need to do in order to establish whether the SM dominates
the observed CP violation, and to put an upper bound on the new
physics contribution to $B^0$–$\overline{B}{}^0$ mixing, is to
project the results of the fit performed in the previous subsection on
the $r_d^2$–$2\theta_d$ plane. If we find that $\theta_d\ll\beta$, then
the SM dominance in the observed CP violation will be established.
The constraints are shown in fig:rdtd(a). Indeed,
[]Constraints in the (a) $r_d^2$–$2\theta_d$ plane, and (b)
$h_d$–$\sigma_d$ plane, assuming that new physics
contributions to tree-level processes are negligible
[14]
An alternative way to present the data is to use the $h_d,\sigma_d$
\begin{equation}
r_d^2e^{2i\theta_d}=1+h_d e^{2i\sigma_d}.
\end{equation}
While the $r_d,\theta_d$ parameters give the relation between the full
mixing amplitude and the SM one, and are convenient to apply to the
measurements, the $h_d,\sigma_d$ parameters give the relation between
the new physics and SM contributions, and are more convenient in
testing theoretical models:
\begin{equation}
\end{equation}
The constraints in the $h_d$–$\sigma_d$ plane are shown in
fig:rdtd(b). We can make the following two statements:
* A new physics contribution to the $B^0$–$\overline{B}^0$ mixing
amplitude that carries a phase that is significantly different
from the KM phase is constrained to lie below the $20$–$30$%
* A new physics contribution to the $B^0$–$\overline{B}^0$ mixing
amplitude which is aligned with the KM phase is constrained to
be at most comparable to the CKM contribution.
One can reformulate these statements as follows:
* The KM mechanism dominates CP violation in
$B^0$–$\overline{B}^0$ mixing.
* The CKM mechanism is a major player in $B^0$–$\overline{B}^0$
§ THE NEW PHYSICS FLAVOUR PUZZLE
It is clear that the Standard Model is not a complete theory of
* It does not include gravity, and therefore it cannot be valid at
energy scales above $m_\text{Planck}\sim10^{19}\UGeV$.
* It does not allow for neutrino masses, and therefore it cannot
be valid at energy scales above
* The fine-tuning problem of the Higgs mass and the puzzle of dark
matter suggest that the scale where the SM is replaced with a
more fundamental theory is actually much lower,
Given that the SM is only an effective low-energy theory,
non-renormalizable terms must be added to $\mathcal{L}_\text{SM}$ of
LagSM. These are terms of dimension higher than four in the
fields which, therefore, have couplings that are inversely
proportional to the scale of new physics $\Lambda_\text{NP}$. For
example, the lowest-dimension non-renormalizable terms are dimension
\begin{equation}\label{Hnint}
\frac{Z_{ij}^\nu}{\Lambda_\text{NP}}L_{Li}^I L_{Lj}^I\phi\phi+\text{h.c.}
\end{equation}
These are the seesaw terms, leading to neutrino masses. We shall
return to the topic of neutrino masses in sec:nu.
Exercise 5: How does the global symmetry breaking pattern
(<ref>) change when (<ref>) is taken into account?
Exercise 6: What is the number of physical lepton flavour
parameters in this case? Identify these parameters in the mass basis.
As concerns quark flavour physics, consider, for example, the following
dimension-six, four-fermion, flavour-changing operators:
\begin{equation}\label{eq:ffll}
\mathcal{L}_{\Delta F=2}=
\frac{z_{sd}}{\Lambda_\text{NP}^2}(\overline{d_L}\gamma_\mu s_L)^2
+\frac{z_{cu}}{\Lambda_\text{NP}^2}(\overline{c_L}\gamma_\mu u_L)^2
+\frac{z_{bd}}{\Lambda_\text{NP}^2}(\overline{d_L}\gamma_\mu b_L)^2
+\frac{z_{bs}}{\Lambda_\text{NP}^2}(\overline{s_L}\gamma_\mu b_L)^2.
\end{equation}
Each of these terms contributes to the mass splitting between the
corresponding two neutral mesons. For example, the term
$\mathcal{L}_{\Delta B=2}\propto(\overline{d_L}\gamma_\mu b_L)^2$
contributes to $\Delta m_B$, the mass difference between the two
neutral $B$-mesons. We use $M_{12}^B=\frac{1}{2m_B}\langle
B^0|\mathcal{L}_{\Delta F=2}|\overline{B}^0\rangle$ and
\begin{equation}
\langle B^0|(\overline{d_{La}}\gamma^\mu
b_{La})(\overline{d_{Lb}}\gamma_\mu b_{Lb})|\overline{B}^0\rangle =
-\frac13 m_B^2f_B^2 B_B.
\end{equation}
Analogous expressions hold for the other neutral mesons[The PDG [12] quotes the following values, extracted
from leptonic charged meson decays: $f_K\approx0.16\UGeV$,
$f_D\approx0.23\UGeV$, $f_B\approx0.18\UGeV$. We further use
This leads to $\Delta
m_B/m_B=2|M_{12}^B|/m_B\sim (|z_{bd}|/3)(f_B/\Lambda_\text{NP})^2$.
Experiments give, for CP conserving observables (the experimental
evidence for $\Delta m_D$ is at the $3\sigma$ level):
\begin{eqnarray}
\Delta m_K/m_K&\sim&7.0\times10^{-15},\nonumber\\
\Delta m_D/m_D&\sim&8.7\times10^{-15},\nonumber\\
\Delta m_B/m_B&\sim&6.3\times10^{-14},\nonumber\\
\Delta m_{B_s}/m_{B_s}&\sim&2.1\times10^{-12},
\end{eqnarray}
and for CP violating ones
\begin{eqnarray}
\epsilon_K&\sim&2.3\times10^{-3},\nonumber\\
A_\Gamma/y_{\rm CP}&\lsim&0.2,\nonumber\\
S_{\psi K_S}&=&0.67\pm0.02,\nonumber\\
\end{eqnarray}
These measurements give then the following constraints:
\begin{equation}
\label{lowlnp1}
\Lambda_{\rm NP}\gsim
\begin{cases}
\sqrt{z_{sd}}\ 1\times10^3\ \textrm{TeV}&\Delta m_K\\
\sqrt{z_{cu}}\ 1\times10^3\ \textrm{TeV}&\Delta m_D\\
\sqrt{z_{bd}}\ 4\times10^2\ \textrm{TeV}&\Delta m_B\\
\sqrt{z_{bs}}\ 7\times10^1\ \textrm{TeV}&\Delta m_{B_s}
\end{cases}
\end{equation}
and, for maximal phases,
\begin{equation}
\label{lowlnp2}
\Lambda_{\rm NP}\gsim
\begin{cases}
\sqrt{z_{sd}}\ 2\times10^4\ \textrm{TeV}&\epsilon_K\\
\sqrt{z_{cu}}\ 3\times10^3\ \textrm{TeV}&A_\Gamma\\
\sqrt{z_{bd}}\ 8\times10^2\ \textrm{TeV}&S_{\psi K}\\
\sqrt{z_{bs}}\ 7\times10^1\ \textrm{TeV}&S_{\psi\phi}
\end{cases}
\end{equation}
If the new physics has a generic flavour structure, that is
$z_{ij}={\cal O}(1)$, then its scale must be above $10^3$–$10^4$ TeV
(or, if the leading contributions involve electroweak loops, above
$10^2$–$10^3$ TeV).[The bounds from the corresponding four-fermi
terms with LR structure, instead of the LL structure of
Eq. (<ref>), are even stronger.]
If indeed $\Lambda_{\rm NP}\gg \textrm{TeV}$, it means
that we have misinterpreted the hints from the fine-tuning problem
and the dark matter puzzle. There is, however, another way to look
at these constraints:
\begin{eqnarray}
\label{zcons1}
z_{sd}&\lsim&8\times10^{-7}\ (\Lambda_{\rm NP}/\textrm{TeV})^2,\nonumber\\
z_{cu}&\lsim&5\times10^{-7}\ (\Lambda_{\rm NP}/\textrm{TeV})^2,\nonumber\\
z_{bd}&\lsim&5\times10^{-6}\ (\Lambda_{\rm NP}/\textrm{TeV})^2,\nonumber\\
z_{bs}&\lsim&2\times10^{-4}\ (\Lambda_{\rm NP}/\textrm{TeV})^2,
\end{eqnarray}
\begin{eqnarray}
\label{zcons2}
z_{sd}^I&\lsim&6\times10^{-9}\ (\Lambda_{\rm NP}/\textrm{TeV})^2,\nonumber\\
z_{cu}^I&\lsim&1\times10^{-7}\ (\Lambda_{\rm NP}/\textrm{TeV})^2,\nonumber\\
z_{bd}^I&\lsim&1\times10^{-6}\ (\Lambda_{\rm NP}/\textrm{TeV})^2,\nonumber\\
z_{bs}^I&\lsim&2\times10^{-4}\ (\Lambda_{\rm NP}/\textrm{TeV})^2.
\end{eqnarray}
It could be that the scale of new physics is of order TeV, but
its flavour structure is far from generic.
One can use that language of effective operators also for the SM,
integrating out all particles significantly heavier than the neutral
mesons (that is, the top, the Higgs, and the weak gauge bosons). Thus
the scale is $\Lambda_\text{SM}\sim m_W$. Since the leading
contributions to neutral meson mixings come from box diagrams, the
$z_{ij}$ coefficients are suppressed by $\alpha_2^2$. To identify the
relevant flavour suppression factor, one can employ the spurion
formalism. For example, the flavour transition that is relevant to
$B^0$–$\overline{B}{}^0$ mixing involves $\overline{d_L}b_L$ which
transforms as $(8,1,1)_{SU(3)_q^3}$. The leading contribution must
then be proportional to $(Y^u Y^{u\dagger})_{13}\propto y_t^2
V_{tb}V_{td}^*$. Indeed, an explicit calculation (using VIA for the
matrix element and neglecting QCD corrections) gives[A detailed derivation can be found in Appendix B of Branco:1999fs.]
\begin{equation} \frac{2M_{12}^B}{m_B}\approx-\frac{\alpha_2^2}{12}
\frac{f_B^2}{m_W^2}S_0(x_t)(V_{tb}V_{td}^*)^2,
\end{equation}
where $x_i=m_i^2/m_W^2$ and
\begin{equation}
x}{2(1-x)}\right]. \end{equation}
Similar spurion analyses, or explicit calculations, allow us to
extract the weak and flavour suppression factors that apply in the
\begin{align}
\mathcal{I}m(z_{sd}^\text{SM})
&\sim \alpha_2^2 y_t^2 |V_{td}V_{ts}|^2\sim1\times10^{-10},\nonumber\\
&\sim \alpha_2^2 y_c^2 |V_{cd}V_{cs}|^2\sim5\times10^{-9},\nonumber\\
&\sim \alpha_2^2 y_t^2 |V_{td}V_{tb}|^2\sim7\times10^{-8},\nonumber\\
&\sim \alpha_2^2 y_t^2 |V_{ts}V_{tb}|^2\sim2\times10^{-6}.
\end{align}
(We did not include $z_{cu}^\text{SM}$ in the list because it requires
a more detailed consideration. The naively leading short distance
contribution is $\propto \alpha_2^2(y_s^4/y_c^2)
|V_{cs}V_{us}|^2\sim5\times10^{-13}$. However, higher dimension terms
can replace a $y_s^2$ factor with $(\Lambda/m_D)^2$
[20]. Moreover, long distance contributions are expected
to dominate. In particular, peculiar phase space effects
[21, 22] have been identified which are expected
to enhance $\Delta m_D$ to within an order of magnitude of its
measured value.)
It is clear then that contributions from new physics at
$\Lambda_\text{NP}\sim1\UTeV$ should be suppressed by factors that are
comparable to or smaller than the SM ones. Why does that happen? This
is the new physics flavour puzzle.
The fact that the flavour structure of new physics at the scale
must be non-generic means that flavour measurements are a good probe of
the new physics. Perhaps the best-studied example is that of
supersymmetry. Here, the spectrum of the superpartners and the
structure of their couplings to the SM fermions will allow us to probe
the mechanism of dynamical supersymmetry breaking.
§ LESSONS FOR SUPERSYMMETRY FROM $D^0$–$\OVERLINE{D}^0$ MIXING
Interesting experimental results concerning $D^0$–$\overline{D}^0$
mixing have recently been achieved by the BELLE and BaBar experiments.
For the first time, there is evidence for width splitting
[23, 24] and mass splitting (of order one
per cent) between the two neutral $D$-mesons. Allowing for indirect CP
violation, the world averages of the mixing parameters are [11]
\begin{align}
\end{align}
It is important to note, however, that there is no evidence for CP
violation in this mixing [11]:
\begin{align}\label{eq:cpvd}
\phi_D &=-0.04\pm0.09.
\end{align}
We use this recent experimental
information to draw important lessons on supersymmetry. This
demonstrates how flavour physics—at the scale—provides a
significant probe of supersymmetry—at the scale.
§.§ Neutral meson mixing with supersymmetry
We consider the contributions from the box diagrams involving the
squark doublets of the first two generations, $\tilde Q_{L1,2}$, to
the $D^0$–$\overline{D}^0$ and $K^0$–$\overline{K}^0$ mixing
amplitudes. The contributions that are relevant to the neutral $D$
system are proportional to $K_{2i}^u K^{u*}_{1i}K_{2j}^u K^{u*}_{1j}$,
where $K^u$ is the mixing matrix of the gluino couplings to a
left-handed up quark and their supersymmetric squark partners. (In the
language of the mass insertion approximation, we calculate here the
contribution that is $\propto[(\delta^u_{LL})_{12}]^2$.) The
contributions that are relevant to the neutral $K$ system are
proportional to $K_{2i}^{d*} K^{d}_{1i}K_{2j}^{d*} K^{d}_{1j}$, where
$K^d$ is the mixing matrix of the gluino couplings to a left-handed
down quark and their supersymmetric squark partners
($\propto[(\delta^d_{LL})_{12}]^2$ in the mass insertion
approximation). We work in the mass basis for both quarks and squarks.
A detailed derivation [26] is given in Appendix
<ref>. It gives
\begin{align}\label{motsusyb}
&=\frac{\alpha_s^2m_Df_D^2B_D\eta_\text{QCD}}{108m_{\tilde u}^2}
[11\tilde f_6(x_u)+4x_uf_6(x_u)]\frac{(\Delta m^2_{\tilde
u})^2}{m_{\tilde u}^4} (K_{21}^uK_{11}^{u*})^2,\\
\label{motsusyc}
&=\frac{\alpha_s^2m_Kf_K^2B_K\eta_\text{QCD}}{108m_{\tilde d}^2}
[11\tilde f_6(x_d)+4x_df_6(x_d)]\frac{(\Delta\tilde m^2_{\tilde
d})^2}{\tilde m_d^4} (K_{21}^{d*}K_{11}^{d})^2.
\end{align}
Here $m_{\tilde u,\tilde d}$ is the average mass of the corresponding
two squark generations, $\Delta m^2_{\tilde u,\tilde d}$ is the
mass-squared difference, and $x_{u,d}=m_{\tilde g}^2/m_{\tilde
u,\tilde d}^2$.
One can immediately identify three generic ways in which
supersymmetric contributions to neutral meson mixing can be
* Heaviness: $m_{\tilde q}\gg1\UTeV$.
* Degeneracy: $\Delta m^2_{\tilde q}\ll m_{\tilde q}^2$.
* Alignment: $K^{d,u}_{21}\ll1$.
When heaviness is the only suppression mechanism, as in split
supersymmetry [27], the squarks are very heavy
and supersymmetry no longer solves the fine tuning problem[When the first two squark generations are mildly heavy and the
third generation is light, as in effective supersymmetry
[28], the fine tuning problem is still solved, but
additional suppression mechanisms are needed.].
If we want to maintain supersymmetry as a solution to the fine tuning
problem, either degeneracy, or alignment, or a combination of both is
needed. This means that the flavour structure of supersymmetry is not
generic, as argued in the previous section.
The $2\times2$ mass-squared matrices for the relevant squarks have the
following form:
\begin{align}\label{mllot}
\tilde M^2_{U_L}
&= \tilde m^2_{Q_L}
+M_u M_u^\dagger,\nonumber\\
\tilde M^2_{D_L}
&= \tilde m^2_{Q_L}
+M_d M_d^\dagger.
\end{align}
We note the following features of the various terms:
* $\tilde m^2_{Q_L}$ is a $2\times2$ Hermitian matrix of soft
supersymmetry breaking terms. It does not break $SU(2)_\text{L}$
and consequently it is common to $\tilde M^2_{U_L}$ and $\tilde
M^2_{D_L}$. On the other hand, it breaks in general the
$SU(2)_Q$ flavour symmetry.
* The terms proportional to $m_Z^2$ are the D terms. They break
supersymmetry (since they involve $D_{T_3}\neq0$ and $D_Y\neq0$)
and $SU(2)_\text{L}$ but conserve $SU(2)_Q$.
* The terms proportional to $M_q^2$ come from the $F_{U_R}$ and
$F_{D_R}$ terms. They break the gauge $SU(2)_\text{L}$ and the
global $SU(2)_Q$ but, since $F_{U_R}=F_{D_R}=0$, conserve
Given that we are interested in squark masses close to the scale
(and the experimental lower bounds are of order $300\UGeV$), the scale of the
eigenvalues of $\tilde m^2_{Q_L}$ is much higher than
$m_Z^2$ which, in turn, is much higher than $m_c^2$, the largest
eigenvalue in $M_q M_q^\dagger$ (in the two-generation framework). We can draw the following conclusions:
* $m_{\tilde u}^2=m_{\tilde d}^2\equiv m_{\tilde q}^2$ up to
effects of order $m_Z^2$, namely to an accuracy of
* $\Delta m^2_{\tilde u}=\Delta m^2_{\tilde d}\equiv \Delta
m^2_{\tilde q}$ up to effects of order $m_c^2$, namely to an
accuracy of $\mathcal{O}(10^{-5})$.
* Since $K_u\simeq V_{uL} \tilde V_L^\dagger$ and $K_d\simeq
V_{dL} \tilde V_L^\dagger$ [the matrices $V_{qL}$ are defined in
diagMq, while $\tilde V_L$ diagonalizes $\tilde
m^2_{Q_L}$], the mixing matrices $K^u$ and $K^d$ are different
from each other, but the following relation to the CKM matrix
holds to an accuracy of $\mathcal{O}(10^{-5})$:
\begin{equation}\label{kkckm}
K^u K^{d\dagger} = V.
\end{equation}
§.§ Non-degenerate squarks at the LHC?
[b]motsusyb and (<ref>) can be translated into our
generic language:
\begin{align}
\Lambda_\text{NP}&= m_{\tilde q},\\
z_{cu} &= z_{12}\sin^2\theta_u,\nonumber\\
z_{sd} &= z_{12}\sin^2\theta_d,\nonumber\\
z_{12} &= \frac{11\tilde f_6(x)+4x f_6(x)}{18}
\alpha_s^2
\left(\frac{\Delta\tilde m_{\tilde q}^2}{m_{\tilde q}^2}\right)^2,
\end{align}
with kkckm giving
\begin{equation}\label{kkckmb}
\sin\theta_u-\sin\theta_d\approx\sin\theta_c=0.23.
\end{equation}
We now ask the following question: Is it possible that the first
two-generation squarks, $\tilde Q_{L1,2}$, are accessible to the LHC
($m_{\tilde q}\lesssim1\UTeV$), and are not degenerate ($\Delta
m^2_{\tilde q}/m_{\tilde q}^2=\mathcal{O}(1)$)?
To answer this question, we use Eqs. (<ref>) and (<ref>). For
$\Lambda_\text{NP}\lesssim1\UTeV$, we have
$z_{cu}\lesssim5\times10^{-7}$ and, for a phase that is $\not\ll0.1$,
$z_{sd}\lesssim6\times10^{-8}$. On the other hand, for non-degenerate
squarks, and, for example, $11\tilde f_6(1)+4f_6(1)=1/6$, we have
$z_{12}=8\times10^{-5}$. Then we need, simultaneously,
$\sin\theta_u\lesssim0.08$ and $\sin\theta_d\lesssim0.03$, but this is
inconsistent with kkckmb.
There are three ways out of this situation:
* The first two generation squarks are quasi-degenerate. The
minimal level of degeneracy is $(\tilde m_2-\tilde m_1)/(\tilde
m_2+\tilde m_1)\lesssim0.1$. It could be the result of RGE
[29]. However, for maximal phases, the bound is even
stronger, of order 0.04 [30], which is difficult to
achieve with just RGE effects.
* The first two generation squarks are heavy. Putting
$\sin\theta_u=0.23$ and $\sin\theta_d\approx0$, as in models of
alignment [31, 32], lowlnp2 leads
\begin{equation}\label{mqali}
m_{\tilde q}\gtrsim3\UTeV\SPp.
\end{equation}
* The ratio $x=\tilde m_g^2/\tilde m_q^2$ is in a fine-tuned
region of parameter space where there are accidental
cancellations in $11\tilde f_6(x)+4xf_6(x)$. For example, for
$x=2.33$, this combination is $\sim0.003$ and the bound
(<ref>) is relaxed by a factor of 7.
Barring accidental cancellations, the model-independent
conclusion is that, if the first two generations of squark doublets
are within the reach of the LHC, they must be quasi-degenerate
[33, 34]. Analogous conclusions can be drawn
for many TeV-scale new physics scenarios: a strong level of degeneracy
is required (for definitions and detailed analysis, see Ref. [30]).
Exercise 7: Does $K_{31}^d\sim|V_{ub}|$ suffice to satisfy the
$\Delta m_B$ constraint with neither degeneracy nor heaviness? (Use
the two-generation approximation and ignore the second generation.)
Is there a natural way to make the squarks degenerate? Examining
Eqs. (<ref>) we learn that degeneracy requires $\tilde
m^2_{Q_L}\simeq\tilde m^2_{\tilde q}\mathbf{1}$. We have mentioned
already that flavour universality is a generic feature of gauge
interactions. Thus the requirement of degeneracy is perhaps a hint
that supersymmetry breaking is gauge mediated to the MSSM
§ FLAVOUR AT THE LHC
The LHC will study the physics of electroweak symmetry breaking. There
are high hopes that it will discover not only the Higgs, but also shed
light on the fine-tuning problem that is related to the Higgs mass.
Here, we focus on the issue of how, through the study of new physics,
the LHC can shed light on the new physics flavour puzzle.
§.§ Minimal flavour violation (MFV)
If supersymmetry breaking is gauge mediated, the squark mass matrices
of mllot, and those for the SU(2)-singlet squarks, have the
following form at the scale of mediation $m_M$:
\begin{align}\label{mllgm}
\tilde M^2_{U_L}(m_M)&= \left(m^2_{\tilde Q_L}+D_{U_L}\right)
\mathbf{1}+M_u M_u^\dagger,\nonumber\\
\tilde M^2_{D_L}(m_M)&= \left(m^2_{\tilde Q_L}+D_{D_L}\right)
\mathbf{1}+M_d M_d^\dagger,\nonumber\\
\tilde M^2_{U_R}(m_M)&= \left(m^2_{\tilde U_R}+D_{U_R}\right)
\mathbf{1}+M_u^\dagger M_u,\nonumber\\
\tilde M^2_{D_R}(m_M)&= \left(m^2_{\tilde D_R}+D_{D_R}\right)
\mathbf{1}+M_d^\dagger M_d,
\end{align}
where $D_{q_A}=(T_3)_{q_A}-(Q_\text{EM})_{q_A}s^2_W m_Z^2\cos2\beta$
are the $D$-term contributions. Here, the only source of the
$SU(3)^3_q$ breaking are the SM Yukawa matrices.
This statement holds also when the renormalization group evolution is
applied to find the form of these matrices at the weak scale. Taking
the scale of the soft breaking terms $m_{\tilde q_A}$ to be somewhat
higher than the electroweak breaking scale $m_Z$ allows us to neglect
the $D_{q_A}$ and $M_q$ terms in (<ref>). Then we obtain
\begin{align}\label{mllrrmz}
\tilde M^2_{Q_L}(m_Z)
&\sim m^2_{\tilde Q_L}\left(r_3\mathbf{1}+c_u
Y_uY_u^\dagger+c_d Y_d Y_d^\dagger\right),\nonumber\\
\tilde M^2_{U_R}(m_Z)
&\sim m^2_{\tilde U_R}\left(r_3\mathbf{1}+c_{uR}
Y_u^\dagger Y_u\right),\nonumber\\
\tilde M^2_{D_R}(m_Z)
&\sim m^2_{\tilde D_R}\left(r_3\mathbf{1}+c_{dR}
Y_d^\dagger Y_d\right).
\end{align}
Here $r_3$ represent the universal RGE contribution that is
proportional to the gluino mass
($r_3=\mathcal{O}(6)\times(M_3(m_M)/m_{\tilde q}(m_M))$) and the
$c$-coefficients depend logarithmically on $m_M/m_Z$ and can be of
$\mathcal{O}(1)$ when $m_M$ is not far below the GUT scale.
Models of gauge mediated supersymmetry breaking (GMSB) provide a
concrete example of a large class of models that obey a simple
principle called minimal flavour violation (MFV)
[35]. This principle guarantees that low-energy
flavour-changing processes deviate only very little from the SM
predictions. The basic idea can be described as follows. The gauge
interactions of the SM are universal in flavour space. The only
breaking of this flavour universality comes from the three Yukawa
matrices, $Y_U$, $Y_D$, and $Y_E$. If this remains true in the
presence of the new physics, namely $Y_U$, $Y_D$, and $Y_E$ are the
only flavour non-universal parameters, then the model belongs to the
MFV class.
Let us now formulate this principle in a more formal way, using the
language of spurions that we presented in sec:spurions.
The Standard Model with vanishing Yukawa couplings has a large global
symmetry of gglobal and (<ref>). In this section we concentrate
only on the quarks. The non-Abelian part of the flavour symmetry for
the quarks is $SU(3)_q^3$ of susuu with the three generations
of quark fields transforming as follows:
\begin{equation}
Q_L(3,1,1),\ \ U_R(1,3,1),\ \ D_R(1,1,3).
\end{equation}
The Yukawa interactions,
\begin{equation}\label{eq:lagy}
\mathcal{L}_Y=\overline{Q_L}Y_D D_R H + \overline{Q_L}Y_U U_R H_c ,
\end{equation}
($H_c=i\tau_2 H^*$) break this symmetry. The Yukawa couplings can thus
be thought of as spurions with the following transformation properties
under $SU(3)_q^3$ [see Gglobq]:
\begin{equation}
Y_U\sim(3,\bar3,1),\qquad Y_D\sim(3,1,\bar3).
\end{equation}
When we say `spurions', we mean that we pretend that the Yukawa
matrices are fields which transform under the flavour symmetry, and
then require that all the Lagrangian terms, constructed
from the SM fields, $Y_{D}$ and $Y_U$, must be (formally)
invariant under the flavour group $SU(3)_q^3$. Of course, in reality,
$\mathcal{L}_Y$ breaks $SU(3)_q^3$ precisely because $Y_{D,U}$ are not fields and do not transform under the symmetry.
The idea of minimal flavour violation is relevant to extensions of the
SM, and can be applied in two ways:
* If we consider the SM as a low-energy effective theory, then all
higher-dimension operators, constructed from SM fields and
$Y$ spurions, are formally invariant under $G_\text{global}$.
* If we consider a full high-energy theory that extends the SM,
then all operators, constructed from SM and the new fields, and
from $Y$ spurions, are formally invariant under
Exercise 8: Use the spurion formalism to argue that, in MFV
models, the $K_L\to\pi^0\nu\bar\nu$ decay amplitude is proportional to
$y_t^2 V_{td}V_{ts}^*$.
Examples of MFV models include models of supersymmetry with gauge- or
anomaly-mediation of its breaking. If the LHC discovers new particles
that couple to the SM fermions, then it will be able to test solutions
to the new physics flavour puzzle such as MFV
[36]. Much of its power to test such frameworks is
based on identifying top and bottom quarks.
To understand this statement, we note that the spurions $Y_U$ and
$Y_D$ can always be written in terms of the two diagonal Yukawa
matrices $\lambda_u$ and $\lambda_d$ and the CKM matrix $V$, see
speint and (<ref>). Thus, the only source of quark
flavour-changing transitions in MFV models is the CKM matrix. Next,
note that to an accuracy that is better than $\mathcal{O}(0.05)$, we
can write the CKM matrix as follows:
\begin{equation}\label{ckmapp}
V=\begin{pmatrix} 1&0.23&0\\ -0.23&1&0\\ 0&0&1\end{pmatrix}\SPp.
\end{equation}
Exercise 9: The approximation (<ref>) should be
intuitively obvious to top-physicists, but definitely
counter-intuitive to bottom-physicists. (Some of them have dedicated a
large part of their careers to experimental or theoretical efforts to
determine $V_{cb}$ and $V_{ub}$.) What does the approximation
imply for the bottom quark? When we take into account that it is
only good to $\mathcal{O}(0.05)$, what would the implications be?
We learn that the third generation of quarks is decoupled, to a good
approximation, from the first two. This, in turn, means that any new
particle that couples to the SM quarks (think, for example, of heavy
quarks in vector-like representations of $G_\text{SM}$), decays into
either a third-generation quark, or into a non-third-generation quark,
but not to both. For example, in Grossman:2007bd, MFV models
with additional charge $-1/3$, $SU(2)_\text{L}$-singlet quarks,
$B^\prime$, were considered. A concrete test of MFV was proposed,
based on the fact that the largest mixing effect involving the third
generation is of order $|V_{cb}|^2\sim0.002$: Is the following
prediction, concerning events of $B^\prime$ pair production,
\begin{equation}
\frac{\Gamma(B^\prime\overline{B^\prime}\to Xq_{1,2}q_3)}
{\Gamma(B^\prime\overline{B^\prime}\to Xq_{1,2}q_{1,2})+
\Gamma(B^\prime\overline{B^\prime}\to Xq_3q_3)}\lesssim10^{-3}.
\end{equation}
If not, then MFV is excluded.
§.§ Supersymmetric flavour at the LHC
One can think of analogous tests in the supersymmetric framework
[37, 38, 39, 40, 41, 42]. Here, there is also a
generic prediction that, in each of the three sectors ($Q_L,U_R,D_R$),
squarks of
the first two generations are quasi-degenerate, and do not decay into
third-generation quarks. Squarks of the third generation can be
separated in mass (though, for small $\tan\beta$, the degeneracy in
the $\tilde D_R$ sector is threefold), and decay only to
third-generation quarks.
It is not necessary, however, that the mediation of supersymmetry
breaking be MFV. Examples of natural and viable solutions to the
supersymmetric flavour problem that are not MFV include the following:
* The leading contribution to the soft supersymmetry breaking
terms is gauge mediated, and therefore MFV, but there are
subleading contributions that are gravity mediated and provide
new sources of flavour and CP violation
[37, 42]. The gravity mediated
contributions could either have some structure (dictated, for
example, by a Froggatt–Nielsen symmetry [37] or by
localization in extra dimensions [43]) or be
anarchical [45].
* The first two sfermion generations are heavy, and their
mixing with the third generation is suppressed (for a recent
analysis, see Ref. [46]). These features can come,
for example, from conformal dynamics [47].
Such frameworks have different predictions concerning the mass
splitting between sfermion generations and the flavour decomposition
of the sfermion mass eigenstates. Note that measurements of
flavour-changing neutral current processes are only sensitive to the products
of the form
\begin{equation}\label{eq:defdel}
\delta_{ij}=\frac{\Delta \tilde m^2_{ij}}{\tilde m^2}\ K_{ij}K_{jj}^*,
\end{equation}
where $\Delta\tilde m^2_{ij}$ is the mass-squared splitting between
the sfermion generations $i$ and $j$, $\tilde m^2$ is their average
mass-squared, and $K$ is the mixing matrix of gaugino couplings to
these sfermions. On the other hand, the LHC experiments—ATLAS and
CMS—can, at least in principle, measure the mass splitting and the
mixing separately [40].
The present situation is depicted schematically in
fig:dmk(a). Flavour factories have provided only upper bounds
on deviations of FCNC processes, such as $\mu\to e\gamma$ or
$D^0$–$\overline{D}^0$ mixing, from the Standard Model
predictions. In the supersymmetric framework, such bounds translate
into an upper bound on a $\delta_{ij}$ parameter of eq:defdel,
corresponding to the blue region in the figure. The supersymmetric
flavour puzzle can be stated as the question of why the region in the
upper right corner—where the flavour parameters are of order
one—is excluded. MFV often puts us in the lower left corner of the
plot, far from the experimental constraints (this is particularly true
for $\delta_{12}$ parameters).
The optimal future situation is depicted schematically in
fig:dmk(b). Imagine that a flavour factory does provide
evidence for new physics, such as observation of $\Gamma(\mu\to
e\gamma)\neq0$ or CP violation in $D^0$–$\overline{D}^0$ mixing. This
will constrain the corresponding $\delta$ parameter, which is shown as
the blue region in the figure. If ATLAS/CMS measure the corresponding
sfermion mass splitting and/or mixing, we shall get a small allowed
region in this flavour plane.
[]Schematic description of the constraints in the plane of
sfermion mass-squared splitting, $\Delta\tilde
m^2_{ij}/\tilde m^2$, and mixing, $K_{ij}K_{jj}^*$: (a)
Upper bounds from not observing any deviation from the SM
predictions in present experiments; (b) Hypothetical future
situation, where deviations have been observed in flavour
factories (such as LHCb, a super-B factory, a $\mu\to
e\gamma$ measurement, etc.) and the mass splitting and
flavour decomposition have been measured by ATLAS/CMS.
If we have at our disposal three such consistent measurements (rate of FCNC
process, spectrum and splitting), then we shall understand the
mechanism by which supersymmetry has its flavour violation
suppressed. This will provide strong hints about the mechanism of
supersymmetry breaking mediation.
If the sfermions are quasi-degenerate, then the mixing is determined
by the small corrections to the unit mass-squared matrix. As mentioned
above, the structure of such corrections may be dictated by the same
symmetry or dynamics that gives the structure of the Yukawa
couplings. If that is the case, then the measurement of the flavour
decomposition might shed light on the Standard Model flavour puzzle.
We conclude that measurements at the LHC related to new particles that
couple to the SM fermions are likely to teach us much more about flavour
§ NEUTRINO ANARCHY VERSUS QUARK HIERARCHY
A detailed presentation of the physics and the formalism of neutrino
flavour transitions is given in Appendix <ref> for both
vacuum oscillations (<ref>) and the matter transitions
(<ref>). It follows Gonzalez-Garcia:2002dz.
Exercise 10: For atmospheric $\nu_\mu$'s with $E\sim1\UGeV$, the
flux coming from above has $P_{\mu\mu}(L\sim10\Ukm)\approx1$, while
the flux from below has $P_{\mu\mu}(L\sim10^4\Ukm)\approx0.5$. Assuming
that for the flux coming from below the oscillations are averaged
out, estimate $\Delta m^2$ and $\sin^22\theta$.
Exercise 11: For solar $\nu_e$'s, the transition
between matter ($\beta_\text{MSW}>1$) and vacuum
($\beta_\text{MSW}<\cos2\theta$) flavour transitions occurs around
$E\sim2\UMeV$. The transition probability is measured to be roughly
$P_{ee}\sim0.30$ for $\beta_\text{MSW}>1$. Estimate $\Delta m^2$ and
$\theta$ and predict $P_{ee}$ for $\beta_\text{MSW}\ll1$.
The derived ranges for the three mixing angles and two mass-squared
differences at $1\sigma$ are [50]
\begin{eqnarray}\label{nupara}
\Delta m^2_{21}&=&(7.9\pm0.3)\times10^{-5}\UeV^2,\ \ \
|\Delta m^2_{32}|=(2.6\pm0.2)\times10^{-3}\UeV^2,\nonumber\\
\sin^2\theta_{12}&=&0.31\pm0.02,\ \ \
\sin^2\theta_{23}=0.47\pm0.07,\ \ \
\sin^2\theta_{13}=0^{+0.008}_{-0.0}.
\end{eqnarray}
The $3\sigma$ range for the matrix elements of $U$ are the following
\begin{equation}\label{uthsi}
\end{pmatrix}\SPp.
\end{equation}
§.§ New physics
The simplest and most straightforward lesson of the evidence for
neutrino masses is also the most striking one: there is new physics
beyond the Standard Model. This is the first experimental result that
is inconsistent with the SM.
Most likely, the new physics is related to the existence of
$G_\text{SM}$-singlet fermions at some high energy scale that induce,
at low energies, the effective terms of Hnint through the
seesaw mechanism. The existence of heavy singlet fermions is predicted
by many extensions of the SM, especially by GUTs [beyond $SU(5)$] and
left–right-symmetric theories. The seesaw mechanism could also be
driven by an $SU(2)_\text{L}$-triplet fermion.
There are other possibilities. In particular, neutrino masses can be
generated without introducing any new fermions beyond those of the SM.
Instead, the existence of a scalar $\Delta_L(1,3)_{+1}$, that is, an
$SU(2)_\text{L}$-triplet, is required. The smallness of the neutrino
masses is related here to the smallness of the vacuum expectation
value $\langle\Delta_L^0\rangle$ (required also by the success of the
$\rho=1$ relation) and does not have a generic natural explanation.
In left–right-symmetric models, however, where the breaking of
$SU(2)_\text{R}\times U(1)_\text{B-L}\to U(1)_\text{Y}$ is induced by
the VEV of an $SU(2)_\text{R}$-triplet, $\Delta_R$, there must exist
also an $SU(2)_\text{L}$-triplet scalar. Furthermore, the Higgs
potential leads to an order of magnitude relation between the various
VEVs, $\langle\Delta_L^0\rangle\langle\Delta_R^0\rangle\sim v^2$, and
the smallness of $\langle\Delta_L^0\rangle$ is correlated with the
high scale of $SU(2)_\text{R}$ breaking. This situation can be thought
of as a seesaw of VEVs. In this model there are, however, also
SM-singlet fermions. The light neutrino masses arise from both the
seesaw mechanism (`type I') and the triplet VEV (`type II').
Neutrino masses could also be of the Dirac type. Here, again, singlet
fermions are introduced, but lepton number is imposed by hand. This
possibility is disfavoured by theorists since it is likely that global
symmetries are violated by gravitational effects. Furthermore, the
lightness of the neutrinos (compared to charged fermions) is
Another possibility is that neutrino masses are generated by mixing
with singlet fermions but the mass scale of these fermions is not
high. Here again the lightness of neutrino masses remains a
puzzle. The best known example of such a scenario is the framework of
supersymmetry without $R$ parity.
Let us emphasize that the seesaw mechanism or, more generally, the
extension of the SM with non-renormalizable terms, is the simplest
explanation of neutrino masses. Models in which neutrino masses are
generated by new physics at low energy imply a much more dramatic
departure from the SM. Furthermore, the existence of seesaw masses is
an unavoidable prediction of various extensions of the SM. In
contrast, many (but not all) of the low-energy mechanisms are
introduced for the specific purpose of generating neutrino masses.
§.§ The scale of new physics
[b]Hnint gives a light neutrino mass matrix:
\begin{equation}\label{seesawmass}
\end{equation}
It is straightforward to use the measured neutrino masses of
nupara in combination with seesawmass to estimate the
scale of new physics that is relevant to their generation. In
particular, if there is no quasi-degeneracy in the neutrino masses,
the heaviest of the active neutrino masses can be estimated:
\begin{equation}\label{mthree}
m_h=m_3\sim\sqrt{\Delta m^2_{32}}\approx0.05\UeV.
\end{equation}
(In the case of inverted hierarchy, the implied scale is
$m_h=m_2\sim\sqrt{\Delta m^2_{32}}\approx0.05\UeV$.) It follows that
the scale in the non-renormalizable terms (<ref>) is given by
\begin{equation}\label{seesawlnp}
\Lambda_\text{NP}\sim v^2/m_h\approx10^{15}\UGeV.
\end{equation}
We should clarify two points regarding seesawlnp:
* There could be some level of degeneracy between the neutrino
masses. In such a case, mthree is modified into a lower
bound on $m_3$ and, consequently, seesawlnp becomes an
upper bound on $\Lambda_\text{NP}$.
* It could be that the $Z_{ij}$ of Hnint are much smaller
than 1. In such a case, again, seesawlnp becomes an upper
bound on the scale of new physics.
On the other hand, in models of approximate flavour symmetries, there
are relations between the structures of the charged lepton and
neutrino mass matrices that give, quite generically, $Z_{33}\gtrsim
m_\tau^2/v^2\sim10^{-4}$. We conclude that the likely range for
$\Lambda_\text{NP}$ is given by
\begin{equation}\label{lnpssfl}
\end{equation}
The estimates (<ref>) and (<ref>) are very
exciting. First, the upper bound on the scale of new physics is well
below the Planck scale. This means that there is new physics in Nature
which is intermediate between the two known scales, the Planck scale,
$m_\text{Pl}\sim10^{19}\UGeV$, and the electroweak breaking scale,
$v\sim 10^2\UGeV$.
Second, the scale $\Lambda_\text{NP}\sim10^{15}\UGeV$ is intriguingly
close to the scale of gauge coupling unification.
Third, the range (<ref>) for the scale of lepton number
breaking is optimal for leptogenesis [51] (for
a recent review, see Davidson:2008bu). If
(i) leptogenesis is generated by the decays of the lightest singlet
neutrino $N_1$, and (ii) the masses of the singlet neutrinos are
hierarchical, $M_1/M_{2,3\ldots}\ll1$ , and (iii) the temperature
when leptogenesis occurs is high enough, $T_\text{LG}>10^{12}\UGeV$,
so that flavour effects are unimportant, then
there is an upper bound on the CP asymmetry in $N_1$ decays
\begin{equation}
\end{equation}
Given that $Y_B^\text{obs}\sim9\times10^{-11}$, and that
$Y_B\sim10^{-3}\eta\epsilon_{N_1}$, where $\eta\lesssim1$ is a washout
factor, we must require $|\epsilon_{N_1}|\gtrsim10^{-7}$. Moreover, we
have $m_3-m_2\leq\sqrt{\Delta m^2_{32}}\sim0.05\UeV$ and therefore
obtain $M_1\gtrsim10^{9}\UGeV$. Violating any of the three conditions
will relax this bound, but typically not by more than about an
order of magnitude.
§.§ The flavour puzzle
In the absence of neutrino masses, there are 13 flavour parameters in
the SM:
\begin{eqnarray}\label{chafla}
y_t&\sim&1,\ \ y_c\sim10^{-2},\ \ y_u\sim10^{-5},\nonumber\\
y_b&\sim&10^{-2},\ \ y_s\sim10^{-3},\ \ y_d\sim10^{-4},\nonumber\\
y_\tau&\sim&10^{-2},\ \ y_\mu\sim10^{-3},\ \ y_e\sim10^{-6},\nonumber\\
|V_{us}|&\sim&0.2,\ \ |V_{cb}|\sim0.04,\ \ |V_{ub}|\sim0.004,\ \
\sin\delta_\text{KM}\sim1.
\end{eqnarray}
These flavour parameters are hierarchical (their magnitudes span six
orders of magnitude), and all but two or three (the top Yukawa, the CP
violating phase, and perhaps the Cabibbo angle) are small. The
unexplained smallness and hierarchy pose the SM flavour puzzle.
Its solution may direct us to physics beyond the Standard Model.
Several mechanisms have been proposed in response to this puzzle. For
example, approximate horizontal symmetries, broken by a small
parameter, can lead to selection rules that explain the hierarchy of
the Yukawa couplings.
In the extension of the SM with three active neutrinos that have
Majorana masses, there are nine new flavour parameters in addition to
those of chafla. These are three neutrino masses, three lepton
mixing angles, and three phases in the mixing matrix. Of the nine new
parameters, four have been measured: two mass-squared differences and
two mixing angles [see nupara]. This adds significantly to the
input data on flavour physics and provides an opportunity to test and
refine flavour models.
If neutrino masses arise from effective terms of the form of
Hnint, then the overall scale of neutrino masses is
related to the scale $\Lambda_\text{NP}$ and, in most cases, does not
tell us anything about flavour physics. More significant information
for flavour models can be written in terms of three dimensionless
parameters whose values can be read from nupara, that is
$\sin\theta_{12}$, $\sin\theta_{23}$ and
\begin{equation}\label{nuflpa}
\Delta m^2_{21}/|\Delta m^2_{32}|=0.030\pm0.003.
\end{equation}
In addition, the upper bound on $\sin\theta_{13}$ often plays a
significant role in flavour model building.
There are several features in the numerical estimates (<ref>)
and (<ref>) that have drawn much attention and have driven
numerous investigations:
(i) Large mixing and strong hierarchy: The mixing angle that is
relevant to the $2$–$3$ sector is large, $\sin\theta_{23}\sim0.7$. On
the other hand, if there is no quasi-degeneracy in the neutrino
masses, the corresponding mass ratio is small, $m_2/m_3\sim0.17$. It
is difficult to explain in a natural way a situation where there is an
$\mathcal{O}(1)$ mixing but the corresponding masses are hierarchical.
(ii) Two large and one small mixing angles: The mixing angles relevant
to the $2$–$3$ sector ($\sin\theta_{23}\sim0.7$) and $1$–$2$ sector
($\sin\theta_{12}\sim0.55$) are large, yet the $1$–$3$ mixing angle is
small ($\sin\theta_{13}\lesssim 0.20$). Such a situation is, again,
difficult—though not impossible—to explain from approximate
symmetries. An example of a symmetry that does predict such a pattern
is that of $L_e$–$L_\mu$–$L_\tau$. This symmetry predicts, however,
$\theta_{12}\simeq\pi/4$, which is experimentally excluded.
(iii) Maximal mixing: The value of $\theta_{23}$ is
intriguingly close to maximal mixing ($\sin^22\theta_{23}=1$). It is
interesting to understand whether a symmetry could explain this
special value.
(iv) Tribimaximal mixing: The mixing matrix (<ref>) has a
structure that is consistent with the following unitary matrix
\begin{equation}
\sqrt{\frac23}&\sqrt{\frac13}&0\\
\sqrt{\frac16}&-\sqrt{\frac13}&\sqrt{\frac{1}{2}}
\end{pmatrix}\SPp.
\end{equation}
It is interesting to understand whether a symmetry could explain this
special structure.
All four features enumerated above are difficult to explain in a large
class of flavour models that do very well in explaining the flavour
features of the quark sector. In particular, models with Abelian
horizontal symmetries (Froggatt–Nielsen type [55])
predict that, in general, $|V_{ub}|\sim|V_{us}V_{cb}|$, $|V_{ij}|\gtrsim
m_i/m_j$ ($i<j$) and $V\sim\mathbf{1}$
[56, 32]. All of these are successful
predictions. At the same time, however, these models predict
[57] that for the neutrinos, in general,
$|U_{ij}|^2\sim m_i/m_j$ and $|U_{e3}|\sim|U_{e2}U_{\mu3}|$, in
contradiction to, respectively, points (i) and (ii) above (and there
is no way to make $\theta_{23}$ parametrically close to $\pi/4$). On
the other hand, there exist very specific models where these features
are related to a symmetry.
It is possible, however, that the above interpretation of the results
is wrong. Indeed, the data can be interpreted in a very different
(v) No small parameters: The two measured mixing angles are
larger than any of the quark mixing angles. Indeed, they are both of
order one. The measured mass ratio, $m_2/m_3\gtrsim0.16$ is larger
than any of the quark and charged lepton mass ratios, and could be
interpreted as an $\mathcal{O}(1)$ parameter (namely, it is
accidentally small, without any parametric suppression). If this is
the correct way of reading the data, the measured neutrino parameters
may actually reflect the absence of any hierarchical structure in the
neutrino mass matrices [58]. The possibility that there
is no structure—neither hierarchy, nor degeneracy—in the neutrino
sector has been called `neutrino mass anarchy'. An important test of
this idea will be provided by the measurement of $|U_{e3}|$. If indeed
the entries in $M_\nu$ have random values of the same order, all three
mixing angles are expected to be of order one. If experiments measure
$|U_{e3}|\sim0.1$, that is, close to the present bound, it can be
argued that its smallness is accidental. The stronger the upper bound
on this angle becomes, the more difficult it will be to maintain this
Neutrino mass anarchy can be accommodated within models of Abelian
flavour symmetries, if the three lepton doublets carry the same
charge. Indeed, consider a supersymmetric model with a $U(1)_H$
symmetry that is broken by a single small spurion $\epsilon_H$ of
charge $-1$. Let us assume that the three fermion generations
contained in the $10$-representation of $SU(5)$ carry charges
$(2,1,0)$, while the three $\bar5$-representations carry charges
$(0,0,0)$. (The Higgs fields carry no $H$ charges.) Such a model
predicts $\epsilon_H^2$ hierarchy in the up sector, $\epsilon_H$
hierarchy in the down and charged lepton sectors, and anarchy in the
neutrino sector.
Exercise 12: The selection rule for this model is that
a term in the superpotential that carries $H$ charge $n\geq0$ is
suppressed by $\epsilon_H^n$. Find the parametric suppression of the
various entries in $M_u,M_d,M_\ell$, and $M_\nu$. Find the parametric
suppression of the mixing angles.
It would be nice if the features of quark mass hierarchy and neutrino
mass anarchy can be traced back to some fundamental principle or to a
stringy origin (see, for example, Antebi:2005hr).
§ CONCLUSIONS
(i) Measurements of CP violating $B$-meson decays have
established that the Kobayashi–Maskawa mechanism is the
dominant source of the observed CP violation.
(ii) Measurements of flavour-changing $B$-meson decays have
established that the Cabibbo–Kobayashi–Maskawa mechanism
is a major player in flavour violation.
(iii) The consistency of all these measurements with the CKM
predictions sharpens the new physics flavour puzzle: If
there is new physics at, or below, the scale,
then its flavour structure must be highly non-generic.
(iv) Measurements of $D^0$–$\overline{D}^0$ mixing imply that
alignment by itself cannot solve the supersymmetric
flavour problem. The first two squark generations must be
(v) Measurements of neutrino flavour parameters have not only
not clarified the Standard Model flavour puzzle, but
actually deepened it. Whether they imply an anarchical
structure, or a tribimaximal mixing, it seems that the
neutrino flavour structure is very different from that of
(vi) If the LHC experiments, ATLAS and CMS, discover new
particles that couple to the Standard Model fermions,
then, in principle, they will be able to measure new
flavour parameters. Consequently, the new physics flavour
puzzle is likely to be understood.
(vii) If the flavour structure of such new particles is
affected by the same physics that sets the flavour
structure of the Yukawa couplings, then the LHC
experiments (and future flavour factories) may be able to
shed light also on the Standard Model flavour puzzle.
The huge progress in flavour physics in recent years has provided
answers to many questions. At the same time, new questions arise. We
look forward to the LHC era for more answers and more questions.
§ ACKNOWLEDGEMENTS
The research of Y. Nir is supported by the Israel Science Foundation;
the United States–Israel Binational Science Foundation (BSF),
Jerusalem, Israel; the German–Israeli Foundation for Scientific
Research and Development (GIF); and the Minerva Foundation.
§ THE CKM MATRIX
The CKM matrix $V$ is a $3\times3$ unitary matrix. Its form, however,
is not unique:
$(i)$ There is freedom in defining $V$ in that we can permute between
the various generations. This freedom is fixed by ordering the up quarks and
the down quarks by their masses, $(u_1,u_2,u_3)\to(u,c,t)$ and
$(d_1,d_2,d_3)\to(d,s,b)$. The elements of $V$ are written as follows:
\begin{equation}\label{defVij}
\end{pmatrix}\SPp.
\end{equation}
$(ii)$ There is further freedom in the phase structure of $V$. This
means that the number of physical parameters in $V$ is smaller than
the number of parameters in a general unitary $3\times3$ matrix which
is nine (three real angles and six phases). Let us define $P_q$
($q=u,d$) to be diagonal unitary (phase) matrices. Then, if instead of
using $V_{qL}$ and $V_{qR}$ for the rotation (<ref>) to the
mass basis we use $\tilde V_{qL}$ and $\tilde V_{qR}$, defined by
$\tilde V_{qL}=P_q V_{qL}$ and $\tilde V_{qR}=P_q V_{qR}$, we still
maintain a legitimate mass basis since $M_q^\text{diag}$ remains
unchanged by such transformations. However, $V$ does change:
\begin{equation}\label{eqphase}
V\to P_u V P_d^*.
\end{equation}
This freedom is fixed by demanding that $V$ has the minimal number of
phases. In the three-generation case $V$ has a single phase. (There
are five phase differences between the elements of $P_u$ and $P_d$ and,
therefore, five of the six phases in the CKM matrix can be removed.) This is
the Kobayashi–Maskawa phase $\delta_\text{KM}$ which is the single source of
CP violation in the quark sector of the Standard Model [1].
The fact that $V$ is unitary and depends on only four independent
physical parameters can be made manifest by choosing a specific
parametrization. The standard choice is [60]
\begin{equation}\label{stapar}
\end{pmatrix}\SPp,
\end{equation}
where $c_{ij}\equiv\cos\theta_{ij}$ and $s_{ij}\equiv\sin\theta_{ij}$.
The $\theta_{ij}$'s are the three real mixing parameters while
$\delta$ is the Kobayashi–Maskawa phase. It is known experimentally
that $s_{13}\ll s_{23}\ll s_{12}\ll1$. It is convenient to choose an
approximate expression where this hierarchy is manifest. This is the
Wolfenstein parametrization, where the four mixing parameters are
$(\lambda,A,\rho,\eta)$ with $\lambda=|V_{us}|=0.23$ playing the role
of an expansion parameter and $\eta$ representing the CP violating
phase [61, 62]:
\begin{equation}\label{wolpar}
1-\frac{1}{2}\lambda^2-\frac18\lambda^4 & \lambda & A\lambda^3(\rho-i\eta)\\
-\lambda +\frac{1}{2}A^2\lambda^5[1-2(\rho+i\eta)] &
1-\frac{1}{2}\lambda^2-\frac18\lambda^4(1+4A^2) & A\lambda^2 \\
-A\lambda^2+\frac{1}{2}A\lambda^4[1-2(\rho+i\eta)]& 1-\frac{1}{2}A^2\lambda^4
\end{pmatrix}\SPp.
\end{equation}
A very useful concept is that of the unitarity triangles. The
unitarity of the CKM matrix leads to various relations among the
matrix elements,
\begin{eqnarray}\label{Unitds}
\label{Unitsb}
\label{Unitdb}
\end{eqnarray}
Each of these three relations requires the sum of three complex
quantities to vanish and so can be geometrically represented in the
complex plane as a triangle. These are `the unitarity triangles',
though the term `unitarity triangle' is usually reserved for the
relation (<ref>) only. The unitarity triangle related to
Unitdb is depicted in fg:tri.
[]Graphical representation of the unitarity constraint
$V_{ud}V_{ub}^*+V_{cd}V_{cb}^*+V_{td}V_{tb}^*=0$ as a
triangle in the complex plane
The rescaled unitarity triangle is derived from (<ref>)
by (a) choosing a phase convention such that $(V_{cd}V_{cb}^*)$
is real, and (b) dividing the lengths of all sides by $|V_{cd}V_{cb}^*|$.
Step (a) aligns one side of the triangle with the real axis, and
step (b) makes the length of this side 1. The form of the triangle
is unchanged. Two vertices of the rescaled unitarity triangle are
thus fixed at (0,0) and (1,0). The coordinates of the remaining
vertex correspond to the Wolfenstein parameters $(\rho,\eta)$.
The area of the rescaled unitarity triangle is $|\eta|/2$.
Depicting the rescaled unitarity triangle in the
$(\rho,\eta)$ plane, the lengths of the two complex sides are
\begin{equation}\label{RbRt}
=\sqrt{\rho^2+\eta^2},\ \ \
\end{equation}
The three angles of the unitarity triangle are defined as follows
[63, 64]:
\begin{equation}\label{abcangles}
\alpha\equiv\arg\left[-\frac{V_{td}V_{tb}^*}{V_{ud}V_{ub}^*}\right],\quad
\beta \equiv\arg\left[-\frac{V_{cd}V_{cb}^*}{V_{td}V_{tb}^*}\right],\quad
\gamma\equiv\arg\left[-\frac{V_{ud}V_{ub}^*}{V_{cd}V_{cb}^*}\right]\SPp.
\end{equation}
They are physical quantities and can be independently measured by CP
asymmetries in $B$ decays. It is also useful to define the two
small angles of the unitarity triangles (<ref>), (<ref>):
\begin{equation}\label{bbangles}
\beta_s\equiv\arg\left[-\frac{V_{ts}V_{tb}^*}{V_{cs}V_{cb}^*}\right],\quad
\beta_K\equiv\arg\left[-\frac{V_{cs}V_{cd}^*}{V_{us}V_{ud}^*}\right]\SPp.
\end{equation}
The $\lambda$ and $A$ parameters are very well determined at present,
see lamaexp. The main effort in CKM measurements is thus
aimed at improving our knowledge of $\rho$ and $\eta$:
\begin{equation}
\rho=0.14^{+0.03}_{-0.02},\ \ \ \eta=0.35\pm0.02.
\end{equation}
The present status of our knowledge is best seen in a plot of the
various constraints and the final allowed region in the $\rho$–$\eta$
plane. This is shown in fg:UT.
§ CP VIOLATION IN NEUTRAL $B$ DECAYS TO FINAL CP EIGENSTATES
We define decay amplitudes of $B$ (which could be charged or neutral)
and its CP conjugate $\Bbar$ to a multiparticle final state $f$ and its
CP conjugate $\fb$ as
\begin{equation}\label{decamp}
A_{\f}=\langle \f|\mathcal{H}|B\rangle\quad , \quad
\overline{A}_{\f}=\langle \f|\mathcal{H}|\Bbar\rangle\quad , \quad
A_{\fb}=\langle \fb|\mathcal{H}|B\rangle\quad , \quad
\overline{A}_{\fb}=\langle \fb|\mathcal{H}|\Bbar\rangle\; ,
\end{equation}
where $\mathcal{H}$ is the Hamiltonian governing
weak interactions. The action of CP on these states introduces
phases $\xi_B$ and $\xi_f$ according to
\begin{eqnarray}\label{eq:phaseconv}
\CP|B\rangle &=& e^{+i\xi_{B}}\,|\Bbar\rangle \quad , \quad
\CP|\f\rangle = e^{+i\xi_{\f}}\,|\fb\rangle \; ,\nonumber\\
\CP|\Bbar\rangle& =& e^{-i\xi_{B}}\,|B\rangle \quad , \quad
\CP|\fb\rangle = e^{-i\xi_{\f}}\,|\f\rangle \ ,
\end{eqnarray}
so that $(\CP)^2=1$. The phases $\xi_B$ and $\xi_f$ are arbitrary and
unphysical because of the flavour symmetry of the strong
interaction. If CP is conserved by the dynamics, $[\CP,\mathcal{H}] =
0$, then $A_f$ and $\overline{A}_{\fb}$ have the same magnitude and an
arbitrary unphysical relative phase
\begin{equation}\label{spupha}
\overline{A}_{\fb} = e^{i(\xi_{\f}-\xi_{B})}\, A_f\; .
\end{equation}
A state that is initially a superposition of $\Bz$ and $\Bzb$, say
\begin{equation}
|\psi(0)\rangle = a(0)|\Bz\rangle+b(0)|\Bzb\rangle \; ,
\end{equation}
will evolve in time acquiring components that describe all possible
decay final states $\{f_1,f_2,\ldots\}$, that is,
\begin{equation}
|\psi(t)\rangle =
\; .
\end{equation}
If we are interested in computing only the values of $a(t)$ and $b(t)$
(and not the values of all $c_i(t)$), and if the times $t$ in which we
are interested are much larger than the typical strong interaction
scale, then we can use a much simplified
formalism [65]. The simplified time evolution is
determined by a $2\times 2$ effective Hamiltonian $\Heff$ that is
not Hermitian, since otherwise the mesons would only oscillate and not
decay. Any complex matrix, such as $\Heff$, can be written in terms of
Hermitian matrices $\Meff$ and $\Geff$ as
\begin{equation}
\Heff = \Meff - \frac{i}{2}\,\Geff \; .
\end{equation}
$\Meff$ and $\Geff$ are associated with
$(\Bz,\Bzb)\leftrightarrow(\Bz,\Bzb)$ transitions via off-shell
(dispersive) and on-shell (absorptive) intermediate states, respectively.
Diagonal elements of $\Meff$ and $\Geff$ are associated with the
flavour-conserving transitions $\Bz\to\Bz$ and $\Bzb\to\Bzb$ while
off-diagonal elements are associated with flavour-changing transitions
The eigenvectors of $\Heff$ have well-defined masses and decay
widths. We introduce complex parameters $p_{L,H}$ and $q_{L,H}$ to
specify the components of the strong interaction eigenstates, $\Bz$ and
$\Bzb$, in the light ($B_L$) and heavy ($B_H$) mass eigenstates:
\begin{equation}\label{defpq}
|B_{L,H}\rangle=p_{L,H}|\Bz\rangle\pm q_{L,H}|\Bzb\rangle
\end{equation}
with the normalization $|p_{L,H}|^2+|q_{L,H}|^2=1$. If either CP or
CPT is a symmetry of $\Heff$ (independently of whether T is conserved or
violated) then $\Meff_{11} = \Meff_{22}$ and $\Geff_{11}=
\Geff_{22}$, and solving the eigenvalue problem for $\Heff$ yields $p_L
= p_H \equiv p$ and $q_L = q_H \equiv q$ with
\begin{equation}
\left(\frac{q}{p}\right)^2=\frac{\Meff_{12}^\ast -
(i/2)\Geff_{12}^\ast}{\Meff_{12}-(i/2)\Geff_{12}}\; .
\end{equation}
From now on we assume that CPT is conserved.
If either CP or T is a symmetry of $\Heff$ (independently of whether
CPT is conserved or violated), then $\Meff_{12}$ and $\Geff_{12}$ are
relatively real, leading to
\begin{equation}
\left(\frac{q}{p}\right)^2 = e^{2i\xi_B} \quad \Rightarrow \quad
\left|\frac{q}{p}\right| = 1 \; ,
\end{equation}
where $\xi_B$ is the arbitrary unphysical phase introduced in
The real and imaginary parts of the eigenvalues of $\Heff$
corresponding to $|B_{L,H}\rangle$ represent their masses and
decay-widths, respectively. The mass difference $\Delta m_B$ and the
width difference $\Delta\Gamma_B$ are defined as follows:
\begin{equation}\label{DelmG}
\Delta m_B\equiv M_H-M_L,\quad\Delta\Gamma_B\equiv\Gamma_H-\Gamma_L\SPp.
\end{equation}
Note that here $\Delta m_B$ is positive by definition, while the sign of
$\Delta\Gamma_B$ is to be experimentally determined.
The average mass and width are given by
\begin{equation}\label{aveMG}
\end{equation}
It is useful to define dimensionless ratios $x$ and $y$:
\begin{equation}\label{defxy}
x\equiv\frac{\Delta m_B}{\Gamma_B},\quad y\equiv\frac{\Delta\Gamma_B}{2\Gamma_B}.
\end{equation}
Solving the eigenvalue equation gives
\begin{equation}\label{eveq}
(\Delta m_B)^2-\frac{1}{4}(\Delta\Gamma_B)^2=(4|M_{12}|^2-|\Gamma_{12}|^2),\ \ \ \
\Delta m_B\Delta\Gamma_B=4\re{M_{12}\Gamma_{12}^*}.
\end{equation}
All CP-violating observables in $B$ and $\Bbar$ decays to final states $f$
and $\fb$ can be expressed in terms of phase-convention-independent
combinations of $A_f$, $\overline{A}_f$, $A_{\overline{f}}$, and
$\overline{A}_{\overline{f}}$, together with, for neutral-meson decays
only, $q/p$. CP violation in charged-meson decays depends only on the
combination $|\overline{A}_{\fb}/A_f|$, while CP violation in
neutral-meson decays is complicated by $\Bz\leftrightarrow\Bzb$
oscillations and depends, additionally, on $|q/p|$ and on $\lambda_f
\equiv (q/p)(\overline{A}_f/A_f)$.
For neutral $D$, $B$, and $B_s$ mesons, $\Delta\Gamma/\Gamma\ll1$ and
so both mass eigenstates must be considered in their evolution. We
denote the state of an initially pure $|\Bz\rangle$ or $|\Bzb\rangle$
after an elapsed proper time $t$ as $|\Bz_{\mathrm{phys}}(t)\rangle$
or $|\Bzb_{\mathrm{phys}}(t)\rangle$, respectively. Using the
effective Hamiltonian approximation, we obtain
\begin{eqnarray}\label{defphys}
- \frac qp\ g_-(t)|\Bzb\rangle,\nonumber\\
- \frac pq\ g_-(t)|\Bz\rangle \; ,
\end{eqnarray}
\begin{equation}
g_\pm(t) \equiv \frac{1}{2}\left(e^{-im_Ht-\frac{1}{2}\Gamma_Ht}\pm
\end{equation}
One obtains the following time-dependent decay rates:
\begin{eqnarray}
\frac{d\Gamma[\Bz_\text{phys}(t)\to f]/dt}{e^{-\Gamma t}\mathcal{N}_f}&=&
\left(|A_f|^2+|(q/p)\overline{A}_f|^2\right)\cosh(y\Gamma t)
+\left(|A_f|^2-|(q/p)\overline{A}_f|^2\right)\cos(x\Gamma t)\nonumber\\
&+&2\,\re{(q/p)A_f^\ast \overline{A}_f}\sinh(y\Gamma t)
-2\,\im{(q/p)A_f^\ast \overline{A}_f}\sin(x\Gamma t)
\label{decratbt1}\;,\\
\frac{d\Gamma[\Bzb_\text{phys}(t)\to f]/dt}{e^{-\Gamma t}\mathcal{N}_f}&=&
\left(|(p/q)A_f|^2+|\overline{A}_f|^2\right)\cosh(y\Gamma t)
-\left(|(p/q)A_f|^2-|\overline{A}_f|^2\right)\cos(x\Gamma t)\nonumber\\
&+&2\,\re{(p/q)A_f\overline{A}^\ast_f}\sinh(y\Gamma t)
-2\,\im{(p/q)A_f\overline{A}^\ast_f}\sin(x\Gamma t)
\label{decratbt2}\; ,
\end{eqnarray}
where $\mathcal{N}_f$ is a common normalization factor. Decay rates to
the CP-conjugate final state $\fb$ are obtained analogously, with
$\mathcal{N}_f = \mathcal{N}_{\fb}$ and the substitutions $A_f\to
A_{\fb}$ and $\overline{A}_f\to\overline{A}_{\fb}$ in
decratbt1 and (<ref>). Terms proportional to
$|A_f|^2$ or $|\overline{A}_f|^2 $ are associated with decays that
occur without any net $B\leftrightarrow\Bbar$ oscillation, while terms
proportional to $|(q/p)\overline{A}_f|^2$ or $|(p/q)A_f|^2$ are
associated with decays following a net oscillation. The $\sinh(y\Gamma
t)$ and $\sin(x\Gamma t)$ terms of decratbt1 and
(<ref>) are associated with the interference between these
two cases. Note that, in multi-body decays, amplitudes are functions
of phase-space variables. Interference may be present in some regions
but not in others, and is strongly influenced by resonant substructure.
One possible manifestation of CP-violating effects in meson decays
[66] is in the interference between a decay without
mixing, $\Bz\to f$, and a decay with mixing, $\Bz\to \Bzb\to f$ (such
an effect occurs only in decays to final states that are common to
$\Bz$ and $\Bzb$, including all CP eigenstates). It is defined by
\begin{equation}\label{cpvint}
\im{\lambda_f}\ne 0 \; ,
\end{equation}
\begin{equation}\label{deflam}
\lambda_f \equiv \frac{q}{p}\frac{\overline{A}_f}{A_f} \; .
\end{equation}
This form of CP violation can be observed, for example, using the
asymmetry of neutral meson decays into final CP eigenstates $f_{\CP}$
\begin{equation}\label{asyfcp}
\mathcal{A}_{f_{\CP}}(t)\equiv\frac{d\Gamma/dt[\Bzb_\text{phys}(t)\to f_{\CP}]-
d\Gamma/dt[\Bz_\text{phys}(t)\to f_{\CP}]}
{d\Gamma/dt[\Bzb_\text{phys}(t)\to f_{\CP}]+d\Gamma/dt[\Bz_\text{phys}(t)\to
f_{\CP}]}\; .
\end{equation}
For $\Delta\Gamma = 0$ and $|q/p|=1$ (which is a good approximation
for $B$ mesons), $\mathcal{A}_{f_{\CP}}$ has a particularly simple form
[67, 68, 69]:
\begin{eqnarray}\label{asyfcpb}
\mathcal{A}_{f}(t)&=&S_f\sin(\Delta mt)-C_f\cos(\Delta mt),\nonumber\\
S_f&\equiv&\frac{2\,\im{\lambda_{f}}}{1+|\lambda_{f}|^2},\ \ \
\end{eqnarray}
Consider the $B\to f$ decay amplitude $A_f$, and the CP conjugate
process $\Bbar\to\fb$ with decay amplitude $\overline{A}_{\fb}$. There
are two types of phases that may appear in these decay amplitudes.
Complex parameters in any Lagrangian term that contributes to the
amplitude will appear in complex conjugate form in the CP-conjugate
amplitude. Thus their phases appear in $A_f$ and
$\overline{A}_{\overline{f}}$ with opposite signs. In the Standard
Model, these phases occur only in the couplings of the $W^\pm$ bosons
and hence are often called `weak phases'. The weak phase of any
single term is convention dependent. However, the difference between
the weak phases in two different terms in $A_f$ is convention
independent. A second type of phase can appear in scattering or decay
amplitudes even when the Lagrangian is real. Their origin is the
possible contribution from intermediate on-shell states in the decay
process. Since these phases are generated by CP-invariant
interactions, they are the same in $A_f$ and
$\overline{A}_{\overline{f}}$. Usually the dominant rescattering is
due to strong interactions and hence the designation `strong phases'
for the phase shifts so induced. Again, only the relative strong
phases between different terms in the amplitude are physically
The `weak' and `strong' phases discussed here appear in addition to
the `spurious' CP transformation phases of spupha. Those
spurious phases are due to an arbitrary choice of phase convention,
and do not originate from any dynamics or induce any violation. For simplicity, we set them to zero from here on.
It is useful to write each contribution $a_i$ to $A_f$ in three parts:
its magnitude $|a_i|$, its weak phase $\phi_i$, and its strong
phase $\delta_i$. If, for example, there are two such
contributions, $A_f = a_1 + a_2$, we have
\begin{eqnarray}\label{weastr}
A_f&=& |a_1|e^{i(\delta_1+\phi_1)}+|a_2|e^{i(\delta_2+\phi_2)},\nonumber\\
\overline{A}_{\overline{f}}&=&
\end{eqnarray}
Similarly, for neutral meson decays, it is useful to write
\begin{equation}\label{defmgam}
\Meff_{12} = |\Meff_{12}| e^{i\phi_M} \quad , \quad
\Geff_{12} = |\Geff_{12}| e^{i\phi_\Gamma} \; .
\end{equation}
Each of the phases appearing in weastr and (<ref>) is
convention dependent, but combinations such as $\delta_1-\delta_2$,
$\phi_1-\phi_2$, $\phi_M-\phi_\Gamma$ and
$\phi_M+\phi_1-\overline{\phi}_1$ (where $\overline{\phi}_1$ is a weak
phase contributing to $\overline{A}_f$) are physical.
In the approximations that only a single weak phase contributes to decay,
$A_f=|a_f|e^{i(\delta_f+\phi_f)}$, and that
$|\Geff_{12}/\Meff_{12}|=0$, we obtain $|\lambda_f|=1$ and
the asymmetries in decays to a final CP
eigenstate $f$ [asyfcp] with eigenvalue $\eta_f= \pm 1$
are given by
\begin{equation}\label{afcth}
\mathcal{A}_{f_{\CP}}(t) = \im{\lambda_f}\; \sin(\Delta m t) \; \
\mathrm{with}\ \
\im{\lambda_f}=\eta_f\sin(\phi_M+2\phi_f).
\end{equation}
Note that the phase so measured is purely a weak phase, and no
hadronic parameters are involved in the extraction of its value from
§ SUPERSYMMETRIC CONTRIBUTIONS TO NEUTRAL MESON MIXING
We consider the squark–gluino box diagram contribution to
$D^0$–$\overline{D}^0$ mixing amplitude that is proportional to
$K_{2i}^u K^{u*}_{1i}K_{2j}^u K^{u*}_{1j}$, where $K^u$ is the mixing
matrix of the gluino couplings to left-handed up quarks and their up
squark partners. (In the language of the mass insertion approximation,
we calculate here the contribution that is $\propto
[(\delta^u_{LL})_{12}]^2$.) We work in the mass basis for both quarks
and squarks.
The contribution is given by
\begin{equation}\label{motsusy}
\sum_{i,j}(K_{2i}^uK_{1i}^{u*}K_{2j}^uK_{1j}^{u*})(11\tilde
I_{4ij}+4\tilde m_g^2I_{4ij})\SPp,
\end{equation}
\begin{eqnarray}
\tilde I_{4ij}&\equiv&\int\frac{d^4p}{(2\pi)^4}\frac{p^2}{(p^2-\tilde
m_g^2)^2(p^2-\tilde m_i^2)(p^2-\tilde m_j^2)}\nonumber\\
&=&\frac{i}{(4\pi)^2}\left[\frac{\tilde m_g^2}
{(\tilde m_i^2-\tilde m_g^2)(\tilde m_j^2-\tilde m_g^2)}\right.\nonumber\\
&& +\left.\frac{\tilde m_i^4}
{(\tilde m_i^2-\tilde m_j^2)(\tilde m_i^2-\tilde
m_g^2)^2}\ln\frac{\tilde m_i^2}{\tilde m_g^2}
+\frac{\tilde m_j^4}
{(\tilde m_j^2-\tilde m_i^2)(\tilde m_j^2-\tilde
m_g^2)^2}\ln\frac{\tilde m_j^2}{\tilde m_g^2}\right],
\end{eqnarray}
\begin{eqnarray}
m_g^2)^2(p^2-\tilde m_i^2)(p^2-\tilde m_j^2)}\nonumber\\
{(\tilde m_i^2-\tilde m_g^2)(\tilde m_j^2-\tilde m_g^2)}\right.\nonumber\\
&& +\left.\frac{\tilde m_i^2}
{(\tilde m_i^2-\tilde m_j^2)(\tilde m_i^2-\tilde
m_g^2)^2}\ln\frac{\tilde m_i^2}{\tilde m_g^2}
+\frac{\tilde m_j^2}
{(\tilde m_j^2-\tilde m_i^2)(\tilde m_j^2-\tilde
m_g^2)^2}\ln\frac{\tilde m_j^2}{\tilde m_g^2}\right].
\end{eqnarray}
We now follow the discussion in Raz:2002zx,Nir:2002ah.
To see the consequences of the super-GIM mechanism, let us expand the
expression for the box integral around some value $\tilde m^2_q$ for
the squark masses-squared:
\begin{eqnarray}
I_4(\tilde m_g^2,\tilde m_i^2,\tilde m_j^2)&=&
I_4(\tilde m_g^2,\tilde m_q^2+\delta\tilde m_i^2,\tilde
m_q^2+\delta\tilde m_j^2)\nonumber\\
&=&I_4(\tilde m_g^2,\tilde m_q^2,\tilde m_q^2)
+(\delta\tilde m_i^2+\delta\tilde m_j^2)I_5(\tilde m_g^2,\tilde
m_q^2,\tilde m_q^2,\tilde m_q^2)\nonumber\\
&+&\frac{1}{2}\left[(\delta\tilde m_i^2)^2+(\delta\tilde
m_j^2)^2+2(\delta\tilde m_i^2)(\delta\tilde m_j^2)\right]I_6(\tilde m_g^2,\tilde
m_q^2,\tilde m_q^2,\tilde m_q^2,\tilde m_q^2)+\cdots
\end{eqnarray}
\begin{equation}
I_n(\tilde m_g^2,\tilde m_q^2,\ldots,\tilde
m_g^2)^2(p^2-\tilde m_q^2)^{n-2}},
\end{equation}
and similarly for $\tilde I_{4ij}$. Note that $I_n\propto(\tilde
m_q^2)^{n-2}$ and $\tilde I_n\propto(\tilde m_q^2)^{n-3}$. Thus, using
$x\equiv\tilde m_g^2/\tilde m_q^2$, it is customary to define
\begin{equation}
I_n\equiv\frac{i}{(4\pi)^2(\tilde m_q^2)^{n-2}}f_n(x),\ \ \ \
\tilde I_n\equiv\frac{i}{(4\pi)^2(\tilde m_q^2)^{n-3}}\tilde f_n(x).
\end{equation}
The unitarity of the mixing matrix implies that
\begin{equation}
\sum_i (K_{2i}^uK_{1i}^{u*}K_{2j}^uK_{1j}^{u*})=
\sum_j (K_{2i}^uK_{1i}^{u*}K_{2j}^uK_{1j}^{u*})=0.
\end{equation}
We learn that the terms that are proportional $f_4,\tilde f_4,f_5$, and
$\tilde f_5$ vanish in their contribution to $M_{12}$. When
$\delta\tilde m_i^2\ll\tilde m_q^2$ for all $i$, the
leading contributions to $M_{12}$ come from $f_6$ and $\tilde f_6$. We
learn that for quasi-degenerate squarks, the leading contribution is
quadratic in the small mass-squared difference. The functions $f_6(x)$
and $\tilde f_6(x)$ are given by
\begin{eqnarray}
f_6(x)&=&\frac{6(1+3x)\ln x+x^3-9x^2-9x+17}{6(1-x)^5},\nonumber\\
\tilde f_6(x)&=&\frac{6x(1+x)\ln x-x^3-9x^2+9x+1}{3(1-x)^5}.
\end{eqnarray}
For example, with $x=1$, $f_6(1)=-1/20$ and $\tilde f_6=+1/30$;
with $x=2.33$, $f_6(2.33)=-0.015$ and $\tilde f_6=+0.013$.
To further simplify things, let us consider a two-generation
case. Then
\begin{eqnarray}
M_{12}^D&\propto& 2(K_{21}^uK_{11}^{u*})^2(\delta\tilde
m_1^2+\delta\tilde m_2^2)^2\nonumber\\
&=&(K^u_{21}K_{11}^{u*})^2(\tilde m_2^2-\tilde m_1^2)^2.
\end{eqnarray}
We thus rewrite motsusy for the case of quasi-degenerate
\begin{equation}\label{motsusyd}
M_{12}^D=\frac{\alpha_s^2m_Df_D^2B_D\eta_\text{QCD}}{108\tilde m_q^2}
[11\tilde f_6(x)+4xf_6(x)]\frac{(\Delta\tilde m^2_{21})^2}{\tilde m_q^4}
\end{equation}
For example, for $x=1$, $11\tilde f_6(x)+4xf_6(x)=+0.17$.
For $x=2.33$, $11\tilde f_6(x)+4xf_6(x)=+0.003$.
§ NEUTRINO FLAVOUR TRANSITIONS
§.§ Neutrinos in vacuum
Neutrino oscillations in vacuum [70] arise since
neutrinos are massive and mix. In other words, the neutrino state that
is produced by electroweak interactions is not a mass eigenstate.
The weak eigenstates $\nu_\alpha$ ($\alpha=e,\mu,\tau$ denotes the
charged lepton mass eigenstates and their neutrino doublet-partners)
are linear combinations of the mass eigenstates $\nu_i$ ($i=1,2,3$):
\begin{equation}
|\nu_\alpha\rangle=U_{\alpha i}^*|\nu_i\rangle.
\end{equation}
After travelling a distance $L$ (or, equivalently for relativistic
neutrinos, time $t$), a neutrino originally produced with a flavour
$\alpha$ evolves as follows:
\begin{equation}
|\nu_\alpha(t)\rangle=U_{\alpha i}^*|\nu_i(t)\rangle.
\end{equation}
It can be detected in the charged-current interaction
$\nu_\alpha(t)N^\prime\to\ell_\beta N$ with a probability
\begin{equation}
\left|\sum_{i=1}^3\sum_{j=1}^3U_{\alpha i}^*U_{\beta
\end{equation}
We follow the analysis of Gonzalez-Garcia:2002dz. We use the
standard approximation that $|\nu\rangle$ is a plane wave,
$|\nu_i(t)\rangle=e^{-iE_it}|\nu_i(0)\rangle$. In all cases of
interest to us, the neutrinos are relativistic:
\begin{equation}
E_i=\sqrt{p_i^2+m_i^2}\simeq p_i+\frac{m_i^2}{2E_i},
\end{equation}
where $E_i$ and $m_i$ are, respectively, the energy and the mass of
the neutrino mass eigenstate. Furthermore, we can assume that
$p_i\simeq p_j\equiv p\simeq E$. Then, we obtain the following
transition probability:
\begin{equation}\label{palbe}
\left(U_{\alpha i}U_{\beta i}^*U_{\alpha j}^*U_{\beta
j}\right)\sin^2 x_{ij},
\end{equation}
where $x_{ij}\equiv\Delta m^2_{ij}L/(4E)$, $\Delta
m^2_{ij}=m_i^2-m_j^2$, and $L=t$ is the distance between the source
(that is, the production point of $\nu_\alpha$) and the detector (that
is, the detection point of $\nu_\beta$). In deriving palbe
we used the orthogonality relation $\langle\nu_j(0)|\nu_i(0)\rangle
=\delta_{ij}$. It is convenient to use the following units:
\begin{equation}
x_{ij}=1.27\ \frac{\Delta m^2_{ij}}{\UeVZ^2}\ \frac{L/E}{\textrm{m}/\UMeVZ}.
\end{equation}
The transition probability [palbe] has an oscillatory
behaviour, with oscillation lengths
\begin{equation}
L_{0,ij}^\text{osc}=\frac{4\pi E}{\Delta m^2_{ij}}
\end{equation}
and amplitude that is proportional to elements of the mixing
matrix. Thus, in order to have oscillations, neutrinos must have
different masses ($\Delta m^2_{ij}\neq0$) and they must mix
($U_{\alpha i}U_{\beta i}\neq 0$).
An experiment is characterized by the typical neutrino energy $E$ and
by the source-detector distance $L$. In order to be sensitive to a
given value of $\Delta m^2_{ij}$, the experiment has to be set up with
$E/L\approx\Delta m^2_{ij}$ ($L\sim L_{0,ij}^\text{osc}$). The typical
values of $L/E$ for different types of neutrino sources and
experiments are summarized in Table <ref>.
Characteristic values of $L$ and $E$ for various neutrino
sources and experiments.
$L~(\UmZ)$ $E~(\UMeVZ)$ $\Delta m^2~(\UeVZ^2)$
$10^{10}$ $1$ $10^{-10}$
$10^4$–$10^7$ $10^2$–$10^5$ $10^{-1}$–$10^{-4}$
$10^2$–$10^3$ $1$ $10^{-2}$–$10^{-3}$
$10^5$ $1$ $10^{-5}$
$10^2$ $10^3$–$10^4$ $\gtrsim10^{-1}$
Long-baseline accelerator
$10^5$–$10^6$ $10^4$ $10^{-2}$–$10^{-3}$
If $(E/L)\gg\Delta m^2_{ij}$ ($L\ll L_{0,ij}^\text{osc}$), the
oscillation does not have time to give an appreciable effect because
$\sin^2x_{ij}\ll1$. The case of $(E/L)\ll\Delta m^2_{ij}$ ($L\gg
L_{0,ij}^\text{osc}$) requires more careful consideration. One must
take into account that, in general, neutrino beams are not
monochromatic. Thus, rather than measuring $P_{\alpha\beta}$, the
experiments are sensitive to the average probability
\begin{equation}
\langle P_{\alpha\beta}\rangle=\delta_{\alpha\beta}
-4\sum_{i=1}^2\sum_{j=i+1}^3\mathcal{R}e \left(U_{\alpha i}U_{\beta
i}^*U_{\alpha j}^*U_{\beta j}\right) \langle\sin^2 x_{ij}\rangle.
\end{equation}
For $L\gg L_{0,ij}^\text{osc}$, the oscillation phase goes through many
cycles before the detection and is averaged to $\langle\sin^2
For a two-neutrino case,
\begin{equation}\label{nuvactwo}
\end{equation}
For averaged oscillations we get, for example,
\begin{equation}
\end{equation}
For a recent careful derivation of the oscillation formulae, see
§.§ Neutrinos in matter
When neutrinos propagate in dense matter, the interactions with the
medium affect their properties. These effects are either coherent or
incoherent. For purely incoherent $\nu$–$p$ scattering, the
characteristic cross-section is very small,
\begin{equation}\label{inccs}
\sigma\sim\frac{G_F^2s}{\pi}\sim10^{-43}\Ucm^2\left(\frac{E}{1\UMeV}\right)^2\SPp.
\end{equation}
The smallness of this cross-section is demonstrated by the fact that
if a beam of $10^{10}$ neutrinos with $E\sim1\UMeV$ was aimed at
Earth, only one would be deflected by the Earth's matter. It may seem
then that for neutrinos matter is irrelevant. However, one must take
into account that inccs does not contain the contribution
from forward elastic coherent interactions. In coherent interactions,
the medium remains unchanged and it is possible to have interference
of scattered and unscattered neutrino waves which enhances the
effect. Coherence further allows one to decouple the evolution
equation of neutrinos from the equations of the medium. In this
approximation, the effect of the medium is described by an effective
potential which depends on the density and composition of the matter
Consider, for example, the effective potential for $\nu_e$
induced by its charged-current interactions with electrons in matter:
\begin{equation}\label{efpoee}
V_C=\langle \nu_e|\int d^3x
\end{equation}
For $\overline{\nu_e}$ the sign of $V$ is reversed. The potential can
also be expressed in terms of the matter density $\rho$:
\begin{equation}
V_C=7.6\ \frac{N_e}{N_p+N_n}\ \frac{\rho}{10^{14}\ \text{g/cm}^3}\UeV\SPp.
\end{equation}
Two examples that are relevant to observations are the following:
* At the Earth's core $\rho\sim10\Ug/\UcmZ^3$ and
* At the solar core $\rho\sim100\Ug/\UcmZ^3$ and
Consider a state that is an admixture of two neutrino species,
$|\nu_e\rangle$ and $|\nu_a\rangle$ or, equivalently, $|\nu_1\rangle$
and $|\nu_2\rangle$. With some approximations, the time evolution
can be written in the following matrix form [72]:
\begin{equation}
-i\frac{\partial}{\partial x} \begin{pmatrix}\nu_e\\\nu_a\end{pmatrix}
=-\frac{1}{2E}M_w^2 \begin{pmatrix}\nu_e\\\nu_a\end{pmatrix}\SPp,
\end{equation}
where we have defined an effective mass matrix in matter,
\begin{equation}\label{hweaknu}
\begin{pmatrix}
m_1^2+m_2^2+4EV_e-\Delta m^2\cos2\theta &\Delta m^2\sin2\theta\\
\Delta m^2\sin2\theta& m_1^2+m_2^2+4EV_a+\Delta m^2\cos2\theta
\end{pmatrix}\SPp,
\end{equation}
with $\Delta m^2=m_2^2-m_1^2$.
We define the instantaneous mass eigenstates in matter, $\nu_i^m$, as
the eigenstates of $M_w$ for a fixed value of $x$. They are related to
the interaction eigenstates by a unitary transformation,
\begin{equation}
\begin{pmatrix}\nu_e \\ \nu_a\end{pmatrix}=U(\theta_m)
\begin{pmatrix}\nu_1^m \\ \nu_2^m\end{pmatrix}=
\begin{pmatrix}\cos\theta_m&\sin\theta_m\\ -\sin\theta_m&\cos\theta_m\end{pmatrix}
\begin{pmatrix}\nu_1^m \\ \nu_2^m\end{pmatrix}\SPp.
\end{equation}
The eigenvalues of $M_w$, that is, the effective masses in matter, are
given by [72, 73]
\begin{equation}
\mu^2_{1,2}=\frac{m_1^2+m_2^2}{2}+E(V_e+V_a)\mp\frac{1}{2}\sqrt{
(\Delta m^2\cos2\theta-A)^2+(\Delta m^2\sin2\theta)^2},
\end{equation}
while the mixing angle in matter is given by
\begin{equation}
\tan2\theta_m=\frac{\Delta m^2\sin2\theta}{\Delta m^2\cos2\theta-A},
\end{equation}
\begin{equation}\label{defa}
\end{equation}
The instantaneous mass eigenstates $\nu_i^m$ are, in general, not
energy eigenstates: they mix in the evolution. The importance of this
effect is controlled by the relative size of $4E\dot\theta_m(t)$ with
respect to $\mu_2^2(t)-\mu_1^2(t)$. When the latter is much larger
than the first, $\nu_i^m$ behave approximately as energy eigenstates
and do not mix during the evolution. This is the adiabatic transition
approximation. The adiabaticity condition reads
\begin{equation}\label{adicon}
\mu_2^2(t)-\mu_1^2(t)\gg 2EA\Delta m^2\sin2\theta\left|\dot
\end{equation}
The transition probability for the adiabatic case is given by
\begin{equation}\label{peeadi}
P_{ee}(t)=\left|\sum_i U_{ei}(\theta)U_{ei}^*(\theta_p)\exp\left(-
\frac{i}{2E}\int_{t_0}^t\mu_i^2(t^\prime)dt^\prime\right)\right|^2,
\end{equation}
where $\theta_p$ is the mixing angle at the production point. For
the case of two-neutrino mixing, peeadi takes the form
\begin{equation}\label{peeadtwo}
\end{equation}
\begin{equation}
\delta(t)=\int_{t_p}^t[\mu_2^2(t^\prime)-\mu_1^2(t^\prime)]dt^\prime.
\end{equation}
For $\mu_2^2(t)-\mu_1^2(t)\gg E$, the last term in
peeadtwo is averaged out and the survival probability
takes the form
\begin{equation}\label{peeadifin}
\end{equation}
The relative importance of the MSW matter term [$A$ of defa]
and the kinematic vacuum oscillation term in the Hamiltonian [the
off-diagonal term in hweaknu] can be parametrized by the
quantity $\beta_\text{MSW}$, which represents the ratio of matter to
vacuum effects (see, for example, Bahcall:2004mz). From
hweaknu we see that the appropriate ratio is
\begin{equation}\label{defbeta}
\beta_\text{MSW}=\frac{2\sqrt{2}G_F n_e E_\nu}{\Delta m^2}.
\end{equation}
The quantity $\beta_\text{MSW}$ is the ratio between the oscillation length in
matter and the oscillation length in vacuum. In convenient units,
$\beta_\text{MSW}$ can be written as
\begin{equation}\label{bequan}
\beta_\text{MSW}=
\left(\frac{\mu_e\rho}{100\Ug\Ucm^{-3}}\right)
\left(\frac{8\times10^{-5}\UeV^2}{\Delta m^2}\right)\SPp.
\end{equation}
Here $\mu_e$ is the electron mean molecular weight
($\mu_e\approx0.5(1+X)$, where $X$ is the mass fraction of hydrogen)
and $\rho$ is the total density. If $\beta_\text{MSW}\lesssim\cos2\theta$,
the survival probability corresponds to vacuum averaged oscillations
[see nuvactwo],
\begin{equation}
P_{ee}=\left(1-\frac{1}{2} \sin^22\theta\right)\ \ \ (\beta_\text{MSW}<\cos2\theta,\
\text{vacuum}).
\end{equation}
If $\beta_\text{MSW}>1$, the survival probability corresponds to
matter-dominated oscillations [see peeadifin],
\begin{equation}
P_{ee}=\sin^2\theta\ \ \ (\beta_\text{MSW}>1,\ \text{MSW}).
\end{equation}
The survival probability is approximately constant in either of the
two limiting regimes, $\beta_\text{MSW}<\cos2\theta$ and
$\beta_\text{MSW}>1$. There is a strong energy dependence only in the
transition region between the limiting regimes.
For the Sun, $N_e(R)=N_e(0)\exp(-R/r_0)$, with $r_0\equiv
R_\odot/10.54=6.6\times10^7\ \text{m}=3.3\times10^{14}\UeV^{-1}$.
Then, the adiabaticity condition for the Sun reads
\begin{equation}
\frac{(\Delta
\end{equation}
[1] M. Kobayashi and T. Maskawa,
Prog. Theor. Phys. 49, 652 (1973).
[2] N. Cabibbo,
Phys. Rev. Lett. 10, 531 (1963).
[3]
A. B. Carter and A. I. Sanda,
Phys. Rev. Lett. 45, 952 (1980);
Phys. Rev. D 23, 1567 (1981).
[5]
I. I. Y. Bigi and A. I. Sanda,
Nucl. Phys. B 193, 85 (1981).
[6]
G. Buchalla, A. J. Buras, and M. E. Lautenbacher,
Rev. Mod. Phys. 68, 1125 (1996)
[7]
Y. Grossman, A. L. Kagan, and Z. Ligeti,
Phys. Lett. B 538, 327 (2002)
[8]
H. Boos, T. Mannel, and J. Reuter,
Phys. Rev. D 70, 036006 (2004)
[9]
H. n. Li and S. Mishima,
JHEP 0703, 009 (2007)
[10]
M. Gronau and J. L. Rosner,
Phys. Lett. B 672, 349 (2009)
[arXiv:0812.4796 [hep-ph]].
[11]
E. Barberio [Heavy Flavor Averaging Group],
arXiv:0808.1297 [hep-ex],
online update at http://www.slac.stanford.edu/xorg/hfag
[12]
C. Amsler et al. [Particle Data Group],
Phys. Lett. B 667, 1 (2008).
[14]
CKMfitter Group (J. Charles ),
Eur. Phys. J. C 41, 1–131 (2005), [hep-ph/0406184],
updated results and plots available at: http://ckmfitter.in2p3.fr
[15]
Y. Nir,
Nucl. Phys. Proc. Suppl. 117, 111 (2003)
[16]
Y. Grossman, Y. Nir, and M. P. Worah,
Phys. Lett. B 407, 307 (1997) [hep-ph/9704287].
[17]
Y. Grossman, Y. Nir, and G. Raz,
Phys. Rev. Lett. 97, 151801 (2006)
[18]
M. Bona et al. [UTfit Collaboration],
JHEP 0803, 049 (2008)
[arXiv:0707.0636 [hep-ph]].
[19]
G. C. Branco, L. Lavoura, and J. P. Silva,
CP Violation
(Clarendon Press, Oxford, 1999).
[20]
I. I. Y. Bigi and N. G. Uraltsev,
Nucl. Phys. B 592, 92 (2001)
[21]
A. F. Falk, Y. Grossman, Z. Ligeti, and A. A. Petrov,
Phys. Rev. D 65, 054034 (2002)
[22]
A. F. Falk, Y. Grossman, Z. Ligeti, Y. Nir, and A. A. Petrov,
Phys. Rev. D 69, 114021 (2004)
[23]
B. Aubert et al. [BaBar Collaboration],
Phys. Rev. Lett. 98, 211802 (2007)
[24]
M. Staric et al. [Belle Collaboration],
Phys. Rev. Lett. 98, 211803 (2007)
[26]
G. Raz,
Phys. Rev. D 66, 037701 (2002)
[27]
N. Arkani-Hamed and S. Dimopoulos,
JHEP 0506, 073 (2005)
[28]
A. G. Cohen, D. B. Kaplan, and A. E. Nelson,
Phys. Lett. B 388, 588 (1996)
[29]
Y. Nir and G. Raz,
Phys. Rev. D 66, 035007 (2002)
[30]
K. Blum, Y. Grossman, Y. Nir and G. Perez,
Phys. Rev. Lett. 102, 211802 (2009)
[arXiv:0903.2118 [hep-ph]].
[31]
Y. Nir and N. Seiberg,
Phys. Lett. B 309, 337 (1993)
[32]
M. Leurer, Y. Nir, and N. Seiberg,
Nucl. Phys. B 420, 468 (1994)
[33]
M. Ciuchini, E. Franco, D. Guadagnoli, V. Lubicz, M. Pierini, V. Porretti, and L. Silvestrini,
Phys. Lett. B 655, 162 (2007)
[34]
Y. Nir,
JHEP 0705, 102 (2007)
[35]
G. D'Ambrosio, G. F. Giudice, G. Isidori, and A. Strumia,
Nucl. Phys. B 645, 155 (2002)
[36]
Y. Grossman, Y. Nir, J. Thaler, T. Volansky, and J. Zupan,
Phys. Rev. D 76, 096006 (2007)
[arXiv:0706.1845 [hep-ph]].
[37]
J. L. Feng, C. G. Lester, Y. Nir, and Y. Shadmi,
Phys. Rev. D 77, 076002 (2008)
[arXiv:0712.0674 [hep-ph]].
[38]
G. Engelhard, J. L. Feng, I. Galon, D. Sanford and F. Yu,
arXiv:0904.1415 [hep-ph].
[39]
J. L. Feng, I. Galon, D. Sanford, Y. Shadmi and F. Yu,
Phys. Rev. D 79, 116009 (2009)
[arXiv:0904.1416 [hep-ph]].
[40]
J. L. Feng, S. T. French, C. G. Lester, Y. Nir and Y. Shadmi,
arXiv:0906.4215 [hep-ph].
[41]
G. Hiller and Y. Nir,
JHEP 0803, 046 (2008)
[arXiv:0802.0916 [hep-ph]].
[42]
G. Hiller, Y. Hochberg, and Y. Nir,
arXiv:0812.0511 [hep-ph].
[43]
Y. Nomura, M. Papucci, and D. Stolarski,
Phys. Rev. D 77, 075006 (2008)
[arXiv:0712.2074 [hep-ph]];
JHEP 0807, 055 (2008)
[arXiv:0802.2582 [hep-ph]].
[45]
G. Hiller, Y. Hochberg, and Y. Nir, work in progress.
[46]
G. F. Giudice, M. Nardecchia, and A. Romanino,
arXiv:0812.3610 [hep-ph].
[47]
A. E. Nelson and M. J. Strassler,
JHEP 0009, 030 (2000)
JHEP 0207, 021 (2002)
[49]
M. C. Gonzalez-Garcia and Y. Nir,
Rev. Mod. Phys. 75, 345 (2003)
[50]
M. C. Gonzalez-Garcia and M. Maltoni,
Phys. Rep. 460, 1 (2008)
[arXiv:0704.1800 [hep-ph]].
[51]
M. Fukugita and T. Yanagida,
Phys. Lett. B 174, 45 (1986).
[52]
S. Davidson, E. Nardi, and Y. Nir,
Phys. Rep. 466, 105 (2008)
[arXiv:0802.2962 [hep-ph]].
[53]
S. Davidson and A. Ibarra,
Phys. Lett. B 535, 25 (2002)
[54]
P. F. Harrison, D. H. Perkins, and W. G. Scott,
Phys. Lett. B 530, 167 (2002)
[55]
C. D. Froggatt and H. B. Nielsen,
Nucl. Phys. B 147, 277 (1979).
[56]
M. Leurer, Y. Nir, and N. Seiberg,
Nucl. Phys. B 398, 319 (1993)
[57]
Y. Grossman and Y. Nir,
Nucl. Phys. B 448, 30 (1995)
[58]
L. J. Hall, H. Murayama, and N. Weiner,
Phys. Rev. Lett. 84, 2572 (2000)
[59]
Y. E. Antebi, Y. Nir, and T. Volansky,
Phys. Rev. D 73, 075009 (2006)
[60]
L. Chau and W. Keung,
Phys. Rev. Lett. 53, 1802 (1984).
[61]
L. Wolfenstein,
Phys. Rev. Lett. 51, 1945 (1983).
[62]
A. J. Buras, M. E. Lautenbacher, and G. Ostermaier,
Phys. Rev. D 50, 3433 (1994)
[63]
C. Dib, I. Dunietz, F. J. Gilman, and Y. Nir,
Phys. Rev. D 41, 1522 (1990).
[64]
J. L. Rosner, A. I. Sanda, and M. P. Schmidt,
[Presented at Workshop on High Sensitivity Beauty Physics, Batavia, IL, Nov 11–14, 1987].
[65]
V. Weisskopf and E. P. Wigner,
Z. Phys. 63, 54 (1930);
Z. Phys. 65, 18 (1930).
[See Appendix A of P. K. Kabir, The CP Puzzle: Strange Decays of the
Neutral Kaon (Academic Press, London, 1968).]
[66]
Y. Nir,
[Lectures given at 20th Summer Institute on Particle Physics:
The Third Family and the Physics of Flavor, Stanford, CA, 1992,
ed. L. Vassilian (SLAC, Stanford, 1993)].
[67]
I. Dunietz and J. L. Rosner,
Phys. Rev. D 34, 1404 (1986).
[68]
Ya. I. Azimov, N. G. Uraltsev, and V. A. Khoze,
Sov. J. Nucl. Phys. 45, 878 (1987)
[Yad. Fiz. 45, 1412 (1987)].
[69]
I. I. Bigi and A. I. Sanda,
Nucl. Phys. B 281, 41 (1987).
[70]
B. Pontecorvo,
Sov. Phys. JETP 6, 429 (1957)
[Zh. Eksp. Teor. Fiz. 33, 549 (1957)].
[71]
A. G. Cohen, S. L. Glashow, and Z. Ligeti,
arXiv:0810.4602 [hep-ph].
[72]
L. Wolfenstein,
Phys. Rev. D 17, 2369 (1978).
[73]
S.P. Mikheyev and A. Yu. Smirnov,
Sov. J. Nucl. Phys. 42, 913 (1985)
[Yad. Fiz. 42, 1441 (1985)].
[74]
J. N. Bahcall and C. Pena-Garay,
New J. Phys. 6, 63 (2004)
|
arxiv-papers
| 2010-10-13T14:16:42 |
2024-09-04T02:49:13.828180
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Y. Nir (Weizmann Institute of Science)",
"submitter": "Scientific Information Service Cern",
"url": "https://arxiv.org/abs/1010.2666"
}
|
1010.2695
|
# Inverse problem for a structural acoustic interaction
Shitao Liu
Department of Mathematics
University of Virginia
Charlottesville, VA 22904, USA
Email: sl3fa@virginia.edu
###### Abstract.
In this work, we consider an inverse problem of determining a source term for
a structural acoustic partial differentia equation (PDE) model, comprised of a
two or three-dimensional interior acoustic wave equation coupled to a Kirchoff
plate equation, with the coupling being accomplished across a boundary
interface. For this PDE system, we obtain the uniqueness and stability
estimate for the source term from a single measurement of boundary values of
the “structure”. The proof of uniqueness is based on Carleman estimate. Then,
by means of an observability inequality and a compactness/uniqueness argument,
we can get the stability result. Finally, an operator theoretic approach gives
us the regularity needed for the initial conditions in order to get the
desired stability estimate.
Keywords: Structural acoustic interaction, inverse problem, Carleman estimate,
continuous observability inequality
## 1\. Introduction and Main Results
### 1.1. Statement of the Problem
Let $\Omega$ be an open bounded subset of $\mathbb{R}^{2}$ or $\mathbb{R}^{3}$
with smooth boundary $\Gamma$ of class $C^{2}$, and we designate a nonempty
simply connected segment of $\Gamma$ as $\Gamma_{0}$ with then
$\Gamma=\Gamma_{0}\cup\Gamma_{1}$ and $\Gamma_{0}\cap\Gamma_{1}=\emptyset$. We
consider here the following system comprised of a “coupling” between a wave
equation and an elastic plate-like equation:
(1.1) $\begin{cases}z_{tt}(x,t)=\Delta z(x,t)+q(x)z(x,t)&\mbox{in
}\Omega\times[0,T]\\\ \frac{\partial z}{\partial\nu}(x,t)=0&\mbox{on
}\Gamma_{1}\times[0,T]\\\
z_{t}(x,t)=-v_{tt}(x,t)-\Delta^{2}v(x,t)-\Delta^{2}v_{t}(x,t)&\mbox{on
}\Gamma_{0}\times[0,T]\\\ v(x,t)=\frac{\partial
v}{\partial\nu}(x,t)=0&\mbox{on }\partial\Gamma_{0}\times[0,T]\\\
\frac{\partial z}{\partial\nu}(x,t)=v_{t}(x,t)&\mbox{on
}\Gamma_{0}\times[0,T]\\\ z(\cdot,\frac{T}{2})=z_{0}(x)&\mbox{in }\Omega\\\
z_{t}(\cdot,\frac{T}{2})=z_{1}(x)&\mbox{in }\Omega\\\
v(\cdot,\frac{T}{2})=v_{0}(x)&\mbox{on }\Gamma_{0}\\\
v_{t}(\cdot,\frac{T}{2})=v_{1}(x)&\mbox{on }\Gamma_{0}\end{cases}$
where the coupling occurs across the boundary interface $\Gamma_{0}$.
$[z_{0},z_{1},v_{0},v_{1}]$ are the given initial conditions and $q(x)$ is a
time-independent unknown coefficient. For this system, notice that the map
$\\{q\\}\to\\{z(q),v(q)\\}$ is nonlinear, therefore we consider the following
nonlinear inverse problem: Let $\\{z=z(q),v=v(q)\\}$ be the weak solution to
system $\eqref{nonlinear}$. Under suitable geometrical conditions on
$\Gamma_{1}=\Gamma\setminus\Gamma_{0}$, is it possible to retrieve $q(x)$,
$x\in\Omega$, from measurement of $v_{tt}(q)$ on $\Gamma_{0}\times[0,T]$? In
other words, is it possible to recover the internal wave potential from the
observation of the acceleration of the elastic plate.
Our emphasis here that we determine the interior acoustic property from
observing the acceleration of the elastic wall (portion of the boundary), is
not only due to physical consideration, but also to the implications of such
inverse type analysis related to the coupling nature of the structural
acoustic flow. In many structural acoustics applications, the problem of
controlling interior acoustic properties is directly correlated with the
problem of controlling structural vibrations since the interior noise fields
are often generated by the vibrations of an enclosing structure. An important
example of this is the problem of controlling interior aircraft cabin noise
which is caused by fuselage vibrations that are induced by the low frequency
high magnitude exterior noise fields generated by the engines.
The primary goal in this paper is to study the uniqueness and stability of the
interior time-independent unknown coefficient $q(x)$ in some appropriate
function space. More precisely, we consider the follow uniqueness and
stability problems:
Uniqueness in the nonlinear inverse problem
Let $\\{z=z(q),v=v(q)\\}$ be the weak solution to system $\eqref{nonlinear}$.
Under geometrical conditions on $\Gamma_{1}$, does the acceleration of the
wall $v_{tt}|_{\Gamma_{0}\times[0,T]}$ determine $q(x)$ uniquely? In other
words, does
$v_{tt}(q)|_{\Gamma_{0}\times[0,T]}=v_{tt}(p)|_{\Gamma_{0}\times[0,T]}$
imply $q(x)=p(x)$ in $\Omega$?
Stability in the nonlinear inverse problem
Let $\\{z(q),v(q)\\}$, $\\{z(p),v(p)\\}$ be weak solutions to system
$\eqref{nonlinear}$ with corresponding coefficients $q(x)$ and $p(x)$. Under
geometric conditions on $\Gamma_{1}$, is it possible to estimate
$\displaystyle\|q-p\|_{L^{2}(\Omega)}$ by some suitable norms of
$\displaystyle(v_{tt}(q)-v_{tt}(p))|_{\Gamma_{0}\times[0,T]}$?
In order to study the nonlinear inverse problem, we first linearize
$\eqref{nonlinear}$ and hence we consider the following system:
(1.2) $\begin{cases}w_{tt}(x,t)-\Delta w(x,t)-q(x)w(x,t)=f(x)R(x,t)&\mbox{in
}\Omega\times[0,T]\\\ \frac{\partial w}{\partial\nu}(x,t)=0&\mbox{on
}\Gamma_{1}\times[0,T]\\\
w_{t}(x,t)=-u_{tt}(x,t)-\Delta^{2}u(x,t)-\Delta^{2}u_{t}(x,t)&\mbox{on
}\Gamma_{0}\times[0,T]\\\ u(x,t)=\frac{\partial
u}{\partial\nu}(x,t)=0&\mbox{on }\partial\Gamma_{0}\times[0,T]\\\
\frac{\partial w}{\partial\nu}(x,t)=u_{t}(x,t)&\mbox{on
}\Gamma_{0}\times[0,T]\\\ w(\cdot,\frac{T}{2})=0&\mbox{in }\Omega\\\
w_{t}(\cdot,\frac{T}{2})=0&\mbox{in }\Omega\\\ u(\cdot,\frac{T}{2})=0&\mbox{on
}\Gamma_{0}\\\ u_{t}(\cdot,\frac{T}{2})=0&\mbox{on }\Gamma_{0}\end{cases}$
where $q\in L^{\infty}(\Omega)$ is given, $R(x,t)$ is fixed suitably while
$f(x)$ is an unknown time-independent coefficient. For this linearized system,
we have the advantage that the map $\\{f\\}\to\\{w(f),u(f)\\}$ is linear,
hence we consider the corresponding linear inverse problem:
Uniqueness in the linear inverse problem
Let $\\{w=w(f),u=u(f)\\}$ be the weak solution to system $\eqref{linear}$.
Under geometrical conditions on $\Gamma_{1}$, does
$u_{tt}|_{\Gamma_{0}\times[0,T]}$ determine $f(x)$ uniquely? In other words,
does
$u_{tt}(f)|_{\Gamma_{0}\times[0,T]}=0$
imply $f(x)=0$ in $\Omega$?
Stability in the linear inverse problem
Let $\\{w=w(f),u=u(f)\\}$ be the weak solution to system $\eqref{linear}$.
Under geometrical conditions on $\Gamma_{1}$, is it possible to estimate
$\displaystyle\|f\|_{L^{2}(\Omega)}$ by some suitable norms of $\displaystyle
u_{tt}|_{\Gamma_{0}\times[0,T]}$?
###### Remark 1.1.
In our models $\eqref{nonlinear}$ and $\eqref{linear}$ we regard
$t=\frac{T}{2}$ as the initial time. This is not essential, because the change
of independent variables $t\to t-\frac{T}{2}$ transforms $t=\frac{T}{2}$ to
$t=0$. However, this is convenient for us to apply the Carleman estimate
established in [29]. In fact, one can keep $t=0$ as initial moment by doing an
even extension of $w$ and $u$ to $\Omega\times[-T,T]$, but then the Carleman
estimate in [29] needs to be modified accordingly.
### 1.2. Literature and Motivation
The PDE system $\eqref{nonlinear}$ is an example of a _structural acoustic
interaction_. It mathematically describes the interaction of a vibrating
beam/plate in an enclosed acoustic field or chamber. In this situation, the
boundary segment $\Gamma_{1}$ represents the “hard” walls of the chamber
$\Omega$, with $\Gamma_{0}$ being the flexible portion of the chamber wall.
The flow with in the chamber is assumed to be of acoustic wave type, and hence
the presence of the wave equation in $\Omega$, satisfied by $z$ in
$\eqref{nonlinear}$, coupled to a structural plate equation (in variable $v$)
on the flexible boundary portion $\Gamma_{0}$. This type of PDE models has
long existed in the literature and has been an object of intensive
experimental and numerical studies at the Nasa Langley Research Center [31, 9,
10]. Moreover, recent innovations in smart material technology and the
potential applications of these innovations in control engineering design have
greatly increased the interest in studying these structural acoustic models.
As a result, there has been a lot of recent contributions to the literature
deal with various topis; e.g., optimal control, stability, controllability,
regularity [1, 2, 3, 4, 5, 6, 7, 8, 14, 23]. However, to the best of our
knowledge, there are no results available in the literature for our inverse
type analysis on the model.
On the other hand, the interest to the inverse problem has been stimulated by
the studies of applied problems such as geophysics, medical imaging,
scattering, nondestructive testing and so on. These problems are of the
determination of unknown coefficients of differential equations which are the
functions depending on the point of the space [11, 15, 16]. For the uniqueness
in multidimensional inverse problem with a single boundary observation, the
pioneering paper by Bukhgeim and Klibanov [12] provides a methodology based on
a type of exponential weighted energy estimate, which is usually referred as
the Carleman estimate since the original work [13] by Carleman. After [12],
several papers concerning inverse problems by using Carleman estimate have
been published (e.g. [17, 21]). In particular, for the inverse hyperbolic type
problems that is related to our concern in this paper, there has been
intensively studies [18, 19, 20, 32, 37]. However, we mentioned again that
there is not any such uniqueness and stability analysis for the structural
acoustic models or even in general coupled PDE systems. This motivates the
work of the present paper.
The usual problem setting for inverse hyperbolic problem includes determining
a coefficient from measurements on the whole boundary or part of the boundary,
either Dirichlet type [12, 20, 32, 37] or Neumann type [18, 19]. Usually the
coefficient describes a physical property of the medium (e.g. the elastic
modulus in Hooke’s law), and the inverse problem is to determine such a
property. In our formulation of the inverse problem, we need to determine the
time-independent wave potential $q(x)$ by observing the acceleration from the
flexible portion of the boundary $\Gamma_{0}$. The mathematical challenge in
this problem stems from the fact that we are dealing with the “coupling” on
the part of the boundary and the main technical difficulty associated with
this structure is the lack of the compactness of the resolvent. As a result,
the space regularity for the solution of the wave equation component is
limited by the structure on the plate and hence this will prevent us going to
higher dimension ($n>7$) no matter how smooth the initial data is. This is a
distinguished feature of this structural acoustic model comparing to the
purely wave equation model as in that case the solution can be as smooth as we
want as long as the initial data is smooth enough. In this present paper, we
prove the cases where the dimension $n=2$ and $3$ (physical meaningful cases)
by using the Carleman estimate for the Neumann problem in [29] and an operator
theoretic formulation. We show that indeed the observation of the acceleration
on the plate can determine the potential $q$ under some restrictions on the
initial data and some geometrical conditions on the boundary. As we mentioned,
the argument will also work for dimension up to $n=7$.
### 1.3. Main Assumptions and Preliminaries
In this section we state the main geometrical assumptions throughout this
paper. These assumptions are essential in order to establish the Carleman
estimate stated in section 2.
Let $\nu=[\nu_{1},\cdots,\nu_{n}]$ be the unit outward normal vector on
$\Gamma$, and let $\frac{\partial}{\partial\nu}=\nabla\cdot\nu$ denote the
corresponding normal derivative Moreover, we assume the following geometric
conditions on $\Gamma_{1}=\Gamma\setminus\Gamma_{0}$:
(A.1) There exists a strictly convex (real-valued) non-negative function
$\displaystyle d:\overline{\Omega}\to\mathbb{R}^{+}$, of class
$C^{3}(\overline{\Omega})$, such that, if we introduce the (conservative)
vector field $h(x)=[h_{1}(x),\cdots,h_{n}(x)]\equiv\nabla d(x),x\in\Omega$,
then the following two properties hold true:
(i)
(1.3) $\frac{\partial d}{\partial\nu}\bigg{|}_{\Gamma_{1}}=\nabla
d\cdot\nu=h\cdot\nu=0;\quad h\equiv\nabla d$
(ii) the (symmetric) Hessian matrix $\mathcal{H}_{d}$ of $d(x)$ [i.e., the
Jacobian matrix $J_{h}$ of $h(x)$] is strictly positive definite on
$\overline{\Omega}$: there exists a constant $\rho>0$ such that for all
$x\in\overline{\Omega}$:
(1.4)
$\mathcal{H}_{d}(x)=J_{h}(x)=\left[\begin{array}[]{ccc}d_{x_{1}x_{1}}&\cdots&d_{x_{1}x_{n}}\\\
\vdots&&\vdots\\\ d_{x_{n}x_{1}}&\cdots&d_{x_{n}x_{n}}\\\
\end{array}\right]=\left[\begin{array}[]{ccc}\frac{\partial h_{1}}{\partial
x_{1}}&\cdots&\frac{\partial h_{1}}{\partial x_{n}}\\\ \vdots&&\vdots\\\
\frac{\partial h_{n}}{x_{1}}&\cdots&\frac{\partial h_{n}}{\partial x_{n}}\\\
\end{array}\right]\geq\rho I$
(A.2) $d(x)$ has no critical point on $\overline{\Omega}$:
(1.5) $\inf_{x\in\Omega}|h(x)|=\inf_{x\in\Omega}|\nabla d(x)|=s>0$
###### Remark 1.2.
One canonical example is that $\Gamma_{1}$ is flat (not the case in our
problem setting here), where then we can take $d(x)=|x-x_{0}|^{2}$, with
$x_{0}$ on the hyperplane containing $\Gamma_{1}$ and outside $\Omega$, then
$h(x)=\nabla d(x)=2(x-x_{0})$ is radial. However, in general $h(x)$ is not
necessary radial. In particularly in our case where $\Gamma_{1}$ is convex,
the corresponding required $d(x)$ can also be explicitly constructed. For more
examples of such function $d(x)$ with different geometries of $\Gamma_{1}$, we
refer to the appendix of [29].
Next we introduce an abstract operator theoretic formulation associated to
$\eqref{nonlinear}$ for which we will need the following facts and
definitions: Let the operator $A$ be
(1.6) $Az=-\Delta z-q(x)z,\quad D(A)=\\{z:\Delta z+q(x)z\in
L^{2}(\Omega),\frac{\partial z}{\partial\nu}\bigg{|}_{\Gamma}=0\\}$
Notice the lower-order part is a perturbation which preserves generation of
the self-adjoint principle part $A_{N}$ (e.g. [27]), where
$A_{N}:L^{2}(\Omega)\supset D(A_{N})\rightarrow L^{2}(\Omega)$ is defined by:
(1.7) $A_{N}z=-\Delta z,\quad D(A_{N})=\\{z:\Delta z\in
L^{2}(\Omega),\frac{\partial z}{\partial\nu}\bigg{|}_{\Gamma}=0\\}$
Then $A_{N}$ is positive self-adjoint and
(1.8) $D(A_{N}^{\frac{1}{2}})=H^{1}_{\Gamma_{1}}(\Omega)=\\{z:z\in
H^{1}(\Omega),\frac{\partial z}{\partial\nu}=0\ \text{on}\ \Gamma_{1}\\}$
Then we define the Neumann map $N$ by:
(1.9) $z=Ng\;\Leftrightarrow\;\begin{cases}\Delta z=0&\text{in}\;\;\;\Omega\\\
\frac{\partial z}{\partial\nu}=0&\text{on}\;\;\;\Gamma_{1}\\\ \frac{\partial
z}{\partial\nu}=g&\text{on}\;\;\;\Gamma_{0}\end{cases}$
By elliptic theory
(1.10) $N\in\mathcal{L}(L^{2}(\Gamma_{0}),H^{3/2}_{\Gamma_{1}}(\Omega))$
Now we define
(1.11) $\mathcal{B}=A_{N}N:L^{2}(\Gamma_{0})\rightarrow D(A_{N}^{1\over
2})^{\prime}$
via the conjugation $\mathcal{B}^{*}=N^{*}A_{N}$. Then with $v\in
L^{2}(\Gamma)$ and for any $y\in D(A_{N}^{1\over 2})$ we have
(1.12)
$-(\mathcal{B}^{*}y,v)_{\Gamma}=-(N^{*}A_{N}y,v)_{\Gamma}=-(A_{N}y,Nv)_{\Omega}=(\Delta
y,Nv)_{\Omega}\\\ =(y,\Delta(Nv))_{\Omega}+(\frac{\partial
y}{\partial\nu},Nv)_{\Gamma}-(y,\frac{\partial(Nv)}{\partial\nu})_{\Gamma}=-(y,v)_{\Gamma_{0}}$
by Green’s theorem, the definition of $N$ and the fact
$\displaystyle\frac{\partial y}{\partial\nu}=0$ on $\Gamma_{1}$ when $y\in
D(A_{N}^{1\over 2})$. In other words, we have
(1.13) $N^{*}A_{N}y=\begin{cases}y,&\text{on}\;\;\;\Gamma_{0}\\\
0,&\text{on}\;\;\;\Gamma_{1}\end{cases}\;\;\;\;\;\text{for}\;y\in
D(A_{N}^{1\over 2})$
i.e. $\mathcal{B}^{*}=N^{*}A_{N}$ is the restriction of the trace map from
$H^{1}(\Omega)$ to $H^{\frac{1}{2}}(\Gamma_{0})$.
Last we set $\textbf{\AA}:L^{2}(\Gamma_{0})\supset D(\textbf{\AA})\rightarrow
L^{2}(\Gamma_{0})$ to be
(1.14) $\textbf{\AA}=\Delta^{2},D(\textbf{\AA})=\\{v\in
H^{2}_{0}(\Gamma_{0}):\Delta^{2}v\in L^{2}(\Gamma_{0})\\}$
where $H^{2}_{0}(\Gamma_{0})=\\{v\in H^{2}(\Omega):v=\frac{\partial
v}{\partial\nu}=0\ \text{on}\ \partial\Gamma_{0}\\}$. Å is self-adjoint,
positive definite, and we have the characterization
(1.15) $D(\textbf{\AA}^{\frac{1}{2}})=H^{2}_{0}(\Gamma_{0})$
Now set
(1.16) $\mathcal{A}=\left[\begin{array}[]{cccc}0&I&0&0\\\
-A_{N}+q&0&0&\mathcal{B}\\\ 0&0&0&I\\\
0&-\mathcal{B}^{*}&-\textbf{\AA}&-\textbf{\AA}\end{array}\right]$
on the energy space
(1.17) $\begin{split}H&=D(A_{N}^{\frac{1}{2}})\times L^{2}(\Omega)\times
D(\textbf{\AA}^{\frac{1}{2}})\times L^{2}(\Gamma_{0})\\\
&=H^{1}_{\Gamma_{1}}(\Omega)\times L^{2}(\Omega)\times
H^{2}_{0}(\Gamma_{0})\times L^{2}(\Gamma_{0})\end{split}$
Then we have the domain of the operator $\mathcal{A}$
(1.18)
$\begin{split}D(\mathcal{A})&=\\{[z_{0},z_{1},v_{0},v_{1}]^{T}\in[D(A_{N}^{\frac{1}{2}})]^{2}\times[D(\textbf{\AA}^{\frac{1}{2}})]^{2}\
\text{such that}\\\ &\qquad-z_{0}+Nv_{1}\in D(A_{N})\ \text{and}\
v_{0}+v_{1}\in D(\textbf{\AA})\\}\\\
&=\\{[z_{0},z_{1},v_{0},v_{1}]^{T}:z_{0}\in
H^{1}_{\Gamma_{1}}(\Omega),z_{1}\in H^{1}_{\Gamma_{1}}(\Omega),v_{0}\in
H^{2}_{0}(\Gamma_{0}),v_{1}\in H^{2}_{0}(\Gamma_{0}),\\\
&\qquad(\Delta+q)z_{0}\in L^{2}(\Omega),\frac{\partial
z_{0}}{\partial\nu}=v_{1}\ \text{on}\ \Gamma_{0}\ \text{and}\ v_{0}+v_{1}\in
D(\textbf{\AA})\\}\\\ &=\\{[z_{0},z_{1},v_{0},v_{1}]^{T}:z_{0}\in
H^{2}_{\Gamma_{1}}(\Omega),z_{1}\in H^{1}_{\Gamma_{1}}(\Omega),v_{0}\in
H^{2}_{0}(\Gamma_{0}),v_{1}\in H^{2}_{0}(\Gamma_{0}),\\\ &\qquad\frac{\partial
z_{0}}{\partial\nu}=v_{1}\ \text{on}\ \Gamma_{0}\ \text{and}\ v_{0}+v_{1}\in
D(\textbf{\AA})\\}\end{split}$
where in the last step we get $z_{0}\in H^{2}(\Omega)$ from $q\in
L^{\infty}(\Omega)$ and $(\Delta+q)z_{0}\in L^{2}(\Omega)$ due to elliptic
theory. Therefore with these notations, the original system
$\eqref{nonlinear}$ becomes to the first order abstract differential equation
(1.19) $\frac{dy}{dt}=\mathcal{A}y$
where $y=[z,z_{t},v,v_{t}]^{T}$. From semigroup theory, when the initial
conditions $[z_{0},z_{1},v_{0},v_{1}]$ are in $D(\mathcal{A})$ we have that
the solution $y$ satisfies
(1.20) $y\in D(\mathcal{A}),\quad y_{t}\in H$
###### Remark 1.3.
The structure of $\mathcal{A}$ reflects the coupled nature of this structural
acoustic system $\eqref{nonlinear}$. One distinguished feature of the system
is that the resolvent of $\mathcal{A}$ is not compact. However, it can still
be shown that $\mathcal{A}$ generates a $C_{0}$-semigroup of contractions
$\\{e^{\mathcal{A}t}\\}_{t\geq 0}$ which establishes the well-posedness of the
system [4].
### 1.4. Main results
For the inverse problems stated in section 1.1, we have the following results:
###### Theorem 1.4.
(Uniqueness for the linear inverse problem) Under the main assumptions (A.1),
(A.2) and let
(1.21) $T>2\sqrt{\max_{x\in\overline{\Omega}}d(x)}$
Moreover, let
(1.22) $R\in W^{3,\infty}(Q)$
and
(1.23) $\bigg{|}R\left(x,\frac{T}{2}\right)\bigg{|}\geq
r_{0}>0,\qquad\bigg{|}R_{t}\left(x,\frac{T}{2}\right)\bigg{|}\geq r_{1}>0$
for some positive constants $r_{0}$, $r_{1}$ and $x\in\overline{\Omega}$. In
addition, let
(1.24) $q\in L^{\infty}(\Omega)$
If the weak solution $\\{w=w(f),u=u(f)\\}$ to system $\eqref{linear}$
satisfies
(1.25) $w,w_{t},w_{tt}\in H^{2}(Q)=H^{2}(0,T^{\prime}L^{2}(\Omega))\cap
L^{2}(0,T;H^{2}(\Omega))$
and
(1.26) $u_{tt}(f)(x,t)=0,\quad x\in\Gamma_{0},t\in[0,T]$
then $f(x)=0$, $x\in\Omega$.
###### Theorem 1.5.
(Uniqueness for the nonlinear inverse problem) Under the main assumptions
(A.1), (A.2), assume $\eqref{time}$ and
(1.27) $q,p\in L^{\infty}(\Omega)$
Let either of $z(q)$ and $z(p)$ satisfy
(1.28) $z\in W^{3,\infty}(Q)$
Moreover, let
(1.29) $|z_{0}(x)|\geq s_{0}>0,\qquad|z_{1}(x)|\geq s_{1}>0$
for some positive constants $s_{0}$, $s_{1}$ and $x\in\overline{\Omega}$. If
the weak solutions $\\{z(q),v(q)\\}$ and $\\{z(p),v(p)\\}$ to system
$\eqref{nonlinear}$ satisfy
(1.30) $z(q)-z(p),z_{t}(q)-z_{t}(p),z_{tt}(q)-z_{tt}(p)\in H^{2}(Q)$
and
(1.31) $v_{tt}(q)(x,t)=v_{tt}(p)(x,t),\quad x\in\Gamma_{0},t\in[0,T]$
then $q(x)=p(x)$, $x\in\Omega$.
###### Theorem 1.6.
(Stability for the linear inverse problem) Under the main assumptions (A.1),
(A.2), assume $\eqref{time}$, $\eqref{regR}$, $\eqref{crucialR}$ and
$\eqref{regq}$. Moreover, let
(1.32) $R_{t}\in H^{\frac{1}{2}+\epsilon}(0,T;L^{\infty}(\Omega))$
for some $0<\epsilon<\frac{1}{2}$. Then there exists a constant
$C=C(\Omega,T,\Gamma_{0},\varphi,q,R)>0$ such that
(1.33) $\|f\|_{L^{2}(\Omega)}\leq
C\left(\|u_{tt}\|_{L^{2}(\Gamma_{0}\times[0,T])}+\|u_{ttt}\|_{L^{2}(\Gamma_{0}\times[0,T])}+\|\Delta^{2}u_{tt}\|_{L^{2}(\Gamma_{0}\times[0,T])}\right)$
for all $f\in L^{2}(\Omega)$.
###### Theorem 1.7.
(Stability for the nonlinear inverse problem) Under the main assumptions
(A.1), (A.2), assume $\eqref{time}$, $\eqref{regqp}$, $\eqref{regw}$ and
$\eqref{crucialz}$. Moreover, let the initial data satisfy the compatibility
condition
1. (1)
When $n=2$, $[z_{0},z_{1},v_{0},v_{1}]\in D(\mathcal{A}^{2})$ where
$\begin{split}D(\mathcal{A}^{2})&=\\{[z_{0},z_{1},v_{0},v_{1}]^{T}:z_{0}\in
H^{3}_{\Gamma_{1}}(\Omega),z_{1}\in H^{2}_{\Gamma_{1}}(\Omega),v_{0}\in
H^{2}_{0}(\Gamma_{0}),v_{1}\in H^{2}_{0}(\Gamma_{0}),\\\
&\qquad\textbf{\AA}(v_{0}+v_{1})+B^{*}z_{1}\in
H^{2}_{0}(\Gamma_{0}),v_{1}-\textbf{\AA}(v_{0}+v_{1})-B^{*}z_{1}\in
D(\textbf{\AA}),\\\ &\qquad\frac{\partial
z_{0}}{\partial\nu}|_{\Gamma_{0}}=v_{1},\frac{\partial
z_{1}}{\partial\nu}|_{\Gamma_{0}}=-\textbf{\AA}(v_{0}+v_{1})-B^{*}z_{1}\\}\end{split}$
2. (2)
When $n=3$, $[z_{0},z_{1},v_{0},v_{1}]\in D(\mathcal{A}^{3})$ where
$\begin{split}D(\mathcal{A}^{3})&=\\{[z_{0},z_{1},v_{0},v_{1}]^{T}:z_{0}\in
H^{\frac{7}{2}}_{\Gamma_{1}}(\Omega),z_{1}\in
H^{3}_{\Gamma_{1}}(\Omega),v_{0}\in H^{2}_{0}(\Gamma_{0}),v_{1}\in
H^{2}_{0}(\Gamma_{0}),\\\ &\textbf{\AA}(v_{0}+v_{1})+B^{*}z_{1}\in
H^{2}_{0}(\Gamma_{0}),\frac{\partial
z_{0}}{\partial\nu}|_{\Gamma_{0}}=v_{1},\frac{\partial
z_{1}}{\partial\nu}|_{\Gamma_{0}}=-\textbf{\AA}(v_{0}+v_{1})-B^{*}z_{1}\\\
&\textbf{\AA}(v_{0}+v_{1})+B^{*}z_{1}+\textbf{\AA}[v_{1}-\textbf{\AA}(v_{0}+v_{1})-B^{*}z_{1}]+B^{*}[(-A_{N}+q)z_{0}+Bv_{1}]\in
D(\textbf{\AA})\\\
&\textbf{\AA}(v_{1}-\textbf{\AA}(v_{0}+v_{1})-B^{*}z_{1})+B^{*}[(-A_{N}+q)z_{0}+Bv_{1}]\in
H^{2}_{0}(\Gamma_{0}),\\\
&\frac{\partial[(-A_{N}+q)z_{0}+Bv_{1}]}{\partial\nu}|_{\Gamma_{0}}=-\textbf{\AA}[v_{1}-\textbf{\AA}(v_{0}+v_{1})-B^{*}z_{1}]-B^{*}[(-A_{N}+q)z_{0}+Bv_{1}]\\}\end{split}$
Then there exists a constant
$C=C(\Omega,T,\Gamma_{0},\varphi,q,p,z_{0},z_{1},v_{0},v_{1})>0$ such that
(1.34) $\|q-p\|_{L^{2}(\Omega)}\leq
C\left(\|v_{tt}(q)-v_{tt}(p)\|_{L^{2}(\Gamma_{0}\times[0,T])}+\|v_{ttt}(q)-v_{ttt}(p)\|_{L^{2}(\Gamma_{0}\times[0,T])}\right.\\\
\left.\qquad+\|\Delta^{2}(v_{tt}(q)-v_{tt}(p))\|_{L^{2}(\Gamma_{0}\times[0,T])}\right)$
for all $q,p\in W^{1,\infty}(\Omega)$ when $n=2$ and all $q,p\in
W^{2,\infty}(\Omega)$ when $n=3$.
The rest of this paper is organized as follows: In section 2 we give the key
Carleman estimate that is used in the proof of uniqueness result. Based on the
same Carleman estimate, we also prove an observability inequality that is
needed in section 5. Section 3 to 6 are devoted to the proofs of our main
results Theorems 1.4 to 1.7. Some concluding remarks will be given in section
7.
## 2\. Carleman estimate and observability inequality
### 2.1. Carleman Estimate
In this section, we state a Carleman estimate result that plays a key role in
the proof of our uniqueness theorem. The result is due to [29].
We first introduce the pseudo-convex function $\varphi(x,t)$ defined by
(2.1) $\varphi(x,t)=d(x)-c\left(t-\frac{T}{2}\right)^{2};\quad
x\in\Omega,t\in[0,T]$
where $T$ is as in $\eqref{time}$ and $0<c<1$ is selected as follows: By
$\eqref{time}$, there exists $\delta>0$ such that
(2.2) $T^{2}>4\max_{x\in\overline{\Omega}}d(x)+4\delta$
For this $\delta>0$, there exists a constant $c$, $0<c<1$, such that
(2.3) $cT^{2}>4\max_{x\in\overline{\Omega}}d(x)+4\delta$
Henceforth, with $T$ and $c$ chosen as described above, this function
$\varphi(x,t)$ has the following properties:
(a) For the constant $\delta>0$ fixed in $\eqref{timesquare}$ and for any
$t>0$
(2.4)
$\varphi(x,t)\leq\varphi(x,\frac{T}{2}),\quad\varphi(x,0)=\varphi(x,T)\leq
d(x)-c\frac{T^{2}}{4}\leq-\delta$
uniformly in $x\in\Omega$.
(b) There are $t_{0}$ and $t_{1}$, with $0<t_{0}<\frac{T}{2}<t_{1}<T$, such
that we have
(2.5) $\min_{x\in\overline{\Omega},t\in[t_{0},t_{1}]}\varphi(x,t)\geq\sigma$
where $0<\sigma<\min_{x\in\overline{\Omega}}d(x)$.
Moreover, if we introduce the space $Q(\sigma)$ that is defined by the
following
(2.6) $Q{(\sigma)}=\\{(x,t)|x\in\Omega,0\leq t\leq
T,\varphi(x,t)\geq\sigma>0\\}$
Then an important property of $Q(\sigma)$ is that (see [29]):
(2.7) $[t_{0},t_{1}]\times\Omega\subset Q(\sigma)\subset[0,T]\times\Omega$
Then for the wave equation of the form
(2.8) $w_{tt}(x,t)-\Delta w(x,t)-q(x)w(x,t)=F(x,t),\quad x\in\Omega,t\in[0,T]$
we have the following Carleman-type estimate:
###### Theorem 2.1.
Under the main assumptions (A.1) and (A.2), with $\varphi(x,t)$ defined in
$\eqref{defphi}$. Let $w\in H^{2}(Q)$ be a solution of the equation
$\eqref{carlemaneqn}$ where $q\in L^{\infty}(\Omega)$ and $F\in L_{2}(Q)$.
Then the following one parameter family of estimates hold true, with $\rho>0$,
$\beta>0$, for all $\tau>0$ sufficiently large and $\epsilon>0$ small:
(2.9)
$BT|_{w}+2\int_{Q}e^{2\tau\varphi}|F|^{2}dQ+C_{1,T}e^{2\tau\sigma}\int_{Q}w^{2}dQ\geq(\tau\epsilon\rho-2C_{T})\int_{Q}e^{2\tau\varphi}\left(w_{t}^{2}+|\nabla
w|^{2}\right)dQ\\\
+\left(2\tau^{3}\beta+\mathcal{O}(\tau^{2})-2C_{T}\right)\int_{Q{(\sigma)}}e^{2\tau\varphi}w^{2}dxdt-
c_{T}\tau^{3}e^{-2\tau\delta}[E_{w}(0)+E_{w}(T)]$
Here $\delta>0$, $\sigma>0$ are the constants in $\eqref{timesquare}$,
$\eqref{propertyb}$, while $C_{T}$, $c_{T}$ and $C_{1,T}$ are positive
constants depending on $T$ and $d$. In addition, the boundary terms $BT|_{w}$
are given explicitly by
(2.10)
$\begin{split}BT|_{w}&=2\tau\int_{0}^{T}\int_{\Gamma_{0}}e^{2\tau\varphi}(w_{t}^{2}-|\nabla
w|^{2})h\cdot\nu d\Gamma dt\\\
&+8c\tau\int_{0}^{T}\int_{\Gamma}e^{2\tau\varphi}(t-\frac{T}{2})w_{t}\frac{\partial
w}{\partial\nu}d\Gamma dt\\\
&+4\tau\int_{0}^{T}\int_{\Gamma}e^{2\tau\varphi}(h\cdot\nabla w)\frac{\partial
w}{\partial\nu}d\Gamma dt\\\
&+4\tau^{2}\int_{0}^{T}\int_{\Gamma}e^{2\tau\varphi}\left(|h|^{2}-4c^{2}(t-\frac{T}{2})^{2}+\frac{\alpha}{2\tau}\right)w\frac{\partial
w}{\partial\nu}d\Gamma dt\\\
&+2\tau\int_{0}^{T}\int_{\Gamma_{0}}e^{2\tau\varphi}\bigg{[}2\tau^{2}\left(|h|^{2}-4c^{2}(t-\frac{T}{2})^{2}\right)\\\
&\quad+\tau(\alpha-\Delta d-2c)\bigg{]}w^{2}h\cdot\nu d\Gamma dt\end{split}$
where $\alpha=\Delta d-2c-1+k$ for $0<k<1$ is a constant and $E_{w}$ is
defined as follows:
(2.11) $E_{w}(t)=\int_{\Omega}[w^{2}(x,t)+w_{t}^{2}(x,t)+|\nabla
w(x,t)|^{2}]d\Omega$
An immediate corollary of the estimate is the following (Theorem 6.1 in [29])
###### Corollary 2.2.
Under the assumptions in Theorem (2.1), the following one-parameter family of
estimates hold true, for all $\tau$ sufficiently large, and for any
$\epsilon>0$ small:
(2.12)
$\overline{BT}|_{w}+2\int_{0}^{T}\int_{\Omega}e^{2\tau\varphi}F^{2}dQ+const_{\varphi}\int_{0}^{T}\int_{\Omega}F^{2}dQ\geq
k_{\varphi}[E_{w}(0)+E_{w}(T)]$
for a constant $k_{\varphi}>0$ while $\overline{BT}|_{w}$ is given by:
(2.13)
$\overline{BT}|_{w}=BT|_{w}+const_{\varphi}\left[\int_{0}^{T}\int_{\Gamma}\bigg{|}\frac{\partial
w}{\partial\nu}w_{t}\bigg{|}d\Gamma
dt+\int_{t_{0}}^{t_{1}}\int_{\Gamma_{0}}w^{2}d\Gamma_{0}dt\right.\\\
+\left.\int_{0}^{T}\int_{\Gamma_{0}}|ww_{t}|d\Gamma_{0}dt\right]$
###### Remark 2.3.
For the proof of the above Carleman estimate and the corollary, we refer to
[29] and we omit the details here.
### 2.2. Continuous Observability Inequality
Using the Carleman estimate in last section, we can prove the following
observability inequality:
###### Theorem 2.4.
Under the main assumptions (A.1) and (A.2), for the following initial boundary
value problem
(2.14) $\begin{cases}w_{tt}(x,t)=\Delta w(x,t)+q(x)w(x,t)&\mbox{in
}\Omega\times[0,T]\\\ w(\cdot,\frac{T}{2})=w_{0}(x)&\mbox{in }\Omega\\\
w_{t}(\cdot,\frac{T}{2})=w_{1}(x)&\mbox{in }\Omega\\\ \frac{\partial
w}{\partial\nu}(x,t)=0&\mbox{on }\Gamma_{1}\times[0,T]\\\ \frac{\partial
w}{\partial\nu}(x,t)=g(x,t)&\mbox{on }\Gamma_{0}\times[0,T]\end{cases}$
where $w_{0}\in H^{1}(\Omega)$, $w_{1}\in L^{2}(\Omega)$, $g\in
L^{2}(\Gamma\times[0,T])$ and $q\in L^{\infty}(\Omega)$. We have the following
continuous observability inequality:
$\|w_{0}\|^{2}_{H^{1}(\Omega)}+\|w_{1}\|^{2}_{L^{2}(\Omega)}\leq
C\left(\|w\|^{2}_{L^{2}(\Gamma_{0}\times[0,T])}+\|w_{t}\|^{2}_{L^{2}(\Gamma_{0}\times[0,T])}+\|g\|^{2}_{L^{2}(\Gamma_{0}\times[0,T])}\right)$
where $T$ is as in $\eqref{time}$ and
$C=C(\Omega,T,\Gamma_{0},\varphi,\tau,q)$ is a positive constant.
###### Proof.
For the case when $g=0$, we refer to [29] where the continuous observability
inequality is established for zero Neumann data on the whole boundary. Here we
give the proof for the case of general $g\in L^{2}(\Gamma_{0}\times[0,T])$,
which is still based on the proof in [29]. We first introduce the following
result that is from the section 7.2 of [28].
###### Lemma 2.5.
Let $w$ be a solution of the equation
(2.15) $w_{tt}(x,t)=\Delta w(x,t)+q(x)w(x,t)+f(x,t)\ \text{in}\
Q=\Omega\times[0,T]$
with $q\in L^{\infty}(\Omega)$ and $w$ in the following class:
(2.16) $\left\\{\begin{aligned} w\in L^{2}(0,T;H^{1}(\Omega))\cap
H^{1}(0,T;L^{2}(\Omega))\\\ w_{t},\frac{\partial w}{\partial\nu}\in
L^{2}(0,T;L^{2}(\Gamma))\end{aligned}\right.$
Given $\epsilon>0$, $\epsilon_{0}>0$ arbitrary, given $T>0$, there exists a
constant $C=C(\epsilon,\epsilon_{0},T)>0$ such that
(2.17) $\int_{\epsilon}^{T-\epsilon}\int_{\Gamma}|\nabla_{tan}w|^{2}d\Gamma
dt\leq C\left(\int_{0}^{T}\int_{\Gamma}w_{t}^{2}+\left(\frac{\partial
w}{\partial\nu}\right)^{2}d\Gamma
dt+\|w\|^{2}_{L^{2}(0,T;H^{\frac{1}{2}+\epsilon_{0}}(\Omega))}\right.\\\
\left.+\|f\|^{2}_{H^{\frac{1}{2}+\epsilon_{0}}(Q)}\right)$
Now to prove $\eqref{observeineq}$, we first establish the following weaker
conclusion under the assumptions (A.1) and (A.2)
(2.18) $E\left(\frac{T}{2}\right)\leq
C\left(\int_{0}^{T}\int_{\Gamma_{0}}[w^{2}+w_{t}^{2}+g^{2}]d\Gamma_{0}dt+\|w\|^{2}_{L^{2}(0,T;H^{\frac{1}{2}+\epsilon_{0}}(\Omega))}\right)$
which is the desired inequality $\eqref{observeineq}$ polluted by the interior
lower order term $\|w\|$. To see this, we introduce a preliminary equivalence
first. Let $u\in H^{1}(\Omega)$, then the following inequality holds true:
there exist positive constants $0<k_{1}<k_{2}<\infty$, independent of $u$,
such that
(2.19) $k_{1}\int_{\Omega}[u^{2}+|\nabla
u|^{2}]d\Omega\leq\int_{\Omega}|\nabla
u|^{2}d\Omega+\int_{\tilde{\Gamma_{0}}}u^{2}d\Gamma\leq
k_{2}\int_{\Omega}[u^{2}+|\nabla u|^{2}]d\Omega$
where $\tilde{\Gamma_{0}}$ is any (fixed) portion of the boundary $\Gamma$
with positive measure. Inequality $\eqref{equivalence}$ is obtained by
combining the following two inequalities:
(2.20) $\int_{\Omega}u^{2}d\Omega\leq c_{1}\left[\int_{\Omega}|\nabla
u|^{2}d\Omega+\int_{\tilde{\Gamma_{0}}}u^{2}d\Gamma\right];\quad\int_{\tilde{\Gamma_{0}}}u^{2}d\Gamma\leq
c_{2}\int_{\Omega}[u^{2}+|\nabla u|^{2}]d\Omega$
The inequality on the left of $\eqref{equivalence1}$ replaces Poincaré’s
inequality, while the inequality on the right of $\eqref{equivalence1}$ stems
from (a conservative version of) trace theory. Thus, for $w\in H^{2}(Q)$, if
we introduce
(2.21) $\varepsilon(t)=\int_{\Omega}\left[|\nabla
w(t)|^{2}+w_{t}^{2}(t)\right]d\Omega+\int_{\Gamma_{0}}w^{2}(t)d\Gamma_{1}$
where $\Gamma_{0}=\Gamma\setminus\Gamma_{1}$ is as defined in the main
assumptions, then $\eqref{equivalence}$ yields the equivalence
(2.22) $aE(t)\leq\varepsilon(t)\leq bE(t)$
for some positive constants $a>0$, $b>0$.
Now in a standard way, we multiply equation $\eqref{wave}$ by $w_{t}$ and
integrate over $\Omega$. After an application of the first Green’s identity,
we have
(2.23) $\frac{1}{2}\frac{\partial}{\partial
t}\left(\int_{\Omega}[w_{t}^{2}+|\nabla
w|^{2}]d\Omega+\int_{\Gamma_{0}}w^{2}d\Gamma_{0}\right)=\int_{\Gamma}\frac{\partial
w}{\partial\nu}w_{t}d\Gamma+\int_{\Gamma_{0}}ww_{t}d\Gamma_{0}\\\
+\int_{\Omega}\left[q(x)+f\right]w_{t}d\Omega$
Notice that on both sides of (2.23) we have added term
$\displaystyle\frac{1}{2}\frac{\partial}{\partial
t}\int_{\Gamma_{0}}w^{2}d\Gamma_{0}=\int_{\Gamma_{0}}ww_{t}d\Gamma_{0}$.
Recalling $\varepsilon(t)$ in $\eqref{equivalence2}$, we integrate (2.23) over
$(s,t)$ and obtain
(2.24)
$\varepsilon(t)=\varepsilon(s)+2\int_{s}^{t}\left[\int_{\Gamma}\frac{\partial
w}{\partial\nu}w_{t}d\Gamma+\int_{\Gamma_{0}}ww_{t}d\Gamma_{0}\right]dr+2\int_{s}^{t}\int_{\Omega}\left[q(x)+f\right]w_{t}d\Omega
dr$
We apply Cauthy-Schwartz inequality on $[q(x)+f]w_{t}$, invoke the left hand
side $\displaystyle E(t)\leq\frac{1}{a}\varepsilon(t)$ of
$\eqref{equivalence}$, and obtain
(2.25)
$\varepsilon(t)\leq[\varepsilon(s)+N(T)]+C_{T}\int^{t}_{s}\varepsilon(r)dr$
(2.26)
$\varepsilon(s)\leq[\varepsilon(t)+N(T)]+C_{T}\int^{t}_{s}\varepsilon(r)dr$
where we have set
(2.27)
$N(T)=\int^{T}_{0}\int_{\Omega}f^{2}dQ+2\int^{T}_{0}\int_{\Gamma}\bigg{|}\frac{\partial
w}{\partial\nu}w_{t}\bigg{|}d\Gamma
dt+2\int^{T}_{0}\int_{\Gamma_{0}}|ww_{t}|d\Gamma_{0}dt$
Gronwall’s inequality applied on $\eqref{varepsilont}$, $\eqref{varepsilons}$
then yields for $0\leq s\leq t\leq T$,
(2.28)
$\varepsilon(t)\leq[\varepsilon(s)+N(T)]e^{C_{T}(t-s)};\quad\varepsilon(s)\leq[\varepsilon(t)+N(T)]e^{C_{T}(t-s)}$
We consider the following three cases here:
Case 1: $0\leq s\leq t\leq\frac{T}{2}$. In this case we set $t=\frac{T}{2}$
and $s=t$ in the first inequality of $\eqref{gronwall}$; and set $s=0$ in the
second inequality of $\eqref{gronwall}$, to obtain
(2.29)
$\varepsilon(\frac{T}{2})\leq[\varepsilon(t)+N(T)]e^{C_{T}\frac{T}{2}};\quad\varepsilon(0)\leq[\varepsilon(t)+N(T)]e^{C_{T}\frac{T}{2}}$
Summing up these two inequalities in $\eqref{case1}$ yields for $0\leq
t\leq\frac{T}{2}$,
(2.30)
$\begin{split}\varepsilon(t)&\geq\frac{\varepsilon(\frac{T}{2})+\varepsilon(0)}{2}e^{-C_{T}\frac{T}{2}}-N(T)\\\
&\geq\frac{a}{2}[E(\frac{T}{2})+E(0)]e^{-C_{T}\frac{T}{2}}-N(T)\end{split}$
after recalling the left hand side of the equivalence in
$\eqref{equivalence2}$.
Case 2: $\frac{T}{2}\leq s\leq t\leq T$. In this case we set $t=T$ and $s=t$
in the first inequality of $\eqref{gronwall}$; and set $s=\frac{T}{2}$ in the
second inequality of $\eqref{gronwall}$, to obtain
(2.31)
$\varepsilon(T)\leq[\varepsilon(t)+N(T)]e^{C_{T}\frac{T}{2}};\quad\varepsilon(\frac{T}{2})\leq[\varepsilon(t)+N(T)]e^{C_{T}\frac{T}{2}}$
Summing up these two inequalities in $\eqref{case2}$ yields for
$\frac{T}{2}\leq t\leq T$,
(2.32)
$\begin{split}\varepsilon(t)&\geq\frac{\varepsilon(\frac{T}{2})+\varepsilon(T)}{2}e^{-C_{T}\frac{T}{2}}-N(T)\\\
&\geq\frac{a}{2}[E(\frac{T}{2})+E(T)]e^{-C_{T}\frac{T}{2}}-N(T)\end{split}$
after recalling the left hand side of the equivalence in
$\eqref{equivalence2}$.
Case 3: $0\leq s\leq\frac{T}{2}\leq t\leq T$. In this case we set $t=0$ and
$s=t$ in the first inequality of $\eqref{gronwall}$; and set $s=\frac{T}{2}$
in the second inequality of $\eqref{gronwall}$, to obtain
(2.33)
$\varepsilon(0)\leq[\varepsilon(t)+N(T)]e^{C_{T}\frac{T}{2}};\quad\varepsilon(\frac{T}{2})\leq[\varepsilon(t)+N(T)]e^{C_{T}\frac{T}{2}}$
Summing up these two inequalities in $\eqref{case3}$ yields for
$\frac{T}{2}\leq t\leq T$,
(2.34)
$\begin{split}\varepsilon(t)&\geq\frac{\varepsilon(\frac{T}{2})+\varepsilon(0)}{2}e^{-C_{T}\frac{T}{2}}-N(T)\\\
&\geq\frac{a}{2}[E(\frac{T}{2})+E(0)]e^{-C_{T}\frac{T}{2}}-N(T)\end{split}$
after recalling the left hand side of the equivalence in
$\eqref{equivalence2}$.
In summary, we get for any $0\leq t\leq T$,
(2.35) $\varepsilon(t)\geq\frac{a}{2}E(\frac{T}{2})e^{-C_{T}\frac{T}{2}}-N(T)$
We now apply the Corollary 2.2 of the Carleman estimate, except on the
interval $[\epsilon,T-\epsilon]$, rather than on $[0,T]$ as in
$\eqref{carleman2}$. Thus, we obtain since $f=0$:
(2.36) $\overline{BT}|_{[\epsilon,T-\epsilon]\times\Gamma}\geq
k_{\varphi}E(\epsilon)$
where $\overline{BT}|_{[\epsilon,T-\epsilon]\times\Gamma}$ is given as in
(2.13). Since we have $\displaystyle\frac{\partial w}{\partial\nu}=0$ on
$\Gamma_{1}\times[0,T]$ and $\displaystyle\frac{\partial
w}{\partial\nu}=g(x,t)$ on $\Gamma_{0}\times[0,T]$ by $\eqref{observe}$, with
the additional information that $h\cdot\nu=0$ on $\Gamma_{1}$ by the
assumption (A.1). Thus, by using the explicit expression $\eqref{boundary}$
for $BT|_{w}$, we have that
$\overline{BT}|_{[\epsilon,T-\epsilon]\times\Gamma}$ is given by:
(2.37)
$\begin{split}\overline{BT}|_{[\epsilon,T-\epsilon]\times\Gamma}&=2\tau\int_{\epsilon}^{T-\epsilon}\int_{\Gamma_{0}}e^{2\tau\varphi}(w_{t}^{2}-|\nabla
w|^{2})h\cdot\nu d\Gamma dt\\\
&+8c\tau\int_{\epsilon}^{T-\epsilon}\int_{\Gamma_{0}}e^{2\tau\varphi}(t-\frac{T}{2})w_{t}gd\Gamma
dt\\\
&+4\tau\int_{\epsilon}^{T-\epsilon}\int_{\Gamma_{0}}e^{2\tau\varphi}(h\cdot\nabla
w)gd\Gamma dt\\\
&+4\tau^{2}\int_{\epsilon}^{T-\epsilon}\int_{\Gamma_{0}}e^{2\tau\varphi}\left(|h|^{2}-4c^{2}(t-\frac{T}{2})^{2}+\frac{\alpha}{2\tau}\right)wgd\Gamma
dt\\\
&+2\tau\int_{\epsilon}^{T-\epsilon}\int_{\Gamma_{0}}e^{2\tau\varphi}\left[2\tau^{2}\left(|h|^{2}-4c^{2}(t-\frac{T}{2})^{2}\right)+\tau(\alpha-\Delta
d-2c)\right]w^{2}h\cdot\nu d\Gamma dt\\\
&+const_{\varphi}\left[\int_{\epsilon}^{T-\epsilon}\int_{\Gamma_{0}}|gw_{t}|d\Gamma
dt+\int_{t_{0}}^{t_{1}}\int_{\Gamma_{0}}w^{2}d\Gamma_{0}dt+\int_{\epsilon}^{T-\epsilon}\int_{\Gamma_{0}}|ww_{t}|d\Gamma_{0}dt\right]\end{split}$
Next, by the right side of equivalences $\eqref{equivalence2}$ and
$\eqref{desired}$, we obtain
(2.38)
$E(\epsilon)\geq\frac{\varepsilon(\epsilon)}{b}\geq\frac{a}{2b}E\left(\frac{T}{2}\right)e^{-C_{T}\frac{T}{2}}-2\int_{0}^{T}\int_{\Gamma}|gw_{t}|d\Gamma
dt-2\int_{0}^{T}\int_{\Gamma_{0}}|ww_{t}|d\Gamma_{0}dt$
recalling $N(T)$ in $\eqref{nt}$. We use $\eqref{finishing}$ in
$\eqref{corollary1}$. Finally, we invoke estimate (2.17) of Lemma 2.5 on the
first and the third integral terms of $\eqref{btbar1}$. This way, we readily
obtain $\eqref{polluted}$, which is our desired inequality polluted by
$\|w\|^{2}_{L^{2}(0,T;H^{\frac{1}{2}+\epsilon_{0}}(\Omega))}$. To eliminate
this interior lower order term, we can apply the standard
compactness/uniqueness argument (e.g.[24]) by invoking the global uniqueness
Theorem 7.1 in [29]. ∎
## 3\. Proof of Theorem 1.4
We let $\bar{w}=\bar{w}(f)=w_{t}(f)$ then from $\eqref{linear}$ we have
$\bar{w}$, $u$ satisfy
(3.1)
$\begin{cases}\bar{w}_{tt}(x,t)-\Delta\bar{w}(x,t)-q(x)\bar{w}(x,t)=f(x)R_{t}(x,t)&\mbox{in
}\Omega\times[0,T]\\\ \frac{\partial\bar{w}}{\partial\nu}(x,t)=0&\mbox{on
}\Gamma_{1}\times[0,T]\\\
\bar{w}(x,t)=-u_{tt}(x,t)-\Delta^{2}u(x,t)-\Delta^{2}u_{t}(x,t)&\mbox{on
}\Gamma_{0}\times[0,T]\\\ u(x,t)=\frac{\partial
u}{\partial\nu}(x,t)=0&\mbox{on }\partial\Gamma_{0}\times[0,T]\\\
\frac{\partial\bar{w}}{\partial\nu}(x,t)=u_{tt}(x,t)&\mbox{on
}\Gamma_{0}\times[0,T]\\\ \bar{w}(\cdot,\frac{T}{2})=0&\mbox{in }\Omega\\\
\bar{w}_{t}(\cdot,\frac{T}{2})=f(x)R(x,\frac{T}{2})&\mbox{in }\Omega\\\
u(\cdot,\frac{T}{2})=0&\mbox{on }\Gamma_{0}\\\
u_{t}(\cdot,\frac{T}{2})=0&\mbox{on }\Gamma_{0}\end{cases}$
Under the assumptions in Theorem 1.4, we can apply the Carleman estimate to
the wave equation in the system $\eqref{lineary}$
$\bar{w}_{tt}(x,t)-\Delta\bar{w}(x,t)-q(x)\bar{w}(x,t)=f(x)R_{t}(x,t)$ and get
$BT|_{\bar{w}}+2\int_{Q}e^{2\tau\varphi}|fR_{t}|^{2}dQ+C_{1,T}e^{2\tau\sigma}\int_{Q}\bar{w}^{2}dQ\geq(\tau\epsilon\rho-2C_{T})\int_{Q}e^{2\tau\varphi}[\bar{w}_{t}^{2}+|\nabla\bar{w}|^{2}]dQ\\\
+[2\tau^{3}\beta+\mathcal{O}(\tau^{2})-2C_{T}]\int_{Q{(\sigma)}}e^{2\tau\varphi}\bar{w}^{2}dxdt-
c_{T}\tau^{3}e^{-2\tau\delta}[E_{\bar{w}}(0)+E_{\bar{w}}(T)]$
where the boundary terms are given explicitly by
(3.2)
$\begin{split}BT|_{\bar{w}}&=2\tau\int_{0}^{T}\int_{\Gamma_{0}}e^{2\tau\varphi}(\bar{w}_{t}^{2}-|\nabla\bar{w}|^{2})h\cdot\nu
d\Gamma dt\\\
&+8c\tau\int_{0}^{T}\int_{\Gamma}e^{2\tau\varphi}(t-\frac{T}{2})\bar{w}_{t}\frac{\partial\bar{w}}{\partial\nu}d\Gamma
dt\\\
&+4\tau\int_{0}^{T}\int_{\Gamma}e^{2\tau\varphi}(h\cdot\nabla\bar{w})\frac{\partial\bar{w}}{\partial\nu}d\Gamma
dt\\\
&+4\tau^{2}\int_{0}^{T}\int_{\Gamma}e^{2\tau\varphi}\left(|h|^{2}-4c^{2}(t-\frac{T}{2})^{2}+\frac{\alpha}{2\tau}\right)\bar{w}\frac{\partial\bar{w}}{\partial\nu}d\Gamma
dt\\\
&+2\tau\int_{0}^{T}\int_{\Gamma_{0}}e^{2\tau\varphi}\bigg{[}2\tau^{2}\left(|h|^{2}-4c^{2}(t-\frac{T}{2})^{2}\right)\\\
&\quad+\tau(\alpha-\Delta d-2c)\bigg{]}\bar{w}^{2}h\cdot\nu d\Gamma
dt\end{split}$
Since we have the extra observation that $u_{tt}(x,t)=0$ on
$\Gamma_{0}\times[0,T]$ and note that the initial conditions
$u(x,\frac{T}{2})=u_{t}(x,\frac{T}{2})=0$ on $\Gamma_{0}$, thus by the
fundamental theorem of calculus we have $u(x,t)=0$ on $\Gamma_{0}\times[0,T]$
and hence from the coupling in the system $\eqref{lineary}$ we get
(3.3) $\bar{w}(x,t)=-u_{tt}(x,t)-\Delta^{2}u(x,t)-\Delta^{2}u_{t}(x,t)=0\
\textrm{on}\ \Gamma_{0}\times[0,T]$
and
(3.4) $\frac{\partial\bar{w}}{\partial\nu}(x,t)=u_{tt}(x,t)=0\ \textrm{on}\
\Gamma_{0}\times[0,T]$
Plugging $\eqref{boundaryterm1}$ and $\eqref{boundaryterm2}$ into
$\eqref{boundaryy}$, note also that
$\displaystyle\frac{\partial\bar{w}}{\partial\nu}=0$ on
$\Gamma_{1}\times[0,T]$, therefore we get $BT|_{\bar{w}}\equiv 0$.
In addition, in view of $\eqref{regR}$, $\eqref{crucialR}$, we have
$|fR_{t}|\leq C|f|$ for some positive constant $C$ depend on $R_{t}$.
Moreover, notice that
$\displaystyle\lim_{\tau\to\infty}\tau^{3}e^{-2\tau\delta}=0$. Hence when
$\tau$ is sufficiently large, the above Carleman estimate can be rewritten as
the following:
(3.5)
$C_{1,\tau}\int_{Q}e^{2\tau\varphi}[\bar{w}_{t}^{2}+|\nabla\bar{w}|^{2}]dQ+C_{2,\tau}\int_{Q(\sigma)}e^{2\tau\varphi}\bar{w}^{2}dxdt\leq
C\int_{Q}e^{2\tau\varphi}|f|^{2}dQ+Ce^{2\tau\sigma}$
where we set
(3.6) $C_{1,\tau}=\tau\epsilon\rho-2C_{T},\quad
C_{2,\tau}=2\tau^{3}\beta+\mathcal{O}(\tau^{2})-2C_{T}$
and $C$ denote generic constants which do not depend on $\tau$ and henceforth
we will use this notation for the rest of this paper. In addition, note that
$f$ is time-independent, so if we differentiate the system $\eqref{lineary}$
in time twice, we can get the following wave equations for $\bar{w_{t}}$ and
$\bar{w_{tt}}$:
(3.7)
$(\bar{w_{t}})_{tt}(x,t)-\Delta\bar{w_{t}}(x,t)-q(x)\bar{w_{t}}(x,t)=f(x)R_{tt}(x,t)$
and
(3.8)
$(\bar{w_{tt}})_{tt}(x,t)-\Delta\bar{w_{tt}}(x,t)-q(x)\bar{w_{tt}}(x,t)=f(x)R_{ttt}(x,t)$
Notice the assumptions $\eqref{regR}$, $\eqref{h2reg}$, therefore we have
similarly as $\eqref{ineq2}$ the following estimates for the two new systems:
(3.9)
$C_{1,\tau}\int_{Q}e^{2\tau\varphi}[\bar{w}_{tt}^{2}+|\nabla\bar{w}_{t}|^{2}]dQ+C_{2,\tau}\int_{Q(\sigma)}e^{2\tau\varphi}\bar{w}_{t}^{2}dxdt\leq
C\int_{Q}e^{2\tau\varphi}|f|^{2}dQ+Ce^{2\tau\sigma}$
and
(3.10)
$C_{1,\tau}\int_{Q}e^{2\tau\varphi}[\bar{w}_{ttt}^{2}+|\nabla\bar{w}_{tt}|^{2}]dQ+C_{2,\tau}\int_{Q(\sigma)}e^{2\tau\varphi}\bar{w}_{tt}^{2}dxdt\leq
C\int_{Q}e^{2\tau\varphi}|f|^{2}dQ+Ce^{2\tau\sigma}$
where $\tau$ is sufficiently large and $C_{1,\tau}$, $C_{2,\tau}$ are defined
as in $\eqref{ctau}$.
Adding $(\ref{ineq2})$, $(\ref{ineq3})$ and $(\ref{ineq4})$ together we then
have
(3.11)
$C_{1,\tau}\int_{Q}e^{2\tau\varphi}[\bar{w}_{t}^{2}+\bar{w}_{tt}^{2}+\bar{w}_{ttt}^{2}+|\nabla\bar{w}|^{2}+|\nabla\bar{w}_{t}|^{2}+|\nabla\bar{w}_{tt}|^{2}]dQ\\\
+C_{2,\tau}\int_{Q(\sigma)}e^{2\tau\varphi}[\bar{w}^{2}+\bar{w}_{t}^{2}+\bar{w}_{tt}^{2}]dxdt\leq
C\left(\int_{Q}e^{2\tau\varphi}|f|^{2}dQ+e^{2\tau\sigma}\right)$
Again we use the wave equation
$\bar{w}_{tt}(x,t)-\Delta\bar{w}(x,t)-q(x)\bar{w}(x,t)=f(x)R_{t}(x,t)$,
plugging in the initial time of $t=\frac{T}{2}$ and use the zero initial
conditions of $\bar{w}(\cdot,\frac{T}{2})=0$, we have
(3.12)
$\bar{w}_{tt}(x,\frac{T}{2})-\Delta\bar{w}(x,\frac{T}{2})-q(x)\bar{w}(x,\frac{T}{2})=\bar{w}_{tt}(x,\frac{T}{2})=f(x)R_{t}(x,\frac{T}{2})$
Since $|R_{t}(x,\frac{T}{2})|\geq r_{1}>0$ from $\eqref{crucialR}$, therefore
we have $|f(x)|\leq C|\bar{w}_{tt}(x,\frac{T}{2})|$ and hence we have the
following estimates on $\displaystyle\int_{Q}e^{2\tau\varphi}|f|^{2}dQ$:
(3.13)
$\begin{split}\int_{Q}e^{2\tau\varphi}|f|^{2}dQ&=\int_{0}^{T}\int_{\Omega}e^{2\tau\varphi(x,t)}|f(x)|^{2}d\Omega
dt\\\ &\leq
C\int_{0}^{T}\int_{\Omega}e^{2\tau\varphi(x,t)}|\bar{w}_{tt}(x,\frac{T}{2})|^{2}d\Omega
dt\\\ &\leq
C\int_{\Omega}e^{2\tau\varphi(x,\frac{T}{2})}|\bar{w}_{tt}(x,\frac{T}{2})|^{2}d\Omega\\\
&=C\left(\int_{\Omega}\int_{0}^{\frac{T}{2}}\frac{d}{ds}(e^{2\tau\varphi(x,s)}|\bar{w}_{tt}(x,s)|^{2})dsd\Omega+\int_{\Omega}e^{2\tau\varphi(x,0)}|\bar{w}_{tt}(x,0)|^{2}d\Omega\right)\\\
&=C\left(4c\tau\int_{\Omega}\int_{0}^{\frac{T}{2}}(\frac{T}{2}-s)e^{2\tau\varphi(x,s)}|\bar{w}_{tt}(x,s)|^{2}dsd\Omega\right.\\\
&\quad+\left.2\int_{\Omega}\int_{0}^{\frac{T}{2}}e^{2\tau\varphi}|\bar{w}_{tt}(x,s)||\bar{w}_{ttt}(x,s)|dsd\Omega+\int_{\Omega}e^{2\tau\varphi(x,0)}|\bar{w}_{tt}(x,0)|^{2}d\Omega\right)\\\
&\leq
C\left(\tau\int_{\Omega}\int^{\frac{T}{2}}_{0}e^{2\tau\varphi}|\bar{w}_{tt}|^{2}dtd\Omega+\int_{\Omega}\int^{\frac{T}{2}}_{0}e^{2\tau\varphi}(|\bar{w}_{tt}|^{2}+|\bar{w}_{ttt}|)^{2}dtd\Omega\right.\\\
&\quad+\left.\int_{\Omega}|\bar{w}_{tt}(x,0)|^{2}d\Omega\right)\\\ &\leq
C\left(\tau\int_{Q}e^{2\tau\varphi}|\bar{w}_{tt}|^{2}dQ+\int_{Q}e^{2\tau\varphi}(|\bar{w}_{tt}|^{2}+|\bar{w}_{ttt}|^{2})dQ\right)\\\
&=C\left((\tau+1)\int_{Q}e^{2\tau\varphi}|\bar{w}_{tt}|^{2}dQ+\int_{Q}e^{2\tau\varphi}|\bar{w}_{ttt}|^{2}dQ\right)\end{split}$
where in the above estimates we use the definition $\eqref{defphi}$ and the
property $\eqref{propertya}$ of $\varphi$ as well as Cauthy-Schwartz
inequality. Collecting $\eqref{mainineq}$ with $(\ref{ineq5})$, we have
(3.14)
$C_{1,\tau}\int_{Q}e^{2\tau\varphi}[\bar{w}_{t}^{2}+\bar{w}_{tt}^{2}+\bar{w}_{ttt}^{2}+|\nabla\bar{w}|^{2}+|\nabla\bar{w}_{t}|^{2}+|\nabla\bar{w}_{tt}|^{2}]dQ\\\
+C_{2,\tau}\int_{Q(\sigma)}e^{2\tau\varphi}[\bar{w}^{2}+\bar{w}_{t}^{2}+\bar{w}_{tt}^{2}]dxdt\leq
C\left((\tau+1)\int_{Q}e^{2\tau\varphi}|\bar{w}_{tt}|^{2}dQ+\int_{Q}e^{2\tau\varphi}|\bar{w}_{ttt}|^{2}dQ+e^{2\tau\sigma}\right)$
Note that in $(\ref{ineq6})$, the right hand side term
$C\int_{Q}e^{2\tau\varphi}|\bar{w}_{ttt}|^{2}dQ$ can be absorbed by the term
$C_{1,\tau}\int_{Q}e^{2\tau\varphi}[\bar{w}_{t}^{2}+\bar{w}_{tt}^{2}+\bar{w}_{ttt}^{2}]dQ$
on the left hand side when $\tau$ is large enough. In addition, since
$e^{2\tau\varphi}<e^{2\tau\sigma}$ on $Q\setminus Q(\sigma)$ by the definition
of $Q(\sigma)$, we have
(3.15)
$\begin{split}C(\tau+1)\int_{Q}e^{2\tau\varphi}|\bar{w}_{tt}|^{2}dQ&=C(\tau+1)\left(\int_{Q(\sigma)}e^{2\tau\varphi}|\bar{w}_{tt}|^{2}dtdx+\int_{Q\setminus
Q(\sigma)}e^{2\tau\varphi}|\bar{w}_{tt}|^{2}dxdt\right)\\\ &\leq
C(\tau+1)\left(\int_{Q(\sigma)}e^{2\tau\varphi}|\bar{w}_{tt}|^{2}dtdx+e^{2\tau\sigma}\int_{Q\setminus
Q(\sigma)}|\bar{w}_{tt}|^{2}dxdt\right)\\\ &\leq
C(\tau+1)\int_{Q(\sigma)}e^{2\tau\varphi}|\bar{w}_{tt}|^{2}dtdx+C(\tau+1)e^{2\tau\sigma}\end{split}$
Again $C(\tau+1)\int_{Q(\sigma)}e^{2\tau\varphi}|\bar{w}_{tt}|^{2}dtdx$ on the
right hand side of $\eqref{absorb}$ can be absorbed by
$C_{2,\tau}\int_{Q(\sigma)}e^{2\tau\varphi}[\bar{w}^{2}+\bar{w}_{t}^{2}+\bar{w}_{tt}^{2}]dxdt$
on the left hand side of $(\ref{ineq6})$ when taking $\tau$ large enough.
Therefore $(\ref{ineq6})$ becomes to
(3.16)
$C_{1,\tau}^{{}^{\prime}}\int_{Q}e^{2\tau\varphi}[\bar{w}_{t}^{2}+\bar{w}_{tt}^{2}+\bar{w}_{ttt}^{2}+|\nabla\bar{w}|^{2}+|\nabla\bar{w}_{t}|^{2}+|\nabla\bar{w}_{tt}|^{2}]dQ\\\
+C_{2,\tau}^{{}^{\prime}}\int_{Q(\sigma)}e^{2\tau\varphi}[\bar{w}^{2}+\bar{w}_{t}^{2}+\bar{w}_{tt}^{2}]dxdt\leq
C\left((\tau+1)e^{2\tau\sigma}+e^{2\tau\sigma}+\tau^{3}e^{-2\tau\delta}\right)$
Where we have
(3.17) $C_{1,\tau}^{{}^{\prime}}=\tau\epsilon\rho-C,\quad
C_{2,\tau}^{{}^{\prime}}=2\tau^{3}\beta+\mathcal{O}(\tau^{2})$
Now we take $\tau$ sufficiently large such that $C_{1,\tau}^{{}^{\prime}}>0$,
$C_{2,\tau}^{{}^{\prime}}>0$. Then in $(\ref{ineq7})$ we can drop the first
term on the left hand side and get
(3.18)
$\begin{split}C_{2,\tau}^{{}^{\prime}}\int_{Q(\sigma)}e^{2\tau\varphi}[\bar{w}^{2}+\bar{w}_{t}^{2}+\bar{w}_{tt}^{2}]dxdt&\leq
C[(\tau+1)e^{2\tau\sigma}+e^{2\tau\sigma}]\\\ &\leq
C(\tau+2)e^{2\tau\sigma}\end{split}$
Note again from $\eqref{qsigma}$ the definition of $Q(\sigma)$, we have
$e^{2\tau\varphi}\geq e^{2\tau\sigma}$ on $Q(\sigma)$, therefore
$(\ref{ineq8})$ implies
(3.19)
$C_{2,\tau}^{{}^{\prime}}\int_{Q(\sigma)}[\bar{w}^{2}+\bar{w}_{t}^{2}+\bar{w}_{tt}^{2}]dxdt\leq
C(\tau+2)$
Divide $\tau+2$ on both sides of $\eqref{ineq9}$, we get
(3.20)
$\frac{C_{2,\tau}^{{}^{\prime}}}{\tau+2}\int_{Q(\sigma)}[\bar{w}^{2}+\bar{w}_{t}^{2}+\bar{w}_{tt}^{2}]dxdt\leq
C$
By $\eqref{ctauprime}$, $\frac{C_{2,\tau}^{{}^{\prime}}}{\tau+2}\to\infty$ as
$\displaystyle\tau\to\infty$, thus $\eqref{ineq10}$ implies that we must have
$\bar{w}\equiv 0$ on $Q(\sigma)$ and hence we have
(3.21)
$f(x)R_{t}(x,t)=\bar{w}_{tt}(x,t)-\Delta\bar{w}(x,t)-q(x)\bar{w}(x,t)=0,\quad(x,t)\in
Q(\sigma)$
Recall again that $|R_{t}(x,\frac{T}{2})|\geq r_{1}>0$ from$\eqref{crucialR}$
and the property that $Q\supset Q(\sigma)\supset[t_{0},t_{1}]\times\Omega$
from $\eqref{qsigmaproperty}$. Thus we have from $\eqref{identity}$ that
$f(x)\equiv 0$, for all $x\in\Omega$. $\qquad\Box$
## 4\. Proof of Theorem 1.5
Setting $f(x)=q(x)-p(x)$, $w(x,t)=z(q)(x,t)-z(p)(x,t)$,
$u(x,t)=v(q)(x,t)-v(p)(x,t)$ and $R(x,t)=z(p)(x,t)$, we then obtain
$\eqref{linear}$ after the subtraction of $\eqref{nonlinear}$ with $p$ from
$\eqref{nonlinear}$ with $q$. Since
$R(x,\frac{T}{2})=z(p)(x,\frac{T}{2})=z_{0}(x)$ and
$R_{t}(x,\frac{T}{2})=z_{t}(p)(x,\frac{T}{2})=z_{1}(x)$, the conditions
$\eqref{crucialz}$ imply $\eqref{crucialR}$. In addition, the condition
$v(q)(x,t)=v(p)(x,t)$, $x\in\Gamma_{0}$, $t\in[0,T]$ implies that $u(x,t)=0$
on $\Gamma_{0}\times[0,T]$ and $\eqref{differenceh2}$ implies $\eqref{h2reg}$.
Therefore from the above Theorem 1.4 we conclude $f(x)=q(x)-p(x)=0$, i.e.,
$q(x)=p(x)$, $x\in\Omega$. $\Box$
## 5\. Proof of Theorem 1.6
In relation with this system $\eqref{lineary}$, we define $\psi$ which
satisfies the following equation
(5.1) $\begin{cases}\psi_{tt}(x,t)=\Delta\psi(x,t)+q(x)\psi(x,t)&\mbox{in
}\Omega\times[0,T]\\\ \frac{\partial\psi}{\partial\nu}(x,t)=0&\mbox{on
}\Gamma_{1}\times[0,T]\\\
\frac{\partial\psi}{\partial\nu}(x,t)=u_{tt}(x,t)&\mbox{on
}\Gamma_{0}\times[0,T]\\\ \psi(\cdot,\frac{T}{2})=0&\mbox{in }\Omega\\\
\psi_{t}(\cdot,\frac{T}{2})=f(x)R(x,\frac{T}{2})&\mbox{in }\Omega\end{cases}$
Set $y=\bar{w}-\psi$, then we have $y$ satisfies the following initial-
boundary value problem
(5.2) $\begin{cases}y_{tt}(x,t)-\Delta
y(x,t)-q(x)y(x,t)=f(x)R_{t}(x,t)&\mbox{in }\Omega\times[0,T]\\\ \frac{\partial
y}{\partial\nu}(x,t)=0&\mbox{on }\Gamma\times[0,T]\\\
y(\cdot,\frac{T}{2})=0&\mbox{in }\Omega\\\ y_{t}(\cdot,\frac{T}{2})=0&\mbox{in
}\Omega\\\ \end{cases}$
It is easy to see that both $\eqref{eqpsi}$ and $\eqref{eqy}$ are well-posed.
For the system $\eqref{eqpsi}$, we apply the continuous observability
inequality in Theorem 2.4 to get
(5.3) $\|fR(\cdot,\frac{T}{2})\|^{2}_{L^{2}(\Omega)}\leq
C\left(\|\psi\|^{2}_{L^{2}(\Gamma_{0}\times[0,T])}+\|\psi_{t}\|^{2}_{L^{2}(\Gamma_{0}\times[0,T])}+\|\frac{\partial\psi}{\partial\nu}\|^{2}_{L^{2}(\Gamma_{0}\times[0,T])}\right)$
Notice that $|R(x,\frac{T}{2})|\geq r_{0}>0$,
$\frac{\partial\psi}{\partial\nu}(x,t)=u_{tt}(x,t)$ on $\Gamma_{0}\times[0,T]$
and $\frac{\partial\psi}{\partial\nu}(x,t)=0$ on $\Gamma_{1}\times[0,T]$,
therefore we have from $\eqref{ineqfr}$
(5.4) $\|f\|_{L^{2}(\Omega)}\leq
C\left(\|\psi\|_{L^{2}(\Gamma_{0}\times[0,T])}+\|\psi_{t}\|_{L^{2}(\Gamma_{0}\times[0,T])}+\|u_{tt}\|_{L^{2}(\Gamma_{0}\times[0,T])}\right)$
On the other hand, for the system $\eqref{eqy}$, we have the following lemma:
###### Lemma 5.1.
Let $q\in L^{\infty}(\Omega)$ and $R(x,t)$ satisfies $R_{t}\in
H^{\frac{1}{2}+\epsilon}(0,T;L^{\infty}(\Omega))$ for some
$0<\epsilon<\frac{1}{2}$ as in Theorem 1.6. If we define the operators $K$ and
$K_{1}$ by $K,K_{1}:L^{2}(\Omega)\rightarrow L^{2}(\Gamma_{0}\times[0,T])$,
such that
(5.5) $(Kf)(x,t)=y(x,t),\quad(K_{1}f)(x,t)=y_{t}(x,t),\quad
x\in\Gamma_{0},t\in[0,T]$
where $y$ is the unique solution of the equation $\eqref{eqy}$. Then $K$ and
$K_{1}$ are both compact operators.
###### Proof.
It suffices to just show that $K_{1}$ is compact, then it follows similarly
that $K$ is also compact. Since $f\in L^{2}(\Omega)$ and $R_{t}\in
H^{\frac{1}{2}+\epsilon}(0,T;L^{\infty}(\Omega))$, we have
(5.6) $fR_{t}\in H^{\frac{1}{2}+\epsilon}(0,T;L^{2}(\Omega))$
Therefore we have the solution $y$ satisfies (e.g. Corollary 5.3 in [27])
(5.7) $y\in C([0,T];H^{\frac{3}{2}+\epsilon}(\Omega)),\quad y_{t}\in
C([0,T];H^{\frac{1}{2}+\epsilon}(\Omega))$
Hence by $\eqref{regfRt}$, $q\in L^{\infty}(\Omega)$ and $y_{tt}=\Delta
y+q(x)y+fR_{t}$ we can get
(5.8) $y_{tt}\in L^{2}(0,T;H^{-\frac{1}{2}+\epsilon}(\Omega))$
In addition, by $\eqref{sharpy}$ and trace theorem we have $y_{t}\in
C([0,T];H^{\epsilon}(\Gamma))$. Since the embedding $H^{\epsilon}(\Gamma)\to
L^{2}(\Gamma)$ is compact, we have by Lions-Aubin’s compactness criterion
(e.g. Proposition III.1.3 in [33]) that the operator $K_{1}$ is a compact
operator. ∎
Now we have that the inequality $\eqref{ineqfr1}$ becomes to
(5.9) $\begin{split}\|f\|_{L^{2}(\Omega)}&\leq
C\left(\|\psi\|_{L^{2}(\Gamma_{0}\times[0,T])}+\|\psi_{t}\|_{L^{2}(\Gamma_{0}\times[0,T])}+\|u_{tt}\|_{L^{2}(\Gamma_{0}\times[0,T])}\right)\\\
&\leq
C\left(\|\bar{w}-y\|_{L^{2}(\Gamma_{0}\times[0,T])}+\|\bar{w}_{t}-y_{t}\|_{L^{2}(\Gamma_{0}\times[0,T])}+\|u_{tt}\|_{L^{2}(\Gamma_{0}\times[0,T])}\right)\\\
&\leq
C\left(\|\bar{w}\|_{L^{2}(\Gamma_{0}\times[0,T])}+\|\bar{w}_{t}\|_{L^{2}(\Gamma_{0}\times[0,T])}+\|u_{tt}\|_{L^{2}(\Gamma_{0}\times[0,T])}\right)\\\
&\qquad+C\|y\|_{L^{2}(\Gamma_{0}\times[0,T])}+C\|y_{t}\|_{L^{2}(\Gamma_{0}\times[0,T])}\\\
&=C\left(\|\bar{w}\|_{L^{2}(\Gamma_{0}\times[0,T])}+\|\bar{w}_{t}\|_{L^{2}(\Gamma_{0}\times[0,T])}+\|u_{tt}\|_{L^{2}(\Gamma_{0}\times[0,T])}\right)\\\
&\qquad+C\|Kf\|_{L^{2}(\Gamma_{0}\times[0,T])}+C\|K_{1}f\|_{L^{2}(\Gamma_{0}\times[0,T])}\\\
&\leq
C\left(\|u_{tt}\|_{L^{2}(\Gamma_{0}\times[0,T])}+\|u_{ttt}\|_{L^{2}(\Gamma_{0}\times[0,T])}+\|\Delta^{2}u_{tt}\|_{L^{2}(\Gamma_{0}\times[0,T])}\right)\\\
&\qquad+C\|Kf\|_{L^{2}(\Gamma_{0}\times[0,T])}+C\|K_{1}f\|_{L^{2}(\Gamma_{0}\times[0,T])}\end{split}$
where in the last step we use the coupling
$\bar{w}(x,t)=-u_{tt}(x,t)-\Delta^{2}u(x,t)-\Delta^{2}u_{t}(x,t)$ on
$\Gamma_{0}\times[0,T]$ from $\eqref{lineary}$ and again the initial
conditions $u(\cdot,\frac{T}{2})=u_{t}(\cdot,\frac{T}{2})=0$ on
$\Gamma_{0}\times[0,T]$ so that by the fundamental theorem of calculus, we
have
(5.10) $\|u\|_{L^{2}(\Gamma_{0}\times[0,T])}\leq
C\|u_{t}\|_{L^{2}(\Gamma_{0}\times[0,T])}\leq
C\|u_{tt}\|_{L^{2}(\Gamma_{0}\times[0,T])}$
To complete the proof, we need to absorb the last two terms in
$\eqref{ineq11}$. To achieve that, we apply the compactness-uniqueness
argument. For simplicity we denote
$\|u\|_{X}=\|u_{tt}\|_{L^{2}(\Gamma_{0}\times[0,T])}+\|u_{ttt}\|_{L^{2}(\Gamma_{0}\times[0,T])}+\|\Delta^{2}u_{tt}\|_{L^{2}(\Gamma_{0}\times[0,T])}$
Suppose contrarily that the inequality $\eqref{stability}$ does not hold. Then
there exists $f_{n}\in L^{2}(\Omega)$, $n\geq 1$ such that
(5.11) $\|f_{n}\|_{L^{2}(\Omega)}=1,\quad n\geq 1$
and
(5.12) $\lim_{n\to\infty}\|u(f_{n})\|_{X}=0$
From $\eqref{contrary1}$, there exists a subsequence, denoted again by
$\\{f_{n}\\}_{n\geq 1}$ such that $f_{n}$ converges to some $f_{0}\in
L^{2}(\Omega)$ weakly in $L^{2}(\Omega)$. Moreover, since $K$ and $K_{1}$ are
compact, we have
(5.13)
$\lim_{m,n\to\infty}\|Kf_{n}-Kf_{m}\|_{L^{2}(\Gamma_{0}\times[0,T])}=0,\quad\lim_{m,n\to\infty}\|K_{1}f_{n}-K_{1}f_{m}\|_{L^{2}(\Gamma_{0}\times[0,T])}=0$
On the other hand, it follows from $\eqref{ineq11}$ that
(5.14) $\begin{split}\|f_{n}-f_{m}\|_{L^{2}(\Omega)}&\leq
C\|u(f_{n})-u(f_{m})\|_{X}+C\|Kf_{n}-Kf_{m}\|_{L^{2}(\Gamma_{0}\times[0,T])}\\\
&\qquad+C\|K_{1}f_{n}-K_{1}f_{m}\|_{L^{2}(\Gamma_{0}\times[0,T])}\\\ &\leq
C\|u(f_{n})\|_{X}+C\|u(f_{m})\|_{X}+C\|Kf_{n}-Kf_{m}\|_{L^{2}(\Gamma_{0}\times[0,T])}\\\
&\qquad+C\|K_{1}f_{n}-K_{1}f_{m}\|_{L^{2}(\Gamma_{0}\times[0,T])}\end{split}$
Thus by $\eqref{contrary2}$ and $\eqref{compactidentity}$, we have that
(5.15) $\lim_{m,n\to\infty}\|f_{n}-f_{m}\|_{L^{2}(\Omega)}=0$
and hence $f_{n}$ converges strongly to $f_{0}$ in $L^{2}(\Omega)$. So by
$\eqref{contrary1}$ we obtain
(5.16) $\|f_{0}\|_{L^{2}(\Omega)}=1$
On the other hand, by $\eqref{crucialR}$ and a usual a-priori estimate, we
have that
(5.17)
$\begin{split}\|\bar{w}(f)\|_{C([0,T];H^{1}(\Omega))}+\|\bar{w}_{t}(f)\|_{C([0,T];L^{2}(\Omega))}&\leq
C\|fR_{t}\|_{L^{1}(0,T;L^{2}(\Omega))}\\\ &\leq
C\|R_{t}\|_{L^{1}(0,T;L^{\infty}(\Omega))}\|f\|_{L^{2}(\Omega)}\end{split}$
Hence trace theorem implies that
(5.18) $\|\bar{w}(f)\|_{L^{2}(\Gamma_{0}\times[0,T])}\leq
C\|f\|_{L^{2}(\Omega)}$
where $C>0$ depends on $\|R_{t}\|_{L^{1}(0,T;L^{\infty}(\Omega))}$. Therefore
by $\eqref{traceineq}$ we have
(5.19)
$\lim_{n\to\infty}\|\bar{w}(f_{n})-\bar{w}(f_{0})\|_{L^{2}(\Gamma_{0}\times[0,T])}\leq
C\lim_{n\to\infty}\|f_{n}-f_{0}\|_{L^{2}(\Omega)}=0$
Moreover, by $\eqref{contrary2}$ and the coupling
$\bar{w}(x,t)=-u_{tt}(x,t)-\Delta^{2}u(x,t)-\Delta^{2}u_{t}(x,t)$ on
$\Gamma_{0}\times[0,T]$, we have
(5.20)
$\lim_{n\to\infty}\|\bar{w}(f_{n})\|_{L^{2}(\Gamma_{0}\times[0,T])}\leq\lim_{n\to\infty}\|u\|_{X}=0$
Thus by $\eqref{ineq12}$ and $\eqref{iden4}$, we obtain
(5.21) $\bar{w}(f_{0})(x,t)=0,\quad x\in\Gamma_{0},t\in[0,T]$
Therefore from $\eqref{lineary}$ we have $u=u(f_{0})$ satisfies the initial
boundary problem:
(5.22)
$\begin{cases}-u_{tt}(x,t)-\Delta^{2}u(x,t)-\Delta^{2}u_{t}(x,t)=0&\mbox{in
}\Gamma_{0}\times[0,T]\\\ u(x,t)=\frac{\partial
u}{\partial\nu}(x,t)=0&\mbox{on }\partial\Gamma_{0}\times[0,T]\\\
u(\cdot,\frac{T}{2})=0&\mbox{in }\Gamma_{0}\\\
u_{t}(\cdot,\frac{T}{2})=0&\mbox{in }\Gamma_{0}\end{cases}$
which has only zero solution, namely, we have
$u(f_{0})(x,t)=0,x\in\Gamma_{0},t\in[0,T]$. Therefore by the uniqueness
theorem 1.4, we have $f_{0}\equiv 0$ in $\Omega$ which contradicts with
$\eqref{iden3}$. Thus we must have
(5.23) $\|f\|_{L^{2}(\Omega)}\leq
C\left(\|u_{tt}\|_{L^{2}(\Gamma_{0}\times[0,T])}+\|u_{ttt}\|_{L^{2}(\Gamma_{0}\times[0,T])}+\|\Delta^{2}u_{tt}\|_{L^{2}(\Gamma_{0}\times[0,T])}\right)$
and the proof of the theorem is complete. $\Box$
## 6\. Proof of Theorem 1.7
We now go back to the original system $\eqref{nonlinear}$.
Case 1: n=2. Let $\bar{z}=z_{t}$ and $\bar{v}=v_{t}$, then the system
$\eqref{nonlinear}$ becomes to
(6.1)
$\begin{cases}\bar{z}_{tt}(x,t)=\Delta\bar{z}(x,t)+q(x)\bar{z}(x,t)&\mbox{in
}\Omega\times[0,T]\\\ \frac{\partial\bar{z}}{\partial\nu}(x,t)=0&\mbox{on
}\Gamma_{1}\times[0,T]\\\
\bar{z}_{t}(x,t)=-\bar{v}_{tt}(x,t)-\Delta^{2}\bar{v}(x,t)-\Delta^{2}\bar{v}_{t}(x,t)&\mbox{on
}\Gamma_{0}\times[0,T]\\\
\bar{v}(x,t)=\frac{\partial\bar{v}}{\partial\nu}(x,t)=0&\mbox{on
}\partial\Gamma_{0}\times[0,T]\\\
\frac{\partial\bar{z}}{\partial\nu}(x,t)=\bar{v}_{t}(x,t)&\mbox{on
}\Gamma_{0}\times[0,T]\\\ \bar{z}(\cdot,\frac{T}{2})=z_{1}(x)&\mbox{in
}\Omega\\\ \bar{z}_{t}(\cdot,\frac{T}{2})=\Delta z_{0}(x)+q(x)z_{0}&\mbox{in
}\Omega\\\ \bar{v}(\cdot,\frac{T}{2})=v_{1}(x)&\mbox{on }\Gamma_{0}\\\
\bar{v}_{t}(\cdot,\frac{T}{2})=-z_{1}(x)-\Delta^{2}v_{0}(x)-\Delta^{2}v_{1}(x)&\mbox{on
}\Gamma_{0}\end{cases}$
By using the similar operator setting as in section 1.2 and notice the new
initial conditions, we can compute the domain of the operator
$\mathcal{A}^{2}$:
(6.2)
$\begin{split}D(\mathcal{A}^{2})&=\\{[z_{0},z_{1},v_{0},v_{1}]^{T}:(z_{1},(-A_{N}+q)z_{0}+Bv_{1},v_{1},-\textbf{\AA}(v_{0}+v_{1})-B^{*}z_{1}\in
D(\mathcal{A})\\}\\\ &=\\{[z_{0},z_{1},v_{0},v_{1}]^{T}:z_{1}\in
H^{2}_{\Gamma_{1}}(\Omega),(-A_{N}+q)z_{0}+Bv_{1}\in
H^{1}_{\Gamma_{1}}(\Omega),v_{0}\in H^{2}_{0}(\Gamma_{0}),\\\ &\qquad v_{1}\in
H^{2}_{0}(\Gamma_{0}),\textbf{\AA}(v_{0}+v_{1})+B^{*}z_{1}\in
H^{2}_{0}(\Gamma_{0}),\\\ &\qquad
v_{1}-\textbf{\AA}(v_{0}+v_{1})-B^{*}z_{1}\in D(\textbf{\AA}),\frac{\partial
z_{1}}{\partial\nu}|_{\Gamma_{0}}=-\textbf{\AA}(v_{0}+v_{1})-B^{*}z_{1}\\}\\\
&=\\{[z_{0},z_{1},v_{0},v_{1}]^{T}:z_{1}\in
H^{2}_{\Gamma_{1}}(\Omega),(\Delta+q)z_{0}\in
H^{1}_{\Gamma_{1}}(\Omega),\frac{\partial
z_{0}}{\partial\nu}|_{\Gamma_{0}}=v_{1},\\\ &\qquad v_{0}\in
H^{2}_{0}(\Gamma_{0}),v_{1}\in
H^{2}_{0}(\Gamma_{0}),\textbf{\AA}(v_{0}+v_{1})+B^{*}z_{1}\in
H^{2}_{0}(\Gamma_{0}),\\\ &\qquad
v_{1}-\textbf{\AA}(v_{0}+v_{1})-B^{*}z_{1}\in D(\textbf{\AA}),\frac{\partial
z_{1}}{\partial\nu}|_{\Gamma_{0}}=-\textbf{\AA}(v_{0}+v_{1})-B^{*}z_{1}\\}\\\
&=\\{[z_{0},z_{1},v_{0},v_{1}]^{T}:z_{0}\in
H^{3}_{\Gamma_{1}}(\Omega),z_{1}\in H^{2}_{\Gamma_{1}}(\Omega),v_{0}\in
H^{2}_{0}(\Gamma_{0}),v_{1}\in H^{2}_{0}(\Gamma_{0}),\\\
&\qquad\textbf{\AA}(v_{0}+v_{1})+B^{*}z_{1}\in
H^{2}_{0}(\Gamma_{0}),v_{1}-\textbf{\AA}(v_{0}+v_{1})-B^{*}z_{1}\in
D(\textbf{\AA}),\\\ &\qquad\frac{\partial
z_{0}}{\partial\nu}|_{\Gamma_{0}}=v_{1},\frac{\partial
z_{1}}{\partial\nu}|_{\Gamma_{0}}=-\textbf{\AA}(v_{0}+v_{1})-B^{*}z_{1}\\}\end{split}$
where in the last step $z_{0}\in H^{3}_{\Gamma_{1}}(\Omega)$ is from elliptic
theory when provided that $q(x)\in W^{1,\infty}(\Omega)$. Therefore when
$z_{0}\in H^{3}_{\Gamma_{1}}(\Omega)$, $z_{1}\in H^{2}_{\Gamma_{1}}(\Omega)$,
$v_{0}\in H^{2}_{0}(\Gamma_{0})$, $v_{1}\in H^{2}_{0}(\Gamma_{0})$ with
compatible conditions as in $D(\mathcal{A}^{2})$ and $q\in
W^{1,\infty}(\Omega)$, then from semigroup theory we have that the solution of
$\eqref{nonlinear2}$ satisfies
(6.3) $\bar{z}_{t}\in C([0,T];H^{1}(\Omega)),\quad\bar{z}_{tt}\in
C([0,T];L^{2}(\Omega))$
Hence we have on the one hand
(6.4) $z_{t}\in H^{1}(0,T;H^{1}(\Omega))$
On the other hand, from $\eqref{barztztt}$ and
$\bar{z}_{tt}(x,t)=\Delta\bar{z}(x,t)+q(x)\bar{z}(x,t)$, we have by elliptic
theory that
(6.5) $z_{t}=\bar{z}\in L^{2}(0,T;H^{2}(\Omega))$
Interpolate between $\eqref{regzt11}$ and $\eqref{regzt02}$, we have for
$0<\epsilon<\frac{1}{2}$,
(6.6) $z_{t}\in
H^{\frac{1}{2}+\epsilon}(0,T;H^{\frac{3}{2}-\epsilon}(\Omega))\subset
H^{\frac{1}{2}+\epsilon}(0,T;L^{\infty}(\Omega))$
where the inclusion is by Sobolev embedding theorem.
Case 2: n=3. We let $\bar{\bar{z}}=z_{tt}$, $\bar{\bar{v}}=v_{tt}$, then we
have $\bar{\bar{z}}$, $\bar{\bar{v}}$ satisfy
(6.7)
$\begin{cases}\bar{\bar{z}}_{tt}(x,t)=\Delta\bar{\bar{z}}(x,t)+q(x)\bar{\bar{z}}(x,t)&\mbox{in
}\Omega\times[0,T]\\\
\frac{\partial\bar{\bar{z}}}{\partial\nu}(x,t)=0&\mbox{on
}\Gamma_{1}\times[0,T]\\\
\bar{\bar{z}}_{t}(x,t)=-\bar{\bar{v}}_{tt}(x,t)-\Delta^{2}\bar{\bar{v}}(x,t)-\Delta^{2}\bar{\bar{v}}_{t}(x,t)&\mbox{on
}\Gamma_{0}\times[0,T]\\\
\bar{\bar{v}}(x,t)=\frac{\partial\bar{\bar{v}}}{\partial\nu}(x,t)=0&\mbox{on
}\partial\Gamma_{0}\times[0,T]\\\
\frac{\partial\bar{\bar{z}}}{\partial\nu}(x,t)=\bar{\bar{v}}_{t}(x,t)&\mbox{on
}\Gamma_{0}\times[0,T]\\\ \bar{\bar{z}}(\cdot,\frac{T}{2})=\Delta
z_{0}(x)+q(x)z_{0}&\mbox{in }\Omega\\\
\bar{\bar{z}}_{t}(\cdot,\frac{T}{2})=\Delta z_{1}(x)+q(x)z_{1}&\mbox{in
}\Omega\\\
\bar{\bar{v}}(\cdot,\frac{T}{2})=-z_{1}(x)-\Delta^{2}v_{0}(x)-\Delta^{2}v_{1}(x)&\mbox{on
}\Gamma_{0}\\\ \bar{\bar{v}}_{t}(\cdot,\frac{T}{2})=-\Delta
z_{0}(x)-q(x)z_{0}(x)-\Delta^{2}v_{1}(x)\\\ \hskip
72.26999pt+\Delta^{2}z_{1}(x)+\Delta^{4}v_{0}(x)+\Delta^{4}v_{1}(x)&\mbox{on
}\Gamma_{0}\end{cases}$
Then still using the similarly operator setting as before we can compute the
domain of $\mathcal{A}^{3}$:
(6.8)
$\begin{split}D(\mathcal{A}^{3})&=\\{[z_{0},z_{1},v_{0},v_{1}]^{T}:(z_{1},(-A_{N}+q)z_{0}+Bv_{1},v_{1},-\textbf{\AA}(v_{0}+v_{1})-B^{*}z_{1}\in
D(\mathcal{A}^{2})\\}\\\ &=\\{[z_{0},z_{1},v_{0},v_{1}]^{T}:z_{1}\in
H^{3}_{\Gamma_{1}}(\Omega),(\Delta+q)z_{0}\in
H^{2}_{\Gamma_{1}}(\Omega),\frac{\partial
z_{0}}{\partial\nu}|_{\Gamma_{0}}=v_{1},\\\ &v_{0}\in
H^{2}_{0}(\Gamma_{0}),v_{1}\in
H^{2}_{0}(\Gamma_{0}),\textbf{\AA}(v_{0}+v_{1})+B^{*}z_{1}\in
H^{2}_{0}(\Gamma_{0}),\\\
&\textbf{\AA}(v_{1}-\textbf{\AA}(v_{0}+v_{1})-B^{*}z_{1})+B^{*}[(-A_{N}+q)z_{0}+Bv_{1}]\in
H^{2}_{0}(\Gamma_{0}),\\\
&\textbf{\AA}(v_{0}+v_{1})+B^{*}z_{1}+\textbf{\AA}[v_{1}-\textbf{\AA}(v_{0}+v_{1})-B^{*}z_{1}]+B^{*}[(-A_{N}+q)z_{0}+Bv_{1}]\in
D(\textbf{\AA})\\\ &\frac{\partial
z_{1}}{\partial\nu}|_{\Gamma_{0}}=-\textbf{\AA}(v_{0}+v_{1})-B^{*}z_{1},\frac{\partial[(-A_{N}+q)z_{0}+Bv_{1}]}{\partial\nu}|_{\Gamma_{0}}=\\\
&-\textbf{\AA}[v_{1}-\textbf{\AA}(v_{0}+v_{1})-B^{*}z_{1}]-B^{*}[(-A_{N}+q)z_{0}+Bv_{1}]\\}\\\
&=\\{[z_{0},z_{1},v_{0},v_{1}]^{T}:z_{0}\in
H^{\frac{7}{2}}_{\Gamma_{1}}(\Omega),z_{1}\in
H^{3}_{\Gamma_{1}}(\Omega),v_{0}\in H^{2}_{0}(\Gamma_{0}),v_{1}\in
H^{2}_{0}(\Gamma_{0}),\\\ &\textbf{\AA}(v_{0}+v_{1})+B^{*}z_{1}\in
H^{2}_{0}(\Gamma_{0}),\frac{\partial
z_{0}}{\partial\nu}|_{\Gamma_{0}}=v_{1},\frac{\partial
z_{1}}{\partial\nu}|_{\Gamma_{0}}=-\textbf{\AA}(v_{0}+v_{1})-B^{*}z_{1}\
\text{on}\ \Gamma_{0},\\\
&\textbf{\AA}(v_{0}+v_{1})+B^{*}z_{1}+\textbf{\AA}[v_{1}-\textbf{\AA}(v_{0}+v_{1})-B^{*}z_{1}]+B^{*}[(-A_{N}+q)z_{0}+Bv_{1}]\in
D(\textbf{\AA})\\\
&\textbf{\AA}(v_{1}-\textbf{\AA}(v_{0}+v_{1})-B^{*}z_{1})+B^{*}[(-A_{N}+q)z_{0}+Bv_{1}]\in
H^{2}_{0}(\Gamma_{0}),\\\
&\frac{\partial[(-A_{N}+q)z_{0}+Bv_{1}]}{\partial\nu}|_{\Gamma_{0}}=-\textbf{\AA}[v_{1}-\textbf{\AA}(v_{0}+v_{1})-B^{*}z_{1}]-B^{*}[(-A_{N}+q)z_{0}+Bv_{1}]\\}\end{split}$
where in the last step $z_{0}\in H^{\frac{7}{2}}_{\Gamma_{1}}(\Omega)$ is from
trace theory of solving $\frac{\partial
z_{0}}{\partial\nu}|_{\Gamma_{0}}=v_{1}\in H^{2}(\Gamma_{0})$. Therefore when
$z_{0}\in H^{\frac{7}{2}}_{\Gamma_{1}}(\Omega)$, $z_{1}\in
H^{3}_{\Gamma_{1}}(\Omega)$, $v_{0}\in H^{2}_{0}(\Gamma_{0})$, $v_{1}\in
H^{2}_{0}(\Gamma_{0})$ with compatible conditions as in $D(\mathcal{A}^{3})$
and $q\in W^{2,\infty}(\Omega)$, then from semigroup theory we have that the
solution of $\eqref{nonlinear2}$ satisfies
(6.9) $\bar{\bar{z}}_{t}\in C([0,T];H^{1}(\Omega)),\quad\bar{\bar{z}}_{tt}\in
C([0,T];L^{2}(\Omega))$
Hence we have on the one hand
(6.10) $z_{t}\in H^{2}(0,T;H^{1}(\Omega))$
On the other hand, from $\eqref{barbarztztt}$ and
$\bar{\bar{z}}_{tt}(x,t)=\Delta\bar{\bar{z}}(x,t)+q(x)\bar{\bar{z}}(x,t)$, we
have by elliptic theory that
(6.11) $z_{tt}=\bar{\bar{z}}\in L^{2}(0,T;H^{2}(\Omega))$
which implies
(6.12) $z_{t}\in H^{1}(0,T;H^{2}(\Omega))$
Now interpolate between $\eqref{regzt21}$ and $\eqref{regzt12}$, we have for
$0<\epsilon<\frac{1}{2}$,
(6.13) $z_{t}\in H^{\frac{3}{2}}(0,T;H^{\frac{3}{2}}(\Omega))\subset
H^{\frac{1}{2}+\epsilon}(0,T;L^{\infty}(\Omega))$
where the inclusion is again by Sobolev embedding.
Hence in either case $n=2$ or $n=3$, under the assumptions on the initial data
$[z_{0},z_{1},v_{0},v_{1}]$ and $q(x)$, $p(x)$ in Theorem 1.7, we have that
$z_{t}\in H^{\frac{1}{2}+\epsilon}(0,T;L^{\infty}(\Omega))$. Thus when we
again set $f(x)=q(x)-p(x)$, $w(x,t)=z(q)(x,t)-z(p)(x,t)$,
$u(x,t)=v(q)(x,t)-v(p)(x,t)$ and $R(x,t)=z(p)(x,t)$ as in section 4, we obtain
$\eqref{regRt}$ that $R_{t}\in
H^{\frac{1}{2}+\epsilon}(0,T;L^{\infty}(\Omega))$ and hence all the
assumptions in theorem 1.6 are satisfied. Therefore, we get the desired
stability $\eqref{nonlinearstability}$ from the stability $\eqref{stability}$
of the linear inverse problem in Theorem 1.6. $\Box$
## 7\. Concluding remark
As we mentioned at the beginning and the calculations of $D(\mathcal{A}^{2})$
and $D(\mathcal{A}^{3})$ show, the lack of compactness of the resolvent limits
the space regularity of the solutions for the wave equation parts since we
always have the elliptic problem for $z$ or $z_{t}$ such that $(\Delta+q)z\in
L^{2}(\Omega)$ with $\frac{\partial z}{\partial\nu}\in H^{2}_{0}(\Gamma_{0})$
provided $q$ in some suitable space. Therefore the best space regularity that
$z$ could get is $2+\frac{3}{2}=\frac{7}{2}$ from elliptic and trace theory.
As a result, our argument of the stability in the nonlinear inverse problem
will only work for dimension up to $n=7$ as we need the Sobolev embedding
$H^{\frac{n}{2}}(\Omega)\subset L^{\infty}(\Omega)$ in order to achieve the
space regularity of $z_{t}$ in $L^{\infty}(\Omega)$ which is needed in the
proof.
## References
* [1] G. Avalos, The exponential stability of a coupled hyperbolic/parabolic system arising in structural acoustics, _Appl. Abstr. Anal._ 1 (2), 203-217.
* [2] G. Avalos, Exact-approximate boundary reachability of thermoelastic plates under variable thermal coupling, _Inverse Problems_ , 16 (2000), 979-996.
* [3] G. Avalos and I. Lasiecka, Differential Riccati equation for the active control of a problem in structural acoustics, _J. Optim. Theory. Appl._ , 91 (3) (2001), 695-728.
* [4] G. Avalos and I. Lasiecka, The strong stability of a semigroup arising from a coupled hyperbolic/parabolic system, _Semigroup Forum_ , Vol. 57 (1998), 278-292.
* [5] G. Avalos and I. Lasiecka, Exact controllability of structrual acoustic interactions, _J. Math. Pures. Appl._ , 82 (2003), 1047-1073.
* [6] G. Avalos and I. Lasiecka, Exact reachability of finite energy states for an acoustic wave/plate interaction under the influence of boundary and localized controls, _IMA Preprint Series $\sharp$2017_, (January, 2005).
* [7] G. Avalos, I. Lasiecka and R. Rebarber, Well-posedness of a structrual acoustics control model with point observation of the pressure, _Journal of Differential Equations_ , 173 (2001), 40-78.
* [8] G. Avalos, I. Lasiecka and R. Rebarber, Boundary controllability of a coupled wave/Kirchoff system, _System and Control Letters_ , 50 (2003), 331-341.
* [9] H. T. Banks, W. Fang, R. J. Silcox and R.C. Smith, Approximation methods for control of acoustic/structure models with piezoceramic actuators, _Contract Report 189578 NASA_ (1991).
* [10] H. T. Banks and R.C. Smith, Feedback control of noise in a 2-D nonlinear structural acoustics model, _Discrete and Continuous Dynamical Systems_ Vol 1, No. 1, (1995), 119-149.
* [11] A. Bukhgeim, _Introduction to the Theory of Inverse Problems_ , VSP, Utrecht, 2000.
* [12] A. Bukhgeim and M. Klibanov, Global uniqueness of a class of multidimensional inverse problem, _Sov. Math.-Dokl._ 24(1981), 244-7.
* [13] T. Carleman, Sur un problème d’unicité pour les systèmes d’équations aux derivées partielles à deux variables independantes, _Ark. Mat. Astr. Fys._ , 2B (1939), 1-9.
* [14] M. Camurdan and R. Triggiani, Sharp regularity of a coupled system of a wave and a Kirchoff plate with point control arising in noise reduction, _Differential and Integral Equations_ , 12 (1999), 101-107.
* [15] V. Isakov, _Inverse Source Problems_ , American Mathematical Society, 2000.
* [16] V. Isakov, _Inverse Problems for Partial Differential Equations_ , Second Edition, Springer, New York, 2006.
* [17] V. Isakov, Uniqueness and stability in multi-dimensional inverse problems, _Inverse Problems_ , 9 (1993), 579-621.
* [18] O. Imanuvilov and M. Yamamoto, Global Lipschitz stability in an inverse hyperbolic problem by interior observations, _Inverse Problems_ , 17(2001), 717-728.
* [19] V. Isakov and M. Yamamoto, Carleman estimate with the Neumann boundary condition and its application to the observability inequality and inverse hyperbolic problems, _Contemp. Math._ , 268(2000), 191-225.
* [20] A. Khaĭdarov, Carleman estimates and inverse problems for second order hyperbolic equations, _Math. USSR Sbornik_ , 58(1987), 267-277.
* [21] M. Klibanov, Inverse problems and Carleman estimates, _Inverse Problems_ , 8(1992), 575-596.
* [22] M. Klibanov, Carleman estimates and inverse problems in the last two decades, _Surveys on Solutions Methods for Inverse Problems_ , Springer, Wien, 2000, pp 119-146.
* [23] I. Lasiecka, Mathematical Control Theory of Coupled PDE’s, _CBMS-NSF Regional Conference Series in Applied Mathematics_ , SIAM Publishing, Philadelphia (2002).
* [24] I. Lasiecka and R. Triggiani, Exact controllability of the wave equation with Neumann boundary control, _Appl. Math. & Optimiz._, 19(1989), 243-290.
* [25] I. Lasiecka and R. Triggiani, Carleman estimates and exact boundary controllability for a system of coupled, nonconservative second order hyperbolic equations, in Partial Differential Equations Methods in Control and Shape Analysis, _Lecture Notes in Pure and Applied Mathematics_ , Marcel Dekker, New York, Vol. 188, 215-243.
* [26] I. Lasiecka and R. Triggiani, Sharp regularity theory for second order hyperbolic equations of Neumann type. Part I. $L_{2}$ Nonhomogeneous data, _Ann. Mat. Pura. Appl. (IV)_ , CLVII(1990), 285-367.
* [27] I. Lasiecka and R. Triggiani, Regularity theory of hyperbolic equations with non-homogeneous Neumann boundary conditions. II. General boundary data, _Journal of Differential Equations_ , 94(1991), 112-164.
* [28] I. Lasiecka and R. Triggiani, Uniform stabilization of the wave equation with Dirichlet or Neumann feedback control without geometrical conditions, _Appl. Math. & Optimiz._, 25(1992), 189-244.
* [29] I. Lasiecka, R. Triggiani and X. Zhang, Nonconservative wave equations with unobserved Neumann B.C.:global uniqueness and observability in one shot, _Contemp. Math._ , 268(2000), 227-325.
* [30] J.L.Lions and E. Magenes, _Non-homogeneous Boundary Value Problems and Applications_ , Vol. I, Springer-Verlag, Berlin, 1972.
* [31] W. Littman and B. Liu, On the spectral properties and stabilization of acoustic flow, _IMA Preprint Series $\sharp$1436_, (November, 1996).
* [32] J-P Puel and M. Yamamoto, On a global estimate in a linear inverse hyperbolic problem, _Inverse Problems_ , 12(1996), 995-1002.
* [33] R.E.Showalter, _Monotone Operators in Banach Space and Nonlinear Partial Differential Equations_ , Mathematical Surveys and Monographs Volume 49, American Mathematical Society, 1997.
* [34] D. Tataru, On the regularity of boundary traces for the wave equation, _Annali Scuola Normale di Pisa_ , Classe Scienze (4), 26(1998), no. 1, 355-387.
* [35] D. Tataru, A priori estimates of Carleman’s type in domains with boundary, _J.Math.Pures Appl_ , (9), 73(1994), no. 4, 185-206.
* [36] R. Triggiani, Wave equation on a bounded domain with boundary dissipation: an operator approach, _Journal of Mathematical Analysis and Applications_ , 137(1989), 438-461.
* [37] M. Yamamoto, Uniqueness and stability in multidimensional hyperbolic inverse problems, _J. Math. Pures Appl._ , 78(1999), 65-98.
|
arxiv-papers
| 2010-10-13T16:28:17 |
2024-09-04T02:49:13.845888
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Shitao Liu",
"submitter": "Shitao Liu",
"url": "https://arxiv.org/abs/1010.2695"
}
|
1010.2696
|
arxiv-papers
| 2010-10-13T16:34:22 |
2024-09-04T02:49:13.857318
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Shitao Liu and Roberto Triggiani",
"submitter": "Shitao Liu",
"url": "https://arxiv.org/abs/1010.2696"
}
|
|
1010.2783
|
# Virus and Warning Spread in Dynamical Networks
Carlos Rodríguez-Lucatero crodriguez@correo.cua.uam.mx &
rbernal@correo.cua.uam.mx Departamento de Tecnologías de la Información,
Universidad Autónoma Metropolitana-Cuajimalpa, Av. Constituyentes 1056,
Col.Lomas Altas, México, D. F., C.P. 11950, México Roberto Bernal-Jaquez
Departamento de Matemáticas Aplicadas y Sistemas,
Universidad Autónoma Metropolitana-Cuajimalpa, Artificios 40, Col. Hidalgo,
Delegación Álvaro Obregón, México, D.F 01120, México
###### Abstract
Recent work on information survival in sensor and human P2P networks, try to
study the datum preservation or the virus spreading in a network under the
dynamical system approach. Some interesting solutions propose to use non-
linear dynamical systems and fixed point stability theorems, providing closed
form formulas that depend on the largest eigenvalue of the dynamic system
matrix. Given that in a the Web there can be messages from one place to
another, and that this messages can be, with some probability, new
unclassified virus warning messages as well as worms or other kind of virus,
the sites can be infected very fast. The question to answer is how and when a
network infection can become global and how it can be controlled or at least
how to stabilize his spreading in such a way that it becomes confined below a
fixed portion of the network. In this paper, we try to make a step ahead in
this direction and apply classic results of the dynamical systems theory to
model the behaviour of a network where warning messages and virus spread.
###### pacs:
89.75.Hc, 05.45.Df
## I Introduction
Recently with the constant augmentation in the number of internet users as
well as the growth in the complexity of such networks, new security problems
have appeared in the scene and there is a lack of adequate security methods
for facing attacks under this new setting. These new environments are for
instance the P2P networks, sensor networks, social nets or wireless networks,
where information is to be stored, generated and retrieved. So under this new
environments it can be very important to study and model how the information
is spread or how to keep the spreading of a virus under control in such a way
that the information still being useful under these vulnerable circumstances.
In Deepa1 it is studied the problem of information survival threshold in
sensor and P2P networks, modeling the problem as a non-linear dynamical system
and using fixed point stability theorems, and obtain a closed form solution
that depends on an additional parameter, the largest eigenvalue of the
dynamical system matrix. In the sensor networks for instance, the nodes can
loss their communications links and the nodes can stop working because of
system failure produced by a virus infection and quarantine process or a
system maintenance procedure. Under such conditions they try to answer the
following question:
PROBLEM: _Under what conditions a datum can survive in a sensor network?_
Given that the nodes as well as the links can fail with some probability the
obvious model can be a Markov chain, but such a model can grow in complexity
very quickly because the number of possible states becomes $3^{N}$ where $N$
is the number of states. To avoid this mathematical problem, one alternative
is to model the system as a non-linear dynamical system. Recently they have
appeared in the conferences and journal articles some very interesting and
relevant research articles about the virus spread behaviour in a P2P network
or in scale free nets such as the Web. In Kempe1 the authors study the
communication mechanisms for gossip based protocols. Another very recent and
interesting work on how to distribute antidotes for controlling the epidemics
spread is presented in chayes1 . In this research the authors analyse the
problem under the approach of contact processes Durrett1 on a finite graph
and obtain very interesting and rigorous results. Concerning the properties
that arise in the random graphs, such as the existence of a _giant component_
, _percolation_ phenomena, node degree distribution and _small world_
phenomena, and that are the base of many recent works on virus spread on
networks, we can mention Barab0 ; Barab1 ; Barab2 as well as Alon1 Bollobas1
Falou1 and Radicchi1 . Concerning the subject on mathematical modeling of
epidemic spreading we should mention the outstanding work done by Romualdo
Pastor-Satorras and Alessandro Vespignani in Satorras1 ; Satorras2 ; Satorras3
. In the present work we will take as source of inspiration Deepa1 . In Deepa1
the authors implement some experiments on several real sensor and P2P networks
(from Intel, MIT, Gnutella, and others) to show the accuracy of their method.
In this work it is claimed that their method is not only applicable to sensor
nets but it is also applicable to many more settings where a piece of
information may be replicated across faulty links and faulty nodes. The
authors establish a survivability condition that produce a bound in the design
of distributed systems, allowing to:
* •
Estimate the most economic retransmission rates for a datum to survive in a
sensor network.
* •
Decide which nodes can be removed while still remaining above the
survivability threshold in a sensor network.
* •
Drive a _virus_ as _datum_ to extinction for anti-virus protection, by
deciding how often to quarantine nodes and how long they should be kept down.
* •
Propagate and maintain information (news, rumors, etc.)
So, this work is closely related with the areas of gossip based protocols,
epidemiology and computer security. Gossip-based protocols on networks, whose
related graphs have dynamic presence of nodes, that keep some level of state
consistency have been proposed in Kempe1 . The basic underlying idea of the
gossip protocols is that at each time step each node $i$ chose to communicate
with a node $j$ generally following a random rule, exchanging information
during a period of time, spreading it in the system in the same way as the
virus are spread. A fundamental issue in this kind of protocols is how the
underlying gossip low level mechanism affects the ability to design efficient
high level gossip protocol algorithms. In Kempe1 the authors show a
fundamental limitation on the power of the commonly used uniform gossip
mechanism for solving nearest-resource location problems. They show as well
that very efficient protocols for this problem can be designed using a non-
uniform spatial gossip mechanism. The gossip-based distributed protocol
algorithms obtained in Kempe1 for complex problems for a set of nodes in
Euclidean space are implemented by constructing an approximated minimum
spanning tree.
#### I.0.1 Previous proposed mathematical model of virus spread.
With the increasing importance and presence of sensor as well as P2P networks,
networks have a high level of congestion and because of that the theory
concerning information survivability becomes very important. One source of
inspiration for mathematical modeling problems of information survivability in
this kind of networks is the _epidemiology_.
The _epidemiology_ community has developed several stochastic models for
studying the spread and die-out of diseases in a population. The two most
relevant ones are the SIR and SIS models. Both are stochastic models of the
spread of disease through a population, where the _susceptible_ nodes can get
_infected_ on contact with _infected_ neighbors. The _infected_ hosts
eventually die (in the SIR model) or recover and become _susceptible_ again
(in the SIS model). The point of view adopted in Deepa1 was the SIS model.
Under this model a node is _susceptible_ to a data item when it is online and
under normal operation. When the nodes start to fail, they become _immune_
during their failure, and later they become _susceptible_ again when they are
back online. Some results obtained in Deepa1 are very useful to analyze the
survival of a infection in a population, based on the graph theory results
similar to the ones mentioned in Barab1 ; Barab2 ; Alon1 . In computer
security one of the important issues that have been studied under SIS and SIR
infection spreading mathematical models are the virus propagation as well as
worms on Internet, from where, the exponential spread of them and the
_epidemic thresholds_ can be estimated Satorras1 ,Satorras2 , Satorras3 . Let
us suppose that we have a sensor/P2P/social network of $N$ nodes (sensors or
computers or people) and $E$ directed links between them. Let us also assume
that we take very small discrete timesteps of size $\Delta t$ where $\Delta
t\rightarrow 0$. The survivability results in Deepa1 apply equally well to
continuous systems. Within a $\Delta t$ time interval, each node $i$ has
probabiity $r_{i}$ of trying to broadcast its information every time step, and
each link $i\rightarrow j$ has a probability $\beta_{i,j}$ of being _up_ , and
thus correctly propagating the information to node $j$. Each node $i$ also has
a node failure probability $\delta_{i}>0$ (e.g., due to battery failure in
sensors). Every dead node $j$ has a rate $\gamma_{j}$ of returning to the _up_
state, but without any information in its memory (e.g., due to the periodic
replacement of dead batteries). These and other symbols are listed in Table 1.
$\begin{array}[]{|l|l|}\hline\cr\mbox{Symbol}&\mbox{Description}\\\ \hline\cr
N&\mbox{Number of nodes in a network}\\\ \beta_{ij}&\mbox{Probability that the
link}\\\ {}{}{}{}{}&i\rightarrow j\mbox{is up}\\\ \delta_{i}&\mbox{Death rate:
Probability that node~{}}i\mbox{~{}dies}\\\ \gamma_{i}&\mbox{Resurrection
rate:}\\\ {}{}{}{}&\mbox{Probability that node~{}}i\mbox{~{}comes back up}\\\
r_{i}&\mbox{Retransmission rate:}\\\ {}{}{}{}{}&\mbox{Probability that
node}i\mbox{~{}broadcasts}\\\ \hline\cr p_{i}(t)&\mbox{Probability that
node}\\\ {}{}{}{}{}&i\mbox{~{} is alive at time}t\mbox{~{}and has info}\\\
q_{i}(t)&\mbox{Probability that node}\\\ {}{}{}{}&i\mbox{ ~{} is alive at
time}t\mbox{~{}but without info}\\\ 1-p_{i}(t)-q_{i}(t)&\mbox{Probability that
node}i\mbox{~{}is dead}\\\ \zeta_{i}&\mbox{Probability that
node}i\mbox{~{}does}\\\ {}{}{}{}&\mbox{not receive info from}\\\
{}{}{}{}&\mbox{any of its neighbors at time~{}}t\\\
\vec{p}(t),\vec{q}(t)&\mbox{Probability column vectors}\\\
f:\Re^{2N}\rightarrow\Re^{2N}&\mbox{Function representing a dynamical
system}\\\ \nabla(f)&\mbox{The Jacobian matrix of~{}}f(.)\\\
S&\mbox{The~{}}N\times N\mbox{~{}system matrix}\\\ \lambda_{S}&\mbox{An
eigenvalue of the~{}}S\mbox{matrix}\\\ \lambda_{1,S}&\mbox{The largest in
magnitude}\\\ {}{}{}{}&\mbox{eigenvalue of the}S\mbox{matrix}\\\
s=|\lambda_{1,S}|&\mbox{Survivability score = Magnitude of}\lambda_{1,S}\\\
\hline\cr\end{array}$
This system can be modeled as a Markov chain, where each node can be in one of
three states: _Has Info_ , _No Info_ or _Dead_ , with transitions between them
as shown in Diagram 1. The full state of the system at any instant consists of
$N$ such states, one for each node. Thus, there are $3^{N}$ system states.
Transitions out of the current system state depend only on the current state
and not on any previous states; thus it is a Markov chain.
The next graph represent the transition that take place in each node.
Resurrected$\gamma_{i}$Prob
$1-p_{i}-q_{i}$$1-\gamma_{i}$Dies$\delta_{i}$HasInfoProb
$p_{i}$$1-\delta_{i}$Dies$\delta_{i}$Receives
Info$1-\zeta_{i}(t)$NoInfo$\zeta_{i}(t)-\delta_{i}$Prob $q_{i}$DeadDiagram 1:
Transitions on each node
It can be pointed out that there is an absorbing set of states where no node
is in _Has Info_ state. Under such circumstances the information dies with
probability $1$ as $t\rightarrow\infty$. Some combination of parameters lend
the system quickly to this state and some other combination does not so in
practice the datum _survives_ for some parameter combination. The question is:
under what conditions does the information survive for a long time, and when
will the information die out quickly? Let $\overline{C}(t)$ denote the
expected number of _carriers_ (nodes in _Has Info_ state) at time $t$. In
general, $\overline{C}(t)$ decays exponentially, polynomially or
logarithmically (with expected extinction time comparable to or larger than
the age of the universe for large graphs), depending on the system being
below, at or above a threshold Durrett1 . Let us focus on the _fast
extinction_ case where $\overline{C}(t)$ decays exponentially.
###### Definition 1.
_Fast extinction_ is the setting where the number of carriers
$\overline{C}(t)$ decays exponentially over time ($\overline{C}(t)\propto
c^{-t},c>1$)
Now, the problem can be formally stated as follows: PROBLEM:
* •
Given: the network topology (link _up_ probabilities) $\beta_{ij}$ the
retransmission rates $r_{i}$, the resurrection rates $\gamma_{i}$ and the
death rates ($\delta_{i}$ $i=1\ldots N,j=1\ldots N$)
* •
Find the condition under which a datum will suffer _fast extinction_.
To simplify the problem and to avoid dependencies on starting conditions, we
consider the case where all nodes are initially in the _have info_ state.
#### I.0.2 Main Idea
Solving this problem for the full Markov chain requires $3^{N}$ variables and
is thus intractable, even for moderate-sized networks. Exact values for the
_fast extinction_ threshold are unavailable even for simpler versions of this
problem. The main contribution of Deepa1 is an accurate approximation, using
a non-linear dynamical system of only $N$ variables. The heart of their
approximation is to consider the states of the two different nodes to be
mutually independent. Let the probability of node $i$ being in the _Has Info_
and _No Info_ states at time $t$ be $p_{i}(t)$ and $q_{i}(t)$ respectively.
Thus, the probability of its being dead is $(1-p_{i}(t)-q_{i}(t))$. Starting
from state _No Info_ at time $t-1$, node $i$ can acquire this information (and
move to state _Has Info_) if it receives a communication from some other node
$j$. Let $\zeta_{i}(t)$ be the probability that node $i$ does not receive the
information from any of its neighbors. Then, assuming the neighbor’s states
are independent:
$\zeta_{i}(t)=\prod_{j=1}^{N}(1-r_{j}\beta_{ji}p_{j}(t-1))$ (1)
For each node $i$, we can use the transition matrix in Diagram 1 to write down
the probabilities of being in each state at time $t$, given the probabilities
at time $t-1$ (recall that we use very small time steps $\Delta t$, and so we
can neglect second-order terms). Thus:
$p_{i}(t)=p_{i}(t-1)(1-\delta_{i})+q_{i}(t-1)(1-\zeta_{i}(t))$ (2)
$q_{i}(t)=q_{i}(t-1)(\zeta_{i}(t)-\delta_{i})+(1-p_{i}(t-1)-q_{i}(t-1))\gamma_{i}$
(3)
#### I.0.3 Main previous Results
In Deepa1 , experimental results have been obtained under fast extinction
conditions that accurately correspond to what their model predicts. The
authors claim that the accuracy of their model predictions are due to the
_mixing properties_ of real networks. For the sake of completeness we will
state the main results in Deepa1 and to show how some of them are obtained
because our own results will be obtained by applying the same procedures.
###### Definition 2.
Define $S$ to be the $N\times N$ system matrix:
$S_{ij}=\left\\{\begin{array}[]{ll}1-\delta_{i}&~{}~{}~{}~{}\text{if~{}}i=j\\\
r_{j}\beta_{ji}\frac{\gamma_{i}}{\gamma_{i}+\delta_{i}}&~{}~{}~{}~{}\mbox{otherwise}\end{array}\right.$
Let $|\lambda_{1,S}|$ be the magnitude of the largest eigenvalue and
$\widehat{C}(t)=\sum_{i=1}^{N}p_{i}(t)$ the expected number of carriers at $t$
of the dynamical system.
###### Theorem 1.
(Condition for fast extinction). Define $s=|\lambda_{1,S}|$ to be the
survivability score for the system. If $s=|\lambda_{1,S}|<1$, then we have
fast extinction in the dynamical system, that is, $\widehat{C}(t)$ decays
exponentially quickly over time.
Where $|\lambda_{i,S}|$ is the magnitude of the largest eigenvalue of $S$,
being $S$ an $N\times N$ system matrix defined as $S_{ij}=1-\delta_{i}$ if
$i=j$ and $S_{ij}=r_{j}\beta_{ji}\frac{\gamma_{i}}{\gamma_{i}+\delta_{i}}$
otherwise, and being $\widehat{C}(t)=\sum_{i=1}^{N}p_{i}(t)$ the expected
number of carriers at time $t$ of the dynamical system. Two additional results
that appears in Deepa1 are the following
###### Lemma 1.
Fixed point. The values
$(p_{i}(t)=0,q_{i}(t)=\frac{\gamma_{i}}{\gamma_{i}+\delta_{i}})$ for all nodes
$i$, are a fixed point of the equations (2) and (3). Proved by a simple
application of the Equations.
###### Theorem 2.
(Stability of the fixed point). The fixed point point of Lemma 1 is
asymptotically if the system is bellow threshold, that is,
$s=|\lambda_{1,S}|<1$
###### Lemma 2.
(From reference [8] of Deepa1 ) Define $\nabla(f)$ (also called the Jacobian
matrix) to be a $2N\times 2N$ matrix such that
$[\nabla(f)]_{ij}=\frac{\partial
f_{i}(\vec{v}(t-1))}{\partial\vec{v}_{j}(t-1)}$ (4)
where $\vec{v}$ is the concatenation of $\vec{p}$ and $\vec{q}$. Then, if the
largest eigenvalue (in magnitude) of $\nabla(f)$ at $\vec{v}_{f}$ (vector
$\vec{v}$ valued at the fixed point) is less than $1$ in magnitude, the system
is asymptotically stable at $\vec{v}_{f}$. Also, if $f$ is linear and the
condition holds, then the dynamical system will exponentially tend to the
fixed point irrespective of initial state.
In Deepa1 the authors apply (2) and obtain the following block matrix
$\nabla(f)|_{\vec{v}_{f}}=\left[\begin{array}[]{lll}S&|&0\\\ \hline\cr
S_{1}&|&S_{2}\end{array}\right]$ (5)
The dimensions of each block matrix are $N\times N$ whose elements are
$S_{ij}=\left\\{\begin{array}[]{ll}1-\delta_{i}&~{}~{}~{}~{}\text{if~{}}i=j\\\
r_{j}\beta_{ji}\frac{\gamma_{i}}{\gamma_{i}+\delta_{i}}&~{}~{}~{}~{}\mbox{otherwise.}\end{array}\right.$
(6)
The others are
$S_{1ij}=\left\\{\begin{array}[]{ll}1-\delta_{i}&~{}~{}~{}~{}\text{if~{}}i=j\\\
-r_{j}\beta_{ji}\frac{\gamma_{i}}{\gamma_{i}+\delta_{i}}&~{}~{}~{}~{}\mbox{otherwise}\end{array}\right.$
(7)
and
$S_{2ij}=\left\\{\begin{array}[]{ll}1-\gamma_{i}-\delta_{i}&~{}~{}~{}~{}\text{if~{}}i=j\\\
0&~{}~{}~{}~{}\mbox{otherwise}\end{array}\right.$ (8)
So the question is _how can be obtained the fixed point of the system?_. In
the following paragraph we will sketch, in an alternative way of the used in
Deepa1 , how it can be done. In dynamical systems theory the _fixed point_ is
called _equilibrium point_ of the system. In this very point the state
probabilities become stable, then $p_{i}(t)=p_{i}(t-1)$ and
$q_{i}(t)=q_{i}(t-1)$. Then simplifying the notation by dropping the subindex
and the time parameter we can state the following equations system:
$\begin{array}[]{l}p=p\cdot(1-\delta)+q\cdot(1-\zeta)\\\
q=q\cdot(\zeta-\delta)+(1-p-q)\cdot\gamma\end{array}$ (9)
after algebraic simplification it can be obtained the following equations
system
$\begin{array}[]{l}-\delta\cdot p+(1-\zeta)\cdot q=0\\\ \gamma\cdot
p+(\zeta-1-\delta-\gamma)\cdot q=q\end{array}$ (10)
Expressing the equations system in matrix form we get
$\left[\begin{array}[]{ll}-\delta&1-\zeta\\\
-\gamma&\zeta-1-\delta-\gamma\end{array}\right]\left[\begin{array}[]{l}p\\\
q\end{array}\right]=\left[\begin{array}[]{l}0\\\ -\gamma\end{array}\right]$
(11)
Solving by Cramer’s method we obtain
$\begin{array}[]{l}p=\frac{\gamma\cdot(1-\zeta)}{\gamma\cdot(1-\zeta)-\delta\cdot(\zeta-1-\delta-\gamma)}\\\
q=\frac{\delta\cdot\gamma}{\gamma\cdot(1-\zeta)-\delta\cdot(\zeta-1-\delta-\gamma)}\end{array}$
(12)
The expressions (12) can be simplified by observing that the stable state _No
Info_ is related with the desired _fast extinction_ condition that is also
related with Markov chain probability condition $(1-\zeta)\rightarrow 0$, that
implies $p\rightarrow 0$. Taking into account this fact, we can rewrite (12)
as follows:
$\begin{array}[]{ll}p=&\frac{\gamma\cdot(1-\zeta)}{\gamma\cdot(1-\zeta)+\delta\cdot(1-\zeta+\delta+\gamma)}\\\
~{}~{}=&0\\\
q=&\frac{\delta\cdot\gamma}{\gamma\cdot(1-\zeta)-\delta\cdot(\zeta-1-\delta-\gamma)}\\\
~{}~{}=&\frac{\delta\cdot\gamma}{\gamma\cdot(1-\zeta)+\delta\cdot(1-\zeta+\delta+\gamma)}\\\
~{}~{}=&\frac{\delta\cdot\gamma}{\delta\cdot(\delta+\gamma)}\\\
~{}~{}=&\frac{\gamma}{\delta+\gamma}\end{array}$ (13)
### I.1 Markovian Analysis
In Deepa1 the Markovian analysis was avoided because of the size of the
resulting configuration space ($3^{N}$ where $N$ correspond to the number of
nodes and $3$ to number of states in the Markov chain). However, it is
interesting to analyse the ergodicity behaviour of the Markov chains
associated inside each node. We have done this by the calculation of the
corresponding Z-transform of the associated matrix and performing the
following steps:
* •
We obtain the transition matrix $P$
$P=\left(\begin{array}[]{ccc}(1-\delta)&0&\delta\\\
(\delta-\zeta)&(1-\zeta)&\delta\\\ (1-\gamma)&0&\gamma\end{array}\right)$ (14)
* •
We calculate the Z-transform of the matrix $M=I-zP$, that is
$M=\left(\begin{array}[]{ccc}1-z(1-\delta)&0&-z\delta\\\
z(\delta-\zeta)&1-z(1-\zeta)&-z\delta\\\
z(-1+\gamma)&0&1-z\gamma\end{array}\right)$ (15)
* •
The inverse matrix $M^{-1}$ of $M$ is obtained
* •
The inverse Z-transform is applied to $M^{-1}$, obtaining
$H(n)={\cal{Z}}^{-1}\\{{M^{-1}}\\}$
This equation can be written as a sum of two matrices Howard , $S$ that
corresponds to the steady behavior and $T$ that represents the transient
behavior of the Markov Chain, that is
$H(n)=k_{1}S+k_{2}(C_{1})^{n}T$,
where $k_{1}$ and $k_{2}$ are constants. Based on this procedure we obtained
the ergodicity condition, that is, $C_{1}<1$ what assures the convergence to a
steady state no matter what the initial state was.
All the calculations were made using MATHEMATICA and the ergodicity condition
expression obtained is given by
$\noindent{C_{1}=\frac{2-\gamma-\delta+\sqrt{\gamma^{2}-6\gamma\delta+\delta^{2}}}{1-\gamma-\delta+2\gamma\delta}}.$
(16)
It’s worthy to notice that the ergodicity of the Markov chain depends on the
values of the transition probabilities involved. We should also mention that
the calculations of this analysis are an additional source of difficulty. The
more states the Markov chain have, the more complex become the algebraic
manipulation and Z-transforms. So, it is not a surprise that the Markovian
analysis was avoided in Deepa1 and the choice was the dynamical systems
approach.
## II Our proposal
The model described in subsection I.0.1 and that constitutes the core of the
results obtained in Deepa1 has as main purpose to estimate the threshold
condition under which the propagation of a virus in a P2P network decays
exponentially. This last question implies that for keeping the network below
the threshold just mentioned, the protocols have to disconnect temporarily
some nodes, fix the problem and reboot them. This method is very efficient for
stopping the propagation of the virus. In this way the number of virus
carriers decays exponentially. Let us assume that at the same time we need to
propagate an alarm signal warning about the presence of a worm virus or an
antidote chayes1 in a P2P network. Then in these cases we need that the
network operates over the estimated threshold. So we are in a situation where
the threshold conditions are antagonist and that can happen in a real world
setting. Under such circumstances the question is _How to keep a datum and at
the same time avoid the virus spreading?_. Our hypothesis is that it will
depend on the proportion of virus messages versus warning messages. For this
reason we will propose a new model that will take into account this situation.
The previous and the new additional symbols are listed in Table 2.
${\scriptsize\begin{array}[]{|l|l|}\hline\cr\mbox{Symbol}&\mbox{Description}\\\
\hline\cr N&\mbox{Number of nodes in a network}\\\
\beta_{ij}&\mbox{Probability that the link}\\\ {}{}{}{}{}&i\rightarrow
j\mbox{is up}\\\ \delta_{i}&\mbox{Death rate: Probability that
node~{}}i\mbox{~{}dies}\\\ \gamma_{i}&\mbox{Resurrection rate:}\\\
{}{}{}{}&\mbox{Probability that node~{}}i\mbox{~{}comes back up}\\\
r_{i}&\mbox{Retransmission rate:}\\\ {}{}{}{}{}&\mbox{Probability that
node}i\mbox{~{}broadcasts}\\\ \hline\cr p_{i}(t)&\mbox{Probability that
node}\\\ {}{}{}{}{}&i\mbox{~{} is infected at time}t\mbox{~{}and has virus
info}\\\ q_{i}(t)&\mbox{Probability that node has no Info}\\\ {}{}{}{}&i\mbox{
~{} is healthy at time}t\mbox{~{}but susceptible}\\\
1-p_{i}(t)-q_{i}(t)-w_{i}(t)&\mbox{Probability that node}i\mbox{~{}is dead}\\\
w_{i}(t)&\mbox{Probability that node has warning Info}\\\ {}{}{}{}&i\mbox{ ~{}
is warned at time}t\\\ \zeta_{i}&\mbox{Probability that
node}i\mbox{~{}does}\\\ {}{}{}{}&\mbox{not receive info from}\\\
{}{}{}{}&\mbox{any of its neighbors at time~{}}t\\\ \nu_{i}&\mbox{Probability
that node}i\\\ {}{}{}{}&\mbox{receive virus}\\\ 1-\nu_{i}&\mbox{Probability
that node}i\\\ {}{}{}{}&\mbox{receive a warning}\\\ \chi_{i}&\mbox{Probability
that node}i\\\ {}{}{}{}&\mbox{applies vaccin}\\\
\vec{p}(t),\vec{q}(t)&\mbox{Probability column vectors}\\\
f:\Re^{2N}\rightarrow\Re^{2N}&\mbox{Function representing a dynamical
system}\\\ \nabla(f)&\mbox{The Jacobian matrix of~{}}f(.)\\\
S&\mbox{The~{}}N\times N\mbox{~{}system matrix}\\\ \lambda_{S}&\mbox{An
eigenvalue of the~{}}S\mbox{matrix}\\\ \lambda_{1,S}&\mbox{The largest in
magnitude}\\\ {}{}{}{}&\mbox{eigenvalue of the}S\mbox{matrix}\\\
s=|\lambda_{1,S}|&\mbox{Survivability score = Magnitude of}\lambda_{1,S}\\\
\hline\cr\end{array}}$
This system can be modeled as well as a Markov chain, where each node can be
in one of three states: _Infected_ ,_Warn Info_ , _No Info_ or _Dead_ , with
transitions between them as shown in Diagram 2. The full state of the system
at any instant consists of $N$ such states, one for each node. Therefore,
there are $4^{N}$ system states. Transitions out of the current system state
depend only on the current state and not on any previous states; then it is a
Markov chain without memory. The next graph represent the transitions that
take place in each node for our model.
Resurrected$\gamma_{i}$Prob
$1-p_{i}-q_{i}-w_{i}$$1-\gamma_{i}$Dies$\delta_{i}$InfecInfoProb
$p_{i}$$1-\delta_{i}$Dies$\delta_{i}$Receives
Virus$(1-\zeta_{i}(t))\nu_{i}$NoInfoWarnInfo
$\zeta_{i}(t)-\delta_{i}$Prob
$q_{i}$DeadWarn$(1-\zeta_{i}(t))$$\cdot(1-\nu_{i})$$\chi_{i}$$1-\chi_{i}-\delta_{i}$$\delta_{i}$Prob
$w_{i}$Diagram 2: Transitions on each node
Making the same node independence probability assumption that is stated in
equation (1) and taking into account the new states and transition
probabilities shown in the Diagram 2, the equations (2) and (3) as well as the
new equation corresponding to $w_{i}$ can be expressed as follows:
$\displaystyle p_{i}(t)$ $\displaystyle=$ $\displaystyle
p_{i}(t-1)(1-\delta_{i})+q_{i}(t-1)(1-\zeta_{i}(t))\nu_{i}$ (17)
$\displaystyle q_{i}(t)$ $\displaystyle=$ $\displaystyle
q_{i}(t-1)(\zeta_{i}(t)-\delta_{i})+$
$\displaystyle(1-p_{i}(t-1)-q_{i}(t-1)-w_{i}(t-1))\gamma_{i}$
$\displaystyle+\chi_{i}w_{i}(t-1)$ $\displaystyle w_{i}(t)$ $\displaystyle=$
$\displaystyle(1-\zeta_{i}(t))(1-\nu_{i})q_{i}(t-1)$
$\displaystyle+(1-\chi_{i}-\delta_{i})w_{i}(t-1)$
Also in our case of study, instead of solving the Markov Chain, that is rather
complicated, we will describe the behaviour of our system considering it as a
dynamical system described by the equations (17), (II) and (II). Following
Hirsch we will calculate the fixed points of the system. As we have stated
before, in these very points the state probabilities become stable, then
$p_{i}(t)=p_{i}(t-1)$, $q_{i}(t)=q_{i}(t-1)$ and $w_{i}(t)=w_{i}(t-1)$. Using
(17), (II) and (II), we can state the following result
###### Lemma 3.
$p_{i}(t)+q_{i}(t)+w_{i}(t)\rightarrow\frac{\gamma_{i}}{\gamma_{i}+\delta_{i}}$
Proof: In the same way that has been done in Deepa1 we can do the subtraction
$1-p_{i}(t)-q_{i}(t)-w_{i}(t)$ and simplify by renaming
$x_{i}(t)=p_{i}(t)+q_{i}(t)+w_{i}(t)$ what give us the following linear system
$x_{i}(t)=(1-\delta_{i}-\gamma_{i})\cdot x_{i}(t-1)+\gamma_{i}$ (20)
In the fixed point $x_{i}(t)=x_{i}(t-1)$ so if we apply this to the last
equation we have that
$x_{i}(t)=\frac{\gamma_{i}}{\gamma_{i}+\delta_{i}}$ (21)
that is,
$p_{i}(t)+q_{i}(t)+w_{i}(t)=\frac{\gamma_{i}}{\gamma_{i}+\delta_{i}}$. Then by
Lemma 2 in Deepa1 , this convergence is exponential.
It is worthy to notice that the same results for the fixed points are obtained
considering the linear behaviour of the system on these points. Using (17),
(II) and (II) and for simplicity dropping the indexes and the the time
dependance, we obtain the equations
$\begin{array}[]{l}-\delta\cdot p+\nu(1-\zeta)\cdot q=0\\\
(-1+\zeta-\delta-\gamma)q+(-\gamma+\chi)w=-\gamma\\\
(1-\zeta-\nu+\zeta\nu)q+(1-\chi-\delta)w=0\\\ \end{array}$ (22)
Whose solutions can be obtained using Cramer’s method
$\begin{array}[]{l}p=-\frac{\gamma\cdot(-1+\zeta)\nu(\delta+\chi)}{(\gamma+\delta)\left(\delta+\delta^{2}-\delta\zeta+\delta\chi+\nu\chi-\zeta\nu\chi\right)}\\\
\\\
q=\frac{\gamma\delta(\delta+\chi)}{(\gamma+\delta)\left(\delta+\delta^{2}-\delta\zeta+\delta\chi+\nu\chi-\zeta\nu\chi\right)}\\\
\\\
w=\frac{\gamma\delta(1-\zeta-\nu+\zeta\nu)}{(\gamma+\delta)\left(\delta+\delta^{2}-\delta\zeta+\delta\chi+\nu\chi-\zeta\nu\chi\right)}\par\end{array}$
(23)
Once more, this expression can be simplified if we observe that the stable
state _No Info_ is related with the desired _fast extinction_ condition that
is also related with Markov chain probability condition $(1-\zeta)\rightarrow
0$, that implies $p\rightarrow 0$, this can be summarized as
$\begin{array}[]{l}p=0\\\ \\\ q=\frac{\gamma}{\gamma+\delta}\\\ \\\
w=0\par\end{array}$ (24)
Given that by hypothesis, the probability of events in each node are
independent, we can try to analyse the problem using the Markov approach. This
will be made in the next subsection.
### II.1 Markovian Analysis
In this section we analyse again the ergodicity behaviour of the Markov chains
associated inside each node under our model.
The calculation steps performed were :
* •
We obtain the transition matrix $P$
$P=\left(\begin{array}[]{ccccc}\zeta_{i}(t)-\delta_{i}&(1-\zeta_{i}(t))\nu_{i}&(1-\zeta_{i}(t))(1-\nu_{i})&\delta_{i}\\\
0&1-\delta_{i}&0&\delta_{i}\\\ \chi_{i}&0&1-\chi_{i}-\delta_{i}&\delta_{i}\\\
\gamma_{i}&0&0&1-\gamma_{i}\par\end{array}\right)$ (25)
* •
we calculate the Z-transform of the matrix $M=I-zP$, that is
$M=\left(\begin{array}[]{cccc}1-z(-\delta+\zeta)&z(-1+\zeta)\nu&z(-1+\zeta)(1-\nu)&-z\delta\\\
0&1-z(1-\delta)&0&-z\delta\\\ -z\chi&0&1-z(1-\delta-\chi)&-z\delta\\\
-z\gamma&0&0&1-z(1-\gamma)\end{array}\right)$ (26)
* •
The inverse matrix $M^{-1}$ of $M$ is obtained
* •
The inverse Z-transform is applied to $M^{-1}$, obtaining
$H(n)={\cal{Z}}^{-1}\\{{M^{-1}}\\}$
This equation can be written as a sum of two matrices Howard , $S$ that
corresponds to the steady behavior and $T$ that represents the transient
behavior of the Markov Chain, that is
$H(n)=k_{1}S+k_{2}(C_{1})^{n}T$,
where $k_{1}$ and $k_{2}$ are constants. Based on this procedure we obtained
the ergodicity condition, that is, $C_{1}<1$ what assures the convergence to a
steady state no matter what the initial state was.
All the calculations were made using MATHEMATICA and the ergodicity condition
expression obtained is given by
$C_{1}=\frac{1-2\delta+\zeta-\chi+\sqrt{1-2\zeta+\zeta^{2}+2\chi-2\zeta\chi-4\nu\chi+4\zeta\nu\chi+\chi^{2}}}{\delta^{2}+\zeta+(-1+\nu)\chi-\zeta\nu\chi+\delta(-1-\zeta+\chi)}.$
(27)
Again, it can be noticed from the calculation above, that the ergodicity of
the associated Markov chain depend on the choice of the transition
probabilities involved. It should be also mentioned that the algebraic
manipulations become even more complex than in the case of the Markov chain of
Deepa1 given that our Markov chain have one state more.
### II.2 Jacobian and fix point
In our case of study, we can proceed as Deepa1 . Firstly, let us define the
column vectors $\vec{\mathbf{p}}(t)=(p_{1}(t),p_{2}(t),\dots,p_{N}(t))$,
$\vec{\mathbf{q}}(t)=(q_{1}(t),q_{2}(t),\dots,q_{N}(t))$ and
$\vec{\mathbf{w}}(t)=(w_{1}(t),w_{2}(t),\dots,w_{N}(t))$. Let the vector
$\vec{\mathbf{v}}(t)=(\vec{\mathbf{p}}(t),\vec{\mathbf{q}}(t),\vec{\mathbf{w}}(t))$
be the concatenation of the previous vectors and let $\vec{\mathbf{v}}_{f}(t)$
be the vector $\vec{\mathbf{v}}(t)$ evaluated at the fixed point. Then, the
entire system can be described by
$\vec{\mathbf{v}}(t)=\mathbf{f}(\vec{\mathbf{v}}(t-1))$ (28)
where
$f_{i}(\vec{\mathbf{v}}(t-1))=\left\\{\begin{array}[]{lllll}p_{i}(t-1)(1-\delta_{i})&~{}~{}\text{if~{}}i\leq
N&\\\ +q_{i}(t-1)(1-\zeta_{i}(t))\nu_{i}\\\ &\\\
q_{i}(t-1)(\zeta_{i}(t)-\delta_{i})\\\
+(1-p_{i}(t-1)-q_{i}(t-1)&~{}~{}\text{if~{}}N<i\leq 2N&\\\
-w_{i}(t-1))\gamma_{i}+\chi_{i}w_{i}(t-1)\\\ &\\\
(1-\zeta_{i}(t))(1-\nu_{i})q_{i}(t-1)\\\
+(1-\chi_{i}-\delta_{i})w_{i}(t-1)&~{}~{}\text{if~{}}2N<i\leq
3N&\end{array}\right.$
Now, following Hirsch , let us define the the Jacobian matrix of the system,
$\nabla\mathbf{f}$ as
$[\nabla\mathbf{f}]_{ij}=\frac{\partial
f_{i}(\vec{\mathbf{v}}(t-1))}{\partial\vec{\mathbf{v}}(t-1)}.$
In order to explore the asymptotic stability of fixed points and according to
Hirsch we will have to take into account the value of the function
$\nabla\mathbf{f}$ in these points In our case, we obtain the $3N\times 3N$
Jacobian matrix
$\nabla(\mathbf{f})|_{\vec{\mathbf{v}}_{f}}=\left[\begin{array}[]{lllll}S&|&0&|&0\\\
\hline\cr S_{1}&|&S_{2}&|&S_{3}\\\ \hline\cr
S_{4}&|&0&|&S_{6}\end{array}\right]$ (29)
where each block $S,S_{1},\dots$ is a $N\times N$ matrix whose elements are
given by
$S_{ij}=\left\\{\begin{array}[]{ll}1-\delta_{i}&~{}~{}~{}~{}\text{if~{}}i=j\\\
r_{j}\beta_{ji}\nu_{i}\frac{\gamma_{i}}{\gamma_{i}+\delta_{i}}&~{}~{}~{}~{}\mbox{otherwise}\end{array}\right.$
(30)
and the others are
$S_{1ij}=\left\\{\begin{array}[]{ll}-\gamma_{i}&~{}~{}~{}~{}\text{if~{}}i=j\\\
-r_{j}\beta_{ji}\frac{\gamma_{i}}{\gamma_{i}+\delta_{i}}&~{}~{}~{}~{}\mbox{otherwise}\end{array}\right.$
(31)
$S_{2ij}=\left\\{\begin{array}[]{ll}1-\gamma_{i}-\delta_{i}&~{}~{}~{}~{}\text{if~{}}i=j\\\
0&~{}~{}~{}~{}\mbox{otherwise}\end{array}\right.$ (32)
$S_{3ij}=\left\\{\begin{array}[]{ll}-\gamma_{i}+\chi_{i}&~{}~{}~{}~{}\text{if~{}}i=j\\\
0&~{}~{}~{}~{}\mbox{otherwise}\end{array}\right.$ (33)
$S_{4ij}=\left\\{\begin{array}[]{ll}0&~{}~{}~{}~{}\text{if~{}}i=j\\\
r_{j}\beta_{ji}\frac{(1-\nu_{i})\gamma_{i}}{\gamma_{i}+\delta_{i}}&~{}~{}~{}~{}\mbox{otherwise}\end{array}\right.$
(34)
$S_{6ij}=\left\\{\begin{array}[]{ll}1-\chi_{i}-\delta_{i}&~{}~{}~{}~{}\text{if~{}}i=j\\\
0&~{}~{}~{}~{}\mbox{otherwise.}\end{array}\right.$ (35)
Once we have calculated the Jacobian matrix of the system we can extend the
results of Lemma 2. That is, if the largest eigenvalue (in magnitude) is less
than one then it is assured that the system is asymptotically stable in the
fixed point $\vec{\mathbf{v}}$ and the dynamical system will exponentially
tend to the fixed point whatever was the initial state. Those interested in
the detailed proof of this Lemma 2 can consult the appendix of Deepa1 .
## III Simulations.
Until now, we have theoretically described the behaviour of the net.
Figure 1: Number of carriers $C(t)$ vs time (simulation epochs) in Chakrabarti
model under the threshold. Values for $\delta=0.1$ and $\gamma=0.01$ Figure 2:
Number of carriers $C(t)$ vs time (simulation epochs) in our model under the
threshold. Values for $\delta=0.1$ and $\gamma=0.01$. Figure 3: Number of
carriers $C(t)$ vs time (simulation epochs) in Chakrabarti model on the
threshold. Values for $\delta=0.07$ and $\gamma=0.004$ Figure 4: Number of
carriers $C(t)$ vs time (simulation epochs) in our model on the threshold.
Values for $\delta=0.07$ and $\gamma=0.004$. Figure 5: Number of carriers
$C(t)$ vs time (simulation epochs) in Chakrabarti model above the threshold.
Values for $\delta=0.01$ and $\gamma=0.01$ Figure 6: Number of carriers $C(t)$
vs time (simulation epochs) in our model above the threshold. Values for
$\delta=0.01$ and $\gamma=0.01$
Using the Dynamical Systems approach we have been able to predict the
conditions (a threshold) under which fast extinction is reached by using a
limited set of parameters that assures the we will converge to a fixed point.
In order to complete the study of our model and compare its performance with
the Chakrabarti model, we have made different simulations corresponding to the
cases in which we are under, on and above the threshold values established in
the previous section.
For this end, we have randomly generated the adjacency matrix corresponding to
a thirty node graph. We have taken, for the sake of comparison, the same set
of parameters that appears in section 4 of Deepa1 , that is, $r=0.1$ and
$\beta_{i,j}=0.1$. Additionally, we used the different values of $\delta$ and
$\gamma$ proposed in Deepa1 corresponding to P2P GNUTELLA data sets. We have
fixed $\nu=0.8$ and $\chi=0.1$ and we have started with six infected nodes
that we choose randomly. The number of time steps have been fixed in our
simulation to one and three hundred.
In Figures 1 and 2 it is shown that if the settings of the parameters values
fulfil the fast extinction condition then the same result is obtained in both
models. We can have a diverse set of parameters values as long as we are under
the threshold condition for achieving fast extinction.
In Figures 3 and 4 we show that if the set of parameters are combined in such
a way that they give exactly the threshold value the fast extinction is
achieved again in both models but in this case the fast extinction is slower
than in the first case. When the parameters values are combined in such a way
that we are beyond the threshold value then fast extinction is no longer
accomplished and the number of carriers grow very fast and eventually the
whole set of nodes could become infected. This behaviour is shown in Figures 5
and 6. In this case it can be observed a slight difference between both models
behaviour due to the presence of the parameter $\nu_{i}$.
## IV Conclusions and Future Work
As we have exposed in the sections corresponding to our proposal, if under our
model $1-\zeta_{i}(t)\rightarrow 0$, our _fix point and fast extinction
condition_ are consistent with those obtained in Deepa1 . In the other side,
if under our model $1-\zeta_{i}(t)>0$, then the _fix point and fast extinction
condition_ mentioned in the appendix section of Deepa1 are not longer valid.
In this last case the dynamical system becomes nonlinear, and the degree of
non-linearity will depend on the topology of the network. In this new setting
our $\nu_{i}$ parameter start to play a rôle in the virus as well as antidote
spreading on the network. In the future we will study this problem. If we take
as starting point this scenario it can be interesting to ask if the system
falls in a chaotic regime and if this is the case then how the stability of
the network can be re-established. If we want to achieve this state of the
system we can recall the synchronization and chaos tools developed in the
research field of automatic control. This is one of the subjects that we will
try to explore in the future.
## References
* (1) R. Albert and A.L. Barabási. Error and attack tolerance of complex networks. Nature, 406, 2000.
* (2) A.L. Barabási and R. Albert. Emergence of scaling in random graphs. Science, 286:509–512, 1999.
* (3) Ayalvadi Ganesh Christian Borgs, Jennifer Chayes and Amin Saberi. How to distribute antidote to control epidemics. Random Structures and Algorithms John Wiley and Sons, Inc. New York, NY, USA, (Volume 37, Issue 2):204–222, 2010.
* (4) R. Durrett and X.-F. Liu. The contact process on a finite set. The Annals of Probability, 16(3):1158–1173, 1988.
* (5) Alain Barrat Filippo Radicchi, Jose J. Ramasco and Santo Fortunato. Complex networks renormalization: Flows and fixed points. Physical Review Letters, (Volume 65):1487011–1487014, 2008.
* (6) M. W. Hirsch and S. Smale. Differential Equations,Dynamical Systems, and Linear Algebra. Academic Press, second edition, 1974.
* (7) Ronald A. Howard. Dynamic Programming and Markov Processes. The MIT Press, fourth edition, 1966.
* (8) C. Faloutsos S. Madden C. Guestrin J. Leskovec, D. Chakrabarti and M. Faloutsos. Information survival threshold in sensor and p2p networks. In IEEE INFOCOM 2007, 2007.
* (9) D. Kempe and J. Kleinberg. Protocols and impossibility results for gossip-based communication mechanisms. In Proceedings of the Symposium on Foundations of Computer Science (FOCS 2002), 2002.
* (10) P. Faloutsos M. Faloutsos and C. Faloutsos. On power-law relationships of the internet topology. In In Proceedings Sigcomm 1999, 1999.
* (11) D. Ben-Avraham A.L. Barabási N. Schwartz, R. Cohen and S. Havlin. Percolation in directed scale-free networks. Physical Review E, 66(1):0151041–0151044, 2002.
* (12) Itai Benjamini Noga Alon and Alan Stacey. Percolation on finite graphs and isoperimetric inequalities. The Annals of Probability, (Volume 32, No.3A):1727–1745, 2004.
* (13) Romualdo Pastor-Satorras and Alessandro Vespignani. Epidemic dynamics and endemic states in complex networks. Physical Review E, (Volume 63, Issue 2):0661171–0661178, 2001.
* (14) Romualdo Pastor-Satorras and Alessandro Vespignani. Epidemic spreading in scale-free networks. Physical Review Letters, (Volume 86, Number 14):3200–3203, 2001\.
* (15) Romualdo Pastor-Satorras and Alessandro Vespignani. Epidemic dynamics in finite size scale-free networks. Physical Review E, (Volume 65):0351081–0351084, 2002.
* (16) Béla Bollobási Paul Balister. Bond percolation with attenuation in high dimensional voronoi tilings. Random Structures and Algorithms, Wiley InterScience, pages 5–10, 2009.
|
arxiv-papers
| 2010-10-13T21:53:02 |
2024-09-04T02:49:13.864213
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Carlos Rodr\\'iguez-Lucatero and Roberto Bernal-Jaquez",
"submitter": "Roberto Bernal-Jaquez",
"url": "https://arxiv.org/abs/1010.2783"
}
|
1010.2813
|
# Engineering Biphoton Wave Packets with an Electromagnetically Induced
Grating
Jianming Wen,1,2111Email Address: jianming.wen@gmail.com Yan-Hua Zhai,3
Shengwang Du,4 and Min Xiao1,2 1National Laboratory of Solid State
Microstructures and Department of Physics, Nanjing University, Nanjing 210093,
China
2Department of Physics, University of Arkansas, Fayetteville, Arkansas 72701,
USA
3Department of Physics, University of Maryland, Baltimore County, Baltimore,
Maryland 21250, USA
4Department of Physics, The Hong Kong University of Science and Technology,
Clear Water Bay, Kowloon, Hong Kong, China
###### Abstract
We propose to shape biphoton wave packets with an electromagnetically induced
grating in a four-level double-$\Lambda$ cold atomic system. We show that the
induced hybrid grating plays an essential role in directing the new fields
into different angular positions, especially to the zeroth-order diffraction.
A number of interesting features appear in the shaped two-photon waveforms.
For example, broadening or narrowing the spectrum would be possible in the
proposed scheme even without the use of a cavity.
###### pacs:
42.50.Dv, 42.65.Lm, 42.50.Ct, 03.65.Ud
## I Introduction
The generation of entangled paired photons with a desired joint spectrum has
become a fascinating conceptual viewpoint for both fundamental and practical
research. This is because the joint spectrum contains the information on
bandwidth, type of frequency correlations, and wave function of the two-photon
state. By manipulating the joint spectrum, one can obtain the most appropriate
form for the specific quantum optics application under consideration. For
instance, biphotons with a narrow bandwidth play a key role in the long-
distance quantum communication protocols based on atom-photon interface
hammerer ; biphotons with a few femtoseconds of correlation time are of
particular interest in the fields of quantum metrology giovannetti and for
some protocols for timing and positioning measurements valencia .
Conventionally, entangled paired photons are produced from the process of
spontaneous parametric down conversion (SPDC) in a nonlinear crystal, where a
pump photon is annihilated and two down-converted daughter photons are
simultaneously emitted spdc . Because of their broad bandwidth and short
coherence time, it is difficult to shape SPDC photon wave packets in the time
domain directly. A number of methods have been proposed and developed to
perform spectral manipulation of the joint spectrum peer ; viciani ; hendrych
or spatial modulation of the nonlinear interaction valencia2 ; harris ; nasr .
Others are to modify the (quasi-)phase matching uren , engineer the dispersive
properties of the nonlinear medium kuzucu , or imprint the spectral and
spatial characteristics of the pump beam into the joint spectrum keller .
A recent demonstration of the generation of narrow-band biphotons in cold
atomic ensembles via spontaneous four-wave mixing (SFWM) balic ; du1 ; du2 ;
wen1 ; wen2 has attracted considerable attention because of their long
coherence time and controllable quantum wave packets. Nonlocal modulation of
temporal correlation has been observed with such narrow-band biphotons sensarn
. In a very recent experiment du3 , shaping of the temporal wave form by
periodically modulating the input driving lasers has confirmed the previous
theoretical prediction du4 , in which the input field profiles can be revealed
in the two-photon correlation measurements. One major advantage over shaping
the SPDC photon temporal wave function is that these narrow-band biphotons
allow further wave-packet modification directly in the time domain.
In this paper, we describe a new way to manipulate paired Stokes and anti-
Stokes wave forms produced from SFWM in a four-level double-$\Lambda$ xiao
cold atomic system with the use of an electromagnetically induced grating
(EIG) xiao1 ; araujo . EIG has been experimentally demonstrated in cold atoms
imoto ; cardoso and has been applied to all optical switching and routing in
hot atomic vapors xiao2 . Here, we show that, by spatially modulating the
control beams, alternating regions of high transmission and absorption can be
created inside the atomic sample that act as an amplitude grating and by which
the joint Stokes and anti-Stokes wave packet can be shaped. Compared with
previous proposals ascribed above, several interesting features appear in the
present one. First, such a medium may exert both amplitude and phase
modulations on biphoton wave packets in much the same way that a hybrid
(amplitude and phase) grating does to the amplitude and phase of an
electromagnetic wave. Second, the spatial modulation of the control fields is
imprinted into both the linear and the nonlinear susceptibilities.
Consequently, this mapping may broaden or narrow the joint spectrum depending
on the system’s parameters. Third, but not least, because of the grating
diffraction interference, the spectral brightness can be improved, and the
emission angle can be confined to some particular angles. For example, the
anti-Stokes field will be mainly directed to the zeroth-order diffraction.
We organize the paper as follows. The basic idea is presented in Sec. II by
considering two-photon temporal correlation measurement. The conclusion is
summarized in Sec. III.
## II Shaping Biphoton Wave Form with EIG
### II.1 EIG
To illustrate the basic idea, we consider a four-level double-$\Lambda$ atomic
system (e.g. 87Rb) depicted in Fig. 1(a), where all the atomic population is
assumed to be in the ground state $|1\rangle$. To ignore the Doppler
broadening, the atoms are laser cooled in a magnetic optical trap. Two strong
control fields ($\omega_{c}$), resonant with the atomic transition
$|2\rangle\rightarrow|3\rangle$ while being symmetrically displaced with
respect to $z$, are incident upon the atomic ensemble at such angles that they
intersect and form a standing wave within the medium [see Fig. 1(c)]. In the
presence of the counterpropagating weak probe field ($\omega_{p}$) far detuned
from the transition $|1\rangle\rightarrow|4\rangle$, phase matched Stokes
($\omega_{s}$) and anti-Stokes ($\omega_{as}$) photons are then spontaneously
generated in opposite directions and are detected by single-photon detectors
D2 and D1, respectively, as shown in Fig. 1(b). Since the linear and nonlinear
optical responses to the generated fields depend on the strength of the
control light, they are expected to change periodically as the standing wave
changes from the nodes to anti-nodes across the $x$ dimension. In the current
configuration, the Stokes photons travel at nearly the speed of light in
vacuum with negligible Raman gain. In contrast, the strong control beams
induce a set of periodic transparency windows to the anti-Stokes field. Thus,
alternatively, a nonmaterial grating is formed in the anti-Stokes channel.
This grating is termed as EIG xiao1 , which will diffract the anti-Stokes
field into some particular angles according to the diffraction orders.
Figure 1: (Color online) Shaping biphoton wave packets with an EIG. (a) The
level structure, where in the presence of a cw probe ($\omega_{p}$) and
control ($\omega_{c}$) fields, paired Stokes ($\omega_{s}$) and anti-Stokes
($\omega_{as}$) photons are spontaneously created from the four-wave mixing
processes in the low-gain regime. (b) The backward generation geometry, where
two strong control beams symmetrically displace with respect to $z$ and form a
standing wave along $x$. (c) The standing wave formed by control fields.
Following the analysis presented in Ref. wen1 , the third-order nonlinear
susceptibility for the generated anti-Stokes field is calculated to be
$\displaystyle\chi^{(3)}_{as}(\omega)=\frac{-N\mu_{13}\mu_{32}\mu_{24}\mu_{41}/[4\hbar^{3}\epsilon_{0}(\Delta_{p}+i\gamma_{41})]}{(\omega-\Omega_{e}+i\gamma_{e})(\omega+\Omega_{e}+i\gamma_{e})},$
(1)
and the linear susceptibilities at the Stokes and anti-Stokes frequencies are,
respectively,
$\displaystyle\chi_{s}(\omega)$ $\displaystyle=$
$\displaystyle\frac{N|\mu_{42}|^{2}(\omega-i\gamma_{31})/(4\hbar\epsilon_{0})}{|\Omega_{c}|^{2}\cos^{2}(\frac{\pi{x}}{d})-(\omega-i\gamma_{31})(\omega-i\gamma_{21})}\frac{|\Omega_{p}|^{2}}{\Delta^{2}_{p}+\gamma^{2}_{41}},$
$\displaystyle\chi_{as}(\omega)$ $\displaystyle=$
$\displaystyle\frac{N|\mu_{31}|^{2}(\omega+i\gamma_{21})/(\hbar\epsilon_{0})}{|\Omega_{c}|^{2}\cos^{2}(\frac{\pi{x}}{d})-(\omega+i\gamma_{31})(\omega+i\gamma_{21})},$
(2)
where $N$ is the atomic density, $\mu_{ij}$ are dipole matrix elements,
$\Omega_{p}$ and $\Omega_{c}$ are the probe/control Rabi frequency,
$\gamma_{ij}$ are the decay or dephasing rate,
$\Delta_{p}=\omega_{p}-\omega_{41}$ is the probe detuning, and
$d=\frac{\pi}{k_{cx}}$ represents the space period, which can be made
arbitrarily larger than the wavelength of the control fields by varying the
angle between their two wave vectors.
$\Omega_{e}=\sqrt{|\Omega_{c}|^{2}\cos^{2}(\frac{\pi{x}}{d})+\gamma_{31}\gamma_{21}}\approx|\Omega_{c}|\cos(\frac{\pi{x}}{d})$
is the effective control Rabi frequency, and
$\gamma_{e}=\frac{\gamma_{31}+\gamma_{21}}{2}$ is the effective dephasing
rate. $\chi^{(3)}_{as}$ in Eq. (1) has two resonances separated by
$\Omega_{e}$ and each is associated with a linewidth of $2\gamma_{e}$. From
Eqs. (1) and (2), it is obvious that the spatial periodic modulation of the
control fields has been mapped into the optical responses to the Stokes and
anti-Stokes fields. Consequently, such a modulation will further modify the
two-photon wave form as will be discussed later. It is known that the linear
susceptibilities determine the transmission bandwidth and dispersion property.
Taking $|\Omega_{p}|\ll\Delta_{p}$, $\chi_{s}$ is approximated as 0, which
means the Stokes photons traverse the medium almost at the speed of light in
vacuum, and the Raman gain is negligible. In contrast, the anti-Stokes photons
may propagate at a lower group velocity, $v_{g}\approx
2\hbar\epsilon_{0}c|\Omega_{c}|^{2}\cos^{2}(\frac{\pi{x}}{d})/N|\mu_{31}|^{2}\omega_{31}=v_{0}\cos^{2}(\frac{\pi{x}}{d})$,
and experience periodic linear loss characterized by
$\alpha=N\sigma_{31}\gamma_{21}\gamma_{31}/2[|\Omega_{c}|^{2}\cos^{2}(\frac{\pi{x}}{d})+\gamma_{21}\gamma_{31}]$,
where $\sigma_{31}=\omega_{31}|\mu_{31}|^{2}/(\hbar\epsilon_{0}c\gamma_{31})$
is the on-resonance absorption cross section in the transition
$|1\rangle\rightarrow|3\rangle$.
Thus, such a periodic linear loss results in an EIG to the anti-Stokes
photons. Figure 2 displays a typical transmission function for the anti-Stokes
light as a function of $x$. It is easy to understand that, at the transverse
locations around the nodes (of the standing wave), the control field
intensities are so weak that the anti-Stokes field is absorbed according to
the usual Beer law. In contrast, since the intensity distribution of the
control fields at the spatial locations around the antinodes is very strong,
the absorption of the anti-Stokes field is greatly suppressed due to the
effect of electromagnetically induced transparency EIT . This leads to a
periodic amplitude modulation across the beam profile of the anti-Stokes
light, a phenomenon reminiscent of the amplitude grating.
Figure 2: A typical transmission profile of the anti-Stokes field as a
function of $x$. Parameters are chosen as $d=2$ $\mu$m, the optical depth
about 5, $\gamma_{31}=2\pi\times 3$ MHz, and
$\gamma_{21}=0.6\times\gamma_{31}$.
### II.2 Shaping two-photon wave form
The paired Stokes and anti-Stokes photon state can be obtained from first-
order perturbation theory du2 ; wen1 ; rubin . For simplicity, we take the
input probe and control beams as classical cw lasers and focus on the two-
photon temporal correlation. The effective interaction length is taken as $L$.
The unnormalized biphoton state at the output surfaces of the sample may be
written as
$\displaystyle|\Psi\rangle=\Psi_{0}\int{d}\omega\Phi(\omega)a^{{\dagger}}_{s}a^{{\dagger}}_{as}|0\rangle,$
(3)
where $\Psi_{0}$ is a grouped constant, and the joint spectral function takes
the form
$\displaystyle\Phi(\omega)=\int^{x_{2}}_{x_{1}}dx\chi^{(3)}_{as}(\omega)\cos\bigg{(}\frac{\pi{x}}{d}\bigg{)}\mathrm{sinc}\bigg{(}\frac{\Delta{k}L}{2}\bigg{)}e^{i\frac{\Delta{k}L}{2}},$
(4)
where the cosine term comes from the standing wave of the control fields, and
the last two terms from the longitudinal phase matching condition with
$\Delta{k}\approx\frac{\omega}{v_{g}}+i\alpha$. In Eq. (4), we have taken the
linear loss into account. It is clear that that the joint spectrum $\Phi$ can
be engineered through $\chi^{(3)}_{as}$ and the phase matching condition.
After, we describe the wave-packet shaping by considering the simple two-
photon temporal correlation measurement in which paired Stokes and anti-Stokes
photons are detected by single-photon detectors D1 and D2 with equal pathways
from the output surfaces of the medium, as shown in Fig. 1(b). Since there are
two characteristic timings embedded in Eq. (4), the resonance linewidth
determined by $\chi^{(3)}_{as}$ and the natural spectral width determined by
the phase matching condition $\frac{L}{v_{0}}$ will be looked at separately by
using the two-photon temporal correlation in which only one characteristic
timing is dominant.
Using the Glauber theory, the two-photon amplitude is
$\displaystyle A=\langle 0|E^{(+)}_{s}E^{(+)}_{as}|\Psi\rangle.$ (5)
The field $E^{(+)}_{j}$ is the positive-frequency part of the free-space
electromagnetic field at position $r_{j}$ and time $t_{j}$. In the far-field
region (Fraunhofer diffraction), the biphoton amplitude (5) over the
diffraction angle $\theta$ (with respect to $z$) can be derived, by following
the procedure done in Refs. wen1 ; wen2 ; wen3 , as
$\displaystyle
A(\tau;\theta)=A\int^{x_{2}}_{x_{1}}dx\cos\bigg{(}\frac{\pi{x}}{d}\bigg{)}e^{ik_{as}x\sin\theta}\int{d}\omega\chi^{(3)}_{as}(\omega)\mathrm{sinc}\bigg{(}\frac{\Delta{k}L}{2}\bigg{)}e^{i(\frac{\Delta{k}L}{2}-\omega\tau)},$
(6)
where $A$ is an integrant-irrelevant constant, $\tau=t_{as}-t_{s}$ is the
relative time delay between two clicks, and $k_{as}$ is the central wave
number of the anti-Stokes photons. Equation (6) can further be recast into a
product of an integral and a geometric series
$\displaystyle
A(\tau;\theta)=A\sum^{M/2}_{n=-M/2}e^{ik_{as}nd\sin\theta}\int^{\frac{d}{2}}_{-\frac{d}{2}}dx\cos\bigg{(}\frac{\pi{x}}{d}\bigg{)}{e}^{ik_{as}x\sin\theta}\int{d}\omega\chi^{(3)}_{as}(\omega)\mathrm{sinc}\bigg{(}\frac{\Delta{k}L}{2}\bigg{)}e^{i(\frac{\Delta{k}L}{2}-\omega\tau)},$
(7)
where $M$ represents the input probe field across $M$ times $d$. This can be
guaranteed by adjusting the diameters of both probe and control fields to
cover $M$ slits. By evaluating the geometric progression in the usual fashion,
Eq. (7) can be written as
$\displaystyle
A(\tau;\theta)=A\frac{\sin\frac{k_{as}Md\sin\theta}{2}}{\sin\frac{k_{as}d\sin\theta}{2}}B(\tau;\theta),$
(8)
with
$\displaystyle
B(\tau;\theta)=\int^{\frac{d}{2}}_{-\frac{d}{2}}dx\cos\bigg{(}\frac{\pi{x}}{d}\bigg{)}e^{ik_{as}x\sin\theta}\int{d}\omega\chi^{(3)}_{as}(\omega)\mathrm{sinc}\bigg{(}\frac{\Delta{k}L}{2}\bigg{)}e^{i(\frac{\Delta{k}L}{2}-\omega\tau)}.$
(9)
Therefore, the diffracted two-photon amplitude is a product of a single slit
Eq. (9) multiplied by the function in Eq. (8). Equations (8) and (9) together
imply that the two correlated Stokes and anti-Stokes photons are
simultaneously produced from any one of the slits, which can be regarded as a
superposition of coherent SFWM subsources. We also notice that the first
integration in Eq. (9) can be visualized as an amplitude grating with a
transmission profile followed by a cosine curvature. The emission angles and
diffraction efficiencies are determined by the ability of the induced grating.
From Eq. (8), it is easy to obtain the diffraction angles of the anti-Stokes
field for different diffraction orders $m$ as
$\displaystyle\sin\theta=m\frac{\lambda_{as}}{d},$ (10)
where $\lambda_{as}=2\pi/k_{as}$. According to the results shown in Ref. xiao1
, the diffraction mainly occurs at the zeroth order. This could be important
to direct the light into a smaller solid angle and, hence, enhance its
spectral brightness at the observation’s location. Equations (8) and (9) are
our starting points to analyze shaping of biphoton wave forms using EIG.
Since, for the anti-Stokes field, the energy is almost emitted toward the
zeroth-order diffraction direction, we assume $\theta=0$ in Eq. (8) to
simplify the analysis in the following.
### II.3 Two-photon coincidence counts
First, let us look at the case in which the coherence time is mainly
determined by the resonance linewidth. In such a case, the natural spectral
width from the phase matching is much greater than the linewidth. Hence, its
effect on two-photon temporal correlation can be ignored. Thus, Eq. (9)
reduces to
$\displaystyle
B(\tau)=\int^{\frac{d}{2}}_{-\frac{d}{2}}dx\cos\bigg{(}\frac{\pi{x}}{d}\bigg{)}\int{d}\omega\chi^{(3)}_{as}(\omega)e^{-i\omega\tau}.$
(11)
Plugging Eq. (1) into Eq. (11) and completing the frequency integral yields
$\displaystyle
B(\tau)=B\int^{\frac{d}{2}}_{-\frac{d}{2}}dx\sin\bigg{[}|\Omega_{c}|\tau\cos\bigg{(}\frac{\pi{x}}{d}\bigg{)}\bigg{]}e^{-\gamma_{e}\tau},$
(12)
where $B$ is a constant. Different from previous findings balic ; du1 ; du2 ;
wen1 ; wen2 ; du3 ; du4 ; wen3 , Eq. (12) clearly shows that the profile of
the biphoton wave form is further manipulated by the periodic modulation of
the control fields. Implementing the integration in Eq. (12) gives
$\displaystyle B(\tau)=BdH_{0}(|\Omega_{c}|\tau)e^{-\gamma_{e}\tau},$ (13)
where $H_{0}(x)$ is the Struve function of order zero. The two-photon
coincidence counting rate equals the square of $A(\tau)$, whose profile is
governed by $H_{0}(x)$ and is manifested by an exponential decay. In Fig. 3,
we have provided two typical simulations of the coincidences using the
parameters in Ref. du2 . We notice that the damped oscillations shown in Fig.
3 do not obey Rabi flopping as previously reported in Refs. balic ; du1 ; du2
; wen2 ; wen3 . The origin of this difference comes from the periodic
modulation of the control fields, which, in turn, modifies the joint-detection
patterns. The minimum coincidences appear at the zero solutions of $H_{0}(x)$.
The lower curve in Fig. 3 gives the over-damped case in which even a single
oscillation is not fully observable because of the fast exponential decay.
Another noticeable feature is that the joint spectrum is broadened in a single
oscillation due to the diffraction interference.
Figure 3: (Color online) Two-photon temporal coincidences exhibit damped and
overdamped oscillations with the space period $d=2$ $\mu$m. Other parameters
are the same as in Ref. du2 .
Next, we look at the two-photon temporal correlation mainly characterized by
the phase-matching condition. That is, the natural spectral width is much
narrower than the resonance linewidth such that the intrinsic mechanism of
biphoton generation is partially or even fully washed out. (The latter case
requires much higher optical depth.) In such a case, Eq. (9) becomes
$\displaystyle
B(\tau)=\int^{\frac{d}{2}}_{-\frac{d}{2}}dx\cos\bigg{(}\frac{\pi{x}}{d}\bigg{)}\int{d}\omega\mathrm{sinc}\bigg{(}\frac{\Delta{k}L}{2}\bigg{)}e^{i(\frac{\Delta{k}L}{2}-\omega\tau)},$
(14)
which can be numerically evaluated. In Fig. 4, we have plotted the coincidence
counting rate with the space period $d=2$ $\mu$m plus taking the third-order
nonlinearity [Eq. (9)] into account. As illustrated in Fig. 4(a), most of the
features appearing in previous studies [for instance, see Fig. 4(b)] can be
observed. For example, the sharp peak in the leading edge of the two-photon
coincidence counts represents the Sommerfeld-Brillouin precursor at the
biphoton level, as report in Ref. du5 . One difference from previous results
in the literature du1 ; du2 ; du4 is that, at the tail in Fig. 4(a), several
small bumps emerge instead of a smooth exponential decay. Another difference
is that the coherence time is extended. In Fig. 4(a), the coherence time is
extended by more than 1 $\mu$s. However, without the induced grating as shown
in Fig. 4(b), the coherence time is only about 800 ns. Alternatively, the
joint spectrum of biphotons is narrowed. This spectrum narrowing is a result
of the spatial modulation of the control fields plus the modulated group
velocities of the anti-Stokes field. Without the use of the cavity, the
spectrum narrowing achieved here is useful for producing narrow-band biphotons
with higher spectral brightness. If the optical depth of the medium could be
made enough high, the two-photon temporal correlation would be closer to a
square-wave pattern as usually observed in the SPDC process. Those small bumps
would become discrete step functions at the tail, which can be verified from
Eq. (14). Since this looks more like an ideal case and might not be detectable
in the experiment, we will not offer further discussions here.
Figure 4: (Color online) Two-photon temporal coincidences: (a) modulated by an
EIG with the space period $d=2$ $\mu$m. Other parameters are chosen as
$L/v_{0}=800$ ns, $\Omega_{c}=5\gamma_{31}$, $\gamma_{31}=2\pi\times 3$ MHz,
and $\gamma_{21}=0.6\gamma_{31}$. (b) without the induced grating. Same
parameters are chosen as in (a).
Before ending the discussions, in Secs. IIA and IIB, we have analyzed how to
shape the entangled Stokes-anti-Stokes temporal wave form with the use of EIG.
The extension of the idea to be used on a nonlinear crystal would be
interesting. Although it is easy to design a diffraction grating within or at
the output surface of the crystal, it is difficult to modulate the dispersion
periodically and spatially. Therefore, it is very challenging to fully recover
the features obtained here in nonlinear crystals.
## III Summary
In summary, here, we have proposed a method to engineer the two-photon
temporal wave packets by utilizing an EIG. The method distinguishes itself
from previous research by the appearance of several features. First, the
induced grating influences both the linear and the nonlinear susceptibilities.
As a consequence, this will shape the biphoton wave packets through both the
dispersive properties of the medium and the periodic nonlinear optical
responses. Second, the induced (hybrid) nonmaterial grating directs the output
anti-Stokes field into different angular positions, especially into its
zeroth-order diffraction. Third, the modulated biphoton wave packets exhibit
different profiles compared with previous studies. For example, the damped
oscillations do not coincide with the Rabi oscillations as observed in Refs.
balic ; du1 ; du2 . The decayed square-wave pattern shows small bumps at the
tail, which, to the best of our knowledge, have never been discovered in the
literature. Fourth, the spectral brightness and emission angle can be further
engineered by the induced grating. This paper is important not only because it
explores another application of the EIG, but also because the shaped biphoton
wave pakckets hold applications in certain protocols of quantum information,
quantum communications, and quantum cryptography. For instance, the properties
ascribed before can be used to direct the propagation of single photons and
improve the efficiency of detecting photons in free space due to the
diffraction. The broadened or narrowed bandwidth could be useful for coherent
absorption and reemission of photons based on the interface between atoms and
photons. The effect of EIG on transverse correlation of entangled photons may
be interesting and worth studying. However, such an issue is beyond the scope
of the current paper and might be addressed somewhere else.
## IV Acknowledgements
We gratefully acknowledge insightful discussions with M. H. Rubin, K.-H. Luo,
and Xiaoshun Jiang. J.W. and M.X. were supported, in part, by the National
Science Foundation (USA). J.W. also acknowledges financial support by 111
Project No. B07026 (China). S.D. was supported by the Hong Kong Research
Grants Council (Project No. HKUST600809).
## References
* (1) K. Hammerer, A. S. Sørensen, and E. Polzik, Rev. Mod. Phys. 82, 1041 (2010).
* (2) V. Giovannetti, S. Lloyd, and L. Maccone, Phys. Rev. Lett. 96, 010401 (2006).
* (3) A. Valencia, G. Scarcelli, and Y.-H. Shih, Appl. Phys. Lett. 85, 2655 (2004).
* (4) Y.-H. Shih, Rep. Prog. Phys. 66, 1009 (2003).
* (5) A. Pe’er, B. Dayan, A. A. Friesem, and Y. Silberberg, Phys. Rev. Lett. 94, 073601 (2005).
* (6) M. Bellini, F. Marin, S. Viciani, A. Zavatta, and F. T. Arecchi, Phys. Rev. Lett. 90, 043602 (2003).
* (7) M. Hendrych, X. Shi, A. Valencia, and J. P. Torres, Phys. Rev. A 79, 023817 (2009).
* (8) A. Valencia, A. Ceré, X. Shi, G. Molina-Terriza, and J. P. Torres, Phys. Rev. Lett. 99, 243601 (2007).
* (9) S. E. Harris, Phys. Rev. A 78, 021807(R) (2008).
* (10) M. B. Nasr, S. Carrasco, B. E. A. Saleh, A. V. Sergienko, M. C. Teich, J. P. Torres, L. Torner, D. S. Hum, and M. M. Fejer, Phys. Rev. Lett. 100, 183601 (2008).
* (11) A. B. U’Ren, R. K. Erdmann, M. de la Cruz-Gutierrez, and I. A. Walmsley, Phys. Rev. Lett. 97, 223602 (2006).
* (12) O. Kuzucu, M. Fiorentino, M. A. Albota, F. N. C. Wong, and F. X. Kärtner, Phys. Rev. Lett. 94, 083601 (2005).
* (13) T. E. Keller and M. H. Rubin, Phys. Rev. A 56, 1534 (1997).
* (14) V. Balić, D. A. Braje, P. Kolchin, G. Y. Yin, and S. E. Harris, Phys. Rev. Lett. 94, 183601 (2005).
* (15) S. Du, P. Kolchin, C. Belthangady, G. Y. Yin, and S. E. Harris, Phys. Rev. Lett. 100, 183603 (2008).
* (16) S. Du, J.-M. Wen, and M. H. Rubin, J. Opt. Soc. Am. B 25, C98 (2008).
* (17) J.-M. Wen and M. H. Rubin, Phys. Rev. A 74, 023808 (2006); 74, 023809 (2006).
* (18) J.-M. Wen, S. Du, and M. H. Rubin, Phys. Rev. A 75, 033809 (2007); S. Du, J.-M. Wen, M. H. Rubin, and G. Y. Yin, Phys. Rev. Lett. 98, 053601 (2007).
* (19) S. Sensarn, G. Y. Yin, and S. E. Harris, Phys. Rev. Lett. 103, 163601 (2009).
* (20) J. F. Chen, S. Zhang, H. Yan, M. M. T. Loy, G. K. L. Wong, and S. Du, Phys. Rev. Lett. 104, 183604 (2010).
* (21) S. Du, J.-M. Wen, and C. Belthangady, Phys. Rev. A 79, 043811 (2009).
* (22) B. L. Lü, W. H. Burkett, and M. Xiao, Opt. Lett. 23, 804 (1998).
* (23) H. Y. Ling, Y.-Q. Li, and M. Xiao, Phys. Rev. A 57, 1338 (1998).
* (24) L. E. E. de Araujo, Opt. Lett. 35, 977 (2010).
* (25) M. Mitsunaga and N. Imoto, Phys. Rev. A 59, 4773 (1999).
* (26) G. C. Cardoso and J. W. R. Tabosa, Phys. Rev. A 65, 033803 (2002).
* (27) A. W. Brown and M. Xiao, Opt. Lett. 30, 699 (2005).
* (28) See, for examples, S. E. Harris, Phys. Today 50 (7), 36 (1997); M. Xiao, Y.-q. Li, S. Z. Jin, and J. Gea-Banacloche, Phys. Rev. Lett. 74, 666 (1995).
* (29) M. H. Rubin, D. N. Klyshko, Y.-H. Shih, and A. V. Sergienko, Phys. Rev. A 50, 5122 (1994).
* (30) J.-M. Wen, S. Du, Y. P. Zhang, M. Xiao, and M. H. Rubin, Phys. Rev. A 77, 033816 (2008).
* (31) S. Du, C. Belthangady, P. Kolchin, G. Y. Yin, and S. E. Harris, Opt. Lett. 33, 2149 (2007).
|
arxiv-papers
| 2010-10-14T03:22:28 |
2024-09-04T02:49:13.873546
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jianming Wen, Yanhua Zhai, Shengwang Du, and Min Xiao",
"submitter": "Jianming Wen",
"url": "https://arxiv.org/abs/1010.2813"
}
|
1010.2860
|
# Electric field response of strongly correlated one-dimensional metals: a
Bethe-Ansatz density functional theory study
A. Akande and S. Sanvito akandea@tcd.ie School of Physics and CRANN, Trinity
College, Dublin 2, Ireland
###### Abstract
We present a theoretical study on the response properties to an external
electric field of strongly correlated one-dimensional metals. Our
investigation is based on the recently developed Bethe-Ansatz local density
approximation (BALDA) to the density functional theory formulation of the
Hubbard model. This is capable of describing both Luttinger liquid and Mott-
insulator correlations. The BALDA calculated values for the static linear
polarizability are compared with those obtained by numerically accurate
methods, such as exact (Lanczos) diagonalization and the density matrix
renormalization group, over a broad range of parameters. In general BALDA
linear polarizabilities are in good agreement with the exact results. The
response of the exact exchange and correlation potential is found to point in
the same direction of the perturbing potential. This is well reproduced by the
BALDA approach, although the fine details depend on the specific
parameterization for the local approximation. Finally we provide a numerical
proof for the non-locality of the exact exchange and correlation functional.
###### pacs:
## I Introduction
Material systems, whose electronic structure cannot be described at a mean
field level, are conventionally named strongly correlated. These display an
enormous variety of properties, which all originate from the interplay between
Coulomb repulsion and kinetic energy, and from their dimensionality. Phenomena
related to electron-electron correlation include metal-insulator transition,
Tomonaga-Luttinger liquid behaviour and superconductivity, just to name a few
Fazeka ; Giamarchi . In particular electron correlations play a fundamental
rôle in one-dimension (1D). In 1D confined structures, electrons cannot avoid
each other and collective excitations emerge over the ground state, so that
the Fermi liquid picture breaks down. In fact one can demonstrate that the
ground state of an interacting 1D object is always a Luttinger liquid
regardless of the strength of the electron-electron interaction Fazeka ;
Giamarchi . Although some aspects are still controversial, experimental
evidence supporting the existence of Luttinger liquids in 1D has been provided
for carbon nanotubes Postma and for atomic wires built of surface terraces
Segovia ; Auslanender .
Strongly correlated systems are regularly modeled by means of effective
Hamiltonians, which usually lack all the details of an ab initio description,
but capture the relevant physical properties arising from electron
correlation. The advantage of dealing with effective Hamiltonians is that they
are commonly mathematically tractable and general enough to be applied to a
variety of problems. Among the many effective Hamiltonians that one can
construct the Hubbard model Gutzwiller ; Hubbard ; Kanamory has enjoyed a
vast popularity since it is simple and still can capture the subtle interplay
between Coulomb repulsion and kinetic energy.
Although exact solutions of the Hubbard model are known in particular limits
Hubbook , a general one for an arbitrary system, which can be finite and
inhomogeneous, requires a numerical treatment. This however represents a
severely demanding task, since the Hilbert space associated to the Hubbard
Hamiltonian for $L$ sites is 4L dimensional, so that exact (Lanczos)
diagonalization (ED) can only handle a relatively small number of sites. Other
many body approaches, such as the density matrix renormalization group (DMRG)
White ; Schollwock , extend the range to a few hundred sites, but little is
possible beyond that limit. It would be then useful to have a method capable
of describing accurately the ground state and still having the computational
overheads of a mean field approach. Such a method is provided by lattice
density functional theory (LDFT).
LDFT was initially proposed by Gunnarsson and Schonhammer Gunnarsson ;
Schonhammer as an extension of standard, ab initio, DFT HK ; KS to lattice
models. The theory essentially reformulates the Hohenberg-Kohn theorem and the
Kohn-Sham construction in terms of the site occupation instead of the electron
density. Although originally introduced with a pedagogical purpose, LDFT has
enjoyed a growing success and it has been already applied to a diverse range
of problems. These include fundamental aspects of DFT and of the Hubbard
model, as the band-gap problem in semiconductors Gunnarsson , the dimerization
of 1D Hubbard chains Lopez and the formation of the Mott-Hubbard gap Capelle1
. LDFT has also been employed for investigating effects at the nanoscale
traceable to strong correlation, like the behavior of impurities Capelle2 ,
spin-density waves Capelle3 and inhomogeneity Silva , as well as more exotic
aspects like the phase diagram of harmonically confined 1D fermions Campo and
that of ultracold fermions trapped in optical lattices Xianlong1 ; Xianlong2 ;
Xianlong3 . More recently LDFT has been extended to the time-dependent domain
Verdozzi , to quantum transport Gross and to response theory Schenk .
As in standard DFT also LDFT is in principle exact. However its practical
implementation is limited by the accuracy of the unknown exchange correlation
(XC) functional, which introduces the many-body effects into the theory. The
construction of an XC functional begins with choosing a reference system, for
which some exact results are known. These impose a number of constraints that
the XC functional must satisfy, as for example its asymptotic behavior or its
scaling properties. Then the functional is built by interpolating and fitting
to known many-body reference results. Such a construction for instance has
been employed in the case of the local density approximation (LDA) in ab
initio DFT. The reference system in two and three dimensions is usually an
electron gas of some kind, since one aims at reproducing a Fermi liquid.
However in 1D the known ground state has a Luttinger-liquid nature, so that
the reference system should be chosen accordingly. In the case of the Hubbard
Hamiltonian in 1D a powerful result is that obtained by Lieb and Wu LiebWu
for the homogeneous case by using the Bethe Ansatz. This is the basis for
constructing an XC functional for the Hubbard model in 1D Capelle1 ; Capelle2
.
In this work we evaluate the ability of a range of known approximations to the
XC functional for the 1D Hubbard model at predicting the electrical response
to an external electric field of finite 1D chains away and in the vicinity of
the Mott transition. This is relevant not just as a test for Hubbard LDFT but
also for understanding real materials, whose electrical response can be
mimicked in terms of the Hubbard model Ishihara1 ; Ishihara2 . The 1D case in
particular can provide important insights into the nonlinear optical
properties of polymers Rojo . Our strategy is that of constantly comparing the
DFT results with those obtained with highly accurate many-body schemes. For
these we use exact diagonalization for small chains and the DMRG method for
larger systems. Our calculations reveal a substantial good agreement between
LDFT and exact results for both the polarizability and the XC potential
response of finite 1D chains. The paper is organized as follows. In the next
section we will briefly review the Hubbard LDFT and the approximations used
for constructing the XC functional. Then we will discuss results, first for
the electrical polarizabilities and then for the response of the XC potential
to an external electric field. Finally we will carry on a numerical
investigation on the validity of the local approximation to the XC functional
and then we will conclude.
## II Theoretical formulation
One-dimensional correlated metals can be described by the homogeneous Hubbard
Hamiltonian, $H_{\mathrm{U}}$. For a 1D chain comprising $L$ sites
$H_{\mathrm{U}}$ writes
$H_{\mathrm{U}}=-t\sum_{i=1,\>\sigma}^{L-1}(c^{\dagger}_{i+1\sigma}c_{i\sigma}+hc)+U\sum_{i=1}^{L}\hat{n}_{i\uparrow}\hat{n}_{i\downarrow},$
(1)
where the first kinetic term describes the hopping of electrons with spin
$\sigma$ ($\sigma=\uparrow,\downarrow$) between nearest neighbour sites with
amplitude $t>0$, while the second accounts for the electrostatic repulsion
$U>0$ of doubly occupied sites. In the equation (1)
$c_{i\sigma}^{\dagger}(c_{i\sigma})$ is the fermion creation (annihilation)
operator for an electron at site $i$ with spin $\sigma$ and the site
occupation operator is written as
$\hat{n}_{i\sigma}=c^{\dagger}_{i\sigma}c_{i\sigma}$. Clearly there is only
one energy scale in the problem so that the ratio $U/t$ determines all the
electronic properties. Note that a second energy scale can be included in the
problem by adding to the Hamiltonian an on-site energy term
$\sum_{i=1,\>\sigma}^{L}\epsilon_{i}\hat{n}_{i\sigma}$ mimicking a ionic
lattice.
As discussed in the introduction the fundamental quantity of LDFT is the site
occupation, $n_{i}$, which is calculated by solving the equivalent Kohn-Sham
problem. This can be generally written as
$\sum_{j=1}^{L}[-t(\delta_{i+1\>j}+\delta_{i-1\>j})+v_{\mathrm{KS}}^{i}]\phi_{j}^{(\alpha)}=\epsilon^{(\alpha)}\phi_{i}^{(\alpha)}\;,$
(2)
where $v_{\mathrm{KS}}^{i}$ is the general Kohn-Sham potential. The occupied
Kohn-Sham eigenvectors, $\phi_{i}^{(\alpha)}$, define $n_{i}$
$n_{i}=\sum_{\alpha}w^{(\alpha)}|\phi_{i}^{(\alpha)}|^{2}\>,$ (3)
where $w^{(\alpha)}$ are the occupation numbers, which satisfy
$\sum_{\alpha}w^{(\alpha)}=N$ with $N$ being the total number of electrons. By
following in the footsteps of standard ab initio DFT the Kohn-Sham potential
can be written as the sum of three terms
$v_{\mathrm{KS}}^{i}=[v_{\mathrm{H}}^{i}+v_{\mathrm{ext}}^{i}+v_{\mathrm{XC}}^{i}]\>,$
(4)
where $v_{\mathrm{H}}=Un_{i}/2$ is the Hartree potential and
$v_{\mathrm{ext}}^{i}$ is the external one. The last term in equation (4) is
the XC potential, which needs to be approximated.
The Kohn-Sham equations simply follow by variational principle from the
minimization of the energy functional. Thus the total energy of the system,
$E$, can be defined as
$E[\\{n_{i}\\}]=\sum_{\alpha}w^{(\alpha)}\epsilon^{(\alpha)}-\sum_{i}v_{\mathrm{XC}}^{i}n_{i}-\sum_{i}\frac{Un_{i}^{2}}{4}+E_{\mathrm{XC}}[\\{n_{i}\\}]\>,$
(5)
where the last term is the XC energy. Note that different values of $U$ and
$t$ define completely the theory, so that one has a different functional for
every value of $U/t$.
We now review the strategy used for constructing a suitable local
$v_{\mathrm{XC}}^{i}$ Capelle1 ; Capelle2 ; Capelle3 . The guiding idea is
that of defining $v_{\mathrm{XC}}^{i}$ as the local counterpart of the Bethe
Ansatz potential for the homogeneous Hubbard model (for an infinite number of
sites), i.e.
$v_{\mathrm{XC}}^{i}|_{\mathrm{BALDA}}=v_{\mathrm{XC}}^{\mathrm{hom}}(n,t,U)|_{n\rightarrow
n_{i}}\>.$ (6)
Here BALDA stands for Bethe Ansatz local density approximation and
$v_{\mathrm{XC}}^{\mathrm{hom}}(n,t,U)$ is the XC potential for the
homogeneous Hubbard model, which is defined only in terms of the band filling
$n=N/L$ (note that for the homogeneous case $n_{i}=n$ for every site $i$).
Formally, and in complete analogy with ab initio DFT,
$v_{\mathrm{XC}}^{\mathrm{hom}}(n,t,U)$ is obtained by functional derivative
of the exact energy density, $e(n,t,U)$, of the reference system (in this case
the homogeneous Hubbard model), after having subtracted the kinetic energy
density of the non-interacting case $e(n,t,U=0)$ and the Hartree energy
density, $e_{\mathrm{H}}(n,U)$. This gives us
$v_{\mathrm{XC}}^{\mathrm{hom}}(n,t,U)=\frac{\partial}{\partial
n}[e(n,t,U)-e(n,t,U=0)-e_{\mathrm{H}}(n,U)]\;.$ (7)
The question is now how to obtain $e(n,t,U)$. Two alternative constructions
have been proposed in the past and here we have adopted and numerically
implemented both. The first one consists in using the analytical
parameterization proposed by Lima et al. Capelle2 ; Capelle3 , which
interpolates the known exact results for: 1) $U\rightarrow 0$ and any $n\leq
1$, 2) $U\rightarrow\infty$ and any $n\leq 1$ and 3) $n=1$ and any $U$. The
resulting XC potential can then be written as
$v_{\mathrm{XC}}^{\mathrm{hom}}(n,t,U)=t\mu\left[2\cos\frac{k\pi}{\beta(U)}-2\cos\frac{k\pi}{2}+\frac{kU}{2}\right],$
(8)
where $k=1-|n-1|$, $\mu=$ sgn$(n-1)$ and $\beta(U)$ is a $U$-dependent
parameter, which can be determined by solving a transcendental equation. The
alternative route is that of employing a direct numerical solution of the
coupled Bethe Ansatz integral equations. This approach has been already used
for the study of ultracold repulsive fermions in 1D optical lattices Xianlong1
.
The first parameterization is known as BALDA/LSOC LSOC and the second as
BALDA/FN (FN = fully numerical). In figure 1 the XC potentials as a function
of the electron filling for the two schemes are shown for different values of
$U$. In the picture (and in the calculations) we always use the particle-hole
symmetry, which imposes
$v_{\mathrm{XC}}^{\mathrm{hom}}(n>1,t,U)=-v_{\mathrm{XC}}^{\mathrm{hom}}(2-n,t,U)$.
From the figure one can immediately observe that the potential in both cases
has a discontinuity in the derivative at half-filling ($n=1$, $N=L$). This
reflects the fact that the underlying homogeneous 1D Hubbard model has a
metal-insulation transition for $n=1$. Such a discontinuity in the derivative
of the potential, as in standard ab initio DFT, is responsible for the opening
of the energy gap. The second observation is that the two parameterizations
always coincide by construction at $n=0$ and $n=2$ but that their agreement
over the entire $n$ range depends on the value of $U$. In particular one can
report a progressively good agreement as $U$ increases. This is not a surprise
since the BALDA/LSOC potential is constructed to exactly reproduce the
$U\rightarrow\infty$ limit.
Figure 1: (Color online) Exchange-correlation potential
$v_{\mathrm{XC}}^{\mathrm{hom}}(n,t,U)$ of the 1D Hubbard model as a function
of the electron filling, $n$, for different values of interaction strength,
$U$. Here we report data for both BALDA/LSOC and BALDA/FN. Note that the
agreement between the two schemes improves as $U$ increases.
## III Polarizabilities
We calculate the electrical polarizability of linear chains with the finite
difference method, i.e. as numerical derivative of calculations performed at
different external electric fields. An external electric field enters into the
problem by adding to the Hubbard Hamiltonian $H_{\mathrm{U}}$ the term
$H_{\cal E}=e{\cal E}\hat{x}=e{\cal
E}\sum_{i=1}^{L}(i-\bar{x})c^{\dagger}_{i}c_{i}\>,$ (9)
where $\bar{x}=\frac{1}{2}(L+1)$ is the middle site position of the chain, $e$
is the electronic charge ($e=-1$) and ${\cal E}$ is the electric field
intensity (the electric field is applied along the chain). In general the
electrical dipole, $P$, induced by an external electric field can be
calculated simply as the expectation value of the dipole operator over the
ground state wave-function $|\Psi_{0}({\cal E})\rangle$ (note that this is a
general definition so that $|\Psi_{0}({\cal E})\rangle$ is not necessarily the
Kohn-Sham ground-state wave-function), i.e.
$P=e\langle\Phi_{0}({\cal
E})|\sum_{i=1}^{L}(i-\bar{x})c^{\dagger}_{i}c_{i}|\Phi_{0}({\cal
E})\rangle=\frac{\mathrm{d}E_{0}({\cal E})}{\mathrm{d}{\cal E}},$ (10)
where $E_{0}$ is the ground state energy. For small fields $P$ can be Taylor
expanded about ${\cal E}=0$ so that the linear polarizability, $\alpha$, is
defined as
$P\sim\alpha{\cal E}+\gamma{\cal E}^{3}+{\cal O}({\cal
E}^{5})\>,\;\;\;\;\;\;\;\;\alpha=\frac{\mathrm{d^{2}}E_{0}({\cal
E})}{\mathrm{d}{\cal E}^{2}}\>.$ (11)
Our calculation then simply proceeds with evaluating $E_{0}({\cal E})$ for
different values of ${\cal E}$ and then by fitting the first derivatives with
respect to the field to the equation (11), as indicated in reference Rojo . We
note that our finite difference scheme is not accurate enough for calculating
the hyper-polarizability, $\gamma$, which then is not investigated here.
It has been already extensively reported that BALDA-LDFT gives a substantial
good agreement with exact calculations in terms of ground state total energy
Capelle1 ; Capelle2 . The polarizability however offers a more stringent test
for the theory since it involves derivative of $E_{0}$. Hence it is important
to compare the various approximations with exact results.
Figure 2: (Color online) Linear polarizability, $\alpha$, as a function of the
Coulomb repulsion $U/t$. Results are presented for BALDA/LSOC and BALDA/FN and
they are compared with those obtained with either exact diagonalization (ED)
or DMRG calculations. In the various panels we show: (a) $L=12$ at quarter
filling ($n=1/2$), (b) $L=16$ at quarter filling, (c) $L=60$ and $N=20$, and
(d) $L=60$ at quarter filling.
For small chains, $L<18$, these are obtained by simply performing ED. However
for the longer chains ED is no longer feasible and we employ instead the DMRG
scheme Schollwock ; DMRG . DMRG has been widely used to investigate one-
dimensional and quasi one-dimensional quantum systems. It usually performs
best with open boundary conditions and utilizes appreciable computational
resources depending on the number of states that are kept for the calculation.
Our DMRG calculations are performed by employing the Algorithms and Libraries
for Physics Simulations (ALPS) ALPS package for strongly correlated quantum
mechanical systems. The DMRG results are obtained by using a cutoff of
$m=350$, i.e. by retaining the dominant 350 density matrix eigenvectors.
Let us start our analysis by looking at the polarizability as a function of
the energy scale $U/t$. Selected results for quarter-filling, $n=1/2$, and for
$n=1/3$ are presented in the various panels of figure 2. Note that throughout
this work we always stay away from the half-filling case ($n=1$), where the
derivative discontinuity of the potential makes the LDFT convergence
problematic.
In general we find that the polarizability decreases monotonically with
increasing the on-site repulsion $U$. This is indeed an expected result since
an increase in on-site repulsion means a suppression of charge fluctuations
and consequently a reduction of $\alpha$. Away from $U=0$ the dependence of
$\alpha$ on $U/t$ can be fitted with
$\alpha(U/t;L,n)=\alpha_{0}(L,n)\left(\frac{U}{t}\right)^{-\xi(L,n)}\>,$ (12)
where all the parameters have a dependance on the length of the chain and on
the band filling. The results of such a fitting procedure are reported in
table 1. Note that in the fit we did not impose any constraints and we have
included only points with $U/t\geq 1$.
Method | $L$ | $N$ | $n$ | $\alpha_{0}$ | $\xi$
---|---|---|---|---|---
ed | 12 | 6 | 1/2 | 59.69 | 0.23
balda/lsoc | | | | 62.06 | 0.27
balda/fn | | | | 59.50 | 0.25
ed | 16 | 8 | 1/2 | 142.88 | 0.27
balda/lsoc | | | | 135.07 | 0.31
balda/fn | | | | 143.56 | 0.30
dmrg | 60 | 30 | 1/2 | 8939.5 | 0.32
balda/lsoc | | | | 8673.1 | 0.33
balda/fn | | | | 8837.8 | 0.31
dmrg | 60 | 20 | 1/3 | 6931.6 | 0.29
balda/lsoc | | | | 6401.0 | 0.30
balda/fn | | | | 6920.7 | 0.29
Table 1: Scaling parameters for $\alpha(U/t;L)$ as obtained by fitting the
data of Fig. 2 to the expression of equation (12). Note that the fit has been
obtained without any constraints and by including data only for $U/t\geq 1$.
From the fit and from figure 2 one can immediately note that both the BALDA
flavors of the exchange and correlation functional reproduce rather well the
exact results, in good agreement with previously published calculations Schenk
. The agreement is particularly good for the FN functional, which matches the
ED/DMRG results almost perfectly over the entire range of $U/t$’s and filling
investigated. A quantitative assessment of goodness of the BALDA results is
provided in figure 3 where the relative error, $\delta$, from the reference
exact calculations is presented. In general, and as expected, we find that the
error grows with $U/t$, i.e. with the system departing from the non-
interacting case. However, there is also a saturation of the error as the
interaction strength increases, reflecting the fact that both the BALDA
potential are exact in the limit of $U\rightarrow\infty$. As a further
consequence of the $U\rightarrow\infty$ limit, we also observe that the
relative error between BALDA/LSOC and BALDA/FN reduces as $U$ grows.
Figure 3: (Color online) Relative error between BALDA calculated
polarizabilities and those obtained with exact methods (either ED or DMRG). In
the panels we show: (a) $L=12$ at quarter filling ($n=1/2$), (b) $L=16$ at
quarter filling, (c) $L=60$ and $N=20$, and (d) $L=60$ at quarter filling.
Given the accuracy of the BALDA/FN scheme we have decided to use the same to
investigate in more details the scaling properties of $\alpha(U/t;L)$. First
we look at the scaling as a function of the interaction strength $U/t$. In
this case we always consider a chain containing $L=60$ sites for which the
deviation from the DMRG results is never larger than 2%. Furthermore this is a
length which allows us to explore a rather large range of electron filling, so
that it allows us to gain a complete understanding of the scaling properties.
Figure 4: (Color online) Polarizability as a function of $U/t$ for a chain of
60 sites and various filling factors, $n$. The figure legend reports the
fitted values for the exponent $\xi$ [see equation (12)]. The symbols
represents the calculated data while the solid lines are just to guide the
eyes. In the inset we present the exponent $\xi$ as a function of the filling
factor $n$.
Our results are presented in figure 4 where we show $\alpha$ as a function of
$U/t$ for different filling factors, we list the values of $\xi$ obtained by
fitting the actual data for $U/t\geq 1$ to the expression in equation (12) and
we provide (inset) the dependence of $\xi$ on $n$.
In general the fit to our data is excellent, suggesting the validity of the
exponential scaling of the polarizability with the interaction strength (away
from half filling). In particular we find that $\xi$ decreases monotonically
with $n$ for $n>0.2$ but it increases for smaller values. This means that
$\xi(n)$ has a maximum just before $n=0.2$, which appears rather sharp (see
inset of figure 4). We are at present uncertain about the precise origin of
such a non-monotonic behavior. However, as we will see in details later on, we
notice that the response of the exchange and correlation potential to the
external electric field has an anomaly for small $U$ and $n$. We believe that
such an anomaly might be the cause of the non-monotonic behaviour of $\xi$.
Next we turn our attention to the scaling of $\alpha$ with the chain length.
In figure 5 we present $\alpha(L)$ for two different filling factors ($n=1/3$
and $1/2$) and different values of $U/t$.
Figure 5: (Color online) Scaling of the polarizability as a function of the
chain length, $L$. Panel (a) and (b) are for $n=1/3$ while (c) and (d) for
$n=1/2$. Note the linear dependence of the $\alpha(L)$ curve when plotted on a
log-log scale, proving the relation $\alpha(L)=\alpha_{1}L^{\gamma}$
Data are plotted both in linear and logarithmic scale, from which a clear
power-law dependance of $\alpha$ on $L$ emerges. A fit to our data provides
the following scaling
$\alpha(U/t;L)=\alpha_{1}L^{\gamma}\>.$ (13)
Importantly this time we find essentially no dependance of both $\alpha_{1}$
and $\gamma$ on either $U/t$ or $n$. The fit reveals a value for the exponent
of $\gamma\sim 3$ (the range is from $\gamma=2.93$ to $\gamma=2.98$). This is
what expected for free electrons in 1D Rojo , and it is substantially
different from the predicted linear scaling at $n=1$. Our results thus
confirms that away from $n=1$ the electrostatic response of the Hubbard model
is similar to that of the non-interacting electron gas. Going in more details
we find a rather small monotonic dependance of $\gamma$ on $U/t$. This however
depends also on $n$ since for $n=1/3$ we find that $\gamma$ reduces as $U/t$
is increased (from 2.98 for $U/t=0.5$ to 2.93 for $U/t=100$), while the
opposite behavior is found for $n=1/2$ ($\gamma=2.94$ for $U/t=0.5$ and 2.96
for $U/t=100$).
## IV Response of the BALDA potential to the external field
In ab initio DFT the failures of local and semi-local XC functionals in
reproducing accurate linear polarizabilities are related to the incorrect
response of the XC potential to the external electric field Gisbergen ; Perdew
, which in turn originates from the presence of the self-interaction error
Kummel ; Das . In particular for ab initio DFT the exact XC potential should
be opposite to the external one, while the LDA/GGA (generalized gradient
approximation, GGA) returns a potential which responds in the same direction.
In order to investigate the same feature for the case of the Hubbard model
LDFT we calculate the potential response
$\Delta v_{\mathrm{XC}}=v_{\mathrm{XC}}^{\cal E}(n_{i})-v_{\mathrm{XC}}^{{\cal
E}=0}(n_{i})\>,$ (14)
where $v_{\mathrm{XC}}^{\cal E}(n_{i})$ is the exchange and correlation
potential at site $i$ in the presence of an electric field ${\cal E}$. Also in
this case we adopt the finite difference method and we use ${\cal E}=0.01$,
after having checked that the trends remaining unchanged irrespectively of the
field strength.
In order to provide a benchmark for our calculations we also need to evaluate
the potential response for the exact Hubbard model. We construct the exact
potential by reverse engineering, a strategy introduced first by Almbladh and
Pedroza Almbladh and by von Barth vbar and then applied to both static and
time dependent LDFT by Verdozzi Verdozzi . This consists in minimizing about
the Kohn-Sham potential the functional ${\cal F}$ (in reality here this is
just a function) defined as
${\cal
F}[v_{\mathrm{XC}}]=\sum_{i}^{L}(n_{i}^{\mathrm{KS}}-n_{i}^{\mathrm{exact}})^{2},$
(15)
where $n_{i}^{\mathrm{exact}}$ is the exact site occupation at site $i$ as
obtained by either ED or the DMRG method, while $n_{i}^{\mathrm{KS}}$ is the
Kohn-Sham one.
Our results are summarized in figures 6 and 7, where we show $\Delta
v_{\mathrm{XC}}$ as a function of the site index for a 60 site chain occupied
respectively with 10 ($n=1/6$) and 30 ($n=1/2$) electrons. The external
electrostatic potential here decreases as the site number increases, i.e. it
has a negative slope. Results are presented for DMRG, BALDA/LSOC and BALDA/FN
and for different values of $U/t$.
Figure 6: (Color online) The difference, $\Delta v_{\mathrm{XC}}$, between the
XC potential calculated at finite electric field and in absence of the field
as a function of the site index. Results are presented for a 60 site chain
with $N=10$ ($n=1/6$). The dots are the calculated data while the lines are a
guide to the eye. The external potential has a negative slope. Figure 7:
(Color online) The difference, $\Delta v_{\mathrm{XC}}$, between the XC
potential calculated at finite electric field and in absence of the field as a
function of the site index. Results are presented for a 60 site chain with
$N=30$ ($n=1/2$). The dots are the calculated data while the lines are a guide
to the eye. The external potential has a negative slope.
In general and in contrast with ab initio DFT, we find that the response of
the exact Hubbard-LDFT XC potential is in the same direction of the external
perturbation for both the filling factors investigated and regardless of the
magnitude of $U/t$. The response however becomes larger as $U/t$ is increased
(the slope of $\Delta v_{\mathrm{XC}}$ is more pronounced), a direct
consequence of the fact that for large $U$’s small deviations from an
homogeneous charge distribution produce large fluctuations in the potential.
Such a behaviour is well reproduced by both the BALDA functionals, with the
BALDA/FN scheme performing marginally better than the BALDA/LSOC one, and
reflecting the same trend already observed for the polarizabilities.
There is however one anomaly in the potential response for the BALDA/LSOC
functional, namely at $n=1/2$ and for small $U/t$ (respectively 2 and 4) the
potential response is actually opposite (positive slope) to that of the DMRG
benchmark. This means that in these particular range of filling and
interaction strength the BALDA/LSOC potential erroneously opposes to the
external perturbation. The anomaly originates from the particular shape of the
BALDA/LSOC potential as a function of $n$ for small $U/t$ (see figure 1). In
fact, $v_{\mathrm{XC}}^{i}$ for BALDA/LSOC has a minimum for both $U/t=2$ and
$U/t=4$ at around $n=1/4$, which means that its slope changes sign when the
occupation sweeps across $n=1/4$. Therefore for those critical interaction
strengths the response is expected to be along the same direction of the
external potential for $n<1/4$ and for $3/4\lesssim n\leq 1$ and opposite to
it for $1/4<n\lesssim 3/4$ (at $n\sim 3/4$ there is a second change in slope).
In the case of the BALDA/FN functional such an anomaly is in general not
expected, except for small $U/t$ and $n$ close to the discontinuity at $n=1$
(see figure 1). This, however, is in the range of occupation not investigated
here. Nevertheless we note that for $n=1/2$ and $U/t=2$ the BALDA/FN
$v_{\mathrm{XC}}$ is almost flat. This feature is promptly mirrored in the
potential response of figure 7, which also shows an almost flat $\Delta
v_{\mathrm{XC}}$, although still with the correct negative slope.
Given the good agreement for both the polarizability and the potential
response between the exact results and those obtained with the BALDA (in
particular with the FN flavour), one can conclude that the local approximation
to the Hubbard-LDFT functional is adequate. Still it is interesting to assess
whether the remaining discrepancies have to do with the particular local
parameterization of $E_{\mathrm{XC}}[\\{n_{i}\\}]$, or with the fact that the
exact XC functional may be intrinsically non-local. In order to answer to this
question we have set a numerical test. We consider a 60 site chain with
$n=1/2$ (this should be long enough to resemble the infinite limit) and we
introduce a local perturbation in half of the chain. This is in the form of a
reduction of the on-site energy of the first 30 sites by $\delta$. We then
calculate the deviation of the XC potential $\delta v$ as a function of the
deviation of the total energy $\delta E_{0}$. These two quantities are defined
respectively as
$\delta
v=\sum_{i}|v_{\mathrm{XC}}^{\delta}(n_{i})-v_{\mathrm{XC}}^{\delta=0}(n_{i})|\>,\;\;\;\;\;\;\;\delta
E_{0}=E_{0}(\delta)-E_{0}(0)\>,$ (16)
with $v_{\mathrm{XC}}^{\delta}$ and $E_{0}(\delta)$ respectively the XC
potential at site $i$ and the total energy calculated for $\delta\neq 0$. One
then expects for a local potential that $\delta v\rightarrow 0$ as $\delta
E_{0}\rightarrow 0$.
Our results are presented in figure 8. These have been obtained for a
relatively small $U/t=2$ by varying $\delta$ in the range $0\leq\delta\leq
0.1$ in steps of 10-5 (this range is used only for small $\delta$, while a
coarse mesh is employed for large $\delta$). Interestingly we note that, after
a steady decrease of $\delta v$ with reducing $\delta E_{0}$, the deviation of
the potential starts to fluctuate independently on the size of $\delta E_{0}$.
We have carefully checked that such fluctuations are well within our numerical
accuracy, so that they should be attributed to the breakdown of the local
approximation. We then conclude that part of the failure of BALDA/FN in
describing the polarizability of finite 1D chains must be ascribed to the
violation of the local approximation.
Figure 8: Variation of the XC potential, $\delta v$, as a function of the
variation of the total energy, $\delta E_{0}$, for a 60 site chain in which
the first 30 sites have an on-site energy lower by $\delta$ with respect to
the remaining 30. The variation are calculated with respect to the homogeneous
case. The inset shows a magnification of the data for small $\delta E_{0}$.
## V Conclusions
In conclusion, we have reported a systematic study of the electrical response
properties of one-dimensional metals described by the Hubbard model. This is
solved within LDFT and local approximations of the exchange and correlation
functional. Whenever possible the calculations are compared with exact results
obtained either by exact diagonalization of with the density matrix
renormalization group approach. In general we find that BALDA functionals
perform rather well in describing the electrical polarizability of finite one-
dimensional chains. The agreement with exact results is particularly good in
the case of numerically evaluated functionals. A similar good agreement is
found for the exchange and correlation potential response. In this case we
obtain the interesting result that the potential response is always along the
same direction of the perturbing potential, in contrast to what happens in ab
initio DFT. Furthermore for small electron filling and weak Coulombic
interaction the commonly used LSOC parameterization is qualitatively incorrect
due to a spurious minimum in the potential as a function of the site
occupation. Finally we provide a numerical test of the breakdown of the local
approximation being the source of the remaining errors.
## VI Acknowledgements
A.A. thanks N. Baadji, I. Rungger and V. L. Campo for useful discussions. This
work is supported by Science Foundation of Ireland under the grant
SFI05/RFP/PHY0062 and 07/IN.1/I945. Computational resources have been provided
by the HEA IITAC project managed by the Trinity Center for High Performance
Computing and by ICHEC.
## References
* (1) P. Fazekas, Lecture Notes on Electron Correlations and Magnetism, Series in Modern Condensed Matter Physics 5, (World Scientific Publishing, Singapore, 1999)
* (2) T. Giamarchi, Quantum Physics in One Dimension, (Oxford University Press, Oxford, 2003).
* (3) H.W.Ch. Postma, T. Teepen, Z. Yao, M. Grifoni and C. Dekker, Science 293, 76 (2001).
* (4) P. Segovia, D. Purdie, M. Hengsberger and Y. Baer, Nature 402, 504 (1999).
* (5) O.M. Auslaender, A. Yacoby, R. de Picciotto, K.W. Baldwin, L.N. Pfeiffer and K.W. West, Phys. Rev. Lett. 84, 1764 (2000).
* (6) M.C. Gutzwiller, Phys. Rev. Lett. 10, 159 (1962).
* (7) J. Hubbard, Proc. Roy. Soc. A 276, 238 (1963).
* (8) J. Kanamory, Prog. Theor. Phys. 30, 275 (1963).
* (9) F.H.L. Essler, H. Frahm, F. Göhmann, A. Klümper and V.E. Korepin, The One-Dimensional Hubbard Model, (Cambridge University Press, Cambridge, 2005).
* (10) S.R. White, Phys. Rev. Lett. 69, 2863 (1992).
* (11) U. Schollwock, Rev. Mod. Phys. 77, 259 (2005).
* (12) O. Gunnarsson and K. Schonhammer, Phys. Rev. Lett. 56, 1968 (1986).
* (13) K. Schonhammer, O. Gunnarsson and R.M. Noack, Phys. Rev. B. 52, 2504 (1995).
* (14) P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964).
* (15) W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965).
* (16) R. Lopez-Sandoval and G.M. Pastor, Phys. Rev. B. 67, 035115 (2003).
* (17) N.A. Lima, L.N. Oliviera and K. Capelle, Euro. Phys. Lett. 60, 601 (2002).
* (18) N.A. Lima, M.F. Silva, L.N. Oliviera and K. Capelle, Phys. Rev. Lett. 90, 146402 (2003).
* (19) K. Capelle, N.A. Lima, M.F. Silva and L.N. Oliviera, in The Fundamentals of Electron Density, Density Matrix and Density Functional theory in Atoms, Molecules and Solids, Kluwer series, “Progress in Theoretical Chemistry and Physics,” edited by N.I. Gidopoulos and S. Wilson (Kluwer, Dordrecht, 2003).
* (20) M.F. Silva, N.A. Lima, A.L. Malvezzi and K. Capelle, Phys. Rev. B. 71, 125130 (2005).
* (21) V.L. Campo Jr. and K. Capelle, Phys. Rev. A. 72, 061602(R) (2005).
* (22) G. Xianlong, M. Polini, M.P. Tosi, V.L. Campo, K. Capelle and M. Rigol, Phys. Rev. B. 73, 165120 (2006).
* (23) G. Xianlong, M. Rizzi, M. Polini, R. Fazio, M.P. Tosi, V.L. Campo and K. Capelle, Phys. Rev. Lett. 98, 030404 (2007).
* (24) G. Xianlong, Phys. Rev. B. 78, 085108 (2008).
* (25) C. Verdozzi, Phys. Rev. Lett. 101, 166401 (2008).
* (26) S. Kurth, G. Stefanucci, E. Khosravi, C. Verdozzi and E.K.U. Gross, Phys. Rev. Lett. 104, 236801 (2010).
* (27) S. Schenk, M. Dzierzawa, P. Schwab and U. Eckern, Phys. Rev. B. 78, 165102 (2008).
* (28) E. H. Lieb and F. Y. Wu, Phys. Rev. Lett. 20, 1445 (1968).
* (29) S. Ishihara, M. Tachiki and T. Egami, Phys. Rev. B 49, 16123 (1994).
* (30) S. Ishihara, M. Tachiki and T. Egami, Phys. Rev. B 53, 15563 (1996).
* (31) A.G. Rojo and G.D. Mahan, Phys. Rev. B 47, 1794 (1993).
* (32) LSOC is after the name of Lima, Silva, Oliviera and Capelle, who proposed the approximation.
* (33) Density-Matrix Renormalization, A New Numerical Method in Physics, edited by I. Peschel, X. Wang, M. Kaulke and K. Hallberg (Springer, Berlin, 1999).
* (34) F. Albuquerque et al., J. Magn. Magn. Mater. 310, 1187 (2007).
* (35) S. J. A. van Gisbergen, P. R. T. Schipper, O. V. Gritsenko, E. J. Baerends, J. G. Snijders, B. Champagne, and B. Kirtman, Phys. Rev. Lett. 83, 694 (1997).
* (36) S. Km̈mel, L. Kronik and J.P. Perdew, Phys. Rev. Lett. 93, 213002 (2004).
* (37) T. Körzdörfer, M. Mundt and S. Kümmel, Phys. Rev. Lett. 100, 133004 (2008).
* (38) C. D. Pemmaraju, S. Sanvito and K. Burke, Phys. Rev. B 77, 121204(R) (2008).
* (39) C. O. Almbladh and A. C. Pedroza, Phys. Rev. A. 29, 2322 (1984).
* (40) U. von Barth, in Many Body Phenomena at Surfaces, D. Langreth and H. Suhl eds., Academic Press (1984)
|
arxiv-papers
| 2010-10-14T09:13:36 |
2024-09-04T02:49:13.882720
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "A. Akande and S. Sanvito",
"submitter": "Akinlolu Akande Mr.",
"url": "https://arxiv.org/abs/1010.2860"
}
|
1010.2942
|
11institutetext: CERN, Geneva, Switzerland
# Trigger and data acquisition
N. Ellis
###### Abstract
The lectures address some of the issues of triggering and data acquisition in
large high-energy physics experiments. Emphasis is placed on hadron-collider
experiments that present a particularly challenging environment for event
selection and data collection. However, the lectures also explain how T/DAQ
systems have evolved over the years to meet new challenges. Some examples are
given from early experience with LHC T/DAQ systems during the 2008 single-beam
operations.
## 0.1 Introduction
These lectures concentrate on experiments at high-energy particle colliders,
especially the general-purpose experiments at the Large Hadron Collider (LHC)
[1]. These experiments represent a very challenging case that illustrates well
the problems that have to be addressed in state-of-the-art high-energy physics
(HEP) trigger and data-acquisition (T/DAQ) systems. This is also the area in
which the author is working (on the trigger for the ATLAS experiment at LHC)
and so is the example that he knows best. However, the lectures start with a
more general discussion, building up to some examples from LEP [2] that had
complementary challenges to those of the LHC. The LEP examples are a good
reference point to see how HEP T/DAQ systems have evolved in the last few
years.
Students at this school come from various backgrounds — phenomenology,
experimental data analysis in running experiments, and preparing for future
experiments (including working on T/DAQ systems in some cases). These lectures
try to strike a balance between making the presentation accessible to all, and
going into some details for those already familiar with T/DAQ systems.
### 0.1.1 Definition and scope of trigger and data acquisition
T/DAQ is the online system that selects particle interactions of potential
interest for physics analysis (trigger), and that takes care of collecting the
corresponding data from the detectors, putting them into a suitable format and
recording them on permanent storage (DAQ). Special modes of operation need to
be considered, the need to calibrate different detectors in parallel outside
of normal data-taking periods. T/DAQ is often taken to include associated
tasks, run control, monitoring, clock distribution and book-keeping, all of
which are essential for efficient collection and subsequent offline analysis
of the data.
### 0.1.2 Basic trigger requirements
As introduced above, the trigger is responsible for selecting interactions
that are of potential interest for physics analysis. These interactions should
be selected with high efficiency, the efficiency should be precisely known
(since it enters in the calculation of cross-sections), and there should not
be biases that affect the physics results. At the same time, a large reduction
of rate from unwanted high-rate processes may be needed to match the
capabilities of the DAQ system and the offline computing system. High-rate
processes that need to be rejected may be instrumental backgrounds or high-
rate physics processes that are not relevant for the analyses that one wants
to make. The trigger system must also be affordable, which implies limited
computing power. As a consequence, algorithms that need to be executed at high
rate must be fast. Note that it is not always easy to achieve the above
requirements (high efficiency for signal, strong background rejection and fast
algorithms) simultaneously.
Trigger systems typically select events111The term ‘event’ will be discussed
in Section 0.3 — for now, it may be taken to mean the record of an
interaction. according to a ‘trigger menu’, a list of selection criteria — an
event is selected if one or more of the criteria are met. Different criteria
may correspond to different signatures for the same physics process —
redundant selections lead to high selection efficiency and allow the
efficiency of the trigger to be measured from the data. Different criteria may
also reflect the wish to concurrently select events for a wide range of
physics studies — HEP ‘experiments’ (especially those with large general-
purpose ‘detectors’ or, more precisely, detector systems) are really
experimental facilities. Note that the menu has to cover the physics channels
to be studied, plus additional data samples required to complete the analysis
(measure backgrounds, and check the detector calibration and alignment).
### 0.1.3 Basic data-acquisition requirements
The DAQ system is responsible for the collection of data from detector
digitization systems, storing the data pending the trigger decision, and
recording data from the selected events in a suitable format. In doing so it
must avoid corruption or loss of data, and it must introduce as little dead-
time as possible (‘dead-time’ refers to periods when interesting interactions
cannot be selected — see below). The DAQ system must, of course, also be
affordable which, for example, places limitations on the amount of data that
can be read out from the detectors.
## 0.2 Design of a trigger and data-acquisition system
In the following a very simple example is used to illustrate some of the main
issues for designing a T/DAQ system. An attempt is made to omit all the detail
and concentrate only on the essentials — examples from real experiments will
be discussed later.
Before proceeding to the issue of T/DAQ system design, the concept of dead-
time, which will be an important element in what follows, is introduced.
‘Dead-time’ is generally defined as the fraction or percentage of total time
where valid interactions could not be recorded for various reasons. For
example, there is typically a minimum period between triggers — after each
trigger the experiment is dead for a short time.
Dead-time can arise from a number of sources, with a typical total of up to
$\mathcal{O}(10\%)$. Sources include readout and trigger dead-time, which are
addressed in detail below, operational dead-time ( time to start/stop data-
taking runs), T/DAQ downtime (following a computer failure), and detector
downtime (following a high-voltage trip). Given the huge investment in the
accelerators and the detectors for a modern HEP experiment, it is clearly very
important to keep dead-time to a minimum.
In the following, the design issues for a T/DAQ system are illustrated using a
very simple example. Consider an experiment that makes a time-of-flight
measurement using a scintillation-counter telescope, read out with time-to-
digital converters (TDCs), as shown in Fig. 1. Each plane of the telescope is
viewed by a photomultiplier tube (PMT) and the resulting electronic signal is
passed to a ‘discriminator’ circuit that gives a digital pulse with a sharp
leading edge when a charged particle passes through the detector. The leading
edge of the pulse appears a fixed time after the particle traverses the
counter. (The PMTs and discriminators are not shown in the figure.)
Two of the telescope planes are mounted close together, while the third is
located a considerable distance downstream giving a measurable flight time
that can be used to determine the particle’s velocity. The trigger is formed
by requiring a coincidence (logical AND) of the signals from the first two
planes, avoiding triggers due to random noise in the photomultipliers — it is
very unlikely for there to be noise pulses simultaneously from both PMTs. The
time of arrival of the particle at the three telescope planes is measured,
relative to the trigger signal, using three channels of a TDC. The pulses
going to the TDC from each of the three planes have to be delayed so that the
trigger signal, used to start the TDC (analogous to starting a stop-watch),
gets there first.
The trigger signal is also sent to the DAQ computer, telling it to initiate
the readout. Not shown in Fig.1 is logic that prevents further triggers until
the data from the TDC have been read out into the computer — the so-called
dead-time logic.
### 0.2.1 Traditional approach to trigger and data acquisition
The following discussion starts by presenting a ‘traditional’ approach to
T/DAQ (as might be implemented using, for example, NIM and CAMAC electronics
modules222NIM [3] and CAMAC [4] modules are electronic modules that conform to
agreed standards — modules for many functions needed in a T/DAQ system are
available commercially., plus a DAQ computer). Note that this approach is
still widely used in small test set-ups. The limitations of this model are
described and ways of improving on it are presented. Of course, a big HEP
experiment has an enormous number of sensor channels [up to
$\mathcal{O}(10^{8})$ at LHC], compared to just three in the example. However,
the principles are the same, as will be shown later.
Limitations of the T/DAQ system shown in 1 are as follows:
1. 1.
The trigger decision has to be made very quickly because the TDCs require a
‘start’ signal that arrives before the signals that are to be digitized (a TDC
module is essentially a multichannel digital stop-watch). The situation is
similar with traditional analog-to-digital converters (ADCs) that digitize the
magnitude of a signal arriving during a ‘gate’ period, the electric charge in
an analog pulse — the gate has to start before the pulse arrives.
2. 2.
The readout of the TDCs by the computer may be quite slow, which implies a
significant dead-time if the trigger rate is high. This limitation becomes
much more important in larger systems, where many channels have to be read out
for each event. For example, if 1000 channels have to be read out with a
readout time of 1 per channel (as in CAMAC), the readout time per event is 1
ms which excludes event rates above 1.
Figure 1: Example of a simple experiment with its T/DAQ system
The ‘readout model’ of this traditional approach to T/DAQ is illustrated in
fig:f2, which shows the sequence of actions — arrival of the trigger, arrival
of the detector signals (followed by digitization and storage in a data
register in the TDC), and readout into the DAQ computer. Since no new trigger
can be accepted until the readout is complete, the readout dead-time is given
by the product of the trigger rate and the readout time.
Figure 2: Readout model in the ‘traditional’ approach
### 0.2.2 Local buffer
The traditional approach described above can be improved by adding a local
‘buffer’ memory into which the data are moved rapidly following a trigger, as
illustrated in 3. This fast readout reduces the dead- time, which is now given
by the product of the trigger rate and the local readout time. This approach
is particularly useful in large systems, where the transfer of data can
proceed in parallel with many local buffers (one local buffer for each crate
of electronics) — local readout can remain fast even in a large system. Also,
the data may be moved more quickly into the local buffer within the crate than
into the DAQ computer. Note that the dashed line in the bottom, right-hand
part of Fig. 1 indicates this extension to the traditional approach — the
trigger signal is used to initiate the local readout within the crate.
Figure 3: Readout system with local buffer memory
The addition of a local buffer reduces the effective readout time, but the
requirement of a fast trigger still remains. Signals have to be delayed until
the trigger decision is available at the digitizers. This is not easy to
achieve, even with very simple trigger logic — typically one relies on using
fast (air-core) cables for trigger signals with the shortest possible routing
so that the trigger signals arrive before the rest of the signals (which
follow a longer routing on slower cables). It is not possible to apply complex
selection criteria on this time-scale.
### 0.2.3 Multi-level triggers
It is not always possible to simultaneously meet the physics requirements
(high efficiency, high background rejection) and achieve an extremely short
trigger ‘latency’ (time to form the trigger decision and distribute it to the
digitizers). A solution is to introduce the concept of multi-level triggers,
where the first level has a short latency and maintains high efficiency, but
only has a modest rejection power. Further background rejection comes from the
higher trigger levels that can be slower. Sometimes the very fast first stage
of the trigger is called the ‘pre-trigger’ — it may be sufficient to signal
the presence of minimal activity in the detectors at this stage.
The use of a pre-trigger is illustrated in fig:f4. Here the pre-trigger is
used to provide the start signal to the TDCs (and to gate ADCs, ), while the
main trigger (which can come later and can therefore be based on more complex
calculations) is used to initiate the readout. In cases where the pre-trigger
is not confirmed by the main trigger, a ‘fast clear’ is used to re-activate
the digitizers (TDCs, ADCs, ).
Figure 4: Readout system with pre-trigger and fast clear
Using a pre-trigger (but without using a local buffer for now), the dead-time
has two components. Following each pre-trigger there is a dead period until
the trigger or fast clear is issued (defined here as the trigger latency). For
the subset of pre-triggers that give rise to a trigger, there is an additional
dead period given by the readout time. Hence, the total dead-time is given by
the product of the pre-trigger rate and the trigger latency, added to the
product of the trigger rate and the readout time.
The two ingredients — use of a local buffer and use of a pre-trigger with fast
clear — can be combined as shown in 5, further reducing the dead-time. Here
the total dead-time is given by the product of the pre-trigger rate and the
trigger latency, added to the product of the trigger rate and the local
readout time.
### 0.2.4 Further improvements
The idea of multi-level triggers can be extended beyond having two levels
(pre-trigger and main trigger). One can have a series of trigger levels that
progressively reduce the rate. The efficiency for the desired physics must be
kept high at all levels since rejected events are lost forever. The initial
levels can have modest rejection power, but they must be fast since they see a
high input rate. The final levels must have strong rejection power, but they
can be slower because they see a much lower rate (thanks to the rejection from
the earlier levels).
In a multi-level trigger system, the total dead-time can be written as the sum
of two parts: the trigger dead-time summed over trigger levels, and the
readout dead-time. For a system with _N_ levels, this can be written
$(\sum^{N}_{i=2}R_{i-1}\times L_{i})+R_{N}\times T_{\mathrm{LRO}}$
where $R_{i}$ is the rate after the $i^{\mathrm{th}}$ trigger level, $L_{i}$
is the latency of the $i^{\mathrm{th}}$ trigger level, and $T_{\mathrm{LRO}}$
is the local readout time. Note that $R_{1}$ corresponds to the pre-trigger
rate.
In the above, two implicit assumptions have been made: (1) that all trigger
levels are completed before the readout starts, and (2) that the pre-trigger
(the lowest-level trigger) is available by the time the first signals from the
detector arrive at the digitizers. The first assumption results in a long dead
period for some events — those that survive the first (fast) levels of
selection. The dead-time can be reduced by moving the data into intermediate
storage after the initial stages of trigger selection, after which further
low-level triggers can be accepted (in parallel with the execution of the
later stages of trigger selection on the first event). The second assumption
can also be avoided, in collider experiments with bunched beams as discussed
below.
In the next section, aspects of particle colliders that affect the design of
T/DAQ systems are introduced. Afterwards, the discussion returns to readout
models and dead-time, considering the example of LEP experiments.
Figure 5: Readout system using both pre-trigger and local buffer
## 0.3 Collider experiments
In high-energy particle colliders (HERA, LEP, LHC, Tevatron), the particles in
the counter-rotating beams are bunched. Bunches of particles cross at regular
intervals and interactions occur only during the bunch crossings. Here the
trigger has the job of selecting the _bunch crossings_ of interest for physics
analysis, those containing interactions of interest.
In the following notes, the term ‘event’ is used to refer to the record of all
the products from a given bunch crossing (plus any activity from other bunch
crossings that gets recorded along with this). Be aware (and beware!) — the
term ‘event’ is not uniquely defined! Some people use the term ‘event’ for the
products of a single interaction between incident particles. Note that many
people use ‘event’ interchangeably to mean different things.
In colliders, the interaction rate is very small compared to the bunch-
crossing rate (because of the low cross-section). Generally, selected events
contain just one interaction — the event is generally a single interaction.
This was the case at LEP and also at the – collider HERA. In contrast, at LHC
with the design luminosity $\mathscr{L}\,\text{\leavevmode\nobreak\
of\leavevmode\nobreak\ }10^{34}\,\text{cm}^{-2}\,\text{s}^{-1}$ for proton
beams, each bunch crossing will contain on average about 25 interactions as
discussed below. This means that an interaction of interest, one that produced
$\PH\rightarrow\PZ\PZ\rightarrow\Pep\Pem\Pep\Pem$, will be recorded together
with 25 other proton–proton interactions that occurred in the same bunch
crossing. The interactions that make up the ‘underlying event’ are often
called ‘minimum-bias’ interactions because they are the ones that would be
selected by a trigger that selects interactions in an unbiased way. The
presence of additional interactions that are recorded together with the one of
interest is sometimes referred to as ‘pile-up’.
A further complication is that particle detectors do not have an infinitely
fast response time — this is analogous to the exposure time of a camera. If
the ‘exposure time’ is shorter than the bunch-crossing period, the event will
contain only information from the selected bunch crossing. Otherwise, the
event will contain, in addition, any activity from neighbouring bunches. In
colliders (LEP) it is very unlikely for there to be any activity in nearby
bunch crossings, which allows the use of slow detectors such as the time
projection chamber (TPC). This is also true at HERA and in the ALICE
experiment [5] at LHC that will study heavy-ion collisions at much lower
luminosities than in the proton–proton case.
The bunch-crossing period for proton–proton collisions at LHC will be only 25
(corresponding to a 40 rate). At the design luminosity the interaction rate
will be $\mathcal{O}(10^{9})$ and, even with the short bunch-crossing period,
there will be an average of about 25 interactions per bunch crossing. Some
detectors, for example the ATLAS silicon tracker, achieve an exposure time of
less than 25, but many do not. For example, pulses from the ATLAS liquid-argon
calorimeter extend over many bunch crossings.
The instrumentation for the LHC experiments is described in the lecture notes
of Jordan Nash from this School [6]. The Particle Data Group’s Review of
Particle Physics [7] includes much useful information, including summaries of
the parameters of various particle colliders.
## 0.4 Design of a trigger and data-acquisition system for LEP
Let us now return to the discussion of designing a T/DAQ system, considering
the case of experiments at LEP (ALEPH [8], DELPHI [9], L3 [10], and OPAL
[11]), and building on the model developed in Section 0.2.
Figure 6: Readout system using bunch-crossing (BC) clock and fast clear
### 0.4.1 Using the bunch-crossing signal as a ‘pre-trigger’
If the time between bunch crossings (BCs) is reasonably long, one can use the
clock that signals when bunches of particles cross as the pre-trigger. The
first-level trigger can then use the time between bunch crossings to make a
decision, as shown in 6. For most crossings the trigger will reject the event
by issuing a fast clear — in such cases no dead-time is introduced. Following
an ‘accept’ signal, dead-time will be introduced until the data have been read
out (or until the event has been rejected by a higher-level trigger). This is
the basis of the model that was used at LEP, where the bunch-crossing interval
of 22 µs (11 µs in eight-bunch mode) allowed comparatively complicated trigger
processing (latency of a few microseconds). Note that there is no first-level
trigger dead-time because the decision is made during the interval between
bunch crossings where no interactions occur. As discussed below, the trigger
rates were reasonably low (very much less than the BC rate), giving acceptable
dead-time due to the second-level trigger latency and the readout.
In the following, the readout model used at LEP is illustrated by
concentrating on the example of ALEPH [8]333The author was not involved in any
of the LEP experiments. In these lectures the example of ALEPH is used to
illustrate how triggers and data acquisition were implemented at LEP; some
numbers from DELPHI are also presented. The T/DAQ systems in all of the LEP
experiments were conceptually similar.. [b] 7 shows the readout model, using
the same kind of block diagram as presented in Section 2\. The BC clock is
used to start the TDCs and generate the gate for the ADCs, and a first-level
(LVL1) trigger decision arrives in less than 5 µs so that the fast clear can
be completed prior to the next bunch crossing. For events retained by LVL1, a
more sophisticated second-level (LVL2) trigger decision is made after a total
of about 50 µs. Events retained by LVL2 are read out to local buffer memory
(within the readout controllers or ‘ROCs’), and then passed to a global
buffer. There is a final level of selection (LVL3) before recording the data
on permanent storage for offline analysis.
Figure 7: LEP readout model (ALEPH)
For readout systems of the type shown in 7, the total dead-time is given by
the sum of two components — the trigger dead-time and the readout dead-time.
The trigger dead-time is evaluated by counting the number of BCs that are lost
following each LVL1 trigger, then calculating the product of the LVL1 trigger
rate, the number of lost BCs and the BC period. Note that the effective LVL2
latency, given by the number of lost BCs and the BC period, is less than (or
equal to) the true LVL2 latency.
The readout dead-time is given by the product of the LVL2 trigger rate and the
time taken to perform local readout into the ROCs. Strictly speaking, one
should also express this dead-time in terms of the number of BCs lost after
the LVL2 trigger, but since the readout time is much longer than the BC period
the difference is unimportant. Note that, as long as the buffers in the ROCs
and the global buffers do not fill up, no additional dead-time is introduced
by the final readout and the LVL3 trigger.
Let us now look quantitatively at the example of the DELPHI experiment for
which the T/DAQ system was similar to that described above for ALEPH. Typical
numbers for LEP-II444LEP-II refers to the period when LEP operated at high
energy, after the upgrade of the RF system. are shown in 1 [9].
### 0.4.2 Data acquisition at LEP
Let us now continue our examination of the example of the ALEPH T/DAQ system.
Following a LVL2 trigger, events were read out locally and in parallel within
the many readout crates — once the data had been transferred within each crate
to the ROC, further LVL1 and LVL2 triggers could be accepted. Subsequently,
the data from the readout crates were collected by the main readout computer,
‘building’ a complete event. As shown in 9, event building was performed in
two stages: an event contained a number of sub-events, each of which was
composed of several ROC data blocks. Once a complete event was in the main
readout computer, the LVL3 trigger made a final selection before the data were
recorded.
Table 1: Typical T/DAQ parameters for the DELPHI experiment at LEP-II Quantity | Value
---|---
LVL1 rate | ~ 500–1000 (instrumental background)
LVL2 rate | 6–8
LVL3 rate | 4–6
LVL2 latency | 38 (1 lost BC $\Rightarrow$ 22 effective)
Local readout time | ~ 2.5
Readout dead-time | ~ 7 × 2.5 $\cdot$ 10-3 = 1.8%
Trigger dead-time | ~ 750 × 22 $\cdot$ 10-6 = 1.7%
Total dead-time | ~ 3–4%
The DAQ system used a hierarchy of computers — the local ROCs in each crate;
event builders (EBs) for sub-events; the main EB; the main readout computer.
The ROCs performed some data processing (applying calibration algorithms to
convert ADC values to energies) in addition to reading out the data from ADCs,
TDCs, (Zero suppression was already performed at the level of the digitizers
where appropriate.) The first layer of EBs combined data read out from the
ROCs of individual sub-detectors into sub-events; then the main EB combined
the sub-events for the different sub-detectors. Finally, the main readout
computer received full events from the main EB, performed the LVL3 trigger
selection, and recorded selected events for subsequent analysis.
As indicated in fig:f9, event building was bus based — each ROC collected data
over a bus from the digitizing electronics; each sub-detector EB collected
data from several ROCs over a bus; the main EB collected data from the sub-
detector EBs over a bus. As a consequence, the main EB and the main readout
computer saw the full data rate prior to the final LVL3 selection. At LEP this
was fine — with an event rate after LVL2 of a few hertz and an event size of
100 kbytes, the data rate was a few hundred kilobytes per second, much less
than the available bandwidth ( 40 Mbytes/s maximum on VME bus [12]).
Figure 8: ALEPH data-acquisition architecture
Figure 9: Event building in ALEPH
### 0.4.3 Triggers at LEP
The triggers at LEP aimed to select any annihilation event with a visible
final state, including events with little visible energy, plus some fraction
of two-photon events, plus Bhabha scattering events. Furthermore, they aimed
to select most events by multiple, independent signatures so as to maximize
the trigger efficiency and to allow the measurement of the efficiency from the
data. The probability for an event to pass trigger A or trigger B is ~
$1-\delta_{\text}{A}\delta_{\text}{B}$, where $\delta_{\text}{A}$ and
$\delta_{\text}{B}$ are the individual trigger inefficiencies, which is very
close to unity for small $\delta$. Starting from a sample of events selected
with trigger A, the efficiency of trigger B can be estimated as the fraction
of events passing trigger B in addition. Note that in the actual calculations
small corrections were applied for correlations between the trigger
efficiencies.
## 0.5 Towards the LHC
In some experiments it is not practical to make a trigger in the time between
bunch crossings because of the short BC period — the BC interval is 396 ns at
Tevatron-II555Tevatron-II refers to the Tevatron collider after the luminosity
upgrade., 96 ns at HERA and 25 ns at LHC. In such cases the concept of
‘pipelined’ readout has to be introduced (also pipelined LVL1 trigger
processing). Furthermore, in experiments at high-luminosity hadron colliders
the data rates after the LVL1 trigger selection are very high, and new ideas
have to be introduced for the high-level triggers (HLTs) and DAQ — in
particular, event building has to be based on data networks and switches
rather than data buses.
### 0.5.1 Pipelined readout
In pipelined readout systems (see 10), the information from each BC, for each
detector element, is retained during the latency of the LVL1 trigger (several
µs). The information may be retained in several forms — analog levels (held on
capacitors); digital values (ADC results); binary values (hit or no hit). This
is done using a logical ‘pipeline’, which may be implemented using a first-in,
first-out (FIFO) memory circuit. Data reaching the end of the pipeline are
either discarded or, in the case of a trigger accept decision, moved to a
secondary buffer memory (small fraction of BCs).
Figure 10: Example of pipelined readout
Pipelined readout systems will be used in the LHC experiments (they have
already been used in experiments at HERA [13, 14] and the Tevatron [15, 16],
but the demands at LHC are even greater because of the short BC period). A
typical LHC pipelined readout system is illustrated in 11, where the digitizer
and pipeline are driven by the 40 BC clock. A LVL1 trigger decision is made
for each bunch crossing (every 25 ns), although the LVL1 latency is several
microseconds — the LVL1 trigger must concurrently process many events (this is
achieved by using pipelined trigger processing as discussed below).
Figure 11: Pipelined readout with derandomizer at the LHC
The data for events that are selected by the LVL1 trigger are transferred into
a ‘derandomizer’ — a memory that can accept the high instantaneous input rate
(one word per 25 ns) while being read out at the much lower average data rate
(determined by the LVL1 trigger rate rather than the BC rate). In principle no
dead-time needs to be introduced in such a system. However, in practice, data
are retained for a few BCs around the one that gave rise to the trigger, and a
dead period of a few BCs is introduced to ensure that the same data do not
have to be accessed for more than one trigger. Dead-time must also be
introduced to prevent the derandomizers from overflowing, where, due to a
statistical fluctuation, many LVL1 triggers arrive in quick succession. The
dead-time from the first of these sources can be estimated as follows (numbers
from ATLAS): taking a LVL1 trigger rate of 75 and 4 dead BCs following each
LVL1 trigger gives $75\UkHz\times 4\times 25\Uns=0.75\%$. The dead-time from
the second source depends on the size of the derandomizer and the speed with
which it can be emptied — in ATLAS the requirements are $<1\%$ dead-time for a
LVL1 rate of 75 ($<6\%$ for 100).
Some of the elements of the readout chain in the LHC experiments have to be
mounted on the detectors (and hence are totally inaccessible during running of
the machine and are in an environment with high radiation levels). This is
shown for the case of CMS in 12.
Figure 12: Location of readout components in CMS
There are a variety of options for the placement of digitization in the
readout chain, and the optimum choice depends on the characteristics of the
detector in question. Digitization may be performed on the detector at 40
rate, prior to a digital pipeline (CMS calorimeter). Alternatively, it may be
done on the detector after multiplexing signals from several analog pipelines
(ATLAS EM calorimeter) — here the digitization rate can be lower, given by the
LVL1 trigger rate multiplied by the number of signals to be digitized per
trigger. Another alternative (CMS tracker) is to multiplex analog signals from
the pipelines over analog links, and then to perform the digitization off-
detector.
### 0.5.2 Pipelined LVL1 trigger
As discussed above, the LVL1 trigger has to deliver a new decision every BC,
although the trigger latency is much longer than the BC period; the LVL1
trigger must concurrently process many events. This can be achieved by
‘pipelining’ the processing in custom trigger processors built using modern
digital electronics. The key ingredients in this approach are to break the
processing down into a series of steps, each of which can be performed within
a single BC period, and to perform many operations in parallel by having
separate processing logic for each calculation. Note that in such a system the
latency of the LVL1 trigger is fixed — it is determined by the number of steps
in the calculation, plus the time taken to move signals and data to, from and
between the components of the trigger system (propagation delays on cables).
Pipelined trigger processing is illustrated in 13 — as will be seen later,
this example corresponds to a (very small) part of the ATLAS LVL1 calorimeter
trigger processor. The drawing on the left of 13 depicts the EM calorimeter as
a grid of ‘towers’ in $\eta\text{--}\phi$ space ($\eta$ is pseudorapidity,
$\phi$ is azimuth angle). The logic shown on the right determines if the
energy deposited in a horizontal or vertical pair of towers in the region [A,
B, C] exceeds a threshold. In each 25 ns period, data from one layer of
‘latches’ (memory registers) are processed through the next step in the
processing ‘pipe’, and the results are captured in the next layer of latches.
Note that, in the real system, such logic has to be performed in parallel for
~ 3500 positions of the reference tower; the tower ‘A’ could be at any
position in the calorimeter. In practice, modern electronics is capable of
doing more than a simple add or compare operation in 25 ns, so there is more
logic between the latches than in this illustration.
Figure 13: Illustration of pipelined processing
The amount of data to be handled varies with depth in the processing pipeline,
as indicated in 14. Initially the amount of data expands compared to the raw
digitization level since each datum typically participates in several
operations — the input data need to be ‘fanned out’ to several processing
elements. Subsequently the amount of data decreases as one moves further down
the processing tree. The final trigger decision can be represented by a single
bit of information for each BC — yes or no (binary 1 or 0). Note that, in
addition to the trigger decision, the LVL1 processors produce a lot of data
for use in monitoring the system and to guide the higher levels of selection.
Although they have not been discussed in these lectures because of time
limitations, some fixed-target experiments have very challenging T/DAQ
requirements. Some examples can be found in Refs. [17, 18].
Figure 14: LVL1 data flow
## 0.6 High-level triggers and data acquisition at the LHC
In the LHC experiments, data are transferred after a LVL1 trigger accept
decision to large buffer memories — in normal operation the subsequent stages
should not introduce further dead-time. At this point in the readout chain,
the data rates are still massive. An event size of ~ 1 Mbyte (after zero
suppression or data compression) at ~ 100 event rate gives a total bandwidth
of ~ 100 Gbytes/s ( ~ 800 Gbits/s). This is far beyond the capacity of the
bus-based event building of LEP. Such high data rates will be dealt with by
using network-based event building and by only moving a subset of the data.
Network-based event building is illustrated in 15 for the example of CMS. Data
are stored in the readout systems until they have been transferred to the
filter systems [associated with high-level trigger (HLT) processing], or until
the event is rejected. Note that no node in the system sees the full data rate
— each readout system covers only a part of the detector and each filter
system deals with only a fraction of the events.
Figure 15: CMS event builder
The LVL2 trigger decision can be made without accessing or processing all of
the data. Substantial rejection can be made with respect to LVL1 without
accessing the inner-tracking detectors — calorimeter triggers can be refined
using the full-precision, full-granularity calorimeter information; muon
triggers can be refined using the high-precision readout from the muon
detectors. It is therefore only necessary to access the inner-tracking data
for the subset of events that pass this initial selection. ATLAS and CMS both
use this sequential selection strategy. Nevertheless, the massive data rates
pose problems even for network-based event building, and different solutions
have been adopted in ATLAS and CMS to address this.
In CMS the event building is factorized into a number of ‘slices’, each of
which sees only a fraction of the total rate (see 16). This still requires a
large total network bandwidth (which has implications for the cost), but it
avoids the need for a very big central network switch. An additional advantage
of this approach is that the size of the system can be scaled, starting with a
few slices and adding more later (as additional funding becomes available).
Figure 16: The CMS slicing concept
In ATLAS the amount of data to be moved is reduced by using the region-of-
interest (RoI) mechanism (see 17). Here, the LVL1 trigger indicates the
geographical location in the detector of candidate objects. LVL2 then only
needs to access data from the RoIs, a small fraction of the total, even for
the calorimeter and muon detectors that participated in the LVL1 selection.
This requires relatively complicated mechanisms to serve the data selectively
to the LVL2 trigger processors.
In the example shown in 17, two muons are identified by LVL1. It can be seen
that only a small fraction of the detector has to be accessed to validate the
muons. In a first step only the data from the muon detectors are accessed and
processed, and many events will be rejected where the more detailed analysis
does not confirm the comparatively crude LVL1 selection (sharper
$p_{\scriptstyle\mathrm{T}}$ cut). For those events that remain, the inner-
tracker data will be accessed within the RoIs, allowing further rejection (of
muons from decays in flight of charged pions and kaons). In a last step,
calorimeter information may be accessed within the RoIs to select isolated
muons (to reduce the high rate of events with muons from bottom and charm
decays, while retaining those from and decays).
Figure 17: The ATLAS region-of-interest concept — example of a dimuon event
(see text)
Concerning hardware implementation, the computer industry is putting on the
market technologies that can be used to build much of the HLT/DAQ systems at
the LHC. Computer network products now offer high performance at affordable
cost. Personal computers (PCs) provide exceptional value for money in
processing power, with high-speed network interfaces as standard items.
Nevertheless, custom hardware is needed in the parts of the system that see
the full LVL1 trigger output rate (~ 100). This concerns the readout systems
that receive the detector data following a positive LVL1 trigger decision, and
(in ATLAS) the interface to the LVL1 trigger that receives the RoI pointers.
Of course, this is in addition to the specialized front-end electronics
associated with the detectors that was discussed earlier (digitization,
pipelines, derandomizers, ).
## 0.7 Physics requirements — two examples
In the following, the physics requirements on the T/DAQ systems at LEP and at
the LHC are examined. These are complementary cases — at LEP precision physics
was the main emphasis, at the LHC discovery physics will be the main issue.
Precision physics at LEP needed accurate determination of the absolute cross-
section (in the determination of the number of light-neutrino species).
Discovery physics at the LHC will require sensitivity to a huge range of
predicted processes with diverse signatures (with very low signal rates
expected in some cases), aiming to be as sensitive as possible to new physics
that has not been predicted (by using inclusive signatures). This has to be
achieved in the presence of an enormous rate of Standard Model physics
backgrounds (the rate of proton–proton collisions at the LHC will be
$\mathcal{O}(10^{9})$ — $\sigma$ ~ 100,
$\mathscr{L}\,\text{\texttildelow}\,10^{34}\,\text{cm}^{-2}\,\text{s}^{-1}$).
### 0.7.1 Physics requirements at LEP
Triggers at LEP aimed to identify all events coming from annihilations with
visible final states. At LEP-I, operating with $\sqrt{s}\sim m_{\PZ}$, this
included $\PZ\rightarrow\text{hadrons}$, $\PZ\rightarrow\Pep\Pem$,
$\PZ\rightarrow\PGmp\PGmm$, and $\PZ\rightarrow\PGtp\PGtm$; at LEP-II,
operating above the WW threshold, this included WW, ZZ and single-boson
events. Sensitivity was required even in cases where there was little visible
energy, in the Standard Model for $\Pep\Pem\rightarrow\PZ\PGg$, with
$\PZ\rightarrow\PGn\PGn$, and in new-particle searches such as
$\Pep\Pem\rightarrow\PSGcp\PSGcm$ for the case of small $\PSGcpm-\PSGcz$ mass
difference that gives only low-energy visible particles ($\PSGcz$ is the
lightest supersymmetric particle). In addition, the triggers had to retain
some fraction of two-photon collision events (used for QCD studies), and
identify Bhabha scatters (needed for precise luminosity determination).
The triggers could retain events with any significant activity in the
detector. Even when running at the peak, the rate of decays was only
$\mathcal{O}(1)$ — physics rate was not an issue. The challenge was in
maximizing the efficiency (and acceptance) of the trigger, and making sure
that the small inefficiencies were very well understood. The determination of
absolute cross-section depends on knowing the integrated luminosity and the
experimental efficiency to select the process in question (the efficiency to
trigger on the specific physics process). Precise determination of the
integrated luminosity required excellent understanding of the trigger
efficiency for Bhabha-scattering events (luminosity determined from the rate
of Bhabha scatters within a given angular range). A major achievement at LEP
was to reach ‘per mil’ precision.
The trigger rates (events per second) and the DAQ rates (bytes per second) at
LEP were modest as discussed in Section 0.4.
### 0.7.2 Physics requirements at the LHC
Triggers in the general-purpose proton–proton experiments at the LHC (ATLAS
[19, 20] and CMS [21, 22]) will have to retain as high as possible a fraction
of the events of interest for the diverse physics programmes of these
experiments. Higgs searches in and beyond the Standard Model will include
looking for $\PH\rightarrow\PZ\PZ\rightarrow\text{leptons}$ and also
$\PH\rightarrow\PQb\PAQb$. Supersymmetry (SUSY) searches will be performed
with and without the assumption of R-parity conservation. One will search for
other new physics using inclusive triggers that one hopes will be sensitive to
unpredicted processes. In parallel with the searches for new physics, the LHC
experiments aim to do precision physics, such as measuring the mass and some
B-physics studies, especially in the early phases of LHC running when the
luminosity is expected to be comparatively low.
In contrast to the experiments at LEP, the LHC trigger systems have a hard job
to reduce the physics event rate to a manageable level for data recording and
offline analysis. As discussed above, the design luminosity
$\mathscr{L}\,\text{\texttildelow}\,10^{34}\,\text{cm}^{-2}\,\text{s}^{-1}$,
together with $\sigma$ ~ 100, implies an $\mathcal{O}(10^{9})$ interaction
rate. Even the rate of events containing leptonic decays of and bosons is
$\mathcal{O}(100)$. Furthermore, the size of the events is very large,
$\mathcal{O}(1)$ Mbyte, reflecting the huge number of detector channels and
the high particle multiplicity in each event. Recording and subsequently
processing offline $\mathcal{O}(100)$ event rate per experiment with an
$\mathcal{O}(1)$ Mbyte event size is considered feasible, but it implies major
computing resources [23]. Hence, only a tiny fraction of proton–proton
collisions can be selected — taking the order-of-magnitude numbers given
above, the maximum fraction of interactions that can be selected is
$\mathcal{O}(10^{-7})$. Note that the general-purpose LHC experiments have to
balance the needs of maximizing physics coverage and reaching acceptable
(affordable) recording rates.
The LHCb experiment [24], which is dedicated to studying B-physics, faces
similar challenges to ATLAS and CMS. It will operate at a comparatively low
luminosity
($\mathscr{L}\,\text{\texttildelow}\,10^{32}\,\text{cm}^{-2}\,\text{s}^{-1}$),
giving an overall proton–proton interaction rate of ~ 20 — chosen to maximize
the rate of single-interaction bunch crossings. The event size will be
comparatively small (~ 100 kbytes) as a result of having fewer detector
channels and of the lower occupancy of the detector (due to the lower
luminosity with less pile-up). However, there will be a very high rate of
beauty production in LHCb — taking $\sigma$ ~ 500 µb, the production rate will
be ~ 100 — and the trigger must search for specific B-decay modes that are of
interest for physics analysis, with the aim of recording an event rate of only
~ 200.
The heavy-ion experiment ALICE [5] is also very demanding, particularly from
the DAQ point of view. The total interaction rate will be much smaller than in
the proton–proton experiments —
$\mathscr{L}\,\text{\texttildelow}\,10^{27}\,\text{cm}^{-2}\,\text{s}^{-1}$ is
predicted to give a rate ~ 8000 for Pb–Pb collisions. However, the event size
will be huge due to the high final-state multiplicity in Pb–Pb interactions at
LHC energy. Up to $\mathcal{O}(10^{4})$ charged particles will be produced in
the central region, giving an event size of up to ~ 40 Mbytes when the full
detector is read out. The ALICE trigger will select ‘minimum-bias’ and
‘central’ events (rates scaled down to a total of about 40), and events with
dileptons (~ 1 with only part of the detector read out). Even compared to the
other LHC experiments, the volume of data to be stored and subsequently
processed offline will be massive, with a data rate to storage of ~ 1 Gbytes/s
(considered to be about the maximum affordable rate).
## 0.8 Signatures of different types of particle
The generic signatures for different types of particle are illustrated in 18.
Moving away from the interaction point (shown as a star on the left-hand side
of Fig. 18), one finds the inner tracking detector (IDET), the electromagnetic
calorimeter (ECAL), the hadronic calorimeter (HCAL) and the muon detectors
(MuDET). Charged particles (electrons, muons and charged hadrons) leave tracks
in the IDET. Electrons and photons shower in the ECAL, giving localized
clusters of energy without activity in the HCAL. Hadrons produce larger
showers that may start in the ECAL but extend into the HCAL. Muons traverse
the calorimeters with minimal energy loss and are detected in the MuDET.
The momenta of charged particles are measured from the radii of curvature of
their tracks in the IDET which is embedded in a magnetic field. A further
measurement of the momenta of muons may be made in the MuDET using a second
magnet system. The energies of electrons, photons and hadrons are measured in
the calorimeters. Although neutrinos leave the detector system without
interaction, one can infer their presence from the momentum imbalance in the
event (sometimes referred to as ‘missing energy’). Hadronic jets contain a
mixture of particles, including neutral pions that decay almost immediately
into photon pairs that are then detected in the ECAL. The jets appear as broad
clusters of energy in the calorimeters where the individual particles will
sometimes not be resolved.
Figure 18: Signatures of different types of particle in a generic detector
## 0.9 Selection criteria and trigger implementations at LEP
The details of the selection criteria and trigger implementations at LEP
varied from experiment to experiment [8, 9, 10, 11]. Discussion of the example
of ALEPH is continued with the aim of giving a reasonably in-depth view of one
system. For triggering purposes, the detector was divided into segments with a
total of 60 regions in $\theta,\phi$ ($\theta$ is polar angle and $\phi$ is
azimuth with respect to the beam axis). Within these segments, the following
trigger objects were identified:
1. 1.
muon — requiring a track penetrating the hadron calorimeter and seen in the
inner tracker;
2. 2.
charged electromagnetic (EM) energy — requiring an EM calorimeter cluster and
a track in the inner tracker;
3. 3.
neutral EM energy — requiring an EM calorimeter cluster (with higher
thresholds than in (2) to limit the rate to acceptable levels).
In addition to the above local triggers, there were total-energy triggers
(applying thresholds on energies summed over large regions — the barrel or a
full endcap), a back-to-back tracks trigger, and triggers for Bhabha
scattering (luminosity monitor).
The LVL1 triggers were implemented using a combination of analog and digital
electronics. The calorimeter triggers were implemented using analog
electronics to sum signals before applying thresholds on the sums. The LVL1
tracking trigger looked for patterns of hits in the inner-tracking chamber
(ITC) consistent with a track with
$p_{\scriptstyle\mathrm{T}}>1\leavevmode\nobreak\ \mathrm{GeV}$ 666Here,
$p_{\scriptstyle\mathrm{T}}$ is transverse momentum (measured with respect to
the beam axis); similarly, $E_{\scriptstyle\mathrm{T}}$ is transverse energy.
— at LVL2 the Time Projection Chamber (TPC) was used instead. The final
decision was made by combining digital information from calorimeter and
tracking triggers, making local combinations within segments of the detector,
and then making a global combination (logical OR of conditions).
## 0.10 Selection criteria at LHC
Features that distinguish new physics from the bulk of the cross-section for
Standard Model processes at hadron colliders are generally the presence of
high-$p_{\scriptstyle\mathrm{T}}$ particles (or jets). For example, these may
be the products of the decays of new heavy particles. In contrast, most of the
particles produced in minimum-bias interactions are soft
($p_{\scriptstyle\mathrm{T}}$ ~ 1 or less). More specific signatures are the
presence of high-$p_{\scriptstyle\mathrm{T}}$ leptons (, , ), photons and/or
neutrinos. For example, these may be the products (directly or indirectly) of
new heavy particles. Charged leptons, photons and neutrinos give a
particularly clean signature (c.f. low-$p_{\scriptstyle\mathrm{T}}$ hadrons in
minimum-bias events), especially if they are ‘isolated’ (not inside jets). The
presence of heavy particles such as and bosons can be another signature for
new physics — they may be produced in Higgs decays. Leptonic and decays give a
very clean signature that can be used in the trigger. Of course it is
interesting to study and boson production $perse$, and such events can be very
useful for detector studies (calibration of the EM calorimeters).
In view of the above, LVL1 triggers at hadron colliders search for the
following signatures (see 18).
* •
High-$p_{\scriptstyle\mathrm{T}}$ muons — these can be identified as charged
particles that penetrate beyond the calorimeters; a
$p_{\scriptstyle\mathrm{T}}$ cut is needed to control the rate of muons from
$\PGppm\rightarrow\PGmpm\PGn$ and $\PKpm\rightarrow\PGmpm\PGn$ decays in
flight, as well as those from semi-muonic beauty and charm decays.
* •
High-$p_{\scriptstyle\mathrm{T}}$ photons — these can be identified as narrow
clusters in the EM calorimeter; cuts are made on transverse energy
($E_{\scriptstyle\mathrm{T}}>\text{threshold}$), and isolation and associated
hadronic transverse energy ($E_{\scriptstyle\mathrm{T}}<\text{threshold}$), to
reduce the rate due to misidentified high-$p_{\scriptstyle\mathrm{T}}$ jets.
* •
High-$p_{\scriptstyle\mathrm{T}}$ electrons — identified in a similar way to
photons, although some experiments require a matching track as early as LVL1.
* •
High-$p_{\scriptstyle\mathrm{T}}$ taus — identified as narrow clusters in the
calorimeters (EM and hadronic energy combined).
* •
High-$p_{\scriptstyle\mathrm{T}}$ jets — identified as wider clusters in the
calorimeters (EM and hadronic energy combined); note that one needs to cut at
very high $p_{\scriptstyle\mathrm{T}}$ to get acceptable rates given that jets
are the dominant high-$p_{\scriptstyle\mathrm{T}}$ process.
* •
Large missing $E_{\scriptstyle\mathrm{T}}$ or scalar
$E_{\scriptstyle\mathrm{T}}$.
Some experiments also search for tracks from displaced secondary vertices at
an early stage in the trigger selection.
The trigger selection criteria are typically expressed as a list of conditions
that should be satisfied — if any of the conditions is met, a trigger is
generated (subject to dead-time requirements, ). In these notes, the list of
conditions is referred to as the ‘trigger menu’, although the name varies from
experiment to experiment. An illustrative example of a LVL1 trigger menu for
high-luminosity running at LHC includes the following (rates [19] are given
for the case of ATLAS at
$\mathscr{L}\,\text{\texttildelow}\,10^{34}\,\text{cm}^{-2}\,\text{s}^{-1}$):
* •
one or more muons with $p_{\scriptstyle\mathrm{T}}>20\UGeV$ (rate ~ 11);
* •
two or more muons each with $p_{\scriptstyle\mathrm{T}}>6\UGeV$ (rate ~ 1);
* •
one or more / with $E_{\scriptstyle\mathrm{T}}>30\UGeV$ (rate ~ 22);
* •
two or more / each with $E_{\scriptstyle\mathrm{T}}>20\UGeV$ (rate ~ 5);
* •
one or more jets with $E_{\scriptstyle\mathrm{T}}>290\UGeV$ (rate ~ 200);
* •
one or more jets with $E_{\scriptstyle\mathrm{T}}>100\UGeV$ and
missing-$E_{\scriptstyle\mathrm{T}}>100\UGeV$ (rate ~ 500);
* •
three or more jets with $E_{\scriptstyle\mathrm{T}}>130\UGeV$ (rate ~ 200);
* •
four or more jets with $E_{\scriptstyle\mathrm{T}}>90\UGeV$ (rate ~ 200).
The above list represents an extract from a LVL1 trigger menu, indicating some
of the most important trigger requirements — the full menu would include many
items in addition (typically more than 100 items in total). The additional
items are expected to include the following:
* •
$\PGt$ (or isolated single-hadron) candidates;
* •
combinations of objects of different types (muon _and_ /);
* •
pre-scaled777Some triggers may be ‘pre-scaled’ — this means that only every
_N_ th event satisfying the relevant criteria is recorded, where _N_ is a
parameter called the pre-scale factor; this is useful for collecting samples
of high-rate triggers without swamping the T/DAQ system. triggers with lower
thresholds;
* •
triggers needed for technical studies and to aid understanding of the data
from the main triggers (trigger on bunch crossings at random to collect an
unbiased data sample).
As for the LVL1 trigger, the HLT has a trigger menu that describes which
events should be selected. This is illustrated in 0.10 for the example of CMS,
assuming a luminosity for early running of
$\mathscr{L}\,\text{\texttildelow}\,10^{33}\,\text{cm}^{-2}\,\text{s}^{-1}$.
The total rate of ~ 100 contains a large fraction of events that are useful
for physics analysis. Lower thresholds would be desirable, but the physics
coverage has to be balanced against considerations of the offline computing
cost. Note that there are large uncertainties on the rate calculations.
Table 2: Estimated high-level trigger rates for
$\mathscr{L}\,\text{\texttildelow}2\times
10^{33}\,\text{cm}^{-2}\,\text{s}^{-1}$ (CMS numbers from Ref. [21])
@X@ r@ Trigger configuration Rate
One or more electrons with $p_{\scriptstyle\mathrm{T}}>29\UGeV$, or two or
more electrons with $p_{\scriptstyle\mathrm{T}}>17\UGeV$ ~ 34
One or more photons with $p_{\scriptstyle\mathrm{T}}>80\UGeV$, or two or more
photons with $p_{\scriptstyle\mathrm{T}}>40,25\UGeV$ ~ 9
One or more muons with $p_{\scriptstyle\mathrm{T}}>19\UGeV$, or two or more
muons with $p_{\scriptstyle\mathrm{T}}>7\UGeV$ ~ 29
One or more taus with $p_{\scriptstyle\mathrm{T}}>86\UGeV$, or two or more
taus with $p_{\scriptstyle\mathrm{T}}>59\UGeV$ ~ 4
One or more jets with $p_{\scriptstyle\mathrm{T}}>180\UGeV$ _and_
missing-$E_{\scriptstyle\mathrm{T}}$$>123\UGeV$ ~ 5
One or more jets with $p_{\scriptstyle\mathrm{T}}>657\UGeV$, or three or more
jets with $p_{\scriptstyle\mathrm{T}}>247\UGeV$, or four or more jets with
$p_{\scriptstyle\mathrm{T}}>113\UGeV$ ~ 9
Others (electron and jet, b-jets, ) ~ 7
A major challenge lies in the HLT/DAQ software. The event-selection algorithms
for the HLT can be subdivided, at least logically, into LVL2 and LVL3 trigger
stages. These might be performed by two separate processor systems (ATLAS), or
in two distinct processing steps within the same processor system (CMS). The
algorithms have to be supported by a software framework that manages the flow
of data, supervising an event from when it arrives at the HLT/DAQ system until
it is either rejected, or accepted and recorded on permanent storage. This
includes software for efficient transfer of data to the algorithms. In
addition to the above, there is a large amount of associated online software
(run control, databases, book-keeping, ).
## 0.11 LVL1 trigger design for the LHC
A number of design goals must be kept in mind for the LVL1 triggers at the
LHC. It is essential to achieve a very large reduction in the physics rate,
otherwise the HLT/DAQ system will be swamped and the dead-time will become
unacceptable. In practice, the interaction rate, $\mathcal{O}(10^{9})$, must
be reduced to less than 100 in ATLAS and CMS. Complex algorithms are needed to
reject the background while keeping the signal events.
Another important constraint is to achieve a short latency — information from
all detector elements ($\mathcal{O}(10^{7}\text{--}10^{8})$ channels!) has to
be held on the detector pending the LVL1 decision. The pipeline memories that
do this are typically implemented in ASICs (application-specific integrated
circuits), and memory size contributes to the cost. Typical LVL1 latency
values are a few microseconds (less than 2.5 µs in ATLAS and less than 3.2 µs
in CMS).
A third requirement is to have flexibility to react to changing conditions (a
wide range of luminosities) and — it is hoped — to new physics! The algorithms
must be programmable, at least at the level of parameters (thresholds, ).
### 0.11.1 Case study — ATLAS / trigger
The ATLAS / trigger algorithm can be used to illustrate the techniques used in
LVL1 trigger systems at LHC. It is based on $4\times 4$ ‘overlapping, sliding
windows’ of trigger towers as illustrated in 20. Each trigger tower has a
lateral extent of $0.1\times 0.1$ in $\eta,\phi$ space, where $\eta$ is
pseudorapidity and $\phi$ is azimuth. There are about 3500 such towers in each
of the EM and hadronic calorimeters. Note that each tower participates in
calculations for 16 windows. The algorithm requires a local maximum in the EM
calorimeter to define the $\eta\text{--}\phi$ position of the cluster and to
avoid double counting of extended clusters (so-called ‘declustering’). It can
also require that the cluster be isolated, little energy surrounding the
cluster in the EM calorimeter or the hadronic calorimeter.
Figure 19: ATLAS / trigger algorithm
Figure 20: Overview of the ATLAS LVL1 calorimeter trigger
The implementation of the ATLAS LVL1 calorimeter trigger [25] is sketched in
20. Analog electronics on the detector sums signals from individual
calorimeter cells to form trigger-tower signals. After transmission to the
‘pre-processor’ (PPr), which is located in an underground room close to the
detector and shielded against radiation, the tower signals are received and
digitized; then the digital data are processed to obtain estimates of
$E_{\scriptstyle\mathrm{T}}$ per trigger tower for each BC. At this point in
the processing chain (at the output of the PPr), there is an
‘$\eta\text{--}\phi$ matrix’ of the $E_{\scriptstyle\mathrm{T}}$ per tower in
each of the EM and hadronic calorimeters that gets updated every 25 ns.
The tower data from the PPr are transmitted to the cluster processor (CP).
Note that the CP is implemented with very dense electronics so that there are
only four crates in total. This minimizes the number of towers that need to be
transmitted (‘fanned out’) to more than one crate. Fan out is required for
towers that contribute to windows for which the algorithmic processing is
implemented in more than one crate. Also, within each CP crate, trigger-tower
data need to be fanned out between electronic modules, and then between
processing elements within each module. Considerations of connectivity and
data-movement drive the design.
In parallel with the CP, a jet/energy processor (JEP) searches for jet
candidates and calculates missing-$E_{\scriptstyle\mathrm{T}}$ and
scalar-$E_{\scriptstyle\mathrm{T}}$ sums. This is not described further here.
A very important consideration in designing the LVL1 trigger is the need to
identify uniquely the BC that produced the interaction of interest. This is
not trivial, especially given that the calorimeter signals extend over many
BCs. In order to assign observed energy deposits to a given BC, information
has to be combined from a sequence of measurements. [b] 21 illustrates how
this is done within the PPr (the logic is repeated ~ 7000 times so that this
is done in parallel for all towers). The raw data for a given tower move along
a pipeline that is clocked by the 40 BC signal. The multipliers together with
the adder tree implement a finite-impulse-response filter whose output is
passed to a peak finder (a peak indicates that the energy was deposited in the
BC currently being examined) and to a look-up table that converts the peak
amplitude to an $E_{\scriptstyle\mathrm{T}}$ value. Special care is taken to
avoid BC misidentification for very large pulses that may get distorted in the
analog electronics, since such signals could correspond to the most
interesting events. The functionality shown in 21 is implemented in ASICs
(four channels per ASIC).
Figure 21: Bunch-crossing identification
The transmission of the data (the $E_{\scriptstyle\mathrm{T}}$ matrices) from
the PPr to the CP is performed using a total of 5000 digital links each
operating at 400 Mbits/s (each link carries data from two towers using a
technique called BC multiplexing [25]). Where fan out is required, the
corresponding links are duplicated with the data being sent to two different
CP crates. Within each CP crate, data are shared between neighbouring modules
over a very high density crate back-plane (~ 800 pins per slot in a 9U crate;
data rate of 160 Mbits/s per signal pin using point-to-point connections). On
each of the modules, data are passed to eight large field-programmable gate
arrays (FPGAs) that perform the algorithmic processing, fanning out signals to
more than one FPGA where required.
As an exercise, it is suggested that students make an order-of-magnitude
estimate of the total bandwidth between the PPr and the CP, considering what
this corresponds to in terms of an equivalent number of simultaneous telephone
calls888One may assume an order-of-magnitude data rate for voice calls of 10
kbits/s — for example, the GSM mobile-phone standard uses a 9600 bit/s digital
link to transmit the encoded voice signal..
The / (together with the /) algorithms are implemented using FPGAs. This has
only become feasible thanks to recent advances in FPGA technology since very
large and very fast devices are needed. Each FPGA handles an area of $4\times
2$ windows, requiring data from $7\times 5$ towers in each of the EM and
hadronic calorimeters. The algorithm is described in a programming language
(VHDL) that can be converted into the FPGA configuration file. This gives
flexibility to adapt algorithms in the light of experience — the FPGAs can be
reconfigured _in situ_. Note that parameters of the algorithms can be changed
easily and quickly, as the luminosity falls during the course of a coast of
the beams in the LHC machine, since they are held in registers inside the
FPGAs that can be modified at run time ( there is no need to change the
‘program’ in the FPGA).
## 0.12 High-level trigger algorithms
There was not time in the lectures for a detailed discussion of the algorithms
that are used in the HLT. However, it is useful to consider the case of the
electron selection that follows after the first-level trigger. The LVL1 /
trigger is already very selective, so it is necessary to use complex
algorithms and full-granularity, full-precision detector data in the HLT.
A calorimeter selection is made applying a sharper
$E_{\scriptstyle\mathrm{T}}$ cut (better resolution than at LVL1) and shower-
shape variables that distinguish between the electromagnetic showers of an
electron or photon on one hand, and activity from jets on the other hand. The
shower-shape variables use both lateral and depth profile information. Then,
for electrons, a requirement is made of an associated track in the inner
detector, matching the calorimeter cluster in space, and with consistent
momentum and energy measurements from the inner detector and calorimeter
respectively.
Much work is going on to develop the algorithms and tune their many parameters
to optimize their signal efficiency and background rejection. So far this has
been done with simulated data, but further optimization will be required once
samples of electrons are available from offline reconstruction of real data.
It is worth noting that the efficiency value depends on the signal definition
as shown in fig:HLTe, an example of a study taken from Ref. [26]. Here the
trigger efficiency is shown, as a function of electron transverse energy,
relative to three different offline selections. With a loose offline
selection, the trigger is comparatively inefficient, whereas it performs much
better relative to the tighter offline cuts. This is related to the
optimization of the trigger both for signal efficiency (where loose cuts are
preferable) and for background rejection (where tighter cuts are required).
Figure 22: Trigger efficiency versus electron $E_{\scriptstyle\mathrm{T}}$ for
three different offline selections of the reference sample
## 0.13 Commissioning of the T/DAQ systems at LHC
Much more detail on the general commissioning of the LHC experiments can be
found in the lectures of Andreas Hoecker at this School [27]. Here an attempt
is made to describe how commissioning of the T/DAQ systems started in
September 2008.
On 10 September 2008 the first beams passed around the LHC in both the
clockwise and anti-clockwise directions, but with only one beam at a time (so
there was no possibility of observing proton–proton collisions). The energy of
the protons was 450 GeV which is the injection energy prior to acceleration;
acceleration to higher energies was not attempted.
As a first step, the beams were brought around the machine and stopped on
collimators such as those upstream of the ATLAS experiment. Given the huge
number of protons per bunch, as well as the sizeable beam energy, extremely
large numbers of secondary particles were produced, including muons that
traversed the experiment depositing energy in all of the detector systems.
Next, the collimators were removed and the beams were allowed to circulate
around the machine for a few turns and, after some tuning, for a few tens of
turns. Subsequently, the beams were captured by the radio-frequency system of
the LHC and circulated for periods of tens of minutes.
The first day of LHC operations was very exciting for all the people working
on the experiments. There was a very large amount of media interest, with
television broadcasts from various control rooms around the CERN site. It was
a particularly challenging time for those working on the T/DAQ systems who
were anxious to see if the first beam-related events would be identified and
recorded successfully. Much to the relief of the author, the online event
display of ATLAS soon showed a spectacular beam-splash event produced when the
beam particles hit the collimator upstream of the experiment. The first ATLAS
event is shown in fig:FirstEv; similar events were seen by the other
experiments.
Figure 23: The first beam-splash event in ATLAS
Analysis of the beam-spash events provided much useful information for
commissioning the detectors and also the trigger. For example, the relative
timing of different detector elements could be measured allowing the
adjustment of programmable delays to the correct settings. The very large
amount of activity in the events had the advantage that signals were seen in
an unusually large fraction of the detector channels.
An example of a very early study done with beam-splash events in shown in
fig:L1calo_splash which plots $E_{\scriptstyle\mathrm{T}}$ versus $\eta$ and
$\phi$ for the ATLAS LVL1 calorimeter trigger readout. The
$E_{\scriptstyle\mathrm{T}}$ values are colour coded; $\eta$ is along the
$x$-axis and $\phi$ is along the $y$-axis. The eight-fold $\phi$ structure of
the ATLAS magnets can be seen, as well as the effects of the tunnel floor and
heavy mechanical support structures that reduced the flux of particles
reaching the calorimeters in the bottom part of the detector
($\phi\leavevmode\nobreak\ \approx$ 270 degrees). The difference in absolute
scale between the left-hand and right-hand sides of the plot is attributed to
the fact that timing of the left-hand side was actually one bunch-crossing
away from ideal when the data were collected; the timing calibration was
subsequently adjusted as a result of these observations.
Figure 24: LVL1 calorimeter trigger energy grid for a beam-splash event
At least in ATLAS, the first beam-spash events were recorded using triggers
that had already been tested, with a free-runing 40 MHz clock, for cosmic-ray
events. This approach was appropriate because of the importance of recording
the first beam-related activity in the detector before the local beam
instrumentation had been calibrated. However, it was crucial to move on as
rapidly as possible to establish a precise and stable time reference.
Once beam-related activity had been seen in all of the LHC experiments,
stopping the beam on the corresponding collimators, all of the collimators
were removed and the beam was allowed to circulate. The first circulating
beams passed around the LHC for only a short period of time, corresponding to
a few turns initially, rising to a few tens of turns. For the 27 km LHC
circumference, the orbit period is about 89 $\mu$s.
Upstream of the LHC detectors (and upstream of the collimators) are passive
beam pick-ups that provide electrical signals induced by the passage of the
proton beams. The photograph in the left-hand side of fig:BPTX shows the beam
pick-up for one of the beams in an LHC experiment. Three of the four cables
that carry the signals can be seen. The analog signals from electrodes above,
below, to the left and to the right of the beam are combined (analog sum). The
resulting signal is fed to an oscilloscope directly and also via a
discriminator (an electronic device that provides a logical output signal when
the analog input signal exceeds a preset threshold, see Section 0.2).
On the right-hand side of fig:BPTX can be seen a plot, from CMS, of the
relative timing of different signals. The upper three traces are ‘orbit’
signals provided by the LHC machine, whereas the bottom trace is the
discriminated beam pick-up signal. As can be seen, the pick-up signal is
present for only four turns and then disappears. The reason for this is that
after a few turns the protons de-bunched and the analog signal from the pick-
ups became too small to fire the discriminator. Similar instrumentation and
timing calibration studies were used in all of the LHC experiments.
Figure 25: Photograph of beam pick-up instrumentation (left) and display of
timing signals recorded on a digital oscilloscope (right). The upper three
traces are ‘orbit’ signals from the LHC machine, whereas the bottom one is the
(inverted) discriminated signal from the beam pick-up.
A key feature of the beam pick-ups is that they provide a stable time
reference with respect to which other signals can be aligned. The time of
arrival of the beam pick-up signal, relative to the moment when the beam
passes through the centre of the LHC detector, depends only on the proton time
of flight from the beam pick-up position to the centre of the detector,
propagation delays of the signal along the electrical cables, and the response
time of the electronic circuits (which is very short).
Thanks to thorough preparations, the beam pick-up signals and their timing
relative to the trigger could be measured as soon as beam was injected.
Programmable delays could then be adjusted to align in time inputs to the
trigger from the beam pick-ups and from other sources. For example, in ATLAS,
the beam pick-up inputs were delayed so that they would have the same timing
as other inputs that had already been adjusted using cosmic-rays.
Once the timing of the beam pick-up inputs to the trigger had been adjusted so
as to initiate the detector readout for the appropriate bunch crossing (BC),
i.e., to read out a time-frame that would contain the detector signals
produced by beam-related activity, they could be used to provide the trigger
for subsequent running.
It is worth noting that the steps described above to set up the timing of the
trigger were completed within just a few hours on the morning of 10 September
2008. From then onwards the beam pick-ups represented a stable time reference
with respect to which other elements in the trigger and in the detector
readout systems could be adjusted.
As already indicated, all of the beam operations in September 2008 were with
just a single beam in the LHC. Operations were performed with beams
circulating in both the clockwise and anti-clockwise directions. Beam activity
in the detectors was produced by beam splash (beam stopped on collimators
upstream of the detectors producing a massive number of secondary particles)
or by beam-halo particles (produced when protons lost from the beam upstream
of the detectors produced one or more high-momentum muons that traversed the
detectors). In both cases one has to take into account the time of flight of
the particles that reach one end of the detector before the other end. In
contrast, beam–beam interactions have symmetric timing for the two ends of the
detector.
The work on timing calibration performed over the days following the LHC start
up can be illustrated by the case of ATLAS. Already on 10 September both sets
of beam pick-ups had been commissioned (with beams circulating in the
clockwise and anti-clockwise directions) giving a fixed time reference with
respect to which the rest of the trigger, and indeed the rest of the
experiment, could be aligned.
The situation on 10 September is summarized in the left-hand plot of
fig:Timing_In. The beam pick-up signal, labelled ‘BPTX’ in the figure, is the
reference. The relative time of arrival of other inputs to the trigger is
shown in units of BC number (i.e. one unit corresponds to 25 ns which is the
nominal bunch-crossing interval at LHC). Although there is a peak at the
nominal timing (bunch-number zero) in the distributions based on different
trigger inputs — the Minimum-Bias Trigger Scintillators (MBTS), the Thin-Gap
Chamber (TGC) forward muon detectors, and the Tau5, J5 and EM3 items from the
calorimeter trigger — the distribution is broad.
Prompt analysis and interpretation of the data allowed the timing to be
understood and calibration corrections to be applied. Issues addressed
included programming delay circuits to correct for time of flight of the
particles according to the direction of the circulating beam and tuning the
relative timing of triggers from different parts of the detector or from
different detector channels.
The situation two days later on 12 September is summarized in the right-hand
plot of fig:Timing_In. It is important to note that the scale is logarithmic —
the vast majority of the triggers are aligned correctly in the nominal bunch
crossing. Although shown in the plot, the input from the Resistive Plate
Chambers (RPC) barrel muon detectors, which see very little beam-halo activity
in single-beam operation, had not been timed-in.
Figure 26: Progress on timing-in ATLAS between 10 and 12 September 2008
As can be seen from the above, very significant progress was made on setting
up the timing of the experiments within the first few days of single-beam
operations at LHC. The experimental teams were eagerly awaiting further beam
time and the first collisions that would have allowed them to continue the
work. However, unfortunately, on 19 September there was a serious accident
with the LHC machine that required a prolonged shutdown for repairs and
improvements. Nevertheless, when the LHC restarts one will be able to build on
the work that was already done (complemented by many further studies that were
done using cosmic rays during the machine shutdown).
A huge amount of work has been done using the beam-related data that were
recorded in September 2008, as discussed in much more detail in the lectures
of Andreas Hoecker at this School [27]. A very important feature of these data
is that activity is seen in the same event in several detector subsystems
which allows one to check the relative timing and spatial alignment. Indeed
the fact that the same event is seen in the different subdetectors is
reassuring — some previous experiments had teething problems where the readout
of some of the subdetectors became desynchronized! A nice example of a beam-
halo event recorded in CMS is shown in fig:BeamHevent. Activity can be seen in
the Cathode-Strip Chamber (CSC) muon detectors at both ends of the experiment
and also in the hadronic calorimeter.
Figure 27: A beam-halo event in CMS
The detectors and triggers that were used in September 2008 were sensitive to
cosmic-ray muons as well as to beam-halo particles when a requirement of a
signal from the beam pick-ups was not made. The presence of beam-halo and
cosmic-ray signals in the data is illustrated in fig:BeamHcosmic which shows
the angular distribution of muons reconstructed in CMS. The shape of the
cosmic-ray distribution, which has a broad peak centred around 0.3–0.4
radians, is known from data collected without beam. The peak at low angles
matches well with the distribution for simulated beam-halo particles.
Figure 28: Angular distribution of muons in CMS recorded with and without
circulating beam. Also shown is the distribution for simulated beam-halo
events.
Before concluding, the author would like to show another example of a study
with single-beam data. Using a timing set-up in the end-cap muon trigger that
would be appropriate for colliding-beam operations, in which the muons emerge
from the centre of the apparatus, the distribution shown in the right-hand
part of fig:TOF_TGC was obtained. The two peaks separated by four bunch
crossings, i.e., 4 $\times$ 25 ns, correspond to triggers seen in the two ends
of the detector system. This is consistent within the resolution with the time
of flight of the beam-halo particles that may trigger the experiment on the
upstream or downstream sides of the detector. As indicated in the left-hand
part of the figure, this is reminiscent of the very simple example that was
introduced early on in the lectures, see 1.
Figure 29: Time of flight of beam-halo muons in ATLAS (one BC is 25 ns)
## 0.14 Concluding remarks
It is hoped that these lectures have succeeded in giving some insight into the
challenges of building T/DAQ systems for HEP experiments. These include
challenges connected with the physics (inventing algorithms that are fast,
efficient for the physics of interest, and that give a large reduction in
rate), and challenges in electronics and computing. It is also hoped that the
lectures have demonstrated how the subject has evolved to meet the increasing
demands, of LHC compared to LEP, by using new ideas based on new technologies.
## Acknowledgements
The author would like to thank the local organizing committee for their
wonderful hospitality during his stay in Colombia. In particular, he would
like to thank Marta Losada and Enrico Nardi who, together, created such a
wonderful atmosphere between all the participants, staff and students alike.
The author would like to thank the following people for their help and advice
in preparing the lectures and the present notes: Bob Blair, Helfried
Burckhart, Vincenzo Canale, Philippe Charpentier, Eric Eisenhandler, Markus
Elsing, Philippe Farthouat, John Harvey, Andreas Hoecker, Jim Linnerman,
Claudio Luci, Jordan Nash, Thilo Pauly, and Wesley Smith.
## References
* [1] L. Evans and P. Bryant (eds.), LHC machine, JINST 3 S08001 (2008).
* [2] R.W. Assmann, M. Lamont, and S. Myers, A brief history of the LEP collider, Nucl. Phys. B, Proc. Suppl. 109 (2002) 17–31, http://cdsweb.cern.ch/record/549223.
* [3] http://en.wikipedia.org/wiki/NIM
* [4] http://en.wikipedia.org/wiki/Computer_Automated_Measurement_and_Control
* [5] J. Schukraft, Heavy-ion physics at the LHC, in Proceedings of the 2003 CERN–CLAF School of High-Energy Physics, San Miguel Regla, Mexico, CERN-2006-001 (2006). ALICE Collaboration, Trigger, Data Acquisition, High Level Trigger, Control System Technical Design Report, CERN-LHCC-2003-062 (2003). The ALICE Collaboration, K. Aamodt et al., The ALICE experiment at the CERN LHC, JINST 3 S08002 (2008) and references therein.
* [6] J. Nash, these proceedings.
* [7] C. Amsler et al. (Particle Data Group), Phys. Lett. B 667 (2008) 1 also available online from http://pdg.lbl.gov/.
* [8] W. von Rueden, The ALEPH data acquisition system, _IEEE Trans. Nucl. Sci_. 36 (1989) 1444–1448. J. F. Renardy et al., Partitions and trigger supervision in ALEPH, _IEEE Trans. Nucl. Sci_. 36 (1989) 1464–1468. A. Belk et al., DAQ software architecture for ALEPH, a large HEP experiment, _IEEE Trans. Nucl. Sci_. 36 (1989) 1534–1539. P. Mato et al., The new slow control system for the ALEPH experiment at LEP, _Nucl. Instrum. Methods A_ 352 (1994) 247–249.
* [9] A. Augustinus et al., The DELPHI trigger system at LEP2 energies, _Nucl. Instrum. Methods A_ 515 (2003) 782–799. DELPHI Collaboration, Internal Notes DELPHI 1999-007 DAS 188 and DELPHI 2000-154 DAS 190 (unpublished).
* [10] B. Adeva et al., The construction of the L3 experiment, _Nucl. Instrum. Methods A_ 289 (1990) 35–102. T. Angelov et al., Performances of the central L3 data acquisition system, _Nucl. Instrum. Methods A_ 306 (1991) 536–539. C. Dionisi et al., The third level trigger system of the L3 experiment at LEP, _Nucl. Instrum. Methods A_ 336 (1993) 78–90 and references therein.
* [11] J.T.M. Baines et al., The data acquisition system of the OPAL detector at LEP, _Nucl. Instrum. Methods A_ 325 (1993) 271–293.
* [12] http://en.wikipedia.org/wiki/VMEbus
* [13] H1 Collaboration, The H1 detector, _Nucl. Instrum. Methods A_ 386 (1997) 310.
* [14] R. Carlin et al., The trigger of ZEUS, a flexible system for a high bunch crossing rate collider, _Nucl. Instrum. Methods A_ 379 (1996) 542–544. R. Carlin et al., Experience with the ZEUS trigger system, _Nucl. Phys. B_ , Proc. Suppl. 44 (1995) 430–434. W.H. Smith et al., The ZEUS trigger system, CERN-92-07, pp. 222–225.
* [15] CDF IIb Collaboration, The CDF IIb Detector: Technical Design Report, FERMILAB-TM-2198 (2003).
* [16] D0 Collaboration, RunIIb Upgrade Technical Design Report, FERMILAB-PUB-02-327-E (2002).
* [17] R. Arcidiacono et al., The trigger supervisor of the NA48 experiment at CERN SPS, _Nucl. Instrum. Methods A_ 443 (2000) 20–26 and references therein.
* [18] T. Fuljahn et al., Concept of the first level trigger for HERA-B, _IEEE Trans. Nucl. Sci_. 45 (1998) 1782–1786. M. Dam et al., Higher level trigger systems for the HERA-B experiment, _IEEE Trans. Nucl. Sci_. 45 (1998) 1787–1792.
* [19] ATLAS Collaboration, First-Level Trigger Technical Design Report, CERN-LHCC-98-14 (1998). ATLAS Collaboration, High-Level Triggers, Data Acquisition and Controls Technical Design Report, CERN-LHCC-2003-022 (2003).
* [20] The ATLAS Collaboration, G. Aad et al., The ATLAS experiment at the CERN Large Hadron Collider, JINST 3 S08003 (2008) and references therein.
* [21] CMS Collaboration, The Level-1 Trigger Technical Design Report, CERN-LHCC-2000-038 (2000). CMS Collaboration, Data Acquisition and High-Level Trigger Technical Design Report, CERN-LHCC-2002-26 (2002).
* [22] The CMS Collaboration, S. Chatrchyan et al., The CMS experiment at the CERN LHC, JINST 3 S08004 (2008) and references therein.
* [23] See, for example, summary talks in Proc. Computing in High Energy and Nuclear Physics, CHEP 2003, http://www.slac.stanford.edu/econf/C0303241/proceedings.html
* [24] LHCb Collaboration, Online System Technical Design Report, CERN-LHCC-2001-040 (2001). LHCb Collaboration, Trigger System Technical Design Report, CERN-LHCC-2003-031 (2003). The LHCb Collaboration, A. Augusto Alves Jr et al., The LHCb detector at the LHC, JINST 3 S08005 (2008) and references therein.
* [25] R. Achenbach et al., The ATLAS level-1 calorimeter trigger, JINST 3 P03001 (2008).
* [26] G. Navara et al., Electron trigger performance of the ATLAS detector, presented at _Signaling the Arrival of the LHC Era_ , Trieste, Italy, 8–13 December 2008.
* [27] A. Hoecker, these proceedings.
|
arxiv-papers
| 2010-10-14T14:47:47 |
2024-09-04T02:49:13.894195
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "N. Ellis (CERN)",
"submitter": "Nicolas Ellis",
"url": "https://arxiv.org/abs/1010.2942"
}
|
1010.2989
|
$Id:espcrc1.tex,v1.22004/02/2411:22:11speppingExp$ Interval total colorings of
graphsP.A. Petrosyan, A.Yu. Torosyan, N.A. Khachatryan
# Interval total colorings of graphs
P.A. Petrosyan, A.Yu. Torosyan[MCSD], N.A. Khachatryan[MCSD] email:
pet_petros@ipia.sci.amemail: arman.yu.torosyan@gmail.comemail:
xachnerses@gmail.com Department of Informatics and Applied Mathematics,
Yerevan State University, 0025, Armenia Institute for Informatics and
Automation Problems,
National Academy of Sciences, 0014, Armenia
###### Abstract
A total coloring of a graph $G$ is a coloring of its vertices and edges such
that no adjacent vertices, edges, and no incident vertices and edges obtain
the same color. An _interval total $t$-coloring_ of a graph $G$ is a total
coloring of $G$ with colors $1,2,\ldots,t$ such that at least one vertex or
edge of $G$ is colored by $i$, $i=1,2,\ldots,t$, and the edges incident to
each vertex $v$ together with $v$ are colored by $d_{G}(v)+1$ consecutive
colors, where $d_{G}(v)$ is the degree of the vertex $v$ in $G$. In this paper
we investigate some properties of interval total colorings. We also determine
exact values of the least and the greatest possible number of colors in such
colorings for some classes of graphs.
Keywords: total coloring, interval coloring, connected graph, regular graph,
bipartite graph
## 1 Introduction
A total coloring of a graph $G$ is a coloring of its vertices and edges such
that no adjacent vertices, edges, and no incident vertices and edges obtain
the same color. The concept of total coloring was introduced by V. Vizing [22]
and independently by M. Behzad [4]. The total chromatic number
$\chi^{\prime\prime}\left(G\right)$ is the smallest number of colors needed
for total coloring of $G$. In 1965 V. Vizing and M. Behzad conjectured that
$\chi^{\prime\prime}\left(G\right)\leq\Delta(G)+2$ for every graph $G$ [4,
22], where $\Delta(G)$ is the maximum degree of a vertex in $G$. This
conjecture became known as Total Coloring Conjecture [10]. It is known that
Total Coloring Conjecture holds for cycles, for complete graphs [5], for
bipartite graphs, for complete multipartite graphs [25], for graphs with a
small maximum degree [11, 12, 18, 21], for graphs with minimum degree at least
$\frac{3}{4}|V(G)|$ [9], and for planar graphs $G$ with $\Delta(G)\neq 6$ [6,
10, 20]. M. Rosenfeld [18] and N. Vijayaditya [21] independently proved that
the total chromatic number of graphs $G$ with $\Delta(G)=3$ is at most $5$. A.
Kostochka in [11] proved that the total chromatic number of graphs with
$\Delta(G)=4$ is at most $6$. Later, also he in [12] proved that the total
chromatic number of graphs with $\Delta(G)=5$ is at most $7$. The general
upper bound for the total chromatic number was obtained by M. Molloy and B.
Reed [15], who proved that
$\chi^{\prime\prime}\left(G\right)\leq\Delta(G)+10^{26}$ for every graph $G$.
The exact value of the total chromatic number is known only for paths, cycles,
complete and complete bipartite graphs [5], $n$-dimensional cubes, complete
multipartite graphs of odd order [8], outerplanar graphs [26] and planar
graphs $G$ with $\Delta(G)\geq 9$ [7, 10, 13, 23].
The key concept discussed in a present paper is the following. Given a graph
$G$, we say that $G$ is interval total colorable if there is $t\geq 1$ for
which $G$ has a total coloring with colors $1,2,\ldots,t$ such that at least
one vertex or edge of $G$ is colored by $i$, $1,2,\ldots,t$, and the edges
incident to each vertex $v$ together with $v$ are colored by $d_{G}(v)+1$
consecutive colors, where $d_{G}(v)$ is the degree of the vertex $v$ in $G$.
The concept of interval total coloring [16, 17] is a new one in graph
coloring, synthesizing interval colorings [1, 2] and total colorings. The
introduced concept is valuable as it connects to the problems of constructing
a timetable without a “gap”and it extends to total colorings of graphs one of
the most important notions of classical mathematics - the one of continuity.
In this paper we investigate some properties of interval total colorings of
graphs. Also, we show that simple cycles, complete graphs, wheels, trees,
regular bipartite graphs and complete bipartite graphs have interval total
colorings. Moreover, we obtain some bounds for the least and the greatest
possible number of colors in interval total colorings of these graphs.
## 2 Definitions and preliminary results
All graphs considered in this work are finite, undirected, and have no loops
or multiple edges. Let $V(G)$ and $E(G)$ denote the sets of vertices and edges
of $G$, respectively. An $(a,b)$-biregular bipartite graph $G$ is a bipartite
graph $G$ with the vertices in one part having degree $a$ and the vertices in
the other part having degree $b$. The degree of a vertex $v\in V(G)$ is
denoted by $d_{G}(v)$, the maximum degree of vertices in $G$ by $\Delta(G)$,
the diameter of $G$ by $diam(G)$, the chromatic number of $G$ by $\chi(G)$ and
the edge-chromatic number of $G$ by $\chi^{\prime}(G)$. A vertex $u$ of a
graph $G$ is universal if $d_{G}(u)=|V(G)|-1$. A _proper edge-coloring_ of a
graph $G$ is a coloring of the edges of $G$ such that no two adjacent edges
receive the same color. For a total coloring $\alpha$ of a graph $G$ and for
any $v\in V(G)$, define the set $S\left[v,\alpha\right]$ as follows:
$S\left[v,\alpha\right]=\left\\{\alpha(v)\right\\}\cup\left\\{\alpha(e)\left|\text{
}e\text{ is incident to }v\right.\right\\}$
Let $\left\lfloor a\right\rfloor$ ($\left\lceil a\right\rceil$) denote the
greatest (the least) integer $\leq a$ ($\geq a$). For two integers $a\leq b$,
the set $\left\\{a,a+1,\ldots,b\right\\}$ is denoted by $\left[a,b\right]$.
An _interval $t$-coloring_ of a graph $G$ is a proper edge-coloring of $G$
with colors $1,2,\ldots,t$ such that at least one edge of $G$ is colored by
$i$, $i=1,2,\ldots,t$, and the edges incident to each vertex $v$ are colored
by $d_{G}(v)$ consecutive colors. A graph $G$ is interval colorable if there
is $t\geq 1$ for which $G$ has an interval $t$-coloring. The set of all
interval colorable graphs is denoted by $\mathfrak{N}$. For a graph
$G\in\mathfrak{N}$, the greatest value of $t$ for which $G$ has an interval
$t$-coloring is denoted by $W\left(G\right)$.
An _interval total $t$-coloring_ of a graph $G$ is a total coloring of $G$
with colors $1,2,\ldots,t$ such that at least one vertex or edge of $G$ is
colored by $i$, $i=1,2,\ldots,t$, and the edges incident to each vertex $v$
together with $v$ are colored by $d_{G}(v)+1$ consecutive colors.
For $t\geq 1$, let $\mathfrak{T}_{t}$ denote the set of graphs which have an
interval total $t$-coloring, and assume: $\mathfrak{T}=\underset{t\geq
1}{\bigcup}\mathfrak{T}_{t}$. For a graph $G\in\mathfrak{T}$, the least and
the greatest values of $t$ for which $G\in\mathfrak{T}_{t}$ are denoted by
$w_{\tau}\left(G\right)$ and $W_{\tau}\left(G\right)$, respectively. Clearly,
$\chi^{\prime\prime}\left(G\right)\leq w_{\tau}\left(G\right)\leq
W_{\tau}\left(G\right)\leq|V(G)|+|E(G)|$ for every graph $G\in\mathfrak{T}$.
Terms and concepts that we do not define can be found in [24, 25].
We will use the following two results.
###### Theorem 1
[1, 2]. If $G$ is a connected triangle-free graph and $G\in\mathfrak{N}$, then
$W(G)\leq|V(G)|-1$.
###### Theorem 2
[3]. If $G$ is a connected $(a,b)$-biregular bipartite graph with $|V(G)|\geq
2(a+b)$ and $G\in\mathfrak{N}$, then
$W(G)\leq|V(G)|-3$.
## 3 Some properties of interval total colorings of graphs
First we prove a simple property of interval total colorings that for any
interval total coloring of a graph $G$ there is an inverse interval total
coloring of the same graph.
###### Proposition 3
If $\alpha$ is an interval total $t$-coloring of a graph $G$, then a total
coloring $\beta$, where
1) $\beta(v)=t+1-\alpha(v)$ for each $v\in V(G)$,
2) $\beta(e)=t+1-\alpha(e)$ for each $e\in E(G)$,
is also an interval total $t$-coloring of a graph $G$.
* Proof.
Clearly, a total coloring $\beta$ contains at least one vertex or edge with
color $i$, $i=1,2,\ldots,t$. Since $S[v,\alpha]$ is an interval for each $v\in
V(G)$, hence $S[v,\alpha]=[a,b]$. By the definition of the coloring $\beta$ it
follows that $S[v,\beta]=[t+1-b,t+1-a]$ for each $v\in V(G)$. $~{}\square$
Next we prove the proposition which implies that in definition of interval
total $t$-coloring, the requirement that every color $i$, $i=1,2,\ldots,t$,
appear in an interval total $t$-coloring isn’t necessary in the case of
connected graphs.
###### Proposition 4
Let $\alpha$ be a total coloring of the connected graph $G$ with colors
$1,2,\ldots,t$ such that the edges incident to each vertex $v\in V(G)$
together with $v$ are colored by $d_{G}(v)+1$ consecutive colors, and
$\min_{v\in V(G),e\in E(G)}\\{\alpha(v),\alpha(e)\\}=1$, $\max_{v\in V(G),e\in
E(G)}\\{\alpha(v),\alpha(e)\\}=t$. Then $\alpha$ is an interval total
$t$-coloring of $G$.
* Proof.
For the proof of the proposition it suffices to show that if $t\geq 3$, then
for color $s$, $1<s<t$, there exists at least one vertex or edge of $G$ which
is colored by $s$. We consider four possible cases.
Case 1: there are vertices $v,v^{\prime}\in V(G)$ such that $\alpha(v)=1$,
$\alpha(v^{\prime})=W_{\tau}(G)$.
Since $G$ is connected, there exists a simple path $P_{1}$ joining $v$ with
$v^{\prime}$, where
$P_{1}=(v_{0},e_{1},v_{1},\ldots,v_{i-1},e_{i},v_{i},\ldots,v_{k-1},e_{k},v_{k})$,
$v_{0}=v$, $v_{k}=v^{\prime}$.
If $\alpha(v_{i})\neq s$, $i=1,2,\ldots,k-1$, and $\alpha(e_{j})\neq s$,
$j=1,2,\ldots,k$, then there exists an index $i_{0}$, $1\leq i_{0}<k$, such
that $\alpha(e_{i_{0}})<s$ and $\alpha(e_{i_{0}+1})>s$. Hence, there is an
edge of $G$ colored by $s$ which is incident to $v_{i_{0}}$. This implies that
for any $s$, $1<s<t$, there is a vertex or an edge with color $s$.
Case 2: there is a vertex $v$ and there is an edge $e^{\prime}$ such that
$\alpha(v)=1$, $\alpha(e^{\prime})=W_{\tau}(G)$.
Let $e^{\prime}=v^{\prime}w$ and $P_{2}$ be a simple path joining $v$ with
$v^{\prime}$, where
$P_{2}=(v_{0},e_{1},v_{1},\ldots,v_{i-1},e_{i},v_{i},\ldots,v_{k-1},e_{k},v_{k})$,
$v_{0}=v$, $v_{k}=v^{\prime}$.
If $\alpha(v_{i})\neq s$, $i=1,2,\ldots,k$, and $\alpha(e_{j})\neq s$,
$j=1,2,\ldots,k$, then there exists an index $i_{1}$, $1\leq i_{1}<k$, such
that $\alpha(e_{i_{1}})<s$ and $\alpha(e_{i_{1}+1})>s$. Hence, there is an
edge of $G$ colored by $s$ which is incident to $v_{i_{1}}$. This implies that
for any $s$, $1<s<t$, there is a vertex or an edge with color $s$.
Case 3: there is an edge $e$ and there is a vertex $v^{\prime}$ such that
$\alpha(e)=1$, $\alpha(v^{\prime})=W_{\tau}(G)$.
Let $e=uv$ and $P_{3}$ be a simple path joining $v$ with $v^{\prime}$, where
$P_{3}=(v_{0},e_{1},v_{1},\ldots,v_{i-1},e_{i},v_{i},\ldots,v_{k-1},e_{k},v_{k})$,
$v_{0}=v$, $v_{k}=v^{\prime}$.
If $\alpha(v_{i})\neq s$, $i=0,1,\ldots,k-1$, and $\alpha(e_{j})\neq s$,
$j=1,2,\ldots,k$, then there exists an index $i_{2}$, $1\leq i_{2}<k$, such
that $\alpha(e_{i_{2}})<s$ and $\alpha(e_{i_{2}+1})>s$. Hence, there is an
edge of $G$ colored by $s$ which is incident to $v_{i_{2}}$. This implies that
for any $s$, $1<s<t$, there is a vertex or an edge with color $s$.
Case 4: there are edges $e,e^{\prime}\in E(G)$ such that $\alpha(e)=1$,
$\alpha(e^{\prime})=W_{\tau}(G)$.
Let $e=uv$, $e=v^{\prime}w$. Without loss of generality we may assume that a
simple path $P_{4}$ joining $e$ with $e^{\prime}$ joins $v$ with $v^{\prime}$,
where
$P_{4}=(v_{0},e_{1},v_{1},\ldots,v_{i-1},e_{i},v_{i},\ldots,v_{k-1},e_{k},v_{k})$,
$v_{0}=v$, $v_{k}=v^{\prime}$.
If $\alpha(v_{i})\neq s$, $i=0,1,\ldots,k$, and $\alpha(e_{j})\neq s$,
$j=1,2,\ldots,k$, then there exists an index $i_{3}$, $1\leq i_{3}<k$, such
that $\alpha(e_{i_{3}})<s$ and $\alpha(e_{i_{3}+1})>s$. Hence, there is an
edge of $G$ colored by $s$ which is incident to $v_{i_{3}}$. This implies that
for any $s$, $1<s<t$, there is a vertex or an edge with color $s$.
$~{}\square$
Now we show that there is an intimate connection between interval total
colorings of graphs and interval edge-colorings of certain bipartite graphs.
Let $G$ be a simple graph with $V(G)=\\{v_{1},v_{2},\ldots,v_{n}\\}$. Define
an auxiliary graph $H$ as follows:
$V(H)=U\cup W$, where
$U=\\{u_{1},u_{2},\ldots,u_{n}\\}$, $W=\\{w_{1},w_{2},\ldots,w_{n}\\}$ and
$E(H)=\left\\{u_{i}w_{j},u_{j}w_{i}|~{}v_{i}v_{j}\in E(G),1\leq i\leq n,1\leq
j\leq n\right\\}\cup\\{u_{i}w_{i}|~{}1\leq i\leq n\\}$.
Clearly, $H$ is a bipartite graph with $|V(H)|=2|V(G)|$.
###### Theorem 5
If $\alpha$ is an interval total $t$-coloring of the graph $G$, then there is
an interval $t$-coloring $\beta$ of the bipartite graph $H$.
* Proof.
For the proof, we define an edge-coloring $\beta$ of the graph $H$ as follows:
(1)
$\beta(u_{i}w_{j})=\beta(u_{j}w_{i})=\alpha(v_{i}v_{j})$ for every edge
$v_{i}v_{j}\in E(G)$,
(2)
$\beta(u_{i}w_{i})=\alpha(v_{i})$ for $i=1,2,\ldots,n$.
It is easy to see that $\beta$ is an interval $t$-coloring of the graph $H$.
$~{}\square$
This theorem shows that any interval total $t$-coloring of a graph $G$ can be
transform into an interval $t$-coloring of the bipartite graph $H$.
###### Corollary 6
If $G$ is a connected graph and $G\in\mathfrak{T}$, then
$W_{\tau}(G)\leq 2|V(G)|-1$.
* Proof.
Let $\alpha$ be an interval total $W_{\tau}(G)$-coloring of the graph $G$. By
Theorem 5, $\beta$ is an interval $W_{\tau}(G)$-coloring of the graph $H$.
Since $H$ is a connected bipartite graph with $|V(H)|=2|V(G)|$ and
$H\in\mathfrak{N}$, by Theorem 1, we have
$W_{\tau}(G)\leq|V(H)|-1=2|V(G)|-1$, thus
$W_{\tau}(G)\leq 2|V(G)|-1$.
$~{}\square$
###### Remark 7
Note that the upper bound in Corollary 6 is sharp for simple paths $P_{n}$,
since $W_{\tau}(P_{n})=2n-1$ for any $n\in\mathbf{N}$.
###### Corollary 8
If $G$ is a connected $r$-regular graph with $|V(G)|\geq 2r+2$ and
$G\in\mathfrak{T}$, then
$W_{\tau}(G)\leq 2|V(G)|-3$.
* Proof.
Let $\alpha$ be an interval total $W_{\tau}(G)$-coloring of the graph $G$. By
Theorem 5, $\beta$ is an interval $W_{\tau}(G)$-coloring of the graph $H$.
Since $H$ is a connected $(r+1)$-regular bipartite graph with $|V(H)|\geq
2(2r+2)$ and $H\in\mathfrak{N}$, by Theorem 2, we have
$W_{\tau}(G)\leq|V(H)|-3=2|V(G)|-3$, thus
$W_{\tau}(G)\leq 2|V(G)|-3$.
$~{}\square$
Next we derive some upper bounds for $W_{\tau}(G)$ depending on degrees and
diameter of a connected graph $G$.
###### Theorem 9
If $G$ is a connected graph and $G\in\mathfrak{T}$, then
$W_{\tau}(G)\leq 1+{\max\limits_{P\in\mathbf{P}}}{\sum\limits_{v\in V(P)}}\
d_{G}(v)$,
where $\mathbf{P}$ is the set of all shortest paths in the graph $G$.
* Proof.
Consider an interval total $W_{\tau}(G)$-coloring $\alpha$ of $G$. We
distinguish four possible cases.
Case 1: there are vertices $v,v^{\prime}\in V(G)$ such that $\alpha(v)=1$,
$\alpha(v^{\prime})=W_{\tau}(G)$.
Let $P_{1}$ be a shortest path joining $v$ with $v^{\prime}$, where
$P_{1}=\left(v_{1},e_{1},v_{2},e_{2},\ldots,v_{i},e_{i},v_{i+1},\ldots,v_{k},e_{k},v_{k+1}\right)$,
$v_{1}=v$, $v_{k+1}=v^{\prime}$.
Note that
$\alpha(e_{1})\leq 1+d_{G}(v_{1})$,
$\alpha(e_{2})\leq\alpha(e_{1})+d_{G}(v_{2})$,
$\cdots\cdots\cdots\cdots\cdots$
$\alpha(e_{i})\leq\alpha(e_{i-1})+d_{G}(v_{i})$,
$\cdots\cdots\cdots\cdots\cdots$
$\alpha(e_{k})\leq\alpha(e_{k-1})+d_{G}(v_{k})$,
$W_{\tau}(G)=\alpha(v^{\prime})=\alpha(v_{k+1})\leq\alpha(e_{k})+d_{G}(v_{k+1})$.
By summing these inequalities, we obtain
$W_{\tau}(G)\leq$ $1+{{\sum\limits_{i=1}^{k+1}d_{G}(v_{i})}}\leq
1+{\max\limits_{P\in\mathbf{P}}}{\sum\limits_{v\in V(P)}}\ d_{G}(v)$.
Case 2: there is a vertex $v$ and there is an edge $e^{\prime}$ such that
$\alpha(v)=1$, $\alpha(e^{\prime})=W_{\tau}(G)$.
Let $e^{\prime}=v^{\prime}w$ and $P_{2}$ be a shortest path joining $v$ with
$v^{\prime}$, where
$P_{2}=\left(v_{1},e_{1},v_{2},e_{2},\ldots,v_{i},e_{i},v_{i+1},\ldots,v_{k},e_{k},v_{k+1}\right)$,
$v_{1}=v$, $v_{k+1}=v^{\prime}$.
Note that
$\alpha(e_{1})\leq 1+d_{G}(v_{1})$,
$\alpha(e_{2})\leq\alpha(e_{1})+d_{G}(v_{2})$,
$\cdots\cdots\cdots\cdots\cdots$
$\alpha(e_{i})\leq\alpha(e_{i-1})+d_{G}(v_{i})$,
$\cdots\cdots\cdots\cdots\cdots$
$\alpha(e_{k})\leq\alpha(e_{k-1})+d_{G}(v_{k})$,
$W_{\tau}(G)=\alpha(e^{\prime})=\alpha(v_{k+1}w)\leq\alpha(e_{k})+d_{G}(v_{k+1})$.
By summing these inequalities, we obtain
$W_{\tau}(G)\leq$ $1+{{\sum\limits_{i=1}^{k+1}d_{G}(v_{i})}}\leq
1+{\max\limits_{P\in\mathbf{P}}}{\sum\limits_{v\in V(P)}}\ d_{G}(v)$.
Case 3: there is an edge $e$ and there is a vertex $v^{\prime}$ such that
$\alpha(e)=1$, $\alpha(v^{\prime})=W_{\tau}(G)$.
Let $e=uv$ and $P_{3}$ be a shortest path joining $v$ with $v^{\prime}$, where
$P_{3}=\left(v_{1},e_{1},v_{2},e_{2},\ldots,v_{i},e_{i},v_{i+1},\ldots,v_{k},e_{k},v_{k+1}\right)$,
$v_{1}=v$, $v_{k+1}=v^{\prime}$.
Note that
$\alpha(e_{1})\leq 1+d_{G}(v_{1})$,
$\alpha(e_{2})\leq\alpha(e_{1})+d_{G}(v_{2})$,
$\cdots\cdots\cdots\cdots\cdots$
$\alpha(e_{i})\leq\alpha(e_{i-1})+d_{G}(v_{i})$,
$\cdots\cdots\cdots\cdots\cdots$
$\alpha(e_{k})\leq\alpha(e_{k-1})+d_{G}(v_{k})$,
$W_{\tau}(G)=\alpha(v^{\prime})=\alpha(v_{k+1})\leq\alpha(e_{k})+d_{G}(v_{k+1})$.
By summing these inequalities, we obtain
$W_{\tau}(G)\leq$ $1+{{\sum\limits_{i=1}^{k+1}d_{G}(v_{i})}}\leq
1+{\max\limits_{P\in\mathbf{P}}}{\sum\limits_{v\in V(P)}}\ d_{G}(v)$.
Case 4: there are edges $e,e^{\prime}\in E(G)$ such that $\alpha(e)=1$,
$\alpha(e^{\prime})=W_{\tau}(G)$.
Let $e=uv$ and $e^{\prime}=v^{\prime}w$. Without loss of generality we may
assume that a shortest path $P_{4}$ joining $e$ and $e^{\prime}$ joins $v$ and
$v^{\prime}$, where
$P_{4}=\left(v_{1},e_{1},v_{2},e_{2},\ldots,v_{i},e_{i},v_{i+1},\ldots,v_{k},e_{k},v_{k+1}\right)$,
$v_{1}=v$, $v_{k+1}=v^{\prime}$.
Note that
$\alpha(e_{1})\leq 1+d_{G}(v_{1})$,
$\alpha(e_{2})\leq\alpha(e_{1})+d_{G}(v_{2})$,
$\cdots\cdots\cdots\cdots\cdots$
$\alpha(e_{i})\leq\alpha(e_{i-1})+d_{G}(v_{i})$,
$\cdots\cdots\cdots\cdots\cdots$
$\alpha(e_{k})\leq\alpha(e_{k-1})+d_{G}(v_{k})$,
$W_{\tau}(G)=\alpha(e^{\prime})=\alpha(v^{\prime}w)\leq\alpha(e_{k})+d_{G}(v_{k+1})$.
By summing these inequalities, we obtain
$W_{\tau}(G)\leq$ $1+{{\sum\limits_{i=1}^{k+1}d_{G}(v_{i})}}\leq
1+{\max\limits_{P\in\mathbf{P}}}{\sum\limits_{v\in V(P)}}\ d_{G}(v)$.
$~{}\square$
###### Corollary 10
If $G$ is a connected graph and $G\in\mathfrak{T}$, then $W_{\tau}(G)\leq
1+(diam(G)+1)\Delta(G)$.
Now we give an upper bound on $W_{\tau}(G)$ for graphs with a unique universal
vertex.
###### Theorem 11
If $G$ is a graph with a unique universal vertex $u$ and $G\in\mathfrak{T}$,
then $W_{\tau}(G)\leq|V(G)|+2k(G)$, where $k(G)={\max}_{v\in V(G)(v\neq
u)}d_{G}(v)$.
* Proof.
Let $\alpha$ be an interval total $W_{\tau}(G)$-coloring of the graph $G$.
Consider the vertex $u$. We show that $1\leq\min S[u,\alpha]\leq k(G)+1$.
Suppose, to the contrary, that $\min S[u,\alpha]\geq k(G)+2$. Since
$d_{G}(v)\leq k(G)$ for any $v\in V(G)(v\neq u)$, then $\min S[v,\alpha]\geq
2$ for any $v\in V(G)(v\neq u)$, which is a contradiction.
Now, we have
$1\leq\min S[u,\alpha]\leq k(G)+1$,
hence,
$|V(G)|\leq\max S[u,\alpha]\leq|V(G)|+k(G)$.
This implies that $\max S[v,\alpha]\leq|V(G)|+2k(G)$ for any $v\in V(G)(v\neq
u)$. $~{}\square$
In the next theorem we prove that regular bipartite graphs, trees and complete
bipartite graphs are interval total colorable.
###### Theorem 12
The set $\mathfrak{T}$ contains all regular bipartite graphs, trees and
complete bipartite graphs.
* Proof.
First we prove that if $G$ is an $r$-regular bipartite graph with bipartition
$(U,V)$, then $G$ has an interval total $(r+2)$-coloring.
Since $G$ is an $r$-regular bipartite graph, we have
$\chi^{\prime}\left(G\right)=\Delta(G)=r$. Let $\alpha$ be a proper edge-
coloring of $G$ with colors $2,3,\ldots,r+1$. Clearly, $S(w,\alpha)=[2,r+1]$
for each $w\in V(G)$.
Define a total coloring $\beta$ of the graph $G$ as follows:
1\. for any $u\in U$, let $\beta(u)=1$;
2\. for any $e\in E(G)$, let $\beta(e)=\alpha(e)$;
3\. for any $v\in V$, let $\beta(v)=r+2$.
It is easy to see that $\beta$ is an interval total $(r+2)$-coloring of $G$.
Next we consider trees. Clearly, $K_{1}$ is a tree and has an interval total
$1$-coloring. Assume that $T$ is a tree and $T\neq K_{1}$. Now we prove that
$T$ has an interval total $(\Delta(T)+2)$-coloring.
We use induction on $|E(T)|$. Clearly, the statement is true for the case
$|E(T)|=1$. Suppose that $|E(T)|=k>1$ and the statement is true for all trees
$T^{\prime}$ with $|E(T^{\prime})|<k$.
Suppose $e=uv\in E(T)$ and $d_{T}(u)=1$. Let $T^{\prime}=T-u$. Since
$|E(T)|>1$, we have $d_{T}(v)\geq 2$. Clearly,
$d_{T^{\prime}}(v)=d_{T}(v)-1,\Delta(T^{\prime})\leq\Delta(T)$ and
$|E(T^{\prime})|=|E(T)|-1<k$. Let $\alpha$ be an interval total
$(\Delta(T^{\prime})+2)$-coloring of the tree $T^{\prime}$ (by induction
hypothesis). Consider the vertex $v$. Let
$S[v,\alpha]=\\{s(1),s(2),\ldots,s(d_{T^{\prime}}(v)+1)\\}$,
where $1\leq s(1)<s(2)<\ldots<s(d_{T^{\prime}}(v)+1)\leq\Delta(T)+2$. We
consider three cases.
Case 1: $s(1)=1$.
Clearly, $s(d_{T^{\prime}}(v)+1)=d_{T^{\prime}}(v)+1=d_{T}(v)$. In this case
we color the edge $e$ with color $d_{T}(v)+1$ and the vertex $u$ with color
$d_{T}(v)+2$. It is easy to see that the obtained coloring is an interval
total $(\Delta(T)+2)$-coloring of the tree $T$.
Case 2: $s(1)=2$.
Subcase 2.1: $\alpha(v)=2$.
Clearly, $s(d_{T^{\prime}}(v)+1)=d_{T}(v)+1$. In this case we color the edge
$e$ with color $d_{T}(v)+2$ and the vertex $u$ with color $d_{T}(v)+1$. It is
easy to see that the obtained coloring is an interval total
$(\Delta(T)+2)$-coloring of the tree $T$.
Subcase 2.2: $\alpha(v)\neq 2$ and $\Delta(T^{\prime})=\Delta(T)$.
We color the edge $e$ with color $1$ and the vertex $u$ with color $2$. It is
easy to see that obtained coloring is an interval total
$(\Delta(T)+2)$-coloring of the tree $T$.
Subcase 2.3: $\alpha(v)\neq 2$ and $\Delta(T^{\prime})<\Delta(T)$.
We define a total coloring $\beta$ of the tree $T^{\prime}$ in the following
way:
1\. $\forall w\in V(T^{\prime})$ $\beta(w)=\alpha(w)+1$;
2\. $\forall e^{\prime}\in E(T^{\prime})$
$\beta(e^{\prime})=\alpha(e^{\prime})+1$.
Now we color the edge $e$ with color $2$ and the vertex $u$ with color $1$. It
is not difficult to see that the obtained coloring is an interval total
$(\Delta(T)+2)$-coloring of the tree $T$.
Case 3: $s(1)\geq 3$.
We color the edge $e$ with color $s(1)-1$ and the vertex $u$ with color
$s(1)-2$. It is easy to see that the obtained coloring is an interval total
$(\Delta(T)+2)$-coloring of the tree $T$.
Finally, we prove that if $K_{m,n}$ is a complete bipartite graph, then it has
an interval total $(m+n+1)$-coloring.
Let $V(K_{m,n})=\\{u_{1},u_{2},\ldots,u_{m},v_{1},v_{2},\ldots,v_{n}\\}$ and
$E(K_{m,n})=\\{u_{i}v_{j}|~{}1\leq i\leq m,1\leq j\leq n\\}$.
Define a total coloring $\gamma$ of the graph $K_{m,n}$ as follows:
1\. for $i=1,2,\ldots,m$, let $\gamma(u_{i})=i$;
2\. for $j=1,2,\ldots,n$, let $\gamma(v_{j})=m+1+j$;
3\. for $i=1,2,\ldots,m$ and $j=1,2,\ldots,n$, let $\gamma(u_{i}v_{j})=i+j$.
It is easy to see that $\gamma$ is an interval total $(m+n+1)$-coloring of
$K_{m,n}$. $~{}\square$
###### Corollary 13
If $G$ is an $r$-regular bipartite graph, then $w_{\tau}(G)\leq r+2$.
###### Corollary 14
If $T$ is a tree, then $w_{\tau}(T)\leq\Delta(T)+2$.
###### Corollary 15
$W_{\tau}(K_{m,n})\geq m+n+1$ for any $m,n\in\mathbf{N}$.
From Corollary 13, we have that $w_{\tau}(G)\leq r+2$ for any $r$-regular
bipartite graph $G$. On the other hand, clearly, $w_{\tau}(G)\geq r+1$. In
[14, 19] it was proved that the problem of determining whether
$\chi^{\prime\prime}\left(G\right)=r+1$ is $NP$-complete even for cubic
bipartite graphs. Thus, we can conclude that the verification whether
$w_{\tau}(G)=r+1$ for any $r$-regular ($r\geq 3$) bipartite graph $G$ is also
$NP$-complete.
## 4 Exact values of $w_{\tau}(G)$ and $W_{\tau}(G)$
In this section we determine the exact values of $w_{\tau}(G)$ and
$W_{\tau}(G)$ for simple cycles, complete graphs and wheels.
In [25] it was proved the following result.
###### Theorem 16
For the simple cycle $C_{n}$,
$\chi^{\prime\prime}(C_{n})=\left\\{\begin{tabular}[]{ll}$3$,&if $n=3k$,\\\
$4$,&if $n\neq 3k$.\\\ \end{tabular}\right.$
###### Theorem 17
For any $n\geq 3$, we have
(1)
$C_{n}\in\mathfrak{T}$,
(2)
$w_{\tau}(C_{n})=\left\\{\begin{tabular}[]{ll}$3$,&if $n=3k$,\\\ $4$,&if
$n\neq 3k$,\\\ \end{tabular}\right.$
(3)
$W_{\tau}(C_{n})=n+2$.
* Proof.
First we prove that $C_{n}$ has either an interval total $3$-coloring or an
interval total $4$-coloring.
Let $V(C_{n})=\\{v_{1},v_{2}\ldots,v_{n}\\}$ and
$E(C_{n})=\\{v_{i}v_{i+1}|~{}1\leq i\leq n-1\\}\cup\\{v_{1}v_{n}\\}$. We
consider three cases.
Case 1: $n=3k$ ($k\in\mathbf{N}$).
Define a total coloring $\alpha$ of the graph $C_{n}$ as follows:
$\alpha\left(v_{i}\right)=\left\\{\begin{tabular}[]{ll}$2$,&if $i\equiv
0\pmod{3}$,\\\ $1$,&if $i\equiv 1\pmod{3}$,\\\ $3$,&if $i\equiv 2\pmod{3}$,\\\
\end{tabular}\right.$
for $i=1,2,\ldots,n$,
$\alpha\left(v_{j}v_{j+1}\right)=\left\\{\begin{tabular}[]{ll}$3$,&if $j\equiv
0\pmod{3}$,\\\ $2$,&if $j\equiv 1\pmod{3}$,\\\ $1$,&if $j\equiv 2\pmod{3}$,\\\
\end{tabular}\right.$
for $j=1,2,\ldots,n-1$, and $\alpha(v_{1}v_{n})=3$.
Case 2: $n\neq 3k$ ($k\in\mathbf{N}$) and $n$ is even.
Define a total coloring $\alpha$ of the graph $C_{n}$ as follows:
$\alpha\left(v_{i}\right)=\left\\{\begin{tabular}[]{ll}$4$,&if $i\equiv
0\pmod{2}$,\\\ $1$,&if $i\equiv 1\pmod{2}$,\\\ \end{tabular}\right.$
for $i=1,2,\ldots,n$,
$\alpha\left(v_{j}v_{j+1}\right)=\left\\{\begin{tabular}[]{ll}$2$,&if $j\equiv
0\pmod{2}$,\\\ $3$,&if $j\equiv 1\pmod{2}$,\\\ \end{tabular}\right.$
for $j=1,2,\ldots,n-1$, and $\alpha(v_{1}v_{n})=2$.
Case 3: $n\neq 3k$ ($k\in\mathbf{N}$) and $n$ is odd.
Define a total coloring $\alpha$ of the graph $C_{n}$ as follows:
$\alpha\left(v_{i}\right)=\left\\{\begin{tabular}[]{ll}$4$,&if $i\equiv
0\pmod{2}$, $i\neq n-1$,\\\ $1$,&if $i\equiv 1\pmod{2}$, $i\neq n$,\\\ $2$,&if
$i=n-1$,\\\ $3$,&if $i=n$,\\\ \end{tabular}\right.$
for $i=1,2,\ldots,n$,
$\alpha\left(v_{j}v_{j+1}\right)=\left\\{\begin{tabular}[]{ll}$2$,&if $j\equiv
0\pmod{2}$, $j\neq n-1$,\\\ $3$,&if $j\equiv 1\pmod{2}$,\\\ $4$,&if
$j=n-1$,\\\ \end{tabular}\right.$
for $j=1,2,\ldots,n-1$, and $\alpha(v_{1}v_{n})=2$.
It is easy to check that $\alpha$ is an interval total $3$-coloring of the
graph $C_{n}$, when $n=3k$, and an interval total $4$-coloring of the graph
$C_{n}$, when $n\neq 3k$. Hence, for any $n\geq 3$, $C_{n}\in\mathfrak{T}$ and
$w_{\tau}(C_{n})\leq 3$ if $n=3k$ and $w_{\tau}(C_{n})\leq 4$ if $n\neq 3k$.
On the other hand, by Theorem 16, and taking into account that
$w_{\tau}(C_{n})\geq\chi^{\prime\prime}(C_{n})$, we have $w_{\tau}(C_{n})\geq
3$ if $n=3k$ and $w_{\tau}(C_{n})\geq 4$ if $n\neq 3k$. Thus, (1) and (2)
hold.
Let us prove (3).
Now we show that $W_{\tau}(C_{n})\geq n+2$ for any $n\geq 3$. For that, we
consider two cases.
Case 1: $n$ is even.
Define a total coloring $\beta$ of the graph $C_{n}$ as follows:
1\. for $i=1,2,\ldots,\frac{n}{2}$, let
$\beta(v_{i})=2i-1$, $\beta(v_{i}v_{i+1})=2i$
2\. for $j=\frac{n}{2}+1,\ldots,n$, let
$\beta(v_{j})=2(n-j)+4$,
3\. for $k=\frac{n}{2}+1,\ldots,n-1$, let
$\beta(v_{k}v_{k+1})=2(n-k)+3$,
and $\beta(v_{1}v_{n})=3$.
Case 2: $n$ is odd.
Define a total coloring $\beta$ of the graph $C_{n}$ as follows:
1\. for $i=1,2,\ldots,\lceil\frac{n}{2}\rceil+1$, let
$\beta(v_{i})=2i-1,$
2\. for $j=\lceil\frac{n}{2}\rceil+2,\ldots,n$, let
$\beta(v_{j})=2(n-j)+4$,
3\. for $k=1,2,\ldots,\lceil\frac{n}{2}\rceil$, let
$\beta(v_{k}v_{k+1})=2k$,
4\. for $l=\lceil\frac{n}{2}\rceil+1,\ldots,n-1$, let
$\beta(v_{l}v_{l+1})=2(n-l)+3$,
and $\beta(v_{1}v_{n})=3$.
It is not difficult to see that $\beta$ is an interval total $(n+2)$-coloring
of the graph $C_{n}$. Thus, $W_{\tau}(C_{n})\geq n+2$ for any $n\geq 3$. On
the other hand, using Corollary 10, and taking into account that
$diam(C_{n})=\lfloor\frac{n}{2}\rfloor$ and $\Delta(C_{n})=2$, it is easy to
show that $W_{\tau}(C_{n})\leq n+2$ for any $n\geq 3$. $~{}\square$
In [5] it was proved the following result.
###### Theorem 18
For the complete graph $K_{n}$,
$\chi^{\prime\prime}(K_{n})=\left\\{\begin{tabular}[]{ll}$n$,&if $n$ is
odd,\\\ $n+1$,&if $n$ is even.\\\ \end{tabular}\right.$
###### Theorem 19
For any $n\in\mathbf{N}$, we have
(1)
$K_{n}\in\mathfrak{T}$,
(2)
$w_{\tau}(K_{n})=\left\\{\begin{tabular}[]{ll}$n$,&if $n$ is odd,\\\
$\frac{3}{2}n$,&if $n$ is even,\\\ \end{tabular}\right.$
(3)
$W_{\tau}(K_{n})=2n-1$.
* Proof.
Let $V(K_{n})=\\{v_{1},v_{2},\ldots,v_{n}\\}$.
First we show that $K_{n}$ has an interval total $(2n-1)$-coloring for any
$n\in\mathbf{N}$. For that, we define a total coloring $\alpha$ of the graph
$K_{n}$ as follows:
1\. for $i=1,2,\ldots,n$, let $\alpha(v_{i})=2i-1$;
2\. for $i=1,2,\ldots,n$ and $j=1,2,\ldots,n$, where $i\neq j$, let
$\alpha(v_{i}v_{j})=i+j-1$.
It is easy to see that $\alpha$ is an interval total $(2n-1)$-coloring of the
graph $K_{n}$. This proves that $K_{n}\in\mathfrak{T}$ and
$W_{\tau}(K_{n})\geq 2n-1$ for any $n\in\mathbf{N}$. On the other hand, using
Corollary 10, and taking into account that $diam(K_{n})=1$ and
$\Delta(K_{n})=n-1$, it is simple to show that $W_{\tau}(K_{n})\leq 2n-1$ for
any $n\in\mathbf{N}$. Thus, (1) and (3) hold.
Let us prove (2). We consider two cases.
Case 1: $n$ is odd.
Since $K_{n}$ is a regular graph, by Theorem 18, we have
$w_{\tau}(K_{n})=\chi^{\prime\prime}(K_{n})=n$.
Case 2: $n$ is even.
Now we show that $w_{\tau}(K_{n})\leq\frac{3}{2}n$.
Define a total coloring $\beta$ of the graph $K_{n}$ as follows:
1\. for $i=1,2,\ldots,\frac{n}{2}$, let
$\beta(v_{i})=i$;
2\. for $j=\frac{n}{2}+1,\ldots,n$, let
$\beta(v_{j})=\frac{n}{2}+j$;
3\. for $i=1,2,\ldots,n$, $j=1,2,\ldots,n$, $i<j$, $i+j$ is odd, and
$i+j-1\leq n$, let
$\beta(v_{i}v_{j})=\frac{n}{2}+\frac{i+j-1}{2}$;
4\. for $i=1,2,\ldots,n$, $j=1,2,\ldots,n$, $i<j$, $i+j$ is odd, and
$i+j-1>n$, let
$\beta(v_{i}v_{j})=\frac{i+j-1}{2}$;
5\. for $i=1,2,\ldots,n$, $j=1,2,\ldots,n$, $i<j$, $i+j$ is even, and $i+j\leq
n$, let
$\beta(v_{i}v_{j})=\frac{i+j}{2}$;
6\. for $i=1,2,\ldots,n$, $j=1,2,\ldots,n$, $i<j$, $i+j$ is even, and $i+j>n$,
let
$\beta(v_{i}v_{j})=\frac{n}{2}+\frac{i+j}{2}$.
Let us show that $\beta$ is an interval total $\frac{3}{2}n$-coloring of the
graph $K_{n}$.
Let $v_{i}\in V(K_{n})$, where $1\leq i\leq n$.
If $i$ is even, by the definition of $\beta$, we have
$\displaystyle S\left[v_{i},\beta\right]$ $\displaystyle=$
$\displaystyle\left({\bigcup\limits_{1\leq
l\leq\frac{n+2-i}{2}}\left\\{\frac{n}{2}+\frac{i+(2l-1)-1}{2}\right\\}}\right)\cup\left({\bigcup\limits_{\frac{n+2-i}{2}<l\leq\frac{n}{2}}\left\\{\frac{i+(2l-1)-1}{2}\right\\}}\right)\cup$
$\displaystyle\left({\bigcup\limits_{1\leq
l\leq\frac{n-i}{2},l\neq\frac{i}{2}}\left\\{\frac{i+2l}{2}\right\\}}\right)\cup\left({\bigcup\limits_{\frac{n-i}{2}<l\leq\frac{n}{2},l\neq\frac{i}{2}}\left\\{\frac{n}{2}+\frac{i+2l}{2}\right\\}}\right)\cup\left\\{i+\frac{n}{2}sg\left(i-\frac{n}{2}\right)\right\\}$
$\displaystyle=$
$\displaystyle\left[\frac{n+i}{2},n\right]\cup\left[\frac{n}{2}+1,\frac{n+i}{2}-1\right]\cup\left(\left[\frac{i}{2}+1,\frac{n}{2}\right]\setminus\\{i\\}\right)\cup$
$\displaystyle\left(\left[n+1,\frac{i}{2}+n\right]\setminus\left\\{\frac{n}{2}+i\right\\}\right)\cup\left\\{i+\frac{n}{2}sg\left(i-\frac{n}{2}\right)\right\\}$
$\displaystyle=$ $\displaystyle\left[\frac{i}{2}+1,\frac{i}{2}+n\right],$
and if $i$ is odd, by the definition of $\beta$, we have
$\displaystyle S\left[v_{i},\beta\right]$ $\displaystyle=$
$\displaystyle\left({\bigcup\limits_{1\leq
l\leq\frac{n+1-i}{2}}\left\\{\frac{n}{2}+\frac{i+2l-1}{2}\right\\}}\right)\cup\left({\bigcup\limits_{\frac{n+1-i}{2}<l\leq\frac{n}{2}}\left\\{\frac{i+2l-1}{2}\right\\}}\right)\cup$
$\displaystyle\left({\bigcup\limits_{1\leq
l\leq\frac{n+1-i}{2},l\neq\frac{i+1}{2}}\left\\{\frac{i+2l-1}{2}\right\\}}\right)\cup\left({\bigcup\limits_{\frac{n+1-i}{2}<l\leq\frac{n}{2},l\neq\frac{i+1}{2}}\left\\{\frac{n}{2}+\frac{i+2l-1}{2}\right\\}}\right)\cup$
$\displaystyle\left\\{i+\frac{n}{2}sg\left(i-\frac{n}{2}\right)\right\\}$
$\displaystyle=$
$\displaystyle\left[\frac{n+i+1}{2},n\right]\cup\left[\frac{n}{2}+1,\frac{n+i-1}{2}\right]\cup\left(\left[\frac{i+1}{2},\frac{n}{2}\right]\setminus\\{i\\}\right)\cup$
$\displaystyle\left(\left[n+1,\frac{i-1}{2}+n\right]\setminus\left\\{\frac{n}{2}+i\right\\}\right)\cup\left\\{i+\frac{n}{2}sg\left(i-\frac{n}{2}\right)\right\\}$
$\displaystyle=$ $\displaystyle\left[\frac{i+1}{2},\frac{i-1}{2}+n\right].$
This shows that $\beta$ is an interval total $\frac{3}{2}n$-coloring of the
graph $K_{n}$.
Next we prove that $w_{\tau}(K_{n})\geq\frac{3}{2}n$.
Suppose, to the contrary, that $\gamma$ is an interval total
$w_{\tau}(K_{n})$-coloring of the graph $K_{n}$, where $n\leq
w_{\tau}(K_{n})\leq\frac{3}{2}n-1$.
Since $w_{\tau}(K_{n})\geq\chi^{\prime\prime}(K_{n})$, by Theorem 18, we have
$n+1\leq w_{\tau}(K_{n})\leq\frac{3}{2}n-1$.
Consider the vertices $v_{1},v_{2},\ldots,v_{n}$. It is clear that
$1\leq\min S[v_{i},\gamma]\leq w_{\tau}(K_{n})-n+1$ for $i=1,2,\ldots,n$.
Hence, $\\{w_{\tau}(K_{n})-n+1,\ldots,n\\}\subseteq S[v_{i},\gamma]$ for
$i=1,2,\ldots,n$. Let us show that none of the vertices
$v_{1},v_{2},\ldots,v_{n}$ is colored by $j$,
$j=w_{\tau}(K_{n})-n+1,\ldots,n$. Suppose that $\gamma(v_{i_{0}})=j_{0}$,
$j_{0}\in\\{w_{\tau}(K_{n})-n+1,\ldots,n\\}$. Clearly, $\gamma(v_{i})\neq
j_{0}$ for $i=1,2,\ldots,n$ and $i\neq i_{0}$. This implies that any vertex
$v_{i}$, except $v_{i_{0}}$, is incident to an edge with color $j_{0}$, which
is a contradiction. The contradiction shows that
$\gamma(v_{i})\notin\\{w_{\tau}(K_{n})-n+1,\ldots,n\\}$ for $i=1,2,\ldots,n$.
Hence,
$\gamma(v_{i})\in\\{1,2,\ldots,w_{\tau}(K_{n})-n\\}\cup\\{n+1,\ldots,w_{\tau}(K_{n})\\}$
for $i=1,2,\ldots,n$.
On the other hand, since $\chi(K_{n})=n$, we have
$|\\{1,2,\ldots,w_{\tau}(K_{n})-n\\}|+|\\{n+1,\ldots,w_{\tau}(K_{n})\\}|\geq
n$,
thus $w_{\tau}(K_{n})\geq\frac{3}{2}n$, which is a contradiction. $~{}\square$
###### Theorem 20
For any $n\in\mathbf{N}$,
(1)
if $2n-1\leq t\leq 4n-3$, then $K_{2n-1}\in\mathfrak{T}_{t}$,
(2)
if $3n\leq t\leq 4n-1$, then $K_{2n}\in\mathfrak{T}_{t}$.
* Proof.
First we prove (1). For that, we transform the interval total
$(4n-3)$-coloring $\alpha$ of the graph $K_{2n-1}$ constructed in the proof of
Theorem 19, into an interval total $t$-coloring $\beta$ of the same graph.
For every $v\in V(K_{2n-1})$, we set:
$\beta(v)=\left\\{\begin{tabular}[]{ll}$\alpha(v)$,&if $1\leq\alpha(v)\leq
t$,\\\ $\alpha(v)-2n+1$,&if $t+1\leq\alpha(v)\leq 4n-3$.\\\
\end{tabular}\right.$
For every $e\in E(K_{2n-1})$, we set:
$\beta(e)=\left\\{\begin{tabular}[]{ll}$\alpha(e)$,&if $1\leq\alpha(e)\leq
t$,\\\ $\alpha(e)-2n+1$,&if $t+1\leq\alpha(e)\leq 4n-3$.\\\
\end{tabular}\right.$
It is easy to see that $\beta$ is an interval total $t$-coloring of the graph
$K_{2n-1}$.
Let us prove (2).
For that, we transform the interval total $3n$-coloring $\beta$ of the graph
$K_{2n}$ constructed in the proof of Theorem 19, into an interval total
$t$-coloring $\gamma$ of the same graph.
Define a total coloring $\gamma$ of the graph $K_{2n}$ as follows:
1\. for $i=1,2,\ldots,2n$, let
$\gamma(v_{i})=\left\\{\begin{tabular}[]{ll}$\beta(v_{i})+t-3n$,&if
$\beta(v_{i})+t-3n\leq 2i-1$,\\\ $2i-1$,&if $\beta(v_{i})+t-3n>2i-1$;\\\
\end{tabular}\right.$
2\. for $i=1,2,\ldots,2n-1$, $j=1,2,\ldots,2n-1$, $i\neq j$, and $i+j-1\leq
2(t-3n)+1$, let
$\gamma(v_{i}v_{j})=i+j-1$;
3\. for $i=1,2,\ldots,2n$, $j=1,2,\ldots,2n$, $i\neq j$, and
$2(t-3n)+1<i+j-1<2n$, let
$\gamma(v_{i}v_{j})=\left\\{\begin{tabular}[]{ll}$\beta(v_{i}v_{j})+t-3n$,&if
$i+j$ is even,\\\ $\beta(v_{i}v_{j})$,&if $i+j$ is odd;\\\
\end{tabular}\right.$
4\. for $i=1,2,\ldots,2n$, $j=1,2,\ldots,2n$, $i\neq j$, and $2n\leq i+j-1\leq
2n+2(t-3n)+1$, let
$\gamma(v_{i}v_{j})=i+j-1$;
5\. for $i=3,4,\ldots,2n$, $j=3,4,\ldots,2n$, $i\neq j$, and
$i+j-1>2n+2(t-3n)+1$, let
$\gamma(v_{i}v_{j})=\left\\{\begin{tabular}[]{ll}$\beta(v_{i}v_{j})+t-3n$,&if
$i+j$ is even,\\\ $\beta(v_{i}v_{j})$,&if $i+j$ is odd;\\\
\end{tabular}\right.$
It can be easily verified that $\gamma$ is an interval total $t$-coloring of
the graph $K_{2n}$. $~{}\square$
Finally, we obtain the exact values of $w_{\tau}(G)$ and $W_{\tau}(G)$ for
wheels. Recall that a wheel $W_{n}$ $(n\geq 4)$ is defined as follows:
$V(W_{n})=\left\\{u,v_{1},v_{2},\ldots,v_{n-1}\right\\}$ and
$E(W_{n})=\left\\{uv_{i}|~{}1\leq i\leq
n-1\right\\}\cup\left\\{v_{i}v_{i+1}|~{}1\leq i\leq
n-2\right\\}\cup\\{v_{1}v_{n-1}\\}$.
###### Lemma 21
For any $n\geq 4$, we have $W_{n}\in\mathfrak{T}$ and
$w_{\tau}(W_{n})=\left\\{\begin{tabular}[]{ll}$n+2$,&if $n=4$,\\\ $n$,&if
$n\geq 5$.\\\ \end{tabular}\right.$
* Proof.
Clearly, $W_{4}=K_{4}$, hence, by Theorem 19, we have $W_{4}\in\mathfrak{T}$
and $w_{\tau}(W_{4})=w_{\tau}(K_{4})=6$.
Assume that $n\geq 5$.
For the proof of the lemma we construct an interval total $n$-coloring of the
graph $W_{n}$. We consider two cases.
Case 1: $n$ is even.
Define a total coloring $\alpha$ of the graph $W_{n}$ as follows:
1) $\alpha(u)=n$, $\alpha(v_{1})=2$ and for $i=2,\ldots,\frac{n}{2}-1$, let
$\alpha\left(v_{i}\right)=2i+1$;
2) $\alpha(v_{\frac{n}{2}})=n-2$, $\alpha(v_{\frac{n}{2}+1})=n-4$, and for
$j=\frac{n}{2}+2,\ldots,n-1$, let $\alpha(v_{j})=2(n-j+1);$
3) for $k=1,2,\ldots,\frac{n}{2}$, let
$\alpha\left(uv_{k}\right)=2k-1$;
4) for $l=\frac{n}{2}+1,\ldots,n-1$, let
$\alpha\left(uv_{l}\right)=2(n-l)$;
5) for $p=1,\ldots,\frac{n}{2}-1$, let
$\alpha\left(v_{p}v_{p+1}\right)=2(p+1)$ and
$\alpha\left(v_{\frac{n}{2}}v_{\frac{n}{2}+1}\right)=n-3$;
6) for $q=\frac{n}{2}+1,\ldots,n-2$, let
$\alpha\left(v_{q}v_{q+1}\right)=2(n-q)+1$ and
$\alpha\left(v_{1}v_{n-1}\right)=3$.
Case 2: $n$ is odd.
Define a total coloring $\beta$ of the graph $W_{n}$ as follows:
1) $\beta(u)=n$, $\beta(v_{1})=2$ and for
$i=2,\ldots,\lfloor\frac{n}{2}\rfloor-1$, let $\beta\left(v_{i}\right)=2i+1$;
2)$\beta(v_{\lfloor\frac{n}{2}\rfloor})=n-4$,
$\beta(v_{\lceil\frac{n}{2}\rceil})=n-2$ and for
$j=\lceil\frac{n}{2}\rceil+1,\ldots,n-1$, let $\beta(v_{j})=2(n-j+1)$;
3) for $k=1,2,\ldots,\lfloor\frac{n}{2}\rfloor$, let
$\beta\left(uv_{k}\right)=2k-1;$
4) for $l=\lceil\frac{n}{2}\rceil,\ldots,n-1$, let
$\beta\left(uv_{l}\right)=2(n-l)$;
5) for $p=1,\ldots,\lfloor\frac{n}{2}\rfloor-1$, let
$\beta\left(v_{p}v_{p+1}\right)=2(p+1)$ and
$\beta\left(v_{\lfloor\frac{n}{2}\rfloor}v_{\lceil\frac{n}{2}\rceil}\right)=n-3$;
6) for $q=\lceil\frac{n}{2}\rceil,\ldots,n-2$, let
$\beta\left(v_{q}v_{q+1}\right)=2(n-q)+1$ and
$\beta\left(v_{1}v_{n-1}\right)=3$.
It is not difficult to check that $\alpha$ is an interval total $n$-coloring
of the graph $W_{n}$, when $n$ is even, and $\beta$ is an interval total
$n$-coloring of the graph $W_{n}$, when $n$ is odd. Hence,
$W_{n}\in\mathfrak{T}$. On the other hand, clearly,
$w_{\tau}(W_{n})\geq\chi^{\prime\prime}(W_{n})=\Delta(W_{n})+1=n$, thus
$w_{\tau}(W_{n})=n$. $~{}\square$
###### Lemma 22
For any $n\geq 5$, we have $W_{n}\in\mathfrak{T}_{n+1}\cap\mathfrak{T}_{n+2}$.
* Proof.
First we show that $W_{n}\in\mathfrak{T}_{n+2}$ for any $n\geq 5$.
Define a total coloring $\alpha$ of the graph $W_{n}$ as follows:
1) $\alpha(u)=1$, $\alpha(v_{1})=3$, $\alpha(v_{\lceil\frac{n}{2}\rceil})=n-1$
and for $i=2,\ldots,\lceil\frac{n}{2}\rceil-1$, let
$\alpha\left(v_{i}\right)=2(i+1)$;
2) for $j=\lceil\frac{n}{2}\rceil+1,\ldots,n-1$, let $\alpha(v_{j})=2(n-j)+3$;
3) for $k=1,2,\ldots,\lfloor\frac{n}{2}\rfloor$, let
$\alpha\left(uv_{k}\right)=2k$;
4) for $l=\lfloor\frac{n}{2}\rfloor+1,\ldots,n-1$, let
$\alpha\left(uv_{l}\right)=2(n-l)+1$;
5) for $p=1,\ldots,\lfloor\frac{n-1}{2}\rfloor$, let
$\alpha\left(v_{p}v_{p+1}\right)=2p+3$;
6) for $q=\lfloor\frac{n-1}{2}\rfloor+1,\ldots,n-2$
$\alpha\left(v_{q}v_{q+1}\right)=2(n-q+1)$ and
$\alpha\left(v_{1}v_{n-1}\right)=4$.
It is easily seen that $\alpha$ is an interval total $(n+2)$-coloring of the
graph $W_{n}$.
Now we show that $W_{n}\in\mathfrak{T}_{n+1}$ for any $n\geq 5$.
Define a total coloring $\beta$ of the graph $W_{n}$ as follows:
1) for $\forall v\in V(W_{n})$, let $\beta(v)=\alpha(v)$;
2) for $\forall e\in E(W_{n})$, let
$\beta(e)=\left\\{\begin{tabular}[]{ll}$\alpha(e)$,&if $\alpha(e)\neq n+2$,\\\
$n-2$,&otherwise.\\\ \end{tabular}\right.$
It is easily seen that $\beta$ is an interval total $(n+1)$-coloring of the
graph $W_{n}$. $~{}\square$
###### Lemma 23
For any $n\geq 4$, we have $W_{\tau}(W_{n})\geq n+3$.
* Proof.
Clearly, for the proof of the lemma it suffices to construct an interval total
$(n+3)$-coloring of the graph $W_{n}$ for $n\geq 4$. We consider two cases.
Case 1: $n$ is even.
Define a total coloring $\alpha$ of the graph $W_{n}$ as follows:
1) for $i=1,2,\ldots,\frac{n}{2}+1$, let $\alpha\left(v_{i}\right)=2i-1$;
2) for $j=\frac{n}{2}+2,\ldots,n-1$, let $\alpha(v_{j})=2(n-j+1)$;
3) for $k=1,2,\ldots,\frac{n}{2}$, let
$\alpha\left(v_{k}v_{k+1}\right)=2k$;
4) for $l=\frac{n}{2}+1,\ldots,n-2$, let
$\alpha\left(v_{l}v_{l+1}\right)=2(n-l)+1$ and
$\alpha\left(v_{1}v_{n-1}\right)=3$;
5) for $p=2,\ldots,\frac{n}{2}$, let
$\alpha\left(uv_{p}\right)=2p+1$ and $\alpha\left(uv_{1}\right)=4$;
6) for $q=\frac{n}{2}+1,\ldots,n-1$, let
$\alpha\left(uv_{q}\right)=2(n-q+2)$ and $\alpha(u)=n+3$.
Case 2: $n$ is odd.
Define a total coloring $\beta$ of the graph $W_{n}$ as follows:
1) for $i=1,2,\ldots,\lfloor\frac{n}{2}\rfloor$, let
$\beta\left(v_{i}\right)=2i-1$, $\beta\left(v_{i}v_{i+1}\right)=2i$;
2) for $j=\lceil\frac{n}{2}\rceil,\ldots,n-1$, let $\beta(v_{j})=2(n-j+1)$;
3) for $k=\lceil\frac{n}{2}\rceil,\ldots,n-2$, let
$\beta\left(v_{k}v_{k+1}\right)=2(n-k)+1$ and
$\beta\left(v_{1}v_{n-1}\right)=3$;
4) for $p=2,3,\ldots,\lceil\frac{n}{2}\rceil$, let
$\beta\left(uv_{p}\right)=2p+1$ and $\beta\left(uv_{1}\right)=4$;
5) for $q=\lceil\frac{n}{2}\rceil+1,\ldots,n-1$, let
$\beta\left(uv_{q}\right)=2(n-q+2)$ and $\beta(u)=n+3$.
It is not difficult to check that $\alpha$ is an interval total
$(n+3)$-coloring of the graph $W_{n}$, when $n$ is even, and $\beta$ is an
interval total $(n+3)$-coloring of the graph $W_{n}$, when $n$ is odd.
$~{}\square$
###### Remark 24
Easy analysis shows that if $4\leq n\leq 8$, then $W_{\tau}(W_{n})=n+3$.
###### Lemma 25
For any $n\geq 9$, we have $W_{\tau}(W_{n})\geq n+4$.
* Proof.
Clearly, for the proof of the lemma it suffices to construct an interval total
$(n+4)$-coloring of the graph $W_{n}$ for $n\geq 9$. We consider two cases.
Case 1: $n$ is even.
Define a total coloring $\alpha$ of the graph $W_{n}$ as follows:
1) $\alpha(u)=7$, $\alpha(v_{1})=1$, $\alpha(v_{2})=6$, $\alpha(v_{3})=8$ and
for $i=4,\ldots,\frac{n}{2}-2$, let $\alpha\left(v_{i}\right)=2i+1$;
2) $\alpha(v_{\frac{n}{2}-1})=n+2$, $\alpha(v_{\frac{n}{2}})=n+4$ and for
$j=\frac{n}{2}+1,\ldots,n-2$, let $\alpha(v_{j})=2(n-j)$, $\alpha(v_{n-1})=3$;
3) $\alpha(uv_{1})=3$, $\alpha(uv_{2})=5$ and for $k=3,\ldots,\frac{n}{2}-1$,
let
$\alpha\left(uv_{k}\right)=2k+3$;
4) for $l=\frac{n}{2},\ldots,n-1$, let
$\alpha\left(uv_{l}\right)=2(n-l+1)$;
5) $\alpha(v_{1}v_{2})=4$, $\alpha(v_{2}v_{3})=7$ and for
$p=3,\ldots,\frac{n}{2}-2$, let
$\alpha\left(v_{p}v_{p+1}\right)=2(p+2)$;
6) for $q=\frac{n}{2}-1,\ldots,n-2$, let
$\alpha\left(v_{q}v_{q+1}\right)=2(n-q)+1$ and
$\alpha\left(v_{1}v_{n-1}\right)=2$.
Case 2: $n$ is odd.
Define a total coloring $\beta$ of the graph $W_{n}$ as follows:
1) $\beta(u)=7$, $\beta(v_{1})=1$, $\beta(v_{2})=6$, $\beta(v_{3})=8$ and for
$i=4,\ldots,\lfloor\frac{n}{2}\rfloor-1$, let $\beta\left(v_{i}\right)=2i+1$;
2) $\beta(v_{\lfloor\frac{n}{2}\rfloor})=n+4$,
$\beta(v_{\lceil\frac{n}{2}\rceil})=n+2$ and for
$j=\lceil\frac{n}{2}\rceil+1,\ldots,n-2$, let $\beta(v_{j})=2(n-j)$,
$\beta(v_{n-1})=3$;
3) $\beta(uv_{1})=3$, $\beta(uv_{2})=5$ and for
$k=3,\ldots,\lfloor\frac{n}{2}\rfloor$, let
$\beta\left(uv_{k}\right)=2k+3$;
4) for $l=\lceil\frac{n}{2}\rceil,\ldots,n-1$, let
$\beta\left(uv_{l}\right)=2(n-l+1)$;
5) $\beta(v_{1}v_{2})=4$, $\beta(v_{2}v_{3})=7$ and for
$p=3,\ldots,\lfloor\frac{n}{2}\rfloor$, let
$\beta\left(v_{p}v_{p+1}\right)=2(p+2)$;
6) for $q=\lceil\frac{n}{2}\rceil,\ldots,n-2$, let
$\beta\left(v_{q}v_{q+1}\right)=2(n-q)+1$ and
$\beta\left(v_{1}v_{n-1}\right)=2$.
It is easy to check that $\alpha$ is an interval total $(n+4)$-coloring of the
graph $W_{n}$, when $n$ is even, and $\beta$ is an interval total
$(n+4)$-coloring of the graph $W_{n}$, when $n$ is odd. $~{}\square$
Figure 1:
###### Lemma 26
For any $n\geq 4$, we have $W_{\tau}(W_{n})\leq n+4$.
* Proof.
First, by Theorem 11, we have $W_{\tau}(W_{n})\leq n+6$ for any $n\geq 4$.
Next we prove that $W_{n}\notin\mathfrak{T}_{n+5}$.
Suppose, to the contrary, that $\alpha$ is an interval total $(n+5)$-coloring
of the graph $W_{n}$ for $n\geq 4$.
Consider the vertex $u$. Clearly,
$1\leq\min S[u,\alpha]\leq 6$,
hence
$n\leq\max S[u,\alpha]\leq n+5$.
Proposition 3 implies that the following three cases are possible:
1) $S[u,\alpha]=[6,n+5]$;
2) $S[u,\alpha]=[5,n+4]$;
3) $S[u,\alpha]=[4,n+3]$.
Case 1: $S[u,\alpha]=[6,n+5]$ or $S[u,\alpha]=[5,n+4]$.
Clearly, $\alpha(uv_{i})\geq 5$ for $i=1,\ldots,n-1$. This implies that $\min
S[v_{i},\alpha]\geq 2$ for $i=1,\ldots,n-1$, which is a contradiction.
Case 2: $S[u,\alpha]=[4,n+3]$.
First we show that $\alpha(u)\neq 4$. Suppose that $\alpha(u)=4$. This implies
that $\alpha(uv_{i})\geq 5$ for $i=1,\ldots,n-1$, which is a contradiction.
Let $e=uv_{1}$ and $\alpha(e)=4$. Note that $\alpha(v_{1})=1$.
Without loss of generality, we may assume that $\alpha(v_{1}v_{2})=2$,
$\alpha(v_{1}v_{n-1})=3$, $\alpha(uv_{2})=5$, $\alpha(uv_{n-1})=6$, and there
is a vertex $v_{k}$ such that either $\alpha(v_{k})=n+5$ or
$\alpha(v_{k}v_{k+1})=n+5$ (see Fig. 1).
Let us consider simple paths
$P_{1}=\left(v_{1},v_{1}v_{2},v_{2},\ldots,v_{k},v_{k}v_{k+1},v_{k+1}\right)$
and
$P_{2}=\left(v_{n-1},v_{n-1}v_{n-2},v_{n-2},\ldots,v_{k+1},v_{k+1}v_{k},v_{k}\right)$,
where $1\leq k\leq n-2$.
Let us show that
1) $\alpha(v_{i})=2i-1$, $\alpha(v_{i}v_{i+1})=2i$, $\alpha(uv_{i})=2i+1$,
2) $\alpha(v_{n+1-i})=2i$, $\alpha(v_{n-i}v_{n+1-i})=2i+1$,
$\alpha(uv_{n+1-i})=2(i+1)$,
for $i=2,\ldots,k$.
We use induction on $i$. For $i=2$, it suffices to prove that
$\alpha(v_{2})=3$, $\alpha(v_{2}v_{3})=4$, $\alpha(v_{n-1})=4$,
$\alpha(v_{n-2}v_{n-1})=5$.
Consider the vertex $v_{2}$. Since $\alpha(v_{1}v_{2})=2$ and
$\alpha(uv_{2})=5$, we have $\min S[v_{2},\alpha]=2$ and $\max
S[v_{2},\alpha]=5$, hence $\\{3,4\\}\subseteq S[v_{2},\alpha]$. If we suppose
that $\alpha(v_{2})=4$, then $\alpha(v_{2}v_{3})=3$ and $\max
S[v_{3},\alpha]<7$, which contradicts $\max S[v_{3},\alpha]\geq 7$. From this
we have $\alpha(uv_{3})=7$ (see Fig. 1).
Now we consider the vertex $v_{n-1}$. Since $\alpha(v_{1}v_{n-1})=3$ and
$\alpha(uv_{n-1})=6$, we have $\min S[v_{n-1},\alpha]=3$ and $\max
S[v_{n-1},\alpha]=6$, hence $\\{4,5\\}\subseteq S[v_{n-1},\alpha]$. If we
suppose that $\alpha(v_{n-1})=5$, then $\alpha(v_{n-2}v_{n-1})=4$ and $\max
S[v_{n-2},\alpha]<8$, which contradicts $\max S[v_{n-2},\alpha]\geq 8$ (see
Fig. 1).
Suppose that the statements 1) and 2) are true for all $i^{\prime}$, $1\leq
i^{\prime}\leq i$. We prove that the statements 1) and 2) are true for the
case $i+1$, that is, $\alpha(v_{i+1})=2i+1$, $\alpha(v_{i+1}v_{i+2})=2i+2$,
$\alpha(uv_{i+1})=2i+3$ and $\alpha(v_{n-i})=2i+2$,
$\alpha(v_{n-i-1}v_{n-i})=2i+3$, $\alpha(uv_{n-i})=2i+4$. From the induction
hypothesis we have:
$1^{\prime}$) $\alpha(v_{j})=2j-1$, $\alpha(v_{j}v_{j+1})=2j$,
$\alpha(uv_{j})=2j+1$,
$2^{\prime}$) $\alpha(v_{n+1-j})=2j$, $\alpha(v_{n-j}v_{n+1-j})=2j+1$,
$\alpha(uv_{n+1-j})=2(j+1)$,
for $j=2,\ldots,i$.
$1^{\prime}$) and $2^{\prime}$) implies that $\alpha(uv_{i+1})=2i+3$ and
$\alpha(uv_{n-i})=2i+4$.
Consider the vertex $v_{i+1}$. Since $\alpha(v_{i}v_{i+1})=2i$ and
$\alpha(uv_{i+1})=2i+3$, we have $\min S[v_{i+1},\alpha]=2i$ and $\max
S[v_{i+1},\alpha]=2i+3$, hence $\\{2i+1,2i+2\\}\subseteq S[v_{i+1},\alpha]$.
If we suppose that $\alpha(v_{i+1})=2i+2$, then $\alpha(v_{i+1}v_{i+2})=2i+1$
and $\max S[v_{i+2},\alpha]<2i+5$, which contradicts $\max
S[v_{i+2},\alpha]\geq 2i+5$. From this we have $\alpha(uv_{i+2})=2i+5$ (see
Fig. 1).
Next we consider the vertex $v_{n-i}$. Since $\alpha(v_{n+1-i}v_{n-i})=2i+1$
and $\alpha(uv_{n-i})=2i+4$, we have $\min S[v_{n-i},\alpha]=2i+1$ and $\max
S[v_{n-i},\alpha]=2i+4$, hence $\\{2i+2,2i+3\\}\subseteq S[v_{n-i},\alpha]$.
If we suppose that $\alpha(v_{n-i})=2i+3$, then
$\alpha(v_{n-i-1}v_{n-i})=2i+2$ and $\max S[v_{n-i-1},\alpha]<2i+6$, which
contradicts $\max S[v_{n-i-1},\alpha]\geq 2i+6$ (see Fig. 1).
By $1^{\prime}$), we have $k\geq\frac{n}{2}+2$.
By $2^{\prime}$), we have $k\leq\frac{n}{2}-1$.
It is easy to see that does not exist such an index $k$, which satisfy the
aforementioned inequalities. This completes the prove of the case 2.
Similarly, it can be shown that $W_{n}\notin\mathfrak{T}_{n+6}$, hence
$W_{\tau}(W_{n})\leq n+4$ for any $n\geq 4$. $~{}\square$
From Lemmas 21-26 and Remark 24, we have the following result:
###### Theorem 27
For $n\geq 4$, we have
(1)
$W_{n}\in\mathfrak{T}$,
(2)
$w_{\tau}(W_{n})=\left\\{\begin{tabular}[]{ll}$n+2$,&if $n=4$,\\\ $n$,&if
$n\geq 5$,\\\ \end{tabular}\right.$
(3)
$W_{\tau}(W_{n})=\left\\{\begin{tabular}[]{ll}$n+3$,&if $4\leq n\leq 8$,\\\
$n+4$,&if $n\geq 9$,\\\ \end{tabular}\right.$
(4)
if $w_{\tau}(W_{n})\leq t\leq W_{\tau}(W_{n})$, then
$W_{n}\in\mathfrak{T}_{t}$.
* Acknowledgement
We would like to thank Rafayel R. Kamalian for his attention to this work.
## References
* [1] A.S. Asratian, R.R. Kamalian, Interval colorings of edges of a multigraph, Appl. Math. 5 (1987) 25-34 (in Russian).
* [2] A.S. Asratian, R.R. Kamalian, Investigation on interval edge-colorings of graphs, J. Combin. Theory Ser. B 62 (1994) 34-43.
* [3] A.S. Asratian, C.J. Casselgren, On interval edge colorings of $(\alpha,\beta)$-biregular bipartite graphs, Discrete Mathematics 307 (2006) 1951-1956.
* [4] M. Behzad, Graphs and their chromatic numbers, Ph.D. thesis, Michigan State University, 1965.
* [5] M. Behzad, G. Chartrand, J.K. Cooper Jr., The colour numbers of complete graphs, J. London Math. Soc. 42 (1967) 226-228.
* [6] O.V. Borodin, On the total colouring of planar graphs, J. Reine Angew. Math. 394 (1989) 180-185.
* [7] O.V. Borodin, A.V. Kostochka, D.R. Woodall, Total colorings of planar graphs with large maximum degree, J. Graph Theory 26 (1997) 53-59.
* [8] K.H. Chew, H.P. Yap, Total chromatic number of complete $r$-partite graphs, J. Graph Theory 16 (1992) 629-634.
* [9] A.J.W. Hilton, H.R. Hind, The total chromatic number of graphs having large maximum degree, Discrete Mathematics 117 (1993) 127-140.
* [10] T.R. Jensen, B. Toft, Graph coloring problems, Wiley Interscience Series in Discrete Mathematics and Optimization, 1995.
* [11] A.V. Kostochka, The total coloring of a multigraph with maximal degree 4, Discrete Mathematics 17 (1977) 161-163.
* [12] A.V. Kostochka, The total chromatic number of any multigraph with maximum degree five is at most seven, Discrete Mathematics 162 (1996) 199-214.
* [13] L. Kowalik, J.-S. Sereni, R. Skrekovski, Total-colouring of plane graphs with maximum degree nine, SIAM J. Discrete Math. 22 (2008) 1462-1479.
* [14] C.J.H. McDiarmid, A. Sanchez-Arroyo, Total colouring regular bipartite graphs is $NP$-hard, Discrete Mathematics 124 (1994) 155-162.
* [15] M. Molloy, B. Reed, A bound on the total chromatic number, Combinatorica 18 (1998) 241-280.
* [16] P.A. Petrosyan, Interval total colorings of complete bipartite graphs, Proceedings of the CSIT Conference (2007) 84-85.
* [17] P.A. Petrosyan, Interval total colorings of certain graphs, Mathematical Problems of Computer Science 31 (2008) 122-129.
* [18] M. Rosenfeld, On the total coloring of certain graphs, Israel J. Math. 9 (1971) 396-402.
* [19] A. Sanchez-Arroyo, Determining the total colouring number is $NP$-hard, Discrete Mathematics 78 (1989) 315-319.
* [20] D.P. Sanders, Y. Zhao, On total 9-coloring planar graphs of maximum degree seven, J. Graph Theory 31 (1999) 67-73.
* [21] N. Vijayaditya, On the total chromatic number of a graph, J. London Math. Soc. (2) 3 (1971) 405-408.
* [22] V.G. Vizing, Chromatic index of multigraphs, Doctoral Thesis, Novosibirsk, 1965 (in Russian).
* [23] W. Wang, Total chromatic number of planar graphs with maximum degree ten, J. Graph Theory 54 (2006) 91-102.
* [24] D.B. West, Introduction to Graph Theory, Prentice-Hall, New Jersey, 1996\.
* [25] H.P. Yap, Total colorings of graphs, Lecture Notes in Mathematics 1623, Springer-Verlag, Berlin, 1996.
* [26] Z. Zhang, J. Zhand, J. Wang, The total chromatic numbers of some graphs, Scientia Sinica A 31 (1988) 1434-1441.
|
arxiv-papers
| 2010-10-14T17:37:33 |
2024-09-04T02:49:13.910582
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "P.A. Petrosyan, A.Yu. Torosyan, N.A. Khachatryan",
"submitter": "Petros Petrosyan",
"url": "https://arxiv.org/abs/1010.2989"
}
|
1010.3157
|
# Lengths matters, periodically.
(The movie)
Fluid Dynamics Video
M. C. Renoult, S. Ferjani, C. Rosenblatt, P. Carles
Fast Lab., UPMC/Paris 6, FRANCE
Physics Dept., CWRU, Cleveland, OH,USA
The video shows the development of the Rayleigh-Taylor Instability between two
immiscible fluids for four distinct initial single-mode perturbations. To
obtain such controlled initial conditions, a Magnetic Levitation technique is
used (see [1]). The principle is explained in the first part of the animation.
A homogeneous magnetic force is produced in opposition to gravity and allows
the stabilization of the dense fluid (paramagnetic aqueous mixture) above the
less dense fluid (hexadecane). In addition, segments of magnetically permeable
wires are placed on the outside of the cell in a precise configuration -
predicted numerically - to achieve almost any desired initial condition (see
[2]). The initial interface thus is no longer flat and the experiment is
started by turning off the magnetic field and allowing the denser fluid to
fall under gravity. Here the wires are periodically aligned along a row, a few
mm above the initial interface position, producing a small amplitude single
mode perturbation of the same wavelength as the wires (but too small to
image). The video presents the Rayleigh-Taylor Instability for four decreasing
wavelengths (15 mm, 12.5, 10 mm and 7.5 mm). In each case, one observes the
different stages of development of the Rayleigh-Taylor Instability from the
early time linear behavior to the late mixing flow. The second instability
also can be observed.
The link for the video is: Video.
## References
* [1] Mahajan M.P., Tsige M., Taylor P.L. and Rosenblatt C., Phys. Fluids 10, 2208 (1998)
* [2] Huang Z., De Luca A., Atherton T.J., Bird M., Rosenblatt C. and Carles P., Phys. Rev. Lett. 204502 (2007)
|
arxiv-papers
| 2010-10-15T13:31:09 |
2024-09-04T02:49:13.925628
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "M.C. Renoult, S. Ferjani, C. Rosenblatt, P. Carles",
"submitter": "Pierre Carles",
"url": "https://arxiv.org/abs/1010.3157"
}
|
1010.3174
|
# The geometry of the disk complex
Howard Masur Department of Mathematics
University of Chicago
Chicago, Illinois 60637 masur@math.uic.edu and Saul Schleimer Department of
Mathematics
University of Warwick
Coventry, CV4 7AL, UK s.schleimer@warwick.ac.uk
###### Abstract.
We give a distance estimate for the metric on the disk complex and show that
it is Gromov hyperbolic. As another application of our techniques, we find an
algorithm which computes the Hempel distance of a Heegaard splitting, up to an
error depending only on the genus.
This work is in the public domain.
###### Contents
1. 1 Introduction
2. 2 Background on complexes
3. 3 Background on coarse geometry
4. 4 Natural maps
5. 5 Holes in general and the lower bound on distance
6. 6 Holes for the non-orientable surface
7. 7 Holes for the arc complex
8. 8 Background on three-manifolds
9. 9 Holes for the disk complex
10. 10 Holes for the disk complex – annuli
11. 11 Holes for the disk complex – compressible
12. 12 Holes for the disk complex – incompressible
13. 13 Axioms for combinatorial complexes
14. 14 Partition and the upper bound on distance
15. 15 Background on Teichmüller space
16. 16 Paths for the non-orientable surface
17. 17 Paths for the arc complex
18. 18 Background on train tracks
19. 19 Paths for the disk complex
20. 20 Hyperbolicity
21. 21 Coarsely computing Hempel distance
## 1\. Introduction
In this paper we initiate the study of the geometry of the disk complex of a
handlebody $V$. The disk complex $\mathcal{D}(V)$ has a natural simplicial
inclusion into the curve complex $\mathcal{C}(S)$ of the boundary of the
handlebody. Surprisingly, this inclusion is not a quasi-isometric embedding;
there are disks which are close in the curve complex yet very far apart in the
disk complex. As we will show, any obstruction to joining such disks via a
short path is a topologically meaningful subsurface of $S=\partial V$. We call
such subsurfaces holes. A path in the disk complex must travel into and then
out of these holes; paths in the curve complex may skip over a hole by using
the vertex representing the boundary of the subsurface. We classify the holes:
###### Theorem 1.1.
Suppose $V$ is a handlebody. If $X\subset\partial V$ is a hole for the disk
complex $\mathcal{D}(V)$ of diameter at least $61$ then:
* •
$X$ is not an annulus.
* •
If $X$ is compressible then there are disks $D,E$ with boundary contained in
$X$ so that the boundaries fill $X$.
* •
If $X$ is incompressible then there is an $I$-bundle $\rho_{F}\colon T\to F$
so that $T$ is a component of $V{\smallsetminus}\partial_{v}T$ and $X$ is a
component of $\partial_{h}T$.
See Theorems 10.1, 11.6 and 12.1 for more precise statements. The $I$–bundles
appearing in the classification lead us to study the arc complex
$\mathcal{A}(F)$ of the base surface $F$. Since the $I$–bundle $T$ may be
twisted the surface $F$ may be non-orientable.
Thus, as a necessary warm-up to the difficult case of the disk complex, we
also analyze the holes for the curve complex of an non-orientable surface, as
well as the holes for the arc complex.
### Topological application
It is a long-standing open problem to decide, given a Heegaard diagram,
whether the underlying splitting surface is reducible. This question has deep
connections to the geometry, topology, and algebra of the ambient three-
manifold. For example, a resolution of this problem would give new solutions
to both the three-sphere recognition problem and the triviality problem for
three-manifold groups. The difficulty of deciding reducibility is underlined
by its connection to the Poincaré conjecture: several approaches to the
Poincaré Conjecture fell at essentially this point. See [10] for a entrance
into the literature.
One generalization of deciding reducibility is to find an algorithm that,
given a Heegaard diagram, computes the distance of the Heegaard splitting as
defined by Hempel [20]. (For example, see [5, Section 2].) The classification
of holes for the disk complex leads to a coarse answer to this question.
In every genus $g$ there is a constant $K=K(g)$ and an algorithm that, given a
Heegaard diagram, computes the distance of the Heegaard splitting with error
at most $K$.
In addition to the classification of holes, the algorithm relies on the Gromov
hyperbolicity of the curve complex [24] and the quasi-convexity of the disk
set inside of the curve complex [26]. However the algorithm does not depend on
our geometric applications of Theorem 1.1.
### Geometric application
The hyperbolicity of the curve complex and the classification of holes allows
us to prove:
The disk complex is Gromov hyperbolic.
Again, as a warm-up to the proof of Theorem 20.3 we prove that
$\mathcal{C}(F)$ and $\mathcal{A}(S)$ are hyperbolic in Corollary 6.4 and
Theorem 20.2. Note that Bestvina and Fujiwara [4] have previously dealt with
the curve complex of a non-orientable surface, following Bowditch [6].
These results cannot be deduced from the fact that $\mathcal{D}(V)$,
$\mathcal{C}(F)$, and $\mathcal{A}(S)$ can be realized as quasi-convex subsets
of $\mathcal{C}(S)$. This is because the curve complex is locally infinite. As
simple example consider the Cayley graph of $\mathbb{Z}^{2}$ with the standard
generating set. Then the cone $C(\mathbb{Z}^{2})$ of height one-half is a
Gromov hyperbolic space and $\mathbb{Z}^{2}$ is a quasi-convex subset. Another
instructive example, very much in-line with our work, is the usual embedding
of the three-valent tree $T_{3}$ into the Farey tessellation.
The proof of Theorem 20.3 requires the distance estimate Theorem 19.1: the
distance in $\mathcal{C}(F)$, $\mathcal{A}(S)$, and $\mathcal{D}(V)$ is
coarsely equal to the sum of subsurface projection distances in holes.
However, we do not use the hierarchy machine introduced in [25]. This is
because hierarchies are too flexible to respect a symmetry, such as the
involution giving a non-orientable surface, and at the same time too rigid for
the disk complex. For $\mathcal{C}(F)$ we use the highly rigid Teichmüller
geodesic machine, due to Rafi [33]. For $\mathcal{D}(V)$ we use the extremely
flexible train track machine, developed by ourselves and Mosher [27].
Theorems 19.1 and 20.3 are part of a more general framework. Namely, given a
combinatorial complex $\mathcal{G}$ we understand its geometry by classifying
the holes: the geometric obstructions lying between $\mathcal{G}$ and the
curve complex. In Sections 13 and 14 we show that any complex $\mathcal{G}$
satisfying certain axioms necessarily satisfies a distance estimate. That
hyperbolicity follows from the axioms is proven in Section 20.
Our axioms are stated in terms of a path of markings, a path in the the
combinatorial complex, and their relationship. For the disk complex the
combinatorial paths are surgery sequences of essential disks while the marking
paths are provided by train track splitting sequences; both constructions are
due to the first author and Minsky [26] (Section 18). The verification of the
axioms (Section 19) relies on our work with Mosher, analyzing train track
splitting sequences in terms of subsurface projections [27].
We do not study non-orientable surfaces directly; instead we focus on
symmetric multicurves in the double cover. This time marking paths are
provided by Teichmüller geodesics, using the fact that the symmetric Riemann
surfaces form a totally geodesic subset of Teichmüller space. The
combinatorial path is given by the systole map. We use results of Rafi [33] to
verify the axioms for the complex of symmetric curves. (See Section 16.)
Section 17 verifies the axioms for the arc complex again using Teichmüller
geodesics and the systole map. It is interesting to note that the axioms for
the arc complex can also be verified using hierarchies or, indeed, train track
splitting sequences.
The distance estimates for the marking graph and the pants graph, as given by
the first author and Minsky [25], inspired the work here, but do not fit our
framework. Indeed, neither the marking graph nor the pants graph are Gromov
hyperbolic. It is crucial here that all holes interfere; this leads to
hyperbolicity. When there are non-interfering holes, it is unclear how to
partition the marking path to obtain the distance estimate.
### Acknowledgments
We thank Jason Behrstock, Brian Bowditch, Yair Minsky, Lee Mosher, Hossein
Namazi, and Kasra Rafi for many enlightening conversations.
We thank Tao Li for pointing out that our original bound inside of Theorem
12.1 of $O(\log g(V))$ could be reduced to a constant.
## 2\. Background on complexes
We use $S_{g,b,c}$ to denote the compact connected surface of genus $g$ with
$b$ boundary components and $c$ cross-caps. If the surface is orientable we
omit the subscript $c$ and write $S_{g,b}$. The complexity of $S=S_{g,b}$ is
$\xi(S)=3g-3+b$. If the surface is closed and orientable we simply write
$S_{g}$.
### 2.1. Arcs and curves
A simple closed curve $\alpha\subset S$ is essential if $\alpha$ does not
bound a disk in $S$. The curve $\alpha$ is non-peripheral if $\alpha$ is not
isotopic to a component of $\partial S$. A simple arc $\beta\subset S$ is
proper if $\beta\cap\partial S=\partial\beta$. An isotopy of $S$ is proper if
it preserves the boundary setwise. A proper arc $\beta\subset S$ is essential
if $\beta$ is not properly isotopic into a regular neighborhood of $\partial
S$.
Define $\mathcal{C}(S)$ to be the set of isotopy classes of essential, non-
peripheral curves in $S$. Define $\mathcal{A}(S)$ to be the set of proper
isotopy classes of essential arcs. When $S=S_{0,2}$ is an annulus define
$\mathcal{A}(S)$ to be the set of essential arcs, up to isotopies fixing the
boundary pointwise. For any surface define
$\mathcal{AC}(S)=\mathcal{A}(S)\cup\mathcal{C}(S)$.
For $\alpha,\beta\in\mathcal{AC}(S)$ the geometric intersection number
$\iota(\alpha,\beta)$ is the minimum intersection possible between $\alpha$
and any $\beta^{\prime}$ equivalent to $\beta$. When $S=S_{0,2}$ we do not
count intersection points occurring on the boundary. If $\alpha$ and $\beta$
realize their geometric intersection number then $\alpha$ is tight with
respect to $\beta$. If they do not realize their geometric intersection then
we may tighten $\beta$ until they do.
Define $\Delta\subset\mathcal{AC}(S)$ to be a multicurve if for all
$\alpha,\beta\in\Delta$ we have $\iota(\alpha,\beta)=0$. Following Harvey [18]
we may impose the structure of a simplical complex on $\mathcal{AC}(S)$: the
simplices are exactly the multicurves. Also, $\mathcal{C}(S)$ and
$\mathcal{A}(S)$ naturally span sub-complexes.
Note that the curve complexes $\mathcal{C}(S_{1,1})$ and
$\mathcal{C}(S_{0,4})$ have no edges. It is useful to alter the definition in
these cases. Place edges between all vertices with geometric intersection
exactly one if $S=S_{1,1}$ or two if $S=S_{0,4}$. In both cases the result is
the Farey graph. Also, with the current definition $\mathcal{C}(S)$ is empty
if $S=S_{0,2}$. Thus for the annulus only we set
$\mathcal{AC}(S)=\mathcal{C}(S)=\mathcal{A}(S)$.
###### Definition 2.1.
For vertices $\alpha,\beta\in\mathcal{C}(S)$ define the distance
$d_{S}(\alpha,\beta)$ to be the minimum possible number of edges of a path in
the one-skeleton $\mathcal{C}^{1}(S)$ which starts at $\alpha$ and ends at
$\beta$.
Note that if $d_{S}(\alpha,\beta)\geq 3$ then $\alpha$ and $\beta$ fill the
surface $S$. We denote distance in the one-skeleton of $\mathcal{A}(S)$ and of
$\mathcal{AC}(S)$ by $d_{\mathcal{A}}$ and $d_{\mathcal{AC}}$ respectively.
Recall that the geometric intersection of a pair of curves gives an upper
bound for their distance.
###### Lemma 2.2.
Suppose that $S$ is a compact connected surface which is not an annulus. For
any $\alpha,\beta\in\mathcal{C}^{0}(S)$ with $\iota(\alpha,\beta)>0$ we have
$d_{S}(\alpha,\beta)\leq 2\log_{2}(\iota(\alpha,\beta))+2$. ∎
This form of the inequality, stated for closed orientable surfaces, may be
found in [20]. A proof in the bounded orientable case is given in [36]. The
non-orientable case is then an exercise. When $S=S_{0,2}$ an induction proves
(2.3) $d_{X}(\alpha,\beta)=1+\iota(\alpha,\beta)$
for distinct vertices $\alpha,\beta\in\mathcal{C}(X)$. See [25, Equation 2.3].
### 2.2. Subsurfaces
Suppose that $X\subset S$ is a connected compact subsurface. We say $X$ is
essential exactly when all boundary components of $X$ are essential in $S$. We
say that $\alpha\in\mathcal{AC}(S)$ cuts $X$ if all representatives of
$\alpha$ intersect $X$. If some representative is disjoint then we say
$\alpha$ misses $X$.
###### Definition 2.4.
An essential subsurface $X\subset S$ is cleanly embedded if for all components
$\delta\subset\partial X$ we have: $\delta$ is isotopic into $\partial S$ if
and only if $\delta$ is equal to a component of $\partial S$.
###### Definition 2.5.
Suppose $X,Y\subset S$ are essential subsurfaces. If $X$ is cleanly embedded
in $Y$ then we say that $X$ is nested in $Y$. If $\partial X$ cuts $Y$ and
also $\partial Y$ cuts $X$ then we say that $X$ and $Y$ overlap.
A compact connected surface $S$ is simple if $\mathcal{AC}(S)$ has finite
diameter.
###### Lemma 2.6.
Suppose $S$ is a connected compact surface. The following are equivalent:
* •
$S$ is not simple.
* •
The diameter of $\mathcal{AC}(S)$ is at least five.
* •
$S$ admits an ending lamination or $S=S_{1}$ or $S_{0,2}$.
* •
$S$ admits a pseudo-Anosov map or $S=S_{1}$ or $S_{0,2}$.
* •
$\chi(S)<-1$ or $S=S_{1,1},S_{1},S_{0,2}$.
Lemma 4.6 of [24] shows that pseudo-Anosov maps have quasi-geodesic orbits,
when acting on the associated curve complex. A Dehn twist acting on
$\mathcal{C}(S_{0,2})$ has geodesic orbits.
Note that Lemma 2.6 is only used in this paper when $\partial S$ is non-empty.
The closed case is included for completeness.
###### Proof sketch of Lemma 2.6.
If $S$ admits a pseudo-Anosov map then the stable lamination is an ending
lamination. If $S$ admits a filling lamination then, by an argument of
Kobayashi [21], $\mathcal{AC}(S)$ has infinite diameter. (This argument is
also sketched in [24], page 124, after the statement of Proposition 4.6.)
If the diameter of $\mathcal{AC}$ is infinite then the diameter is at least
five. To finish, one may check directly that all surfaces with $\chi(S)>-2$,
other than $S_{1,1}$, $S_{1}$ and the annulus have $\mathcal{AC}(S)$ with
diameter at most four. (The difficult cases, $S_{012}$ and $S_{003}$, are
discussed by Scharlemann [35].) Alternatively, all surfaces with $\chi(S)<-1$,
and also $S_{1,1}$, admit pseudo-Anosov maps. The orientable cases follow from
Thurston’s construction [38]. Penner’s generalization [32] covers the non-
orientable cases. ∎
### 2.3. Handlebodies and disks
Let $V_{g}$ denote the handlebody of genus $g$: the three-manifold obtained by
taking a closed regular neighborhood of a polygonal, finite, connected graph
in $\mathbb{R}^{3}$. The genus of the boundary is the genus of the handlebody.
A properly embedded disk $D\subset V$ is essential if $\partial
D\subset\partial V$ is essential.
Let $\mathcal{D}(V)$ be the set of essential disks $D\subset V$, up to proper
isotopy. A subset $\Delta\subset\mathcal{D}(V)$ is a multidisk if for every
$D,E\in\Delta$ we have $\iota(\partial D,\partial E)=0$. Following McCullough
[28] we place a simplical structure on $\mathcal{D}(V)$ by taking multidisks
to be simplices. As with the curve complex, define $d_{\mathcal{D}}$ to be the
distance in the one-skeleton of $\mathcal{D}(V)$.
### 2.4. Markings
A finite subset $\mu\subset\mathcal{AC}(S)$ fills $S$ if for all
$\beta\in\mathcal{C}(S)$ there is some $\alpha\in\mu$ so that
$\iota(\alpha,\beta)>0$. For any pair of finite subsets
$\mu,\nu\subset\mathcal{AC}(S)$ we extend the intersection number:
$\iota(\mu,\nu)=\sum_{\alpha\in\mu,\beta\in\nu}\iota(\alpha,\beta).$
We say that $\mu,\nu$ are $L$–close if $\iota(\mu,\nu)\leq L$. We say that
$\mu$ is a $K$–marking if $\iota(\mu,\mu)\leq K$. For any $K,L$ we may define
$\mathcal{M}_{K,L}(S)$ to be the graph where vertices are filling $K$–markings
and edges are given by $L$–closeness.
As defined in [25] we have:
###### Definition 2.7.
A complete clean marking $\mu=\\{\alpha_{i}\\}\cup\\{\beta_{i}\\}$ consists of
* •
A collection of base curves $\operatorname{base}(\mu)=\\{\alpha_{i}\\}$: a
maximal simplex in $\mathcal{C}(S)$.
* •
A collection of transversal curves $\\{\beta_{i}\\}$: for each $i$ define
$X_{i}=S{\smallsetminus}\bigcup_{j\neq i}\alpha_{j}$ and take
$\beta_{i}\in\mathcal{C}(X_{i})$ to be a Farey neighbor of $\alpha_{i}$.
If $\mu$ is a complete clean marking then $\iota(\mu,\mu)\leq
2\xi(S)+6\chi(S)$. As discussed in [25] there are two kinds of elementary
moves which connected markings. There is a twist about a pants curve $\alpha$,
replacing its transversal $\beta$ by a new transversal $\beta^{\prime}$ which
is a Farey neighbor of both $\alpha$ and $\beta$. We can flip by swapping the
roles of $\alpha_{i}$ and $\beta_{i}$. (In the case of the flip move, some of
the other transversals must be cleaned.)
It follows that for any surface $S$ there are choices of $K,L$ so that
$\mathcal{M}(S)$ is non-empty and connected. We use $d_{\mathcal{M}}(\mu,\nu)$
to denote distance in the marking graph.
## 3\. Background on coarse geometry
Here we review a few ideas from coarse geometry. See [8], [12], or [15] for a
fuller discussion.
### 3.1. Quasi-isometry
Suppose $r,s,A$ are non-negative real numbers, with $A\geq 1$. If $s\leq
A\cdot r+A$ then we write $s\mathbin{\leq_{A}}r$. If $s\mathbin{\leq_{A}}r$
and $r\mathbin{\leq_{A}}s$ then we write $s\mathbin{=_{A}}r$ and call $r$ and
$s$ quasi-equal with constant $A$. We also define the cut-off function
$[r]_{c}$ where $[r]_{c}=0$ if $r<c$ and $[r]_{c}=r$ if $r\geq c$.
Suppose that $(\mathcal{X},d_{\mathcal{X}})$ and
$(\mathcal{Y},d_{\mathcal{Y}})$ are metric spaces. A relation
$f\colon\mathcal{X}\to\mathcal{Y}$ is an $A$–quasi-isometric embedding for
$A\geq 1$ if, for every $x,y\in\mathcal{X}$,
$d_{\mathcal{X}}(x,y)\mathbin{=_{A}}d_{\mathcal{Y}}(f(x),f(y)).$
The relation $f$ is a quasi-isometry, and $\mathcal{X}$ is quasi-isometric to
$\mathcal{Y}$, if $f$ is an $A$–quasi-isometric embedding and the image of $f$
is $A$–dense: the $A$–neighborhood of the image equals all of $\mathcal{Y}$.
### 3.2. Geodesics
Fix an interval $[u,v]\subset\mathbb{R}$. A geodesic, connecting $x$ to $y$ in
$\mathcal{X}$, is an isometric embedding $f\colon[u,v]\to\mathcal{X}$ with
$f(u)=x$ and $f(v)=y$. Often the exact choice of $f$ is unimportant and all
that matters are the endpoints $x$ and $y$. We then denote the image of $f$ by
$[x,y]\subset\mathcal{X}$.
Fix now intervals $[m,n],[p,q]\subset\mathbb{Z}$. An $A$–quasi-isometric
embedding $g\colon[m,n]\to\mathcal{X}$ is called an $A$–quasi-geodesic in
$\mathcal{X}$. A function $g\colon[m,n]\to\mathcal{X}$ is an
$A$–unparameterized quasi-geodesic in $\mathcal{X}$ if
* •
there is an increasing function $\rho\colon[p,q]\to[m,n]$ so that
$g\circ\rho\colon[p,q]\to\mathcal{X}$ is an $A$–quasi-geodesic in
$\mathcal{X}$ and
* •
for all $i\in[p,q-1]$,
$\operatorname{diam}_{\mathcal{X}}\left(g\left[\rho(i),\rho(i+1)\right]\right)\leq
A$.
(Compare to the definition of $(K,\delta,s)$–quasi-geodesics found in [24].)
A subset $\mathcal{Y}\subset\mathcal{X}$ is $Q$–quasi-convex if every
$\mathcal{X}$–geodesic connecting a pair of points of $\mathcal{Y}$ lies
within a $Q$–neighborhood of $\mathcal{Y}$.
### 3.3. Hyperbolicity
We now assume that $\mathcal{X}$ is a connected graph with metric induced by
giving all edges length one.
###### Definition 3.1.
The space $\mathcal{X}$ is $\delta$–hyperbolic if, for any three points
$x,y,z$ in $\mathcal{X}$ and for any geodesics $k=[x,y]$, $g=[y,z]$,
$h=[z,x]$, the triangle $ghk$ is $\delta$–slim: the $\delta$–neighborhood of
any two sides contains the third.
An important tool for this paper is the following theorem of the first author
and Minsky [24]:
###### Theorem 3.2.
The curve complex of an orientable surface is Gromov hyperbolic. ∎
For the remainder of this section we assume that $\mathcal{X}$ is
$\delta$–hyperbolic graph, $x,y,z\in\mathcal{X}$ are points, and
$k=[x,y],g=[y,z],h=[z,x]$ are geodesics.
###### Definition 3.3.
We take $\rho_{k}\colon\mathcal{X}\to k$ to be the closest points relation:
$\rho_{k}(z)=\big{\\{}w\in k\mathbin{\mid}\mbox{ for all $v\in k$,
$d_{\mathcal{X}}(z,w)\leq d_{\mathcal{X}}(z,v)$ }\big{\\}}.$
We now list several lemmas useful in the sequel.
###### Lemma 3.4.
There is a point on $g$ within distance $2\delta$ of $\rho_{k}(z)$. The same
holds for $h$. ∎
###### Lemma 3.5.
The closest points $\rho_{k}(z)$ have diameter at most $4\delta$. ∎
###### Lemma 3.6.
The diameter of $\rho_{g}(x)\cup\rho_{h}(y)\cup\rho_{k}(z)$ is at most
$6\delta$. ∎
###### Lemma 3.7.
Suppose that $z^{\prime}$ is another point in $\mathcal{X}$ so that
$d_{\mathcal{X}}(z,z^{\prime})\leq R$. Then
$d_{\mathcal{X}}(\rho_{k}(z),\rho_{k}(z^{\prime}))\leq R+6\delta.$ ∎
###### Lemma 3.8.
Suppose that $k^{\prime}$ is another geodesic in $X$ so that the endpoints of
$k^{\prime}$ are within distance $R$ of the points $x$ and $y$. Then
$d_{X}(\rho_{k}(z),\rho_{k^{\prime}}(z))\leq R+11\delta$. ∎
We now turn to a useful consequence of the Morse stability of quasi-geodesics
in hyperbolic spaces.
###### Lemma 3.9.
For every $\delta$ and $A$ there is a constant $C$ with the following
property: If $\mathcal{X}$ is $\delta$–hyperbolic and
$g\colon[0,N]\to\mathcal{X}$ is an $A$–unparameterized quasi-geodesic then for
any $m<n<p$ in $[0,N]$ we have:
$d_{\mathcal{X}}(x,y)+d_{\mathcal{X}}(y,z)<d_{\mathcal{X}}(x,z)+C$
where $x,y,z=g(m),g(n),g(p)$. ∎
### 3.4. A hyperbolicity criterion
Here we give a hyperbolicity criterion tailored to our setting. We thank Brian
Bowditch for both finding an error in our first proof of Theorem 3.11 and for
informing us of Gilman’s work [13, 14].
Suppose that $\mathcal{X}$ is a graph with all edge-lengths equal to one.
Suppose that $\gamma\colon[0,N]\to\mathcal{X}$ is a loop in $\mathcal{X}$ with
unit speed. Any pair of points $a,b\in[0,N]$ gives a chord of $\gamma$. If
$a<b$, $N/4\leq b-a$ and $N/4\leq a+(N-b)$ then the chord is $1/4$–separated.
The length of the chord is $d_{\mathcal{X}}(\gamma(a),\gamma(b))$.
Following Gilman [13, Theorem B] we have:
###### Theorem 3.10.
Suppose that $\mathcal{X}$ is a graph with all edge-lengths equal to one. Then
$\mathcal{X}$ is Gromov hyperbolic if and only if there is a constant $K$ so
that every loop $\gamma\colon[0,N]\to\mathcal{X}$ has a $1/4$–separated chord
of length at most $N/7+K$. ∎
Gilman’s proof goes via the subquadratic isoperimetric inequality. We now give
our criterion, noting that it is closely related to another paper of Gilman
[14].
###### Theorem 3.11.
Suppose that $\mathcal{X}$ is a graph with all edge-lengths equal to one. Then
$\mathcal{X}$ is Gromov hyperbolic if and only if there is a constant $M\geq
0$ and, for all unordered pairs $x,y\in\mathcal{X}^{0}$, there is a connected
subgraph $g_{x,y}$ containing $x$ and $y$ with the following properties:
* •
(Local) If $d_{\mathcal{X}}(x,y)\leq 1$ then $g_{x,y}$ has diameter at most
$M$.
* •
(Slim triangles) For all $x,y,z\in\mathcal{X}^{0}$ the subgraph $g_{x,y}$ is
contained in an $M$–neighborhood of $g_{y,z}\cup g_{z,x}$.
###### Proof.
Suppose that $\gamma\colon[0,N]\to\mathcal{X}$ is a loop. If $\epsilon$ is the
empty string let $I_{\epsilon}=[0,N]$. For any binary string $\omega$ let
$I_{\omega 0}$ and $I_{\omega 1}$ be the first and second half of
$I_{\omega}$. Note that if $|\omega|\geq\lceil\log_{2}N\rceil$ then
$|I_{\omega}|\leq 1$.
Fix a string $\omega$ and let $[a,b]=I_{\omega}$. Let $g_{\omega}$ be the
subgraph connecting $\gamma(a)$ to $\gamma(b)$. Note that $g_{0}=g_{1}$
because $\gamma(0)=\gamma(N)$. Also, for any binary string $\omega$ the
subgraphs $g_{\omega},g_{\omega 0},g_{\omega 1}$ form an $M$–slim triangle. If
$|\omega|\leq\lceil\log_{2}N\rceil$ then every $x\in g_{\omega}$ has some
point $b\in I_{\omega}$ so that
$d_{\mathcal{X}}(x,\gamma(b))\leq M(\lceil\log_{2}N\rceil-|\omega|)+2M.$
Since $g_{0}$ is connected there is a point $x\in g_{0}$ that lies within the
$M$–neighborhoods both of $g_{00}$ and of $g_{01}$. Pick some $b\in I_{1}$ so
that $d_{\mathcal{X}}(x,\gamma(b))$ is bounded as in the previous paragraph.
It follows that there is a point $a\in I_{0}$ so that $a,b$ are
$1/4$–separated and so that
$d_{\mathcal{X}}(\gamma(a),\gamma(b))\leq 2M\lceil\log_{2}N\rceil+2M.$
Thus there is an additive error $K$ large enough so that $\mathcal{X}$
satisfies the criterion of Theorem 3.10 and we are done. ∎
## 4\. Natural maps
There are several natural maps between the complexes and graphs defined in
Section 2. Here we review what is known about their geometric properties, and
give examples relevant to the rest of the paper.
### 4.1. Lifting, surgery, and subsurface projection
Suppose that $S$ is not simple. Choose a hyperbolic metric on the interior of
$S$ so that all ends have infinite areas. Fix a compact essential subsurface
$X\subset S$ which is not a peripheral annulus. Let $S^{X}$ be the cover of
$S$ so that $X$ lifts homeomorphically and so that
$S^{X}\mathrel{\cong}{\operatorname{interior}}(X)$. For any
$\alpha\in\mathcal{AC}(S)$ let $\alpha^{X}$ be the full preimage.
Since there is a homeomorphism between $X$ and the Gromov compactification of
$S^{X}$ in a small abuse of notation we identify $\mathcal{AC}(X)$ with the
arc and curve complex of $S^{X}$.
###### Definition 4.1.
We define the cutting relation
$\kappa_{X}\colon\mathcal{AC}(S)\to\mathcal{AC}(X)$ as follows:
$\alpha^{\prime}\in\kappa_{X}(\alpha)$ if and only if $\alpha^{\prime}$ is an
essential non-peripheral component of $\alpha^{X}$.
Note that $\alpha$ cuts $X$ if and only if $\kappa_{X}(\alpha)$ is non-empty.
Now suppose that $S$ is not an annulus.
###### Definition 4.2.
We define the surgery relation
$\sigma_{X}\colon\mathcal{AC}(S)\to\mathcal{C}(S)$ as follows:
$\alpha^{\prime}\in\sigma_{S}(\alpha)$ if and only if
$\alpha^{\prime}\in\mathcal{C}(S)$ is a boundary component of a regular
neighborhood of $\alpha\cup\partial S$.
With $S$ and $X$ as above:
###### Definition 4.3.
The subsurface projection relation
$\pi_{X}\colon\mathcal{AC}(S)\to\mathcal{C}(X)$ is defined as follows: If $X$
is not an annulus then define $\pi_{X}=\sigma_{X}\circ\kappa_{X}$. When $X$ is
an annulus $\pi_{X}=\kappa_{X}$.
If $\alpha,\beta\in\mathcal{AC}(S)$ both cut $X$ we write
$d_{X}(\alpha,\beta)=\operatorname{diam}_{X}(\pi_{X}(\alpha)\cup\pi_{X}(\beta))$.
This is the subsurface projection distance between $\alpha$ and $\beta$ in
$X$.
###### Lemma 4.4.
Suppose $\alpha,\beta\in\mathcal{AC}(S)$ are disjoint and cut $X$. Then
$\operatorname{diam}_{X}(\pi_{X}(\alpha)),d_{X}(\alpha,\beta)\leq 3$. ∎
See Lemma 2.3 of [25] and the remarks in the section Projection Bounds in
[29].
###### Corollary 4.5.
Fix $X\subset S$. Suppose that $\\{\beta_{i}\\}_{i=0}^{N}$ is a path in
$\mathcal{AC}(S)$. Suppose that $\beta_{i}$ cuts $X$ for all $i$. Then
$d_{X}(\beta_{0},\beta_{N})\leq 3N+3$. ∎
It is crucial to note that if some vertex of $\\{\beta_{i}\\}$ misses $X$ then
the projection distance $d_{X}(\beta_{0},\beta_{n})$ may be arbitrarily large
compared to $n$. Corollary 4.5 can be greatly strengthened when the path is a
geodesic [25]:
###### Theorem 4.6.
[Bounded Geodesic Image] There is constant $M_{0}$ with the following
property. Fix $X\subset S$. Suppose that $\\{\beta_{i}\\}_{i=0}^{n}$ is a
geodesic in $\mathcal{C}(S)$. Suppose that $\beta_{i}$ cuts $X$ for all $i$.
Then $d_{X}(\beta_{0},\beta_{n})\leq M_{0}$. ∎
Here is a converse for Lemma 4.4.
###### Lemma 4.7.
For every $a\in\mathbb{N}$ there is a number $b\in\mathbb{N}$ with the
following property: for any $\alpha,\beta\in\mathcal{AC}(S)$ if
$d_{X}(\alpha,\beta)\leq a$ for all $X\subset S$ then $\iota(\alpha,\beta)\leq
b$.
Corollary D of [11] gives a more precise relation between projection distance
and intersection number.
###### Proof of Lemma 4.7.
We only sketch the contrapositive: Suppose we are given a sequence of curves
$\alpha_{n},\beta_{n}$ so that $\iota(\alpha_{n},\beta_{n})$ tends to
infinity. Passing to subsequences and applying elements of the mapping class
group we may assume that $\alpha_{n}=\alpha_{0}$ for all $n$. Setting
$c_{n}=\iota(\alpha_{0},\beta_{n})$ and passing to subsequences again we may
assume that $\beta_{n}/c_{n}$ converges to $\lambda\in\mathcal{PML}(S)$, the
projectivization of Thurston’s space of measured laminations. Let $Y$ be any
connected component of the subsurface filled by $\lambda$, chosen so that
$\alpha_{0}$ cuts $Y$. Note that $\pi_{Y}(\beta_{n})$ converges to
$\lambda|_{Y}$. Again applying Kobayashi’s argument [21], the distance
$d_{Y}(\alpha_{0},\beta_{n})$ tends to infinity. ∎
### 4.2. Inclusions
We now record a well known fact:
###### Lemma 4.8.
The inclusion $\nu\colon\mathcal{C}(S)\to\mathcal{AC}(S)$ is a quasi-isometry.
The surgery map $\sigma_{S}\colon\mathcal{AC}(S)\to\mathcal{C}(S)$ is a quasi-
inverse for $\nu$.
###### Proof.
Fix $\alpha,\beta\in\mathcal{C}(S)$. Since $\nu$ is an inclusion we have
$d_{\mathcal{AC}}(\alpha,\beta)\leq d_{S}(\alpha,\beta)$. In the other
direction, let $\\{\alpha_{i}\\}_{i=0}^{N}$ be a geodesic in $\mathcal{AC}(S)$
connecting $\alpha$ to $\beta$. Since every $\alpha_{i}$ cuts $S$ we apply
Corollary 4.5 and deduce $d_{S}(\alpha,\beta)\leq 3N+3$.
Note that the composition
$\sigma_{S}\circ\nu=\operatorname{Id}|\mathcal{C}(S)$. Also, for any arc
$\alpha\in\mathcal{A}(S)$ we have
$d_{\mathcal{AC}}(\alpha,\nu(\sigma_{S}(\alpha)))=1$. Finally,
$\mathcal{C}(S)$ is $1$–dense in $\mathcal{AC}(S)$, as any arc $\gamma\subset
S$ is disjoint from the one or two curves of $\sigma_{S}(\gamma)$. ∎
Brian Bowditch raised the question, at the Newton Institute in August 2003, of
the geometric properties of the inclusion $\mathcal{A}(S)\to\mathcal{AC}(S)$.
The natural assumption, that this inclusion is again a quasi-isometric
embedding, is false. In this paper we will exactly characterize how the
inclusion distorts distance.
We now move up a dimension. Suppose that $V$ is a handlebody and $S=\partial
V$. We may take any disk $D\in\mathcal{D}(V)$ to its boundary $\partial
D\in\mathcal{C}(S)$, giving an inclusion
$\nu\colon\mathcal{D}(V)\to\mathcal{C}(S)$. It is important to distinguish the
disk complex from its image $\nu(\mathcal{D}(V))$; thus we will call the image
the disk set.
The first author and Minsky [26] have shown:
###### Theorem 4.9.
The disk set is a quasi-convex subset of the curve complex. ∎
It is natural to ask if this map is a quasi-isometric embedding. If so, the
hyperbolicity of $\mathcal{C}(V)$ immediately follows. In fact, the inclusion
again badly distorts distance and we investigate exactly how, below.
### 4.3. Markings and the mapping class group
Once the connectedness of $\mathcal{M}(S)$ is in hand, it is possible to use
local finiteness to show that $\mathcal{M}(S)$ is quasi-isometric to the
Cayley graph of the mapping class group [25].
Using subsurface projections the first author and Minsky [25] obtained a
distance estimate for the marking complex and thus for the mapping class
group.
###### Theorem 4.10.
There is a constant ${C_{0}}={C_{0}}(S)$ so that, for any $c\geq{C_{0}}$ there
is a constant $A$ with
$d_{\mathcal{M}}(\mu,\mu^{\prime})\,\,\mathbin{=_{A}}\,\,\sum[d_{X}(\mu,\mu^{\prime})]_{c}$
independent of the choice of $\mu$ and $\mu^{\prime}$. Here the sum ranges
over all essential, non-peripheral subsurfaces $X\subset S$.
This, and their similar estimate for the pants graph, is a model for the
distance estimates given below. Notice that a filling marking
$\mu\in\mathcal{M}(S)$ cuts all essential, non-peripheral subsurfaces of $S$.
It is not an accident that the sum ranges over the same set.
## 5\. Holes in general and the lower bound on distance
Suppose that $S$ is a compact connected surface. In this paper a combinatorial
complex $\mathcal{G}(S)$ will have vertices being isotopy classes of certain
multicurves in $S$. We will assume throughout that vertices of
$\mathcal{G}(S)$ are connected by edges only if there are representatives
which are disjoint. This assumption is made only to simplify the proofs — all
arguments work in the case where adjacent vertices are allowed to have
uniformly bounded intersection. In all cases $\mathcal{G}$ will be connected.
There is a natural map $\nu\colon\mathcal{G}\to\mathcal{AC}(S)$ taking a
vertex of $\mathcal{G}$ to the isotopy classes of the components. Examples in
the literature include the marking complex [25], the pants complex [9] [2],
the Hatcher-Thurston complex [19], the complex of separating curves [7], the
arc complex and the curve complexes themselves.
For any combinatorial complex $\mathcal{G}$ defined in this paper other than
the curve complex we will denote distance in the one-skeleton of $\mathcal{G}$
by $d_{\mathcal{G}}(\cdot,\cdot)$. Distance in $\mathcal{C}(S)$ will always be
denoted by $d_{S}(\cdot,\cdot)$.
### 5.1. Holes, defined
Suppose that $S$ is non-simple. Suppose that $\mathcal{G}(S)$ is a
combinatorial complex. Suppose that $X\subset S$ is an cleanly embedded
subsurface. A vertex $\alpha\in\mathcal{G}$ cuts $X$ if some component of
$\alpha$ cuts $X$.
###### Definition 5.1.
We say $X\subset S$ is a hole for $\mathcal{G}$ if every vertex of
$\mathcal{G}$ cuts $X$.
Almost equivalently, if $X$ is a hole then the subsurface projection
$\pi_{X}\colon\mathcal{G}(S)\to\mathcal{C}(X)$ never takes the empty set as a
value. Note that the entire surface $S$ is always a hole, regardless of our
choice of $\mathcal{G}$. A boundary parallel annulus cannot be cleanly
embedded (unless $S$ is also an annulus), so generally cannot be a hole. A
hole $X\subset S$ is strict if $X$ is not homeomorphic to $S$.
We now classify the holes for $\mathcal{A}(S)$.
###### Example 5.2.
Suppose that $S=S_{g,b}$ with $b>0$ and consider the arc complex
$\mathcal{A}(S)$. The holes, up to isotopy, are exactly the cleanly embedded
surfaces which contain $\partial S$. So, for example, if $S$ is planar then
only $S$ is a hole for $\mathcal{A}(S)$. The same holds for $S=S_{1,1}$. In
these cases it is an exercise to show that $\mathcal{C}(S)$ and
$\mathcal{A}(S)$ are quasi-isometric. In all other cases the arc complex
admits infinitely many holes.
###### Definition 5.3.
If $X$ is a hole and if $\pi_{X}(\mathcal{G})\subset\mathcal{C}(X)$ has
diameter at least $R$ we say that the hole $X$ has diameter at least $R$.
###### Example 5.4.
Continuing the example above: Since the mapping class group acts on the arc
complex, all non-simple holes for $\mathcal{A}(S)$ have infinite diameter.
Suppose now that $X,X^{\prime}\subset S$ are disjoint holes for $\mathcal{G}$.
In the presence of symmetry there can be a relationship between
$\pi_{X}|\mathcal{G}$ and $\pi_{X^{\prime}}|\mathcal{G}$ as follows:
###### Definition 5.5.
Suppose that $X,X^{\prime}$ are holes for $\mathcal{G}$, both of infinite
diameter. Then $X$ and $X^{\prime}$ are paired if there is a homeomorphism
$\tau\colon X\to X^{\prime}$ and a constant $L_{4}$ so that
$d_{X^{\prime}}(\pi_{X^{\prime}}(\gamma),\tau(\pi_{X}(\gamma)))\leq L_{4}$
for every $\gamma\in\mathcal{G}$. Furthermore, if $Y\subset X$ is a hole then
$\tau$ pairs $Y$ with $Y^{\prime}=\tau(Y)$. Lastly, pairing is required to be
symmetric; if $\tau$ pairs $X$ with $X^{\prime}$ then $\tau^{-1}$ pairs
$X^{\prime}$ with $X$.
###### Definition 5.6.
Two holes $X$ and $Y$ interfere if either $X\cap Y\neq\emptyset$ or $X$ is
paired with $X^{\prime}$ and $X^{\prime}\cap Y\neq\emptyset$.
Examples arise in the symmetric arc complex and in the discussion of twisted
$I$–bundles inside of a handlebody.
### 5.2. Projection to holes is coarsely Lipschitz
The following lemma is used repeatedly throughout the paper:
###### Lemma 5.7.
Suppose that $\mathcal{G}(S)$ is a combinatorial complex. Suppose that $X$ is
a hole for $\mathcal{G}$. Then for any $\alpha,\beta\in\mathcal{G}$ we have
$d_{X}(\alpha,\beta)\leq 3+3\cdot d_{\mathcal{G}}(\alpha,\beta).$
The additive error is required only when $\alpha=\beta$.
###### Proof.
This follows directly from Corollary 4.5 and our assumption that vertices of
$\mathcal{G}$ connected by an edge represent disjoint multicurves. ∎
### 5.3. Infinite diameter holes
We may now state a first answer to Bowditch’s question.
###### Lemma 5.8.
Suppose that $\mathcal{G}(S)$ is a combinatorial complex. Suppose that there
is a strict hole $X\subset S$ having infinite diameter. Then
$\nu\colon\mathcal{G}\to\mathcal{AC}(S)$ is not a quasi-isometric embedding. ∎
This lemma and Example 5.2 completely determines when the inclusion of
$\mathcal{A}(S)$ into $\mathcal{AC}(S)$ is a quasi-isometric embedding. It
quickly becomes clear that the set of holes tightly constrains the intrinsic
geometry of a combinatorial complex.
###### Lemma 5.9.
Suppose that $\mathcal{G}(S)$ is a combinatorial complex invariant under the
natural action of $\mathcal{MCG}(S)$. Then every non-simple hole for
$\mathcal{G}$ has infinite diameter. Furthermore, if $X,Y\subset S$ are
disjoint non-simple holes for $\mathcal{G}$ then there is a quasi-isometric
embedding of $\mathbb{Z}^{2}$ into $\mathcal{G}$. ∎
We will not use Lemmas 5.8 or 5.9 and so omit the proofs. Instead our interest
lies in proving the far more powerful distance estimate (Theorems 5.10 and
13.1) for $\mathcal{G}(S)$.
### 5.4. A lower bound on distance
Here we see that the sum of projection distances in holes gives a lower bound
for distance.
###### Theorem 5.10.
Fix $S$, a compact connected non-simple surface. Suppose that $\mathcal{G}(S)$
is a combinatorial complex. Then there is a constant ${C_{0}}$ so that for all
$c\geq{C_{0}}$ there is a constant $A$ satisfying
$\sum[d_{X}(\alpha,\beta)]_{c}\mathbin{\leq_{A}}d_{\mathcal{G}}(\alpha,\beta).$
Here $\alpha,\beta\in\mathcal{G}$ and the sum is taken over all holes $X$ for
the complex $\mathcal{G}$. ∎
The proof follows the proof of Theorems 6.10 and 6.12 of [25], practically
word for word. The only changes necessary are to
* •
replace the sum over all subsurfaces by the sum over all holes,
* •
replace Lemma 2.5 of [25], which records how markings differing by an
elementary move project to an essential subsurface, by Lemma 5.7 of this
paper, which records how $\mathcal{G}$ projects to a hole.
One major goal of this paper is to give criteria sufficient obtain the reverse
inequality; Theorem 13.1.
## 6\. Holes for the non-orientable surface
Fix $F$ a compact, connected, and non-orientable surface. Let $S$ be the
orientation double cover with covering map $\rho_{F}\colon S\to F$. Let
$\tau\colon S\to S$ be the associated involution; so for all $x\in S$,
$\rho_{F}(x)=\rho_{F}(\tau(x))$.
###### Definition 6.1.
A multicurve $\gamma\subset\mathcal{AC}(S)$ is symmetric if
$\tau(\gamma)\cap\gamma=\emptyset$ or $\tau(\gamma)=\gamma$. A multicurve
$\gamma$ is invariant if there is a curve or arc $\gamma^{\prime}\subset F$ so
that $\gamma=\rho_{F}^{-1}(\gamma^{\prime})$. The same definitions holds for
subsurfaces $X\subset S$.
###### Definition 6.2.
The invariant complex $\mathcal{C}^{\tau}(S)$ is the simplicial complex with
vertex set being isotopy classes of invariant multicurves. There is a
$k$–simplex for every collection of $k+1$ distinct isotopy classes having
pairwise disjoint representatives.
Notice that $\mathcal{C}^{\tau}(S)$ is simplicially isomorphic to
$\mathcal{C}(F)$. There is also a natural map
$\nu\colon\mathcal{C}^{\tau}(S)\to\mathcal{C}(S)$. We will prove:
###### Lemma 6.3.
$\nu\colon\mathcal{C}^{\tau}(S)\to\mathcal{C}(S)$ is a quasi-isometric
embedding.
It thus follows from the hyperbolicity of $\mathcal{C}(S)$ that:
###### Corollary 6.4 ([4]).
$\mathcal{C}(F)$ is Gromov hyperbolic. ∎
We begin the proof of Lemma 6.3: since $\nu$ sends adjacent vertices to
adjacent edges we have
(6.5) $d_{S}(\alpha,\beta)\leq d_{\mathcal{C}^{\tau}}(\alpha,\beta),$
as long as $\alpha$ and $\beta$ are distinct in $\mathcal{C}^{\tau}(S)$. In
fact, since the surface $S$ itself is a hole for $\mathcal{C}^{\tau}(S)$ we
may deduce a slightly weaker lower bound from Lemma 5.7 or indeed from Theorem
5.10.
The other half of the proof of Lemma 6.3 consists of showing that $S$ is the
only hole for $\mathcal{C}^{\tau}(S)$ with large diameter. After a discussion
of Teichmüller geodesics we will prove:
There is a constant $K$ with the following property: Suppose that
$\alpha,\beta$ are invariant multicurves in $S$. Suppose that $X\subset S$ is
an essential subsurface where $d_{X}(\alpha,\beta)>K$. Then $X$ is symmetric.
From this it follows that:
###### Corollary 6.6.
With $K$ as in Lemma 16.4: If $X\subset S$ is a hole for
$\mathcal{C}^{\tau}(S)$ with diameter greater than $K$ then $X=S$.
###### Proof.
Suppose that $X\subset S$ is a strict subsurface, cleanly embedded. Suppose
that $\operatorname{diam}_{X}(\mathcal{C}^{\tau}(S))>K$. Thus $X$ is
symmetric. It follows that $\partial X{\smallsetminus}\partial S$ is also
symmetric. Since $\partial X$ does not cut $X$ deduce that $X$ is not a hole
for $\mathcal{C}^{\tau}(S)$. ∎
This corollary, together with the upper bound (Theorem 13.1), proves Lemma
6.3.
## 7\. Holes for the arc complex
Here we generalize the definition of the arc complex and classify its holes.
###### Definition 7.1.
Suppose that $S$ is a non-simple surface with boundary. Let $\Delta$ be a non-
empty collection of components of $\partial S$. The arc complex
$\mathcal{A}(S,\Delta)$ is the subcomplex of $\mathcal{A}(S)$ spanned by
essential arcs $\alpha\subset S$ with $\partial\alpha\subset\Delta$.
Note that $\mathcal{A}(S,\partial S)$ and $\mathcal{A}(S)$ are identical.
###### Lemma 7.2.
Suppose $X\subset S$ is cleanly embedded. Then $X$ is a hole for
$\mathcal{A}(S,\Delta)$ if and only if $\Delta\subset\partial X$. ∎
This follows directly from the definition of a hole. We now have an straight-
forward observation:
###### Lemma 7.3.
If $X,Y\subset S$ are holes for $\mathcal{A}(S,\Delta)$ then $X\cap
Y\neq\emptyset$. ∎
The proof follows immediately from Lemma 7.2. Lemma 5.9 indicates that Lemma
7.3 is essential to proving that $\mathcal{A}(S,\Delta)$ is Gromov hyperbolic.
In order to prove the upper bound theorem for $\mathcal{A}$ we will use pants
decompositions of the surface $S$. In an attempt to avoid complications in the
non-orientable case we must carefully lift to the orientation cover.
Suppose that $F$ is non-simple, non-orientable, and has non-empty boundary.
Let $\rho_{F}\colon S\to F$ be the orientation double cover and let
$\tau\colon S\to S$ be the induced involution. Fix
$\Delta^{\prime}\subset\partial F$ and let
$\Delta=\rho_{F}^{-1}(\Delta^{\prime})$.
###### Definition 7.4.
We define $\mathcal{A}^{\tau}(S,\Delta)$ to be the invariant arc complex:
vertices are invariant multi-arcs and simplices arise from disjointness.
Again, $\mathcal{A}^{\tau}(S,\Delta)$ is simplicially isomorphic to
$\mathcal{A}(F,\Delta^{\prime})$. If $X\cap\tau(X)=\emptyset$ and
$\Delta\subset X\cup\tau(X)$ then the subsurfaces $X$ and $\tau(X)$ are paired
holes, as in Definition 5.5. Notice as well that all non-simple symmetric
holes $X\subset S$ for $\mathcal{A}^{\tau}(S,\Delta)$ have infinite diameter.
Unlike $\mathcal{A}(F,\Delta^{\prime})$ the complex
$\mathcal{A}^{\tau}(S,\Delta)$ may have disjoint holes. Nonetheless, we have:
###### Lemma 7.5.
Any two non-simple holes for $\mathcal{A}^{\tau}(S,\Delta)$ interfere.
###### Proof.
Suppose that $X,Y$ are holes for the $\tau$–invariant arc complex,
$\mathcal{A}^{\tau}(S,\Delta)$. It follows from Lemma 16.4 that $X$ is
symmetric with $\Delta\subset X\cup\tau(X)$. The same holds for $Y$. Thus $Y$
must cut either $X$ or $\tau(X)$. ∎
## 8\. Background on three-manifolds
Before discussing the holes in the disk complex, we record a few facts about
handlebodies and $I$–bundles.
Fix $M$ a compact connected irreducible three-manifold. Recall that $M$ is
irreducible if every embedded two-sphere in $M$ bounds a three-ball. Recall
that if $N$ is a closed submanifold of $M$ then $\operatorname{fr}(N)$, the
frontier of $N$ in $M$, is the closure of $\partial N{\smallsetminus}\partial
M$.
### 8.1. Compressions
Suppose that $F$ is a surface embedded in $M$. Then $F$ is compressible if
there is a disk $B$ embedded in $M$ with $B\cap\partial M=\emptyset$, $B\cap
F=\partial B$, and $\partial B$ essential in $F$. Any such disk $B$ is called
a compression of $F$.
In this situation form a new surface $F^{\prime}$ as follows: Let $N$ be a
closed regular neighborhood of $B$. First remove from $F$ the annulus $N\cap
F$. Now form $F^{\prime}$ by gluing on both disk components of $\partial
N{\smallsetminus}F$. We say that $F^{\prime}$ is obtained by compressing $F$
along $B$. If no such disk exists we say $F$ is incompressible.
###### Definition 8.1.
A properly embedded surface $F$ is boundary compressible if there is a disk
$B$ embedded in $M$ with
* •
${\operatorname{interior}}(B)\cap\partial M=\emptyset$,
* •
$\partial B$ is a union of connected arcs $\alpha$ and $\beta$,
* •
$\alpha\cap\beta=\partial\alpha=\partial\beta$,
* •
$B\cap F=\alpha$ and $\alpha$ is properly embedded in $F$,
* •
$B\cap\partial M=\beta$, and
* •
$\beta$ is essential in $\partial M{\smallsetminus}\partial F$.
A disk, like $B$, with boundary partitioned into two arcs is called a bigon.
Note that this definition of boundary compression is slightly weaker than some
found in the literature; the arc $\alpha$ is often required to be essential in
$F$. We do not require this additional property because, for us, $F$ will
usually be a properly embedded disk in a handlebody.
Just as for compressing disks we may boundary compress $F$ along $B$ to obtain
a new surface $F^{\prime}$: Let $N$ be a closed regular neighborhood of $B$.
First remove from $F$ the rectangle $N\cap F$. Now form $F^{\prime}$ by gluing
on both bigon components of $\operatorname{fr}(N){\smallsetminus}F$. Again,
$F^{\prime}$ is obtained by boundary compressing $F$ along $B$. Note that the
relevant boundary components of $F$ and $F^{\prime}$ cobound a pair of pants
embedded in $\partial M$. If no boundary compression exists then $F$ is
boundary incompressible.
###### Remark 8.2.
Recall that any surface $F$ properly embedded in a handlebody $V_{g}$, $g\geq
2$, is either compressible or boundary compressible.
Suppose now that $F$ is properly embedded in $M$ and $\Gamma$ is a multicurve
in $\partial M$.
###### Remark 8.3.
Suppose that $F^{\prime}$ is obtained by a boundary compression of $F$
performed in the complement of $\Gamma$. Suppose that $F^{\prime}=F_{1}\cap
F_{2}$ is disconnected and each $F_{i}$ cuts $\Gamma$. Then $\iota(\partial
F_{i},\Gamma)<\iota(\partial F,\Gamma)$ for $i=1,2$.
It is often useful to restrict our attention to boundary compressions meeting
a single subsurface of $\partial M$. So suppose that $X\subset\partial M$ is
an essential subsurface. Suppose that $\partial F$ is tight with respect to
$\partial X$. Suppose $B$ is a boundary compression of $F$. If $B\cap\partial
M\subset X$ we say that $F$ is boundary compressible into $X$.
###### Lemma 8.4.
Suppose that $M$ is irreducible. Fix $X$ a connected essential subsurface of
$\partial M$. Let $F\subset M$ be a properly embedded, incompressible surface.
Suppose that $\partial X$ and $\partial F$ are tight and that $X$ compresses
in $M$. Then either:
* •
$F\cap X=\emptyset$,
* •
$F$ is boundary compressible into $X$, or
* •
$F$ is a disk with $\partial F\subset X$.
###### Proof.
Suppose that $X$ is compressible via a disk $E$. Isotope $E$ to make $\partial
E$ tight with respect to $\partial F$. This can be done while maintaining
$\partial E\subset X$ because $\partial F$ and $\partial X$ are tight. Since
$M$ is irreducible and $F$ is incompressible we may isotope $E$, rel
$\partial$, to remove all simple closed curves of $F\cap E$. If $F\cap E$ is
non-empty then an outermost bigon of $E$ gives the desired boundary
compression lying in $X$.
Suppose instead that $F\cap E=\emptyset$ but $F$ does cut $X$. Let
$\delta\subset X$ be a simple arc meeting each of $F$ and $E$ in exactly one
endpoint. Let $N$ be a closed regular neighborhood of $\delta\cup E$. Note
that $\operatorname{fr}(N){\smallsetminus}F$ has three components. One is a
properly embedded disk parallel to $E$ and the other two $B,B^{\prime}$ are
bigons attached to $F$. At least one of these, say $B^{\prime}$ is trivial in
the sense that $B^{\prime}\cap\partial M$ is a trivial arc embedded in
$\partial M{\smallsetminus}\partial F$. If $B$ is non-trivial then $B$
provides the desired boundary compression.
Suppose that $B$ is also trivial. It follows that $\partial E$ and one
component $\gamma\subset\partial F$ cobound an annulus $A\subset X$. So
$D=A\cup E$ is a disk with $(D,\partial D)\subset(M,F)$. As $\partial
D=\gamma$ and $F$ is incompressible and $M$ is irreducible deduce that $F$ is
isotopic to $E$. ∎
### 8.2. Band sums
A band sum is the inverse operation to boundary compression: Fix a pair of
disjoint properly embedded surfaces $F_{1},F_{2}\subset M$. Let
$F^{\prime}=F_{1}\cup F_{2}$. Fix a simple arc $\delta\subset\partial M$ so
that $\delta$ meets each of $F_{1}$ and $F_{2}$ in exactly one point of
$\partial\delta$. Let $N\subset M$ be a closed regular neighborhood of
$\delta$. Form a new surface by adding to $F^{\prime}{\smallsetminus}N$ the
rectangle component of $\operatorname{fr}(N){\smallsetminus}F^{\prime}$. The
surface $F$ obtained is the result of band summing $F_{1}$ to $F_{2}$ along
$\delta$. Note that $F$ has a boundary compression dual to $\delta$ yielding
$F^{\prime}$: that is, there is a boundary compression $B$ for $F$ so that
$\delta\cap B$ is a single point and compressing $F$ along $B$ gives
$F^{\prime}$.
### 8.3. Handlebodies and I-bundles
Recall that handlebodies are irreducible.
Suppose that $F$ is a compact connected surface with at least one boundary
component. Let $T$ be the orientation $I$–bundle over $F$. If $F$ is
orientable then $T\mathrel{\cong}F\times I$. If $F$ is not orientable then $T$
is the unique $I$–bundle over $F$ with orientable total space. We call $T$ the
$I$–bundle and $F$ the base space. Let $\rho_{F}\colon T\to F$ be the
associated bundle map. Note that $T$ is homeomorphic to a handlebody.
If $A\subset T$ is a union of fibers of the map $\rho_{F}$ then $A$ is
vertical with respect to $T$. In particular take
$\partial_{v}T=\rho_{F}^{-1}(\partial F)$ to be the vertical boundary of $T$.
Take $\partial_{h}T$ to be the union of the boundaries of all of the fibers:
this is the horizontal boundary of $T$. Note that $\partial_{h}T$ is always
incompressible in $T$ while $\partial_{v}T$ is incompressible in $T$ as long
as $F$ is not homeomorphic to a disk.
Note that, as $|\partial_{v}T|\geq 1$, any vertical surface in $T$ can be
boundary compressed. However no vertical surface in $T$ may be boundary
compressed into $\partial_{h}T$.
We end this section with:
###### Lemma 8.5.
Suppose that $F$ is a compact, connected surface with $\partial
F\neq\emptyset$. Let $\rho_{F}\colon T\to F$ be the orientation $I$–bundle
over $F$. Let $X$ be a component of $\partial_{h}T$. Let $D\subset T$ be a
properly embedded disk. If
* •
$\partial D$ is essential in $\partial T$,
* •
$\partial D$ and $\partial X$ are tight, and
* •
$D$ cannot be boundary compressed into $X$
then $D$ may be properly isotoped to be vertical with respect to $T$. ∎
## 9\. Holes for the disk complex
Here we begin to classify the holes for the disk complex, a more difficult
analysis than that of the arc complex. To fix notation let $V$ be a
handlebody. Let $S=S_{g}=\partial V$. Recall that there is a natural inclusion
$\nu\colon\mathcal{D}(V)\to\mathcal{C}(S)$.
###### Remark 9.1.
The notion of a hole $X\subset\partial V$ for $\mathcal{D}(V)$ may be phrased
in several different ways:
* •
every essential disk $D\subset V$ cuts the surface $X$,
* •
$\overline{S{\smallsetminus}X}$ is incompressible in $V$, or
* •
$X$ is disk-busting in $V$.
The classification of holes $X\subset S$ for $\mathcal{D}(V)$ breaks roughly
into three cases: either $X$ is an annulus, is compressible in $V$, or is
incompressible in $V$. In each case we obtain a result:
Suppose $X$ is a hole for $\mathcal{D}(V)$ and $X$ is an annulus. Then the
diameter of $X$ is at most $5$.
Suppose $X$ is a compressible hole for $\mathcal{D}(V)$ with diameter at least
$15$. Then there are a pair of essential disks $D,E\subset V$ so that
* •
$\partial D,\partial E\subset X$ and
* •
$\partial D$ and $\partial E$ fill $X$.
Suppose $X$ is an incompressible hole for $\mathcal{D}(V)$ with diameter at
least $61$. Then there is an $I$–bundle $\rho_{F}\colon T\to F$ embedded in
$V$ so that
* •
$\partial_{h}T\subset S$,
* •
$X$ is isotopic in $S$ to a component of $\partial_{h}T$,
* •
some component of $\partial_{v}T$ is boundary parallel into $S$,
* •
$F$ supports a pseudo-Anosov map.
As a corollary of these theorems we have:
###### Corollary 9.2.
If $X$ is hole for $\mathcal{D}(V)$ with diameter at least $61$ then $X$ has
infinite diameter.
###### Proof.
If $X$ is a hole with diameter at least $61$ then either Theorem 11.6 or
Theorem 12.1 applies.
If $X$ is compressible then Dehn twists, in opposite directions, about the
given disks $D$ and $E$ yields an automorphism $f\colon V\to V$ so that $f|X$
is pseudo-Anosov. This follows from Thurston’s construction [38]. By Lemma 2.6
the hole $X$ has infinite diameter.
If $X$ is incompressible then $X\subset\partial_{h}T$ where $\rho_{F}\colon
T\to F$ is the given $I$–bundle. Let $f\colon F\to F$ be the given pseudo-
Anosov map. So $g$, the suspension of $f$, gives a automorphism of $V$. Again
it follows that the hole $X$ has infinite diameter. ∎
Applying Lemma 5.8 we find another corollary:
###### Theorem 9.3.
If $S=\partial V$ contains a strict hole with diameter at least $61$ then the
inclusion $\nu\colon\mathcal{D}(V)\to\mathcal{C}(S)$ is not a quasi-isometric
embedding. ∎
## 10\. Holes for the disk complex – annuli
The proof of Theorem 10.1 occupies the rest of this section. This proof shares
many features with the proofs of Theorems 11.6 and 12.1. However, the
exceptional definition of $\mathcal{C}(S_{0,2})$ prevents a unified approach.
Fix $V$, a handlebody.
###### Theorem 10.1.
Suppose $X$ is a hole for $\mathcal{D}(V)$ and $X$ is an annulus. Then the
diameter of $X$ is at most $5$.
We begin with:
###### Claim.
For all $D\in\mathcal{D}(V)$, $|D\cap X|\geq 2$.
###### Proof.
Since $X$ is a hole, every disk cuts $X$. Since $X$ is an annulus, let
$\alpha$ be a core curve for $X$. If $|D\cap X|=1$, then we may band sum
parallel copies of $D$ along an subarc of $\alpha$. The resulting disk misses
$\alpha$, a contradiction. ∎
Assume, to obtain a contradiction, that $X$ has diameter at least $6$. Suppose
that $D\in\mathcal{D}(V)$ is a disk chosen to minimize $D\cap X$. Among all
disks $E\in\mathcal{D}(V)$ with $d_{X}(D,E)\geq 3$ choose one which minimizes
$|D\cap E|$. Isotope $D$ and $E$ to make the boundaries tight and also tight
with respect to $\partial X$. Tightening triples of curves is not canonical;
nonetheless there is a tightening so that $S{\smallsetminus}(\partial
D\cup\partial E\cup X)$ contains no triangles. See Figure 1.
$\begin{array}[]{ccc}\includegraphics[height=3.5cm]{badtriangles}&&\includegraphics[height=3.5cm]{goodtriangles}\\\
\end{array}$ Figure 1. Triangles outside of $X$ (see the left side) can be
moved in (see the right side). This decreases the number of points of $D\cap
E\cap(S{\smallsetminus}X)$.
After this tightening we have:
###### Claim.
Every arc of $\partial D\cap X$ meets every arc of $\partial E\cap X$ at least
once.
###### Proof.
Fix components arcs $\alpha\subset D\cap X$ and $\beta\subset E\cap X$. Let
$\alpha^{\prime},\beta^{\prime}$ be the corresponding arcs in $S^{X}$ the
annular cover of $S$ corresponding to $X$. After the tightening we find that
$|\alpha\cap\beta|\geq|\alpha^{\prime}\cap\beta^{\prime}|-1.$
Since $d_{X}(D,E)\geq 3$ Equation 2.3 implies that
$|\alpha^{\prime}\cap\beta^{\prime}|\geq 2$. Thus $|\alpha\cap\beta|\geq 1$,
as desired. ∎
###### Claim.
There is an outermost bigon $B\subset E{\smallsetminus}D$ with the following
properties:
* •
$\partial B=\alpha\cup\beta$ where $\alpha=B\cap D$, $\beta=\partial
B{\smallsetminus}\alpha\subset\partial E$,
* •
$\partial\alpha=\partial\beta\subset X$, and
* •
$|\beta\cap X|=2$.
Furthermore, $|D\cap X|=2$.
See the lower right of Figure 2 for a picture.
###### Proof.
Consider the intersection of $D$ and $E$, thought of as a collection of arcs
and curves in $E$. Any simple closed curve component of $D\cap E$ can be
removed by an isotopy of $E$, fixed on the boundary. (This follows from the
irreducibility of $V$ and an innermost disk argument.) Since we have assumed
that $|D\cap E|$ is minimal it follows that there are no simple closed curves
in $D\cap E$.
So consider any outermost bigon $B\subset E{\smallsetminus}D$. Let
$\alpha=B\cap D$. Let $\beta=\partial B{\smallsetminus}\alpha=B\cap\partial
V$. Note that $\beta$ cannot completely contain a component of $E\cap X$ as
this would contradict either the fact that $B$ is outermost or the claim that
every arc of $E\cap X$ meets some arc of $D\cap X$. Using this observation,
Figure 2 lists the possible ways for $B$ to lie inside of $E$.
2pt $E$ [bl] at 22 2 $\alpha$ [tl] at 78 30
$\begin{array}[]{cc}\includegraphics[height=3.2cm]{possiblealphas1}\end{array}$
Figure 2. The arc $\alpha$ cuts a bigon $B$ off of $E$. The darker part of
$\partial E$ are the arcs of $E\cap X$. Either $\beta$ is disjoint from $X$,
$\beta$ is contained in $X$, $\beta$ meets $X$ in a single subarc, or $\beta$
meets $X$ in two subarcs.
Let $D^{\prime}$ and $D^{\prime\prime}$ be the two essential disks obtained by
boundary compressing $D$ along the bigon $B$. Suppose $\alpha$ is as shown in
one of the first three pictures of Figure 2. It follows that either
$D^{\prime}$ or $D^{\prime\prime}$ has, after tightening, smaller intersection
with $X$ than $D$ does, a contradiction. We deduce that $\alpha$ is as
pictured in lower right of Figure 2.
Boundary compressing $D$ along $B$ still gives disks
$D^{\prime},D^{\prime\prime}\in\mathcal{D}(V)$. As these cannot have smaller
intersection with $X$ we deduce that $|D\cap X|\leq 2$ and the claim holds. ∎
Using the same notation as in the proof above, let $B$ be an outermost bigon
of $E{\smallsetminus}D$. We now study how $\alpha\subset\partial B$ lies
inside of $D$.
###### Claim.
The arc $\alpha\subset D$ connects distinct components of $D\cap X$.
###### Proof.
Suppose not. Then there is a bigon $C\subset D{\smallsetminus}\alpha$ with
$\partial C=\alpha\cup\gamma$ and $\gamma\subset\partial D\cap X$. The disk
$C\cup B$ is essential and intersects $X$ at most once after tightening,
contradicting our first claim. ∎
We finish the proof of Theorem 10.1 by noting that $D\cup B$ is homeomorphic
to $\Upsilon\times I$ where $\Upsilon$ is the simplicial tree with three edges
and three leaves. We may choose the homeomorphism so that $(D\cup B)\cap
X=\Upsilon\times\partial I$. It follows that we may properly isotope $D\cup B$
until $(D\cup B)\cap X$ is a pair of arcs. Recall that $D^{\prime}$ and
$D^{\prime\prime}$ are the disks obtained by boundary compressing $D$ along
$B$. It follows that one of $D^{\prime}$ or $D^{\prime\prime}$ (or both) meets
$X$ in at most a single arc, contradicting our first claim. ∎
## 11\. Holes for the disk complex – compressible
The proof of Theorem 11.6 occupies the second half of this section.
### 11.1. Compression sequences of essential disks
Fix a multicurve $\Gamma\subset S=\partial V$. Fix also an essential disk
$D\subset V$. Properly isotope $D$ to make $\partial D$ tight with respect to
$\Gamma$.
If $D\cap\Gamma\neq\emptyset$ we may define:
###### Definition 11.1.
A compression sequence $\\{\Delta_{k}\\}_{k=1}^{n}$ starting at $D$ has
$\Delta_{1}=\\{D\\}$ and $\Delta_{k+1}$ is obtained from $\Delta_{k}$ via a
boundary compression, disjoint from $\Gamma$, and tightening. Note that
$\Delta_{k}$ is a collection of exactly $k$ pairwise disjoint disks properly
embedded in $V$. We further require, for $k\leq n$, that every disk of
$\Delta_{k}$ meets some component of $\Gamma$. We call a compression sequence
maximal if either
* •
no disk of $\Delta_{n}$ can be boundary compressed into
$S{\smallsetminus}\Gamma$ or
* •
there is a component $Z\subset S{\smallsetminus}\Gamma$ and a boundary
compression of $\Delta_{n}$ into $S{\smallsetminus}\Gamma$ yielding an
essential disk $E$ with $\partial E\subset Z$.
We say that such maximal sequences end essentially or end in $Z$,
respectively.
All compression sequences must end, by Remark 8.3. Given a maximal sequence we
may relate the various disks in the sequence as follows:
###### Definition 11.2.
Fix $X$, a component of $S{\smallsetminus}\Gamma$. Fix $D_{k}\in\Delta_{k}$. A
disjointness pair for $D_{k}$ is an ordered pair $(\alpha,\beta)$ of essential
arcs in $X$ where
* •
$\alpha\subset D_{k}\cap X$,
* •
$\beta\subset\Delta_{n}\cap X$, and
* •
$d_{\mathcal{A}}(\alpha,\beta)\leq 1$.
If $\alpha\neq\alpha^{\prime}$ then the two disjointness pairs
$(\alpha,\beta)$ and $(\alpha^{\prime},\beta)$ are distinct, even if $\alpha$
is properly isotopic to $\alpha^{\prime}$. A similar remark holds for the
second coordinate.
The following lemma controls how subsurface projection distance changes in
maximal sequences.
###### Lemma 11.3.
Fix a multicurve $\Gamma\subset S$. Suppose that $D$ cuts $\Gamma$ and choose
a maximal sequence starting at $D$. Fix any component $X\subset
S{\smallsetminus}\Gamma$. Fix any disk $D_{k}\in\Delta_{k}$. Then either
$D_{k}\in\Delta_{n}$ or there are four distinct disjointness pairs
$\\{(\alpha_{i},\beta_{i})\\}_{i=1}^{4}$ for $D_{k}$ in $X$ where each of the
arcs $\\{\alpha_{i}\\}$ appears as the first coordinate of at most two pairs.
###### Proof.
We induct on $n-k$. If $D_{k}$ is contained in $\Delta_{n}$ there is nothing
to prove. If $D_{k}$ is contained in $\Delta_{k+1}$ we are done by induction.
Thus we may assume that $D_{k}$ is the disk of $\Delta_{k}$ which is boundary
compressed at stage $k$. Let $D_{k+1},D_{k+1}^{\prime}\in\Delta_{k+1}$ be the
two disks obtained after boundary compressing $D_{k}$ along the bigon $B$. See
Figure 3 for a picture of the pair of pants cobounded by $\partial D_{k}$ and
$\partial D_{k+1}\cup\partial D_{k+1}^{\prime}$.
2pt $\delta$ [bl] at 82 55.5 $D_{k}$ [bl] at 184 95 $D_{k+1}$ [bl] at 40 49
$D^{\prime}_{k+1}$ [bl] at 148.5 47.5 $\Gamma$ [l] at 120 28
$\begin{array}[]{c}\includegraphics[height=3.5cm]{pants}\end{array}$
Figure 3. All arcs connecting $D_{k}$ to itself or to $D_{k+1}\cup
D^{\prime}_{k+1}$ are arcs of $\Gamma\cap P$. The boundary compressing arc
$B\cup S$ meets $D_{k}$ twice and is parallel to the vertical arcs of
$\Gamma\cap P$.
Let $\delta$ be a band sum arc dual to $B$ (the dotted arc in Figure 3). We
may assume that $|\Gamma\cap\delta|$ is minimal over all arcs dual to $B$. It
follows that the band sum of $D_{k+1}$ with $D_{k+1}^{\prime}$ along $\delta$
is tight, without any isotopy. (This is where we use the fact that $B$ is a
boundary compression in the complement of $\Gamma$, as opposed to being a
general boundary compression of $D_{k}$ in $V$.)
There are now three possibilities: neither, one, or both points of
$\partial\delta$ are contained in $X$.
First suppose that $X\cap\partial\delta=\emptyset$. Then every arc of
$D_{k+1}\cap X$ is parallel to an arc of $D_{k}\cap X$, and similarly for
$D_{k+1}^{\prime}$. If $D_{k+1}$ and $D_{k+1}^{\prime}$ are both components of
$\Delta_{n}$ then choose any arcs $\beta,\beta^{\prime}$ of $D_{k+1}\cap X$
and of $D_{k+1}^{\prime}\cap X$. Let $\alpha,\alpha^{\prime}$ be the parallel
components of $D_{k}\cap X$. The four disjointness pairs are then
$(\alpha,\beta)$, $(\alpha,\beta^{\prime})$, $(\alpha^{\prime},\beta)$,
$(\alpha^{\prime},\beta^{\prime})$. Suppose instead that $D_{k+1}$ is not a
component of $\Delta_{n}$. Then $D_{k}$ inherits four disjointness pairs from
$D_{k+1}$.
Second suppose that exactly one endpoint $x\in\partial\delta$ meets $X$. Let
$\gamma\subset D_{k+1}$ be the component of $D_{k+1}\cap X$ containing $x$.
Let $X^{\prime}$ be the component of $X\cap P$ that contains $x$ and let
$\alpha,\alpha^{\prime}$ be the two components of $D_{k}\cap X^{\prime}$. Let
$\beta$ be any arc of $D_{k+1}^{\prime}\cap X$.
If $D_{k+1}\mathbin{\notin}\Delta_{n}$ and $\gamma$ is not the first
coordinate of one of $D_{k+1}$’s four pairs then $D_{k}$ inherits disjointness
pairs from $D_{k+1}$. If $D_{k+1}^{\prime}\mathbin{\notin}\Delta_{n}$ then
$D_{k}$ inherits disjointness pairs from $D_{k+1}^{\prime}$.
Thus we may assume that both $D_{k+1}$ and $D_{k+1}^{\prime}$ are in
$\Delta_{n}$ or that only $D_{k+1}^{\prime}\in\Delta_{n}$ while $\gamma$
appears as the first arc of disjointness pair for $D_{k+1}$. In case of the
former the required disjointness pairs are $(\alpha,\beta)$,
$(\alpha^{\prime},\beta)$, $(\alpha,\gamma)$, and $(\alpha^{\prime},\gamma)$.
In case of the latter we do not know if $\gamma$ is allowed to appear as the
second coordinate of a pair. However we are given four disjointness pairs for
$D_{k+1}$ and are told that $\gamma$ appears as the first coordinate of at
most two of these pairs. Hence the other two pairs are inherited by $D_{k}$.
The pairs $(\alpha,\beta)$ and $(\alpha^{\prime},\beta)$ give the desired
conclusion.
Third suppose that the endpoints of $\delta$ meet $\gamma\subset D_{k+1}$ and
$\gamma^{\prime}\subset D_{k+1}^{\prime}$. Let $X^{\prime}$ be a component of
$X\cap P$ containing $\gamma$. Let $\alpha$ and $\alpha^{\prime}$ be the two
arcs of $D_{k}\cap X^{\prime}$. Suppose both $D_{k+1}$ and $D_{k+1}^{\prime}$
lie in $\Delta_{n}$. Then the desired pairs are $(\alpha,\gamma)$,
$(\alpha^{\prime},\gamma)$, $(\alpha,\gamma^{\prime})$, and
$(\alpha^{\prime},\gamma^{\prime})$. If $D_{k+1}^{\prime}\in\Delta_{n}$ while
$D_{k+1}$ is not then $D_{k}$ inherits two pairs from $D_{k+1}$. We add to
these the pairs $(\alpha,\gamma^{\prime})$, and
$(\alpha^{\prime},\gamma^{\prime})$. If neither disk lies in $\Delta_{n}$ then
$D_{k}$ inherits two pairs from each disk and the proof is complete. ∎
Given a disk $D\in\mathcal{D}(V)$ and a hole $X\subset S$ our Lemma 11.3
allows us to adapt $D$ to $X$.
###### Lemma 11.4.
Fix a hole $X\subset S$ for $\mathcal{D}(V)$. For any disk
$D\in\mathcal{D}(V)$ there is a disk $D^{\prime}$ with the following
properties:
* •
$\partial X$ and $\partial D^{\prime}$ are tight.
* •
If $X$ is incompressible then $D^{\prime}$ is not boundary compressible into
$X$ and $d_{\mathcal{A}}(D,D^{\prime})\leq 3$.
* •
If $X$ is compressible then $\partial D^{\prime}\subset X$ and
$d_{\mathcal{AC}}(D,D^{\prime})\leq 3$.
Here $\mathcal{A}=\mathcal{A}(X)$ and $\mathcal{AC}=\mathcal{AC}(X)$.
###### Proof.
If $\partial D\subset X$ then the lemma is trivial. So assume, by Remark 9.1,
that $D$ cuts $\partial X$. Choose a maximal sequence with respect to
$\partial X$ starting at $D$.
Suppose that the sequence is non-trivial ($n>1$). By Lemma 11.3 there is a
disk $E\in\Delta_{n}$ so that $D\cap X$ and $E\cap X$ contain disjoint arcs.
If the sequence ends essentially then choose $D^{\prime}=E$ and the lemma is
proved. If the sequence ends in $X$ then there is a boundary compression of
$\Delta_{n}$, disjoint from $\partial X$, yielding the desired disk
$D^{\prime}$ with $\partial D^{\prime}\subset X$. Since $E\cap
D^{\prime}=\emptyset$ we again obtain the desired bound.
Assume now that the sequence is trivial ($n=1$). Then take $E=D\in\Delta_{n}$
and the proof is identical to that of the previous paragraph. ∎
###### Remark 11.5.
Lemma 11.4 is unexpected: after all, any pair of curves in $\mathcal{C}(X)$
can be connected by a sequence of band sums. Thus arbitrary band sums can
change the subsurface projection to $X$. However, the sequences of band sums
arising in Lemma 11.4 are very special. Firstly they do not cross $\partial X$
and secondly they are “tree-like” due to the fact every arc in $D$ is
separating.
When $D$ is replaced by a surface with genus then Lemma 11.4 does not hold in
general; this is a fundamental observation due to Kobayashi [21] (see also
[17]). Namazi points out that even if $D$ is only replaced by a planar surface
Lemma 11.4 does not hold in general.
### 11.2. Proving the theorem
We now prove:
###### Theorem 11.6.
Suppose $X$ is a compressible hole for $\mathcal{D}(V)$ with diameter at least
$15$. Then there are a pair of essential disks $D,E\subset V$ so that
* •
$\partial D,\partial E\subset X$ and
* •
$\partial D$ and $\partial E$ fill $X$.
###### Proof.
Choose disks $D^{\prime}$ and $E^{\prime}$ in $\mathcal{D}(V)$ so that
$d_{X}(D^{\prime},E^{\prime})\geq 15$. By Lemma 11.4 there are disks $D$ and
$E$ so that $\partial D,\partial E\subset X$, $d_{X}(D^{\prime},D)\leq 6$, and
$d_{X}(E^{\prime},E)\leq 6$. It follows from the triangle inequality that
$d_{X}(D,E)\geq 3$. ∎
## 12\. Holes for the disk complex – incompressible
This section classifies incompressible holes for the disk complex.
###### Theorem 12.1.
Suppose $X$ is an incompressible hole for $\mathcal{D}(V)$ with diameter at
least $61$. Then there is an $I$–bundle $\rho_{F}\colon T\to F$ embedded in
$V$ so that
* •
$\partial_{h}T\subset\partial V$,
* •
$X$ is a component of $\partial_{h}T$,
* •
some component of $\partial_{v}T$ is boundary parallel into $\partial V$,
* •
$F$ supports a pseudo-Anosov map.
Here is a short plan of the proof: We are given $X$, an incompressible hole
for $\mathcal{D}(V)$. Following Lemma 11.4 we may assume that $D,E$ are
essential disks, without boundary compressions into $X$ or
$S{\smallsetminus}X$, with $d_{X}(D,E)>43$. Examine the intersection pattern
of $D$ and $E$ to find two families of rectangles $\mathcal{R}$ and
$\mathcal{Q}$. The intersection pattern of these rectangles in $V$ will
determine the desired $I$–bundle $T$. The third conclusion of the theorem
follows from standard facts about primitive annuli. The fourth requires
another application of Lemma 11.4 as well as Lemma 2.6.
### 12.1. Diagonals of polygons
To understand the intersection pattern of $D$ and $E$ we discuss diagonals of
polygons. Let $D$ be a $2n$ sided regular polygon. Label the sides of $D$ with
the letters $X$ and $Y$ in alternating fashion. Any side labeled $X$ (or $Y$)
will be called an $X$ side (or $Y$ side).
###### Definition 12.2.
An arc $\gamma$ properly embedded in $D$ is a diagonal if the points of
$\partial\gamma$ lie in the interiors of distinct sides of $D$. If $\gamma$
and $\gamma^{\prime}$ are diagonals for $D$ which together meet three
different sides then $\gamma$ and $\gamma^{\prime}$ are non-parallel.
###### Lemma 12.3.
Suppose that $\Gamma\subset D$ is a collection of pairwise disjoint non-
parallel diagonals. Then there is an $X$ side of $D$ meeting at most eight
diagonals of $\Gamma$.
###### Proof.
A counting argument shows that $|\Gamma|\leq 4n-3$. If every $X$ side meets at
least nine non-parallel diagonals then $|\Gamma|\geq\frac{9}{2}n>4n-3$, a
contradiction. ∎
### 12.2. Improving disks
Suppose now that $X$ is an incompressible hole for $\mathcal{D}(V)$ with
diameter at least $61$. Note that, by Theorem 10.1, $X$ is not an annulus. Let
$Y=\overline{S{\smallsetminus}X}$.
Choose disks $D^{\prime}$ and $E^{\prime}$ in $V$ so that
$d_{X}(D^{\prime},E^{\prime})\geq 61$. By Lemma 11.4 there are a pair of disks
$D$ and $E$ so that both are essential in $V$, cannot be boundary compressed
into $X$ or into $Y$, and so that $d_{\mathcal{A}(X)}(D^{\prime},D)\leq 3$ and
$d_{\mathcal{A}(X)}(E^{\prime},E)\leq 3$. Thus $d_{X}(D^{\prime},D)\leq 9$ and
$d_{X}(E^{\prime},E)\leq 9$ (Lemma 5.7). By the triangle inequality
$d_{X}(D,E)\geq 61-18=43$.
Recall, as well, that $\partial D$ and $\partial E$ are tight with respect to
$\partial X$. We may further assume that $\partial D$ and $\partial E$ are
tight with respect to each other. Also, minimize the quantities
$|X\cap(\partial D\cap\partial E)|$ and $|D\cap E|$ while keeping everything
tight. In particular, there are no triangle components of $\partial
V{\smallsetminus}(D\cup E\cup\partial X)$. Now consider $D$ and $E$ to be
even-sided polygons, with vertices being the points $\partial D\cap\partial X$
and $\partial E\cap\partial X$ respectively. Let $\Gamma=D\cap E$. See Figure
4 for one a priori possible collection $\Gamma\subset D$.
$\begin{array}[]{c}\includegraphics[height=3.5cm]{rectanglewithbadarcs}\end{array}$
Figure 4. In fact, $\Gamma\subset D$ cannot contain simple closed curves or
non-diagonals.
From our assumptions and the irreducibility of $V$ it follows that $\Gamma$
contains no simple closed curves. Suppose now that there is a
$\gamma\subset\Gamma$ so that, in $D$, both endpoints of $\gamma$ lie in the
same side of $D$. Then there is an outermost such arc, say
$\gamma^{\prime}\subset\Gamma$, cutting a bigon $B$ out of $D$. It follows
that $B$ is a boundary compression of $E$ which is disjoint from $\partial X$.
But this contradicts the construction of $E$. We deduce that all arcs of
$\Gamma$ are diagonals for $D$ and, via a similar argument, for $E$.
Let $\alpha\subset D\cap X$ be an $X$ side of $D$ meeting at most eight
distinct types of diagonal of $\Gamma$. Choose $\beta\subset E\cap X$
similarly. As $d_{X}(D,E)\geq 43$ we have that $d_{X}(\alpha,\beta)\geq
43-6=37$.
Now break each of $\alpha$ and $\beta$ into at most eight subarcs
$\\{\alpha_{i}\\}$ and $\\{\beta_{j}\\}$ so that each subarc meets all of the
diagonals of fixed type and only of that type. Let $R_{i}\subset D$ be the
rectangle with upper boundary $\alpha_{i}$ and containing all of the diagonals
meeting $\alpha_{i}$. Let $\alpha_{i}^{\prime}$ be the lower boundary of
$R_{i}$. Define $Q_{j}$ and $\beta_{j}^{\prime}$ similarly. See Figure 5 for a
picture of $R_{i}$.
2pt $R_{i}$ [l] at 92 61 $\alpha_{i}$ [bl] at 93 146 $\alpha_{i}^{\prime}$
[tr] at 63 3
$\begin{array}[]{c}\includegraphics[height=3.5cm]{rectangle}\end{array}$
Figure 5. The rectangle $R_{i}\subset D$ is surrounded by the dotted line. The
arc $\alpha_{i}$ in $\partial D\cap X$ is indicated. In general the arc
$\alpha^{\prime}_{i}$ may lie in $X$ or in $Y$.
Call an arc $\alpha_{i}$ large if there is an arc $\beta_{j}$ so that
$|\alpha_{i}\cap\beta_{j}|\geq 3$. We use the same notation for $\beta_{j}$.
Let $\Theta$ be the union of all of the large $\alpha_{i}$ and $\beta_{j}$.
Thus $\Theta$ is a four-valent graph in $X$. Let $\Theta^{\prime}$ be the
union of the corresponding large $\alpha_{i}^{\prime}$ and
$\beta_{i}^{\prime}$.
###### Claim 12.4.
The graph $\Theta$ is non-empty.
###### Proof.
If $\Theta=\emptyset$, then all $\alpha_{i}$ are small. It follows that
$|\alpha\cap\beta|\leq 128$ and thus $d_{X}(\alpha,\beta)\leq 16$, by Lemma
2.2. As $d_{X}(\alpha,\beta)\geq 37$ this is a contradiction. ∎
Let $Z\subset\partial V$ be a small regular neighborhood of $\Theta$ and
define $Z^{\prime}$ similarly.
###### Claim 12.5.
No component of $\Theta$ or of $\Theta^{\prime}$ is contained in a disk
$D\subset\partial V$. No component of $\Theta$ or of $\Theta^{\prime}$ is
contained in an annulus $A\subset\partial V$ that is peripheral in $X$.
###### Proof.
For a contradiction suppose that $W$ is a component of $Z$ contained in a
disk. Then there is some pair $\alpha_{i},\beta_{j}$ having a bigon in
$\partial V$. This contradicts the tightness of $\partial D$ and $\partial E$.
The same holds for $Z^{\prime}$.
Suppose now that some component $W$ is contained in an annulus $A$, peripheral
in $X$. Thus $W$ fills $A$. Suppose that $\alpha_{i}$ and $\beta_{j}$ are
large and contained in $W$. By the classification of arcs in $A$ we deduce
that either $\alpha_{i}$ and $\beta_{j}$ form a bigon in $A$ or $\partial X$,
$\alpha_{i}$ and $\beta_{j}$ form a triangle. Either conclusion gives a
contradiction. ∎
###### Claim 12.6.
The graph $\Theta$ fills $X$.
###### Proof.
Suppose not. Fix attention on any component $W\subset Z$. Since $\Theta$ does
not fill, the previous claim implies that there is a component
$\gamma\subset\partial W$ that is essential and non-peripheral in $X$. Note
that any large $\alpha_{i}$ meets $\partial W$ in at most two points, while
any small $\alpha_{i}$ meets $\partial W$ in at most $32$ points. Thus
$|\alpha\cap\partial W|\leq 256$ and the same holds for $\beta$. Thus
$d_{X}(\alpha,\beta)\leq 36$ by the triangle inequality. As
$d_{X}(\alpha,\beta)\geq 37$ this is a contradiction. ∎
The previous two claims imply:
###### Claim 12.7.
The graph $\Theta$ is connected. ∎
There are now two possibilities: either $\Theta\cap\Theta^{\prime}$ is empty
or not. In the first case set $\Sigma=\Theta$ and in the second set
$\Sigma=\Theta\cup\Theta^{\prime}$. By the claims above, $\Sigma$ is connected
and fills $X$. Let $\mathcal{R}=\\{R_{i}\\}$ and $\mathcal{Q}=\\{Q_{j}\\}$ be
the collections of large rectangles.
### 12.3. Building the I-bundle
We are given $\Sigma$, $\mathcal{R}$ and $\mathcal{Q}$ as above. Note that
$\mathcal{R}\cup\mathcal{Q}$ is an $I$–bundle and $\Sigma$ is the component of
its horizontal boundary meeting $X$. See Figure 6 for a simple case.
2pt $R_{i}$ [l] at 312 173 $Q_{j}$ [l] at 388 371
$\begin{array}[]{c}\includegraphics[height=4.5cm]{rectangles}\end{array}$
Figure 6. $\mathcal{R}\cup\mathcal{Q}$ is an $I$–bundle: all arcs of
intersection are parallel.
Let $T_{0}$ be a regular neighborhood of $\mathcal{R}\cup\mathcal{Q}$, taken
in $V$. Again $T_{0}$ has the structure of an $I$–bundle. Note that
$\partial_{h}T_{0}\subset\partial V$, $\partial_{h}T_{0}\cap X$ is a component
of $\partial_{h}T_{0}$, and this component fills $X$ due to Claim 12.6. We
will enlarge $T_{0}$ to obtain the correct $I$–bundle in $V$.
Begin by enumerating all annuli $\\{A_{i}\\}\subset\partial_{v}T_{0}$ with the
property that some component of $\partial A_{i}$ is inessential in $\partial
V$. Suppose that we have built the $I$–bundle $T_{i}$ and are now considering
the annulus $A=A_{i}$. Let $\gamma\cup\gamma^{\prime}=\partial
A\subset\partial V$ with $\gamma$ inessential in $\partial V$. Let
$B\subset\partial V$ be the disk which $\gamma$ bounds. By induction we assume
that no component of $\partial_{h}T_{i}$ is contained in a disk embedded in
$\partial V$ (the base case holds by Claim 12.5). It follows that $B\cap
T_{i}=\partial B=\gamma$. Thus $B\cup A$ is isotopic, rel $\gamma^{\prime}$,
to be a properly embedded disk $B^{\prime}\subset V$. As $\gamma^{\prime}$
lies in $X$ or $Y$, both incompressible, $\gamma^{\prime}$ must bound a disk
$C\subset\partial V$. Note that $C\cap T_{i}=\partial C=\gamma^{\prime}$,
again using the induction hypothesis.
It follows that $B\cup A\cup C$ is an embedded two-sphere in $V$. As $V$ is a
handlebody $V$ is irreducible. Thus $B\cup A\cup C$ bounds a three-ball
$U_{i}$ in $V$. Choose a homeomorphism $U_{i}\mathrel{\cong}B\times I$ so that
$B$ is identified with $B\times\\{0\\}$, $C$ is identified with
$B\times\\{1\\}$, and $A$ is identified with $\partial B\times I$. We form
$T_{i+1}=T_{i}\cup U_{i}$ and note that $T_{i+1}$ still has the structure of
an $I$–bundle. Recalling that $A=A_{i}$ we have
$\partial_{v}T_{i+1}=\partial_{v}T_{i}{\smallsetminus}A_{i}$. Also
$\partial_{h}T_{i+1}=\partial_{h}T_{i}\cup(B\cup C)\subset\partial V$. It
follows that no component of $\partial_{h}T_{i+1}$ is contained in a disk
embedded in $\partial V$. Similarly, $\partial_{h}T_{i+1}\cap X$ is a
component of $\partial_{h}T_{i+1}$ and this component fills $X$.
After dealing with all of the annuli $\\{A_{i}\\}$ in this fashion we are left
with an $I$–bundle $T$. Now all components of $\partial\partial_{v}T$ [sic]
are essential in $\partial V$. All of these lying in $X$ are peripheral in
$X$. This is because they are disjoint from $\Sigma\subset\partial_{h}T$,
which fills $X$, by induction. It follows that the component of
$\partial_{h}T$ containing $\Sigma$ is isotopic to $X$.
This finishes the construction of the promised $I$–bundle $T$ and demonstrates
the first two conclusions of Theorem 12.1. For future use we record:
###### Remark 12.8.
Every curve of $\partial\partial_{v}T=\partial\partial_{h}T$ is essential in
$S=\partial V$.
### 12.4. A vertical annulus parallel into the boundary
Here we obtain the third conclusion of Theorem 12.1: at least one component of
$\partial_{v}T$ is boundary parallel in $\partial V$.
Fix $T$ an $I$–bundle with the incompressible hole $X$ a component of
$\partial_{h}T$.
###### Claim 12.9.
All components of $\partial_{v}T$ are incompressible in $V$.
###### Proof.
Suppose that $A\subset\partial_{v}T$ was compressible. By Remark 12.8 we may
compress $A$ to obtain a pair of essential disks $B$ and $C$. Note that
$\partial B$ is isotopic into the complement of $\partial_{h}T$. So
$\overline{S{\smallsetminus}X}$ is compressible, contradicting Remark 9.1. ∎
###### Claim 12.10.
Some component of $\partial_{v}T$ is boundary parallel.
###### Proof.
Since $\partial_{v}T$ is incompressible (Claim 12.9) by Remark 8.2, we find
that $\partial_{v}T$ is boundary compressible in $V$. Let $B$ be a boundary
compression for $\partial_{v}T$. Let $A$ be the component of $\partial_{v}T$
meeting $B$. Let $\alpha$ denote the arc $A\cap B$.
The arc $\alpha$ is either essential or inessential in $A$. Suppose $\alpha$
is inessential in $A$. Then $\alpha$ cuts a bigon, $C$, out of $A$. Since $B$
was a boundary compression the disk $D=B\cup C$ is essential in $V$. Since $B$
meets $\partial_{v}T$ in a single arc, either $D\subset T$ or
$D\subset\overline{V{\smallsetminus}T}$. The former implies that
$\partial_{h}T$ is compressible and the latter that $X$ is not a hole. Either
gives a contradiction.
It follows that $\alpha$ is essential in $A$. Now carefully boundary compress
$A$: Let $N$ be the closure of a regular neighborhood of $B$, taken in
$V{\smallsetminus}A$. Let $A^{\prime}$ be the closure of $A{\smallsetminus}N$
(so $A^{\prime}$ is a rectangle). Let $B^{\prime}\cup B^{\prime\prime}$ be the
closure of $\operatorname{fr}(N){\smallsetminus}A$. Both $B^{\prime}$ and
$B^{\prime\prime}$ are bigons, parallel to $B$. Form $D=A^{\prime}\cup
B^{\prime}\cup B^{\prime\prime}$: a properly embedded disk in $V$. If $D$ is
essential then, as above, either $D\subset T$ or
$D\subset\overline{V{\smallsetminus}T}$. Again, either gives a contradiction.
It follows that $D$ is inessential in $V$. Thus $D$ cuts a closed three-ball
$U$ out of $V$. There are two final cases: either $N\subset U$ or $N\cap
U=B^{\prime}\cup B^{\prime\prime}$. If $U$ contains $N$ then $U$ contains $A$.
Thus $\partial A$ is contained in the disk $U\cap\partial V$. This contradicts
Remark 12.8. Deduce instead that $W=U\cup N$ is a solid torus with meridional
disk $B$. Thus $W$ gives a parallelism between $A$ and the annulus $\partial
V\cap\partial W$, as desired. ∎
###### Remark 12.11.
Similar considerations prove that the multicurve
$\\{\partial A\mathbin{\mid}\mbox{$A$ is a boundary parallel component of
$\partial_{v}T$}\\}$
is disk-busting for $V$.
### 12.5. Finding a pseudo-Anosov map
Here we prove that the base surface $F$ of the $I$–bundle $T$ admits a pseudo-
Anosov map.
As in Section 12.2, pick essential disks $D^{\prime}$ and $E^{\prime}$ in $V$
so that $d_{X}(D^{\prime},E^{\prime})\geq 61$. Lemma 11.4 provides disks $D$
and $E$ which cannot be boundary compressed into $X$ or into
$\overline{S{\smallsetminus}X}$ – thus $D$ and $E$ cannot be boundary
compressed into $\partial_{h}T$. Also, as above, $d_{X}(D,E)\geq 61-18=43$.
After isotoping $D$ to minimize intersection with $\partial_{v}T$ it must be
the case that all components of $D\cap\partial_{v}T$ are essential arcs in
$\partial_{v}T$. By Lemma 8.5 we conclude that $D$ may be isotoped in $V$ so
that $D\cap T$ is vertical in $T$. The same holds of $E$. Choose $A$ and $B$,
components of $D\cap T$ and $E\cap T$. Each are vertical rectangles. Since
$\operatorname{diam}_{X}(\pi_{X}(D))\leq 3$ (Lemma 4.4) we now have
$d_{X}(A,B)\geq 43-6=37$.
We now begin to work in the base surface $F$. Recall that $\rho_{F}\colon T\to
F$ is an $I$–bundle. Take $\alpha=\rho_{F}(A)$ and $\beta=\rho_{F}(B)$. Note
that the natural map $\mathcal{C}(F)\to\mathcal{C}(X)$, defined by taking a
curve to its lift, is distance non-increasing (see Equation 6.5). Thus
$d_{F}(\alpha,\beta)\geq 37$. By Theorem 10.1 the surface $F$ cannot be an
annulus. Thus, by Lemma 2.6 the subsurface $F$ supports a pseudo-Anosov map
and we are done.
### 12.6. Corollaries
We now deal with the possibility of disjoint holes for the disk complex.
###### Lemma 12.12.
Suppose that $X$ is a large incompressible hole for $\mathcal{D}(V)$ supported
by the $I$–bundle $\rho_{F}\colon T\to F$. Let
$Y=\partial_{h}T{\smallsetminus}X$. Let
$\tau\colon\partial_{h}T\to\partial_{h}T$ be the involution switching the ends
of the $I$–fibres. Suppose that $D\in\mathcal{D}(V)$ is an essential disk.
* •
If $F$ is orientable then $d_{\mathcal{A}(F)}(D\cap X,D\cap Y)\leq 6$.
* •
If $F$ is non-orientable then $d_{X}(D,\mathcal{C}^{\tau}(X))\leq 3$.
###### Proof.
By Lemma 11.4 there is a disk $D^{\prime}\subset V$ which is tight with
respect to $\partial_{h}T$ and which cannot be boundary compressed into
$\partial_{h}T$ (or into the complement). Also, for any component
$Z\subset\partial_{h}T$ we have $d_{\mathcal{A}(Z)}(D,D^{\prime})\leq 3$.
Properly isotope $D^{\prime}$ to minimize $D^{\prime}\cap\partial_{v}T$. Then
$D^{\prime}\cap\partial_{v}T$ is properly isotopic, in $\partial_{v}T$, to a
collection of vertical arcs. Let $E\subset D^{\prime}\cap T$ be a component.
Lemma 8.5 implies that $E$ is vertical in $T$, after an isotopy of
$D^{\prime}$ preserving $\partial_{h}T$ setwise. Since $E$ is vertical, the
arcs $E\cap\partial_{h}T\subset D^{\prime}$ are $\tau$–invariant. The
conclusion follows. ∎
Recall Lemma 7.3: all holes for the arc complex intersect. This cannot hold
for the disk complex. For example if $\rho_{F}\colon T\to F$ is an $I$–bundle
over an orientable surface then take $V=T$ and notice that both components of
$\partial_{h}T$ are holes for $\mathcal{D}(V)$. However, by the first
conclusion of Lemma 12.12, $X$ and $Y$ are paired holes, in the sense of
Definition 5.5. So, as with the invariant arc complex (Lemma 7.5), all holes
for the disk complex interfere:
###### Lemma 12.13.
Suppose that $X,Z\subset\partial V$ are large holes for $\mathcal{D}(V)$. If
$X\cap Z=\emptyset$ then there is an $I$–bundle $T\mathrel{\cong}F\times I$ in
$V$ so that $\partial_{h}T=X\cup Y$ and $Y\cap Z\neq\emptyset$.
###### Proof.
Suppose that $X\cap Z=\emptyset$. It follows from Remark 9.1 that both $X$ and
$Z$ are incompressible. Let $\rho_{F}\colon T\to F$ be the $I$–bundle in $V$
with $X\subset\partial_{h}T$, as provided by Theorem 12.1. We also have a
component $A\subset\partial_{v}T$ so that $A$ is boundary parallel. Let $U$ be
the solid torus component of $V{\smallsetminus}A$. Note that $Z$ cannot be
contained in $\partial U{\smallsetminus}A$ because $Z$ is not an annulus
(Theorem 10.1).
Let $\alpha=\rho_{F}(A)$. Choose any essential arc $\delta\subset F$ with both
endpoints in $\alpha\subset\partial F$. It follows that
$\rho_{F}^{-1}(\delta)$, together with two meridional disks of $U$, forms an
essential disk $D$ in $V$. Let $W=\partial_{h}T\cup(U{\smallsetminus}A)$ and
note that $\partial D\subset W$.
If $F$ is non-orientable then $Z\cap W=\emptyset$ and we have a contradiction.
Deduce that $F$ is orientable. Now, if $Z$ misses $Y$ then $Z$ misses $W$ and
we again have a contradiction. It follows that $Z$ cuts $Y$ and we are done. ∎
## 13\. Axioms for combinatorial complexes
The goal of this section and the next is to prove, inductively, an upper bound
on distance in a combinatorial complex $\mathcal{G}(S)=\mathcal{G}$. This
section presents our axioms on $\mathcal{G}$: sufficient hypotheses for
Theorem 13.1. The axioms, apart from Axiom 13.2, are quite general. Axiom 13.2
is necessary to prove hyperbolicity and greatly simplifies the recursive
construction in Section 14.
###### Theorem 13.1.
Fix $S$ a compact connected non-simple surface. Suppose that
$\mathcal{G}=\mathcal{G}(S)$ is a combinatorial complex satisfying the axioms
of Section 13. Let $X$ be a hole for $\mathcal{G}$ and suppose that
$\alpha_{X},\beta_{X}\in\mathcal{G}$ are contained in $X$. For any constant
$c>0$ there is a constant $A$ satisfying:
$d_{\mathcal{G}}(\alpha_{X},\beta_{X})\mathbin{\leq_{A}}\sum[d_{Y}(\alpha_{X},\beta_{X})]_{c}$
where the sum is taken over all holes $Y\subseteq X$ for $\mathcal{G}$.
The proof of the upper bound is more difficult than that of the lower bound,
Theorem 5.10. This is because naturally occurring paths in $\mathcal{G}$
between $\alpha_{X}$ and $\beta_{X}$ may waste time in non-holes. The first
example of this is the path in $\mathcal{C}(S)$ obtained by taking the short
curves along a Teichmüller geodesic. The Teichmüller geodesic may spend time
rearranging the geometry of a subsurface. Then the systole path in the curve
complex must be much longer than the curve complex distance between the
endpoints.
In Sections 16, 17, 19 we will verify these axioms for the curve complex of a
non-orientable surface, the arc complex, and the disk complex.
### 13.1. The axioms
Suppose that $\mathcal{G}=\mathcal{G}(S)$ is a combinatorial complex. We begin
with the axiom required for hyperbolicity.
###### Axiom 13.2 (Holes interfere).
All large holes for $\mathcal{G}$ interfere, as given in Definition 5.6.
Fix vertices $\alpha_{X},\beta_{X}\in\mathcal{G}$, both contained in a hole
$X$. We are given $\Lambda=\\{\mu_{n}\\}_{n=0}^{N}$, a path of markings in
$X$.
###### Axiom 13.3 (Marking path).
We require:
1. (1)
The support of $\mu_{n+1}$ is contained inside the support of $\mu_{n}$.
2. (2)
For any subsurface $Y\subseteq X$, if $\pi_{Y}(\mu_{k})\neq\emptyset$ then for
all $n\leq k$ the map $n\mapsto\pi_{Y}(\mu_{n})$ is an unparameterized quasi-
geodesic with constants depending only on $\mathcal{G}$.
The second condition is crucial and often technically difficult to obtain.
We are given, for every essential subsurface $Y\subset X$, a perhaps empty
interval $J_{Y}\subset[0,N]$ with the following properties.
###### Axiom 13.4 (Accessibility).
The interval for $X$ is $J_{X}=[0,N]$. There is a constant ${B_{3}}$ so that
1. (1)
If $m\in J_{Y}$ then $Y$ is contained in the support of $\mu_{m}$.
2. (2)
If $m\in J_{Y}$ then $\iota(\partial Y,\mu_{m})<{B_{3}}$.
3. (3)
If $[m,n]\cap J_{Y}=\emptyset$ then $d_{Y}(\mu_{m},\mu_{n})<{B_{3}}$.
There is a combinatorial path
$\Gamma=\\{\gamma_{i}\\}_{i=0}^{K}\subset\mathcal{G}$ starting with
$\alpha_{X}$ ending with $\beta_{X}$ and each $\gamma_{i}$ is contained in
$X$. There is a strictly increasing reindexing function $r\colon[0,K]\to[0,N]$
with $r(0)=0$ and $r(K)=N$.
###### Axiom 13.5 (Combinatorial).
There is a constant ${C_{2}}$ so that:
* •
$d_{Y}(\gamma_{i},\mu_{r(i)})<{C_{2}}$, for every $i\in[0,K]$ and every hole
$Y\subset X$,
* •
$d_{\mathcal{G}}(\gamma_{i},\gamma_{i+1})<{C_{2}}$, for every $i\in[0,K-1]$.
###### Axiom 13.6 (Replacement).
There is a constant ${C_{4}}$ so that:
1. (1)
If $Y\subset X$ is a hole and $r(i)\in J_{Y}$ then there is a vertex
$\gamma^{\prime}\in\mathcal{G}$ so that $\gamma^{\prime}$ is contained in $Y$
and $d_{\mathcal{G}}(\gamma_{i},\gamma^{\prime})<{C_{4}}$.
2. (2)
If $Z\subset X$ is a non-hole and $r(i)\in J_{Z}$ then there is a vertex
$\gamma^{\prime}\in\mathcal{G}$ so that
$d_{\mathcal{G}}(\gamma_{i},\gamma^{\prime})<{C_{4}}$ and so that
$\gamma^{\prime}$ is contained in $Z$ or in $X{\smallsetminus}Z$.
There is one axiom left: the axiom for straight intervals. This is given in
the next subsection.
### 13.2. Inductive, electric, shortcut and straight intervals
We describe subintervals that arise in the partitioning of $[0,K]$. As
discussed carefully in Section 13.3, we will choose a lower threshold
${L_{1}}(Y)$ for every essential $Y\subset X$ and a general upper threshold,
${L_{2}}$.
###### Definition 13.7.
Suppose that $[i,j]\subset[0,K]$ is a subinterval of the combinatorial path.
Then $[i,j]$ is an inductive interval associated to a hole $Y\subsetneq X$ if
* •
$r([i,j])\subset J_{Y}$ (for paired $Y$ we require $r([i,j])\subset J_{Y}\cap
J_{Y^{\prime}}$) and
* •
$d_{Y}(\gamma_{i},\gamma_{j})\geq{L_{1}}(Y)$.
When $X$ is the only relevant hole we have a simpler definition:
###### Definition 13.8.
Suppose that $[i,j]\subset[0,K]$ is a subinterval of the combinatorial path.
Then $[i,j]$ is an electric interval if $d_{Y}(\gamma_{i},\gamma_{j})<{L_{2}}$
for all holes $Y\subsetneq X$.
Electric intervals will be further partitioned into shortcut and straight
intervals.
###### Definition 13.9.
Suppose that $[p,q]\subset[0,K]$ is a subinterval of the combinatorial path.
Then $[p,q]$ is a shortcut if
* •
$d_{Y}(\gamma_{p},\gamma_{q})<{L_{2}}$ for all holes $Y$, including $X$
itself, and
* •
there is a non-hole $Z\subset X$ so that $r([p,q])\subset J_{Z}$.
###### Definition 13.10.
Suppose that $[p,q]\subset[0,K]$ is a subinterval of the combinatorial path
and is contained in an electric interval $[i,j]$. Then $[p,q]$ is a straight
interval if $d_{Y}(\mu_{r(p)},\mu_{r(q)})<{L_{2}}$ for all non-holes $Y$.
Our final axiom is:
###### Axiom 13.11 (Straight).
There is a constant $A$ depending only on $X$ and $\mathcal{G}$ so that for
every straight interval $[p,q]$:
$d_{\mathcal{G}}(\gamma_{p},\gamma_{q})\mathbin{\leq_{A}}d_{X}(\gamma_{p},\gamma_{q})$
### 13.3. Deductions from the axioms
Axiom 13.3 and Lemma 3.9 imply that the reverse triangle inequality holds for
projections of marking paths.
###### Lemma 13.12.
There is a constant ${C_{1}}$ so that
$d_{Y}(\mu_{m},\mu_{n})+d_{Y}(\mu_{n},\mu_{p})<d_{Y}(\mu_{m},\mu_{p})+{C_{1}}$
for every essential $Y\subset X$ and for every $m<n<p$ in $[0,N]$. ∎
We record three simple consequences of Axiom 13.4.
###### Lemma 13.13.
There is a constant ${C_{3}}$, depending only on ${B_{3}}$, with the follow
properties:
* (i)
If $Y$ is strictly nested in $Z$ and $m\in J_{Y}$ then $d_{Z}(\partial
Y,\mu_{m})\leq{C_{3}}$.
* (ii)
If $Y$ is strictly nested in $Z$ then for any $m,n\in J_{Y}$,
$d_{Z}(\mu_{m},\mu_{n})<{C_{3}}$.
* (iii)
If $Y$ and $Z$ overlap then for any $m,n\in J_{Y}\cap J_{Z}$ we have
$d_{Y}(\mu_{m},\mu_{n}),d_{Z}(\mu_{m},\mu_{n})<{C_{3}}$.
###### Proof.
We first prove conclusion (i): Since $Y$ is strictly nested in $Z$ and since
$Y$ is contained in the support of $\mu_{m}$ (part (1) of Axiom 13.4), both
$\partial Y$ and $\mu_{m}$ cut $Z$. By Axiom 13.4, part (2), we have that
$\iota(\partial Y,\mu_{m})\leq{B_{3}}$. It follows that $\iota(\partial
Y,\pi_{Z}(\mu_{m}))\leq 2{B_{3}}$. By Lemma 2.2 we deduce that $d_{Z}(\partial
Y,\mu_{m})\leq 2\log_{2}{B_{3}}+3$. We take ${C_{3}}$ larger than this right
hand side.
Conclusion (ii) follows from a pair of applications of conclusion (i) and the
triangle inequality.
For conclusion (iii): As in (ii), to bound $d_{Z}(\mu_{m},\mu_{n})$ it
suffices to note that $\partial Y$ cuts $Z$ and that $\partial Y$ has bounded
intersection with both of $\mu_{m},\mu_{n}$. ∎
We now have all of the constants ${C_{1}},{C_{3}},{C_{2}},{C_{4}}$ in hand.
Recall that $L_{4}$ is the pairing constant of Definition 5.5 and that $M_{0}$
is the constant of 4.6. We must choose a lower threshold ${L_{1}}(Y)$ for
every essential $Y\subset X$. We must also choose the general upper threshold,
${L_{2}}$ and general lower threshold ${L_{0}}$. We require, for all essential
$Z,Y$ in $X$, with $\xi(Z)<\xi(Y)\leq\xi(X)$:
(13.14) $\displaystyle{L_{0}}>{C_{3}}+2{C_{2}}+2L_{4}$ (13.15)
$\displaystyle{L_{2}}>{L_{1}}(X)+2L_{4}+6{C_{1}}+2{C_{2}}+14{C_{3}}+10$
(13.16) $\displaystyle{L_{1}}(Y)>M_{0}+2{C_{3}}+4{C_{2}}+2L_{4}+{L_{0}}$
(13.17) $\displaystyle{L_{1}}(X)>{L_{1}}(Z)+2{C_{3}}+4{C_{2}}+4L_{4}$
## 14\. Partition and the upper bound on distance
In this section we prove Theorem 13.1 by induction on $\xi(X)$. The first
stage of the proof is to describe the inductive partition: we partition the
given interval $[0,K]$ into inductive and electric intervals. The inductive
partition is closely linked with the hierarchy machine [25] and with the
notion of antichains introduced in [34].
We next give the electric partition: each electric interval is divided into
straight and shortcut intervals. Note that the electric partition also gives
the base case of the induction. We finally bound
$d_{\mathcal{G}}(\alpha_{X},\beta_{X})$ from above by combining the
contributions from the various intervals.
### 14.1. Inductive partition
We begin by identifying the relevant surfaces for the construction of the
partition. We are given a hole $X$ for $\mathcal{G}$ and vertices
$\alpha_{X},\beta_{X}\in\mathcal{G}$ contained in $X$. Define
$B_{X}=\\{Y\subsetneq X\mathbin{\mid}\mbox{$Y$ is a hole
and~{}}d_{Y}(\alpha_{X},\beta_{X})\geq{L_{1}}(X)\\}.$
For any subinterval $[i,j]\subset[0,K]$ define
$B_{X}(i,j)=\\{Y\in
B_{X}\mathbin{\mid}d_{Y}(\gamma_{i},\gamma_{j})\geq{L_{1}}(X)\\}.$
We now partition $[0,K]$ into inductive and electric intervals. Begin with the
partition of one part $\mathcal{P}_{X}=\\{[0,K]\\}$. Recursively
$\mathcal{P}_{X}$ is a partition of $[0,K]$ consisting of intervals which are
either inductive, electric, or undetermined. Suppose that
$[i,j]\in\mathcal{P}_{X}$ is undetermined.
###### Claim.
If $B_{X}(i,j)$ is empty then $[i,j]$ is electric.
###### Proof.
Since $B_{X}(i,j)$ is empty, every hole $Y\subsetneq X$ has either
$d_{Y}(\gamma_{i},\gamma_{j})<{L_{1}}(X)$ or $Y\mathbin{\notin}B_{X}$. In the
former case, as ${L_{1}}(X)<{L_{2}}$, we are done.
So suppose the latter holds. Now, by the reverse triangle inequality (Lemma
13.12),
$d_{Y}(\mu_{r(i)},\mu_{r(j)})<d_{Y}(\mu_{0},\mu_{N})+2{C_{1}}.$
Since $r(0)=0$ and $r(K)=N$ we find:
$d_{Y}(\gamma_{i},\gamma_{j})<d_{Y}(\alpha_{X},\beta_{X})+2{C_{1}}+4{C_{2}}.$
Deduce that
$d_{Y}(\gamma_{i},\gamma_{j})<{L_{1}}(X)+2{C_{1}}+4{C_{2}}<{L_{2}}.$
This completes the proof. ∎
Thus if $B_{X}(i,j)$ is empty then $[i,j]\in\mathcal{P}_{X}$ is determined to
be electric. Proceed on to the next undetermined element. Suppose instead that
$B_{X}(i,j)$ is non-empty. Pick a hole $Y\in B_{X}(i,j)$ so that $Y$ has
maximal $\xi(Y)$ amongst the elements of $B_{X}(i,j)$
Let $p,q\in[i,j]$ be the first and last indices, respectively, so that
$r(p),r(q)\in J_{Y}$. (If $Y$ is paired with $Y^{\prime}$ then we take the
first and last indices that, after reindexing, lie inside of $J_{Y}\cap
J_{Y^{\prime}}$.)
###### Claim.
The indices $p,q$ are well-defined.
###### Proof.
By assumption $d_{Y}(\gamma_{i},\gamma_{j})\geq{L_{1}}(X)$. By Equation 13.14,
${L_{1}}(X)>{C_{3}}+2{C_{2}}.$
We deduce from Axiom 13.4 and Axiom 13.5 that $J_{Y}\cap r([i,j])$ is non-
empty. Thus, if $Y$ is not paired, the indices $p,q$ are well-defined.
Suppose instead that $Y$ is paired with $Y^{\prime}$. Recall that measurements
made in $Y$ and $Y^{\prime}$ differ by at most the pairing constant $L_{4}$
given in Definition 5.5. By (13.16),
${L_{1}}(X)>{C_{3}}+2{C_{2}}+2L_{4}.$
We deduce again from Axiom 13.4 that $J_{Y^{\prime}}\cap r([i,j])$ is non-
empty.
Suppose now, for a contradiction, that $J_{Y}\cap J_{Y^{\prime}}\cap r([i,j])$
is empty. Define
$h=\max\\{\ell\in[i,j]\mathbin{\mid}r(\ell)\in J_{Y}\\},\quad
k=\min\\{\ell\in[i,j]\mathbin{\mid}r(\ell)\in J_{Y^{\prime}}\\}$
Without loss of generality we may assume that $h<k$. It follows that
$d_{Y^{\prime}}(\gamma_{i},\gamma_{h})<{C_{3}}+2{C_{2}}$. Thus
$d_{Y}(\gamma_{i},\gamma_{h})<{C_{3}}+2{C_{2}}+2L_{4}$. Similarly,
$d_{Y}(\gamma_{h},\gamma_{j})<{C_{3}}+2{C_{2}}$. Deduce
$d_{Y}(\gamma_{i},\gamma_{j})<2{C_{3}}+4{C_{2}}+2L_{4}<{L_{1}}(X),$
the last inequality by (13.16). This is a contradiction to the assumption. ∎
###### Claim.
The interval $[p,q]$ is inductive for $Y$.
###### Proof.
We must check that $d_{Y}(\gamma_{p},\gamma_{q})\geq{L_{1}}(Y)$. Suppose first
that $Y$ is not paired. Then by the definition of $p,q$, (2) of Axiom 13.4,
and the triangle inequality we have
$d_{Y}(\mu_{r(i)},\mu_{r(j)})\leq d_{Y}(\mu_{r(p)},\mu_{r(q)})+2{C_{3}}.$
Thus by Axiom 13.5,
$d_{Y}(\gamma_{i},\gamma_{j})\leq
d_{Y}(\gamma_{p},\gamma_{q})+2{C_{3}}+4{C_{2}}.$
Since by (13.17),
${L_{1}}(Y)+2{C_{3}}+4{C_{2}}<{L_{1}}(X)\leq d_{Y}(\gamma_{i},\gamma_{j})$
we are done.
When $Y$ is paired the proof is similar but we must use the slightly stronger
inequality ${L_{1}}(Y)+2{C_{3}}+4{C_{2}}+4L_{4}<{L_{1}}(X)$. ∎
Thus, when $B_{X}(i,j)$ is non-empty we may find a hole $Y$ and indices $p,q$
as above. In this situation, we subdivide the element
$[i,j]\in\mathcal{P}_{X}$ into the elements $[i,p-1]$, $[p,q]$, and $[q+1,j]$.
(The first or third intervals, or both, may be empty.) The interval
$[p,q]\in\mathcal{P}_{X}$ is determined to be inductive and associated to $Y$.
Proceed on to the next undetermined element. This completes the construction
of $\mathcal{P}_{X}$.
As a bit of notation, if $[i,j]\in\mathcal{P}_{X}$ is associated to $Y\subset
X$ we will sometimes write $I_{Y}=[i,j]$.
### 14.2. Properties of the inductive partition
###### Lemma 14.1.
Suppose that $Y,Z$ are holes and $I_{Z}$ is an inductive element of
$\mathcal{P}_{X}$ associated to $Z$. Suppose that $r(I_{Z})\subset J_{Y}$ (or
$r(I_{Z})\subset J_{Y}\cap J_{Y^{\prime}}$, if $Y$ is paired). Then
* •
$Z$ is nested in $Y$ or
* •
$Z$ and $Z^{\prime}$ are paired and $Z^{\prime}$ is nested in $Y$.
###### Proof.
Let $I_{Z}=[i,j]$. Suppose first that $Y$ is strictly nested in $Z$. Then by
(ii) of Lemma 13.13, $d_{Z}(\mu_{r(i)},\mu_{r(j)})<{C_{3}}$. Then by Axiom
13.5
$d_{Z}(\gamma_{i},\gamma_{j})<{C_{3}}+2{C_{2}}<{L_{1}}(Z),$
a contradiction. We reach the same contradiction if $Y$ and $Z$ overlap using
(iii) of Lemma 13.13.
Now, if $Z$ and $Y$ are disjoint then there are two cases: Suppose first that
$Y$ is paired with $Y^{\prime}$. Since all holes interfere, $Y^{\prime}$ and
$Z$ must meet. In this case we are done, just as in the previous paragraph.
Suppose now that $Z$ is paired with $Z^{\prime}$. Since all holes interfere,
$Z^{\prime}$ and $Y$ must meet. If $Z^{\prime}$ is nested in $Y$ then we are
done. If $Y$ is strictly nested in $Z^{\prime}$ then, as $r([i,j])\subset
J_{Y}$, we find that as above by Axioms 13.5 and (ii) of Lemma 13.13 that
$d_{Z^{\prime}}(\gamma_{i},\gamma_{j})<{C_{3}}+2{C_{2}}$
and so $d_{Z}(\gamma_{i},\gamma_{j})<{C_{3}}+2{C_{2}}+2L_{4}<{L_{1}}(Z)$, a
contradiction. We reach the same contradiction if $Y$ and $Z^{\prime}$
overlap. ∎
###### Proposition 14.2.
Suppose $Y\subsetneq X$ is a hole for $\mathcal{G}$.
1. (1)
If $Y$ is associated to an inductive interval $I_{Y}\in\mathcal{P}_{X}$ and
$Y$ is paired with $Y^{\prime}$ then $Y^{\prime}$ is not associated to any
inductive interval in $\mathcal{P}_{X}$.
2. (2)
There is at most one inductive interval $I_{Y}\in\mathcal{P}_{X}$ associated
to $Y$.
3. (3)
There are at most two holes $Z$ and $W$, distinct from $Y$ (and from
$Y^{\prime}$, if $Y$ is paired) such that
* •
there are inductive intervals $I_{Z}=[h,i]$ and $I_{W}=[j,k]$ and
* •
$d_{Y}(\gamma_{h},\gamma_{i}),d_{Y}(\gamma_{j},\gamma_{k})\geq{L_{0}}$.
###### Remark 14.3.
It follows that for any hole $Y$ there are at most three inductive intervals
in the partition $\mathcal{P}_{X}$ where $Y$ has projection distance greater
than ${L_{0}}$.
###### Proof of Proposition 14.2.
To prove the first claim: Suppose that $I_{Y}=[p,q]$ and
$I_{Y^{\prime}}=[p^{\prime},q^{\prime}]$ with $q<p^{\prime}$. It follows that
$[r(p),r(q^{\prime})]\subset J_{Y}\cap J_{Y^{\prime}}$. If $q+1=p^{\prime}$
then the partition would have chosen a larger inductive interval for one of
$Y$ or $Y^{\prime}$. It must be the case that there is an inductive interval
$I_{Z}\subset[q+1,p^{\prime}-1]$ for some hole $Z$, distinct from $Y$ and
$Y^{\prime}$, with $\xi(Z)\geq\xi(Y)$. However, by Lemma 14.1 we find that $Z$
is nested in $Y$ or in $Y^{\prime}$. It follows that $Z=Y$ or $Y$, a
contradiction.
The second statement is essentially similar.
Finally suppose that $Z$ and $W$ are the first and last holes, if any,
satisfying the hypotheses of the third claim. Since
$d_{Y}(\gamma_{h},\gamma_{i})\geq{L_{0}}$ we find by Axiom 13.5 that
$d_{Y}(\mu_{r(h)},\mu_{r(i)})\geq{L_{0}}-2{C_{2}}.$
By (13.14), ${L_{0}}-2{C_{2}}>{C_{3}}$ so that
$J_{Y}\cap r(I_{Z})\neq\emptyset.$
If $Y$ is paired then, again by (13.14) we have
${L_{0}}>{C_{3}}+2{C_{2}}+2L_{4}$, we also find that $J_{Y^{\prime}}\cap
r(I_{Z})\neq\emptyset$. Symmetrically, $J_{Y}\cap r(I_{W})$ (and
$J_{Y^{\prime}}\cap r(I_{W})$) are also non-empty.
It follows that the interval between $I_{Z}$ and $I_{W}$, after reindexing, is
contained in $J_{Y}$ (and $J_{Y^{\prime}}$, if $Y$ is paired). Thus for any
inductive interval $I_{V}=[p,q]$ between $I_{Z}$ and $I_{W}$ the associated
hole $V$ is nested in $Y$ (or $V^{\prime}$ is nested in $Y$), by Lemma 14.1.
If $V=Y$ or $V=Y^{\prime}$ there is nothing to prove. Suppose instead that $V$
(or $V^{\prime}$) is strictly nested in $Y$. It follows that
$d_{Y}(\gamma_{p},\gamma_{q})<{C_{3}}+2{C_{2}}<{L_{0}}.$
Thus there are no inductive intervals between $I_{Z}$ and $I_{W}$ satisfying
the hypotheses of the third claim. ∎
The following lemma and proposition bound the number of inductive intervals.
The discussion here is very similar to (and in fact inspired) the antichains
defined in [34, Section 5]. Our situation is complicated by the presence of
non-holes and interfering holes.
###### Lemma 14.4.
Suppose that $X,\alpha_{X},\beta_{X}$ are given, as above. For any
$\ell\geq(3\cdot{L_{2}})^{\xi(X)}$, if $\\{Y_{i}\\}_{i=1}^{\ell}$ is a
collection of distinct strict sub-holes of $X$ each having
$d_{Y_{i}}(\alpha_{X},\beta_{X})\geq{L_{1}}(X)$ then there is a hole
$Z\subseteq X$ such that $d_{Z}(\alpha_{X},\beta_{X})\geq{L_{2}}-1$ and $Z$
contains at least ${L_{2}}$ of the $Y_{i}$. Furthermore, for at least
${L_{2}}-4({C_{1}}+{C_{3}}+2{C_{3}}+2)$ of these $Y_{i}$ we find that
$J_{Y_{i}}\subsetneq J_{Z}$. (If $Z$ is paired then $J_{Y_{i}}\subsetneq
J_{Z}\cap J_{Z^{\prime}}$.) Each of these $Y_{i}$ is disjoint from a distinct
vertex $\eta_{i}\in[\pi_{Z}(\alpha_{X}),\pi_{Z}(\beta_{X})]$.
###### Proof.
Let $g_{X}$ be a geodesic in $\mathcal{C}(X)$ joining $\alpha_{X},\beta_{X}$.
By the Bounded Geodesic Image Theorem (Theorem 4.6), since ${L_{1}}(X)>M_{0}$,
for every $Y_{i}$ there is a vertex $\omega_{i}\in g_{X}$ such that
$Y_{i}\subset X{\smallsetminus}\omega_{i}$. Thus $d_{X}(\omega_{i},\partial
Y_{i})\leq 1$. If there are at least ${L_{2}}$ distinct $\omega_{i}$,
associated to distinct $Y_{i}$, then
$d_{X}(\alpha_{X},\beta_{X})\geq{L_{2}}-1$. In this situation we take $Z=X$.
Since $J_{X}=[0,N]$ we are done.
Thus assume there do not exist at least ${L_{2}}$ distinct $\omega_{i}$. Then
there is some fixed $\omega$ among these $\omega_{i}$ such that at least
$\frac{\ell}{{L_{2}}}\geq 3(3\cdot{L_{2}})^{\xi(X)-1}$ of the $Y_{i}$ satisfy
$Y_{i}\subset(X{\smallsetminus}\omega).$
Thus one component, call it $W$, of $X{\smallsetminus}\omega$ contains at
least $(3\cdot{L_{2}})^{\xi(X)-1}$ of the $Y_{i}$. Let $g_{W}$ be a geodesic
in $\mathcal{C}(W)$ joining $\alpha_{W}=\pi_{W}(\alpha_{X})$ and
$\beta_{W}=\pi_{W}(\beta_{X})$. Notice that
$d_{Y_{i}}(\alpha_{W},\beta_{W})\geq d_{Y_{i}}(\alpha_{X},\beta_{X})-8$
because we are projecting to nested subsurfaces. This follows for example from
Lemma 4.4. Hence $d_{Y_{i}}(\alpha_{W},\beta_{W})\geq{L_{1}}(W)$.
Again apply Theorem 4.6. Since ${L_{1}}(W)>M_{0}$, for every remaining $Y_{i}$
there is a vertex $\eta_{i}\in g_{W}$ such that
$Y_{i}\subset(W{\smallsetminus}\eta_{i})$
If there are at least ${L_{2}}$ distinct $\eta_{i}$ then we take $Z=W$.
Otherwise we repeat the argument. Since the complexity of each successive
subsurface is decreasing by at least $1$, we must eventually find the desired
$Z$ containing at least ${L_{2}}$ of the $Y_{i}$, each disjoint from distinct
vertices of $g_{Z}$.
So suppose that there are at least ${L_{2}}$ distinct $\eta_{i}$ associated to
distinct $Y_{i}$ and we have taken $Z=W$. Now we must find at least
${L_{2}}-4({C_{1}}+{C_{3}}+2{C_{3}}+2)$ of these $Y_{i}$ where
$J_{Y_{i}}\subsetneq J_{Z}$.
To this end we focus attention on a small subset
$\\{Y^{j}\\}_{j=1}^{5}\subset\\{Y_{i}\\}$. Let $\eta_{j}$ be the vertex of
$g_{Z}=g_{W}$ associated to $Y^{j}$. We choose these $Y^{j}$ so that
* •
the $\eta_{j}$ are arranged along $g_{Z}$ in order of index and
* •
$d_{Z}(\eta_{j},\eta_{j+1})>{C_{1}}+{C_{3}}+2{C_{3}}+2$, for $j=1,2,3,4$.
This is possible by (13.15) because
${L_{2}}>4({C_{1}}+{C_{3}}+2{C_{3}}).$
Set $J_{j}=J_{Y^{j}}$ and pick any indices $m_{j}\in J_{j}$. (If $Z$ is paired
then $Y^{j}$ is as well and we pick $m_{j}\in J_{Y^{j}}\cap
J_{(Y^{j})^{\prime}}$.) We use $\mu(m_{j})$ to denote $\mu_{m_{j}}$. Since
$\partial Y^{j}$ is disjoint from $\eta_{j}$, Axiom 13.4 and Lemma 2.2 imply
(14.5) $d_{Z}(\mu(m_{j}),\eta_{j})\leq{C_{3}}+1.$
Since the sequence $\pi_{Z}(\mu_{n})$ satisfies the reverse triangle
inequality (Lemma 13.12), it follows that the $m_{j}$ appear in $[0,N]$ in
order agreeing with their index. The triangle inequality implies that
$d_{Z}(\mu(m_{1}),\mu(m_{2}))>{C_{3}}.$
Thus Axiom 13.4 implies that $J_{Z}\cap[m_{1},m_{2}]$ is non-empty. Similarly,
$J_{Z}\cap[m_{4},m_{5}]$ is non-empty. It follows that $[m_{2},m_{4}]\subset
J_{Z}$. (If $Z$ is paired then, after applying the symmetry $\tau$ to $g_{Z}$,
the same argument proves $[m_{2},m_{4}]\subset J_{Z^{\prime}}$.)
Notice that $J_{2}\cap J_{3}=\emptyset$. For if $m\in J_{2}\cap J_{3}$ then by
(14.5) both $d_{Z}(\mu_{m},\eta_{2})$ and $d_{Z}(\mu_{m},\eta_{3})$ are
bounded by ${C_{3}}+1$. It follows that
$d_{Z}(\eta_{2},\eta_{3})<2{C_{3}}+2,$
a contradiction. Similarly $J_{3}\cap J_{4}=\emptyset$. We deduce that
$J_{3}\subsetneq[m_{2},m_{4}]\subset J_{Z}$. (If $Z$ is paired $J_{3}\subset
J_{Z}\cap J_{Z^{\prime}}$.) Finally, there are at least
${L_{2}}-4({C_{1}}+{C_{3}}+2{C_{3}}+2)$
possible $Y_{i}$’s which satisfy the hypothesis on $Y^{3}$. This completes the
proof. ∎
Define
$\mathcal{P}_{\text{ind}}=\\{I\in\mathcal{P}_{X}\mathbin{\mid}\mbox{ $I$ is
inductive}\\}.$
###### Proposition 14.6.
The number of inductive intervals is a lower bound for the projection distance
in $X$:
$d_{X}(\alpha_{X},\beta_{X})\geq\frac{|\mathcal{P}_{\text{ind}}|}{2(3\cdot{L_{2}})^{\xi(X)-1}+1}-1.$
###### Proof.
Suppose, for a contradiction, that the conclusion fails. Let $g_{X}$ be a
geodesic in $\mathcal{C}(X)$ connecting $\alpha_{X}$ to $\beta_{X}$. Then, as
in the proof of Lemma 14.4, there is a vertex $\omega$ of $g_{X}$ and a
component $W\subset X{\smallsetminus}\omega$ where at least
$(3\cdot{L_{2}})^{\xi(X)-1}$ of the inductive intervals in $I_{X}$ have
associated surfaces, $Y_{i}$, contained in $W$.
Since $\xi(X)-1\geq\xi(W)$ we may apply Lemma 14.4 inside of $W$. So we find a
surface $Z\subseteq W\subsetneq X$ so that
* •
$Z$ contains at least ${L_{2}}$ of the $Y_{i}$,
* •
$d_{Z}(\alpha_{X},\beta_{X})\geq{L_{2}}$, and
* •
there are at least ${L_{2}}-4({C_{1}}+{C_{3}}+2{C_{3}}+2)$ of the $Y_{i}$
where $J_{Y_{i}}\subsetneq J_{Z}$.
Since $Y_{i}\subsetneq Z$ and $Y_{i}$ is a hole, $Z$ is also a hole. Since
${L_{2}}>{L_{1}}(X)$ it follows that $Z\in B_{X}$. Let
$\mathcal{Y}=\\{Y_{i}\\}$ be the set of $Y_{i}$ satisfying the third bullet.
Let $Y^{1}\in\mathcal{Y}$ and $\eta_{1}\in g_{Z}$ satisfy $\partial
Y^{1}\cap\eta_{1}=\emptyset$ and $\eta_{1}$ is the first such. Choose $Y^{2}$
and $\eta_{2}$ similarly, so that $\eta_{2}$ is the last such. By Lemma 14.4
(14.7) $d_{Z}(\eta_{1},\eta_{2})\geq L_{2}-4({C_{1}}+{C_{3}}+2{C_{3}}+2).$
Let $p=\min I_{Y^{1}}$ and $q=\max I_{Y^{2}}$. Note that $[p,q]\subset J_{Z}$.
(If $Z$ is paired with $Z^{\prime}$ then $[p,q]\subset J_{Z}\cap
J_{Z^{\prime}}$.) Again by (1) of Axiom 13.4, and Lemma 2.2,
$d_{Z}(\mu_{r(p)},\partial Y^{1})<{C_{3}}.$
It follows that
$d_{Z}(\mu_{r(p)},\eta_{1})\leq{C_{3}}+1$
and the same bound applies to $d_{Z}(\mu_{r(q)},\eta_{2})$. Combined with
(14.7) we find that
$d_{Z}(\mu_{r(p)},\mu_{r(q)})\geq{L_{2}}-4{C_{1}}-4{C_{3}}-10{C_{3}}-10.$
By the reverse triangle inequality (Lemma 13.12), for any $p^{\prime}\leq
p,q\leq q^{\prime}$,
$d_{Z}(\mu_{r(p^{\prime})},\mu_{r(q^{\prime})})\geq{L_{2}}-6{C_{1}}-4{C_{3}}-10{C_{3}}-10.$
Finally by Axiom 13.5 and the above inequality we have
$d_{Z}(\gamma_{p^{\prime}},\gamma_{q^{\prime}})\geq{L_{2}}-6{C_{1}}-4{C_{3}}-10{C_{3}}-10-2{C_{2}}.$
By (13.15) the right-hand side is greater than ${L_{1}}(X)+2L_{4}$ so we
deduce that $Z\in B_{X}(p^{\prime},q^{\prime})$, for any such
$p^{\prime},q^{\prime}$. (When $Z$ is paired deduce also that $Z^{\prime}\in
B_{X}(p^{\prime},q^{\prime})$.)
Let $I_{V}$ be the first inductive interval chosen by the procedure with the
property that $I_{V}\cap[p,q]\neq\emptyset$. Note that, since $I_{Y^{1}}$ and
$I_{Y^{2}}$ will also be chosen, $I_{V}\subset[p,q]$. Let
$p^{\prime},q^{\prime}$ be the indices so that $V$ is chosen from
$B_{X}(p^{\prime},q^{\prime})$. Thus $p^{\prime}\leq p$ and $q\leq
q^{\prime}$. However, since $I_{V}\subset[p,q]\subset J_{Z}$, Lemma 14.1
implies that $V$ is strictly nested in $Z$. (When pairing occurs we may find
instead that $V\subset Z^{\prime}$ or $V^{\prime}\subset Z$.) Thus
$\xi(Z)>\xi(V)$ and we find that $Z$ would be chosen from
$B_{X}(p^{\prime},q^{\prime})$, instead of $V$. This is a contradiction. ∎
### 14.3. Electric partition
The goal of this subsection is to prove:
###### Proposition 14.8.
There is a constant $A$ depending only on $\xi(X)$, so that: if
$[i,j]\subset[0,K]$ is a electric interval then
$d_{\mathcal{G}}(\gamma_{i},\gamma_{j})\mathbin{\leq_{A}}d_{X}(\gamma_{i},\gamma_{j}).$
We begin by building a partition of $[i,j]$ into straight and shortcut
intervals. Define
$C_{X}=\\{Y\subsetneq X\mathbin{\mid}\mbox{$Y$ is a non-hole and
}d_{Y}(\mu_{r(i)},\mu_{r(j)})\geq{L_{1}}(X)\\}.$
We also define, for all $[p,q]\subset[i,j]$
$C_{X}(p,q)=\\{Y\in C_{X}\mathbin{\mid}J_{Y}\cap[r(p),r(q)]\neq\emptyset\\}.$
Our recursion starts with the partition of one part,
$\mathcal{P}(i,j)=\\{[i,j]\\}$. Recursively $\mathcal{P}(i,j)$ is a partition
of $[i,j]$ into shortcut, straight, or undetermined intervals. Suppose that
$[p,q]\in\mathcal{P}(i,j)$ is undetermined.
###### Claim.
If $C_{X}(p,q)$ is empty then $[p,q]$ is straight.
###### Proof.
We show the contrapositive. Suppose that $Y$ is a non-hole with
$d_{Y}(\mu_{r(p)},\mu_{r(q)})\geq{L_{2}}$. Since ${L_{2}}>{C_{3}}$, Axiom 13.4
implies that $J_{Y}\cap[r(p),r(q)]$ is non-empty. Also, the reverse triangle
inequality (Lemma 13.12) gives:
$d_{Y}(\mu_{r(p)},\mu_{r(q)})<d_{Y}(\mu_{r(i)},\mu_{r(j)})+2{C_{1}}.$
Since ${L_{2}}>{L_{1}}(X)+2{C_{1}}$, we find that $Y\in C_{X}$. It follows
that $Y\in C_{X}(p,q)$. ∎
So when $C_{X}(p,q)$ is empty the interval $[p,q]$ is determined to be
straight. Proceed onto the next undetermined element of $\mathcal{P}(i,j)$.
Now suppose that $C_{X}(p,q)$ is non-empty. Then we choose any $Y\in
C_{X}(p,q)$ so that $Y$ has maximal $\xi(Y)$ amongst the elements of
$C_{X}(p,q)$. Notice that by the accessibility requirement that
$J_{Y}\cap[r(p),r(q)]$ is non-empty.
There are two cases. If $J_{Y}\cap r([p,q])$ is empty then let
$p^{\prime}\in[p,q]$ be the largest integer so that $r(p^{\prime})<\min
J_{Y}$. Note that $p^{\prime}$ is well-defined. Now divide the interval
$[p,q]$ into the two undetermined intervals $[p,p^{\prime}]$,
$[p^{\prime}+1,q]$. In this situation we say $Y$ is associated to a shortcut
of length one and we add the element $[p^{\prime}+\frac{1}{2}]$ to
$\mathcal{P}(i,j)$.
Next suppose that $J_{Y}\cap r([p,q])$ is non-empty. Let
$p^{\prime},q^{\prime}\in[p,q]$ be the first and last indices, respectively,
so that $r(p^{\prime}),r(q^{\prime})\in J_{Y}$. (Note that it is possible to
have $p^{\prime}=q^{\prime}$.) Partition
$[p,q]=[p,p^{\prime}-1]\cup[p^{\prime},q^{\prime}]\cup[q^{\prime}+1,q]$. The
first and third parts are undetermined; either may be empty. This completes
the recursive construction of the partition.
Define
$\mathcal{P}_{\text{short}}=\\{I\in\mathcal{P}(i,j)\mathbin{\mid}\mbox{$I$ is
a shortcut}\\}$
and
$\mathcal{P}_{\text{str}}=\\{I\in\mathcal{P}(i,j)\mathbin{\mid}\mbox{$I$ is
straight}\\}.$
###### Proposition 14.9.
With $\mathcal{P}(i,j)$ as defined above,
$d_{X}(\gamma_{i},\gamma_{j})\geq\frac{|\mathcal{P}_{\text{short}}|}{2(3\cdot{L_{2}})^{\xi(X)-1}+1}-1.$
###### Proof.
The proof is identical to that of Proposition 14.6 with the caveat that in
Lemma 14.4 we must use the markings $\mu_{r(i)}$ and $\mu_{r(j)}$ instead of
the endpoints $\gamma_{i}$ and $\gamma_{j}$. ∎
Now we “electrify” every shortcut interval using Theorem 13.1 recursively.
###### Lemma 14.10.
There is a constant ${L_{3}}={L_{3}}(X,\mathcal{G})$, so that for every
shortcut interval $[p,q]$ we have
$d_{\mathcal{G}}(\gamma_{p},\gamma_{q})<{L_{3}}$.
###### Proof.
As $[p,q]$ is a shortcut we are given a non-hole $Z\subset X$ so that
$r([p,q])\subset J_{Z}$. Let $Y=X{\smallsetminus}Z$. Thus Axiom 13.6 gives
vertices $\gamma_{p}^{\prime},\gamma_{q}^{\prime}$ of $\mathcal{G}$ lying in
$Y$ or in $Z$, so that
$d_{\mathcal{G}}(\gamma_{p},\gamma_{p}^{\prime}),d_{\mathcal{G}}(\gamma_{q},\gamma_{q}^{\prime})\leq{C_{4}}$.
If one of $\gamma_{p}^{\prime},\gamma_{q}^{\prime}$ lies in $Y$ while the
other lies in $Z$ then
$d_{\mathcal{G}}(\gamma_{p},\gamma_{q})<2{C_{4}}+1.$
If both lie in $Z$ then, as $Z$ is a non-hole, there is a vertex
$\delta\in\mathcal{G}(S)$ disjoint from both of $\gamma_{p}^{\prime}$ and
$\gamma_{q}^{\prime}$ and we have
$d_{\mathcal{G}}(\gamma_{p},\gamma_{q})<2{C_{4}}+2.$
If both lie in $Y$ then there are two cases. If $Y$ is not a hole for
$\mathcal{G}(S)$ then we are done as in the previous case. If $Y$ is a hole
then by the definition of shortcut interval, Lemma 5.7, and the triangle
inequality we have
$d_{W}(\gamma_{p}^{\prime},\gamma_{q}^{\prime})<6+6{C_{4}}+{L_{2}}$
for all holes $W\subset Y$. Notice that $Y$ is strictly contained in $X$. Thus
we may inductively apply Theorem 13.1 with $c=6+6{C_{4}}+{L_{2}}$. We deduce
that all terms on the right-hand side of the distance estimate vanish and thus
$d_{\mathcal{G}}(\gamma_{p}^{\prime},\gamma_{q}^{\prime})$ is bounded by a
constant depending only on $X$ and $\mathcal{G}$. The same then holds for
$d_{\mathcal{G}}(\gamma_{p},\gamma_{q})$ and we are done. ∎
We are now equipped to give:
###### Proof of Proposition 14.8.
Suppose that $\mathcal{P}(i,j)$ is the given partition of the electric
interval $[i,j]$ into straight and shortcut subintervals. As a bit of
notation, if $[p,q]=I\in\mathcal{P}(i,j)$, we take
$d_{\mathcal{G}}(I)=d_{\mathcal{G}}(\gamma_{p},\gamma_{q})$ and
$d_{X}(I)=d_{X}(\gamma_{p},\gamma_{q})$. Applying Axiom 13.5 we have
(14.11) $\displaystyle d_{\mathcal{G}}(\gamma_{i},\gamma_{j})$
$\displaystyle\leq\sum_{I\in\mathcal{P}_{\text{str}}}d_{\mathcal{G}}(I)+\sum_{I\in\mathcal{P}_{\text{short}}}d_{\mathcal{G}}(I)+{C_{2}}|\mathcal{P}(i,j)|$
The last term arises from connecting left endpoints of intervals with right
endpoints. We must bound the three terms on the right.
We begin with the third; recall that
$|\mathcal{P}(i,j)|=|\mathcal{P}_{\text{short}}|+|\mathcal{P}_{\text{str}}|$,
that $|\mathcal{P}_{\text{str}}|\leq|\mathcal{P}_{\text{short}}|+1$, and that
$|\mathcal{P}_{\text{short}}|\mathbin{\leq_{A}}d_{X}(\gamma_{i},\gamma_{j})$.
The second inequality follows from the construction of the partition while the
last is implied by Proposition 14.9. Thus the third term of Equation 14.11 is
quasi-bounded above by $d_{X}(\gamma_{i},\gamma_{j})$.
By Lemma 14.10, the second term of Equation 14.11 at most
${L_{3}}|\mathcal{P}_{\text{short}}|$. Finally, by Axiom 13.11, for all
$I\in\mathcal{P}_{\text{str}}$ we have
$d_{\mathcal{G}}(I)\mathbin{\leq_{A}}d_{X}(I),$
Also, it follows from the reverse triangle inequality (Lemma 13.12) that
$\sum_{I\in\mathcal{P}_{\text{str}}}d_{X}(I)\leq
d_{X}(\gamma_{i},\gamma_{j})+(2{C_{1}}+2{C_{2}})|\mathcal{P}_{\text{str}}|+2{C_{2}}.$
We deduce that $\sum_{I\in\mathcal{P}_{\text{str}}}d_{\mathcal{G}}(I)$ is also
quasi-bounded above by $d_{X}(\gamma_{i},\gamma_{j})$. Thus for a somewhat
larger value of $A$ we find
$d_{\mathcal{G}}(\gamma_{i},\gamma_{j})\mathbin{\leq_{A}}d_{X}(\gamma_{i},\gamma_{j}).$
This completes the proof. ∎
### 14.4. The upper bound
We will need:
###### Proposition 14.12.
For any $c>0$ there is a constant $A$ with the following property. Suppose
that $[i,j]=I_{Y}$ is an inductive interval in $\mathcal{P}_{X}$. Then we
have:
$d_{\mathcal{G}}(\gamma_{i},\gamma_{j})\mathbin{\leq_{A}}\sum_{Z}[d_{Z}(\gamma_{i},\gamma_{j})]_{c}$
where $Z$ ranges over all holes for $\mathcal{G}$ contained in $X$.
###### Proof.
Axiom 13.6 gives vertices $\gamma^{\prime}_{i}$,
$\gamma^{\prime}_{j}\in\mathcal{G}$, contained in $Y$, so that
$d_{\mathcal{G}}(\gamma_{i},\gamma^{\prime}_{i})\leq{C_{4}}$ and the same
holds for $j$. Since projection to holes is coarsely Lipschitz (Lemma 5.7) for
any hole $Z$ we have $d_{Z}(\gamma_{i},\gamma^{\prime}_{i})\leq 3+3{C_{3}}$.
Fix any $c>0$. Now, since
$\displaystyle d_{\mathcal{G}}(\gamma_{i},\gamma_{j})$ $\displaystyle\leq
d_{\mathcal{G}}(\gamma^{\prime}_{i},\gamma^{\prime}_{j})+2{C_{3}}$
to find the required constant $A$ it suffices to bound
$d_{\mathcal{G}}(\gamma^{\prime}_{i},\gamma^{\prime}_{j})$. Let
$c^{\prime}=c+6{C_{3}}+6$. Since $Y\subsetneq X$, induction gives us a
constant $A$ so that
$\displaystyle d_{\mathcal{G}}(\gamma^{\prime}_{i},\gamma^{\prime}_{j})$
$\displaystyle\mathbin{\leq_{A}}\sum_{Z}[d_{Z}(\gamma^{\prime}_{i},\gamma^{\prime}_{j})]_{c^{\prime}}$
$\displaystyle\leq\sum_{Z}[d_{Z}(\gamma_{i},\gamma_{j})+6{C_{3}}+6]_{c^{\prime}}$
$\displaystyle<(6{C_{3}}+6)N+\sum_{Z}[d_{Z}(\gamma_{i},\gamma_{j})]_{c}$
where $N$ is the number of non-zero terms in the final sum. Also, the sum
ranges over sub-holes of $Y$. We may take $A$ somewhat larger to deal with the
term $(6{C_{3}}+6)N$ and include all holes $Z\subset X$ to find
$\displaystyle d_{\mathcal{G}}(\gamma_{i},\gamma_{j})$
$\displaystyle\mathbin{\leq_{A}}\sum_{Z}[d_{Z}(\gamma_{i},\gamma_{j})]_{c}$
where the sum is over all holes $Z\subset X$. ∎
### 14.5. Finishing the proof
Now we may finish the proof of Theorem 13.1. Fix any constant $c\geq 0$.
Suppose that $X$, $\alpha_{X}$, $\beta_{X}$ are given as above. Suppose that
$\Gamma=\\{\gamma_{i}\\}_{i=0}^{K}$ is the given combinatorial path and
$\mathcal{P}_{X}$ is the partition of $[0,K]$ into inductive and electric
intervals. So we have:
(14.13) $\displaystyle d_{\mathcal{G}}(\alpha_{X},\beta_{X})$
$\displaystyle\leq\sum_{I\in\mathcal{P}_{\text{ind}}}d_{\mathcal{G}}(I)+\sum_{I\in\mathcal{P}_{\text{ele}}}d_{\mathcal{G}}(I)+{C_{2}}|\mathcal{P}_{X}|$
Again, the last term arises from adjacent right and left endpoints of
different intervals.
We must bound the terms on the right-hand side; begin by noticing that
$|\mathcal{P}_{X}|=|\mathcal{P}_{\text{ind}}|+|\mathcal{P}_{\text{ele}}|$,
$|\mathcal{P}_{\text{ele}}|\leq|\mathcal{P}_{\text{ind}}|+1$ and
$|\mathcal{P}_{\text{ind}}|\mathbin{\leq_{A}}d_{X}(\alpha_{X},\beta_{X})$. The
second inequality follows from the way the partition is constructed and the
last follows from Proposition 14.6. Thus the third term of Equation 14.13 is
quasi-bounded above by $d_{X}(\alpha_{X},\beta_{X})$.
Next consider the second term of Equation 14.13:
$\displaystyle\sum_{I\in\mathcal{P}_{\text{ele}}}d_{\mathcal{G}}(I)$
$\displaystyle\mathbin{\leq_{A}}\sum_{I\in\mathcal{P}_{\text{ele}}}d_{X}(I)$
$\displaystyle\leq
d_{X}(\alpha_{X},\beta_{X})+(2{C_{1}}+2{C_{2}})|\mathcal{P}_{\text{ele}}|+2{C_{2}}$
with the first inequality following from Proposition 14.8 and the second from
the reverse triangle inequality (Lemma 13.12).
Finally we bound the first term of Equation 14.13. Let $c^{\prime}=c+{L_{0}}$.
Thus,
$\displaystyle\sum_{I\in\mathcal{P}_{\text{ind}}}d_{\mathcal{G}}(I)$
$\displaystyle\leq\sum_{I_{Y}\in\mathcal{P}_{\text{ind}}}\left(A^{\prime}_{Y}\left(\sum_{Z\subsetneq
Y}[d_{Z}(I_{Y})]_{c^{\prime}}\right)+A^{\prime}_{Y}\right)$ $\displaystyle\leq
A^{\prime\prime}\left(\sum_{I\in\mathcal{P}_{\text{ind}}}\sum_{Z\subsetneq
X}[d_{Z}(I)]_{c^{\prime}}\right)+A^{\prime\prime}\cdot|\mathcal{P}_{\text{ind}}|$
$\displaystyle\leq A^{\prime\prime}\left(\sum_{Z\subsetneq
X}\sum_{I\in\mathcal{P}_{\text{ind}}}[d_{Z}(I)]_{c^{\prime}}\right)+A^{\prime\prime}\cdot|\mathcal{P}_{\text{ind}}|$
Here $A^{\prime}_{Y}$ and the first inequality are given by Proposition 14.12.
Also $A^{\prime\prime}=\max\\{A^{\prime}_{Y}\mathbin{\mid}Y\subsetneq X\\}$.
In the last line, each sum of the form
$\sum_{I\in\mathcal{P}_{\text{ind}}}[d_{Z}(I)]_{c^{\prime}}$ has at most three
terms, by Remark 14.3 and the fact that $c^{\prime}>{L_{0}}$. For the moment,
fix a hole $Z$ and any three elements
$I,I^{\prime},I^{\prime\prime}\in\mathcal{P}_{\text{ind}}$.
By the reverse triangle inequality (Lemma 13.12) we find that
$d_{Z}(I)+d_{Z}(I^{\prime})+d_{Z}(I^{\prime\prime})<d_{Z}(\alpha_{X},\beta_{X})+6{C_{1}}+8{C_{2}}$
which in turn is less than $d_{Z}(\alpha_{X},\beta_{X})+{L_{0}}$.
It follows that
$[d_{Z}(I)]_{c^{\prime}}+[d_{Z}(I^{\prime})]_{c^{\prime}}+[d_{Z}(I^{\prime\prime})]_{c^{\prime}}<[d_{Z}(\alpha_{X},\beta_{X})]_{c}+{L_{0}}.$
Thus,
$\displaystyle\sum_{Z\subsetneq
X}\sum_{I\in\mathcal{P}_{\text{ind}}}[d_{Z}(I)]_{c^{\prime}}$
$\displaystyle\leq{L_{0}}\cdot N+\sum_{Z\subsetneq
X}[d_{Z}(\alpha_{X},\beta_{X})]_{c}$
where $N$ is the number of non-zero terms in the final sum. Also, the sum
ranges over all holes $Z\subsetneq X$.
Combining the above inequalities, and increasing $A$ once again, implies that
$d_{\mathcal{G}}(\alpha_{X},\beta_{X})\mathbin{\leq_{A}}\sum_{Z}[d_{Z}(\alpha_{X},\beta_{X})]_{c}$
where the sum ranges over all holes $Z\subseteq X$. This completes the proof
of Theorem 13.1. ∎
## 15\. Background on Teichmüller space
Our goal in Sections 16, 17 and 19 will be to verify the axioms stated in
Section 13 for the complex of curves of a non-orientable surface, for the arc
complex, and for the disk complex. Here we give the necessary background on
Teichmüller space.
Fix now a surface $S=S_{g,n}$ of genus $g$ with $n$ punctures. Two conformal
structures on $S$ are equivalent, written $\Sigma\sim\Sigma^{\prime}$, if
there is a conformal map $f\colon\Sigma\to\Sigma^{\prime}$ which is isotopic
to the identity. Let $\mathcal{T}=\mathcal{T}(S)$ be the Teichmüller space of
$S$; the set of equivalence classes of conformal structures $\Sigma$ on $S$.
Define the Teichmüller metric by,
$d_{\mathcal{T}}(\Sigma,\Sigma^{\prime})=\inf_{f}\left\\{\frac{1}{2}\log
K(f)\right\\}$
where the infimum ranges over all quasiconformal maps
$f\colon\Sigma\to\Sigma^{\prime}$ isotopic to the identity and where $K(f)$ is
the maximal dilatation of $f$. Recall that the infimum is realized by a
Teichmüller map that, in turn, may be defined in terms of a quadratic
differential.
### 15.1. Quadratic differentials
###### Definition 15.1.
A quadratic differential $q(z)\,dz^{2}$ on $\Sigma$ is an assignment of a
holomorphic function to each coordinate chart that is a disk and of a
meromorphic function to each chart that is a punctured disk. If $z$ and
$\zeta$ are overlapping charts then we require
$q_{z}(z)=q_{\zeta}(\zeta)\left(\frac{d\zeta}{dz}\right)^{2}$
in the intersection of the charts. The meromorphic function $q_{z}(z)$ has at
most a simple pole at the puncture $z=0$.
At any point away from the zeroes and poles of $q$ there is a natural
coordinate $z=x+iy$ with the property that $q_{z}\equiv 1$. In this natural
coordinate the foliation by lines $y=c$ is called the horizontal foliation.
The foliation by lines $x=c$ is called the vertical foliation.
Now fix a quadratic differential $q$ on $\Sigma=\Sigma_{0}$. Let $x,y$ be
natural coordinates for $q$. For every $t\in\mathbb{R}$ we obtain a new
quadratic differential $q_{t}$ with coordinates
$x_{t}=e^{t}x,\qquad y_{t}=e^{-t}y.$
Also, $q_{t}$ determines a conformal structure $\Sigma_{t}$ on $S$. The map
$t\mapsto\Sigma_{t}$ is the Teichmüller geodesic determined by $\Sigma$ and
$q$.
### 15.2. Marking coming from a Teichmüller geodesic
Suppose that $\Sigma$ is a Riemann surface structure on $S$ and $\sigma$ is
the uniformizing hyperbolic metric in the conformal class of $\Sigma$. In a
slight abuse of terminology, we call the collection of shortest simple non-
peripheral closed geodesics the systoles of $\sigma$. Fix a constant
$\epsilon$ smaller than the Margulis constant. The $\epsilon$–thick part of
Teichmüller space consists of those Riemann surfaces such that the hyperbolic
systole has length at least $\epsilon$.
We define $P=P(\sigma)$, a Bers pants decomposition of $S$, as follows: pick
$\alpha_{1}$, any systole for $\sigma$. Define $\alpha_{i}$ to be any systole
of $\sigma$ restricted to
$S{\smallsetminus}(\alpha_{1}\cup\ldots\cup\alpha_{i-1})$. Continue in this
fashion until $P$ is a pants decomposition. Note that any curve with length
less than the Margulis constant will necessarily be an element of $P$.
Suppose that $\Sigma,\Sigma^{\prime}\in\mathcal{T}(S)$. Suppose that
$P,P^{\prime}$ are Bers pants decompositions with respect to $\Sigma$ and
$\Sigma^{\prime}$. Suppose also that
$d_{\mathcal{T}}(\Sigma,\Sigma^{\prime})\leq 1$. Then the curves in $P$ have
uniformly bounded lengths in $\Sigma^{\prime}$ and conversely. By the Collar
Lemma, the intersection $\iota(P,P^{\prime})$ is bounded, solely in terms of
$\xi(S)$.
Suppose now that $\\{\Sigma_{t}\mathbin{\mid}t\in[-M,M]\\}$ is the Teichmüller
geodesic defined by the quadratic differentials $q_{t}$. Let $\sigma_{t}$ be
the hyperbolic metric uniformizing $\Sigma_{t}$. Let $P_{t}=P(\sigma_{t})$ be
a Bers pants decomposition.
We now find transversals in order to complete $P_{t}$ to a Bers marking
$\nu_{t}$. Suppose that $P_{t}=\\{\alpha_{i}\\}$. For each $i$, let $A^{i}$ be
the annular cover of $S$ corresponding to $\alpha_{i}$. Note that $q_{t}$
lifts to a singular Euclidean metric $q^{i}_{t}$ on $A^{i}$. Let $\alpha^{i}$
be a geodesic representative of the core curve of $A^{i}$ with respect to the
metric $q_{t}^{i}$. Choose $\gamma_{i}\in\mathcal{C}(A^{i})$ to be any
geodesic arc, also with respect to $q^{i}_{t}$, that is perpendicular to
$\alpha^{i}$. Let $\beta_{i}$ be any curve in
$S{\smallsetminus}(\\{\alpha_{j}\\}_{j\neq i})$ which meets $\alpha_{i}$
minimally and so that $d_{A_{i}}(\beta_{i},\gamma_{i})\leq 3$. (See the
discussion after the proof of Lemma 2.4 in [25].) Doing this for each $i$
gives a complete clean marking $\nu_{t}=\\{\alpha_{i}\\}\cup\\{\beta_{i}\\}$.
We now have:
###### Lemma 15.2.
[33, Remark 6.2 and Equation (3)] There is a constant $B_{0}=B_{0}(S)$ with
the following property. For any Teichmüller geodesic and for any time $t$,
there is a constant $\delta>0$ so that if $|t-s|\leq\delta$ then
$\iota(\nu_{t},\nu_{s})<B_{0}.$
Suppose that $\Sigma_{t}$ and $\Sigma_{s}$ are surfaces in the
$\epsilon$–thick part of $\mathcal{T}(S)$. We take $B_{0}$ sufficiently large
so that if $\iota(\nu_{t},\nu_{s})\geq B_{0}$ then
$d_{\mathcal{T}}(\Sigma_{t},\Sigma_{s})\geq 1$.
### 15.3. The marking axiom
We construct a sequence of markings $\mu_{n}$, for
$n\in[0,N]\subset\mathbb{N}$, as follows. Take $\mu_{0}=\nu_{-M}$. Now suppose
that $\mu_{n}=\nu_{t}$ is defined. Let $s>t$ be the first time that there is a
marking with $\iota(\nu_{t},\nu_{s})\geq B_{0}$, if such a time exists. If so,
let $\mu_{n+1}=\nu_{s}$. If no such time exists take $N=n$ and we are done.
We now show that $\mu_{n}=\nu_{t}$ and $\mu_{n+1}=\nu_{s}$ have bounded
intersection. By the above lemma there is a marking $\nu_{r}$ with $t\leq r<s$
and
$\iota(\nu_{r},\nu_{s})\leq B_{0}.$
By construction
$\iota(\nu_{t},\nu_{r})<B_{0}.$
Since intersection number bounds distance in the marking complex we find that
by the triangle inequality, $\nu_{t}$ and $\nu_{s}$ are bounded distance in
the marking complex. Conversely, since distance bounds intersection in the
marking complex we find that $\iota(\mu_{n},\mu_{n+1})$ is bounded. It follows
that $d_{Y}(\mu_{n},\mu_{n+1})$ is uniformly bounded, independent of $Y\subset
S$ and of $n\in[0,N]$.
It now follows from Theorem 6.1 of [33] that, for any subsurface $Y\subset S$,
the sequence $\\{\pi_{Y}(\mu_{n})\\}\subset\mathcal{C}(Y)$ is an
unparameterized quasi-geodesic. Thus the marking path $\\{\mu_{n}\\}$
satisfies the second requirement of Axiom 13.3. The first requirement is
trivial as every $\mu_{n}$ fills $S$.
### 15.4. The accessibility axiom
We now turn to Axiom 13.4. Since $\mu_{n}$ fills $S$ for every $n$, the first
requirement is a triviality.
In Section 5 of [33] Rafi defines, for every subsurface $Y\subset S$, an
interval of isolation $I_{Y}$ inside of the parameterizing interval of the
Teichmüller geodesic. Note that $I_{Y}$ is defined purely in terms of the
geometry of the given quadratic differentials. Further, for all $t\in I_{Y}$
and for all components $\alpha\subset\partial Y$ the hyperbolic length of
$\alpha$ in $\Sigma_{t}$ is less than the Margulis constant. Furthermore, by
Theorem 5.3 [33], there is a constant ${B_{3}}$ so that if $[s,t]\cap
I_{Y}=\emptyset$ then
$d_{Y}(\nu_{s},\nu_{t})\leq{B_{3}}.$
So define $J_{Y}\subset[0,N]$ to be the subinterval of the marking path where
the time corresponding to $\mu_{n}$ lies in $I_{Y}$. The third requirement
follows. Finally, if $m\in J_{Y}$ then $\partial Y$ is contained in
$\operatorname{base}(\mu_{m})$ and thus $\iota(\partial Y,\mu_{m})\leq
2\cdot|\partial Y|$.
### 15.5. The distance estimate in Teichmüller space
We end this section by quoting another result of Rafi:
###### Theorem 15.3.
[33, Theorem 2.4] Fix a surface $S$ and a constant $\epsilon>0$. There is a
constant ${C_{0}}={C_{0}}(S,\epsilon)$ so that for any $c>{C_{0}}$ there is a
constant $A$ with the following property. Suppose that $\Sigma$ and
$\Sigma^{\prime}$ lie in the $\epsilon$–thick part of $\mathcal{T}(S)$. Then
$d_{\mathcal{T}}(\Sigma,\Sigma^{\prime})\mathbin{=_{A}}\sum_{X}[d_{X}(\mu,\mu^{\prime})]_{c}+\sum_{\alpha}[\log
d_{\alpha}(\mu,\mu^{\prime})]_{c}$
where $\mu$ and $\mu^{\prime}$ are Bers markings on $\Sigma$ and
$\Sigma^{\prime}$, $Y\subset S$ ranges over non-annular surfaces and $\alpha$
ranges over vertices of $\mathcal{C}(S)$. ∎
## 16\. Paths for the non-orientable surface
Fix $F$ a compact, connected, and non-orientable surface. Let $S$ be the
orientation double cover with covering map $\rho_{F}\colon S\to F$. Let
$\tau\colon S\to S$ be the associated involution. Note that
$\mathcal{C}(F)=\mathcal{C}^{\tau}(S)$. Let
$\mathcal{C}^{\tau}(S)\to\mathcal{C}(S)$ be the relation sending a symmetric
multicurve to its components.
Our goal for this section is to prove Lemma 16.4, the classification of holes
for $\mathcal{C}(F)$. As remarked above, Lemma 6.3 and Corollary 6.4 follow,
proving the hyperbolicity of $\mathcal{C}(F)$.
### 16.1. The marking path
We will use the extreme rigidity of Teichmüller geodesics to find
$\tau$–invariant marking paths. We first show that $\tau$–invariant Bers pants
decompositions exist.
###### Lemma 16.1.
Fix a $\tau$–invariant hyperbolic metric $\sigma$. Then there is a Bers pants
decomposition $P=P(\sigma)$ which is $\tau$–invariant.
###### Proof.
Let $P_{0}=\emptyset$. Suppose that $0\leq k<\xi(S)$ curves have been chosen
to form $P_{k}$. By induction we may assume that $P_{k}$ is $\tau$–invariant.
Let $Y$ be a component of $S{\smallsetminus}P_{k}$ with $\xi(Y)\geq 1$. Note
that since $\tau$ is orientation reversing, $\tau$ does not fix any boundary
component of $Y$.
Pick any systole $\alpha$ for $Y$.
###### Claim.
Either $\tau(\alpha)=\alpha$ or $\alpha\cap\tau(\alpha)=\emptyset$.
###### Proof.
Suppose not and take $p\in\alpha\cap\tau(\alpha)$. Then
$\tau(p)\in\alpha\cap\tau(\alpha)$ as well, and, since $\tau$ has no fixed
points, $p\neq\tau(p)$. The points $p$ and $\tau(p)$ divide $\alpha$ into
segments $\beta$ and $\gamma$. Since $\tau$ is an isometry, we have
$\ell_{\sigma}(\tau(\alpha))=\ell_{\sigma}(\alpha)\quad\mbox{and}\quad\ell_{\sigma}(\tau(\beta))=\ell_{\sigma}(\beta).$
Now concatenate to obtain (possibly immersed) loops
$\beta^{\prime}=\beta*\tau(\beta)\quad\mbox{and}\quad\gamma^{\prime}=\gamma*\tau(\gamma).$
If $\beta^{\prime}$ is null-homotopic then $\alpha\cup\tau(\alpha)$ cuts a
monogon or a bigon out of $S$, contradicting our assumption that $\alpha$ was
a geodesic. Suppose, by way of contradiction, that $\beta^{\prime}$ is
homotopic to some boundary curve $b\subset\partial Y$. Since
$\tau(\beta^{\prime})=\beta^{\prime}$, it follows that $\tau(b)$ and
$\beta^{\prime}$ are also homotopic. Thus $b$ and $\tau(b)$ cobound an
annulus, implying that $Y$ is an annulus, a contradiction. The same holds for
$\gamma^{\prime}$.
Let $\beta^{\prime\prime}$ and $\gamma^{\prime\prime}$ be the geodesic
representatives of $\beta^{\prime}$ and $\gamma^{\prime}$. Since $\beta$ and
$\tau(\beta)$ meet transversely, $\beta^{\prime\prime}$ has length in $\sigma$
strictly smaller than $2\ell_{\sigma}(\beta)$. Similarly the length of
$\gamma^{\prime\prime}$ is strictly smaller than $2\ell_{\sigma}(\gamma)$.
Suppose that $\beta^{\prime\prime}$ is shorter then $\gamma^{\prime\prime}$.
It follows that $\beta^{\prime\prime}$ strictly shorter than $\alpha$. If
$\beta^{\prime\prime}$ is embedded then this contradicts the assumption that
$\alpha$ was shortest. If $\beta^{\prime\prime}$ is not embedded then there is
an embedded curve $\beta^{\prime\prime\prime}$ inside of a regular
neighborhood of $\beta^{\prime\prime}$ which is again essential, non-
peripheral, and has geodesic representative shorter than
$\beta^{\prime\prime}$. This is our final contradiction and the claim is
proved. ∎
Thus, if $\tau(\alpha)=\alpha$ we let $P_{k+1}=P_{k}\cup\\{\alpha\\}$ and we
are done. If $\tau(\alpha)\neq\alpha$ then by the above claim
$\tau(\alpha)\cap\alpha=\emptyset$. In this case let
$P_{k+2}=P_{k}\cup\\{\alpha,\tau(\alpha)\\}$ and Lemma 16.1 is proved. ∎
Transversals are chosen with respect to a quadratic differential metric.
Suppose that $\alpha,\beta\in\mathcal{C}^{\tau}(S)$. If $\alpha$ and $\beta$
do not fill $S$ then we may replace $S$ by the support of their union.
Following Thurston [38] there exists a square-tiled quadratic differential $q$
with squares associated to the points of $\alpha\cap\beta$. (See [6] for
analysis of how the square-tiled surface relates to paths in the curve
complex.) Let $q_{t}$ be image of $q$ under the Teichmüller geodesic flow. We
have:
###### Lemma 16.2.
$\tau^{*}q_{t}=q_{t}$.
###### Proof.
Note that $\tau$ preserves $\alpha$ and also $\beta$. Since $\tau$ permutes
the points of $\alpha\cap\beta$ it permutes the rectangles of the singular
Euclidean metric $q_{t}$ while preserving their vertical and horizontal
foliations. Thus $\tau$ is an isometry of the metric and the conclusion
follows. ∎
We now choose the Teichmüller geodesic
$\\{\Sigma_{t}\mathbin{\mid}t\in[-M,M]\\}$ so that the hyperbolic length of
$\alpha$ is less than the Margulis constant in $\sigma_{-M}$ and the same
holds for $\beta$ in $\sigma_{M}$. Also, $\alpha$ is the shortest curve in
$\sigma_{-M}$ and similarly for $\beta$ in $\sigma_{M}$
###### Lemma 16.3.
Fix $t$. There are transversals for $P_{t}$ which are close to being quadratic
perpendicular in $q_{t}$ and which are $\tau$–invariant.
###### Proof.
Let $P=P_{t}$ and fix $\alpha\in P$. Let
$X=S{\smallsetminus}(P{\smallsetminus}\alpha)$. There are two cases: either
$\tau(X)\cap X=\emptyset$ or $\tau(X)=X$. Suppose the former. So we choose any
transversal $\beta\subset X$ close to being $q_{t}$–perpendicular and take
$\tau(\beta)$ to be the transversal to $\tau(\alpha)$.
Suppose now that $\tau(X)=X$. It follows that $X$ is a four-holed sphere. The
quotient $X/\tau$ is homeomorphic to a twice-holed $\mathbb{RP}^{2}$.
Therefore there are only four essential non-peripheral curves in $X/\tau$. Two
of these are cores of Möbius bands and the other two are their doubles. The
cores meet in a single point. Perforce $\alpha$ is the double cover of one
core and we take $\beta$ the double cover of the other.
It remains only to show that $\beta$ is close to being $q_{t}$–perpendicular.
Let $S^{\alpha}$ be the annular cover of $S$ and lift $q_{t}$ to $S^{\alpha}$.
Let $\perp$ be the set of $q_{t}^{\alpha}$–perpendiculars. This is a
$\tau$-invariant diameter one subset of $\mathcal{C}(S^{\alpha})$. If
$d_{\alpha}(\perp,\beta)$ is large then it follows that
$d_{\alpha}(\perp,\tau(\beta))$ is also large. Also, $\tau(\beta)$ twists in
the opposite direction from $\beta$. Thus
$d_{\alpha}(\beta,\tau(\beta))-2d_{\alpha}(\perp,\beta)=O(1)$
and so $d_{\alpha}(\beta,\tau(\beta))$ is large, contradicting the fact that
$\beta$ is $\tau$–invariant. ∎
Thus $\tau$–invariant markings exist; these have bounded intersection with the
markings constructed in Section 15. It follows that the resulting marking path
satisfies the marking path and accessibility requirements, Axioms 13.3 and
13.4.
### 16.2. The combinatorial path
As in Section 15 break the interval $[-M,M]$ into short subintervals and
produce a sequence of $\tau$-invariant markings $\\{\mu_{n}\\}_{n=0}^{N}$. To
choose the combinatorial path, pick
$\gamma_{n}\in\operatorname{base}(\mu_{n})$ so that $\gamma_{n}$ is a
$\tau$–invariant curve or pair of curves and so that $\gamma_{n}$ is shortest
in $\operatorname{base}(\mu_{n})$.
We now check the combinatorial path requirements given in Axiom 13.5. Note
that $\gamma_{0}=\alpha$, $\gamma_{N}=\beta$; also the reindexing map is the
identity. Since
$\iota(\gamma_{n},\mu_{r(n)})=\iota(\gamma_{n},\mu_{n})=2$
the first requirement is satisfied. Since $\mu_{n}$ and $\mu_{n+1}$ have
bounded intersection, the same holds for $\gamma_{n}$ and $\gamma_{n+1}$.
Projection to $F$, surgery, and Lemma 2.2 imply that
$d_{\mathcal{C}^{\tau}}(\gamma_{n},\gamma_{n+1})$ is uniformly bounded. This
verifies Axiom 13.5.
### 16.3. The classification of holes
We now finish the classification of large holes for $\mathcal{C}^{\tau}(S)$.
Fix $L_{0}>3{C_{3}}+2{C_{2}}+2{C_{1}}$. Note that these constants are
available because we have verified the axioms that give them.
###### Lemma 16.4.
Suppose that $\alpha,\beta\in\mathcal{C}^{\tau}(S)$. Suppose that $X\subset S$
has $d_{X}(\alpha,\beta)>L_{0}$. Then $X$ is symmetric.
###### Proof.
Let $(\Sigma_{t},q_{t})$ be the Teichmüller geodesic defined above and let
$\sigma_{t}$ be the uniformizing hyperbolic metric. Since
$L_{0}>{C_{3}}+2{C_{2}}$ it follows from the accessibility requirement that
$J_{X}=[m,n]$ is non-empty. Now for all $t$ in the interval of isolation
$I_{X}$
$\ell_{\sigma_{t}}(\delta)<\epsilon,$
where $\delta$ is any component of $\partial X$ and $\epsilon$ is the Margulis
constant. Let $Y=\tau(X)$. Since $\tau$ is an isometry (Lemma 16.2) and since
the interval of isolation is metrically defined we have $I_{Y}=I_{X}$ and thus
$J_{Y}=J_{X}$. Deduce that $\partial Y$ is also short in $\sigma_{t}$. This
implies that $\partial X\cap\partial Y=\emptyset$. If $X$ and $Y$ overlap then
by (iii) of Lemma 13.13 we have
$d_{X}(\mu_{m},\mu_{n})<{C_{3}}$
and so by the triangle inequality, two applications of (2) of Axiom 13.4, we
have
$d_{X}(\mu_{0},\mu_{N})<3{C_{3}}.$
By the combinatorial axiom it follows that
$d_{X}(\alpha,\beta)<3{C_{3}}+2{C_{2}}$
a contradiction. Deduce that either $X=Y$ or $X\cap Y=\emptyset$ as desired. ∎
As noted in Section 6 this shows that the only hole for
$\mathcal{C}^{\tau}(S)$ is $S$ itself. Thus all holes trivially interfere,
verifying Axiom 13.2.
### 16.4. The replacement axiom
We now verify Axiom 13.6 for subsurfaces $Y\subset S$ with
$d_{Y}(\alpha,\beta)\geq L_{0}$. (We may ignore all subsurfaces with smaller
projection by taking ${L_{1}}(Y)>L_{0}$.)
By Lemma 16.4 the subsurface $Y$ is symmetric. If $Y$ is a hole then $Y=S$ and
the first requirement is vacuous. Suppose that $Y$ is not a hole. Suppose that
$\gamma_{n}$ is such that $n\in J_{Y}$. Thus
$\gamma_{n}\in\operatorname{base}(\mu_{n})$. All components of $\partial Y$
are also pants curves in $\mu_{n}$. It follows that we may take any symmetric
curve in $\partial Y$ to be $\gamma^{\prime}$ and we are done.
### 16.5. On straight intervals
Lastly we verify Axiom 13.11. Suppose that $[p,q]$ is a straight interval. We
must show that $d_{\mathcal{C}^{\tau}}(\gamma_{p},\gamma_{q})\leq
d_{S}(\gamma_{p},\gamma_{q})$. Suppose that $\mu_{p}=\nu_{s}$ and
$\mu_{q}=\nu_{t}$; that is, $s$ and $t$ are the times when $\mu_{p},\mu_{q}$
are short markings. Thus $d_{X}(\mu_{p},\mu_{q})\leq{L_{2}}$ for every
$X\subsetneq S$. This implies that the Teichmüller geodesic, along the
straight interval, lies in the thick part of Teichmüller space.
Notice that $d_{\mathcal{C}^{\tau}}(\gamma_{p},\gamma_{q})\leq{C_{2}}|p-q|$,
since for all $i\in[p,q-1]$,
$d_{\mathcal{C}^{\tau}}(\gamma_{i},\gamma_{i+1})\leq{C_{2}}$. So it suffices
to bound $|p-q|$. By our choice of $B_{0}$ and because the Teichmüller
geodesic lies in the thick part we find that $|p-q|\leq
d_{\mathcal{T}}(\Sigma_{s},\Sigma_{t})$. Rafi’s distance estimate (Theorem
15.3) gives:
$d_{\mathcal{T}}(\Sigma_{s},\Sigma_{t})\mathbin{=_{A}}d_{S}(\nu_{s},\nu_{t}).$
Since $\nu_{s}=\mu_{p}$, $\nu_{t}=\mu_{q}$, and since
$\gamma_{p}\in\operatorname{base}(\mu_{p})$,
$\gamma_{q}\in\operatorname{base}(\mu_{q})$ deduce that
$d_{S}(\mu_{p},\mu_{q})\leq d_{S}(\gamma_{p},\gamma_{q})+4.$
This verifies Axiom 13.11. Thus the distance estimate holds for
$\mathcal{C}^{\tau}(S)=\mathcal{C}(F)$. Since there is only one hole for
$\mathcal{C}(F)$ we deduce that the map $\mathcal{C}(F)\to\mathcal{C}(S)$ is a
quasi-isometric embedding. As a corollary we have:
###### Theorem 16.5.
The curve complex $\mathcal{C}(F)$ is Gromov hyperbolic. ∎
## 17\. Paths for the arc complex
Here we verify that our axioms hold for the arc complex
$\mathcal{A}(S,\Delta)$. It is worth pointing out that the axioms may be
verified using Teichmüller geodesics, train track splitting sequences, or
resolutions of hierarchies. Here we use the former because it also generalizes
to the non-orientable case; this is discussed at the end of this section.
First note that Axiom 13.2 follows from Lemma 7.3.
### 17.1. The marking path
We are given a pair of arcs $\alpha,\beta\in\mathcal{A}(X,\Delta)$. Recall
that $\sigma_{S}\colon\mathcal{A}(X)\to\mathcal{C}(X)$ is the surgery map,
defined in Definition 4.2. Let $\alpha^{\prime}=\sigma_{S}(\alpha)$ and define
$\beta^{\prime}$ similarly. Note that $\alpha^{\prime}$ cuts a pants off of
$S$. As usual, we may assume that $\alpha^{\prime}$ and $\beta^{\prime}$ fill
$X$. If not we pass to the subsurface they do fill.
As in the previous sections let $q$ be the quadratic differential determined
by $\alpha^{\prime}$ and $\beta^{\prime}$. Exactly as above, fix a marking
path $\\{\mu_{n}\\}_{n=0}^{N}$. This path satisfies the marking and
accessibility axioms (13.3, 13.4).
### 17.2. The combinatorial path
Let $Y_{n}\subset X$ be any component of
$X{\smallsetminus}\operatorname{base}(\mu_{n})$ meeting $\Delta$. So $Y_{n}$
is a pair of pants. Let $\gamma_{n}$ be any essential arc in $Y_{n}$ with both
endpoints in $\Delta$. Since
$\alpha^{\prime}\subset\operatorname{base}(\mu_{0})$ and
$\beta^{\prime}\subset\operatorname{base}(\mu_{N})$ we may choose
$\gamma_{0}=\alpha$ and $\gamma_{N}=\beta$.
As in the previous section the reindexing map is the identity. It follows
immediately that $\iota(\gamma_{n},\mu_{n})\leq 4$. This bound, the bound on
$\iota(\mu_{n},\mu_{n+1})$, and Lemma 4.7 imply that
$\iota(\gamma_{n},\gamma_{n+1})$ is likewise bounded. The usual surgery
argument shows that if two arcs have bounded intersection then they have
bounded distance. This verifies Axiom 13.5.
### 17.3. The replacement and the straight axioms
Suppose that $Y\subset X$ is a subsurface and $\gamma_{n}$ has $n\in J_{Y}$.
Let $\mu_{n}=\nu_{t}$; that is $t$ is the time when $\mu_{n}$ is a short
marking. Thus $\partial Y\subset\operatorname{base}(\mu_{n})$ and so
$\gamma_{n}\cap\partial Y=\emptyset$. So regardless of the hole-nature of $Y$
we may take $\gamma^{\prime}=\gamma_{n}$ and the axiom is verified.
Axiom 13.11 is verified exactly as in Section 16.
### 17.4. Non-orientable surfaces
Suppose that $F$ is non-orientable and $\Delta_{F}$ is a collection of
boundary components. Let $S$ be the orientation double cover and $\tau\colon
S\to S$ the involution so that $S/\tau=F$. Let $\Delta$ be the preimage of
$\Delta_{F}$. Then $\mathcal{A}^{\tau}(S,\Delta)$ is the invariant arc
complex.
Suppose that $\alpha_{F},\beta_{F}$ are vertices in
$\mathcal{A}(F,\Delta^{\prime})$. Let $\alpha,\beta$ be their preimages. As
above, without loss of generality, we may assume that $\sigma_{F}(\alpha_{F})$
and $\sigma_{F}(\beta_{F})$ fill $F$. Note that $\sigma_{F}(\alpha_{F})$ cuts
a surface $X$ off of $F$. The surface $X$ is either a pants or a twice-holed
$\mathbb{RP}^{2}$. When $X$ is a pants we define $\alpha^{\prime}\subset S$ to
be the preimage of $\sigma_{F}(\alpha_{F})$. When $X$ is a twice-holed
$\mathbb{RP}^{2}$ we take $\gamma_{F}$ to be a core of one of the two Möbius
bands contained in $X$ and we define $\alpha^{\prime}$ to be the preimage of
$\gamma_{F}\cup\sigma_{F}(\alpha_{F})$. We define $\beta^{\prime}$ similarly.
Notice that $\alpha$ and $\alpha^{\prime}$ meet in at most four points.
We now use $\alpha^{\prime}$ and $\beta^{\prime}$ to build a $\tau$–invariant
Teichmüller geodesic. The construction of the marking and combinatorial paths
for $\mathcal{A}^{\tau}(S,\Delta)$ is unchanged. Notice that we may choose
combinatorial vertices because $\operatorname{base}(\mu_{n})$ is
$\tau$–invariant. There is a small annoyance: when $X$ is a twice-holed
$\mathbb{RP}^{2}$ the first vertex, $\gamma_{0}$, is disjoint from but not
equal to $\alpha$. Strictly speaking, the first and last vertices are
$\gamma_{0}$ and $\gamma_{N}$; our constants are stated in terms of their
subsurface projection distances. However, since
$\alpha\cap\gamma_{0}=\emptyset$, and the same holds for $\beta$,
$\gamma_{N}$, their subsurface projection distances are all bounded.
## 18\. Background on train tracks
Here we give the necessary definitions and theorems regarding train tracks.
The standard reference is [31]. See also [30]. We follow closely the
discussion found in [27].
### 18.1. On tracks
A generic train track $\tau\subset S$ is a smooth, embedded trivalent graph.
As usual we call the vertices switches and the edges branches. At every switch
the tangents of the three branches agree. Also, there are exactly two incoming
branches and one outgoing branch at each switch. See Figure 7 for the local
model of a switch.
2pt incoming [bl] at 1 74 incoming [tl] at 1 1 outgoing [bl] at 145 38
$\begin{array}[]{c}\includegraphics[height=2cm]{ttmodel}\end{array}$
Figure 7. The local model of a train track.
Let $\mathcal{B}(\tau)$ be the set of branches. A transverse measure on $\tau$
is function $w\colon\mathcal{B}\to\mathbb{R}_{\geq 0}$ satisfying the switch
conditions: at every switch the sum of the incoming measures equals the
outgoing measure. Let $P(\tau)$ be the projectivization of the cone of
transverse measures. Let $V(\tau)$ be the vertices of $P(\tau)$. As discussed
in the references, each vertex measure gives a simple closed curve carried by
$\tau$.
For every track $\tau$ we refer to $V(\tau)$ as the marking corresponding to
$\tau$ (see Section 2.4). Note that there are only finitely many tracks up to
the action of the mapping class group. It follows that $\iota(V(\tau))$ is
uniformly bounded, depending only on the topological type of $S$.
If $\tau$ and $\sigma$ are train tracks, and $Y\subset S$ is an essential
surface, then define
$d_{Y}(\tau,\sigma)=d_{Y}(V(\tau),V(\sigma)).$
We also adopt the notation $\pi_{Y}(\tau)=\pi_{Y}(V(\tau))$.
A train track $\sigma$ is obtained from $\tau$ by sliding if $\sigma$ and
$\tau$ are related as in Figure 8. We say that a train track $\sigma$ is
obtained from $\tau$ by splitting if $\sigma$ and $\tau$ are related as in
Figure 9.
$\begin{array}[]{cc}\includegraphics[height=1.5cm]{slide}&\includegraphics[height=1.5cm]{slide2}\end{array}$
Figure 8. All slides take place in a small regular neighborhood of the
affected branch.
$\begin{array}[]{cc}\includegraphics[height=1.5cm]{split}&\includegraphics[height=1.5cm]{split2}\\\
\includegraphics[height=1.5cm]{split3}&\includegraphics[height=1.5cm]{split4}\\\
\end{array}$ Figure 9. There are three kinds of splitting: right, left, and
central.
Again, since the number of tracks is bounded (up to the action of the mapping
class group) if $\sigma$ is obtained from $\tau$ by either a slide or a split
we find that $\iota(V(\tau),V(\sigma))$ is uniformly bounded.
### 18.2. The marking path
We will use sequences of train tracks to define our marking path.
###### Definition 18.1.
A sliding and splitting sequence is a collection $\\{\tau_{n}\\}_{n=0}^{N}$ of
train tracks so that $\tau_{n+1}$ is obtained from $\tau_{n}$ by a slide or a
split.
The sequence $\\{\tau_{n}\\}$ gives a sequence of markings via the map
$\tau_{n}\mapsto V_{n}=V(\tau_{n})$. Note that the support of $V_{n+1}$ is
contained within the support of $V_{n}$ because every vertex of $\tau_{n+1}$
is carried by $\tau_{n}$. Theorem 5.5 of [27] verifies the remaining half of
Axiom 13.3.
###### Theorem 18.2.
Fix a surface $S$. There is a constant $A$ with the following property.
Suppose that $\\{\tau_{n}\\}_{n=0}^{N}$ is a sliding and splitting sequence in
$S$ of birecurrent tracks. Suppose that $Y\subset S$ is an essential surface.
Then the map $n\mapsto\pi_{Y}(\tau_{n})$, as parameterized by splittings, is
an $A$–unparameterized quasi-geodesic. ∎
Note that, when $Y=S$, Theorem 18.2 is essentially due to the first author and
Minsky; see Theorem 1.3 of [26].
In Section 5.2 of [27], for every sliding and splitting sequence
$\\{\tau_{n}\\}_{n=0}^{N}$ and for any essential subsurface $X\subsetneq S$ an
accessible interval $I_{X}\subset[0,N]$ is defined. Axiom 13.4 is now verified
by Theorem 5.3 of [27].
### 18.3. Quasi-geodesics in the marking graph
We will also need Theorem 6.1 from [27]. (See [16] for closely related work.)
###### Theorem 18.3.
Fix a surface $S$. There is a constant $A$ with the following property.
Suppose that $\\{\tau_{n}\\}_{n=0}^{N}$ is a sliding and splitting sequence of
birecurrent tracks, injective on slide subsequences, where $V_{N}$ fills $S$.
Then $\\{V(\tau_{n})\\}$ is an $A$–quasi-geodesic in the marking graph. ∎
## 19\. Paths for the disk complex
Suppose that $V=V_{g}$ is a genus $g$ handlebody. The goal of this section is
to verify the axioms of Section 13 for the disk complex $\mathcal{D}(V)$ and
so complete the proof of the distance estimate.
###### Theorem 19.1.
There is a constant ${C_{0}}={C_{0}}(V)$ so that, for any $c\geq{C_{0}}$ there
is a constant $A$ with
$d_{\mathcal{D}}(D,E)\mathbin{=_{A}}\sum[d_{X}(D,E)]_{c}$
independent of the choice of $D$ and $E$. Here the sum ranges over the set of
holes $X\subset\partial V$ for the disk complex.
### 19.1. Holes
The fact that all large holes interfere is recorded above as Lemma 12.13. This
verifies Axiom 13.2.
### 19.2. The combinatorial path
Suppose that $D,E\in\mathcal{D}(V)$ are disks contained in a compressible hole
$X\subset S=\partial V$. As usual we may assume that $D$ and $E$ fill $X$.
Recall that $V(\tau)$ is the set of vertices for the track $\tau\subset X$. We
now appeal to a result of the first author and Minsky, found in [26].
###### Theorem 19.2.
There exists a surgery sequence of disks $\\{D_{i}\\}_{i=0}^{K}$, a sliding
and splitting sequence of birecurrent tracks $\\{\tau_{n}\\}_{n=0}^{N}$, and a
reindexing function $r\colon[0,K]\to[0,N]$ so that
* •
$D_{0}=D$,
* •
$E\in V_{N}$,
* •
$D_{i}\cap D_{i+1}=\emptyset$ for all $i$, and
* •
$\iota(\partial D_{i},V_{r(i)})$ is uniformly bounded for all $i$.
∎
###### Remark 19.3.
For the details of the proof we refer to [26]. Note that the double-wave curve
replacements of that paper are not needed here; as $X$ is a hole, no curve of
$\partial X$ compresses in $V$. It follows that consecutive disks in the
surgery sequence are disjoint (as opposed to meeting at most four times).
Also, in the terminology of [27], the disk $D_{i}$ is a wide dual for the
track $\tau_{r(i)}$. Finally, recurrence of $\tau_{n}$ follows because $E$ is
fully carried by $\tau_{N}$. Transverse recurrence follows because $D$ is
fully dual to $\tau_{0}$.
Thus $V_{n}$ will be our marking path and $D_{i}$ will be our combinatorial
path. The requirements of Axiom 13.5 are now verified by Theorem 19.2.
### 19.3. The replacement axiom
We turn to Axiom 13.6. Suppose that $Y\subset X$ is an essential subsurface
and $D_{i}$ has $r(i)\in J_{Y}$. Let $n=r(i)$. From Theorem 19.2 we have that
$\iota(\partial D_{i},V_{n})$ is uniformly bounded. By Axiom 13.4 we have
$Y\subset\operatorname{supp}(V_{n})$ and $\iota(\partial Y,\mu_{n})$ is
bounded. It follows that there is a constant $K$ depending only on $\xi(S)$ so
that
$\iota(\partial D_{i},\partial Y)<K.$
Isotope $D_{i}$ to have minimal intersection with $\partial Y$. As in Section
11.1 boundary compress $D_{i}$ as much as possible into the components of
$X{\smallsetminus}\partial Y$ to obtain a disk $D^{\prime}$ so that either
* •
$D^{\prime}$ cannot be boundary compressed any more into
$X{\smallsetminus}\partial Y$ or
* •
$D^{\prime}$ is disjoint from $\partial Y$.
We may arrange matters so that every boundary compression reduces the
intersection with $\partial Y$ by at least a factor of two. Thus:
$d_{\mathcal{D}}(D_{i},D^{\prime})\leq\log_{2}(K).$
Suppose now that $Y$ is a compressible hole. By Lemma 8.4 we find that
$\partial D^{\prime}\subset Y$ and we are done.
Suppose now that $Y$ is an incompressible hole. Since $Y$ is large there is an
$I$-bundle $T\to F$, contained in the handlebody $V$, so that $Y$ is a
component of $\partial_{h}T$. Isotope $D^{\prime}$ to minimize intersection
with $\partial_{v}T$. Let $\Delta$ be the union of components of
$\partial_{v}T$ which are contained in $\partial V$. Let
$\Gamma=\partial_{v}T{\smallsetminus}\Delta$. Notice that all intersections
$D^{\prime}\cap\Gamma$ are essential arcs in $\Gamma$: simple closed curves
are ruled out by minimal intersection and inessential arcs are ruled out by
the fact that $D^{\prime}$ cannot be boundary compressed in the complement of
$\partial Y$. Let $D^{\prime\prime}$ be a outermost component of
$D^{\prime}{\smallsetminus}\Gamma$. Then Lemma 8.5 implies that
$D^{\prime\prime}$ is isotopic in $T$ to a vertical disk.
If $D^{\prime\prime}=D^{\prime}$ then we may replace $D_{i}$ by the arc
$\rho_{F}(D^{\prime})$. The inductive argument now occurs inside of the arc
complex $\mathcal{A}(F,\rho_{F}(\Delta))$.
Suppose that $D^{\prime\prime}\neq D^{\prime}$. Let $A\in\Gamma$ be the
vertical annulus meeting $D^{\prime\prime}$. Let $N$ be a regular neighborhood
of $D^{\prime\prime}\cup A$, taken in $T$. Then the frontier of $N$ in $T$ is
again a vertical disk, call it $D^{\prime\prime\prime}$. Note that
$\iota(D^{\prime\prime\prime},D^{\prime})<K-1$. Finally, replace $D_{i}$ by
the arc $\rho_{F}(D^{\prime\prime\prime})$.
Suppose now that $Y$ is not a hole. Then some component $S{\smallsetminus}Y$
is compressible. Applying Lemma 8.4 again, we find that either $D^{\prime}$
lies in $Z=X{\smallsetminus}Y$ or in $Y$. This completes the verification of
Axiom 13.6.
### 19.4. Straight intervals
We end by checking Axiom 13.11. Suppose that $[p,q]\subset[0,K]$ is a straight
interval. Recall that $d_{Y}(\mu_{r(p)},\mu_{r(q)})<{L_{2}}$ for all strict
subsurfaces $Y\subset X$. We must check that
$d_{\mathcal{D}}(D_{p},D_{q})\mathbin{\leq_{A}}d_{X}(D_{p},D_{q})$. Since
$d_{\mathcal{D}}(D_{p},D_{q})\leq{C_{2}}|p-q|$ it is enough to bound $|p-q|$.
Note that $|p-q|\leq|r(p)-r(q)|$ because the reindexing map is increasing.
Now, $|r(p)-r(q)|\mathbin{\leq_{A}}d_{\mathcal{M}(X)}(\mu_{r(p)},\mu_{r(q)})$
because the sequence $\\{\mu_{n}\\}$ is a quasi-geodesic in $\mathcal{M}(X)$
(Theorem 18.3). Increasing $A$ as needed and applying Theorem 4.10 we have
$d_{\mathcal{M}}(\mu_{r(p)},\mu_{r(q)})\mathbin{\leq_{A}}\sum_{Y}[d_{Y}(\mu_{r(p)},\mu_{r(q)})]_{L_{2}}$
and the right hand side is thus less than $d_{X}(\mu_{r(p)},\mu_{r(q)})$ which
in turn is less than $d_{X}(D_{p},D_{q})+2{C_{2}}$. This completes our
discussion of Axiom 13.11 and finishes the proof of Theorem 19.1.
## 20\. Hyperbolicity
The ideas in this section are related to the notion of “time-ordered domains”
and to the hierarchy machine of [25] (see also Chapters 4 and 5 of Behrstock’s
thesis [1]). As remarked above, we cannot use those tools directly as the
hierarchy machine is too rigid to deal with the disk complex.
### 20.1. Hyperbolicity
We prove:
###### Theorem 20.1.
Fix $\mathcal{G}=\mathcal{G}(S)$, a combinatorial complex. Suppose that
$\mathcal{G}$ satisfies the axioms of Section 13. Then $\mathcal{G}$ is Gromov
hyperbolic.
As corollaries we have
###### Theorem 20.2.
The arc complex is Gromov hyperbolic. ∎
###### Theorem 20.3.
The disk complex is Gromov hyperbolic. ∎
In fact, Theorem 20.1 follows quickly from:
###### Theorem 20.4.
Fix $\mathcal{G}$, a combinatorial complex. Suppose that $\mathcal{G}$
satisfies the axioms of Section 13. Then for all $A\geq 1$ there exists
$\delta\geq 0$ with the following property: Suppose that $T\subset\mathcal{G}$
is a triangle of paths where the projection of any side of $T$ into into any
hole is an $A$–unparameterized quasi-geodesic. Then T is $\delta$–slim.
###### Proof of Theorem 20.1.
As laid out in Section 14 there is a uniform constant $A$ so that for any pair
$\alpha,\beta\in\mathcal{G}$ there is a recursively constructed path
$\mathcal{P}=\\{\gamma_{i}\\}\subset\mathcal{G}$ so that
* •
for any hole $X$ for $\mathcal{G}$, the projection $\pi_{X}(\mathcal{P})$ is
an $A$–unparameterized quasi-geodesic and
* •
$|\mathcal{P}|\mathbin{=_{A}}d_{\mathcal{G}}(\alpha,\beta)$.
So if $\alpha\cap\beta=\emptyset$ then $|\mathcal{P}|$ is uniformly short.
Also, by Theorem 20.4, triangles made of such paths are uniformly slim. Thus,
by Theorem 3.11, $\mathcal{G}$ is Gromov hyperbolic. ∎
The rest of this section is devoted to proving Theorem 20.4.
### 20.2. Index in a hole
For the following definitions, we assume that $\alpha$ and $\beta$ are fixed
vertices of $\mathcal{G}$.
For any hole $X$ and for any geodesic $k\in\mathcal{C}(X)$ connecting a point
of $\pi_{X}(\alpha)$ to a point of $\pi_{X}(\beta)$ we also define
$\rho_{k}\colon\mathcal{G}\to k$ to be the relation
$\pi_{X}|\mathcal{G}\colon\mathcal{G}\to\mathcal{C}(X)$ followed by taking
closest points in $k$. Since the diameter of $\rho_{k}(\gamma)$ is uniformly
bounded, we may simplify our formulas by treating $\rho_{k}$ as a function.
Define $\operatorname{index}_{X}\colon\mathcal{G}\to\mathbb{N}$ to be the
index in $X$:
$\operatorname{index}_{X}(\sigma)=d_{X}(\alpha,\rho_{k}(\sigma)).$
###### Remark 20.5.
Suppose that $k^{\prime}$ is a different geodesic connecting $\pi_{X}(\alpha)$
to $\pi_{X}(\beta)$ and $\operatorname{index}^{\prime}_{X}$ is defined with
respect to $k^{\prime}$. Then
$|\operatorname{index}_{X}(\sigma)-\operatorname{index}^{\prime}_{X}(\sigma)|\leq
17\delta+4$
by Lemma 3.7 and Lemma 3.8. After permitting a small additive error, the index
depends only on $\alpha,\beta,\sigma$ and not on the choice of geodesic $k$.
### 20.3. Back and sidetracking
Fix $\sigma,\tau\in\mathcal{G}$. We say $\sigma$ precedes $\tau$ by at least
$K$ in $X$ if
$\operatorname{index}_{X}(\sigma)+K\leq\operatorname{index}_{X}(\tau).$
We say $\sigma$ precedes $\tau$ by at most $K$ if the inequality is reversed.
If $\sigma$ precedes $\tau$ then we say $\tau$ succeeds $\sigma$.
Now take $\mathcal{P}=\\{\sigma_{i}\\}$ to be a path in $\mathcal{G}$
connecting $\alpha$ to $\beta$. Recall that we have made the simplifying
assumption that $\sigma_{i}$ and $\sigma_{i+1}$ are disjoint.
We formalize a pair of properties enjoyed by unparameterized quasi-geodesics.
The path $\mathcal{P}$ backtracks at most $K$ if for every hole $X$ and all
indices $i<j$ we find that $\sigma_{j}$ precedes $\sigma_{i}$ by at most $K$.
The path $\mathcal{P}$ sidetracks at most $K$ if for every hole $X$ and every
index $i$ we find that
$d_{X}(\sigma_{i},\rho_{k}(\sigma_{i}))\leq K,$
for some geodesic $k$ connecting a point of $\pi_{X}(\alpha)$ to a point of
$\pi_{X}(\beta)$.
###### Remark 20.6.
As in Remark 20.5, allowing a small additive error makes irrelevant the choice
of geodesic in the definition of sidetracking. We note that, if $\mathcal{P}$
has bounded sidetracking, one may freely use in calculation whichever of
$\sigma_{i}$ or $\rho_{k}(\sigma_{i})$ is more convenient.
### 20.4. Projection control
We say domains $X,Y\subset S$ overlap if $\partial X$ cuts $Y$ and $\partial
Y$ cuts $X$. The following lemma, due to Behrstock [1, 4.2.1], is closely
related to the notion of time ordered domains [25]. An elementary proof is
given in [23, Lemma 2.5].
###### Lemma 20.7.
There is a constant ${M_{1}}={M_{1}}(S)$ with the following property. Suppose
that $X,Y$ are overlapping non-simple domains. If $\gamma\in\mathcal{AC}(S)$
cuts both $X$ and $Y$ then either $d_{X}(\gamma,\partial Y)<{M_{1}}$ or
$d_{Y}(\partial X,\gamma)<{M_{1}}$. ∎
We also require a more specialized version for the case where $X$ and $Y$ are
nested.
###### Lemma 20.8.
There is a constant ${M_{2}}={M_{2}}(S)$ with the following property. Suppose
that $X\subset Y$ are nested non-simple domains. Fix
$\alpha,\beta,\gamma\in\mathcal{AC}(S)$ that cut $X$. Fix
$k=[\alpha^{\prime},\beta^{\prime}]\subset\mathcal{C}(Y)$, a geodesic
connecting a point of $\pi_{Y}(\alpha)$ to a point of $\pi_{Y}(\beta)$. Assume
that $d_{X}(\alpha,\beta)\geq M_{0}$, the constant given by Theorem 4.6. If
$d_{X}(\alpha,\gamma)\geq{M_{2}}$ then
$\operatorname{index}_{Y}(\partial X)-4\leq\operatorname{index}_{Y}(\gamma).$
Symmetrically, we have
$\operatorname{index}_{Y}(\gamma)\leq\operatorname{index}_{Y}(\partial X)+4$
if $d_{X}(\gamma,\beta)\geq{M_{2}}$. ∎
### 20.5. Finding the midpoint of a side
Fix $A\geq 1$. Let $\mathcal{P},\mathcal{Q},\mathcal{R}$ be the sides of a
triangle in $\mathcal{G}$ with vertices at $\alpha,\beta,\gamma$. We assume
that each of $\mathcal{P}$, $\mathcal{Q}$, and $\mathcal{R}$ are
$A$–unparameterized quasi-geodesics when projected to any hole.
Recall that $M_{0}=M_{0}(S)$, ${M_{1}}={M_{1}}(S)$, and ${M_{2}}={M_{2}}(S)$
are functions depending only on the topology of $S$. We may assume that if
$T\subset S$ is an essential subsurface, then $M_{0}(S)>M_{0}(T)$.
Now choose ${K_{1}}\geq\max\\{M_{0},4{M_{1}},{M_{2}},8\\}+8\delta$
sufficiently large so that any $A$–unparameterized quasi-geodesic in any hole
back and side tracks at most ${K_{1}}$.
###### Claim 20.9.
If $\sigma_{i}$ precedes $\gamma$ in $X$ and $\sigma_{j}$ succeeds $\gamma$ in
$Y$, both by at least $2{K_{1}}$, then $i<j$.
###### Proof.
To begin, as $X$ and $Y$ are holes and all holes interfere, we need not
consider the possibility that $X\cap Y=\emptyset$. If $X=Y$ we immediately
deduce that
$\operatorname{index}_{X}(\sigma_{i})+2{K_{1}}\leq\operatorname{index}_{X}(\gamma)\leq\operatorname{index}_{X}(\sigma_{j})-2{K_{1}}.$
Thus
$\operatorname{index}_{X}(\sigma_{i})+4{K_{1}}\leq\operatorname{index}_{X}(\sigma_{j})$.
Since $\mathcal{P}$ backtracks at most ${K_{1}}$ we have $i<j$, as desired.
Suppose instead that $X\subset Y$. Since $\sigma_{i}$ precedes $\gamma$ in $X$
we immediately find $d_{X}(\alpha,\beta)\geq 2{K_{1}}\geq M_{0}$ and
$d_{X}(\alpha,\gamma)\geq 2{K_{1}}-2\delta\geq{M_{2}}$. Apply Lemma 20.8 to
deduce $\operatorname{index}_{Y}(\partial
X)-4\leq\operatorname{index}_{Y}(\gamma)$. Since $\sigma_{j}$ succeeds
$\gamma$ in $Y$ it follows that $\operatorname{index}_{Y}(\partial
X)-4+2{K_{1}}\leq\operatorname{index}_{Y}(\sigma_{j})$. Again using the fact
that $\sigma_{i}$ precedes $\gamma$ in $X$ we have that
$d_{X}(\sigma_{i},\beta)\geq{M_{2}}$. We deduce from Lemma 20.8 that
$\operatorname{index}_{Y}(\sigma_{i})\leq\operatorname{index}_{Y}(\partial
X)+4$. Thus
$\operatorname{index}_{Y}(\sigma_{i})-8+2{K_{1}}\leq\operatorname{index}_{Y}(\sigma_{j}).$
Since $\mathcal{P}$ backtracks at most ${K_{1}}$ in $Y$ we again deduce that
$i<j$. The case where $Y\subset X$ is similar.
Suppose now that $X$ and $Y$ overlap. Applying Lemma 20.7 and breaking
symmetry, we may assume that $d_{X}(\gamma,\partial Y)<{M_{1}}$. Since
$\sigma_{i}$ precedes $\gamma$ we have $\operatorname{index}_{X}(\gamma)\geq
2{K_{1}}$. Lemma 3.7 now implies that $\operatorname{index}_{X}(\partial
Y)\geq 2{K_{1}}-{M_{1}}-6\delta$. Thus,
$d_{X}(\alpha,\partial Y)\geq 2{K_{1}}-{M_{1}}-8\delta\geq{M_{1}}$
where the first inequality follows from Lemma 3.4.
Applying Lemma 20.7 again, we find that $d_{Y}(\alpha,\partial X)<{M_{1}}$.
Now, since $\sigma_{j}$ succeeds $\gamma$ in $Y$, we deduce that
$\operatorname{index}_{Y}(\sigma_{j})\geq 2{K_{1}}$. So Lemma 3.4 implies that
$d_{Y}(\alpha,\sigma_{j})\geq 2{K_{1}}-2\delta$. The triangle inequality now
gives
$d_{Y}(\partial X,\sigma_{j})\geq 2{K_{1}}-{M_{1}}-2\delta\geq{M_{1}}.$
Applying Lemma 20.7 one last time, we find that $d_{X}(\partial
Y,\sigma_{j})<{M_{1}}$. Thus $d_{X}(\gamma,\sigma_{j})\leq 2{M_{1}}$. Finally,
Lemma 3.7 implies that the difference in index (in $X$) between $\sigma_{i}$
and $\sigma_{j}$ is at least $2{K_{1}}-2{M_{1}}-6\delta$. Since this is
greater than the backtracking constant, ${K_{1}}$, it follows that $i<j$. ∎
Let $\sigma_{\alpha}\in\mathcal{P}$ be the last vertex of $\mathcal{P}$
preceding $\gamma$ by at least $2{K_{1}}$ in some hole. If no such vertex of
$\mathcal{P}$ exists then take $\sigma_{\alpha}=\alpha$.
###### Claim 20.10.
For every hole $X$ and geodesic $h$ connecting $\pi_{X}(\alpha)$ to
$\pi_{X}(\beta)$:
$d_{X}(\sigma_{\alpha},\rho_{h}(\gamma))\leq 3{K_{1}}+6\delta+1$
###### Proof.
Since $\sigma_{i}$ and $\sigma_{i+1}$ are disjoint we have
$d_{X}(\sigma_{i},\sigma_{i+1})\geq 3$ and so Lemma 3.7 implies that
$|\operatorname{index}_{X}(\sigma_{i+1})-\operatorname{index}_{X}(\sigma_{i})|\leq
6\delta+3.$
Since $\mathcal{P}$ is a path connecting $\alpha$ to $\beta$ the image
$\rho_{h}(\mathcal{P})$ is $6\delta+3$–dense in $h$. Thus, if
$\operatorname{index}_{X}(\sigma_{\alpha})+2{K_{1}}+6\delta+3<\operatorname{index}_{X}(\gamma)$
then we have a contradiction to the definition of $\sigma_{\alpha}$.
On the other hand, if
$\operatorname{index}_{X}(\sigma_{\alpha})\geq\operatorname{index}_{X}(\gamma)+2{K_{1}}$
then $\sigma_{\alpha}$ precedes and succeeds $\gamma$ in $X$. This directly
contradicts Claim 20.9.
We deduce that the difference in index between $\sigma_{\alpha}$ and $\gamma$
in $X$ is at most $2{K_{1}}+6\delta+3$. Finally, as $\mathcal{P}$ sidetracks
by at most ${K_{1}}$ we have
$d_{X}(\sigma_{\alpha},\rho_{h}(\gamma))\leq 3{K_{1}}+6\delta+3$
as desired. ∎
We define $\sigma_{\beta}$ to be the first $\sigma_{i}$ to succeed $\gamma$ by
at least $2{K_{1}}$ — if no such vertex of $\mathcal{P}$ exists take
$\sigma_{\beta}=\beta$. If $\alpha=\beta$ then
$\sigma_{\alpha}=\sigma_{\beta}$. Otherwise, from Claim 20.9, we immediately
deduce that $\sigma_{\alpha}$ comes before $\sigma_{\beta}$ in $\mathcal{P}$.
A symmetric version of Claim 20.10 applies to $\sigma_{\beta}$: for every hole
$X$
$d_{X}(\rho_{h}(\gamma),\sigma_{\beta})\leq 3{K_{1}}+6\delta+3.$
### 20.6. Another side of the triangle
Recall now that we are also given a path $\mathcal{R}=\\{\tau_{i}\\}$
connecting $\alpha$ to $\gamma$ in $\mathcal{G}$. As before, $\mathcal{R}$ has
bounded back and sidetracking. Thus we again find vertices $\tau_{\alpha}$ and
$\tau_{\gamma}$ the last/first to precede/succeed $\beta$ by at least
$2{K_{1}}$. Again, this is defined in terms of the closest points projection
of $\beta$ to a geodesic of the form $h=[\pi_{X}(\alpha),\pi_{X}(\gamma)]$. By
Claim 20.10, for every hole $X$, $\tau_{\alpha}$ and $\tau_{\gamma}$ are close
to $\rho_{h}(\beta)$.
By Lemma 3.6, if $k=[\pi_{X}(\alpha),\pi_{X}(\beta)]$, then
$d_{X}(\rho_{k}(\gamma),\rho_{h}(\beta))\leq 6\delta$. We deduce:
###### Claim 20.11.
$d_{X}(\sigma_{\alpha},\tau_{\alpha})\leq 6{K_{1}}+18\delta+2$. ∎
This claim and Claim 20.10 imply that the body of the triangle
$\mathcal{P}\mathcal{Q}\mathcal{R}$ is bounded in size. We now show that the
legs are narrow.
###### Claim 20.12.
There is a constant ${N_{2}}={N_{2}}(S)$ with the following property. For
every $\sigma_{i}\leq\sigma_{\alpha}$ in $\mathcal{P}$ there is a
$\tau_{j}\leq\tau_{\alpha}$ in $\mathcal{R}$ so that
$d_{X}(\sigma_{i},\tau_{j})\leq{N_{2}}$
for every hole $X$.
###### Proof.
We only sketch the proof, as the details are similar to our previous
discussion. Fix $\sigma_{i}\leq\sigma_{\alpha}$.
Suppose first that no vertex of $\mathcal{R}$ precedes $\sigma_{i}$ by more
than $2{K_{1}}$ in any hole. So fix a hole $X$ and geodesics
$k=[\pi_{X}(\alpha),\pi_{X}(\beta)]$ and
$h=[\pi_{X}(\alpha),\pi_{X}(\gamma)]$. Then $\rho_{h}(\sigma_{i})$ is within
distance $2{K_{1}}$ of $\pi_{X}(\alpha)$. Appealing to Claim 20.11, bounded
sidetracking, and hyperbolicity of $\mathcal{C}(X)$ we find that the initial
segments
$[\pi_{X}(\alpha),\rho_{k}(\sigma_{\alpha})],\quad[\pi_{X}(\alpha),\rho_{h}(\tau_{\alpha})]$
of $k$ and $h$ respectively must fellow travel. Because of bounded
backtracking along $\mathcal{P}$, $\rho_{k}(\sigma_{i})$ lies on, or at least
near, this initial segment of $k$. Thus by Lemma 3.8 $\rho_{h}(\sigma_{i})$ is
close to $\rho_{k}(\sigma_{i})$ which in turn is close to
$\pi_{X}(\sigma_{i})$, because $\mathcal{P}$ has bounded sidetracking. In
short, $d_{X}(\alpha,\sigma_{i})$ is bounded for all holes $X$. Thus we may
take $\tau_{j}=\tau_{0}=\alpha$ and we are done.
Now suppose that some vertex of $\mathcal{R}$ precedes $\sigma_{i}$ by at
least $2{K_{1}}$ in some hole $X$. Take $\tau_{j}$ to be the last such vertex
in $\mathcal{R}$. Following the proof of Claim 20.9 shows that $\tau_{j}$
comes before $\tau_{\alpha}$ in $\mathcal{R}$. The argument now required to
bound $d_{X}(\sigma_{i},\tau_{j})$ is essentially identical to the proof of
Claim 20.10. ∎
By the distance estimate, we find that there is a uniform neighborhood of
$[\sigma_{0},\sigma_{\alpha}]\subset\mathcal{P}$, taken in $\mathcal{G}$,
which contains $[\tau_{0},\tau_{\alpha}]\subset\mathcal{P}$. The slimness of
$\mathcal{P}\mathcal{Q}\mathcal{R}$ follows directly. This completes the proof
of Theorem 20.4. ∎
## 21\. Coarsely computing Hempel distance
We now turn to our topological application. Recall that a Heegaard splitting
is a triple $(S,V,W)$ consisting of a surface and two handlebodies where
$V\cap W=\partial V=\partial W=S$. Hempel [20] defines the quantity
$d_{S}(V,W)=\min\big{\\{}d_{S}(D,E)\mathbin{\mid}D\in\mathcal{D}(V),E\in\mathcal{D}(W)\big{\\}}$
and calls it the distance of the splitting. Note that a splitting can be
completely determined by giving a pair of cut systems: simplices
$\mathbb{D}\subset\mathcal{D}(V)$, $\mathbb{E}\subset\mathcal{D}(W)$ where the
corresponding disks cut the containing handlebody into a single three-ball.
The triple $(S,\mathbb{D},\mathbb{E})$ is a Heegaard diagram. The goal of this
section is to prove:
###### Theorem 21.1.
There is a constant $R_{1}=R_{1}(S)$ and an algorithm that, given a Heegaard
diagram $(S,\mathbb{D},\mathbb{E})$, computes a number $N$ so that
$|d_{S}(V,W)-N|\leq R_{1}.$
Let $\rho_{V}\colon\mathcal{C}(S)\to\mathcal{D}(V)$ be the closest points
relation:
$\rho_{V}(\alpha)=\big{\\{}D\in\mathcal{D}(V)\mathbin{\mid}\mbox{ for all
$E\in\mathcal{D}(V)$, $d_{S}(\alpha,D)\leq d_{S}(\alpha,E)$ }\big{\\}}.$
Theorem 21.1 follows from:
###### Theorem 21.2.
There is a constant $R_{0}=R_{0}(V)$ and an algorithm that, given an essential
curve $\alpha\subset S$ and a cut system $\mathbb{D}\subset\mathcal{D}(V)$,
finds a disk $C\in\mathcal{D}(V)$ so that
$d_{S}(C,\rho_{V}(\alpha))\leq R_{0}.$
###### Proof of Theorem 21.1.
Suppose that $(S,\mathbb{D},\mathbb{E})$ is a Heegaard diagram. Using Theorem
21.2 we find a disk $D$ within distance $R_{0}$ of $\rho_{V}(\mathbb{E})$.
Again using Theorem 21.2 we find a disk $E$ within distance $R_{0}$ of
$\rho_{W}(D)$. Notice that $E$ is defined using $D$ and not the cut system
$\mathbb{D}$.
Since computing distance between fixed vertices in the curve complex is
algorithmic [22, 37] we may compute $d_{S}(D,E)$. By the hyperbolicity of
$\mathcal{C}(S)$ (Theorem 3.2) and by the quasi-convexity of the disk set
(Theorem 4.9) this is the desired estimate. ∎
Very briefly, the algorithm asked for in Theorem 21.2 searches an
$R_{2}$–neighborhood in $\mathcal{M}(S)$ about a splitting sequence from
$\mathbb{D}$ to $\alpha$. Here are the details.
###### Algorithm 21.3.
We are given $\alpha\in\mathcal{C}(S)$ and a cut system
$\mathbb{D}\subset\mathcal{D}(V)$. Build a train track $\tau$ in $S=\partial
V$ as follows: make $\mathbb{D}$ and $\alpha$ tight. Place one switch on every
disk $D\in\mathbb{D}$. Homotope all intersections of $\alpha$ with $D$ to run
through the switch. Collapse bigons of $\alpha$ inside of
$S{\smallsetminus}\mathbb{D}$ to create the branches. Now make $\tau$ a
generic track by combing away from $\mathbb{D}$ [31, Proposition 1.4.1]. Note
that $\alpha$ is carried by $\tau$ and so gives a transverse measure $w$.
Build a splitting sequence of measured tracks $\\{\tau_{n}\\}_{n=0}^{N}$ where
$\tau_{0}=\tau$, $\tau_{N}=\alpha$, and $\tau_{n+1}$ is obtained by splitting
the largest switch of $\tau_{n}$ (as determined by the measure imposed by
$\alpha$).
Let $\mu_{n}=V(\tau_{n})$ be the vertices of $\tau_{n}$. For each filling
marking $\mu_{n}$ list all markings in the ball
$B(\mu_{n},R_{2})\subset\mathcal{M}(S)$, where $R_{2}$ is given by Lemma 21.5
below. (If $\mu_{0}$ does not fill $S$ then output $\mathbb{D}$ and halt.)
For every marking $\nu$ so produced we use Whitehead’s algorithm (see Lemma
21.4) to try and find a disk meeting some curve $\gamma\in\nu$ at most twice.
For every disk $C$ found compute $d_{S}(\alpha,C)$ [22, 37]. Finally, output
any disk which minimizes this distance, among all disks considered, and halt.
We use the following form of Whitehead’s algorithm [3]:
###### Lemma 21.4.
There is an algorithm that, given a cut system $\mathbb{D}\subset V$ and a
curve $\gamma\subset S$, outputs a disk $C\subset V$ so that
$\iota(\gamma,\partial C)=\min\\{\iota(\gamma,\partial
E)\mathbin{\mid}E\in\mathcal{D}(V)\\}$. ∎
We now discuss the constant $R_{2}$. We begin by noticing that the track
$\tau_{n}$ is transversely recurrent because $\alpha$ is fully carried and
$\mathbb{D}$ is fully dual. Thus by Theorem 18.2 and by Morse stability, for
any essential $Y\subset S$ there is a stability constant $M_{3}$ for the path
$\pi_{Y}(\mu_{n})$. Let $\delta$ be the hyperbolicity constant for
$\mathcal{C}(S)$ (Theorem 3.2) and let $Q$ be the quasi-convexity constant for
$\mathcal{D}(V)\subset\mathcal{C}(S)$ (Theorem 4.9).
Since $\iota(\mathbb{D},\mu_{0})$ is bounded we will, at the cost of an
additive error, identify their images in $\mathcal{C}(S)$. Now, for every $n$
pick some $E_{n}\in\rho_{V}(\mu_{n})$.
###### Lemma 21.5.
There is a constant $R_{2}$ with the following property. Suppose that $n<m$,
$d_{S}(\mu_{n},E_{n}),d_{S}(\mu_{m},E_{m})\leq M_{3}+\delta+Q$, and
$d_{S}(\mu_{n},\mu_{m})\geq 2(M_{3}+\delta+Q)+5$. Then there is a marking
$\nu\in B(\mu_{n},R_{2})$ and a curve $\gamma\in\nu$ so that either:
* •
$\gamma$ bounds a disk in $V$,
* •
$\gamma\subset\partial Z$, where $Z$ is a non-hole or
* •
$\gamma\subset\partial Z$, where $Z$ is a large hole.
###### Proof of Lemma 21.5.
Choose points $\sigma,\sigma^{\prime}$ in the thick part of $\mathcal{T}(S)$
so that all curves of $\mu_{n}$ have bounded length in $\sigma$ and so that
$E_{n}$ has length less than the Margulis constant in $\sigma^{\prime}$. As in
Section 15 there is a Teichmüller geodesic and associated markings
$\\{\nu_{k}\\}_{k=0}^{K}$ so that $d_{\mathcal{M}}(\nu_{0},\mu_{n})$ is
bounded and $E_{n}\in\operatorname{base}(\nu_{K})$.
We say a hole $X\subset S$ is small if
$\operatorname{diam}_{X}(\mathcal{D}(V))<61$.
###### Claim.
There is a constant $R_{3}$ so that for any small hole $X$ we have
$d_{X}(\mu_{n},\nu_{K})<R_{3}$.
###### Proof.
If $d_{X}(\mu_{n},\nu_{K})\leq M_{0}$ then we are done. If the distance is
greater than $M_{0}$ then Theorem 4.6 gives a vertex of the
$\mathcal{C}(S)$–geodesic connecting $\mu_{n}$ to $E_{n}$ with distance at
most one from $\partial X$. It follows from the triangle inequality that every
vertex of the $\mathcal{C}(S)$–geodesic connecting $\mu_{m}$ to $E_{m}$ cuts
$X$. Another application of Theorem 4.6 gives
$d_{X}(\mu_{m},E_{m})<M_{0}.$
Since $X$ is small $d_{X}(E_{m},\mathbb{D}),d_{X}(E_{n},\mathbb{D})\leq 60$.
Since $\iota(\nu_{K},E_{n})=2$ the distance $d_{X}(\nu_{K},E_{n})$ is bounded.
Finally, because $p\mapsto\pi_{X}(\mu_{p})$ is an $A$–unparameterized quasi-
geodesic in $\mathcal{C}(X)$ it follows that $d_{X}(\mathbb{D},\mu_{n})$ is
also bounded and the claim is proved. ∎
Now consider all strict subsurfaces $Y$ so that
$d_{Y}(\mu_{n},\nu_{M})\geq R_{3}.$
None of these are small holes, by the claim above. If there are no such
surfaces then Theorem 4.10 bounds $d_{\mathcal{M}}(\mu_{n},\nu_{M})$: taking
the cutoff constant larger than
$\max\\{R_{3},{C_{0}},M_{3}+\delta+Q\\}$
ensures that all terms on the right-hand side vanish. In this case the
additive error in Theorem 4.10 is the desired constant $R_{2}$ and the lemma
is proved.
If there are such surfaces then choose one, say $Z$, that minimizes $\ell=\min
J_{Z}$. Thus $d_{Y}(\mu_{n},\nu_{\ell})<{C_{3}}$ for all strict non-holes and
all strict large holes. Since $d_{S}(\mu_{n},E_{n})\leq M_{3}+\delta+Q$ and
$\\{\nu_{m}\\}$ is an unparameterized quasi-geodesic [33, Theorem 6.1] we find
that $d_{S}(\mu_{n},\nu_{l})$ is uniformly bounded. The claim above bounds
distances in small holes. As before we find a sufficiently large cutoff so
that all terms on the right-hand side of Theorem 4.10 vanish. Again the
additive error of Theorem 4.10 provides the constant $R_{2}$. Since $\partial
Z\subset\operatorname{base}(\nu_{\ell})$ the lemma is proved. ∎
To prove the correctness of Algorithm 21.3 it suffices to show that the disk
produced is close to $\rho_{V}(\alpha)$. Let $m$ be the largest index so that
for all $n\leq m$ we have
$d_{S}(\mu_{n},E_{n})\leq M_{3}+\delta+Q.$
It follows that $\mu_{m+1}$ lies within distance $M_{3}+\delta$ of the
geodesic $[\alpha,\rho_{V}(\alpha)]$. Recall that
$d_{S}(\mu_{n},\mu_{n+1})\leq{C_{1}}$ for any value of $n$. A shortcut
argument shows that
$d_{S}(\mu_{m},\rho_{V}(\alpha))\leq 2{C_{1}}+3M_{3}+3\delta+Q.$
Let $n\leq m$ be the largest index so that
$2(M_{3}+\delta+Q)+5\leq d_{S}(\mu_{n},\mu_{m}).$
If no such $n$ exists then take $n=0$. Now, Lemma 21.5 implies that there is a
disk $C$ with $d_{S}(C,\mu_{n})\leq 4R_{2}$ and this disk is found during the
running of Algorithm 21.3. It follows from the above inequalities that
$d_{S}(C,\alpha)\leq
4R_{2}+5M_{3}+5\delta+3Q+5+2{C_{1}}+d_{S}(\alpha,\rho_{V}(\alpha)).$
So the disk $C^{\prime}$, output by the algorithm, is at least this close to
$\alpha$ in $\mathcal{C}(S)$. Examining the triangle with vertices
$\alpha,\rho_{V}(\alpha),C^{\prime}$ and using a final short-cut argument
gives
$d_{S}(C^{\prime},\rho_{V}(\alpha))\leq 4R_{2}+5M_{3}+9\delta+5Q+5+2{C_{1}}.$
This completes the proof of Theorem 21.2. ∎
## References
* [1] Jason Behrstock. Asymptotic geometry of the mapping class group and Teichmüller space. PhD thesis, SUNY Stony Brook, 2004. http://www.math.columbia.edu/$\sim$jason/thesis.pdf.
* [2] Jason Behrstock, Cornelia Drutu, and Lee Mosher. Thick metric spaces, relative hyperbolicity, and quasi-isometric rigidity. arXiv:math/0512592.
* [3] John Berge. Heegaard documentation. documentation of computer program available at computop.org.
* [4] Mladen Bestvina and Koji Fujiwara. Quasi-homomorphisms on mapping class groups. Glas. Mat. Ser. III, 42(62)(1):213–236, 2007. arXiv:math/0702273.
* [5] Joan S. Birman. The topology of 3-manifolds, Heegaard distance and the mapping class group of a 2-manifold. In Problems on mapping class groups and related topics, volume 74 of Proc. Sympos. Pure Math., pages 133–149. Amer. Math. Soc., Providence, RI, 2006. http://www.math.columbia.edu/$\sim$jb/papers.html.
* [6] Brian H. Bowditch. Intersection numbers and the hyperbolicity of the curve complex. J. Reine Angew. Math., 598:105–129, 2006. bhb-curvecomplex.pdf.
* [7] Tara E. Brendle and Dan Margalit. Commensurations of the Johnson kernel. Geom. Topol., 8:1361–1384 (electronic), 2004. arXiv:math/0404445.
* [8] Martin R. Bridson and André Haefliger. Metric spaces of non-positive curvature. Springer-Verlag, Berlin, 1999.
* [9] Jeffrey F. Brock. The Weil-Petersson metric and volumes of 3-dimensional hyperbolic convex cores. J. Amer. Math. Soc., 16(3):495–535 (electronic), 2003. arXiv:math/0109048.
* [10] Alberto Cavicchioli and Fulvia Spaggiari. A note on irreducible Heegaard diagrams. Int. J. Math. Math. Sci., pages Art. ID 53135, 11, 2006.
* [11] Young-Eun Choi and Kasra Rafi. Comparison between Teichmüller and Lipschitz metrics. J. Lond. Math. Soc. (2), 76(3):739–756, 2007. math.GT/0510136.
* [12] M. Coornaert, T. Delzant, and A. Papadopoulos. Géométrie et théorie des groupes. Springer-Verlag, Berlin, 1990. Les groupes hyperboliques de Gromov.
* [13] Robert H. Gilman. The geometry of cycles in the Cayley diagram of a group. In The mathematical legacy of Wilhelm Magnus: groups, geometry and special functions (Brooklyn, NY, 1992), volume 169 of Contemp. Math., pages 331–340. Amer. Math. Soc., Providence, RI, 1994.
* [14] Robert H. Gilman. On the definition of word hyperbolic groups. Math. Z., 242(3):529–541, 2002.
* [15] Mikhael Gromov. Hyperbolic groups. In Essays in group theory, pages 75–263. Springer, New York, 1987\.
* [16] Ursula Hamenstädt. Geometry of the mapping class groups. I. Boundary amenability. Invent. Math., 175(3):545–609, 2009.
* [17] Kevin Hartshorn. Heegaard splittings of Haken manifolds have bounded distance. Pacific J. Math., 204(1):61–75, 2002. http://nyjm.albany.edu:8000/PacJ/2002/204-1-5nf.htm.
* [18] Willam J. Harvey. Boundary structure of the modular group. In Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978), pages 245–251, Princeton, N.J., 1981. Princeton Univ. Press.
* [19] A. Hatcher and W. Thurston. A presentation for the mapping class group of a closed orientable surface. Topology, 19(3):221–237, 1980.
* [20] John Hempel. 3-manifolds as viewed from the curve complex. Topology, 40(3):631–657, 2001. arXiv:math/9712220.
* [21] Tsuyoshi Kobayashi. Heights of simple loops and pseudo-Anosov homeomorphisms. In Braids (Santa Cruz, CA, 1986), pages 327–338. Amer. Math. Soc., Providence, RI, 1988.
* [22] Jason Leasure. Geodesics in the complex of curves of a surface. Ph.D. thesis. http://repositories.lib.utexas.edu/bitstream/handle/2152/1700/leasurejp46295.pdf.
* [23] Johanna Mangahas. Uniform uniform exponential growth of subgroups of the mapping class group. Geom. Funct. Anal., 19(5):1468–1480, 2010. arXiv:0805.0133.
* [24] Howard A. Masur and Yair N. Minsky. Geometry of the complex of curves. I. Hyperbolicity. Invent. Math., 138(1):103–149, 1999. arXiv:math/9804098.
* [25] Howard A. Masur and Yair N. Minsky. Geometry of the complex of curves. II. Hierarchical structure. Geom. Funct. Anal., 10(4):902–974, 2000. arXiv:math/9807150.
* [26] Howard A. Masur and Yair N. Minsky. Quasiconvexity in the curve complex. In In the tradition of Ahlfors and Bers, III, volume 355 of Contemp. Math., pages 309–320. Amer. Math. Soc., Providence, RI, 2004\. arXiv:math/0307083.
* [27] Howard A. Masur, Lee Mosher, and Saul Schleimer. On train track splitting sequences. arXiv:1004.4564.
* [28] Darryl McCullough. Virtually geometrically finite mapping class groups of $3$-manifolds. J. Differential Geom., 33(1):1–65, 1991.
* [29] Yair Minsky. The classification of Kleinian surface groups. I. Models and bounds. Ann. of Math. (2), 171(1):1–107, 2010. arXiv:math/0302208.
* [30] Lee Mosher. Train track expansions of measured foliations. 2003\. http://newark.rutgers.edu/$\sim$mosher/.
* [31] R. C. Penner and J. L. Harer. Combinatorics of train tracks, volume 125 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, 1992.
* [32] Robert C. Penner. A construction of pseudo-Anosov homeomorphisms. Trans. Amer. Math. Soc., 310(1):179–197, 1988.
* [33] Kasra Rafi. Relative hyperbolicity in Teichmüller space, 2010. Preprint. http://www.math.ou.edu/$\sim$rafi/research/Fellow.pdf.
* [34] Kasra Rafi and Saul Schleimer. Covers and the curve complex. Geom. Topol., 13(4):2141–2162, 2009. arXiv:math/0701719.
* [35] Martin Scharlemann. The complex of curves on nonorientable surfaces. J. London Math. Soc. (2), 25(1):171–184, 1982.
* [36] Saul Schleimer. Notes on the curve complex. http://www.warwick.ac.uk/ masgar/Maths/notes.pdf.
* [37] Kenneth J. Shackleton. Tightness and computing distances in the curve complex. arXiv:math/0412078.
* [38] William P. Thurston. On the geometry and dynamics of diffeomorphisms of surfaces. Bull. Amer. Math. Soc. (N.S.), 19(2):417–431, 1988.
|
arxiv-papers
| 2010-10-15T14:06:44 |
2024-09-04T02:49:13.938020
|
{
"license": "Public Domain",
"authors": "Howard Masur and Saul Schleimer",
"submitter": "Saul Schleimer",
"url": "https://arxiv.org/abs/1010.3174"
}
|
1010.3213
|
# Monte Carlo simulation of growth of hard-sphere crystals on a square pattern
Atsushi Mori Institute of Technology and Science, The University of
Tokushima, Tokushima 770-8506, Japan
###### Abstract
Monte Carlo simulations of the colloidal epitaxy of hard spheres (HSs) on a
square pattern have been performed. This is an extension of previous
simulations; we observed a shrinking intrinsic stacking fault running in an
oblique direction through the glide of a Shockley partial dislocation
terminating its lower end in fcc (001) stacking [Mori et al., Molec. Phys. 105
(2007) 1377], which was an answer to a question why the defect in colloidal
crystals reduced by gravity [Zhu et al., Nature 387 (1997) 883]. We have
resolved one of shortcomings of the previous simulations; the driving force
for fcc (001) stacking, which was stress from a small periodic boundary
simulation box, has been replaced with the stress from a pattern on the
bottom. We have observed disappearance of stacking fault in this realizable
condition. Sinking of the center of gravity has been smooth and of a single
relaxation mode under the condition that the gravitational energy $mg\sigma$
is slightly less than the thermal energy $k_{\mbox{\scriptsize B}}T$. In the
snapshots tetrahedral structures have appeared often, suggesting formation of
staking fault tetrahedra.
###### keywords:
A1 Computer simulation; A1 Planar defect; B1 Polymer; A3 Colloidal epitaxy; A1
Alder transition (Kirkwood-Alder-Wainwright transition)
††journal: Journal of Crystal Growth
## 1 Introduction
In 1957 the crystalline phase transition was discovered in the hard-sphere
(HS) system by a Monte Carlo (MC) simulation [1] and a molecular dynamics (MD)
simulation [2]. Their results were surprising because the phase transition
occurred in a pure repulsive system. In 1960-70s colloidal crystallizations
were extensively studied as the HS crystalline phase transition in reality.
For historical details, see, for example, the introduction of a review [3].
Recent situation of studies on the colloidal crystal is different from that in
those days; so-called HS suspensions are synthesized [4], which exhibit a HS
nature in the crystal-fluid phase transition [5, 6, 7, 8]. There is another
trend of studies of the colloidal crystals in recent days. Because in
colloidal crystals a periodic structure of dielectric constant with the
periodicity of the same order of optical wavelength, the colloidal crystals
can be used as photonic crystals [9, 10, 11]. As compared to micro
manufacturing technologies of fabricating the photonic crystals, the colloidal
crystallization is of low cost in introducing equipment and less time
consuming in the fabrication. One of shortcomings of the colloidal
crystallization is that the colloidal crystals contain many crystal defects.
From fundamental as well as application point of view, the defect in the
photonic crystal should be reduced. The photonic band cannot be opened unless
the defect is reduced.
In relation to the reduction of the crystal defect in the colloidal crystals,
in 1997 Zhu et al. [12] found an effect of gravity that reduces the stacking
disorder in the HS colloidal crystals. They found that the colloidal particles
formed a random hexagonal close pack (rhcp) structure under micro gravity. On
the other hand, the sediment is rhcp/face-center cubic (fcc) mixture under
normal gravity [13]. The mechanism of the reduction of the stacking disorder
under gravity was so far unresolved until the present author and coworkers
found a glide mechanism of the disappearance of a stacking fault [14]. Viewing
$\langle$111$\rangle$ fcc is characterized by a stacking of the ABCABC$\cdots$
sequence, where A, B, and C distinguish hexagonal planes on the basis of the
positions of the particles in the hexagonal plane. On the other hand,
hexagonal close pack (hcp) structure is given by ABAB$\cdots$ stacking and
rhcp by a random sequence of A, B, and C. The stacking disorder is the
disorder in the sequence of A, B, and C. For example, an intrinsic stacking
fault is given by a sequence such as ABABC$\cdots$; here the third C plane has
been removed from ABCABC$\cdots$. We note that even if the stacking is out of
order, the particle number density remains unchanged. In this respect, the
varieties of stacking sequence are not affected by gravity. So, the mechanism
of the reduction of the stacking disorder due to gravity was a long standing
problem.
In Ref. [14] looking into the evolution of snapshots of MC simulations of HSs
[15], in which transformation from a defective crystal into a less-defective
crystal under gravity was observed, we found that a glide of a Shockley
partial dislocation terminating an intrinsic stacking fault shrunk the
stacking fault in fcc (001) stacking. The key is the fcc (001) stacking; in
those simulations this stacking was forced due to a stress from a small
periodic boundary simulation box. In contrast, in the colloidal
crystallization patterned bottom walls are sometimes used; the fcc (001)
stacking is forced due to the stress from the pattern on the bottom. Use of
the patterned bottom wall is called a colloidal epitaxy. In 1997 van Blaaderen
et al. succeeded in the fcc (001) stacking using a fcc (001) pattern [16]. The
basic idea of the colloidal epitaxy is that the stacking sequence is unique in
$\langle$100$\rangle$. The finding of Ref. [14] is that in the fcc (001)
stacking, even if an intrinsic stacking fault running along oblique {111}
plane is introduced, through the glide of a Shockley partial dislocation
terminating the lower end of the stacking fault the stacking fault shrinks. In
other words, Ref. [14] points out a superiority of the colloidal epitaxy other
than the unique stacking sequence. We note here that this glide mechanism is
merely one of mechanisms. The intrinsic stacking fault is mere one of
metastable configurations. Therefore, there exist mechanisms connecting other
metastable configurations. Moreover, we have already found a configuration
which was succeeded in newly grown crystal in the fluid phase in simulations
of the same condition [17]. In addition, we confirmed that a coherent growth
occurred in the simulations [18]. Complementarily to the simulations, we have
given elastic energy calculations to understand the driving force of upward
move of the Shockley partial dislocation [19, 20].
The purpose of the present simulation is to resolve the shortcoming of
previous simulations [14, 15, 17, 18]. In those simulation fcc (001) stacking
was forced due to the stress from a small periodic boundary simulation box.
This artifact should be resolved. Of course, the same stress can be provided
by the patterned substrate (the colloidal epitaxy). However, the system size
cannot be systematically enlarged in the previous simulations. As already
shown [15] fcc {111} stacking occurs for a large lateral system size. In the
present simulation we use a square pattern. An advantage of the square pattern
is that matching between the crystal grown and the substrate on the lattice
line, not only on the lattice point, is possible [21].
## 2 Simulation method
HSs (diameter $\sigma$) under gravity (the acceleration due to gravity $g$)
were confined in a simulation box with the periodic boundary condition in
horizontal direction and a top flat and bottom square-patterned hard walls
(Fig. 1 of Ref. [3]). The groove width was $0.707106781\sigma$. So, the
diagonal distance of the intersection of the longitudinal and transverse
grooves was $0.707106781\sigma\times\sqrt{2}=0.9999999997\sigma$. Thus, a HS
located on the lattice point of the bottom square lattice fell into the
intersection of the grooves by almost the half of HS diameter. The separation
between neighboring groove edges was $0.338\sigma$. Accordingly, the
periodicity of the lattice was $1.045\sigma$ and the diagonal distance
$1.478\sigma$; i.e., we set the bottom lattice so as to coincide with the
bottom (001) layer of the fcc crystal of the previous flat wall simulation
[18].
In this paper we shall take some close looks into the simulation results of
two lateral system sizes, $L_{x}$ = $L_{y}$ = $12.55\sigma$ and $L_{x}$ =
$L_{y}$ = $25.09\sigma$. The vertical system size was fixed at $L_{z}$ =
$200\sigma$; this size was enough large so that at the initial ($g^{*}$ = 0.0)
the HSs were dispersed randomly. To prepare an initial state we ran a MC
simulation for 2$\times 10^{7}$ MC cycle (MCC). Here, one MCC was define so
that it contains $N$ MC particle moves, i.e., every particle undergoes one
particle move on average in one MCC. The maximum displacement was fixed at
$\Delta r_{\mbox{\scriptsize max}}$ = $0.06\sigma$ throughout. Therefore, as
the density changes the acceptance ratio changes; as a result the time
corresponding to one MCC varied as the simulation proceeded. Nevertheless, the
states emerges in a course of a simulation are arranged as a time series. The
numbers of particles were $N$ = 6656 and 26624. These numbers of particle,
$N$s, were selected such that the numbers of particles laid on the bottom per
unit area, $n_{\mbox{\scriptsize s}}$, became the same value as that of
previous simulations [15].
In those simulations the gravitational number $g^{*}\equiv
mg\sigma/k_{\mbox{\scriptsize B}}T$ was increased stepwise to avoid the
trapping of the system by a metastable state such as polycrystalline state
[15]. Here, $m$ is the mass of a particle, $k_{\mbox{\scriptsize B}}T$ the
temperature multiplied by Boltzmann’s constant. If gravity such as $g^{*}$ =
0.9 is suddenly switched on for the flat wall case, the system is trapped into
a polycrystalline metastable state [22]. We note here that an effective
control of $g^{*}$ can be done in a centrifugation method [23], in comparison
to a gravitational sedimentation. Parameters of the stepwise control of
$g^{*}$ was $\Delta t$ = $2\times 10^{5}$MCC for $N$ = 6656 system and $\Delta
t$ = $8\times 10^{5}$MCC for $N$ = 26624; $\Delta g^{*}$ = 0.1 for both. In
Ref. [15] at first we kept $g^{*}$ at 0 for $\Delta t$ and then increased
$g^{*}$ by $\Delta g^{*}$. On the other hand, in this simulation we kept
$g^{*}$ at $\Delta g^{*}$ at first for $\Delta g$ and then increased. Although
several combinations were tested, we will not present on the optimization of
$g^{*}$ control in this paper for the limitation of pages. Without any
optimization we have got results, which were enough to complement the
shortcoming of the previous studies. With an optimization we will look at
detailed processes of the defect disappearance. With other optimizations
disappearance of several types of defects must be observed.
## 3 Results and discussions
We have preformed seven simulations for $N$ = 6656 system and three for $N$ =
26624 with different series of random numbers. In two of these simulations for
$N$ = 6656 defect disappearance at $g^{*}$ less than 0.9 was observed; at
$g^{*}$=0.9 shrinking of an intrinsic stacking fault occurred for a flat
bottom wall case [14]. In four of these simulations for $N$ = 6656 defect
disappearance occurred at $g^{*}$ greater than 0.9. For remainder one defect
disappearance was not appreciable. For $N$ = 26624 system in all three
simulation defect disappearance was observed at $g^{*}$ less than 0.9. In two
of three the defect disappearance occurred during $g^{*}$ = 0.5 and in the
remainder one during $g^{*}$ = 0.7. What was notable for $N$ = 26624 system
was that increase of disorder in appearance was seen at large $g^{*}$. In a
case this phenomenon was observed in one of projected snapshot and, on the
other hand, defect disappearance was observed in the other projection. Let us
postpone the detail analysis and discussion after looking at snapshots.
Snapshots and evolutions of the center of gravity will be shown in section 3.1
for $N$ = 6656 system and in section 3.2 for $N$ = 26624.
### 3.1 $N$=6656 system
Figure 1: Snapshots projected on $xz$ and $yx$ planes at (a) $g^{*}$=0.7, (b)
0.8, (c) 0.9, and (d) 1.0 for a case that the defect disappearance occurred at
$g^{*}$ less than 0.9 for $N$ = 6656 system. Defect existed in
$7.5<z/\sigma<13.5$, as indicated by $yz$ projection of (a), disappeared
during $g^{*}$ = 0.8. Also defect in $13.5<z/\sigma<17.5$ in $xz$ projection
of (c) disappeared during $g^{*}$ = 1.0. Defects around
$(x/\sigma,y/\sigma,z/\sigma)$ = $(-2,-3,6)$ and $(-5,-4,3)$ did not
disappear.
Snapshots at $g^{*}$ = 0.7-1.0 are shown in Fig. 1 for a case that the defect
disappearance occurred at $g^{*}$ less than 0.9. Though defects in the lower
portion remained, a defect in appearance expanded over the middle portion
disappeared during $g^{*}$ = 0.8 and then that in top portion disappeared
during $g^{*}$ = 1.0 again. In fcc (001) stacking, if a single stacking fault
runs along one of {111}, [110] ([1$\bar{1}$0]) lattice line makes an array of
two separated points in (110) [(1$\bar{1}$0)] projection. And, on the other
projection we can observe a fault directly. To understand Fig. 1 we must take
into account the fact that $x$ and $y$ directions correspond to $\langle$110]
(see Ref. [15]). In Fig. 1 (a) splittings are observed in both $xz$ and $yz$
projections and the both splittings disappeared in Fig. 1 (b). Therefore, we
cannot conjecture those as involving shrinking of a single stacking fault as
for a flat wall case [14]. If two stacking faults along, e.g., (111) and
(11$\bar{1}$), coexist, then splitting on (110) projection and two
intersecting fault on (1$\bar{1}$0) projection are seen. So, two stacking
faults along, e.g., (111) and (1$\bar{1}$1), must coexist. To observe
intersections between (110) or (1$\bar{1}$0) and stacking faults by making
three-dimensional (3D) view may give an answer. The surface structure of the
3D snapshot was, however, complicated as imagined geometrically. Although a
crossing two faults was seen, we cannot successfully follow the evolution as
previously done [14] because of the complexity. Let us postpone this complex
analysis as a future research. On the other hand, comparing Fig. 1 (c) and (d)
we find that splitting of $xz$ projection of lattice lines disappeared during
$g^{*}$ = 1.0. That the splitting in $yz$ projection did not disappear
suggests that a single stacking fault such as running along, e.g.,
(1$\bar{1}$1), remained. In other words, disappearance of a stacking fault
along (111) or (11$\bar{1}$) is deduced.
We discuss about the possibility of staking fault tetrahedra. We observe
downward triangles in $xz$ projection and upward triangles in $yz$ projection
at the defect around $(x/\sigma,y/\sigma,z/\sigma)$ = $(-2,-3,6)$ in Fig. 1
(b)-(d). Those triangles are seen if we make projections of a tetrahedron
surrounded by {111} onto (110) and (1$\bar{1}$0). Regarding the defect around
$(x/\sigma,y/\sigma,z/\sigma)$ = $(-5,-4,3)$ we cannot identify the three
dimensional shape. Those defects seem to be sessile, because they remain for a
long time.
Figure 2: Evolution of the center of gravity during (a) $g^{*}$=0.8 and (b)
1.0 for a case that the defect disappearance occurred at $g^{*}$ less than 0.9
for $N$ = 6656 system. The curve in (a) is more smooth as compared to that in
Ref. [14]. That in (b), however, exhibits a multiple relaxation manner as in
Ref. [14]. Statistical errors are within $0.011\sigma$ for (a) and
$0.008\sigma$ for (b).
Figure 2 is the evolution of the center of gravity for a case that the defect
disappearance occurred at $g^{*}$ less than 0.9. For the flat wall case we
observed some plateaus in the evolution of the center of gravity. In contrast,
the evolution of the center of gravity in the preset case during $g^{*}$ = 0.8
is more smooth and nearly of a single relaxation mode. On the other hand, that
during $g^{*}$ = 1.0 implies trapping at a metastable configuration during
defect disappearance. The relaxation in Fig. 2 (b) is of two step manner. In
1.87-1.9 $\times 10^{6}$MCC a settlement at a metastable configuration
occurred and then a relaxation started again.
Figure 3: Snapshots projected on $xz$ and $yx$ planes at (a) $g^{*}$=0.7, (b)
0.8, (c) 0.9, (d) 1.0, (e) 1.3, and (f) 1.4 for a case that the defect
disappearance occurred at $g^{*}$ greater than 0.9 for $N$ = 6656 system.
Whereas no defect disappearance occurred in (a)-(d), defect in $2<z/\sigma<9$
in $xz$ projection of (e) disappeared in (f) during $g^{*}$=1.4.
Let us look at snapshots for a case that the defect disappearance occurred at
$g^{*}$ greater than 0.9. Snapshots are shown in Fig. 3. At first, we note
that two layer defect free region are formed at the bottom. We observed
formation of a few crystalline layers commonly for seven simulations performed
at low $g^{*}$ such as 0.5. Looking at snapshots (not shown) of those layers
parallel to the bottom wall, we find that the bottom layers match to the
pattern on the bottom wall on the lattice point. From density profiles (not
shown) we find that layering of two layers, which possessed no significant
interlayer ordering, at the bottom occurred even at $g^{*}$ = 0.2. This
layering phenomena are common to flat wall cases [24, 25]. Defect
disappearance did not occurred up to $g^{*}$ = 1.4. During $g^{*}$ = 1.4
defect shown in $xz$ projection disappeared. We confirm a stacking fault in
the right-lower region in $xz$ projection in Fig. 3 (e). In addition,
following a lattice line horizontally we find a step on a lattice line in the
left-lower region. This is characteristic of a stacking fault. Accordingly,
there exist two stacking faults of different directions. At
($x/\sigma$,$z/\sigma$) $\sim$ (2,7.5) those two staking faults meet. There is
a possibility of a star-rod partial dislocation there.
Let us look at the portion $9<z/\sigma<17$, e.g., in Fig. 3 (f). We observe a
downward triangle in $xz$ projection and an upward triangle in $yx$ projection
at the middle. Taking into account the periodic boundary condition, both sides
of these triangles make triangles upward and downward. It is suggested that an
upward tetrahedron and downward tetrahedron fills a part of the space.
However, this situation takes place so as to match the periodic boundary
condition. So, if the system size is large enough, a configuration such that a
tetrahedron is embedded in a defect free matrix as Fig. 1 (b)-(d) must be
seen. Why tetrahedral configuration did not appeared in the flat wall cases
might be due to the system size.
Figure 4: Evolution of the center of gravity during (a) $g^{*}$=1.0 and (b)
1.4 for a case that the defect disappearance occurred at $g^{*}$ greater than
0.9 for $N$ = 6656 system. The curve in (a) is of a single relaxation. On the
other hand, that in (b) exhibits a multiple relaxation manner as in Ref. [14].
Statistical errors are within $0.008\sigma$ for (a) and $0.005\sigma$ for (b).
We have compared the evolution of the center of gravity for this case and that
for a case that the defect disappearance occurred at $g^{*}$ less than 0.9
during $g^{*}$ = 0.8. There is no significant difference, though, if the
defect disappearance did not occur due to trapping at the metastable
configuration, plateaus indicating this trap were expected. We speculate that
upward growth of nucleated crystalline layers on the bottom, which occurred up
to $g^{*}\sim 0.4$ as mentioned above, is involved in this smooth relaxation.
Figure 4 is the evolution of the center of gravity for a case that the defect
disappearance occurred at $g^{*}$ greater than 0.9. It is interesting that
both in Fig. 1 (d) and Fig. 3 (f) the relaxation is multiple manner when the
process is from a state including “multiple” stacking faults to that including
a “single” stacking fault.
### 3.2 $N$=26624 system
Figure 5: Snapshots projected on $xz$ and $yx$ planes at (a) $g^{*}$=0.4, (b)
0.5, (c) 0.8, and (d) 0.9 for $N$ = 26624 system. Defect disappearance
occurred in $xz$ projection of (a)-(b) and (c)-(d), and increasing of disorder
in appearance is observed in $yz$ projection of (c)-(d).
Snapshots at $g^{*}$ = 0.4-0.5 and 0.8-0.9 are shown in Fig. 5. Looking at the
$yz$ projection of Fig. 5 (a) and (b) we see that defect in $3<z/\sigma<9$
disappeared during $g^{*}$ = 0.5. Splitting of the projection of lattice lines
in $xz$ projection of Fig. 5 (b) in this region implies that lattice lines
along $y$ axis are crossing with faults. Indeed, we can confirm steps on a
lattice line traversing along the lattice line horizontally in $yx$ projection
of Fig. 5 (b). As for the $N$ = 6656 system disappearance of stacking faults
in one direction is suggested. Let us compare $xz$ projections of Fig. 5 (b)
and (c). We find that some splittings of projections of lattice lines
disappeared, indicating that disappearance of the stacking fault in the
corresponding direction. We note that new defects such as at
$(y/\sigma,z/\sigma)$ = $(-6,10)$-$(-4,13)$, around $(y/\sigma,z/\sigma)$ =
$(-3,5)$-$(-1,8)$ formed. The former may be a stacking fault. Indeed splitting
of projections of lattice lines is seen in $xz$ projection of Fig. 5 (c) over
the levels same at this defect in $yz$ projection. On the other hand, the
latter defect is somewhat widened. We cannot identify only from the projected
snapshots.
Let us look at Fig. 5 (d). A defect is expanded over a wide region in $yz$
projection. This is a newly formed defect. Correspondingly, we see a downward
thick triangle structure and an upward thick triangle structure in $xz$
projection. If a downward (upward) triangle in one direction corresponds to an
upward (downward) triangle in the other projection, a stacking fault
tetrahedron is suggested. Simultaneously, we observe defect disappearance
around $(x/\sigma,z\sigma)$ = $(11,6)$ in $xz$ projection.
Figure 6: Evolution of the center of gravity during (a) $g^{*}$=0.5 and (b)
0.9 for $N$ = 26624 system. The both curves in (a) and (b) are of a single
relaxation mode. In (a) relaxation has not reached to the equilibrium yet. On
the other hand, in (b) the lowering of the center of gravity has reached to
equilibrium and is fluctuating around it. Statistical errors are within
$0.014\sigma$ for (a) and $0.008\sigma$ for (b).
Let us look at the evolution of the center of the gravity (Fig. 6). Unlike the
$N$ = 6656 system, multiple relaxations are not appreciable. The evolution of
the center of gravity during $g^{*}$ = 0.4 is of a single relaxation mode and
does not reached at equilibrium. Also, that during $g^{*}$ = 0.9 is of a
single relaxation mode. In this case, however, it has reached at equilibrium
and is fluctuating around the equilibrium. Formation of a defect structure
during $g^{*}$ = 0.9 mentioned above may occur in an equilibrium fluctuation.
However, in a magnification of the equilibrium fluctuation of the evolution of
the center of gravity, we can find a correlation of the structural change and
the evolution. Some detail analysis will be given in a future research.
## 4 Concluding remarks
We successfully performed Monte Carlo simulations of a colloidal epitaxy on a
square pattern using hard spheres. In other words, we have succeeded in
replacing the artificial stress, which is a driving force for fcc (001)
stacking, with realizable one. Moreover, we would say that the system size can
be enlarged systematically. For a large system, however, a number of defects
running along deferent directions occurred in a system. It makes analyses
complicated.
In a case that defect disappearance was observed at lower $g^{*}$ than that
for the flat bottom wall cases, the sinking of the center of gravity of the
system was smooth and of a single relaxation mode. Also for a large system, it
was smooth and of a single relaxation mode. That is, in this case the
shrinking of the defect was not trapped temporarily at a metastable
configuration. On the other hand, at $g^{*}$ greater than the value at which
the defect disappearance and temporal stopping of the lower end of an
intrinsic stacking fault occurred for the flat wall cases (at $g^{*}$ = 0.9),
the temporal stopping of the sinking of the center of gravity was observed.
For large system, such temporal stopping was not appreciable.
In the snapshots tetrahedral structures appeared often, suggesting staking
fault tetrahedra being sessile. Observation of the tetrahedral configuration
for more large systems are in progress.
To accomplish complicated analyses to observe the manner of defect
disappearance and identify the structure of defects are left as future
researches. System size is to be systematically enlarged. The way of
controlling $\Delta g^{*}$ should be optimized both to observe the details of
the defect disappearance and to efficiently erase the defects in reality.
## References
* [1] W. W. Wood and J. D. Jacobson, J. Chem. Phys. 27 (1957) 1207.
* [2] B. J. Alder and T. E. Wainwright, J. Chem. Phys. 27 (1957) 1208.
* [3] A. Mori, in Theory and Applications of Monte Carlo Simulations (INTECH), in press.
* [4] L. Antl, J.W. Goodwin, R. D. Hill, R. H. Ottewill, S. M. Owens, S. P. Papworth, and J. A. Waters, Colloids Surf. 17 (1986) 67.
* [5] P. N. Pusey and W. van Megen, Nature 320 (1986) 340.
* [6] S. E. Paulin and B. J. Ackerson, Phys. Rev. Lett. 64 (1990) 2663; errata ibid., 65 (1990) 668.
* [7] S. M. Underwood, J. R. Taylor, and W. van Megen, Langmuir 10 (1994) 3550.
* [8] S. E. Phan, W. B. Russel, Z. Cheng, J. Zhu, P. M. Chaikin, J. H. Dunsmur, and R. H. Ottewill, Phys. Rev. E 54 (1996) 6633.
* [9] K. Ohtaka, Phys. Rev. B 19 (1979) 5057.
* [10] E. Yablonovitch, Phys. Rev. Lett. 58 (1987) 2059.
* [11] S. John, Phys. Rev. Lett. 58 (1987) 2486.
* [12] J. Zhu, M. Li, R. Rogers, W. Meyer, R. Ottewill, STS-73 Space Shuttle Crew, W. Russel, and P. M. Chaikin, Nature 387 (1997) 883.
* [13] P. N. Pusey, W. van Megen R. Bartlett, B. J. Ackerson, J. G. Parity, and S. M. Underwood, Phys. Rev. Lett. 63 (1989) 2753.
* [14] A. Mori, Y. Suzuki, S.-i. Yanagiya, T. Sawada, and K. Ito, Molec. Phys. 105 (2007) 1377; errata ibid 106 (2008) 187.
* [15] A. Mori, S.-i. Yanagiya, Y. Suzuki, T. Sawada, and K. Ito, J. Chem. Phys. 124 (2006) 174507.
* [16] A. van Blaaderen, R. Ruel, and R. Wiltzius, Nature 385 (1997) 321.
* [17] A. Mori, Y. Suzuk, and S.-i. Yanagiya, Fluid Phase Equil. 257 (2007) 131.
* [18] A. Mori, S.-i. Yanagiya, Y. Suzuki, T. Sawada, and K. Ito, Sci. Technol. Adv. Mater. 7 (2006) 296.
* [19] A. Mori, Y. Suzuki, and S. Matsuo, Prog. Theor. Phys. Suppl. 178 (2009) 33.
* [20] A. Mori and Y. Suzuki, Molec. Phys., 108 (2010) 1731.
* [21] K.-h. Lin, J. C. Crocker, V. Parasad, A. Schofield, D. A. Weitz, T. C.Lubensky, and Y. G. Yodh, Phys. Rev. Lett. 85 (2000) 1770.
* [22] S.-i. Yanagiya, A. Mori, Y. Suzuki, Y. Miyoshi, M. Kasuga, T. Sawada, K. Ito, and T. Inoue, Jpn. J. Appl. Phys., Part. 1 44 (2005) 5113.
* [23] Y. Suzuki, T. Sawada, A. Mori, and K. Tamura, Kobunshi Ronbunshu 64 (2007) 161 [in Japanese].
* [24] T. Biben, R. Ohnesorge, and H. Löwen, Europhys. Lett. 28 (1994) 665.
* [25] M. Marechal and M. Dijkstra Phys. Rev. E 75 (2007) 061404.
|
arxiv-papers
| 2010-10-15T16:43:46 |
2024-09-04T02:49:13.963226
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Atsushi Mori",
"submitter": "Atsushi Mori",
"url": "https://arxiv.org/abs/1010.3213"
}
|
1010.3221
|
# Acoustically bound crystals
P. Marmottant D. Rabaud P. Thibault M. Mathieu
Laboratoire de Spectrométrie Physique
CNRS and University of Grenoble,
Grenoble, France
###### Abstract
In these fluid dynamics videos, we show how bubbles flowing in a thin
microchannel interact under an acoustic field. Because of acoustic
interactions without direct contact, bubbles self-organize into periodic
patterns, and spontaneously form acoustically bound crystals. We also present
the interaction with boundaries, equivalent to the interaction with image
bubbles, and unravel the peculiar vibration modes of the confined bubbles.
We generate microbubbles, with a diameter ranging from 20 to 50 micrometer,
confined within thin 25 micrometers high elastomeric channels made of
polydimethylsiloxane. The microbubbles are generated by flow-focusing a gas
jet with a solution of surfactants. The bubbles are highly confined in between
the top and bottom walls, and have therefore the shape a squeezed sphere shape
with two flat faces (white on images) and a curved perimeter (black on
images). We then apply an acoustic field by molding a glass plate into the
elastomer , just above the microchannel, separated by only 145 micrometers. We
set a standing wave in the glass rod, and the sound is emitted through the
elastomer, that transmits efficiently sound to the channel.
We observe that bubbles vibrate and interact, with acoustic forces called
secondary Bjerknes forces. This interaction has a specificity: it presents a
minimum at a finite distance, and a minimum at contact. Bubbles either keep a
fixed distance or either agglomerate without coalescing, because surfactants
impede the contact of interfaces. Spontaneous patterns therefore occur.
The interaction is believed to be mediated by surface waves on the elastomer,
which is comforted by the fact that they have a very slow velocity and
therefore a small wavelength comparable to the observed distances equilibrium
distances (the wavelength of sound in water is 50 times larger). Another fact
comforting this scenario, is that it explains why bubbles are attracted to
walls or keep a fixed distance. Indeed surface waves reflect on boundaries,
which is equivalent to placing an image bubble symmetrically to the boundary.
Bubbles then interact with their own images.
The vibration mode of the bubbles is not always an axisymmetric pulsation. At
large sound amplitude we observe undulations, that are due to a surface
instability at the free surface of the bubble. It creates standing waves on
the perimeter. Large bubbles present a larger number of crests.
In conclusion, bubbles can form acoustically bound crystals, which are
autonomous structures flowing with the liquid.
|
arxiv-papers
| 2010-10-15T16:59:37 |
2024-09-04T02:49:13.970101
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "P. Marmottant, D. Rabaud, P. Thibault and M. Mathieu",
"submitter": "Philippe Marmottant Philippe Marmottant",
"url": "https://arxiv.org/abs/1010.3221"
}
|
1010.3284
|
# Rigidity of Polyhedral Surfaces, III
Feng Luo Department of Mathematics, Rutgers University, New Brunswick, New
Jersey 08854 fluo@math.rutgers.edu To Dennis Sullivan on the occasion of his
seventieth birthday
(Date: Oct. 1, 2010.)
###### Abstract.
This paper investigates several global rigidity issues for polyhedral surfaces
including inversive distance circle packings. Inversive distance circle
packings are polyhedral surfaces introduced by P. Bowers and K. Stephenson in
[2] as a generalization of Andreev-Thurston’s circle packing. They conjectured
that inversive distance circle packings are rigid. Using a recent work of R.
Guo [9] on variational principle associated to the inversive distance circle
packing, we prove rigidity conjecture of Bowers-Stephenson in this paper. We
also show that each polyhedral metric on a triangulated surface is determined
by various discrete curvatures introduced in [12], verifying a conjecture in
[12]. As a consequence, we show that the discrete Laplacian operator
determines a Euclidean polyhedral metric up to scaling.
###### Key words and phrases:
polyhedral metrics, discrete curvatures, rigidity
###### 1991 Mathematics Subject Classification:
Primary 54C40, 14E20; Secondary 46E25, 20C20
The work is supported in part by a NSF Grant.
## 1\. Introduction
### 1.1.
This is a continuation of the study of polyhedral surfaces [12], [13]. The
paper focuses on inversive distance circle packings introduced by Bowers and
Stephenson and several other rigidity issues. Using a recent work of Ren Guo
[9], we prove a conjecture of Bowers-Stephenson that inversive distance circle
packings are rigid. Namely, a Euclidean inversive distance circle packing on a
compact surface is determined up to scaling by its discrete curvature. This
generalizes an earlier result of Andreev [1] and Thurston [17] on the rigidity
of circle packing with acute intersection angles. In [12], using 2-dimensional
Schlaefli formulas, we introduced two families of discrete curvatures for
polyhedral surfaces and conjectured that each of one these discrete curvatures
determines the polyhedral metric (up to scaling in the Euclidean case). We
verify this conjecture in the paper. One consequence is that for a Euclidean
or spherical polyhedral metric on a surface, the cotangent discrete Laplacian
operator determines the metric (up to scaling in the case of Euclidean
metric). The theorems are proved using variational principles and are based on
the work of [9] and [12]. The main idea of the paper comes from reading of
[4], [7] and [15].
### 1.2.
Recall that a Euclidean (or spherical or hyperbolic) polyhedral surface is a
triangulated surface with a metric, called a polyhedral metric, so that each
triangle in the triangulation is isometric to a Euclidean (or spherical or
hyperbolic) triangle. To be more precise, let $\mathbb{E}^{2}$,
$\mathbb{S}^{2}$ and $\mathbb{H}^{2}$ be the Euclidean, the spherical and the
hyperbolic 2-dimensional geometries. Suppose $(S,T)$ is a closed triangulated
surface so that $T$ is the triangulation, $E$ and $V$ are the sets of all
edges and vertices. A $K^{2}$ ($K^{2}$ = $\mathbb{E}^{2}$, or
$\mathbb{S}^{2}$, or $\mathbb{H}^{2}$) polyhedral metric on $(S,T)$ is a map
$l:E\to\mathbb{R}$ so that whenever $e_{i},e_{j},e_{k}$ are three edges of a
triangle in $T$, then
$l(e_{i})+l(e_{j})>l(e_{k}),$
and if $K^{2}=\mathbb{S}^{2}$, in addition to the inequalities above, one
requires
$l(e_{i})+l(e_{j})+l(e_{k})<2\pi.$
Given $l:E\to\mathbb{R}$ satisfying the inequalities above, there is a metric
on the surface $S$, called a polyhedral metric, so that the restriction of the
metric to each triangle is isometric to a triangle in $K^{2}$ geometry and the
length of each edge $e$ in the metric is $l(e)$. We also call
$l:E\to\mathbb{R}$ the edge length function. For instance, the boundary of a
generic convex polytope in the 3-dimensional space $\mathbb{E}^{3}$, or
$\mathbb{S}^{3}$ or $\mathbb{H}^{3}$ of constant curvature $0,1,$ or $-1$ is a
polyhedral surface. The discrete curvature $k$ of a polyhedral surface is a
function $k:V\to\mathbb{R}$ so that $k(v)=2\pi-\sum_{i=1}^{m}\theta_{i}$ where
$\theta_{i}$’s are the angles at the vertex $v$. See figure 1.
Since the discrete curvature is built from inner angles of triangles, we
consider inner angles of triangles as the basic unit of measurement of
curvature. Using inner angles, we introduce three families of curvature like
quantities in [12]. The relationships between the polyhedral metrics and
curvatures are the focus of the study in this paper.
###### Definition 1.1.
([12]) Let $h\in\mathbb{R}$. Given a $K^{2}$ polyhedral metric on $(S,T)$
where $K^{2}$ $=\mathbb{E}^{2}$, or $\mathbb{S}^{2}$ or $\mathbb{H}^{2}$, the
$\phi_{h}$ curvature of a polyhedral metric is the function
$\phi_{h}:E\to\mathbb{R}$ sending an edge $e$ to:
(1.1)
$\phi_{h}(e)=\int_{\pi/2}^{a}\sin^{h}(t)dt+\int_{\pi/2}^{a^{\prime}}\sin^{h}(t)dt$
where $a,a^{\prime}$ are the inner angles facing the edge $e$. See figure 1.
The $\psi_{h}$ curvature of the metric $l$ is the function
$\psi_{h}:E\to\mathbb{R}$ sending an edge $e$ to
(1.2)
$\psi_{h}(e)=\int^{\frac{b+c-a}{2}}_{0}\cos^{h}(t)dt+\int^{\frac{b^{\prime}+c^{\prime}-a^{\prime}}{2}}_{0}\cos^{h}(t)dt$
where $b,b^{\prime},c,c^{\prime}$ are inner angles adjacent to the edge $e$
and $a,a^{\prime}$ are the angles facing the edge $e$. See figure 1.
Figure 1.
The curvatures $\phi_{0}$ and $\psi_{0}$ were first introduced by I. Rivin
[Ri] and G. Leibon [Le] respectively. If the surface $S=\mathbb{S}^{2}$, then
these curvatures are essentially the dihedral angles of the associated
3-dimensional hyperbolic polyhedra at edges. The curvature
$\phi_{-2}(e)=-\cot(a)-\cot(a^{\prime})$ is the discrete (cotangent) Laplacian
operator on a polyhedral surface derived from the finite element approximation
of the smooth Beltrami Laplacian on Riemannian manifolds.
One of the remarkable theorems proved by Rivin [15] is that a Euclidean
polyhedral metric on a triangulated surface is determined up to scaling by its
$\phi_{0}$ discrete curvature. In particular, he proved that an ideal convex
hyperbolic polyhedron is determined up to isometry by its dihedral angles.
We prove,
###### Theorem 1.2.
Let $(S,T)$ be a closed triangulated connected surface. Then for any
$h\in\mathbb{R}$,
(1) a Euclidean polyhedral metric on $(S,T)$ is determined up to isometry and
scaling by its $\phi_{h}$ curvature.
(2) a spherical polyhedral metric on $(S,T)$ is determined up to isometry by
its $\phi_{h}$ curvature.
(3) a hyperbolic polyhedral surface is determined up to isometry by its
$\psi_{h}$ curvature.
We remark that theorem 1.2(1) for $h=0$ was aforementioned Rivin’s theorem.
However, our proof of Rivin’s theorem is different from that in [15] and we
use the variational principle established by Cohen-Kenyon-Propp [5]. Theorem
1.2(3) for $h=0$ was first proved by Leibon [11]. Theorem 1.2(2) for $h=0$ was
proved in [14] and theorem 1.2(2) and (3) for $h\leq-1$ or $h\geq 0$ was
proved in [12].
Take $h=-2$ in theorem 1.2, we obtain,
###### Corollary 1.3.
(1) A connected Euclidean polyhedral surface is determined up to scaling by
its discrete Laplacian operator.
(2) A spherical polyhedral surface is determined by its discrete Laplacian
operator.
Note that for a Euclidean polyhedral surface, $\phi_{h}=\psi_{h}$. There
remain two questions on whether $\phi_{h}$ curvature determines a hyperbolic
polyhedral surface or whether $\psi_{h}$ curvature determines a spherical
polyhedral surface. It seems the results may still be true in these cases.
### 1.3.
Inversive distance circle packings are polyhedral metrics on a triangulated
surface introduced by Bowers and Stephenson in [2]. An expansion of the
discussion of [2] is in [3]. See also [16]. They are generalizations of
Andreev and Thurston’s circle packings. Unlike the case of Andreev and
Thurston where adjacent circles are intersecting, Bowers and Stephenson allow
adjacent circles to be disjoint and measure their relative positions by the
inversive distance. As observed in [2], this relaxation of intersection
condition is very useful for practical applications of circle packing to many
fields, including medical imaging and computer graphics. Based on extensive
numerical evidences, they conjectured the rigidity and convergence of
inversive distance circle packings in [2]. Our result shows that Bowers-
Stephenson’s rigidity conjecture holds. The proof is based on a recent work of
Ren Guo [9] which established a variational principle for inversive distance
circle packings. A very nice geometric interpretation of the variational
principle was given in [8].
We begin with a brief recall of the inversive distance in Euclidean,
hyperbolic and spherical geometries. See [3] for a more detailed discussion.
Let $K^{2}$ be $\mathbb{E}^{2}$, or $\mathbb{H}^{2}$ or $\mathbb{S}^{2}$.
Given two circles $C_{1},C_{2}$ in $K^{2}$ centered at $v_{1},v_{2}$ of radii
$r_{1}$ and $r_{2}$ so that $v_{1},v_{2}$ are of distance $l$ apart, the
inversive distance $I=I(C_{1},C_{2})$ between the circles is given by
(1.3) $I=\frac{l^{2}-r_{1}^{2}-r_{2}^{2}}{2r_{1}r_{2}}$
in the Euclidean plane,
(1.4) $I=\frac{\cosh(l)-\cosh(r_{1})\cosh(r_{2})}{\sinh(r_{1})\sinh(r_{2})}$
in the hyperbolic plane and
(1.5) $I=\frac{\cos(l)-\cos(r_{1})\cos(r_{2})}{\sin(r_{1})\sin(r_{2})}$
in the 2-sphere. See [9] for more details on (1.4) and (1.5). If one considers
$\mathbb{E}^{2}$, $\mathbb{H}^{2}$ and $\mathbb{S}^{2}$ as appeared in the
infinity of the hyperbolic 3-space $\mathbb{H}^{3}$, then $C_{1}$ and $C_{2}$
are the boundary of two totally geodesic hyperplanes $D_{1}$ and $D_{2}$. The
inversive distance $I$ is essentially the hyperbolic distance (or the
intersection angle) between $D_{1}$ and $D_{2}$. In particular, for the
Euclidean plane $\mathbb{E}^{2}$, the inversive distance $I(C_{1},C_{2})$ is
invariant under the inversion and hence the name.
Bowers and Stephenson’s construction of an inversive distance circle packing
with prescribed inversive distance on a triangulated surface $(S,T)$ is as
follows. Fix once and for all a vector $I\in\mathbb{[}-1,\infty)^{E}$, called
the inversive distance.
In the Euclidean case, for any $r\in\mathbb{R}_{>0}^{V}$, called the radius
vector, define the edge length function $l\in\mathbb{R}_{>0}^{E}$ by the
formula
(1.6) $l(e)=\sqrt{r(v)^{2}+r(u)^{2}+2r(v)r(u)I(e)}$
where the end points of the edge $e$ is $\\{u,v\\}$. If $l(e)$’s satisfy the
triangular inequalities that
(1.7) $l(e_{i})+l(e_{j})>l(e_{k})$
for three edges $e_{i},e_{j},e_{k}$ of each triangle in $T$, then the length
function $l:E\to\mathbb{R}$ sending $e$ to $l(e)$ defines a Euclidean
polyhedral metric on $(S,T)$ called the inversive distance circle packing with
inversive distance $I(e)$ at edge $e$. Note that if $I(e)\in[0,1]$ for all
$e$, the polyhedral metric is the circle packing investigated by Andreev and
Thurston where the intersection angle between two circles at the end points of
an edge is $\arccos(I(e))$.
In the hyperbolic geometry, one uses
(1.8) $l(e)=\cosh^{-1}(\cosh(r(v))\cosh(r(u))+I(e)\sinh(r(v))\sinh(r(u))$
as the length of an edge. If (1.7) holds, then the lengths $l(e)$’s define a
hyperbolic inversive distance circle packing with inversive distances $I$ on
$(S,T)$. The spherical inversive distance circle packing is defined similarly
with additional condition on $l(e)$’s that
$l(e_{i})+l(e_{j})+l(e_{k})<2\pi$
for each triangle with edges $e_{i},e_{j},e_{k}$.
The geometric meaning of these polyhedral metrics is the following. In each
metric, if one draws a circle of radius $r(v)$ at each vertex $v$, then
inversive distance of two circles at the end points of an edge $e$ is the
given number $I(e)$.
Our result which solves Bowers-Stephenson’s rigidity conjecture is the
following.
###### Theorem 1.4.
Given a closed triangulated connected surface $(S,T)$ with the set of edges
$E$ and $I\in\mathbb{R}_{\geq 0}^{E}$ considered as the inversive distance,
(1) a hyperbolic inversive distance circle packing metric on $(S,T)$ of
inversive distance $I$ is determined by its discrete curvature
$k:V\to\mathbb{R}$.
(2) an Euclidean inversive distance circle packing metric on $(S,T)$ of
inversive distance $I$ is determined by its discrete curvature
$k:V\to\mathbb{R}$ up to scaling.
Note that for $I\in[0,1]^{E}$, the above result was Andreev-Thurston’s
rigidity for circle packing with intersection angles between $[0,\pi/2]$. It
seems the similar result may be true for $I\in[-1,\infty)^{E}$.
### 1.4.
The paper is organized as follows. In §2, we prove an extension lemma for
angles of triangles. We also establish a criterion for extending a locally
convex function to convex function. In §3, we prove theorem 1.4. Theorem 1.2
is proved in §4.
The following notations and conventions will be used in the paper. We use
$\mathbb{R}$, $\mathbb{R}_{>0}$, $\mathbb{R}_{\geq 0}$, $\mathbb{R}_{<0}$ to
denote the sets of all real numbers, positive real numbers, non-negative real
numbers, and negative real numbers respectively. If $X$ is a set,
$\mathbb{R}^{X}=\\{f:X\to\mathbb{R}$} is the vector space of all functions on
$X$. If $A$ is a subspace of a topological space $X$, then the closure of $A$
in $X$ is denoted by $\bar{A}$.
We thank Ren Guo for comments and careful reading of the manuscript.
## 2\. Convex Extension of Locally Convex Functions
### 2.1. Continuous extension by constants
###### Definition 2.1.
Suppose $A$ is a subspace of a topological space $X$ and $f:A\to Y$ is
continuous. If there exists a continuous function $F:X\to Y$ so that
$F|_{A}=f$ and $F$ is a constant function on each connected component of
$X-A$, then we say $f$ can be extended continuously by constant functions to
$X$.
Note that if each connected component of $X-A$ intersects the closure of $A$,
then the extension function $F$ is uniquely determined by $f$.
The key observation of the paper is the following simple lemma.
###### Lemma 2.2.
Suppose $\Delta$ is a triangle in the Euclidean plane $\mathbb{E}^{2}$, or the
hyperbolic plane $\mathbb{H}^{2}$, or the 2-sphere $\mathbb{S}^{2}$ so that
its edge lengths are $l_{1},l_{2},l_{3}$ and its inner angles are
$\theta_{1},\theta_{2},\theta_{3}$. Assume that $\theta_{i}$’s angle is
opposite to the edge of length $l_{i}$ for each $i$. Consider
$\theta_{i}=\theta_{i}(l)$ as a function of $l=(l_{1},l_{2},l_{3})$.
1. (1)
If $\Delta$ is Euclidean or hyperbolic, the angle function $\theta_{i}$
defined on
$\Omega=\\{(l_{1},l_{2},l_{3})\in\mathbb{R}^{3}|l_{1}+l_{2}>l_{3},l_{2}+l_{3}>l_{1},l_{3}+l_{1}>l_{2}\\}$
can be extended continuously by constant functions to a function
$\tilde{\theta_{i}}$ on $\mathbb{R}^{3}_{>0}$.
2. (2)
If $\Delta$ is spherical, the angle function $\theta_{i}$ defined on
$\Omega=\\{(l_{1},l_{2},l_{3})\in\mathbb{R}^{3}|l_{1}+l_{2}>l_{3},l_{2}+l_{3}>l_{1},l_{3}+l_{1}>l_{2},l_{1}+l_{2}+l_{3}<2\pi\\}$
can be extended continuously by constant functions to a function
$\tilde{\theta_{i}}$ on $\mathbb{(}0,\pi)^{3}$.
We call the set $\Omega$ in the lemma the natural domain of the length
vectors.
###### Proof.
In the case (1), the extension function $\tilde{\theta_{i}}$ of $\theta_{i}$
is given by $\tilde{\theta_{i}}=\pi$ when $l_{i}\geq l_{j}+l_{k}$, and
$\tilde{\theta_{i}}=0$ when $l_{j}\geq l_{i}+l_{k}$. It remains to verify the
continuity of $\tilde{\theta_{i}}$ on $\mathbb{R}_{>0}^{3}$. It is based on
the cosine law. Given a point $L=(L_{1},L_{2},L_{3})$ in the boundary
$\bar{\Omega}-\Omega$ of $\Omega$ inside $\mathbb{R}^{3}_{>0}$, we may assume
without loss of generality that $L_{1}=L_{2}+L_{3}$. The continuity of
$\tilde{\theta_{i}}$ follows from
$\lim_{l\to L}\theta_{1}(l)=\pi,\quad\lim_{l\to L}\theta_{j}(l)=0,\quad
j=2,3.$
Indeed, the cosine law says, in the case of $\Delta\subset\mathbb{E}^{2}$,
that
(2.1) $\cos(\theta_{i})=\frac{l_{j}^{2}+l_{k}^{2}-l_{i}^{2}}{2l_{j}l_{k}}.$
One sees easily that when $l$ tends to $L$, then the right-hand-side of (2.1)
tends to $1$ if i=2,3 and $-1$ if $i=1$. This verifies the continuity in the
Euclidean case. In the hyperbolic case, the cosine law says
(2.2)
$\cos(\theta_{i})=\frac{\cosh(l_{j})\cosh(l_{k})-\cosh(l_{i})}{\sinh(l_{j})\sinh(l_{k})}.$
Thus one sees that as $l$ tends to $L=(L_{1},L_{2},L_{3})$ with $L_{j}>0$, the
right-hand-side of (2.2) tends to $1$ if $i=2,3$ and to $-1$ if $i=1$. Thus
$\tilde{\theta_{i}}$ is continuous.
To see (2), recall that the cosine law for spherical triangle says
(2.3)
$\cos(\theta_{i})=\frac{\cos(l_{i})-\cos(l_{j})\cos(l_{k})}{\sin(l_{j})\sin(l_{k})}.$
If $l$ tends to $L$ where $L_{1}=L_{2}+L_{3}$ with $L_{i}\in(0,\pi)$, then
$\lim_{l\to L}\cos(\theta_{1})=-1$ and $\lim_{l\to L}\cos(\theta_{i})=1$ when
$i=2,3$. On the other hand, if $L_{1}+L_{2}+L_{3}=2\pi$ for $L_{i}\in(0,\pi)$,
then the cosine law implies that $\lim_{l\to L}\cos(\theta_{i})=-1$ for all
$i$, i.e., all inner angles are $\pi$ in this case. Thus by setting the
extended function $\tilde{\theta_{i}}$ in $(0,\pi)^{3}$ to be
$\tilde{\theta_{i}}(l)=\pi$ if $l_{i}\geq l_{j}+l_{k}$,
$\tilde{\theta_{i}}(l)=0$ if $l_{j}\geq l_{i}+l_{k}$, and
$\tilde{\theta_{i}}(l)=\pi$ if $l_{i}+l_{j}+l_{k}\geq\pi$, (
$\\{i,j,k\\}=\\{1,2,3\\}$), we see that $\tilde{\theta_{i}}$ is continuous.
∎
### 2.2. Continuous extension of 1-forms and of locally convex functions
We establish some simple facts on extending closed 1-forms and locally convex
functions to convex functions in this subsection.
###### Definition 2.3.
A differential 1-form $w=\sum_{i=1}^{n}a_{i}(x)dx_{i}$ in an open set
$U\subset\mathbb{R}^{n}$ is said to be continuous if each $a_{i}(x)$ is a
continuous function on $U$. A continuous 1-form $w$ is called closed if
$\int_{\partial\tau}w=0$ for each Euclidean triangle $\tau$ in $U$.
By the standard approximation theory, if $w$ is closed and $\gamma$ is a
piecewise smooth null homologous loop in $U$, then $\int_{\gamma}w=0$. If $U$
is simply connected, then the integral $F(x)=\int_{a}^{x}w$ is well defined,
independent of the choice of piecewise smooth paths in $U$ from $a$ to $x$.
The function $F(x)$ is $C^{1}$-smooth so that $\partial F(x)/\partial
x_{i}=a_{i}(x)$.
###### Proposition 2.4.
Suppose $X$ is an open set in $\mathbb{R}^{n}$ and $A\subset X$ is an open
subset bounded by a smooth (n-1)-dimensional submanifold in $X$. If
$w=\sum_{i=1}^{n}a_{i}(x)dx_{i}$ is a continuous 1-form on $X$ so that
$w|_{A}$ and $w|_{X-\bar{A}}$ are closed where $\overline{A}$ is the closure
of $A$ in $X$, then $w$ is closed in X.
###### Proof.
Since closedness is a local property and is invariant under smooth change of
coordinates in $X$, we may assume that $X=\mathbb{R}^{n}$ and
$A=\\{(x_{1},...,x_{n})\in\mathbb{R}^{n}|x_{n}>0\\}$. Take a Euclidean
triangle $\tau\subset X$. To verify $\int_{\partial\tau}w=0$, we may assume
that $\tau$ is not in $\overline{A}$ or $X-A$ since otherwise
$\int_{\partial\tau}w=0$ follows from the assumption and the standard
approximation theorem. In the remaining case, $\tau$ intersects both $A$ and
$X-A$. The plane $x_{n}=0$ cuts the triangle $\tau$ into a triangle
$\gamma_{1}$ and a quadrilateral $\gamma_{2}$ so that $\gamma_{1}$ and
$\gamma_{2}$ are in the closure of $A$ and $X-A$. We can express, in the
singular chain level, $\partial\tau=\partial\gamma_{1}+\partial\gamma_{2}$. By
definition, $\int_{\partial\gamma_{i}}w=0$ for each $i$. Thus
$\int_{\partial\tau}w=\int_{\partial\gamma_{1}}w+\int_{\partial\gamma_{2}}w=0$.
∎
A real analytic codimension-1 submanifold $Y$ in an open set $X$ in
$\mathbb{R}^{n}$ is a smooth submanifold so that locally $Y$ is defined by
$k(x)=0$ for a non-constant real analytic function $k$. Note that if $L$ is a
(compact) line segment in $X$, then either $L\subset Y$ or $L\cap Y$ is a
finite set. This is due to the fact that a non-constant real analytic function
on an open interval has isolated zeros.
Recall that a function $f$ defined on a convex set $X\subset\mathbb{R}^{n}$ is
called convex if for all $p,q\in X$ and all $t\in[0,1]$, $tf(p)+(1-t)f(q)\geq
f(tp+(1-t)q)$. It is called strictly convex if for all $p\neq q$ in $X$ and
all $t\in(0,1)$, $tf(p)+(1-t)f(q)>f(tp+(1-t)q)$. A function $f$ defined in an
open set $U\subset\mathbb{R}^{n}$ is said to be locally convex (or locally
strictly convex ) if it is convex (or strictly convex) in a convex
neighborhood of each point.
###### Proposition 2.5.
Suppose $X\subset\mathbb{R}^{n}$ is an open convex set and $A\subset X$ is an
open subset of $X$ bounded by a codimension-1 real analytic submanifold in
$X$. If $w=\sum_{i=1}^{n}a_{i}(x)dx_{i}$ is a continuous closed 1-form on $X$
so that $F(x)=\int^{x}_{a}w$ is locally convex in $A$ and in $X-\overline{A}$,
then $F(x)$ is convex in $X$.
###### Proof.
Since $X$ is simply connected, the function $F$ is well defined. To verify
convexity, take $p,q\in X$ and consider $f(t)=F(tp+(1-t)q)$ for $t\in[0,1]$.
It suffices to show that $f(t)$ is convex in $t$. Since $F$ is $C^{1}$-smooth,
$f$ is $C^{1}$-smooth. Let $\partial A=\bar{A}-A$ and $L$ be the line segment
from $p$ to $q$. Since $\partial A$ is real analytic, either $L$ intersects
$\partial A$ in a finite set of points, or $L$ is in $\partial A$. In the
first case, let $0=t_{0}<t_{1}<...,t_{n}=1$ be the partition of $[0,1]$ so
that the line segment $tp+(1-t)q$ for $t\in(t_{i},t_{i+1})$ is either in $A$
or in $X-\overline{A}$. By definition, $f(t)$ is convex in $[t_{i},t_{i+1}]$,
i.e., $f^{\prime}(t)$ is increasing in $[t_{i},t_{i+1}]$ for $i=0,...,n-1$.
Since $f^{\prime}(t)$ is continuous in $[0,1]$, this implies that
$f^{\prime}(t)$ is increasing in $[0,1]$, i.e., $f(t)$ is convex in $[0,1]$.
In the second case that $L\subset\partial A$, we take two sequences of points
$p_{m}$ and $q_{m}$ converging to $p$ and $q$ respectively in $X$ so that
$p_{m},q_{m}$ are not in $\partial A$. Then by the case just proved, the
functions $f_{m}(t)=F(tp_{m}+(1-t)q_{m})$ are convex in $t$. Furthermore,
$f_{m}$ converges to $f$. Thus $f$ is convex. ∎
###### Corollary 2.6.
Suppose $X\subset\mathbb{R}^{n}$ is an open convex set and $A\subset X$ is an
open subset of $X$ bounded by a real analytic codimension-1 submanifold in
$X$. If $w=\sum_{i=1}^{n}a_{i}(x)dx_{i}$ is a continuous closed 1-form on $A$
so that $F(x)=\int^{x}_{a}w$ is locally convex on $A$ and each $a_{i}$ can be
extended continuously to $X$ by constant functions to a function
$\tilde{a_{i}}$ on $X$, then
$\tilde{F}(x)=\int^{x}_{a}\sum_{i=1}^{n}\tilde{a_{i}}dx_{i}$ is a
$C^{1}$-smooth convex function on $X$ extending $F$.
We remark that the real analytic assumption in the proposition 2.5 can be
relaxed to $C^{2}$ smooth.
## 3\. A Proof of Bowers-Stephenson’s Rigidity Conjecture
We begin by recalling Guo’s work on a variational principle associated to
inversive distance circle packings and then prove theorem 1.4. We will work in
Euclidean and hyperbolic geometries only.
### 3.1. Guo’s variational principle for inversive distance circle packing
Suppose $\Delta$ is a triangle with vertices $v_{1},v_{2},v_{3}$ and edges
$e_{ij}=v_{i}v_{j}$, $i\neq j$. Fix once and for all an inversive distance
$I_{ij}\in[0,\infty)$ at each edge $e_{ij}$. Then for each assignment of
positive number $r_{i}$ at $v_{i}$ for $i=1,2,3$, let
(3.1) $l_{k}=\sqrt{r_{i}^{2}+r_{j}^{2}+2r_{i}r_{j}I_{ij}}$
for Euclidean geometry and
(3.2)
$l_{k}=\cosh^{-1}(\cosh(r_{i})\cosh(r_{j})+I_{ij}\sinh(r_{i})\sinh(r_{j}))$
for hyperbolic geometry where $\\{i,j,k\\}=\\{1,2,3\\}$.
Let
$\Omega=\\{(x_{1},x_{2},x_{3})\in\mathbb{R}_{>0}^{3}|x_{i}+x_{j}>x_{k},\\{i,j,k\\}=\\{1,2,3\\}\\}$.
If $(l_{1},l_{2},l_{3})$ is in $\Omega$, then we construct a Euclidean
triangle $\Delta$ with length $l_{k}$ of $e_{ij}$ given by (3.1) and a
hyperbolic triangle, still denoted by $\Delta$, with length $l_{k}$ of
$e_{ij}$ given by (3.2). Suppose the angle of the triangle at $v_{i}$ is
$\theta_{i}$ and consider $\theta_{i}$ as a function of $(r_{1},r_{2},r_{3})$.
Guo proved the following theorem in [9].
###### Theorem 3.1.
(Guo [9]) Fix any $(I_{12},I_{23},I_{31})\in[0,\infty)^{3}$.
(1) For Euclidean triangles, let $u_{i}=\ln r_{i}$, then the differential
1-form $w=\sum_{i=1}^{3}\theta_{i}du_{i}$ is closed in the open subset of
$\mathbb{R}^{3}$ where it is defined. The integral $F(u)=\int_{0}^{u}w$ is a
locally concave function in $u=(u_{1},u_{2},u_{3})$ and is strictly locally
concave in $u_{1}+u_{2}+u_{3}=0$. Furthermore, if $c\in\mathbb{R}$ and $F(u)$
is defined, then $F(u+(c,c,c))=F(u)$.
(2) For hyperbolic triangles, let $u_{i}=\ln(\tanh(r_{i}/2))$, then the
differential 1-form $w=\sum_{i=1}^{3}\theta_{i}du_{i}$ is closed in the open
subset of $\mathbb{R}_{<0}^{3}$ where it is defined. Furthermore, the integral
$F(u)=\int_{-(1,1,1)}^{u}w$ is a strictly locally concave function in
$u=(u_{1},u_{2},u_{3})$.
It is also proved in [9] that the open sets where the 1-forms $w$ are defined
in theorem 3.1 are connected and simply connected. Theorem 3.1 is a
generalization of an earlier result obtained in [6]. Guo proved a local and
infinitesimal rigidity theorem for inversive distance circle packing using
theorem 3.1. It says that a Euclidean inversive distance circle packing is
locally determined, up to scaling, by the discrete curvature of the underlying
polyhedral surface. He also proved the local and infinitesimal rigidity for
hyperbolic inversive distance circle packings.
### 3.2. Concave extension of Guo’s action functional
Our main observation is that Guo’s differential 1-forms
$w=\sum_{i=1}^{3}\theta_{i}du_{i}$ can be extended to a closed 1-form on
$\mathbb{R}^{3}$ in the Euclidean case and on $\mathbb{R}^{3}_{<0}$ in the
hyperbolic case so that the integrations of the extended 1-forms are still
concave.
###### Proposition 3.2.
Let $w$ be the 1-forms defined in theorem 3.1.
(a) In the case of Euclidean triangles, the 1-form $w$ can be extended to a
continuous closed 1-form $\tilde{w}$ on $\mathbb{R}^{3}$ so that the
integration $\tilde{F}(u)=\int^{u}_{0}\tilde{w}$ is a $C^{1}$-smooth concave
function.
(b) In the case of hyperbolic triangles, the 1-form $w$ can be extended to a
continuous closed 1-form $\tilde{w}$ on $\mathbb{R}_{<0}^{3}$ so that the
integration $\tilde{F}(u)=\int^{u}_{-(1,1,1)}\tilde{w}$ is a $C^{1}$-smooth
concave function.
We begin by focusing the 1-forms in its radius coordinate
$r=(r_{1},r_{2},r_{3})\in\mathbb{R}_{>0}^{3}$. In this case, the 1-forms are
given by $w=\sum_{i=1}^{3}\theta_{i}\frac{dr_{i}}{r_{i}}$ and
$w=\sum_{i=1}^{3}\theta_{i}\frac{dr_{i}}{\sinh(r_{i})}$. The 1-form $w$ is
defined on the open set $U$ of $\mathbb{R}_{>0}^{3}$ where
(3.3)
$U=\\{(r_{1},r_{2},r_{3})\in\mathbb{R}_{>0}^{3}|l_{i}+l_{j}>l_{k},\\{i,j,k\\}=\\{1,2,3\\}\\},$
where $l_{i}=l_{i}(r_{1},r_{2},r_{3})$ is defined on $\mathbb{R}_{>0}^{3}$.
(Note that for hyperbolic and Euclidean geometries, the sets $U$ are different
due to (3.1) and (3.2)). The extension of the 1-form $w$ is the natural one.
Namely, we replace $\theta_{i}$ in $w$ by $\tilde{\theta_{i}}$ appeared in
lemma 2.1. Thus the extended 1-form is
$\tilde{w}=\sum_{i=1}^{3}\tilde{\theta_{i}}\frac{dr_{i}}{r_{i}}$ or
$\tilde{w}=\sum_{i=1}^{3}\tilde{\theta_{i}}\frac{dr_{i}}{\sinh(r_{i})}.$
It remains to show that $\tilde{w}$ is continuous and closed in
$\mathbb{R}_{>0}^{3}$ so that its pull back to the $u$-coordinate has a
concave integration. To this end, we prove,
###### Lemma 3.3.
Let $\bar{U}$ be the closure of $U$ in $\mathbb{R}_{>0}^{3}$. Then,
(1) $\theta_{i}$ is a constant function on each connected component of
$\bar{U}-U$, and
(2) for each connected component $V$ of $\mathbb{R}_{>0}^{3}-U$, the
intersection $V\cap\bar{U}$ is a connected component of $\bar{U}-U$.
###### Proof.
By (3.3), the boundary $\partial U=\bar{U}-U$ is given by
$\cup_{i=1}^{3}\partial_{i}U$ where
$\partial_{i}U=\\{(r_{1},r_{2},r_{3})\in\mathbb{R}_{>0}^{3}|l_{i}=l_{j}+l_{k},\\{j,k\\}=\\{1,2,3\\}-\\{i\\}\\}$.
Furthermore, $\mathbb{R}_{>0}^{3}-U=\cup_{i=1}^{3}V_{i}$ where
$V_{i}=\\{(r_{1},r_{2},r_{3})\in\mathbb{R}_{>0}^{3}|l_{i}\geq
l_{j}+l_{k},\\{j,k\\}=\\{1,2,3\\}-\\{i\\}\\}.$
First, we note that if $I_{ij}\leq 1$, then $\partial_{k}U=\emptyset$ and
$V_{k}=\emptyset$. Indeed, if $I_{ij}\leq 1$, then by (3.1) and (3.2),
$l_{k}\leq r_{i}+r_{j}.$
But due to $I_{ab}\geq 0$, (3.1) and (3.2), $r_{j}<l_{i}$ and $r_{i}<l_{j}$.
Therefore, $l_{k}<l_{i}+l_{j}$. This implies that $\partial_{k}U=\emptyset$
and $V_{k}=\emptyset$.
Next $\partial_{i}U\cap\partial_{j}U=\emptyset$ and $V_{i}\cap
V_{j}=\emptyset$ for $i\neq j$. Indeed, if
$r\in\partial_{i}U\cap\partial_{j}U$ or $r\in V_{i}\cap V_{j}$, then
$l_{i}\geq l_{j}+l_{k}$ and $l_{j}\geq l_{i}+l_{k}$. Thus $l_{k}=0$. But
$l_{k}>r_{i}>0$.
We claim that if $I_{ij}>1$, then both $V_{k}$ and $\partial_{k}U$ are non-
empty and connected. Assume the claim, then the lemma follows. Indeed, since
$l_{s}>0$ for all indices $s$, it follows, by lemma 2.1, that $\theta_{i}$ is
either $0$ or $\pi$ in $\partial_{s}U$, i.e., (1) holds. Next, $V_{s}$’s are
the connected components of $\mathbb{R}_{>0}^{3}-U$ so that
$V_{s}\cap\bar{U}=\partial_{s}U$. Thus (2) holds.
To see the claim, it suffices to show that there is a smooth function
$f(r_{i},r_{j})$ defined on $\mathbb{R}_{>0}^{3}$ so that its graph is
$\partial_{k}U$ and
$V_{k}=\\{(r_{1},r_{2},r_{3})\in\mathbb{R}_{>0}^{3}|0<r_{3}\leq
f(r_{1},r_{2})\\}$.
To this end, consider the equation
(3.4) $l_{k}=l_{i}+l_{j},$
and let the right-hand-side of (3.4) be $g(r_{k},r_{i},r_{j})$. We will deal
with the Euclidean and hyperbolic geometry separately.
CASE 1 Euclidean triangles. In this case, the function $g(r_{k},r_{i},r_{j})$
is given by
(3.5)
$g(r_{k},r_{i},r_{j})=\sqrt{r_{k}^{2}+r_{j}^{2}+2I_{kj}r_{k}r_{j}}+\sqrt{r_{i}^{2}+r_{k}^{2}+2I_{ik}r_{i}r_{k}}$
Evidently, for a fixed $(r_{i},r_{j})\in\mathbb{R}^{2}_{>0}$,
$g(r_{k},r_{i},r_{j})$ is a strictly increasing function of
$r_{k}\in\mathbb{R}_{>0}$ so that
$g(0,r_{i},r_{j})=r_{i}+r_{j}<\sqrt{r_{i}^{2}+r_{j}^{2}+2I_{ij}r_{i}r_{j}}$
(due to $I_{ij}>1$) and $\lim_{r_{k}\to\infty}g(r_{k},r_{i},r_{k})=\infty$. By
the mean-value theorem, there exists a unique positive number $f(r_{i},r_{j})$
so that
$g(f(r_{i},r_{j}),r_{i},r_{j})=\sqrt{r_{i}^{2}+r_{j}^{2}+2r_{i}r_{j}I_{ij}}=l_{k}$.
The smoothness of $f(r_{i},r_{j})$ follows from the implicit function theorem
applied to (3.4). Indeed,
$\frac{\partial g}{\partial
r_{k}}=\frac{r_{k}+2I_{kj}r_{j}}{l_{i}}+\frac{r_{k}+2I_{ik}r_{i}}{l_{j}}>0.$
Thus, $f(r_{i},r_{j})$ is smooth.
This shows $\partial_{k}U$ is the graph of the smooth function $f$ defined on
$\mathbb{R}_{>0}^{2}$, i.e.,
$\partial_{k}U=\\{(r_{1},r_{2},r_{3})\in\mathbb{R}_{>0}^{3}|r_{k}=f(r_{i},r_{j})\\}.$
Thus it is connected. Since $g(r_{k},r_{i},r_{j})$ is an increasing function
of $r_{k}$, $V_{k}=\\{r\in R_{>0}^{3}|0<r_{k}\leq
f(r_{i},r_{j}),\\{i,j\\}=\\{1,2,3\\}-\\{k\\}\\}$. Thus $V_{k}$ is connected.
CASE 2 hyperbolic triangles. By the same argument as in case 1, it suffices to
show the same properties established in case 1 hold for $g(r_{k},r_{i},r_{j})$
given by
(3.6)
$\cosh^{-1}(\cosh(r_{i})\cosh(r_{k})+I_{ik}\sinh(r_{i})\sinh(r_{k}))+\cosh^{-1}(\cosh(r_{k})\cosh(r_{j})+I_{kj}\sinh(r_{k})\sinh(r_{j})).$
Fix $(r_{i},r_{j})\in\mathbb{R}^{2}_{>0}$. Then the function
$g(r_{k},r_{i},r_{j})$ is clearly strictly increasing in
$r_{k}\in\mathbb{R}_{>0}$ so that
$\lim_{r_{k}\to\infty}g(r_{k},r_{i},r_{j})=\infty$ and due to $I_{ij}>1$,
$g(0,r_{i},r_{j})=r_{i}+r_{j}$ $=\cosh^{-1}(\cosh(r_{i}+r_{j}))$
$=\cosh^{-1}(\cosh(r_{i})\cosh(r_{j})+\sinh(r_{i})\sinh(r_{j}))$
$<\cosh^{-1}(\cosh(r_{i})\cosh(r_{j})+I_{ij}\sinh(r_{i})\sinh(r_{j}))=l_{k}.$
By the mean value theorem, there exists a unique positive number
$f(r_{i},r_{j})$ so that $g(f(r_{i},r_{j}),r_{i},r_{j})=l_{k}$. The smoothness
of $f(r_{i},r_{j})$ follows form the implicit function theorem that
$\frac{\partial g}{\partial
r_{k}}=\frac{\cosh(r_{i})\sinh(r_{k})+I_{ik}\sinh(r_{i})\cosh(r_{k})}{\sqrt{(\cosh(r_{i})\cosh(r_{k})+I_{ik}\sinh(r_{i})\sinh(r_{k}))^{2}-1}}$
$+\frac{\cosh(r_{j})\sinh(r_{k})+I_{jk}\sinh(r_{j})\cosh(r_{k})}{\sqrt{(\cosh(r_{j})\cosh(r_{k})+I_{jk}\sinh(r_{j})\sinh(r_{k}))^{2}-1}}$
$>0.$
By the same argument as in case 1, we see that $\partial_{k}U$, being the
graph of the smooth function $f$, is connected and $V_{k}$, being the region
below the positive function $f$ over $\mathbb{R}^{2}_{>0}$, is also connected.
∎
Now back to the proof of proposition 3.2, for part (1), consider the real
analytic diffeomorphism $u=u(r):\mathbb{R}^{3}_{>0}\to\mathbb{R}^{3}$ where
$u_{i}=\ln r_{i}$. The differential 1-form
$w=\sum_{i=1}^{3}\theta_{i}\frac{dr_{i}}{r_{i}}$ pulls back (via
$r=u^{-1}(r)$) to $w=\sum_{i=1}^{3}\theta_{i}du_{i}$ as appeared in theorem
3.1. By lemma 3.3, the extension
$\tilde{w}=\sum_{i=1}^{3}\tilde{\theta_{i}}du_{i}$ is obtained from $w$ by
extending each coefficient $\theta_{i}$ by constant functions on
$\mathbb{R}^{3}-u^{-1}(U)$. Thus, by corollary 2.6, the function
$\tilde{F}(u)=\int^{u}_{0}\tilde{w}$ is a $C^{1}$-smooth concave function in
$u\in\mathbb{R}^{3}$ so that
(3.7) $\partial\tilde{F}/\partial u_{i}=\tilde{\theta_{i}}.$
The same argument also works for part (2) since $u=u(r)$ with
$u_{i}=\ln\tanh(r_{i})$ is a real analytic diffeomorphism from
$\mathbb{R}_{>0}^{3}$ onto $\mathbb{R}_{<0}^{3}$.
### 3.3. A proof of theorem 1.4 for Euclidean inversive distance circle
packing
Suppose otherwise that there exist two inversive circle packing metrics
$d_{1},d_{2}$ on $(S,T)$ with the same inversive distance
$I\in\mathbb{[}0,\infty)^{E}$ so that their discrete curvatures are the same
and $d_{1}\neq\lambda d_{2}$ for any $\lambda$. Let $a\in\mathbb{R}^{V}$ be
their common discrete curvature.
We will use the notation that if $i\in V$ and $x\in\mathbb{R}^{V}$, then
$x_{i}=x(i)$ below. Let $T^{(2)}$ be the set of all triangles in $T$. If a
triangle $s\in T^{(2)}$ has vertices $i,j,k\in V$, then we denote the triangle
by $s=\\{i,j,k\\}$. For circle packing metrics of radii
$r\in\mathbb{R}_{>0}^{V}$ with a given inversive distance $I$, we use
$u\in\mathbb{R}^{V}$ to denote their logarithm coordinate where $u_{i}=\ln
r_{i}$. Thus, there are two points $p,q$ in $\mathbb{R}^{V}$ as the
logarithmic coordinates of $d_{1}$ and $d_{2}$ so that their discrete
curvatures are $a\in\mathbb{R}^{V}$ and $p-q\neq\lambda(1,1,1,..,1)$ for any
$\lambda$.
We will derive a contradiction by using the locally concave functions $F$ and
its concave extension $\tilde{F}=\int^{u}_{0}\tilde{w}$ appeared in
proposition 3.2 associated to theorem 3.1(1).
Define a $C^{1}$-smooth function $W:\mathbb{R}^{V}\to\mathbb{R}$ by
(3.8) $W(u)=-\sum_{s\in T^{(2)},s=\\{i,j,k\\},i,j,k\in
V}\tilde{F}(u_{i},u_{j},u_{k})+\sum_{i\in V}(2\pi-a_{i})u_{i}.$
The function $W$ is convex since it is a summation of convex functions.
Furthermore, by the definition of $W$, (3.7), the definition of discrete
curvature $(a_{i})$, $p$ and $q$ are both critical points of $W$. Since $W$ is
convex in $\mathbb{R}^{V}$, $p$ and $q$ are both minimal points of $W$.
Furthermore, for all $t\in[0,1]$, $tp+(1-t)q$ are minimal points of $W$. In
particular,
$W(tp+(1-t)q)=W(p)$
for all $t\in[0,1]$. Since
$W(tp+(1-t)q)=\sum_{s\in T^{(2)},s=\\{i,j,k\\},i,j,k\in
V}f_{ijk}(t)+\sum_{i\in E}(2\pi-a_{i})(tp_{i}+(1-t)q_{i})$
where the function
(3.9)
$f_{ijk}(t)=-\tilde{F}(tp_{i}+(1-t)q_{i},tp_{j}+(1-t)q_{j},tp_{k}+(1-t)q_{k})$
is convex, it follows that $f_{ijk}(t)$ is linear in $t\in[0,1]$ for all
triangle $s$ with vertices $i,j,k$. This is due to the simple fact that a
summation of a convex function with a strictly convex function is strictly
convex. By the assumption that $p-q\neq c(1,1,....,1)$ in $\mathbb{R}^{V}$ and
the surface is connected, there exists a triangle $s$ with vertices $i,j,k\in
V$ so that $(p_{i},p_{j},p_{k})-(q_{i},q_{j},q_{k})\neq(c,c,c)$ for all
$c\in\mathbb{R}$. By the given assumption, $(p_{i},p_{j},p_{k})$ and
$(q_{i},q_{j},q_{k})$ are in the domain of definition of $w$ in theorem 3.1.
Thus for $t\in[0,1]$ close to $0$ or $1$, by theorem 3.1 on the local strictly
convexity of $-F(u_{1},u_{2},u_{3})$ on $u_{1}+u_{2}+u_{3}=0$ and
$F(u+(c,c,c))=F(u)$, $f_{ijk}(t)$ is strictly convex in $t$ near $0,1$. This
is a contradiction to the linearity of $f_{ijk}(t)$.
### 3.4. A proof of theorem 1.4 for hyperbolic inversive distance circle
packing
The proof is essentially the same as in §3.3 and is simpler. For any
$r\in\mathbb{R}_{>0}^{V}$, define $u=u(r)\in\mathbb{R}_{<0}^{V}$ by
$u_{i}=\ln\tanh(r_{i}/2))$. For a circle packing with radii
$r\in\mathbb{R}_{>0}^{V}$, let $u=u(r)$ and call it the $u$-coordinate of the
circle packing metric.
We use the same notation as in §3.3. Suppose the result does not hold and let
$p\neq q\in\mathbb{R}_{<0}^{V}$ be the $u$-coordinates of the two distinct
hyperbolic circle packing metrics having the same hyperbolic inversive
distance $I\in\mathbb{R}_{\geq 0}^{E}$ and the same discrete curvature
$a=(a_{i})\in\mathbb{R}^{V}$. Define the action functional $W$ on
$\mathbb{R}_{<0}^{V}$ by the same formula (3.8) where $\tilde{F}$ is the
concave function in proposition 3.2 associated to theorem 3.1(2). Then the
same proof goes through as in §3.3 since in this case, one of $f_{ijk}(t)$ is
strictly convex for $t$ near 0 and 1.
## 4\. 2-dimensional Schlaefli Type Action Functionals and Their Extensions
The following was proved in [12]. The proof is a straight forward calculation.
###### Theorem 4.1.
Suppose $\Delta$ is a triangle in the Euclidean plane $\mathbb{E}^{2}$, or the
hyperbolic plane $\mathbb{H}^{2}$, or the 2-sphere $\mathbb{S}^{2}$ so that
its edge lengths are $l_{1},l_{2},l_{3}$ and its inner angles are
$\theta_{1},\theta_{2},\theta_{3}$ where $\theta_{i}$’s angle is opposite to
the edge of length $l_{i}$. Let $h\in\mathbb{R}$ and let $\Omega$ be the
natural domain for length vectors appeared in lemma 2.2.
1. (1)
For a Euclidean triangle,
$w_{h}=\sum_{i=1}^{3}\frac{\int^{\theta_{i}}_{\pi/2}\sin^{h}(t)dt}{l_{i}^{h+1}}dl_{i}$
is a closed 1-form on $\Omega$. The integral $\int^{u}_{-(h,h,h)}w_{h}$ is
locally convex in variable $u=(u_{1},u_{2},u_{3})$ where $u_{i}=\ln l_{i}$ for
$h=0$ and $u_{i}=-\frac{l_{i}^{-h}}{h}$ for $h\neq 0$. Furthermore,
$\int^{u}_{-(h,h,h)}w_{h}$ is locally strictly convex in hypersurface
$u_{1}+u_{2}+u_{3}=0$.
2. (2)
For a spherical triangle,
$w_{h}=\sum_{i=1}^{3}\frac{\int^{\theta_{i}}_{\pi/2}\sin^{h}(t)dt}{\sin^{h+1}(l_{i})}dl_{i}$
is a closed 1-form on $\Omega$. The integral $\int^{u}_{0}w_{h}$ is locally
strictly convex in $u=(u_{1},u_{2},u_{3})$ where $u_{i}=\int^{l_{i}}_{\pi/2}$
$\sin^{-h-1}(t)dt$.
3. (3)
For a hyperbolic triangle,
$w_{h}=\sum_{i=1}^{3}\frac{\int^{\theta_{i}}_{\pi/2}\sin^{h}(t)dt}{\sinh^{h+1}(l_{i})}dl_{i}$
is a closed 1-form.
4. (4)
For a hyperbolic triangle,
$w_{h}=\sum_{i=1}^{3}\frac{\int^{\frac{1}{2}(\theta_{i}-\theta_{j}-\theta_{k})}_{0}\cos^{h}(t)dt}{\coth^{h+1}(l_{i}/2)}dl_{i}$
is a closed 1-form. The integral $\int^{u}_{0}w_{h}$ is locally strictly
convex in $u=(u_{1},u_{2},u_{3})$ where $u_{i}=\int^{l_{i}}_{1}$
$\coth^{-h-1}(t/2)dt$.
### 4.1.
Recall that the natural domain $\Omega$ of the edge length vectors is given by
$\Omega=\\{(l_{1},l_{2},l_{3})\in\mathbb{R}_{>0}^{3}|l_{i}+l_{j}>l_{k},\\{i,j,k\\}=\\{1,2,3\\}\\}$
for Euclidean and hyperbolic triangles and
$\Omega=\\{(l_{1},l_{2},l_{3})\in\mathbb{R}_{>0}^{3}|l_{i}+l_{j}>l_{k},l_{1}+l_{2}+l_{3}<2\pi,\\{i,j,k\\}=\\{1,2,3\\}\\}$.
Let $J$ be the natural interval for each individual length $l_{i}$, i.e.,
$J=\mathbb{R}_{>0}$ for Euclidean and hyperbolic triangles and $J=(0,\pi)$ for
spherical triangles. In each case of theorem 4.1, there exists a real analytic
diffeomorphism $g:J\to g(J)$ from $J$ onto the open interval $g(J)$ so that
$u_{i}=g(l_{i})$. To be more precise, $g(t)=\ln t$ in the case of $h=0$ of
theorem 4.1(1), $g(t)=-\frac{t^{-h}}{h}$ ($h\neq 0$) in the case of $h\neq 0$
in theorem 4.1(1), $g(t)=\int^{t}_{\pi/2}\sin^{-h-1}(x)dx$ in the case (2) of
theorem 4.1, $g(t)=\int^{t}_{1}\sinh^{-h-1}(x)dx$ in the case (3) of theorem
4.1 and $g(t)=\int^{t}_{1}\coth^{-h-1}(x)dx$ in the case of (4). The real
analytic diffeomorphism $u(l_{1},l_{2},l_{3})=(u_{1},u_{2},u_{3})$ where
$u_{i}=g(l_{i})$ sends $J^{3}$ onto the open cube $g(J)^{3}$ in
$\mathbb{R}^{3}$.
By lemma 2.2, each of the angle function $\theta_{i}(l):\Omega\to\mathbb{R}$
can be extended by constant functions to a continuous function
$\tilde{\theta_{i}}(l):J^{3}\to\mathbb{R}$. Define a continuous 1-form
$\tilde{w_{h}}$ on $J^{3}$ by replacing $\theta_{i}$ in the definition of
$w_{h}$ in theorem 4.1 by $\tilde{\theta_{i}}$.
###### Lemma 4.2.
The continuous differential 1-form $\tilde{w_{h}}$ is closed in $J^{3}$.
###### Proof.
By proposition 2.4 where we take $X=J^{3}$ and $A=\Omega$, it suffices to show
that $\tilde{w_{h}}$ is closed in each connected component $U$ of
$J^{3}-\overline{\Omega}$. By theorem 4.1 $\tilde{w}|_{A}$ is closed, the
restriction of $\tilde{w_{h}}$ to $U$ is of the form
$\sum_{i=1}^{3}c_{i}du_{i}$ where $u_{i}=g(l_{i})$ and $c_{i}$ is a constant.
Thus $\tilde{w_{h}}|_{U}$ is closed. ∎
###### Proposition 4.3.
The pull back 1-form $(u^{-1})^{*}(\tilde{w_{h}})$ on $g(J)^{3}$ is a closed
1-form. Furthermore, if $F(u)=\int^{u}w_{h}$ is locally convex in $u(\Omega)$
(i.e, in the case (1),(2), (4) of theorem 4.1), then
$\tilde{F}(u)=\int^{u}(u^{-1})^{*}(\tilde{w_{h}})$ is convex in $u$ in
$g(J)^{3}$.
Note that by the construction, if $u\in u(\Omega)$ and
$w_{h}=\sum_{i=1}^{3}\alpha_{i,h}(u)du_{i}$ (as shown in theorem 4.1) then
(4.1) $\frac{\partial\tilde{F}(u)}{\partial u_{i}}=\alpha_{i,h}(u).$
Furthermore, by definition, the $\phi_{h}$ and $\psi_{h}$ curvatures are sum
of two of $a_{i,h}(u)$’s.
###### Proof.
By corollary 2.6 where we take $X=g(J)^{3}$ and $A=u(\Omega)$, it suffices to
show that $u(\Omega)$ is bounded by a real analytic surface in $X$ and
$\tilde{F}(u)$ is convex in $u(\Omega)$ and in each component of
$g(J)^{3}-\overline{u(\Omega)}$.
Since $\Omega$ in $J^{3}$ is bounded by hyperplanes and
$u(l)=(g(l_{1}),g(l_{2}),g(l_{3}))$ is a real analytic diffeomorphism, it
follows that $u(\Omega)$ is bounded by a real analytic surface in $g(J)^{3}$.
By the assumption $\tilde{F}(u)$ is convex in $u(\Omega)$. If $U$ is a
connected component of $g(J)^{3}-\overline{u(\Omega)}$, then $\tilde{F}(u)$ is
linear on $U$ since its partial derivatives are constants on $U$ by the
construction. Thus by corollary 2.6, the result follows. ∎
## 5\. A Proof of Theorem 1.2
The argument is essentially the same as that in §3.3. Recall that $E$ is the
set of all edges in the triangulated surface $(S,T)$. If $x\in\mathbb{R}^{E}$
and $i\in E$, we use $x_{i}$ to denote $x(i)$. If $s\in T^{(2)}$ is a triangle
with edges $i,j,k\in E$, we denote it by $s=\\{i,j,k\\}$.
### 5.1. A proof of theorem 1.2(3)
Suppose otherwise that there exist two distinct hyperbolic polyhedral metrics
on $(S,T)$ so that their $\psi_{h}$ curvatures are the same. Let
$a=(a_{i})\in\mathbb{R}^{E}$ be their common $\psi_{h}$ curvature.
Recall that a polyhedral metric on $(S,T)$ is given by its edge length map
$l:E\to\mathbb{R}_{>0}$. In using the variational principle in theorem 4.1(4),
the natural variable is given by $u:E\to\mathbb{R}$ where $u(e)=g(l(e))$ with
$g(t)=\int^{t}_{1}\coth^{h+1}(s/2)ds$. We call it the $u$-coordinate of the
polyhedral metric $l$ and we will use the $u$-coordinate to set up the
variational principle. Therefore there are two distinct points (as
$u$-coordinates) $p\neq q\in g(\mathbb{R}_{>0})^{E}$ so that their
corresponding $\psi_{h}$ curvatures are the same $a\in\mathbb{R}^{E}$. We will
derive a contradiction by using the locally strictly convex functions $F$ and
its convex extension $\tilde{F}$ introduced in proposition 4.3 (associated to
theorem 4.1(4)).
Define a $C^{1}$-smooth function $W:g(\mathbb{R}_{>0})^{E}\to\mathbb{R}$ by
$W(u)=\sum_{s\in T^{(2)},s=\\{i,j,k\\},i,j,k\in
E}\tilde{F}(u_{i},u_{j},u_{k})-\sum_{i\in E}a_{i}u_{i}.$
The function $W$ is convex since it is a summation of convex functions.
Furthermore, by the definition of $W$, (4.1), the definition of $\psi_{h}$ and
$(a_{i})$, $p$ and $q$ are both critical points of $W$. Since $W$ is convex,
$p$ and $q$ are both minimal points of $W$. Furthermore, for all $t\in[0,1]$,
$tp+(1-t)q$ are minimal points of $W$. In particular,
$W(tp+(1-t)q)=W(p)$
for all $t\in[0,1]$. Since
$W(tp+(1-t)q)=\sum_{i,j,k\in E,\\{i,j,k\\}\in T^{(2)}}f_{ijk}(t)-\sum_{i\in
E}a_{i}(tp_{i}+(1-t)q_{i})$
where the function
(5.1)
$f_{ijk}(t)=\tilde{F}(tp_{i}+(1-t)q_{i},tp_{j}+(1-t)q_{j},tp_{k}+(1-t)q_{k})$
is convex, it follows that $f_{ijk}(t)$ is linear in $t\in[0,1]$. Since $p\neq
q$, there exists a triangle with edges $i,j,k\in E$ so that
$(p_{i},p_{j},p_{k})\neq(q_{i},q_{j},q_{k})$. Thus for $t\in[0,1]$ close to
$0$ or $1$, by theorem 4.1 on the local strictly convexity, $f_{ijk}(t)$ is
strictly convex in $t$ near $0,1$. This is a contradiction to the linearity of
$f_{ijk}(t)$.
### 5.2. A proof of theorem 1.2(2)
The proof is exactly the same as above using the extended convex function
$\tilde{F}$ in proposition 4.3 associated to theorem 4.1(2).
### 5.3. A proof of theorem 1.2(1)
The proof is the same as that in §5.1 using the similarly defined function
$W$. To be more precise, let $g(t)=-\frac{t^{-h}}{h}$ for $h\neq 0$ and
$g(t)=\ln t$. By the same set up as in §5.1, we conclude that $f_{ijk}(t)$
given by (5.1) is linear in $t$. We claim this implies that the two Euclidean
polyhedral metrics $u^{-1}(p)$ and $u^{-1}(q)$ differ by a scalar
multiplication. There are two cases to be discussed depending on $h=$ or
$h\neq 0$.
CASE 1. $h=0$. In this case, $p-q\neq c(1,1,...,1)$ in $\mathbb{R}^{E}$ for
any constant $c$. By the connectivity of the surface $S$, there exists a
triangle with edges $i,j,k\in E$ so that
$(p_{i},p_{j},p_{k})-(q_{i},q_{j},q_{k})\neq(c,c,c)$ for any constant $c$. On
the other hand, by theorem 4.1(1), the action functional $F$ is strictly
locally convex in the hyperplane $u_{1}+u_{2}+u_{3}=0$ and $F(u+(c,c,c)=F(u)$
for a scalar $c$ and $u\in u(\Omega)$. In particular, this implies that the
function $f_{ijk}(t)$ is strictly convex in $t\in[0,1]$ for $t$ close to $0$
or $1$. This contradicts the linearity of the function $f_{ijk}(t)$.
CASE 2. $h\neq 0$. In this case, $p\neq cq$ for any constant $c$. In
particular, there exists a triangle with three edges $i,j,k\in E$ so that
$(p_{i},p_{j},p_{k})\neq c(q_{i},q_{j},q_{k})$ for any $c\in\mathbb{R}$. By
theorem 4.1(1), the function $f_{ijk}(t)$ is strictly convex in $t\in[0,1]$
for $t$ close to $0$ or $1$. This contradicts the linearity of the function
$f_{ijk}(t)$.
Thus the two polyhedral metrics differ by a scaling.
## References
* [1] Andreev, E. M., Convex polyhedra in Lobachevsky spaces. (Russian) Mat. Sb. (N.S.) 81 (123) 1970 445–478.
* [2] Bowers, Philip L.; Stephenson, Kenneth, Uniformizing dessins and Belyi (maps via circle packing). Mem. Amer. Math. Soc. 170 (2004), no. 805, xii+97 pp.
* [3] Bowers, Philip L.; Hurdal, Monica K. Planar conformal mappings of piecewise flat surfaces. Visualization and mathematics III, 3 34, Math. Vis., Springer, Berlin, 2003,
* [4] Bobenko, Alexander; Pinkall, Ulrich; Springborn Boris, Discrete conformal maps and ideal hyperbolic polyhedra, http://front.math.ucdavis.edu/1005.2698
* [5] Cohn, Henry; Kenyon, Richard; Propp, James, A variational principle for domino tiling. J. Amer. Math. Soc. 14 (2001), no. 2, 297–346.
* [6] Chow, Bennett; Luo, Feng, Combinatorial Ricci flows on surfaces. J. Differential Geom. 63 (2003), no. 1, 97–129.
* [7] Colin de Verdiere, Yves, Un principe variationnel pour les empilements de cercles. Invent. Math. 104 (1991), no. 3, 655–669.
* [8] Glickenstein, David, Discrete conformal variations and scalar curvature on piecewise flat two and three dimensional manifolds, to appear in J. of Diff. Geom., http://front.math.ucdavis.edu/0906.1560.
* [9] Guo, Ren, Local rigidity of inversive distance circle packing, to appear in Trans AMS, http://front.math.ucdavis.edu/0903.1401.
* [10] Guo, Ren; Luo, Feng, Rigidity of polyhedral surfaces, II, Geom. Topol. 13 (2009), no. 3, 1265 1312
* [11] Leibon, Gregory, Characterizing the Delaunay decompositions of compact hyperbolic surfaces. Geom. Topol. 6 (2002), 361–391.
* [12] Luo, Feng, rigidity of polyhedral surfaces, to appear in J. Differential Geom. http://front.math.ucdavis.edu/0612.5714.
* [13] Luo, Feng, On Teichmüller spaces of Surfaces with boundary, Duke Journal of Math, 139, no. 3 (2007), 463-482.
* [14] Luo, Feng, A characterization of spherical polyhedral surfaces. J. Differential Geom. 74 (2006), no. 3, 407 424.
* [15] Rivin, Igor, Euclidean structures on simplicial surfaces and hyperbolic volume. Ann. of Math. (2) 139 (1994), no. 3, 553–580.
* [16] Stephenson, Kenneth, Introduction to circle packing. The theory of discrete analytic functions. Cambridge University Press, Cambridge, 2005.
* [17] Thurston, William, Geometry and topology of 3-manifolds, lecture notes, Math Dept., Princeton University, 1978, at www.msri.org/publications/books/gt3m/
|
arxiv-papers
| 2010-10-15T21:23:22 |
2024-09-04T02:49:13.977345
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Feng Luo",
"submitter": "Luo",
"url": "https://arxiv.org/abs/1010.3284"
}
|
1010.3289
|
# Experimental Evidence for a Dynamical Non-Locality Induced Effect in Quantum
Interference Using Weak Values
S. E. Spence and A. D. Parks Quantum Processing Group, Electromagnetic and
Sensor Systems Department, Naval Surface Warfare Center, Dahlgren, VA 22448
USA
(Experimental / Draft Completion……January 19, / March 12, 2010 )
###### Abstract
The quantum theoretical concepts of modular momentum and dynamical non-
locality, which were introduced four decades ago, have recently been used to
explain single particle quantum interference phenomena. Although the non-local
exchange of modular momentum associated with such phenomena cannot be directly
observed, it has been suggested that effects induced by this exchange can be
measured experimentally using weak measurements of pre- and post-selected
ensembles of particles. This paper reports on such an optical experiment that
yielded measured weak values that were consistent with the theoretical
prediction of an effect induced by a non-local exchange of modular momentum.
quantum interference, quantum non-locality, weak values, weak measurement,
two-slit experiment, twin Mach-Zehnder interferometer
###### pacs:
03.65-w, 03.65.Ta, 03.65.Ud, 07.05.Fb, 07.60.Ly
LABEL:FirstPage1 LABEL:LastPage#1102
## I Introduction
Because it differs fundamentally from the interference phenomena of classical
physics, quantum interference has remained a continuing topic for discussion
and debate since quantum theory’s early days. The essence of this difference
is exhibited by the two-slit experiment. From both the classical and
Schrödinger wave perspectives, the two slit interference pattern is easily
described in terms of the overlapping contributions of the wave which have
passed through each slit. The wave perspective also explains the disappearance
of the interference pattern when one of the slits is closed.
However, interference experiments using low intensity electron or photon beams
in which only one particle at a time passes through a two-slit apparatus have
shown that the accumulated effect when both slits are open is an interference
pattern like that produced by higher intensity ensembles and that the pattern
likewise disappears when one slit is closed, e.g. Ton . This peculiar behavior
necessitates an answer to the question ”how does a particle passing through
one slit sense that the other slit is open or closed?” when interference is
considered from the perspective of a single quantum particle.
Although this question concerning single particle behaviour has been answered
and explained theoretically in terms of a non-local exchange of modular
momentum APP1 ; APP2 , there have been no direct experimental observations of
such an exchange to support this explanation. This lack of observations is due
to the fact that the conditions required to observe a non-local exchange of
modular momentum are precisely those that make the associated modular variable
completely uncertain and unobservable. Recently, however, it was suggested
that an experimental methodology using weak measurements performed on a pre-
and post-selected ensemble of particles could be exploited in order to observe
an effect _induced_ by a non-local exchange of modular momentum. This
methodology was illustrated by a gedanken experiment which used a twin Mach-
Zehnder interferometer to duplicate relevant aspects of the two-slit
interference experiment TACKN .
This paper reports the results of an optical twin Mach-Zehnder interferometer
experiment similar to that described in the above gedanken experiment. This
experiment yielded measured weak values that were consistent with the
associated theoretical prediction describing the effect induced by a non-local
exchange of modular momentum. The remainder of this paper is organized as
follows: in the next section the theories of modular momentum, dynamical non-
locality, weak measurements, and weak values are briefly summarized. A
description of the experimental apparatus and an overview of the experiment
are presented in section III. The experimental results are discussed in
section IV. Concluding remarks comprise the final section of this paper.
## II Summary of the Theories
### II.1 Modular Momentum and Dynamical Non-locality
Consider a quantum particle propagating in the positive $y$-direction
perpendicular to the plane of two symmetric slits which are separated by a
distance $\ell$ in the $x$-direction (the slit at $x-\ell$ will be referred to
as the left slit). At time $t$ after the particle passes through the slits its
wavefunction is the superposition
$\psi\left(x,y,z,t\right)=\frac{1}{\sqrt{2}}\left\\{\varphi\left(x-\ell,y,z,t\right)+e^{i\alpha}\varphi\left(x,y,z,t\right)\right\\},$
(1)
where the $\varphi$’s are assumed to be identical ”wave packets” which do not
overlap at $t=0$ and $\alpha$ is their relative phase. Although information
about $\alpha$ can be obtained from the spatial interference pattern
$\left|\psi\left(x,d,z,\tau\right)\right|^{2}$ produced by an ensemble of such
particles on a screen parallel to and at an appropriate distance $d$ from the
plane of the slits at time $\tau>0$, there are no local measurements using
operators of the form $\widehat{x}^{j}\widehat{p}_{x}^{k}$, where $j$ and $k$
are non-negative integers, that can be performed upon the initial non-
overlapping wave packets that will determine $\alpha$. The relative phase
$\alpha$ is thus a non-local feature of quantum mechanics.
The induced momentum uncertainty and the Heisenberg uncertainty principle are
traditionally used to explain the loss of the interference pattern when one
slit is closed. However, measuring which slit the particle passes through does
not necessarily increase the momentum uncertainty. This - along with the fact
that position and momentum observables and their moments are not sensitive to
relative phase (prior to wave packet overlap) - suggests that these
observables, as well as the Heisenberg uncertainty principle, are not the
appropriate physical concepts for describing quantum interference phenomena.
The (modular) operator $e^{-\frac{i}{\hbar}\widehat{p}_{x}\ell}$ and its
modular property, however, do provide an alternative physical basis for the
rational description of quantum interference. Unlike the operators
$\widehat{x}^{j}\widehat{p}_{x}^{k}$, the expectation value of the operator
$e^{-\frac{i}{\hbar}\widehat{p}_{x}\ell}$ with respect to
$\psi\left(x,y,z,t\right)$ is sensitive to $\alpha$ \- even when the two
wavepackets don’t overlap. This sensitivity results from the action of
$e^{-\frac{i}{\hbar}\widehat{p}_{x}\ell}$ upon $\varphi\left(x,y,z,t\right)$
which overlaps the two wavepackets in eq. (1) by translating
$\varphi\left(x,y,z,t\right)$ to $\varphi\left(x-\ell,y,z,t\right)$. Also,
since $e^{-\frac{i}{\hbar}\widehat{p}_{x}\ell}$ is invariant under the
replacement $\widehat{p}_{x}\rightarrow\widehat{p}_{x}-n\frac{h}{\ell}$,
$n=0,\pm 1,\pm 2,\cdots$, (because
$e^{-\frac{i}{\hbar}\left(-n\frac{h}{\ell}\right)\ell}=e^{2in\pi}=1$), it
depends upon values of the _modular momentum_
$p_{x}\operatorname{mod}\left(\frac{nh}{\ell}\right)\equiv
p_{x,\operatorname{mod}}\in I\equiv[0,\frac{h}{\ell})$ instead of those of
$p_{x}$. This modular property establishes a fundamental relationship between
modular momentum uncertainty and quantum interference via the complete
uncertainty principle: ”$\widehat{p}_{x,\operatorname{mod}}$ is completely
uncertain (i.e. all its values are uniformly distributed over $I$) if and only
if $\left\langle
e^{-\frac{i}{\hbar}n\widehat{p}_{x}\frac{\ell}{h}}\right\rangle=0$ for every
positive integer $n$”. When this principle is applied to the two slit case, it
is found that while the required expectation value with respect to
$\psi\left(x,y,z,t\right)$ does not vanish for $n=1$, it does vanish for every
$n$ when the expectation value is with respect to
$\varphi\left(x,y,z,t\right)$. Thus, when the left slit is closed, i.e. it is
known that the particle passed through the right slit, then
$\widehat{p}_{x,\operatorname{mod}}$ becomes completely uncertain so that all
knowledge about $p_{x,\operatorname{mod}}$ is lost and the interference
pattern vanishes.
The Heisenberg equation of motion provides the formalism for describing and
understanding the notion of _dynamical non-locality_. Within the context of
two slits, the Heisenberg equation of motion for
$e^{-\frac{i}{\hbar}\widehat{p}_{x}\ell}$ is given by
$\frac{d}{dt}e^{-\frac{i}{\hbar}\widehat{p}_{x}\ell}=\frac{i}{\hbar}\left[\widehat{H},e^{-\frac{i}{\hbar}\widehat{p}_{x}\ell}\right]=\frac{i}{\hbar}\left(\widehat{V}\left(x\right)-\widehat{V}\left(x-\ell\right)\right)e^{-\frac{i}{\hbar}\widehat{p}_{x}\ell},$
(2)
where $\widehat{H}=\frac{\widehat{p}_{x}^{2}}{2m}+\widehat{V}\left(x\right)$
is the system Hamiltonian and $\widehat{V}\left(x\right)$ (
$\widehat{V}\left(x-\ell\right)$ ) is the potential operator for the right
(left) slit. This is a non-local equation of motion and therefore has no
classical analogue: only the potential at each slit is involved in the rate of
change of $e^{-\frac{i}{\hbar}\widehat{p}_{x}\ell}$ \- i.e. there are no
forces involved \- and the potential at the left slit influences this rate of
change even if $\psi\left(x,y,z,0\right)=\varphi\left(x,y,z,0\right)$ \- i.e.
when the particle is initially localized at the right slit. Consequently, the
effect of closing the left slit produces non-locally a change in modular
momentum while leaving the expectation values of the associated moments of
momentum unchanged. More specifically, the modular operator is conserved when
both slits are open since
$\widehat{V}\left(x\right)=\widehat{V}\left(x-\ell\right)$ so that
$\frac{d}{dt}e^{-\frac{i}{\hbar}\widehat{p}_{x}\ell}=0$. However, if the left
slit is closed and the particle is localized at the right slit, then
$\left(\widehat{V}\left(x\right)-\widehat{V}\left(x-\ell\right)\right)\neq 0$
$\neq\frac{d}{dt}e^{-\frac{i}{\hbar}\widehat{p}_{x}\ell}$ and the modular
momentum is changed non-locally as a result of the change in the potential at
the left slit. Interference is destroyed and the modular momentum becomes
completely uncertain - thereby rendering it unobservable. In fact, it is in
this manner that the complete uncertainty principle also reconciles dynamical
non-locality with causality.
For additional details concerning the theory of modular momentum and dynamical
non-locality the reader is invited to consult references APP1 ; APP2 ; TACKN ;
AR .
### II.2 Weak Measurements and Weak Values
Although the exchange of modular momentum is not directly observable, it has
been suggested that dynamical non-locality induces effects which can be
observed using weak measurements of pre- and post-selected ensembles of
particles. Weak measurements arise in the von Neumann description of a quantum
measurement at time $t_{0}$ of a time-independent observable $\widehat{A}$
that describes a quantum system in an initial fixed pre-selected state
$\left|\psi_{i}\right\rangle=\sum_{J}c_{j}\left|a_{j}\right\rangle$ at
$t_{0}$, where the set $J$ indexes the eigenstates $\left|a_{j}\right\rangle$
of $\widehat{A}$. In this description the Hamiltonian for the interaction
between the measurement apparatus and the quantum system is
$\widehat{H}=\gamma(t)\widehat{A}\widehat{p}.$
Here $\gamma\left(t\right)=\gamma\delta\left(t-t_{0}\right)$ defines the
strength of the impulsive measurement interaction at $t_{0}$ and $\widehat{p}$
is the momentum operator for the pointer of the measurement apparatus which is
in the initial state $\left|\phi\right\rangle$. Let $\widehat{q}$ be the
pointer’s position operator that is conjugate to $\widehat{p}$ and assume that
$\left\langle q\right|\left.\phi\right\rangle\equiv\phi\left(q\right)$ is real
valued with $\left\langle
q\right\rangle\equiv\left\langle\phi\right|\widehat{q}\left|\phi\right\rangle=0$.
Prior to the measurement the pre-selected system and the pointer are in the
tensor product state $\left|\psi_{i}\right\rangle\left|\phi\right\rangle$.
Immediately following the measurement the combined system is in the state
$\left|\Phi\right\rangle=e^{-\frac{i}{\hbar}\int\widehat{H}dt}\left|\psi_{i}\right\rangle\left|\phi\right\rangle={\displaystyle\sum\nolimits_{J}}c_{j}e^{-\frac{i}{\hbar}\gamma
a_{j}\widehat{p}}\left|a_{j}\right\rangle\left|\phi\right\rangle,$
where use has been made of the fact that
$\int\widehat{H}dt=\gamma\widehat{A}\widehat{p}$. The exponential factor in
this equation is the translation operator $\widehat{S}\left(\gamma
a_{j}\right)$ for $\left|\phi\right\rangle$ in its $q$-representation. It is
defined by the action $\left\langle q\right|\widehat{S}\left(\gamma
a_{j}\right)\left|\phi\right\rangle=\left\langle q-\gamma
a_{j}\right|\left.\phi\right\rangle\equiv\phi\left(q-\gamma a_{j}\right)$
which translates the pointer’s wavefunction over a distance $\gamma a_{j}$
parallel to the $q$-axis. The $q$-representation of the combined system and
pointer state is
$\left\langle
q\right|\left.\Phi\right\rangle={\displaystyle\sum\nolimits_{J}}c_{j}\left\langle
q\right|\widehat{S}\left(\gamma
a_{j}\right)\left|\phi\right\rangle\left|a_{j}\right\rangle.$
When the measurement interaction is strong, the quantum system is appreciably
disturbed and its state ”collapses” to an eigenstate
$\left|a_{n}\right\rangle$ leaving the pointer in the state $\left\langle
q\right|\widehat{S}\left(\gamma a_{n}\right)\left|\phi\right\rangle$ with
probability $\left|c_{n}\right|^{2}$. Strong measurements of an ensemble of
identically prepared systems yield $\gamma\left\langle
A\right\rangle\equiv\gamma\left\langle\psi_{i}\right|\widehat{A}\left|\psi_{i}\right\rangle$
as the centroid of the pointer probability distribution
$\left|\left\langle
q\right|\left.\Phi\right\rangle\right|^{2}={\displaystyle\sum\nolimits_{J}}\left|c_{j}\right|^{2}\left|\left\langle
q\right|\widehat{S}\left(\gamma
a_{j}\right)\left|\phi\right\rangle\right|^{2}$ (3)
with $\left\langle A\right\rangle$ as the measured value of $\widehat{A}$.
A _weak measurement_ of $\widehat{A}$ occurs when the interaction strength
$\gamma$ is sufficiently small so that the system is essentially undisturbed
and the uncertainty $\Delta q$ is much larger than $\widehat{A}$’s eigenvalue
separation. In this case, eq.(3) is the superposition of broad overlapping
$\left|\left\langle q\right|\widehat{S}\left(\gamma
a_{j}\right)\left|\phi\right\rangle\right|^{2}$ terms. Although a single
measurement provides little information about $\widehat{A}$, many repetitions
allow the centroid of eq.(3) to be determined to any desired accuracy.
If a system state is post-selected after a weak measurement is performed, then
the resulting pointer state is
$\left|\Psi\right\rangle=\left\langle\psi_{f}\right|\left.\Phi\right\rangle={\displaystyle\sum\nolimits_{J}}c_{j}^{\prime\ast}c_{j}\widehat{S}\left(\gamma
a_{j}\right)\left|\phi\right\rangle,$
where
$\left|\psi_{f}\right\rangle={\displaystyle\sum\nolimits_{J}}c_{j}^{\prime}\left|a_{j}\right\rangle$,
$\left\langle\psi_{f}\right|\left.\psi_{i}\right\rangle\neq 0$, is the post-
selected state at $t_{0}$. Since
$\widehat{S}\left(\gamma a_{j}\right)=\sum_{m=0}^{\infty}\frac{\left[-i\gamma
a_{j}\widehat{p}/\hbar\right]^{m}}{m!},$
then
$\left|\Psi\right\rangle={\displaystyle\sum\nolimits_{J}}c_{j}^{\prime\ast}c_{j}\left\\{1-\frac{i}{\hbar}\gamma
A_{w}\widehat{p}+\sum_{m=2}^{\infty}\frac{\left[-i\gamma\widehat{p}/\hbar\right]^{m}}{m!}\left(A^{m}\right)_{w}\right\\}\left|\phi\right\rangle\approx\left\\{{\displaystyle\sum\nolimits_{J}}c_{j}^{\prime\ast}c_{j}\right\\}e^{-\frac{i}{\hbar}\gamma
A_{w}\widehat{p}}\left|\phi\right\rangle$
in which case
$\left|\Psi\right\rangle\approx\left\langle\psi_{f}\right|\left.\psi_{i}\right\rangle\widehat{S}\left(\gamma
A_{w}\right)\left|\phi\right\rangle$ (4)
so that
$\left|\left\langle
q\right|\left.\Psi\right\rangle\right|^{2}\approx\left|\left\langle\psi_{f}\right|\left.\psi_{i}\right\rangle\right|^{2}\left|\left\langle
q\right|\widehat{S}\left(\gamma\operatorname{Re}A_{w}\right)\left|\phi\right\rangle\right|^{2}$
or
$\left|\Psi\left(q\right)\right|^{2}\approx\left|\left\langle\psi_{f}\right|\left.\psi_{i}\right\rangle\right|^{2}\left|\phi\left(q-\gamma\operatorname{Re}A_{w}\right)\right|^{2}.$
(5)
Here
$\left(A^{m}\right)_{w}=\frac{{\displaystyle\sum\nolimits_{J}}c_{j}^{\prime\ast}c_{j}a_{j}^{m}}{{\displaystyle\sum\nolimits_{J}}c_{j}^{\prime\ast}c_{j}}=\frac{\left\langle\psi_{f}\right|\widehat{A}^{m}\left|\psi_{i}\right\rangle}{\left\langle\psi_{f}\right|\left.\psi_{i}\right\rangle},$
with the _weak value_ $A_{w}$ of $\widehat{A}$ defined by
$A_{w}\equiv\left(A^{1}\right)_{w}=\frac{\left\langle\psi_{f}\right|\widehat{A}\left|\psi_{i}\right\rangle}{\left\langle\psi_{f}\right|\left.\psi_{i}\right\rangle}.$
(6)
From this expression it is obvious that $A_{w}$ is - in general - a complex
valued quantity that can be calculated directly from theory. Since
$\phi\left(q\right)$ is real valued, then eq.(5) corresponds to a broad
pointer position distribution with a single peak at $\left\langle
q\right\rangle=\gamma\operatorname{Re}A_{w}$ with $\operatorname{Re}A_{w}$ as
the measured value of $\widehat{A}$. This condition occurs when both of the
following inequalities relating $\gamma$ and the pointer momentum uncertainty
$\Delta p$ are satisfied DSS ; PCS :
$\Delta p\ll\frac{\hbar}{\gamma}\left|A_{w}\right|^{-1}\text{ and }\Delta
p\ll\underset{(m=2,3,\cdots)}{\min}\frac{\hbar}{\gamma}\left|\frac{A_{w}}{\left(A^{m}\right)_{w}}\right|^{\frac{1}{m-1}}.$
(7)
It is important to keep in mind that although the weak measurement of
$\widehat{A}$ occurs at time $t_{0}$ so that $\left|\psi_{i}\right\rangle$ and
$\left|\psi_{f}\right\rangle$ are states at $t_{0}$, these states result from
states that are pre-selected and post-selected at times $t_{i}<t_{0}$ and
$t_{f}>t_{0}$, respectively. Therefore it is necessary to propagate the pre-
selected state forward in time from $t_{i}$ to $t_{0}$ and the post-selected
state backward in time from $t_{f}$ to $t_{0}$ in order to calculate $A_{w}$
at $t_{0}$.
The reader is invited to consult references TACKN ; AR ; AAV ; DSS ; AV ; RSH
; PCS ; HK ; DSJH for additional details concerning the theoretical and
experimental aspects of weak measurements and weak values.
## III The Experiment
### III.1 Apparatus
As mentioned above, the setup for this experiment follows that of the optical
gedanken experiment discussed in TACKN where a twin Mach-Zehnder
interferometer is used to replicate aspects of the two-slit interference
experiment. A schematic of the apparatus used in this experiment is shown in
figure 1. Here the paths followed by photons have been labeled using the
traditional ”right” ($R$) and ”left” ($L$) notation
$R1,R2,\cdots,R6,L2,L3,\cdots,L6$. For future reference an overlay of the
”metaphorical” two slits emulated by the twin Mach-Zehnder interferometer is
also provided in this figure. Note that paths $R4$ and $L4$ correspond to
photon paths through the right and left slits, respectively. Thus, blocking
path $L4$ corresponds to closing the left slit.
Since photons do not interact with one another, it is not necessary to perform
the experiment in such a manner that only one photon at a time traverses the
interferometer. Accordingly, large ensembles of photons of wavelength $637.2$
$nm$ produced by a classically intense laser diode source were used in this
experiment. A $150$ $\mu m$ diameter pinhole spatially filtered the photon
beam into a smooth Gaussian-like shape. The exiting beam had an optical power
of $24.5$ $\mu W$ ($\sim 7.9\times 10^{13}$ photons/s) and was collimated with
a $200$ $mm$ focal length lens. A mirror launched the collimated beam into the
interferometer via the input path $R1$. Three identical non-polarizing cube
50/50 beam-splitters - labeled BS1, BS2, BS3 in figure 1 - along with four
identical mirrors - labeled M1, M2, M3, M4 in figure 1 - formed the basic
architecture of the interferometer (the collection BS1, M1, M2, and BS2 (BS2,
M3, M4, and BS3) is hereafter referred to as ” _the first (second) Mach-
Zehnder_ ”). The beam emerging along path $R6$ was neutral density filtered
before reaching a $640\times 480$ pixel resolution machine vision camera which
recorded the beam’s two dimensional intensity distribution. The optical power
of the beam reaching the camera was approximately four to five orders of
magnitude smaller than that exiting the pinhole. Each camera pixel had a size
$7.4$ $\mu m$ $\times$ $7.4$ $\mu m$ and a $0-255$ digital intensity range.
The pixel saturation level exceeded the measured maximum pixel intensity level
of the images obtained from this experiment.
The gedanken experiment utilized slightly tilted thin glass plates placed at
locations in paths $R2$ and $L2$ to perform weak measurements of the
projection operators $\left|R2\right\rangle\left\langle R2\right|$ and
$\left|L2\right\rangle\left\langle L2\right|$ by producing transverse spatial
shifts in the photon paths that were small relative to the uncertainty in the
transverse position of a photon. The theoretically predicted change in the
weak values of these operators when path $L4$ is blocked was interpreted as an
observable effect induced in the first Mach-Zehnder by an associated non-local
exchange of modular momentum produced by blocking path $L4$ in the second
Mach-Zehnder (direct measurement of the modular momentum exchange is not
possible because blocking path $L4$ in the second Mach-Zehnder makes the
modular variable completely uncertain - thereby destroying all information
about the modular momentum).
In this experiment, however, a piezoelectrically driven computer controlled
stage was used instead to produce small changes in the location of mirror M1
(in the direction shown in figure 1) in order to produce a series of
transverse spatial shifts in the photon beam that could be made small compared
to the uncertainty in a photon’s transverse position. This approach proved
more efficient than the tilted plate method and was equivalent to performing
weak measurements of the projection operator
$\left|L2\right\rangle\left\langle L2\right|$ located in path $L2$. As shown -
both theoretically and experimentally - below, the weak value of
$\left|L2\right\rangle\left\langle L2\right|$ changes in accordance with the
gedanken experiment when path $L4$ is blocked. This change can also be
interpreted as a dynamical non-locality induced effect.
By avoiding the use of micro-positioners as much as possible, the setup was
passively stable for several tens of minutes. The entire apparatus was also
enclosed in a $1$ $m$ $\times$ $1$ $m$ covered box to provide additional
isolation from the environment. In order that the box not have to be uncovered
during a measurement data run, electromagnetic shutters were used as much as
possible to block and unblock photon paths and the piezoelectric stage and
camera were computer controlled using data collected by the camera. Because of
these features, all required measurement data were collected before opto-
mechanical instability occurred using only one initial fine alignment. A data
analysis and graphing software tool was developed and used to automatically
process the camera images.
### III.2 Overview
The essence of this experiment involved comparing the measured weak values of
the operator $\left|L2\right\rangle\left\langle L2\right|\equiv\widehat{N}$
for two distinct (data) classes of weak measurements. For each of these weak
measurement classes the pre-selected state prior to the time of
$\widehat{N}$’s measurement was the spatial mode $\left|R1\right\rangle$ and
the post-selected state after $\widehat{N}$’s measurement time was the spatial
mode $\left|R6\right\rangle$. Also, for each of these classes the path lengths
in the first Mach-Zehnder were arranged so that photons effectively only
emerged from BS2 along path $R4$ in spatial mode $-\left|R4\right\rangle$.
Thus, paths $R4$ and $L4$ will be referred to as the ”bright” and ”dark”
paths, respectively. Arranging the first Mach-Zehnder in this way corresponded
to localizing a photon at the right slit of a two slit screen prior to its
traversing the screen. Weak measurements of $\widehat{N}$ for both measurement
classes were made while the apparatus was in this configuration - except that
a shutter blocked path $L4$ for the second measurement class. Blocking path
$L4$ in this manner corresponded to closing the left slit in a two slit screen
while the photon is localized at the right slit.
If $N_{w,1}$ and $N_{w,2}$ correspond to the weak values of $\widehat{N}$ for
the first and second measurement classes, respectively, then - since $L4$ is a
dark path - it might be expected that blocking path $L4$ should have no effect
upon the weak measurement of $\widehat{N}$ in $L2$, in which case
$N_{w,1}=N_{w,2}$. However, when eq.(6) is used to calculate these weak values
it is found that for the first measurement class (which corresponds to both
slits being open) $N_{w,1}=+1$ and for the second measurement class (which
corresponds to closing the left slit) $N_{w,2}=+\frac{1}{2}$. More
specifically, for the first measurement class, forward propagation of the pre-
selected state $\left|R1\right\rangle$ and backward propagation of the post-
selected state $\left|R6\right\rangle$ through the interferometer to where
$\widehat{N}$ is measured yields the states
$\frac{1}{\sqrt{2}}\left(i\left|L2\right\rangle+\left|R2\right\rangle\right)$
and $i\left|L2\right\rangle$, respectively, so that
$N_{w,1}=\frac{\left[-i\left\langle
L2\right|\right]\widehat{N}\left[\frac{1}{\sqrt{2}}\left(i\left|L2\right\rangle+\left|R2\right\rangle\right)\right]}{\left[-i\left\langle
L2\right|\right]\left[\frac{1}{\sqrt{2}}\left(i\left|L2\right\rangle+\left|R2\right\rangle\right)\right]}=+1$
(note that the theoretical weak value of $\left|R2\right\rangle\left\langle
R2\right|$ is $0$). Similarly, for the second measurement class - with the
dark path $L4$ blocked - forward propagation of the pre-selected state
$\left|R1\right\rangle$ and backward propagation of the post-selected state
$\left|R6\right\rangle$ through the interferometer to where $\widehat{N}$ is
measured yields the states
$\frac{1}{\sqrt{2}}\left(i\left|L2\right\rangle+\left|R2\right\rangle\right)$
and $\frac{1}{2}\left(i\left|L2\right\rangle+\left|R2\right\rangle\right)$,
respectively, so that
$N_{w,2}=\frac{\left[\frac{1}{2}\left(-i\left\langle L2\right|+\left\langle
R2\right|\right)\right]\widehat{N}\left[\frac{1}{\sqrt{2}}\left(i\left|L2\right\rangle+\left|R2\right\rangle\right)\right]}{\left[\frac{1}{2}\left(-i\left\langle
L2\right|+\left\langle
R2\right|\right)\right]\left[\frac{1}{\sqrt{2}}\left(i\left|L2\right\rangle+\left|R2\right\rangle\right)\right]}=+\frac{1}{2}$
(note that the theoretical weak value of $\left|R2\right\rangle\left\langle
R2\right|$ is also $+\frac{1}{2}$).
Thus, $N_{w,1}\neq N_{w,2}$ so that - similar to the gedanken experiment -
weak value theory applied to this experiment predicts that blocking path $L4$
produces a dramatic observable change in the weak value of $\widehat{N}$ when
there are effectively no photons along path $L4$. Following TACKN and using
the two-slit case along with eq.(2) as guides, $N_{w,1}\neq N_{w,2}$ _has an
interpretation as being an effect induced in the first Mach-Zehnder by the
non-local exchange of modular momentum that results from a change in the
potential associated with blocking the dark_ $L4$_path in the second Mach-
Zehnder._
A third class of weak measurements of $\widehat{N}$ designated by the weak
value $N_{w,0}$ was used for the purpose of order compliance. For this
measurement class the configuration of the first Mach-Zehnder was the same as
for the other two classes so that forward propagation of the pre-selected
state $\left|R1\right\rangle$ through the first Mach-Zehnder yielded the state
$-\left|R4\right\rangle$. Here, however, a relative (to the other two classes)
phase shift of $\pi$ $rad$ was introduced into path $R5$ so that backward
propagation of the post-selected state $\left|R6\right\rangle$ backwards
through the interferometer gives $\left|R2\right\rangle$ as the state where
the measurement is made. Again using
$\frac{1}{\sqrt{2}}\left(i\left|L2\right\rangle+\left|R2\right\rangle\right)$
as the forward propagated pre-selected state yields the weak value
$N_{w,0}=\frac{\left[\left\langle
R2\right|\right]\widehat{N}\left[\frac{1}{\sqrt{2}}\left(i\left|L2\right\rangle+\left|R2\right\rangle\right)\right]}{\left[\left\langle
R2\right|\right]\left[\frac{1}{\sqrt{2}}\left(i\left|L2\right\rangle+\left|R2\right\rangle\right)\right]}=0$
(note that the theoretical weak value of $\left|R2\right\rangle\left\langle
R2\right|$ is $+1$). This class of measurements served as a data consistency
check by demonstrating that the weak values $N_{w,0}$, $N_{w,1}$, and
$N_{w,2}$ measured by this experiment were compliant with the theoretical
ordering requirement
$N_{w,0}<N_{w,2}<N_{w,1}.$ (8)
## IV Results
In order to experimentally demonstrate this induced $N_{w,1}\neq N_{w,2}$
effect, three sequences of weak measurements of $\widehat{N}$ \- one sequence
for each of the $N_{w,0}$, $N_{w,1}$, and $N_{w,2}$ measurement classes - were
generated following the configuration prescriptions outlined in the above
overview of the experiment. Different interaction strength ($\gamma$) values
were produced for each sequence by varying the M1 position via controlling
that of the piezoelectric stage. The photon beam intensity served as the
measurement pointer for the apparatus and its image was recorded by the
machine vision camera for each M1 position used in the measurement sequences.
As indicated on figure 1, the associated movement of the pointer in the image
plane was horizontal (i.e. in the plane of the apparatus). For each M1
position $x$, the analysis software tool used the associated pointer image to
locate the pointer position as the intensity averaged horizontal pixel number
$\overline{y}$. Each such measurement was represented as the pair
$\left(x,\overline{y}\right)$. Let $S_{i}$ be the set of such measurement
pairs for the $N_{w,i}$ measurement class, $i=0,1,2$.
To calibrate the experimental data, a fourth sequence of measurements was made
to relate M1 positions to pointer pixel positions. Here, paths $L3$ and $R4$
were blocked by shutters and a sequence of M1 positions were used to sweep the
beam emerging along path $R6$ across the image plane of the camera. As was the
case for the previous sequences of measurements, the beam’s intensity averaged
horizontal pixel number was determined from each M1 position image and
represented as an ordered pair $\left(x,\overline{y}\right)$. Let $S_{3}$ be
the set of these ordered pairs of calibration measurements.
Fourteen M1 positions equally spaced over a $1300$ $\mu m$ range were used to
generate fourteen ordered pairs of measurements in each set $S_{k}$, $k\in
K\equiv\left\\{0,1,2,3\right\\}$. These M1 positions were identical for each
of the four measurement sequences (i.e. for every
$\left(x,\overline{y}\right)\in S_{k}$, $k\in K$, there is exactly one
$\left(x^{\prime},\overline{y}^{\prime}\right)\in S_{j}$, $j\in
K-\left\\{k\right\\}$, such that $x=x^{\prime}$). Examination of the
measurement pairs in $S_{1}$ and $S_{2}$ revealed the existence of a data
crossing point located between the middle two M1 positions $x_{7}$ and
$x_{8}$. The pair $\left(x_{0},y_{0}\right)$ defined by the intersection of
the line containing the middle two measurement pairs in $S_{1}$ with that
containing the middle two measurement pairs in $S_{2}$ was selected as the
estimate of this crossing point.
Recall from eq.(5) that in the weak measurement regime defined by inequalities
(7) the ordinates in each data pair in the sets $S_{i}$ effectively record the
measured quantity $\gamma N_{w,i}$, $i=0,1,2$. Thus, at the crossing point the
condition $\gamma N_{w,1}=\gamma N_{w,2}$ must hold true. Since $N_{w,1}=+1$
and $N_{w,2}=+\frac{1}{2}$, this condition can only be satisfied if
$\gamma=0$. This identified $\left(x_{0},y_{0}\right)$ as the point where the
interaction strength $\gamma$ vanishes and defined it as the origin of the
Cartesian reference frame $\mathcal{F}$ which has as its abscissa axis M1
displacements in $\mu m$ referenced to $x_{0}$ and as its ordinate axis
pointer pixel displacements referenced to $y_{0}$. Let
$\left(x^{\prime},\overline{y}^{\prime}\right)\in S_{k}^{\prime}$ be
$\left(x,\overline{y}\right)\in S_{k}$, $k\in\left\\{0,1,2\right\\}$,
transformed into $\mathcal{F}$ according to $x^{\prime}=x-x_{0}$ and
$\overline{y}^{\prime}=\overline{y}-y_{0}$.
As anticipated - the calibration measurement pairs in $S_{3}$ were linear. The
associated slope which relates pointer pixel positions to M1 positions in $\mu
m$ was $-0.198$. This slope defined the calibration line
$\overline{y}^{\prime}=-0.198x^{\prime}$ in $\mathcal{F}$. Multiplying the
slope of this equation by the pixel size $7.4$ $\mu m$ (the camera rated
distance between consecutive pixels) yielded the equation
$\gamma\left(x^{\prime}\right)=-1.5x^{\prime}$ in which both $\gamma$ and
$x^{\prime}$ are in $\mu m$. The ordinate $\overline{y}^{\prime}$ is relabeled
as $\gamma\left(x^{\prime}\right)$ in this equation because it now directly
relates the interaction strengths of measurements to the displacement of M1
(inspection of the argument of the operator $\widehat{S}$ in eq.(4) reveals
that $\gamma$ is a distance since $N_{w,i}$ is a dimensionless quantity). Thus
- for this experiment - the ”ideal” pointer displacements
$\rho_{i}\left(x^{\prime}\right)\equiv\gamma\left(x^{\prime}\right)N_{w,i}$ in
$\mu m$ as functions of M1 displacements in $\mu m$ and $N_{w,i}$ values are
represented by the lines
$\rho_{i}\left(x^{\prime}\right)=-1.5x^{\prime}N_{w,i}\text{, }i=0,1,2.$ (9)
This result is useful for estimating the boundaries of the weak measurement
regime for this experiment in terms of $x^{\prime}$. Since $\widehat{N}$ is a
projection operator then $\widehat{N}^{m}=\widehat{N}$, $m\geq 1$, so that
$\left(N^{m}\right)_{w,i}=N_{w,i}$ and inequalities (7) become $\Delta
p\ll\frac{\hbar}{\gamma N_{w,i}}$, $i\neq 0$, and $\Delta
p\ll\frac{\hbar}{\gamma}$. Application of the uncertainty relation $\Delta
q\cdot\Delta p\geq\frac{\hbar}{2}$ yields $\gamma\ll\frac{2\Delta
q}{N_{w,i}}$, $i\neq 0$, and $\gamma\ll 2\Delta q$. Both of these inequalities
are satisfied by $\gamma\ll 2\Delta q$ when $i=1,2$. Using the pinhole
diameter as the uncertainty in a photon’s tranverse position, i.e. $\Delta
q\approx 150$ $\mu m$, defines $\left|\gamma\right|\ll 300$ $\mu m$ as the
estimated weak measurement regime for the interaction strength
($\left|\gamma\right|$ is used since in this experiment $\gamma$ can be a
positive or a negative distance). Using this range in eq.(9) with $N_{w,1}=+1$
gives $\rho_{1}\left(x^{\prime}\right)=\gamma\left(x^{\prime}\right)$ and
yields
$\left|x^{\prime}\right|\ll 200\text{ }\mu m$ (10)
as the estimated weak measurement regime for M1 displacement.
A plot of the measurement pairs in sets $S_{i}^{\prime}$, $i=0,1,2$ is
presented in figure 2. Here the ordinate of each measurement pair has been
scaled by the pixel distance of $7.4$ $\mu m$ in order to express the pointer
displacements in $\mu m$. Also shown as dashed lines are graphs of the three
ideal pointer displacement lines $\rho_{i}\left(x^{\prime}\right)$, $i=0,1,2$,
given by eq.(9) and as a boxed region the estimated weak measurement regime
defined by inequality (10). Inspection of figure 2 (where $\gamma N_{w,i}$
data points are labeled ”$\gamma N$ class $i$” and $\rho_{i}$ is labeled
”$\rho$ class $i$”) reveals good agreement within (and slightly outside) the
weak measurement regime between the measured pointer displacements $\gamma
N_{w,1}$ (corresponding to the measurement pairs in set $S_{1}^{\prime}$) and
$\rho_{1}$ and between the measured pointer displacements $\gamma N_{w,2}$
(corresponding to the measurement pairs in set $S_{2}^{\prime}$) and
$\rho_{2}$. It is also clear that - except at $x_{8}^{\prime}$ \- the measured
quantities within the weak measurement regime are compliant with the
theoretical ordering requirement (8). It is noted that the $\sim 75$ $\mu
m-100$ $\mu m$ offsets of the measured pointer displacements $\gamma N_{w,0}$
(corresponding to the measurement pairs in set $S_{0}^{\prime}$) from
$\rho_{0}$ in the weak measurement regime are likely due to complicated
intensity profile inversions introduced by the phase window during this
sequence of measurements. Interestingly, if these offsets are treated as a
constant bias, then removal of the bias from the measurement pairs in
$S_{0}^{\prime}$ not only produces complete compliance with (8) in the weak
measurement regime - but it also provides more overall symmetry in the data,
as well as good agreement between the measured pointer displacements $\gamma
N_{w,0}$ and $\rho_{0}$ in the weak measurement regime.
As expected, the further the M1 displacement is outside the weak measurement
regime the ”stronger” the measurement becomes and the greater the discrepancy
between the $S_{0}^{\prime}$ data and $\rho_{0}$ and between the
$S_{1}^{\prime}$ data and $\rho_{1}$. However, except for the data asymmetry
associated with negative M1 displacements (likely introduced by the
complicated optical properties of the apparatus), the agreement between the
$S_{2}^{\prime}$ data and $\rho_{2}$ remains good over the entire range of M1
displacements while the $S_{0}^{\prime}$ and $S_{1}^{\prime}$ data converge to
$\rho_{2}$. This feature in the data is completely consistent with the fact
that in the limit of ”strong collapsing” measurements, the measurement pointer
is displaced by $\gamma\left\langle N\right\rangle=\frac{1}{2}\gamma$ since
$\left\langle N\right\rangle=\frac{1}{\sqrt{2}}\left[-i\left\langle
L2\right|+\left\langle
R2\right|\right]\widehat{N}\left[i\left|L2\right\rangle+\left|R2\right\rangle\right]\frac{1}{\sqrt{2}}=+\frac{1}{2}$
(refer to the discussion surrounding eq.(3)).
## V Concluding Remarks
This experiment used weak measurements of pre- and post-selected ensembles of
photons in a twin Mach-Zehnder interferometer to observe an effect
theoretically predicted to be induced in the first Mach-Zehnder by the non-
local exchange of modular momentum produced by blocking the dark path in the
second Mach-Zehnder (it is intended that a second ”follow up” paper be written
which will detail the novel aspects of the apparatus and techniques used in
this experiment). This effect is manifested as a dramatic change in the
associated weak values. The attendant weak values measured by this experiment
changed in complete accordance with the theoretical predictions. Consequently,
the results of this experiment support both the existence of such an effect
and the authenticity of dynamical non-locality as its cause.
Before closing, it is noted that - although this experiment was specifically
designed for the purpose of confirming or denying the $N_{w,1}\neq N_{w,2}$
effect - it was observed that - for the weakest measurements with abscissa
$x_{8}\simeq 37$ $\mu m$ \- the ratio of the number of camera pixels excited
by the associated $S_{2}^{\prime}$ measurement to that excited by the
associated $S_{1}^{\prime}$ measurement was $0.6$. The drop in this excitation
ratio was $4$ to $5$ times greater than expected based upon the alignment
contrast ratios for the apparatus. This informal observation provides
additional credence to dynamical non-locality as inducing the $N_{w,1}\neq
N_{w,2}$ effect and suggests a future experiment that could further examine
dynamical non-locality from this perspective.
###### Acknowledgements.
The authors thank Yakir Aharonov and Jeff Tollaksen for suggesting this
experiment; John Gray and James Troupe for constructive technical discussions;
and David Niemi for his efforts in the instrumentation of this experiment.
Special thanks are given to Susan Hudson, Electromagnetic and Sensor Systems
Department Head, for her commitment to this research. This work was supported
in part by a grant from the NSWCDD ILIR program sponsored by the Office of
Naval Research.
## References
* (1) Tonomura A, Endo J, Matsuda T, Kawasaki T and Ezawa H 1989 Am. J. Phys. 57 117
* (2) Aharonov Y, Pendelton H and Peterson A 1969 Int. J. Theor. Phys. 3 213
* (3) Aharonov Y, Pendelton H and Peterson A 1970 Int. J. Theor. Phys. 3 443
* (4) Tollaksen J, Aharonov Y, Casher A, Kaufherr T and Nussinov S 2010 New J. Phys. 12 013023
* (5) Aharonov Y and Rohrlich D 2005 Quantum Paradoxes: Quantum Theory for the Perplexed (Weinheim : Wiley-VCH) p 67, p 225
* (6) Aharonov Y, Albert D and Vaidman L 1988 Phys. Rev. Lett. 60 1351
* (7) Duck I, Stevenson P and Sudarshan E 1989 Phys. Rev. D 40 2112
* (8) Aharonov Y and Vaidman L 1990 Phys. Rev. A 41 11
* (9) Ritchie N, Story J and Hulet R 1991 Phys. Rev. Lett. 66 1107
* (10) Parks A, Cullin D and Stoudt D 1998 Proc. Roy. Soc. Lond. A 454 2997
* (11) Hosten O and Kwiat P 2008 Science 319 787
* (12) Dixon P, Starling D, Jordan A and Howell J 2009 Phys. Rev. Lett 102 173601
$\begin{array}[c]{cccccccccccccccccc}&&&&&&&&&&&&&&&&&\\\ &&&&&&&&&&&&&&&&&\\\
&&&&&&&&&\cdot&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&Phase&\\\
&&&&&&&&&\vdots&&&&&&&window&\\\ &&&&&&&&&\vdots&&&\uparrow&&&&&\\\
&&&&&&&&&\vdots&&&L6&&&&&\\\ &&&&&&&&&\vdots&&&\uparrow&&&&&\\\
&&&&&&&M3&\diagup&\cdots&R5&\cdots&\diagup&\longrightarrow&R6&\longrightarrow&\updownarrow\text{
}Camera&\\\ &&&&&&&\diagdown&\vdots&&&BS3&\uparrow&&&&with\text{
}directions&\\\ &&&&&&&&L4&&&&L5&&&&for\text{ }pointer&\\\
&&&\nwarrow\searrow&&&&&\vdots&\diagdown&&&\uparrow&&&&movement&\\\
&&&M1&\diagup&\longrightarrow&R3&\longrightarrow&\diagup&\longrightarrow&R4&\longrightarrow&\diagup&M4&&&&\\\
&&&&\uparrow&&&BS2&\uparrow&&&\diagdown&\cdots&\cdots&\cdots&\cdots&Metaphorical&\\\
&&&&L2&&&&L3&&&&&&&&two-slits&\\\ &&&&\uparrow&&&&\uparrow&&&&&&&&&\\\
Laser&\longrightarrow&R1&\longrightarrow&\diagup&\longrightarrow&R2&\longrightarrow&\diagup&M2&&&&&&&&\\\
beam&&&BS1&&&&&&&&&&&&&&\\\ &&&&&&&&&&&&&&&&&\end{array}$ (Figure 1.
Apparatus, best available diagram for electronic publishing)
|
arxiv-papers
| 2010-10-15T22:24:44 |
2024-09-04T02:49:13.986851
|
{
"license": "Public Domain",
"authors": "S. E. Spence and A. D. Parks",
"submitter": "Scott Spence",
"url": "https://arxiv.org/abs/1010.3289"
}
|
1010.3425
|
# Identifying the consequences of dynamic treatment strategies:
A decision-theoretic overview
A. Philip Dawidlabel=e1]apd@statslab.cam.ac.uklabel=e2 [[
url]tinyurl.com/2maycn Centre for Mathematical Sciences
University of Cambridge
Wilberforce Road
Cambridge CB3 0WB
UK
Vanessa Didelezlabel=e3]vanessa.didelez@bristol.ac.uk label=e4 [[
url]tinyurl.com/2uuteo8 Department of Mathematics
University of Bristol
University Walk
Bristol BS8 1TW
UK
(0000; 10 2010)
###### Abstract
We consider the problem of learning about and comparing the consequences of
dynamic treatment strategies on the basis of observational data. We formulate
this within a probabilistic decision-theoretic framework. Our approach is
compared with related work by Robins and others: in particular, we show how
Robins’s ‘$G$-computation’ algorithm arises naturally from this decision-
theoretic perspective. Careful attention is paid to the mathematical and
substantive conditions required to justify the use of this formula. These
conditions revolve around a property we term stability, which relates the
probabilistic behaviours of observational and interventional regimes. We show
how an assumption of ‘sequential randomization’ (or ‘no unmeasured
confounders’), or an alternative assumption of ‘sequential irrelevance’, can
be used to infer stability. Probabilistic influence diagrams are used to
simplify manipulations, and their power and limitations are discussed. We
compare our approach with alternative formulations based on causal DAGs or
potential response models. We aim to show that formulating the problem of
assessing dynamic treatment strategies as a problem of decision analysis
brings clarity, simplicity and generality.
,
62C05,
62A01,
Causal inference,
$G$-computation,
Influence diagram,
Observational study,
Potential response,
Sequential decision theory,
Stability,
###### doi:
10.1214/154957804100000000
###### keywords:
[class=AMS]
###### keywords:
††volume: 0
and
###### Contents
1. 1 Introduction
1. 1.1 Conditional independence
2. 1.2 Overview
2. 2 A multistage decision problem
3. 3 Regimes and consequences
1. 3.1 Inference across regimes
4. 4 Evaluation of consequences
5. 5 Identifying the ingredients
1. 5.1 Control strategies
2. 5.2 Stability
1. 5.2.1 Some comments
2. 5.2.2 Positivity
3. 5.3 $G$-recursion
6. 6 Extended stability
1. 6.1 Preliminaries
2. 6.2 Stability regained
1. 6.2.1 Sequential randomization
2. 6.2.2 Sequential irrelevance
7. 7 Influence diagrams
1. 7.1 Semantics
2. 7.2 Extended stability
1. 7.2.1 Sequential randomization
2. 7.2.2 Sequential irrelevance
3. 7.2.3 Further examples
4. 7.2.4 Positivity
8. 8 A more general approach
1. 8.1 $G$-recursion: General conditions
2. 8.2 Extended stability
1. 8.2.1 Graphical check
3. 8.3 Examples
1. 8.3.1 Stability
2. 8.3.2 $G$-recursion without stability
9. 9 Constructing an admissible sequence
1. 9.1 Finding a better sequence
2. 9.2 Admissible orderings of ${\cal A}$
10. 10 Potential response models
1. 10.1 Potential responses and stability
1. 10.1.1 Connexions
2. 10.2 Potential responses without stability
1. 10.2.1 Connexions
11. 11 Discussion
1. 11.1 What has been achieved?
2. 11.2 Syntax and semantics
3. 11.3 Statistical inference
4. 11.4 Optimal dynamic treatment strategies
5. 11.5 Complete identifiability
6. 11.6 Other problems
12. A Two lemmas on DAG-separation
## 1 Introduction
Many important practical problems involve sequential decisions, each chosen in
the light of the information available at the time, including in particular
the observed outcomes of earlier decisions. As an example, consider long-term
anticoagulation treatment, as often given after events such as stroke,
pulmonary embolism or deep vein thrombosis. The aim is to ensure that the
patient’s prothrombin time (INR) is within a target range (which may depend on
the diagnosis). Patients on this treatment are monitored regularly, and when
their INR is outside the target range the dose of anticoagulant is increased
or decreased, so that the dose at any given time is a function of the previous
INR observations. Despite the availability of limited guidelines for adjusting
the dose, the quality of anticoagulation control achieved is often poor
Rosthøj et al. (2006). Another example is the question of when to initiate
antiretroviral therapy for an HIV-1-infected patient. The CD4 cell count at
which therapy should be started is a central unresolved issue. Preliminary
findings indicate that treatment should be initiated when the CD4 cell count
drops below a certain level, i.e. treatment should be a function of the
patient’s previous CD4 count history Sterne et al. (2009).
In general, any well-specified way of adjusting the choice of the next
decision (treatment or dose to administer) in the light of previous
information constitutes a dynamic decision (or treatment) strategy. There will
typically be an enormous number of strategies that could be thought of.
Researchers would like to be able to evaluate and compare these and, ideally,
choose a strategy that is optimal according to a suitable criterion Murphy
(2003). In many applications, such as the examples given above, it is unlikely
that we will have access to large random samples of patients treated under
each one of the strategies under consideration. At best, the data available
will have been gathered in controlled clinical trials, but often we will have
to content ourselves with data from uncontrolled observational studies, with,
for example, the treatments being selected by doctors according to informal
criteria that we do not know. The key question we address in the present paper
is: Under what conditions, and how, could the available data be used to
evaluate, compare, and hence choose among, the various decision strategies?
When a given strategy can be evaluated from available data it will be termed
identifiable.
In principle, our problem can be formulated, represented and solved using the
machinery of sequential decision theory, including decision trees and
influence diagrams Raiffa (1968); Oliver and Smith (1990) — and this is indeed
the approach that we shall take in this paper. However, this machinery does
not readily provide us with an answer to the question of when data obtained,
for example, from an observational study will be sufficiently informative to
identify a given strategy. Here, we shall be concerned only with issues around
potential biases in the data, rather than their completeness. Thus wherever
necessary we suppose that the quantity of data available is sufficient to
estimate, to any desired precision, the parameters of the process that
actually produced those data. However, that process might still differ from
that in the new decision problem at hand. We shall therefore propose simple
and empirically meaningful conditions (which can thus be meaningfully
criticised) under which it is appropriate and possible to make use of the
available parameter estimates, and we shall develop formulae for doing this.
These conditions will be termed stability due to the way they relate
observational and interventional regimes. We shall further discuss how one
might justify this stability condition by including unobservable variables
into the decision theoretic framework, and by using influence diagrams.
Our proposal is closely related to the seminal work of Robins Robins (1986,
1987, 1989, 1997). Much of Robins (1986) takes an essentially decision
theoretic approach, while also using the framework of structured tree graphs
as well as potential responses (and later using causal direct acyclic graphs
(DAGs), see Robins (1997)). He shows that under conditions linking
hypothetical studies, where the different treatment strategies to be compared
are applied, identifiability can be achieved. Robins calls these conditions
sequential randomization (and later no unmeasured confounding, see e.g. Robins
(1992)). While these are often formalised using potential responses, a closer
inspection of Robins (1986) (or especially Robins (1997)) reveals that all
that is needed is an equality of conditional distributions under different
regimes, which is what our stability conditions state explicitly. Furthermore,
Robins (1986) introduces the $G$-computation algorithm as a method to evaluate
a sequential strategy, and contrasts it with traditional regression approaches
that yield biased results even when stability or sequential randomization
holds Robins (1992). We shall demonstrate below that, assuming stability, this
$G$-computation algorithm arises naturally out of our decision-theoretic
analysis, where it can be recognized as a version of the fundamental ‘backward
induction’ recursion algorithm of dynamic programming.
### 1.1 Conditional independence
The technical underpinning for our decision-theoretic formulation is the
application of the language and calculus of conditional independence Dawid
(1979, 2002) to relate observable variables of two types: ‘random’ variables
and ‘decision’ (or ‘intervention’) variables. This formalism is used to
express relationships that may be assumed between the probabilistic behaviour
of random variables under differing regimes (e.g., observational and
interventional). Nevertheless, although it does greatly clarify and simplify
analysis, this particular language is not indispensable: everything we do
could, if so desired, be expressed directly in terms of relationships between
probability distributions for observable variables. Thus no essential
additional ingredients are being added to the standard formulation of
statistical decision theory.
In many cases the conditional independence relations we work with can be
represented by means of a graphical display: the influence diagram (ID). Once
again, although enormously helpful this is, in a formal sense, only an
optional extra. Moreover, although we pay special attention to problems that
can be represented by influence diagrams, there are yet others, still falling
under our general approach, where this is not possible.
Inessential though these ingredients are, we nevertheless suggest that it is
well worth the effort of mastering the basic language and properties, both
algebraic and graphical, of conditional independence. In particular, these
allow very simple derivations of the logical consequences of assumptions made
Dawid (1979); Lauritzen et al. (1990).
### 1.2 Overview
In §§ 2 and 3 we set out the basic ingredients of our problem and our
notation. Section 4 identifies a simple recursion that can be used to
calculate the consequence of applying a given treatment regime when the
appropriate probabilistic ingredients are available. In § 5 we consider how
these ingredients might be come by, and show that the simple stability
condition mentioned above allows estimation of these ingredients — and thus,
by application of the procedure of $G$-recursion, of the overall consequence.
In §§ 6 and 7 we consider how one might justify this stability condition,
starting from a position (‘extended stability’) that might sometimes be more
defensible, and relate various sets of sufficient conditions for this to
properties of influence diagrams. Section 8 develops more general conditions,
similar to Robins (1987) and Robins (1997), under which $G$-recursion can be
justified, while § 9 addresses the question of finding an ordering of the
involved variables suitable to carry out $G$-recursion. Finally §10 shows how
analyses based on the alternative formalism of potential responses can be
related mathematically to our own development.
## 2 A multistage decision problem
We are concerned with a sequential data-gathering and decision-making process,
progressing through a discrete sequence of stages. The archetypical context is
that of a sequence of medical treatments applied to a patient over time, each
taking into account any interim responses or adverse reactions to earlier
treatments, such as the anticoagulation treatment for stroke patients or the
decision of when to start antiretroviral therapy for HIV patients. We shall
sometimes use this language.
Associated with each patient are two sets of variables: ${\cal L}$, the set of
observable variables, and ${\cal A}$, the set of action variables. The
variables in ${\cal A}$ can, in principle, be manipulated by external
intervention, while those in ${\cal L}$ are generated and revealed by Nature.
The variables in ${\cal L}\cup{\cal A}$ are termed domain variables. There is
a distinguished variable $Y\in{\cal L}$, the response variable, of special
concern.
A specified sequence ${\cal I}:=(L_{1},A_{1},\ldots,L_{N},A_{N},L_{N+1}\equiv
Y)$, where $A_{i}\in{\cal A}$ and the $L_{i}$ are disjoint subsets of ${\cal
L}$, defines the information base. The interpretation is that the variables
arise or are observed in that order; $L_{i}$ represents (possibly
multivariate, generally time-dependent) patient characteristics or other
variables over which we have no control, observable between times $i-1$ and
$i$; $A_{i}$ describes the treatment action applied to the patient at time
$i$; and $Y$ is the final ‘response variable’ of primary interest.
For simplicity we suppose throughout that all these variables exist and can be
observed for every patient. Thus we do not directly consider cases where,
e.g., $Y$ is time to death, which might occur before some of the $L$’s and
$A$’s have had a chance to materialize. However our analyses could readily be
elaborated to handle such extensions.
When the aim is to control $Y$ through appropriate choices for the action
variables $(A_{i})$, any principled approach will involve making comparisons,
formal or informal, between the implied distributions of $Y$ under a variety
of possible strategies for choosing the $(A_{i})$. For example, we might have
specified a loss $L(y)$ associated with each outcome $y$ of $Y$, and desire to
minimise its expectation ${\mbox{E}}\\{L(Y)\\}$.111 Realistically the loss
could also depend on the values of intermediate variables, e.g. if these
relate to adverse drug reactions. Such problems can be treated by redefining
$Y$ as the overall loss suffered (at any rate so long as this loss does not
depend on other, unobserved, variables.) Any such decision problem can be
solved as soon as we know the relevant distributions for $Y$ (Dawid, 2000,
Section 6).
The simplest kind of strategy is to apply some fixed pre-defined sequence of
actions, irrespective of any observations on the patient: we call this a
static or unconditional strategy (Pearl (2009) terms it atomic). However in
realistic contexts static strategies, which do not take any account of
accruing information, will be of little interest. In particular, under a
decision-theoretically optimal strategy the action to be taken at any stage
must typically be chosen to respond appropriately to the data available at
that stage Robins (1989); Murphy (2003).
A non-randomized dynamic treatment strategy (with respect to a given
information base ${\cal I}$) is a rule that determines, for each stage $i$ and
each configuration (or partial history)
$h_{i}:=(l_{1},a_{1},\ldots,a_{i-1},l_{i})$ for the variables
$(L_{1},A_{1},\ldots,A_{i-1},L_{i})$ available prior to that stage, the value
$a_{i}$ of $A_{i}$ that is then to be applied.
Any decision-theoretically optimal strategy can always be chosen to be non-
randomized. Nevertheless, for added generality we shall also consider
randomized222More correctly, these correspond to what are termed behavioral
rules in decision theory Ferguson (1967) dynamic treatment strategies. Such a
strategy determines, for each stage $i$ and associated partial history
$h_{i}$, a probability distribution for $A_{i}$, describing the random way in
which the next action $A_{i}$ is to be generated. When every such
randomization distribution is degenerate at a single action this reduces to a
non-randomized strategy.
Suppose now we wish to compare a number of such strategies. If we knew or
could estimate the full probabilistic structure of all the variables under
each of these, we could simply calculate and compare directly the various
distributions for the response $Y$. As outlined in the introduction, our
principal concern in this paper is how to obtain such distributional
knowledge, when in many cases the only data available will have been gathered
under purely observational or other circumstances that might be very different
from the strategies we want to compare. To clarify the potential difficulties,
consider a statistician or scientist S, who has obtained data on a collection
of variables for a large number of patients. She wishes to use her data, if
possible, to identify and compare the consequences of various treatment
interventions or policies that might be contemplated for some new patient. A
major complication, and the motivation for much work in this area, is that S’s
observational data will often be subject to ‘confounding’. For example, S’s
observations may include actions $(A_{i})$ that have been determined by a
doctor D, partly on the basis of additional private information D has about
the patient, over and above the variables S has measured. Then knowledge of
the fact that D has selected an act $A_{i}=a_{i}$, by virtue of that being
correlated with unobserved private information D has that may also be
predictive of the response $Y$, could affect the distribution of $Y$ in this
observational regime in a way different from what would occur if D had no such
private information, or if S had herself chosen the value of $A_{i}$. In
particular, without giving careful thought to the matter we cannot simply
assume that probabilistic behaviour seen under the observational regime will
be directly relevant to other, e.g. interventional, regimes of interest.
## 3 Regimes and consequences
In general, we consider the distribution of all the variables in the problem
under a variety of different regimes, possibly but not necessarily involving
external intervention. For example, these might describe different locations,
time-periods, or contexts in which observations can be made. For simplicity we
suppose that the domain variables are the same for all regimes. Formally, we
introduce a regime indicator, $\sigma$, taking values in some set ${\cal S}$,
which specifies which regime is under consideration — and thus which (known or
unknown) joint distribution over the domain variables ${\cal L}\cup{\cal A}$
is operating. Thus $\sigma$ has the logical status of a parameter or decision
variable, rather than a random variable. We think of the value $s$ of $\sigma$
as being determined externally, before any observations are made; all
probability statements about the domain variables must then be explicitly or
implicitly conditional on the value of $\sigma$. We use e.g. $p(y\mid
x\,;\,s)$ to denote the conditional density for $Y$, at $y$, given $X=x$,
under regime $\sigma=s$. In order to side-step measure-theoretic subtleties,
we shall confine attention to the case that all variables considered are
discrete; in particular, the terms ‘distribution’ or ‘density’ should be
interpreted as denoting a probability mass function. However, the basic logic
of our arguments does extend to more general cases (albeit with some non-
trivial technical complications to handle null events.)
If we know $p(y;s)$ for all $y$, we can determine, for any function
$k(\cdot)$, the expectation ${\mbox{E}}\\{k(Y);s\\}$. Often we shall be
interested in one or a small number of such functions, e.g. a loss function
$k(y)\equiv L(y)$. For definiteness we henceforth consider a fixed given
function $k(Y)$, and use the term consequence of $s$ to denote the expectation
${\mbox{E}}\\{k(Y);s\\}$ of $k(Y)$ when regime $s$ is followed.
More generally we might wish to focus attention on a subgroup (typically
defined in terms of the pre-treatment information $L_{1}$), and compare the
various ‘conditional consequences’, given membership of the subgroup. Although
we do not address this directly here, it is straightforward to extend our
unconditional analysis to this case.
### 3.1 Inference across regimes
In the most usual and useful situation, ${\cal S}=\\{o\\}\cup{\cal S}^{*}$,
where $o$ is a particular observational regime under which data have been
gathered, and ${\cal S}^{*}$ is a collection of contemplated interventional
strategies with respect to the information base
$(L_{1},A_{1},\ldots,L_{N},A_{N},Y)$. We wish to use data collected under the
observational regime $o$ to identify the consequence of following any of the
strategies $e\in{\cal S}^{*}$. This means we need to make inference strictly
beyond the available data to what would happen, in future cases, under regimes
that we have not been able to observe in the past.
It should be obvious, but nonetheless deserves emphasis, that we can not begin
to address this problem without assuming some relationships between the
probabilistic behaviour of the variables across the differing regimes, both
observed and unobserved. Inferences across regimes will typically be highly
sensitive to the assumptions made, and the validity of our conclusions will
depend on their reasonableness. Although in principle any such assumptions are
open to empirical test, using data gathered under all the regimes involved,
this will often be impossible in practice. In this case, while it is easy to
make assumptions, it can be much harder to justify them. Any justification
must involve context-dependent considerations, which we can not begin to
address here. Instead we simply aim to understand the logical consequences of
making certain assumptions. One message that could be drawn is: if you don’t
like the consequences, rethink your assumptions.
## 4 Evaluation of consequences
Writing e.g. $(L_{1},L_{2})$ for $L_{1}\cup L_{2}$, we denote
$(L_{1},\ldots,L_{i})$ by $\overline{L}_{i}$, with similar conventions for
other variables in the problem.
For any fixed regime $s$, we can specify the joint distribution of
$(L_{1},A_{1},\ldots,L_{N},A_{N},Y)$, when ${\sigma}={s}$, in terms of its
sequential conditional distributions for each variable, given all earlier
variables. These comprise:
1. (i).
$p(l_{i}\mid\overline{l}_{i-1},\overline{a}_{i-1}\,;\,s)$ for $i=1,\ldots,N$.
2. (ii).
$p(a_{i}\mid\overline{l}_{i},\overline{a}_{i-1}\,;\,s)$ for $i=1,\ldots,N$.
3. (iii).
$p(y\mid\overline{l}_{N},\overline{a}_{N}\,;\,s)$.
Note that (iii) can also be considered as the special case of (i) for $i=N+1$.
With $l_{N+1}\equiv y$, we can factorize the overall joint density as:
$\displaystyle
p(y,\overline{l},\overline{a}\,;\,{s})=\left\\{\prod_{i=1}^{N+1}p(l_{i}\mid\overline{l}_{i-1},\overline{a}_{i-1}\,;\,{s})\right\\}\times\left\\{\prod_{i=1}^{N}p(a_{i}\mid\overline{l}_{i},\overline{a}_{i-1}\,;\,{s})\right\\}.$
(1)
If we know all the terms in (1), we can simply sum out over all variables but
$l_{N+1}\equiv y$ to obtain the desired distribution $p(y;s)$ of $Y$ under
regime $s$, from which we can in turn compute the consequence
${\mbox{E}}\\{k(Y);s\\}$.
Alternatively, and more efficiently, this calculation can be implemented
recursively, as follows. Let $h$ denote a partial history, of the form
$(\overline{l}_{i},\overline{a}_{i-1})$ or
$(\overline{l}_{i},\overline{a}_{i})$ ($0\leq i\leq N)$. We also include the
‘null’ history $\emptyset$, and ‘full’ histories
$(\overline{l}_{N},\overline{a}_{N},y)$. We denote the set of all partial
histories by ${\cal H}$. Fixing the regime $s$, define a function $f$ on
${\cal H}$ by:
$f(h):={\mbox{E}}\\{k(Y)\mid h\,;\,s\\}.$ (2)
Simple application of the laws of probability yields:
$\displaystyle f(\overline{l}_{i},\overline{a}_{i-1})$ $\displaystyle=$
$\displaystyle\sum_{a_{i}}p(a_{i}\mid\overline{l}_{i},\overline{a}_{i-1}\,;\,s)\times
f(\overline{l}_{i},\overline{a}_{i})$ (3) $\displaystyle
f(\overline{l}_{i-1},\overline{a}_{i-1})$ $\displaystyle=$
$\displaystyle\sum_{l_{i}}p(l_{i}\mid\overline{l}_{i-1},\overline{a}_{i-1}\,;\,s)\times
f(\overline{l}_{i},\overline{a}_{i-1}).$ (4)
For $h$ a full history $(\overline{l}_{N},\overline{a}_{N},y)$, we have
$f(h)=k(y)$. Using these as starting values, by successively implementing (3)
and (4) in turn, starting with (4) for $i=N+1$ and ending with (4) for $i=1$,
we step down through ever shorter histories until we have computed
$f(\emptyset)={\mbox{E}}\\{k(Y)\,;\,s\\}$, the consequence of regime
$s$.333More generally (see footnote 1), we could consider a function $Y^{*}$
of $(\overline{L}_{N},\overline{A}_{N},Y)$. Starting now with
$f(\overline{l}_{N},\overline{a}_{N},y):=Y^{*}(\overline{l}_{N},\overline{a}_{N},y)$,
we can apply the identical steps to arrive at
$f(\emptyset)={\mbox{E}}\\{Y^{*}\,;\,s\\}$. In particular we can evaluate the
expected overall loss under $s$, even when the loss function depends on the
full sequence of variables.
The recursion expressed by (3) and (4) is exactly that underlying the
‘extensive form’ analysis of sequential decision theory (see e.g. Raiffa
(1968)). In particular, under suitable further conditions we can combine this
recursive method for evaluation of consequences with the selection of an
optimal strategy, when it becomes dynamic programming. This ‘step-down
histories’ approach also applies just as readily to more general probability
or decision trees, where the length of the history, and even the variables
entering into it, can vary with the path followed. We do not consider such
extensions here, but they raise no new issues of principle.
When $s$ is a non-randomized strategy, the distribution of $A_{i}$ given
$\overline{L}_{i}=\overline{l}_{i}$, when $\sigma=s$, is degenerate, at
$a_{i}=g_{i}=g_{i}(\overline{l}_{i}\,;\,s)$, say, and the only randomness left
is for the variables $(L_{1},\ldots,L_{N},Y)$. We can now consider $f(h)$ as a
function of only the $(l_{i})$ appearing in $h$, since, under $s$, these then
determine the $(a_{i})$. Then (3) holds automatically, while (4) becomes:
$f(\overline{l}_{i-1})=\sum_{l_{i}}\,p(l_{i}\mid\overline{l}_{i-1},\overline{g}_{i-1};s)\times
f(\overline{l}_{i}).$ (5)
When, further, the regime $s$ is static, each $g_{i}$ in the above expressions
reduces to the fixed action $a_{i}^{*}$ specified by $s$.
We remark that the conditional distributions in (i)–(iii) and (2) are
undefined when the conditioning event has probability 0 under $s$. The overall
results of recursive application of (3) and (4) will not depend on how such
ambiguities are resolved. However, for later convenience we henceforth assume
that $f(h)$ in (2) is defined as 0 whenever $p(h\,;\,s)=0$. Note that this
property is preserved under (3) and (4).
## 5 Identifying the ingredients
In order for the statistician S to be able to apply the above recursive method
to calculate the consequence of some contemplated regime $s$, she needs to
know all the ingredients (i), (ii) and (iii). How might such knowledge be
attained?
### 5.1 Control strategies
Consider first the term
$p(a_{i}\mid\overline{l}_{i},\overline{a}_{i-1}\,;\,s)$ in (ii), as needed for
(3). It will often be the case that for the regimes $s$ of interest this is
known a priori to the statistician S for all $i$. For instance we might be
interested in strategies for initiating antiretroviral treatment of HIV
patients as soon as the CD4 count has dropped below a given value $c$. The
strategy therefore fully determines the value of the binary $A_{i}$ given the
previous covariate history $\overline{l}_{i}$ as long as this includes
information on the CD4 counts. In such a case we shall call $s$ a control
strategy (with respect to the information base ${\cal
I}=(L_{1},A_{1},\ldots,L_{N},A_{N},Y)$). In particular this will typically be
the case when $s$ is a (possibly randomized) dynamic strategy, as introduced
in § 2.
### 5.2 Stability
More problematic is the source of knowledge of the conditional density
$p(l_{i}\mid\overline{l}_{i-1},\overline{a}_{i-1}\,;\,s)$ in (i) as required
for (4) (including, as a special case, that of
$p(y\mid\overline{l}_{N},\overline{a}_{N}\,;\,s)$ in (iii)).
If we observed many instances of regime $s$, we may be able to estimate this
directly; but typically we will be interested in assessing the consequences of
various contemplated regimes (e.g. control strategies) that we have never yet
observed. The problem then becomes: under what conditions can we use
probability distributions assessed under one regime to deduce the required
conditional probabilities, (i) and (iii), under another?
In the application of most interest, we have ${\cal S}=\\{o\\}\cup{\cal
S}^{*}$, where $o$ is an observational regime under which data have been
gathered, and ${\cal S}^{*}$ is a collection of contemplated interventional
strategies. If we can use data collected under the observational regime $o$ to
identify the consequence of following any of the strategies $e\in{\cal
S}^{*}$, we will be in a position to compare the consequences of different
interventional strategies (and thus, if desired, choose an optimal one) on the
basis of data collected in the single regime $o$.
In general, the distribution of $L_{i}$ given
$(\overline{L}_{i-1},\overline{A}_{i-1})$ will depend on which regime is in
operation. Even application of a control strategy might well have effects on
the joint distribution of all the variables, beyond the behaviour it directly
specifies for the actions. For example, consider an educational experiment in
which we can select certain pupils to undergo additional home tutoring. Such
an intervention can not be imposed without subjecting the pupil and his family
to additional procedures and expectations, which would probably be different
if the decision to undergo extra tutoring had come directly from the pupil,
and possibly different again if it had come from the parents. Consequently we
can not necessarily assume that the distribution of $L_{i}$ given
$(\overline{L}_{i-1},\overline{A}_{i-1})$ assessed under the observational
regime will be the same as that for an interventional strategy, or that it
would be the same for different interventional strategies.
It will clearly be helpful when we can impose this assumption — and so be able
to identify the required interventional distributions of $L_{i}$ given
$(\overline{L}_{i-1},\overline{A}_{i-1})$ with those assessed under the
observational regime. We formalize this assumption as follows:
###### Definition 1.
We say that the problem exhibits simple stability, with respect to the
information base ${\cal I}=(L_{1},A_{1},\ldots,L_{N},A_{N},Y)$ and the set
${\cal S}$ of regimes if, with $\sigma$ denoting the non-random regime
indicator taking values in ${\cal S}$:
$\mbox{$L_{i}\,\mbox{$\perp\\!\\!\\!\perp$}\,\sigma\mid(\overline{L}_{i-1},\overline{A}_{i-1})$}\quad(i=1,\ldots,N+1).$
(6)
Here and throughout, we use the notation and theory of conditional
independence introduced by Dawid (1979), as generalized as in Dawid (2002) to
apply also to problems involving decision or parameter variables. In words,
condition (6) asserts that the stochastic way in which $L_{i}$ arises, given
the previous values of the $L$’s and $A$’s, should be the same, irrespective
of which regime in ${\cal S}$ is in operation. More precisely, expressed in
terms of densities, (6) requires that, for each $i=1,\ldots,N+1$, there exist
some common conditional density specification
$q(L_{i}=l_{i}\mid\overline{L}_{i-1}=\overline{l}_{i-1},\overline{A}_{i-1}=\overline{a}_{i-1})$
such that, for each $s\in{\cal S}$,
$p(L_{i}=l_{i}\mid\overline{L}_{i-1}=\overline{l}_{i-1},\overline{A}_{i-1}=\overline{a}_{i-1};s)=q(L_{i}=l_{i}\mid\overline{L}_{i-1}=\overline{l}_{i-1},\overline{A}_{i-1}=\overline{a}_{i-1})$
(7)
whenever the conditioning event has positive probability under regime $s$.
As will be described further in § 7 below, it is often helpful (though never
essential) to represent conditional independence properties graphically, using
the formalism of influence diagrams (IDs): such diagrams have very specific
semantics, and can facilitate logical arguments by displaying implied
properties in a particularly transparent form Dawid (2002). The appropriate
graphical encoding of property (6) for $i=$ 1, 2 and 3 is shown in Figure 1.
The specific property (6) is represented by the absence of arrows from
$\sigma$ to $L_{1}$, $L_{2}$, and $Y\equiv L_{3}$. For general $N$ we simply
supplement the complete directed graph on $(L_{1},A_{1},\ldots,L_{N},A_{N},Y)$
with an additional regime node $\sigma$, and an arrow from $\sigma$ to each
$A_{i}$.
Figure 1: Influence diagram: stability
#### 5.2.1 Some comments
An important question is how we should assess whether property (6) holds in
any given situation. It could in principle be tested empirically, if we could
collect data under all regimes. In practice this is usually impossible, and
other arguments for or against its appropriateness would be brought to bear.
Whether or not the simple stability property can be regarded as appropriate in
any application will depend on the overall context of the problem. In
particular, it will depend on the specific information base involved. For
example, if $e$ is a control strategy with respect to S’s information base,
and $o$ an observational regime under which the doctor D chooses the $(A_{i})$
on the basis of private information not represented in S’s information base,
possibly associated with $L_{i}$, then, for ${\cal S}=\\{o,e\\}$, we might
well expect (6) to be violated. This is often described as (potential)
confounding.
The simple stability property (6) is our version of a condition termed
‘sequential randomization’ Robins (1986, 1997) or ‘no unmeasured confounding’
Robins (1992); Robins, Hernán and Brumback (2000) or ‘sequential ignorability’
Robins (2000). The connexions become particularly clear when comparing (6)
with the equalities derived in Theorem 3.1 of Robins (1997), which we consider
in more detail in § 10.1.1 below. These alternative names suggest particular
situations where stability should be satisfied, such as when the data have
been gathered under an observational regime where the actions were indeed
physically sequentially randomized; or when S’s information base contains all
the information the doctor D has used in choosing the $(A_{i})$. However, we
emphasise that our property (6) can be meaningfully considered even without
referring to any ‘potential confounder’ variables; and that if (as in § 6
below) we do choose to introduce such further variables to help us assess
whether (6) holds, nevertheless the property itself must hold or fail quite
independently of which additional variables (if any) are considered.
In any case, because stability is a property of the relationship between
different regimes, it can never be empirically established on the basis of
data collected under only one (e.g., observational) regime, nor can it be
deduced from properties assumed to hold for just one such regime.
#### 5.2.2 Positivity
The purpose of invoking simple stability (with respect to ${\cal
S}=\\{o\\}\cup{\cal S}^{*}$) is to get a handle on (4) for an unobserved
interventional strategy $s=e\in{\cal S}^{*}$, using data obtained in the
observational regime $o$. Intuitively, under simple stability we can replace
$p(l_{i}\mid\overline{l}_{i-1},\overline{a}_{i-1};e)$ by
$p(l_{i}\mid\overline{l}_{i-1},\overline{a}_{i-1};o)$, which is estimable from
the observational data. However, some care is needed on account of the
positivity qualification following (7). If, for example, we want to assess the
consequence of a static interventional strategy $e$, which always applies some
pre-specified action sequence $\overline{a}^{*}$, we clearly will be unable to
do so using data from an observational regime in which the probability of
obtaining that particular sequence of actions is zero. (Pragmatically it may
still be difficult to do so if that probability is non-zero but so small that
we are unable to estimate it well from available observational data. However
we ignore that difficulty here, supposing that the data are sufficiently
extensive that we can indeed get good estimates of all probabilities under
$o$).
In order to avoid this problem, we impose the positivity (absolute continuity)
condition:
###### Definition 2.
We say the problem exhibits positivity if, for any $e\in{\cal S}^{*}$, the
joint distribution of $(\overline{L}_{N},\overline{A}_{N},Y)$ under $P_{e}$ is
absolutely continuous with respect to that under $P_{o}$, i.e.
$p(E;e)>0\Rightarrow p(E;o)>0$ (8)
for any event $E$ defined in terms of $(\overline{L}_{N},\overline{A}_{N},Y)$.
We write this as $P_{e}\ll P_{o}$.
In our discrete set-up, it is clearly enough to demand (8) whenever $E$
comprises a single sequence $(\overline{l}_{N},\overline{a}_{N},y)$. Denoting
by ${\cal O}$, ${\cal E}$ the sets of partial histories having positive
probability under, respectively, regimes $o$ and $e$, we can restate (8) as
${\cal E}\subseteq{\cal O}.$ (9)
### 5.3 $G$-recursion
Let $e\in{\cal S}^{*}$. Given enough data collected under $o$ we can identify
$p(l_{i}\mid\overline{l}_{i-1},\overline{a}_{i-1}\,;\,o)$ ($i=1,\ldots,N+1$)
for $(\overline{l}_{i-1},\overline{a}_{i-1})\in{\cal O}$. Under simple
stability (7) and positivity (9), this will also give us
$p(l_{i}\mid\overline{l}_{i-1},\overline{a}_{i-1}\,;\,e)$ ($i=1,\ldots,N+1$)
for all $(\overline{l}_{i-1},\overline{a}_{i-1})\in{\cal E}$. If, further, $e$
is a control strategy, then using the known form for
$p(a_{i}\mid\overline{l}_{i},\overline{a}_{i-1}\,;\,e)$
($(\overline{l}_{i},\overline{a}_{i})\in{\cal E}$), we have all the
ingredients to apply (3) and (4) and thus identify the consequence of regime
$e$ from data collected under $o$.
Specifically, we have
$\displaystyle f(\overline{l}_{i},\overline{a}_{i-1})$ $\displaystyle=$
$\displaystyle\sum_{a_{i}}p(a_{i}\mid\overline{l}_{i},\overline{a}_{i-1}\,;\,e)\times
f(\overline{l}_{i},\overline{a}_{i})$ (10) $\displaystyle
f(\overline{l}_{i-1},\overline{a}_{i-1})$ $\displaystyle=$
$\displaystyle\sum_{l_{i}}p(l_{i}\mid\overline{l}_{i-1},\overline{a}_{i-1}\,;\,o)\times
f(\overline{l}_{i},\overline{a}_{i-1}).$ (11)
We start the recursion with
$f(\overline{l}_{N},\overline{a}_{N})\equiv{\mbox{E}}\\{k(Y)\mid\overline{l}_{N},\overline{a}_{N}\,;\,e\\}=\left\\{\begin{array}[c]{ll}{\mbox{E}}\\{k(Y)\mid\overline{l}_{N},\overline{a}_{N}\,;\,o\\}&\mbox{if
}(\overline{l}_{N},\overline{a}_{N})\in{\cal E}\\\
0&\mbox{otherwise}\end{array}\right.$
(using simple stability for $i=N+1$), and exit with the desired interventional
consequence $f(\emptyset)\equiv{\mbox{E}}\\{k(Y)\,;\,e\\}$.
We refer to the above method as $G$-recursion.444Cases in which simple
stability may not hold but we can nevertheless still apply $G$-recursion are
considered in Section 8.
For the case that $e$ is a non-randomized strategy, $G$-recursion can be based
on (5), becoming
$f(\overline{l}_{i-1})=\sum_{l_{i}}\,p(l_{i}\mid\overline{l}_{i-1},\overline{g}_{i-1};o)\times
f(\overline{l}_{i}),$ (12)
starting with
$f(\overline{l}_{N})={\mbox{E}}\\{k(Y)\mid\overline{l}_{N},\overline{g}_{N}\,;\,o\\}$.
The $G$-computation formula Robins (1986) is the algebraic formula for
$f(\emptyset)$ in terms of $f(\overline{l}_{N})$ that results when we write
out explicitly the successive substitutions required to perform this
recursion.
Finally we remark that, when the simple stability property (6) holds for
$(L_{1},A_{1},\ldots,L_{N},A_{N},Y)$, it also holds for
$(L_{1},A_{1},\ldots,L_{N},A_{N},Y^{*})$, where $Y^{*}$ is any function of
$(L_{1},A_{1},\ldots,L_{N},A_{N},Y)$. For $i\leq N$ there is nothing new to
show, while (6) for $i=N+1$ follows easily for $Y^{*}$ when it holds for $Y$,
using general properties of conditional independence Dawid (1979). It is also
easy to see that when positivity, Definition 2, holds for
$(\overline{L}_{N},\overline{A}_{N},Y)$ it likewise holds for
$(\overline{L}_{N},\overline{A}_{N},Y^{*})$. Consequently, under the same
conditions that allow $G$-recursion to compute the interventional distribution
of $Y$, we can use it to compute that of $Y^{*}$. In particular (see footnote
1), this will allow us to evaluate the expected loss of applying $e$, even
when the loss function depends on all of
$(\overline{L}_{N},\overline{A}_{N},Y)$.
## 6 Extended stability
We have already alluded to the possibility that, in many applications, the
simple stability assumption (6) might not be easy to justify directly. This
might be the case, in particular, when we are concerned about the possibility
of ‘confounding effects’ due to unobserved influential variables.
In such a case we might proceed by constructing a more detailed model,
incorporating a collection ${\cal U}$ of additional, possibly unobserved,
variables; and investigate its implications. These unobserved variables might
be termed ‘sequential (potential) confounders’. Under certain additional
assumptions to be discussed below, we might then be able to deduce that simple
stability does, after all, apply. This programme can be helpful when the
assumptions involving the additional variables are easier to justify than
assumptions referring only to the variables of direct interest. We here
initially express these additional assumptions purely algebraically, in terms
of conditional independence; in § 7 we shall conduct a parallel analysis
utilising influence diagrams to facilitate the expression and manipulation of
the relevant conditional independencies.
Reasoning superficially similar to ours has been conducted by Pearl and Robins
(1995) and Robins (1997). However, that is mostly based on the assumed
existence of a ‘causal DAG’ representation of the problem. We once again
emphasise that the simple stability property (6) is always meaningful of
itself, and its truth or falsity can not rely on the possibility of carrying
out such a programme of reduction from a more complex model including
unobservable variables.
### 6.1 Preliminaries
We shall specifically investigate models having a property we term extended
stability. Such a model again involves a collection ${\cal L}$ of observable
domain variables (including a response variable $Y$) and a collection ${\cal
A}$ of action domain variables, together with a regime indicator variable
$\sigma$ taking values in ${\cal S}=\\{o\\}\cup{\cal S}^{*}$. But now we also
have the collection ${\cal U}$ of unobservable domain variables (for
simplicity we suppose throughout that which variables are observed or
unobserved is the same under all regimes considered). Let ${\cal I}^{\prime}$
denote an ordering of all these observable and unobservable domain variables
(typically, though not necessarily, their time-ordering). As before we assume
that $A_{i-1}$ comes before $A_{i}$ in this ordering. We term ${\cal
I}^{\prime}$ an extended information base. Let $L_{i}\subseteq{\cal L}$
[resp., $U_{i}\subseteq{\cal U}$] denote the set of observed [resp.,
unobserved] variables between $A_{i-1}$ and $A_{i}$.
###### Definition 3.
We say that the problem exhibits extended stability with respect to the
extended information base ${\cal I}^{\prime}$ and the set ${\cal S}$ of
regimes if, for $i=1,\ldots,N+1$,
$\mbox{$(U_{i},L_{i})\,\mbox{$\perp\\!\\!\\!\perp$}\,\sigma\mid($}{\overline{U}_{i-1},\overline{L}_{i-1},\overline{A}_{i-1}}).$
(13)
(If the ($U_{i}$) were observable, this would be identical with the definition
of simple stability.)
Under extended stability the marginal distribution of $U_{1}$ is supposed the
same in both regimes, as is the conditional distribution of $U_{2}$ given
$(U_{1},L_{1},A_{1})$, etc. Similarly, the distributions of $L_{1}$ given
$U_{1}$, of $L_{2}$ given $(U_{1},L_{1},A_{1},U_{2})$,…, and finally of $Y$
($=L_{N+1}$) given $(U_{1},L_{1},A_{1},\ldots,U_{N},L_{N},A_{N})$, are all
supposed to be independent of the regime operating.
There is a corresponding extension of Definition 2:
###### Definition 4.
We say the problem exhibits extended positivity if, for any $e\in{\cal
S}^{*}$, $P_{e}\ll P_{o}$ as distributions over
$(\overline{L}_{N},\overline{U}_{N},\overline{A}_{N},Y)$; that is,
$p(E;e)>0\Rightarrow p(E;o)>0$ and any event $E$ defined in terms of
$(\overline{L}_{N},\overline{U}_{N},\overline{A}_{N},Y)$.
In many problems, though by no means universally, an extended stability
assumption might be regarded as more reasonable and defensible than simple
stability — so long as appropriate unobserved variables ${\cal U}$ are taken
into account. For example, this might be the case if we believed that, in the
observational regime, the actions were chosen by a decision-maker who had been
able to observe, in sequence, some or all of the variables in the problem,
including possibly the $U$’s; and was then operating a control strategy with
respect to this extended information base, so that, when choosing each action,
he was taking account of all previous variables in this extended sequence, but
nothing else. But even then, as discussed in § 5.2, the extended stability
property is a strong additional assumption, that needs to be justified in any
particular problem. And again, because it involves the relationships between
distributions under different regimes, it can not be justified on the basis of
considerations or findings that apply only to one regime.
Unobservable variables can assist in modelling the observational regime and
its relationship with the interventional control regimes under consideration.
But, because they are unobserved, they can not form part of the information
taken into account by such control regimes. Thus we shall still be concerned
with evaluating — using $G$-recursion when possible — a regime $e$ that is a
control strategy with respect to the observable information base ${\cal
I}=(L_{1},A_{1},\ldots,L_{N},A_{N},Y)$ as introduced in § 5.1. More
specifically, in this more general context we define:
###### Condition 6.1 (Control strategy)
The regime $e$ is a control strategy if, for $i=1,\ldots,N$,
$A_{i}\,\mbox{$\perp\\!\\!\\!\perp$}\,\overline{U}_{i}\mid(\overline{L}_{i},\overline{A}_{i-1}\,;\,e)$
(14)
and, in addition, the conditional distribution of $A_{i}$, given
$(\overline{L}_{i},\overline{A}_{i-1})$, under regime $e$, is known to the
analyst.
Condition 6.1 expresses the property that, under regime $e$, the randomization
distribution or other sources of uncertainty about $A_{i}$, given all earlier
variables, does not in fact depend on the earlier unobserved variables; and
that this conditional distribution is known. The condition will hold, in
particular, in the important common case that, under $e$, $A_{i}$ is fully
specified as a function of previous observables.
### 6.2 Stability regained
When there are unobservables in the problem, the extended positivity property
of Definition 4 will clearly imply the simple positivity property of
Definition 2. However, even when extended stability holds, the simple
stability property, with respect to the observable information base
$(L_{1},A_{1},\ldots,L_{N},A_{N},Y)$ from which (as is a pragmatic necessity)
we have had to exclude the unobserved variables, will typically fail. But we
can sometimes incorporate additional background knowledge, most usefully
expressed in terms of conditional independence, to show that it does, after
all, hold.
We now describe two sets of additional sufficient (though not necessary)
conditions, either of which will, when appropriate, allow us to deduce the
simple stability property (6) — and with it, the possibility of applying
$G$-recursion (ignoring the unobservable variables), as set out in § 5.3. The
results in this section can be regarded as extending the analysis of Dawid
(2002) § 8.3 (see also Guo and Dawid (2010)) to the sequential setting.
#### 6.2.1 Sequential randomization
It has frequently been proposed (e.g., Robins (1986, 1997)) that when, under
an observational regime, the actions $(A_{i})$ have been physically
(sequentially) randomized, then simple stability (6) will hold. Indeed, our
concept of simple stability has also been termed ‘sequential randomization’
Robins (1986). However we shall be more specific and restrict the term
sequential randomization to the special case that we have extended stability
and, in addition, Condition 6.2 below holds. We shall show that these
properties are indeed sufficient to imply simple stability — but they are by
no means necessary.
So consider now the following condition:
###### Condition 6.2
$\mbox{$A_{i}\,\mbox{$\perp\\!\\!\\!\perp$}\,\overline{U}_{i}\mid(\overline{L}_{i},\overline{A}_{i-1}\,;\,\sigma)$}\quad(i=1,\ldots,N).$
(15)
This is essentially a discrete-time version of Definition 2 (ii) of Arjas and
Parner (2004), but with the additional vital requirement that the unobservable
variables ${\cal U}$ involved already be such as to allow us to assume the
extended stability property (13). (Without such an underlying assumption there
can be no way of relating different regimes together.)
Condition 6.2 requires that, for each regime, any earlier unobserved variables
in the extended information base ${\cal I}^{\prime}$ can have no further
effect on the distribution of $A_{i}$, once the earlier observed variables are
taken into account. This will certainly be the case when, under each regime,
treatment assignment, at any stage, is determined by some deterministic or
randomizing device that only has the values of those earlier observed
variables as inputs. While this will necessarily hold for a control strategy
with respect to the observed information base, whether or not it is a
reasonable requirement for the observational regime will depend on deeper
consideration of the specific context and circumstances. It will typically do
so if all information available to and utilised by the decision-maker (the
doctor, for instance) in the observational regime is included in
$\overline{L}_{i}$, or, indeed, if the actions $(A_{i})$ have been physically
randomized within levels of $(\overline{L}_{i},\overline{A}_{i-1})$.
###### Theorem 6.1.
Suppose our model exhibits extended stability. If in addition Condition 6.2
holds, then we shall also have the simple stability property (6).
###### Proof 6.2.
Our proof will be based on universal general properties of conditional
independence, as described by Dawid (1979, 1998).
Let $E_{i}$, $R_{i}$, $H_{i}$ denote, respectively, the following assertions:
$\displaystyle E_{i}$ $\displaystyle:$
$(L_{i},U_{i})\,\mbox{$\perp\\!\\!\\!\perp$}\,\sigma\mid(\overline{L}_{i-1},\overline{U}_{i-1},\overline{A}_{i-1})$
$\displaystyle R_{i}$ $\displaystyle:$
$A_{i}\,\mbox{$\perp\\!\\!\\!\perp$}\,\overline{U}_{i}\mid(\overline{L}_{i},\overline{A}_{i-1};\sigma)$
$\displaystyle H_{i}$ $\displaystyle:$
$(L_{i},\overline{U}_{i})\,\mbox{$\perp\\!\\!\\!\perp$}\,\sigma\mid(\overline{L}_{i-1},\overline{A}_{i-1})$
Extended stability is equivalent to $E_{i}$ holding for all $i$, so we assume
that; while $R_{i}$ is just Condition 6.2, which we are likewise assuming for
all $i$. We shall show that these assumptions imply $H_{i}$ for all $i$, which
in turn implies
$L_{i}\,\mbox{$\perp\\!\\!\\!\perp$}\,\sigma\mid(\overline{L}_{i-1},\overline{A}_{i-1})$,
i.e., simple stability.
We proceed by induction. Since $E_{1}$ and $H_{1}$ are both equivalent to
$(L_{1},U_{1})\,\mbox{$\perp\\!\\!\\!\perp$}\,\sigma$, $H_{1}$ holds.
Suppose now $H_{i}$ holds. Conditioning on $L_{i}$ yields
$\mbox{$\overline{U}_{i}\,\mbox{$\perp\\!\\!\\!\perp$}\,\sigma\mid(\overline{L}_{i},\overline{A}_{i-1})$},$
(16)
and this together with $R_{i}$ is equivalent to
$\overline{U}_{i}\,\mbox{$\perp\\!\\!\\!\perp$}\,(A_{i},\sigma)\mid(\overline{L}_{i},\overline{A}_{i-1})$,
which on conditioning on $A_{i}$ then yields
$\mbox{$\overline{U}_{i}\,\mbox{$\perp\\!\\!\\!\perp$}\,\sigma\mid(\overline{L}_{i},\overline{A}_{i})$}.$
(17)
Also, by $E_{i+1}$ we have
$\mbox{$(L_{i+1},U_{i+1})\,\mbox{$\perp\\!\\!\\!\perp$}\,\sigma\mid(\overline{L}_{i},\overline{U}_{i},\overline{A}_{i})$}.$
(18)
Taken together, (17) and (18) are equivalent to $H_{i+1}$, so the induction is
established.
#### 6.2.2 Sequential irrelevance
Another possible condition is:
###### Condition 6.3
$\mbox{$L_{i}\,\mbox{$\perp\\!\\!\\!\perp$}\,\overline{U}_{i-1}\mid(\overline{L}_{i-1},\overline{A}_{i-1}\,;\,\sigma)$}\quad(i=1,\ldots,N+1).$
(19)
In contrast to (15), (19) does permit the unobserved variables to date,
$\overline{U}_{i}$, to influence the next action $A_{i}$ (which can however
only happen in the observational regime), as well as the current observable
$L_{i}$; but they do not affect the subsequent development of the $L$’s
(including, in particular, the response variable $Y$).
###### Theorem 6.3.
Suppose:
1. (i).
Extended stability, (13), holds.
2. (ii).
Sequential irrelevance, Condition 6.3, holds for the observational regime
$\sigma=o$:
$\mbox{$L_{i}\,\mbox{$\perp\\!\\!\\!\perp$}\,\overline{U}_{i-1}\mid(\overline{L}_{i-1},\overline{A}_{i-1};\sigma=o)$}\quad(i=1,\ldots,N+1).$
(20)
3. (iii).
Extended positivity, as in Definition 4, holds.
Then we shall have simple stability:
$\mbox{$L_{i}\,\mbox{$\perp\\!\\!\\!\perp$}\,\sigma\mid(\overline{L}_{i-1},\overline{A}_{i-1})$}\quad(i=1,\ldots,N+1).$
(21)
Moreover, sequential irrelevance holds under any regime:
$\mbox{$L_{i}\,\mbox{$\perp\\!\\!\\!\perp$}\,\overline{U}_{i-1}\mid(\overline{L}_{i-1},\overline{A}_{i-1};\sigma)$}\quad(i=1,\ldots,N+1).$
(22)
###### Proof 6.4.
Let $k(L_{i})$ be a bounded real function of $L_{i}$, and, for each regime
$s\in{\cal S}$, let
$h(\overline{U}_{i-1},\overline{L}_{i-1},\overline{A}_{i-1};s)$ be a version
of
${\mbox{E}}\\{k(L_{i})\mid\overline{U}_{i-1},\overline{L}_{i-1},\overline{A}_{i-1};s\\}$.
By (20) there exists $f(\overline{L}_{i-1},\overline{A}_{i-1})$ such that
$h(\overline{U}_{i-1},\overline{L}_{i-1},\overline{A}_{i-1};o)=f(\overline{L}_{i-1},\overline{A}_{i-1})\quad\mbox{a.s.
[$P_{o}$]}$ (23)
whence, from (8), for all $s\in{\cal S}$,
$h(\overline{U}_{i-1},\overline{L}_{i-1},\overline{A}_{i-1};o)=f(\overline{L}_{i-1},\overline{A}_{i-1})\quad\mbox{a.s.
[$P_{s}$]}.$ (24)
Also, from (13),
$L_{i}\,\mbox{$\perp\\!\\!\\!\perp$}\,\sigma\mid\overline{U}_{i-1},\overline{L}_{i-1},\overline{A}_{i-1}$
(25)
and so there exists
$g(\overline{U}_{i-1},\overline{L}_{i-1},\overline{A}_{i-1})$ such that, for
all $s\in{\cal S}$,
$h(\overline{U}_{i-1},\overline{L}_{i-1},\overline{A}_{i-1};s)=g(\overline{U}_{i-1},\overline{L}_{i-1},\overline{A}_{i-1})\quad\mbox{a.s.
[$P_{s}$]}.$ (26)
In particular,
$h(\overline{U}_{i-1},\overline{L}_{i-1},\overline{A}_{i-1};o)=g(\overline{U}_{i-1},\overline{L}_{i-1},\overline{A}_{i-1})\quad\mbox{a.s.
[$P_{o}$]},$ (27)
so that, again using (8),
$h(\overline{U}_{i-1},\overline{L}_{i-1},\overline{A}_{i-1};o)=g(\overline{U}_{i-1},\overline{L}_{i-1},\overline{A}_{i-1})\quad\mbox{a.s.
[$P_{s}$]}.$ (28)
Combining (24), (26) and (28), we obtain
$h(\overline{U}_{i-1},\overline{L}_{i-1},\overline{A}_{i-1};s)=f(\overline{L}_{i-1},\overline{A}_{i-1})\quad\mbox{a.s.
[$P_{s}$]}.$ (29)
Since this property holds for all $s\in{\cal S}$ and every bounded real
function $k(L_{i})$, we deduce
$L_{i}\,\mbox{$\perp\\!\\!\\!\perp$}\,(\overline{U}_{i-1},\sigma)\mid(\overline{L}_{i-1},\overline{A}_{i-1})$
(30)
from which both (21) and (22) follow.
It is worth noting that we do not need the full force of extended stability
for the above proof, but only (25). In particular, we could allow arbitrary
dependence of $U_{i}$ on any earlier variables, including $\sigma$. We note
further that the above proof makes essential use of the extended positivity
property of Definition 4: (21) can not be deduced from extended stability and
Condition 6.3 making use of the standard conditional independence axioms Dawid
(1998); Pearl and Paz (1987); Dawid (2001) alone.
Although we can certainly deduce simple stability when we can assume the
conditions of either Theorem 6.1 or Theorem 6.3, it can also arise our of
extended stability in other ways. For example, this can be so when Condition
6.2 holds for some subsets of $\overline{U}_{i}$, while Condition 6.3 holds
for some subsets of $\overline{U}_{i-1}$. Such cases are addressed by
Corollaries 4.1 and 4.2 of Robins (1997); we give examples in § 7.2.3 below.
## 7 Influence diagrams
As previously mentioned, it is often helpful (though never essential) to
represent and manipulate conditional independence properties graphically,
using the formalism of influence diagrams (IDs). In particular, when including
unobserved variables $\cal U$ and assuming extended stability, we can often
deduce directly from graph-theoretic separation properties whether simple
stability holds.
### 7.1 Semantics
Here we very briefly describe the semantics of IDs, and show how they can
facilitate logical arguments by displaying implied properties in a
particularly transparent form. We shall use the theory and notation of Cowell
et al. (1999) and Dawid (2002) in relation to directed acyclic graphs (DAGs)
and IDs, and their application to probability and decision models. The reader
is referred to these sources for more details.
For any DAG or ID ${\cal D}$, its moral graph, or moralization, is the
undirected graph ${\rm mo}({\cal D})$ in which first an edge is inserted
between any unlinked parents of a common child in ${\cal D}$, and then all
directions are ignored. For any set $S$ of nodes of ${\cal D}$ we denote the
smallest ancestral subgraph of ${\cal D}$ containing $S$ by ${\rm an}_{\cal
D}(S)$, and its moralization by ${\rm man}_{\cal D}(S)$ (we may omit the
specification of ${\cal D}$ when this is clear). For sets $A,B,C$ of nodes of
${\cal D}$ we write $A\,\mbox{$\perp\\!\\!\\!\perp$}_{\mbox{\scriptsize${\cal
D}$}}\,B\mid C$, and say $C$ separates $A$ from $B$ (with respect to ${\cal
D}$) to mean that, in ${\rm man}(A\cup B\cup C)$, every path joining $A$ to
$B$ intersects $C$. Let ${\rm nd}(V)$ and ${\rm pa}(V)$ denote the non-
descendants and parents of a random node $V$, then it can be shown Lauritzen
et al. (1990); Dawid (2002) that, whenever a probability distribution or
decision problem is represented by ${\cal D}$, in the sense that for any such
$V$ the probabilistic conditional independence
$V\,\mbox{$\perp\\!\\!\\!\perp$}\,{\rm nd}(V)\mid{\rm pa}(V)$ holds, we have
$\mbox{$A\,\mbox{$\perp\\!\\!\\!\perp$}_{\mbox{\scriptsize${\cal D}$}}\,B\mid
C$}\Rightarrow\mbox{$A\,\mbox{$\perp\\!\\!\\!\perp$}\,B\mid C$}.$ (31)
This moralization criterion thus allows us to infer probabilistic independence
properties from purely graph-theoretic separation properties.555An
alternative, and entirely equivalent, approach can be based on the
‘$d$-separation criterion’ Verma and Pearl (1990); Pearl (2009). We have found
(31) more straightforward to understand and apply.
While the above allows us to read off conditional independencies from a DAG,
we can, conversely, construct an ID ${\cal D}$ from a given collection of
joint distributions over the domain variables (one for each regime) in the
following way.
The node-set is given by ${\cal V}=\\{\sigma\\}\cup{\cal L}\cup{\cal
U}\cup{\cal A}$. The graph has random (round) nodes for all the domain
variables, and a founder decision (square) node for $\sigma$. The ordering
given by the extended information base ${\cal I}^{\prime}$ induces an ordering
on ${\cal V}$ such that any nodes in the (possibly empty) sets $L_{i}$,
$U_{i}$ come after $A_{i-1}$ and before $A_{i}$, and $L_{N+1}\equiv Y$ is
last. In addition we require the node $\sigma$ to be prior to any domain
variables in this ordering. With each node $\nu\in{\cal V}_{0}:={\cal
V}\setminus\\{\sigma\\}$ is associated its collection of conditional
distributions, given values for all its predecessors, ${\rm pre}(\nu)$, in the
ordering (including, in particular, specification of the relevant regime).
For each such $\nu$ we will have a conditional independence (CI) property of
the form:
$C(\nu):\mbox{$\nu\,\mbox{$\perp\\!\\!\\!\perp$}\,{\rm pre}(\nu)\mid{\rm
pa}(\nu)$}$
where ${\rm pa}(\nu)$ is some given subset of ${\rm pre}(\nu)$. Thus $C(\nu)$
asserts that the distributions of $\nu$, given all its predecessors, in fact
only depends on the values of those in ${\rm pa}(\nu)$. Note that property
$C(\nu)$ will be vacuous, and can be omitted, when ${\rm pa}(\nu)={\rm
pre}(\nu)$. Such a collection, ${\cal C}$ say, of CI properties is termed
recursive. We represent ${\cal C}$ graphically by drawing an arrow into each
node $\nu\in{\cal V}_{0}$ from each member of its parent set ${\rm pa}(\nu)$,
and we associate with $\nu$ the ‘parent-child’ conditional probabilities of
the form $p(\nu=\nu^{*}\mid{\rm pa}(\nu)=pa^{*})$. The ID constructed in this
way will ensure that the joint distribution of the domain variables, in each
regime, satisfies any conditional independencies obtained by applying the
moralization criterion (31).
From this point on, when we use the terms ‘parents’, ‘ancestors’ etc., the
regime node $\sigma$ will be excluded from these sets. Also, while in general
the terms $L_{i}$, $U_{i}$ could each refer to a collection of variables, for
simplicity we shall consider only the case in which they represent just one
(or sometimes none), and so can be modelled (if present at all) by a single
node in the graph.
We emphasise that IDs are related to but distinct from ‘causal DAGs’ Spirtes,
Glymour and Scheines (2000); Pearl (1995). For a discussion see Dawid (2010)
and Didelez, Kreiner and Keiding (2010).
### 7.2 Extended stability
The extended stability property (13) embodies a recursive collection of CI
properties with respect to the ordering induced by the extended information
base. Consequently it can be faithfully expressed by an ID ${\cal D}$
satisfying:
###### Condition 7.1
The only arrows out of $\sigma$ in ${\cal D}$ are into ${\cal A}$.
Figure 2: Unobserved variables: $N=2$
For $N=2$ this is depicted in Figure 2. Note that the subgraph corresponding
to the domain variables is complete.
#### 7.2.1 Sequential randomization
With the ordering induced by the extended information base ${\cal
I}^{\prime}$, (13) and (15) together form a recursive collection ${\cal C}$ of
CI properties. Therefore the conditions of Theorem 6.1 can be faithfully
represented graphically in an ID ${\cal D}$, in which, for extended stability,
the only arrows out of $\sigma$ are into the $A$’s, while also, for sequential
randomization, there are no arrows into the $A$’s from the $U$’s. Thus
starting from Figure 2, for example, we simply delete all the arrows from a
$U$ to an $A$, so obtaining Figure 3.
Figure 3: ID showing sequential randomization.
We can now verify Theorem 6.1 using only graphical manipulations, as follows.
Since, under (13), the only children of $\sigma$ are action variables, and
under (15) no action variable can be a child of any unobservable variable, it
follows that in ${\rm man}(\sigma,\overline{L}_{i},\overline{A}_{i-1})$ there
will be no direct link between $\sigma$ and any $U\in{\cal U}$. A similar
argument shows that (13) implies that there is no direct link in ${\rm
man}(\sigma,\overline{L}_{i},\overline{A}_{i-1})$ between $\sigma$ and
$L_{i}$. It follows that every path from $L_{i}$ to $\sigma$ must pass through
one of the remaining variables, i.e.
$(\overline{L}_{i-1},\overline{A}_{i-1})$, demonstrating that
$L_{i}\,\mbox{$\perp\\!\\!\\!\perp$}_{\mbox{\scriptsize${\cal
D}$}}\,\sigma\mid(\overline{L}_{i-1},\overline{A}_{i-1})$ for
$i=1,\ldots,N+1$. Simple stability (6) now follows from (31).
#### 7.2.2 Sequential irrelevance
The case of sequential irrelevance is more subtle. This is because when we
combine extended stability (13) with sequential irrelevance (19) we do not
obtain a recursive collection of CI properties. Consequently this combined
collection of conditional independencies cannot be faithfully represented by
any ID.
It might be thought that, starting with an ID representing extended stability,
we could operate on it to incorporate (19) also simply by deleting all arrows
from $U_{i}$ into $L_{j}$ for $j>i$. Doing this to Figure 2 yields the ID of
Figure 4. However, that ID also represents the stronger property (30) (shown
by the absence of edges from $\sigma$ and $\overline{U}_{i-1}$ into $L_{i}$),
which does not follow from (13) and (19) without imposing further, non-
graphical conditions (as was done in Theorem 6.3). We can indeed read off the
stability property (6) from Figure 4, but while that graph thus displays
clearly the conclusion of Theorem 6.3, it does not supply an alternative
graphical proof.
Figure 4: ID implying sequential irrelevance.
By omitting some of the nodes and/or arrows in an ID, such as Figure 3 or
Figure 4, that already embodies either sequential randomization or sequential
irrelevance, we obtain simpler special cases with the same property. Two such
examples, starting from Figure 4, are given in Figure 5.
Figure 5: Specialisations of Figure 4
#### 7.2.3 Further examples
As mentioned before, we can have simple stability even when both sequential
randomization and sequential irrelevance (or more precisely, the conditions of
Theorems 6.1 and 6.3) fail. Two examples are given by the IDs of Figure 6.
Applying the moralisation criterion to the graphs, we verify, for example,
that in both IDs of Figure 6 simple stability is satisfied.
Figure 6: Alternative IDs displaying stability
In full generality it is easy to see, using Condition 7.1, that application of
the moralization criterion to ${\cal D}$ to check the simple stability
condition (6) is equivalent to checking that, for each $i$,
$\overline{L}_{i-1}$ satisfies Pearl’s back-door criterion Pearl (1995)
relative to $(\overline{A}_{i-1},L_{i})$. (Pearl only considers atomic
interventions, but our analysis shows that this condition also allows
identification of conditional interventions.)
#### 7.2.4 Positivity
Suppose that (whether by appealing to sequential randomization, or to
sequential irrelevance, or the back-door criterion, or otherwise) we have been
able to demonstrate simple stability with respect to an observable information
base. Suppose further that $e$ is a control strategy in the sense of Condition
6.1. It will now follow that we can use $G$-recursion, exactly as in § 5.3, to
identify the consequence of regime $e$ from data gathered under regime $o$ —
so long only as we can also ensure the positivity constraint of Definition 2.
It is easy to see that a sufficient condition for Definition 2 to hold is:
###### Condition 7.2 (Parent-child positivity)
For each $A\in{\cal A}$, and each configuration $(a,pa^{*})$ of $(A,{\rm
pa}_{{\cal D}}{(A)})$, $p(a\mid pa^{*};e)>0\Rightarrow p(a\mid pa^{*};o)>0$.
More generally, suppose that we specify, for each entry in each parent-child
conditional probability table for the ID ${\cal D}$, whether it is zero or
non-zero. We can then apply constraint propagation algorithms Dechter (2003)
to determine ${\cal E}$ and ${\cal O}$. One such method Dawid (1992) uses an
analogue of the computational method of probability propagation Cowell et al.
(1999). This generates a collection of ‘cliques’ (subsets of the variables)
with, for each clique, an assignment of 1 (meaning possible) or 0 (impossible)
to each configuration of its variables. Definition 2 will then hold if and
only if, for each clique containing $\sigma$, no entry changes from 0 to 1
when we change the value of $\sigma$ from $o$ to $e$.
## 8 A more general approach
The simple stability condition (6) requires that, for each $i$, the
conditional distribution of $L_{i}$, given the earlier variables
$(\overline{L}_{i-1},\overline{A}_{i-1})$, should be the same under both
regimes $o$ and $e$ — a strong assumption that, in certain problems, one might
be unwilling to accept directly, and unable to deduce, as in § 6.2, from more
acceptable assumptions. However, while we have shown that stability (together
with Definition 2) is sufficient to support $G$-recursion, it turns out not to
be necessary.
In this section we first give some very general conditions under which
$G$-recursion can be justified; then we consider their specific application to
models incorporating extended stability. Our analysis parallels parts of
Robins (1987) (see also Section 3.4 of Robins (1997)), in which the
‘sequential randomization’ assumption is relaxed. We consider the relation
between the two approaches in more detail in § 10.2.
Rather than work directly with (10) and (11), we combine them into the
following form:
$f(\overline{l}_{i-1},\overline{a}_{i-1})=\sum_{l_{i}}\sum_{a_{i}}\,p(l_{i}\mid\overline{l}_{i-1},\overline{a}_{i-1};o)\times
p(a_{i}\mid\overline{l}_{i},\overline{a}_{i-1}\,;\,e)\times
f(\overline{l}_{i},\overline{a}_{i}).$ (32)
To justify $G$-recursion it is enough to demonstrate the applicability of
(32).
### 8.1 $G$-recursion: General conditions
A primitive building block of our model is the specification of the
interventional conditional probabilities
$p(a_{i}\mid\overline{l}_{i},\overline{a}_{i-1}\,;\,e)$. We suppose that this
is well-defined (e.g. by deterministic functions or specified randomization)
at least for all $(\overline{l}_{i},\overline{a}_{i-1})\in{\cal O}$ ($1\leq
i\leq N$), even if $(\overline{l}_{i},\overline{a}_{i-1})\not\in{\cal E}$.
We introduce a function $\gamma:{\cal H}\rightarrow\\{0,1\\}$ defined by:
$\gamma(h):=\left\\{\begin{array}[c]{ll}1&\mbox{if }h\in{\cal O}\mbox{ and
}\prod_{j=1}^{i}p(a_{j}\mid\overline{l}_{j},\overline{a}_{j-1}\,;\,e)>0\\\
0&\mbox{otherwise.}\end{array}\right.$ (33)
In (33), $i$ is the highest index of an action variable appearing in $h$, i.e.
$h=(\overline{l}_{i},\overline{a}_{i})$ or
$(\overline{l}_{i+1},\overline{a}_{i})$. Note that if $h$ is an initial
segment of $h^{\prime}$, then $\gamma(h)=0\Rightarrow\gamma(h^{\prime})=0$.
We define:
$\Gamma:=\\{h\in{\cal H}:\gamma(h)=1\\}$ (34)
(so that, in particular, $\Gamma\subseteq{\cal O}$).
We now impose the following positivity condition in place of Definition 2:
###### Condition 8.1
For $1\leq i\leq N$, if $(\overline{l}_{i},\overline{a}_{i-1})$ is in
${\Gamma}$ and $p(a_{i}\mid\overline{l}_{i},\overline{a}_{i-1}\,;\,e)>0$, then
$(\overline{l}_{i},\overline{a}_{i})$ is in ${\cal O}$ (and thus in
${\Gamma}$).
This requires that, subsequent to any partial history
$(\overline{l}_{i},\overline{a}_{i-1})$ in ${\Gamma}$, if some value of the
next action variable can be generated by intervention, it can also arise
observationally.
Our approach now involves the construction, if possible, of a sequence of
joint distributions $p_{i}(\,\cdot\,)$ ($i=0,\ldots,N$) for all the variables
in the problem, such that
$p_{0}(y)\equiv p(y\,;\,e),$ (35)
and certain further properties hold, as described below. For maximum
applicability these are stated here in a very abstract and general form. Some
concrete cases where we can specify suitable $(p_{i})$ and verify that they
have the requisite properties are treated in § 8.2 and § 10.2 below.
Let the class of partial histories $h\in{\cal H}$ having positive probability
under $p_{i}$ be denoted by ${\cal B}_{i}$, and let ${\Gamma}_{i}:={\cal
B}_{i}\cap{\Gamma}$.
We require the following positivity property:
$(\overline{l}_{i},\overline{a}_{i})\in{\cal
B}_{i}\Leftrightarrow(\overline{l}_{i},\overline{a}_{i})\in{\cal O}.$ (36)
Since $\Gamma\subseteq{\cal O}$, from ‘$\Leftarrow$’ in (36) we readily deduce
$(\overline{l}_{i},\overline{a}_{i})\in\Gamma\Leftrightarrow(\overline{l}_{i},\overline{a}_{i})\in\Gamma_{i}.$
(37)
More substantively we require:
$\displaystyle p_{i-1}(l_{i}\mid\overline{l}_{i-1},\overline{a}_{i-1})$
$\displaystyle=$ $\displaystyle
p(l_{i}\mid\overline{l}_{i-1},\overline{a}_{i-1}\,;\,o)\quad(i=1,\ldots,N+1)$
(38) $\displaystyle p_{i-1}(a_{i}\mid\overline{l}_{i},\overline{a}_{i-1})$
$\displaystyle=$ $\displaystyle
p(a_{i}\mid\overline{l}_{i},\overline{a}_{i-1}\,;\,e)\quad(i=1,\ldots,N+1)$
(39) $\displaystyle p_{i-1}(y\mid\overline{l}_{i},\overline{a}_{i})$
$\displaystyle=$ $\displaystyle
p_{i}(y\mid\overline{l}_{i},\overline{a}_{i})\quad(i=1,\ldots,N)$ (40)
whenever, in each case, the conditioning partial history on the left-hand side
is in ${\Gamma}_{i-1}$ (in which case the conditional probabilities on both
sides are unambiguously defined).
Suppose now that such a collection of distributions $(p_{i})$ can be found.
Let ${\cal H}_{0}$ denote the set of all partial histories of the form
$(\overline{l}_{i},\overline{a}_{i})$ for some $i$. We define a function
$f:{\cal H}_{0}\rightarrow\Re$ by:
$f(h):=\gamma(h)\times{\mbox{E}}_{i}\\{k(Y)\mid h\\},$ (41)
for $h=(\overline{l}_{i},\overline{a}_{i})$, where ${\mbox{E}}_{i}$ denotes
expectation under $p_{i}$. We note that $f$ is well-defined, since
$\gamma(h)\neq 0\Rightarrow h\in{\cal O}$, whence $h\in{\cal B}_{i}$ by (36).
For $h=(\overline{l}_{N},\overline{a}_{N})$, if $\gamma(h)\neq 0$ then by (37)
$h\in\Gamma_{N}$, so that we can apply (38) for $i=N+1$ to see that:
$f(\overline{l}_{N},\overline{a}_{N})=\left\\{\begin{array}[c]{ll}{\mbox{E}}\\{k(Y)\mid\overline{l}_{N},\overline{a}_{N}\,;\,o\\}&\mbox{if
}(\overline{l}_{N},\overline{a}_{N})\in\Gamma\\\
0&\mbox{otherwise.}\end{array}\right.$ (42)
Also, by (35),
$f(\emptyset)=p(y\,;\,e).$ (43)
###### Lemma 5.
Under Condition 8.1 and properties (35)–(40), the $G$-recursion (32) holds for
the interpretation (41).
###### Proof 8.1.
If $\gamma(\overline{l}_{i-1},\overline{a}_{i-1})=0$ then both sides of (32)
are 0.
Otherwise $(\overline{l}_{i-1},\overline{a}_{i-1})$ is in $\Gamma$ and so, by
(37), in $\Gamma_{i-1}$. We have:
$\displaystyle f(\overline{l}_{i-1},\overline{a}_{i-1})$ $\displaystyle=$
$\displaystyle{\mbox{E}}_{i-1}\\{k(Y)\mid\overline{l}_{i-1},\overline{a}_{i-1}\\}$
$\displaystyle=$
$\displaystyle\sum_{l_{i}}\,\sum_{a_{i}}\,\,p_{i-1}(l_{i}\mid\overline{l}_{i-1},\overline{a}_{i-1})\times
p_{i-1}(a_{i}\mid\overline{l}_{i},\overline{a}_{i-1})\times{\mbox{E}}_{i-1}\\{k(Y)\mid\overline{l}_{i},\overline{a}_{i}\\}.$
(44)
Denote the three terms on the right-hand side of (44) by $T_{l}$, $T_{a}$,
$T_{y}$, respectively. By (38)
$T_{l}=p(l_{i}\mid\overline{l}_{i-1},\overline{a}_{i-1}\,;\,o)$. We do not
need to consider the other terms when $T_{l}$ is 0. Otherwise,
$(\overline{l}_{i},\overline{a}_{i-1})$ is in $\Gamma_{i-1}$. By (39), we now
have $T_{a}=p(a_{i}\mid\overline{l}_{i},\overline{a}_{i-1}\,;\,e)$. Again we
do not have to worry about $T_{y}$ unless $T_{a}$ is non-zero. In that case
$(\overline{l}_{i},\overline{a}_{i})$ is in ${\cal B}_{i-1}$ and also, by
Condition 8.1, in $\Gamma$, hence in $\Gamma_{i-1}$. We can now use (40) to
replace $T_{y}$ by
${\mbox{E}}_{i}\\{k(Y)\mid\overline{l}_{i},\overline{a}_{i}\\}=f(\overline{l}_{i},\overline{a}_{i})$,
and the result follows.
Starting from (42), we can thus apply $G$-recursion as given by (32), or
equivalently by (10) and (11), to compute $f(\emptyset)$ — which, by (43), is
just the desired consequence of regime $e$. In this computation we only need
consider partial histories in $\Gamma$. When $e$ is a deterministic strategy
we recover the form (12) of $G$-recursion.
Note that, for histories of intermediate length, the function $f$ defined by
(41) involves the constructed distributions $(p_{i})$, which need not have any
real-world interpretation. Note further that, in contrast to the case when
stability applies, even when we can use the above construction to compute the
marginal interventional distribution of the response variable $Y$, there is no
guarantee that we can identify the full joint interventional distribution of
$(\overline{L}_{N},\overline{A}_{N},Y)$. In particular, if the loss function
depends on variables other than $Y$ we may not be able to estimate the
expected loss of an interventional strategy on the basis of observational
data.
### 8.2 Extended stability
We now specialize the general approach of § 8.1 to problems exhibiting
extended stability, as in (13). This can be regarded as extending the analysis
of Pearl and Robins (1995) to handle dynamic regimes, as also considered by
Robins (1997).666Both these papers refer for the details to an unpublished
paper, Robins and Pearl (1996).
We aim to identify a graphical counterpart to the conditions of § 8.1, that
would allow us to apply $G$-recursion to this extended information base so as
to identify the effect of regime $e$ from observations made under $o$.
For the remainder of this section we consider a given information base ${\cal
I}^{\prime}$ that induces an ordering of the nodes of the influence diagram
${\cal D}$; in § 9 we consider the converse, i.e. how to find an ordering of
the information base from a given influence diagram ${\cal D}$ such that the
graphical check of § 8.2.1 succeeds.
We impose Condition 7.2. It is then easy to see that Condition 8.1 will hold
(and in fact $\Gamma={\cal E}$). We also impose Condition 6.1 on the control
strategy $e$.
For each $i=0,\ldots,N$, we now construct an artificial joint distribution
$p_{i}(\,\cdot\,)$ for all the domain variables as follows. The distribution
$p_{i}$ factors according to the ID ${\cal D}^{\prime}={\cal D}$ with the node
$\sigma$ removed. The parent-child tables for any variable $V\in{\cal
L}\cup{\cal U}$ are unchanged from the original ones for ${\cal D}$ (which do
not involve $\sigma$). That for any action variable $A_{j}$ for $j\leq i$ is
the same as for ${\cal D}$, conditional on $\sigma=o$; while that for $A_{j}$
($j>i$) is the same as for ${\cal D}$, conditional on $\sigma=e$.
With this definition, $p_{0}(\,\cdot\,)\equiv p(\,\cdot\,;\,e)$, so that (35)
holds. Properties (36), and (38) for $i\leq N$, hold because the joint
distribution of all variables up to and including $L_{i}$ is the same under
$p_{i-1}$ as under $p(\,\cdot\,;\,o)$; for (38) with $i=N+1$, when
$L_{N+1}\equiv Y$, we also use the fact that extended stability, i.e.
Condition 7.1, implies that the distribution of $Y$ given all earlier domain
variables is the same under both $e$ and $o$.
Finally (39) holds because, by construction, the parent-child distribution for
$A_{i}$ has the same specification for $p_{i-1}(\cdot)$ as for
$p(\,\cdot\,;e)$ — and, by Condition 6.1, ${\rm
pa}(A_{i})\subseteq(\overline{L}_{i},\overline{A}_{i-1})$.
#### 8.2.1 Graphical check
We have shown that, under Conditions 6.1 and 7.2, properties (35)–(39) hold
automatically for our above construction of $(p_{i})$. However, whether or not
(40) holds will depend on more specific conditional independence properties of
the problem under study. We now describe a graphical method based on IDs for
checking this property.
For each action node $A\in{\cal A}$ we identify two subsets, ${\rm
pa}_{o}{(A)}$ and ${\rm pa}_{e}{(A)}$, of ${\rm pa}_{{\cal D}}{(A)}$, such
that, when $\sigma=o$ [resp. $e$], the conditional distribution of $A$, given
its domain parents, can be chosen to depend only on ${\rm pa}_{o}{(A)}$ [resp.
${\rm pa}_{e}{(A)}$].
To ensure Condition 6.1, we suppose:
###### Condition 8.2
${\rm pa}_{e}{(A)}\subseteq{\cal L}\cup{\cal A}$.
In order to investigate (40) for a specific value of $i$, we now construct,
for $0\leq i\leq N+1$, a new ID ${\cal D}_{i}$ on ${\cal V}$, as follows. The
only arrow out of $\sigma$ (again a founder node) is now into $A_{i}$. For
$j<i$, the parent set of $A_{j}$ is ${\rm pa}_{o}{(A_{j})}$ with conditional
distributions determined as under $o$; for $j>i$ it is ${\rm
pa}_{e}{(A_{j})}$, with conditional distributions determined as under $e$;
finally, for $A_{i}$ it is $({\rm pa}(A_{i})\,;\,\sigma)$, with conditional
distributions exactly as in ${\cal D}$. Any domain variable $V\in{\cal
L}\cup{\cal U}$ has the same parent set ${\rm pa}(V)$ (which will not include
$\sigma$) and conditional distributions as in ${\cal D}$. We shall use ${\rm
an}_{i}(\cdot)$ to denote a minimal ancestral set in ${\cal D}_{i}$, with
similar usages of ${\rm nd}_{i}$, $\mbox{$\perp\\!\\!\\!\perp$}_{i}$, etc.
It is easy to see that the joint density of all the domain variables in ${\cal
D}_{0}={\cal D}_{e}$ is $p_{0}=p_{e}$; in ${\cal D}_{N+1}={\cal D}_{o}$ it is
$p_{N+1}=p_{o}$; while in ${\cal D}_{i}$, given $\sigma=o$ it is $p_{i-1}$,
and given $\sigma=e$ it is $p_{i}$. Thus (40) will certainly hold if
$Y\,\mbox{$\perp\\!\\!\\!\perp$}_{\mbox{\scriptsize$i$}}\,\sigma\mid(\overline{L}_{i},\overline{A}_{i})$
(45)
holds. We can easily check (45) by inspection of the graph ${\cal D}_{i}$.
Note that ${\cal D}_{0}$ is similar to the ‘manipulated’ DAG of Spirtes,
Glymour and Scheines (2000).
In summary we have shown the following:
###### Theorem 8.2.
Under Conditions 7.2 and 8.2, if the graphical separation property (45) holds
for each $i$, then we can compute the consequence of regime $e$ from data
gathered under regime $o$ by means of the $G$-recursion (32), starting with
$f_{N}$ as in (42), and ending with $f_{0}=p(y\,;\,e)$.
A variant of this approach is described in Robins (1997), and works as
follows. Let ${\cal D}_{i}^{\prime}$ be obtained from ${\cal D}_{i}$ by
omitting the node $\sigma$, and deleting all arrows out of $A_{i}$. Because
moralization links in ${\cal D}_{i}$ involving $\sigma$ can only be to
predecessors of $A_{i}$, it is not difficult to see there exists a path from
$Y$ to $\sigma$ avoiding $(\overline{L}_{i},\overline{A}_{i})$ in ${\rm
man}_{{\cal D}_{i}}(Y,\overline{L}_{i},\overline{A}_{i})$ if and only if there
exists such a path from $Y$ to ${\rm pa}(A_{i})$ in ${\rm man}_{{\cal
D}_{i}^{\prime}}(Y,\overline{L}_{i},\overline{A}_{i})$. And the latter
condition can in turn be seen to be equivalent to the existence, in that
graph, of a path from $Y$ to ${A_{i}}$ avoiding
$(\overline{L}_{i},\overline{A}_{i-1})$. Thus
$Y\,\mbox{$\perp\\!\\!\\!\perp$}_{\mbox{\scriptsize$i$}}\,\sigma\mid(\overline{L}_{i},\overline{A}_{i})$
if and only if $Y\,\mbox{$\perp\\!\\!\\!\perp$}_{\mbox{\scriptsize${\cal
D}_{i}^{\prime}$}}\,A_{i}\mid(\overline{L}_{i},\overline{A}_{i-1})$. Hence we
can prove (40) by demonstrating the latter property.
It is shown in Dawid and Didelez (2008) that, under certain further conditions
— informally, that each intermediate variable has some influence on the
response under the interventional regime — when the graphical method described
above succeeds we can deduce that the problem in fact exhibits simple
stability with respect to the observed information base.
### 8.3 Examples
#### 8.3.1 Stability
We first show that the conditions of § 5.2 are a special case of those of §
8.1, by verifying that the construction of § 8.2.1 works for the case of
simple stability, as represented by Figure 1. In this case the $(U_{i})$ are
absent, and, for each domain variable $V$, ${\rm pa}_{e}{(V)}={\rm
pa}_{o}{(V)}={\rm pre}(V)$. Thus ${\cal D}_{i}$ consists of the complete
directed graph on $(L_{1},A_{1},\ldots,L_{N},A_{N},Y)$, together with an
additional regime node $\sigma$ and an arrow from $\sigma$ to $A_{i}$. Figure
7 shows these graphs for the case $N=2$, and Figure 8 the corresponding graphs
${\cal D}_{i}^{\prime}$.
Figure 7: Influence diagrams ${\cal D}_{1}$, ${\cal D}_{2}$ for stability
($N=2$) Figure 8: Influence diagrams ${\cal D}_{1}^{\prime}$, ${\cal
D}_{2}^{\prime}$ for stability ($N=2$)
Since, after moralization of ${\cal D}_{i}$, $\sigma$ has direct links only
into $(\overline{L}_{i},\overline{A}_{i})$, any path in this moral graph
joining $Y$ to $\sigma$ must intersect $(\overline{L}_{i},\overline{A}_{i})$,
whence we deduce (40). Equivalently, there is no path in ${\cal
D}_{i}^{\prime}$ from $Y$ to $A_{i}$ avoiding
$(\overline{L}_{i},\overline{A}_{i-1})$. Hence we have confirmed that, when
stability holds, it is possible to construct a sequence of joint densities
$p_{i}$ satisfying (38)–(40).
#### 8.3.2 $G$-recursion without stability
More interesting is the possibility of applying the construction of § 8.2 to
justify $G$-recursion even in cases where simple stability does not hold. This
is illustrated by the following example, based on Pearl and Robins (1995) (and
see Robins (1987) and Robins (1997) for description of medical scenarios that
are reasonably captured by this example).
###### Example 8.1
Figure 9 shows a specific model incorporating extended stability for the
information base $(U_{1},A_{1},U_{2},L_{2},A_{2},Y)$ (with $L_{1}=\emptyset$).
Note that this does not embody simple stability, since moralization would
create a direct link between $\sigma$ and $U_{1}$, and hence a path
$L_{2}$—$U_{1}$—$\sigma$ that avoids $A_{1}$. We thus can not deduce
$L_{2}\,\mbox{$\perp\\!\\!\\!\perp$}\,\sigma\mid A_{1}$, as would be required
for simple stability.
Figure 9: An ID displaying non-stability
We use stippled arrows to represent independence under the control regime $e$.
Thus the stippled arrow from $U_{1}$ to $A_{1}$ in Figure 9 represents the
property
$\mbox{$A_{1}\,\mbox{$\perp\\!\\!\\!\perp$}\,U_{1}\mid\sigma=e$},$ (46)
which is (14) for $i=1$. (The equivalent property for $i=2$ is already implied
by the lack of any arrows from $U_{1}$ and $U_{2}$ to $A_{2}$).
The stippled arrow from $L_{2}$ to $A_{2}$ embodies an additionally assumed
property:
$\mbox{$A_{2}\,\mbox{$\perp\\!\\!\\!\perp$}\,L_{2}\mid(A_{1}\,;\,\sigma=e)$}.$
(47)
That is, we are supposing that interventional assignment of $A_{2}$ can only
depend (deterministically or stochastically) on the value chosen for the
previous treatment, $A_{1}$. This is a restriction on the type of
interventional strategy $e$ that we are considering. It will turn out that we
can identify the causal effect of $e$ from the observational data gathered
under $o$, using $G$-recursion, only for strategies $e$ of this special type.
In this problem we thus have ${\rm pa}_{o}{(A_{1})}=U_{1}$, ${\rm
pa}_{e}{(A_{1})}=\emptyset$, ${\rm pa}_{o}{(A_{2})}=(A_{1},L_{2})$, ${\rm
pa}_{e}{(A_{2})}=A_{1}$. The constructed IDs ${\cal D}_{1}$ and ${\cal D}_{2}$
are shown in Figure 10, and the variant forms ${\cal D}_{1}^{\prime}$ and
${\cal D}_{2}^{\prime}$ (Pearl and Robins, 1995, Figure 2) in Figure 11.
Figure 10: Influence diagrams ${\cal D}_{1}$, ${\cal D}_{2}$ for Figure 9
Figure 11: Influence diagrams ${\cal D}_{1}^{\prime}$, ${\cal D}_{2}^{\prime}$
for Figure 9 Figure 12: Relevant moral ancestral graphs, for ${\cal D}_{1}$
and ${\cal D}_{1}^{\prime}$
We first examine ${\cal D}_{1}$ to see if
$Y\,\mbox{$\perp\\!\\!\\!\perp$}_{\mbox{\scriptsize${\cal
D}_{1}$}}\,\sigma\mid A_{1}$. The relevant moral ancestral graph (see Figure
12) is easily seen to have the desired separation property: thus we have shown
(40) for $i=1$. Alternatively, from examination of the relevant moral
ancestral graph based on ${\cal D}_{1}^{\prime}$ we readily see the desired
property $Y\,\mbox{$\perp\\!\\!\\!\perp$}_{{\cal D}_{1}^{\prime}}A_{1}$. (Note
that this approach does not succeed if we allow $A_{2}$ to depend on $L_{2}$
under $e$, thus retaining an arrow from $L_{2}$ to $A_{2}$ and so making
$L_{2}$ an ancestor of $Y$ in ${\cal D}_{1}$: in the now larger relevant moral
ancestral graph formed from ${\cal D}_{1}$ we could then trace a path
$Y$—$U_{2}$—$U_{1}$—$\sigma$ from $Y$ to $\sigma$ avoiding $A_{1}$.)
Finally, since in ${\cal D}_{2}$ neither $U_{1}$ nor $U_{2}$ is a parent of
$A_{2}$, even after moralization there will be no direct link from $\sigma$ to
either $U_{1}$ or $U_{2}$: consequently any path from $Y$ to $\sigma$ will
have to intersect $(A_{1},L_{2},A_{2})$. Equivalently, we see that in ${\cal
D}_{2}^{\prime}$, after moralization (which adds a futher link between $U_{1}$
and $U_{2}$) every path from $Y$ to $A_{2}$ intersects $(A_{1},L_{2})$. We
deduce $Y\,\mbox{$\perp\\!\\!\\!\perp$}\,\sigma\mid(A_{1},L_{2},A_{2})$, i.e.
(40) for $i=2$.
If we now assume Conditions 7.2 and 6.1 then, all the required conditions
being satisfied, we will have justified use of $G$-recursion to identify the
consequences of an interventional regime $e$ of the specified form, from data
collected under the observational regime $o$. $\Box$
The graphical check illustrated above simplifies considerably in the case of
an unconditional interventional strategy $e$, where the values of the action
variables are determined in advance, as considered by Pearl and Robins (1995).
In this case ${\rm pa}_{e}{(A_{i})}=\emptyset$ for all $i$, and ${\cal D}_{i}$
is obtained from ${\cal D}$ by deleting all arrows into every $A_{j}$ with
$j>i$. Then ${\cal D}_{i}^{\prime}$ is obtained by further deleting $\sigma$
and all arrows out of $A_{i}$. However, if our aim is to compare strategies,
and ideally find an optimal one, it is necessary also to consider dynamic
strategies.
## 9 Constructing an admissible sequence
In order to apply the graphical check of § 8.2.1 we need to have the variables
already completely ordered. More generally, we could ask whether there exists
an ordering $(A_{1},\ldots,A_{N})$ of ${\cal A}$, and $(L_{1},\ldots,L_{N})$
of disjoint subsets of ${\cal L}$, such that we can apply the construction of
§ 8.2.1 to show (45). Somewhat more restricted, we might suppose an ordering
$(A_{1},\ldots,A_{N})$ already given, and look for a sequence
$(L_{1},\ldots,L_{N})$ to satisfy (45). Such a sequence will be termed
admissible. In this section we assume that a graphical representation of the
problem in form of an ID is given, and we note that by definition an
admissible sequence has to satisfy $\overline{L}_{i}\subseteq{\rm
nd}(A_{i},\ldots,A_{N})$. Below, we give conditions under which we can
determine whether such an admissible sequence exists, and construct one if it
does. We shall need some general properties of directed-graph separation from
Appendix A.
We impose the following conditions:
###### Condition 9.1
For all $i$,
${\rm pa}_{e}{(A_{i})}\subseteq{\rm pa}_{o}{(A_{i})}.$
This can always be ensured by redefining, if necessary, ${\rm
pa}_{o}{(A_{i})}$ as ${\rm pa}_{o}{(A_{i})}\cup{\rm pa}_{e}{(A_{i})}$, with
any added parents having no effect on the conditional probabilities for
$A_{i}$ under $o$.
###### Condition 9.2
Each action variable $A\in{\cal A}$ is an ancestor of $Y$ in ${\cal D}_{e}$.
In typical contexts Condition 9.2 will hold, since we would not normally
contemplate an intervention that has no effect on the response. Clearly when
Conditions 9.1 and 9.2 both hold every $A\in{\cal A}$ is also an ancestor of
$Y$ in ${\cal D}_{o}={\cal D}$.
Define, for $i=1,\ldots,N$:
$M_{i}:={\cal L}\cap{\rm nd}_{e}(A_{i},A_{i+1},\ldots,A_{N})\cap{\rm
an}_{i}(Y).$ (48)
We note that $M_{i-1}\subseteq M_{i}$. This follows from ${\rm
an}_{i-1}(Y)\subseteq{\rm an}_{i}(Y)$ which in turn holds because, by
Condition 9.1, the edge set of ${\cal D}_{i-1}$ is a subset of that of ${\cal
D}_{i}$.
Now let
$L^{*}_{i}:=M_{i}\setminus M_{i-1},$ (49)
so that $M_{i}=\bar{L}^{*}_{i}$. For the information sequence $(L^{*}_{i})$,
the total information taken into account up to time $i$, $M_{i}$, consists of
just those variables in ${\cal L}$ that are ancestors of $Y$ in ${\cal
D}_{i}$, but are not descendants of $A_{i}$ or any later actions.
The sequence $(L^{*}_{1},\ldots,L^{*}_{N})$ will be admissible if, for
$i=1,\ldots,N$,
$\mbox{$Y\,\mbox{$\perp\\!\\!\\!\perp$}_{\mbox{\scriptsize$i$}}\,\sigma\mid(M_{i},\overline{A}_{i})$}.$
(50)
Taking into account Condition 9.2 and (48), (50) requires that
$M_{i}\cup\overline{A}_{i}$ separate $Y$ from $\sigma$ in the undirected graph
${\cal G}_{i}$ obtained by moralizing the ancestral set of $Y$ in ${\cal
D}_{i}$. It is thus straightforward to check whether or not it holds. When it
does we shall call $i$ admissible.
The following result can be regarded as simultaneously simplifying,
generalizing, and rendering more operational that of Pearl and Robins (1995).
In particular, it supplies an explicit construction, while allowing for
conditional interventions.
###### Theorem 9.1.
Under Conditions 9.1 and 9.2, if any admissible sequence exists then
$(L_{1}^{*},\ldots,L_{N}^{*})$ is admissible.
That is: There exists an admissible sequence if and only if every $i$ is
admissible. In this case $(L^{*}_{1},\ldots,L^{*}_{N})$ is an admissible
sequence.
###### Proof 9.2.
Suppose that there exists some admissible sequence $(L_{1},\ldots,L_{N})$.
Then, for each $i$,
$\mbox{$Y\,\mbox{$\perp\\!\\!\\!\perp$}_{\mbox{\scriptsize$i$}}\,\sigma\mid\overline{L}_{i}\cup\overline{A}_{i}$}.$
(51)
By Lemma A.3, this graph-theoretical separation continues to hold if we
intersect the conditioning set with $({\rm an}_{i}(Y),\sigma)$. Since, by
Condition 9.2, $\overline{A}_{i}\subseteq{\rm an}_{i}(Y)$, we obtain
$\mbox{$Y\,\mbox{$\perp\\!\\!\\!\perp$}_{\mbox{\scriptsize$i$}}\,\sigma\mid\left(\overline{L}_{i}\cap{\rm
an}_{i}(Y)\right)\cup\overline{A}_{i}$}.$ (52)
But, $\overline{L}_{i}\subseteq{\cal L}\cap{\rm nd}_{e}(A_{i},\ldots,A_{N})$;
and thus $\overline{L}_{i}\cap{\rm an}_{i}(Y)\subseteq M_{i}$. Hence, by Lemma
A.3, (50) holds, and the result follows.
###### Example 9.1
(We are indebted to Susan Murphy for this example.) In the problem represented
in Figure 13, it may be checked that the ‘obvious’ choice
$L_{1}=\\{X\\},L_{2}=\\{Z\\}$ is not an admissible sequence. Using the method
above yields $L^{*}_{1}=\\{X,Z\\},L^{*}_{2}=\emptyset$, which is admissible
(indeed, yields simple stability, as may either be checked directly, or
deduced from Theorem 2 in Dawid and Didelez (2008)).
Figure 13: Finding an admissible sequence
$\Box$
### 9.1 Finding a better sequence
While the above procedure will always construct an admissible sequence
$(L_{1},\ldots,L_{N})$ when one exists, that might not be the best possible.
Thus in Figure 14, with ${\cal L}=\\{X,Z\\}$, we find
$L_{1}^{*}=\\{Z\\},L_{2}^{*}=\\{X\\}$. These satisfy (50), so that the
sequence $\\{L_{1}^{*},L_{2}^{*}\\}$ is admissible. However a smaller
admissible sequence is given by $L_{1}=\emptyset,L_{2}=\\{X\\}$.
Figure 14: A choice of admissible sequences
If we had initially regarded $Z$ as unobservable, so taking ${\cal
L}=\\{X\\}$, we would have found this smaller sequence. However in general we
would need hindsight or good fortune to start off with such a minimal
specification of ${\cal L}$.
Even without redefining ${\cal L}$, however, we can often improve on the
sequence given by (49). At each stage $i$ we first check (50). If this fails
we abort the process. Otherwise, sequentially choose $L_{i}$ to be any subset
of $M_{i}$, disjoint from $\overline{L}_{i-1}$, such that (51) holds. (Since,
by (50), (51) holds for the choice $L_{i}=M_{i}\setminus\overline{L}_{i-1}$,
such a set must exist.) Then (if the process is never aborted) we shall have
constructed an admissible sequence $(L_{i})$, improving on $(L^{*}_{i})$ in
the sense that $\overline{L}_{i}\subseteq\overline{L}^{*}_{i}$.
Ideally we would want the set $L_{i}$ to be small. When each $L_{i}$ is
minimal, in the sense that no proper subset of $L_{i}$ satisfies (51), we
obtain a generalization of the method of Pearl and Robins (1995) for
constructing a minimal admissible sequence. However in large problems the
search for such a minimal $L_{i}$ can be computationally non-trivial, and we
may have to be satisfied with some other choices for the $(L_{i})$. Minimality
is in any case not a requirement for admissibility.
### 9.2 Admissible orderings of ${\cal A}$
In general there will be several orderings of ${\cal A}$ possible. It can then
happen that an admissible sequence $(L_{1},\ldots,L_{N})$ exists for one
ordering of ${\cal A}$ (which we may then likewise call admissible), but not
for another.
###### Example 9.2
In the ID of Figure 15, ${\cal U}=\\{U\\}$, ${\cal L}=\\{L\\}$, ${\cal
A}=\\{A,B\\}$. Note that $A\,\mbox{$\perp\\!\\!\\!\perp$}\,B$ under either
regime. Both $A_{1}=A,A_{2}=B$ and $A_{1}=B,A_{2}=A$ are possible orderings of
${\cal A}$. For the former choice we find $M_{1}=\emptyset$; then (50) for
$i=1$ becomes $Y\,\mbox{$\perp\\!\\!\\!\perp$}_{\mbox{\scriptsize${\cal
D}_{A}$}}\,\sigma\mid A$, where ${\cal D}_{A}$ is ${\cal D}$ with the arrow
from $\sigma$ to $B$ removed. Since this is easily seen to fail (moralization
creates a link between $U$ and $\sigma$), Theorem 9.1 implies that there can
be no admissible sequence to support $G$-recursion. However if we take
$A_{1}=B,A_{2}=A$, we obtain $M_{1}=\emptyset$, $M_{2}=\\{L\\}$, and (50)
becomes $Y\,\mbox{$\perp\\!\\!\\!\perp$}_{\mbox{\scriptsize${\cal
D}_{B}$}}\,\sigma\mid B$ for $i=1$,
$Y\,\mbox{$\perp\\!\\!\\!\perp$}_{\mbox{\scriptsize${\cal
D}_{A}$}}\,\sigma\mid(B,L,A)$ for $i=2$, where ${\cal D}_{B}$ is ${\cal D}$
with the arrows into $A$ from both $\sigma$ and $U$ removed. Both of these
properties are easily confirmed to hold. We can thus (under suitable
positivity conditions) apply $G$-recursion with respect to the admissible
ordering $(B,L,A)$. $\Box$
As yet we do not have a method that will automatically identify an admissible
ordering of ${\cal A}$ when one exists.
Figure 15: Unordered actions
## 10 Potential response models
In this section, we examine the relationship between the potential response
(PR) approach to dynamic treatments and our own decision-theoretic one.
The PR approach typically confines attention to non-randomized, though
possibly dynamic, strategies. Such a strategy is defined by a function $g$ on
the set of all ‘partial $L$-histories’ of the form $(\overline{l}_{i})$
($1\leq i\leq N)$, such that, for each $i$, $g(\overline{l}_{i})$ is one of
the available options for $A_{i}$. We shall write
$\overline{g}(\overline{l}_{i})$ for the sequence
$(g(l_{1}),g(l_{1},l_{2}),\ldots,g(\overline{l}_{i}))$. Under this strategy,
if at time $i$ we have observed $\overline{L}_{i}=\overline{l}_{i}$, the next
action will be $A_{i}=g(\overline{l}_{i})$.
We henceforth confine attention to a pair of regimes ${\cal S}=\\{o,e\\}$,
where $o$ is observational, while $e$ is a non-randomized strategy, determined
by a given function $g$ as described above.
### 10.1 Potential responses and stability
We first interpret and analyse the model introduced by Robins (1986) (see also
Robins (1997), Section 3.3; Robins (2000); Murphy (2003)).
We need to introduce, for each regime $s\in\\{o,e\\}$, a collection of
‘potential variables’
$\Pi_{s}:=(L_{s,1},A_{s,1},\ldots,L_{s,N},A_{s,N},L_{s,N+1}\equiv Y_{s})$. It
is supposed that, when regime $s$ is operating, the actual observable
variables in the problem, $(L_{1},A_{1},\ldots,L_{N},A_{N},Y)$, will be those
in $\Pi_{s}$.
Note that, by the definition of $e$, we have the functional constraint
$A_{e,i}=g(\overline{L}_{e,i})\quad(i=1,\ldots,N).$ (53)
All the potential variables, across both regimes, are regarded as having
simultaneous existence, their values being unaffected by which regime is
actually followed.777Note, as a matter of logic, that if we follow $e$ we
shall not be able to observe e.g. $Y_{o}$ (though see the note after Condition
10.2 below). This is a version of the so-called ‘fundamental problem of causal
inference’ Holland (1986) which has been critically discussed by Dawid (2000).
The effect of following regime $s$ is thus to uncover the values of some of
these, viz. those in $\Pi_{s}$, while hiding others.
This collection of all potential observables across both regimes is further
considered to have a joint distribution (respecting the logical constraints
(53)), whose density we denote by $p(\cdot)$. This distribution is supposed
unaffected by which regime is in operation: all this can do is change the
relationship between potential and actual variables.
Since, under $e$, $Y\equiv Y_{e}$, the consequence of the interventional
strategy $e$ is simply the marginal distribution of $Y_{e}$. Our aim is to
identify this distribution from observations made under regime $o$.
It can be shown directly that this can be effected by means of the
$G$-recursion formula under the following conditions:
###### Condition 10.1 (Positivity)
Whenever $p(\overline{L}_{o,N}=\overline{l}_{N})>0$,
$p(\overline{A}_{o,N}=\overline{g}(\overline{l}_{N})\mid\overline{L}_{o,N}=\overline{l}_{N})>0.$
That is, in the observational regime, for any set of values $\overline{l}_{N}$
of the variables $\overline{L}_{N}$ that can arise with positive probability,
there is a positive probability that the actions taken will be those specified
by $e$.
###### Condition 10.2 (Consistency)
If $\overline{A}_{o,i}=\overline{g}(\overline{L}_{o,i})$, then
$L_{o,i+1}=L_{e,i+1}$ $(i=0,\ldots,N)$.
(Note that for $i=0$ the antecedent of this condition is vacuously satisfied,
while for $i=N$ its conclusion is $Y_{o}=Y_{e}$.)
That is, if, in the observational regime, we happen to obtain a partial
history $(\overline{l}_{i},\overline{a}_{i})$ that could also be obtained
under the operation of $e$, then we will next observe the identical variable
$L_{e,i+1}$ that would have been observed if we had been operating $e$. (This
condition of course imposes further logical constraints on the joint
distribution $p$).
###### Condition 10.3 (Sequential ignorability)
Whenever $p(\overline{L}_{o,i}=\overline{l}_{i})>0,$
$\mbox{$A_{o,i}\,\mbox{$\perp\\!\\!\\!\perp$}\,\overline{L}_{e}^{i+1}\mid(\overline{L}_{o,i}=\overline{l}_{i},\overline{A}_{o,i-1}=\overline{g}(\overline{l}_{i-1}))$}\quad(i=1,\ldots,N),$
(where $\overline{L}_{e}^{j}:=(L_{e,j},L_{e,j+1},\ldots,L_{e,N},Y_{e})$).
That is, in the observational regime, given any partial history consistent
with the operation of $e$, the next action is independent of all the future
potential observables associated with $e$.888This is sometimes expressed in a
stronger form that drops the restriction to future variables, so replacing
$L_{e}^{i+1}$ by $(\overline{L}_{e,N},Y_{e})$ Robins (2000).
#### 10.1.1 Connexions
We now consider the relationship between the above approach and that of § 5.2,
which founds $G$-recursion on the stability property (6). We will show that
Conditions 10.1, 10.2 and 10.3 imply our conditions in § 5.2. Our reasoning
is, in spirit, very similar to Theorem 3.1 of Robins (1997) (see also Robins
(1986), Theorem 4.1).
###### Lemma 6.
If Conditions 10.2 and 10.3 hold, then for any sequence
$\overline{l}_{N+1}=(l_{1},\ldots,l_{N},y)$ such that
$p(\overline{L}_{e,N}=\overline{l}_{N})>0$,
$p(\overline{L}_{e}^{i+1}=\overline{l}^{i+1}\mid\overline{L}_{e,i}=\overline{l}_{i},\overline{A}_{o,i}=\overline{g}(\overline{l}_{i}))=p(\overline{L}_{e}^{i+1}=\overline{l}^{i+1}\mid\overline{L}_{e,i}=\overline{l}_{i})$
(54)
for $i=0,\ldots,N$.
###### Proof 10.1.
First note that, from Condition 10.2, when
$\overline{A}_{o,i}=\overline{g}(\overline{l}_{i})$,
$\overline{L}_{o,i+1}=\overline{l}_{i+1}$ is equivalent to
$\overline{L}_{e,i+1}=\overline{l}_{i+1}$. So from Condition 10.3
$\mbox{$A_{o,i+1}\,\mbox{$\perp\\!\\!\\!\perp$}\,\overline{L}_{e}^{i+2}\mid(\overline{L}_{e,i+1}=\overline{l}_{i+1},\overline{A}_{o,i}=\overline{g}(\overline{l}_{i}))$}.$
(55)
We now show (54) by induction on $i$.
It holds trivially for $i=0$. Suppose then it holds for $i$. Conditioning both
sides on $L_{e,i+1}=l_{i+1}$ then yields
$p(\overline{L}_{e}^{i+2}=\overline{l}^{i+2}\mid\overline{L}_{e,i+1}=\overline{l}_{i+1},\overline{A}_{o,i}=\overline{g}(\overline{l}_{i}))=p(\overline{L}_{e}^{i+2}=\overline{l}^{i+2}\mid\overline{L}_{e,i+1}=\overline{l}_{i+1}).$
But from (55) we have
$p(\overline{L}_{e}^{i+2}=\overline{l}^{i+2}\mid\overline{L}_{e,i+1}=\overline{l}_{i+1},\overline{A}_{o,i+1}=\overline{g}(\overline{l}_{i+1}))$
$=p(\overline{L}_{e}^{i+2}=\overline{l}^{i+2}\mid\overline{L}_{e,i+1}=\overline{l}_{i+1},\overline{A}_{o,i}=\overline{g}(\overline{l}_{i})).$
Hence (54) holds with $i$ replaced by $i+1$ and the induction proceeds.
###### Theorem 6.
If Conditions 10.2 and 10.3 hold, then so does the stability condition (6).
###### Proof 10.3.
Because of (53), and the restriction immediately below the density
interpretation (7) of (6), it is enough to show that
$p(L_{e,i+1}=l_{i+1}\mid\overline{L}_{e,i}=\overline{l}_{i},\overline{A}_{e,i}=\overline{g}(\overline{l}_{i}))=p(L_{o,i+1}=l_{i+1}\mid\overline{L}_{o,i}=\overline{l}_{i},\overline{A}_{o,i}=\overline{g}(\overline{l}_{i}))$.
But, again by (53),
$p(L_{e,i+1}=l_{i+1}\mid\overline{L}_{e,i}=\overline{l}_{i},\overline{A}_{e,i}=\overline{g}(\overline{l}_{i}))=p(L_{e,i+1}=l_{i+1}\mid\overline{L}_{e,i}=\overline{l}_{i})$.
By Lemma 6, this is the same as
$p(L_{e,{i+1}}=l_{i+1}\mid\overline{L}_{e,i}=\overline{l}_{i},\overline{A}_{o,i}=\overline{g}(\overline{l}_{i}))$,
and by Condition 10.2 this is in turn the same as
$p(L_{o,i+1}=l_{i+1}\mid\overline{L}_{e,i}=\overline{l}_{i},\overline{A}_{o,i}=\overline{g}(\overline{l}_{i}))$.
Finally, in the light of (53), it is easy to see that Condition 10.1 implies
positivity as given by Definition 2.
In summary, whenever the conditions usually used to justify $G$-recursion in
the potential response framework hold, so will our own (as in § 5.2). But our
conditions are more general in that they do not require the existence of, let
alone any probabilistic relationships between, potential responses under
different regimes; and can, moreover, just as easily handle randomized
interventional strategies, which are more problematic for the PR approach.
### 10.2 Potential responses without stability
A more general approach Robins (1987, 1989); Robins, Hernán and Brumback
(2000); Gill and Robins (2001); Lok et al. (2004) within the potential
response framework replaces Conditions 10.2 and 10.3 with the following
variants:
###### Condition 10.4
If $\overline{A}_{o,N}=\overline{g}(\overline{L}_{o,N})$, then $Y_{o}=Y_{e}$.
That is, if in the observational regime we happen to observe a complete
history that could have arisen under the operation of $e$, then the response
will be identical to what we would have observed had we been operating $e$.
###### Condition 10.5
$\mbox{$A_{o,i}\,\mbox{$\perp\\!\\!\\!\perp$}\,Y_{e}\mid(\overline{L}_{o,i}=\overline{l}_{i},\overline{A}_{o,i-1}=\overline{g}(\overline{l}_{i-1}))$}\quad(i=1,\ldots,N).$
That is, if, in the observational strategy, we happen to observe a partial
history that could have arisen under the operation of $e$, then the next
action is independent of the potential response under $e$.
Condition 10.4 implies, and can in fact be replaced by:
###### Condition 10.6
Given
$(\overline{L}_{o,N}=\overline{l}_{N},\overline{A}_{o,N}=\overline{g}(\overline{l}_{N}))$,
$Y_{o}$ and $Y_{e}$ have the same conditional distribution.
The deterministic strategy $e$ is termed evaluable if, for each $i$:
###### Condition 10.7
$p\left(\overline{L}_{o,i}=\overline{l}_{i},\overline{A}_{o,i-1}=g(\overline{l}_{i-1})\right)>0\Rightarrow
p\left(\overline{L}_{o,i}=\overline{l}_{i},\overline{A}_{o,i}=g(\overline{l}_{i})\right)>0.$
Note that Conditions 10.4–10.7 make no mention of potential intermediate
variables $(\overline{L}_{e,N},\overline{A}_{e,N})$ under $e$ — though they do
involve both versions, $Y_{o}$ and $Y_{e}$, of the response. The relevant
variables in the problem can thus be taken as
$(\overline{L}_{o,N},\overline{A}_{o,N},Y_{o},Y_{e})$, having a joint
distribution $p$ say.
Conditions 10.5 and 10.6 are weaker than those of § 10.1 as none of the
variables under strategy $e$ other than $Y_{e}$ are involved. Note that, for
example, it is not required that, when an observational partial history could
have arisen under $e$, that is the history that would have so arisen; but even
so, constraints on $Y_{e}$ are then imposed.
#### 10.2.1 Connexions
It is straightforward to show directly that, when Conditions 10.5, 10.6 and
10.7 hold, the marginal distribution of $Y_{e}$, or the interventional
consequence ${\mbox{E}}\\{k(Y_{e})\\}$, can be identified by the $G$-recursion
(12). We now show how this approach can be related to our own decision-
theoretic one. Specifically, we shall show that, when the above conditions
hold, so do those of § 8.1 (see also Theorem 3.2 of Robins (1997)).
Condition 10.7 is just Condition 8.1 specialized to the case of the
deterministic strategy $e$.
To continue, we construct a fictitious distribution $p_{i}(\cdot)$
$(i=0,\ldots,N)$, for variables $(L_{1},A_{1},\ldots,L_{N},A_{N},Y)$, as
follows.
###### Definition 7.
The distribution $p_{i}$ of $(L_{1},A_{1},\ldots,L_{N},A_{N},Y)$ is defined as
the distribution under $p$ of
$(L_{o,1},A_{o,1},\ldots,L_{o,i},A_{o,i},L_{o,i+1},g(\overline{L}_{o,i+1}),\ldots,L_{o,N},g(\overline{L}_{o,N}),Y_{e})$.
Thus
$\displaystyle
p_{i}(\overline{L}_{N}=\overline{l}_{N},\overline{A}_{N}=\overline{a}_{N},Y=y):=$
(58)
$\displaystyle\left\\{\begin{array}[c]{ll}p(\overline{L}_{o,N}=\overline{l}_{N},\overline{A}_{o,i}=\overline{a}_{i},Y_{e}=y)&\mbox{if
}a_{i+1}=g(\overline{l}_{i+1}),\ldots,a_{N}=g(\overline{l}_{N})\\\
0&\mbox{otherwise.}\end{array}\right.$
Note that this construction of $p_{i}$ is quite different from that developed,
in a different context, in § 8.2. In particular, the marginal joint
distribution of $(\overline{L}_{N})$ is, for every $p_{i}$, the same as under
$p_{o}$.
Equation (35) follows trivially from Definition 7.
As in § 8.2, Properties (36), and (38) for $i\leq N$, hold because the joint
distribution of all variables up to and including $L_{i}$ is the same under
$p_{i-1}$ as under $p(\,\cdot\,;\,o)$; while for (38) with $i=N+1$, when
$L_{N+1}\equiv Y$, we also use Condition 10.6.
Equation (39) holds since the distribution on either side is concentrated on
$g(\overline{l}_{i})$.
Finally we show (40).
We only need this for $(\overline{l}_{i},\overline{a}_{i})\in\Gamma_{i-1}$.
Since then $(\overline{l}_{i},\overline{a}_{i})\in\Gamma$, we must by (33)
have $p(a_{j}\mid\overline{l}_{j},\overline{a}_{j-1}\,;\,e)>0$ ($1\leq j\leq
i$), which in virtue of the deterministic nature of strategy $e$ requires
$\overline{a}_{i}=\overline{g}(\overline{l}_{i})$; and then the additional
condition $(\overline{l}_{i},\overline{a}_{i})\in{\cal O}$ becomes
$p(\overline{L}_{o,i}=\overline{l}_{i},\overline{A}_{o,i}=\overline{g}(\overline{l}_{i}))>0$.
So in this case (40) becomes:
$p(Y_{e}=y\mid\overline{L}_{o,i}=\overline{l}_{i},\overline{A}_{o,i-1}=\overline{g}(\overline{l}_{i-1}))=p(Y_{e}=y\mid\overline{L}_{o,i}=\overline{l}_{i},\overline{A}_{o,i}=\overline{g}(\overline{l}_{i})).$
(59)
But this is an immediate consequence of Condition 10.5.
In summary, we have shown:
###### Theorem 7.
Under Conditions 10.1, 10.6 and 10.5, and defining $p_{i}(\cdot)$ by
Definition 7, Conditions 2 and 8.1 and equations (35)–(40) are all satisfied.
From Lemma 5 we now deduce:
###### Corollary 8.
Under Conditions 10.5–10.7, the consequence of strategy $e$ can be recovered
using the $G$-recursion (12).
## 11 Discussion
### 11.1 What has been achieved?
In this work we have described and developed a fully decision-theoretic
approach to the problem of dynamic treatment assignment. The central issue
identified and addressed is the transfer of probabilistic information between
differing regimes. When justified, this can allow future policy analysis to
take appropriate account of previously gathered data.
Out of this approach we have developed an alternative derivation and
interpretation of Robins’s $G$-computation algorithm, relating it to the
fundamental ‘backward induction’ recursion algorithm of dynamic programming.
Moreover we have shown that this is applicable more generally, including to
problems involving randomized treatment decisions.
We have devoted some attention to the question of how one might justify the
simple stability property (6), or the more general conditions of Lemma 5. One
can attempt this by including unobservable variables into one’s reasoning, and
using influence diagram to check the desired properties by simple graphical
manipulations. However, as discussed in § 7.2.2, the graphical approach
sometimes imposes more restrictions than necessary, and an algebraic approach
based on manipulations of conditional independence properties can be more
general and powerful.
We have also broadened the application of the graphical approach of Pearl and
Robins (1995) to allow assessment of the effects of conditional interventions,
that are allowed to depend on the values of other variables in the problem.
This is a particularly natural requirement when we contemplate sequential
interventions, where it is clearly desirable to be able to respond
appropriately to the information obtained to date, and so naturally to
consider dynamic strategies. We have noted that the graphical expression of
condition (6) for simple stability is equivalent to sequential application of
Pearl’s back-door criterion, and that this allows identification by
$G$-recursion of the consequences of conditional interventions, not only for
the ultimate response $Y$ but also for every intermediate covariate $L_{i}$.
We have further noted that our graphical check for the more general case of §
8 is equivalent to that suggested by Robins (1997).
### 11.2 Syntax and semantics
An important pragmatic aspect of our approach is that, in order to apply it
sensibly, we have to be very clear about the real-world meaning of all the
variables (whether ‘random’ or ‘decision’) appearing in our formulae. Thus,
when considering some interventional regime, we need to understand exactly
what real-world interventions are involved: we can not assume that setting a
variable to a specific value in different ways, or in different contexts, will
have the same overall effects on the system studied — see Hernán and Taubman
(2008) for a discussion of these issues in the context of a specific
application. Whenever we consider arguments in favour of or against accepting
a condition such as stability or extended stability, we must do so in full
appreciation of the applied context and circumstances — there can be no purely
formal way of addressing such issues.
This emphasis on the semantics of our representations contrasts with that of
other popular approaches, such as causal interpretation of DAGs or the do-
calculus Pearl (2009), which appear to operate purely syntactically. However
that is an illusion, since those interpretations and manipulations are always
grounded in an already assumed formal representation of the problem (e.g. as a
DAG, or a set of structural equations). So until we have satisfied ourselves
that this representation truly does capture our understanding of the real-
world behaviour of our problem — in particular, that it correctly describes
the effects of the interventions we care about — there can be no reason to
have any faith in the results of any formal manipulations on it.
### 11.3 Statistical inference
We have not directly addressed problems of statistical inference. One might
want to estimate the consequences of some proposed sequential strategy, or
test a null hypothesis that no strategy is effective in controlling the
outcome. In principle one can estimate the ingredients of the $G$-recursion
formula, either parametrically or non-parametrically, from the available data,
and then (assuming simple stability, or the more general conditions of Lemma
5) apply it to supply estimates or tests of the effects of strategies of
interest. The proposal by Arjas and Saarela (2010) can be regarded as a
Bayesian version of $G$-computation. However, as pointed out by Robins and
Wasserman (1997), naïve use of parametric models for the required conditional
distributions can lead to a ‘null-paradox’, rendering it impossible to
discover that different strategies have the same consequences. Also, when
continuous variables are included, $G$-recursion can involve a large number of
nested integrals and become computationally impossible to implement. Hence we
find only a few instances where $G$-computation has been used for practical
data analysis Robins, Greenland and Hu (1999); Taubman et al. (2009). The
problems in applying $G$-recursion are exacerbated by the need, in many
practical applications, to choose a large set of covariates ${\cal L}$ so as
to justify the stability assumption. This makes the modelling task more
difficult and raises issues of robustness to misspecification. Such
considerations have motivated the introduction of marginal or nested
‘structural models’ Robins (1998); Robins, Hernán and Brumback (2000), as well
as doubly-robust methods Kang and Schafer (2007), avoiding the null–paradox.
Note that while $G$-recursion provides a likelihood-based approach to the
estimation of the consequence of a given strategy, these latter methods rely
on estimating equations. It should be straightforward to reinterpret these
models and analyses within a fully decision-theoretic framework, by
appropriate modelling of the intervention distributions $p(\,\cdot\,;s)$.
### 11.4 Optimal dynamic treatment strategies
Our work is motivated in part by the desire to compare a variety of sequential
treatment strategies so as to identify the best one. Recall that our set of
regimes is given by ${\cal S}=\\{o\\}\cup{\cal S}^{*}$, where $o$ is the
observational regime, and ${\cal S}^{*}$ is the set of interventional
strategies that we want to compare. If we want to apply $G$-recursion,
justifying it by simple stability as in § 5.3 or by the more general
conditions of Lemma 5, we need to ensure that the respective conditions hold
for all strategies $e\in{\cal S}^{*}$ that we want to compare. As we saw in §
8.3.2, this is not trivial: if ${\cal S}^{*}$ contains static as well as
dynamic strategies, in some situations the former may be identified while the
latter are not. In fact it follows from Dawid and Didelez (2008) that if want
to find an optimal strategy among all dynamic regimes, we will usually need
the restrictive requirement of simple stability to hold for all $e\in{\cal
S}^{*}$.
As mentioned in § 4, the standard dynamic programming routine for identifying
an optimal strategy can be regarded as a combination of $G$-recursion and
stagewise optimisation. Under conditions allowing $G$-recursion, this can in
principle be put directly into effect, after estimating all the required
distributional ingredients from the available data. In practice (as pointed
out by Robins (1986) and many others since), this quickly becomes infeasible,
especially if one wants to avoid parametric restrictions. This is because the
number of possible histories for which the optimal next decision has to be
determined at each stage of the backward induction recursion can grow
extremely rapidly with increasing number $N$ of time points and levels of
$(\overline{l}_{i},\overline{a}_{i-1})$.
Alternative approaches to the optimisation problem to sidestep this
computational complexity have been suggested. Murphy (2003) introduces a
method based on regret functions (see the discussion and application in
Rosthøj et al. (2006)), which is closely related to the structural nested
models of Robins (2004) (see Moodie, Richardson and Stephens (2007) for a
comparison of these two approaches). Henderson, Ansel and Alshibani (2010)
modify Murphy’s approach so as to be amenable to standard statistical model
checking procedures. However, all these alternative methods for finding
optimal dynamic treatments rely on the same identification conditions
underlying $G$-computation, as well as on various additional (semi-)parametric
assumptions.
### 11.5 Complete identifiability
Simple stability, or the alternative conditions of Lemma 5, are sufficient
conditions allowing the use of $G$-recursion, and thereby identification of
the consequences of a given strategy. In recent years the Artificial
Intelligence community has devoted some effort to finding necessary as well as
sufficient conditions for the identifiability of consequences of interventions
Huang and Valtorta (2006); Shpitser and Pearl (2006a, b). These results rely
heavily on the assumptions encoded in causal DAGs or semi-Markovian causal
models. Even within this more restricted framework, the general question of
identifiability of dynamic treatment strategies seems still to be an open
problem (but see Tian (2008)).
### 11.6 Other problems
Many problems in causal inference, previously tackled using potential response
or causal DAG formulations, gain in clarity, simplicity and generality when
reformulated as problems of decision analysis. Specific topics that have been
fruitfully treated in this way include: confounding Dawid (2002); partial
compliance Dawid (2003); direct and indirect effects Didelez, Dawid and
Geneletti (2006); Geneletti (2007); identification of the effect of treatment
on the treated Geneletti and Dawid (2010); Mendelian randomization Didelez and
Sheehan (2007); Granger causality Eichler and Didelez (2010); and causal
inference under outcome-dependent sampling Didelez, Kreiner and Keiding
(2010). However there still remains a wide range of other issues in ‘causal
inference’ that we believe would benefit from the application of the decision-
theoretic viewpoint.
## Acknowledgment
We are indebted to Susan Murphy for stimulating this work and for many
valuable comments. We also want to thank Jamie Robins for helpful discussions.
Financial support from MRC Collaborative Project Grant G0601625 is gratefully
acknowledged.
## APPENDIX
## Appendix A Two lemmas on DAG-separation
Here we prove generalised versions of equations (8) and (9) (Lemma 1) of Pearl
and Robins (1995).
Let ${\cal D}$ be a DAG.
###### Lemma A.1.
$\mbox{$Y\,\mbox{$\perp\\!\\!\\!\perp$}_{\mbox{\scriptsize${\cal D}$}}\,X\mid
Z$}\Rightarrow\mbox{$Y\,\mbox{$\perp\\!\\!\\!\perp$}_{\mbox{\scriptsize${\cal
D}$}}\,X\mid Z^{*}$}$ (60)
whenever $Z\subseteq Z^{*}\subseteq{\rm an}(X\cup Y\cup Z)$.
###### Proof A.2.
Let ${\cal G}:={\rm man}(X\cup Y\cup Z)$; then also ${\cal G}={\rm man}(X\cup
Y\cup Z^{*})$. The left-hand side of (60) says that any path from $Y$ to $X$
in ${\cal G}$ intersects $Z$, whence it must also intersect the larger set
$Z^{*}$.
###### Lemma A.3.
$\mbox{$Y\,\mbox{$\perp\\!\\!\\!\perp$}_{\mbox{\scriptsize${\cal D}$}}\,X\mid
Z$}\Rightarrow\mbox{$Y\,\mbox{$\perp\\!\\!\\!\perp$}_{\mbox{\scriptsize${\cal
D}$}}\,X\mid Z^{*}$}$ (61)
whenever $Z^{*}=Z\cap A$ for $A$ an ancestral set in ${\cal D}$ containing
$X\cup Y$.
###### Proof A.4.
We first note that $(X\cup Y)\cup Z^{*}$ is a subset of $A$, since both its
terms are. Since $A$ is ancestral, it follows that
${\rm an}(X\cup Y\cup Z^{*})\subseteq A.$ (62)
Define ${\cal G}$ as above, and ${\cal G}^{\prime}:={\rm man}(X\cup Y\cup
Z^{*})$. Then both the node-set and edge-set for ${\cal G}^{\prime}$ are
subsets of the corresponding sets for ${\cal G}$, and hence the same property
holds for the path-set. Suppose the right-hand side of (61) fails. Then there
exists a path $\pi$ in ${\cal G}^{\prime}$ connecting $Y$ and $X$ and avoiding
$Z^{*}$; then $\pi$ is a path in ${\cal G}$ with the same property. Since
$\pi\subseteq{\cal G}^{\prime}$, if it intersects $Z$ anywhere it can only do
so at a point of ${\rm an}(X\cup Y\cup Z^{*})$ — and thus, by (62), at a point
in $A$, and hence in $Z^{*}$. Since this has been excluded, the result
follows.
## References
* Arjas and Parner (2004) [author] Arjas, EliasE. and Parner, JanJ. (2004). Causal reasoning from longitudinal data. Scandinavian Journal of Statistics 31 171–187. 2066247
* Arjas and Saarela (2010) [author] Arjas, EljaE. and Saarela, OlliO. (2010). Optimal dynamic regimes: Presenting a case for predictive inference. The International Journal of Biostatistics 6.
http://tinyurl.com/33dfssf 2602553
* Cowell et al. (1999) [author] Cowell, Robert G.R. G., Dawid, A. PhilipA. P., Lauritzen, Steffen LilholtS. L. and Spiegelhalter, David J.D. J. (1999). Probabilistic Networks and Expert Systems. Springer, New York. 1697175
* Dawid (1979) [author] Dawid, Alexander PhilipA. P. (1979). Conditional independence in statistical theory (with Discussion). Journal of the Royal Statistical Society, Series B 41 1–31. 0535541
* Dawid (1992) [author] Dawid, A. P.A. P. (1992). Applications of a general propagation algorithm for probabilistic expert systems. Statistics and Computing 2 25–36.
* Dawid (1998) [author] Dawid, Alexander PhilipA. P. (1998). Conditional independence. In Encyclopedia of Statistical Science (Update Volume 2) (SamuelS. Kotz, Campbell B.C. B. Read and David L.D. L. Banks, eds.) 146–155. Wiley-Interscience, New York.
* Dawid (2000) [author] Dawid, Alexander PhilipA. P. (2000). Causal inference without counterfactuals (with Discussion). Journal of the American Statistical Association 95 407–448. 1803167
* Dawid (2001) [author] Dawid, Alexander PhilipA. P. (2001). Separoids: A mathematical framework for conditional independence and irrelevance. Annals of Mathematics and Artificial Intelligence 32 335–372. 1859870
* Dawid (2002) [author] Dawid, Alexander PhilipA. P. (2002). Influence diagrams for causal modelling and inference. International Statistical Review 70 161–189. Corrigenda, ibid., 437\.
* Dawid (2003) [author] Dawid, Alexander PhilipA. P. (2003). Causal inference using influence diagrams: The problem of partial compliance (with Discussion). In Highly Structured Stochastic Systems (Peter J.P. J. Green, Nils LidN. L. Hjort and SylviaS. Richardson, eds.) 45–81. Oxford University Press. 2082406
* Dawid (2010) [author] Dawid, Alexander PhilipA. P. (2010). Beware of the DAG! In Proceedings of the NIPS 2008 Workshop on Causality. Journal of Machine Learning Research Workshop and Conference Proceedings (D.D. Janzing, II. Guyon and B.B. Schölkopf, eds.) 6 59–86. http://tinyurl.com/33va7tm
* Dawid and Didelez (2008) [author] Dawid, Alexander PhilipA. P. and Didelez, VanessaV. (2008). Identifying optimal sequential decisions. In Proceedings of the Twenty-Fourth Annual Conference on Uncertainty in Artificial Intelligence (UAI-08) (DavidD. McAllester and AnnA. Nicholson, eds.). 113-120. AUAI Press, Corvallis, Oregon. http://tinyurl.com/3899qpp
* Dechter (2003) [author] Dechter, RinaR. (2003). Constraint Processing. Morgan Kaufmann Publishers.
* Didelez, Dawid and Geneletti (2006) [author] Didelez, VanessaV., Dawid, Alexander PhilipA. P. and Geneletti, Sara GisellaS. G. (2006). Direct and indirect effects of sequential treatments. In Proceedings of the Twenty-Second Annual Conference on Uncertainty in Artificial Intelligence (UAI-06) (RinaR. Dechter and ThomasT. Richardson, eds.). 138-146. AUAI Press, Arlington, Virginia. http://tinyurl.com/32w3f4e
* Didelez, Kreiner and Keiding (2010) [author] Didelez, VanessaV., Kreiner, SvendS. and Keiding, NielsN. (2010). Graphical models for inference under outcome dependent sampling. Statistical Science (to appear).
* Didelez and Sheehan (2007) [author] Didelez, VanessaV. and Sheehan, Nuala S.N. S. (2007). Mendelian randomisation: Why epidemiology needs a formal language for causality. In Causality and Probability in the Sciences, (F.F. Russo and J.J. Williamson, eds.). Texts in Philosophy Series 5 263–292. College Publications, London.
* Eichler and Didelez (2010) [author] Eichler, MichaelM. and Didelez, VanessaV. (2010). Granger-causality and the effect of interventions in time series. Lifetime Data Analysis 16 3–32. 2575937
* Ferguson (1967) [author] Ferguson, Thomas S.T. S. (1967). Mathematical Statistics: A Decision Theoretic Approach. Academic Press, New York, London. 0215390
* Geneletti (2007) [author] Geneletti, Sara GisellaS. G. (2007). Identifying direct and indirect effects in a non–counterfactual framework. Journal of the Royal Statistical Society: Series B 69 199 -215. 2325272
* Geneletti and Dawid (2010) [author] Geneletti, Sara GisellaS. G. and Dawid, Alexander PhilipA. P. (2010). Defining and identifying the effect of treatment on the treated. In Causality in the Sciences (P. McKayP. M. Illari, F.F. Russo and J.J. Williamson, eds.) Oxford University Press (to appear).
* Gill and Robins (2001) [author] Gill, Richard D.R. D. and Robins, James M.J. M. (2001). Causal inference for complex longitudinal data: The continuous case. Annals of Statistics 29 1785–1811. 1891746
* Guo and Dawid (2010) [author] Guo, HuiH. and Dawid, Alexander PhilipA. P. (2010). Sufficient covariates and linear propensity analysis. In Proceedings of the Thirteenth International Workshop on Artificial Intelligence and Statistics, (AISTATS) 2010, Chia Laguna, Sardinia, Italy, May 13-15, 2010. Journal of Machine Learning Research Workshop and Conference Proceedings (Yee WhyeY. W. Teh and D. MichaelD. M. Titterington, eds.) 9 281–288. http://tinyurl.com/33lmuj7
* Henderson, Ansel and Alshibani (2010) [author] Henderson, RobinR., Ansel, PhilP. and Alshibani, DeyadeenD. (2010). Regret-regression for optimal dynamic treatment regimes. Biometrics (to appear).
doi:10.1111/j.1541-0420.2009.01368.x
* Hernán and Taubman (2008) [author] Hernán, Miguel A.M. A. and Taubman, Sarah L.S. L. (2008). Does obesity shorten life? The importance of well defined interventions to answer causal questions. International Journal of Obesity 32 S8–S14.
* Holland (1986) [author] Holland, Paul W.P. W. (1986). Statistics and causal inference (with Discussion). Journal of the American Statistical Association 81 945–970. 0867618
* Huang and Valtorta (2006) [author] Huang, YiminY. and Valtorta, MarcoM. (2006). Identifiability in causal Bayesian networks: A sound and complete algorithm. In AAAI’06: Proceedings of the 21st National Conference on Artificial Intelligence 1149–1154. AAAI Press.
* Kang and Schafer (2007) [author] Kang, Joseph D. Y.J. D. Y. and Schafer, Joseph L.J. L. (2007). Demystifying double robustness: A comparison of alternative strategies for estimating a population mean from incomplete data. Statistical Science 22 523–539. 2420458
* Lauritzen et al. (1990) [author] Lauritzen, Steffen LilholtS. L., Dawid, Alexander PhilipA. P., Larsen, B. N.B. N. and Leimer, H. G.H. G. (1990). Independence properties of directed Markov fields. Networks 20 491–505. 1064735
* Lok et al. (2004) [author] Lok, JudithJ., Gill, RichardR., van der Vaart, AadA. and Robins, JamesJ. (2004). Estimating the causal effect of a time-varying treatment on time-to-event using structural nested failure time models. Statistica Neerlandica 58 271–295. 2157006
* Moodie, Richardson and Stephens (2007) [author] Moodie, Erica M.E. M., Richardson, Thomas S.T. S. and Stephens, David A.D. A. (2007). Demystifying optimal dynamic treatment regimes. Biometrics 63 447–455. 2370803
* Murphy (2003) [author] Murphy, Susan A.S. A. (2003). Optimal dynamic treatment regimes (with Discussion). Journal of the Royal Statistical Society, Series B 65 331-366. 1983752
* Oliver and Smith (1990) [author] Oliver, R. M.R. M. and Smith, J. Q.J. Q., eds. (1990). Influence Diagrams, Belief Nets and Decision Analysis. John Wiley and Sons, Chichester, United Kingdom. 1056324
* Pearl (1995) [author] Pearl, JudeaJ. (1995). Causal diagrams for empirical research (with Discussion). Biometrika 82 669-710. 1380809
* Pearl (2009) [author] Pearl, JudeaJ. (2009). Causality: Models, Reasoning and Inference, Second ed. Cambridge University Press, Cambridge. 2548166
* Pearl and Paz (1987) [author] Pearl, JudeaJ. and Paz, AzariaA. (1987). Graphoids: A graph-based logic for reasoning about relevance relations. In Advances in Artificial Intelligence (D.D. Hogg and L.L. Steels, eds.) II 357–363. North-Holland, Amsterdam.
* Pearl and Robins (1995) [author] Pearl, JudeaJ. and Robins, JamesJ. (1995). Probabilistic evaluation of sequential plans from causal models with hidden variables. In Proceedings of the 11th Conference on Uncertainty in Artificial Intelligence (PhilippeP. Besnard and SteveS. Hanks, eds.) 444–453. Morgan Kaufmann Publishers, San Francisco. 1615028
* Raiffa (1968) [author] Raiffa, HowardH. (1968). Decision Analysis. Addison-Wesley, Reading, Massachusetts.
* Robins (1986) [author] Robins, James M.J. M. (1986). A new approach to causal inference in mortality studies with sustained exposure periods—Application to control of the healthy worker survivor effect. Mathematical Modelling 7 1393–1512. 0877758
* Robins (1987) [author] Robins, James M.J. M. (1987). Addendum to “A new approach to causal inference in mortality studies with sustained exposure periods—Application to control of the healthy worker survivor effect”. Computers & Mathematics with Applications 14 923–945. 0922792
* Robins (1989) [author] Robins, James M.J. M. (1989). The analysis of randomized and nonrandomized AIDS treatment trials using a new approach to causal inference in longitudinal studies. In Health Service Research Methodology: A Focus on AIDS (L.L. Sechrest, H.H. Freeman and A.A. Mulley, eds.) 113–159. NCSHR, U.S. Public Health Service.
* Robins (1992) [author] Robins, James M.J. M. (1992). Estimation of the time-dependent accelerated failure time model in the presence of confounding factors. Biometrika 79 321–324. 1185134
* Robins (1997) [author] Robins, James M.J. M. (1997). Causal inference from complex longitudinal data. In Latent Variable Modeling and Applications to Causality, (M.M. Berkane, ed.). Lecture Notes in Statistics 120 69–117. Springer-Verlag, New York. 1601279
* Robins (1998) [author] Robins, James M.J. M. (1998). Structural nested failure time models. In Survival Analysis, (P. K.P. K. Andersen and N.N. Keiding, eds.). Encyclopedia of Biostatistics 6 4372–4389. John Wiley and Sons, Chichester, UK.
* Robins (2000) [author] Robins, James M.J. M. (2000). Robust estimation in sequentially ignorable missing data and causal inference models. In Proceedings of the American Statistical Association Section on Bayesian Statistical Science 1999 6–10.
* Robins (2004) [author] Robins, James M.J. M. (2004). Optimal structural nested models for optimal sequential decisions. In Proceedings of the Second Seattle Symposium on Biostatistics (D. Y.D. Y. Lin and P.P. Heagerty, eds.) 189–326. Springer, New York. 2129402
* Robins, Greenland and Hu (1999) [author] Robins, James M.J. M., Greenland, SanderS. and Hu, F. C.F. C. (1999). Estimation of the causal effect of a time-varying exposure on the marginal mean of a repeated binary outcome. Journal of the American Statistical Association 94 687–700. 1723276
* Robins, Hernán and Brumback (2000) [author] Robins, James M.J. M., Hernán, Miguel ÁngelM. A. and Brumback, BabetteB. (2000). Marginal structural models and causal inference in epidemiology. Epidemiology 11 550–560.
* Robins and Wasserman (1997) [author] Robins, J. M.J. M. and Wasserman, L. A.L. A. (1997). Estimation of effects of sequential treatments by reparameterizing directed acyclic graphs. In Proceedings of the 13th Annual Conference on Uncertainty in Artificial Intelligence (DanD. Geiger and PrakashP. Shenoy, eds.) 409-420. Morgan Kaufmann Publishers, San Francisco. http://tinyurl.com/33ghsas
* Rosthøj et al. (2006) [author] Rosthøj, SusanneS., Fullwood, CatherineC., Henderson, RobinR. and Stewart, SydS. (2006). Estimation of optimal dynamic anticoagulation regimes from observational data: A regret-based approach. Statistics in Medicine 25 4197–4215. 2307585
* Shpitser and Pearl (2006a) [author] Shpitser, IlyaI. and Pearl, JudeaJ. (2006a). Identification of conditional interventional distributions. In Proceedings of the 22nd Annual Conference on Uncertainty in Artificial Intelligence (UAI-06) (RinaR. Dechter and ThomasT. Richardson, eds.). 437–444. AUAI Press, Corvallis, Oregon. http://tinyurl.com/2um8w47
* Shpitser and Pearl (2006b) [author] Shpitser, IlyaI. and Pearl, JudeaJ. (2006b). Identification of joint interventional distributions in recursive semi-Markovian causal models. In Proceedings of the Twenty-First National Conference on Artificial Intelligence 1219–1226. AAAI Press, Menlo Park, California.
* Spirtes, Glymour and Scheines (2000) [author] Spirtes, PeterP., Glymour, ClarkC. and Scheines, RichardR. (2000). Causation, Prediction and Search, Second ed. Springer-Verlag, New York.
* Sterne et al. (2009) [author] Sterne, J. A. C.J. A. C., May, M.M., Costagliola, D.D., de Wolf, F.F., Phillips, A. N.A. N., Harris, R.R., Funk, M. J.M. J., Geskus, R. B.R. B., Gill, J.J., Dabis, F.F., Miro, J. M.J. M., Justice, A. C.A. C., Ledergerber, B.B., Fatkenheuer, G.G., Hogg, R. S.R. S., D’Arminio-Monforte, A.A., Saag, M.M., Smith, C.C., Staszewski, S.S., Egger, M.M., Cole, S. R.S. R. and When To Start Consortium (2009). Timing of initiation of antiretroviral therapy in AIDS-Free HIV-1-infected patients: A collaborative analysis of 18 HIV cohort studies. Lancet 373 1352–1363.
* Taubman et al. (2009) [author] Taubman, Sarah L.S. L., Robins, James M.J. M., Mittleman, Murray A.M. A. and Hernán, Miguel A.M. A. (2009). Intervening on risk factors for coronary heart disease: An application of the parametric $g$-formula. International Journal of Epidemiology 38 1599–1611.
* Tian (2008) [author] Tian, JinJ. (2008). Identifying dynamic sequential plans. In Proceedings of the Twenty-Fourth Annual Conference on Uncertainty in Artificial Intelligence (UAI-08) (DavidD. McAllester and AnnA. Nicholson, eds.). 554–561. AUAI Press, Corvallis, Oregon. http://tinyurl.com/36ufx2h
* Verma and Pearl (1990) [author] Verma, T.T. and Pearl, J.J. (1990). Causal networks: Semantics and expressiveness. In Uncertainty in Artificial Intelligence 4 (R. D.R. D. Shachter, T. S.T. S. Levitt, L. N.L. N. Kanal and J. F.J. F. Lemmer, eds.) 69–76. North-Holland, Amsterdam. 1166827
|
arxiv-papers
| 2010-10-17T16:02:58 |
2024-09-04T02:49:14.000806
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "A. Philip Dawid and Vanessa Didelez",
"submitter": "Philip Dawid",
"url": "https://arxiv.org/abs/1010.3425"
}
|
1010.3505
|
# Adiabatic quantum state transfer in non-uniform triple-quantum-dot system
Bing Chen, Wei Fan, and Yan Xu College of Science, Shandong University of
Science and Technology, Qingdao 266510, China
###### Abstract
We introduce an adiabatic quantum state transfer scheme in a non-uniform
coupled triple-quantum-dot system. By adiabatically varying the external gate
voltage applied on the sender and receiver, the electron can be transferred
between them with high fidelity. By numerically solving the master equation
for a system with _always-on_ interaction, it is indicated that the transfer
fidelity depends on the ration between the peak voltage and the maximum
coupling constants. The effect of coupling mismatch on the transfer fidelity
is also investigated and it is shown that there is a relatively large
tolerance range to permit high fidelity quantum state transfer.
###### pacs:
03.65.-w, 03.67.Hk, 73.23.Hk
## I Introduction
In quantum information science, quantum state transfer (QST), as the name
suggests, refers to the transfer of an arbitrary quantum state from one qubit
to another. Recently, there are two major mechanisms for QST. The first
approaches are usually characterized by preparing the quantum channel with an
always-on interaction where QST is equivalent to the time evolution of the
quantum state in data bus Bose1 ; Song ; Christandle1 . However, these
approaches require precise control of distance and timing. Any deviation may
leads to significant errors. The other approaches have paid much attention to
adiabatic passage for coherent QST in time-evolving quantum systems. The most
well known example of these is the so-called Stimulated Raman Adiabatic
Passage (STIRAP) technique, which is used to produce a complete population
transfer between two internal quantum states of an atom or molecule Shore .
Such methods are relatively insensitive to gate errors and other external
noises and do not require an accurate control of the system parameters, thus
can realize high-fidelity QST.
Due to the potential scalability and long decoherence times, the applications
of adiabatic passage have been widely investigated in solid-state systems
Vitanov ; TB ; Eckert ; Zhang ; GT1 ; GT2 ; GT3 ; BEC1 ; BEC2 ; BEC4 ; DAS ;
McE . Eckert et al. Eckert have introduced an implementation of the STIRAP in
the three-trap potential array. By coherently manipulating the trap separation
between each two traps, the neutral atoms can be transferred in the
millisecond range. Zhang et al. Zhang have describe a scheme for using an
all-electrical, adiabatic population transfer between two spatially separated
dots in a triple-quantum-dot (TQD) system by adiabatically engineering
external gate voltage. In ref. GT1 , A. D. Greentree et al. have described a
method of coherent electronic transport through a triple-well system by
adiabatically following a particular energy eigenstate of the system. By
adiabatically modulating coherent tunneling rates between nearest neighbor
dots, it can realize a high fidelity transfer. This method was termed Coherent
Tunneling by Adiabatic Passage (CTAP) which was demonstrated experimentally
very recently via optical waveguide Longhi . Since then, adiabatic passage has
also been used to transport quantum information from a single sender to
multiple receivers, which relates to a quantum wire or fan-out GT3 . Following
a different perspective, there have been several recent proposals to
coherently manipulate BECs BEC1 ; BEC2 ; BEC4 in triple-well potentials. Ref.
DAS has analytically derived the condition for coherent tunneling via
adiabatic passage in a triple-well system with negligible central-well
population at all times during the transfer.
In CTAP technique GT1 , the basic idea is to use the existence of a spatial
dark state which is a coherent superposition state of two “distant” spatial
trapping sites,
$\left|D_{0}\right\rangle=\cos\theta_{1}\left|L\right\rangle+0\left|M\right\rangle-\sin\theta_{1}\left|R\right\rangle,$
where the mixing angle $\theta_{1}$ is defined as
$\tan\theta_{1}=\Omega^{LM}/\Omega^{MR}$ with $\Omega^{LM}$ ($\Omega^{MR}$)
denoting the tunneling rate between nearest-neighbor dots. By maintaining the
system in state $\left|D_{0}\right\rangle$ and adiabatically manipulating the
tunneling rates, it is possible to achieve coherent population transfer from
site $\left|L\right\rangle$ to $\left|R\right\rangle$ without any probability
being in the state $\left|M\right\rangle$. In this paper we consider a
different adiabatic protocol to achieve population transfer between two
spatially separated dots. We introduce a non-uniform coupled triple-quantum-
dot array which can be manipulated by external gate voltage applied on the two
external dots (sender and receiver). Through maintaining the system in the
ground state we show that the electron initially in the left dot can be
transferred to the right dot occupation with high fidelity. Furthermore, we
study in details the dynamic competition between the adiabatic QST and the
decoherence. There are two time scales $T_{1}$ and $T_{2}$ depicting such
competition, where $T_{1}$ represents the adiabatic time limited by the
adiabatic conditions and $T_{2}$ represents the decoherence time.
The paper is organized as follows. In Sec. II we setup the model and we
describe the adiabatic transfer of an electron between quantum dots. We also
derive a perturbative, analytical expression of fidelity. In Sec. III we show
numerical results that substantiate the analytical results. The last section
is the summary and discussion of this paper.
## II Model setup
Figure 1: (Color online) Schematic illustrations of quantum state transfer in
non-uniform triple-dot system: (a) the system is controlled by gates voltage
$\mu_{\alpha}(t)$ $(\alpha=L,$ $R)$; (b) by adiabatically change the
$\mu_{\alpha}(t)$ $(\alpha=L,$ $R)$ one can achieve QST from
$\left|L\right\rangle$ to $\left|R\right\rangle$.
In this section, we first introduce the isolated (no coupling to the leads)
TQD array $\left|L,\sigma\right\rangle$, $\left|M,\sigma\right\rangle$,
$\left|R,\sigma\right\rangle$ ($\sigma=\uparrow,\downarrow$), where
$\left|L,\sigma\right\rangle=c_{L,\sigma}^{{\dagger}}\left|\text{vac}\right\rangle$
($\left|M\right\rangle=c_{M,\sigma}^{{\dagger}}\left|\text{vac}\right\rangle$,
$\left|R\right\rangle=c_{R,\sigma}^{{\dagger}}\left|\text{vac}\right\rangle$)
corresponds to an electron in the left (center, right) dot with spin $\sigma$.
The scheme is schematically shown in Fig. 1(a). Specifically, we consider the
interactions between nearest-neighbor quantum dots are timeless and slightly
different. We term this model non-uniform triple-quantum-dot (NUTQD) system.
The system are controlled by external time-varying gates voltage
$\mu_{\alpha}(t)$ $(\alpha=L,$ $R)$, which control the site energies of two
end of the array. In this proposal we will show that the information encoded
in electronic spin can be transported from
$\cos\theta\left|L,\uparrow\right\rangle+\sin\theta\left|L,\downarrow\right\rangle$
to
$\cos\theta\left|R,\uparrow\right\rangle+\sin\theta\left|R,\downarrow\right\rangle$.
Notice that the polarization of the spin of an electron is not changed as time
evolves. Then the problem about the quantum information transfer (QIT) can be
reduced to the issue of QST and a complete QST can achieve perfect QIT. In
this sense, we can ignore spin degrees of freedom to illustrate the principles
of QST from $\left|L\right\rangle$ to $\left|R\right\rangle$.
We use
$\left\\{\left|L\right\rangle,\left|M\right\rangle,\left|R\right\rangle\right\\}$
as basis of the Hilbert space, the Hamiltonian for NUTQD system in matrix form
reads as
$H=\left[\begin{array}[]{ccc}\mu_{L}(t)&J_{1}&0\\\ J_{1}&0&J_{2}\\\
0&J_{2}&\mu_{R}(t)\end{array}\right],$ (1)
where $J_{i}\ (i=1,2)$ is the fixed coupling constant between nearest-neighbor
dots, assumed to be real (negative) for the sake of simplicity. The on-site
energies $\mu_{L}(t)$ and $\mu_{R}(t)$ are modulated in Gaussian pulses to
realize the adiabatic transfer, according to (shown in Fig. 2)
$\displaystyle\mu_{L}(t)$
$\displaystyle=-\mu_{L}^{\max}\exp\left[-\frac{1}{2}\alpha^{2}t^{2}\right],$
(2a) $\displaystyle\mu_{R}(t)$
$\displaystyle=-\mu_{R}^{\max}\exp\left[-\frac{1}{2}\alpha^{2}\left(t-\tau\right)^{2}\right],$
(2b) where $\tau$ and $\alpha$ are the total adiabatic evolution time and
standard deviation of the control pulse modulating the chemical potential of
states $\left|L\right\rangle$ and $\left|R\right\rangle$. For simplicity we
set the peak voltage of each dot to be equal
$\mu_{L}^{\max}=\mu_{R}^{\max}=\mu_{0}$ and satisfy
$\mu_{0}\gg\left|J_{i}\right|$ $(i=1,2)$. We will see below that this
simplicity has no relevance to the problem.
Figure 2: Gate voltages as a function of time (in units of $\tau$) ,
$\mu_{L}(t)$ is the solid line and $\mu_{R}(t)$ is the dash line.
At time $t=t_{0}$, the Hamiltonian $H(t_{0})$ has eigenvectors
$\left|\psi_{n}(t_{0})\right\rangle$ ($n=0,1,2$) which are superpositions of
$\left|L\right\rangle,$ $\left|M\right\rangle,$ $\left|R\right\rangle$ and the
eigenvalues are denoted by $\varepsilon_{n}(t_{0})$, sorting in ascending
order $\varepsilon_{0}<\varepsilon_{1}<\varepsilon_{2}$. Under adiabatic
evolution, these eigenstates evolve continuously to
$\left|\psi_{n}(t)\right\rangle$. The instantaneous Hamiltonian’s eigen
equation is
$H(t)\left|\psi_{n}(t)\right\rangle=\varepsilon_{n}(t)\left|\psi_{n}(t)\right\rangle.$
(3)
In this proposal, we use ground state $\left|\psi_{0}(t)\right\rangle$ of Eq.
(3) to induce population transfer from state $\left|L\right\rangle$ to
$\left|R\right\rangle$ (see Fig. 1(b)). One advantage of this proposal is that
there can be no heat dissipation during the transfer.
Starting from $t=0$, the Hamiltonian is approximate separable in the case
$\mu_{0}\gg\left|J_{i}\right|$:
$H(t=0)\simeq H_{L}\oplus H_{MR},$ (4)
with
$\displaystyle H_{L}$ $\displaystyle=-\mu_{0}\left|L\right\rangle\left\langle
L\right|,$ (5a) $\displaystyle H_{MR}$
$\displaystyle=J_{2}\left(\left|M\right\rangle\left\langle
R\right|+\left|R\right\rangle\left\langle M\right|\right).$ (5b) This
Hamiltonian has the eigenstates
$\displaystyle\left|\psi_{\pm}\left(t=0\right)\right\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(\left|M\right\rangle\pm\left|R\right\rangle\right),$
$\displaystyle\left|\psi_{0}\left(t=0\right)\right\rangle$ $\displaystyle=$
$\displaystyle\left|L\right\rangle,$ (6)
the energies of these states are
$\varepsilon_{\pm}=\pm J_{2},\text{ }\varepsilon_{0}=-\mu_{0}.$ (7)
Our aim is to induce population transfer from state $\left|L\right\rangle$ to
$\left|R\right\rangle$ by maintaining the system in ground state. Now we will
show that an adiabatic change of $\mu_{L}(t)$ and $\mu_{R}(t)$ will realize
the QST.
In the adiabatic limit, $t\rightarrow\tau$, the parameter $\mu_{L}(t)$ goes to
zero and $\mu_{R}(t)$ goes to $-\mu_{0}$. The Hamiltonian adiabatically
evolves to
$H(t=\tau)\simeq H_{LM}\oplus H_{R},$ (8)
with
$\displaystyle H_{LM}$
$\displaystyle=J_{1}\left(\left|L\right\rangle\left\langle
M\right|+\left|M\right\rangle\left\langle L\right|\right),$ (9a)
$\displaystyle H_{R}$ $\displaystyle=-\mu_{0}\left|R\right\rangle\left\langle
R\right|,$ (9b) the corresponding eigenstate are
$\displaystyle\left|\psi_{\pm}\left(t=\tau\right)\right\rangle$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(\left|L\right\rangle\pm\left|M\right\rangle\right),$
$\displaystyle\left|\psi_{0}\left(t=\tau\right)\right\rangle$ $\displaystyle=$
$\displaystyle\left|R\right\rangle.$ (10)
and then the ground state evolves to be $\left|R\right\rangle$.
Providing adiabaticity is satisfied Blum
$\left|\varepsilon_{m}-\varepsilon_{n}\right|\gg|\langle\psi_{m}|\dot{\psi}_{n}\rangle|,$
(11)
the overall system will remain in its instantaneous ground state. At $t=0$,
the system is prepared in state
$\left|\psi_{0}\left(t=0\right)\right\rangle=\left|L\right\rangle$, then the
adiabatic theorem states that the system will stay in
$\left|\psi_{0}\left(t\right)\right\rangle$. Note that $\left|L\right\rangle$
and $\left|R\right\rangle$ denote the states in which the electron is on the
left and right QD, respectively. Therefore, we can see that an electron
starting in $\left|L\right\rangle$ will end up in $\left|R\right\rangle$.
Providing the length of time $\tau$ is too large, that is, the time-dependent
change is introduced slowly enough, the fidelity of QST is also determined by
peak gate voltage $\mu_{0}$. Notice that the square of the module of fidelity
$\left|F(t)\right|^{2}=\left|\left\langle
R\right.\left|\psi_{0}\left(t\right)\right\rangle\right|^{2}$ denotes the
probability of finding $\left|R\right\rangle$ in the ground state
$\left|\psi_{0}\left(t\right)\right\rangle$. Now we suppose to get analytical
expression of fidelity using first order perturbation theory. We start from
Eq. (1) at $t=\tau$ and consider the coupling term
$J_{2}\left(\left|R\right\rangle\left\langle
M\right|+\left|M\right\rangle\left\langle R\right|\right)$ as a weak
perturbation. The Hamiltonian
$H(t=\tau)=H_{0}+H_{I},$ (12)
contains two parts
$\displaystyle H_{0}$
$\displaystyle=J_{1}\left(\left|L\right\rangle\left\langle
M\right|+\left|M\right\rangle\left\langle
L\right|\right)-\mu_{0}\left|R\right\rangle\left\langle R\right|,$ (13a)
$\displaystyle H_{I}$
$\displaystyle=J_{2}\left(\left|R\right\rangle\left\langle
M\right|+\left|M\right\rangle\left\langle R\right|\right).$ (13b)
Our aim is to find the approximate expression for the ground state
$\left|\psi_{0}\right\rangle$ of the perturbed Hamiltonian $H(t=\tau)$. The
eigenfunctions of unperturbed Hamiltonian $H_{0}$ is
$\displaystyle|\psi_{0}^{(0)}\rangle$ $\displaystyle=$
$\displaystyle|R\rangle,$ $\displaystyle|\psi_{\pm}^{(0)}\rangle$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(\left|L\right\rangle\pm\left|M\right\rangle\right).$
(14)
In the picture of
$\left\\{|\psi_{-}^{(0)}\rangle,|\psi_{+}^{(0)}\rangle,|\psi_{0}^{(0)}\rangle\right\\}$,
The Hamiltonian $H_{0}$ can be diagonalized as
$H_{0}=\left[\begin{array}[]{ccc}-J_{1}&0&0\\\ 0&J_{1}&0\\\
0&0&-\mu_{0}\end{array}\right].$
As the first order perturbation, we have the corrected ground state to be
$\displaystyle\left|\psi_{0}\right\rangle$ $\displaystyle=$
$\displaystyle|\psi_{0}^{(0)}\rangle+\sum_{\eta=\pm}\frac{\langle\psi_{\eta}^{(0)}|H_{I}|\psi_{0}^{(0)}\rangle}{E_{0}^{(0)}-E_{\eta}^{(0)}}|\psi_{\eta}^{(0)}\rangle$
(15) $\displaystyle=$
$\displaystyle\frac{J_{1}J_{2}}{\mu_{0}^{2}-J_{1}^{2}}|L\rangle-\frac{\mu_{0}J_{2}}{\mu_{0}^{2}-J_{1}^{2}}|M\rangle+|R\rangle.$
So the transfer fidelity of adiabatic QST at $t=\tau$ is
$\displaystyle\left|F(\tau)\right|^{-2}$ $\displaystyle=$ $\displaystyle
1+\left(\frac{J_{1}J_{2}}{\mu_{0}^{2}-J_{1}^{2}}\right)^{2}+\left(\frac{\mu_{0}J_{2}}{\mu_{0}^{2}-J_{1}^{2}}\right)^{2}$
(16) $\displaystyle=$ $\displaystyle
1+\frac{J_{2}^{2}\left(\mu_{0}^{2}+J_{1}^{2}\right)}{\left(\mu_{0}^{2}-J_{1}^{2}\right)^{2}},$
which shows that the peak voltage $\mu_{0}$ determined the fidelity of QST. As
$\mu_{0}\gg\left|J_{i}\right|$ is satisfied, the fidelity is near to unity.
## III Numerical Simulations
The analysis above is based on the assumption that the adiabaticity is
satisfied. In order to demonstrate the QST in the system (1) and to show how
exact the approximation is, in this section we numerically solve the master
equation and the above central conclusion can be get confirmed. The main goal
of this section is to analyze the parameters which influence the fidelity of
adiabatic QST and find the proper matching relation between them.
First, initialize electron in the left dot, i.e., the total initial state is
$\left|\Psi\left(0\right)\right\rangle=\left|L\right\rangle$, the time
evolution creates a coherent superposition:
$\left|\Psi\left(t\right)\right\rangle=c_{1}(t)\left|L\right\rangle+c_{2}(t)\left|M\right\rangle+c_{3}(t)\left|R\right\rangle.$
(17)
with this notation we assume the initial condition $c_{1}(0)=1$, and the other
two equal zero. In order to proceed, we numerically solve the master equations
for the density matrix $\rho$. The master equation is written as Blum
(assuming $\hbar=1$)
$i\frac{d\rho\left(t\right)}{dt}=\left[H,\rho\left(t\right)\right],$ (18)
where
$\rho\left(t\right)=\left|\Psi\left(t\right)\right\rangle\left\langle\Psi\left(t\right)\right|$.
With the basis state ordering
$\left\\{\left|L\right\rangle,\left|M\right\rangle,\left|R\right\rangle\right\\}$,
the density matrix can be written as
$\rho\left(t\right)=\left[\begin{array}[]{ccc}\left|c_{1}(t)\right|^{2}&c_{1}(t)c_{2}^{\ast}(t)&c_{1}(t)c_{3}^{\ast}(t)\\\
c_{2}(t)c_{1}^{\ast}(t)&\left|c_{2}(t)\right|^{2}&c_{2}(t)c_{3}^{\ast}(t)\\\
c_{3}(t)c_{1}^{\ast}(t)&c_{3}(t)c_{2}^{\ast}(t)&\left|c_{3}(t)\right|^{2}\end{array}\right]\;.$
According to the definition of fidelity, we can see that
$\left|F(t)\right|^{2}=\left|c_{3}(t)\right|^{2}$. The crucial requirement for
adiabatic evolution is Eq. (11). Firstly, one must to make sure that no level
crossings occur, i.e., $\varepsilon_{0}(t)-\varepsilon_{j}(t)<0$. To calculate
the energies is generally only possible numerically. In Fig. 3(a) we present
the results showing the eigenenergy gap
$\Delta(t)=\varepsilon_{1}(t)-\varepsilon_{0}(t)$ between the first-excited
state and ground state of the NUTQD system undergoing evolution due to
modulation of the gate voltage according to pulse Eq. (1) for $\mu_{0}=20$,
$J_{1}=0.8$, $J_{2}=1.0$, $\tau=10\mu_{0}/J_{1}^{2}$ and $\alpha=3/\tau$,
$4/\tau$, $5/\tau$, $6/\tau$. It shows that for the given evolution time
$\tau=400$ the minimum of the energy gap decrease as standard deviation
$\alpha$ increasing. The slower Hamiltonian (1) varies, the closer adiabatic
theorem holds. In Fig. 3 we also show the numerically computed behavior of the
populations $|c_{1}(t)|^{2}$, $|c_{2}(t)|^{2}$ and $|c_{3}(t)|^{2}$ on the
three quantum dots as a function of time with $\alpha=4/\tau$ and
$\alpha=5/\tau$. Note that for $\alpha=4/\tau$ transfer, as illustrated in
Fig. 3(b), the population on state $\left|R\right\rangle$ is decoupled and
stays constant 0.92. The fraction of population left in states
$\left|L\right\rangle$ and $\left|M\right\rangle$ is
$\left|c_{1}(\tau)\right|^{2}+\left|c_{2}(\tau)\right|^{2}=0.08$ and executes
Rabi oscillations because the quantum dots $L$ and $M$ are coupled with
$J_{1}=0.8$. Whereas for $\alpha=5/\tau$ case, shown in Fig. 3(c), one can see
that the fidelity of adiabatic QST has been improved considerably by this
slight change. The fidelity of QST achieve 0.995 and only 0.5% of population
remains in states $\left|L\right\rangle$ and $\left|M\right\rangle$. This is
consistent with the results shown in Fig. 3(a) because the eigenenergy gap
plays opposite role for transition probability.
Figure 3: (Color online) (a) The energy gap
$\Delta(t)=\varepsilon_{1}(t)-\varepsilon_{0}(t)$ between the first-excited
state and ground state of the triple-dot system undergoing evolution due to
modulation of the gate volgate according to pulse Eq. (3) for $\mu_{0}=20$,
$J_{1}=0.8$, $J_{2}=1.0$, $\tau=10\mu_{0}/J_{1}^{2}$ and $\alpha=3/\tau$,
$4/\tau$, $5/\tau$, $6/\tau$. The time evolution of the probabilities induced
by the pulses in Fig. 2 for (b) $\alpha=4/\tau$ and (c) $\alpha=5/\tau$.
Initially the population is on left qubit (black line) and finally mainly on
right qubit (red line). The population on the intermediate qubit is shown as a
blue line.
The fidelity of population transfer will be very high as long as the
Hamiltonian evolves sufficiently slowly in time (as determined by criteria for
the applicability of the theorem). In practice the maximum possible transfer
rates will be a few times greater than $\mu_{0}/J_{1}^{2}$ which is
illustrated in Fig. 4. Note that the transfer fidelity becomes stable when the
total evolution time satisfy $\tau\geq 4\mu_{0}/J_{1}^{2}$.
Figure 4: Fidelity as a function of total adiabatic evolution time $\tau$ (in
units of $\mu_{0}/J_{1}^{2}$). When $\tau\geq 4\mu_{0}/J_{1}^{2}$, the
fidelity of QST becomes stable.
The preceding discussion is based on the assumption that the system parameters
are setup with arbitrary precision that is the system is coupled with
$J_{1}=0.8$ and $J_{2}=1.0$. However, it is difficult to fabricate such
precise Hamiltonian in experiment. Next we will show that the adiabatic
passage like us is relatively insensitive to the system parameters. From the
analytical results, the fidelity of adiabatic QST depends on the contrast
ratio between peak voltage $\mu_{0}$ and coupling constants $J_{i}$. To
determine the parameter range needed to achieve high fidelity transfer, we
numerically integrate the density matrix equations of motion, with varying the
peak voltage $\mu_{0}$. In Fig. 5(a) we present results showing the square of
fidelity $\left|F(\tau)\right|^{2}=|c_{3}(\tau)|^{2}$ as a function of
$\mu_{0}$ with $J_{1}=0.8$, $J_{2}=1.0$, $\tau=375$ and $\alpha=5/\tau$. We
can see that the population transfer is close to one
($\left|F(\tau)\right|^{2}\geq 0.99$) and stable when $\mu_{0}$ is achieved
for $|\mu_{0}/J_{2}|\geq 14$. The plot in Fig. 5(a) is in agreement with the
analytical results Eq. (16) with high accuracy. On the other hand, the
difference between $J_{1}$ and $J_{2}$ has a little effect upon transfer
fidelity within certain range. We have illustrated this in Fig. 5(b) where the
effects of mismatch between $J_{1}\ $and $J_{2}$ have been modeled. Here we
show $\left|F(\tau)\right|^{2}$ as a function of $J_{1}/J_{2}$ for peak
voltage $\mu_{0}=20$ to simulate the effect of a systematic error in the
coupling constants. Note that the ratio as much as $0.35$ still permits
$\left|F(\tau)\right|^{2}\approx 0.994$.
Figure 5: The plot of the square of fidelity $\left|F(\tau)\right|^{2}$ as a
function of system paremeters: (a) the peak voltage $\mu_{0}$ and (b) the
ratio $J_{1}/J_{2}$. If the condition is satisfied when $|\mu_{0}/J_{max}|\geq
14$ and $J_{1}/J_{2}\geq 0.4$, the transfer fidelity is near to one.
## IV Summary and discussion
In summary, we have introduced a method of coherent QST through a NUTQD system
by adiabatic passage. This scheme is realized by modulation of gate voltage of
QDs. Different from the CTAP Scheme, our method is to induce population
transfer by maintaining the system in its ground state which is more stable
than dark state. We have studied the adiabatic QST through a NTQD system by
theoretical analysis and numerical simulations of the ground state evolution
of NTQD model. The result shows that it is a high fidelity process for a
proper choose of standard deviation and peak voltage.
In order to investigate the relation between the fidelity of quantum state
transfer $\left|F(\tau)\right|^{2}$ and peak voltage $\mu_{0}$, we have
numerically solve the master equation under different peak voltage. The
numerical result shows that if we want to achieve a high fidelity more than
99.5% we require the ratio of $\left|\mu_{0}/J_{2}\right|\geq 14$. We also
show that the sight difference between $J_{1}$ and $J_{2}$ does small
influence on the fidelity.
It is worthwhile to discuss the applicability of the scheme presented above.
In a real system, quantum decoherence is the main obstacle to the experimental
implementation of quantum information. For coupled QDs, experiments T1 show
that the coupling strength $J$ is about 0.25 meV while $\mu_{0}\sim 20J$. we
can estimate a time of $\sim$ 50 ps required for adiabatic operation. On the
other hand, the typical decoherence time $T_{2}$ for electron-spin has been
indicated experimentally T2 to be longer than 80$\pm$9 $\mu$s at 2.5 K which
is much longer than adiabatic operation time. So our scheme has applicability
in practice.
###### Acknowledgements.
One of the authors (BC) thanks Z. Song for discussions and encouraging
comments. We also acknowledge the support of the NSF of China (Grant Nos.
10847150, 61071016) and Shandong Provincial Natural Science Foundation, China
(Grant No. ZR2009AM026).
## References
* (1) S. Bose, Phys. Rev. lett. 91, 207901 (2003).
* (2) Z. Song and C.P. Sun, Low Temperature Physics 31, 686 (2005).
* (3) M. Christandl, N. Datta, A. Ekert and A.J. Landahl, Phys. Rev. Lett. 92, 187902 (2004); M. Christandl, N. Datta, T. C. Dorlas, A. Ekert, A. Kay and A.J. Landahl, Phys. Rev. A 71, 032312 (2005).
* (4) K. Bergmann, H. Theuer, and B. Shore, Rev. Mod. Phys. 70, 1003 (1998).
* (5) N. V. Vitanov, T. Halfmann, B. W. Shore, and K. Bergmann, Annu. Rev. Phys. Chem. 52, 763 (2001); A. D. Greentree, J. H. Cole, A. R. Hamilton, and Lloyd C. L. Hollenberg, Phys. Rev. B 70, 235317 (2004).
* (6) T. Brandes and T. Vorrath, Phys. Rev. B 66, 075341 (2002).
* (7) K. Eckert, M. Lewenstein, R. Corbalan, G. Birkl, W. Ertmer, and J. Mompart, Phys. Rev. A 70, 023606 (2004).
* (8) P. Zhang, Q. K. Xue, X. G. Zhao, and X. C. Xie, Phys. Rev. A 69, 042307 (2004).
* (9) S. Longhi, G. Della Valle, M. Ornigotti, and P. Laporta, Phys. Rev. B 76, 201101(R) (2007).
* (10) A. D. Greentree, J. H. Cole, A. R. Hamilton, and L. C. L. Hollenberg, Phys. Rev. B 70, 235317 (2004).
* (11) J. H. Cole, A. D. Greentree, L. C. L. Hollenberg, and S. Das Sarma, Phys. Rev. B 77, 235418 (2008).
* (12) A. D. Greentree, S. J. Devitt, and L. C. L. Hollenberg, Phys. Rev. A 73, 032319 (2006).
* (13) E. M. Graefe, H. J. Korsch, and D. Witthaut, Phys. Rev. A 73, 013617 (2006).
* (14) M. Rab, J. H. Cole, N. G. Parker, A. D. Greentree, L. C. L. Hollenberg, and A. M. Martin, Phys. Rev. A 77, 061602(R) (2008).
* (15) V. O. Nesterenko, A. N. Nikonov, F. F. de Souza Cruz, and E. L. Lapolli, Laser Phys. 19, 616 (2009).
* (16) Tomás̆ Opatrný and Kunal K. Das, Phys. Rev. A 79, 012113 (2009).
* (17) S. McEndoo, S. Croke, J. Brophy, and Th. Busch, Phys. Rev. A 81, 043640 (2010).
* (18) A.Messiah, Quantum Mechanics Vol. II (North-Holland, Amsterdam, 1961).
* (19) K. Blum, Density Matrix Theory and Applications (Plenum, NewYork, 1996).
* (20) Guido Burkard, Daniel Loss, and D. P. DiVincenzo, Phys. Rev. B 59, 2070 (1999); A. V. Onufriev and J. B. Marston, ibid. 59, 12573 (1999); W Gvander Wiel et al., New J. Phys. 8, 28 (2006).
* (21) A. M. Tyryshkin, S. A. Lyon, A. V. Astashkin, A. M. Raitsimring, Phys. Rev. B 68, 193207 (2003).
|
arxiv-papers
| 2010-10-18T07:35:29 |
2024-09-04T02:49:14.020513
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Bing Chen, Wei Fan, and Yan Xu",
"submitter": "Bing Chen",
"url": "https://arxiv.org/abs/1010.3505"
}
|
1010.3711
|
Construction a new generating function of Bernstein type polynomials
Yilmaz Simsek
Department of Mathematics, Faculty of Art and Science
University of Akdeniz
TR-07058 Antalya, Turkey
E-mail: ysimsek@akdeniz.edu.tr
Dedicated to Professor H. M. Srivastava on the occasion of his seventieth
birth anniversary
Abstract
> Main purpose of this paper is to reconstruct generating function of the
> Bernstein type polynomials. Some properties this generating functions are
> given. By applying this generating function, not only derivative of these
> polynomials but also recurrence relations of these polynomials are found.
> Interpolation function of these polynomials is also constructed via Mellin
> Transformation. This function interpolates these polynomials at negative
> integers which are given explicitly. Moreover, relations between these
> polynomials, the generalized Stirling numbers, and Bernoulli polynomials of
> higher order are given. Furthermore some applications associated with
> B´ezier curve are given.
2010 Mathematics Subject Classification. Primary 11B68, 11M06, 33B15 ;
Secondary 33B15, 65D17.
Key Words and Phrases. Generating function, Bernstein polynomials, Bernoulli
polynomials of higher-order, Stirling numbers of second kind, interpolation
function, Mellin transformation, Gamma function, beta function and B´ezier
curve.
## 1\. Introduction, Definitions and Preliminaries
The Bernstein polynomials, recently, have been defined by many different ways,
for examples in $q$-series, by complex function and many algorithms. These
polynomials are used not only approximations of functions in various, but also
in the other fields such as smoothing in statistics, numerical analysis, the
solution of the differential equations, and constructing B´ezier curve and in
Computer Aided Design cf. ([2], [8], [3], [4], [7], [10], [1]), and see also
the references cited in each of these earlier works.
By the same motivation of Ozden’ [6] paper, which is related to the
unification of the Bernoulli, Euler and Genocchi polynomials, we, in this
paper, construct a generating function of the Bernstein polynomials which
unify generating function in [10], [1].
## 2\. Construction generating functions of Bernstein type polynomials
In this section we unify generating function of the Bernstein polynomials. We
define
$\mathcal{F}(t,b,s:x)=\frac{2^{b}x^{bs}\left(\frac{t}{2}\right)^{bs}e^{t(1-x)}}{\left(bs\right)!}$
where $b,s\in\mathbb{Z}^{+}:=\\{1,2,3,\cdots\\}$, $t\in\mathbb{C}$ and
$x\in\left[0,1\right]$. This function is generating function of the
polynomials $\mathfrak{S}_{n}(bs,x)$:
$\mathcal{F}(t,b,s:x)=\sum_{n=0}^{\infty}\mathfrak{S}_{n}(bs,x)\frac{t^{n}}{n!},$
(2.1)
where $\mathfrak{S}_{0}(bs,x)=\cdots=\mathfrak{S}_{bs-1}(bs,x)=0$.
Remark 1. If we set $s=1$ in (2.1), we obtain
$\frac{\left(xt\right)^{b}e^{t(1-x)}}{b!}=\sum_{n=0}^{\infty}B_{n}(b,x)\frac{t^{n}}{n!},$
and $\mathfrak{S}_{n}(b,x)=B_{n}(b,x)$, which denotes the Bernstein
polynomials cf. ([2], [3], [4], [8], [10], [1]).
By using Taylor expansion of $e^{t}$ in (2.1), we arrive at the following
theorem:
###### Theorem 1.
Let $x,y\in[0,1]$. Let $b,$ $n$ and $s\ $be nonnegative integers. If $n\geq
bs$, then we have
$\mathfrak{S}_{n}(bs,x)=\left(\begin{array}[]{c}n\\\
bs\end{array}\right)\frac{x^{bs}(1-x)^{n-bs}}{2^{b(s-1)}}.$
Remark 1. Setting $s=1$ in Theorem 1, one can see that the polynomials
$\mathfrak{S}_{n}(b,x)=\left(\begin{array}[]{c}n\\\
b\end{array}\right)x^{b}(1-x)^{n-b},$
which give us the Bernstein polynomials cf. ([10], [1]). Consequently, the
polynomials $\mathfrak{S}_{n}(bs,x)$ are unification of the Bernstein
polynomials.
By using Theorem 1, we easily obtain the following results.
###### Corollary 1.
Let $b$, $n$ and $s$ be nonnegative integers with $n\geq bs$. Then we have
$\left(\begin{array}[]{c}n\\\
bs\end{array}\right)\mathfrak{S}_{n-bs}(bs;x)=\left(\begin{array}[]{c}n+bs\\\
n\end{array}\right)\mathfrak{S}_{n}(bs;x).$
Setting
$\mathfrak{g}_{n}(bs,x)=2^{b(s-1)}\mathfrak{S}_{n}(bs,x),$
where, for $bs=j$,
$\sum_{j=0}^{n}\mathfrak{g}_{n}(j,x)=1.$
Let $f$ be a continuous function on $\left[0,1\right]$. Then we define
unification Bernstein type operator as follows:
$\mathbb{S}_{n}\left(f(x)\right)=\sum_{j=0}^{n}f\left(\frac{j}{n}\right)\mathfrak{g}_{n}(j;x),$
(2.2)
where $x\in[0,1]$, $n$ is positive integer.
Setting $f(x)=x$ in (2.2), then we have
$\mathbb{S}_{n}\left(x\right)=\sum_{j=0}^{n}\frac{j}{n}\left(\begin{array}[]{c}n\\\
j\end{array}\right)x^{j}(1-x)^{n-j}.$
From the above, we get
$\mathbb{S}_{n}\left(x\right)=x\sum_{j=0}^{n}\mathfrak{g}_{n-1}(j-1,x).$
## 3\. Fundamental relations of the polynomials $\mathfrak{S}_{n}(bs,x)$
By using generating function of $\mathfrak{S}_{n}(bs,x)$, in this section we
give derivative of $\mathfrak{S}_{n}(bs,x)$ and recurrence relation of
$\mathfrak{S}_{n}(bs,x)$.
###### Theorem 2.
Let $x\in[0,1]$. Let $b$, $n$ and $s$ be nonnegative integers with $n\geq bs$.
Then we have
$\frac{d}{dx}\mathfrak{S}_{n}(bs,x)=n\left(\mathfrak{S}_{n-1}(bs-1,x)-\mathfrak{S}_{n}(bs,x)\right).$
(3.1)
###### Proof.
By using the partial derivative of a function in (2.1) with respect to the
variable $x$, we have
$\sum_{n=0}^{\infty}\frac{\partial}{\partial
x}\left(\mathfrak{S}_{n}(bs,x)\right)\frac{t^{n}}{n!}=t\sum_{n=0}^{\infty}\mathfrak{S}_{n}(bs-1,x)\frac{t^{n}}{n!}-t\sum_{n=0}^{\infty}\mathfrak{S}_{n}(bs,x)\frac{t^{n}}{n!}.$
From the above, we obtain
$\sum_{n=0}^{\infty}\left(\frac{d}{dx}\mathfrak{S}_{n}(bs,x)\right)\frac{t^{n}}{n!}=\sum_{n=0}^{\infty}n\mathfrak{S}_{n-1}(bs-1,x)\frac{t^{n}}{n!}-\sum_{n=0}^{\infty}n\mathfrak{S}_{n-1}(bs,x)\frac{t^{n}}{n!}.$
By using the partial derivative of a function in (2.1) with respect to the
variable $t$, we arrive at the following theorem:
###### Theorem 3.
Let $x\in[0,1]$. Let $b$, $n$ and $s$ be nonnegative integers with $n\geq bs$.
Then we have
$\mathfrak{S}_{n}(bs,x)=x\mathfrak{S}_{n-1}(bs-1,x)+(1-x)\mathfrak{S}_{n-1}(bs,x).$
(3.2)
Remark 3. If setting $s=1$, then (3.2) reduces to a recursive relation of the
Bernstein polynomials
$B_{n}(b,x)=(1-x)B_{n-1}(b,x)+xB_{n-1}(b-1,x)$
and (3.1) reduces to derivative of the Bernstein polynomials
$\frac{d}{dx}B_{n}(j,x)=n\left(B_{n-1}(j-1,x)-B_{n-1}(j,x)\right),$
respectively.
By the umbral calculus convention in (2.1), we get
$\frac{2^{b}x^{bs}\left(\frac{t}{2}\right)^{bs}}{\left(bs\right)!}=e^{\left(\mathfrak{S}(bs,x)-(1-x)\right)t},$
where $\mathfrak{S}^{n}(bs;x)$ is replaced by $\mathfrak{S}_{n}(bs;x)$. After
some elementary calculation, we arrive at the following theorem.
###### Theorem 4.
If $n=bs$, then we have
$2^{b(1-s)}x^{bs}=\sum_{j=0}^{bs}\left(\begin{array}[]{c}bs\\\
j\end{array}\right)(-1)^{bs-j}\left(1-x\right)^{bs-j}\mathfrak{S}_{j}(bs,x).$
If $n>bs$, then we have
$\sum_{j=bs+1}^{n}\left(\begin{array}[]{c}n\\\
j\end{array}\right)(-1)^{n-j}\left(1-x\right)^{n-j}\mathfrak{S}_{j}(bs,x)=0.$
Relations between the polynomials the polynomial $\mathfrak{S}_{n}(bs,x)$,
Bernoulli polynomial of higher order and Stirling numbers of second kind is
given by the following theorem:
###### Theorem 5.
Let $b$, $n$ and $s$ be nonnegative integers with $n\geq bs$. Then we have
$\mathfrak{S}_{n}(bs,x)=2^{b(1-s)}x^{bs}\sum_{j=0}^{n}\left(\begin{array}[]{c}n\\\
j\end{array}\right)S(j,bs)B_{n-j}^{(bs)}(1-x),$
where $B_{n}^{(v)}(x)$ and $S(n,j)$ denote Bernoulli polynomial of higher
order and Stirling numbers of second kind, which are given by means of the
following generating function, respectively
$\frac{t^{v}e^{xt}}{\left(e^{t}-1\right)^{v}}=\sum_{n=0}^{\infty}B_{n}^{(v)}(x)\frac{t^{n}}{n!},\text{
}(\left|t\right|<2\pi)$
and
$(-1)^{v}\frac{\left(1-e^{t}\right)^{v}}{v!}=\sum_{n=0}^{\infty}S(n,v)\frac{t^{n}}{n!}.$
###### Proof.
By (2.1), we have
$2^{b(1-s)}x^{bs}\left(\frac{(-1)^{bs}(e^{t}-1)^{bs}}{\left(bs\right)!}\right)\left(\frac{t^{bs}e^{(1-x)t}}{\left(e^{t}-1\right)^{bs}}\right)=\sum_{n=0}^{\infty}\mathfrak{S}_{n}(bs,x)\frac{t^{n}}{n!}.$
From the above, we have
$\sum_{n=0}^{\infty}\mathfrak{S}_{n}(bs,x)\frac{t^{n}}{n!}=2^{b(1-s)}x^{bs}\left(\sum_{n=0}^{\infty}B_{n}^{(bs)}(1-x)\frac{t^{n}}{n!}\right)\left(\sum_{n=0}^{\infty}S(n,k)\frac{t^{n}}{n!}\right).$
By Cauchy product in the above, after some calculation, we find the desired
result.
By using same method of Lopez and Temme’ [5], we give contour integral
representation of $\mathfrak{S}_{n}(bs,x)$ as follows:
$\mathfrak{S}_{n}(bs,x)=\frac{\Gamma(m+1)}{\Gamma(k+1)}\frac{1}{2\pi
i}\int_{\mathcal{C}}\mathcal{F}(t,b,s:x)\frac{dz}{z^{m+1}},$
where $\mathcal{C}$ is a circle around the origin and the integration is in
positive direction.
## 4\. Interpolation Function of the polynomials $\mathfrak{S}_{n}(bs,x)$
In this section, we construct meromorphic function. This function interpolates
$\mathfrak{S}_{n}(bs;x)$ at negative integers. These values are given
explicitly in Theorem 6.
For $z\in\mathbb{C}$, by applying the Mellin transformation to (2.1), we
obtain
$\mathfrak{B}(z,bs;x)=\frac{1}{\Gamma(z)}\int_{0}^{\infty}t^{z-1}\mathcal{F}(-t,b,s:x)dt,$
where $\Gamma(z)$ is Euler gamma function. From the above, we define the
following interpolation function.
###### Definition 1.
Let $z\in\mathbb{C}$ with $\Re(z)>0$ and $x\neq 1$. Let $b$ and $s$ be
nonnegative integers. Then we define
$\mathfrak{B}(z,bs;x)=(-1)^{bs}\frac{\Gamma(z+bs)}{\Gamma(bs+1)\Gamma(z)}\frac{2^{b(1-s)}x^{bs}}{\left(1-x\right)^{z+bs}},$
(4.1)
Remark 4. By the well-known identity $\Gamma(bs+1)=bs\Gamma(bs)$, for
$\Re(z)>0$ we have
$\mathfrak{B}(z,k;x)=\frac{(-1)^{bs}2^{b(1-s)}x^{bs}}{bsB(z,k)\left(1-x\right)^{z+bs}},$
where $B(z,k)$ denotes the beta function. Observe that if $x=1$, then
$\mathfrak{B}(z,bs,1)=\infty.$
###### Theorem 6.
Let $b$, $n$ and $s$ be nonnegative integers with $n\geq bs$ and $x\in[0,1]$.
Then we have
$\mathfrak{B}(-n,bs;x)=\mathfrak{S}_{n}(bs,x).$
###### Proof.
Let $n$ and $b$, and $s$ be positive integers with $bs\leq n$. $\Gamma(z)$ has
simple poles at $z=-n=0,-1,-2,-3,\cdots$. The residue of $\Gamma(z)$ is
$Res(\Gamma(z),-n)=\frac{(-1)^{n}}{n!}.$
Taking $z\rightarrow-n$ into (4.1) and using the above relations, the desired
result can be obtained.
Observe that if we set $s=1$ in Theorem 6, we arrive at
$\mathfrak{B}(-n,b;x)=B_{n}(b,x).$
## 5\. Further Remarks on B´ezier curves
The Bernstein polynomials are used to construct B´ezier curves. B´ezier was an
engineer with the Renault car company and set out in the early 1960’s to
develop a curve formulation which would lend itself to shape design. Engineers
may find it most understandable to think of B´ezier curves in terms of the
center of mass of a set of point masses cf. [13], for example, consider the
four masses $m_{0}$, $m_{1}$, $m_{2}$, and $m_{3}$ located at points $P_{0}$,
$P_{1}$, $P_{2}$, $P_{3}$. The center of mass of these four point masses is
given by the equation
$P=\frac{m_{0}P_{0}+m_{1}P_{1}+m_{2}P_{2}+m_{3}P_{3}}{m_{0}+m_{1}+m_{2}+m_{3}}.$
Next, imagine that instead of being fixed, constant values, each mass varies
as a function of some parameter $x$. In specific case, let $m_{0}=(1-x)^{3}$,
$m_{1}=3t(1-x)^{2}$, $m_{2}=3t^{2}(1-x)$ and $m_{3}=x^{3}$. The values of
these masses are a function of $x$. For each value of $x$, the masses assume
different weights and their center of mass changes continuously. As $x$ varies
between $0$ and $1$, a curve is swept out by the center of masses. This curve
is a cubic B´ezier curve. For any value of $x$, this B´ezier curve is
$P=m_{0}P_{0}+m_{1}P_{1}+m_{2}P_{2}+m_{3}P_{3},$
where $m_{0}+m_{1}+m_{2}+m_{3}\equiv 1$. These variable masses $m_{i}$ are
normally called blending functions and their locations $P_{i}$ are known as
control points or B´ezier points. The blending functions, in the case of
B´ezier curves, are known as Bernstein polynomials. This curve is used in
computer graphics and related fields and also in the time domain, particularly
in animation and interface design cf. ([3], [4], [13]).
The B´ezier curve of degree $n$ can be generalized as follows. Given points
$P_{0}$, $P_{1}$, $P_{2}$,$\cdots$, $P_{n}$ the B´ezier curve is
$B(x)=\sum_{k=0}^{n}P_{k}B_{n}(k,x),$ (5.1)
where $x\in[0,1]$ and $B_{n}(k,t)$ denotes Bernstein polynomials.
We now unify the B´ezier curve in (5.1) by the polynomials
$\mathfrak{g}_{n}(bs,x)$ as follows
$\mathbb{B}_{n}(x,y)=\sum_{k=0}^{n}P_{k}\mathfrak{g}_{n}(k;x),$
with control points $P_{k}$.
###### Acknowledgement 1.
The present investigation was supported by the Scientific Research Project
Administration of Akdeniz University.
## References
* [1] M. Acikgoz and S. Aracı. On generating function of the Bernstein polynomials, Numer. Anal. Appl. AIP Conf. ICNAAM 2010.
* [2] S. N. Bernstein, Démonstration du théorème de Weierstrass fondée sur la calcul des probabilités. Comm. Soc. Math. Charkow Sér. 2 t. 13, 1-2 (1912-1913).
* [3] L. Busé, Goldman, R.: Division algorithms for Bernstein polynomials, Computer Aided Geometric Design, 25(9), 850-865 (2008).
* [4] G. Morin and R. Goldman, On the smooth convergence of subdivision and degree elevation for B´ezier curves. Computer Aided Geometric Design. 18(7), 657-666 (2001).
* [5] L. Lopez and N.. M. Temme, Hermite polynomials in asymptotic representations of generalized Bernoulli, Euler, Bessel and Buchholz polynomials, Modelling. Analysis and Simulation (MAS), MAS-R9927 September 30, (1999).
* [6] H. Ozden, Unification of generating function of the Bernoulli, Euler and Genocchi numbers and polynomials, to appear in Numerical analysis and applied mathematics, AIP Conference Proceedings.
* [7] S. Ostrovska, The unicity theorems for the limit $q$-Bernstein opera, Appl. Analy. 88(2), 161–167 (2009).
* [8] G. M. Phillips, Interpolation and approximation by polynomials, CMS Books in Mathematics/ Ouvrages de Mathématiques de la SMC, 14. Springer-Verlag, New York, (2003).
* [9] Y. Simsek, Twisted $(h,q)$-Bernoulli numbers and polynomials related to twisted $(h,q)$-zeta function and $L$-function, J. Math. Anal. Appl. 324 (2006), 790-804.
* [10] Y. Simsek and M. Acikgoz, A new generating function of ($q$-) Bernstein-type polynomials and their interpolation function, Abstract and Applied Analysis, vol. 2010, Article ID 769095, 12 pages, 2010. doi:10.1155/2010/769095.
* [11] H. M. Srivastava, Some formulas for the Bernoulli and Euler polynomials at rational arguments, Math. Proc. Cambridge Philos. Soc. 129 (2000), 77–84.
* [12] H. M. Srivastava and J. Choi, Series Associated with the Zeta and Related Functions, Kluwer Acedemic Publishers, Dordrecht, Boston and London, 2001.
* [13] Sederberg, T.: BYU B´ezier curves, http://www.tsplines.com/resources/class_notes/B’ezier_curves.pdf.
|
arxiv-papers
| 2010-10-18T20:11:37 |
2024-09-04T02:49:14.031387
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yilmaz Simsek",
"submitter": "Yilmaz Simsek",
"url": "https://arxiv.org/abs/1010.3711"
}
|
1010.3876
|
# On minimal non-$CL$-groups
Daniele Ettore Otera Département de Mathématique
Batiment 425, Faculté de Science d’Orsay
Université Paris-Sud 11
F-91405, Orsay Cedex, France daniele.otera@math-psud.fr and Francesco G.
Russo Mathematics Department
University of Palermo
via Archirafi 14
90123, Palermo, Italy francescog.russo@yahoo.com
###### Abstract.
If $m$ is a positive integer or infinity, the $m$-layer (or briefly, the
layer) of a group $G$ is the subgroup $G_{m}$ generated by all elements of $G$
of order $m$. This notion goes back to some contributions of R. Baer and Ya.D.
Polovickii of almost 60 years ago and is often investigated, because the
presence of layers influences the group structure. If $G_{m}$ is finite for
all $m$, $G$ is called $FL$-group (or $FO$-group). A generalization is given
by $CL$-groups, that is, groups in which $G_{m}$ is a Chernikov group for all
$m$. By working on the notion of $CL$-group instead of that of $FL$-group, we
extend a recent result of Z. Zhang, describing the structure of a group which
is not a $CL$-group, but all whose proper subgroups are $CL$-groups.
###### Key words and phrases:
$CL$-groups, minimal non-$CL$-groups, locally graded groups, locally finite
groups, Chernikov layers.
###### 2010 Mathematics Subject Classification:
20F24, 20F15, 20E34, 20E45.
## 1\. Introduction
A group $G$ has Chernikov conjugacy classes, or briefly is a CC-group, if
$G/C_{G}(\langle x\rangle^{G})$ is a Chernikov group for all $x\in G$, where
$C_{G}(\langle x\rangle^{G})$ denotes the centralizer of the normal closure
$\langle x\rangle^{G}$ of $\langle x\rangle$ in $G$. These groups were
introduced by Ya.D. Polovickii in [6] and generalize the groups with finite
conjugacy classes, also known as FC-groups. A classic reference is [10] for
the study of $FC$-groups. Among these groups, there is a special subclass,
which has received attention in [11] and will be the subject of the present
work.
From [7, pp. 133–134] we recall that, if $m$ is a positive integer or
$\infty$, the m-layer (or briefly, the layer) of a group $G$ is the subgroup
$G_{m}$ generated by all elements of $G$ of order $m$. If $G_{m}$ is finite
for all $m$, $G$ is called FL-group (or FO-group). An $FL$-group is
characterized to have only a finite number of elements of each order,
including $\infty$. This justifies the terminology $FO$-group, used by some
authors. Of course, $FL$-groups are $FC$-groups, but their structure can be
described more accurately with respect to that of $FC$-groups. In order to do
this, we recall some notions from [7]. Following [7, p.135], a direct product
of groups is called prime-thin if for each prime $p$ at most a finite number
of the direct factors contain elements of order $p$. Successively, we recall
that a group $G$ is central-by-finite, if its center $Z(G)$ has finite index
in $G$. A group $G$ is said to be locally normal and finite if each finite
subset of $G$ is contained in a finite normal subgroup of $G$. Similarly, a
group $G$ is said to be locally normal and Chernikov if each finite subset of
$G$ is contained in a Chernikov normal subgroup of $G$. Finally, we recall
that a group $G$ satisfies min-$ab$, if it satisfies the minimal condition on
its abelian subgroups.
Now we are able to state the main characterizations of $FL$-groups.
###### Theorem 1.1 (See [7], Theorem 4.43).
The following properties of a group $G$ are equivalent.
* (i)
$G$ is an $FL$-group.
* (ii)
$G$ is a locally normal and finite group and each Sylow subgroup satisfies
min-$ab$.
* (iii)
$G$ is isomorphic with a subgroup of a prime-thin direct product of central-
by-finite Chernikov groups.
A group $G$ in which $G_{m}$ is a Chernikov group for all $m$ is said to be a
$CL$-group. Of course, $FL$-groups are $CL$-groups. There is not a rich
literature in English language on $FL$-groups and $CL$-groups and [1] may help
the reader, who is interested to investigate the relations among ascending
chains of $CC$-groups and the structure of $CL$-groups. However, weakening
Theorem 1.1, $CL$-groups may be characterized analogously.
###### Theorem 1.2 (See [7], Theorem 4.42).
The following properties of a group $G$ are equivalent.
* (i)
$G$ is a $CL$-group.
* (ii)
$G$ is a locally normal and Chernikov group and each Sylow subgroup satisfies
min-$ab$.
* (iii)
$G$ is isomorphic with a subgroup of a prime-thin direct product of Chernikov
groups.
If $\mathcal{X}$ is an arbitrary class of groups, $G$ is said to be a minimal
non-$\mathcal{X}$-group, or briefly an $MNX$-group, if it is not an
$\mathcal{X}$-group but all of whose proper subgroups are
$\mathcal{X}$-groups. Many results have been obtained on $MNX$-groups, for
various choices of $\mathcal{X}$. If $\mathcal{X}$ is the class of
$FC$-groups, we find the $MNFC$-groups characterized by V.V. Belyaev and N.F.
Sesekin in [10, Section 8]. They proved that an $MNFC$-group is a finite
cyclic extension of a divisible $p$-group of finite rank ($p$ a prime). If
$\mathcal{X}$ is the class of $CC$-groups, J. Otál and J. M. Peña proved in
[5, Theorem, p.1232] that there are no $MNCC$-groups which have a non-trivial
finite or abelian factor group. Similar subjects have been investigated in [2,
3, 4, 8, 9]. More recently, Z. Zhang choose $\mathcal{X}$ to be the class of
$FL$-groups, proving in [11, Theorem 2.5] that all $MNFL$-groups are
$MNFC$-groups. Consequently, these groups may be described by the quoted
classification of $MNFC$-groups.
In this paper we extend the results of Z. Zhang to $MNCL$-groups. In Section 2
we show that all $MNCL$-groups are $MNCC$-groups and this allows us to reduce
the classification of $MNCL$-groups to that in [5]. In Section 3 we
characterize $MNCL$-groups and draw some conclusions on the perfect case.
## 2\. $MNCL$-groups
An easy consequence of Theorem 1.2 is listed below.
###### Corollary 2.1 (See [7], p.134).
$CL$-groups are countable and locally finite.
The structure of a $CC$-group is well–known and described in [7, Theorem
4.36]. A consequence, which we will use in several arguments, is expressed
below.
###### Corollary 2.2 (See [5], p. 1234).
The set of all elements of finite order of a $CC$-group $G$ is a locally
normal and Chernikov characteristic subgroup of $G$. In particular, a periodic
$CC$-group is a locally normal and Chernikov group.
Torsion-free groups should be avoided in our investigations.
###### Lemma 2.3.
Let $G$ be an $MNCL$-group. Then $G$ is periodic.
###### Proof.
Assume that this is false and let $x$ be an element of infinite order. For any
positive integer $n$, the subgroup $\langle x\rangle^{n}$ is a torsion-free
proper subgroup of $G$. At the same time $\langle x\rangle^{n}$ is a
$CL$-group and then it is periodic by Corollary 2.1. This contradiction
implies the result. ∎
An important role is played by the normal subgroups whose factors are
Chernikov groups.
###### Lemma 2.4.
Let $G$ be a $CC$-group and $H$ be a normal subgroup of $G$ such that $G/H$ is
a Chernikov group. Then $G$ is a $CL$-group if and only if $H$ is a
$CL$-group.
###### Proof.
If $G$ is a $CL$-group, then $H$ is of course a $CL$-group. Conversely, assume
that $H$ is a $CL$-group. From Corollary 2.1 $H$ is a periodic group, but also
$G/H$ is a periodic group. Since the class of periodic groups is closed with
respect to forming extensions of its members (see [7, p.34]), we conclude that
$G$ is a periodic group. Now $G$ is a periodic $CC$-group and Corollary 2.2
implies that $G$ is a locally normal and Chernikov group. By Theorem 1.2, it
remains to prove that each Sylow subgroup of $G$ satisfies min-$ab$. Let $P$
be a Sylow subgroup of $G$. $P\cap H$ is contained in some Sylow $p$-subgroup
of $H$, which is a $CL$-group and has all its Sylow subgroups satysfying
min-$ab$ by Theorem 1.2. Therefore $P\cap H$ satisfies min-$ab$. $P/(P\cap
H)\simeq PH/H\leq G/H$ is a Chernikov group and also satisfies min-$ab$. We
conclude that $P$ is an extension of two groups with min-$ab$ and then it
satisfies min-$ab$. The result follows. ∎
Also the subgroups of finite index play an important role.
###### Lemma 2.5.
Let $G$ be a $CC$-group and $H$ be a subgroup of $G$ of finite index. Then $G$
is a $CL$-group if and only if $H$ is a $CL$-group.
###### Proof.
If $G$ is a $CL$-group, then $H$ is of course a $CL$-group. Conversely, assume
that $H$ is a $CL$-group. From Corollary 2.1 $H$ is a periodic group and so is
$G$. Denoting with $H_{G}$ the core of $H$ in $G$, $|G:H_{G}|\leq|G:H|$ is
finite and then there is no loss of generality in assuming that $H$ is a
normal subgroup of $G$. Now $G$ is a periodic $CC$-group and Corollary 2.2
implies that $G$ is a locally normal and Chernikov group. By Theorem 1.2, it
remains to prove that each Sylow subgroup of $G$ satisfies min-$ab$. Let $P$
be a Sylow subgroup of $G$. $P\cap H$ is contained in a some Sylow
$p$-subgroup of $H$, which is a $CL$-group and has all its Sylow subgroups
satysfying min-$ab$ by Theorem 1.2. Therefore $P\cap H$ satisfies min-$ab$.
Since $|P:P\cap H|\leq|PH:H|\leq|G:H|$ is finite, we conclude that $P$ is a
finite extension of a group with min-$ab$. Then it satisfies min-$ab$ and the
result follows. ∎
The subgroups of $MNCL$-groups are subject of severe restrictions.
###### Lemma 2.6.
Let $G$ be a $CC$-group. If $G$ is an $MNCL$-group, then there is no proper
normal subgroup $H$ such that $G/H$ is a Chernikov group.
###### Proof.
Suppose that $H$ is a proper normal subgroup of $G$ such that $G/H$ is a
Chernikov group. Then $H$ is a $CL$-group. From Lemma 2.4 $G$ is a $CL$-group,
against the assumption. ∎
###### Lemma 2.7.
Let $G$ be a $CC$-group. If $G$ is an $MNCL$-group, then there is no proper
subgroup $H$ of finite index.
###### Proof.
Suppose that $H$ is a proper subgroup of $G$ of finite index. Then $H$ is a
$CL$-group. From Lemma 2.5 $G$ is a $CL$-group, against the assumption. ∎
Now we prove the main result of the present section.
###### Theorem 2.8.
All $MNCL$-groups are $MNCC$-groups.
###### Proof.
Assume that $G$ is an $MNCL$-group. All proper subgroups of $G$ are
$CL$-groups and then $CC$-groups. In order to complete the proof, it is enough
to prove that $G$ is not a $CC$-group.
Assume that $G$ is a $CC$-group. For any element $x$ of $G$, the centralizer
$C_{G}(\langle x\rangle^{G})$ is a normal subgroup of $G$ such that
$G/C_{G}(\langle x\rangle^{G})$ is a Chernikov group. Therefore it must be
trivial by Lemma 2.6 and so $G=C_{G}(\langle x\rangle^{G})$ for all $x$ in
$G$. This means that $G$ is an abelian group. On another hand, Lemma 2.3
implies that $G$ is periodic, then $G$ is a periodic abelian group.
If $G$ is not an abelian $p$-group for some prime $p$, then each Sylow
subgroup $P$ of $G$ is a proper subgroup of $G$ and hence a $CL$-group.
Theorem 1.2 implies that $P$ must be a Chernikov group. On another hand, $G$
is periodic abelian, then a locally normal and Chernikov group and by Theorem
1.2 it should be a $CL$-group, which is a contradiction.
Therefore we may assume that $G$ is an abelian $p$-group. However it cannot
contain any proper subgroup of finite index by Lemma 2.7, hence it should be
divisible, that is, the direct product of $m$ quasicyclic $p$-groups. If $m$
is finite, then $G$ is a Chernikov group, which is in contradiction with the
fact that $G$ is not a $CL$-group. If $m$ is infinite, then a proper subgroup
$H$ of $G$ which is a direct product of an infinite number of quasicyclic
$p$-groups cannot be a $CL$-group by Theorem 1.2. Then a proper subgroup $H$
of $G$ should be a direct product of a finite number of quasicyclic
$p$-groups, then $H$ would be a Chernikov group, still against Lemma 2.6.
It follows that $G$ cannot be a $CC$-group, as claimed. ∎
## 3\. Consequences
[11, Theorem 2.3] can be found as a special case of Theorem 2.8. We need to
recall that the finite residual $G^{*}$ of a group $G$ is the intersection of
all normal subgroups of $G$ of finite index. $G$ is said to be residually
finite, if $G^{*}$ is trivial.
###### Corollary 3.1.
Let $G$ be a group in which the layers of the proper subgroups are of finite
exponent. If $G$ is an $MNCL$-group, then $G$ is an $MNFC$-group.
###### Proof.
All $MNFL$-groups are $MNFC$-groups by [11, Theorem 2.3] and it is enough to
prove that, if $G$ is an $MNCL$-group, then it is an $MNFL$-group.
A proper subgroup $H$ of $G$ has its layers $H_{m}$ which are Chernikov groups
of finite exponent. Then $H_{m}$ are finite groups for all $m\geq 1$.
Consequently, $H$ is an $FL$-group. Since the choice of $H$ was aribitrary,
the same is true for all proper subgroups of $G$ and then all proper subgroups
of $G$ are $FL$-groups. On another hand, if $G$ is an $FL$-group, then it is a
$CL$-group, against the assumption. Then $G$ is an $MNFL$-group, as claimed. ∎
Corollary 3.1 allows us to apply the classification of V.V. Belyaev and N.F.
Sesekin [10, Theorem 8.13]. This is shown in the next two results.
###### Corollary 3.2.
Assume that the layers of the proper subgroups of a group $G$ are of finite
exponent. $G$ is a nonperfect $MNCL$-group if and only if $G$ satisfies the
following conditions:
* (i)
$G^{\prime}=G^{*}$; $G=\langle G^{*},x\rangle$, where $x^{p^{n}}\in G^{*}$,
$x^{p}\in Z(G)$, $p$ is a prime and $n$ is a positive integer;
* (ii)
$G^{*}$ can be expressed as a direct product of finitely many quasicyclic
$q$-groups, where $q$ is a prime;
* (iii)
There is no proper $G$-admissible subgroup in $G^{*}$.
[11, Corollary 3.2] shows the equivalence of the first four conditions in the
next corollary and the fifth condition is due to Corollary 3.2. We should also
mention that the $MNFA$-groups and the $MNCF$-groups, which we are going to
characterize, are exactly the groups studied in [9].
###### Corollary 3.3.
Assume that the layers of the proper subgroups of a nonperfect group $G$ are
of finite exponent. Then the following conditions are equivalent:
* (i)
$G$ is an $MNFC$-group;
* (ii)
$G$ is an $MNFA$-group;
* (iii)
$G$ is an $MNCF$-group;
* (iv)
$G$ is an $MNFL$-group;
* (v)
$G$ is an $MNCL$-group.
We recall that a group $G$ is called locally graded if every finitely
generated subgroup of $G$ has a proper subgroup of finite index. As usual, the
imposition of this condition is to avoid from our treatment the Tarski groups,
that is, infinite non-abelian groups whose proper subgroups are finite. For
the case of $CC$-groups we know as follows.
###### Theorem 3.4 (See [5], Corollary, p.1232).
If $G$ is a locally graded minimal non-$CC$-group, then $G$ is locally finite
and countable. Furthermore, $G=G^{*}=G^{\prime}$. In particular, $G$ is
perfect.
Therefore we may conclude the next result.
###### Theorem 3.5.
If $G$ is a locally graded $MNCL$-group, then $G$ is locally finite and
countable. Furthermore, $G=G^{*}=G^{\prime}$. In particular, $G$ is perfect.
###### Proof.
It is enough to combine Theorems 2.8 and 3.4. ∎
The importance of Theorem 3.5 is due to the fact that it describes a
situation, which is completely different from that of $MNFL$-groups. In fact,
[11, Corollary 4.2] states that there are no locally graded perfect
$MNFL$-groups, while Theorem 3.5 has just illustrated that all locally graded
$MNCL$-groups are perfect. Two more properties of $MNCL$-groups are summarized
below.
###### Corollary 3.6.
If $G$ is a locally graded $MNCL$-group, then $G$ has no non-trivial finite
factor groups.
###### Proof.
From Theorem 3.5, $G=G^{*}$ implies that there are no non-trivial finite
factor groups. ∎
###### Corollary 3.7.
Assume that $G$ is an $MNCL$-group. If $G$ is locally graded, then $G$ is not
finitely generated.
###### Proof.
Assume that $G$ is a finitely generated locally graded $MNCL$-group. Then $G$
must have finite factor groups, against Theorem 2.8, which implies $G=G^{*}$.
We conclude that $G$ cannot be finitely generated. ∎
We end with two remarks which hide some deep open questions, related to the
efforts in [2, 3, 4].
###### Remark 3.8.
From [11, Remark], it is not known whether there exists a perfect 2-generated
$MNFL$-group. Probably this is due to the fact that there are no examples of
perfect $MNFL$-groups, which are not $MNFC$-groups. Note that [11, Theorem
4.1] states that there are no locally finite perfect $MNFL$-groups. Then, if
the desired examples exist, then they should be periodic but not locally
finite. On another hand, the absence of such examples makes plausible that
also the converse of [11, Theorem 2.3] would be true: one may expect that, not
only all $MNFL$-groups are $MNFC$-groups, but that also the contrary is true.
In fact, this is proved in the nonperfect case in [11, Corollary 3.2].
###### Remark 3.9.
Similarly as in Remark 3.8, it is not known whether all $MNCC$-groups are
$MNCL$-groups.
## Acknowledgements
We are grateful to Professor Z. Zhang, who noted some weak points in the
original version of the manuscript.
## References
* [1] J.C. Beidleman, A. Galoppo and C. Manfredino, On $PC$-hypercentral and $CC$-hypercentral groups, Comm. Algebra 26 (1998), 3045–3055.
* [2] B. Bruno and R. E. Phillips, On minimal conditions related to Miller-Moreno type groups, Rend. Sem. Mat. Univ. Padova 69 (1983), 153–168.
* [3] M. Kuzucuoǧlu and R. E. Phillips, Locally finite minimal non-$FC$-groups, Math. Proc. Cambridge Philos. Soc. 105 (3) (1989), 417–420.
* [4] F. Leinen, A reduction theorem for perfect locally finite minimal non-$FC$-groups, Glasgow. Math. J. 41 (1999), 81–83.
* [5] J. Otál and M. Peña, Minimal non-$CC$-groups, Comm. Algebra 16 (1988), 1231–1242.
* [6] Ya.D. Polovickii, Groups with extremal classes of conjugate elements, Siberian Math. J. 5 (1964), 891–895.
* [7] D.J. Robinson, Finiteness Conditions and Generalized Soluble Groups, vol. I, Springer, Heidelberg, 1970.
* [8] F.G. Russo and N. Trabelsi, Minimal non-$PC$-groups, Ann. Math. Blaise Pascal 16 (2009), 277–286.
* [9] K.P. Shum and Z. Zhang, Minimal non-$CF$-groups, SEA Bull. Math. 13 (1994), 183–186.
* [10] M.J. Tomkinson, FC-groups, Pitman Publishing, London, 1984.
* [11] Z. Zhang, Minimal non-$FO$-groups, Comm. Algebra 38 (2010), 1983–1987.
|
arxiv-papers
| 2010-10-19T12:32:55 |
2024-09-04T02:49:14.043526
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Daniele Ettore Otera (Universite' Paris-Sud 11, Orsay Cedex, France)\n and Francesco G. Russo (Universita' degli Studi di Palermo, Palermo, Italy)",
"submitter": "Francesco G. Russo",
"url": "https://arxiv.org/abs/1010.3876"
}
|
1010.3893
|
2010 Vol. 10 No. XX, 000–000
11institutetext: North West University (Potchefstroom Campus), School of
Physics (Unit for Space Research), Private Bag X6001, Potchefstroom $2531$,
Republic of South Africa, Email: gadzirai@gmail.com
Received 2010 May 14; accepted 2010 June 27
# Bipolar Outflows as a Repulsive Gravitational Phenomenon
– Azimuthally Symmetric Theory of Gravitation (II)
Golden Gadzirayi Nyambuya00footnotetext: ∗Supported by the Republic of South
Africa’s National Research Foundation and the North West University, and
Germany’s DAAD Programme via the University of K$\ddot{\rm{o}}$ln.
###### Abstract
This reading is part in a series on the Azimuthally Symmetric Theory of
Gravitation (ASTG) set-out in Nyambuya ([$2010a$]). This theory is built on
Laplace-Poisson’s well known equation and it has been shown therein (Nyambuya
[$2010a$]) that the ASTG is capable of explaining – from a purely classical
physics standpoint; the precession of the perihelion of solar planets as being
a consequence of the azimuthal symmetry emerging from the spin of the Sun.
This symmetry has and must have an influence on the emergent gravitational
field. We show herein that the emergent equations from the ASTG – under some
critical conditions determined by the spin – do possess repulsive
gravitational fields in the polar regions of the gravitating body in question.
This places the ASTG on an interesting pedal to infer the origins of outflows
as a repulsive gravitational phenomena. Outflows are an ubiquitous phenomena
found in star forming systems and their true origins is a question yet to be
settled. Given the current thinking on their origins, the direction that the
present reading takes is nothing short of an asymptotic break from
conventional wisdom; at the very least, it is a complete paradigm shift as
gravitation is not at all associated; let alone considered to have anything to
do with the out-pour of matter but is thought to be an all-attractive force
that tries only to squash matter together into a single point. Additionally,
we show that the emergent Azimuthally Symmetric Gravitational Field from the
ASTG strongly suggests a solution to the supposed Radiation Problem that is
thought to be faced by massive stars in their process of formation. That is,
at $\sim 8-10\,\mathcal{M}_{\odot}$, radiation from the nascent star is
expected to halt the accretion of matter onto the nascent star. We show that
in-falling material will fall onto the equatorial disk and from there, this
material will be channelled onto the forming star via the equatorial plane
thus accretion of mass continues well past the curtain value of $\sim
8-10\,\mathcal{M}_{\odot}$ albeit via the disk. Along the equatorial plane,
the net force (with the radiation force included) on any material there-on
right up-till the surface of the star, is directed toward the forming star,
hence accretion of mass by the nascent star is un-hampered.
PACS (2010): $97.10.$Bt, $97.10.$Gz, $97.10.$Fy
###### keywords:
stars: formation – stars: mass-loss – stars: winds, outflows – ISM: jets and
outflows.
## 1 Introduction
Champagne like bipolar molecular outflows are an unexpected natural phenomenon
that grace the star formation podium. Bipolar molecular outflows are the most
spectacular physical phenomenon intimately associated with newly formed stars.
Studies of bipolar outflows reveal that they [bipolar outflows] are ubiquitous
toward High Mass Star (HMS) forming regions. These outflows in HMS forming
regions are far more massive and energetic than those found associated with
Low Mass Stars (LMS) forming regions (see e.g. Shepherd & Churchwell
[$1996a$]; Shepherd & Churchwell [$1996b$]; Zhang et al. [$2001$]; Zhang et
al. [$2005$]; Beuther [$2002$]). Obviously, this points to a correlation
between the mass of the star and the outflow itself. Independent studies have
established the existence of such a correlation. The mass outflow rate
$\dot{\mathcal{M}}_{out}$ has been shown to be related to the bolometric
luminosity $\mathcal{L}$ by the relationship:
$\dot{\mathcal{M}}_{out}\propto\mathcal{L}^{0.60}_{star}$, and this is for
stars in the luminosity range:
${0.30}\mathcal{L}_{\odot}\leq\mathcal{L}_{star}\leq{10}^{5}\mathcal{L}_{\odot}$
(we shall use the term luminosity to mean bolometric luminosity). Another
curious property of outflows is that the mass-flow rate,
$\dot{\mathcal{M}}_{out}$, is related to the speed of the molecular outflow
$\dot{\mathcal{M}}_{out}\propto V^{-\gamma}_{out}$ where $\gamma\sim{1.80}$
and $V_{out}$ is the speed of the outflow. How and why outflows come to
exhibit these properties is an interesting field of research that is not part
of the present reading. However, we shall show that these relationships do
emerge from our proposed ASTG Outflow Model. In the present, we simple want to
show that an outflow model emerges from the ASTG model. We set herein the
mathematical foundations for such a model. Once we have a fully-fledged
mathematical model, we shall move on to building a numerical model (i.e.
computer code). Once this computer code is available, an endeavor to answer
the above and other questions surrounding the nature of outflows will be made.
Pertaining to their association with star formation activity, it is believed
that molecular outflows are a necessary part of the star formation process
because their existence may explain the apparent angular momentum imbalance.
It is well known that the amount of initial angular momentum in a typical
star-forming molecular cloud core is several orders of magnitude too large to
account for the observed angular momentum found in formed or forming stars
(see e.g. Larson [$2003b$]). The sacrosanct Law of Conservation of angular
momentum informs us that this angular momentum can not just disappear into the
oblivion of interstellar spacetime. So, the question is where does this
angular momentum go to? It is here that outflows are thought to come to the
rescue as they can act as a possible agent that carries away the excess
angular momentum. This angular moment, if it where to remain as part of the
nascent star, it would, via the strong centrifugal forces, tear the star
apart. This however does not explain, why they exist and how they come to
exist but simply posits them as a vehicle needed to explain the mystery of
“The Missing Angular Momentum Problem” in star forming systems and the
existence of stars in their intact and compact form as stable firery balls of
gas.
In the existing literature, viz the question why and how molecular outflows
exist, there are about four proposed leading models that endeavor to explain
the aforesaid. These four major proposals are:
Wind Driven Outflow Model: In this model, a wide-angle radial wind blows into
the stratified surrounding ambient material, forming a thin swept-up shell
that can be identified as the outflow shell (see Shu et al. [$1991$]; Li & Shu
[$1996$]; Matzner & McKee [$1999$]).
Jet Driven Bow Shocks Model: In this model, a highly collimated jet propagates
into the surrounding ambient material producing a thin outflow shell around
the jet (see Raga et al. [$1993a$]; Masson & Chernin [$1993$]).
Jet Driven Turbulent Outflow Model: In this model, Kelvin-Helmholtz
instabilities along the jet and or environmental boundary leading to the
formation of a turbulent viscous mixing layer, through which the molecular
cloud gas in entrained (see Cantó & Raga [$1991$]; Raga et al. [$1993b$];
Stahler [$1994$]; Lizano & Giovanardi [$1995$]; Cantó et al. [$2003$]).
Circulation Flows Model: In this model, the molecular outflow is not entrained
by an underlying wind jet but is rather formed by in-falling matter that is
deflected away from the protostar in the central torus of high magneto-
hydrodynamic pressure through a quadrupolar circulation pattern around the
protostar and is accelerated above escape speeds by local heating (see Fiege &
Henriksen [$1996a$]; Fiege & Henriksen [$1996a$]).
All these ad hoc models and some that are not mentioned here explain outflows
as a feedback effect. The endeavor of the work presented in this reading is to
make an alternative suggestion albeit a complete, if not a radical departure
from the already existing models briefly mentioned above. Our model flows
naturally from the Laplace-Poison equation, namely from the Azimuthally
Symmetric Theory of Gravitation (ASTG) laid down in Nyambuya ([$2010a$])
(hereafter Paper I). This model is new and has never before appeared in the
literature. Because we are at the stage of setting this model, we see no need
to get into the details of the existing models as this would lead to an
unnecessary digression, confusion, and an un-called for lengthy reading.
Our model is a complete departure from the already existing models because, of
all the agents that could lead to outflows, gravitation is not even considered
to be a possible agent because it is thought of as, or assumed to be, an all-
attractive force. Actually, the idea of a gravitating body such as a star
producing a repulsive gravitational field, is at the very least unthinkable.
Contrary to this, we show here that an azimuthally symmetric gravitational
system does in-principle give rise to a bipolar repulsive gravitational field
and this – in our view, clearly suggests that these regions of repulsive
gravitation, possibly are the actual driving force of the bipolar molecular
outflows. We also see that the ASTG provides a neat solution (possibly and
very strongly so) to the so-called Radiation Problem thought to bedevil and
bewilder the formation of HMSs (see Larson & Starrfield [$1971$]; Kahn
[$1974$]; Bonnell et al. [$1998$]; Bonnell & Bate [$2002$]; Palla & Stahler
[$1993$]) and as-well the observed Ring of Masers (Bartkiewicz et al.
[$2008$], [$2009$]).
We need to reiterate this so as to make it clear to our reader, that, the work
presented in this reading is meant to lay down the mathematical foundations of
the outflow model emergent from the ASTG. It is not a comparative study of
this outflow model with those currently in existence. We believe we have to
put thrust on lying down these ideas and only worry about their plausibility,
i.e. whether or not they correspond with experience and only thereafter make a
literature wide comparative study. Given that this model flows naturally from
a well accepted equation (the Poisson-Laplace equation), against the
probability of all unlikelihood, this model should have a bearing with
reality. If it does not have a bearing with reality, then, at the very least,
it needs to be investigated since this solution of the Poisson-Laplace
equation has not been explored anywhere in the literature111In our exhaustive
survey of the accessible literature, we have not come across a treatment of
the Poisson-Laplace equation as is done in the present, hence our proclamation
that this solution of the Poisson-Laplace equation is the first such..
Also, we should say that as we build this model, we are doing this with
expediency, that is, watchful of what experience dictates, at the end of the
day, if our efforts are to bear any fruits, our model must correspond with
reality. This literature wide comparative study is expected to be done once a
mathematical model of our proposed outflow model is in full-swing. This
mathematical model is expected to form part of the future works where only-
after that, it would make sense then to embark on this literature wide
comparative study. How does one compare a baby human-being to a human-embryo?
It does not make sense, does it? Should not the baby be born first and only
thereafter a comparative study be conducted of this baby with those babies
already in existence? We hope the reader concurs with us that this is perhaps
the best way to set into motion a new idea amid a plethora of ideas that
champion a similar if not the same endeavor.
Further, we need to say this; that, as already stated above, the direction
that the present reading takes is nothing short of an asymptotic break from
conventional wisdom; at the very least, it is a complete paradigm shift as
gravitation is not at all associated; let alone considered to have anything to
do with the out-pour of matter but is thought to be an all-attractive force
that tries only to squash matter together into a single point. Because of this
reason, that, the present is “nothing short of an asymptotic break from
conventional wisdom” and that “at the very least, it is a complete paradigm
shift”, we strongly believe that this is enough to warrant the reader’s
attention to this seemingly seminal theoretical discovery.
The synopsis of this reading is as follows. In the subsequent section, we
present the theory to be used in setting up the proposed ASTG Outflow Model.
In §($3$), we revisit the persistent problem of the ASTG model, that of “The
ASTG’s Undetermined Parameter Problem”. Therein, we present what we believe
may be a solution to this problem. As to what really these parameters may be,
this is still an open question subject to debate. In §($4$), we present the
main findings of the present reading, that is. the repulsive bipolar
gravitational field and therein we argue that this field fits the description
of outflows. We present this for both the empty and non-empty space solutions
of the Poisson-Laplace equation. In §($5$), we look at the anatomy of the
outflow model, i.e. the switching on and off outflows, the nature of the
repulsive polar field, the emergent shock rings and the collimation factor of
these outflows. In §($6$), we show that the ASTG model posits what strongly
appears to be a perdurable solution to the so-called Radiation Problem that is
thought to be faced by massive stars during their formation process. Lastly,
in §($7$), we give a general discussion and make conclusion that cane be drawn
from this reading.
Lastly, it is important that we mention here in the penultimate of this
introductory section that this reading is fundamental in nature and because of
this, we shall seek to begin whatever argument we seek to rise, from the soils
of its very basic and fundamental level. This is done so that we are at the
same level of understanding with the reader. With the aforesaid approach, if
at any point we have errored, it would be easy to know and understand where
and how we have errored.
## 2 Theory
Newton’s Law of universal gravitation can be written in a more general and
condensed form as Poisson’s Law, i.e.:
$\vec{\nabla}^{2}\Phi=4\pi G\rho,$ (1)
where $\rho$ is the density of matter and $G=6.667\times
10^{-11}\textrm{kg}^{-1}\textrm{ms}^{-2}$ is Newton’s universal constant of
gravitation and the operator $\vec{\nabla}^{2}$ written for a spherical
coordinate system [see figure (1) for the coordinate setup] is given by:
$\vec{\nabla}^{2}=\frac{1}{r^{2}}\frac{\partial}{\partial
r}\left(r^{2}\frac{\partial}{\partial
r}\right)+\frac{1}{r^{2}\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta\frac{\partial}{\partial\theta}\right)+\frac{1}{r^{2}\sin^{2}\theta}\frac{\partial^{2}}{\partial\varphi^{2}},$
(2)
where the symbols have their usual meanings. For a spherically symmetric
setting, the solution to Poisson’s equation outside the vacuum space (where
$\rho=0$) of a central gravitating body of mass $\mathcal{M}_{star}$ is given
by the traditional inverse distance Newtonian gravitational potential which is
given by:
$\Phi(r)=-\frac{G\mathcal{M}_{star}}{r},$ (3)
where $r$ is the radial distance from the center of the gravitating body. The
Poisson equation for the case $(\rho=0)$ is known as the Laplace equation. The
Poisson equation is an extension of the Laplace equation. Because of this, we
shall generally refer to the Poisson equation as the Poisson-Laplace equation.
In the case where there is material surrounding this central mass, that is
$\mathcal{M}=\mathcal{M}(r)$, where:
$\mathcal{M}(r)=\int^{r}_{0}\int^{2\pi}_{0}\int^{2\pi}_{0}r^{2}\rho(r,\theta,\varphi)\sin\theta
d\theta d\varphi dr,$ (4)
we must – in (3), make the replacement:
$\mathcal{M}_{star}\longmapsto\mathcal{M}(r)$. As already argued in Paper I,
if the gravitating body in question is spinning, we ought to consider an
Azimuthally Symmetric Gravitational Field (ASGF). Thus, we shall solve the
azimuthally symmetric setting of (1) for both cases of empty and non-empty
space and show from these solutions that Poisson’s equation entails a
repulsive bipolar gravitational field. We shall assume that if one has the
empty space solution, to obtain the non-empty space solutions, one has to make
the replacement: $\mathcal{M}_{star}\longmapsto\mathcal{M}(r)$, just as is
done in Newtonian gravitation. This is a leaf that we shall take from
spherically symmetric Newtonian gravitation into the ASTG model.
Figure (1): This figure shows a generic spherical coordinate system, with the
radial coordinate denoted by $r$, the zenith (the angle from the North Pole;
the co-latitude) denoted by $\theta$, and the azimuth (the angle in the
equatorial plane; the longitude) by $\varphi$.
### 2.1 Empty Space Solutions
As already argued in Paper I, for a scenario or setting that exhibits
azimuthal symmetry such as a spinning gravitating body as the Sun and also the
stars that populate the heavens (where the unexpected and spectacular
champagne like bipolar molecular outflows are the observed); we must have:
$\Phi=\Phi(r,\theta)$. There-in Paper I, the Poisson equation for empty space
has been “solved” for a spinning gravitating system and the solution to it is:
$\Phi(r,\theta)=-\sum^{\infty}_{\ell=0}\left[\lambda_{\ell}c^{2}\left(\frac{G\mathcal{M}_{star}}{rc^{2}}\right)^{\ell+1}P_{\ell}(\cos\theta)\right],$
(5)
where $\lambda_{\ell}$ is an infinite set of dimensionless parameters with
$\lambda_{0}=1$ and the rest of the parameters $\lambda_{\ell}$ for
$(\ell>1)$, generally take values different from unity. There-in Paper I, a
suggestion as to what these parameters may be has been made. In §($3$) we go
further and suggest a form for these parameters. This suggestion, if correct,
puts the ASTG on a pedestal to make predictions without first seeking these
values (i.e. the $\lambda_{\ell}$’s) from observations. We will show that
there lays embedded in (5) a solution that is such that the polar regions of
the gravitating central body will exhibit a repulsive gravitational field. It
is this repulsive gravitational field that we shall propose as the driving
force causing the emergence of outflows. But, we must bare in mind that
outflows are seen in regions in which the central gravitating body is found in
the immensement of ambient circumstellar material, thus we must – for the
azimuthally symmetric case (where the central gravitating body is spinning),
solve the Poisson-Laplace equation for the setting $(\rho\neq 0)$.
### 2.2 Non-Empty Space Solutions
Clearly, in the event that $(\rho\neq 0)$ for the azimuthally symmetric case,
we must have $\rho=\rho(r,\theta)$. In Paper I, an argument has been advanced
in support of this claim that: $\Phi(r,\theta)\Rightarrow\rho(r,\theta)$.
Taking this as given, the question we wish to answer is; what form does
$\Phi(r,\theta)$ take for a given mass distribution $\rho(r,\theta)$? or the
reverse, what form does $\rho(r,\theta)$ take for a given $\Phi(r,\theta)$? It
is reasonable and most logical to assume that the gravitational field is what
influences the distribution of mass and not the other way round. Taking this
as the case, then, we must have $\rho(r,\theta)=\rho(\Phi)$, i.e. the
distribution of the matter in any mass distribution must be a function of the
gravitational field. We find that the form for $\rho(r,\theta)$ that meets the
requirement: $\rho(r,\theta)=\rho(\Phi)$, and most importantly the requirement
that to obtain the non-empty space solution from the empty space solution one
simply makes the replacement: $\mathcal{M}_{star}\longmapsto\mathcal{M}(r)$,
is:
$\rho(r,\theta)=-\frac{1}{4\pi G}\left[\frac{2}{r}\frac{\partial}{\partial
r}-\frac{1}{r^{2}}\right]\frac{\partial\Phi(r,\theta)}{\partial\theta}.$ (6)
How did we arrive at this? We have to answer this question. To make life very
easy for us to arrive at the answer, we shall write Poisson’s equation in
rectangular coordinates, i.e.:
$\left(\sum_{j=1}^{3}\frac{\partial^{2}}{\partial
x^{2}_{j}}\right)\Phi(x,y,z)=4\pi G\rho(x,y,z),$ (7)
where $x_{1}=x,x_{2}=y,x_{3}=z$. Now suppose we had a function $F(x,y,z)$ such
that:
$\left(\sum_{j=1}^{3}\frac{\partial}{\partial x_{j}}\right)^{2}F(x,y,z)=0.$
(8)
This equation can be written as:
$\left(\sum_{j=1}^{3}\frac{\partial^{2}}{\partial
x^{2}_{j}}\right)F(x,y,z)=-\left(\sum_{j}^{3}\sum_{i\neq
j}^{3}\frac{\partial^{2}}{\partial x_{i}\partial x_{j}}\right)F(x,y,z).$ (9)
Now, if and only if the gravitational potential did satisfy (7), then,
comparison of (7) with (9) requires the identification: $\Phi(x,y,z,)\equiv
F(x,y,z)$, and as-well the identification:
$\rho(x,y,z)=-\frac{1}{4\pi G}\left(\sum_{j}^{3}\sum_{i\neq
j}^{3}\frac{\partial^{2}}{\partial x_{i}\partial x_{j}}\right)\Phi(x,y,z).$
(10)
What this means is that the non-linear terms of (7) come about because of the
presence of matter. Now, if we transform to spherical coordinates, it is now
understood as to why and how we came to the choice of $\rho$ given in (6). At
the end of the day, what this means is that we can choose whatever form for
$\Phi$, the density $\rho$ will have to conform and prefigure to this setting
of the gravitational field via (10). Only and only after accepting (10), do we
have the mathematical legitimacy to choose to maintain the form (5) which we
found for the case of empty space such that in the place of
$\mathcal{M}_{star}$ we now can put $\mathcal{M}(r)$, hence thus in the case
where a central gravitating condensation of mass is in the immensement of
ambient circumstellar material, we must have:
$\Phi(r,\theta)=-\sum^{\infty}_{\ell=0}\lambda_{\ell}c^{2}\left(\frac{G\mathcal{M}(r)}{rc^{2}}\right)^{\ell+1}P_{\ell}(\cos\theta),$
(11)
where $\mathcal{M}(r)$ is given in (4). We believe this answers the question
“What form does $\rho(r,\theta)$ take for a given $\Phi(r,\theta)$?” and at
the same-time we have justified (6) viz how we have come to it. Importantly,
it should be noted that the observed radial density profile is maintained by
the choice (10), i.e.
$\rho(r)=\int^{r}_{0}\int^{2\pi}_{0}r^{2}\rho(r,\theta)\sin\theta d\theta
dr\propto r^{-\alpha_{\rho}}$. Also important to state clearly is that, all
the above implies that the gravitational field is what influences the
distribution of matter – this, in our view, resonates both with logic and
intuition. We shall demonstrated the assertion that:
$\rho(r)=\int^{r}_{0}\int^{2\pi}_{0}r^{2}\rho(r,\theta)\sin\theta d\theta
dr\propto r^{-\alpha_{\rho}}$. We know that:
$\int^{r}_{0}\int^{2\pi}_{0}r^{2}\rho(r,\theta)\sin\theta d\theta
dr=\int^{r}_{0}r^{2}\left(\int^{2\pi}_{0}\rho(r,\theta)\sin\theta
d\theta\right)dr=4\pi\int^{r}_{0}r^{2}\rho(r)dr$ (12)
this means:
$\rho(r)=\frac{1}{4\pi}\int^{2\pi}_{0}\rho(r,\theta)\sin\theta d\theta.$ (13)
Our claim is that if $\rho(r,\theta)$ is given by (6) such that
$\Phi(r,\theta)$ is given by (11), where $\mathcal{M}(r)$ in (11) is such that
$\mathcal{M}(r)\propto r^{\alpha}$ for some constant $\alpha$, then:
$\rho(r)=\frac{1}{4\pi}\int^{2\pi}_{0}\rho(r,\theta)\sin\theta d\theta\propto
r^{\alpha_{\rho}},$ (14)
where $\alpha_{\rho}$ is some constant. We know that:
$\frac{1}{4\pi}\int^{2\pi}_{0}\rho(r,\theta)\sin\theta
d\theta=-\frac{1}{16\pi^{2}G}\int^{2\pi}_{0}\left(\frac{2}{r}\frac{\partial}{\partial
r}-\frac{1}{r^{2}}\right)\frac{\partial\Phi(r,\theta)}{\partial\theta}\sin\theta
d\theta.$ (15)
We have substituted $\rho(r,\theta)$ in (6) into the above. This simplifies
to:
$\frac{1}{4\pi}\int^{2\pi}_{0}\rho(r,\theta)\sin\theta
d\theta=-\frac{1}{16\pi^{2}G}\left(\frac{2}{r}\frac{\partial}{\partial
r}-\frac{1}{r^{2}}\right)\int^{2\pi}_{0}\frac{\partial\Phi(r,\theta)}{\partial\theta}\sin\theta
d\theta.$ (16)
From (11), we know that:
$\frac{\partial\Phi(r,\theta)}{\partial\theta}=\sum^{\infty}_{\ell=0}\lambda_{\ell}c^{2}\left(\frac{G\mathcal{M}(r)}{rc^{2}}\right)^{\ell+1}\sin\theta\frac{\partial
P_{\ell}(\cos\theta)}{\partial(\cos\theta)}$ (17)
and this implies:
$\frac{1}{4\pi}\int^{2\pi}_{0}\rho(r,\theta)\sin\theta
d\theta=-\frac{c^{2}}{16\pi^{2}G}\left(\frac{2}{r}\frac{\partial}{\partial
r}-\frac{1}{r^{2}}\right)\sum^{\infty}_{\ell=0}\lambda_{\ell}\int^{2\pi}_{0}\left(\frac{G\mathcal{M}(r)}{rc^{2}}\right)^{\ell+1}\sin^{2}\theta\frac{\partial
P_{\ell}(\cos\theta)}{\partial(\cos\theta)}d\theta$ (18)
this simplifies to:
$\frac{1}{4\pi}\int^{2\pi}_{0}\rho(r,\theta)\sin\theta
d\theta=-\frac{c^{2}}{16\pi^{2}G}\left(\frac{2}{r}\frac{\partial}{\partial
r}-\frac{1}{r^{2}}\right)\sum^{\infty}_{\ell=0}\lambda_{\ell}\left(\frac{G\mathcal{M}(r)}{rc^{2}}\right)^{\ell+1}\overbrace{\int^{2\pi}_{0}\sin^{2}\theta\frac{dP_{\ell}(\cos\theta)}{d(\cos\theta)}d\theta}^{\textrm{Let}\,\,\textrm{this}\,\,\textrm{be:}\,\,\textrm{I}_{\ell}\textrm{(}\theta\textrm{)}}$
(19)
where $I_{\ell}(\theta)$ is as defined above. It should not be difficult to
see that $I_{0}(\theta)=0$, $I_{1}(\theta)=1$ and that $I_{\ell}(\theta)\equiv
0$ for all $\ell\geq 2$. From this, it follows that:
$\rho(r)=\frac{1}{4\pi}\int^{2\pi}_{0}\rho(r,\theta)\sin\theta
d\theta=\left(\frac{\lambda_{1}c^{2}}{16\pi^{2}G}\right)\left(\frac{2}{r}\frac{\partial}{\partial
r}-\frac{1}{r^{2}}\right)\left(\frac{G\mathcal{M}(r)}{rc^{2}}\right)^{2},$
(20)
Now, if $\mathcal{M}(r)\propto r^{\alpha}$ this means
$\mathcal{M}(r)=kr^{\alpha}$ for some adjustable constant $k$. Plugging this
into the above, one obtains:
$\rho(r)=\left(\frac{\left(2\alpha-1\right)\lambda_{1}c^{2}}{16\pi^{2}G}\right)\left(\frac{Gk}{c^{2}}\right)^{2}r^{2\alpha-4}.$
(21)
This222Under the prescribed conditions $\mathcal{M}(r)\propto r^{\alpha}$
leads to $\rho(r)\propto r^{2\alpha-4}$. While
$\mathcal{M}(r)=\int^{r}_{0}\int^{2\pi}_{0}\rho(r,\theta)\sin\theta d\theta
dr$, the basic definition $\mathcal{M}(r)=4\pi r^{3}\rho(r)/3$ must hold too,
since $\mathcal{M}(r)$ is the amount of mass enclosed in volume sphere of
radius $r$ and $\rho(r)$, is the mass-density of material in this volume
sphere. These two definitions must lead to identical formulas. If this is to
be so – then; one is lead to the conclusion that $\alpha=1$, and this means
$\mathcal{M}(r)\propto r$ and $\rho(r)\propto r^{-2}$. In the face of
observations, the later result is very interesting since MCs seem to favor
this density profile. verifies our claim in (14). As already said, all the
above implies that the gravitational field is what influences the distribution
of matter. Co-joining this result with the result $(0\leq\alpha_{\rho}<3)$ in
Nyambuya ([$2010c$]) (hereafter Paper III), it follows that
$(0.5\leq\alpha<2)$. Further, a deduction to be made from the above result is
that the spin does control the mass distribution via the term $\lambda_{1}$.
## 3 The Undetermined Constants $\lambda_{\ell}$
Again, as already stated in Paper I, one of the draw backs of the ASTG is that
it is heavily dependent on observations for the values of $\lambda_{\ell}$
have to be determined from observations. Without knowledge of the
$\lambda_{\ell}^{\prime}s$, one is unable to produce the hard numbers required
to make any numerical quantifications. Clearly, a theory incapable of making
any numerical quantifications is – in the physical realm, useless. To avert
this, already in Paper I and as-well in Nyambuya ([$2010b$]) (hereafter Paper
II) an effort to solve this problem has been made. In Paper I, a reasonable
suggestion was made to the effect that:
$\lambda_{\ell}=\left(\frac{(-1)^{\ell+1}}{\left(\ell^{\ell}\right)!\left(\ell^{\ell}\right)}\right)\lambda_{1}.$
(22)
This suggestion meets the intuitive requirements stated there-in Paper I. If
these $\lambda$’s are to be given by (22), then, there is just one unknown
parameter and this parameter is $\lambda_{1}$. The question is what does this
depend on? We strongly feel/believe that $\lambda_{1}$ is dependent on the
spin angular frequency and the radius of the gravitating body in question and
our reasons are as follows.
The ASTG will be shown shortly to be able to explain outflows as a
gravitational phenomenon. Pertaining to their association with star formation
activity, it is believed that molecular outflows are a necessary part of the
star formation process because their existence may explain the apparent
angular momentum imbalance. It is well known that the amount of initial
angular momentum in a typical star-forming cloud core is several orders of
magnitude too large to account for the observed angular momentum found in
formed or forming stars (see e.g. Larson [$2003b$]). The sacrosanct Law of
Conservation of angular momentum informs us that this angular momentum can not
just disappear into the oblivion of interstellar spacetime. So, the question
is where does this angular momentum go to? It is here that outflows are
thought to come to the rescue as they can act as a possible agent that carries
away the excess angular momentum. Whether or not this assertion is true or may
have a bearing with reality, no one really knows.
This angular momentum, if it where to remain as part of the nascent star, it
would, via the strong centrifugal forces (the centrifugal acceleration is
given by: $a_{c}=\omega^{2}_{star}\mathcal{R}_{star}$), tear the star apart.
This however does not explain, why they [outflows] exist and how they come to
exist but simple posits them as a vehicle needed to explain the mystery of
“The Missing Angular Momentum Problem” in star forming systems and the
existence of stars in their intact and compact form as firery balls of gas.
In Paper II, guided more by intuition than anything else, it was drawn from
the tacit thesis “that outflows possibly save the star from the detrimental
centrifugal forces”, the suggestion that
$\lambda_{1}\propto(a_{c})^{\zeta_{0}}$ where $\zeta_{0}$ is a pure constant
that must be universal, that is, it must be the same for all spinning
gravitating systems. This suggestion, if correct leads us to:
$\lambda_{\ell}=\left(\frac{(-1)^{\ell+1}}{\left(\ell^{\ell}\right)!\left(\ell^{\ell}\right)}\right)\left(\frac{a_{c}}{a_{*}}\right)^{\zeta_{0}}.$
(23)
Knowing the solar values of $\lambda_{1}$ and as-well the value of
$\zeta_{0}$, one is lead to:
$a_{*}=\omega^{2}_{\odot}\mathcal{R}_{\odot}(\lambda_{1}^{\odot})^{-\frac{1}{\zeta_{0}}}$.
As will be demonstrated soon, the term $\lambda_{1}$ controls outflows. Given
that $\lambda_{1}$ controls outflows and that outflows possibly aid the star
in shedding off excess spin angular momentum, the best choice333We speak of
“choice” here as though the decision is ours on what this parameter must be.
No, the decision was long made by Nature, ours is to find out what choice
Nature has made. That said, we should say that, this “choice” is made with
expediency – i.e., this choice which is based on intuition, is to be measured
against experience. for this parameter is one that leads to these outflows
responding to the spin of the star and as well the centrifugal forces
generated by this spin in such a way that the star is able to shed off this
excess spin angular momentum. So, what led to this proposal
$\lambda_{1}\propto(a_{c})^{\zeta_{0}}$ is the aforesaid. Now, we shall revise
this suggestion by advancing what we believe is a far much better argument.
If outflows are there to save the nascent star from the ruthlessness of the
centrifugal forces, then, it is logical to imagine that at the moment the
centrifugal forces are about to rip the star apart, outflows will switch-on,
thus shedding off this excess spin angular momentum. The centrifugal forces
have their maximum toll on the equatorial surface of the star hence if the
centrifugal forces are to rip the nascent star apart, this would start at the
equator of the nascent star. The centrifugal force on the surface of the star
acting on a particle of mass $m$ is
$F_{c}=m\omega^{2}_{star}\mathcal{R}_{star}=ma_{c}$ and the gravitational
force on the same particle is
$F_{g}=G\mathcal{M}m/\mathcal{R}_{star}^{2}=mg_{star}$. Now lets define the
quotient $\mathcal{Q}=F_{c}/F_{g}=a_{c}/g_{star}$. If the particle where to
stay put on the surface of the star, then we will have
$F_{c}-F_{g}<0\Rightarrow\mathcal{Q}<1$; and if the particle where to fly off
the surface, we will have $F_{c}-F_{g}>0\Rightarrow\mathcal{Q}>1$. The
critical condition before the star begins to be torn apart is
$F_{c}-F_{g}=0\Rightarrow\mathcal{Q}=1$. All the above can be summarized as:
$\mathcal{Q}:=\left\\{\begin{array}[]{l l l}<1&&\textrm{No\,\,
Outflow\,\,Activity}\\\ =1&&\textrm{Critical\,\, Condition}\\\ >1&&\
textrm{Outflow\,\,Activity}\end{array}\right..$ (24)
Lets call this quotient, the Outflow Control Quotient (OCQ). Clearly, the OCQ
determines the necessary conditions for outflows to switch on. Given this, and
as-well the thinking that $\lambda_{1}$ controls outflows, the suggestion is
clear that $\lambda_{1}\propto\mathcal{Q}^{\zeta_{0}}$. If this is correct,
then:
$\lambda_{1}=\zeta\mathcal{Q}^{\zeta_{0}}.$ (25)
We shall take this as our proposal for $\lambda_{1}$ and this means we must
determine $(\zeta,\zeta_{0})$. From the above, it follows that:
$\frac{\lambda_{1}^{\oplus}}{\lambda_{1}^{\odot}}=\left(\frac{\mathcal{Q}_{\oplus}}{\mathcal{Q}_{\odot}}\right)^{\zeta_{0}},$
(26)
where $\mathcal{Q}_{\oplus}=a_{c}^{\oplus}/g_{\oplus}$ and $a_{c}^{\oplus}$ is
the centripetal acceleration generated by the Earth’s spin at the equator and
$g_{\oplus}$ is the gravitational field strength at the Earth equator.
Likewise, $\mathcal{Q}_{\odot}=a_{c}^{\odot}/g_{\odot}$, is the solar outflow
quotient where $a_{c}^{\odot}$ is the centripetal acceleration generated by
the Sun’s spin at the solar equator and $g_{\odot}$ is the gravitational field
strength at the solar equator. Given that: $(\omega_{\oplus}=7.27\times
10^{-5}\,\textrm{Hz}$ and $\omega_{\odot}=2.04\times 10^{-5}\,\textrm{Hz})$,
$(\mathcal{R}_{\oplus}=6.40\times 10^{6}\,\textrm{m}$ and
$\mathcal{R}_{\odot}=6.96\times 10^{8}\,\textrm{m})$ and
$(g_{\oplus}=9.80\,\textrm{ms}^{-2}$ and $g_{\odot}=27.9g_{\oplus})$. From
this data, it follows that:
$\frac{\mathcal{Q}_{\oplus}}{\mathcal{Q}_{\odot}}=169.$ (27)
Now, in Paper II, we did show that depending on how one interprets the flyby
equation, one obtains two values of $\lambda_{1}^{\oplus}$, i.e.
$\lambda_{1}^{\oplus}=(2.00\pm 0.80)\times 10^{3}$ and
$\lambda_{1}^{\oplus}=(1.50\pm 0.70)\times 10^{4}$. If the spin of the Earth
is significantly variable during the course of its orbit around the Sun, we
will have $\lambda_{1}^{\oplus}=(2.00\pm 0.80)\times 10^{3}$ and if the spin
is not significantly variable, then, $\lambda_{1}^{\oplus}=(1.50\pm
0.70)\times 10^{4}$. If $\lambda_{1}^{\oplus}=(2.00\pm 0.80)\times 10^{3}$,
then:
$\frac{\lambda_{1}^{\oplus}}{\lambda_{1}^{\odot}}=\frac{15000\pm
7000}{21.00\pm 4.00}=800\pm 500,$ (28)
and from this it follows that $800\pm 500=169.19^{\zeta_{0}}$, hence
$\zeta_{0}=1.30\pm 0.10$. If $\lambda_{1}^{\oplus}=(1.50\pm 0.70)\times
10^{4}$, then:
$\frac{\lambda_{1}^{\oplus}}{\lambda_{1}^{\odot}}=\frac{2000\pm 800}{21.00\pm
4.00}=100\pm 60,$ (29)
and from this it follows that $100\pm 60=169.19^{\zeta_{0}}$ hence
$\zeta_{0}=0.90\pm 0.10$.
If $\zeta_{\oplus}$ and $\zeta_{\odot}$ are the $\zeta$-values for the Earth
and the Sun respectively, then, for $\lambda_{1}^{\oplus}=15000\pm 7000$, we
will have $\zeta_{\oplus}=(3.40\pm 2.70)\times 10^{7}$ and
$\zeta_{\odot}=(8.00\pm 4.00)\times 10^{10}$; and for
$\lambda_{1}^{\oplus}=2000\pm 800$, we will have $\zeta_{\oplus}=(3.40\pm
2.70)\times 10^{7}$ and $\zeta_{\odot}=(8.00\pm 4.00)\times 10^{10}$. Table
(I) is a self explanatory summary of all the above calculations. The mean
values of $\zeta$ for the case $\lambda_{1}^{\oplus}=(2.00\pm 0.80)\times
10^{3}$ is $\zeta=(8.00\pm 1.00)\times 10^{5}$ and for the case
$\lambda_{1}^{\oplus}=(1.50\pm 0.70)\times 10^{4}$ is $\zeta=(4.00\pm
2.00)\times 10^{7}$. These mean values have been obtained by taking the values
of $\zeta_{\oplus}$ and $\zeta_{\odot}$ where they intersect in their error
margins.
Table (I): : The $(\zeta_{0},\zeta)$ Values for the Two Different Values of $\lambda_{1}^{\oplus}$. $\lambda_{1}^{\oplus}$ | $\lambda_{1}^{\odot}$ | $\zeta_{0}$ | $\zeta_{\odot}$ | $\zeta_{\oplus}$
---|---|---|---|---
($10^{3}$) | | | ($10^{5}$) | ($10^{5}$)
$\,\,\,2.00\pm 0.80$ | $21.00\pm 4.00$ | $0.90\pm 0.10$ | $13.00\pm 6.00$ | $5.00\pm 3.00$
$15.00\pm 7.00$ | $21.00\pm 4.00$ | $1.30\pm 0.10$ | $500\pm 400$ | $400\pm 200$
As argued in Paper II, the value $\lambda_{1}^{\oplus}=(2.00\pm 0.80)\times
10^{3}$ has been obtained from the assumption that the spin of the Earth
varies widely during its course on its orbit around the Sun. This is not
supported by observations thus we are not persuaded to take-up/recommend this
value of $\lambda_{1}^{\oplus}=(2.00\pm 0.80)\times 10^{3}$. Also, as argued
in Paper II, the value $\lambda_{1}^{\oplus}=(1.50\pm 0.70)\times 10^{4}$ is
obtained from the assumption that the spin of the Earth does not vary widely
during its course on its orbit. Thus, we shall adopt the values of
$(\zeta_{0},\zeta)$ that conform with $\lambda_{1}^{\oplus}=(1.50\pm
0.70)\times 10^{4}$ and $\lambda_{1}^{\odot}=21.0\pm 0.40$, hence:
$\lambda_{1}=(4.00\pm 2.00)\times
10^{7}\left(\frac{a_{c}}{g_{star}}\right)^{1.30\pm 0.10}.$ (30)
Obviously, the greatest criticism against this result is that it is obtained
from just two data points. To obtain something more reliable, one needs more
data points. This is something that a future study must handle, at present, we
simple want to set-up the mathematical model from the little available data
and when data becomes available, amendments are made accordingly. While we
have used the minimal possible data points, one thing that can be deduced from
this data is that this result obtained points to a correlation as proposed in
(25) – otherwise, if there was no correlation as proposed, the values of
$(\zeta_{0},\zeta)$ obtained the two values of $\lambda_{1}$ do not vary
widely as is expected if the proposed relationship (25) did not hold at all.
## 4 Outflows as a Gravitational Phenomenon
We shall look into the empty and non-empty space solution of the solution of
the Poisson-Laplace equation and show that both these solutions exhibit a
repulsive bipolar gravitational field and that this repulsive gravitational
field is controlled by the parameter $\lambda_{1}$.
### 4.1 Non-Empty Space Solutions
Now, if one accepts what has been presented thus far – as will be shown in
this section; it follows that outflows may-well be a gravitational phenomena.
First, from the previous section, it follows that we must take the ASTG only
up to second order, i.e.:
$\Phi=-\frac{G\mathcal{M}(r)}{r}\left[1+\frac{\lambda_{1}G\mathcal{M}(r)\cos\theta}{rc^{2}}+\lambda_{2}\left(\frac{G\mathcal{M}(r)}{rc^{2}}\right)^{2}\left(\frac{3\cos^{2}\theta-1}{2}\right)\right].$
(31)
We know that the gravitational field intensity:
$\vec{\textbf{g}}(r,\theta)=-\mbox{\boldmath$\nabla$}\Phi(r,\theta)=g_{r}(r,\theta)\hat{\textbf{r}}+g_{\theta}(r,\theta)\hat{\mbox{\boldmath$\theta$}}$,
this means:
$g_{r}=g_{N}\left[\overbrace{1+\frac{2\lambda_{1}G\mathcal{M}(r)\cos\theta}{rc^{2}}}^{\textbf{Term
I}}+\overbrace{3\lambda_{2}\left(\frac{G\mathcal{M}(r)}{rc^{2}}\right)^{2}\left(\frac{3\cos^{2}\theta-1}{2}\right)}^{\textbf{Term
II}}\right],$ (32)
where: $g_{N}=-G\mathcal{M}(r)/r^{2}$, is the Newtonian gravitational field
intensity and:
$g_{\theta}=g_{N}r^{2}\sin\theta\left[\frac{\lambda_{1}G\mathcal{M}(r)}{rc^{2}}+9\lambda_{2}\left(\frac{G\mathcal{M}(r)}{rc^{2}}\right)^{2}\cos\theta\right].$
(33)
For gravitation to be exclusively attractive (as is expected), we must have:
[$g_{r}(r,\theta)>0$] and [$g_{\theta}(r,\theta)~{}>~{}0$]. From (32) and
(33), it is clear that regions of exclusively repulsive gravitation will exist
and these will occur in the region where: [$g_{r}(r,\theta)<0$] and
[$g_{\theta}(r,\theta)~{}<~{}0$]. This region where gravity is exclusively
repulsive is the region where it is not attractive, it is the negated region
of the region of attractive gravitation: [i.e.
$\left\\{g_{r}(r,\theta)>0\right\\}$ and
$\left\\{g_{\theta}(r,\theta)~{}>~{}0\right\\}$]. Let us start by treating the
case: [$g_{r}(r,\theta)~{}<~{}0$]. From (32), if: [$g_{r}(r,\theta)<0$], then:
(Term I $<0$) and (Term II $<0$), as well. The condition: (Term I $<0$),
implies:
$r<-\lambda_{1}\left(\frac{2G\mathcal{M}(r)}{c^{2}}\right)\cos\theta=\lambda_{1}\left(\frac{2G\mathcal{M}(r)}{c^{2}}\right)\cos\theta,$
(34)
(NB: $\cos\theta\equiv-\cos\theta$) and if one where to take $r$ such that it
only takes positive values, then, (34) must be written in the equivalent form:
$r<\lambda_{1}\left(\frac{2G\mathcal{M}(r)}{c^{2}}\right)\left|\cos\theta\right|,$
(35)
where the brackets $\left|[]\right|$ represents the absolute value. We have to
explain this, i.e. why we concealed the negative sign in (34) and inserted the
absolute value operator in (35). From (34), it is seen that this inequality
includes negative values of $r$ and to avoid any confusion as to what these
negative values of $r$ really mean, this needs to be explained for failure to
do so or failure by the reader to understand this means they certainly will be
unable to agree with the outflow “picture” laid down herein. This explanation
is important in order to understand the morphology of the outflow and as-well
the ASGF.
For a moment, imagine a flat Euclidean plane and on this plane let O, A and P
be distinct and separate points on this plane with O and A being fixed and P
is a variable point. In polar coordinates, as in the present case, a point P
is characterized by two numbers: the distance $(r\geq{0})$ to the fixed pole
or origin O, and the angle $\theta$ the line OP makes with the fixed reference
line OA. The angle $\theta$ is only defined up to a multiple of
${360}\hbox{${}^{\circ}$}$ (or ${2}\pi\,\textrm{rad}$, in radians). This is
the conventional definition. Sometimes it is convenient as in the present case
to relax the condition $(r\geq 0)$ and allow $r$ to be assigned a negative
value such that the point $(r,\theta)$ and
$(-r,\theta+{180}\hbox{${}^{\circ}$})$ represent the same-point, hence thus
when ever we have $(-r,\theta)$ this must be replaced by
$(r,\theta-{180}\hbox{${}^{\circ}$})$. It is easier for us to always think of
$r$ as always being positive. To achieve this, given the fact that
$(-r,\theta)\equiv(r,\theta-{180}\hbox{${}^{\circ}$})$, we must write (34) as
has been done in (35), hence (35) finds justification. This explanation can be
found in any good mathematics textbook that deals extensively with polar
coordinates. Hereafter, whenever a similar scenario arises where negative
values of $r$ emerge, we will automatically and without notification assume
that $(-r,\theta)$ is $(r,\theta-{180}\hbox{${}^{\circ}$})$ and this will come
with the introduction of the absolute value sign as has been done in (35).
Now, proceeding from where we left. As has already been explained at the
beginning of this section, we have to substitute the Mass Distribution
Function (MDF) $\mathcal{M}(r)$ into (35) and having done so we would have to
make $r$ the subject. It has been argued in equation $24$ of Paper III, that
for a MC that exhibits a density profile: $\rho(r)\propto r^{-\alpha_{\rho}}$,
where $\alpha_{\rho}$ is the density index, that the MDF is given by:
$\mathcal{M}(r)=\overbrace{\mathcal{M}_{csl}\left(\frac{r^{3-\alpha_{\rho}}-\mathcal{R}_{star}^{3-\alpha_{\rho}}}{\mathcal{R}^{3-\alpha_{\rho}}_{core}-\mathcal{R}_{star}^{3-\alpha_{\rho}}}\right)}^{\textrm{\tiny\bf{Circumstellar}\,{Mass}\,{Inside}\,{Region}\,{of
\,\,Radius}\,{r}}}+\overbrace{\mathcal{M}_{star}}^{\textbf{ {\tiny
Nascent\,Star's\,Mass}}}\,\,\,\,\,\,\,\,\,\textrm{for}\,\,\,\,\,\,r\geq\mathcal{R}_{star},$
(36)
where $\mathcal{M}_{csl}$ is total mass of the circumstellar material at any
given time, $\mathcal{R}_{star}$ is the radius of the nascent star at any
given time, $\mathcal{R}_{core}$ is the radius at any given time of the
gravitationally bound core from which the star is forming.
Now, substituting the MDF (given above) into (35) and thereafter making $r$
the subject of the formula would lead to a horribly complicated inequality
that would require the use of the Newton-Ralphson approach to solve. Since
ours in the present is but a qualitative analysis, we can make some very
realistic simplifying assumptions that can make our life much easier. If the
spatial extent of the star is small compared to that of the core i.e.:
$(\mathcal{R}_{star}\ll\mathcal{R}_{core}\Rightarrow\mathcal{R}_{core}^{3-\alpha_{\rho}}-\mathcal{R}_{star}^{3-\alpha_{\rho}}\simeq\mathcal{R}_{core}^{3-\alpha_{\rho}})$
and the mass of the star is small compared to the mass of the core i.e.:
$(\mathcal{M}_{star}\ll\mathcal{M}_{core}\Rightarrow\mathcal{M}_{csl}\simeq\mathcal{M}_{core})$,
then, the MDF simplifies to:
$\mathcal{M}(r)\simeq\mathcal{M}_{core}\left(\frac{r}{\mathcal{R}_{core}}\right)^{3-\alpha_{\rho}}.$
(37)
Inserting this into (35) and thereafter performing some basic algebraic
computations that see $r$ as the subject of the formula, one is lead to:
$r<\left[\lambda_{1}\left(\frac{2G\mathcal{M}_{core}}{c^{2}\mathcal{R}_{core}}\right)\mathcal{R}_{core}\right]\left|\cos\theta\right|^{\frac{1}{2-\alpha_{\rho}}}.$
(38)
Now, if we set:
$\epsilon_{1}^{core}=\left(\left[\lambda_{1}\left(\frac{2G\mathcal{M}_{core}}{c^{2}\mathcal{R}_{core}}\right)\right]^{\frac{1}{2-\alpha_{\rho}}}\right)\left(\frac{\mathcal{R}_{core}}{\mathcal{R}_{star}}\right),$
(39)
then (38) reduces to:
$r<\epsilon_{1}^{core}\mathcal{R}_{star}\left|\cos\theta\right|^{\frac{1}{2-\alpha_{\rho}}}=l_{max}\left|\cos\theta\right|^{\frac{1}{2-\alpha_{\rho}}},$
(40)
where: $l_{max}=\epsilon_{1}^{core}\mathcal{R}_{star}$. On the xy-plane as
shown in figure (2), the equation:
$r=l_{max}\left|\cos\theta\right|^{\frac{1}{2-\alpha_{\rho}}}$, describes two
lobs. For the purposes of this reading, let the volume of revolution of the
lob be called a loboid, and the loboid above the x-axis shall be called the
upper loboid, and likewise the loboid below the x-axis shall be called the
lower loboid.
Now, the condition: (Term II $<0$), implies: $[\theta~{}<~{}\cos^{-1}(\pm
1/\sqrt{3})]$, which means: $(-54.7<\theta<54.7)$. Now, for the azimuthal
component to be repulsive, we must have: $[g_{\theta}(r,\theta)>0]$, we will
have from (33), the condition:
$r>-\left(\frac{9\lambda_{2}}{2\lambda_{1}^{2}}\right)\left(\frac{2\lambda_{1}G\mathcal{M}(r)}{c^{2}}\right)\cos\theta.$
(41)
Now going through the same procedure as above, (41) can be written as:
$r>l_{min}\left|\cos\theta\right|^{\frac{1}{2-\alpha_{\rho}}},$ (42)
where:
$l_{min}=\left(\left|\frac{9\lambda_{2}}{2\lambda_{1}^{2}}\right|^{\frac{1}{2-\alpha_{\rho}}}\right)l_{max}.$
(43)
Thus, coalescing the results, invariably, one is led to conclude that the
region of repulsive gravitation is:
$\left[l_{min}<r<l_{max}\right]\,\textit{\&}\,\left[\cos^{-1}\left(-\frac{1}{\sqrt{3}}\right)<\theta<\cos^{-1}\left(\frac{1}{\sqrt{3}}\right)\right].$
(44)
In the region described above, the gravitational field is both radially and
azimuthally repulsive, that is, there is complete gravitational repulsion in
this region. Pictorially, a summary of the emergent picture of the repulsive
gravitational field in shown in figure (2). This picture – in our view, fits
the description of outflows, the limiting factors are the sizes of $l_{max}$
and $l_{min}$, these values all depend on the one parameter $\lambda_{1}$,
hence thus, this parameter is the crucial parameter which determines the
properties of outflows. Shortly, we will discuss this picture but before this,
it is necessary that we go through the empty space solutions as-well.
Figure (2): This figure illustrates the emergent picture from the azimuthally
symmetric considerations of the Poisson equation. While fanning out matter in
the region of repulsive gravitation, the rotating star is surrounded by an
equatorial disk; once the outflow switches-on, this disk is the only channel
via which the mass of the star feeds. The disk is not affected by radiation in
the sense that some of its material close to the nascent star will be swept
away by the radiation field, no! The force of gravity along this disk is
purely radial and is directed toward the nascent.
### 4.2 Empty Space Solutions
As will be demonstrated in this section, the picture imaging from the empty
space solution is not different from that of the non-empty space solution.
However, there is an important difference between these two pictures and this
difference need to be stated. If our spinning gravitating body is not giving
off material like the Sun, then the region of repulsive gravitation will occur
inside the this body. We shall consider the star to be a point mass i.e., all
of its mass is concentrated at the star’s center of mass.
As before, from (32) and (33), it is clear that regions of repulsive
gravitation will exist and these will occur where [$g_{r}(r,\theta)<0$] and or
[$g_{\theta}(r,\theta)~{}<~{}0$]. We shall as before start by treating the
case [$g_{r}(r,\theta)~{}<~{}0$]. From (32), if [$g_{r}(r,\theta)<0$], then
(Term I $<0$) and (Term II $<0$) as well. The condition (Term I $<0$) implies:
$r<-\lambda_{1}\left(\frac{2G\mathcal{M}}{c^{2}}\right)\cos\theta,$ (45)
where in the present case $\mathcal{M}(r)$ must be replaced by
$\mathcal{M}_{star}$ and this can be written in the equivalent form:
$r<\lambda_{1}\left(\frac{2G\mathcal{M}_{star}}{c^{2}}\right)\left|\cos\theta\right|.$
(46)
Now, if we set:
$\epsilon_{1}^{star}=\lambda_{1}\left(\frac{\mathcal{R}_{star}^{s}}{\mathcal{R}_{star}}\right),$
(47)
where $\mathcal{R}_{star}^{s}=2G\mathcal{M}_{star}/c^{2}$ is the Schwarzchild
radius of the star, then (46) reduces to:
$r<\epsilon_{1}^{star}\mathcal{R}_{star}|\cos\theta|=l_{max}|\cos\theta|.$
(48)
Now, the condition (Term II $<0$), as before, implies:
$[\theta~{}<~{}\cos^{-1}(\pm 1/\sqrt{3})]$ , which means:
$(-54.7<\theta<54.7)$. Again as before, for the azimuthal component to be
repulsive, we must have: $[g_{\theta}(r,\theta)>0]$, we will have from (33),
that:
$r>\left(\frac{9\lambda_{2}}{2\lambda_{1}^{2}}\right)\left(\frac{2\lambda_{1}G\mathcal{M}_{star}}{c^{2}}\right)\left|\cos\theta\right|,$
(49)
and we need not explain anymore why the above can be written as:
$r>l_{min}|\cos\theta|,$ (50)
where this time:
$l_{min}=\left|\frac{9\lambda_{2}}{2\lambda_{1}^{2}}\right|l_{max}.$ (51)
Coalescing the results, invariably, one is led to conclude that the region of
repulsive gravitation is:
$\left[l_{min}<r<l_{max}\right]\,\textit{\&}\,\left[\cos^{-1}\left(-\frac{1}{\sqrt{3}}\right)<\theta<\cos^{-1}\left(\frac{1}{\sqrt{3}}\right)\right].$
(52)
As in the case of the non-empty space, in the region described above, the
gravitational field is both radially and azimuthally repulsive, hence there is
complete gravitational repulsion in this region. The emergent picture is no
different from that of the case of non-empty space. The important difference
is that the region of gravitational repulsion is confined in the interior of
the star if $(\epsilon_{1}<1)$, it is not visible outside. If
$(\epsilon_{1}<1)$, there will exist no repulsive bipolar gravitational field
that is visible to beyond the surface of the spinning star. In the interior of
the star, the solutions obtained for the case of non-empty space is what must
apply.
## 5 ASGF of a Spinning Core with an Embedded Spinning Star
Central to the ASTG is that the material under consideration possesses some
finite spin angular momentum. In the case of a nascent star embedded inside a
gravitationally bound core, we are going to have the star’s spin angular
frequency being different to that of the circumstellar material; because, in
the early stages when the nascent star is forming, the spin angular frequency
of the circumstellar material and the star will, on the average, be the same
since it is expected that circumstellar material and the star will co-rotate;
but, because of the increasing mass and spin angular momentum of the nascent
star due to the accretion of material, at some-point, the star must break-off
from this co-rotational motion and spin independently of the circumstellar
material, thus in the end, the star will have a different spin angular
frequency to that of the circumstellar material. The different spin angular
momentum of the nascent star and the circumstellar material will come along
with different $\lambda$-values. Assuming the circumstellar material is co-
rotating with itself, it must have its own $\lambda$-value, let us call this
$\lambda_{\ell}^{csl}$ and that for the star be $\lambda_{\ell}^{star}$.
If there is a way of calculating the ASGF of the star at point $(r,\theta)$
and that of the circumstellar material at that same point $(r,\theta)$, then
one will be able to calculate the resultant ASGF at any point $(r,\theta)$
because the gravitational field is here a scalar. Let $\Phi_{star}$ be the
Azimuthally Symmetric Gravitational Potential (ASGP) of the star and that of
the circumstellar material be $\Phi_{csl}$. Knowing $\Phi_{star}$ and
$\Phi_{csl}$, clearly the resultant ASGP $\Phi_{eff}$ at any point
$(r,\theta)$ is $\Phi_{eff}=\Phi_{star}+\Phi_{csl}$, hence one will be able to
obtain the resultant ASGF. The ASGF of the star is not difficult to obtain, we
already know that it must be given by:
$\Phi_{star}=-\frac{G\mathcal{M}_{star}}{r}\left[1+\frac{\lambda_{1}^{star}G\mathcal{M}_{star}\cos\theta}{rc^{2}}+\lambda_{2}^{star}\left(\frac{G\mathcal{M}_{star}}{rc^{2}}\right)^{2}\left(\frac{3\cos^{2}\theta-1}{2}\right)\right].$
(53)
Now, we have to obtain the ASGF of a spinning core. The gravitational
potential (31) is the potential of star that is co-rotating with the
circumstellar material. If we remove the central star from this gravitational
potential what remains is the gravitational potential of a spinning core.
Removing the central star from this potential means set
$\mathcal{M}_{star}=0$, hence, the gravitational potential of a spinning core
must be:
$\Phi_{csl}=-\frac{G\mathcal{M}_{csl}(r)}{r}\left[1+\frac{\lambda_{1}^{csl}G\mathcal{M}_{csl}(r)\cos\theta}{rc^{2}}+\lambda_{2}^{csl}\left(\frac{G\mathcal{M}_{csl}(r)}{rc^{2}}\right)^{2}\frac{3\cos^{2}\theta-1}{2}\right],$
(54)
where $\lambda_{\ell}^{csl}$ is the $\lambda_{\ell}$-value for the spinning
circumstellar material and:
$\mathcal{M}_{csl}(r)=\mathcal{M}_{csl}\left(\frac{r^{3-\alpha_{\rho}}-\mathcal{R}_{cav}^{3-\alpha_{\rho}}(t)}{\mathcal{R}^{3-\alpha_{\rho}}_{core}(t)-\mathcal{R}_{cav}^{3-\alpha_{\rho}}(t)}\right)\,\,\textrm{for}\,\,r\geq\mathcal{R}_{cav}(t),$
(55)
is the circumstellar material enclosed in radius $r$. Now, as argued already:
$\Phi_{eff}=\Phi_{star}+\Phi_{csl}$, thus adding these two potentials (i.e. 53
& 54), one obtains:
$\Phi_{eff}(r,\theta)=-\sum^{\infty}_{\ell=0}c^{2}\left(\frac{G\left\\{\lambda_{\ell}^{star}\mathcal{M}_{star}^{\ell+1}+\lambda_{\ell}^{csl}\mathcal{M}_{csl}^{\ell+1}(r)\right\\}^{\frac{1}{{}^{\ell+1}}}}{rc^{2}}\right)^{\ell+1}P_{\ell}(\cos\theta).$
(56)
This is the ASGP of a star that spins independently from its core. For
convenience, we can write:
$\mathcal{M}^{eff}_{\ell}(r)=\left\\{\lambda_{\ell}^{star}\mathcal{M}_{star}^{\ell+1}+\lambda_{\ell}^{csl}\mathcal{M}_{csl}^{\ell+1}(r)\right\\}^{\frac{1}{{}^{\ell+1}}}$,
and call this the effective gravitational mass for the $\ell^{th}$
gravitational-pole. By $\ell^{th}$ gravitational-pole, it shall be understood
to mean the $\ell^{th}$-term in the gravitational potential term. This means
the above can be written in the clearer and simpler form:
$\Phi_{eff}(r,\theta)=-\sum^{\infty}_{\ell=0}c^{2}\left(\frac{G\mathcal{M}^{eff}_{\ell}(r)}{rc^{2}}\right)^{\ell+1}P_{\ell}(\cos\theta).$
(57)
To second order approximation, this potential is given by:
$\Phi_{eff}=-\left(\frac{G\mathcal{M}_{0}^{eff}(r)}{r}\right)\left[1+\gamma_{1}\lambda_{1}^{star}\left(\frac{G\mathcal{M}_{1}^{eff}(r)\cos\theta}{rc^{2}}\right)+\gamma_{2}\lambda_{2}^{star}\left(\frac{G\mathcal{M}_{2}^{eff}(r)}{rc^{2}}\right)^{2}\left(\frac{3\cos^{2}\theta-1}{2}\right)\right],$
(58)
where: $\gamma_{\ell}=\mathcal{M}_{\ell}^{eff}(r)/\mathcal{M}_{0}^{eff}(r)$.
We shall assume this ASGP for a star that spins independently from its core.
## 6 Outflow Power
Clearly, we do have from the ASTG regions of repulsive gravitation whose shape
is similar to that seen in outflows. If these outflows are really powered by
gravity, the question is: does the gravitational field have that much energy
to drive these and if so, where does this energy come from? To answer this
question, one will need to know the dominant radial component of the
gravitational force since outflows dominantly operate along the radial
direction. Clearly, one of the new extra poles in the gravitational field must
be the cause of the outflows since without them, there are no outflow. For our
investigations, the correct gravitational potential to use is (57) and of
interest in this potential is the gravitational potential of the star. This
invariably means we are looking at (53). So doing, one sees that the first
order term (involving $\lambda_{1}$) is an all-repulsive term as already
argued while the second order them (involving $\lambda_{2}$) is repulsive and
attractive, it depends on the region under consideration.
Now, to ask what powers outflows amounts to asking: “What is their energy
source?”. If this energy source is the gravitational field, then, we know that
the energy stored in the gravitational field whose potential is described by
$\Phi(r,\theta)$, is given by:
$\mathcal{E}^{star}_{gpe}(r)=\int^{\mathcal{M}_{star}}_{0}\int^{\Phi(\infty,2\pi)}_{\Phi(r,0)}d\Phi(r,\theta)d\mathcal{M},$
(59)
and plugging into the above the ASGP, and thereafter performing the
integration, one is led to:
$\mathcal{E}^{star}_{gpe}(r)=-\frac{G\mathcal{M}_{star}^{2}}{2r}\left[1+\frac{\lambda_{1}G\mathcal{M}_{star}}{rc^{2}}+\lambda_{2}\left(\frac{G\mathcal{M}_{star}}{rc^{2}}\right)^{2}\right],$
(60)
and using the fact that
$\mathcal{L}_{star}=\mathcal{L}_{\odot}\left(\mathcal{M}_{star}/\mathcal{M}_{\odot}\right)^{3}$,
one is further led to:
$\mathcal{E}^{star}_{gpe}(r)=-\frac{G\mathcal{M}_{\odot}^{2}}{2r}\left(\frac{\mathcal{L}_{star}}{\mathcal{L}_{\odot}}\right)^{\frac{2}{3}}\left[1+\frac{\lambda_{1}G\mathcal{M}_{\odot}}{rc^{2}}\left(\frac{\mathcal{L}_{star}}{\mathcal{L}_{\odot}}\right)^{\frac{1}{3}}\right].$
(61)
If $\mathcal{M}_{out}$ is the mass of the outflow at position $r$ and
$V_{out}$ is the speed of this outflow at this position and
$\mathcal{K}_{out}$ is the kinetic energy, we know that:
$\left<\frac{d\mathcal{M}_{out}(r)}{dt}\right>=\frac{1}{V_{out}^{2}}\frac{d\left[\mathcal{M}_{out}(r)V_{out}^{2}\right]}{dt}=\frac{2}{V_{out}^{2}}\frac{d\mathcal{K}_{out}}{dt},$
(62)
where the bracket $\left<[]\right>$ tells us that we are looking at the
average. Now if the gravitational energy $\mathcal{E}^{star}_{gpe}(r)$ is
equal to the kinetic energy of the outflow, then, from the above and, coupled
with the said, one is led to:
$\left<\frac{d\mathcal{M}_{out}(r)}{dt}\right>=-\frac{\tau_{G}G\mathcal{M}_{\odot}^{2}V_{out}^{-2}}{r}\left(\frac{\mathcal{L}_{star}}{\mathcal{L}_{\odot}}\right)^{\frac{2}{3}}\left[1+\frac{2\lambda_{1}G\mathcal{M}_{\odot}}{rc^{2}}\left(\frac{\mathcal{L}_{star}}{\mathcal{L}_{\odot}}\right)^{\frac{1}{3}}\right],$
(63)
where $\tau_{G}=\dot{G}/G$ and $\dot{G}$ is the time derivative of the
Newton’s gravitational constant. In the derivation of the above, we have
considered only first order terms and we have assumed that the gravitational
constant is not a constant. Evidence that the gravitational constant maybe
changing exists e.g. see Pitjeva ([$2005$]) and references therein. The ASTG
also points to a variation of the gravitational constant and the details of
this are being worked out444We are at an advanced stage of preparation of this
work and it will soon be archived on viXra.org: check Golden Gadzirayi
Nyambuya’s profile. Title of the Paper: A Foundational Basis for Variable-G
and Variable-c Theories. and we give in the subsequent paragraphs how this
comes about.
As it stands, the Poisson equation ($\vec{\nabla}^{2}\Phi=4\pi G\rho$) for a
time varying $\Phi$ & $\rho$, is not in conformity with the Relativity
Principle. According to our current understanding of physics and Nature, the
seemingly sacrosanct Relativity Principle is a symmetry that every Law of
Physics must fulfill. The Relativity Principle states that Laws of Physics
must be independent of the observer’s state of motion and as-well of the
coordinate system used to formulate them. If the Poisson equation is to be a
Law of Nature, then, it must successfully fulfill the Relativity Principle.
This means we must extend the Poisson equation to meet this requirement and
the most natural and readily available such is:
$\vec{\nabla}^{2}\Phi-\frac{1}{c^{2}}\frac{\partial^{2}\Phi}{\partial
t^{2}}=4\pi G\rho,$ (64)
where $t$ is the time coordinate. This equation satisfies the Relativity
Principle simply because it directly emerges from Einstein’s equation of the
General Theory of Relativity (GTR). We know Einstein’s GTR, specifically the
Law of Gravitation relating matter to the curvature of spacetime, does satisfy
the Relativity Principle; hence (64) too, satisfies the Relativity Principle.
This equation (i.e. 64) is what we are working out, we shall show that it
leads to a time variable $G$. So, as will be shown in the near future, the
time variable $G$ in (63) is not without a basis.
Now, from (63), one sees that: $\dot{\mathcal{M}}_{out}(r)\propto
V_{out}^{-2}(r)\mathcal{L}^{2/3}_{star}$. Given as stated in the introduction
that observations find: $\dot{\mathcal{M}}_{out}(r)\propto
V_{out}^{-1.8}\mathcal{L}^{0.6}_{star}$, which is close to what we have
deduced here; this points to the fact that the thesis leading to our
deduction: $\dot{\mathcal{M}}_{out}(r)\propto
V_{out}^{-2}(r)\mathcal{L}^{2/3}_{star}$, may very well be on the right path
of discovery. This clearly points to the need to look into these matters
deeper than has been done here.
From the above, clearly – a meticulous study of outflows should be able to
measure the time variation in the gravitational constant $G$ and this hinges
on the corrects of the ASTG. This would require higher resolution observations
to measure the mass outflow rate [$\dot{\mathcal{M}}_{out}(r)$] at position
$r$ from the star and as well the speed of the outflow at that point and
knowing the mass or luminosity of the central driving source, a graph of,
e.g.: $r\dot{\mathcal{M}}_{out}(r)$ vs
$V_{out}^{-2}(r)\mathcal{L}^{2/3}_{star}$, should in accordance with the ideas
above, produce a straight line graph whose slope is
$\tau_{G}G\mathcal{M}_{\odot}^{2}$. This kind of work, if it where possible,
it would help in making an independent confirmation of the measured time
variation of Newton’s constant of gravitation and it would act as further
testing grounds for the falsification of the ASTG.
## 7 Outflow Anatomy
Briefly, we shall look into the anatomy of the outflow. We say “briefly”
because each of the issues we shall look into requires a separate reading to
fully address them. First, before we do that, it is important to find out when
does the outflow switch-on and also when does it switch-off. That at some
point in time in the evolution of a star, outflows switch-on and off is not
debatable. So, before we even look into them, it makes perfect sense to
investigate this. From figure (2), we see that the anatomy of the outflow has
been identified with four regions, i.e. the Outflow Feed Region, the Outflow
Region and the Shock Ring. After investigating the switching-on and off of the
outflow, we will look into the nature of these regions. Our analysis is
qualitative rather than quantitative. We believe a quantitative analysis will
require a fully-fledged numerical code. Work on this numerical code is
underway.
### 7.1 Switching-on of Outflows
Let us call the loboid described by (40) the outflow loboid and likewise the
loboid described by (42) the outflow feed loboid. From the preceding section,
it is abundantly clear that we are going to have repulsive bipolar regions
whose surface is described by a cone and a outflow loboid section. From this,
we know that the maximum spacial extend of the repulsive gravitational field
region will be given by the maximum spatial length of the lobes which occurs
when: $\cos\theta=1$, i.e. $l_{max}=\epsilon_{1}^{star}\mathcal{R}_{star}$.
Now, to ask the question when does the outflow switch-on amounts to asking
when is $l_{max}$ equal to the radius of the star? because the repulsive
gravitational field will only manifest beyond the surface of the star if and
only if the maximum spatial extent of the region of repulsive gravitation is
at least equal to the radius of the star, i.e.:
$l_{max}~{}\geq~{}\mathcal{R}_{star}$, this means,
$l_{max}~{}=~{}\epsilon_{1}^{star}\mathcal{R}_{star}$; clearly, this will
occur when: $(\epsilon_{1}^{star}=1)$. Therefore, outflows will switch-on when
the condition: $(\epsilon_{1}=1)$, is reached, otherwise when:
$(\epsilon_{1}^{star}<1)$, the repulsive gravitational field is confined
inside the star.
This strongly suggests that if we are to use the ASGT to model outflows, then
we must think of $\epsilon_{1}$ (hence $\lambda_{1}$) as an evolutionary
parameter of the star i.e., this value starts of from a given absolute minimum
value $($say $\epsilon_{1}^{star}=0)$, and as the star evolves, this value
gets larger and larger until such a time that the repulsive gravitational
field is switched on when: $(\epsilon_{1}^{star}=1)$, and thereafter it
continues to grow and as it grows so does the spatial extend of the outflow
(since this parameter controls the spatial size of the region of the repulsive
gravitational field).
If the outflow switches on – as it must, the question is: “Why does it switch
on at that moment when it switches on and not at any other moment? What is so
special about that moment when it switches on that triggers it [outflows] to
switch on?” As we have already argued, this special moment is when
$(\epsilon_{1}^{core}=1)$ for a star that co-rotates with its parent core and
$(\epsilon_{1}^{star}=1)$ for a star that rotates independently of its parent
core. From equation (39 and 47), this means we must have:
$\epsilon_{1}^{core}=\left[\zeta\left(\frac{4\pi^{2}\mathcal{R}_{core}^{3}}{G\mathcal{M}_{core}\mathcal{T}_{core}^{2}}\right)^{\zeta_{0}}\left(\frac{2G\mathcal{M}_{core}}{c^{2}\mathcal{R}_{core}}\right)\right]^{\frac{1}{2-\alpha_{\rho}}}\left(\frac{\mathcal{R}_{core}}{\mathcal{R}_{star}}\right)=1,$
(65)
and for a star that rotates independently of its core:
$\epsilon_{1}^{star}=\zeta\left(\frac{4\pi^{2}\mathcal{R}_{star}^{3}}{G\mathcal{M}_{star}\mathcal{T}_{star}^{2}}\right)^{\zeta_{0}}\left(\frac{2G\mathcal{M}_{star}}{c^{2}\mathcal{R}_{star}}\right)=1,$
(66)
where $(\mathcal{T}_{core},\mathcal{T}_{star})$ are the period of the spin of
the core and the star respectively. If $\mathcal{T}_{core}^{on}$ is the period
of the core’s spin when the outflow switches on and $\mathcal{T}_{star}^{on}$
is the period of the spin when the outflow switches on, then, from the above
equations, it follows that:
$\mathcal{T}^{on}_{core}=\left(\frac{\pi}{c}\right)\left[\zeta\left(\frac{2G\mathcal{M}_{core}}{c^{2}}\right)^{1-\zeta_{0}}\left(\mathcal{R}_{star}^{on}\right)^{\alpha_{\rho}-1}(\mathcal{R}_{core}^{on})^{3\zeta_{0}-\alpha_{\rho}+1}\right]^{\frac{1}{2\zeta_{0}}},$
(67)
$\mathcal{T}^{on}_{star}=\left(\frac{\pi}{c}\right)\left[\zeta\left(\frac{2G\mathcal{M}_{star}^{on}}{c^{2}}\right)^{1-\zeta_{0}}\left(\mathcal{R}_{star}^{on}\right)^{3\zeta_{0}-1}\right]^{\frac{1}{2\zeta_{0}}},$
(68)
where
$(\mathcal{M}_{star}^{on},\mathcal{R}^{on}_{star},\mathcal{R}^{on}_{core})$
are the mass and radius of the star and core at the time the outflow switches
on respectively. From this, it follows that if the Sun were to spin on its
axis once in every $7.70\pm 0.40\,\textrm{hrs}$ (i.e. $39.0\pm
2.00\,\mu\textrm{Hz}$), the bipolar repulsive gravitational field must switch
on and for the Earth, it would require it to spin once on its axis in every
$10.00\pm 2.00\,\textrm{min}$ (i.e. $1.80\pm 0.50\,\textrm{mHz}$). If the
above is correct, then the Earth must spin about one hundred and forty four
times its current spin in order to achieve the bipolar repulsive gravitational
field while the Sun must spin about five thousand six hundred its current spin
rate to achieve a bipolar repulsive gravitation. The spin rate of the Earth is
far less than that needed to cause the bipolar repulsive gravitational to
switch on thus polar bears can smile knowing they will not fly off into space
anytime soon.
We know that outflows are not always present, at some-point in the evolution
of the star, they switch-off. What could cause them to do so? Given the
reality that within the outflow loboid, there is the outflow feed loboid; this
too, grows in size as the outflow loboid grows; at some-point the outflow and
the outflow feed loboid will become equal – leaving the outflow with no feed
point. At this point when the outflow and outflow feed loboids become equal,
clearly, the outflow must switch-off. This occurs when $l_{max}=l_{min}$ and
from (43) this means the condition for this to occur is
$|\lambda_{2}|=2\lambda_{1}^{2}/9$ and given that
$\lambda_{2}=-\lambda_{1}/96$, this means $\lambda_{1}^{off}=9/192$. From
(39), it follows that:
$\lambda_{1}^{off}=\zeta\left(\frac{4\pi^{2}\mathcal{R}_{star}^{3}}{G\mathcal{M}_{star}\mathcal{T}_{star}^{2}}\right)^{\zeta_{0}}\left(\frac{2G\mathcal{M}_{star}}{c^{2}\mathcal{R}_{star}}\right)=\frac{9}{192},$
(69)
this implies:
$\mathcal{T}^{off}_{star}=\left(\frac{192}{9}\right)^{\frac{1}{2\zeta_{0}}}\left(\frac{\pi}{c}\right)\left[\zeta\left(\frac{2G\mathcal{M}_{star}^{off}}{c^{2}}\right)^{1-\zeta_{0}}\left(\mathcal{R}_{star}^{off}\right)^{3\zeta_{0}-1}\right]^{\frac{1}{2\zeta_{0}}},$
(70)
where likewise $(\mathcal{M}_{star}^{off},\mathcal{R}^{off}_{star})$ are the
mass and the radius of the star at the time when the outflow switches off. We
expect that $\mathcal{T}^{on}_{star}>\mathcal{T}^{off}_{star}$. If this is to
hold, then:
$\left(\frac{\mathcal{M}_{star}^{off}}{\mathcal{M}_{star}^{on}}\right)^{1-\zeta_{0}}\left(\frac{\mathcal{R}^{off}_{star}}{\mathcal{R}^{on}_{star}}\right)^{3\zeta_{0}-1}<\left(\frac{9}{192}\right)^{\frac{2}{5}}=0.30.$
(71)
Hence, outflow activity will take place when:
$(\mathcal{T}^{off}_{star}\leq\mathcal{T}_{star}\leq\mathcal{T}^{on}_{star})$.
When $\mathcal{T}_{star}=\mathcal{T}^{off}_{star}$, we have:
$\epsilon_{1}=9\mathcal{R}_{star}^{s}/192\mathcal{R}_{star}$. Using the
approximate relation for an accreting star:
$\mathcal{R}_{star}~{}\sim~{}61\mathcal{R}_{\odot}\left(\mathcal{M}_{star}/\mathcal{M}_{\odot}\right)$,
one arrives at: $\epsilon_{1}=3.32\times 10^{6}$. This means:
$(\epsilon_{1}^{on}=1)$, and: $\epsilon_{1}^{off}=3.32\times 10^{6}$, where
$\epsilon_{1}^{on}$ and $\epsilon_{1}^{off}$, are the values of $\epsilon_{1}$
when the outflow switches on and off respectively, hence thus outflow activity
will take place during which period when:
$1\leq\epsilon_{1}<3.32\times 10^{6}.$ (72)
The emerging picture is that $\mathcal{T}$ gets larger and larger as the star
accretes more and more matter until a peak moment is reached (most probably
when the star stops growing in mass) where upon the spin begins to slow down,
in which process of slowing down the inner cavity inside the lob of the
outflow is created. This inner cavity grows bigger and bigger as the star’s
spin slows down, until such a time when the spatial dimensions of this cavity
is equal to the outflow lobe itself. Once this state is attained, the outflow
switches off because the growing cavity has – eaten up from within, all the
outflow region.
Clearly, the above picture suggests that the spin of a star is what controls
outflows, at some specific state, the outflow switch’s-on; it evolves to some
peak spin-value; thereafter, its spin slows down. This means that during the
outflow process after the begins to slow down, the star loses some spin
angular momentum. This idea resonates with the long held suggestion discussed
earlier that outflows are thought to exist as one means to tame the spin
angular momentum of a star (see e.g. Larson $2003b$). We will not go deeper
than this in our analysis. The aim has been to show that the emergent picture
of outflows from the ASTG is capable (in principle) to answer such questions.
This means in a future study, these are the things to look forward to.
### 7.2 Outflow Feed Region
In the Outflow Feed Region – i.e. the region in figure (2) described OEF and
OGH, clearly, any material that enters this region is going to be channeled
into the Outflow Region because the repulsive radial component of the
gravitational field (aided by the radiation field) is going to channel this
matter radially outward while the azimuthal component is going to going to
channel this outward radially moving material toward the spin axis, hence it
is expected that most of the matter will enter the Outflow Region along the
the spin axis of the star. It is important to state that no matter the
radiation from the star, there will be no reversal of in-falling matter
outside the region of repulsive gravitation due to the radiation field of the
nascent star – we shall discuss this in §$(8)$.
### 7.3 Outflow Region
The Outflow Region is comprised of a section of a cone (OAB & OCD), the
outflow loboid minus the Outflow Feed Region. In this Outflow Region, the
gravitational force is both radially azimuthally repulsive i.e.,
$(g_{\theta}>0)$ and $(g_{r}>0)$. This means, once the repulsive gravitational
force is switched-on and it is in a fully fledged phase, all material found in
this region is going to be channeled out of this region radially along with
most of the matter concentrated along the spin axis. The material will be
concentrated along the spin axis because the repulsive azimuthally
gravitational component will channel toward the spin axis. The repulsive
radial component pushes the material out radially, while the repulsive
azimuthal component of the gravitational force draws this material close the
spin axis hence the bulk of the outflow material must be found along the edge
spin axis.
Where the cone meets the outflow loboid i.e., along AB and CD, there is going
to be rings. Considering the ring AB, it is clear that this ring (as CD) must
be a shock front since on this ring, along the radial line OA, the in-coming
material will meet the outgoing material with equation but opposite radial
forces. This equal and but opposite forces must create (radially) a stationery
shock. This shock is going to have a ring structure – let us call this the
Shock Ring. As the rings AB & CD, EF & GH will be rings too, but not shock
things. These rings EF & GH are the mouth of the outflow and matter enters in
to the outflow region via this opening.
### 7.4 Shock Rings and Methanol Masers
Given that $(1):$ AB & CD are shock rings, $(2):$ that methanol masers
(amongst other pumping mechanisms) are thought to arise in shock regions and
$(3):$ the observations of Bartkiewicz et al. ([$2005$]) where these authors
discovered a ring distribution of $6.7\,\textrm{GHz}$ methanol masers, it is
logical to assume that this shock ring may well be a hub of methanol masers
arising from the shock present on this ring. Recent and further work by these
authors strongly suggests that a Ring of Masers is a natural occurrence in
star forming regions as (Bartkiewicz et al. [$2009$]).
This ring distribution of masers components, they believe strongly suggests
the existence of a central source – this is the case here, the central source
must exists and it is the forming star. They found an infrared object
coinciding with the center of the ring of masers within $78\,mas$ and this
source is cataloged in the 2MASS survey as 2MASS183451.56-08182114. They
believe this is an evolving evolving protostar driving this masers via
circular shocks – this is in line with the the present. Very strongly, the
Bartkiewicz Ring of Masers suggests – in our opinion that; our outflow model
may very well contain an element of truth, that our model contains the
possible seeds of resolution of this puzzling occurrence of Ring Masers.
About this shock ring; when viewed from the projection as shown in figure (2),
the distance of the shock ring from the star will be:
$l_{sh}=l_{max}(3)^{-\frac{3-\alpha_{\rho}}{4-2\alpha_{\rho}}},$ (73)
and the radius of this shock ring will be:
$\mathcal{R}_{ring}=l_{max}(1.5)^{-\frac{3-\alpha_{\rho}}{4-2\alpha_{\rho}}}.$
(74)
Clearly, for an isolated system, depending on the orientation relative to the
observer, this ring can appear as a linear structure, a circular or an
elliptical ring.
At present more than $500$ $6.7\,\textrm{GHz}$ methanol masers sources are
known to exist (Malyshev & Sobolev [$2003$]; Pestalozzi et al. [$2005$]; Xu et
al. [$2003$]) and are associated with a very early evolutionary phase of high
mass star formation. The methanol maser emitting at the $6.7\,\textrm{GHz}$
frequency first discovered by [$1991$] is the second strongest centimeter
masing transition of any molecule (after the $22\,\textrm{GHz}$ water
transition) and is commonly found toward star formation regions. It is
typically stronger than $12.2\,\textrm{GHz}$ methanol masers (discovered by
Batrla et al. [$1987$]) observed toward the same region. Methanol masers have
become well established tracers or sign spots of high mass star formation
regions. It is thought that methanol masers occur in the very early stages of
massive star formation.
While methanol masers are found in regions of massive star formation, some
have been found with no associated high mass star formation actively (see e.g.
Ellingsen et al. [$1996$], Szymczak et al. [$2002$]. Besides this non-
association, some methanol masers are and have been observed to exist in close
spatial proximity of massive stars. This has lead to the classification of
methanol masers into Class I and Class II. Class I masers emit at the
frequencies $25.0$, $44.0$, $36.0\,\textrm{GHz}$ etc while class II methanol
masers emit at $6.7$, $12.2$, $157.0\,\textrm{GHz}$ etc methanol masers is
classified as Class II. Class I methanol masers are often observed to exist
apart from the continuum sources , while Class II are observed to exist very
close, albeit, both classes often co-exist in the same star forming region
inside an HII regions (e.g. Sobolev et al. [$2004$]). Clearly,
$l_{sh}=l_{sh}(t)$ and $\mathcal{R}_{ring}=\mathcal{R}_{ring}(t)$ and as the
star evolves, $l_{sh}$ and $\mathcal{R}_{ring}$ get larger. This means in the
case of young stars, if this ring is a hub of methanol masers, it is expected
that methanol masers will be found closer to the star for young HMS and
likewise, for more evolved massive stars, methanol masers will be found
further from the nascent star. If this is correct, then it may explain the
aforesaid; why Class II methanol masers are mostly found close to the nascent
star and why Class I methanol masers are found existing further from the
nascent star.
High resolution imaging of the $6.7$ and $12.2\,\textrm{GHz}$ methanol masers
has found that many exhibit a simple elongated linear or curved spatial
morphologies (Norris et al. [$1988$]; Norris et al. [$1993$]; Minier et al.
[$2000$]) and as already stated, depending on the orientation of the observer
relative to the star forming system, the ring may appear as a linear
structure. These linear structures have lengths of $50$ to
$1300\,\textrm{AU}$. Because of this, one of the possible interpretations that
has been entertained for sometime is that the masers originate in the
circumstellar accretion disc surrounding the newly formed star (Edris et al.
[$2005$]) and besides this; because of their strong association with outflows
(see e.g. Plambeck & Menten [$1990$]; Kalenskii et al. [$1992$]; Bachiller et
al. [$1995$]; Johnston et al. [$1992$]), other than originating from the
circumstellar disk, also, it has been entertained that methanol maser may
originate from outflows (see e.g. Pratap & Menten [$1992$]; de Buizer et al.
[$2000$]). Clearly, the outflow origin of methanol masers resonates with the
present ideas. If the ideas herein are correct, then, this reading would of
value to researchers seeking an outflow origin of methanol masers.
Further, if viewed from the same view as in figure (2), and if as argued above
that masers are found on the ring, one will expect to observe a linear
alignment of masers above and below the the nascent star. This would explain
the observed linear alignment of methanol masers and also the observed linear
alignment of masers above an below the IRAS source found in molecular cloud
G69.489-0.785 (see Fish [$2007$]). Given Fish’s observations of blue and red-
shifted masers in the ON1-region (Fish [$2007$]), the suggested model of this
ring of masers is interesting as it may offer an explanation of this
unexplained and puzzle of red and blue-shifted masers at opposite sides of the
IRAS source associated with ON1.
### 7.5 Collimation Factor
We can calculate the collimation factor of the outflow since we know the
extent ($l_{max}$) and the breath of the outflow which is the size of the
shock rings i.e., the collimation factor could be:
$q_{col}=\mathcal{R}_{ring}/l_{max}$, which can also be written as:
$q_{col}=(1.5)^{\frac{3-\alpha_{\rho}}{4-2\alpha_{\rho}}},$ (75)
(this has been deduced from equation 74). Now, it is believed that the most
stable density profile is one with a density index $(\alpha_{\rho}=2)$, this
means molecular clouds in a state different from this density profile will
tend to it. Using this assumption, we see that as: $(\alpha_{\rho}\longmapsto
2)$ from $(\alpha_{\rho}=0)$, i.e. $(\alpha_{\rho}:0\longmapsto 2)$, then we
will have: $(q_{col}\longmapsto\infty)$. For this setting, generally:
$(q_{col}>1)$. We also realize that now as: $(q_{col}\longmapsto\infty)$,
when: $(\alpha_{\rho}:3\longmapsto 2)$, then: $(q_{col}>1)$, and if:
$(\alpha_{\rho}:0\longmapsto 2)$. For this setting, generally: $(q_{col}\geq
1.36)$. This means we are going to have two categories of collimation factor
value i.e. $(1<q_{col}<1.36)$ for $(\alpha_{\rho}:0\longmapsto 2)$ and
$(q_{col}\geq 1.36)$ for $(\alpha_{\rho}:3\longmapsto 2)$. Because of
projection effects, it is very difficult to measure the true collimation
factor.
Also, because of projection effects, the collimation factor that we measure in
real life is not the actual collimation factor but the projected collimation
factor. If we know the actual collimation factor, we will be able to know the
density index since from (75) we can deduce that:
$\alpha_{\rho}=2-\left(\frac{\log q_{col}^{2}}{\log
1.5}-1\right)^{-1}=\frac{\log(q_{col}^{4}/8)}{\log(1.5)}.$ (76)
LMSs are known to have relatively low outflow collimation factors
($q_{col}<2$) while HMSs have significantly high outflow collimation factors
($2<q_{col}<10$), sometimes reach $q_{col}\sim 20$. From (76) the aforesaid
implies, assuming these collimation factors are a good representation of the
real collimation factor, that LMSs cores have density index
$\alpha_{\rho}=1.56$ and HMS cores have density index $\alpha_{\rho}=1.98$.
This is not unreasonable but very much expected. The fact that for HMS forming
cores, we have $\alpha_{\rho}=1.98$ and for LMS forming cores we have
$\alpha_{\rho}=1.56$, means HMS cores are much more dense compared to LMS
forming cores.
## 8 Radiation Problem
While the main thrust and focus of this reading is not on the Radiation
Problem associated with massive stars, but on the polar repulsive
gravitational field and its possible association with the observed bipolar
molecular outflows, we find that the ASTG affords us a window of opportunity
to visit this problem. This so-called radiation problem associated with
massive stars has been well articulated in Paper III. There is no need for us
to go through the details of this same problem here but we shall direct the
reader to Paper III for an exposition of the radiation problem. In the
subsequent paragraphs, we shall – for the sack of achieving a smooth
continuous reading; present the findings of Paper III in nutshell.
In general, a massive star is defined to be one with mass greater than $\sim
8-10\mathcal{M}_{\odot}$ and central to the on-going debate on how these
objects [massive stars] come into being is this so-called radiation problem.
For nearly forty years, it has been argued that the radiation field emanating
from massive stars is high enough to cause a global reversal of direct radial
in-fall of material onto the nascent star. In Paper III, it is argued that
only in the case of a non-spinning isolated star does the gravitational field
of the nascent star overcome the radiation field. An isolated non-spinning
star is a non-spinning star without any circumstellar material around it, and
the gravitational field beyond its surface is described exactly by Newton’s
inverse square law. The supposed fact that massive stars have a gravitational
field that is much stronger than their radiation field is drawn from the
analysis of a non-spinning isolated massive star. In this case, the
gravitational field is (correctly) much stronger than the radiation field.
This conclusion has been erroneously extended to the case of non-spinning
massive stars enshrouded in gas and dust.
It is argued there, in Paper III, that, for the case of a non-spinning
gravitating body where the circumstellar material is taken into consideration,
that at $\sim 8-10\mathcal{M}_{\odot}$, the radiation field will not reverse
the radial in-fall of matter, but rather a stalemate between the radiation and
gravitational field will be achieved, i.e. in-fall is halted but not reversed.
Any further mass growth is stymied and the star’s mass stays constant at $\sim
8-10\mathcal{M}_{\odot}$. This picture is very different from the common
picture that is projected and accepted in the wider literature where at $\sim
8-10\mathcal{M}_{\odot}$, all the circumstellar material, from the surface of
the star right up to the edge of the molecular core, is expected to be swept
away by the all-marauding and pillaging radiation field. There in Paper III,
it is argued that massive stars should be able to start their normal stellar
processes if the molecular core from which they form has some rotation,
because a rotating core exhibits an ASGF which causes there to be an accretion
disk and along this disk the radiation is not powerful enough to pillage the
in-falling material. We show here that in the region:
($\theta:\,[125.3<\theta<54.7]$ & $[234.7<\theta<305.3]$), around a spinning
star the gravitational field in the face of the radiation field, will never be
overcome by the radiation field hence in-fall reversal does not take place in
this region and this region is the region via which the nascent massive star
forms once the repulsive outflow field and the star’s mass has surpassed the
critical $8-10\,\mathcal{M}_{\odot}$. Reiterating, in this region i.e.
($\theta:\,[125.3<\theta<54.7]$ & $[234.7<\theta<305.3]$), infall is never
halted but continues unaborted and unabated.
There are three cases of an embedded spinning nascent star $(1):$ Where the
nascent star is spinning and the circumstellar material is not spinning or
where the spin of the circumstellar material is so small compared to the star
so much that the circumstellar material can be considered to be not spinning.
$(2):$ Where the nascent star is spinning independently of the circumstellar
material which is itself spinning. $(3):$ Where the nascent star is co-
spinning or co-rotating with the circumstellar material. It should suffice to
consider one case because the procedure to show that in the region:
($\theta:\,[305.3<\theta<54.7]$ & $[234.7<\theta<125.3]$), infall is never
halted but continues unaborted and unabated, is the same. Of the three cases
stated, the most likely scenario in Nature is the second case i.e., where the
nascent star is spinning independently of the circumstellar material which is
itself spinning. We shall consider this case.
The ASGP for the case of a star that is spinning independently of its core has
be argued to be given by (58) and in the face of radiation field, the
resultant radial component of the gravitational field intensity is given by:
$g_{r}(r,\theta)=-\frac{G\mathcal{M}_{0}^{eff}}{r^{2}}\left[1-\frac{\kappa\mathcal{L}_{star}}{4\pi
G\mathcal{M}_{0}^{eff}c}+\frac{2\lambda_{1}^{star}\gamma_{1}G\mathcal{M}_{1}^{eff}\cos\theta}{rc^{2}}+3\lambda_{2}^{star}\gamma_{2}\left(\frac{G\mathcal{M}_{2}^{eff}}{rc^{2}}\right)^{2}\left(\frac{3\cos^{2}\theta-1}{2}\right)\right].$
(77)
For the radiation component to be attractive, we must have:
$[g_{r}(r,\theta)<0]$, and for this to be so, the term in the square brackets
must be greater than zero, this implies:
$\left[1-\frac{\kappa\mathcal{L}_{star}}{4\pi
G\mathcal{M}_{0}^{eff}c}\right]r^{2}+\left[\frac{2\lambda_{1}^{star}\gamma_{1}G\mathcal{M}_{1}^{eff}\cos\theta}{c^{2}}\right]r+\left[3\lambda_{2}^{star}\gamma_{2}\left(\frac{G\mathcal{M}_{2}^{eff}}{c^{2}}\right)^{2}\left(\frac{3\cos^{2}\theta-1}{2}\right)\right]>0.$
(78)
This inequality is quadratic in $r$ and can be written as: $(Ar^{2}+Br+C>0)$,
where: $A,B,$ and $C$, can easily be obtained by making a comparison.
Since555In Paper I, we argued that, $r$ can take both negative and positive
values, and further argued that the set up of the coordinate system of the
ASGF is such that [$r>0$ & $\cos\theta>0$] and [$r<0$ $\cos\theta<0$], hence
$r\cos\theta>0$, which implies $(Br>0)$.: $(Br>0)$, for: $(Ar^{2}+Br+C>0)$, to
hold absolutely, we must have: $(Ar^{2}>0\Rightarrow A>0)$ and $(C>0)$. The
condition: $(A>0)$, implies:
$\mathcal{M}(r)>\frac{\kappa_{eff}\mathcal{L}_{star}}{4\pi Gc}.$ (79)
To arrive at the above one must remember that:
$\mathcal{M}_{0}^{eff}=\mathcal{M}_{star}+\mathcal{M}_{csl}(r)=\mathcal{M}(r)$.
As shown in Paper II (see equation $5$ & §$5$ of Paper II), the condition (79)
for: $\mathcal{M}_{star}>8-10\,\mathcal{M}_{\odot}$, leads to the formation of
a cavity inside the star forming core. In this cavity, the radiation field in
powerful enough to halt infall reversal but outside of it, it is not.
Now, for the condition: $(C>0)$, to hold (remember $\lambda^{star}_{2}<0$),
this means: $(3\cos^{2}\theta-1<0)$, hence: ($\theta:\,[125.3<\theta<54.7]$ &
$[234.7<\theta<305.3]$). The result just obtained invariably means inside the
cavity created by the radiation field, the region:
($\theta:\,[125.3<\theta<54.7]$ & $[234.7<\theta<305.3]$) will have an
attractive gravitational field, hence matter will still be able to fall onto
the nascent star via this region and this in-falling of matter is completely
independent of the opacity of the material of the core! Hence we expect
spinning massive stars to face no radiation problem at all. Clearly, if:
$(\lambda_{2}>0)$, then in the region: ($\theta:\,[125.3<\theta<54.7]$ &
$[234.7<\theta<305.3]$), the gravitational field was going to cause in-fall
reversal in the cavity hence disallowing for the star to continues is
accretion. This obviously would have been at odds with experience hence thus
we have the strongest reason for setting: $(\lambda_{2}>0)$, otherwise the
ASTG would be seriously at odds with physical and natural reality as we know
it. Beside, the condition $(\lambda_{2}>0)$ is supported by the solar data
(see Paper I). The fact that in the region; ($\theta:\,[125.3<\theta<54.7]$ &
$[234.7<\theta<305.3]$), is a region of attractive gravitation, it is clearly
that the ASGF will form a disk around the nascent star. Although no detailed
study of accretion disks has been made (Brogen et al. [$2007$]; Araya et al.
[$2008$]) and this being due technological challenges in obtaining must higher
resolution observations on the scale of these accretion disks, it has long
been thought that the accretion disk is a means by which accretion of matter
on the nascent stars continues soon after radiation has (significantly)
sounded her presence on the star formation podium (see e.g. Chini et al.
[$2004$]; Beltr$\acute{\textrm{a}}$n et al. [$2004$]). If our investigation
prove correct, as we believe they will, then, researchers have been right to
think that that accretion disk serves a platform for further accretion of mass
by the nascent star.
## 9 Discussion and Conclusion
This reading should be taken more as a genesis that lays down the mathematical
foundations that seek to lead to the resolution of the problem of outflows,
vis, what their origin is. Also, we should say that, if this reading is
anything go by i.e., if it proves itself to have a real direct correspondence
with the experience of physical reality, then not only have we laid down the
mathematical foundations that may lead to the understanding of outflows; but
we have laid a three fold foundation that could lead to the resolution of
three problems, and these problems are:
> 1. (1).
>
> The Origins and Nature of Outflows
>
> 2. (2).
>
> The Radiation Problem thought to exist for HMS.
>
> 3. (3).
>
> The Origin of Linear & Ring Structures of Methanol Masers.
>
>
All this we have arrived at after the consideration of the azimuthal symmetry
arising from the spin of a gravitating body. This symmetry has been applied to
the gravitational field and where upon we have come up with the ASTG. In Paper
I, we did show that the ASTG can explain the perihelion shift of planets in
the solar system and therein, the ASTG as it lays there, suffers the setback
that the “constants” $\lambda_{\ell}$ are unknown. We have gone so far in the
present as to suggest a way to solve this problem but this suggestion is
subject to revision pending any new data.
It should be said that, to the best of what we can remember ever-since we
learnt that the force of gravity is what causes an apple to fall to the ground
and that the very same force causes the moon and the planets to stay in their
orbs; we have never really convinced of gravitation as being a repulsive
force, let alone that it possibly can have anything to do with the power
behind outflows. Just as anyone would find these ideas in violation of their
intuition, we find our-self in the same bracket. But one thing is clear, the
picture emerging from the mathematics thereof, is hard to dismiss. It calls
one to make a closer look at the what the Poisson equation is “saying to us”.
In closing, allow me to say that as things stand in the present – while we
firmly believe we have discovered something worthwhile; it is difficult to
make any bold conclusions. Perhaps we should only mention that work has began
on a numerical model of outflows based on what we have discovered herein. Only
then – we believe; it will be possible to make any bold conclusions.
## Acknowledgments
I am grateful to my brother George and his wife Samantha for their kind
hospitality they offered while working on this reading and to Mr. Isak D.
Davids & Ms. M. Christina Eddington for proof reading the grammar and spelling
and Mr. M. Donald Ngobeni for the magnanimous support.
## References
* [$2008$] Araya E., Hofner P., Kurtz S., Olmi L. & Linz H., $2008$, ApJ, $675$, $420$.
* [$1995$] Bachiller R., Liechti S., Walmsley C. M. & Colomer F., $1995$, A&A, $295$, L$51$.
* [$2005$] Bartkiewicz A., Szymczak M. & van Langevelde H. J., $2005$, A&A, $442$, L$62$.
* [$2008$] Bartkiewicz A., Brunthaler A., M. Szymczak, van Langevelde H. J. & Reid M. J., $2008$, A&A, $490$, $787$.
* [$2009$] Bartkiewicz A., Brunthaler A., Szymczak M., van Langevelde H. J. & Reid M. J., $2009$, A&A $(arXiv:0905.3469v1)$.
* [$1987$] Batrla W., Matthews H. E., Menten K. M. & Walmsley C. M., $1987$, Nature, $326$, $49$.
* [$2004$] Beltr$\acute{\textrm{a}}$n M. T., Cesaroni R., Neri R., Codella C., Furuya R. S., Testi L. & Olmi L., $2004$, AJ, $60$, $187$.
* [$2002$] Beuther H., Schilke1 P., Gueth F., McCaughrean M., Andersen M., Sridharan T. K. & Menten K. M., $2002$, A&A, $387$, $931$.
* [$1998$] Bonnell I. A., Bate M. & Zinnecker H., $1998$, MNRAS, $298$, $93$.
* [$2001$] Bonnell I. A., Clarke C. J., Bate M. R. & Pringle J. E., $2001$, MNRAS, $324$, $573$.
* [$2002$] Bonnell I. A. & Bate M. R., $2002$, MNRAS, $336$, $659$.
* [$2004$] Bonnell I. A, Vine S. G. & Bate M. R., $2004$, MNRAS, $349$, $735$.
* [$2006$] Bonnell I. A., Clarke C. J. & Bate M. R., $2006$, MNRAS, $368$, $1296$.
* [$2007$] Bonnell I. A., Larson R. B. & Zinnecker H., $2007$, $(arXiv:astro-ph/0603447v1)$.
* [$2007$] Brogan C. L., Chandler C. J., Hunter T. R., Shirley Y. L. & Sarma A. P., $2007$, ApJ, $660$, L$133$.
* [$1991$] Cantó J. & Raga A. C., $1991$, AJ, $372$, $646$.
* [$2003$] Cantó, J., Raga A. C. & Riera A., $2003$, RMxAA, $39$, $207$.
* [$2004$] Chini R., Hoffemeister V., Kimeswenger S., Nielbock M., N$\ddot{\textrm{u}}$rnberger D., Schmidtobreick & Sterzik M., $2004$, Nature, $429$, $155$.
* [$2000$] Clarke C. J., Bonnell I. A. & Hillenbrand L. A., $2000$, Protostars and Planets IV, $151$.
* [$2000$] de Buizer J. M., Piña R. K. & Telesco C. M., $2000$, AJSS, $130$, $437$.
* [$1994$] Diamond J. P., Kemball. A. J., Junor W., Zensus A., Benson J. & Dhawan V., $2000$, Protostars and Planets IV, $151$.
* [$2005$] Edris K. A., Fuller G. A., Cohen R. J. & Etoka S., $2005$, A&A, $434$, $213$.
* [$1996$] Ellingsen S. P., von Bibra M. L., McCulloch P. M., et al., $1996$, MNRAS, $280$, $378$ $(arXiv:astro-ph/9601016)$.
* [$1996a$] Fiege J. D. & Henriksen R. N., $1996a$, MNRAS, $281$, $1038$.
* [$1996b$] Fiege J. D. & Henriksen R. N., $1996b$, MNRAS, $311$, $105$.
* [$2007$] Fish V. L., $2007$, ApJ, $669$, L$81$ $(arXiv:0710.1310v1)$.
* [$1997$] Hillenbrand L. A., $1997$, $AJ$, $113$, $1733$.
* [$1992$] Johnston K. J., Gaume R., Stolovy S., Wilson T. L., Walmsley C. M. & Menten K. M., $1992$, AJ, $385$, $232$.
* [$1974$] Kahn F. D., $1974$, A&A, $37$, $149$.
* [$1992$] Kalenskii S. V., Bachiller R., Berulis I. I., Valtts I. E., Gomez-Gonzales J., Martin-Pintado J., Rodriguez-Franco A. & Slysh V. I., $1992$, Soviet Astronomy, $36$, $517$.
* [$2005$] Krumholz M. R., Klein R. I. & McKee C. F., $2005$, Proc. of the IAU (Cambridge Univ. Press), $1$, $231$.
* [$1971$] Larson R. B. & Starrfield S., $1971$, A&A, $13$, $190$.
* [$2003a$] Larson R. B., $2003a$, ASP Conf. Ser. (Eds: de Buizer, J. M. & van der Bliek, N. S.), $287$, $11$.
* [$2003b$] Larson R. B., $2003b$, Reports on Progress in Physics, $66$, $1651$ $(arXiv:astro-ph/0306595)$.
* [$1996$] Li Z. & Shu F. H., $1996$, AJ, $468$, $261$.
* [$1995$] Lizano S. & Giovanardi C., $1995$, AJ, $447$, $742$.
* [$2002$] M$\ddot{\textrm{a}}$der A. & Behrend R., $2002$, Hot Star Workshop III, ASP Conf. Series, $267$, $179$.
* [$2003$] Malyshev A. V. & A. M. Sobolev A. M., $2003$, Astronomical & Astrophysical Transactions, $22$, $1$.
* [$1993$] Masson C. R. & Chernin L. M., $1993$, AJ, $414$, $230$.
* [$1999$] Matzner C. D. & McKee C. F., $1999$, AJ, $526$, L$109$.
* [$2007$] McKee C. F. & Ostriker E. C., $2007$, ARA&A, $45$, $565$ $(arXiv:0707.3514)$.
* [$1991$] Menten K. M., $1991$, AJL, $380$, L$75$.
* [$2000$] Minier V., Booth R. S. & Conway J. E. $2000$, A&A, $362$, $1093$.
* [$1988$] Norris R. P., Byleveld S. E., Diamond P. J., Ellingsen S. P., Ferris R. H., Gough R. G., Kesteven M. J., McCulloch P. M., Phillips C. J., Reynolds J. E., Tzioumis A. K., Takahashi Y., Troup E. R. & Wellington K. J., $1998$, AJ, $508$, $275$ $(arXiv:astro-ph/9806284)$.
* [$1993$] Norris R. P., Byleveld S. E., Diamond P. J., Ellingsen S. P., Ferris R. H., Gough R. G., Kesteven M. J., McCulloch P. M., Phillips C. J., Reynolds J. E., Tzioumis A. K., Takahashi Y., Troup E. R. & Wellington K. J., $1998$, AJ, $508$, $275$. $(arXiv:astro-ph/9806284)$.
* [$2010a$] Nyambuya G. G., $2010a$, MNRAS, $403$, $1381$ $(arXiv:0912.2966$, $viXra:0911.0013)$ (Paper I).
* [$2010b$] Nyambuya G. G., $2010b$, $(arXiv:0803.1370)$ (Paper II).
* [$2010c$] Nyambuya G. G., $2010c$, RAA (Research in Astronomy and Astrophysics), Vol. 10 No. $11$, $1137$-$1150$ (Paper III).
* [$1993$] Palla F. & Stahler S. W., $1993$, ApJ, $418$, $414$.
* [$2005$] Pestalozzi M. R., Minier V., & Booth R. S., $2005$, A&A, $432$, $737$.
* [$2005$] Pitjeva, E. V. $2005$, Solar Syst. Res., $39$, $176$.
* [$1990$] Plambeck R. L. & Menten K. M., $1990$, AJ, $364$, $555$.
* [$1992$] Pratap P. & Menten K., $1992$, Bulletin of the American Astronomical Society, $24$, $1157$.
* [$1993a$] Raga A. & Cabrit S., $1993a$, A&A , $278$, $267$.
* [$1993b$] Raga A. C., Canto J., Calvet N., Rodriguez L. F. & Torrelles J. M., $1993b$, A&A, $276$, $539$.
* [$1996a$] Shepherd D. S. & Churchwell E., $1996a$, AJ, $472$, $225$.
* [$1996b$] Shepherd D. S. & Churchwell E. $1996b$, AJ, $457$, $267$.
* [$1977$] Shu F. H., $1977$, ApJ, $214$, $488$.
* [$1991$] Shu F. H., Ruden S. P., Lada C. J. & Lizano S., $1991$, AJL, $370$,L$31$.
* [$2004$] Sobolev A. M., Ellingsen S., Ostrovskii A. & Alakoz A., $2004$, Kluwer Academic Publishers.
* [$1994$] Stahler S. W., $1994$, AJ, $422$, $616$.
* [$2002$] Szymczak M., Kus A. J., Hrynek G., Kepa A. & Pazderski, E., $2002$, A&A, $392$, $277$.
* [$2002$] Yorke H. W., $2002$, Hot Star Workshop III, ASP Conf. Series, $267$, $165$.
* [$2003$] Yorke H. W., $2003$, Star Formation at High Angular Resolution, ASP Conference Series, Ed.: Jayawardhana R., Burton M. G. & Bourke T. L., Vol. S-$221$.
* [$2001$] Xu Y., $2005$, Chin. J. A&A, $1$, $389$.
* [$2003$] Xu Y., Zheng X. -W. & Jiang D.-R., $2003$, Chin. J. A&A, $3$, $49$.
* [$2007$] Zinnecker H. & Yorke H. W., ARA&A, $45$, $481$ $(arXiv:0707.1279)$.
* [$2001$] Zhang Q., Hunter T. R. & Shridaran T. K., Molinari T. K., Kramer M. A. & Cesaroni R., $2001$, AJ, $552$, $167$.
* [$2005$] Zhang Q., Hunter T. R., Brand J., Sridharan T. K., Cesaroni R., Molinari S., Wang J. & Kramer M., $2005$, AJ, $625$, $864$.
|
arxiv-papers
| 2010-10-18T06:27:12 |
2024-09-04T02:49:14.050989
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "G. G. Nyambuya",
"submitter": "Golden Gadzirayi Nyambuya Mr.",
"url": "https://arxiv.org/abs/1010.3893"
}
|
1010.3898
|
arxiv-papers
| 2010-10-19T13:17:39 |
2024-09-04T02:49:14.067642
|
{
"license": "Public Domain",
"authors": "Ranjeet Devarakonda, Giri Palanisamy, Bruce Wilson",
"submitter": "R Devarakonda",
"url": "https://arxiv.org/abs/1010.3898"
}
|
|
1010.3907
|
# Steady state of tapped granular polygons.
Carlos M Carlevaro1,2 and Luis A Pugnaloni1 1 Instituto de Física de Líquidos
y Sistemas Biológicos (CONICET La Plata, UNLP), Casilla de Correo 565, 1900,
La Plata, Argentina.
2 Universidad Tecnológica Nacional - FRBA, UDB Física, Mozart 2300, C1407IVT
Buenos Aires, Argentina. manuel@iflysib.unlp.edu.ar (C M Carlevaro)
###### Abstract
The steady state packing fraction of a tapped granular bed is studied for
different grain shapes via a discrete element method. Grains are monosized
regular polygons, from triangles to icosagons. Comparisons with disk packings
show that the steady state packing fraction as a function of the tapping
intensity presents the same general trends in polygon packings. However,
better packing fractions are obtained, as expected, for shapes that can
tessellate the plane (triangles, squares and hexagons). In addition, we find a
sharp transition for packings of polygons with more than 13 vertices signaled
by a discontinuity in the packing fraction at a particular tapping intensity.
Density fluctuations for most shapes are consistent with recent experimental
findings in disk packing; however, a peculiar behavior is found for triangles
and squares.
## 1 Introduction
Granular materials settle under gravity and come to mechanical equilibrium
unless an external excitation is provided. The properties of such static
packings are difficult to predict, since the history of preparation of the
sample is important. However, there exist different protocols to prepare a
granular bed in a well defined macroscopic state. In such state, the packing
fraction (and other macroscopic observables such as the pressure on the
container) are reproducible if the given protocol is followed. A canonical
example of this is the steady states obtained by tapping the sample with a
given intensity [1]. After a suitable annealing, tapping at a constant
intensity produces mechanically stable configurations (inherent states, or
microstates) whose ensemble has well defined mean values of all macroscopic
observables.
In recent years, the dependency of the steady state packing fraction, $\phi$,
on the tapping intensity, $\Gamma$, has been shown to be nonmonotonic;
presenting a minimum at relatively high values of $\Gamma$ for disks and
spheres [2, 3], and a maximum at very low $\Gamma$ for spheres [4]. In
general, the symbol $\Gamma$ is used for the reduced peak acceleration given
to the system during a tap. However, we will use $\Gamma$ in what follows to
refer to any suitable parameter that characterizes the tapping intensity.
On the one hand, there exist some studies on the response to tapping of non-
spherical particles [5, 6, 7, 8], however these do not consider polygonal
particles. On the other hand, the are some investigations on polygon packings
[9, 10, 11]. These latter studies, however, do not focus on the steady state
obtained after a repeated pulse excitation. Inspired by previous works on
pentagon packings [15, 16], we investigate the $\phi$–$\Gamma$ tapping curve
in the steady state for monosized regular polygons with different number $N$
of vertices; from triangles ($N=3$) to icosagons ($N=20$). As the number of
vertices grows, we expect polygon packings to approach the properties of disk
packings. Since depending on the number of vertices these particles may or may
not tessellate the plane, we also expect strong deviations from the general
trends for some grain shapes.
In this paper, we compare the general features found in the $\phi$–$\Gamma$
curve of disk packings with those of regular polygons. Although some general
trends are conserved, new phenomenology emerges.
In Section 2 we present the simulation technique and the model particles. In
Section 3.1 we analyze the behavior of polygons with fewer than ten vertices.
In Section 3.2 we present results for polygons of up to twenty vertices.
Section 3.3 is devoted to the study of the density fluctuations. Finally, we
draw the conclusions in Section 4 and point out some interesting areas of
research suggested by the new results.
## 2 Simulation
We perform molecular dynamic type simulations by solving the Newton–Euler
equations of motion for rigid bodies confined on a vertical plane. Gravity
acts on the negative vertical direction. The bodies (particles) are placed in
a rectangular box which is confined to move in the vertical direction. This
box is high enough to avoid particles to contact the ceiling during the
simulations. We prepare nineteen samples that consist of 500 monosized regular
polygons of a single type (from triangles to icosagons) or monosized disks.
Particles, initially placed at random without overlaps in the box, are let to
settle until they come to rest in order to prepare the initial packing. Then,
the same tapping protocol is applied to each sample.
We set the particle–particle interactions to yield a normal restitution
coefficient $\epsilon=0.058$ and a static and dynamic friction coefficient
$\mu_{s}=\mu_{d}=0.5$. The confining box is $24.8r$ wide and $2000r$ tall
(with $r$ the radius of the particles). The particle–box friction coefficient
is $\mu_{s}=\mu_{d}=0.07$ and the restitution coefficient is as in the
particle–particle interaction. All polygons have the same radius and material
density. Therefore, the actual weight of a particle depends on the number of
vertices. We use as unit mass, $m$, the mass of a disk; as unit length $r$;
and the unit time is $(r/g)^{1/2}$, with $g$ the acceleration of gravity.
Tapping is simulated by giving the box an impulse. In practice, we set the
initial velocity $v_{0}$ of the box (originally at rest after deposition) to a
given positive value and restart the dynamics. In doing so, the box and its
filling move upward and fall back on top of a zero restitution base. While the
box dissipate all its kinetic energy on contacting the base, particles inside
the box bounce against the box walls and floor until they fully settle. After
all particles come to rest a new tap is applied. The intensity of the taps is
measured by the initial velocity imposed to the confining box at each tap
(i.e. $\Gamma=v_{0}$). A similar parameter (the lift-off velocity) has been
recently proposed as a suitable measure of the tap intensity [17].
The tapping protocol consist in a series of $50000$ taps. Every $250$ taps we
change the value of $\Gamma$ by a small amount $\Delta\Gamma$. We initially
decrease $\Gamma$ from $\approx 15.0(rg)^{1/2}$ down to a very low value and
then increase it back to its initial high value. At each value of $\Gamma$ the
last $150$ taps are used to average the packing fraction in order to plot the
$\phi$–$\Gamma$ curve.
The simulations were implemented by means of the Box2D library [18]. Box2D
uses a constraint solver to handle hard bodies. At each time step of the
dynamics a series of iterations (typically 20) are used to resolve
penetrations between bodies through a Lagrange multiplier scheme [19]. After
resolving penetrations, the inelastic collision at each contact (a contact is
defined by a manifold in the case of polygons) is solved and new linear and
angular velocities are assigned. The equations of motion are integrated
through a symplectic Euler algorithm. The time step $\delta t$ used to
integrate the equations of motion is $0.025\sqrt{d/g}$. Solid friction is also
handled by means of a Lagrange multiplier scheme that implements the Coulomb
criterion. This library achieves a high performance when handling complex
bodies such as polygons.
## 3 Results
### 3.1 From triangles to nonagons
Figure 1: Mean packing fraction $\phi$ as a function of tapping intensity
$\Gamma$ for triangles (violet), squares (red), pentagons (green), hexagons
(blue), heptagons (yellow), octagons (cyan), nonagons (magenta), and disks
(black). Except for disks, all curves correspond to a progressive decrease of
$\Gamma$ followed by an increase back to high values. For disks only the
decreasing part has been carried out. Error bars correspond to the estimated
error of the mean.
The steady state packing fraction as a function of the tapping intensity for
triangles, squares, pentagons, hexagons, heptagons, octagons and nonagons is
presented in Fig. 1 alongside with the results for disks. Packing fraction is
estimated from the number density measured in a rectangular slab of half the
packing hight at the middle of the sample. The fact that the same
$\phi$-$\Gamma$ curve is obtained for decreasing and increasing $\Gamma$
indicates that these states are reversible and that $\phi$ is uniquely defined
for each $\Gamma$. From this results we can see that polygon packings present
similar features to those observed in disk packings. At low tapping
intensities, a decrease of $\phi$ is observed for increasing $\Gamma$ down to
a minimum packing fraction $\phi_{min}$. A further increase of $\Gamma$
induces an increase of $\phi$ until a plateau is reached at a packing fraction
somewhat lower than the maximum obtained for the lowest values of $\Gamma$.
Disks also show a not very pronounced maximum at low $\Gamma$ which is not
observed in polygon packings. This maximum has been recently observed in
sphere packings [4]. Another overall trend is that the range of packing
fractions attained by disk packings is narrower than for polygons.
Beyond these general features, there are some peculiarities associated to the
ability of a given polygonal shape to tessellate the plane. As is to be
expected, triangles, squares and hexagons can reach packing fractions of
nearly $1$ at the lower tapping intensities. All other shapes reach packing
fractions similar to disk packings at low $\Gamma$. It is important to notice
at this point that our results differ from those obtained by Vidales _et al._
[15, 16] in the case of pentagons in the framework of a pseudo-dynamic
algorithm. In Refs. [15, 16] the $\phi$–$\Gamma$ curve does not present any
minimum of the packing fraction.
In Fig. 2, we plot the minimum steady state density, $\phi_{min}$, as a
function of the number of vertices of the polygon. As the number of vertices
is increased, a consistent increase of $\phi_{min}$ is found for all polygons
with the exception of triangles, squares and hexagons. As we mentioned, these
three polygons can tessellate the plane. Correspondingly, triangles, squares
and hexagons present higher densities than expected by the trend showed by all
other polygons. We have seen that the position, $\Gamma_{min}$, of the minimum
is independent of the number of vertices. The existence of $\phi_{min}$ has
been associated to a competition between arch formation and arch breaking [2].
The position $\Gamma_{min}$ of such minimum signals the crossover between a
regime where arches cannot form due to the particles settling one by one (in a
sequential manner) at very high $\Gamma$, and a regime where arches do form
but are “melted down” in successive taps creating a dynamic equilibrium. The
fact that $\Gamma_{min}$ is the same for all shapes is a clear indication that
arching is not favored (nor prevented) by any particular shape at these
intermediate values of $\Gamma$.
Figure 2: Minimum packing fraction $\phi_{min}$ as a function of the number of
vertices. The blue line is drawn only to guide the eye.
### 3.2 An unforeseen sharp transition for triskaidecagons and beyond
Figure 3: Mean packing fraction $\phi$ as a function of tapping intensity
$\Gamma$ for nonagons, decagons, … and icosagons. The brown data in the lower
right panel correspond to disks. A progressive decrease (red data) of $\Gamma$
is followed by an increase (blue data) back to the high initial values. Error
bars as in Fig. 1. Figure 4: Mean packing fraction $\phi$ as a function of
tapping intensity $\Gamma$ for tetrakaidecagons. Panel (a), increasing
$\Gamma$. Panel (b), decreasing $\Gamma$. The red and blue data correspond to
independent realizations of the tapping protocol. The black data correspond to
the ones presented in Fig 3 for tetrakaidecagons, where larger steps in
$\Gamma$ are taken. The full and dashed black lines are to guide the eye.
Error bars as in Fig. 1.
We now focus on the behavior of polygons with larger number of vertices (from
nonagons up to icosagons). Figure 3 shows the $\phi$–$\Gamma$ curves for each
shape. One might have expected that a smooth change would appear in these
curves as the number of vertices is increased up to a point where the behavior
of the n-vertex polygon will converge to the one shown by disk packings.
However, a sudden change is found as we move from dodecagons to
triskaidecagons. While a continuous $\phi$–$\Gamma$ curve is observed for
polygons with up to $12$ vertices, a sharp discontinuity in $\phi$ is present
in all packings with polygons of $13$ vertices or more. A gap of “forbidden”
values of $\phi$ appears between roughly $0.80$ and $0.83$ in all these
polygon packings with more than $12$ vertices. It is important to mention that
fluctuations are rather large, and configurations (microstates) with
$0.80<\phi<0.83$ are rather common. It is the mean values that present a gap.
A similar discontinuity has been seen in tapped disk packings simulated under
a pseudo-dynamic algorithm [20]. However, this is not observed in our
simulations of disks (see brown data in the lower right panel in Fig. 3) nor
in previous molecular dynamic simulations where the same region of $\Gamma$
was explored [21]. The pseudo-dynamic algorithm [20] conducts a deposition of
disks that roll on top of each other without sliding. This might mimic, rather
realistically, the behavior of regular polygons with a large number of
vertices. These polygons behave like gears in the sense that they interlock
very easily just as if they were infinitely rough disks. We presume this basic
characteristic shared by polygons with many vertices and disk that roll
without sliding is the underlying phenomenon that leads to the emergence of a
discontinuous $\phi$–$\Gamma$ curve. We mention in pass that, although it is
difficult to relate with the static packings studied here, a similar
discontinuity has been reported in an oscillation experiment of a 2D granular
sample [22].
In order to have a rough indication of the nature of the transition, we have
made a more detailed simulation for tetrakaidecagons ($N=14$). In Fig. 4, the
steady state value of $\phi$ is plotted for $\Gamma$ in the interval
$[2.8,4.0]$ with a smaller $\Delta\Gamma$ step. In panel (a), we plot two
independent experiments obtained by increasing $\Gamma$ alongside with the
corresponding results from Fig. 3 (where a larger $\Delta\Gamma$ was used).
The results for the reversed protocol in which $\Gamma$ is decreased is
presented in panel (b) of Fig. 4. In Fig. 4(a), the system seems to present a
first order type transition where metastable branches are explored. Since
fluctuations are rather large for this small system sizes, the system may
explore microstates compatible with both “coexisting” phases. Nevertheless, in
Fig. 4(b), where the protocol corresponds to decreasing $\Gamma$, the
transition looks much smoother if the rate $\Delta\Gamma$ is reduced. Although
the data is noisy, we can see that the width of transition region is rate
dependent.
### 3.3 Density fluctuations
Density fluctuations have recently received renewed interest as a way to
measure configurational temperature (as defined by Edwards [23]) and entropy
[24]. It was in a fluidization experiment that a nonmonotonic dependence of
the fluctuations $\Delta\phi$ as a function of $\phi$ in the steady state was
first reported [25]. In that work, Schroter _et al._ found a minimum in the
density fluctuations for spheres. However, a recent study on disks reported a
maximum in fluctuations from both, experiments and simulations [3].
Figure 5: Standard deviation $\Delta\phi$ of the packing fraction in the
steady state as a function of $\phi$. The red line is a simple running average
to guide the eye. The arrows indicate the direction of increasing $\Gamma$.
In Fig. 5 we show the steady state density fluctuations $\Delta\phi$ as
measured by the standard deviation as a function of $\phi$ for several
polygons and disks. The results for disks are entirely in agreement with Ref.
[3]. A clear maximum in $\Delta\phi$ appears for disks. One can also see that
states of equal $\phi$ at each side of $\phi_{min}$ present slightly different
fluctuations. This indicates that these states are not equivalent and that
$\phi$ is not sufficient to characterize the macroscopic state. A more
detailed analysis of this can be found in Ref. [3] where the force moment
tensor is found to be a suitable extra macroscopic variable in accordance with
theoretical suggestions [26].
The behavior of the density fluctuations in the polygon packings show the
signal of the transition for shapes with $N>12$. However the same general
trends as those seen for disks are observed. Interestingly, a peculiar
behavior appears for triangles, squares and pentagons. Pentagons present the
same fluctuations at both sides of the $\phi$ minimum whereas triangles and
squares present a reversed situation where fluctuations are larger for large
$\Gamma$, instead of smaller as seen in all other shapes. This change in trend
should have an important impact in the calculation of configurational
temperature and entropy. We will pursue this point further elsewhere.
## 4 Conclusions
We have carried out simulations of the tapping of assemblies of regular
polygonal grains and studied the steady state of such systems. The comparison
with more widely studied disk packings has shown some general similarities but
also remarkable new phenomenology.
On the one hand, beyond the expectable result for triangles, squares and
hexagons that cover the space if gently tapped, polygons with $N>12$ show a
sharp transition with a clear density gap. On the other hand, triangles and
squares present density fluctuations that are larger at large tapping
intensities in contrast with all other shapes (including disks).
A number of questions arise from this study that can lead future research.
Some of these questions are:
1. 1.
What is the true nature of the transition for polygons with a large number of
vertices? Can this transition be effectively found in infinite rough disks?
Can the low density coexisting phase be related with the so called _random
close packing_ state [12, 13, 14, 10].
2. 2.
Given that fluctuations have a different trend, is the granular
(configurational) temperature in the case of triangles and squares radically
different from that of other shape packings?
3. 3.
Given that for pentagons the fluctuations are equivalent for states at each
side of $\phi_{min}$ obtained with different $\Gamma$, which suggest that the
states are equivalent, is the force moment tensor equivalent?
We thank Ana María Vidales and Irene Ippolito for valuable discussions. This
work has been supported by CONICET and ANPCyT (Argentina).
## References
## References
* [1] E. R. Nowak, J. B. Knight, M. Povinelli, H. M. Jaeger, and S. R. Nagel, Powder Technol. 94,79 (1997).
* [2] L. A. Pugnaloni, M. Mizrahi, M. C. Carlevaro, F. Vericat, Phys. Rev. E 78, 051305 (2008).
* [3] L. A. Pugnaloni, D. Maza, I. Sánchez, P. A. Gago, J. Damas, I. Zuriguel, arXiv:1002.3264 (2010)
* [4] A. D. Rosato, O. Dybenko, D. J. Horntrop, V. Ratnaswamy, L. Kondic, Phys. Rev. E 81, 061301 (2010).
* [5] I. C. Rankenburg, R. J. Zieve, Phys. Rev. E 63, 061303 (2001).
* [6] F. X. Villarruel, B. E. Lauderdale, D. M. Mueth, H. M. Jaeger, Phys. Rev. E 61, 6914 (2000).
* [7] G. Lumay, N. Vandewalle, Phys. Rev. E 74, 021301 ̵͑(2006͒).
* [8] M. Ramaioli, L. Pournin, Th. M. Liebling, Phys. Rev. E 76, 021304 ̵͑(2007͒).
* [9] R. Cruz Hidalgo, I. Zuriguel, D. Maza, I. Pagonabarraga, J. Stat. Mech. P06025 (2010).
* [10] Y. Limon Duparcmeur, J. P. Troadec, A. Gervois, J. Phys. I (France) 7, 1181 (1997).
* [11] M. Ammi, D. Bideau, J. P. Troadec, J. Phys. D 20, 424 (1987).
* [12] C. Radin, J. Stat. Phys. 131, 567-573 (2008).
* [13] D. Aristoff, C. Radin, arXiv:0909.2608 (2009).
* [14] Y. Jin, H. A. Makse, arXiv:1001.5287 (2010).
* [15] A. M. Vidales, L. A. Pugnaloni and I. Ippolito, Phys. Rev. E 77, 051305 (2008)
* [16] A. M. Vidales, L. A. Pugnaloni and I. Ippolito, Gran. Matter 11, 53 (2009)
* [17] J. A. Dijksman, M. van Hecke, Eur. Phys. Lett. 88, 44001 (2009).
* [18] Box2D Physics Engine, www.box2d.org
* [19] E. Catto, Iterative dynamics with temporal coherence (2005), available at http://box2d.googlecode.com/files/GDC2005_ErinCatto.zip (retrived on October 2010).
* [20] L. A. Pugnaloni, M. G. Valluzzi, L. G. Valluzzi, Phys. Rev. E 73, 051302 (2006).
* [21] R. Arévalo, D. Maza and L. A. Pugnaloni, Phys. Rev. E 74, 021303 (2006).
* [22] M. D. Shattuck, arXiv:cond-mat/0610839 (2006).
* [23] S. F. Edwards, R. B. S. Oakeshott, Physica A 157, 1080(1989).
* [24] S. McNamara, P. Richard, S. Kiesgen de Richter, G. Le Caër, R. Delannay Phys. Rev. E 80, 031301 (2009).
* [25] M. Schröter, D. I. Goldman, H. L. Swinney, Phys. Rev. E. 71, 030301(R) (2005).
* [26] R. Blumenfeld, S. F. Edwards, J. Phys. Chem. B 113, 3981 (2009).
|
arxiv-papers
| 2010-10-19T13:37:59 |
2024-09-04T02:49:14.074267
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Carlos M. Carlevaro and Luis A. Pugnaloni",
"submitter": "Luis Ariel Pugnaloni",
"url": "https://arxiv.org/abs/1010.3907"
}
|
1010.4149
|
Thermodynamics and Energy Technology Laboratory (ThEt),
University of Paderborn, Warburger Str. 100, 33098 Paderborn, Germany
# Research on the behavior of liquid
fluids atop superhydrophobic gas-bubbled surfaces
Gerrit C. Lehmann Frithjof Dubberke Martin Horsch Yow-Lin Huang (bkai黃佑霖)
Svetlana Miroshnichenko Rüdiger Pflock Gerrit Sonnenrein & Jadran
Vrabec111Corresponding author: Prof. Dr.-Ing. habil. Jadran Vrabec
$<$jadran.vrabec@upb.de$>$, phone +49 5251 602 421.
###### Abstract
Abstract: Superhydrophobic surfaces play an important role in the development
of new product coatings such as cars, but also in mechanical engineering,
especially design of turbines and compressors. Thus a vital part of the design
of these surfaces is the computational simulation of such with a special
interest on variation of shape and size of minor pits grooved into plane
surfaces. In the present work, the dependence of the contact angle on the
fluid-wall dispersive energy is determined by molecular simulation and static
as well as dynamic properties of unpolar fluids in contact with extremely
rough surfaces are obtained.
###### keywords:
Keywords: Contact angle, superhydrophobicity, molecular dynamics
Fluid flow over extremely rough surfaces is governed by non-trivial boundary
conditions which can be related to the contact angle as discussed by Voronov
et al. (2008). Boundary slip is most relevant for microscopic and nanoscopic
flow, while the influence of surface roughness on the contact angle becomes
extreme in case of superhydrophobic surfaces. For nanoscopic channel
dimensions as well as roughness on the molecular length scale, the accuracy of
simulation results can be optimized by applying molecular dynamics (MD), since
this approach reflects the actual structure of the material more directly than
higher-level methods that rely on aggregated models and properties.
As long as no hydrogen bonds are formed between the wall and the fluid, the
interfacial properties mainly depend on the fluid-wall dispersive interaction,
even for hydrogen bonding fluids. The truncated and shifted Lennard-Jones
(LJTS) potential with a cutoff radius of $r_{\mathrm{c}}$ = 2.5 $\sigma$
accurately reproduces the dispersive interaction if adequate values for the
size and energy parameters $\sigma$ and $\epsilon$ are specified, cf. Vrabec
et al. (2006).
Fluid-wall interactions can be represented by Lennard-Jones-12-6 effective
potentials, acting between fluid particles and the atoms of the solid, cf.
Battezzati et al. (1975). Following this approach, the LJTS potential with the
size and energy parameters $\sigma_{\mathrm{fw}}=\sigma$ as well as
$\epsilon_{\mathrm{fw}}=\mathnormal{W}\epsilon$ was applied for the unlike
interaction using the same cutoff radius as for the fluid. The wall was
modeled as a system of coupled harmonic oscillators with different spring
constants for transverse and longitudinal motion, adjusted to simulation
results for graphite with a rescaled variant of the Tersoff (1988) potential.
Massively parallel MD simulations were conducted with the program ls1 mardyn,
cf. Bernreuther et al. (2009). A periodic boundary condition was applied to
the system, leaving a channel with a diameter of 27 $\sigma$ between the wall
and its periodic image, cf. Fig. 1.
Figure 1: Simulation snaphots for a smooth surface with a reduced fluid-wall
dispersive energy $\mathnormal{W}$ of 0.09 (left) and 0.16 (right) at a
temperature of 0.73 $\epsilon/k$. The upper half is reproduced in the bottom
to illustrate the effect of the periodic boundary condition.
The contact angle was determined from the density profiles by averaging over
at least 800 ps after equilibration. A circle was adjusted to the positions of
the interface in the bins corresponding to distances between 3 and 11 $\sigma$
from the wall, and the tangent to this circle at a distance of 1 $\sigma$ from
the wall was consistently used to determine the contact angle.
A contact angle – as opposed to total dewetting or wetting – appears only for
a relatively narrow range of $\mathnormal{W}$ values. As the temperature
increases and the vapor-liquid surface tension decreases, the contact angle
reaches more extreme values, leading to the well-known phenomenon
characterized by Cahn (1977) as crticial point wetting, cf. Fig. 2. This plot
agrees qualitatively with the results of Giovambattista et al. (2007)
regarding the influence of the polarity of hydroxylated silica surfaces on the
contact angle formed with water.
Figure 2: MD simulation results and correlation for the contact angle of the
LJTS fluid on a smooth surface in dependence of the temperature with reduced
fluid-wall dispersive energy $\mathnormal{W}$ values of 0.09 ($\Delta$ / —),
0.10 (/ – –), 0.12 ($\bullet$ / —) as well as 0.14 ($\nabla$ /
$\cdot\cdot\cdot$). The entire range between triple point and critical
temperature is shown.
For a constant value $\mathnormal{W}=0.09$ of the reduced fluid-wall energy,
corresponding to a contact angle of about ${110}^{\circ}$ for moderate as well
as low temperatures, the surface shape and roughness was varied in further
simulations, cf. Fig. 3. The stability of the Cassie state as well as the
influence of the surface shape on dynamic properties such as the boundary slip
length and slip velocity in nanoscopic Poiseuille flow were studied by MD
simulation.
Figure 3: Left: Rectangular elementary cell of a pit grid with rectangular
pit for simulation with gaseous and liquid fluids. Right: Rectangular
elementary cell (prototype version) with cylindrical bore for simulation of
streaming fluids.
The simulation results regard the length scale between 1 and 100 nm and can be
reliably extrapolated to the characteristic system dimensions corresponding to
typical superhydrophobic surfaces, e.g. about one micron in case of the
material manufactured by Steinberger et al. (2008). Thereby, the experimental
point of view can be complemented by a theoretical treatment, applying the
variant of computational fluid dynamics that is best suited for the
investigation of nanopatterned surfaces: MD simulation.
The authors would like to thank the German Science Foundation (DFG) for
funding SFB 716 and M. Heitzig (Copenhagen), J. Harting (Eindhoven), and D.
Vollmer (Mainz) as well as M. Bernreuther, C. Dan, and M. Hecht (Stuttgart)
for technical support and fruitful discussions. The presented research was
carried out under the auspices of the Boltzmann-Zuse Society of Computational
Molecular Engineering (BZS) and the simulations were performed on the XC 4000
supercomputer at the Steinbuch Centre of Computing, Karlsruhe, under the grant
LAMO.
## References
* Battezzati, L., Pisani, C. & Ricca, F. 1975 Equilibrium conformation and surface motion of hydrocarbon molecules physisorbed on graphit. J. Chem. Soc. Faraday Trans. 2, Vol. 71, pp. 1629-1639.
* [1] Bernreuther, M., Niethammer, C., Horsch, M., Vrabec, J., Deublein, S., Hasse, H. & Buchholz, M. 2009 Innovative HPC methods and application to highly scalable molecular simulation. Innov. Supercomp. Deutschl., Vol. 7, pp. 50-53.
* [2] Cahn, J. W. 1977 Critical point wetting. J. Chem. Phys., Vol. 66, pp. 3667-3672.
* [3] Steinberger, A., Cottin-Bizonne, C., Kleimann, P. & Charlaix, E. 2008 Nanoscale flow on a bubble mattress: Effect of surface elasticity. Phys. Rev. Lett., Vol. 100, no. 134501.
* [4] Tersoff, J. 1988 Empiric interatomic potential for carbon, with applications to amorphous carbon. Phys. Rev. Lett., Vol. 61, pp. 2879-2882.
* [5] Voronov, R. S., Papavassiliou, D. V. & Lee, L. L. 2008 Review of fluid slip over superhydrophobic surfaces and its dependence on the contact angle. Ind. Eng. Chem. Res., Vol. 47, pp. 2455-2477.
* [6] Vrabec, J., Kedia, G. K., Fuchs, G. & Hasse, H. 2006 Comprehensive study on vapour-liquid coexistence of the truncated and shifted Lennard-Jones fluid including planar and spherical interface properties. Molec. Phys., Vol. 104, pp. 1509-1527.
|
arxiv-papers
| 2010-10-20T10:23:21 |
2024-09-04T02:49:14.093267
|
{
"license": "Public Domain",
"authors": "Gerrit C. Lehmann and Frithjof Dubberke and Martin Horsch and Yow-Lin\n Huang and Svetlana Miroshnichenko and R\\\"udiger Pflock and Gerrit Sonnenrein\n and Jadran Vrabec",
"submitter": "Martin Horsch",
"url": "https://arxiv.org/abs/1010.4149"
}
|
1010.4152
|
Thermodynamics and Energy Technology Laboratory (ThEt),
University of Paderborn, Warburger Str. 100, 33098 Paderborn, Germany
# Molecular simulation of fluid dynamics on the nanoscale
Jadran Vrabec111Corresponding author: Prof. Dr.-Ing. habil. Jadran Vrabec
$<$jadran.vrabec@upb.de$>$, phone +49 5251 602 421. Elmar Baumhögger Andreas
Elsner Martin Horsch
Zheng Liu (gbsn刘峥) Svetlana Miroshnichenko Azer Nazdrajić & Thorsten
Windmann
###### Abstract
Abstract: Molecular dynamics simulation is applied to Poiseuille flow of
liquid methane in planar graphite channels, covering channel diameters between
3 and 135 nm. On this length scale, a transition is found between the regime
where local ordering induced by the wall dominates the entire system and
larger channel diameters where the influence of boundary slip is still
present, but of a more limited extent. The validity of Darcy’s law for
pressure-driven flow through porous media is not affected by the transition
between these regimes.
###### keywords:
Keywords: Molecular dynamics, boundary slip, microporous and nanoporous media
On the nanometer length scale, continuum approaches like the Navier-Stokes
equation break down, cf. Karnidiakis et al. (2005). Therefore, the study of
nanoscopic transport processes requires a molecular point of view and
preferably the application of molecular dynamics (MD) simulation.
In the past, MD could be applied to small systems with a few thousand
particles only, due to the low capacity of computing equipment. Consequently,
a large gap between MD simulation results on the one hand and experimental
results as well as calculations based on continuum methods was present. The
constant increase in available computational power is eliminating this
barrier, and the characteristic length of the systems accessible to MD
simulation approaches micrometers, see also Bernreuther and Vrabec (2005) for
a discussion of efficient massively parallel simulation algorithms and their
implementation.
The present work deals with the flow behavior of liquid methane, modeled by
the truncated and shifted Lennard-Jones (LJTS) potential, cf. Allen and
Tildesley (1987), confined between graphite walls. While the LJTS potential
can also be applied to the interaction between the solid wall and the fluid,
the carbon structure itself is modeled using a rescaled variant of the Tersoff
(1988) multi-body potential. The unlike interaction parameters of the LJTS
potential acting between methane and carbon were determined according to the
Lorentz-Berthelot combination rule with the Lennard-Jones parameters of Wang
et al. (2000) for ‘pure’ sp2 configured carbon.
MD simulations of methane confined between graphite walls with up to 4,800,000
interaction sites, i.e. carbon atoms and methane molecules, were conducted
while the channel diameter was varied to include both the boundary-dominated
regime and the transition to the continuum regime. A pressure gradient was
induced by an external gravitation-like acceleration acting on all methane
molecules and a force in the opposite direction acting on the carbon atoms of
the graphite wall. The flow was regulated using a proportional-integral
controller such that the wall velocity was zero while the fluid reached a
specified average velocity.
The fluid-solid interaction induces a local ordering in the vicinity of the
wall. For channel diameters below 5 nm, cf. Fig. 1, this effect determines the
structure of the entire system. The resulting velocity profile is affected by
the local structure and therefore does not exhibit an exactly parabolic shape.
However, aggregated quantities such as the slip velocity and the slip length,
serving as boundary conditions for higher-order CFD methods and in particular
for Navier-Stokes solvers, can be determined by extrapolating a parabolic fit
as shown in Fig. 2.
Figure 1: Density profile for liquid methane at a temperature of 175 K in
confined within a planar graphite channel with a diameter of 3 nm from MD
simulation, averaged over the time intervals from 60 to 120 ps (– –) and from
420 to 480 ps (—) after simulation onset. Figure 2: MD simulation results
$\mathrm{(\textnormal{---})}$ with a parabolic fit ($\cdot\cdot\cdot$) for the
velocity profile during Poiseuille flow of liquid methane through a planar
graphite channel with a diameter of 8.5 nm for an average flow velocity of 50
m/s at a density of 19 $\pm$ 1 mol/l in the central region of the channel and
a temperature of 166.3 K.
For channel diameters between 20 and 50 nm, the boundary slip undergoes a
qualitative transition, cf. Fig. 3. In an extremely narrow channel the regular
ordering of the fluid molecules due to the vicinity of the wall entirely
dominates not only the static structure, but also the fluid dynamics. With
respect to the characteristic direction of the system, this highly ordered
structure does not support the extreme velocity gradient that would be implied
by the no-slip condition.
However, down to molecular length scales, the pressure drop
$-\Delta\mathnormal{p}/\Delta\mathnormal{z}$ is approximately proportional to
the average velocity $\bar{\mathnormal{v}}_{\mathnormal{z}}$ and inversely
proportional to the cross-sectionional area of the channel, cf. Fig. 3, in
agreement with Darcy’s law. Therefore, it can be concluded that the
qualitative transition between boundary-dominated laminar flow and laminar
flow which is only influenced by boundary slip to a certain extent is not
reflected by a corresponding change for the effective adhesive forces acting
between the fluid and the solid.
Figure 3: Pressure drop $-\Delta\mathnormal{p}$ in terms of
$\bar{\mathnormal{v}}_{\mathnormal{z}}$ and the channel length
$\Delta\mathnormal{z}$ (top) as well as slip velocity in terms of
$\bar{\mathnormal{v}}_{\mathnormal{z}}$ (bottom), for Poiseuille flow of
saturated liquid methane at a temperature of $\mathnormal{T}$ = 166.3 K and
average velocities $\bar{\mathnormal{v}}_{\mathnormal{z}}$ of 10 m/s (circles)
and 30 m/s (bullets), in dependence of the channel width; solid line: Darcy’s
law.
The authors would like to thank M. Bernreuther (Stuttgart), M. Buchholz
(Munich), J. Harting (Eindhoven), H. R. Hasse (Kaiserslautern), S. Jakirlić
(Darmstadt), and T.-H. Yen (Tainan) for technical support as well as fruitful
discussions and the German Federal Ministry of Education and Research (BMBF)
for funding the project IMEMO. The presented research was carried out under
the auspices of the Boltzmann-Zuse Society of Computational Molecular
Engineering (BZS) and the simulations were performed on the Nehalem cluster
laki at the High Performance Computing Center Stuttgart (HLRS) under the grant
MMSTP.
## References
* Allen, M. P. & Tildesley, D. J. 1987 Computer Simulation of Liquids, Clarendon, Oxford.
* [1] Bernreuther, M. & Vrabec, J. 2005 Molecular simulation of fluids with short range potentials. High Performance Computing on Vector Systems, Springer, Heidelberg, pp. 187-195.
* [2] Giovambattista, N., Debenedetti, P. G. & Rossky, P. J. 2007 Effect of surface polarity on water contact angle and interfacial hydration structure. J. Phys. Chem. B, Vol. 111, pp. 9581-9587.
* [3] Karnidiakis, G., Beskok, A. & Aluru, N. 2005 Microflows and Nanoflows: Fundamentals and Simulation, Springer, New York.
* [4] Tersoff, J. 1988 Empiric interatomic potential for carbon, with applications to amorphous carbon. Phys. Rev. Lett., Vol. 61, pp. 2879-2882.
* [5] Wang, Y., Scheerschmidt, K. & Gösele, U. 2000 Theoretical investigations of bond properties in graphite and graphitic silicon. Phys. Rev. B, Vol. 61, pp. 12864-12869.
|
arxiv-papers
| 2010-10-20T10:28:39 |
2024-09-04T02:49:14.099203
|
{
"license": "Public Domain",
"authors": "Jadran Vrabec and Elmar Baumh\\\"ogger and Andreas Elsner and Martin\n Horsch and Zheng Liu and Svetlana Miroshnichenko and Azer Nazdraji\\'c and\n Thorsten Windmann",
"submitter": "Martin Horsch",
"url": "https://arxiv.org/abs/1010.4152"
}
|
1010.4195
|
# Oil filaments produced by an impeller in a water stirred thank
Fluid Dynamics Videos
Rene Sanjuan-Galindo, Enrique Soto, Gabriel Ascanio and Roberto Zenit
Universidad Nacional Autonoma de Mexico, Mexico, Distrito Federal, Mexico
###### Abstract
In this video
(http://ecommons.library.cornell.edu/bitstream/1813/8237/2/LIFTED_H2_EMS
T_FUEL.mpgVideo 1 and
http://ecommons.library.cornell.edu/bitstream/1813/8237/4/LIFTED_H2_IEM
_FUEL.mpgVideo 2), the mechanism followed to disperse an oil phase in water
using a Scaba impeller in a cylindrical tank is presented. Castor oil
(viscosity = 500 mPas) is used and the Reynolds number was fixed to 24,000.
The process was recorded with a high-speed camera. Initially, the oil is at
the air water interface. At the beginning of the stirring, the oil is dragged
into the liquid bulk and rotates around the impeller shaft, then is pushed
radially into the flow ejected by the impeller. In this region, the flow is
turbulent and exhibits velocity gradients that contribute to elongate the oil
phase. Viscous thin filaments are generated and expelled from the impeller.
Thereafter, the filaments are elongated and break to form drops. This process
is repeated in all the oil phase and drops are incorporated into the
dispersion. Two main zones can be identified in the tank: the impeller
discharge characterized by high turbulence and the rest of the flow where low
velocity gradients appear. In this region surface forces dominate the inertial
ones, and drops became spheroidal.
|
arxiv-papers
| 2010-10-18T18:35:42 |
2024-09-04T02:49:14.105470
|
{
"license": "Public Domain",
"authors": "Rene Sanjuan-Galindo, Enrique Soto, Gabriel Ascanio and Roberto Zenit",
"submitter": "Roberto Zenit",
"url": "https://arxiv.org/abs/1010.4195"
}
|
1010.4196
|
# Formation and displacement of bubbles in a packed bed
Fluid Dynamics Videos
Enrique Soto, Alicia Aguilar-Corona* and Roberto Zenit
Universidad Nacional Autonoma de Mexico, Mexico
*Universidad Michoacana de San Nicolas de Hidalgo, Morelia, Mich, Mexico
###### Abstract
The fluid dynamics video show a gas stream which is injected into a packed bed
immersed in water and fluid dynamcis video present the dynamics
involved
(http://ecommons.library.cornell.edu/bitstream/1813/8237/2/LIFTED_H2_EMS
T_FUEL.mpgVideo 1 and
http://ecommons.library.cornell.edu/bitstream/1813/8237/4/LIFTED_H2_IEM
_FUEL.mpgVideo 2). The refractive index of the water an the packed bed are
quite similar and the edges of the spherical
particles can be seen. Two distinctive regimens can be observed. The first
one, for low air flow rates, which is
characterized by the percolation of the air thought the interstitial space
among particles. And the second one,
for high air flow rates, which is characterized by the accumulation of air
inside the packed bed without
percolation, it can be observed that the bubble pull apart the particles
apart. Furthermore, for the first case
the position of the particles remains constant while for the second one a
circulation of particles is induced by
the bubbles flow.
## 1 References
1. 1.
Gostiaux, L., Gayvallet,H. and G minard, J.-C. 2002 Dynamics of a gas bubble
rising through a thin immersed layer of granular material: an experimental
study. GranularMatter, 4,39-44.
|
arxiv-papers
| 2010-10-18T18:13:57 |
2024-09-04T02:49:14.110791
|
{
"license": "Public Domain",
"authors": "Enrique Soto, Alicia Aguilar-Corona and Roberto Zenit",
"submitter": "Roberto Zenit",
"url": "https://arxiv.org/abs/1010.4196"
}
|
1010.4360
|
# The Early Evolution of Primordial Pair-Instability Supernovae
C.C. Joggerst11affiliation: Department of Astronomy and Astrophysics,
University of California at Santa Cruz, Santa Cruz, CA 95060\.
Email:cchurch@ucolick.org 22affiliation: Nuclear and Particle Physics,
Theoretical Astrophysics and Cosmology (T-2), Los Alamos National Laboratory,
Los Alamos, NM 87545 and Daniel J. Whalen33affiliation: McWilliams Fellow,
Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213
###### Abstract
The observational signatures of the first cosmic explosions and their chemical
imprint on second-generation stars both crucially depend on how heavy elements
mix within the star at the earliest stages of the blast. We present numerical
simulations of the early evolution of Population III pair-instability
supernovae with the new adaptive mesh refinement code CASTRO. In stark
contrast to 15 - 40 M⊙ core-collapse primordial supernovae, we find no mixing
in most 150 - 250 M⊙ pair-instability supernovae out to times well after
breakout from the surface of the star. This may be the key to determining the
mass of the progenitor of a primeval supernova, because vigorous mixing will
cause emission lines from heavy metals such as Fe and Ni to appear much sooner
in the light curves of core-collapse supernovae than in those of pair-
instability explosions. Our results also imply that unlike low-mass Pop III
supernovae, whose collective metal yields can be directly compared to the
chemical abundances of extremely metal-poor stars, further detailed numerical
simulations will be required to determine the nucleosynthetic imprint of very
massive Pop III stars on their direct descendants.
## 1\. Introduction
The first stars in the universe form at $z\sim$ 20 and are likely very
massive, 30 - 500 M⊙ (Bromm et al., 1999; Abel et al., 2000, 2002; Bromm et
al., 2002; Nakamura & Umemura, 2001; O’Shea & Norman, 2007). The fates of
these stars depend on their masses: 15 - 50 M⊙ stars die in core collapse
supernovae (CC SNe), 140 - 260 M⊙ stars explode in far more energetic
thermonuclear pair-instability supernovae (PISNe Heger & Woosley, 2002), and
40 - 60 M⊙ stars may die as hypernovae, whose explosion mechanisms are not yet
understood but are thought to have energies intermediate to those of CC and
pair-instability SNe (Iwamoto et al., 2005). Most Pop III SNe (Bromm et al.,
2003; Kitayama & Yoshida, 2005; Greif et al., 2007) occur in low densities
(0.1 - 1 cm-1) because UV radiation from the star sweeps most of the baryons
from the dark matter halo in which it resides (Whalen et al., 2004; Kitayama
et al., 2004; Alvarez et al., 2006; Abel et al., 2007; Wise & Abel, 2008a).
Metals from Pop III SNe determine the character of second-generation stars and
the primeval galaxies they populate by enhancing cooling in the gas in which
such stars form, and therefore the mass scales on which it fragments. The
manner in which the first metals contaminate pristine gas in the primordial
IGM crucially depends on mixing processes that have only begun to be studied
numerically. Preliminary calculations indicate that metals from Pop III SNe
mix with gas in a halo on two characteristic spatial scales prior to their
emergence into cosmological flows on kpc scales (Whalen et al., 2008): 10 - 15
pc, when a reverse shock forms in the remnant, and 150 - 200 pc, when the
remnant collides with the dense shell of the relic H II region of the
progenitor. New simulations now prove that violent mixing can occur within the
star prior to shock breakout from its surface, if the star is 15 - 40 M⊙
(Joggerst et al., 2010a, hereafter JET10). Capturing mixing on the smallest
scales is prerequisite to following its cascade out to the largest ones, and
hence to determining the colors and morphologies of primitive galaxies. Early
mixing also governs which elements are imprinted on next-generation stars,
whose chemical abundances may impose indirect constraints on the masses of the
stars that enriched them. Low-mass remnants of this generation are now sought
in ongoing surveys of ancient, dim metal-poor stars in the Galactic halo
(Beers & Christlieb, 2005; Frebel et al., 2005). Since mixing in Pop III SNe
in part sets elemental abundances in second-generation stars, it is integral
to future measures of the primordial IMF.
Early mixing is also key to computing the light curves and spectra of the
first cosmic explosions, whose detection could yield the first direct measure
of the primordial IMF. Optical and UV radiation breaks free of the SN shock
when the ejecta reaches the outer envelope of the star and is exposed to the
IGM. In the frame of the shock, the photosphere from which photons escape into
space descends deeper into the ejecta as it expands outward because of the
spherical dilution of the ejecta. If mixing precedes radiation breakout, it
determines the elements that the photosphere encounters as it sinks deeper
into the ejecta, and therefore the emission lines that propagate into the IGM
over time. Accurate multidimensional models of the explosion from its earliest
stages are therefore necessary to compute lines in primordial SN light curves
and spectra.
Mixing in galactic core-collapse supernovae has been studied for over twenty
years, particularly in connection with SN 1987A (e.g. Fryer et al., 2007, and
references therein). A core objective of these studies is to understand the
premature appearance of ${}^{56}\mathrm{Ni}$ in the spectra of 1987A, whose
emission lines appear much earlier than a simple picture of segregated,
spherically-symmetric expanding mass shells would predict. Mixing in Pop III
core-collapse SNe was first examined by Joggerst et al. (2009), who found that
both vigorous mixing and fallback onto the compact remnant in 15 - 40 M⊙ Pop
III SNe govern which metals escape into the IGM at high redshifts.
PISNe may have been commonplace in the primeval universe, but their enrichment
of the early IGM is yet to be understood. To investigate the propagation of
metals into the IGM by such explosions, constrain their nucleosynthetic
imprint on second-generation stars, and to evaluate the impact of mixing on
PISN spectra, we have performed two-dimensional simulations of the explosions
of 150 - 250 M⊙ stars. In $\S\,2$ we review how blast profiles from the KEPLER
code were ported as initial conditions to CASTRO and discuss what factors
govern the presupernova structure of the star. We examine the outcomes of the
explosions in $\S\,3$ and in $\S\,4$ we conclude.
## 2\. Models
### 2.1. KEPLER
As in JET10, the simulations in our PISN survey were carried out in two
stages. First, primordial stars were evolved through all stages of stable
nuclear burning from the zero-age main sequence to initial collapse via the
pair instability in the one-dimensional Lagrangian stellar evolution code
KEPLER (Weaver et al., 1978; Woosley et al., 2002). The PISN is triggered when
this collapse incites explosive O, and some cases Si, burning. Unlike the
simulations of JET10, in which the blast is artificially launched with a
piston, the pair instability and subsequent collapse that triggers these
explosions are an emergent feature of the stellar evolution calculation. They
are genuinely spherical, barring (magneto)rotational effects, and their
energies are set by O and Si burning. The blast was followed until the end of
all nuclear burning, $\sim$ 20 s after the start of the explosion. The energy
generated was computed with a 19-isotope network up to the point of oxygen
depletion in the core of the star and with a 128-isotope quasi-equilibrium
network thereafter.
Table 1PISNe Models: Properties at time models were mapped to 2D model | $M_{He}$ ($\mathrm{M}_{\odot}$) | $M_{N}$ ($\mathrm{M}_{\odot}$) | $M_{{{}^{56}\mathrm{Ni}}}$ ($\mathrm{M}_{\odot}$) | $M_{final}$ ($\mathrm{M}_{\odot}$) | $R$ ($10^{13}$ cm) | $E_{kin}$ ($10^{51}$ ergs)
---|---|---|---|---|---|---
u150 | 41 | 9.1(-5) | 0.079 | 143 | 16 | 3.7
u175 | 46 | 1.1(-4) | 0.72 | 164 | 18 | 9.5
u200 | 50 | 1.3(-4) | 5.2 | 183 | 19 | 17
u225 | 54 | 1.1(-3) | 17 | 200 | 33 | 28
u250 | 68 | 1.7(-4) | 38 | 238 | 23 | 34
z175 | 53 | 1.6(-5) | 0.24 | 175 | 4.2 | 4.7
z200 | 60 | 1.8(-5) | 2.0 | 200 | 4.5 | 12
z225 | 67 | 1.7(-5) | 8.8 | 225 | 4.9 | 17
z250 | 74 | 1.6(-5) | 23 | 250 | 6.2 | 29
### 2.2. CASTRO
The one-dimensional explosion profiles were then mapped onto two-dimensional
$RZ$ axisymmetric grids in CASTRO (Compressible ASTROphysics), a multi-
dimensional Eulerian AMR code with a high-order unsplit Godunov hydro solver
(Almgren et al., 2010). Each explosion was evolved past breakout from the
surface of the star until all mixing ceased and each element in the ejecta was
expanding homologously in mass coordinate. We smoothly join the density at the
surface of the star to a uniform circumstellar medium of 1 cm-3 with an
$r^{-3.1}$ power law, in keeping with the usual assumption of a low-density H
II region around the progenitor star with no wind-blown shell or prior mass
ejection. The medium beyond the star has no effect on the dynamics within the
star if its density falls more steeply than $r^{-3}$. In mapping the radial
profile onto the $RZ$ grid in CASTRO, care was taken to resolve the key
elements of the explosion: the shock, the shells of elements, and the high-
density core. In particular, both the ${}^{56}\mathrm{Ni}$ core and the O
shell were resolved with a minimum of 16 cells. Our mapping excludes
departures from spherical symmetry due to O burning, but such perturbations
would likely be high mode and low amplitude and therefore have minimal effect
on the evolution of instabilities. Our models thus only capture later
asymmetries of mode greater than $l=1$ or 2.
As in JET10, we adopt a monopole approximation for self-gravity. We first
compute a radial average of the density from the $RZ$ grid to create a one-
dimensional density profile. We then compute a one-dimensional gravitational
potential from this profile and map it back onto the $RZ$ grid. Since
departures from spherical symmetry in the densities are minor, this
approximation has a negligible effect on the final state of the explosion.
Because the PISN completely disperses the star, there is no compact remnant,
fallback, or thus any need to include a point potential centered at the origin
of the coordinate mesh.
We follow 15 chemical elements as individual species, each with their own
continuity equation, and calculate local energy deposition due to radioactive
decay of ${}^{56}\mathrm{Ni}$ in the same manner as JET10. However, the
formation of a nearly degenerate core at white dwarf densities in the
progenitor necessitates the use of the Helmholtz equation of state (EOS) at
early stages of the explosion. As the ejecta expands and cools, we transition
back to the ideal EOS used in JET10, which assumes that the gas is fully
ionized and includes contributions from both radiation and ideal gas pressure.
The base grid is 10242, with the initial outer boundaries set so that the
inner portion of the star is resolved as described above using no more than 6
levels of refinement. The star is centered at the lower left corner of the
mesh. We apply reflecting and outflow boundary conditions to the inner and
outer boundaries of the grid, respectively. Our refinement criteria are the
same as those in JET10. When the shock nears the edge of the grid, the
simulation is stopped, the grid is doubled, and the calculation is then
restarted, subtracting or adding levels of refinement as needed. This
procedure is repeated up to 12 times, depending on the model. We halt the
simulation when all chemical species are expanding homologously, which always
occurs by the time the ejecta has propagated a short distance into the uniform
circumstellar density.
### 2.3. Progenitor Models
JET10 found that mixing in low-mass Pop III core-collapse explosions primarily
depends on the internal structure of the progenitor. We likewise expect the
early evolution of pair-production explosions to be determined by the envelope
of the star, which is determined its mass, internal convective mixing over its
lifetime, and by its metallicity. Capturing the full range of structures for
these stars is essential to a comprehensive survey of early mixing in Pop III
supernovae.
#### 2.3.1 Convective Mixing
The initial absence of metals and the large contribution of radiation to the
pressure in massive Pop III stars promotes convection within them. The CNO
cycle cannot begin in primordial stars until a threshold mass fraction of
${}^{12}\mathrm{C}$ is first created by the triple $\alpha$ process. This
trace ${}^{12}\mathrm{C}$ sets the entropy of the hydrogen layer to be just
above that in the core, without the sharp entropy gradient in the upper layer
of the helium shell that is usually present in He burning stars with metals.
This plus radiation pressure facilitates convection. In 140 - 260 M⊙ stars,
the central convection zone can approach, come in contact with, or even reach
into the lower hydrogen layers, mixing them with carbon brought up from the
core during helium burning. When these two high-temperature components mix,
they burn vigorously, elevating energy release rates in the H shell by up to
several orders of magnitude (the so-called hydrogen boost).
Convection affects the structure of the star in two ways. First, since it
raises energy production (and hence opacities) in the lower hydrogen layers,
the star can puff up by more than an order of magnitude in radius and acquire
a red supergiant structure. Second, if convection is extreme the transport of
material out of the core could reduce its size and explosion energy in
comparison to modest convection. Unfortunately, one-dimensional stellar
evolution models cannot predict these inherently three-dimensional processes
from first principles. Instead, they parametrize them with semi-convection
coefficients.
#### 2.3.2 Metallicity
Gas in high-redshift halos that is enriched to metallicities below 10-3.5 Z⊙
fragments on mass scales that are essentially identical to those of pristine
gas and still forms very massive stars (e. g. Bromm et al., 2001; Mackey et
al., 2003; Smith & Sigurdsson, 2007). However, such small metal fractions are
more than enough to enhance CNO reaction and energy generation rates in the
hydrogen burning layers of the star, enlarging it in the same manner as
convective mixing (Hirschi, 2007; Ekström et al., 2008). Hence, we would
expect a strong degeneracy between the influence of metals and convection on
the envelope of the star, and that the full range of mixing in PISNe can be as
easily spanned by metallicity as by convective overshoot.
We considered 150, 175, 200, 225 and 250 M⊙ non-rotating progenitors at $Z=0$
(the z-series) and $Z=10^{-4}Z_{\odot}$ (the u-series), which we summarize in
Table 1. The $Z=0$ 150 M⊙ star collapses to a black hole without exploding, so
we exclude it from our CASTRO models. We employ metallicity rather than
convective overshoot to cover the range of plausible progenitor structures in
our study, because there is uncertainty about how much semi-convection there
is in a given star but none about the range of metallicities over which it can
form ($Z$ = 0 and 10-4. Comparison of Table 1 with Table 1 of Scannapieco et
al. (2005) confirms that for a given progenitor mass these two metallicities
do yield upper and lower limits to stellar radius similar to those for all
reasonable values of semi-convection coefficients.
## 3\. Results
As expected, the u-series models die as red giants, with radii more than an
order of magnitude larger than those of the z-series, which die as blue
giants. As shown in Table 1, the u-series models in general have more
energetic explosions than the z-series models for a given mass. As we show in
Figure 1, convective mixing has completely disrupted the helium layer in model
u225 and mixed it with the hydrogen envelope. The helium layer has also been
mixed with the hydrogen envelope, although to a lesser extent, in model u200
and to an even slighter degree in model u250. These u-series models also
experienced some mass loss due to pulsations.
The $\rho r^{3}$ values through which the shock passes are an effective
predictor of post-explosion hydrodynamics. In regions where $\rho r^{3}$
increases, the shock must decelerate, and a reverse shock forms that inverts
the pressure gradient and induces Rayleigh-Taylor (RT) instabilities. Models
with steeper values of $\rho r^{3}$ will experience more mixing because the
forward shock slows down more abruptly, which leads to a stronger reverse
shock. We show $\rho r^{3}$ values for all models in Figure 1, scaled to the
maximum value in model u225. It is clear from this figure that models u225 and
u200 exhibit the largest increases in $\rho r^{3}$ near the outer edge of the
star. This is due to slight bumps in density that are connected to the
pulsations that ejected mass from these stars earlier.
Figure 1.— The structure of helium and oxygen shells (delineated by dashed
lines for clarity) in all the progenitors, with $\rho r^{3}$ superimposed on
them. The $\rho r^{3}$ profiles have all been scaled to the maximum $\rho
r^{3}$ in model u225. $\rho r^{3}$ increases dramatically near the edge of
some models that die as red giants, while none of the blue giants show a
similar structure. The plateaus in helium abundance above cosmological values,
which are most apparent in model u225, denote convective regions in which the
hydrogen envelope mixed with a portion of the helium layer. This process
completely disrupted the helium layer in model u225, and mixed a smaller
fraction of the helium layer with the hydrogen envelope in model u200. A small
amount of the helium shell convectively mixed with the hydrogen envelope in
model u250. Figure 2.— Images of density and oxygen abundance for all models
after the forward shock exits the star or the reverse shock (if one forms)
traverses the star, whichever is later. Density is scaled to the minimum and
maximum values within a given simulation. A dense shell formed in all models
with masses greater than 200 $\mathrm{M}_{\odot}$, but the RT instability only
grows appreciably in the u200 and u225 models. These were the largest
progenitors in radius and had the steepest $\rho r^{3}$ curves. Only slight RT
growth is evident in model u250.
We show final states of mixing for the PISNe in Figure 2. The models are shown
at different times, but always after the shock emerged from the surface of the
star and the reverse shock, if one formed, dissipated. Density is shown on the
left in each panel, with values scaled to the maximum and minimum densities in
the simulation, while oxygen is shown on the right, with the values indicated
in the color bar. Some similarities between the models exist across all
stellar structures. In the higher-mass compact models (z225 and z250) and in
the red u-series models above 150 solar masses, the bulk of the oxygen layer
was swept up into a shell of higher density than the material on either side.
In general, however, the z-series explosions evolved quite differently than
the u-series SNe. In the z-series, no reverse shock formed. A reverse shock
formed in all the u-series models, but dissipated by the time it reached the
density contrast at the oxygen layer (u150 and u175) or was too weak by the
time it reached this dense shell to induce rapid RT growth (u250). In two
models, a reverse shock formed that was strong enough to drive rapid growth of
RT instabilities (u200 and u225). The reverse shock was strongest in model
u225, which had the steepest increase in $\rho r^{3}$ and the second highest
explosion energy, after model u250. The RT instability is clearly visible in
the u200 and u225 panels in Figure 2. In these models the instability became
nonlinear because individual RT fingers strongly interacted. In model u250,
the RT instability only reached the early linear phase before its growth was
halted.
The extent to which the structure of the PISN at the time of explosion was
preserved or disrupted by mixing is shown in Figure 3. We plot the initial
shell structure of the PISN at the time of mapping into CASTRO as a dashed
line, and overlay a solid line indicating the abundances of these elements
after the shock exited the star or the reverse shock traversed the star,
whichever was later. The slight smoothing in the final profiles in comparison
to the initial ones is due to numerical diffusion over the course of the
simulation. The u200 and u225 panels demonstrate the extent of mixing in these
explosions. The oxygen layer has been completely disrupted, and mixing has
penetrated just to the top of the silicon layer (in u200) or through the
silicon layer (in u225). The most vigorous mixing occured in model u225, where
hydrogen shell boost led to convection that completely mixed the helium shell
and the hydrogen envelope. The only other models to experience RT mixing,
models u200 and u250, are also the only other models that show evidence of
convective mixing in the outer region of the star prior to explosion. Model
u200 manifests more RT mixing and a larger region (in radius and mass) in
which convection mixed part of the helium layer with the hydrogen envelope
prior to explosion. Model u250, which has the smallest amount of convection
extending into the helium layer, also exhibits the least RT growth. Models
with no convective mixing between the helium layer and the hydrogen envelope
manifest no RT mixing during the explosion, so mixing seems tied to the depth
to which this convective envelope has penetrated the helium layer of the star.
The deeper the convective envelope, the more RT mixing occurs during the
supernova shock.
Figure 3.— Initial and final abundances of He, O, Si, and Fe for all models.
The RT instability is only present in models u200, u225, and to a small degree
in u250. The slight disagreement between initial and final profiles for models
which experienced no mixing is due to numerical diffusion over the evolution
times of these models. In none of the mixed models did mixing break through
the Si layer of the star, so ${}^{56}\mathrm{Ni}$ never reaches the surface of
the explosion as it does in core-collapse supernova explosions.
RT-induced mixing in PISNe, if it occurs at all, is unlikely to leave the
remnant in a state that resembles a core-collapse supernova remnant. In
particular, it is unlikely to dredge up significant amounts of oxygen, let
alone silicon or ${}^{56}\mathrm{Ni}$, into the outer regions of the star, or
draw much hydrogen back towards the center of the blast. The most vigorous
mixing occurs in the model with the helium shell that has been completely
disrupted by convective mixing with the hydrogen envelope, and likely
represents the most mixing that can occur in a PISN. Even in this model,
${}^{56}\mathrm{Ni}$ cannot reach the upper layers of lighter elements.
As noted earlier, our CASTRO simulations exclude the first stages of the
explosion, in which oxygen burning drives convective mixing that may perturb
the star (Chen et al., 2010). Since these perturbations are expected to be of
low mode and high amplitude, it is unlikely they would alter the essential
conclusions of our survey. Reprising our two-dimensional calculations in three
dimensions is unlikely to yield significantly different results. The RT
instability initially grows about $30\%$ faster in three dimensions than in
two because of artificial drag forces arising from two-dimensional geometries
(Hammer et al., 2010). However, once the fingers of the instability begin to
interact with one another, they mix more efficiently in three dimensions than
in two, reducing the Atwood number and hence their growth rate (Joggerst et
al., 2010b). These two effects cancel each other, and the width of the mixed
region in two and three dimensions is the same. For spherical simulations like
ours in which the RT fingers significantly interact, two-dimensional and
three-dimensional simulations would exhibit comparable degrees of mixing.
Also, the RT instability will not appear in three dimensions when it is not
manifest in two.
## 4\. Discussion and Conclusions
Unlike 15 - 40 M⊙ core-collapse Pop III SNe, 150 - 250 M⊙ Pop III PISNe
experience either no internal mixing prior to shock breakout from surface of
the star or only modest mixing between the O and He shells. Minor mixing
occurs in only two of the explosions and is due to the formation of a reverse
shock that is strong enough to trigger the RT instability at the dense shell
created by the forward shock at the top of the oxygen layer. The degree of
mixing is principally a function of how well hydrogen shell boost mixes the
helium shell with the hydrogen envelope. The model in which the helium shell
is completely mixed with the hydrogen envelope exhibits the most mixing. The
general lack of internal mixing in PISNe has several consequences for early
chemical enrichment of the IGM, the formation of second generation stars, and
the observational signatures of such explosions.
First, the elements that are expelled by low-mass Pop III SNe (which are
governed by both mixing and fallback onto the compact remnant) are later
imprinted on new stars in essentially the proportions in which they are
created by the explosions, regardless of how and where the stars form. This is
because the metals are already highly mixed by the time the shock exits the
star and are merely diluted upon further expansion into the halo and IGM,
where new stars might form. This implies that IMF averages of nucleosynthetic
yields of primordial core-collapse SNe can be directly compared to chemical
abundances in ancient metal-poor stars, without regard for any intervening
hydrodynamical processes. Indeed, the fact that elemental yields from Salpeter
IMF averages of 15 - 40 M⊙ progenitors in JET10 match those found in two
observational surveys of extremely metal-poor (EMP) stars suggests that the
bulk of early chemical enrichment may have been due to low-mass Pop III stars.
This is at odds with the current state of the art in Pop III star formation
simulations, which suggest that the first stars were predominantly 100 - 500
M⊙.
Determining the nucleosynthetic imprint of PISNe on second-generation stars is
much more problematic because their metals may have differentially
contaminated new stars. This is because metals in PISNe mixed with each other
and with the surrounding IGM on much larger spatial scales, at radii where new
stars may have formed. Except in models u200 and u225 at the interface of the
O and He shells, and in model u225 at the interface of the O and Si shells,
the shells of elements in the ejecta of other PISNe expand homologously until
they sweep up their own mass in the ambient H II region, at radii of 10 - 15
pc. At this point, a reverse shock detaches from the forward shock and a
contact discontinuity forms between them (the Chevalier phase– Whalen et al.,
2008). Both the reverse shock and contact discontinuity are prone to dynamical
instabilities that will mix elements from the interior of the remnant with the
surrounding medium. Later, on scales of 100 \- 200 pc, further mixing will
occur upon collision of the remnant with the dense shell of the relic H II
region.
In this case, the elements that are taken up into new stars depends on how and
where the stars form. If metals migrate out into cosmological flows and then
fall back into the the halo via accretion and mergers on timescales of 50 -
100 Myr, they likely will be well-mixed, with all elements formed in the
explosion appearing in the new star (Greif et al., 2007; Wise & Abel, 2008b;
Greif et al., 2010). However, new stars may form at much earlier times in the
SN remnant at radii where mixing takes place. One way this could happen is if
metals at the interface between the ejecta and swept-up shell suffuse into and
cool the shell, causing it to fragment into clumps that are unstable to
gravitational collapse (e.g. Mackey et al., 2003). In this scenario, such
clumps would be enriched only by the elements residing in the relatively
narrow zone in which the stars form. This still does not explain why some
hyper metal-poor stars have such high [C, N, and O/Fe] ratios. While C and O
would be preferentially deposited on the clumps in which such stars form
because they are predominant in the outer layers of the ejecta, PISNe do not
produce enough N to account for measured abundances in these stars. However,
little iron from deep in the interior of the remnant would reach the clumps
because it is not mixed, which is consistent with the abundances measured in
EMP stars to date. The failure to uncover the characteristic odd-even
nucleosynthetic signature of PISNe predicted by Heger & Woosley (2002) metal-
poor star surveys thus far has led some to suppose that primordial stars may
not have been very massive, but it is possible that such signatures have been
masked by observational selection effects (Karlsson et al., 2008). Unlike for
low-mass stars, detailed numerical simulations that follow differential
enrichment are required to determine the true chemical imprint of PISNe on
second-generation stars.
Second, our results imply that, unlike in SN 1987A or low-mass Pop III SNe, Ni
and Fe emission lines will not appear immediately after radiation breakout
from the shock because there is not enough mixing to transport these elements
out to the photosphere of the fireball. This may be key to distinguishing
between Pop III core-collapse explosions and PISNe. The light curves of these
supernovae are characterized by a sharp intense initial transient that decays
over several hours into a dimmer extended plateau that persists for 2 - 3
months in core-collapse events and 2 - 3 years in PISNe (Fryer et al., 2010;
Whalen & Fryer, 2010). The inital pulse is powered by the thermal energy of
the shock and the plateau is energized by radioactive decay of
${}^{56}\mathrm{Ni}$ (the long life of the plateau is due to the longer
radiation diffusion timescales through the massive ejecta). Preliminary
radiation hydrodynamical calculations of PISN light curves and spectra
indicate that their peak bolometric luminosities are similar to those of Type
Ia and core-collapse SNe, making determination of the mass of the progenitor
from the magnitude of the initial transient problematic. If, however, Ni and
Fe are detected just after radiation breakout, one can be confident that the
explosion is due to a low-mass primordial star, and a Pop III IMF could begin
to be built up by samples of such detections.
Our models also demonstrate that one-dimensional radiation hydrodynamical
models are sufficient to capture most features of PISN light curves and
spectra because the mixing such calculations exclude, which would alter the
order in which emission lines appear over time, is minor. The picture for
core-collapse Pop III SNe is quite different because vigorous mixing prior to
the eruption of the shock from the surface of the star mandates its inclusion
in light curve models. Two-dimensional multigroup radiation hydrodynamical
calculations of such spectra lie within the realm of current petascale
platforms, but a less costly approach can incorporate mixing in one-
dimensional models. If most mixing in low-mass Pop III explosions occurs
before radiation breaks free from the shock, two-dimensional models such as
those in JET10 can be used to compute the distribution of elements in the
ejecta just before breakout. These explosions can then be azimuthally averaged
onto the one-dimensional grid of the light curve calculation and evolved to
compute spectra. On average, along any given line of sight out of the SN, this
method will produce emission lines in the likely order they would be observed.
These simulations, together with our previous survey of mixing and fallback in
low-mass Pop III SNe, are the first of a numerical campaign to model the
chemical enrichment of the early cosmos from its smallest relevant spatial
scales. The eventual goal of this campaign is to understand the contribution
of the first SNe to the formation of new stars and the assembly of primeval
galaxies, which will soon be probed by the James Webb Space Telescope (JWST)
and the Atacama Large Millimeter Array (ALMA). The next stage of these
numerical simulations will incorporate mixing on subparsec scales to determine
if new stars directly form in the remnants of the first supernova explosions
and follow their congregation into the first galaxies.
The authors thank Stan Woosley for helpful discussions and the use of his
KEPLER progenitor models. CCJ was supported in part by the SciDAC Program
under contract DE-FC02-06ER41438. DJW acknowledges support from the Bruce and
Astrid McWilliams Center for Cosmology at Carnegie Mellon University. Work at
LANL was done under the auspices of the National Nuclear Security
Administration of the U.S. Department of Energy at Los Alamos National
Laboratory under Contract No. DE-AC52-06NA25396. All simulations were
performed on the open cluster Coyote at Los Alamos National Laboratory.
## References
* Abel et al. (2000) Abel, T., Bryan, G. L., & Norman, M. L. 2000, ApJ, 540, 39
* Abel et al. (2002) —. 2002, Science, 295, 93
* Abel et al. (2007) Abel, T., Wise, J. H., & Bryan, G. L. 2007, ApJ, 659, L87
* Almgren et al. (2010) Almgren, A. S., Beckner, V. E., Bell, J. B., Day, M. S., Howell, L. H., Joggerst, C. C., Lijewski, M. J., Nonaka, A., Singer, M., & Zingale, M. 2010, ApJ, 715, 1221
* Alvarez et al. (2006) Alvarez, M. A., Bromm, V., & Shapiro, P. R. 2006, ApJ, 639, 621
* Beers & Christlieb (2005) Beers, T. C. & Christlieb, N. 2005, ARA&A, 43, 531
* Bromm et al. (1999) Bromm, V., Coppi, P. S., & Larson, R. B. 1999, ApJ, 527, L5
* Bromm et al. (2002) —. 2002, ApJ, 564, 23
* Bromm et al. (2001) Bromm, V., Ferrara, A., Coppi, P. S., & Larson, R. B. 2001, MNRAS, 328, 969
* Bromm et al. (2003) Bromm, V., Yoshida, N., & Hernquist, L. 2003, ApJ, 596, L135
* Chen et al. (2010) Chen, K., Heger, A., & Almgren, A. 2010, ArXiv e-prints
* Ekström et al. (2008) Ekström, S., Meynet, G., Chiappini, C., Hirschi, R., & Maeder, A. 2008, A&A, 489, 685
* Frebel et al. (2005) Frebel, A., Aoki, W., Christlieb, N., Ando, H., Asplund, M., Barklem, P. S., Beers, T. C., Eriksson, K., Fechner, C., Fujimoto, M. Y., Honda, S., Kajino, T., Minezaki, T., Nomoto, K., Norris, J. E., Ryan, S. G., Takada-Hidai, M., Tsangarides, S., & Yoshii, Y. 2005, Nature, 434, 871
* Fryer et al. (2007) Fryer, C. L., Hungerford, A. L., & Rockefeller, G. 2007, International Journal of Modern Physics D, 16, 941
* Fryer et al. (2010) Fryer, C. L., Whalen, D. J., & Frey, L. 2010, ArXiv e-prints
* Greif et al. (2010) Greif, T. H., Glover, S. C. O., Bromm, V., & Klessen, R. S. 2010, ApJ, 716, 510
* Greif et al. (2007) Greif, T. H., Johnson, J. L., Bromm, V., & Klessen, R. S. 2007, ApJ, 670, 1
* Hammer et al. (2010) Hammer, N. J., Janka, H., & Müller, E. 2010, ApJ, 714, 1371
* Heger & Woosley (2002) Heger, A. & Woosley, S. E. 2002, ApJ, 567, 532
* Hirschi (2007) Hirschi, R. 2007, A&A, 461, 571
* Iwamoto et al. (2005) Iwamoto, N., Umeda, H., Tominaga, N., Nomoto, K., & Maeda, K. 2005, Science, 309, 451
* Joggerst et al. (2010a) Joggerst, C. C., Almgren, A., Bell, J., Heger, A., Whalen, D., & Woosley, S. E. 2010a, ApJ, 709, 11
* Joggerst et al. (2010b) Joggerst, C. C., Almgren, A., & Woosley, S. E. 2010b, ArXiv e-prints
* Joggerst et al. (2009) Joggerst, C. C., Woosley, S. E., & Heger, A. 2009, ApJ, 693, 1780
* Karlsson et al. (2008) Karlsson, T., Johnson, J. L., & Bromm, V. 2008, ApJ, 679, 6
* Kitayama & Yoshida (2005) Kitayama, T. & Yoshida, N. 2005, ApJ, 630, 675
* Kitayama et al. (2004) Kitayama, T., Yoshida, N., Susa, H., & Umemura, M. 2004, ApJ, 613, 631
* Mackey et al. (2003) Mackey, J., Bromm, V., & Hernquist, L. 2003, ApJ, 586, 1
* Nakamura & Umemura (2001) Nakamura, F. & Umemura, M. 2001, ApJ, 548, 19
* O’Shea & Norman (2007) O’Shea, B. W. & Norman, M. L. 2007, ApJ, 654, 66
* Scannapieco et al. (2005) Scannapieco, E., Madau, P., Woosley, S., Heger, A., & Ferrara, A. 2005, ApJ, 633, 1031
* Smith & Sigurdsson (2007) Smith, B. D. & Sigurdsson, S. 2007, ApJ, 661, L5
* Weaver et al. (1978) Weaver, T. A., Zimmerman, G. B., & Woosley, S. E. 1978, ApJ, 225, 1021
* Whalen et al. (2004) Whalen, D., Abel, T., & Norman, M. L. 2004, ApJ, 610, 14
* Whalen et al. (2008) Whalen, D., van Veelen, B., O’Shea, B. W., & Norman, M. L. 2008, ApJ, 682, 49
* Whalen & Fryer (2010) Whalen, D. J. & Fryer, C. 2010, ArXiv e-prints
* Wise & Abel (2008a) Wise, J. H. & Abel, T. 2008a, ApJ, 684, 1
* Wise & Abel (2008b) —. 2008b, ApJ, 685, 40
* Woosley et al. (2002) Woosley, S. E., Heger, A., & Weaver, T. A. 2002, Reviews of Modern Physics, 74, 1015
|
arxiv-papers
| 2010-10-21T03:59:10 |
2024-09-04T02:49:14.120005
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "C. C. Joggerst and Daniel Whalen",
"submitter": "Candace Joggerst",
"url": "https://arxiv.org/abs/1010.4360"
}
|
1010.4375
|
# An arbitrary Lagrangian-Eulerian formulation for the numerical simulation of
flow patterns generated by the hydromedusa Aequorea victoria
Mehmet SAHIN and Kamran MOHSENI
Department of Aerospace Engineering Sciences,
University of Colorado, Boulder,
Colorado, 80309, USA
###### Abstract
A new geometrically conservative arbitrary Lagrangian-Eulerian (ALE)
formulation is presented for the moving boundary problems in the swirl-free
cylindrical coordinates. The governing equations are multiplied with the
radial distance and integrated over arbitrary moving Lagrangian-Eulerian
quadrilateral elements. Therefore, the continuity and the geometric
conservation equations take very simple form similar to those of the Cartesian
coordinates. The continuity equation is satisfied exactly within each element
and a special attention is given to satisfy the geometric conservation law
(GCL) at the discrete level. The equation of motion of a deforming body is
solved in addition to the Navier-Stokes equations in a fully-coupled form. The
mesh deformation is achieved by solving the linear elasticity equation at each
time level while avoiding remeshing in order to enhance numerical robustness.
The resulting algebraic linear systems are solved using an ILU(k)
preconditioned GMRES method provided by the PETSc library. The present ALE
method is validated for the steady and oscillatory flow around a sphere in a
cylindrical tube and applied to the investigation of the flow patterns around
a free-swimming hydromedusa Aequorea victoria (crystal jellyfish). The
calculations for the hydromedusa indicate the shed of the opposite signed
vortex rings very close to each other and the formation of large induced
velocities along the line of interaction while the ring vortices moving away
from the hydromedusa. In addition, the propulsion efficiency of the free-
swimming hydromedusa is computed and its value is compared with values from
the literature for several other species. The fluid dynamics video presented
here shows the time variation of the instantaneous three-dimensional vorticity
isosurfaces around a free-swimming hydromedusa Aequorea victoria.
## 1 Introduction
The present dynamics video presented here shows the time variation of the
instantaneous three-dimensional vorticity isosurfaces around a free-swimming
hydromedusa Aequorea victoria. To solve the flow pattern around the free-
swimming hydromedusa Aequorea victoria, a new geometrically conservative
arbitrary Lagrangian-Eulerian (ALE) formulation presented in [1] has been
used. The maximum bell radius of the medusa is approximately $2.3$ cm and the
period of one cycle T is being approximately equal to 1.16 seconds. To compute
the velocity of the medusa, the equation of motion is solved in addition to
the Navier Stokes equations in a fully coupled form. The calculations are
carried out on high resolution computational meshes: a coarse mesh M1 with
63,099 vertices and 62,610 quadrilateral elements and a fine mesh M2 with
205,714 vertices and 204,784 quadrilateral elements. The computed average
swimming velocity is computed to be $1.453$ cm/s and $1.462$ cm/s for the
meshes M1 and M2, respectively. Based on the average medusa velocity on mesh
M2 and the maximum bell diameter, the dimensionless parameters Reynolds and
Strouhal numbers are computed to be 672 and 8.47, respectively. The details of
the present work can be found in the papers listed in the references.
## References
* [1] M. Sahin and K. Mohseni, An Arbitrary Lagrangian-Eulerian Formulation for the Numerical Simulation of Flow Patterns Generated by the Hydromedusa Aequorea Victoria. J. Comput. Phys., (2009), 228:4588-4605.
* [2] M. Sahin, K. Mohseni and S. Colin, The Numerical Comparison of Flow Patterns and Propulsive Performances for the Hydromedusae Sarsia Tubulosa and Aequorea Victoria. J. Exp. Biol., (2009), 212:2656-2667.
[a]
[b] Figure 1: The mesh convergence is given for the wake structure behind a
free-swimming hydromedusa Aequorea victoria on meshes M1 [a] and M2 [b].
|
arxiv-papers
| 2010-10-21T05:46:17 |
2024-09-04T02:49:14.129317
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Mehmet Sahin and Kamran Mohseni",
"submitter": "Mehmet Sahin Dr.",
"url": "https://arxiv.org/abs/1010.4375"
}
|
1010.4403
|
arxiv-papers
| 2010-10-21T09:35:07 |
2024-09-04T02:49:14.135375
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yue-Jun Peng and Yong-Fu Yang",
"submitter": "Yongfu Yang",
"url": "https://arxiv.org/abs/1010.4403"
}
|
|
1010.4666
|
11institutetext: Institut für Quanteninformationsverarbeitung Albert-Einstein-
Allee 11 D-89069 Ulm, Germany.
Quantum Information Complex systems Decision theory
# Quantum Game of Life
D. Bleh T. Calarco S. Montangero
###### Abstract
We introduce a quantum version of the Game of Life and we use it to study the
emergence of complexity in a quantum world. We show that the quantum evolution
displays signatures of complex behaviour similar to the classical one, however
a regime exists, where the quantum Game of Life creates more complexity, in
terms of diversity, with respect to the corresponding classical reversible
one.
###### pacs:
03.67.-a
###### pacs:
89.75.-k
###### pacs:
02.50.Le
The Game of Life (GoL) has been proposed by Conway in 1970 as a wonderful
mathematical game which can describe the appearance of complexity and the
evolution of “life” under some simple rules [1]. Since its introduction it has
attracted a lot of attention, as despite its simplicity, it can reveal complex
patterns with unpredictable evolution: From the very beginning a lot of
structures have been identified, from simple blinking patterns to complex
evolving figures which have been named “blinkers”, “gliders” up to
“spaceships” due to their appearance and/or dynamics [2]. The classical GoL
has been the subject of many studies: It has been shown that cellular automata
defined by the GoL have the power of a Universal Turing machine, that is,
anything that can be computed algorithmically can be computed within Conway’s
GoL [3, 4]. Statistical analysis and analytical descriptions of the GoL have
been performed; many generalisations or modifications of the initial game have
been introduced as, for example, a simplified one dimensional version of the
GoL and a semi-quantum version [5, 6, 7]. Finally, to allow a statistical
mechanics description of the GoL, stochastic components have been added [8].
In this letter, we bridge the field of complex systems with quantum mechanics
introducing a purely quantum GoL and we investigate its dynamical properties.
We show that it displays interesting features in common with its classical
counterpart, in particular regarding the variety of supported dynamics and
different behaviour. The system converges to a quasi-stationary configuration
in terms of macroscopic variables, and these stable configurations depend on
the initial state, e.g. the initial density of “alive” sites for random
initial configurations. We show that simple, local rules support complex
behaviour and that the diversity of the structures formed in the steady state
resembles that of the classical GoL, however a regime exists where quantum
dynamics allows more diversity to be created than possibly reached by the
classical one.
Figure 1: Example of the evolution of the GoL described by Hamiltonian (1) for
a simple initial configuration. Empty (blue) squares are “dead” sites,
coloured (red) ones are “alive”.
The universe of the original GoL is an infinite two-dimensional orthogonal
grid of square cells with coordination number eight, each of them in one of
two possible states, alive or dead [1]. At each step in time, the pattern
present on the grid evolves instantaneously following simple rules: any dead
cell with exactly three live neighbours comes to life; any live cell with less
than two or more than three live neighbours dies as if by loneliness or
overcrowding. As already pointed out in [7], the rules of the GoL are
irreversible, thus their generalisation to the quantum case implies rephrasing
them to make them compatible with a quantum reversible evolution. The system
under study is a collection of two-level quantum systems, with two possible
orthogonal states, namely the state “dead” ($|0\rangle$) and “alive”
($|1\rangle$). Clearly, differently from the classical case, a site can be
also in a superposition of the two possible classical states.
Figure 2: Colour on-line. From left to right: Countour plot of the time
evolution of the populations $\langle n_{i}(t)\rangle$ (column 1), visibility
$v_{i}(t)$ (column 2), discretized populations $\mathcal{D}_{i}(t)$ (column 3)
and clustering $\mathcal{C}(\ell,t)$ (column 4) for three different initial
configurations: four alive sites separated by two dead ones (A), twenty-four
alive sites grouped together (B) and a random initial configuration (C). Time
is reported on the x-axis (in arbitrary units), and position (cluster size)
$i=1,\dots,L$ on the y-axis in columns one to three (four). Arrows in panel 3A
highlight the three subsequent generations of a “blinker” reported
schematically in Fig. 3. The colour code goes from zero to $M=1$ ($M=4$ for
the clustering and to $M=.1$ for the visibility), from blue through green to
red.
The dynamics is defined as follows in terms of the GoL language: a site with
two or three neighbouring alive sites is active, where active means that it
will come to life and eventually die on a typical timescale $T$ (setting the
problem timescale, or time between subsequent generations). That is, if
maintained active by the surrounding conditions, the site will complete a full
rotation, if not, it is “frozen” in its state. Stretching the analogy with
Conway’s GoL to the limit, we are describing the evolution of a Virus culture:
each individual undergoes its life cycle if the environment allows it,
otherwise it hibernates in its current state and waits for conditions to
change such that the site may become active again. This slight modification
allows us to recover the reversibility of the dynamics and to introduce a
quantum model that, as we shall see, reproduces most of the interesting
complex behaviour of the classical GoL from the point of view of a classical
observer. However, its evolution is purely quantum and thus we are introducing
a tool that will allow to study the emergence of complexity from the quantum
world.
## 1 Model
The Hamiltonian describing the aforementioned model is given by
$H=\sum_{i=3}^{L-2}(b_{i}+b_{i}^{\dagger})\cdot\left(\mathcal{N}^{3}_{i}+\mathcal{N}^{2}_{i}\right)$
(1)
where $L$ is the number of sites; $b$ and $b^{\dagger}$ are the usual
annihilation and creation operators ($\hbar=1$); the operators
$\mathcal{N}^{2}_{i}=\sum_{P}n_{\alpha}n_{\beta}\bar{n}_{\gamma}\bar{n}_{\delta}$
and
$\mathcal{N}^{3}_{i}=\sum_{P^{\prime}}n_{\alpha}n_{\beta}n_{\gamma}\bar{n}_{\delta}$
($n=b^{\dagger}b$, $\bar{n}=1-n$, the indices $\alpha,\beta,\gamma,\delta$
label the four neighbouring sites) count the population present in the four
neighbouring sites (the sum runs on every possible permutation $P$ and
$P^{\prime}$ of the positions of the $n$ and $\bar{n}$ operators) and
$\mathcal{N}^{2}$ ($\mathcal{N}^{3}$) gives the null operator if the
population is different from two (three), the identity otherwise. For
classical states, as for example an initial random configuration of dead and
alive states, the Hamiltonian (1) is, at time zero,
$H_{Active}=b_{i}+b_{i}^{\dagger}$ on the sites with two or three alive
neighbours and $H_{Hibernate}=0$ otherwise. If the Hamiltonian would remain
constant, every active site would oscillate forever while the hibernated ones
would stand still. On the contrary as soon as the evolution starts, the state
evolves into a superposition of possible classical configurations, resulting
in a complex dynamics as shown below and the interaction between sites starts
to play a role. Thus, the Hamiltonian introduced in Eq. (1) induces a quantum
dynamics that resembles the rules of the GoL: a site with less than two or
more than three alive neighbouring sites ”freezes” while, on the contrary, it
“lives”. The difference with the classical game – connected to the
reversibility of quantum dynamics – is that “living” means oscillating with a
typical timescale between two possible classical states (see e.g. Fig. 1.
Figure 3: Schematic representation of a one-dimensional time-evolution of the
discretized population $\mathcal{D}_{i}(t)$ of a “blinker” (case A of Fig. 2).
From left to right the states of subsequent generations are sketched. Empty
(blue) squares are “dead” sites, coloured (red) ones are “alive”
## 2 Dynamics
To study the quantum GoL dynamics we employ the time dependent Density Matrix
Renormalization group (DMRG). Originally developed to investigate condensed
matter systems, the DMRG and its time dependent extension have been proven to
be a very powerful method to numerically investigate many-body quantum systems
[9, 11, 10, 12]. As it is possible to use it efficiently only in one-
dimensional systems, we concentrate to the one-dimensional version of the
Hamiltonian (1): the operators $\mathcal{N}^{2}$ and $\mathcal{N}^{3}$ count
the populated sites on the nearest-neighbour and next-nearest-neighbour sites
and thus $\alpha=i-2,\beta=i-1,\gamma=i+1,\delta=i+2$. Note that it has been
shown that the main statistical properties of the classical GoL are the same
in both two- and one-dimensional versions [6].
To describe the system dynamics we introduce different quantities that
characterise in some detail the system evolution. We first concentrate on the
population dynamics, measuring the expectation values of the number operator
at every site $\langle n_{i}(t)\rangle$. This clearly gives a picture of the
“alive” and “dead” sites as a function of time, as it gives the probability of
finding a site in a given state when measured. That is, if we observe the
system at some final time $T_{f}$ we will find dead or alive sites according
to these probabilities. In Fig. 2 we show three typical evolutions (leftmost
pictures): configuration $A$ corresponds to a “blinker” where two couples of
nearest-neighbour sites oscillate regularly between dead and alive states (a
schematic representation of the resulting dynamics of the discretised
population $\mathcal{D}_{i}(t)$ is reproduced also in Fig. 3); configuration
$B$ is a typical overcrowded scenario where twenty-four “alive” sites
disappear leaving behind only some residual activity; finally a typical
initial random configuration ($C$) is shown. Notice that in all configurations
it is possible to identify the behaviour of the wave function tails that
propagate and generate interference effects. These effects can be highlighted
by computing the visibility of the dynamics, the maximum variation of the
populations within subsequent generations, defined as:
$v_{i}(t)=|\max_{t^{\prime}}n_{i}(t^{\prime})-\min_{t^{\prime}}n_{i}(t^{\prime})|;t^{\prime}\in[t-\frac{T}{2};t+\frac{T}{2}];$
(2)
that is, the visibility at time $t$ reports the maximum variation of the
population in the time interval of length $T$ centered around $t$. The
visibility clearly follows the preceding dynamics (see Fig. 2, second column)
and identifies the presence of “activity” in every site.
Figure 4: Left: Average population $\rho(t)$ (upper) and diversity $\Delta(t)$
(lower) as a function of time for different initial population density
$\rho_{0}$. Right: Equilibrium average population $\rho$ (upper) and diversity
$\Delta$ (lower) for the quantum (blue squares) and classical (red circles)
GoL as a function of the initial population density $\rho_{0}$. Simulations
are performed with a t-DMRG at third order, Trotter step $\delta t=10^{-2}$,
truncation dimension $m=30$, size $L=32$, averaged over up to thirty different
initial configurations.
To stress the connections and comparisons with the original GoL we introduce a
classical figure of merit (shown in the third column of Fig. 2): we report a
discretized version of the populations as a function of time
($\mathcal{D}_{i}(t)=1$ for $n_{i}(t)>0.5$ and $\mathcal{D}_{i}(t)=0$
otherwise). Notice that $\mathcal{D}_{i}(t)$ gives the most probable
configuration of the system after a measurement on every site in the basis
$\\{|0\rangle,|1\rangle\\}$. Thus, we recover a “classical” view of the
quantum GoL with the usual definition of site status. For example,
configuration $A$ is a “blinker” that changes status at every generation (see
Fig.2 and 3). More complex configurations appear in the other two cases. The
introduction of the discretized populations $\mathcal{D}_{i}$ can also be
viewed as a new definition of “alive” and “dead” sites from which we could
have started from the very beginning to introduce a stochastic component as
done in [8]. This quantity allows analysis to be performed as usually done on
the classical GoL and to stress the similarities between the quantum and the
classical GoL. Following the literature to quantify such complexity, we
compute the clustering function $\mathcal{C}(\ell,t)$ that gives the number of
clusters of neighbouring “alive” sites of size $\ell$ as a function of time
[6]. For example, the function $\mathcal{C}(\ell,t)$ for a uniform
distribution of “alive” sites would be simply $\mathcal{C}(L)=1$ and zero
otherwise while a random pattern would result in a random cluster function.
This function characterises the complexity of the evolving patterns, e.g. it
is oscillating between zero- and two-size clusters for the initial condition
$A$, while it is much more complex for the random configuration $C$ (see Fig.
2, rightmost column).
## 3 Statistics
To characterise the statistical properties of the quantum GoL we study the
time evolution of different initial random configurations as a function of the
initial density of alive sites. We concentrate on two macroscopic quantities:
the density of the sites that if measured would with higher probability result
in “alive” states
$\rho(t)=\sum_{i}\mathcal{D}_{i}(t)/L;$ (3)
and the diversity
$\Delta(t)=\sum_{\ell}\mathcal{C}(\ell,t),$ (4)
the number of different cluster sizes that are present in the systems, that
quantifies the complexity of the generated dynamics [6, 8]. Typical results,
averaged over different initial configurations, are shown in Fig. 4 (left). As
it can be clearly seen the system equilibrates and the density of states as
well as the diversity reach a steady value. This resembles the typical
behaviour of the classical GoL where any typical initial random configuration
eventually equilibrates to a stable configuration. Moreover, we compare the
quantum GoL with a classical reversible version of GoL corresponding to that
introduced here: at every step a cell changes its status if and only if within
the first four neighbouring cells only three or two are alive. Notice that,
the evolution being unitary and thus reversible, the equilibrium state locally
changes with time, however the macroscopic quantities reach their equilibrium
values that depend non trivially only on the initial population density. In
fact, for the classical game, we were able to check that the final population
density is independent of the system size while the final diversity scales as
$L^{1/2}$ (up to $2^{10}$ sites, data not shown). Moreover, the time needed to
reach equilibrium is almost independent of the system size and initial
population density. These results on the scaling of classical system
properties support the conjecture that our findings for the quantum case will
hold in general, while performing the analysis for bigger system sizes is
highly demanding. A detailed analysis of the size scaling of the system
properties will be presented elsewhere. In Fig. 4 we report the final
(equilibrium) population density (right upper) and diversity (right lower) as
a function of the initial population density for both the classical and the
quantum GoL for systems of $L=32$ cells. The equilibrium population density
$\rho$ is a non linear function of the initial one $\rho_{0}$ in both cases:
the classical one has an initial linear dependence up to half-filling where a
plateau is present up to the final convergence to unit filling for
$\rho_{0}=1$. Indeed, the all-populated configuration is a stable system
configuration. The quantum GoL follows a similar behaviour, with a more
complex pattern. Notice that here a first signature of quantum behaviour is
present: the steady population density reached by the quantum GoL is always
smaller than its classical counterpart. This is probably due to the fact that
the evolution is not completely captured by this classical quantity: the sites
with population below half filling, i.e. the tails of the wave functions, are
described as unpopulated by $\mathcal{D}_{i}$. However, this missing
population plays a role in the evolution: within the overall superposition of
basis states, a part of the probability density (corresponding to the states
where the sites are populated) undergoes a different evolution than the
classical one. In general, the quantum system is effectively more populated
than the classical $\rho$ indicates. This difference in the quantum and
classical dynamics is even more evident in the dependence of the equilibrium
diversity on the initial population density $\rho_{0}$. In the classical case
the maximum diversity is slightly above three: on average, in the steady
state, there are no more than about three different cluster sizes present in
the system independently of the initial configuration. On the contrary –in the
quantum case– the maximal diversity is about four, increasing the information
content (the complexity) generated by the evolution by about $10-20\%$.
These findings are a signature of the difference between quantum and classical
GoL. In particular we have shown that the quantum GoL has a higher capacity of
generating diversity than the corresponding classical one. This property
arises from the possibility of having quantum superpositions of states of
single sites. Whether purely quantum correlations (entanglement) play a
crucial role is under investigation. Similarly, as there is some arbitrariness
in our definition of the quantum GoL, the investigation of possible variations
is left for future work.
The investigation presented here fits perfectly as a subject of study for
quantum simulators, like for example cold atoms in optical lattices. Indeed,
the five-body Hamiltonian (1) can be written in pseudo spin-one-half operators
(Pauli matrices) and thus it can be simulated along the lines presented in
[13]. In particular, these simulations would give access to investigations in
two and three dimensions that are not feasible by means of t-DMRG [10].
In conclusion we note that this is one of the few available simulations of a
many-body quantum game scalable in the number of sites [14, 15, 16]. With a
straightforward generalisation (adding more than one possible strategy defined
in Eq. (1)) one could study also different many-player quantum games. This
approach will allow different issues to be studied related to many-player
quantum games such as the appearance of new equilibria and their
thermodynamical properties. Moreover, the approach introduced here shows that
one might investigate many different aspects of many-body quantum systems with
the tools developed in the field of complexity and dynamical systems: In
particular, the relations with Hamiltonian quantum cellular automata in one
dimension and quantum games [14, 17]. Finally, the search for the possible
existence of self-organised criticality in these systems along the lines of
similar investigations in the classical GoL [18], if successful, would be the
first manifestation of such effect in a quantum system and might have
intriguing implications in quantum gravity [19, 20].
After completing this work we became aware of another work on the same subject
[21].
We acknowledge interesting discussions and support by R. Fazio and M.B.
Plenio, the SFB-TRR21, the EU-funded projects AQUTE, PICC for funding, the BW-
Grid for computational resources, and the PwP project for the t-DMRG code
(www.dmrg.it).
## References
* [1] M. Gardner, Sci. Am. 223, 120 (1970).
* [2] “Winning ways for your mathematical plays” J. Conway et.al., A K Peters/CRC Press (1982).
* [3] ”Game of Life cellular Automata”, Andrew Adamatzky (ed.), Springer 2010.
* [4] “Collision-Based Computing”, A. Adamatzky (ed.), Springer (2002).
* [5] M. Dresden and D. Wong, Proc. Nat. Acad. Sci. USA 72, 956 (1975).
* [6] T.R.M. Sales, J.Phys. A 26, 6187 (1993).
* [7] “Quantum aspects of life”, A.P. Flitney and D. Abbot (ed.), Imperial College Press (2008).
* [8] L.S. Schulman and P.E. Seiden, J. Stat. Phys. 19, 293 (1978).
* [9] S.R. White and A.E. Feiguin, Phys. Rev. Lett. 93, 076401 (2004).
* [10] U. Schollwöck Rev. Mod. Phys. 77, 259 (2005); K. Hallberg Adv. Phys. 55, 477 (2006).
* [11] A.J. Daley, C. Kollath, U. Schollwöck and G. Vidal, J. Stat. Mech.: Theor. Exp. P04005 (2004).
* [12] G. De Chiara, M. Rizzi, D. Rossini, and S. Montangero, J. Comput. Theor. Nanosc. 5, 1277 (2008).
* [13] M. Lewenstein et. al., Adv. Phys. 56, 243 (2007); E. Jané et. al., Quantum Inf. Comput. 3, 15 (2003); J.J. Garcia-Ripoll, A.M. Martin-Delgado, and J.I. Cirac, Phys. Rev. Lett. 93, 250405 (2004); L.M. Duan, E. Demler, and M.D. Lukin, Phys. Rev. Lett. 91, 090402 (2003).
* [14] J. Eisert, M. Wilkens, and M. Lewenstein, Phys. Rev. Lett. 83, 3077 (1999).
* [15] S.C. Benjamin and P.M. Hayden, Phys. Rev. A 64, 030301 (2001).
* [16] Q. Chen, Y.Wang, J-T Liu, and K-L Wang, Phys. Lett. A 327, 98 (2004).
* [17] D. Nagaj and P. Wocjan, Phys. Rev. A 78, 032311 (2008).
* [18] P. Bak, C. Tang, and K. Wiesenfeld, Phys. Rev. Lett. 59, 381 (1987). P. Bak, K. Chen, and M. Creutz, Nature 342, 780 (1989). C. Bennet and M.S. Bowutschky, Nature 350, 468 (1991).
* [19] M.H. Ansari and L. Smolin, Class. Quantum Grav. 25, 095016 (2008).
* [20] R. Borissov and S. Gupta, Phys. Rev. D 60, 024002 (1999).
* [21] P. Arrighi and J. Grattage, Proceedings of JAC 2010 - Journées Automates Cellulaires 2010, Finland (2010).
|
arxiv-papers
| 2010-10-22T10:40:46 |
2024-09-04T02:49:14.145634
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "D. Bleh, T. Calarco, S. Montangero",
"submitter": "Simone Montangero",
"url": "https://arxiv.org/abs/1010.4666"
}
|
1010.4744
|
# Optimal Variational Principle for Backward Stochastic Control Systems
Associated with Lévy Processes ††thanks: This work is partially supported by
the National Basic Research Program of China (973 Program) (Grant
No.2007CB814904), the National Natural Science Foundation of China (Grants
No.10325101, 11071069), the Specialized Research Fund for the Doctoral Program
of Higher Education of China (Grant No.20090071120002) and the Innovation Team
Foundation of the Department of Education of Zhejiang Province (Grant
No.T200924).
Maoning Tanga Qi Zhangb
aDepartment of Mathematical Sciences, Huzhou University, Zhejiang 313000,
China
Email: tmorning@hutc.zj.cn
bSchool of Mathematical Sciences, Fudan University, Shanghai 200433, China
Email: qzh@fudan.edu.cn
###### Abstract
The paper is concerned with optimal control of backward stochastic
differential equation (BSDE) driven by Teugel’s martingales and an independent
multi-dimensional Brownian motion, where Teugel’s martingales are a family of
pairwise strongly orthonormal martingales associated with Lévy processes (see
Nualart and Schoutens [14]). We derive the necessary and sufficient conditions
for the existence of the optimal control by means of convex variation methods
and duality techniques. As an application, the optimal control problem of
linear backward stochastic differential equation with a quadratic cost
criteria (called backward linear-quadratic problem, or BLQ problem for short)
is discussed and characterized by stochastic Hamilton system.
Keywords: stochastic control, stochastic maximum principle, Lévy processes,
Teugel’s martingales, backward stochastic differential equations
## 1 Introduction
It is well known that the maximum principle for a stochastic optimal control
problem involves the so-called adjoint processes which solve the corresponding
adjoint equation. In fact, the adjoint equation is in general a linear
backward stochastic differential equation (BSDE) with a specified a random
terminal condition on the state. Unlike a forward stochastic differential
equation, the solution of a BSDE is a pair of adapted solutions. Thus, in
order to obtain the maximum principle, we need first obtain the existence and
uniqueness theorem for the pair of adapted solutions of adjoint equation.
The linear BSDE was first proposed by Bismut [4] in 1973. This research field
developed fast after the pioneer work of Pardoux and Peng [16] in 1990 got the
existence and uniqueness theorem for the solution of nonlinear BSDE driven by
Brownian motion under Lipschitz condition. Now BSDE theory has been playing a
key role not only in dealing with stochastic optimal control problems, but in
mathematical finance, particularly in hedging and nonlinear pricing theory for
imperfect market (see e.g. [7]).
As for BSDE driven by the non-continuous martingale, Tang and Li [20] first
discussed the existence and uniqueness theorem of the solution of BSDE driven
by Poisson point process and consequently proved the maximum principle for
optimal control of stochastic systems with random jumps. In 2000, Nualart and
Schoutens [14] got a martingale representation theorem for a type of Lévy
processes through Teugel’s martingales, where Teugel’s martingales are a
family of pairwise strongly orthonormal martingales associated with Lévy
processes. Later, they proved in [15] the existence and uniqueness theory of
BSDE driven by Teugel’s martingales. The above results are further extended to
the one-dimensional BSDE driven by Teugel’s martingales and an independent
multi-dimensional Brownian motion by Bahlali et al [1]. One can refer to [8,
9, 17, 18] for more results on such kind of BSDEs.
In the mean time, the stochastic optimal control problems related to Teugel’s
martingales were studied. In 2008, a stochastic linear-quadratic problem with
Lévy processes was considered by Mitsui and Tabata [13], in which they
established the closeness property of multi-dimensional backward stochastic
Riccati differential equation(BSRDE) with Teugel’s martingales and proved the
existence and uniqueness of solution to such kind of one-dimensional BSRDE,
moreover, in their paper an application of BSDE to a financial problem with
full and partial observations was demonstrated. Motivated by [13], Meng and
Tang [12] studied the general stochastic optimal control problem for the
forward stochastic systems driven by Teugel’s martingales and an independent
multi-dimensional Brownian motion, of which the necessary and sufficient
optimality conditions in the form of stochastic maximum principle with the
convex control domain are obtained.
However, [12] and [13] are only concerned with the optimal control problem of
the forward controlled stochastic system. Since a BSDE is a well-defined
dynamic system itself and has important applications in mathematical finance,
it is necessary and natural to consider the optimal control problem of BSDE.
Actually, there has been much literature on BSDE control system driven by
Brownian motion (see e.g. [2, 3, 5, 11, 10]). But to our best knowledge, there
is no discussion on the optimal control problem of BSDE driven by Teugel
martingales and an independent Brownian motion, which motives us to write this
paper.
In this paper, by means of convex variation methods and duality techniques, we
will give the necessary and sufficient conditions for the existence of the
optimal control for BSDE system driven by Teugel martingales and an
independent multi-dimensional Brownian motion. As an application, the optimal
control for linear backward stochastic differential equation with a quadratic
cost criteria or called backward linear-quadratic (BLQ) problem is discussed
in details. The optimal control of BLQ problem will be characterized by
stochastic Hamilton systems. In this case, the stochastic Hamilton system is a
linear forward-backward stochastic differential equation driven by Teugel’s
martingales and an independent multi-dimensional Brownian motion, consisting
of the state equation, the adjoint equation and the dual presentation of the
optimal control.
The rest of this paper is organized as follows. In section 2, we introduce
useful notation and some existing results on stochastic differential equations
(SDEs) and BSDEs driven by Teugel’s martingales. In section 3, we state the
optimal control problem we study, give needed assumptions and prove some
preliminary results on variational equation and variational inequality. In
section 4, we prove the necessary and sufficient optimality conditions for the
optimal control problem put forward in section 3. As an application, the
optimal control for BLQ problem is discussed in section 5.
## 2 Notation and preliminaries
Let $(\Omega,\mathscr{F},\\{\mathscr{F}_{t}\\}_{0\leq t\leq T},P)$ be a
complete probability space. The filtration $\\{\mathscr{F}_{t}\\}_{0\leq t\leq
T}$ is right-continuous and generated by a $d$-dimensional standard Brownian
motion $\\{W(t),0\leq t\leq T\\}$ and a one-dimensional Lévy process
$\\{L(t),0\leq t\leq T\\}$. It is known that $L(t)$ has a characteristic
function of the form
$Ee^{i\theta L(t)}=\exp\bigg{[}ia\theta t-{1\over
2}\sigma^{2}\theta^{2}t+t\int_{\mathbb{R}^{1}}(e^{i\theta x}-1-i\theta
xI_{\\{|x|<1\\}})v(dx)\bigg{]},$
where $a\in\mathbb{R}^{1}$, $\sigma>0$ and $v$ is a measure on
$\mathbb{R}^{1}$ satisfying (i)$\displaystyle\int_{0}^{T}(1\wedge
x^{2})v(dx)<\infty$ and (ii) there exists $\varepsilon>0$ and $\lambda>0$,
s.t.
$\displaystyle\int_{\\{-\varepsilon,\varepsilon\\}^{c}}e^{\lambda|x|}v(dx)<\infty$.
These settings imply that the random variables $L(t)$ have moments of all
orders. Denote by $\mathscr{P}$ the predictable sub-$\sigma$ field of
$\mathscr{B}([0,T])\times\mathscr{F}$, then we introduce the following
notation used throughout this paper.
$\bullet$ $H$: a Hilbert space with norm $\|\cdot\|_{H}$.
$\bullet$ $\langle\alpha,\beta\rangle:$ the inner product in
$\mathbb{R}^{n},\forall\alpha,\beta\in\mathbb{R}^{n}.$
$\bullet$ $|\alpha|=\sqrt{\langle\alpha,\alpha\rangle}:$ the norm of
$\mathbb{R}^{n},\forall\alpha\in\mathbb{R}^{n}.$
$\bullet$ $\langle A,B\rangle=tr(AB^{T}):$ the inner product in
$\mathbb{R}^{n\times m},\forall A,B\in\mathbb{R}^{n\times m}.$
$\bullet$ $|A|=\sqrt{tr(AA^{T})}:$ the norm of $\mathbb{R}^{n\times m},\forall
A\in\mathbb{R}^{n\times m}$.
$\bullet$ $l^{2}$: the space of all real-valued sequences $x=(x_{n})_{n\geq
0}$ satisfying
$\|x\|_{l^{2}}\triangleq\sqrt{\displaystyle\sum_{i=1}^{\infty}x_{i}^{2}}<+\infty.$
$\bullet$ $l^{2}(H):$ the space of all H-valued sequence $f=\\{f^{i}\\}_{i\geq
1}$ satisfying
$\|f\|_{l^{2}(H)}\triangleq\sqrt{\displaystyle\sum_{i=1}^{\infty}||f^{i}||_{H}^{2}}<+\infty.$
$\bullet$ $l_{\mathscr{F}}^{2}(0,T,H):$ the space of all $l^{2}(H)$-valued and
${\mathscr{F}}_{t}$-predictable processes $f=\\{f^{i}(t,\omega),\
(t,\omega)\in[0,T]\times\Omega\\}_{i\geq 1}$ satisfying
$\|f\|_{l_{\mathscr{F}}^{2}(0,T,H)}\triangleq\sqrt{E\displaystyle\int_{0}^{T}\sum_{i=1}^{\infty}||f^{i}(t)||_{H}^{2}dt}<\infty.$
$\bullet$ $M_{\mathscr{F}}^{2}(0,T;H):$ the space of all $H$-valued and
${\mathscr{F}}_{t}$-adapted processes $f=\\{f(t,\omega),\
(t,\omega)\in[0,T]\times\Omega\\}$ satisfying
$\|f\|_{M_{\mathscr{F}}^{2}(0,T;H)}\triangleq\sqrt{E\displaystyle\int_{0}^{T}\|f(t)\|_{H}^{2}dt}<\infty.$
$\bullet$ $S_{\mathscr{F}}^{2}(0,T;H):$ the space of all $H$-valued and
${\mathscr{F}}_{t}$-adapted càdlàg processes $f=\\{f(t,\omega),\
(t,\omega)\in[0,T]\times\Omega\\}$ satisfying
$\|f\|_{S_{\mathscr{F}}^{2}(0,T;H)}\triangleq\sqrt{E\displaystyle\sup_{0\leq
t\leq T}\|f(t)\|_{H}^{2}dt}<+\infty.$
$\bullet$ $L^{2}(\Omega,{\mathscr{F}},P;H):$ the space of all $H$-valued
random variables $\xi$ on $(\Omega,{\mathscr{F}},P)$ satisfying
$\|\xi\|_{L^{2}(\Omega,{\mathscr{F}},P;H)}\triangleq E\|\xi\|_{H}^{2}<\infty.$
We denote by $\\{H^{i}(t),0\leq t\leq T\\}_{i=1}^{\infty}$ the Teugel’s
martingales associated with the Lévy process $\\{L(t),0\leq t\leq T\\}$.
$H^{i}(t)$ is given by
$H^{i}(t)=c_{i,i}Y^{(i)}(t)+c_{i,i-1}Y^{(i-1)}(t)+\cdots+c_{i,1}Y^{(1)}(t),$
where $Y^{(i)}(t)=L^{(i)}(t)-E[L^{(i)}(t)]$ for all $i\geq 1$, $L^{(i)}(t)$
are so called power-jump processes with $L^{(1)}(t)=L(t)$,
$L^{(i)}(t)=\displaystyle\sum_{0<s\leq t}(\Delta L(s))^{i}$ for $i\geq 2$ and
the coefficients $c_{ij}$ correspond to the orthonormalization of polynomials
$1,x,x^{2},\cdots$ w.r.t. the measure
$\mu(dx)=x^{2}v(dx)+\sigma^{2}\delta_{0}(dx)$. The Teugel’s martingales
$\\{H^{i}(t)\\}_{i=1}^{\infty}$ are pathwise strongly orthogonal and their
predictable quadratic variation processes are given by
$\langle H^{(i)}(t),H^{(j)}(t)\rangle=\delta_{ij}t$
For more details of Teugel’s martingales, we invite the reader to consult
Nualart and Schoutens [14, 15].
In what follows, we will state some basic results on SDE and BSDE driven by
Teugel’s martingales $\\{H^{i}(t),0\leq t\leq T\\}_{i=1}^{\infty}$ and the
$d$-dimensional Brownian motion $\\{W(t),0\leq t\leq T\\}.$
Consider SDE:
$\begin{array}[]{ll}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}X(t)=&a+\displaystyle\int_{0}^{t}b(s,X(s))ds+\sum_{i=1}^{d}\int_{0}^{t}g^{i}(s,X(s))dW^{i}(s)\\\
&+\displaystyle\sum_{i=1}^{\infty}\int_{0}^{t}\sigma^{i}(s,X(s-))dH^{i}(s),\ \
t\in[0,T],\end{array}$ (2.1)
where $(a,b,g,\sigma)$ are given mappings satisfying the assumptions below.
###### Assumption 2.1.
Random variable $a$ is ${\mathscr{F}}_{0}$-measurable and $(b,g,\sigma)$ are
three random mappings
$b:[0,T]\times\Omega\times\mathbb{R}^{n}\longrightarrow\mathbb{R}^{n},$
$g\equiv(g^{1},g^{2},\cdots,g^{d}):[0,T]\times\Omega\times\mathbb{R}^{n}\longrightarrow\mathbb{R}^{n\times
d},$
$\sigma\equiv{(\sigma^{i})}_{i=1}^{\infty}:[0,T]\times\Omega\times\mathbb{R}^{n}\longrightarrow
l^{2}(\mathbb{R}^{n})$
satisfying
(i) $b,g$ and $\sigma$ are
${\mathscr{P}}\bigotimes{\mathscr{B}}(\mathbb{R}^{n})$ measurable with
$b(\cdot,0)\in M_{\mathscr{F}}^{2}(0,T;\mathbb{R}^{n})$, $g(\cdot,0)\in
M_{\mathscr{F}}^{2}(0,T;\mathbb{R}^{n\times d})$ and $\sigma(\cdot,0)\in
l_{\mathscr{F}}^{2}(0,T;\mathbb{R}^{n}).$
(ii) $b,g$ and $\sigma$ are uniformly Lipschitz continuous w.r.t. $x$, i.e.
there exists a constant $C>0$ s.t. for all
$(t,x,\bar{x})\in[0,T]\times\mathbb{R}^{n}\times\mathbb{R}^{n}$ and a.s.
$\omega\in\Omega$,
$\begin{array}[]{ll}|b(t,x)-b(t,\bar{x})|+|g(t,x)-g(t,\bar{x})|+||\sigma(t,x)-\sigma(t,\bar{x})||_{l^{2}(\mathbb{R}^{n})}\leq
C|x-\bar{x}|.\end{array}$
###### Lemma 2.1 ([19], Existence and Uniqueness Theorem of SDE).
If coefficients $(a,b,g,\sigma)$ satisfy Assumption 2.1, then SDE (2.1) has a
unique solution $x(\cdot)\in S_{\mathscr{F}}^{2}(0,T;\mathbb{R}^{n})$.
###### Lemma 2.2 ([12], Continuous Dependence Theorem of SDE).
Assume coefficients $(a,b,g,\sigma)$ and
$(\bar{a},\bar{b},\bar{g},\bar{\sigma})$ satisfy Assumption 2.1. If $x(\cdot)$
and $\bar{x}(\cdot)$ are the solutions to SDE (2.1) corresponding to
$(a,b,g,\sigma)$ and $(\bar{a},\bar{b},\bar{g},\bar{\sigma})$, respectively,
then we have
$\begin{array}[]{ll}~{}E\displaystyle\sup_{0\leq t\leq
T}|x(t)-\bar{x}(t)|^{2}\leq&K\bigg{[}|a-\bar{a}|^{2}+E\displaystyle\int_{0}^{T}|b(t,\bar{x}(t))-\bar{b}(t,\bar{x}(t))|^{2}dt\\\
&+E\displaystyle\int_{0}^{T}|g(t,\bar{x}(t))-\bar{g}(t,\bar{x}(t))|^{2}dt\\\
&+E\displaystyle\int_{0}^{T}||\sigma(t,\bar{x}(t))-\bar{\sigma}(t,\bar{x}(t))||_{l^{2}(\mathbb{R}^{n})}^{2}dt\bigg{]},\end{array}$
where $K$ is a positive constant depending only on $T$ and the Lipschitz
constant $C$.
In particular, for $(\bar{a},\bar{b},\bar{g},\bar{\sigma})=(0,0,0,0),$ we have
$\begin{array}[]{ll}&E\displaystyle\sup_{0\leq t\leq T}|x(t)|^{2}\\\
\leq&K\bigg{[}|a|^{2}+E\displaystyle\int_{0}^{T}|b(t,0)|^{2}dt+E\displaystyle\int_{0}^{T}|g(t,0)|^{2}dt+E\displaystyle\int_{0}^{T}||\sigma(t,0)||_{l^{2}(\mathbb{R}^{n})}^{2}dt\bigg{]}<+\infty.\end{array}$
Now we consider BSDE:
$\displaystyle\begin{split}y(t)=&\xi+\displaystyle\int_{t}^{T}f(s,y{(s)},q(s),z(s))ds-\displaystyle\sum_{i=1}^{d}\int_{t}^{T}q^{i}(s)dW^{i}(s)\\\
&-\displaystyle\sum_{i=1}^{\infty}\int_{t}^{T}z^{i}(s)dH^{i}(s),\ \
t\in[0,T],\end{split}$ (2.2)
where coefficients $(\xi,f)$ are given mappings satisfying the assumptions
below.
###### Assumption 2.2.
The terminal value $\xi\in L^{2}(\Omega,{\mathscr{F}}_{T},P;\mathbb{R}^{n})$
and $f$ is a random mapping
$f:[0,T]\times\Omega\times\mathbb{R}^{n}\times\mathbb{R}^{n\times d}\times
l^{2}(\mathbb{R}^{n})\longrightarrow\mathbb{R}^{n}$
satisfying
(i) $f$ is
${\mathscr{P}}\bigotimes{\mathscr{B}}(\mathbb{R}^{n})\bigotimes{\mathscr{B}}(\mathbb{R}^{n\times
d})\bigotimes{\mathscr{B}}(l^{2}(\mathbb{R}^{n}))$ measurable with
$f(\cdot,0,0,0)\in M_{\mathscr{F}}^{2}(0,T;\mathbb{R}^{n})$.
(ii) $f$ is uniformly Lipschitz continuous w.r.t. $(y,q,z)$, i.e. there exists
a constant $C>0$ s.t. for all
$(t,y,q,z,\bar{y},\bar{q},\bar{z})\in[0,T]\times\mathbb{R}^{n}\times\mathbb{R}^{n\times
d}\times l^{2}(\mathbb{R}^{n})\times\mathbb{R}^{n}\times\mathbb{R}^{n\times
d}\times l^{2}(\mathbb{R}^{n})$ and a.s. $\omega\in\Omega$,
$\begin{array}[]{ll}&|f(t,y,q,z)-f(t,\bar{y},\bar{q},\bar{z})|\leq
C\bigg{[}|y-\bar{y}|+|q-\bar{q}|+\|z-\bar{z}\|_{l^{2}({\mathbb{R}^{n}})}\bigg{]}.\end{array}$
###### Lemma 2.3 ([1], Existence and Uniqueness of BSDE).
If coefficients $(\xi,f)$ satisfy Assumption 2.2, then BSDE (2.2) has a unique
solution
$(y(\cdot),q(\cdot),z(\cdot))\in S_{\mathscr{F}}^{2}(0,T;\mathbb{R}^{n})\times
M_{\mathscr{F}}^{2}(0,T;\mathbb{R}^{n\times d})\times
l_{\mathscr{F}}^{2}(0,T;\mathbb{R}^{n}).$
###### Lemma 2.4 ([1], Continuous Dependence Theorem of BSDE).
Assume that coefficients $(\xi,f)$ and $(\bar{\xi},\bar{f})$ satisfy
Assumption 2.2. If $(y(\cdot),q(\cdot),z(\cdot))$ and
$(\bar{y}(\cdot),\bar{q}(\cdot),\bar{z}(\cdot))$ are the solutions to BSDE
(2.2) corresponding to $(\xi,f)$ and $(\bar{\xi},\bar{f})$, respectively, then
we have
$\begin{array}[]{ll}{}&E\displaystyle\sup_{0\leq t\leq
T}|y(t)-\bar{y}(t)|^{2}+E\int_{0}^{T}|q(t)-\bar{q}(t)|^{2}dt+E\int_{0}^{T}||z(t)-\bar{z}(t)||^{2}_{l^{2}({\mathbb{R}^{n}})}dt\\\
\leq&K\bigg{[}E|\xi-\bar{\xi}|^{2}+E\displaystyle\int_{0}^{T}|f(t,\bar{y}{(t)},\bar{q}(t),\bar{z}(t))-\bar{f}(t,\bar{y}{(t)},\bar{q}(t),\bar{z}(t))|^{2}dt\bigg{]},\end{array}$
where $K$ is a positive constant depending only on $T$ and the Lipschitz
constant $C$.
In particular, if $(\bar{\xi},\bar{f})=(0,0)$, we have
$\begin{array}[]{ll}&E\displaystyle\sup_{0\leq t\leq
T}|y(t)|^{2}+E\int_{0}^{T}|q(t)|^{2}dt+E\int_{0}^{T}||z(t)||^{2}_{l^{2}({\mathbb{R}^{n}})}dt\\\
\leq&K\bigg{[}E|\xi|^{2}+E\displaystyle\int_{0}^{T}|f(t,0,0,0)|^{2}dt\bigg{]}.\end{array}$
(2.3)
In view of Assumptions 2.1-2.2, Lemmas 2.1-2.4 follow from an application of
Itô’s formula, Gronwall’s inequality and Burkholder-Davis-Gundy inequality.
One can refer to [1], [12] and [19] for details.
## 3 Formulation of the problem and preliminary lemmas
Let the admissible control set $U$ be a nonempty convex subset of
$\mathbb{R}^{m}$. An admissible control process $u(\cdot)$ is defined as a
${\mathscr{F}}_{t}$-predictable process with values in $U$ s.t.
$E\displaystyle\int_{0}^{T}|u(t)|^{2}dt<+\infty$. We denote by ${\mathcal{A}}$
the set including all admissible control processes.
For any given admissible control $u(\cdot)\in{\mathcal{A}}$, we consider the
following controlled nonlinear BSDE driven by multi-dimensional Brownian
motion $W$ and Teugel’s martingales $\\{H^{i}\\}_{i=1}^{\infty}$:
$\displaystyle\begin{split}y(t)=&\xi+\displaystyle\int_{t}^{T}f(s,y{(s)},q(s),z(s),u(s))ds\\\
&-\displaystyle\sum_{i=1}^{d}\int_{t}^{T}q^{i}(s)dW^{i}(s)-\displaystyle\sum_{i=1}^{\infty}\int_{t}^{T}z^{i}(s)dH^{i}(s),\
\ t\in[0,T]\end{split}$ (3.1)
with the cost functional
$J(u(\cdot))=E\displaystyle\bigg{[}\int_{0}^{T}l(t,y(t),q(t),z(t),u(t))dt+\Phi(y(0))\bigg{]},$
(3.2)
where
$\xi:\Omega\longrightarrow\mathbb{R}^{n},$
$f:[0,T]\times\Omega\times\mathbb{R}^{n}\times\mathbb{R}^{n\times d}\times
l^{2}(\mathbb{R}^{n})\times U\longrightarrow\mathbb{R}^{n},$
$l:[0,T]\times\Omega\times\mathbb{R}^{n}\times\mathbb{R}^{n\times d}\times
l^{2}(\mathbb{R}^{n})\times U\longrightarrow\mathbb{R}^{1}$
and
$\phi:\Omega\times\mathbb{R}^{n}\longrightarrow\mathbb{R}^{1}$
are given coefficients.
Throughout this paper, we introduce the following basic assumptions on
coefficients $(\xi,f,l,\phi)$.
###### Assumption 3.1.
The terminal value $\xi\in L^{2}(\Omega,{\mathscr{F}}_{T},P;\mathbb{R}^{n})$
and the random mapping $f$ is
${\mathscr{P}}\bigotimes{\mathscr{B}}(\mathbb{R}^{n})\bigotimes{\mathscr{B}}(\mathbb{R}^{n\times
d})\bigotimes{\mathscr{B}}(l^{2}(\mathbb{R}^{n}))\bigotimes{\mathscr{B}}(U)$
measurable with $f(\cdot,0,0,0,0)\in M^{2}(0,T;\mathbb{R}^{n})$. For almost
all $(t,\omega)\in[0,T]\times\Omega$, $f(t,\omega,y,p,z,u)$ is Fréchet
differentiable w.r.t. $(y,p,z,u)$ and the corresponding Fréchet derivatives
$f_{y},f_{p},f_{z},f_{u}$ are continuous and uniformly bounded.
###### Assumption 3.2.
The random mapping $l$ is
${\mathscr{P}}\bigotimes{\mathscr{B}}(\mathbb{R}^{n})\bigotimes{\mathscr{B}}(\mathbb{R}^{n\times
d})\bigotimes{\mathscr{B}}(l^{2}(\mathbb{R}^{n}))\bigotimes{\mathscr{B}}(U)$
measurable and for almost all $(t,\omega)\in[0,T]\times\Omega$, $l$ is Fréchet
differentiable w.r.t. $(y,p,z,u)$ with continuous Fréchet derivatives
$l_{y},l_{q},l_{z},l_{u}$. The random mapping $\phi$ is
${\mathscr{F}}_{T}\bigotimes{\mathscr{B}}(\mathbb{R}^{n})$ measurable and for
almost all $(t,\omega)\in[0,T]\times\Omega$, $\phi$ is Fréchet differentiable
w.r.t. $y$ with continuous Fréchet derivative $\phi_{y}$. Moreover, for almost
all $(t,\omega)\in[0,T]\times\Omega$, there exists a constant $C$ s.t. for all
$(p,q,z,u)\in\mathbb{R}^{n}\times\mathbb{R}^{n\times d}\times
l^{2}(\mathbb{R}^{n})\times U$,
$|l|\leq C(1+|y|^{2}+|q|^{2}+|z|^{2}+|u|^{2}),\ \ |\phi|\leq C(1+|y|^{2}),$
$|l_{y}|+|l_{q}|+|l_{z}|+|l_{u}|\leq C(1+|y|+|q|+|z|+|u|)\ and\ |\phi_{y}|\leq
C(1+|y|).$
Under Assumption 3.1, we can get from Lemma 2.3 that for each
$u(\cdot)\in{\mathcal{A}}$, the system (3.1) admits a unique strong solution.
We denote the strong solution of (3.1) by
$(y^{u}(\cdot),q^{u}(\cdot),z^{u}(\cdot))$, or $(y(\cdot),q(\cdot),z(\cdot))$
if its dependence on admissible control $u(\cdot)$ is clear from context. Then
we call $(y(\cdot),q(\cdot),z(\cdot))$ the state processes corresponding to
the control process $u(\cdot)$ and call
$(u(\cdot);y(\cdot),q(\cdot),z(\cdot))$ the admissible pair. Furthermore, by
Assumption 3.2 and a priori estimate (2.3), it is easy to check that
$|J(u(\cdot))|<\infty.$
Then we put forward the optimal control problem we study.
###### Problem 3.1.
Find an admissible control $\bar{u}(\cdot)$ such that
$J(\bar{u}(\cdot))=\displaystyle\inf_{u(\cdot)\in{\mathcal{A}}}J(u(\cdot)).$
Any $\bar{u}(\cdot)\in{\mathcal{A}}$ satisfying above is called an optimal
control process of Problem 3.1 and the corresponding state processes
$(\bar{y}(\cdot),\bar{q}(\cdot),\bar{z}(\cdot))$ are called the optimal state
processes. Correspondingly
$(\bar{u}(\cdot);\bar{y}(\cdot),\bar{q}(\cdot),\bar{z}(\cdot))$ is called an
optimal pair of Problem 3.1.
Before we deduce the necessary and sufficient conditions for the optimal
control of Problem 3.1, we need do some preparations. Since the control domain
$U$ is convex, the classical method to get necessary conditions for optimal
control processes is the so-called convex perturbation method. More precisely,
assuming that $(\bar{u}(\cdot);\bar{y}(\cdot),\bar{q}(\cdot),\bar{z}(\cdot))$
is an optimal pair of Problem 3.1, for any given admissible control
${u}(\cdot)$, we define an admissible control in the form of convex variation
$u^{\varepsilon}(\cdot)=\bar{u}(\cdot)+\varepsilon(u(\cdot)-\bar{u}(\cdot)),$
where $\varepsilon>0$ can be chosen sufficiently small. Denoting by
$(y^{\varepsilon}(\cdot),q^{\varepsilon}(\cdot),z^{\varepsilon}(\cdot))$ the
state processes of the control system (3.1) corresponding to the control
process $u^{\varepsilon}(\cdot)$, we obtain the variational inequality
$J(u^{\varepsilon}(\cdot))-J(\bar{u}(\cdot))\geq 0.$
In what follows, we do some estimates on the optimal pair and the convex
variable pair.
###### Lemma 3.2.
Under Assumptions 3.1-3.2, we have
$\displaystyle\begin{split}E\sup_{0\leq t\leq
T}|y^{\varepsilon}(t)-\bar{y}(t)|^{2}+E\int_{0}^{T}|q^{\varepsilon}(t)-\bar{q}(t)|^{2}dt+E\int_{0}^{T}||z^{\varepsilon}(t)-\bar{z}(t)||^{2}_{l^{2}({\mathbb{R}^{n}})}dt=O(\varepsilon^{2}).\end{split}$
###### Proof.
By continuous dependence theorem of BSDE (Lemma 2.4) and the uniformly bounded
property of Fréchet derivative $f_{u}$, we have
$\displaystyle\begin{split}&E\sup_{0\leq t\leq
T}|y^{\varepsilon}(t)-\bar{y}(t)|^{2}+E\int_{0}^{T}|q^{\varepsilon}(t)-\bar{q}(t)|^{2}dt+E\int_{0}^{T}||z^{\varepsilon}(t)-\bar{z}(t)||^{2}_{l^{2}({\mathbb{R}^{n}})}dt\\\
\leq&KE\displaystyle\int_{0}^{T}|f(t,\bar{y}(t),\bar{q}(t),\bar{z}(t),{u}^{\varepsilon}(t))-f(t,\bar{y}(t),\bar{q}(t),\bar{z}(t),\bar{u}(t))\big{|}^{2}dt\\\
\leq&KE\displaystyle\int_{0}^{T}|u^{\varepsilon}(t)-\bar{u}(t)|^{2}dt\\\
=&KE\displaystyle\int_{0}^{T}|(\bar{u}(t)+\varepsilon(u(t)-\bar{u}(t))-\bar{u}(t))|^{2}dt\\\
=&K\varepsilon^{2}E\displaystyle\int_{0}^{T}|u(t)-\bar{u}(t)|^{2}dt=O(\varepsilon^{2}).\end{split}$
Here and in the rest of this paper, $K$ is a generic positive constant and
might change from line to line. ∎
Then we consider the following linear BSDE served as a variational equation:
$\displaystyle
dY_{t}=-\bigg{[}f_{y}(t,\bar{y}(t),\bar{q}(t),\bar{z}(t),\bar{u}_{t})Y(t)+f_{q}(t,\bar{y}(t),\bar{q}(t),\bar{z}(t),\bar{u}_{t})Q_{t}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+f_{z}(t,\bar{y}(t),\bar{q}(t),\bar{z}(t),\bar{u}_{t})Z(t)+f_{u}(t,\bar{y}(t),\bar{q}(t),\bar{z}(t),\bar{u}_{t})(u(t)-\bar{u}(t))\bigg{]}dt$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}\displaystyle\sum_{i=1}^{d}\int_{t}^{T}Q^{i}(s)dW^{i}(s)+\displaystyle\sum_{i=1}^{\infty}Z^{i}(t)dH^{i}(t)$
(3.3) $\displaystyle Y(T)=0.$
Under Assumption 3.1, by Lemma 2.3 we know that BSDE (3) has a unique solution
$(Y,Q,Z)\in S_{\mathscr{F}}^{2}(0,T;\mathbb{R}^{n})\times
M_{\mathscr{F}}^{2}(0,T;\mathbb{R}^{n\times d})\times
l_{\mathscr{F}}^{2}(0,T;\mathbb{R}^{n}).$
###### Lemma 3.3.
Under Assumptions 3.1-3.2, it follows that
$\displaystyle\begin{split}&E\displaystyle\sup_{0\leq t\leq
T}|y^{\varepsilon}(t)-\bar{y}(t)-\varepsilon
Y(t)|^{2}+E\int_{0}^{T}|q^{\varepsilon}(t)-\bar{q}(t)-\varepsilon
Q(t)|^{2}dt\\\ &+E\int_{0}^{T}||z^{\varepsilon}(t)-\bar{z}(t)-\varepsilon
Z(t)||^{2}_{l^{2}({\mathbb{R}^{n}})}dt=o(\varepsilon^{2}).\end{split}$
###### Proof.
Firstly, one can check that
$\begin{array}[]{ll}&y^{\varepsilon}(t)-\bar{y}(t)\\\
=&\displaystyle\int_{t}^{T}\bigg{[}{f}_{y}^{\varepsilon}(s)(y^{\varepsilon}(s)-\bar{y}(s))+{f}_{q}^{\varepsilon}(s)(q^{\varepsilon}(s)-\bar{q}(s))\\\
&\ \ \ \ \ \ \ \
+{f}_{z}^{\varepsilon}(s)(z^{\varepsilon}(s)-\bar{z}(s))+{f}_{u}^{\varepsilon}(s)(u^{\varepsilon}(s)-\bar{u}(s))\bigg{]}ds\\\
&\ \ \ \ \
-\displaystyle\sum_{i=1}^{d}\int_{t}^{T}\big{(}q^{i\varepsilon}(s)-\bar{q}^{i}(s)\big{)}dW^{i}(s)-\displaystyle\sum_{i=1}^{\infty}\int_{t}^{T}\big{(}z^{i\varepsilon}(s)-\bar{z}^{i}(s)\big{)}dH^{i}(s)\end{array}$
and
$\displaystyle\begin{split}\varepsilon
Y(t)=&\displaystyle\displaystyle\int_{t}^{T}\bigg{[}{f}_{y}(s)\varepsilon{Y}(s)++{f}_{q}(s)\varepsilon
Q(s)+{f}_{z}(s)\varepsilon
Z(s)+{f}_{u}(s)\varepsilon(u(s)-\bar{u}(s))\bigg{]}ds\\\ &\ \ \ \ \
-\displaystyle\sum_{i=1}^{d}\int_{t}^{T}\varepsilon
Q^{i}(s)dW^{i}(s)-\displaystyle\sum_{i=1}^{\infty}\int_{t}^{T}\varepsilon
Z^{i}(s)dH^{i}(s),\end{split}$
where we have used the abbreviations for $\varphi=f,l$ as follows:
$\displaystyle\varphi_{y}(t)=\varphi_{y}(t,\bar{y}(t),\bar{q}(t),\bar{z}(t),\bar{u}_{t}),$
$\displaystyle\varphi_{z}(t)=\varphi_{z}(t,\bar{y}(t),\bar{q}(t),\bar{z}(t),\bar{u}_{t}),$
$\displaystyle\varphi_{q}(t)=\varphi_{q}(t,\bar{y}(t),\bar{q}(t),\bar{z}(t),\bar{u}_{t}),$
$\displaystyle\varphi_{u}(t)=\varphi_{u}(t,\bar{y}(t),\bar{q}(t),\bar{z}(t),\bar{u}_{t}),$
(3.4)
$\displaystyle\tilde{\varphi}_{y}^{\varepsilon}(t)=\displaystyle\int_{0}^{1}\varphi_{y}(t,\bar{y}(t)+\lambda(y^{\varepsilon}(t)-\bar{y}(t)),\bar{z}(t)+\lambda(z^{\varepsilon}(t)-\bar{z}(t)),\bar{u}(t)+\lambda(u^{\varepsilon}(t)-u(t)))d\lambda,$
$\displaystyle\tilde{\varphi}_{z}^{\varepsilon}(t)=\displaystyle\int_{0}^{1}\varphi_{z}(t,\bar{y}(t)+\lambda(y^{\varepsilon}(t)-\bar{y}(t)),\bar{z}(t)+\lambda(z^{\varepsilon}(t)-\bar{z}(t)),\bar{u}(t)+\lambda(u^{\varepsilon}(t)-u(t)))d\lambda,$
$\displaystyle\tilde{\varphi}_{q}^{\varepsilon}(t)=\displaystyle\int_{0}^{1}\varphi_{q}(t,\bar{y}(t)+\lambda(q^{\varepsilon}(t)-\bar{y}(t)),\bar{z}(t)+\lambda(q^{\varepsilon}(t)-\bar{z}(t)),\bar{u}(t)+\lambda(u^{\varepsilon}(t)-u(t)))d\lambda,$
$\displaystyle\tilde{\varphi}_{u}^{\varepsilon}(t)=\displaystyle\int_{0}^{1}\varphi_{u}(t,\bar{y}(t)+\lambda(y^{\varepsilon}(t)-\bar{y}(t)),\bar{z}(t)+\lambda(z^{\varepsilon}(t)-\bar{z}(t)),\bar{u}(t)+\lambda(u^{\varepsilon}(t)-u(t)))d\lambda.$
Thus by Lemma 2.4 again, we get
$\displaystyle\begin{array}[]{ll}&E\displaystyle\sup_{0\leq t\leq
T}|y^{\varepsilon}(t)-\bar{y}(t)-\varepsilon
Y(t)|^{2}+E\int_{0}^{T}|q^{\varepsilon}(t)-\bar{q}(t)-\varepsilon
Q(t)|^{2}dt\\\
&~{}+E\displaystyle\int_{0}^{T}||z^{\varepsilon}(t)-\bar{z}(t)-\varepsilon
Z(t)||^{2}_{l^{2}({\mathbb{R}^{n}})}dt\\\
\leq&K\varepsilon^{2}\bigg{[}E\displaystyle\int_{0}^{T}\bigg{|}(\tilde{f}^{\varepsilon}_{y}(t)-f_{y}(t))Y(t)+(\tilde{f}^{\varepsilon}_{q}(t)-f_{q}(t))Q(t)+(\tilde{f}^{\varepsilon}_{z}(t)-f_{z}(t))Z(t)\\\
&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+(\tilde{f}^{\varepsilon}_{u}(t)-f_{u}(t))(u(t)-\bar{u}(t))\bigg{|}^{2}dt\bigg{]}\\\
=&K\varepsilon^{2}\cdot\alpha(\varepsilon),\end{array}$ (3.10)
where
$\displaystyle\begin{split}\alpha(\varepsilon)=E\displaystyle&\int_{0}^{T}\bigg{|}(\tilde{f}^{\varepsilon}_{y}(t)-f_{y}(t))Y(t)+(\tilde{f}^{\varepsilon}_{q}(t)-f_{q}(t))Q(t)\\\
&\ \ \ \ \ \ \
+(\tilde{f}^{\varepsilon}_{z}(t)-f_{z}(t))Z(t)+(\tilde{f}^{\varepsilon}_{u}(t)-f_{u}(t))(u(t)-\bar{u}(t))\bigg{|}^{2}dt.\end{split}$
Consequently, using Lemma 3.2 and Assumption 3.1, by the dominated convergence
theorem we can deduce
$\displaystyle\lim_{\varepsilon\rightarrow 0}\alpha(\varepsilon)=0.$
Then the lemma follows from above and (3.10). ∎
###### Lemma 3.4.
Under Assumptions 3.1-3.2, using the abbreviations (3) we have
$\begin{array}[]{ll}J(u^{\varepsilon}(\cdot))-J(\bar{u}(\cdot))=&\varepsilon
E\phi_{y}(\bar{y}(0))Y(0)+\varepsilon
E\displaystyle\int_{0}^{T}l_{y}(t)Y(t)dt+\varepsilon
E\displaystyle\int_{0}^{T}l_{q}(t)Q(t)dt\\\ &+\varepsilon
E\displaystyle\int_{0}^{T}l_{z}(t)Z(t)dt+\varepsilon
E\displaystyle\int_{0}^{T}l_{u}(t)(u(t)-\bar{u}(t))dt+o(\varepsilon).\end{array}$
###### Proof.
After a first order development, we have
$\displaystyle J(u^{\varepsilon}(\cdot))-J(\bar{u}(\cdot))$ $\displaystyle=$
$\displaystyle
E\displaystyle\int_{0}^{1}\phi_{y}(\bar{y}(0)+\lambda(y^{\varepsilon}(0)-\bar{y}(0)))(y^{\varepsilon}(0)-\bar{y}(0))d\lambda$
$\displaystyle+E\displaystyle\int_{0}^{T}\tilde{l}_{y}^{\varepsilon}(t)(y^{\varepsilon}(t)-\bar{y}(t))dt+E\displaystyle\int_{0}^{T}\tilde{l}_{q}^{\varepsilon}(t)(q^{\varepsilon}(t)-\bar{q}(t))dt$
$\displaystyle+E\displaystyle\int_{0}^{T}\tilde{l}_{z}^{\varepsilon}(t)(z^{\varepsilon}(t)-\bar{z}(t))dt+E\displaystyle\int_{0}^{T}\tilde{l}_{u}^{\varepsilon}(t)(u^{\varepsilon}(t)-\bar{u}(t))dt$
$\displaystyle=$ $\displaystyle\varepsilon
E\phi_{y}(\bar{y}(0))Y(0)+E\phi_{y}(\bar{y}(0))(y^{\varepsilon}(0)-\bar{y}(0)-\varepsilon
Y(0))$
$\displaystyle+E\displaystyle\int_{0}^{1}\bigg{[}\phi_{y}(\bar{y}(0)+\lambda(y^{\varepsilon}(0)-\bar{y}(0)))-\phi_{y}(\bar{y}(0))\bigg{]}(y^{\varepsilon}(0)-\bar{y}(0))d\lambda$
$\displaystyle+\varepsilon
E\displaystyle\int_{0}^{T}l_{y}(t)Y(t)dt+E\displaystyle\int_{0}^{T}l_{y}(t)(y^{\varepsilon}(t)-\bar{y}(t)-\varepsilon
Y(t))dt$
$\displaystyle+E\displaystyle\int_{0}^{T}(\tilde{l}_{y}^{\varepsilon}(t)-l_{y}(t))(y^{\varepsilon}(t)-\bar{y}(t))dt$
$\displaystyle+\varepsilon
E\displaystyle\int_{0}^{T}l_{q}(t)q(t)dt+E\displaystyle\int_{0}^{T}l_{q}(t)(q^{\varepsilon}(t)-\bar{q}(t)-\varepsilon
Q(t))dt$
$\displaystyle+E\displaystyle\int_{0}^{T}(\tilde{l}_{q}^{\varepsilon}(t)-l_{q}(t))(q^{\varepsilon}(t)-\bar{q}(t))dt$
$\displaystyle+\varepsilon
E\displaystyle\int_{0}^{T}l_{z}(t)Z(t)dt+E\displaystyle\int_{0}^{T}l_{z}(t)(z^{\varepsilon}(t)-\bar{z}(t)-\varepsilon
Z(t))dt$
$\displaystyle+E\displaystyle\int_{0}^{T}(\tilde{l}_{z}^{\varepsilon}(t)-l_{z}(t))(z^{\varepsilon}(t)-\bar{z}(t))dt$
$\displaystyle+E\displaystyle\int_{0}^{T}l_{u}(t)\varepsilon(u(t)-\bar{u}(t))dt+E\displaystyle\int_{0}^{T}(\tilde{l}_{u}^{\varepsilon}(t)-l_{u}(t))\varepsilon(u(t)-\bar{u}(t))dt$
$\displaystyle=$ $\displaystyle\varepsilon
E\phi_{y}(\bar{y}(0))Y(0)+\varepsilon
E\displaystyle\int_{0}^{1}l_{y}(t)Y(t)dt+\varepsilon
E\displaystyle\int_{0}^{1}l_{q}(t)Q(t)dt$ $\displaystyle+\varepsilon
E\displaystyle\int_{0}^{1}l_{z}(t)Z(t)dt+\varepsilon
E\displaystyle\int_{0}^{1}l_{u}(t)(u(t)-\bar{u}(t))dt+\beta(\varepsilon),$
where $\beta(\varepsilon)$ is given by
$\begin{array}[]{ll}\beta(\varepsilon)=&E\phi_{y}(\bar{y}(0))(y^{\varepsilon}(0)-\bar{y}(0)-\varepsilon
Y(0))\\\
&+E\displaystyle\int_{0}^{1}\bigg{[}\phi_{y}(\bar{y}(0)+\lambda(y^{\varepsilon}(0)-\bar{y}(0)))-\phi_{y}(\bar{y}(0))\bigg{]}(y^{\varepsilon}(0)-\bar{y}(0))d\lambda\\\
&+E\displaystyle\int_{0}^{T}l_{y}(t)(y^{\varepsilon}(t)-\bar{y}(t)-\varepsilon
Y(t))dt+E\displaystyle\int_{0}^{T}(\tilde{l}_{y}^{\varepsilon}(t)-l_{y}(t))(y^{\varepsilon}(t)-\bar{y}(t))dt\\\
&+E\displaystyle\int_{0}^{T}l_{q}(t)(q^{\varepsilon}(t)-\bar{q}(t)-\varepsilon
Q(t))dt+E\displaystyle\int_{0}^{T}(\tilde{l}_{q}^{\varepsilon}(t)-l_{q}(t))(q^{\varepsilon}(t)-\bar{q}(t))dt\\\
&+E\displaystyle\int_{0}^{T}l_{z}(t)(z^{\varepsilon}(t)-\bar{z}(t)-\varepsilon
Z(t))dt+E\displaystyle\int_{0}^{T}(\tilde{l}_{z}^{\varepsilon}(t)-l_{z}(t))(z^{\varepsilon}(t)-\bar{z}(t))dt\\\
&+E\displaystyle\int_{0}^{T}(\tilde{l}_{u}^{\varepsilon}(t)-l_{u}(t))\varepsilon(u(t)-\bar{u}(t))dt.\end{array}$
Thus combining Lemma 3.2, Lemma 3.4 and Assumption 3.2, by the dominated
convergence theorem we conclude that $\beta(\varepsilon)=o(\varepsilon)$. ∎
By Lemma 3.4 and the fact that $\displaystyle\lim_{\varepsilon\rightarrow
0^{+}}\frac{J(u^{\varepsilon})-J(\bar{u})}{\varepsilon}\geq 0$, we can further
deduce
###### Corollary 3.5.
Under Assumptions 3.1-3.2, we have the variation inequality below
$\begin{array}[]{ll}&E\phi_{y}(\bar{y}(0))Y(0)+E\displaystyle\int_{0}^{T}l_{y}(t)Y(t)dt+E\displaystyle\int_{0}^{T}l_{y}(t)Y(t)dt\\\
&+E\displaystyle\int_{0}^{T}l_{z}(t)Z(t)dt+E\displaystyle\int_{0}^{T}l_{u}(t)(u(t)-\bar{u}(t))dt\geq
0.\end{array}$ (3.11)
## 4 Necessary and sufficient optimality conditions
We first introduce the adjoint equation corresponding to the variational
equation (3):
$\displaystyle
dk(t)=-\bigg{[}-f_{y}^{*}(t)k(t)+l_{y}(t)\bigg{]}dt-\displaystyle\sum_{i=1}^{d}\bigg{[}-f_{q^{i}}^{*}(t)k(t)+l_{q^{i}}(t)\bigg{]}dW^{i}(t)$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \
-\displaystyle\sum_{i=1}^{\infty}\bigg{[}-f_{z^{i}}^{*}(t)k(t)+l_{z^{i}}(t)\bigg{]}dH^{i}(t)$
(4.1) $\displaystyle k(0)=-\phi_{y}(\bar{y}(0)),~{}~{}~{}~{}0\leq t\leq T,$
where $f_{y}^{*},f_{q^{i}}^{*}$and $f_{z^{i}}^{*}$ are the dual operators of
$f_{y},f_{q^{i}}$ and $f_{z^{i}}$, respectively.
Under Assumptions 3.1-3.2, by Lemma 2.1 it is easy to see that the above
adjoint equation has a unique solution
$k(\cdot)\in{\mathcal{S}}^{2}_{\mathscr{F}}(0,T;\mathbb{R}^{n})$. Then we
define the Hamiltonian function
$H:[0,T]\times\mathbb{R}^{n}\times\mathbb{R}^{n\times d}\times
l^{2}(\mathbb{R}^{n})\times U\times\mathbb{R}^{n}\rightarrow\mathbb{R}^{1}$ by
$\begin{array}[]{ll}\displaystyle H(t,y,q,z,u,k)=\langle
k,-f(t,y,q,z,u)\rangle+l(t,y,q,z,u)\end{array}$ (4.2)
and rewrite the adjoint equation in the Hamiltonian system form:
$\left\\{\begin{array}[]{ll}dk(t)=-H_{y}(t,\bar{y}(t),\bar{q}(t),\bar{z}(t),\bar{u}(t),k(t))dt\\\
\ \ \ \ \ \ \ \ \ \ \
-\displaystyle\sum_{i=1}^{d}H_{q}^{i}(t,\bar{y}(t),\bar{q}(t),\bar{z}(t),\bar{u}(t),k(t))dW^{i}(t)\\\
\ \ \ \ \ \ \ \ \ \ \
-\displaystyle\sum_{i=1}^{\infty}H_{z^{i}}(t,\bar{y}(t),\bar{q}(t),\bar{z}(t),\bar{u}(t),k(t))dH^{i}(t)\\\
k(0)=-\phi_{y}(\bar{y}(0)).\end{array}\right.$ (4.3)
Now we are ready to give the necessary conditions for an optimal control of
Problem 3.1.
###### Theorem 4.1.
Under Assumptions 3.1-3.2, if
$(\bar{u}(\cdot);\bar{y}(\cdot),\bar{q}(\cdot),\bar{z}(\cdot))$ is an optimal
pair of Problem 3.1, then we have
$H_{u}(t,\bar{y}(t-),\bar{q}(t),\bar{z}(t),\bar{u}(t),k(t-))(u-\bar{u}(t))\geq
0,~{}\forall u\in U,\ \ a.e.\ a.s.,$ (4.4)
where $k(\cdot)$ is the solution to the adjoint equation (4).
###### Proof.
By (3) and (4), applying Itô formula to $\langle Y(t),k(t)\rangle$ we have
$\begin{array}[]{ll}&E\phi_{y}(\bar{y}(0))Y(0)+E\displaystyle\int_{0}^{T}l_{y}(t)Y(t)dt+E\displaystyle\int_{0}^{T}l_{z}(t)Z(t)dt+E\displaystyle\int_{0}^{T}l_{u}(t)(u(t)-\bar{u}(t))dt\\\
=&-E\displaystyle\int_{0}^{T}\langle
k(t),f_{u}(t,\bar{y}(t),\bar{q}(t),\bar{z}(t),\bar{u}_{t})(u(t)-\bar{u}(t))\rangle
dt+E\displaystyle\int_{0}^{T}l_{u}(t)(u(t)-\bar{u}(t))dt.\end{array}$
Then noticing the definition of Hamilton function (4.2) and the variational
inequality (3.11), for any $u(\cdot)\in{\mathcal{A}}$, we have
$E\displaystyle\int_{0}^{T}H_{u}(t,\bar{y}(t),\bar{q}(t),\bar{z}(t),\bar{u}(t),k(t))(u(t)-\bar{u}(t))dt\geq
0,$
which implies (4.4). ∎
We then consider the sufficient conditions for an optimal control of Problem
3.1.
###### Theorem 4.2.
Under Assumptions 3.1-3.2, let
$(\bar{u}(\cdot);\bar{y}(\cdot),\bar{q}(\cdot),\bar{z}(\cdot))$ be an
admissible pair and ${k}(\cdot)$ be the unique solution of the corresponding
adjoint equation (4.3). Assume that for almost all
$(t,\omega)\in[0,T]\times\Omega$ , $H(t,y,q,z,u,{k}(t))$ and $\phi(y)$ are
convex w.r.t. $(y,q,z,u)$ and $y$, respectively, and the optimality condition
$H(t,\bar{y}(t),\bar{q}(t),\bar{z}(t),\bar{u}(t),k(t))=\displaystyle\min_{u\in
U}H(t,\bar{y}(t),\bar{q}(t),\bar{z}(t),u,k(t))$
holds, then $(\bar{u}(\cdot);\bar{y}(\cdot),\bar{q}(\cdot),\bar{z}(\cdot))$ is
an optimal pair of Problem 3.1.
###### Proof.
Let $(u(\cdot);y(\cdot),q(\cdot),z(\cdot))$ be an arbitrary admissible pair.
It follows from the form of the cost functional (3.2) that
$\displaystyle J(u(\cdot))-J(\bar{u}(\cdot))$ (4.5) $\displaystyle=$
$\displaystyle
E\displaystyle\int_{0}^{T}\bigg{[}l(t,y(t),q(t),z(t),u(t))-l(t,\bar{y}(t),\bar{q}(t),\bar{z}(t),\bar{u}(t))\bigg{]}dt+E\bigg{[}\phi(y(0))-\phi(\bar{y}(0))\displaystyle\bigg{]}$
$\displaystyle=$ $\displaystyle I_{1}+I_{2},$
where
$\displaystyle
I_{1}=E\int_{0}^{T}\bigg{[}l(t,y(t),q(t),z(t),u(t))-l(t,\bar{y}(t),\bar{q}(t),\bar{z}(t),\bar{u}(t))\bigg{]}dt$
and
$I_{2}=E\bigg{[}\phi(y(0))-\phi(\bar{y}(0))\bigg{]}.$
Due to the convexity of $\phi$, applying Itô formula to
$\langle{k}(t),y(t)-\bar{y}(t)\rangle$, we have
$\displaystyle\begin{split}I_{2}=&E[\phi(y(0))-\phi(\bar{y}(0))]\geq
E[\langle\phi_{y}(\bar{y}(0)),y(0)-\bar{y}(0)\rangle]=-E[\langle{k}(0),y(0)-\bar{y}(0)\rangle]\\\
=&-E\displaystyle\int_{0}^{T}\langle
H_{y}(t,\bar{y}(t),\bar{q}(t),\bar{z}(t),\bar{u}(t),{k}(t)),y(t)-\bar{y}(t)\rangle
dt\\\ &-\sum_{i=1}^{d}E\displaystyle\int_{0}^{T}\langle
H_{q}^{i}(t,\bar{y}(t),\bar{q}(t),\bar{z}(t),\bar{u}(t),{k}(t)),q^{i}(t)-\bar{q}^{i}(t)\rangle
dt\\\ &-\displaystyle\sum_{i=1}^{\infty}E\displaystyle\int_{0}^{T}\langle
H_{z}^{i}(t,\bar{y}(t),\bar{q}(t),\bar{z}(t),\bar{u}(t),{k}(t)),z^{i}(t)-\bar{z}^{i}(t)\rangle
dt\\\ &-E\displaystyle\int_{0}^{T}\langle
f(t,{y}(t),{q}(t),{z}(t),{u}(t))-f(t,\bar{y}(t),\bar{q}(t),\bar{z}(t),\bar{u}(t)),k(t)\rangle
dt\\\ =&-J_{1}+J_{2},\end{split}$ (4.6)
where
$\begin{array}[]{ll}J_{1}=&E\displaystyle\int_{0}^{T}\langle
H_{y}(t,\bar{y}(t),\bar{q}(t),\bar{z}(t),\bar{u}(t),{k}(t)),y(t)-\bar{y}(t)\rangle
dt\\\ &+\displaystyle\sum_{i=1}^{d}E\displaystyle\int_{0}^{T}\langle
H_{q}^{i}(t,\bar{y}(t),\bar{q}(t),\bar{z}(t),\bar{u}(t),{k}(t)),q^{i}(t)-\bar{q}^{i}(t)\rangle
dt\\\ &+\displaystyle\sum_{i=1}^{\infty}E\displaystyle\int_{0}^{T}\langle
H_{z}^{i}(t,\bar{y}(t),\bar{q}(t),\bar{z}(t),\bar{u}(t),{k}(t)),z^{i}(t)-\bar{z}^{i}(t)\rangle
dt\end{array}$
and
$J_{2}=-E\displaystyle\int_{0}^{T}\langle
f(t,{y}(t),{q}(t),{z}(t),{u}(t))-f(t,\bar{y}(t),\bar{q}(t),\bar{z}(t),\bar{u}(t)),k(t)\rangle
dt.$
Using the definition of the Hamiltonian function (4.2) again, we have
$\begin{array}[]{ll}I_{1}&=E\displaystyle\int_{0}^{T}\bigg{[}l(t,y(t),q(t),z(t),u(t))-l(t,\bar{y}(t),\bar{q}(t),\bar{z}(t),\bar{u}(t))\bigg{]}dt\\\
&=E\displaystyle\int_{0}^{T}\bigg{[}H(t,y(t),q(t),z(t),u(t),k(t))-H(t,\bar{y}(t),\bar{q}(t),\bar{z}(t),\bar{u}(t),k(t))\bigg{]}dt\\\
&\ \ \ +E\displaystyle\int_{0}^{T}\langle
f(t,{y}(t),{q}(t),{z}(t),{u}(t))-f(t,\bar{y}(t),\bar{q}(t),\bar{z}(t),\bar{u}(t)),k(t)\rangle
dt\\\ &=J_{3}-J_{2},\end{array}$ (4.7)
where
$\begin{array}[]{ll}J_{3}=E\displaystyle\int_{0}^{T}\bigg{[}H(t,y(t),q(t),z(t),u(t),k(t))-H(t,\bar{y}(t),\bar{q}(t),\bar{z}(t),\bar{u}(t),k(t))\bigg{]}dt.\end{array}$
(4.8)
Since $H(t,y,q,z,u,{k}(t))$ is convex w.r.t. $(y,q,z,u)$ for almost all
$(t,\omega)\in[0,T]\times\Omega$, it turns out that
$\displaystyle\begin{split}&H(t,y(t),q(t),z(t),u(t),{k}(t))-H(t,\bar{y}(t),\bar{q}(t),\bar{z}(t),\bar{u}(t),{k}(t))\\\
\geq&\langle
H_{y}(t,\bar{y}(t),\bar{q}(t),\bar{z}(t),\bar{u}(t),{k}(t)),y(t)-\bar{y}(t)\rangle\\\
&+\sum_{i=1}^{d}\langle
H_{q}^{i}(t,\bar{y}(t),\bar{q}(t),\bar{z}(t),\bar{u}(t),{k}(t)),q^{i}(t)-\bar{q}^{i}(t)\rangle\\\
&+\sum_{i=1}^{\infty}\langle
H_{z}^{i}(t,\bar{y}(t),\bar{q}(t),\bar{z}(t),\bar{u}(t),{k}(t)),z^{i}(t)-\bar{z}^{i}(t)\rangle\\\
&+\langle
H_{u}(t,\bar{y}(t),\bar{q}(t),\bar{z}(t),\bar{u}(t),{k}(t)),u(t)-\bar{u}(t)\rangle,\
\ a.s.\ a.e.\end{split}$ (4.9)
On the other hand, for almost all $(t,\omega)\in[0,T]\times\Omega$,
$u\rightarrow H(t,\bar{y}(t),\bar{q}(t),\bar{z}(t),u,k(t))$ takes its minimal
value at $\bar{u}(t)$ in the domain $U$, thus
$\begin{array}[]{ll}\langle
H_{u}(t,\bar{y}(t),\bar{q}(t),\bar{z}(t),\bar{u}(t),k(t)),u(t)-\bar{u}(t)\rangle\geq
0,\ \ a.s.\ a.e.\end{array}$ (4.10)
Therefore, by (4.8)–(4.10) we first have
$\begin{array}[]{ll}J_{3}\geq J_{1}.\end{array}$ (4.11)
By (4.11), together with (4.5)–(4.7), it follows that
$\begin{array}[]{ll}J(u(\cdot))-J(\bar{u}(\cdot))=I_{1}+I_{2}=(J_{3}-J_{2})+(-J_{1}+J_{2})\geq(J_{1}-J_{2})+(-J_{1}+J_{2})=0.\end{array}$
Due to the arbitrariness of $u(\cdot)$, we conclude that $\bar{u}(\cdot)$ is
an optimal control process and thus
$(\bar{u}(\cdot);\bar{y}(\cdot),\bar{q}(\cdot),\bar{z}(\cdot))$ is an optimal
pair. ∎
## 5 Applications in BLQ problems
In this section, we will apply our stochastic maximum principle to the so-
called BLQ problem, i.e. minimize the following quadratic cost functional over
$u(\cdot)\in\mathcal{A}$:
$\displaystyle\begin{split}J(u(\cdot)):=&E\langle
My(0),y(0)\rangle+E\displaystyle\int_{0}^{T}\langle E(s)y(s),y(s)\rangle
ds+\sum_{i=1}^{d}E\displaystyle\int_{0}^{T}\langle
F^{i}(s)q^{i}(s),q^{i}(s)\rangle ds\\\
&+\sum_{i=1}^{\infty}E\displaystyle\int_{0}^{T}\langle
G^{i}(s)z^{i}(s),z^{i}(s)\rangle ds+E\displaystyle\int_{0}^{T}\langle
N(s)u(s),u(s)\rangle ds,\end{split}$ (5.1)
where the state processes $(y(\cdot),q(\cdot),z(\cdot))$ are the solution to
the controlled linear backward stochastic system as follows:
$\displaystyle
dy(t)=-\bigg{[}A(t)y(t)+\displaystyle\sum_{i=1}^{d}B^{i}(t)q^{i}(t)+\displaystyle\sum_{i=1}^{\infty}C^{i}(t)z^{i}(t)+D(t)u(t)\bigg{]}dt$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \
+\displaystyle\sum_{i=1}^{d}q^{i}dW^{i}(t)+\displaystyle\sum_{i=1}^{\infty}z^{i}dH^{i}(t)$
(5.2) $\displaystyle y(T)=\xi.$
To study this problem, we need the assumptions on the coefficients below.
###### Assumption 5.1.
The $\\{{\mathscr{F}}_{t},0\leq t\leq T\\}$-predictable matrix processes
$A:[0,T]\times\Omega\rightarrow\mathbb{R}^{n\times
n},B^{i}:[0,T]\times\Omega\rightarrow\mathbb{R}^{n\times
n},i=1,2,\cdots,d,C^{i}:[0,T]\times\Omega\rightarrow\mathbb{R}^{n\times
n},i=1,2,\cdots,D:[0,T]\times\Omega\rightarrow\mathbb{R}^{n\times
m},E:[0,T]\times\Omega\rightarrow\mathbb{R}^{n\times
n},F^{i}:[0,T]\times\Omega\rightarrow\mathbb{R}^{n\times n},i=1,2,\cdots
d,G^{i}:[0,T]\times\Omega\rightarrow\mathbb{R}^{n\times
n},i=1,2,\cdots,N:[0,T]\times\Omega\rightarrow\mathbb{R}^{m\times m}$ and the
${\mathscr{F}}_{T}$-measurable random matrix
$M:\Omega\rightarrow\mathbb{R}^{n\times n}$ are uniformly bounded.
###### Assumption 5.2.
The state weighting matrix processes $E$, $F^{i}$, $G^{i}$, the control
weighting matrix process $N$ and the random matrix $M$ are a.e. a.s. symmetric
and nonnegative. Moreover, $N$ is a.e. a.s. uniformly positive, i.e.
$N\geq\delta I$ for some positive constant $\delta$ a.e. a.s.
###### Assumption 5.3.
There is no further constraint imposed on the control processes, i.e.
$\mathcal{A}=\bigg{\\{}u(\cdot)|u(\cdot)\ is\ \mathscr{F}_{t}-predictable\
with\ values\ in\ \mathbb{R}^{m}\ and\
E\displaystyle\int_{0}^{T}|u(t)|^{2}dt<\infty\\}.$
From Assumption 5.3, we know that $\mathcal{A}$ is a Hilbert space. If we
denote the norm of $\mathcal{A}$ by $\|\cdot\|_{\mathcal{A}}$, then for any
control process $u(\cdot)\in\mathcal{A}$,
$\|u(\cdot)\|_{\mathcal{A}}=E\displaystyle\sqrt{\int_{0}^{T}|u(t)|^{2}dt}$.
Under Assumptions 5.1, by Lemma 2.3 we first know that the linear BSDE (5) in
BLQ problem has a unique solution and thus the BLQ problem is well-defined.
Then, under Assumptions 5.1-5.3, we will demonstrate that BLQ problem has a
unique optimal control.
###### Lemma 5.1.
Under Assumptions 5.1-5.3, the cost functional $J$ is strictly convex over
$\mathcal{A}$ and
$\displaystyle\lim_{\|u(\cdot)\|_{\mathcal{A}}{\rightarrow\infty}}J(u(\cdot))=\infty.$
###### Proof.
The convexity of the cost functional $J$ over $\mathcal{A}$ is obvious.
Actually, since the weighting matrix process $N$ is uniformly positive, $J$ is
strictly convex. In view of the nonnegative property of $M,E,F^{i},G^{i}$ and
the strictly positive property of $N$, we have
$J(u(\cdot))\geq\delta
E\displaystyle\int_{0}^{T}|u(t)|^{2}dt=\delta\|u(\cdot)\|^{2}_{\mathcal{A}}.$
Therefore,
$\displaystyle\lim_{\|u(\cdot)\|_{\mathcal{A}}{\rightarrow\infty}}J(u(\cdot))=\infty.$
∎
###### Lemma 5.2.
Under Assumptions 5.1-5.3, the cost functional $J$ is Fréchet differentiable
over $\mathcal{A}$ and its Fréchet derivative $J^{\prime}$ at any admissible
control process $u(\cdot)\in{\mathcal{A}}$ is given by
$\displaystyle\begin{split}\langle
J^{\prime}(u(\cdot)),v(\cdot)\rangle=&2E\int_{0}^{T}\langle
E(t)y^{u}(t),Y^{v}(t)\rangle dt+2\sum_{i=1}^{d}E\int_{0}^{T}\langle
F^{i}(t)q^{iu}(t),Q^{iv}(t)\rangle dt\\\
&+2\sum_{i=1}^{\infty}E\int_{0}^{T}\langle G^{i}(t)z^{iu}(t),Z^{iv}(t)\rangle
dt+2E\int_{0}^{T}\langle N(t)u(t),v(t)\rangle dt\\\ &+2E\langle
My^{u}(0),Y^{v}(0)\rangle,\end{split}$ (5.3)
where $v(\cdot)\in\mathcal{A}$ is arbitrary, $(Y^{v},Q^{v},Z^{v})$ is the
solution of BSDE (5) corresponding to the control process
$v(\cdot)\in\mathcal{A}$ and the terminal value $0$, and
$(y^{u}(\cdot),q^{u}(\cdot),z^{u}(\cdot))$ are the state processes
corresponding to the control process $u(\cdot)$.
###### Proof.
For any $v(\cdot)\in\mathcal{A}$, we set
$\displaystyle\begin{split}\Delta
J=&J(u(\cdot)+v(\cdot))-J(u(\cdot))-2E\int_{0}^{T}\langle
E(t)y^{u}(t),Y^{v}(t)\rangle dt\\\ &-2\sum_{i=1}^{d}E\int_{0}^{T}\langle
F^{i}(t)q^{iu}(t),Q^{iv}(t)\rangle dt-2\sum_{i=1}^{\infty}E\int_{0}^{T}\langle
G^{i}(t)z^{iu}(t),Z^{iv}(t)\rangle dt\\\ &-2E\int_{0}^{T}\langle
N(t)u(t),v(t)\rangle dt-2E\langle My^{u}(0),Y^{v}(0)\rangle.\end{split}$
By the definition of cost functional (5.1), we have
$\displaystyle\begin{split}\Delta J=&E\langle
MY^{v}(0),Y^{v}(0)\rangle+E\displaystyle\int_{0}^{T}\langle
E(s)Y^{v}(s),Y^{v}(s)\rangle
ds+\sum_{i=1}^{d}E\displaystyle\int_{0}^{T}\langle
F^{i}(s)Q^{iv}(s),Q^{iv}(s)\rangle ds\\\
&+\sum_{i=1}^{\infty}E\displaystyle\int_{0}^{T}\langle
G^{i}(s)Z^{iv}(s),Z^{iv}(s)\rangle ds+E\displaystyle\int_{0}^{T}\langle
N(s)v(s),v(s)\rangle ds.\end{split}$
Then it follows from Assumption 5.1 and a priori estimate (2.3) that
$\displaystyle|\Delta J|$ $\displaystyle\leq$ $\displaystyle
K\bigg{[}E\sup_{0\leq t\leq
T}|Y^{v}(t)|^{2}+E\int_{0}^{T}|Q^{v}(t)|^{2}dt+E\int_{0}^{T}\|Z^{v}(t)\|^{2}_{l^{2}(\mathbb{R}^{N})}dt+E\int_{0}^{T}|v(t)|^{2}dt\bigg{]}$
$\displaystyle\leq$ $\displaystyle
KE\int_{0}^{T}|v(t)|^{2}dt=K\|v(\cdot)\|^{2}_{\mathcal{A}}.$
Consequently, we have
$\displaystyle\lim_{\|v(\cdot)\|_{\mathcal{A}}{\rightarrow 0}}\frac{|\Delta
J|}{\|v(\cdot)\|_{\mathcal{A}}}=0,$
which implies that $J$ is Fréchet differentiable and its Fréchet derivative
$J^{\prime}$ is given by (5.3). ∎
The strict convexity and the Fréchet differentiability of $J$ deduced from
Lemmas 5.1-5.2 lead to the lower semi-continuity of $J$, thus the following
lemma is applicable to $J$ and $\mathcal{A}$ in our BLQ problem.
###### Lemma 5.3.
(Proposition 1.2 of Chapter II in [6]) Let $\mathcal{A}$ be a reflexive Banach
space and $J:\mathcal{A}\mapsto\mathbb{R}^{1}$ be a convex function. Assume
that $J$ is lower semi-continuous and proper, and consider the minimization
problem
$\inf_{u\in\mathcal{A}}J(u).$
If the function $J$ is coercive over $\mathcal{A}$, i.e.
$\lim_{\|u\|_{\mathcal{A}}\to\infty}J(u)=\infty,$
then the minimization problem has at least one solution. Moreover, if $J$ is
strictly convex over $\mathcal{A}$, then the minimization problem has a unique
solution.
By Lemma 5.3 we can immediately conclude
###### Theorem 5.4.
Under Assumptions 5.1-5.3, BLQ problem has a unique optimal control.
In what follows, we will utilize the stochastic maximum principle to study the
dual representation of the optimal control to BLQ problem and construct its
stochastic Hamilton system. As in section 4, we first introduce the adjoint
forward equation corresponding to an admissible pair
$(u(\cdot);y(\cdot),q(\cdot),z(\cdot))$:
$\displaystyle
dk(t)=\bigg{(}A^{*}(t)k(t)-2E(t)y(t)\bigg{)}dt+\displaystyle\sum_{i=1}^{d}\bigg{(}B^{i*}(t)k^{i}(t)-2F^{i}(t)q^{i}(t)\bigg{)}dW^{i}(t)$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \
+\displaystyle\sum_{i=1}^{\infty}\bigg{(}C^{i*}(t)k(t)-2G^{i}(t)z^{i}(t)\bigg{)}dH^{i}(t)$
(5.4) $\displaystyle k(0)={-2My(0)}.$
Also we define the Hamiltonian function
$H:[0,T]\times\Omega\times\mathbb{R}^{n}\times\mathbb{R}^{n\times d}\times
l^{2}(\mathbb{R}^{n})\times U\times\mathbb{R}^{n}\rightarrow\mathbb{R}^{1}$ by
$\displaystyle\displaystyle H(t,y,q,z,u,k)$ $\displaystyle=$
$\displaystyle-\bigg{\langle}k,A(t)y+\displaystyle\sum_{i=1}^{d}B^{i}(t)q^{i}+\displaystyle\sum_{i=1}^{\infty}C^{i}(t)z^{i}+D(t)u\bigg{\rangle}$
$\displaystyle+\langle E(t)y,y\rangle+\displaystyle\sum_{i=1}^{d}\langle
F^{i}(t)q^{i},q^{i}\rangle+\displaystyle\sum_{i=1}^{\infty}\langle
G^{i}(t)z^{i},z^{i}\rangle+\langle N(t)u,u\rangle.$
Then the adjoint equation can be rewritten as a Hamiltonian form:
$\displaystyle
dk(t)=-H_{y}(t,{y}(t),{q}(t),{z}(t),{u}(t),k(t))dt-\displaystyle\sum_{i=1}^{d}H_{q}^{i}(t,{y}(t),{q}(t),{z}(t),{u}(t),k(t))dB^{i}(t)$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \
-\displaystyle\sum_{i=1}^{\infty}H_{z^{i}}(t,{y}(t-),{q}(t),{z}(t),{u}(t),k(t))dH^{i}(t)$
(5.6) $\displaystyle k(0)=-2My(0).$
Under Assumption 5.1, for each admissible pair
$({u}(\cdot);{y}(\cdot),{q}(\cdot),{z}(\cdot))$, by Lemma 2.1 the adjoint
equation (5) has a unique solution $k(\cdot)$.
It is time to give the the dual characterization of the optimal control.
###### Theorem 5.5.
Under Assumptions 5.1-5.3, BLQ problem has a unique optimal control and the
optimal control is given by
$\displaystyle\begin{split}u(t)=-\frac{1}{2}N^{-1}(t)D^{*}(t)k(t-),\ \ a.e.\
a.s.,\end{split}$ (5.7)
where $k(\cdot)$ is the unique solution of the adjoint equation (5) (or
equivalently, (5)) corresponding to the optimal pair
$(u(\cdot);y(\cdot),q(\cdot),z(\cdot))$.
###### Proof.
By Theorem 5.4, we know the existence and uniqueness of optimal control to BLQ
problem and denote the optimal control by $u(\cdot)$. We only need to prove
$u$ has an expression as in (5.7). For this, let
$(y(\cdot),q(\cdot),z(\cdot))$ be the optimal state processes corresponding to
$u(\cdot)$ and $k(\cdot)$ be the unique solution of the adjoint equation (5)
corresponding to the optimal pair $(u(\cdot);y(\cdot),q(\cdot),z(\cdot))$. By
the necessary optimality condition (4.4) and Assumption 5.3, we have
$H_{u}(t,y(t-),q(t),z(t),u(t),k(t-))=0,\ \ a.e.\ a.s.$
Noticing the definition of $H$ in (5), we get
$2N(t)u(t)+D^{*}(t)k(t-)=0,\ \ a.e.\ a.s.$
Then the claim that the unique optimal control $u(\cdot)$ satisfies (5.7)
follows. ∎
Finally we introduce the so-called stochastic Hamilton system which consists
of the state equation (5), the adjoint equation (5) (or equivalently, (5)) and
the dual representation (5.7):
$\displaystyle
dy(t)=-\bigg{(}A(t)y(t)+\sum_{i=1}^{d}B^{i}(t)q^{i}(t)+\sum_{i=1}^{\infty}C^{i}(t)z^{i}(t)+D(t)u(t)\bigg{)}dt$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \
+\displaystyle\sum_{i=1}q^{i}dW^{i}(t)+\displaystyle\sum_{i=1}^{\infty}z^{i}dH^{i}(t)$
$\displaystyle y(T)=\xi,$ $\displaystyle
dk(t)=\bigg{(}A^{*}(t)k(t)-2E(t)y(t)\bigg{)}dt+\displaystyle\sum_{i=1}^{d}\bigg{(}B^{i*}(t)k^{i}(t)-2F^{i}(t)q^{i}(t)\bigg{)}dW^{i}(t)$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \
+\displaystyle\sum_{i=1}^{\infty}\bigg{(}C^{i*}(t)k(t)-2G^{i}(t)z^{i}(t)\bigg{)}dH^{i}(t)$
(5.8) $\displaystyle k(0)=-2My(0),$ $\displaystyle
u_{t}=-\frac{1}{2}N^{-1}(t)D^{*}(t)k(t-).$
Clearly this is a fully coupled forward-backward stochastic differential
equation (FBSDE) driven by $d$-dimensional Brownian motion $W$ and Teugel’s
martingales $\\{H^{i}\\}_{i=1}^{\infty}$, and its solution is a stochastic
processes quaternary $(k(\cdot),y(\cdot),q(\cdot),z(\cdot))$.
###### Theorem 5.6.
Under Assumptions 5.1-5.3, the stochastic Hamilton system (5) has a unique
solution $(k(\cdot),y(\cdot),q(\cdot),z(\cdot))\in
S_{\mathscr{F}}^{2}(0,T;\mathbb{R}^{n})\times
S_{\mathscr{F}}^{2}(0,T;\mathbb{R}^{n})\times
M_{\mathscr{F}}^{2}(0,T;\mathbb{R}^{n\times d})\times
l_{\mathscr{F}}^{2}(0,T;\mathbb{R}^{n})$, where $u(\cdot)$ is the optimal
control of BLQ problem and $(y(\cdot),q(\cdot),z(\cdot))$ are its
corresponding optimal state. Moreover,
$\displaystyle\begin{split}&\displaystyle E\sup_{0\leqslant t\leqslant
T}|k(t)|^{2}+E\displaystyle\sup_{0\leq t\leq
T}|y(t)|^{2}+E\int_{0}^{T}|q(t)|^{2}dt+E\int_{0}^{T}||z(t)||^{2}_{l^{2}({\mathbb{R}^{n}})}dt\leqslant
KE{{|\xi|^{2}}}.\end{split}$ (5.9)
###### Proof.
The existence result follows from Theorem 5.5 and the uniqueness result is
obvious once a priori estimate (5.9) holds. But noticing Assumptions 5.1-5.3
and using Lemmas 2.2 and 2.4, we can deduce (5.9) immediately. ∎
In summary, the stochastic Hamilton system (5) completely characterize the
optimal control of BLQ problem in this section. Therefore, solving BLQ problem
is equivalent to solving the stochastic Hamilton system, moreover, the unique
optimal control of the stochastic Hamilton system can be given explicitly by
(5.7).
## References
* [1] K. Bahlali, M. Eddahbi and E. Essaky. BSDE associated with Lévy processes and application to PDIE. J. Appl. Math. Stochastic Anal., 16:1–17, 2003.
* [2] K. Bahlali, B. Gherbal and B. Mezerdi. Existence and optimality conditions in stochastic control of linear BSDEs. Random Oper. Stoch. Equ., 18:185–197, 2010.
* [3] S. Bahlali. Stochastic controls of backward systems. Random Oper. Stoch. Equ., 18:125–140, 2010.
* [4] J.-M. Bismut. Conjugate convex functions in optimal stochastic control. J. Math. Anal. Appl., 44:384–404, 1973.
* [5] N. Dokuchaev and X. Y. Zhou. Stochastic controls with terminal contingent conditions. J. Math. Anal. Appl., 238:143–165, 1999.
* [6] I. Ekeland and R. Témam. Convex Analysis and Variational Problems. Amsterdam: North-Holland, 1976.
* [7] N. El Karoui, S. Peng and M. C. Quenez. Backward stochastic differential equations in finance. Math. Finance, 7:1–71, 1997.
* [8] M. El Otmani. Generalized BSDE driven by a Lévy process. J. Appl. Math. Stoch. Anal., 25pp, 2006.
* [9] M. El Otmani. Backwark stochastic differnetial equations associated with Lévy processes and partial integro-differential eqiations. Commun. Stoch. Anal., 2:277–288, 2008.
* [10] A. E. B. Lim and X. Y. Zhou. Linear-quadratic control of backward stochastic differential equations. SIAM J. Control Optim., 40:450–474, 2001.
* [11] A. E. B. Lim and X. Y. Zhou. Optimal control of linear backward stochastic differential equations with a quadratic cost criterion. Stochastic Theory and Control, Lecture Notes in Control and Inform. Sci., 280:301–317, Berlin: Springer, 2002.
* [12] Q. Meng and M. Tang. Necessary and sufficient conditions for optimal control of stochastic systems associated with Lévy processes. Sci. China Ser. F, 52:1982–1992, 2009.
* [13] K. Mitsui and Y. Tabata. A stochastic linear-quadratic problem with Lévy processes and its application to finance. Stochastic Proecss. Appl., 118:120–152, 2008.
* [14] D. Nualart and W. Schoutens. Chaotic and predicatable representation for Lévy processes. Stochastic Process. Appl., 90:109–122, 2000.
* [15] D. Nualart and W. Schoutens. Backward stochastic differential equations and Feynman-Kac formula for Lévy processes with applications in finance. Bernouli, 7:761–776, 2001.
* [16] E. Pardoux and S. Peng. Adapted solution of a backward stochastic differential equation. Systems Control Lett., 14:55–61, 1990.
* [17] Y. Ren and X. L. Fan. Refelected backward stochastic differential equations driven by a Lévy process. ANZIAM J., 50:486–500, 2009.
* [18] Y. Ren and M. El Otmani. Generalized reflected BSDEs driven by a Lévy process and an obstacle problem for PDIEs with a nonlinear Neumann boundary condition. J. Comput. Appl. Math., 233:2027–2043, 2010.
* [19] H. Tang and Z. Wu. Stochastic differential equations and stochastic linear quadratic optiaml control problem with Lévy processes. J. Syst. Sci. Complex., 22:122–136, 2009.
* [20] S. Tang and X. Li. Necessary conditions for optimal control of stochastic systems with random jumps. SIAM J. Control Optim., 32:1447–1475, 1994.
|
arxiv-papers
| 2010-10-22T15:45:35 |
2024-09-04T02:49:14.159113
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Maoning Tang and Qi Zhang",
"submitter": "Meng Qingxin",
"url": "https://arxiv.org/abs/1010.4744"
}
|
1010.5002
|
# Talbot Workshop 2010 Talk 2: K-theory and Index Theory
Chris Kottke ckottke@math.brown.edu
From the point of view of an analyst, one of the most delightful things about
complex K-theory is that it has a nice realization by analytical objects,
namely (pseudo)differential operators and their Fredholm indices. This
connection allows quite a bit of interesting information to flow both ways:
from analysis to topology and vice versa.
This talk will try and give a sketch of this picture, and consists of three
parts or themes. The first is “the Gysin map as the index,” describing the
families index theorem of Atiyah and Singer, and how the pushforward along a
fibration in K-theory can be realized as the index of a family of operators.
The second is “spinc as an orientation,” in which I discuss Clifford algebras,
spin and spinc structures, Dirac operators and the analytic realization of the
Thom isomorphism for complex K-theory. Finally (I did not have time to get to
this part in the actual Talbot talk) I will discuss the constructions leading
to higher index maps (i.e. $K^{1}$ instead of $K^{0}$; of course this is more
interesting for real K-theory than for complex K-theory), namely Clifford-
linear differential and Fredholm operators. The best reference for almost
everything in this talk is the wonderful book [LM89] by Lawson and Michelson.
I cannot recommend this book highly enough. I will also try and give
references to original sources (essentially all of which involve Atiyah as an
author).
## 1\. Some notation and facts
First let us get down some notation and facts about K-theory that will be of
use in the following. Let $V\longrightarrow X$ be a (not necessarily complex)
vector bundle. There are many ways of constructing the Thom space of $V$,
which will be denoted $X^{V}$:
$X^{V}:=DV/SV=\overline{V}/\partial V=\mathbb{P}(V\oplus 1)/\infty.$
The first space denotes the unit disk bundle of $V$ (with respect to some
choice of metric) quotiented out by the unit sphere bundle; the second denotes
the radial compactification of $V$ quotiented out by its boundary; the third
denotes the projective bundle111Note that this constructs the fiberwise one
point compactification of $V$. associated to $V\oplus 1$ (where $1$ is the
trivial complex line bundle), modulo the section at infinity. Indeed we could
take any compactification of $V$ modulo the points added at infinity.
In any cohomology theory, the reduced cohomology of $X^{V}$ is of interest. In
index theory, however, we prefer to think of the K-theory of $X^{V}$ as
compactly supported K-theory222Though we shall only need it for vector
bundles, by compactly supported cohomology of any space $M$ can be thought of
as the relative cohomology of $(M^{+},\infty)$ where $M^{+}$ denotes the (one-
point or otherwise) compactification of $M$. This is consistent if we agree to
take $M^{+}=M\sqcup\left\\{pt\right\\}$ for compact $M$. of the space $V$:
$K^{\ast}_{c}(V):=\widetilde{{K}}^{\ast}(X^{V})=K^{\ast}(\overline{V},\partial\overline{V}).$
There is a convenient representation of relative even K-theory of a pair
$(X,A)$, where $A\subset X$ is a nice enough subset, known as the difference
bundle construction:
$K^{0}(X,A)=\left\\{E,F,\sigma\;;\;\sigma_{|A}:E\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}F\right\\}/\sim,$
where $E,F\longrightarrow X$ are vector bundles and $\sigma:E\longrightarrow
F$ is a bundle map covering the identity on $X$, which restricts to an
isomorphism over $A$. The equivalence relations amount to stabilization and
homotopy. Intuitively this should be clear; if $E$ and $F$ are isomorphic over
$A$, then, $[E]-[F]$ should be trivial in K-theory when restricted to $A$.
Unpacking this in the case of compactly supported K-theory for $V$, we
conclude that we can represent $K^{0}_{c}(V)$ by
$K^{0}_{c}(V)=\left\\{[\pi^{\ast}E,\pi^{\ast}F,\sigma]\;;\;\sigma_{V\setminus
0}:\pi^{\ast}E\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\pi^{\ast}F\right\\},$
where $\pi:V\longrightarrow X$ is the projection and $\mathbf{0}$ denotes the
zero section. Indeed, the pair $(\overline{V},\partial\overline{V})$ is
homotopy equivalent to $(V,V\setminus\mathbf{0})$, and by contractibility of
the fibers of $\overline{V}$, any vector bundles over $V$ are homotopic to
ones pulled up from the base, i.e. of the form $\pi^{\ast}E$.
Let $V\longrightarrow X$ now be a complex vector bundle. The Thom isomorphism
in K-theory states that $V$ has a K-theory orientation, so that
$\widetilde{{K}}^{\ast}(X^{V})=K^{\ast}(X)$. Specifically,
$\widetilde{{K}}^{\ast}(X^{V})=K^{\ast}_{c}(V)$ is a freely generated, rank
one module over $K^{\ast}(X)$, and in the representation of compactly
supported K-theory discussed above, the generator, or Thom class333We’ll
discuss orientation classes more generally in section 3, and we’ll interpret
the Thom class in terms of spinc structures in section 7. $\mu\in
K^{0}_{c}(V)$ has the following nice description.
###### Proposition.
The Thom class $\mu\in K^{0}_{c}(V)$ for a complex vector bundle
$V\longrightarrow X$ can be represented as the element
$\mu=[\pi^{\ast}{\textstyle{\bigwedge}}^{\mathrm{even}}V,\pi^{\ast}{\textstyle{\bigwedge}}^{\mathrm{odd}}V,\mathrm{c}\ell]\in
K^{0}_{c}(V),\qquad\mathrm{c}\ell(v)\cdot=v{\scriptstyle{\wedge}}\cdot-v^{\ast}\lrcorner\cdot,$
where the isomorphism off $0$ is given by
$\sigma(v)=\mathrm{c}\ell(v)=v{\scriptstyle{\wedge}}\cdot-v^{\ast}\lrcorner\cdot$,
the first term denoting exterior product with $v$ and the second denoting the
contraction with $v^{\ast}$ (equivalently, the inner product with $v$), with
respect to any choice of metric.
The isomorphism $\mathrm{c}\ell(v)$ is an example of Clifford multiplication,
about which we will have much more to say in section 5.
Finally, a bit about Fredholm operators. Let $H$ be a separable, infinite
dimensional Hilbert space, and recall that a bounded linear operator $P$ is
Fredholm if it is invertible modulo compact operators, which in turn are those
operators in the norm closure of the finite rank operators. Thus $P$ is
Fredholm iff there exists an operator $Q$ such that $PQ-\mathrm{Id}$ and
$QP-\mathrm{Id}$ are compact. One upshot of this is that
$\mathrm{ker}(P)\text{ and
}\mathrm{coker}(P)=\mathrm{ker}(P^{\ast})\quad\text{are finite dimensional.}$
The relationship between Fredholm operators and K-theory starts with the
observation of Atiyah [Ati67] that the space of Fredholm operators on $H$
classifies $K^{0}$:
###### Proposition (Atiyah).
$[X,\mathrm{Fred}(H)]=K^{0}(X),$
where the left hand side denotes homotopy classes of maps
$X\longrightarrow\mathrm{Fred}(H)$, the latter given the operator
topology.444The precise topology one should take on $\mathrm{Fred}(H)$ becomes
a little difficult in twisted K-theory, but (I guess!) not here.
Morally, the idea is to take a map $P:X\longrightarrow\mathrm{Fred}(H)$, and
examine the vector bundles $\mathrm{ker}(P)$ and $\mathrm{coker}(P)$, whose
fibers at a point $x\in X$ are the finite dimensional vector spaces
$\mathrm{ker}(P(x))$ and $\mathrm{coker}(P(x))$, respectively. Of course this
is a bit of a lie, since the ranks of these bundles will generally jump around
as $x$ varies; nevertheless, it is possible to stabilize the situation and see
that the class
$[\mathrm{ker}(P)]-[\mathrm{coker}(P)]\in K^{0}(X)$
is well-defined.
## 2\. Differential operators and families
One of the most important sources of such maps
$X\longrightarrow\mathrm{Fred}(H)$ are families of differential operators on
$X$. Let’s start with differential operators themselves. A working definition
of the differential operators of order $k$,
$\mathrm{Diff}^{k}(X;E,F):C^{\infty}(X;E)\longrightarrow C^{\infty}(X;F)$,
where $E$ and $F$ are vector bundles over $X$ is the following local
definition.
$\mathrm{Diff}^{k}(X;E,F)\ni
P\stackrel{{\scriptstyle\text{locally}}}{{=}}\sum_{\left|\alpha\right|\leq
k}a_{\alpha}(x)\partial_{x}^{\alpha},\quad
a_{\alpha}(x)\in\mathrm{Hom}(E_{x},F_{x}),$
where we’re employing multi-index notation:
$\alpha=(\alpha_{1},\ldots,\alpha_{n})\in\mathbb{N}^{n}$,
$\left|\alpha\right|=\sum_{i}\alpha_{i}$,
$\partial_{x}^{\alpha}=\partial_{x_{1}}^{\alpha_{1}}\cdots\partial_{x_{n}}^{\alpha_{n}}$,
where $x=(x_{1},\ldots,x_{n})$ are local coordinates on $X$.
This local expression for $P$ does not transform well under changes of
coordinates; however, the highest order terms (those with
$\left|\alpha\right|=k$) do behave well. If we consider
$\partial_{x}^{\alpha}\in\mathrm{Sym}^{\left|\alpha\right|}T_{x}X$, we can
view it as a monomial map $T_{x}^{\ast}X\longrightarrow\mathbb{R}$ of order
$\left|\alpha\right|$. If $x=(x_{1},\ldots,x_{n})$ are coordinates on $X$
inducing coordinates $(x,\xi)=(x_{1},\ldots,x_{n},\xi_{1},\ldots,\xi_{n})$ on
$T^{\ast}X$, the monomial obtained is just
$\partial_{x}^{\alpha}=\xi^{\alpha}=\xi_{1}^{\alpha_{1}}\cdots\xi_{n}^{\alpha_{n}}.$
Summing up all the terms of order $k$ gives us a homogeneous polynomial of
order $k$, which because of the $a_{\alpha}$ term is a homogeneous polynomial
on $T_{x}^{\ast}X$ valued in $\mathrm{Hom}(E_{x},F_{x})$. The claim is that
this principal symbol
$\sigma(P)(x,\xi)=\sum_{\left|\alpha\right|=k}a_{\alpha}(x)\xi^{\alpha}\in
C^{\infty}(T^{\ast}X;\mathrm{Hom}(\pi^{\ast}E,\pi^{\ast}F))$
is well-defined.
An operator is elliptic if its principal symbol is invertible away from the
zero section $\mathbf{0}\in T^{\ast}X$. The canonical example of an elliptic
operator is $\Delta$, the Laplacian (on functions, say), a second order
operator whose principal symbol is $\sigma(\Delta)(\xi)=\left|\xi\right|^{2}$,
where $\xi\in T^{\ast}X$ and the norm comes from a Riemannian metric. The
canonical non-example on $\mathbb{R}\times X$ is
$\Box=\partial_{t}^{2}-\Delta$, the D’Alembertian or wave-operator, whose
principal symbol is $\sigma(\Box)=\tau^{2}-\left|\sigma\right|^{2}$, where
$(\tau,\sigma)\in T^{\ast}\mathbb{R}\times T^{\ast}X$, which vanishes on the
(light) cones $\left\\{\tau=\pm\left|\xi\right|\right\\}$.
The reader whose was paying particularly close attention earlier will note
that the symbol of an elliptic differential operator is just the right kind of
object to represent an element in the compactly supported K-theory555In fact,
any element of $K_{c}^{0}(T^{\ast}X)$ can be represented as the symbol of an
elliptic pseudodifferential operator, though we shall not discuss these here.
of $T^{\ast}X$ since it is invertible away from $\mathbf{0}\subset T^{\ast}X$:
$P\text{ elliptic}\implies[\pi^{\ast}E,\pi^{\ast}F,\sigma(P)]\in
K_{c}^{0}(T^{\ast}X).$
For our purposes, the other important feature of an elliptic operator $P$ on a
compact manifold is that it extends to a Fredholm operator
$P:L^{2}(X;E)\longrightarrow L^{2}(X;F)$. Actually, this is a bit of a lie,
since if $k>0$, $P\in\mathrm{Diff}^{k}(X;E,F)$ is unbounded on $L^{2}(X;E)$,
and we should really consider it acting on its maximal domain in $L^{2}(X;E)$,
which is the Sobolev space $H^{k}(X;E)$ which itself has a natural Hilbert
space structure. However, since the order of operators is immaterial as far as
index theory is concerned, we will completely ignore this issue for the rest
of this note, pretending all operators in sight are of order zero666In fact it
is always possible to compose $P$ with an invertible pseudodifferential
operator (of order $-k$) so that the composite has order zero, without
altering the index of $P$. One such choice is $(1+\Delta)^{-k/2}$, another is
$(1+P^{\ast}P)^{-1/2}$., which act boundedly on $L^{2}$.
Now let $X\longrightarrow Z$ be a fibration of compact manifolds with fibers
$X_{z}\cong Y$. A family of differential operators with respect to
$X\longrightarrow Z$, is just a set of differential operators on (vector
bundles over) the fibers $X_{z}$, parametrized smoothly by the base $Z$. For a
formal definition, take the principal $\mathrm{Diffeo}(Y)$ bundle
$\mathcal{P}\longrightarrow Z$ such that
$X=\mathcal{P}\times_{\mathrm{Diffeo}(Y)}Y$; then the differential operator
families of order $k$ are obtained as777I will be sloppy about distinguishing
between families vector bundles on the fibers and vector bundles on the total
space $X$. In fact they are the same.
$\mathrm{Diff}^{k}(X/Z;E_{1},E_{2})=\mathcal{P}\times_{\mathrm{Diffeo}(Y)}\mathrm{Diff}^{k}(Y;E_{1},E_{2}).$
As before, there is a principal symbol map
$\mathrm{Diff}^{k}(X/Z;E_{1},E_{2})\ni P\longmapsto\sigma(P)\in
C^{\infty}(T^{\ast}(X/Z);\mathrm{Hom}(\pi^{\ast}E_{1},\pi^{\ast}E_{2})),$
where $T^{\ast}(X/Z)$ denotes the vertical (a.k.a. fiber) cotangent bundle.
Once again $P$ is elliptic if $\sigma(P)$ is invertible away from the zero
section; if this is the case, $P$ extends to a family of Fredholm operators on
the Hilbert space bundles
$\mathcal{H}_{i}\longrightarrow Z\quad i=1,2\quad\text{with fiber
$(\mathcal{H}_{i})_{z}=L^{2}(X_{z};E_{i})$.}$
By Kuiper’s theorem that the unitary group of an infinite dimensional Hilbert
space is contractible, the bundles $\mathcal{H}_{i}$ are trivializable, so
trivializing and identifying $\mathcal{H}_{1}$ and $\mathcal{H}_{2}$ (all
separable, infinite dimensional Hilbert spaces are isomorphic), we obtain a
map
$P:Z\longrightarrow\mathrm{Fred}(H),\quad H\cong L^{2}(Y)$
which must therefore have an index in the even K-theory of $Z$:
$\mathrm{Diff}^{\ast}(X/Z;E,F)\ni P\text{
elliptic}\implies\mathrm{ind}(P)=[\mathrm{ker}(P)]-[\mathrm{coker}(P)]\in
K^{0}(Z).$
Just as in the case of the single operator, the principal symbol of the family
$P$ represents a class in compactly supported K-theory of $T^{\ast}(X/Z)$:
$[\pi^{\ast}E,\pi^{\ast}F,\sigma(P)]\in K^{0}_{c}(T^{\ast}(X/Z)).$
We will come back to the relationship between these two objects in a moment;
for now you should think of the index as an assignment which maps
$[\pi^{\ast}E,\pi^{\ast}F,\sigma(P)]\in K^{0}_{c}(T^{\ast}(X/Z))$ to
$\mathrm{ind}(P)=[\mathrm{ker}(P)]-[\mathrm{coker}(P)]\in K^{0}(Z)$. This is
well-defined since any two elliptic operators with the same principal symbol
are homotopic through elliptic (hence Fredholm) operators, and since the index
is homotopy invariant, any two choices of operators $P,P^{\prime}$ with the
same symbol $\sigma(P)=\sigma(P^{\prime})$ will have the same index in
$K^{0}(Z)$.
## 3\. Gysin maps
In the first Talbot talk, Jesse Wolfson discussed the Gysin map in K-theory
associated to an embedding. We will need a similar kind of Gysin map
associated to fibrations. Let $p:X\longrightarrow Z$ be a smooth fibration
with $Z$ compact but not necessarily having compact fibers. By the theorem of
Whitney, we can embed any manifold into $\mathbb{R}^{N}$ for sufficiently
large $N$, and since $Z$ is compact, this can be done fiberwise to obtain an
embedding of fibrations from $X$ into a trivial fibration:
$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{Z\times\mathbb{R}^{N}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{pr}_{1}}$$\textstyle{Z}$
Let $\nu\longrightarrow X$ denote the normal bundle to $X$ with respect to
this embedding; by the collar neighborhood theorem it is isomorphic to an open
neighborhood of $X$ in $Z\times\mathbb{R}^{N}$. The open embedding
$i:\nu\hookrightarrow Z\times\mathbb{R}^{N}$ induces a wrong way map
$\tilde{i}:\Sigma^{N}Z\longrightarrow X^{\nu}$
by adding points at infinity and considering the quotient map
$Z\times\mathbb{R}^{N}/\infty\longrightarrow\nu/\infty$. Thus we obtain a
Gysin (a.k.a. “wrong way,” “umkehr,” “pushforward,” “shriek”) map
$\tilde{i}^{\ast}:\widetilde{{h}}^{\ast}(X^{\nu})\longrightarrow\widetilde{{h}}^{\ast}(\Sigma^{N}Z)=h^{\ast-N}(Z)$
in any generalized cohomology theory $h^{\ast}(\cdot)$. If the fibration is
oriented (in a sense defined below), we will actually obtain a map from the
cohomology of $X$ to that of $Z$.
Let $V\longrightarrow X$ be a vector bundle. We say that $V$ has an
orientation for the cohomology theory $h^{\ast}$ (which we are assuming is
multiplicative) if there is a global (Thom) class
$\mu\in\widetilde{{h}}^{n}\left(X^{V}\right)$ which restricts to the
multiplicative unit
$\mu_{x}\in\widetilde{{h}}^{n}(X_{x}^{V})=\widetilde{{h}}^{n}(S^{n})=\widetilde{{h}}^{0}(\mathrm{pt})$
(here $n$ is the rank of the vector bundle); if such a class exists, we have a
Thom isomorphism
$h_{c}^{\ast}(X)\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\widetilde{{h}}^{\ast+n}\left(X^{V}\right).$
Specifically, $\widetilde{{h}}^{\ast}(X^{V})$ is a freely generated module
over $h_{c}^{\ast}(X)$ with generator $\mu$.
We say a fibration $X\longrightarrow Z$ is oriented with respect to the
cohomology theory if $T(X/Z)\longrightarrow X$ has an $h^{\ast}$ orientation.
Indeed, if this is the case, the orientation on $T(X/Z)\longrightarrow X$
induces888It is a theorem that if $\alpha$ and $\beta$ are vector bundles over
$X$, than orientability of any two of $\alpha,\beta,\alpha\oplus\beta$ implies
orientability of the third. one on $\nu\longrightarrow X$, and we obtain the
Gysin map associated to an oriented fibration
$p_{!}:h_{c}^{\ast}(X)\longrightarrow h^{\ast-n}(Z)$
via the composition
$h_{c}^{\ast}(X)\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\widetilde{{h}}^{\ast+(N-n)}(X^{\nu})\stackrel{{\scriptstyle\tilde{i}^{\ast}}}{{\longrightarrow}}h^{\ast-n}(Z)$.
Note in particular that the degree shifts by $N$ cancel; indeed the Gysin map
is completely independent of the choice of embedding $X\longrightarrow
Z\times\mathbb{R}^{N}$.
## 4\. The Gysin map as the index
Now let us return to families of elliptic differential operators. Given
$P\in\mathrm{Diff}^{k}(X/Z;E,F)$, we have the element
$[\pi^{\ast}E,\pi^{\ast}F,\sigma(P)]\in K^{0}_{c}(T^{\ast}(X/Z))$, which maps
to the index $\mathrm{ind}(P)=[\mathrm{ker}(P)]-[\mathrm{coker}(P)]\in
K^{0}(Z)$. The famous index theorem of Atiyah and Singer [AS68] [AS71] can now
be stated quite simply.
###### Theorem (Atiyah-Singer).
The index map
$\mathrm{ind}:[\pi^{\ast}E,\pi^{\ast}F,\sigma(P)]\in
K^{0}_{c}(T^{\ast}(X/Z))\longrightarrow[\mathrm{ker}(P)]-[\mathrm{coker}(P)]\in
K^{0}(Z)$
coincides with the Gysin map $K_{c}^{0}(T^{\ast}(X/Z))\longrightarrow
K^{0}(Z)$ associated to the oriented fibration999We’ll see below why this
fibration is canonically oriented.
$p:T^{\ast}(X/Z)\longrightarrow Z.$
In short,
$\mathrm{ind}=p_{!}:K^{0}_{c}(T^{\ast}(X/Z))\longrightarrow K^{0}(Z).$
In particular, we can recover the integer index
$\mathrm{ind}(P)=\mathrm{dim}\;\mathrm{ker}(P)-\mathrm{dim}\;\mathrm{coker}(P)\in\mathbb{Z}$
of a single operator on a compact manifold $X$ from the case $Z=\mathrm{pt}$;
from the unique map $X\longrightarrow\mathrm{pt}$, we get an oriented
fibration $T^{\ast}X\longrightarrow\mathrm{pt}$ and a Gysin map
$p_{!}:K^{0}_{c}(T^{\ast}X)\longrightarrow K^{0}(\mathrm{pt})=\mathbb{Z}$.
Let us unpack this a bit. We have the fibration $T^{\ast}(X/Z)\longrightarrow
Z$ which factors as $T^{\ast}(X/Z)\longrightarrow X\longrightarrow Z$. As
noted above there is always an embedding of $X$ into a trivial Euclidean
bundle $Z\times\mathbb{R}^{N}\longrightarrow Z$. This induces an embedding
$T^{\ast}(X/Z)\hookrightarrow Z\times
T^{\ast}\mathbb{R}^{N}=Z\times\mathbb{R}^{2N}$, so we have the following
situation
---
$\textstyle{T^{\ast}(X/Z)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Z\times\mathbb{R}^{2N}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Z\times\mathbb{R}^{N}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Z}$
The point is that, because the embedding $T^{\ast}(X/Z)\hookrightarrow
Z\times\mathbb{R}^{2N}$ comes from an embedding of $X$, the normal bundle
$\nu\longrightarrow T^{\ast}(X/Z)$ carries a canonical complex structure.
Indeed, if we denote by $NX\longrightarrow X$ the normal bundle of $X$ with
respect to $X\hookrightarrow Z\times\mathbb{R}^{N}$, then $\nu$ is isomorphic
to two copies of $NX$:
$\nu\cong NX\oplus NX\cong NX\otimes\mathbb{C},$
one copy representing the normal to the base $X$, and the other copy
representing the normal to the fiber $T^{\ast}(X/Z)_{x},x\in X$. Thus we
conclude that, whether or not the fibration $X\longrightarrow Z$ has a
K-theory orientation, $T^{\ast}(X/Z)\longrightarrow Z$ always has a K-theory
orientation, so we have a Gysin map
$p_{!}:K^{0}_{c}(T^{\ast}(X/Z))\longrightarrow K^{0}(Z),$
which coincides with the index map on elements in $K^{0}_{c}(T^{\ast}(X/Z))$
which represent symbols of elliptic differential operators101010As remarked in
previous footnotes, it is desirable to broaden one’s focus to include
pseudodifferential operators, for then every element of
$K^{0}_{c}(T^{\ast}(X/Z))$ can be represented as the symbol of an elliptic
pseudodifferential operator which extends to a Fredholm operator, so the index
map extends to all of $K^{0}_{c}(T^{\ast}(X/Z))$ and is equal to the Gysin
map.. Note that there is no degree shift since $T^{\ast}(X/Z)$ has even
dimensional fibers over $Z$ and K-theory is 2-periodic.
In the next sections we shall discuss the conditions necessary for
$X\longrightarrow Z$ to have a K-theory orientation, and how to realize the
Gysin map $K^{0}(X)\longrightarrow K^{0}(Z)$ in terms of elliptic differential
operators and their indices; this will involve a digression through Clifford
algebras, spin groups, spin structures and Dirac operators. Later we will see
how to deal with objects in odd K-theory.
## 5\. Clifford algebras
Let $(V,q)$ be a finite dimensional vector space over $\mathbb{R}$ with $q$ a
non-degenerate quadratic form. The (real) Clifford algebra
$\mathrm{C}\ell(V,q)$ is the universal object with respect to maps
$f:V\longrightarrow A$, where $A$ is an associative algebra with unit,
satisfying $f(v)\cdot f(v)=-q(v)1$. It can be constructed as a quotient of the
tensor algebra:
$\mathrm{C}\ell(V,q)=\bigoplus_{n=0}^{\infty}V^{\otimes
n}/\mathcal{I},\quad\mathcal{I}=\left\langle v\otimes v+q(v)1\right\rangle$
where $\mathcal{I}$ is the ideal generated by all elements of the form
$v\otimes v+q(v)1$.
As a vector space (but not as an algebra unless $q\equiv 0$!)
$\mathrm{C}\ell(V,q)$ is isomorphic to the exterior algebra
$\bigoplus_{n=0}^{\mathrm{dim}}(V)\bigwedge^{n}V$; in particular if
$\left\\{e_{i}\right\\}$ is a basis of $V$, then $\left\\{e_{i_{1}}\cdots
e_{i_{k}}\;;\;i_{1}<\cdots<i_{k}\right\\}$ form a basis for
$\mathrm{C}\ell(V,q)$, which under multiplication are subject to the relation
$e_{i}e_{j}=-e_{j}e_{i}-2q(e_{i},e_{j})$
where we denote also by $q$ the bilinear form associated to $q$. Computations
are easiest when $\left\\{e_{i}\right\\}$ is an orthonormal basis, whence the
multiplication simplifies to the rules $e_{i}e_{j}=-e_{j}e_{i},i\neq j$ and
$e_{i}^{2}=-1$.
In fact $\mathrm{C}\ell(V,q)$ is a $\mathbb{Z}_{2}$-graded algebra. Indeed, if
we let $\mathrm{C}\ell^{0}(V,q)$ and $\mathrm{C}\ell^{1}(V,q)$ be the images
of $\bigwedge^{\mathrm{even}}V$ and $\bigwedge^{\mathrm{odd}}V$, respectively,
under the vector space isomorphism with the exterior algebra, it is easy to
check that
$\mathrm{C}\ell(V,q)=\mathrm{C}\ell^{0}(V,q)\oplus\mathrm{C}\ell^{1}(V,q)\quad\text{and}\quad\mathrm{C}\ell^{i}(V,q)\cdot\mathrm{C}\ell^{j}(V,q)\subset\mathrm{C}\ell^{(i+j)\mathrm{\;mod\;}2}(V,q).$
Alternatively, the involution $\alpha:V\longrightarrow V:v\longmapsto-v$
extends multiplicatively to an involution on all of $\mathrm{C}\ell(V,q)$:
$\alpha:\mathrm{C}\ell(V,q)\longrightarrow\mathrm{C}\ell(V,q),\quad\alpha^{2}=\mathrm{Id},\quad\alpha:V\ni
v\longmapsto-v\in V$
and we can define $\mathrm{C}\ell^{0}(V,q)$ and $\mathrm{C}\ell^{1}(V,q)$ as
the positive and negative eigenspaces of $\alpha$, respectively. Note in
particular that $V\subset\mathrm{C}\ell^{1}(V,q)$ as a vector subspace.
We can also form the complex Clifford algebra $\mathbb{C}\ell(V,q)$ by
tensoring up with $\mathbb{C}$:
$\mathbb{C}\ell(V,q):=\mathrm{C}\ell(V\otimes\mathbb{C},q_{\mathbb{C}})\cong\mathrm{C}\ell(V,q)\otimes\mathbb{C}.$
Complex Clifford algebras are those of primary importance for this talk, since
it concerns (mostly) complex K-theory. There is a parallel relationship
between real Clifford algebras and real K-theory.
We will denote the Clifford algebra of Euclidean $n$-space by
$\mathrm{C}\ell_{n}:=\mathrm{C}\ell(\mathbb{R}^{n},\left\langle\cdot,\cdot\right\rangle)$
and call it the (real) Clifford algebra of dimension $n$. Similarly, we will
denote the complex Clifford algebra of dimension $n$ by
$\mathbb{C}\ell_{n}:=\mathbb{C}\ell(\mathbb{R}^{n},\left\langle\cdot,\cdot\right\rangle)=\mathrm{C}\ell(\mathbb{C}^{n},\left\langle\cdot,\cdot\right\rangle).$
It is a rather nice fact that complex Clifford algebras are isomorphic to
matrix algebras111111In fact real Clifford algebras are also isomorphic to
matrix algebras over $\mathbb{R}$, $\mathbb{C}$ or $\mathbb{H}$, or a direct
sum of two such, with an 8-periodic pattern related to Bott periodicity in the
real setting..
###### Proposition.
$\mathbb{C}\ell_{2n}\cong
M(2^{n},\mathbb{C})\quad\text{and}\quad\mathbb{C}\ell_{2n+1}\cong
M(2^{n},\mathbb{C})\oplus M(2^{n},\mathbb{C})$
where $M(k,\mathbb{C})$ denotes the algebra of $k\times k$ complex matrices.
As a consequence of this, the representations of $\mathbb{C}\ell_{n}$ are easy
to classify: $\mathbb{C}\ell_{2n}$ has a unique irreducible representation of
(complex) dimension $2^{n}$ given by the obvious action of
$M(2^{n},\mathbb{C})$ on $\mathbb{C}^{2^{n}}$; and $\mathbb{C}\ell_{2n+1}$ has
two distinct irreps of dimension $2^{n}$ corresponding to action of one or the
other of the factors of $M(2^{n},\mathbb{C})$.
The last tidbit we shall need is the algebra isomorphism
$\mathbb{C}\ell_{n-1}\cong\mathbb{C}\ell^{0}_{n}$
(note that $\mathbb{C}\ell^{0}_{n}$ is a subalgebra of $\mathbb{C}\ell_{n}$).
This is obtained by considering the map
$f:\mathbb{R}^{n-1}\longrightarrow\mathbb{C}\ell^{0}_{n}$ given on basis
vectors by
$\mathbb{R}^{n-1}\ni e_{i}\longmapsto
f(e_{i})=e_{i}\,e_{n}\in\mathbb{C}\ell^{0}_{n}.$
This satisfies $f(v)\cdot f(v)=v\,e_{n}\,v\,e_{n}=-q(v)1$ and thus generates
an algebra map $\mathbb{C}\ell_{n-1}\longrightarrow\mathbb{C}\ell^{0}_{n}$ by
the universal property, which is easily seen to be bijective.
This isomorphism leads to an equivalence between graded $\mathbb{C}\ell_{n}$
modules and ungraded $\mathbb{C}\ell_{n-1}$ modules. In the one direction, if
$M=M^{0}\oplus M^{1}$ is a graded module over $\mathbb{C}\ell_{n}$, then
$M^{0}$ and $M^{1}$ are (possibly inequivalent!) modules over
$\mathbb{C}\ell_{n-1}\cong\mathbb{C}\ell^{0}_{n}$. In the other direction,
given a $\mathbb{C}\ell_{n-1}$-module $M$, the we can form
$M\otimes_{\mathbb{C}\ell^{0}_{n}}\mathbb{C}\ell_{n}$, which is a graded
module over $\mathbb{C}\ell_{n}$.
Putting this fact together with the classification of irreps above, we see
that, for even Clifford algebras, there is a unique irreducible
$\mathbb{C}\ell_{2n}$ module
$\mathbb{S}_{2n}=\mathbb{S}^{+}_{2n}\oplus\mathbb{S}^{-}_{2n}$
which splits as the two inequivalent irreps of
$\mathbb{C}\ell_{2n}^{0}\cong\mathbb{C}\ell_{2n-1}$. Hence it has two
inequivalent gradings, either
$\mathbb{S}^{0}\oplus\mathbb{S}^{1}=\mathbb{S}^{+}\oplus\mathbb{S}^{-}$ or
$\mathbb{S}^{0}\oplus\mathbb{S}^{1}=\mathbb{S}^{-}\oplus\mathbb{S}^{+}$. On
the other hand, for odd Clifford algebras, there is a unique graded
$\mathbb{C}\ell_{2n+1}$ module
$\mathbb{S}_{2n+1}=\mathbb{S}^{+}_{2n+1}\oplus\mathbb{S}^{-}_{2n+1}$
since both $\mathbb{S}^{+}_{2n+1}$ and $\mathbb{S}^{-}_{2n+1}$ must be
equivalent to the unique irrep of
$\mathbb{C}\ell^{0}_{2n+1}\cong\mathbb{C}\ell_{2n}$.
## 6\. Spin and Spinc groups
Given $(V,q)$, the group $\mathrm{Spin}(V,q)$ is the universal cover of the
special orthogonal group $\mathrm{SO}(V,q)$. We can find it inside the
Clifford algebra of $V$ as follows. Let $\mathrm{C}\ell(V,q)^{\times}$ be the
group of units inside $\mathrm{C}\ell(V,q)$. This group acts on
$\mathrm{C}\ell(V,q)$ by a twisted conjugation:
$\mathrm{C}\ell(V,q)^{\times}\times\mathrm{C}\ell(V,q)\ni(x,v)\longmapsto
x\,v\,\alpha(x)^{-1}$
where $\alpha_{|\mathrm{C}\ell^{i}(V,q)}=(-1)^{i}\mathrm{Id}$ is the
involution from earlier. The Clifford group
$\Gamma\subset\mathrm{C}\ell(V,q)^{\times}$ is the subgroup which fixes the
subspace $V\subset\mathrm{C}\ell(V,q)$; it also preserves the quadratic form
$q$ and hence maps to the orthogonal group $\mathrm{O}(V,q)$ with kernel
$\mathbb{R}^{\times}$:
$1\longrightarrow\mathbb{R}^{\times}\longrightarrow\Gamma\longrightarrow\mathrm{O}(V,q)\longrightarrow
1.$
Up to a scalar factor, there is a natural choice of multiplicative norm
$\left|\cdot\right|:\Gamma\longrightarrow\mathbb{R}^{\times}$, and the Spin
group of $(V,q)$, $\mathrm{Spin}(V,q)$, is defined to be the subgroup of norm
1 elements covering121212The subgroup of norm 1 elements covering
$\mathrm{O}(V,q)$ is called $\mathrm{Pin}(V,q)$, a joke which is apparently
due to Serre. the special orthogonal group $\mathrm{SO}(V,q)$:
$\mathrm{Spin}(V,q):=\left\\{u\in\Gamma\;;\;\left|u\right|=1,u\text{ maps to
}\mathrm{SO}(V,q)\right\\}\subset\Gamma.$
Alternatively, it can be defined as the subgroup of
$\mathrm{C}\ell(V,q)^{\times}$ generated by finite products of the form
$v_{1}\cdots v_{2n}$, $v_{i}\in V$, $q(v_{i})=1$ with an even number of
factors. We have the exact sequence
$1\longrightarrow\left\\{\pm
1\right\\}\longrightarrow\mathrm{Spin}(V,q)\longrightarrow\mathrm{SO}(V,q)\longrightarrow
1$
and $\mathrm{Spin}(V,q)$ is compact if $q$ has positive signature. It also
lies in the 0-graded component of $\mathrm{C}\ell(V,q)$:
$\mathrm{Spin}(V,q)\subset\mathrm{C}\ell^{0}(V,q)$
The spin group of $(\mathbb{R}^{n},\left\langle\cdot,\cdot\right\rangle)$ will
simply be called the spin group of dimension $n$, and denoted
$\mathrm{Spin}_{n}:=\mathrm{Spin}(\mathbb{R}^{n},\left\langle\cdot,\cdot\right\rangle).$
From now on, we focus on the even dimensional case. Since
$\mathrm{Spin}_{2n}\subset\mathrm{C}\ell_{2n}$, we have a complex
representation coming from the irreducible
$\mathbb{C}\ell_{2n}=\mathrm{C}\ell_{2n}\otimes\mathbb{C}$ module
$\mathbb{S}_{2n}$. In fact, since
$\mathrm{Spin}_{2n}\subset\mathrm{C}\ell^{0}_{2n}$, this splits as two
inequivalent, irreducible half spin representations
$\mathbb{S}_{2n}=\mathbb{S}^{+}_{2n}\oplus\mathbb{S}^{-}_{2n}.$
We denote these fundamental representations by
$\rho_{1/2}^{\pm}:\mathrm{Spin}_{2n}\longrightarrow\mathrm{GL}(\mathbb{S}^{\pm}_{2n}).$
The representation
$\rho:=\rho^{+}_{1/2}\oplus\rho^{-}_{1/2}:\mathrm{Spin}_{2n}\longrightarrow\mathrm{GL}(\mathbb{S}_{2n})=\mathrm{GL}(\mathbb{S}^{+}_{2n}\oplus\mathbb{S}^{-}_{2n})$
is called the fundamental spin representation and vectors in $\mathbb{S}_{2n}$
are called spinors. We’ll see the importance of spinors in defining K-theory
orientation classes in section 7.
As we are primarily interested in complex representations, there is another
group inside $\mathbb{C}\ell_{n}$ to discuss. The Spinc group associated to
$(V,q)$ is the quotient
$\mathrm{Spin}^{c}(V,q)=\mathrm{Spin}(V,q)\times_{\mathbb{Z}_{2}}\mathrm{U}_{1}$
where the $\mathbb{Z}_{2}$ is generated by the element
$(-1,-1)\in\mathrm{Spin}(V,q)\times\mathrm{U}_{1}$. We have the exact sequence
$1\longrightarrow\left\\{\pm
1\right\\}\longrightarrow\mathrm{Spin}^{c}(V,q)\longrightarrow\mathrm{SO}(V,q)\times\mathrm{U}_{1}\longrightarrow
1$
As with the spin group, $\mathrm{Spin}^{c}(V,q)$ sits inside
$\mathbb{C}\ell(V,q)$,
$\mathrm{Spin}^{c}(V,q)\subset\mathbb{C}\ell^{0}(V,q)\subset\mathbb{C}\ell(V,q)=\mathrm{C}\ell(V,q)\otimes\mathbb{C}$
and, for the canonical spinc groups of even dimension,
$\mathrm{Spin}^{c}_{2n}:=\mathrm{Spin}^{c}(\mathbb{R}^{2n},\left\langle\cdot,\cdot\right\rangle)$,
we have a fundamental spinc representation
$\rho:=\rho^{+}_{1/2}\oplus\rho^{-}_{1/2}:\mathrm{Spin}^{c}_{2n}\longrightarrow\mathrm{GL}(\mathbb{S}_{2n})=\mathrm{GL}(\mathbb{S}^{+}_{2n}\oplus\mathbb{S}^{-}_{2n})$
on the spinors $\mathbb{S}_{2n}$, where again, $\mathbb{S}_{2n}$ is the unique
irreducible $\mathbb{C}\ell_{2n}$ module.
## 7\. Spin(c) structures
Let us now transfer this to a manifold setting. We’ll define spin and spinc
structures on a manifold and see that they produce K-theory orientations on
the tangent bundle. Given a Riemannian manifold $(X,g)$, we can form the
(complex) Clifford bundle131313Of course we also have the real Clifford bundle
$\mathrm{C}\ell(X)\longrightarrow X$, but we shall not need it for our
applications. More generally, we can define Clifford bundles
$\mathrm{C}\ell(V)\longrightarrow X$ and $\mathbb{C}\ell(V)\longrightarrow X$
whenever $V\longrightarrow X$ is a vector bundle with inner product.
$\mathbb{C}\ell(X)\longrightarrow
X,\quad\mathbb{C}\ell(X)_{x}:=\mathbb{C}\ell(T_{x}X,g_{x}),\text{ for all
$x\in X$,}$
which is a bundle of complex Clifford algebras of dimension $n=\dim(X)$. A
complex vector bundle $E\longrightarrow X$ is called a Clifford module if it
carries a fiberwise action
$\mathrm{c}\ell:\mathbb{C}\ell(X)\longrightarrow\mathrm{End}(E).$
Such an action, if it exists, will be called Clifford multiplication.
Of course, over each $x\in X$, any Clifford module decomposes as a direct sum
of irreducible modules over $\mathbb{C}\ell(X)_{x}$, but this is not
necessarily true globally. This leads us to the notion of spin and spinc
structures on $X$.
Let $P_{\mathrm{SO}}(X)\longrightarrow X$ be the frame bundle of $X$, i.e. the
principal $\mathrm{SO}_{n}$ bundle to which $TX$ is associated:
$TX=P_{\mathrm{SO}}(X)\times_{\mathrm{SO}_{n}}\mathbb{R}^{n}.$
$X$ is called a spin manifold if there exists a principal $\mathrm{Spin}_{n}$
bundle $P_{\mathrm{Spin}}(X)$ and a bundle map
$\textstyle{\mathrm{Spin}_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{SO}_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P_{\mathrm{Spin}}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P_{\mathrm{SO}}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X}$
which is a 2-sheeted cover of $P_{\mathrm{SO}}(X)$. $P_{\mathrm{Spin}}(X)$ is
called a spin structure on $X$.141414Similarly, a general vector bundle with
inner product $V\longrightarrow X$ admits a spin structure whenever
$P_{\mathrm{SO}}(V)$ admits a 2-sheeted cover $P_{\mathrm{Spin}}(V)$. The
obstruction to obtaining such a cover of $P_{\mathrm{SO}}(X)$ is the second
Stiefel-Whitney class151515This is straightforward to see by trying to patch
$P_{\mathrm{Spin}}(X)$ together over a trivializing cover. In order to do so,
we must have a Cech “cohomology class” (I’m using quotes since the
coefficients are in a nonabelian group; nevertheless
$H^{1}(X;\mathrm{SO}_{n})$ is a based set) in $H^{1}(X;\mathrm{SO}_{n})$ which
is the image of a class in $H^{1}(X;\mathrm{Spin}_{n})$. Using the long exact
sequence associated to
$1\longrightarrow\mathbb{Z}_{2}\longrightarrow\mathrm{Spin}_{n}\longrightarrow\mathrm{SO}_{n}\longrightarrow
1,$ the image of this class in $H^{2}(X;\mathbb{Z}_{2})$ is exactly
$w_{2}(X)\in H^{2}(X;\mathbb{Z}_{2})$. of $X$:
$X\text{ is spin iff }w_{2}(X)\equiv 0,$
and if $X$ is spin, the possible spin structures of $X$ are parametrized by
$H^{1}(X,\mathbb{Z}_{2})$.
A spinc structure on $X$ consists of a complex line bundle $L\longrightarrow
X$ and a lift
$\textstyle{\mathrm{Spin}^{c}_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{SO}_{n}\times\mathrm{U}_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P_{\mathrm{Spin}^{c}}(X,L)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P_{\mathrm{SO}}(X)\times
P_{U_{1}}(L)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X}$
which is a 2-sheeted covering of $P_{\mathrm{SO}}(X)\times P_{U_{1}}(L)$,
where $P_{U_{1}}(L)$ is the structure bundle of $L$. $X$ is called a spinc
manifold if such a lift exists. The obstruction to obtaining
$P_{\mathrm{Spin}^{c}}(X,L)$ is the class $w_{2}(X)+c_{1}(L)(\text{mod }2)$;
thus
$X\text{ is spin${}^{c}$ iff }w_{2}(X)=\alpha(\text{mod }2)\text{ for some
}\alpha\in H^{2}(X,\mathbb{Z})$
Being spinc is a weaker condition than being spin:
###### Proposition.
If $X$ is spin, then it has a canonical spinc structure associated to the
trivial line bundle, so
$X\text{ spin }\implies X\text{ spin${}^{c}$.}$
Additionally, any (almost) complex manifold has a canonical spinc structure.
###### Proposition.
If $X$ is an almost complex manifold, then $w_{2}(X)=c_{1}(X)(\text{mod }2)$,
and $X$ has a canonical spinc structure associated to the determinant line
bundle ${\textstyle{\bigwedge}}^{n}_{\mathbb{C}}TX$ (which satisfies
$c_{1}\left({\textstyle{\bigwedge}}^{n}TX\right)=c_{1}(X)$).
Note that if $X$ is both spin and almost complex, the spinc structure coming
from the spin structure is generally not the same as the one coming from the
almost complex structure.
The importance of spinc structures is the following proposition, which says
that, given a spinc structure, Clifford modules are globally reducible, and in
bijection with the set of $\mathbb{C}\ell_{n}$ modules.
###### Proposition.
If $X$ is spinc, then every Clifford module $E\longrightarrow X$ has the form
$E=P_{\mathrm{Spin}^{c}}(X,L)\times_{\sigma}F,$
where $\sigma:\mathrm{Spin}^{c}_{n}\longrightarrow\mathrm{GL}(F)$ is a
representation of $\mathrm{Spin}^{c}_{n}$ which extends to a representation of
$\mathbb{C}\ell_{n}$ (here $n=\dim(X)$).
Turning the construction around, we obtain Clifford modules over a spinc
manifold $X$ for every representation of $\mathbb{C}\ell_{n}$; in particular
for $\dim(X)=2n$, we have the complex spinor bundle
$\mathbb{S}(X)=\mathbb{S}^{+}(X)\oplus\mathbb{S}^{-}(X)=P_{\mathrm{Spin}^{c}}(X,L)\times_{\rho^{+}_{1/2}\oplus\rho^{-}_{1/2}}\mathbb{S}^{+}\oplus\mathbb{S}^{-}$
with the (graded) action
$\mathrm{c}\ell:\mathbb{C}\ell(X)\longrightarrow\mathrm{End}_{\mathrm{gr}}(\mathbb{S}^{+}(X)\oplus\mathbb{S}^{-}(X)).$
Finally, we can get to the main point about spinc structures, which is that
they allow us to construct K-theory orientation classes161616In fact the
proposition is valid for general vector bundles $V\longrightarrow X$ with
spinc structure; the analogous element
$\mu=[\pi^{\ast}\mathbb{S}^{+}(V),\pi^{\ast}\mathbb{S}^{-}(V),\mathrm{c}\ell]\in
K^{0}_{c}(V)$ is a Thom class. for $T^{\ast}X\longrightarrow X$.
###### Proposition.
If an even dimensional manifold $X$ has a spinc structure and
$\mathbb{S}(X)=\mathbb{S}^{+}(X)\oplus\mathbb{S}^{-}(X)$ is the bundle of
spinors, then
$\mu=[\pi^{\ast}\mathbb{S}^{+}(X),\pi^{\ast}\mathbb{S}^{-}(X),\mathrm{c}\ell]\in
K^{0}_{c}(T^{\ast}X)$
is an orientation/Thom class for complex K-theory, so
$K^{\ast}_{c}(T^{\ast}X)$ is freely generated by $\mu$ as a module over
$K^{\ast}(X)$, and
$K^{\ast}_{c}(T^{\ast}X)\cong K^{\ast}(X).$
Note that $T^{\ast}X\subset\mathbb{C}\ell^{1}(X)$, so
$\mathrm{c}\ell(\xi):\mathbb{S}^{\pm}(X)_{x}\longrightarrow\mathbb{S}^{\mp}(X)_{x}$
for $(x,\xi)\in T^{\ast}X$; moreover, this multiplication is invertible with
inverse $\left|\xi\right|^{-2}\mathrm{c}\ell(\xi)$ provided $\xi\neq 0$.
As a side remark, let me point out that an analogous theorem is true for spin
structures and real K-theory: If $X$ is spin and $8n$ dimensional, then
$[\pi^{\ast}\mathbb{S}^{+}(X),\pi^{\ast}\mathbb{S}^{-}(X),\mathrm{c}\ell]\in
KO^{0}_{c}(T^{\ast}X)$ is an orientation class, and $KO^{\ast}(X)\cong
KO_{c}^{\ast}(T^{\ast}X)$.
Note that if $X$ is almost complex, with the corresponding spinc structure,
then we can identify $\mathbb{S}(X)$ with
${\textstyle{\bigwedge}}^{\ast}_{\mathbb{C}}T^{\ast}X$; in this case,
$\mathbb{S}^{\pm}(X)\cong{\textstyle{\bigwedge}}^{\mathrm{even/odd}}_{\mathbb{C}}T^{\ast}X$
and
$\mathrm{c}\ell(\xi)=\xi{\scriptstyle{\wedge}}\cdot-\xi^{\ast}\lrcorner\cdot$
under this identification. Thus we recover the Thom element
$\mu=[\pi^{\ast}{\textstyle{\bigwedge}}^{\mathrm{even}}V,\pi^{\ast}{\textstyle{\bigwedge}}^{\mathrm{odd}}V,\mathrm{c}\ell]\in
K^{0}_{c}(V)$
for complex bundles, and we see that the Thom isomorphism for such bundles can
be thought of as a special case of the isomorphism for spinc bundles.
Finally, let us briefly discuss what this looks like in the setting of a
fibration $X\longrightarrow Z$. In this case the relevant Clifford bundle is
$\mathbb{C}\ell(X/Z)=\mathbb{C}\ell(T(X/Z),g)\longrightarrow X,$
and the fibration is oriented as long as $T(X/Z)\longrightarrow X$ admits a
spinc structure. Indeed, if it does, we have the orientation class
$\mu=[\pi^{\ast}\mathbb{S}^{+}(X/Z),\pi^{\ast}\mathbb{S}^{-}(X/Z),\mathrm{c}\ell]\in
K^{0}_{c}(T^{\ast}(X/Z))$
constructed from the bundles of spinors $\mathbb{S}^{\pm}(X/Z)\longrightarrow
X$.
## 8\. Dirac operators
In this section we’ll see that the orientation classes discussed above are in
fact the symbols of particularly nice (families of) elliptic differential
operators. Let $E\longrightarrow X$ be any Clifford module over $X$. Suppose
$E$ is endowed with a connection $\nabla:C^{\infty}(X;E)\longrightarrow
C^{\infty}(X;T^{\ast}X\otimes E)$ such that
$\nabla(\mathrm{c}\ell(\xi)s)=\mathrm{c}\ell(\nabla^{\mathrm{LC}}\xi)s+\mathrm{c}\ell(\xi)\nabla
s$
where $\nabla^{\mathrm{LC}}$ is the Levi-Civita connection, which extends
canonically to $\mathbb{C}\ell(X)\longrightarrow X$. Such a connection (which
always exists) is called a Clifford connection on $E$.
Given such data, we can construct a canonical first order, elliptic,
differential operator $D\in\mathrm{Diff}^{1}(X;E)$ called a Dirac operator; at
a point $x\in X$, $D$ is defined by
$D_{p}=\sum_{i}\mathrm{c}\ell(e_{i})\nabla_{e_{i}},\quad\text{ for an
orthonormal basis $\left\\{e_{i}\right\\}$ of $T_{x}X$.}$
###### Proposition.
Such a Dirac operator is an elliptic, essentially self adjoint (on
$L^{2}(X;E)$) operator with principal symbol
$\sigma(D)(\xi)=i\mathrm{c}\ell(\xi).$
If $E=E^{+}\oplus E^{-}$ is a graded $\mathbb{C}\ell(X)$ module, then $D$ has
the form
$D=\begin{pmatrix}0&D^{-}\\\ D^{+}&0\end{pmatrix}$
with $D^{+}$ and $D^{-}$ mutual adjoints.
Note that $\sigma(D^{2})=\sigma(D)^{2}=\left|\xi\right|^{2}\mathrm{Id}$, so
$D^{2}$ is a Laplacian operator on $E$.
If $X$ is a spinc manifold, we can form a canonical spinc Dirac operator
${/}\\!\\!\\!\\!D=\begin{pmatrix}0&{/}\\!\\!\\!\\!D^{-}\\\
{/}\\!\\!\\!\\!D^{+}&0\end{pmatrix}\in\mathrm{Diff}^{1}(X;\mathbb{S}^{+}(X)\oplus\mathbb{S}^{-}(X))$
acting on the spinors $\mathbb{S}(X)$, and it follows that
$[\pi^{\ast}\mathbb{S}^{+}(X),\pi^{\ast}\mathbb{S}^{-}(X),\sigma({/}\\!\\!\\!\\!D^{+})]\in
K^{0}_{c}(T^{\ast}X)$
is an orientation class for K-theory. This realizes the Thom isomorphism as
follows. We can always twist ${/}\\!\\!\\!\\!D$ by a vector bundle
$E\longrightarrow X$ by trivially extending the Clifford action to the bundle
$\mathbb{S}(X)\otimes E$ and taking a product connection to get
${/}\\!\\!\\!\\!D_{E}\in\mathrm{Diff}^{1}(X;\mathbb{S}(X)\otimes E)$. Then for
an element
$[E]-[F]\in K^{0}(X),$
the image under the Thom isomorphism $K^{0}(X)\longrightarrow
K^{0}_{c}(T^{\ast}X)$ is the element171717The better way to write this is to
use $\mathbb{Z}_{2}$ gradings everywhere. Let $\mathbb{E}=E\oplus F$
considered as a graded vector bundle and form
$\mathbb{S}(X)\hat{\otimes}\mathbb{E}$, where $\hat{\otimes}$ denotes the
graded tensor product. Then ${/}\\!\\!\\!\\!D$ extends to a graded, twisted
operator ${/}\\!\\!\\!\\!D_{\mathbb{E}}$, and the orientation class is given
by
$[\pi^{\ast}(\mathbb{S}(X)\hat{\otimes}\mathbb{E})^{+},\pi^{\ast}(\mathbb{S}(X)\hat{\otimes}\mathbb{E})^{-},\sigma({/}\\!\\!\\!\\!D_{\mathbb{E}}^{+})]\in
K^{0}_{c}(T^{\ast}X)$. We’ll talk more about gradings in section 9.
$[\pi^{\ast}\mathbb{S}^{+}(X)\otimes E,\pi^{\ast}\mathbb{S}^{-}(X)\otimes
E,\sigma({/}\\!\\!\\!\\!D_{E})]-[\pi^{\ast}\mathbb{S}^{+}(X)\otimes
F,\pi^{\ast}\mathbb{S}^{-}(X)\otimes F,\sigma({/}\\!\\!\\!\\!D_{F})]\in
K^{0}_{c}(T^{\ast}X).$
For a complex manifold $X$, you have probably already met the canonical spinc
Dirac operator. Indeed, using the identifications
$\mathbb{S}(X)\cong{\textstyle{\bigwedge}}^{\ast}_{\mathbb{C}}T^{\ast}X\cong{\textstyle{\bigwedge}}^{0,\ast}T^{\ast}X$,
one can easily see that
${/}\\!\\!\\!\\!D^{+}=\overline{\partial}+\overline{\partial}^{\ast}\in\mathrm{Diff}^{1}(X;{\textstyle{\bigwedge}}^{0,\mathrm{even}}T^{\ast}X,{\textstyle{\bigwedge}}^{0,\mathrm{odd}}T^{\ast}X)$
is just the Dolbeault operator acting from even to odd harmonic forms.
Finally, in the case of an oriented fibration $X\longrightarrow Z$, we
construct in precisely the same way the canonical family of spinc Dirac
operators
${/}\\!\\!\\!\\!D\in\mathrm{Diff}^{1}(X/Z;\mathbb{S}^{+}(X/Z)\oplus\mathbb{S}^{-}(X/Z))$
and of course
$[\pi^{\ast}\mathbb{S}^{+}(X/Z),\pi^{\ast}\mathbb{S}^{-}(X/Z),\sigma({/}\\!\\!\\!\\!D)]\in
K^{0}_{c}(T^{\ast}(X/Z))$
is the Thom class.
This gives the analytical realization of the Gysin map
$K^{0}(X)\longrightarrow K^{0}(Z)$; namely, it coincides with the analytical
index of the family of spinc Dirac operators, twisted by the given element in
$K^{0}(X)$:
$K^{0}(X)\ni[E]-[F]\longmapsto\mathrm{ind}({/}\\!\\!\\!\\!D_{E}-{/}\\!\\!\\!\\!D_{F})\in
K^{0}(Z).$
## 9\. Higher Index
We will develop two pictures of the higher $K$-groups of a manifold $X$, in
analogy to the two we’ve developed for $K^{0}(X)$, namely, the Grothendieck
group of vector bundles, and the classifying space consisting of Fredholm
operators. Really this whole story is a bit more interesting in the case of
real K-theory, and much of what we describe below will be valid if one
replaces $K^{\ast}(X)$ by $KO^{\ast}(X)$ and $\mathbb{C}\ell_{\ast}$ by
$\mathrm{C}\ell_{\ast}$ (with the obvious exception of 2 periodicity of
$\mathbb{C}\ell$ modules, which would be replaced by an analogous 8-fold
periodicity of $\mathrm{C}\ell$ modules).
Fix $k$ for a moment, and consider the semigroup of $\mathbb{C}\ell_{k}$
modules. Of course this can be completed to a group by the usual Grothendieck
construction, and we denote the Grothendieck group of $\mathbb{C}\ell_{k}$
modules by $\mathcal{M}_{k}$. Now, the inclusion
$i:\mathbb{R}^{k}\hookrightarrow\mathbb{R}^{k+1}$ induces an injective algebra
homomorphism $i:\mathbb{C}\ell_{k}\hookrightarrow\mathbb{C}\ell_{k+1}$, which
gives a restriction operation
$i^{\ast}:\mathcal{M}_{k+1}\longrightarrow\mathcal{M}_{k}$
on Clifford modules. It turns out that the interesting object to consider is
$\mathcal{M}_{k}/i^{\ast}\mathcal{M}_{k+1}$.
Actually, it is more convenient at this point to work in terms of graded
modules. Thus, let $\widehat{\mathcal{M}}_{k}$ denote the Grothendieck group
of graded $\mathbb{C}\ell_{k}$ modules. Again we have a restriction
$i^{\ast}:\widehat{\mathcal{M}}_{k+1}\longrightarrow\widehat{\mathcal{M}}_{k},$
and from the equivalence between graded $\mathbb{C}\ell_{k}$ modules and
ungraded $\mathbb{C}\ell_{k-1}$ modules, we have
$\widehat{\mathcal{M}}_{k}/i^{\ast}\widehat{\mathcal{M}}_{k+1}\cong\mathcal{M}_{k-1}/i^{\ast}\mathcal{M}_{k}.$
Furthermore, it is easy to check using the representation theory of complex
Clifford algebras, that we have the following periodicity (related of course
to Bott periodicity)
$\widehat{\mathcal{M}}_{k}/i^{\ast}\widehat{\mathcal{M}}_{k+1}\cong\mathcal{M}_{k-1}/i^{\ast}\mathcal{M}_{k}=\begin{cases}\mathbb{Z}&\text{if
$k$ is even}\\\ 0&\text{if $k$ is odd.}\end{cases}$
Let $W=W^{0}\oplus W^{1}\in\mathcal{M}_{k}$, and form the trivial bundles
$E^{i}=D^{k}\times W^{i}$ over the unit disk $D^{k}\subset\mathbb{R}^{k}$. We
can form the element
$\left\\{E^{0},E^{1},\mathrm{c}\ell(\cdot)\right\\}\in K^{0}(D^{k},S^{k})$
where
$\mathrm{c}\ell(\cdot):S^{k}\subset\mathbb{R}^{k}\setminus\left\\{0\right\\}\subset\mathbb{C}\ell_{k}\longrightarrow\mathrm{Iso}(W^{0},W^{1})$.
This bundle isomorphism over $S^{k}$ can be shown to extend over $D^{k}$ if
and only if $W$ actually comes from a $\mathbb{C}\ell_{k+1}$ module. Thus one
obtains the celebrated result of Atiyah, Bott and Shapiro [ABS64].
###### Theorem (Atiyah-Bott-Shapiro).
The above construction gives a graded ring isomorphism
$\widehat{\mathcal{M}}_{\ast}/i^{\ast}\widehat{\mathcal{M}}_{\ast+1}\cong
K^{0}(D^{\ast},S^{\ast})=K^{-\ast}(\mathrm{pt}).$
[ABS64] contains an analogous result for real K-theory and $\mathrm{C}\ell$
modules. While the above looks like an appealing way to prove Bott periodicity
from the more obvious periodicity of Clifford modules, it is not actually so.
Indeed, the ABS result uses periodicity of K-theory in the proof.
This leads to the analogue of the vector bundle representation of $K^{0}(X)$.
Namely, elements of $K^{k}(X)$ can be represented181818I’m not sure of a good
reference for this explicit representation of higher K-theory. It is implicit
in Karoubi’s formulation, but he takes the algebraic approach, with projective
$C^{0}(X)$ modules instead of vector bundles. as (isomorphism classes of)
bundles of graded $\mathbb{C}\ell_{k}$ modules, modulo those which admit a
graded $\mathbb{C}\ell_{k+1}$ action:
$K^{k}(X)=\left\\{V^{0}\oplus V^{1}\longrightarrow
X\;;\;\mathbb{C}\ell_{k}\longrightarrow\mathrm{End}_{\mathrm{gr}}(V^{0}\oplus
V^{1})\right\\}/\left\\{\mathbb{C}\ell_{k+1}\longrightarrow\mathrm{End}_{\mathrm{gr}}(V^{0}\oplus
V^{1})\right\\}$
It is an instructive exercise to recover the vector bundle representation of
$K^{0}(X)$ from this definition. Indeed, since $\mathbb{C}\ell_{0}=\mathbb{C}$
with the trivial grading and $\mathbb{C}\ell_{1}=\mathbb{C}\oplus\mathbb{C}$,
we see that graded $\mathbb{C}\ell_{0}$ modules are just vector bundles of the
form $E\oplus F$, which extend to $\mathbb{C}\ell_{1}$ modules only if $E\cong
F$ (since the action of the generator of the 1-graded part of
$\mathbb{C}\ell_{1}$ must be an isomorphism: $0\oplus 1:E\oplus
F\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}F\oplus E$). Thus we have
the equation $E\oplus E=0\in K^{0}(X)$ or equivalently $-E\oplus 0=0\oplus E$,
and we see that we can identify $[E]-[F]$ in the old representation with
$E\oplus F$ in this new representation.
Next we can generalize the Fredholm operator representation of $K^{0}(X)$,
following Atiyah and Singer’s paper [AS69]. Let $H=H^{0}\oplus H^{1}$ be a
graded, separable, infinite dimensional Hilbert space, and assume $H$ is a
module over $\mathbb{C}\ell_{k}$ (for a given $k$, but then the action can be
extended for all $k$). Let
$\mathrm{Fred}_{k}(H)=\left\\{P\in\mathrm{Hom}(H^{i};H^{i+1})\;;\;P\text{
Fredholm and commutes with $\mathbb{C}\ell_{k}$}\right\\}$
be the space of graded (i.e. acting as 1-graded elements) Fredholm
operators191919Actually this is a bit of an oversimplification. When $k$ is
odd, $\mathrm{Fred}_{k}(H)$ can be separated into open components
$\mathrm{Fred}_{k}^{+}$, $\mathrm{Fred}_{k}^{-}$ and
$\mathrm{Fred}_{k}^{\ast}$ consisting of operators which are essentially
positive (meaning positive off of a finite dimensional subspace), essentially
negative, or neither. The first two are contractible, and we take
$\mathrm{Fred}_{k}^{\ast}(H)$ in this case. commuting (in the graded sense)
with $\mathbb{C}\ell_{k}$. We will call these the $\mathbb{C}\ell_{k}$-linear
Fredholm operators. These form a classifying space for $K^{-k}(\ast)$:
###### Theorem (Atiyah-Singer).
There is an explicit homotopy equivalence
$\mathrm{Fred}_{k}(H)\simeq\Omega\mathrm{Fred}_{k-1}(H)$
for all $k$, and therefore
$[X,\mathrm{Fred}_{k}(H)]=K^{-k}(X)$
Again, for $P\in[X,\mathrm{Fred}_{k}(H)]$ one can morally take $[\ker P]\in
K^{-k}(X)$, since at each point $\ker P$ is a graded $\mathbb{C}\ell_{k}$
module. A stabilization procedure would be required to make this precise, and
I have to admit I’ve never seen it written down, though I’m sure it’s
possible.
Finally, one can make the Atiyah-Singer index construction go through in this
case (again, I’ve not seen this written explicitly, but reliable sources
assure me it’s true!). Namely, given a fibration $X\longrightarrow Z$, if one
has a $\mathbb{C}\ell_{k}$ linear family of elliptic differential operators
$P\in\mathrm{Diff}^{l}(X/Z;E^{0}\oplus E^{1}),$
meaning that $P=\begin{pmatrix}0&P_{1}\\\ P_{0}&0\end{pmatrix}$ is graded and
commutes in the graded sense with an action
$\mathbb{C}\ell_{k}\longrightarrow\mathrm{End}_{\mathrm{gr}}(E^{0}\oplus
E^{1})$, then
$\mathrm{ind}(P)=[\ker P_{0}\oplus\ker P_{1}]\in K^{k}(Z)$
and that this index coincides with the Gysin map
$\mathrm{ind}=p_{!}:K^{k}_{c}(T^{\ast}X/Z)\longrightarrow K^{k}(Z).$
Of course, since $K^{1}(\mathrm{pt})=0$, operators on a manifold $X$ never
have any interesting odd index (however a family of operators might, provided
$K^{1}(Z)\neq 0$). For real K-theory, however, this can be an interesting and
useful concept. For instance, the Kervaire semicharacteristic on a ($4k+1$)
manifold $X$ can be computed (see [LM89]) as the odd index in
$KO^{1}(\mathrm{pt})=\mathbb{Z}_{2}$ of a $\mathrm{C}\ell_{1}$ linear elliptic
differential operator on $X$!202020Sorry about all the footnotes.
## References
* [ABS64] M.F. Atiyah, R. Bott, and A. Shapiro, _Clifford modules_ , Topology 3 (1964), no. 3, 38.
* [AS68] M.F. Atiyah and I. Singer, _Index theorem of elliptic operators, I, III_ , Annals of Mathematics 87 (1968), 484–530.
* [AS69] by same author, _Index theory for skew-adjoint Fredholm operators_ , Publications Mathématiques de l’IHÉS 37 (1969), no. 1, 5–26.
* [AS71] by same author, _The index of elliptic operators: IV_ , Annals of Mathematics (1971), 119–138.
* [Ati67] M.F. Atiyah, _K-theory: lectures by MF Atiyah; Notes by DW Anderson_.
* [LM89] H.B. Lawson and M.L. Michelsohn, _Spin geometry_ , Princeton University Press, 1989.
|
arxiv-papers
| 2010-10-24T21:05:40 |
2024-09-04T02:49:14.183608
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Chris Kottke",
"submitter": "Chris Kottke",
"url": "https://arxiv.org/abs/1010.5002"
}
|
1010.5291
|
11institutetext: 1 National Key Lab. of ISN, Xidian University , Xi’an 710071,
P.R.China
11email: wp_ma@mail.xidian.edu.cn
2 Datang Mobile Communications Equipment Co.,Lid,Beijing 100083, P.R.China
# New Class of Optimal Frequency-Hopping Sequences by Polynomial Residue Class
Rings
Wenping Ma and Shaohui Sun 1122
###### Abstract
In this paper, using the theory of polynomial residue class rings, a new
construction is proposed for frequency hopping patterns having optimal Hamming
autocorrelation with respect to the well-known $Lempel$-$Greenberger$ bound.
Based on the proposed construction, many new $Peng$-$Fan$ optimal families of
frequency hopping sequences are obtained. The parameters of these sets of
frequency hopping sequences are new and flexible.
Index Terms:Autocorrelation Functions, Cross-correlation functions, Frequency
Hopping sequences, Hamming Correlation, lower bounds.
## 1 Introduction
Let $\mathcal{F}=\\{f_{0},f_{1},\cdots,f_{l-1}\\}$ be a set of available
frequencies, called an $alphabet$. Let S be the set of all sequences of length
$\nu$ over $\mathcal{F}$. Any element of S is called a frequency-hopping
sequence of length $\nu$ over $\mathcal{F}$. Given any two frequency hopping
sequences $X,Y\in\textit{S}$, we define their Hamming correlation $H_{X,Y}$ to
be
$H_{X,Y}(t)=\sum^{\nu-1}_{i=0}h[x_{i},y_{i+t}],0\leq t<\nu,$
where $h[a,b]=1$ if $a=b$, and 0, otherwise, and all operations among the
position indices are performed modulo $\nu$. For any distinct
$X,Y\in\textit{S}$, we define
$H(X)=\max_{1\leq t<\nu}{\\{H_{X,X}(t)\\}}$ $H(X,Y)=\max_{0\leq
t<\nu}{\\{H_{X,Y}(t)\\}}$ $M(X,Y)=\max{\\{H(X),H(Y),H(X,Y)\\}}.$
Lempel and Greenberger[2] developed the following lower bound for $H(X)$.
Lemma 1: For every frequency hopping sequence X of length $\nu$ over an
alphabet of size $l$ , we have
$H(X)\geq{\biggl{\lfloor}{(\nu-\varepsilon)(\nu+\varepsilon-l)\over{l(\nu-1)}}\biggr{\rfloor}}$
where $\varepsilon$ is the least nonnegative residue of $\nu$ modulo $l$.
Corollary 1(6): For any single frequency hopping sequence of length $\nu$ over
an alphabet of size $l$ , we have
$H(X)\geq\left\\{\begin{array}[]{ll}$$k,if\thinspace\thinspace\nu\neq l$$\\\
$$0,if\thinspace\thinspace\nu=l$$\end{array}\right.$
where $\nu=kl+\varepsilon$, $0\leq\varepsilon<l$.
Let $\Gamma$ be a subset of S containing $N$ sequences. We define the maximum
nontrivial Hamming correlation of the sequence set $\Gamma$ as
$M(\Gamma)=\max\\{\max_{X\in\Gamma}H(X),\max_{X,Y\in\Gamma,X\neq Y}H(X,Y)\\}$
$H_{a}(\Gamma)=\max_{X\in\Gamma}H(X)$ $H_{c}(\Gamma)=\max_{X,Y\in\Gamma,X\neq
Y}H(X,Y)$
``Throughout this paper, we use $(\nu,N,l,\lambda)$ to denote a set of $N$
frequency hopping sequences $\Gamma$ of length $\nu$ over an alphabet of size
$l$, where $\lambda=M(\Gamma)$ .
Peng and Fan[3] developed the following bound on $H_{a}(\Gamma)$and
$H_{c}(\Gamma)$, which take into consideration the number of sequences in the
family.
Lemma 2:For any family of frequency hopping sequences $\Gamma$ , with length
$\nu$, an alphabet of size $l$ , and $|\Gamma|=N$ , we have
$(\nu-1)NH_{a}(\Gamma)+(N-1)N\nu H_{c}(\Gamma)\geq 2I\nu N-(I+1)Il$
where $\verb+ +\displaystyle I=\lfloor{\nu N\over{l}}\rfloor.$
Lemma 3(6): For any pair of distinct frequency hopping sequences $X,Y$, with
$|\mathcal{F}|=l$, we have
$M(X,Y)\geq{{4I\nu-(I+1)Il}\over{4\nu-2}}$
where $2\nu=Il+r$ and $0\leq r<l$.
###### Definition 1
(1) A sequence $X\in\textit{S}$ is called optimal if the
$Lempel$-$Greenberger$ bound in Lemma 1 is met.
(2) A subset $\Gamma\subset\textit{S}$ is an optimal set if the $Peng-Fan$
bound in Lemma 2 is met.
(3) Any pair of distinct frequency hopping sequence
$\\{X,Y\\}\subset\textit{S}$ constitute a $Lempel$-$Greenberger$ optimal pair
of frequency hopping sequences if the bound in Lemma 3 is met .
Lempel and Greenberger[2] defined optimality for both single sequences and
sets of sequences in other ways. A set of frequency hopping sequences meeting
the $Peng$-$Fan$ bound in Lemma 2 must be optimal in the $Lempel$ and
$Greenberger$ sense.
In modern radar and communication systems, frequency hopping spread-spectrum
techniques have been popular, such as frequency hopping code division multiple
access and “Bluetooth” technologies[7, 8].
The objective of this paper is to present a new method to construct new family
of frequency hopping sequences. Both individual optimal frequency-hopping
sequences and optimal families of frequency hopping sequences are presented.
## 2 Polynomial Residue Class Rings Preliminary
In the following, we introduce in brief polynomial residue class rings
preliminary. For details on polynomial residue class rings, we refer to [1]
###### Definition 2
Let $p$ be a prime, $GF(p)$ be a finite field, $GF(p)[\xi]$ be the ring of all
polynomials over $GF(p)$, and $\omega(\xi)$ be an irreducible polynomial of
degree $m$ over $GF(p)$, where $m\geq 1$. Then $\Re$ is defined as the
quotient ring generated by $\omega(\xi)^{k}$ in $GF(p)[\xi]$, $k\geq 1$ .
$\Re=GF(p)[\xi]\Bigl{/}(\omega(\xi)^{k})$
We have a natural homomorphic mapping, $\mu$ from $\Re$ to its residue field
$F=GF(p)[\xi]\Bigl{/}(\omega(\xi))$. Define $\mu:\Re\rightarrow F$ by
$\mu(a)=a\verb+ +mod\verb+ +\omega(\xi)$ . It is easy to verify that the
elements in the set
$\\{1,\omega(\xi),\omega^{2}(\xi),\cdots,\omega^{k-1}(\xi)\\}$ are linearly
independent over $F$ and hence constitute a basis of $\Re$ over $F$. Thus any
element $a\in\Re$ can be represented uniquely as
$a=a_{0}+a_{1}\omega(\xi)+\cdots+a_{k-1}\omega_{k-1}(\xi),a_{i}\in
F,i=0,1,2,\cdots,k-1.$
Thus $\Re$ can be written as
$\Re=F+F\omega+F\omega^{2}+\cdots+F\omega^{k-1}\verb+ +(1)$
The group of units $\Re^{*}$ of $\Re$ is given by the direct product of two
group $G_{PRC}$ and $G_{PRA}$, $\Re^{*}=G_{PRC}\times G_{PRA}$ , where
$G_{PRC}$ is a cyclic group of order $p^{m}-1$ and $G_{PRA}$ is an Abelian
group of order $p^{m(k-1)}$.
Lemma 4:The set $\\{G_{PRC},0\\}$ is isomorphic to residue field $F$ and is
also a subspace of $\Re$. Thus the set $\\{G_{PRC},0\\}$ is a subring of $\Re$
.
From now on, we will omit the indeterminate $\xi$ from the representation.
Let $\Re[x]$ be the ring of polynomials over $\Re$ . We extend the homomorphic
mapping $\mu$ on $\Re$ to polynomial reduction mapping :
$\hat{\mu}:\Re[x]\rightarrow F[x]$ in the obvious way
$f(x)=\sum^{r}_{i=0}a_{i}x^{i}\ext@arrow
0099{\arrowfill@-->}{}{\displaystyle\hat{\mu}\hskip
10.00002pt}\sum^{r}_{i=0}\mu(a_{i})x^{i}$
A polynomial $f(x)\in\Re[x]$ is a basic irreducible if $\mu(f(x))$ is
irreducible in $F[x]$; it is monic if its leading coefficient is 1\.
###### Definition 3
The Galois ring of $\Re$ denoted as $GR(\Re,r)$ is defined as
$\Re[x]\Bigl{/}(f(x))$, where $f(x)$ is a basic monic irreducible polynomial
of degree $r$ over $\Re$ .
The group of units of $GR(\Re,r)$ denoted by $GR^{*}(\Re,r)$ is given by a
direct product of two groups:
$GR^{*}(\Re,r)=G_{C}\times G_{A}$
where $G_{C}$ is a cyclic group of order $p^{mr}-1$ and $G_{A}$ is an Abelian
group of order $p^{m(k-1)r}$ . On the lines of Lemma 4, it is easy to show
that the set $\\{G_{C},0\\}$is a field of order $p^{mr}$ . This is denoted by
$GF(p^{mr})$. Thus like the representation (1) for $\Re$, we have
$GR(\Re,r)=GF(p^{mr})+\omega
GF(p^{mr})+\omega^{2}GF(p^{mr})+\cdots+\omega^{k-1}GF(p^{mr}),$
hence, any element $\alpha\in GR(\Re,r)$ can be uniquely expressed as
$\alpha=\alpha_{0}+\omega\alpha_{1}+\omega^{2}\alpha_{2}+\cdots+\omega^{k-1}\alpha_{k-1}$,$\alpha_{i}\in
GF(p^{mr})$, $i=0,1,\cdots,k-1.$` `(2)
The elements of $G_{A}$ are of the form $1+\omega(x)A^{\prime}$ , where
$A^{\prime}\in GR(\Re,r)$. From (2), the elements of $G_{A}$ are given by the
set
$\\{(1+\omega\gamma),\gamma=\gamma_{0}+\omega\gamma_{1}+\cdots+\omega^{k-2}\gamma_{k-2},\gamma_{i}\in
GF(p^{mr}\\}\verb+ +(3)$
The Galois automorphism group of $GR(\Re,r)$ over its intermediate subring
$GR(\Re,s)$, where $s$ divides $r$ is cyclic of order $(r/s)$ generated by the
Frobenius map $\sigma^{s}$ defined by
$\sigma^{s}(\alpha)=(\alpha_{0})^{p^{s}}+(\alpha_{1})^{p^{s}}\omega+\cdots+(\alpha_{k-1})^{p^{s}}\omega^{k-1}$
where $\alpha$ is as in (2). When $s=1$ , the above Frobenius map generates
Galois group over $\Re$. Using the automorphisms given above, we define below
generalized trace functions which map elements of $GR(\Re,r)$ to its
intermediate subrings $GR(\Re,s)$ where $s$ divides $r$. They are given by
$Tr^{r}_{s}(\alpha)=\sum^{(r/s-1)}_{i=0}[(\alpha_{0})^{p^{si}}+(\alpha_{1})^{p^{si}}\omega+(\alpha_{2})^{p^{si}}\omega^{2}+\cdots+(\alpha_{k-1})^{p^{si}}\omega^{k-1}]$
where $\alpha\in GR(\Re,r)$.The above trace function is the generalization of
trace function defined for finite fields. Like their counterparts in finite
fields, the trace functions satisfy the following properties:
$Tr^{r}_{s}(\alpha)=Tr^{r}_{s}(\sigma^{si}(\alpha)),for\verb+ +all\verb+ +i.$
$Tr^{r}_{s}(a\alpha+b\beta)=aTr^{r}_{s}(\alpha)+bTr^{r}_{s}(\beta)$; $\forall
a,b\in GR(\Re,s)$ and $\forall\alpha,\beta\in GR(\Re,r)$.
For any fixed $b$ of $GR(\Re,s)$, the equation $Tr^{r}_{s}(\alpha)=b$, has
exactly $p^{mk(r-s)}$ solutions in $GR(\Re,r)$.
$Tr^{r}_{1}(\alpha)=Tr^{s}_{1}(Tr^{r}_{s}(\alpha)).$
###### Theorem 2.1
[1] Every $m$-sequence over $\Re$ has a unique trace representation given by
$\\{s^{\gamma}_{i}\\}^{\infty}_{i=0}=Tr^{r}_{1}(\gamma\alpha^{i})$, where
$\gamma\in GR(\Re,r)$ and $\alpha$ is a primitive root of $f(x)$ and belongs
to $G_{C}$.
We shall denote $S^{*}(f)$ as the set of sequences which contains not all zero
divisors. By using the structure of group of units $GR^{*}(\Re,r)=G_{C}\times
G_{A}$ and (3), all $m$-sequence in $S^{*}(f)$ are given by the set
$\\{(s^{\gamma}_{i})^{\infty}_{0},\gamma=(1+\omega(\gamma_{0}+\gamma_{1}\omega+\cdots+\gamma_{k-2}\omega^{k-2})),where\verb++\gamma_{j}\in
GF(p^{mr}),j=0,1,\cdots,k-2\\}.$
###### Definition 4
Let $\alpha\in GR(\Re,r)$ as in (2) be equal to
$\alpha_{0}+\alpha_{1}\omega+\cdots+\alpha_{k-1}\omega^{k-1}$,$\alpha_{i}\in
GF(p^{mr})$. Then, let $M_{\alpha}$ be a matrix over $F$ of dimension $r\times
k$ formed by placing together $k$ elements
$\alpha_{0},\alpha_{1},\cdots,\alpha_{k-1}$ as columns of M. Then the rank
number $\kappa(\alpha)$ of $\alpha$ is defined as the rank of matrix
$M_{\alpha}$ over $F$.
###### Definition 5
Given a sequence $\mathcal{S}$ and an element $s$ of $\Re$, we define
$W_{s}(\mathcal{S})$ as the number of occurrences of the element s in
$\mathcal{S}$ within its one period length.
###### Theorem 2.2
[1] Let $\\{s^{\gamma}_{i}\\}^{\infty}_{0}$ be an $m$-sequence with
$\kappa(\gamma)=\rho$. Then,$W_{0^{k}}(s^{\gamma})=p^{m(r-\rho)}-1$, and
$W_{s}(s^{\gamma})=p^{m(r-\rho)}$, for $s\neq 0^{k}$.
###### Definition 6
The Trace Image of an $m$-sequence, $s^{\gamma}$ is defined as the set of
distinct elements in $s^{\gamma}$. The cardinality of the Trace Image is given
by $p^{m\rho}$.
## 3 New Optimal Frequency Hopping Sequences from Residue Class Rings
Let$\verb+ +q=p^{m},$ $z$` ` is a positive integer satisfying $\verb+
+z|(q-1),n={{q^{r}-1}\over{z}}$, $r$ is a positive integer, in this paper, we
suppose $gcd(\frac{q^{r}-1}{q-1},z)=1$, $\alpha$ be a primitive generator of
$G_{C}$ present in $GR^{*}(\Re,r)$,$\gamma\in G_{A}$ with
$\kappa(\gamma)=\rho$.
Let $s$ be an integer with $gcd(s,q^{r}-1)=1$ , and define
$\beta={\alpha}^{zs}$ . It is easy to check that the minimal positive integer
$d$ satisfying ${\beta}^{q^{d}-1}=1$ is $r$ , thus
$1,\beta,\beta^{2},\cdots,\beta^{r-1}$ is linear independent over $G_{PRC}$.
We define the following sequence:
$s^{(\gamma,g)}_{i}=Tr^{r}_{1}(\gamma g\beta^{i}),i=0,1,\cdots,k,\cdots,g\in
G_{C}$
It is easy to check that $s^{(\gamma,g)}_{i}=s^{(\gamma,g)}_{i+n}$ , then
$(s^{(\gamma,g)}_{i})^{\infty}_{0}$ is a sequence of period $n$.
We define the following sequences set:
$\Gamma=\\{(s^{(\gamma,\alpha^{sk})}_{i})^{\infty}_{0}:0\leq k<z\\}\verb+
+(4)$
It is obvious that $|\Gamma|=z$.
###### Definition 7
Two sequences ${(s^{(\gamma,g)}_{i}})_{0}^{\infty}$ and
$(s^{(\gamma,g^{\prime})}_{i})^{\infty}_{0}$ are called projectively
cyclically equivalent if there exist an integer $t$ and a nonzero scalar
$\lambda\in G_{PRC}$
$s^{(\gamma,g)}_{i}=\lambda
s^{(\gamma,g^{\prime})}_{i+t},i=0,1,2,\cdots.\verb+ +(5)$
We wish to count the number of inequivalent in $\Gamma$ using (5) as the
definition of equivalence.
###### Theorem 3.1
For any two sequences $(s^{(\gamma,g)}_{i})^{\infty}_{0}$ and
$(s^{(\gamma,g^{\prime})}_{i})^{\infty}_{0}$ belonging to $\Gamma$, they are
projectively cyclically equivalent.
Proof:Formula (5) can be written as
$Tr^{r}_{1}(\gamma g\beta^{i})=Tr_{1}^{r}(\gamma\lambda
g^{\prime}\beta^{(t+i)}),i\geq 0$ $Tr^{r}_{1}[\gamma(g-\lambda
g^{\prime}\beta^{t})\beta^{i}]=0,i\geq 0$
It follows that Formula (5) is equivalent to
${g\over{g^{\prime}}}=\lambda\beta^{t}\verb+ +(6)$
The set of elements in $G_{C}$ of the form $\lambda\beta^{t}$ where
$\lambda\in G_{PRC}$ is a subgroup of the multiplicative group of nonzero
elements of $G_{C}$. What (6) says is that $g$ and $g^{\prime}$ are equivalent
if and only if $g$ and $g^{\prime}$ lie in the same coset of this subgroup. It
follows that the number of inequivalent $g$’s is equal to the number of such
cosets, viz.
$N_{1}={(q^{r}-1)\over{|G|}},$
where $G$ is the subgroup of elements of the form $\\{\lambda\beta^{i}\\}$. It
remains to calculate $|G|$. Now $G$ is the direct product of the two groups
$G_{PRC}$ and $A=\\{1,\beta,\cdots,\beta^{n-1}\\}$. From elementary group
theory we have
$|G|={|A|\cdot|G_{PRC}|\over{|G_{PRC}\cap A|}}.$
To calculate $|G_{PRC}\cap A|$ we note that this number is just the number of
distinct powers of $\beta$, which are elements of $G_{PRC}$ . But
$\beta^{i}\in G_{PRC}$ if and only if $\beta^{i(q-1)}=1$. Since
$ord(\beta)=n$, this is equivalent to $n|i(q-1),i.e,$
${n\over{gcd(n,q-1)}}|i$
Thus if we define
$e=gcd(n,q-1)$ $d={n\over{e}}.$
Because $e=gcd(n,q-1)=gcd(\frac{q^{r}-1}{q-1}\frac{q-1}{z},z\frac{q-1}{z})$
and $\displaystyle gcd(z,\frac{q^{r}-1}{q-1})=1$, then $\displaystyle
e=\frac{q-1}{z}$.
We see that $\beta^{i}\in G_{PRC}$ iff $i=0,d,2d,\cdots,(e-1)d$, hence
$|G_{PRC}\bigcap A|=e$, and we have
$|G|=n(q-1)\Bigl{/}e=q^{r}-1,$ $N_{1}=1.$
###### Theorem 3.2
$W_{0^{k}}((s^{(\gamma,g)}_{i})^{\infty}_{0})={q^{r-\rho}-1\over{z}}.$
Proof:Let $1,\alpha^{s},\cdots,\alpha^{s(z-1)}$ be a complete set of
representatives for the cosets of $\\{1,\beta,\cdots,\beta^{n-1}\\}$ in the
multiplicative group $G_{C}$. Every nonzero element $\theta\in G_{C}$ can be
written as $\theta=\alpha^{si}\beta^{j}$ for a unique pair $(i,j),0\leq i\leq
z-1$,$0\leq j\leq n-1$. Now consider the following $z\times n$ array, which we
call Array 1:
$\begin{array}[]{ccccc}1&\beta&\beta^{2}&\cdots&\beta^{n-1}\\\
\alpha^{s}&\alpha^{s}\beta&\alpha^{s}\beta^{2}&\cdots&\alpha^{s}\beta^{n-1}\\\
\alpha^{2s}&\alpha^{2s}\beta&\alpha^{2s}\beta^{2}&\cdots&\alpha^{2s}\beta^{n-1}\\\
\vdots&\vdots&\vdots&\vdots&\vdots\\\
\alpha^{(z-1)s}&\alpha^{(z-1)s}\beta&\alpha^{(z-1)s}\beta^{2}&\cdots&\alpha^{(z-1)s}\beta^{n-1}\end{array}$
Now let $s_{ij}=Tr^{r}_{1}(\alpha^{is}\beta^{j})$ and consider this
array,which we call Array 2:
$\begin{array}[]{ccccc}s_{00}&s_{01}&s_{02}&\cdots&s_{0(n-1)}\\\
s_{10}&s_{11}&s_{12}&\cdots&s_{1(n-1)}\\\
s_{20}&s_{21}&s_{22}&\cdots&s_{2(n-1)}\\\
\vdots&\vdots&\vdots&\vdots&\vdots\\\
s_{(z-1)0}&s_{(z-1)1}&s_{(z-1)2}&\cdots&s_{(z-1)(n-1)}\\\ \end{array}$
Since Array 2 is the “trace” of Array 1, and since every nonzero element of
$G_{C}$ appears exactly once in Array 1, It follows that $0$ appears exactly
$q^{(r-\rho)}-1$ times in Array 2. Finally, since $N_{1}=1$ , we know that
every row of Array 2 can be obtained from the first row by shifting and
multiplying by scalars. Thus 0 appears the same number of times in each row of
Array 2. Since there are $z$ rows in the array, and 0 appears $q^{(r-\rho)}-1$
time altogether, each row contains exactly
$\displaystyle{q^{r-\rho}-1\over{z}}$ 0.
###### Theorem 3.3
$\\{s^{(\gamma,g)}_{i}\\}^{\infty}_{0}$ is an optimal frequency hopping
sequence with parameters
$\displaystyle({q^{r}-1\over{z}},q^{\rho},{q^{r-\rho}-1\over{z}}).$
Proof:Because
$\displaystyle{q^{r}-1\over{z}}=q^{\rho}\cdot{q^{r-\rho}-1\over{z}}+{q^{\rho}-1\over{z}}$
, the conclusion follows from Lemma 1 and Corollary 1.
###### Theorem 3.4
if $g,g^{\prime}$ belong to distinct cyclotomic classes of order $z$ in
$G_{C}$ , then $((s^{(\gamma,g)}_{i})^{\infty}_{0})$ and
${(s^{(\gamma,g^{\prime})}_{i})}^{\infty}_{0}$ constitute a $Lempel-
Greenberger$ optimal pair of frequency hopping sequences.
Proof: By Theorem 5,
$H_{a}((s^{(\gamma,g)}_{i})^{\infty}_{0})=H_{a}((s^{(\gamma,g^{\prime})}_{i})^{\infty}_{0})$
. Now we compute the cross-correlation values of
$(s^{(\gamma,g)}_{i})^{\infty}_{0}$ and
$(s^{(\gamma,g^{\prime})}_{i})^{\infty}_{0}$. From the definition of
$s^{(\gamma,g^{\prime})}_{i}$, we know that for any
$t\in\\{0,1,\cdots,n-1\\}$, if we cyclically shift
$s^{(\gamma,g^{\prime})}_{i}$ to the left for $t$ time, we obtain
$s^{(\gamma,g^{\prime})}_{i+t}=Tr_{1}^{r}(\gamma
g^{\prime}\beta^{t}\beta^{i}),i=0,1,2,\cdots,$ then, by noting that
$s^{(\gamma,g)}_{i}-s^{(\gamma,g^{\prime})}_{i+t}=Tr^{r}_{1}[\nu(g-g^{\prime}\beta^{t})\beta^{i}],i=1,2,\cdots$
. Since $g,g^{\prime}$ are in distinct cyclotomic classes of order $z$ in
$G_{C}$ , $g-g^{\prime}\beta^{t}$ can never be zero. It then follows from
Theorem 4 that
$H_{(s^{(\gamma,g)}_{i})^{\infty}_{0},(s^{(\gamma,g^{\prime})}_{i})^{\infty}_{0}}(t)=\displaystyle{{q^{r-\rho}-1}\over{z}}.$
For any $t\in\\{0,1,\cdots,n-1\\}$ . Therefore we can conclude that
$H((s^{(\gamma,g)}_{i})^{\infty}_{0},(s^{(\gamma,g^{\prime})}_{i})^{\infty}_{0})=\displaystyle{q^{r-\rho}-1\over{z}}$
. We claim that $(s^{(\gamma,g)}_{i})^{\infty}_{0}$ and
$(s^{(\gamma,g^{\prime})}_{i})^{\infty}_{0}$ constitute a $Lempel-Greenberger$
optimal pair of frequency hopping sequences, if $g,g^{\prime}$ belong to
distinct cyclotomic classes of order $z\geq 2$ in $G_{C}$. In fact, for any
two $q^{\rho}$-ary sequences $(s^{(\gamma,g)}_{i})^{\infty}_{0}$ and
$(s^{(\gamma,g^{\prime})}_{i})^{\infty}_{0}$ of length
$\displaystyle{q^{r}-1\over{z}}$, since
$\displaystyle{{(q^{r}-1)\over{z}}={{q^{r-\rho}-1\over{z}}q^{\rho}+{q^{\rho}-1\over{z}}}}$,
we put $\displaystyle{d={q^{r-\rho}-1\over{z}}}$ and
$e=\displaystyle{q^{\rho}-1\over{z}}$, then by Lemma 3, we have
$M((s^{(\gamma,g)}_{i})^{\infty}_{0},(s^{(\gamma,g^{/})}_{i})^{\infty}_{0})\geq{4I\nu-(I+1)Il\over{4\nu-2}}={2d\nu-\nu+2de+e\over{2\nu-1}}$
$=d-{\nu-2de-e-d\over{2\nu-1}}$ $=d-{de(z-2)\over{2\nu-1}}$
This implies that
$M((s^{(\gamma,g)}_{i})^{\infty}_{0},(s^{(\gamma,g^{/})}_{i})^{\infty}_{0})\geq
d={q^{r-\rho}-1\over{z}}.$
###### Theorem 3.5
The $\Gamma$ of (4) is a
$\displaystyle({q^{r}-1\over{z}},z,q^{\rho},{q^{r-\rho}-1\over{z}})$ set of
frequency hopping sequence, meeting the $Peng-Fan$ bound.
Proof: We apply Lemma 2, where $\displaystyle I=\lfloor{\nu
z/q^{\rho}}\rfloor=q^{r-\rho}-1$,
$(\nu-1)zH_{a}(\Gamma)+(z-1)z\nu H_{c}({\Gamma})$
$=({q^{r}-1\over{z}}-1)z{q^{r-\rho}-1\over{z}}+(z-1)z{q^{r}-1\over{z}}{q^{r-\rho}-1\over{z}}$
$=(q^{r}-z-1){q^{r-\rho}-1\over{z}}+(z-1)(q^{r}-1){q^{r-\rho}-1\over{z}}$
$=(q^{r}-2)(q^{r-\rho}-1)$
and
$2I\nu z-(I+1)Iq^{\rho}$
$=2(q^{r-\rho}-1){q^{r}-1\over{z}}z-q^{r-\rho}(q^{r-\rho}-1)q^{\rho}$
$=(q^{r}-2)(q^{r-\rho}-1).$
We know that
$(\nu-1)zHa({\Gamma})+(z-1)z\nu H_{c}({\Gamma})=2I\nu z-(I+1)Iq^{\rho}$
which means that
$\displaystyle\\{H_{a}({\Gamma})={q^{r-\rho}-1\over{z}},H_{c}({\Gamma})={q^{r-\rho}-1\over{z}}\\}$
is a pair of the minimum integer solutions of the inequality described in
Lemma 2, that is, ${\Gamma}$ is a $Peng-Fan$ optimal family of frequency
hopping sequences.
## 4 Conlusion
In this paper, new optimal frequency hopping sequences are constructed from
polynomial residue class rings. When $\rho=1$, our construction is same with
the related constructions in [4, 5, 6], thus our construction can be take as
an extension of the related constructions in [4, 5, 6]. Our construction
posses the following advantages: (1) the parameters of the construction are
new and flexible, (2) by choose different parameter $\gamma$ , one can
construct many different $Peng-Fan$ optimal frequency hopping sequence
families.
## References
* [1] P.Udaya and M.U.Siddiqi, Optimal large linear complexity frequency hopping patterns derived from polynomial residue class rings, IEEE Transactions on Information Theory, Vol.44, No.4, July 1998.
* [2] Abraham Lempel, and Haim Greenberger, Families of sequences with optimal Hamming correlation properties, IEEE Transactions on Information Theory, Vol.20, No.1, January 1974.
* [3] Daiyuan Peng and Pingzhi Fan, Lower bounds on the Hamming Auto- and Cross correlations of Frequency-Hopping sequences, IEEE Transactions on Information Theory, Vol.50, No.9, September, 2004.
* [4] Cunsheng Ding, Marko J. Moisio, and Jin Yuan, Algebraic constructions of optimal frequency hopping sequences, IEEE Transactions on Information Theory, Vol.53, No.7, July 2007.
* [5] Cunsheng Ding, Jianxing Yin, Sets of optimal frequency hopping sequences, IEEE Transactions on Information Theory, Vol.54, No.8, August 2008.
* [6] Gennian Ge, Ying Miao, and Zhongxiang Yao, Optimal frequency hopping sequences: Auto-and Cross correlation properties, IEEE Transactions on Information Theory, Vol.55, No.2, February 2008.
* [7] R.A.Scholtz, ”The spread spectrum concept,” IEEE Trans. Commun. Vol.25, No.8, pp.748-755, Aug.1977.
* [8] Specification of the Bluetooth systems-Core.The Bluetooth special interest Group. Available:http://www.bluetooth.com/
* [9] Robert J.EcEliece, Finite fields for computer scientists and engineers, Kluwer Academic Publishers, 1987.
|
arxiv-papers
| 2010-10-26T00:44:54 |
2024-09-04T02:49:14.208232
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Wenping Ma and Shaohui Sun",
"submitter": "Wenping Ma",
"url": "https://arxiv.org/abs/1010.5291"
}
|
1010.5297
|
# Phase transitions and thermodynamics of the two-dimensional Ising model on a
distorted Kagomé lattice
Wei Li1, Shou-Shu Gong1, Yang Zhao1, Shi-Ju Ran1, Song Gao2, and Gang Su1,∗
1College of Physical Sciences, Graduate University of Chinese Academy of
Sciences, P. O. Box 4588, Beijing 100049, People’s Republic of China
2College of Chemistry and Molecular Engineering, State Key Laboratory of Rare
Earth Materials Chemistry and Applications, Peking University, Beijing 100871,
People’s Republic of China
###### Abstract
The two-dimensional Ising model on a distorted Kagomé lattice is studied by
means of exact solutions and the tensor renormalisation group (TRG) method.
The zero-field phase diagrams are obtained, where three phases such as
ferromagnetic, ferrimagnetic and paramagnetic phases, along with the second-
order phase transitions, have been identified. The TRG results are quite
accurate and reliable in comparison to the exact solutions. In a magnetic
field, the magnetization ($m$), susceptibility and specific heat are studied
by the TRG algorithm, where the $m=1/3$ plateaux are observed in the
magnetization curves for some couplings. The experimental data of
susceptibility for the complex Co(N3)2(bpg)$\cdot$ DMF4/3 are fitted with the
TRG results, giving the couplings of the complex $J=22K$ and $J^{\prime}=33K$.
###### pacs:
75.10.Hk, 75.40.Cx, 75.50.Xx, 64.70.qd
## I Introduction
Kagomé lattice, one of the most interesting frustrated spin lattices, has
attracted much attention both experimentally and theoretically in recent
years. The Heisenberg model on a Kagomé lattice is likely a candidate for
finding a spin liquid state in two-dimensional (2D) spin systems. Some
numerical works have revealed that the system possesses a magnetic disordered
ground state.Lecheminant ; Jiang0 Nevertheless, the nature of its ground
state is still an open question.Richter Recently, a number of spin systems,
such as volborthite Cu3V2O7(OH)${}_{2}\cdot$2H2O (Refs. Wang, ; Schnyder, ;
Yoshida, ; Yoshida2, ),
[H3N(CH2)2NH2(CH2)2(NH3]4[Fe${}^{\textrm{II}}_{9}$F18(SO4)6]$\cdot$9H2O (Ref.
Behera, ) and Co(N3)2(bpg)$\cdot$ DMF4/3 (Ref. Gao, ), are found to form a
Kagomé lattice with distortions, where the structural distortions give rise to
two different exchange couplings $J$ and $J^{\prime}$. Such a spatially bond
anisotropic spin lattice can be called a distorted Kagomé (DK) lattice, as
schematically depicted in Fig. 1. A distortion-induced magnetization step at
small fields and the $1/3$ magnetization plateau on the DK lattice have been
observed in experiments. Yoshida2 In order to explain the experimental
results, the quantum and classical Heisenberg models on the DK lattice are
considered.Hida ; Kaneko In most cases, the interactions between spins are
usually of Heisenberg type on a DK lattice Kaneko . However, in some materials
the spin-spin couplings are anisotropic, and even of Ising-type in particular
situations. For instance, in complex Co(N3)2(bpg)$\cdot$ DMF4/3, Co+2 ions
form a spin-1/2 DK lattice and might be coupled by Ising-type interactions at
low temperatures.Carlin Therefore, it should also be necessary to pay more
attention on the 2D Ising model with the DK lattice, especially when an
external magnetic field is present, where the works in the literature are
still sparse.
Figure 1: (Color online) (a) The distorted Kagomé lattice, where $J$ and
$J^{\prime}$ denote the different nearest neighbor couplings. The dots in the
center of small triangles and lines connecting them form a tensor network
presented by Eq. (3). (b) and (c) show two degenerate spin configurations of
the ferrimagnetic structure, where up and down arrows represent the spin-up
and spin-down states, respectively.
In this article, we shall focus on the Ising model on the DK lattice, where
the thermodynamics and magnetic properties will be carefully studied by means
of exact solutions and a numerical method. In the present model, the quantum
fluctuations are completely suppressed, and only the thermal fluctuations are
considered. It should be stressed that the present model for $h=0$ can be
exactly solved, but for $h\neq 0$ the exact solution is not available at
present and the numerical method should be involved in. For this reason, a
recently developed tensor renormalization group (TRG) method Levin ; Jiang ;
Gu is employed to investigate the thermodynamic properties of the system. The
cases with different couplings $J$ and $J^{\prime}$ and both with and without
a magnetic field $h$ will be discussed. The TRG results, being consistent with
the exact solutions, reveal that the system has ferromagnetic, ferrimagnetic,
and paramagnetic phases in the phase diagram, where the paramagnetic phase can
exist at $T=0$ in the strongly frustrated region. Moreover, the magnetic
order-disorder phase transitions separating these three phases are disclosed.
In ferrimagnetic and paramagnetic phases, the $1/3$ magnetization plateaux are
seen. A magnetization step at an infinitesimal field is observed for the
paramagnetic phase at low temperature. The specific heat no longer possesses a
divergent peak once the external field is switched on, implying the absence of
phase transitions at $h\neq 0$. In addition, we shall also make an attempt to
fit the experimental data of susceptibility for the complex
Co(N3)2(bpg)$\cdot$ DMF4/3 with the TRG results so as to estimate the exchange
couplings in this complex.
The other parts of this article are organized as follows. In Sec. II, the
exact solutions and phase diagrams are presented. The TRG method is introduced
in Sec. III. The specific heat without a magnetic field is explored in Sec.
IV. Magnetization, susceptibility, and a comparison to the experimental data
are shown in Sec. V. Sec. VI contains the results of specific heat in an
external magnetic field. The summary and discussions are given finally.
## II Exact solutions in zero magnetic field
The Hamiltonian of the system under interest has the form of
$H=J^{\prime}\sum_{<i\in\alpha,j\in\beta>}{S_{i}S_{j}}+J^{\prime}\sum_{<i\in\alpha,j\in\gamma>}{S_{i}S_{j}}+J\sum_{<i\in\beta,j\in\gamma>}{S_{i}S_{j}}-h\sum_{i}{S_{i}},$
(1)
where $S_{i}$ is an Ising spin with two discrete values $\pm 1$, and the whole
lattice can be divided into three sublattices labeled as $\alpha$, $\beta$,
and $\gamma$. The spin-spin coupling terms are restricted to nearest neighbor
sites, $J$ and $J^{\prime}$ are different nearest neighboring couplings as
shown in Fig. 1, where $J(J^{\prime})>0$ and $<0$ represent antiferromagnetic
and ferromagnetic couplings, respectively, $h$ is the uniform external
magnetic field, and $g\mu_{B}=1$ is assumed.
Figure 2: (Color online) The zero-field phase diagrams for different
couplings: (a) $J>0$ and (b) $J<0$. The phase boundaries are determined by Eq.
(2). In the phase diagrams, FM means ferromagnetic phase, FI represents
ferrimagnetic phase, and PM is paramagnetic phase.
Let us first utilize the exact mapping of 2D Ising model onto the 16-vertex
model to present the exact solutions on the DK lattice. Following Refs. Diep,
; Diep1, ; Diep2, , for $h=0$, we can write down the free energy per triangle
in the thermodynamic limit as
$\displaystyle f$ $\displaystyle=$
$\displaystyle-\frac{T}{16\pi^{2}}\int_{-\pi}^{\pi}d\theta\int_{-\pi}^{\pi}d\phi\,\ln[a+2b\cos(\theta)$
$\displaystyle+$ $\displaystyle
2c\cos(\phi)+2d\cos(\theta-\phi)+2e\cos(\theta+\phi)],\text{ \ \ \ \ \ \ }(2)$
where
$\displaystyle a$ $\displaystyle=$
$\displaystyle\omega_{1}^{2}+\omega_{2}^{2}+\omega_{3}^{2}+\omega_{4}^{2},$
$\displaystyle b$ $\displaystyle=$
$\displaystyle\omega_{1}\omega_{3}-\omega_{2}\omega_{4},$ $\displaystyle c$
$\displaystyle=$ $\displaystyle\omega_{1}\omega_{4}-\omega_{2}\omega_{3},$
$\displaystyle d$ $\displaystyle=$
$\displaystyle\omega_{3}\omega_{4}-\omega_{7}\omega_{8},$ $\displaystyle e$
$\displaystyle=$ $\displaystyle\omega_{3}\omega_{4}-\omega_{5}\omega_{6},$
$\displaystyle\omega_{1}$ $\displaystyle=$ $\displaystyle
2\exp(-2J/T)[1+\exp(2J/T)\cosh(2J^{\prime}/T)]^{2},$ $\displaystyle\omega_{2}$
$\displaystyle=$ $\displaystyle\omega_{1}-8\cosh(2J^{\prime}/T),$
$\displaystyle\omega_{3}$ $\displaystyle=$
$\displaystyle\omega_{4}=\omega_{5}=\omega_{6}=\exp(2J/T)\cosh(4J^{\prime}/T)-\exp(2J/T),$
$\displaystyle\omega_{7}$ $\displaystyle=$
$\displaystyle\omega_{8}=\omega_{1}-4\exp(-2J/T),$
where $k_{B}=1$ is presumed. It is straightforward to readily verify that
these $\omega$’s satisfy the free-fermion conditions,Diep showing that the
Ising model on the DK lattice is exactly solvable. The critical temperature
$T_{c}$ at which the phase transition takes place is determined by the
following equation
$1+4\exp(2J/T_{c})\cosh(2J^{\prime}/T_{c})-\cosh(4J^{\prime}/T_{c})=0.$ (2)
Two different cases are presented in Figs. 2 (a) and (b). When $J>0$, the
magnetic ordered phases appear when $|J^{\prime}/J|>1$, which can be
recognized as a ferrimagnetic phase for $J^{\prime}/J>1$, and a ferromagnetic
phase for $J^{\prime}/J<-1$. This is obtained by checking the magnitude of
spontaneous magnetization [see Fig.3 (c) below]. The disordered paramagnetic
phase separates the two ordered phases, where the phase boundaries are
determined by Eq. (2). For a small $T_{c}$, Eq. (2) can lead to a simple
expression $T_{c}/J\approx 2(|J^{\prime}/J|-1)/\ln 4$, that gives the straight
phase boundaries in Fig. 2 (a). Notice that the paramagnetic phase exists even
at zero temperature owing to the frustration. When $J>0$ and
$|J^{\prime}/J|\in[0,1]$, the spin surrounded by $J^{\prime}$ couplings on
$\alpha$ sublattice is free to flop up or down without costing energy.
Meanwhile, the ground-state spin configurations on $\beta$ and $\gamma$
sublattices are highly degenerate. Hence, the total degeneracy is
$K=2^{N/3+N_{c}}$, with $N$ the number of total sites, and $N_{c}$ the number
of chains consisting of spins on $\beta$ and $\gamma$ sublattices (the
horizontal lines in Fig. 1). This superdegenerate state at zero temperature
connects continuously with the paramagnetic phase at finite temperatures. No
phase transition appears, and the system is disordered at all temperatures.
This observation can also be manifested in Fig. 3 (a), where no singularities
exist in the specific heat for $|J^{\prime}/J|=0.4,0.8,1.0$. However, for
$|J^{\prime}/J|>1$, as Fig. 2 (a) indicates, there exists a ferromagnetic
($J^{\prime}<0$) or ferrimagnetic ($J^{\prime}>0$) state at low temperatures,
and these ordered phases would be destroyed through an order-disorder phase
transition with increasing temperature. Correspondingly, the specific heat for
$|J^{\prime}/J|=1.2$ in Fig. 3 (a) shows a divergent peak, which is a typical
character of second-order phase transition. When $J<0$ [Fig. 2 (b)], the
situations are similar, and the system is ordered (ferromagnetic or
ferrimagnetic) at low temperatures except for the case $J=-1,J^{\prime}=0$,
where the model is degenerated into the decoupled one-dimensional Ising
chains, which has $T_{c}=0$ and thus is disordered at any finite temperature.
Figure 3: (Color online) Temperature dependence of specific heat and
magnetization for different coupling ratio $J^{\prime}/J$ at $h=0$. The TRG
results (open symbols) along with the exact solutions (solid and dashed lines)
are presented for (a) $J>0$ and (b) $J<0$. In (c) and (d), the magnetization
$m$ is plotted for $J>0$ and $J<0$, respectively, where a magnetic order-
disorder transition is clearly seen. The insets illustrate the critical
behaviors of $m$ near $T_{c}$.
## III TRG algorithm
Exact solutions can offer us a reliable phase diagram of the model. However,
some other quantities such as the magnetization $m$ and specific heat in
nonzero magnetic fields cannot be obtained within the above framework. Hence,
we adopt the recently proposed TRG numerical algorithm. The TRG method is
first introduced to calculate the 2D classical models,Levin ; Chang and then
generalized to study 2D quantum spin models.Jiang ; Li ; Gu ; Chen The
principal idea of TRG algorithm is to express the partition function (or the
expectation value of quantum operators) as a tensor network, and then utilizes
the coarse-graining and decimation procedures to approximately obtain the
results. TRG is an efficient method both for classical and quantum spin
models.
The first step is to replace each triangle on Kagomé lattice by a tensor, as
shown in Fig. 1 (a). The energy of each triangle in an external magnetic field
$h$ is
$\varepsilon_{\triangle}(s_{1},s_{2},s_{3})=J^{\prime}s_{1}s_{2}+J^{\prime}s_{1}s_{3}+Js_{2}s_{3}-\frac{1}{2}h(s_{1}+s_{2}+s_{3})$.
We introduce a three-order tensor
$T^{A/B}_{s_{1},s_{2},s_{3}}=\exp(-\varepsilon_{\triangle}(s_{1},s_{2},s_{3})/T)$,
where A(B) means down (up)-pointing triangle in Fig. 1 (a). These tensors form
a honeycomb lattice, and the partition function can be expressed as
$\displaystyle Z$ $\displaystyle=$
$\displaystyle\sum_{s_{1},s_{2},s_{3},...=-1,1}\exp\\{-[\varepsilon_{\triangle}(s_{1},s_{2},s_{3})+\varepsilon_{\triangle}(s_{1},s_{4},s_{5})+...]/T\\}$
(3) $\displaystyle=$
$\displaystyle\sum_{s_{1},s_{2},s_{3},...=-1,1}T^{A}_{s_{1},s_{2},s_{3}}T^{B}_{s_{1},s_{4},s_{5}}...=tTr(T^{A}T^{B}...),$
where $tTr$ represents the tensor trace. Hence the problem of solving the
partition function of Ising model on a DK lattice is equivalently transformed
into a honeycomb tensor network problem, which can be efficiently evaluated
through the rewiring and coarse-graining iterations (see the details in Ref.
Jiang, ). Upon obtaining the partition function $Z$, other thermodynamic
quantities can be evaluated straightforwardly. Alternatively, we can also
introduce some impurity tensors in the tensor networks to achieve this goal.
For example, in order to calculate the magnetization $m$, an impurity tensor
$T_{s_{1},s_{2},s_{3}}^{Im}=(\frac{s_{1}+s_{2}+s_{3}}{3})\exp[-1/T\varepsilon_{\triangle}(s_{1},s_{2},s_{3})]$
can be introduced. By replacing one tensor $T^{A/B}$ in Eq. (3), we can get
the magnetization per site
$m=\frac{tTr(T^{Im}T^{B}T^{A}...)}{Z}.$ (4)
In the following, the second scheme is adopted for evaluating the
thermodynamical quantities, such as the magnetization $m$, energy per site
$e$, etc.
In our calculations, the number of coarse-graining iterations is generally
taken as 20, i.e., the total sites of DK lattice under investigation is
$3^{22}\thickapprox 3\times 10^{10}$, which is close to the thermodynamic
limit. In addition, the periodic boundary conditions are adopted during the
simulations. The initial bond dimension D of tensor $T$ is chosen as 2 owing
to the two states (spin-up and -down) of the Ising spins. With the coarse-
graining procedure, the bond dimension will increase dramatically, and hence
we have to make a truncation and reserve a finite dimension $D_{c}$. In our
calculations, $D_{c}$ is taken as high as 18, and the convergence with various
$D_{c}$ has always been checked.
Figure 4: (Color online) The specific heat and magnetization $m$ as functions
of temperature with $|J^{\prime}/J|=0.04$ at $h=0$. (a) Specific heat, where
three peaks appear, one of which is divergent; (b) Magnetization $m$ and
sublattice magnetization $m_{\alpha}$, where $|m_{\alpha}|$ decreases rapidly
(but does not vanish) around the peak position of the specific heat at low
temperature, which is also revealed as a local maximum of $dm_{\alpha}/dT$ in
the inset.
## IV Specific Heat and Phase Transitions
When $h=0$, both exact solution and TRG method can be utilized to evaluate the
specific heat. In Figs. 3 (a) and (b), the TRG results are plotted by symbols,
while the exact results by lines. Excellent agreement can be observed, except
for the region around the critical point where a divergent peak occurs.
Another character is that the specific heat at zero field is independent of
the sign of coupling $J^{\prime}$, but is relevant to the magnitude of
$|J^{\prime}|$. In Fig. 3 (a), when $|J^{\prime}|$ is small, there is only one
round peak in the specific heat. By tuning $|J^{\prime}|$ to approach $J$ from
below (e.g. $|J^{\prime}|/J=0.8$), a new round peak appears at low
temperature, which disappears when $|J^{\prime}|=J$ and the system again
exhibits a single round peak. These observations imply that there exist no
phase transitions when $|J^{\prime}|\leq J$, which is owing to the strong
frustration, and is in accordance with the phase diagram in Fig. 2 (a).
Furthermore, if $|J^{\prime}|$ exceeds $J$ [as $|J^{\prime}/J|=1.2$ in the
Fig. 3 (a)], a divergent peak emerges, implying the occurrence of phase
transition. In Fig. 3 (b), a typical curve of specific heat with
$|J^{\prime}/J|=1$ is shown. A divergent peak occurs at the transition
temperature $T_{c}\approx 2.14$, again in agreement with the exact solution
($T_{c}=4/\ln(3+2\sqrt{3})$). Moreover, the specific heat is logarithmically
divergent at the critical point because of
$2\exp(2J/T_{c})+\cosh(4J^{\prime}/T_{c})-1\neq 0$, as shown in Figs. 3 (a)
and (b).
The phase transitions can also be verified by studying the order parameter,
i.e., the magnetization per site $m$ defined in Eq. (4). In Figs. 3 (c) and
(d), when $J=1,J^{\prime}/J>1$ or $J=-1,J^{\prime}>0$, $m=1/3$ at $T=0$ and
remains finite at small temperatures. This nonzero spontaneous magnetization
implies the existence of a ferrimagnetic phase; while $J=1,J^{\prime}/J<-1$ or
$J=-1,J^{\prime}<0$, the magnetization starts from $m=1$, and the system is in
a ferromagnetic phase when $T$ is smaller than the critical temperature
$T_{c}$. With increasing temperature, $m$ decreases steeply to zero in the
vicinity of $T_{c}$, showing a order-disorder phase transition happens. In
addition, the critical behavior of $m$ near $T_{c}$ has been investigated. In
the insets of Figs. 3 (c) and (d), in terms of $m\propto(T_{c}-T)^{\eta}$, the
fittings in different cases coincidentally give $\eta\simeq 1/8$, which is the
same as that of Ising model on a square lattice.Yang The phase transition
occurring at $T_{c}$ probably falls into the universality class of 2D Ising
models.
Another interesting case is shown in Fig. 4 (a), where the temperature
dependence of the specific heat is presented for $J<0$ and
$|J^{\prime}|\ll|J|$. One may note that there are three peaks, including two
round peaks and a divergent one. Both exact solutions and TRG method give the
same results. In order to investigate the origin of each peak, the TRG method
is utilized to calculate the magnetization $m$ and sublattice magnetization
$m_{\alpha}$. As shown in Fig. 4 (b), $m$ behaves rather differently for
$J^{\prime}=0.04$ and $-0.04$, although the specific heat coincides for both.
When $J^{\prime}=-0.04$, the ground state is ferromagnetic, and $m$ decreases
monotonously with increasing temperature and vanishes sharply at critical
temperature $T_{c}$. The case with $J^{\prime}=0.04$ is more interesting,
where the system possesses a ferrimagnetic ground state with $m=1/3$ at $T=0$.
With increasing temperature, $m$ first increases until the temperature is
close to the critical point $T_{c}$, and then goes down steeply to zero. In
order to understand this peculiar behavior, we have plotted the sublattice
magnetization $m_{\alpha}$ as a function of $T$ in Fig. 4 (b). In the
ferrimagnetic case, $m_{\alpha}$ is aligned anti-parallel with the spins on
the other two sublattices, and its magnitude decreases rapidly with increasing
temperature owing to the coupling $J^{\prime}$ weak. Hence,
$m=(-|m_{\alpha}|+m_{\beta}+m_{\gamma})/3$ would first increase until the
temperature approaches to $T_{c}$, where $m_{\alpha}$, $m_{\beta}$, and
$m_{\gamma}$ disappear simultaneously. Moreover, as the inset shows, the
first-order derivative $dm_{\alpha}/dT$ has a round peak at the temperature
$T_{r}$, which coincides with the low temperature peak position of specific
heat. Although $m_{\alpha}$ decreases rapidly around $T_{r}$, it does not
vanish. It should be pointed out that $T_{r}$ is not a critical point, as the
specific heat shows only a round peak and never diverges at $T_{r}$.
## V Magnetization and Susceptibility
### V.1 Magnetization Plateaux and Ground State Phase Diagrams
When the external magnetic field is switched on, the exact solution in Sec. II
no longer works. The TRG method, which has been verified to be accurate and
reliable in the previous sections, is utilized to study the response of the
system to an external magnetic field.
Figure 5: (Color online) In (a) and (b), the magnetic curves for different
couplings for $J>0$ and $J<0$ are shown, respectively, where $|T/J|=0.2$ and
$D_{c}$=18. Inset in (b) presents the magnetic curves with different
temperatures below and above $T_{c}$. In (c) and (d), the ground-state phase
diagrams on $J^{\prime}-h$ plane are presented.
Let us first focus on the magnetization, where the $1/3$ magnetization
plateaux are obtained, as shown in Fig. 5. When $J>0$ and
$J^{\prime}/J\in[-1,1]$ in Fig. 5 (a), the system is in a paramagnetic phase
at all temperatures. An infinitesimal small magnetic field can polarize the
free spin on $\alpha$ sublattice at $T=0$, and hence drives the ground state
to a ferrimagnetic state with $m=1/3$ after a magnetization jump, and the
$1/3$ plateaux appear in the magnetization curves. At finite but small
temperatures, these plateaux are still present. This field-induced $1/3$
plateau ferrimagnetic phase is highly degenerate, and the degeneracy is
$K=2^{N_{c}}$, where $N_{c}$ is the number of independent spin chains in the
system. One of the degenerate spin configurations is shown in Fig.1 (b). When
the field is larger than a critical field $h_{c}$, the spins on $\beta$ and
$\gamma$ sublattices align parallel instead of antiparallel, and the system
has the saturated magnetization $m=1$, leading to a ferromagnetic spin
configuration. The energy difference per site between the polarized
ferromagnetic and the plateau ferrimagnetic state is $\delta
e=\frac{4}{3}(J^{\prime}+J)$. The Zeeman energy $\delta e_{z}=-\frac{2}{3}h$
at the critical magnetic field $h_{c}$ has to compensate this energy
difference. $\delta e_{z}+\delta e=0$ leads to $h_{c}=2(J+J^{\prime})$, as
verified in Fig. 5 (a), where the critical field $h_{c}$ increases with
enhancing the coupling $J^{\prime}$. In addition, there exists a notable
difference between the magnetization curves of $J^{\prime}=1.2$ and others in
Fig. 5 (a). The former starts from a nonzero spontaneous magnetization
($m=1/3$) owing to its ferrimagnetic ground state instead of a paramagnetic
one. The ground state spin configuration for $J^{\prime}>1$ is illustrated in
Fig. 1 (c), which is ferrimagnetically ordered. Considering the spontaneously
broken $Z_{2}$ symmetry, this ground state is no longer degenerate. Hence, the
width of $1/3$ plateau does not obey the relation mentioned above, but has
another relation in the ferrimagnetic case to be discussed below.
Figure 6: (Color online) Temperature dependence of zero-field susceptibility
for different $J$ and $J^{\prime}$. (a) $\chi$ diverges at $T=0$ and $\chi T$
converges to 1/3 as the inset shows; (b) $\chi$ shows divergent peaks at the
critical temperature $T_{c}$, where the magnitude of $\chi$ with
$J=J^{\prime}=-1$ has been divided by two. The susceptibility is calculated at
$h=0.01$ and, the convergence with various small fields has been checked.
For $J<0$, there exist ferromagnetic ($J^{\prime}<0$) and ferrimagnetic
($J^{\prime}>0$) ground states. The cases with $J^{\prime}>0$ possess $m=1/3$
plateaux, as seen in Fig. 5 (b). Comparing with the former case $J>0$, the
width of $1/3$ plateaux has a different relation with couplings $J$ and
$J^{\prime}$. By identifying $\delta e=\frac{8}{3}J^{\prime}$ and $\delta
e_{z}=-\frac{2}{3}h$, $h_{c}=4J^{\prime}$ is obtained, that is independent of
$J$. Here, the spin configuration on $m=1/3$ plateau is illustrated in Fig. 1
(c). The order parameter characterizing this phase is the spontaneous
magnetization $m|_{h=0}$, which implies the breaking of $Z_{2}$ symmetry. By
increasing temperature, the $Z_{2}$ symmetry will eventually recover above the
critical temperature $T_{c}$, and the spontaneous magnetization will vanish.
As shown in the inset of Fig. 5 (b), the magnetization at $T>T_{c}$ starts
from $m=0$, and the $1/3$ plateau is smeared and finally destroyed by strong
thermal fluctuations.
In order to look at the effects of external magnetic fields, the phase
diagrams at zero temperature are plotted in Figs. 5(c) and (d). For $J>0$, as
shown in Fig. 5 (c), there are four different phases including ferromagnetic,
ferrimagnetic, plateau ferrimagnetic, and disorder phase that only exists in
the $h=0$ line. It is worthwhile emphasizing that although the magnetization
in the $1/3$ plateau ferrimagnetic phase has the same value as that in the
ferrimagnetic phase at $T=0$, they are quite different in nature. The former
is induced by a magnetic field and highly degenerate with the degeneracy
$K=2^{N_{c}}$, while the latter is an spontaneously ordered phase with the
$Z_{2}$ symmetry breaking. As indicated in Fig. 5 (d), for $J<0$, only a
ferrimagnetic phase and a ferromagnetic phase exist. Here we would like to
stress that the phase transitions between these different phases only occur at
$T=0$, and the temperature would then blur the transitions. In fact, in the
presence of an external magnetic field, there are no thermodynamic phase
transitions at finite temperatures, which will be discussed in Sec. VI.
### V.2 Susceptibility
In order to understand the magnetic response of the present system to an
external magnetic field, the zero-field susceptibility $\chi$ is obtained by
$\chi=[m(h)-m(h=0)]/h$ for a small magnetic field. In Fig. 6 (a), where $J=1$
and $J^{\prime}\in(0,1)$, the ground state is disordered, and the spins on one
($\alpha$) sublattice are free to flip up or down without an energy cost due
to the frustration effect. $\chi$ diverges, obeying Curie law, i.e.,
$\chi\propto 1/T$ as $T$ approaches zero. This result is validated in the
inset of Fig. 6 (a), where the $\chi T$ curves converge to a constant $1/3$ at
low temperatures, which is independent of $J^{\prime}$. The specific value
$1/3$ can be attributed to the free spins on one of three sublattices. On the
other hand, in the high temperature limit, $\chi$ decays with the Curie-Weiss
law, which can be fitted by $\chi T=\frac{T}{T+\theta}$ up to $T/J=100$ (Note
that Fig. 6 shows only to $T/J\simeq 9$). It is straightforward to use the
mean-field approximation to obtain the Curie-Weiss temperature $\theta$ as
$(8J^{\prime}+4J)/3$, and $\theta=1.867$ and $2.933$ for $J^{\prime}=0.2$ and
$0.6$, respectively. The fittings in the inset of Fig. 6 (a) agree with the
mean-field predictions, that further validates our TRG results. In Fig. 6 (a),
it is interesting to notice that there exists a turning point at an
intermediate temperature in the crossover region, which separates the low $T$
Curie behavior and high $T$ Curie-Weiss behavior, as shown in the inset of
Fig. 6(a). Quite differently, as seen in Fig. 6 (b), when the ground state is
ferrimagnetically or ferromagnetically ordered, the susceptibility has a
divergent peak at $T_{c}$ where the magnetic ordering is destroyed by thermal
fluctuations. This again certifies the existence of magnetic order-disorder
phase transitions.
Figure 7: (Color online) A comparison of TRG results to the experiment, where
the experimental data are taken from Ref. Gao, . The TRG result is calculated
at a small field $h/J=0.05$. The high temperature fittings with the Curie-
Weiss law to both experimental and TRG results are also shown. The inset
predicts a divergent peak in the specific heat around $T=20$K.
### V.3 Comparison to Experiments
The complex Co(N3)2(bpg)$\cdot$ DMF4/3 reported in Ref. Gao, is a molecular
magnetic material, in which the Co+2 ions form a distorted Kagomé layer.
Experimentally, the susceptibility does not go to zero as $T$ approaches zero
(see Fig. 7), which is unusual for an isotropic Heisenberg antiferromagnetic
system. In fact, the Co+2 ions are believed to have effective spin-1/2 when
$T\leq$ 20K, with anisotropic Lande $g$ factors ( $g_{\parallel}\neq 0$,
$g_{\perp}\approx 0$), which implies that in this compound the Ising-type
couplings may be dominant between Co+2 ions.Carlin ; xyWang Here, we try to
use our TRG results to fit the experimental data of susceptibility (especially
for the low $T$ region) for this complex. To be consistent with the
experimental convention, the definition of susceptibility $\chi=m(h)/h$ is
adopted. As shown in Fig. 7, $\chi$ decreases steeply around the transition
temperature and, one may see that the fittings agree rather well with the
experimental data at low temperatures. The exchange coupling constants for
this compound are estimated through the fittings as $J=22K$ and
$J^{\prime}=33K$. According to our study on the Ising DK lattice with the
parameters $J>0$ and $J^{\prime}/J=1.5$, the system has a ferrimagnetic phase
at low temperatures. It is thus not difficult to understand why the low
temperature susceptibility goes to a finite value instead of zero for this
compound. At high temperatures, by fitting the TRG results with the Curie-
Weiss law $\chi=C^{\prime}/(T+\theta)$, we find that the Curie-Weiss
temperature $\theta\thickapprox 161.3K$, which agrees well with that of
experimental estimation ($\theta\thickapprox 165.8K$, see online supporting
material of Ref. Gao, ). Besides, we find that the ratio of the experimental
susceptibility to the result from the Ising model equals a constant $R\approx
5.4$ in the high temperature limit. This constant ratio may be ascribed to the
fact that in the material at $T>$20K the effective spin of Co+2 ions may no
longer be 1/2 and also, the other interactions such as XY couplings may
intervene, giving rise to that the Ising model is insufficient to describe the
behaviors of this complex. Surely, more experimental results towards this
direction are needed. In addition, we have calculated the specific heat based
on the Ising model with the couplings given above, and found that a divergent
peak exists around $T=20$K, as depicted in the inset of Fig. 7, suggesting
that this compound may undergo a phase transition at low temperature.
Experimental studies on the specific heat and other quantities for this
compound will be carried out in near future.
Figure 8: (Color online) The specific heat $C$ and the magnetization $m$ as
functions of temperature in different magnetic fields. (a), (b) and (e)
illustrate the ferrimagnetic case $J>0,J^{\prime}/J=1.5$, (c) and (f) are for
the paramagnetic case $J>0,J^{\prime}/J=0.5$, and (d) depicts the
ferromagnetic case $J>0,J^{\prime}/J=-1.5$.
## VI Specific Heat and magnetization in a Magnetic Field
Next, we will study the effect of an external magnetic field on the specific
heat. Three typical cases will be studied: ferrimagnetic
($J>0,J^{\prime}/J=1.5$), paramagnetic ($J>0,J^{\prime}/J=-0.5$), and
ferromagnetic ($J>0,J^{\prime}/J=-1.5$) cases.
In Fig. 8, the specific heat in the presence of an external field for the
ferrimagnetic case ($J>0$, $J^{\prime}/J>1$) is shown. Fig. 8(a) shows at
small fields with $h\leq 2J$, the peak of specific heat moves towards high
temperatures, and its height firstly decreases, and then increases with
enhancing the field until it approaches the spin flop critical field $h_{c}$,
which polarizes all spins. In Fig. 8(b), when the field keeps increasing, the
peak of specific heat becomes dulled, and then splits into double peaks
(except for the point $h=h_{c}$), which can be viewed as a field-induced
splitting. Similar phenomena have also been observed in other Ising and
Heisenberg spin systems.Gong ; Li2 The double peak scenario will eventually
be spoiled by further increasing $h$. When $h\gg h_{c}$, the specific heat
will again be single-peaked. It is notable that the divergent peaks at zero
field disappear immediately when the field is switched on, which means that
the phase transitions are absent and the system remains in the ferrimagnetic
phase at all temperatures. When $h=0$, the ferrimagnetic ordered phase
spontaneously breaks the $Z_{2}$ symmetry contained in the Hamiltonian (see
Eq. 1), and possesses a nonzero order parameter. When $T>T_{c}$, the thermal
fluctuations will destroy the magnetic order, while $Z_{2}$ symmetry will be
recovered, and $m$ vanishes immediately [the solid line in Fig. 8 (e)].
However, the external field explicitly breaks the $Z_{2}$ symmetry in the
Hamiltonian, and $m$ is nonzero even at high temperature $T>T_{c}$ [the symbol
lines in Fig. 8 (e)]. Therefore, no phase transition occurs in the presence of
a magnetic field. In Fig. 8(e), according to the magnetization $m$ at zero
temperature, the curves can be classified into three classes. When $h<h_{c}$,
the curves start from $m=1/3$; while $h>h_{c}$, the spins are polarized and
$m=1$ at $T=0$; when $h=h_{c}$, the case is of a little subtlety, where $m$
equals to the statistical mean value $2/3$ at zero temperature.
In Figs. 8 (c) and (f), the paramagnetic case $J>0,J^{\prime}/J=0.5$ is
studied. The field will firstly promote the peak height of the specific heat,
and moves the peak to the high temperature side. When $h$ is close to the
critical field, the specific heat will again be dulled, where the height is
decreasing, and the peak splits into two sub peaks except at the point
$h=h_{c}$. When the field $h\gg h_{c}$ a single peak of the specific heat
recurs. The $m-T$ curves in Fig. 8 (f) are quite similar to those in Fig. 8
(e) and can be classified analogously. At last, the ferromagnetic case
$J>0,J^{\prime}/J<-1$ is shown in Fig. 8 (d). The divergent peak for the
ferromagnetic-paramagnetic phase transition disappears owing to the same
reason in the ferrimagnetic case as mentioned above. The specific heat reveals
a round peak, which moves towards the high temperature side. By continuously
enhancing the field, the height of the peak decreases down firstly and then
goes slowly up.
Besides, we have also studied other situations with different couplings, and
found that they can be ascribed into the above three classes. For instance,
other ferrimagnetic cases with $J<0,J^{\prime}>0$ and ferromagnetic cases with
all ferromagnetic couplings ($J,J^{\prime}<0$) behave similarly with those
presented in Fig. 8.
## VII Summary and discussion
In this article, we have systematically studied the thermodynamics and
magnetic properties of Ising model on a DK lattice by exact solutions and the
TRG numerical method. It is shown that the phase diagrams are composed of
three phases including ferromagnetic, ferrimagnetic, and paramagnetic phases.
Phase transitions between them are identified by studying the specific heat
and magnetization. The critical exponent $\eta$ of $m$ near $T_{c}$ is
determined as $1/8$, which appears to fall into the universality of the 2D
Ising models. The TRG results of zero-field specific heat agree very well with
the exact solutions, showing that TRG is an efficient and accurate tool in
dealing with 2D Ising models. The TRG method is also utilized to study the
properties in the presence of a magnetic field. In the magnetization curves,
$1/3$ plateaux at low $T$ are identified and, the relations of the plateau
width with coupling constants $J,J^{\prime}$ are obtained. In addition, the
zero temperature $J^{\prime}-h$ phase diagrams are presented to clarify the
various ground state phases in external magnetic field. The zero-field
susceptibility $\chi$ of the paramagnetic case ($J>0,|J^{\prime}/J|\leq 1$) is
found to obey Curie law at low $T$ and Curie-Weiss law at high $T$. While in
the ferrimagnetic or ferromagnetic case, the divergent peak of $\chi$ is found
at the critical temperature. Moreover, the specific heat under different
magnetic fields is also investigated. It is uncovered that the phase
transitions are absent immediately when a magnetic field is switched on, and
the field-induced peak splitting of the specific heat is recognized when $h$
is close to the critical field. We have also fitted the experimental data of
susceptibility of the complex Co(N3)2(bpg)$\cdot$ DMF4/3 with the TRG results,
and obtained the couplings $J=22$K and $J^{\prime}=33$K. Based on TRG
calculations, a ferrimagnetic-paramagnetic phase transition is expected to
occur at about $T=20$K in this complex, which will be studied in future. The
present study offers a systematic understanding for physical properties of the
2D Ising model on the DK lattice, and will be useful for analyzing future
experimental observations in related magnetic materials with DK lattices.
###### Acknowledgements.
We are indebted to Z. Y. Chen, Y. T. Hu, X. L. Sheng, Z. C. Wang, X. Y. Wang,
B. Xi, Q. B. Yan, F. Ye, and Q. R. Zheng for helpful discussions. This work is
supported in part by the NSFC (Grants No. 10625419, No. 10934008, No.
90922033) and the Chinese Academy of Sciences.
## References
* (1) P. Lecheminant, B. Bernu, C. Lhuillier, L. Pierre, and P. Sindzingre, Phys. Rev. B 56, 2521 (1997).
* (2) H. C. Jiang, Z. Y. Weng, and D. N. Sheng, Phys. Rev. Lett. 101, 117203 (2008).
* (3) See, for instance, U. Schollwök, J. Richter, and D. J. J. Farnell, and R. F. Bishop, Quantum Magnetism, Lect. Notes Phys. 645 (Springer, Berlin, 2004), Chapter 2, and references therein.
* (4) Fa Wang, Ashvin Vishwanath, and Yong Baek Kim, Phys. Rev. B 76, 094421 (2007).
* (5) Andreas P. Schnyder, Oleg A. Starykh, and Leon Balents, Phys. Rev. B 78, 174420 (2008).
* (6) M. Yoshida, M. Takigawa, H. Yoshida, Y. Okamoto, and Z. Hiroi, Phys. Rev. Lett. 103, 077207(2009).
* (7) H. Yoshida, Y. Okamoto, T. Tayama, et.al., J. Phys. Soc. Jpn. 78, 043704 (2009).
* (8) J. N. Behera and C. N. R. Rao, Inorg. Chem. 45, 9475 (2006).
* (9) Xin-Yi Wang, Lu Wang, Zhe-Ming Wang, and Song Gao, J. Am. Chem. Soc. 128, 674 (2006).
* (10) R. Kaneko, T. Misawa, and M. Imada, arXiv:1004.2401v1 (2010).
* (11) K. Hida, J. Phys. Soc. Jpn. 70, 3673 (2001).
* (12) Richard L. Carlin, Magnetochemistry, (Springer-Verlag, Berlin, 1986), P29-30, P65-69, and references therein.
* (13) M. Levin and C. P. Nave, Phys. Rev. Lett. 99, 120601 (2007).
* (14) H.C. Jiang, Z.Y. Weng, and T. Xiang, Phys. Rev. Lett. 101 090603 (2008); Z.Y. Xie, H.C. Jiang, Q.N. Chen, Z.Y.Weng, and T. Xiang, Phys. Rev. Lett. 103, 160601 (2009); H. H. Zhao, Z. Y. Xie, Q. N. Chen, Z. C. Wei, J. W. Cai, and T. Xiang, Phys. Rev. B 81, 174411 (2010).
* (15) Z. C. Gu, M. Levin, and X. G. Wen, Phys. Rev. B 78, 205116 (2008).
* (16) H. T. Diep and H. Giacomini, Frustrate Spin Systems, ed. H. T. Diep (World Scientific, Singapore, 2005) p. 1.
* (17) P. Azaria, H. T. Diep, and H. Giacomini, Phys. Rev. Lett. 59, 1629 (1987).
* (18) M. Debauche, H. T. Diep, P. Azaria, and H. Giacomini, Phys. Rev. B 44, 2369 (1991).
* (19) Ming-Che Chang, Min-Fong Yang, Phys. Rev. B 79, 104411 (2009).
* (20) Wei Li, Shou-Shu Gong, Yang Zhao, and Gang Su, Phys. Rev. B 81, 184427 (2010).
* (21) P. Chen, C.Y. Lai, and M.F. Yang, J. Stat. Mech. P10001 (2009).
* (22) C. N. Yang, Phys. Rev. 85, 808 (1952).
* (23) We thank X. Y. Wang for discussions about the low temperature magnetic properties of Co+2 ions.
* (24) Shou-Shu Gong, Song Gao, and Gang Su, Phys. Rev. B 80, 014413 (2009).
* (25) Wei Li, Shou-Shu Gong, Yang Zhao, Ziyu Chen, and Gang Su, Phys. Lett. A 374, 2589 (2010).
|
arxiv-papers
| 2010-10-26T02:01:28 |
2024-09-04T02:49:14.216723
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Wei Li, Shou-Shu Gong, Yang Zhao, Shi-Ju Ran, Song Gao, and Gang Su",
"submitter": "Wei Li",
"url": "https://arxiv.org/abs/1010.5297"
}
|
1010.5337
|
††footnotetext: File: main.tex, printed: 2024-08-27, 15.56
# On Kaluza’s sign criterion for reciprocal power series
Árpád Baricz , Jetro Vesti and Matti Vuorinen Department of Economics,
Babeş-Bolyai University, Cluj-Napoca 400591, Romania bariczocsi@yahoo.com
Department of Mathematics, University of Turku, Turku 20014, Finland
jejove@utu.fi Department of Mathematics, University of Turku, Turku 20014,
Finland vuorinen@utu.fi
###### Abstract.
T. Kaluza has given a criterion for the signs of the power series of a
function that is the reciprocal of another power series. In this note the
sharpness of this condition is explored and various examples in terms of the
Gaussian hypergeometric series are given. A criterion for the monotonicity of
the quotient of two power series due to M. Biernacki and J. Krzyż is applied.
###### Key words and phrases:
Power series; Log-convexity; Hypergeometric functions; Trigonometric
functions.
###### 2000 Mathematics Subject Classification:
30B10, 33C05, 33B10
## 1\. Introduction
In this paper we are mainly interested on the class of Maclaurin series
$\sum_{n\geq 0}a_{n}x^{n},$ which are convergent for $x\in\mathbb{R}$ such
that $|x|<r.$ Throughout in the paper $\\{a_{n}\\}_{n\geq 0}$ is a sequence of
real numbers and $r>0$ is the radius of convergence. Note that if
$f(x)=\sum_{n\geq 0}a_{n}x^{n}$ and $g(x)=\sum_{n\geq 0}b_{n}x^{n}$ are two
Maclaurin series with radius of convergence $r,$ then their product
$h(x)=f(x)g(x)=\sum_{n\geq 0}c_{n}x^{n}$ has also radius of convergence $r$
and Cauchy’s product rule gives the coefficients $c_{n}$ of $h(x)$ as
(1.1) $c_{n}=\sum_{k=0}^{n}a_{k}b_{n-k},$
known as the convolution of $a_{n}$ and $b_{n}.$ If $g(x)$ never vanishes,
also the quotient $q(x)=f(x)/g(x)=\sum_{n\geq 0}q_{n}x^{n}$ is convergent with
radius of convergence $r$ and we obtain the rule for the coefficients $q_{n}$
by interchanging $a$ and $c$ in (1.1)
$q_{n}=(a_{n}-\sum_{k=0}^{n-1}q_{k}b_{n-k})/b_{0}.$
We note that a special case of the above relation when $a_{0}=1$ and
$0=a_{1}=a_{2}=\dots$ yields the following result.
###### Proposition 1.2.
Suppose that $g(x)=\sum_{n\geq 0}b_{n}x^{n}$ with $b_{0}\neq 0$ and
${1}/{g(x)}=\sum_{n\geq 0}q_{n}x^{n}.$ In order to solve $q_{n}$ we need to
know $b_{0},b_{1},b_{2},\dots,b_{n}$.
###### Proof.
Since
$\frac{1}{b_{0}+b_{1}x+b_{2}x^{2}+{\dots}+b_{n}x^{n}+{\dots}}=q_{0}+q_{1}x+q_{2}x^{2}+{\dots}+q_{n}x^{n}+{\dots},$
we just need to solve the linear equations
$\left\\{\begin{array}[]{ll}1=b_{0}q_{0}\\\ 0=b_{1}q_{0}+b_{0}q_{1}\\\
0=b_{2}q_{0}+b_{1}q_{1}+b_{0}q_{2}\\\ \vdots\\\
0=\sum_{k=0}^{n}b_{k}q_{n-k}\\\
\end{array}\right.\Longleftrightarrow\left\\{\begin{array}[]{ll}q_{0}={1}/{b_{0}}\\\
q_{1}=(-b_{1}q_{0})/b_{0}\\\ q_{2}=(-b_{2}q_{0}-b_{1}q_{1})/b_{0}\\\ \vdots\\\
q_{n}=(-\sum_{k=1}^{n}b_{k}q_{n-k})/b_{0}\end{array}\right..$
Thus, $q_{n}=\phi(b_{0},b_{1},\dots,b_{n}),$ where $\phi$ is some function. ∎
In 1928 Theodor Kaluza111In passing we remark that he was a German
mathematician interested in Physics, where his name is associated with so
called Kaluza-Klein theory. [15] proved the following theorem.
###### Theorem 1.3.
Let $f(x)=\sum_{n\geq 0}a_{n}x^{n}$ be a convergent Maclaurin series with
radius of convergence $r>0.$ If $a_{n}>0$ for all $n\in\\{0,1,\dots\\}$ and
the sequence $\\{a_{n}\\}_{n\geq 0}$ is log-convex, that is, for all
$n\in\\{1,2,\dots\\}$
(1.4) $a_{n}^{2}\leq a_{n-1}a_{n+1},$
then the coefficients $b_{n}$ of the reciprocal power series
$1/f(x)=\sum_{n\geq 0}b_{n}x^{n}$ have the following properties:
$b_{0}=1/a_{0}>0$ and $b_{n}\leq 0$ for all $n\in\\{1,2,\dots\\}.$
In what follows we say that a power series has the Kaluza sign property if the
coefficients of its reciprocal power series are all non-positive except the
constant term. Theorem 1.3 then says that if the power series $f(x)$ has
positive and log-convex coefficients, then $f(x)$ has the Kaluza sign
property. For a short proof of Theorem 1.3 see [7]. This result is also cited
in [10, p. 68] and [12, p. 13]. Note that Theorem 1.3 in Jurkat’s paper [14]
is attributed to Kaluza and Szegő, however Szegő [19] attributes this result
to Kaluza. We also note that this result implies, in particular, that the
function $x\mapsto 1/f(x)$ is decreasing on $(0,r).$ This observation is also
clear because $x\mapsto f(x)$ is increasing on $(0,r).$ It is also important
to note here that Kaluza’s result is useful in the study of renewal sequences,
which are frequently applied in probability theory. For more details we refer
to the papers [9, 13, 16, 17] and to the references contained therein.
We will next look at the condition (1.4) from the point of view of power
means. For fixed $a,b,t>0,$ we define the power mean by
$m(a,b,t)=\left(\frac{a^{t}+b^{t}}{2}\right)^{1/t}.$
It is well-known (see for example [4]) that $\lim\limits_{t\to
0}m(a,b,t)=\sqrt{ab}$ and the function $t\mapsto m(a,b,t)$ is increasing on
$(0,\infty)$ for all fixed $a,b>0.$ Therefore for all $u>t>0$ we have
$\sqrt{ab}\leq m(a,b,t)\leq m(a,b,u).$
By observing that (1.4) is the same as $a_{n}\leq\lim\limits_{t\to
0}m(a_{n-1},a_{n+1},t)$ we can prove that (1.4) is sharp in the following
sense.
###### Theorem 1.5.
Suppose that in the above theorem all the hypotheses except (1.4) are
satisfied and (1.4) is replaced with
(1.6) $a_{n}\leq m(a_{n-1},a_{n+1},t)$
where $n\in\\{1,2,\dots\\}$ and $t\geq 1/100.$ Then the conclusion of Theorem
1.3 is no longer true.
###### Proof.
The monotonicity with respect to $t$ yields for all $n\in\\{1,2,\dots\\}$ and
$u\geq t>0$
$\left(\frac{a_{n-1}^{t}+a_{n+1}^{t}}{2}\right)^{1/t}\leq\left(\frac{a_{n-1}^{u}+a_{n+1}^{u}}{2}\right)^{1/u}.$
The series $q(x)=1.999+\sum_{n\geq 1}{x^{n}}/n$ satisfies all the hypotheses
that were made:
$1<\left(\frac{1.999^{1/100}+0.5^{1/100}}{2}\right)^{100}(\approx
1.00215)\leq\left(\frac{1.999^{t}+0.5^{t}}{2}\right)^{1/t}$
for all $t\geq 1/100$ and generally when $n\in\\{2,3,\dots\\}$
$\frac{1}{n}<\sqrt{\frac{1}{(n-1)(n+1)}}\leq\left(\frac{\left(\frac{1}{n-1}\right)^{t}+\left(\frac{1}{n+1}\right)^{t}}{2}\right)^{1/t}$
for all $t\geq 1/100.$ Because the series
$\frac{1}{q(x)}=0.50025-0.25025x+0.000062594x^{2}-\ldots$
has a positive coefficient different from a constant term we get our claim. ∎
Theorem 1.5 shows that it is not possible to replace the hypothesis (1.4) with
(1.6), at least if $t\geq{1}/{100}.$ Moreover, we note that it is easy to
reduce the number ${1}/{100}$. To that end, it is enough to replace the
constant $1.999$ of the Maclaurin series $q(x)$ in the proof of Theorem 1.5
with another constant in $(1.999,2).$
## 2\. Remarks on the Kaluza sign property
In this section we will make some general observations about power series and
Kaluza’s Theorem 1.3. The Gaussian hypergeometric series is often useful for
illustration purposes and it is available at the Mathematica(R) software
package which is used for the examples. For $a,b,c$ real numbers and $|x|<1,$
it is defined by
${}_{2}F_{1}(a,b;c;x)=\sum_{n\geq 0}\frac{(a,n)(b,n)}{(c,n)n!}x^{n},$
where $(a,n)=a(a+1)...(a+n-1)=\Gamma(a+n)/\Gamma(a)$ for $n\in\\{1,2,\dots\\}$
and ${(a,0)}=1,$ is the rising factorial and it is required that $c\neq
0,-1,\dots$ in order to avoid division by zero. Some basic properties of this
series may be found in standard handbooks, see for example [18].
We begin with an example which is related to Proposition 1.2.
###### Example 2.1.
Let
$f(x)=\cosh{x}=\sum_{n\geq 0}\frac{1}{(2n)!}x^{2n}$
and
$g(x)=\cos{x}=\sum_{n\geq 0}\frac{(-1)^{n}}{(2n)!}x^{2n}.$
Then
$\frac{1}{f(x)}=1-\frac{x^{2}}{2}+\frac{5x^{4}}{24}-\frac{61x^{6}}{720}+\frac{277x^{8}}{8064}-\frac{50521x^{10}}{3628800}+\mathcal{O}\left(x^{11}\right)$
and
$\frac{1}{g(x)}=1+\frac{x^{2}}{2}+\frac{5x^{4}}{24}+\frac{61x^{6}}{720}+\frac{277x^{8}}{8064}+\frac{50521x^{10}}{3628800}+\mathcal{O}\left(x^{11}\right).$
Observe the similarities in the coefficients. Similarly, if
$f(x)=\frac{\sinh(x)}{x}=\sum_{n\geq 0}\frac{1}{(2n+1)!}x^{2n}$
and
$g(x)=\frac{\sin(x)}{x}=\sum_{n\geq 0}\frac{(-1)^{n}}{(2n+1)!}x^{2n},$
then
$\frac{1}{f(x)}=1-\frac{x^{2}}{6}+\frac{7x^{4}}{360}-\frac{31x^{6}}{15120}+\frac{127x^{8}}{604800}-\frac{73x^{10}}{3421440}+\mathcal{O}\left(x^{11}\right)$
and
$\frac{1}{g(x)}=1+\frac{x^{2}}{6}+\frac{7x^{4}}{360}+\frac{31x^{6}}{15120}+\frac{127x^{8}}{604800}+\frac{73x^{10}}{3421440}+\mathcal{O}\left(x^{11}\right).$
These observations are special cases of the following result.
###### Proposition 2.2.
Let
$f(x)=\sum_{n\geq 0}a_{2n}x^{2n}\ \ \mbox{and}\ \ g(x)=\sum_{n\geq
0}(-1)^{n}a_{2n}x^{2n},$
where $a_{2n}>0$ for all $n\in\\{0,1,\dots\\}.$ Then the coefficients of the
reciprocal power series
$\frac{1}{f(x)}=\sum_{n\geq 0}b_{n}x^{n}\ \ \mbox{and}\ \
\frac{1}{g(x)}=\sum_{n\geq 0}c_{n}x^{n}$
satisfy $b_{2n+1}=c_{2n+1}=0$ and $b_{2n}=(-1)^{n}c_{2n}$ for all
$n\in\\{0,1,\dots\\}.$
###### Proof.
From the equation
$\displaystyle 1$
$\displaystyle=(a_{0}+a_{2}x^{2}+a_{4}x^{4}+\dots)(b_{0}+b_{1}x+b_{2}x^{2}+\dots)$
$\displaystyle=a_{0}b_{0}+a_{0}b_{1}x+(b_{0}a_{2}+b_{2}a_{0})x^{2}+{\dots}$
$\displaystyle+\left(\sum_{k=0}^{n}b_{2k}a_{2(n-k)}\right)x^{2n}+\left(\sum_{k=0}^{n}b_{2k+1}a_{2(n-k)}\right)x^{2n+1}+\dots$
we get inductively for all $n\in\\{0,1,\dots\\}$
$b_{1}=b_{3}={\dots}=b_{2n+1}=0$
and
$b_{0}=\frac{1}{a_{0}},b_{2}=\frac{1}{a_{0}}(-b_{0}a_{2}),\dots,b_{2n}=\frac{1}{a_{0}}\left(-\sum_{k=0}^{n-1}b_{2k}a_{2(n-k)}\right).$
Similarly, for all $n\in\\{0,1,\dots\\}$ we get
$c_{1}=c_{3}={\dots}=c_{2n+1}=0$
and
$c_{0}=\frac{1}{a_{0}},c_{2}=\frac{1}{a_{0}}(c_{0}a_{2}),\dots,c_{2n}=\frac{1}{a_{0}}\left(-\sum_{k=0}^{n-1}c_{2k}(-1)^{n-k}a_{2(n-k)}\right).$
From these we get our claim: $b_{2n+1}=0=c_{2n+1}$ is clear and
$b_{2n}=(-1)^{n}c_{2n}$ follows by induction. ∎
In the next proposition we show that log-convex sequences can be classified
into two types.
###### Proposition 2.3.
If the positive sequence $\\{a_{n}\\}_{n\geq 0}$ is log-convex, then the
following assertions are true:
1. (1)
If $a_{0}\leq a_{1},$ then $a_{0}\leq a_{1}\leq a_{2}\leq\dots;$
2. (2)
If $a_{1}\leq a_{0},$ then $a_{0}\geq a_{1}\geq a_{2}\geq\dots$ or there
exists $k>0$ such that $a_{0}\geq a_{1}\geq a_{2}\geq\dots\geq a_{k-1}\geq
a_{k}$ and $a_{k}\leq a_{k+1}\leq{\dots}.$
###### Proof.
(1) First suppose that $a_{0}\leq a_{1}.$ Then we have $a_{1}^{2}\leq
a_{0}a_{2}\leq a_{1}a_{2},$ which implies that $a_{1}\leq a_{2}.$ Suppose that
$a_{k-1}\leq a_{k}$ holds for all $k\in\\{1,2,\dots,n\\}.$ Again from
hypothesis we get $a_{k}^{2}\leq a_{k-1}a_{k+1}\leq a_{k}a_{k+1},$ which
implies that $a_{k}\leq a_{k+1}.$ Thus, the first claim follows by induction.
(2) Secondly, suppose that $a_{1}\leq a_{0}.$ If there exists an index $k>0$
such that $a_{k}\leq a_{k+1}$ and does not exist $s<k$ such that $a_{s}\leq
a_{s+1},$ then we get from hypothesis that $a_{k+1}^{2}\leq a_{k}a_{k+2}\leq
a_{k+1}a_{k+2},$ which implies that $a_{k+1}\leq a_{k+2}.$ By induction for
all $n\geq k$ we have that $a_{n}\leq a_{n+1}.$ We also have $a_{n}^{2}\leq
a_{n-1}a_{n+1}\leq a_{n-1}a_{n}$ for all $n<k,$ which implies that $a_{n}\leq
a_{n-1}$ for all $n<k.$ From these we get the last case.
If there does not exists an index $k>0$ such that $a_{k}\leq a_{k+1},$ then we
get the former case by the same way: for all $n\in\\{1,2,\dots\\}$ we have
$a_{n}^{2}\leq a_{n-1}a_{n+1}\leq a_{n-1}a_{n},$ which implies that $a_{n}\leq
a_{n-1}$ for all $n\in\\{1,2,\dots\\}.$ ∎
It should be mentioned here that the previous result is related to the
following well-known result: log-concave sequences are unimodal. Note that a
sequence $\\{a_{n}\\}_{n\geq 0}$ is said to be log-concave if for all $n\geq
1$ we have $a_{n}^{2}\geq a_{n-1}a_{n+1}$ and by definition a sequence
$\\{a_{n}\\}_{n\geq 0}$ is said to be unimodal if its members rise to a
maximum and then decrease, that is, there exists an index $k>0$ such that
$a_{0}\leq a_{1}\leq a_{2}\leq{\dots}\leq a_{k}$ and $a_{k}\geq a_{k+1}\
\geq{\dots}\geq a_{n}\geq{\dots}.$
We now illustrate our previous result by giving some examples.
###### Example 2.4.
The power series
$f_{1}(x)=\sum_{n\geq
0}\frac{2^{n}+1}{2}x^{n}=1+\frac{3}{2}x+\frac{5}{2}x^{2}+\frac{9}{2}x^{3}+\dots$
is of type (1) considered in Proposition 2.3 since
$1<\frac{3}{2}<\frac{5}{2}<\frac{9}{2}<{\dots}.$
###### Example 2.5.
The power series
$f_{2}(x)={}_{2}F_{1}(1,1;2;x)=-\frac{\log(1-x)}{x}=1+\frac{x}{2}+\frac{x^{2}}{3}+\frac{x^{3}}{4}+\frac{x^{4}}{5}+\dots$
and
$f_{3}(x)={}_{2}F_{1}\left(\frac{1}{2},\frac{1}{2};1;x\right)=\sum_{n\geq
0}\frac{\left(\frac{1}{2},n\right)\left(\frac{1}{2},n\right)}{(1,n)n!}x^{n}=1+\frac{1}{4}x+\frac{9}{64}x^{2}+\frac{25}{256}x^{3}+\dots$
are of type (2) considered in Proposition 2.3 since
$1>\frac{1}{2}>\frac{1}{3}>\frac{1}{4}>\dots\ \ \ \mbox{and}\ \ \
1>\frac{1}{4}>\frac{9}{64}>\frac{25}{256}>{\dots}.$
###### Example 2.6.
The power series
$f_{4}(x)=1+\frac{77}{80}x+\frac{19}{20}x^{2}+\frac{3}{2}x^{3}+\frac{5}{2}x^{4}+\frac{9}{2}x^{5}+\sum_{n\geq
6}\frac{2^{n-2}+1}{2}x^{n}$
is of type (2) considered in Proposition 2.3 since
$1>\frac{77}{80}>\frac{19}{20}<\frac{3}{2}<\frac{5}{2}<\frac{9}{2}<{\dots}.$
Now, let us recall some simple properties of log-convex sequences: the product
and sum of log-convex sequences are also log-convex. Moreover, it is easy to
see that log-convexity is stable under term by term integration in the
following sense: if the coefficients of the power series $f(x)=\sum_{n\geq
0}a_{n}x^{n}$ form a log-convex sequence, then coefficients of the series
$g(x)=\frac{1}{x}\int_{0}^{x}f(t)dt=\sum_{n\geq 0}\frac{1}{n+1}a_{n}x^{n}$
also form a log-convex sequence and in view of Theorem 1.3 this implies that
the power series $g(x)$ has also the Kaluza sign property. On the other hand
this is not true about differentiation: if the coefficients of the series
$f(x)=\sum_{n\geq 0}a_{n}x^{n}$ form a log-convex sequence, then the
coefficients of the power series
$f^{\prime}(x)=\sum_{n\geq 0}(n+1)a_{n+1}x^{n}$
do not form necessarily a log-convex sequence. Moreover, it can be shown that
if the above power series $f(x)$ has the Kaluza sign property, then the power
series $f^{\prime}(x)$ does not need to have the Kaluza sign property.
###### Example 2.7.
The hypergeometric series
$f_{2}(x)=1+\frac{x}{2}+\frac{x^{2}}{3}+\frac{x^{3}}{4}+\frac{x^{4}}{5}+\dots$
has Kaluza’s sign property but the series
$f_{2}^{\prime}(x)=\frac{1}{2}+\frac{2}{3}x+\frac{3}{4}x^{2}+\frac{4}{5}x^{3}+\dots$
does not since
$\frac{1}{f_{2}^{\prime}(x)}=2-\frac{8}{3}x+\frac{5}{9}x^{2}+{\dots}.$
All the same, the power series
$\frac{1}{x}\int_{0}^{x}f_{2}(t)dt=1+\frac{x}{4}+\frac{x^{2}}{9}+\frac{x^{3}}{16}+\frac{x^{4}}{25}+{\dots}$
has the Kaluza sign property.
The following examples show that if the power series $f(x)$ and $g(x)$ have
Kaluza’s sign property, then in general it is not true that the series
$f(x)g(x)$ or the quotient $f(x)/g(x)$ would also have Kaluza’s sign property.
Furthermore, if the series $f(x)$ has the Kaluza sign property, then in
general the series $\left[f(x)\right]^{\alpha}$ does not have the Kaluza sign
property if $\alpha>1.$
###### Example 2.8.
Let $f_{1}(x),f_{2}(x)$ be as earlier. The series $f_{1}(x)f_{2}(x)$ and
$f_{2}(x)/f_{1}(x)$ do not have the Kaluza sign property because
$\frac{1}{f_{1}(x)f_{2}(x)}=1-2x+\frac{5}{12}x^{2}-\frac{1}{6}x^{3}-\dots$
and
$\frac{1}{f_{2}(x)/f_{1}(x)}=1+x+\frac{5}{3}x^{2}+\frac{37}{12}x^{3}+\dots$
###### Example 2.9.
The series $\left[f_{1}(x)\right]^{3}$ and $\left[f_{2}(x)\right]^{1.8}$ do
not have the Kaluza sign property because
$\frac{1}{\left[f_{1}(x)\right]^{3}}=1-\frac{9}{2}x+6x^{2}-\frac{9}{4}x^{3}+\dots$
and
$\frac{1}{\left[f_{2}(x)\right]^{1.8}}=1-0.9x+0.03x^{2}-0.009x^{3}-{\dots}.$
###### Example 2.10.
We note that if the sequence $\\{a_{n}\\}_{n\geq 0}$ is log-convex and either
$a_{0}\leq a_{1}\leq a_{2}\leq\dots$ or $a_{0}\geq a_{1}\geq a_{2}\geq\dots,$
then the sequence $\\{a_{n}^{\alpha}\\}_{n\geq 0}$ would seem to be also log-
convex if $0<\alpha\leq 1.$ However, if there exists an index $k\geq 1$ such
that $a_{0}\geq a_{1}\geq a_{2}\geq{\dots}\geq a_{k}\leq a_{k+1}\leq\dots$
then generally the sequence $\\{a_{n}^{\alpha}\\}_{n\geq 0}$ is not log-convex
if $0<\alpha<1.$ The series $f_{1}(x),f_{2}(x)$ and $f_{3}(x)$ are all either
of type $a_{0}<a_{1}<a_{2}<\dots$ or of type $a_{0}>a_{1}>a_{2}>{\dots}.$
Numerical experiments show that the series
$[f_{1}(x)]^{\alpha},[f_{2}(x)]^{\alpha}$ and $[f_{3}(x)]^{\alpha}$ have the
Kaluza sign property at least for the first 20 terms when $\alpha=0.05k+0.05$
and $k\in\\{0,1,\dots,19\\}.$
The series $f_{4}(x)$ is of type
$a_{0}>a_{1}>a_{2}>\dots>a_{k}<a_{k+1}<{\dots}.$ The series
$\left[f_{4}(x)\right]^{1/2}$ does not have the log-convexity property because
$\frac{1}{\left[f_{4}(x)\right]^{1/2}}=1+\frac{77}{160}x+\frac{18391}{51200}x^{2}+\frac{4727893}{8192000}x^{3}+\frac{190367203}{209715200}x^{4}+\cdots$
and $a_{3}^{2}>a_{2}a_{4}\,.$
Finally, we note that the coefficients of the Maclaurin series
$f_{5}(x)=1+\sum_{n\geq 1}\frac{x^{n}}{n}$
satisfy (1.4) for all $n\in\\{2,3,\dots\\},$ but the reciprocal power series
has a positive coefficient, that is,
$\frac{1}{f_{5}(x)}=1-x+\frac{1}{2}x^{2}-\frac{1}{3}x^{3}+{\dots}.$
Thus, for the Kaluza sign property it is not enough that (1.4) holds starting
from some index $n_{0}\in\\{2,3,\dots\\}.$ Moreover, it is not easy to find a
series $f(x)$ whose coefficients would not form a log-convex sequence and in
the series $1/f(x)$ all the coefficients except the constant would be
negative. Hence it seems that log-convexity is near of being necessary.
Motivated by the above discussion we present the following result.
###### Theorem 2.11.
Let $f(x)=\sum_{n\geq 0}a_{n}x^{n}$ and $g(x)=\sum_{n\geq 0}b_{n}x^{n}$ be two
convergent power series such that $a_{n},b_{n}>0$ for all
$n\in\\{0,1,\dots\\}$ and the sequences $\\{a_{n}\\}_{n\geq 0},$
$\\{b_{n}\\}_{n\geq 0}$ are log-convex. Then the following power series have
the Kaluza sign property:
1. (1)
the scalar multiplication $\alpha f(x)=\sum_{n\geq 0}(\alpha a_{n})x^{n},$
where $\alpha>0;$
2. (2)
the sum $f(x)+g(x)=\sum_{n\geq 0}(a_{n}+b_{n})x^{n};$
3. (3)
the linear combination $\alpha f(x)+\beta g(x)=\sum_{n\geq 0}(\alpha
a_{n}+\beta b_{n})x^{n},$ where $\alpha,\beta>0;$
4. (4)
the Hadamard (or convolution) product $f(x)*g(x)=\sum_{n\geq
0}a_{n}b_{n}x^{n};$
5. (5)
$u(x)=\sum_{n\geq 0}u_{n}x^{n},$ where
$u_{n}=\sum_{k=0}^{n}C_{n}^{k}a_{k}b_{n-k};$
6. (6)
$v(x)=\sum_{n\geq 0}v_{n}x^{n},$ where
$v_{n}=\sum_{k=0}^{n}\frac{(\alpha,k)(\beta,n-k)}{k!(n-k)!}a_{k}b_{n-k}$ and
$\alpha,\beta>0$ such that $\alpha+\beta=1.$
###### Proof.
Since the sequences $\\{a_{n}\\}_{n\geq 0}$ and $\\{b_{n}\\}_{n\geq 0}$ are
positive and log-convex, clearly the sequences $\\{\alpha a_{n}\\}_{n\geq 0},$
$\\{a_{n}+b_{n}\\}_{n\geq 0},$ $\\{\alpha a_{n}+\beta b_{n}\\}_{n\geq 0}$ and
$\\{a_{n}b_{n}\\}_{n\geq 0}$ are also positive and log-convex. Moreover, due
to Davenport and Pólya [8] we know that the binomial convolution
$\\{u_{n}\\}_{n\geq 0}$, and the sequence $\\{v_{n}\\}_{n\geq 0}$ are also
log-convex. Thus, applying Kaluza’s Theorem 1.3, the proof is complete. ∎
We note that some related results were proved by Lamperti [17], who proved
among others that if the power series $f(x)$ and $g(x)$ in Theorem 2.11 have
the Kaluza sign property, then the power series $f(x)*g(x)$ and $u(x)$ in
Theorem 2.11 have also Kaluza sign property. With other words the convolution
and the binomial convolution preserve the Kaluza sign property. Lamperti’s
approach is different from Kaluza’s approach and provides a necessary and
sufficient condition for a power series (with the aid of infinite matrixes) to
have the Kaluza sign property.
## 3\. Kaluza’s criterion and the hypergeometric series
In this section we give examples of cases of hypergeometric series when the
Kaluza sign property either holds or fails. We shall use the notation
${}_{2}F_{1}(a,b;c;x)=\sum_{n\geq 0}\alpha_{n}x^{n},$
where
$\alpha_{n}=\frac{(a,n)(b,n)}{(c,n)n!}.$
###### Theorem 3.1.
If $a,b,c>0,$ $2ab(c+1)\leq(a+1)(b+1)c$ and $c\geq a+b-1,$ then the sequence
$\\{\alpha_{n}\\}_{n\geq 0}$ is positive and log-convex, and then the Gaussian
hypergeometric series ${}_{2}F_{1}(a,b;c;x)$ has the Kaluza sign property.
###### Proof.
To show that the sequence $\\{\alpha_{n}\\}_{n\geq 0}$ is log-convex we just
need to prove that for all $n\in\\{1,2,\dots\\}$
$\frac{(a,n)^{2}(b,n)^{2}}{(c,n)^{2}{(n!)}^{2}}\leq\frac{(a,n-1)(b,n-1)}{(c,n-1)(n-1)!}\frac{(a,n+1)(b,n+1)}{(c,n+1)(n+1)!}$
or equivalently
$\frac{(a+n-1)(b+n-1)}{(c+n-1)n}<\frac{(a+n)(b+n)}{(c+n)(n+1)}.$
Now, this is equivalent to the inequality for the second degree polynomial
$W(n)=w_{1}n^{2}+w_{2}n+w_{3}\geq 0,$
where
$\left\\{\begin{array}[]{ll}w_{1}=c+1-a-b\\\ w_{2}=a+b+c-2ab-1\\\ w_{3}=ac+bc-
abc-c\end{array}\right.$
and $n\in\\{1,2,\dots\\}.$ If $w_{1}\geq 0,$ i.e. $c\geq a+b-1,$ then in view
of $n^{2}\geq 2n-1,$ we obtain that
$W(n)\geq(3c-a-b-2ab+1)n+(ac+bc-abc-2c+a+b-1).$
Observe that if we suppose $a+b-1-ab>0,$ then $c\geq a+b-1>(a+b+2ab-1)/3$ and
this together with $2ab(c+1)\leq(a+1)(b+1)c$ imply
(3.2) $W(n)\geq c(a+b-ab+1)-2ab\geq 0.$
On the other hand, if we have $a+b-1-ab\leq 0,$ then because of
$2ab(c+1)\leq(a+1)(b+1)c$ we obtain $a+b+1-ab\geq 2ab/c>0$ and then
$c\geq\frac{2ab}{a+b+1-ab}\geq ab\geq\frac{a+b+2ab-1}{3},$
which implies again (3.2). This completes the proof. ∎
The next result shows that the condition $2ab(c+1)\leq(a+1)(b+1)c$ in the
above theorem is not only sufficient, but even necessary.
###### Theorem 3.3.
If $a,b,c>0$ and $2ab(c+1)>(a+1)(b+1)c,$ then the hypergeometric series
${}_{2}F_{1}(a,b;c;x)$ does not have the Kaluza sign property.
###### Proof.
Suppose that the coefficient $a_{n}$ are defined by
$\frac{1}{\sum_{n\geq
0}\frac{(a,n)(b,n)}{(c,n)n!}x^{n}}=1+a_{1}x+a_{2}x^{2}+a_{3}x^{3}+{\dots}.$
Then
$a_{1}=-\frac{ab}{c},\
a_{2}=-\frac{ab}{c}a_{1}-\frac{a(a+1)b(b+1)}{c(c+1)2}=\frac{ab}{c}\left(\frac{ab}{c}-\frac{(a+1)(b+1)}{(c+1)2}\right).$
We shall only look at the sign of $a_{2}.$ If $a_{2}>0$ then
${}_{2}F_{1}(a,b;c;x)$ does not have Kaluza’s sign property. With this the
proof is complete. ∎
For Theorem 3.3 we now give an illuminating example.
###### Example 3.4.
If we consider the hypergeometric series
${}_{2}F_{1}(3,3;6;x)=1+\frac{3}{2}x+\frac{12}{7}x^{2}+\frac{25}{14}x^{3}+\frac{25}{14}x^{4}+\dots$
and look at its reciprocal series we get a positive coefficient different from
a constant term
$\frac{1}{{}_{2}F_{1}(3,3;6;x)}=1-\frac{3}{2}x+\frac{15}{28}x^{2}-\frac{1}{56}x^{3}+{\dots}.$
Next we are going to present a counterpart of Theorem 3.1. To do this we first
recall the following result of Jurkat [14].
###### Theorem 3.5.
Let us consider the power series $p(x)=\sum_{n\geq 0}p_{n}x^{n}$ and
$q(x)=\sum_{n\geq 0}q_{n}x^{n},$ where $p_{0}>0$ and the sequence
$\\{p_{n}\\}_{n\geq 0}$ is decreasing. If for all $n\in\\{1,2,\dots\\}$
(3.6) $\overline{\Delta}q_{n}\geq\frac{q_{0}}{p_{0}}\overline{\Delta}p_{n},$
where $\overline{\Delta}a_{n}=a_{n}-a_{n-1}$ for all $n\in\\{1,2,\dots\\},$
$\overline{\Delta}a_{0}=a_{0},$ then the coefficients of the power series
$k(x)=q(x)/p(x)=\sum_{n\geq 0}k_{n}x^{n}$ satisfies $k_{n}\geq 0$ for all
$n\in\\{1,2,\dots\\}.$ Moreover, if (3.6) is reversed, then $k_{n}\leq 0$ for
all $n\in\\{1,2,\dots\\}.$
Note that the first part of the above result is [14, Theorem 4], while the
second is [14, Theorem 5]. First, let us consider in the above theorem
$q_{0}=1$ and $q_{n}=0$ for all $n\in\\{1,2,\dots\\}$ to have $k(x)=1/p(x),$
as in [14, Theorem 3]. Then the condition
$q_{n}-q_{n-1}\geq(q_{0}/p_{0})(p_{n}-p_{n-1}),$ i.e. (3.6) for $n=1$ means
that $p_{1}\leq 0$ and for $n\in\\{2,3,\dots\\}$ means that $p_{n}\leq
p_{n-1}.$ Thus, we obtain the following result.
###### Proposition 3.7.
If $a_{0}>0\geq a_{1}\geq a_{2}\geq{\dots}\geq a_{n}\geq{\dots},$ then the
reciprocal of the power series $f(x)=\sum_{n\geq 0}a_{n}x^{n}$ has all
coefficients non-negative. More precisely, if $1/f(x)=\sum_{n\geq
0}b_{n}x^{n},$ then $b_{n}\geq 0$ for all $n\in\\{0,1,\dots\\}.$
By using the above result we may get the following.
###### Theorem 3.8.
If $a,b,c>-1,$ $c\neq 0,$ $ab/c\leq 0,$ and $c\leq\min\\{a+b-1,ab\\},$ then
the reciprocal of the series ${}_{2}F_{1}(a,b;c;x)$ has all coefficients non-
negative, that is, we have $1/{}_{2}F_{1}(a,b;c;x)=1+\sum_{n\geq
1}\beta_{n}x^{n}$ with $\beta_{n}\geq 0$ for all $n\in\\{1,2,\dots\\}.$
###### Proof.
Clearly $\alpha_{0}=1>0$ and $\alpha_{1}=ab/c\leq 0.$ The condition
$\alpha_{n}\geq\alpha_{n+1}$ holds for all $n\in\\{1,2,\dots\\}$ if and only
if we have
$\frac{(a,n)(b,n)}{(c,n+1)(n+1)!}\left((c+n)(n+1)-(a+n)(b+n)\right)\geq 0$
for all $n\in\\{0,1,\dots\\}.$ Now, because $a,b,c>-1,$ $c\neq 0$ and
$ab/c\leq 0,$ for all $n\in\\{0,1,\dots\\}$ we should have
$(a+b-c-1)n+ab-c\geq 0$
Applying Proposition 3.7, the result follows. ∎
Now, let us focus on the second part of Theorem 3.5, i.e. [14, Theorem 5].
Consider again $q_{0}=1$ and $q_{n}=0$ for all $n\in\\{1,2,\dots\\}$ to have
$k(x)=1/p(x),$ as above. Then the condition
$q_{n}-q_{n-1}\leq(q_{0}/p_{0})(p_{n}-p_{n-1})$ for $n=1$ means that
$p_{1}\geq 0$ and for $n\in\\{2,3,\dots\\}$ means that $p_{n}\geq p_{n-1},$
which contradicts condition [14, Eq. (6)], i.e. the hypothesis that the
sequence $\\{p_{n}\\}_{n\geq 0}$ is decreasing. However, following the proof
of [14, Theorem 4], it is easy to see that to have a correct version of [14,
Theorem 5] we need to assume that the sequence $\\{q_{n}\\}_{n\geq 0}$ is
strictly decreasing. More precisely, with the notation of Theorem 3.5 we have
$q_{n}=\sum_{i=0}^{n}k_{i}p_{n-i},$
and then
$q_{n}-q_{n-1}=k_{0}(p_{n}-p_{n-1})+\sum_{i=1}^{n-1}k_{i}(p_{n-i}-p_{n-i-1})+k_{n}p_{0}.$
which can be rewritten in the form
$k_{n}p_{0}=\overline{\Delta}q_{n}-\frac{q_{0}}{p_{0}}\overline{\Delta}p_{n}-\sum_{i=1}^{n-1}k_{i}(p_{n-i}-p_{n-i-1}).$
Now, suppose that $k_{1},k_{2},\dots,k_{n-1}\leq 0.$ Since $\\{p_{n}\\}_{n\geq
0}$ is decreasing, we obtain
$k_{n}p_{0}\leq\overline{\Delta}q_{n}-\frac{q_{0}}{p_{0}}\overline{\Delta}p_{n}$
which is clearly non-positive if the reversed form of (3.6) holds. However,
here it is very important to note that if $\overline{\Delta}q_{n}\geq 0,$ then
the right-hand side of the above expression is non-negative. Summarizing, in
the second part of Theorem 3.5 we need to suppose that the sequence
$\\{q_{n}\\}_{n\geq 0}$ is strictly decreasing.
## 4\. The monotonicity of the quotient of two hypergeometric series
The next result, due to M. Biernacki and and J. Krzyż, has found numerous
applications during the past decade. For instance in [11] the authors give a
variant of Theorem 4.1 where the numerator and denominator Maclaurin series
are replaced with polynomials of the same degree. See also [2] for an
alternative proof of Theorem 4.1 and [3] for some interesting applications.
###### Theorem 4.1.
Suppose that the power series $f(x)=\sum_{n\geq 0}a_{n}x^{n}$ and
$g(x)=\sum_{n\geq 0}b_{n}x^{n}$ have the radius of convergence $r>0$ and
$b_{n}>0$ for all $n\in\\{0,1,\dots\\}.$ Then the function
$x\mapsto{f(x)}/{g(x)}$ is increasing (decreasing) on $(0,r)$ if the sequence
$\\{a_{n}/b_{n}\\}_{n\geq 0}$ is increasing (decreasing).
Now, with the help of Theorem 4.1 we prove the following, which completes [11,
Theorem 3.8].
###### Theorem 4.2.
Let $a_{1},a_{2},b_{1},b_{2},c_{1},c_{2}$ be positive numbers. Then the series
$x\mapsto
q(x)=\frac{{}_{2}F_{1}(a_{1},b_{1};c_{1};x)}{{}_{2}F_{1}(a_{2},b_{2};c_{2};x)}=\frac{r_{0}+r_{1}x+r_{2}x^{2}+\dots}{s_{0}+s_{1}x+s_{2}x^{2}+\dots}$
is increasing on $(0,1)$ if one of the following conditions holds
1. (1)
$a_{1}\geq a_{2},$ $b_{1}\geq b_{2}$ and $c_{2}\geq c_{1}.$
2. (2)
$a_{1}+b_{1}\geq a_{2}+b_{2},$ $c_{2}\geq c_{1}$ and $a_{2}\leq a_{1}\leq
b_{1}\leq b_{2}.$
3. (3)
$a_{1}+b_{1}\geq a_{2}+b_{2},$ $c_{2}\geq c_{1}$ and $a_{1}b_{1}\geq
a_{2}b_{2}.$
Moreover, if the above inequalities are reversed, then the function $x\mapsto
q(x)$ is decreasing on $(0,1).$
###### Proof.
We prove only the part when $x\mapsto q(x)$ is increasing. The other case is
similar, so we omit the details. Observe that the sequence
$\\{r_{n}/s_{n}\\}_{n\geq 0}$ is increasing if and only if for all
$n\in\\{0,1,\dots\\}$ we have
$\frac{r_{n}}{s_{n}}=\frac{\frac{(a_{1},n)(b_{1},n)}{(c_{1},n)n!}}{\frac{(a_{2},n)(b_{2},n)}{(c_{2},n)n!}}\leq\frac{\frac{(a_{1},n+1)(b_{1},n+1)}{(c_{1},n+1)(n+1)!}}{\frac{(a_{2},n+1)(b_{2},n+1)}{(c_{2},n+1)(n+1)!}}=\frac{r_{n+1}}{s_{n+1}}$
or equivalently
(4.3) $(a_{2}+n)(b_{2}+n)(c_{1}+n)\leq(a_{1}+n)(b_{1}+n)(c_{2}+n).$
(1) By using the previous theorem we get both cases of the first claim.
(2) For the second claim we only need to prove that
$(a_{2}+n)(b_{2}+n)\leq(a_{1}+n)(b_{1}+n)$ for all $n\in\\{0,1,\dots\\}.$ We
can reduce $a_{1}$ and $b_{1}$ into $a_{1}^{\prime}$ and $b_{1}^{\prime}$ so
that $a_{1}^{\prime}+b_{1}^{\prime}=a_{2}+b_{2}$ and $0<a_{2}\leq
a_{1}^{\prime}\leq b_{1}^{\prime}\leq b_{2}$ still holds. Now we get both
cases of the second claim by noticing that the graph of the function
$f(t)=(a_{2}+b_{2}+n-t)(n+t)$ is a parabola which gets its maximum value in
$(a_{2}+b_{2})/2$ and that $f(a_{2})\leq f(a_{1}^{\prime}).$
(3) Observe that if $a_{1}b_{1}\geq a_{2}b_{2}$ and $a_{1}+b_{1}\geq
a_{2}+b_{2},$ then
$n^{2}+(a_{1}+b_{1})n+a_{1}b_{1}\geq n^{2}+(a_{2}+b_{2})n+a_{2}b_{2}$
or equivalently
$(a_{2}+n)(b_{2}+n)\leq(a_{1}+n)(b_{1}+n)$
for all $n\in\\{0,1,\dots\\}.$ ∎
Now, we would like to study the sign of the coefficients of the power series
$q(x)$ in Theorem 4.2. However, it is not easy to use Jurkat’s result in
Theorem 3.5, since it is difficult to verify for what
$a_{1},b_{1},c_{1},a_{2},b_{2}$ and $c_{2}$ is valid the inequality
$r_{n}-r_{n-1}\geq s_{n}-s_{n-1}$ or its reverse for all
$n\in\\{1,2,\dots\\}.$ All the same, there is another useful result of Jurkat
[14], which generalizes Kaluza’s Theorem 1.3 and it is strongly related to
Theorem 4.1 of Biernacki and Krzyż.
###### Theorem 4.4.
Let us consider the power series $f(x)=\sum_{n\geq 0}a_{n}x^{n}$ and
$g(x)=\sum_{n\geq 0}b_{n}x^{n},$ where $b_{n}>0$ for all $n\in\\{0,1,\dots\\}$
and the sequence $\\{b_{n}\\}_{n\geq 0}$ is log-convex. If the sequence
$\\{a_{n}/b_{n}\\}_{n\geq 0}$ is increasing (decreasing), then the
coefficients of the power series $q(x)=f(x)/g(x)=\sum_{n\geq 0}q_{n}x^{n}$
satisfies $q_{n}\geq 0$ ($q_{n}\leq 0$) for all $n\in\\{1,2,\dots\\}.$
It is important to note here that if the radius of convergence of the above
power series is $r,$ as above, then clearly the conditions of the above
theorem imply the monotonicity of the quotient $q.$ Thus, combining Theorem
3.1 with Theorem 4.4 we obtain the following result.
###### Theorem 4.5.
Suppose that all the hypotheses of Theorem 4.2 are satisfied and in addition
$2a_{2}b_{2}(c_{2}+1)\leq(a_{2}+1)(b_{2}+1)c_{2}$ and $c_{2}\geq
a_{2}+b_{2}-1.$ Then the coefficients of the quotient
$x\mapsto
q(x)=\frac{{}_{2}F_{1}(a_{1},b_{1};c_{1};x)}{{}_{2}F_{1}(a_{2},b_{2};c_{2};x)}=\frac{r_{0}+r_{1}x+r_{2}x^{2}+\dots}{s_{0}+s_{1}x+s_{2}x^{2}+\dots}=q_{0}+q_{1}x+q_{2}x^{2}+\dots$
satisfy $q_{n}\geq 0$ for all $n\in\\{1,2,\dots\\}.$ Moreover, if the
inequalities in Theorem 4.2 are reversed, then $q_{n}\leq 0$ for all
$n\in\\{1,2,\dots\\}.$
Rational expressions involving hypergeometric functions occur in many contexts
in classical analysis. For instance [1, Theorem 3.21] states some properties
such as monotonicity or convexity of several functions of this type. But much
stronger conclusions might be true. In fact, in [1, p. 466] it is suggested
that several of the functions in the long list of [1, Theorem 3.21] might have
Maclaurin series with coefficients of the same sign (except possibly the
leading coefficient). This topic remains widely open since there does not seem
to exist a method for approaching this type of questions.
Finally, let us mention another result, which is also strongly related to
Biernacki and Krzyż criterion and is useful in actuarial sciences in the study
of the non-monotonic ageing property of residual lifetime.
###### Theorem 4.6.
Suppose that the power series $f(x)=\sum_{n\geq 0}a_{n}x^{n}$ and
$g(x)=\sum_{n\geq 0}b_{n}x^{n}$ have the radius of convergence $r>0.$ If the
sequence $\\{a_{n}/b_{n}\\}_{n\geq 0}$ satisfies $a_{0}/b_{0}\leq
a_{1}/b_{1}\leq\dots\leq a_{n_{0}}b_{n_{0}}$ and $a_{n_{0}}b_{n_{0}}\geq
a_{n_{0}+1}b_{n_{0}+1}\geq\dots\geq a_{n}b_{n}\geq\dots$ for some
$n_{0}\in\\{0,1,\dots,n\\},$ then there exists an $x_{0}\in(0,r)$ such that
the function $x\mapsto{f(x)}/{g(x)}$ is increasing on $(0,x_{0})$ and
decreasing on $(x_{0},r).$
Note the a variant of the above result appears recently in [5, Lemma 6.4] with
$a_{n}$ and $b_{n}$ replaced with $a_{n}/n!$ and $b_{n}/n!$ and the proof is
based on the so-called variation diminishing property of totally positive
functions in the sense of Karlin.
### Acknowledgments
The research of Árpád Baricz was supported by the János Bolyai Research
Scholarship of the Hungarian Academy of Sciences and by the Romanian National
Council for Scientific Research in Education CNCSIS-UEFISCSU, project number
PN-II-RU-PD 388/2011. The research of Matti Vuorinen was supported by the
Academy of Finland, Project 2600066611. The authors are indebted to the
referee for his/her constructive comments and helpful suggestions, which
improved the first draft of this paper.
## References
* [1] G.D. Anderson, M.K. Vamanamurthy, M. Vuorinen: Conformal Invariants, Inequalities and Quasiconformal Maps, John Wiley & Sons, New York, 1997.
* [2] Á. Baricz: Generalized Bessel functions of the first kind, Lecture Notes in Mathematics, vol. 1994, Springer-Verlag, Berlin, 2010.
* [3] Á. Baricz: Bounds for modified Bessel functions of the first and second kinds, Proc. Edinb. Math. Soc. 53(3) (2010) 575–599.
* [4] E.F. Beckenbach, R. Bellman: Inequalities, Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 30, Springer-Verlag, Berlin, 1961.
* [5] F. Belzunce, E.M. Ortega, J.M. Ruiz: On non-monotonic ageing properties from the Laplace transform, with actuarial applications, Insurance: Mathematics and Economics 40 (2007) 1–14.
* [6] M. Biernacki, J. Krzyż: On the monotonicity of certain functionals in the theory of analytic functions, Ann. Univ. Mariae Curie-Skłodowska. Sect. A. 9 (1955) 135–147.
* [7] L. Carlitz: Advanced Problems and Solutions: Solutions: 4803, Amer. Math. Monthly 66(5) (1959) 430.
* [8] H. Davenport, G. Pólya: On the product of two power series, Canadian J. Math. 1 (1949) 1–5.
* [9] B.G. Hansen, F.W. Steutel: On moment sequences and infinitely divisible sequences, J. Math. Anal. Appl. 136 (1988) 304–313.
* [10] G.H. Hardy: Divergent Series, With a preface by J. E. Littlewood and a note by L. S. Bosanquet. Reprint of the revised (1963) edition. Éditions Jacques Gabay, Sceaux, 1992.
* [11] V. Heikkala, M.K. Vamanamurthy, M. Vuorinen: Generalized elliptic integrals, Comput. Methods Funct. Theory 9(1) (2009) 75–109.
* [12] P. Henrici: Applied and computational complex analysis, Vol. 1. Power series – integration – conformal mapping – location of zeros, Reprint of the 1974 original, John Wiley & Sons, Inc., New York, 1988.
* [13] R.A. Horn: On moment sequences and renewal sequences, J. Math. Anal. Appl. 31 (1970) 130–135.
* [14] W.B. Jurkat: Questions of signs in power series, Proc. Amer. Math. Soc. 5(6) (1954) 964–970.
* [15] T. Kaluza: Über die Koeffizienten reziproker Potenzreihen, Math. Z. 28 (1928) 161–170.
* [16] D.G. Kendall: Renewal sequences and their arithmetic, Proceedings of Loutraki Symposium on Probability Methods in Analysis, Lecture Notes in Mathematics, Vol. 31, pp. 147–175, Springer-Verlag, New York/Berlin, 1967.
* [17] J. Lamperti: On the coefficients of reciprocal power series, Amer. Math. Monthly 65(2) (1958) 90–94.
* [18] F.W.J. Olver, D.W. Lozier, R.F. Boisvert, C.W. Clark, eds.: _NIST Handbook of Mathematical Functions,_ Cambridge Univ. Press, 2010.
* [19] G. Szegő: Bemerkungen zu einer Arbeit von Herrn Fejér über die Legendreschen Polynome, Math. Z. 25 (1926) 172–187.
|
arxiv-papers
| 2010-10-26T08:59:22 |
2024-09-04T02:49:14.227364
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "\\'Arp\\'ad Baricz, Jetro Vesti, Matti Vuorinen",
"submitter": "Matti Vuorinen",
"url": "https://arxiv.org/abs/1010.5337"
}
|
1010.5395
|
# Open Quantum Systems in Noninertial Frames
Salman Khan and M. K. Khan
Department of Physics, Quaid-i-Azam University,
Islamabad 45320, Pakistan sksafi@phys.qau.edu.pk
###### Abstract
We study the effects of decoherence on the entanglement generated by Unruh
effect in noninertial frames by using bit flip, phase damping and depolarizing
channels. It is shown that decoherence strongly influences the initial state
entanglement. The entanglement sudden death can happens irrespective of the
acceleration of the noninertial frame under the action of phase flip and phase
damping channels. It is investigated that an early sudden death happens for
large acceleration under the depolarizing environment. Moreover, the
entanglement increases for a highly decohered phase flip channel.
PACS: 03.65.Ud; 03.65.Yz; 03.67.Mn;04.70.Dy
Keywords: Entanglement; Decoherence; Noninertial frames.
## 1 Introduction
Entanglement is one of the potential sources of quantum theory. It is the key
concept and major resource for quantum communication and computation [1]. In
the last few years, enormous efforts has been made to investigate various
aspects of quantum entanglement and its benefits in a number of setups, such
as teleportation of unknown states [2] , quantum key distribution [3], quantum
cryptography [4] and quantum computation [5, 6]. Recently, the study of
quantum entanglement of various fields has been extended to the relativistic
setup [7, 8, 9, 10, 11, 12] and interesting results about the behavior of
entanglement have been obtained. The study of entanglement in the relativistic
framework is important not only from quantum information perspective but also
to understand deeply the black hole thermodynamics [13, 14] and the black hole
information paradox [15, 16].
The earlier investigations on quantum entanglement in the relativistic
framework is mainly focused by considering isolated quantum systems. In fact,
no quantum system can be completely isolated from its environment and may
results in a non-unitary dynamics of the system. Therefore, it is important to
study the effect of environment on the entanglement in an initial state of a
quantum system during its evolution. The interaction between an environment
and a quantum system leads to the phenomenon of decoherence and it gives rise
to an irreversible transfer of information from the system to the environment
[17, 18, 19].
\put(-350.0,220.0){} | |
---|---|---
Figure 1: Rindler spacetime diagram: A uniformly accelerated observer Rob (R)
moves on a hyperbola with acceleration $a$ in region $I$ and is causally
disconnected from region $II$.
In this paper we work out the effect of decoherence on the entanglement of
Dirac field in a noninertial system. Alsing et al [7] have shown that the
entanglement between two modes of a free Dirac field is degraded by the Unruh
effect and asymptotically reaches a nonvanishing minimum value in the infinite
acceleration. We investigate that how the loss of entanglement through Unruh
effect is influenced in the presence of decoherence by using a phase flip, a
phase damping and a depolarizing channel in the Kraus operators formalism. The
effect of amplitude damping channel on Dirac field in a noninertial system is
recently studied by Wang and Jing [20]. We consider two observers, Alice and
Rob, that share a maximally entangled initial state of two qubits at a point
in flat Minkowski spacetime. Then Rob moves with a uniform acceleration and
Alice stays stationary. To achieve our goal, we consider two cases. In one
instance we allow only Rob’s qubit to interact with a noisy environment and in
the second instance both qubits of the two observers interact with a noisy
environment. Let the two modes of Minkowski spacetime that correspond to Alice
and Rob are, respectively, given by $|n\rangle_{A}$ and $|n\rangle_{R}$.
Moreover, we assume that the observers are equipped with detectors that are
sensitive only to their respective modes and share the following maximally
entangled initial state
$|\psi\rangle_{A,R}=\frac{1}{\sqrt{2}}\left(|00\rangle_{A,R}+|11\rangle_{A,R}\right),$
(1)
where the first entry in each ket corresponds to Alice and the second entry
corresponds to Rob. From the accelerated Rob’s frame, the Minkowski vacuum
state is found to be a two-mode squeezed state [7],
$|0\rangle_{M}=\cos r|0\rangle_{I}|0\rangle_{II}+\sin
r|1\rangle_{I}|1\rangle_{II},$ (2)
where $\cos r=\left(e^{-2\pi\omega c/a}+1\right)^{-1/2}$. The constant
$\omega$, $c$ and $a$, in the exponential stand, respectively, for Dirac
particle’s frequency, light’s speed in vacuum and Rob’s acceleration. In Eq.
(2) the subscripts $I$ and $II$ of the kets represent the Rindler modes in
region $I$ and $II$, respectively, in the Rindler spacetime diagram (see Fig.
(1)). The excited state in Minkowski spacetime is related to Rindler modes as
follow [7]
$|1\rangle_{M}=|1\rangle_{I}|0\rangle_{II}.$ (3)
In terms of Minkowski modes for Alice and Rindler modes for Rob, the maximally
entangled initial state of Eq. (1) by using Eqs. (2) and (3) becomes
$|\psi\rangle_{A,I,II}=\frac{1}{\sqrt{2}}\left(\cos
r|0\rangle_{A}|0\rangle_{I}|0\rangle_{II}+\sin
r|0\rangle_{A}|1\rangle_{I}|1\rangle_{II}+|1\rangle_{A}|1\rangle_{I}|0\rangle_{II}\right).$
(4)
Since Rob is causally disconnected from region $II$, we must take trace over
all the modes in region $II$. This leaves the following mixed density matrix
between Alice and Rob, that is,
$\displaystyle\rho_{A,I}$ $\displaystyle=$
$\displaystyle\frac{1}{2}[\cos^{2}r|00\rangle_{A,I}\langle 00|+\cos
r(|00\rangle_{A,I}\langle 11|+|11\rangle_{A,I}\langle 00|)$ (5)
$\displaystyle\sin^{2}r|01\rangle_{A,I}\langle 01|+|11\rangle_{A,I}\langle
11|].$
Table 1: A single qubit Kraus operators for phase flip channel, phase damping channel and depolarizing channel. phase flip channel | $E_{o}=\sqrt{1-p}\left(\begin{array}[]{cc}1&0\\\ 0&1\end{array}\right),\qquad E_{1}=\sqrt{p}\left(\begin{array}[]{cc}1&0\\\ 0&-1\end{array}\right)$
---|---
phase damping channel | $E_{o}=\left(\begin{array}[]{cc}1&0\\\ 0&\sqrt{1-p}\end{array}\right),\qquad E_{1}=\left(\begin{array}[]{cc}0&0\\\ 0&\sqrt{p}\end{array}\right)$
depolarizing channel | $\begin{array}[]{c}E_{o}=\sqrt{1-p}\left(\begin{array}[]{cc}1&0\\\ 0&1\end{array}\right),\qquad E_{1}=\sqrt{p/3}\left(\begin{array}[]{cc}0&1\\\ 1&0\end{array}\right),\\\ E_{2}=\sqrt{p/3}\left(\begin{array}[]{cc}0&-i\\\ i&0\end{array}\right),\qquad E_{3}=\sqrt{p/3}\left(\begin{array}[]{cc}1&0\\\ 0&-1\end{array}\right)\end{array}$
[htb]
## 2 single qubit in a noisy environment
In this section we consider that only the Rob’s qubit is coupled to a noisy
environment. The final density matrix of the system in the Kraus operators
representation becomes
$\rho_{f}=\sum_{i}\left(\sigma_{o}\otimes
E_{i}\right)\rho_{A,I}\left(\sigma_{o}\otimes E_{i}^{{\dagger}}\right),$ (6)
where $\rho_{A,I}$ is the initial density matrix of the system given by Eq.
(5), $\sigma_{o}$ is the single qubit identity matrix and $E_{i}$ are a single
qubit Kraus operators of the channel under consideration. The Kraus operators
of the channels we use are given in Table $1$. The spin-flip matrix of the
final density matrix of Eq. (6) is defined as
$\tilde{\rho}_{f}=\left(\sigma_{2}\otimes\sigma_{2}\right)\rho_{f}\left(\sigma_{2}\otimes\sigma_{2}\right)$,
where $\sigma_{2}$ is the Pauli matrix. The degree of entanglement in the two
qubits mixed state in a noisy environment can be quantified conveniently by
concurrence $C$, which is given as [21, 22]
$C=\max\left\\{0,\sqrt{\lambda_{1}}-\sqrt{\lambda_{2}}-\sqrt{\lambda_{3}}-\sqrt{\lambda_{4}}\right\\}\qquad\lambda_{i}\geq\lambda_{i+1}\geq
0,$ (7)
where $\lambda_{i}$ are the eigenvalues of the matrix
$\rho_{f}\tilde{\rho}_{f}$. The eigenvalues under the action of phase-flip
channel becomes
\put(-350.0,220.0){} | |
---|---|---
Figure 2: The concurrence $C$ under the action of phase flip channel is
plotted against decoherence parameter $p$ for the case when only Rob’s qubit
is coupled to a noisy environment.
$\displaystyle\lambda_{1}^{\mathrm{PF}}$ $\displaystyle=$
$\displaystyle(1-2p+p^{2})\cos^{2}r,$ $\displaystyle\lambda_{2}^{\mathrm{PF}}$
$\displaystyle=$ $\displaystyle p^{2}\cos^{2}r,$
$\displaystyle\lambda_{3}^{\mathrm{PF}}$ $\displaystyle=$
$\displaystyle\lambda_{4}^{\mathrm{PF}}=0,$ (8)
where the superscript PF corresponds to phase flip channel. Similarly, the
eigenvalues under the action of phase damping and depolarizing channels are,
respectively, given by
$\displaystyle\lambda_{1,2}^{\mathrm{PD}}$ $\displaystyle=$
$\displaystyle\frac{1}{4}(2-p\pm 2\sqrt{1-p})\cos^{2}r,$
$\displaystyle\lambda_{3}^{\mathrm{PD}}$ $\displaystyle=$
$\displaystyle\lambda_{4}^{\mathrm{PD}}=0,$ (9)
$\displaystyle\lambda_{1}^{\mathrm{DP}}$ $\displaystyle=$
$\displaystyle(-1+p)^{2}\cos^{2}r,$ $\displaystyle\lambda_{2}^{\mathrm{DP}}$
$\displaystyle=$
$\displaystyle\lambda_{3}^{\mathrm{DP}}=\lambda_{4}^{\mathrm{DP}}=\frac{1}{9}p^{2}\cos^{2}r,$
(10)
where the superscripts PD and DP stand for phase damping and depolarizing
channels, respectively. In all these equations $p\in\left[0,1\right]$ is the
decoherence parameter. The upper and lower values of $p$ correspond to
undecohered and fully decohered case of the channels, respectively. The
concurrence under the action of every channel reduces to the result of Ref.
[7] when the decoherence parameter $p=0$.
To see how the concurrence and hence the entanglement is influenced by
decoherence parameter $p$ in the presence of Unruh effect, we plot the
concurrence for each channel against $p$ for various values of $r$. In Fig.
(2), the concurrence under the action of phase flip channel is plotted against
$p$. The figure shows that for smaller values of $p$, the entanglement is
strongly acceleration dependent, such that for large values of Rob’s
acceleration (the value of $r$) it gets weakened. However, as $p$ increases
the dependence of entanglement on acceleration decreases and the increasing
value of $p$ causes a rapid loss of entanglement.
\put(-350.0,220.0){} | |
---|---|---
Figure 3: The concurrence $C$ under the action of phase damping channel is
plotted against decoherence parameter $p$ for the case when only Rob’s qubit
is coupled to a noisy environment.
The entanglement sudden death happens irrespective of the acceleration of
Rob’s frame for a $50\%$ decoherence. Fig. (3) shows the effect of decoherence
on the concurrence under the action of phase damping channel. In this case,
the degradation of entanglement due to decoherence is smaller as compare to
the the degradation in the case of phase flip. The entanglement vanishes for
all values of acceleration only when the channel is fully decohered. The
concurrence under the action of the depolarizing channel is exactly equal to
the one for phase flip channel. Hence it influences the entanglement in a way
exactly similar to the phase flip channel as shown in Fig. ($2$).
## 3 Both qubits in a noisy environment
In this section we consider that both Alice’s and Rob’s qubits are influenced
simultaneously by a noisy environment. The final density matrix in this case
can be written in the Kraus operators formalism as follows
$\rho_{f}=\sum_{k}E_{k}\rho_{A,I}E_{k}^{{\dagger}},$ (11)
where $\rho_{A,I}$ is given by Eq. (5) and $E_{k}$ are the Kraus operators for
a two qubit system, satisfying the completeness relation
$\sum_{k}E_{k}E_{k}=I$ and are constructed from a single qubit Kraus operators
of a channel by taking tensor product of all the possible combinations in the
following way
$E_{k}=\sum_{i,j}E_{i}\otimes E_{j},$ (12)
where $E_{i,j}$ are the single qubit Kraus operators of a channel given in
Table $1$. We consider that both Alice’s and Bob’s qubits are influenced by
the same environment, that is, the decoherence parameter $p$ for both qubits
is same. Proceeding in a similar way like the case of single qubit coupled to
the environment, the eigenvalues of the matrix $\rho_{f}\tilde{\rho}_{f}$
under the action of phase flip channel become
$\displaystyle\lambda_{1}^{\mathrm{PF}}$ $\displaystyle=$
$\displaystyle(1+2(-1+p)p)^{2}\cos^{2}r,$
$\displaystyle\lambda_{2}^{\mathrm{PF}}$ $\displaystyle=$ $\displaystyle
4(-1+p)^{2}p^{2}\cos^{2}r,$ $\displaystyle\lambda_{3}^{\mathrm{PF}}$
$\displaystyle=$ $\displaystyle\lambda_{4}^{\mathrm{PF}}=0,$ (13)
Likewise the eigenvalues for phase damping and depolarizing channels,
respectively, becomes
$\displaystyle\lambda_{1}^{\mathrm{PD}}$ $\displaystyle=$
$\displaystyle\frac{1}{4}(-2+p)^{2}\cos^{2}r,$
$\displaystyle\lambda_{2}^{\mathrm{PD}}$ $\displaystyle=$
$\displaystyle\frac{1}{4}p^{2}\cos^{2}r,$
$\displaystyle\lambda_{3}^{\mathrm{PD}}$ $\displaystyle=$
$\displaystyle\lambda_{4}^{\mathrm{PD}}=0,$ (14)
$\displaystyle\lambda_{1,3}^{\mathrm{DP}}$ $\displaystyle=$
$\displaystyle\frac{1}{1296}[324+p(-3+2p)(387+152p(-3+2p))$
$\displaystyle+4(3-4p)^{2}(9+5p(-3+2p))\cos 2r$
$\displaystyle+(3-4p)^{2}p(-3+2p)\cos 4r\pm 4(3-4p)^{2}\cos r$
$\displaystyle\times\\{3(54+p(-3+2p)(33+8p(-3+2p)))$
$\displaystyle+(3-4p)^{2}(2(9-6p+4p^{2})\cos 2r+p(-3+2p)\cos 4r)\\}^{1/2}],$
$\displaystyle\lambda_{2}^{\mathrm{DP}}$ $\displaystyle=$
$\displaystyle\lambda_{4}^{\mathrm{DP}}=\frac{1}{648}p(-3+2p)(-9+4p$ (15)
$\displaystyle+(-3+4p)\cos 2r)(3+4p+(-3+4p)\cos 2r),$
The $"\pm"$ sign in Eq. (15), correspond to the eigenvalues $\lambda_{1}$, and
$\lambda_{3}$ respectively. It is necessary to point out here that the
concurrence under the action of each channel reduces to the result of Ref. [7]
when we set the decoherence parameter $p=0$.
\put(-350.0,220.0){} | |
---|---|---
Figure 4: The concurrence $C$ under the action of phase flip channel is
plotted against decoherence parameter $p$ for the case when both qubits are
coupled to a noisy environment.
To see how the entanglement behaves when both the qubits are coupled to the
noisy environment, we plot the concurrence against the decoherence parameter
$p$ for different values of $r$ under the action of each channel separately.
Fig. (4) shows the dependence of concurrence on decoherence parameter $p$
under the action of phase flip channel. The dependence of entanglement on
acceleration of Rob’s frame is obvious in the region of lower values of $p$.
However, this dependence diminishes as $p$ increases and a rapid decrease in
the degree of entanglement develops. At a $50\%$ decoherence level, the
entanglement sudden death occurs irrespective of Rob’s acceleration. It’s
interesting to see that beyond this point onward, the entanglement regrows as
$p$ increases. The dependence of entanglement on acceleration of the Rob’s
frame reemerges and the entanglement reaches to the corresponding undecohered
maximum value for a fully decohered case. The concurrence varies as a
parabolic function of decoherence parameter $p$ with its vertex at $p=0.5$.
\put(-350.0,220.0){} | |
---|---|---
Figure 5: The concurrence $C$ under the action of phase damping channel is
plotted against decoherence parameter $p$ for the case when both qubits are
coupled to a noisy environment.
The dependence of entanglement on $p$ under the action of phase damping
channel is shown in Fig. (5). In this case the entanglement decreases linearly
as $p$ increases and the dependence on acceleration diminishes. Whatever the
acceleration of Rob’s frame may be, the entanglement sudden death occurs when
the channel is fully decohered. The influence of depolarizing channel on the
entanglement is shown in Fig. (6). Unlike the other two channels, the
depolarizing channel does not diminish the effect of acceleration on the
entanglement as the $p$ increases. However a rapid decrease in entanglement
appears which leads to entanglement sudden death at different values of
decoherence parameter for different acceleration of Rob’s frame. The larger
the acceleration the earlier the entanglement sudden death occurs.
\put(-350.0,220.0){} | |
---|---|---
Figure 6: The concurrence $C$ under the action of depolarizing channel is
plotted against decoherence parameter $p$ for the case when both qubits are
coupled to a noisy environment.
If we compare the single qubit and the both qubits decohering situations, it
becomes obvious that the entanglement loss is rapid when both the qubits are
coupled to the noisy environment. For example, in the case of bit flip channel
the concurrence behaves as a linear function of $p$ for single qubit
decohering case whereas in the case of both qubits decohering case it varies
as a parabolic function. Nevertheless, the sudden death happens at the same
value of $p$, irrespective of the acceleration, for both cases under the
action of bit flip and phase damping channels. For depolarizing channel,
however, this is not true.
## 4 Conclusion
In conclusion, we have investigated that the entanglement in Dirac fields is
strongly dependent on coupling with a noisy environment. This result is
contrary to the case of an isolated system in which the entanglement of Dirac
fields survives even in the limit of infinite acceleration of Rob’s frame. In
the presence of decoherence, the entanglement rapidly decreases and
entanglement sudden death occurs even for zero acceleration. Under the action
of phase flip channel, the entanglement can regrow when both qubits are
coupled to a noisy environment in the limit of large values of decoherence
parameter. The entanglement disappears, irrespective of the acceleration,
under the action of phase damping channel only when the channel is fully
decohered both for single qubit and the two qubits decohering cases. However,
under the action of depolarizing channel an early sudden death occurs for
larger acceleration when both qubits are coupled to the environment. In
summary, the entanglement generated by Unruh effect in noninertial frame is
strongly influenced by decoherence.
## References
* [1] The Physics of Quantum Information, D. Bouwmeester, A. Ekert, A. Zeilinger (Springer-Verlag, Berlin, 2000)
* [2] C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, and W. K. Wootters, Phys. Rev. Lett. 70, 1895 (1993).
* [3] A. Ekert, Phys. Rev. Lett. 67, 661 (1991).
* [4] C.H. Bennett, G. Brassard, N.D. Mermin, Phys. Rev. Lett. 68, 557 (1992).
* [5] L.K. Grover, Phys. Rev. Lett. 79, 325 (1997).
* [6] D.P. DiVincenzo, Science 270, 255 (1995)
* [7] P.M. Alsing, I. Fuentes-Schuller, R. B.Mann, and T. E. Tessier, Phys. Rev. A 74, 032326 (2006).
* [8] Yi Ling et al, J. Phys. A: Math. Theor. 40, 9025 (2007).
* [9] R. M. Gingrich and C. Adami Phys. Rev. Lett. 89, 270402 (2002).
* [10] Q. Pan and J. Jing, Phys. Rev. A 77, 024302 (2008).
* [11] I. Fuentes-Schuller and R. B. Mann, Phys. Rev. Lett. 95, 120404 (2005).
* [12] H. Terashima and M. Ueda, Int. J. Quantum Inf. 1, 93 (2003).
* [13] L. Bombelli, R. K. Koul, J. Lee, and R. Sorkin, Phys. Rev. D 34, 373 (1986).
* [14] C. Callen and F. Wilzcek, Phys. Lett. B 333, 55 (1994).
* [15] S. W. Hawking, Commun. Math. Phys. 43, 199 (1975); Phys. Rev. D 14, 2460 (1976).
* [16] H. Terashima, Phys. Rev. D 61, 104016 (2000).
* [17] Zurek W H et al. Phys. Today 44, 36 (1991).
* [18] H. P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford University Press, Oxford), 2002; H. Carmichael, An Open Systems Approach to Quantum Optics (Springer, Berlin, 1993).
* [19] W. H. Zurek, Rev. Mod. Phys. 75, 715 (2003).
* [20] J. Wang and J. Jing, arxiv: 1005.2865v4 (2010).
* [21] W. K. Wootters, Phys. Rev. Lett. 80, 2245 (1998).
* [22] V. Coffman, J. Kundu, and W. K. Wootters, Phys. Rev. A 61, 052306 (2000).
|
arxiv-papers
| 2010-10-26T13:48:25 |
2024-09-04T02:49:14.237770
|
{
"license": "Public Domain",
"authors": "Salman Khan, M. K. Khan",
"submitter": "Salman Khan",
"url": "https://arxiv.org/abs/1010.5395"
}
|
1010.5503
|
# A GMBCG Galaxy Cluster Catalog of 55,424 Rich Clusters from SDSS DR7
Jiangang Hao11affiliation: Center for Particle Astrophysics, Fermi National
Accelerator Laboratory, Batavia, IL 60510 , Timothy A. McKay22affiliation:
Department of Physics, University of Michigan, Ann Arbor, MI 48109
33affiliation: Department of Astronomy, University of Michigan, Ann Arbor, MI
48109 , Benjamin P. Koester44affiliation: Department of Astronomy and
Astrophysics, The University of Chicago, Chicago, IL 60637 , Eli S.
Rykoff55affiliation: TABASGO Fellow, Physics Department, University of
California at Santa Barbara, Santa Barbara, CA 93106 66affiliation: Physics
Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720 , Eduardo
Rozo77affiliation: Einstein and KICP Fellow, Kavli Institute for Cosmological
Physics, The University of Chicago, Chicago, IL 60637 , James
Annis11affiliation: Center for Particle Astrophysics, Fermi National
Accelerator Laboratory, Batavia, IL 60510 , Risa H. Wechsler88affiliation:
Kavli Institute for Particle Astrophysics & Cosmology, Physics Department, and
SLAC National Accelerator Laboratory, Stanford University, Stanford, CA 94305
, August Evrard22affiliation: Department of Physics, University of Michigan,
Ann Arbor, MI 48109 33affiliation: Department of Astronomy, University of
Michigan, Ann Arbor, MI 48109 , Seth R. Siegel22affiliation: Department of
Physics, University of Michigan, Ann Arbor, MI 48109 , Matthew
Becker1010affiliation: Department of Physics, The University of Chicago,
Chicago, IL 60637 , Michael Busha88affiliation: Kavli Institute for Particle
Astrophysics & Cosmology, Physics Department, and SLAC National Accelerator
Laboratory, Stanford University, Stanford, CA 94305 , David
Gerdes22affiliation: Department of Physics, University of Michigan, Ann Arbor,
MI 48109 , David E. Johnston11affiliation: Center for Particle Astrophysics,
Fermi National Accelerator Laboratory, Batavia, IL 60510 and Erin
Sheldon1111affiliation: Brookhaven National Laboratory, Upton, New York 11973
###### Abstract
We present a large catalog of optically selected galaxy clusters from the
application of a new Gaussian Mixture Brightest Cluster Galaxy (GMBCG)
algorithm to SDSS Data Release 7 data. The algorithm detects clusters by
identifying the red sequence plus Brightest Cluster Galaxy (BCG) feature,
which is unique for galaxy clusters and does not exist among field galaxies.
Red sequence clustering in color space is detected using an Error Corrected
Gaussian Mixture Model. We run GMBCG on 8240 square degrees of photometric
data from SDSS DR7 to assemble the largest ever optical galaxy cluster
catalog, consisting of over 55,000 rich clusters across the redshift range
from $0.1<z<0.55$. We present Monte Carlo tests of completeness and purity and
perform cross-matching with X-ray clusters and with the maxBCG sample at low
redshift. These tests indicate high completeness and purity across the full
redshift range for clusters with 15 or more members. The catalog can be
accessed from the following website:
http://home.fnal.gov/~jghao/gmbcg_sdss_catalog.html.
Galaxies: clusters, Catalog- Cosmology: observations - Methods: Data analysis,
Gaussian Mixture
99affiliationtext: Department of Physics, California Institute of Technology,
Pasadena CA 91125
## 1 Introduction
One of the most exciting discoveries in physics and astronomy over the past
decade is the accelerating expansion of the Universe (Perlmutter et al., 1999;
Riess et al., 1998), which has been more recently confirmed by a series of
independent experiments (Spergel et al., 2003, 2007; Tegmark et al., 2004;
Eisenstein et al., 2005). This cosmic acceleration cannot be explained without
exotic physics, for example, modifications to General Relativity (GR), a
cosmological constant, or an additional energy component with negative
pressure adequate to drive acceleration. Perhaps the simplest possibility, a
cosmological constant, is consistent with all available data, although the
theoretical challenges with this explanation have not been resolved. If the
framework of GR is retained without a cosmological constant, something like
dark energy must exist. In an effort to distinguish between these
possibilities, studies of expansion history and the growth of structure have
become central research topics in physics and astronomy.
One way to test theories of expansion and the growth of structure is to
measure the abundance and properties of galaxy clusters. Clusters are the
largest peaks in the density field. Their abundance and spatial distribution
encode rich information about the Universe (Evrard, 1989; Oukbir & Blanchard,
1992), making them sensitive probes for cosmology (Majumdar & Mohr, 2004; Hu,
2003; Lima & Hu, 2004, 2005). Cosmological constraints from optically selected
galaxy clusters have been carried out recently by Gladders et al. (2007) based
on the RCS cluster catalog (Gladders & Yee, 2005a), by Rozo et al. (2007b, a,
2010), based on the maxBCG catalog (Koester et al., 2007a, b) and by Wen et
al. (2010) based on a cluster catalog assembled based on photometric redshift
(Wen et al., 2009).
Galaxy clusters are observationally rich as well. They can be detected based
on their properties determined using a number of different observables,
including X-ray emission from and the Sunyaev-Zeldovich decrement caused by
hot intracluster gas, optical and NIR emission from stars in cluster galaxies,
and the gravitational lensing distortions imposed on background galaxy images
by the total cluster gravitational potential. Each probe relies on different
aspects of cluster physics and provides different, though often correlated,
information about cluster mass and structure. For cluster detection, the
different probes have complementary virtues. Cluster X-ray emission and the SZ
decrement both require the presence of very hot intracluster gas. This can
only be present in very deep potential wells, so these methods only detect the
highest mass systems, but are consequently relatively free from projection
contamination. Unfortunately, neither very naturally provides information
about cluster redshift, so optical follow-up is required. Cluster searches
using optical data are more able to identify clusters in three dimensions,
obtaining distances as part of cluster detection. Optical selection can
identify systems corresponding to much lower mass dark matter halos than
methods based on the intracluster gas, but this also results in more serious
projection effects. Cluster detection in the optical also benefits from the
high signal to noise for individual galaxy detection and large data volumes
available in optical surveys.
The existence of a uniformly old stellar population in many cluster galaxies
gives them remarkably similar spectral energy distributions which include a
strong 4000 Åbreak. As a result, galaxies within clusters are tightly
clustered in color as well as space. When the cluster redshift increases, this
break shifts across the optical filters, creating a strong correlation between
cluster galaxy color and redshift. It has been shown that red-sequence
galaxies exist in clusters of varied richness and extend to redshift $z\sim
1.6$ (Bower et al., 1992; Smail et al., 1998; Barrientos, 1999; Mullis et al.,
2005; Eisenhardt et al., 2005; Papovich et al., 2010). Red sequence galaxies
are a very prominent feature of galaxy clusters and thus provide a very
powerful means for removing projected field galaxies during cluster detection.
As these red sequence galaxies have mostly E and S0 morphologies, dominate the
bright end of the cluster luminosity function (Sandage et al., 1985; Barger et
al., 1998), and exhibit narrow color scatter ( $\sim 0.05$ in $g-r$ and $r-i$
colors in the redshift range we probe), they are also referred to as the E/S0
ridgeline (Visvanathan & Sandage, 1977; Annis et al., 1999). For reviews of
red sequence galaxies in clusters, refer to Gladders & Yee (2000), Hao et al.
(2009) and references therein.
In this paper, we extend the use of red sequence galaxies and brightest
cluster galaxies (BCG) for cluster detection, and develop an efficient cluster
finding algorithm which we name the Gaussian Mixture Brightest Cluster Galaxy
(GMBCG) method. The algorithm uses the Error Corrected Gaussian Mixture Model
(ECGMM) algorithm (Hao et al., 2009) to identify the BCG plus red sequence
feature and convolves the identified red sequence galaxies with a spatial
smoothing kernel to measure the clustering strength of galaxies around BCGs.
We apply this technique to the Data Release 7 of Sloan Digital Sky Survey and
assemble a catalog of over 55,000 rich galaxy clusters in a redshift range
extending from $0.1<z<0.55$. The catalog is approximately volume limited up to
redshift $z\sim 0.4$ and shows high purity and completeness when tested
against a mock catalog. The algorithm is very efficient, producing a cluster
catalog for the full SDSS DR7 data ($\sim$ 8,000 ${\rm deg}^{2}$) within 23
hours on a single modern desktop computer.
Cluster finding algorithms are closely related to the properties of the data
they are applied to. Therefore, we begin with a general description of the
GMBCG algorithm, then add additional features that are particular to its
application to the SDSS data. The paper is organized as follows: in § 2, we
review de-projection, the major challenge of optical cluster detection,
summarizing the de-projection methods used in previous cluster finding
algorithms and demonstrating why red sequence color outperforms the others. In
§ 3, we introduce the major steps of the GMBCG algorithm and compare it with
the maxBCG algorithm. In § 4, we introduce the cluster catalog we constructed
from the SDSS DR7 using the GMBCG algorithm. In § 5, we evaluate this new DR7
catalog by matching it to catalogs of known X-ray clusters and previously
published maxBCG clusters. The completeness and purity of the GMBCG catalog
are then also tested against a mock catalog. We conclude with a summary of the
properties of the GMBCG catalog, along with a discussion of the prospects for
using this method on future optical surveys.
By convention, we use a $\Lambda$CDM cosmology with $h=1$, $\Omega_{m}=0.3$
and $\Omega_{\Lambda}=0.7$ throughout this paper. Also, we will omit the
$h^{-1}$ when describing distances, i.e., we will use Mpc directly instead of
$h^{-1}$Mpc.
## 2 Optical Galaxy Cluster Detection and De-projection
Our goal is to detect galaxies clustered in three spatial dimensions, but we
have precise information in only two: RA and DEC. Large uncertainties in
galaxy position along the line of sight leads to projections which contaminate
richness estimates for all clusters and confuse cluster detection at low
richness. Therefore, every optical cluster finding algorithm needs to
effectively de-project field galaxies before calculating overdensities in the
RA/DEC plane.
The ability to locate the positions of galaxies along the line of sight is
limited by the technology available. Over the past 60 years, various
algorithms for optical galaxy cluster detection based on photometric data have
been employed (Abell, 1957; Huchra & Geller, 1982; Davis et al., 1985;
Shectman, 1985; Efstathiou et al., 1988; Couch et al., 1991; Lidman &
Peterson, 1996; Postman et al., 1996; Kepner et al., 1999; Annis et al., 1999;
Gladders & Yee, 2000, 2005b; Gal et al., 2000, 2003; Kim et al., 2002; Goto et
al., 2002; Ramella et al., 2002; Lopes et al., 2004; Botzler et al., 2004;
Koester et al., 2007b; Li & Yee, 2008; Wen et al., 2009).111When spectroscopic
redshifts are available, other algorithms have been developed, for example,
Berlind et al. (2006); Yang et al. (2007); Miller et al. (2005). In this
paper, we will mainly consider the algorithms based on photometric data. For a
recent review of the cluster finding algorithms, see Gal (2006). Though these
methods differ in many detailed respects, we can roughly classify them
according to the de-projection methods they use. In Table. 1, we list the
cluster finding algorithms for photometric data of the past two decades and
the de-projection methods used.
Table 1: Summary of optical cluster finding algorithms for photometric data Algorithm | Type of data applied | De-projection method
---|---|---
Percolation222 Huchra & Geller (1982); Davis et al. (1985); Efstathiou et al. (1988); Ramella et al. (2002) | Single band/Simulation | Magnitude/photo-$z$
Smoothing Kernels333 Shectman (1985) | Single band | Magnitude
Adaptive Kernel444 Gal et al. (2000, 2003) | Single band | Magnitude
Matched Filter 555 Postman et al. (1996) | Single band | Magnitude
Hybrid and Adaptive Matched Filter666 Kepner et al. (1999); Kim et al. (2002); Dong et al. (2008) | Single band | Magnitude/photo-$z$
Voronoi Tessellation777 Kim et al. (2002); Lopes et al. (2004) | Single band | Magnitude
Cut-and-Enhance 888 Goto et al. (2002) | Single band | Magnitude
Modified Friends of Friends999 Li & Yee (2008) | Multi-band | Photo-$z$
C4101010 Miller et al. (2005) | Multi-band | All Colors
Percolation with Spectroscopic redshift111111 Berlind et al. (2006) | Multi-Band | Spectroscopic Redshift
Cluster Red Sequence121212 Gladders & Yee (2000, 2005b) | Multi-band | Red sequence
MaxBCG 131313 Annis et al. (1999); Koester et al. (2007a, b) | Multi-band | Red sequence
WHL141414 Wen et al. (2009) | Multi-band | Photo-$z$
GMBCG151515 Hao & Mckay (2008, 2009); Hao (2009) | Multi-band | Red sequence
The de-projection method used by each algorithm is often determined by the
properties of the data for which the algorithm was developed. When only single
band data were available the major de-projection methods were all magnitude
based. However, the broad luminosity function of galaxies makes magnitude a
poor indicator of galaxy position along the line of sight. Even so, these
methods are quite effective for detecting massive clusters. Unfortunately,
they cannot maintain good purity and completeness for clusters with low or
intermediate richness. Moreover, the contamination of cluster richness induced
by projection also creates large scatter in the richness-mass relations
derived from these methods.
Multi-band digital imaging technology greatly alleviates the projection
effects that plagued optical galaxy cluster detection for decades. In a
precise multi-band sky survey, we have magnitude information from more than
one band, allowing better reconstruction of the galaxy spectra. Even the crude
Spectral Energy Distribution (SED) information provided by colors provides
very effective information for locating galaxies along the line of sight.
The red sequence, or E/S0 ridgeline, which defines cluster galaxies, has a
very narrow color scatter ( $\sim 0.05$ in $g-r$ and $r-i$ colors) and a
slightly tilted color magnitude relation, the study of which has a long
history, e.g. (Visvanathan & Sandage, 1977; Bower et al., 1992; Gladders et
al., 1998; López-Cruz et al., 2004; Blakeslee et al., 2003, 2006; De Lucia et
al., 2007; Stott et al., 2009; Mei et al., 2009; Hao et al., 2009). This color
information is the primary tool to determine the position of galaxies along
the line of sight.
There are basically two ways to de-project galaxies using multi-color data:
use the colors to obtain photometric redshifts and then de-project using these
redshifts, or use the red sequence to detect clustering directly in color
space. The first approach is straightforward in principle, but more complex in
practice. There are many machine learning algorithms (Oyaizu et al., 2007;
Gerdes et al., 2009) that can be used to assign photo-$z$s based on the multi-
band colors/magnitudes. However, these methods are limited by the available
training set of spectroscopic redshifts. For galaxies that are similar to the
training set, reconstructed photo-$z$s can reach a precision of $\sim 0.03$
(Oyaizu et al., 2007). However, for galaxies that are not represented in the
training set, photo-$z$s can be very imprecise and biased.
To get a sense of how photo-$z$s perform for all galaxies (up to 21 magnitude
in I-band), we can simply compare the results of two different estimators.
Take the neural network photo-$z$s for SDSS data (Oyaizu et al., 2007) as an
example. There are two well-tested estimators provided in the SDSS catalogs,
labeled photo-$z$d1 and photo-$z$cc2. The photo-$z$d1 is obtained by training
only on magnitudes, while photo-$z$cc2 is obtained by training only on colors.
In Figure 1, we compare photo-$z$s based on these two estimators. The
difference of the two photo-$z$s has a standard deviation of $\sim 0.1$. For a
typical cluster, with a velocity dispersion of 900 km $s^{-1}$, the dispersion
between galaxy redshifts is $\pm 0.003$, much smaller than the precision
possible from photo-$z$s alone. Therefore, though it is a lot better than the
magnitude based de-projection, photo-$z$ de-projection will still be
insufficient to remove projection effects, especially when we probe slightly
fainter cluster populations.
Figure 1: Scatter between two well-trained photo-$z$ estimators. Two neural
network algorithms from Oyaizu et al. (2007) (photo-$z$d1, which uses
magnitudes only, and photo-$z$cc2, which uses galaxy colors) applied to SDSS
DR6 data are compared. Left panel plots the two estimators against each other
for the full sample, right panel shows the scatter between the two estimators.
Although the algorithms are well tuned using existing spectroscopic data, the
two photo-$z$s have an rms difference of $\sim 0.1$.
As an alternative, we may stay closer to the data and look for clustering
directly in color space. Red sequence galaxies in low redshift clusters
display a scatter in $g-r$ color of $\sim 0.05$. Most importantly, a tight
cluster red sequence accompanied by a BCG presents a pattern exhibited only by
clusters and not found in field galaxies. Therefore, directly looking for the
red sequence plus BCG feature provides a powerful way to improve cluster
detections. It is this approach which we follow in the GMBCG method. 161616One
may wonder why do red sequence colors do better than photo-$z$s that are
essentially derived from colors. In particular, photo-$z$s are obtained by
using multi-color/magnitudes while the ridgeline color is only one color. This
would suggest that photo-$z$s should do better than red sequence colors.
However, looking at the problem closely, one can immediately realize that
there are two additional information associated with red sequence color de-
projection. The first is the spatial proximity/clustering and the second is
the discrimination of red and blue galaxies. For this combination of reasons,
de-projection using red sequence colors out-performs de-projection using
photo-$z$s. As a result, we can push the cluster detection to lower richness
limits than we can do using photo-$z$s. For very big clusters, one can find
them with any means. But for lower richness systems, appropriate de-projection
is crucial for detection and richness measurement. For cosmology, clusters
with a wide mass and redshift ranges will provide substantially more leverage
on the constraints on cosmological parameters.
## 3 Details of the GMBCG Algorithm for Optical Cluster Detection
### 3.1 Overview
As pointed out in the previous section, the BCG plus red sequence pattern is a
unique feature of galaxy clusters. We therefore make identifying this feature
a key step in our cluster finding algorithm. The distribution of galaxy colors
in a cluster can be well approximated by a mixture of two Gaussian
distributions(Hao et al., 2009). The redder and narrower Gaussian distribution
corresponds to the cluster’s red sequence, while the bluer and wider one
includes both foreground and background galaxies along with the “blue cloud”
cluster members. In Figure 2, we show the galaxy color distribution around two
real clusters and the corresponding color magnitude relation. If there is no
cluster, then the color distribution in a given patch of sky will be well
represented by a single Gaussian with a wide width. Fitting the color
distribution with mixture of Gaussian distributions is well suited for our
purpose. A complication in our case is that the measurement errors of the
colors are not negligible and proper modelling of them is essential for the
detection of red sequence. The traditional Gaussian Mixture Model (GMM) does
not consider the measurement errors and we therefore use an error corrected
GMM to developed in our earlier work (Hao et al., 2009).
As long as we effectively isolate red sequence galaxies, we reduce the problem
of cluster finding to a clustering analysis on the ra/dec plane. One can then
use either parametric (such as convolving with a model kernel) or non-
parametric (such as Voronoi Tessellation) methods to analyze the strength of
the clustering signal. When we apply such a scheme to data spanning a wide
redshift range there are three other complications to consider.
Figure 2: Color distributions and color-magnitude relations around two
representative clusters. Top Left Galaxy $g-r$ color distribution around a
cluster overlaid with a model constructed of a mixture of two Gaussian
distributions. The red curve corresponds to the red sequence component while
the blue one corresponds to the sum of background galaxies and blue cluster
members. The green vertical line indicates the color of the BCG. $\mu$ and
$\sigma$ are the means and standard deviations of the two Gaussian components.
Top right Color-magnitude relation for the same galaxies. Galaxies within the
2$\sigma$ clip of the red sequence component are shown with red points; the
green line indicates the best fit slope and intercept of this red sequence.
The left most red point is the BCG. The bottom two panels shown the same plots
for a second, higher redshift cluster, where the color used is $r-i$ instead
of $g-r$.
First, as redshift increases the red sequence shows up in different colors.
This is mainly a result of the 4000 Å break shifting across the filters.
Because of this effect, the most informative color will vary as redshift
increases. For the set of SDSS filters, the relation between red sequence
color and redshift is given in Table 2.
Table 2: Red sequence color in different redshift ranges for SDSS filters Ridgeline color | Redshift range
---|---
$g-r$ | 0.0 $\sim$ 0.43171717Although the 4000 Åbreak starts shifting into SDSS r band at $z\sim 0.36$, we observed that $g-r$ color is still better than $r-i$ color for detecting red sequence up to redshift 0.43.
$r-i$ | 0.43 $\sim$ 0.70
$i-z$ | 0.70 $\sim$ 1.0
Beyond $z\sim 1.0$, one needs near infra-red color information, from bands
like Y, J, H, or K. Therefore, when detecting clusters in data spanning a wide
redshift range, it is necessary to determine which ridgeline color we should
examine. Since we will be searching for the red sequence around candidate
BCGs, we adopt the BCG’s photo-$z$ as a good estimator of cluster’s redshift.
BCGs are bright, making their photo-$z$s generally much better determined than
more typical galaxies. As we discuss in §3.3, the precision of BCG photo-$z$s
is sufficient to determine which red sequence color should be examined around
a given BCG. This does require a determination of the photo-$z$ for every
candidate BCG before proceeding.
A second complication for cluster finding across a broad redshift range is the
increased chance of overlapping clusters, one at low redshift and another at
relatively high redshift. Such an overlap will complicate the distribution in
color space, turning it from bimodal to tri-modal or even more. To reduce the
possibility of this occurring, we apply a broad photo-$z$ window (such as
$\pm$ 0.25 in photo-$z$) to select potential member galaxies before searching
the color distribution. The available photo-$z$ precision is adequate for this
purpose. In addition to photo-$z$ clips, we also apply luminosity cuts and
require the potential member galaxies to be brighter than 0.4 $L^{*}$, where
$L^{*}$ is the characteristic luminosity in the Schechter luminosity function.
For our application, the $i$-band apparent magnitude corresponding to
$0.4L^{*}$ as a function of redshift is shown in the lower right panel of
Figure 3. We adopted this from Annis et al. (1999) and Koester et al. (2007a).
Selecting potential member galaxies by cutting on photo-$z$ and luminosity is
very effective at simplifying the color space structure around the target
galaxies. In addition to this, the $0.4L^{*}$ cut allows us to measure a
consistent richness at different redshift 181818For the SDSS DR7 data, the
$0.4L^{*}$ can keep a consistent richness up to redshift 0.4..
Figure 3: The top two and bottom left panels are the color evolution based on
a color model of the red sequence galaxies (Koester et al., 2007a). The bottom
right panel is the I band apparent magnitude corresponding to $0.4L^{*}$ at
different redshifts.
The third complication concerns defining a consistent measurement of richness
across more than one color. Red sequence galaxies selected from different
color bands have different degrees of contamination from the background. This
is a fundamental limit of all color-based red sequence selection methods,
though it has a relatively minor effect on our cluster detection. Once a
cluster catalog is produced, we will need to further calibrate the richness
measured from different color bands using other means, such as gravitational
lensing analysis (Sheldon et al., 2007; Johnston et al., 2007). In the present
work, we just adjust the richness definitions to result in a smooth transition
between filters.
### 3.2 Brightest Cluster Galaxies as Cluster Centers
Brightest cluster galaxies (BCGs) reside near the cluster center of mass, and
provide important clues to other observational features of clusters. Choosing
the BCG as the center in a cluster finding algorithm has good physical,
algorithmic, and computational motivations. The major physical motivation for
focusing on the BCG is that the central galaxy in a cluster (the one which
resides near the bottom of the cluster potential well) is very often the
brightest galaxy in the cluster. This BCG is then coincident with the region
with the deepest potential traditionally identified in theory as the center of
a cluster. To the extent that this is true, using the BCG as the cluster
center simplifies precise comparisons between observations and theory,
although the extent to which the brightest galaxy is always at the center, and
the extent to which the most central galaxy is at the center of the dark
matter potential well, are still areas of investigation.
In an algorithmic sense, the BCG helps to distinguish among the bright
galaxies typically found near the cluster center. Such galaxies are all
similarly clustered, and the choice of a cluster center is thus somewhat
dominated by noise. The uniqueness of BCGs, including their often cD-like
morphologies, acts as a “noise damper” for positioning cluster center.
Computationally, BCGs are bright and have well-determined photo-$z$s, and the
combination of these phenomena boosts the efficiency of cluster detection by
omitting searches around intrinsically faint galaxies that dominate the
luminosity function. These motivating factors underscore the fact that while
BCGs do not drive the identification of clusters in the current algorithm,
they play an important fine-tuning role that minimizes the need for downstream
modelling in cosmological analyses.
### 3.3 Red Sequence Color Selection
A filter combination tuned to the selection of red-sequence galaxies at a
given redshift is of utmost importance. In our algorithm, we use the photo-$z$
of the BCG to determine which color to choose. For SDSS filters, we list the
corresponding red shift ranges for different colors in Table 2. In principle,
the wrong color can be chosen for a cluster due to an inaccurate BCG
photo-$z$. In practice this is not a serious problem; the photo-$z$s for BCGs
are usually well determined ($\sim$ 0.015 for SDSS DR7, see § 4.2.2).
Redshifts that place the 4000 Å break near the border of the filters are also
a cause for concern, as they can confuse the filter choice. However, near the
filter transitions, the BCG plus red sequence pattern is apparent in both
adjacent colors. For example, for a rich cluster located at $z=0.42$, which
falls in the transition region from the SDSS $g$ band to the $r$ band, the
combined red sequence and BCG features can be still be easily captured in
either the $g-r$ or $r-i$. This ambiguity can impact the richness estimates
for clusters near the transition between filters (see § 4.2), but does not
result in issues for cluster detection for the richness range considered in
the current work.
### 3.4 Red Sequence Detection
#### 3.4.1 Cluster Member Galaxy Selection
The sizes of clusters are varied, increasing substantially with mass.
Therefore, using a scaled aperture is preferred for keeping a consistent
richness estimation. Ideally for a candidate cluster, a series of different
aperture radii should be examined and chosen by maximizing S/N. However, this
can be computationally expensive. As a substitute, we take a two-step approach
similar to Koester et al. (2007b), which attempts to deal with this fact:
first, we measure the richness of the cluster using a fixed metric aperture;
then we scale the radius based on our measured fixed aperture richness and
remeasure everything using the scaled aperture size. The following describes
the exact implementation for the current work.
#### 3.4.2 Fixed Aperture Membership and Richness
For a candidate BCG, we identify cluster members using a multi-step process.
We draw a 0.5 Mpc circle around the candidate BCG at its photo-$z$191919The
BCG’s photo-$z$ is a good estimator of the cluster redshift (see Figure 11).
One might use the weighted average of the member galaxy photo-$z$s in the
expectation that the $\sqrt{N}$ averaging would provide a more accurate
estimated redshift. This is true, however, only when there is no systematic
bias in the member photo-$z$s. In current practice there are often systematic
errors in these members related to their being fainter and yet just as red as
the BCG. There will always be the issue that they have lower signal/noise than
the BCG. and select all galaxies fainter than the candidate BCG, but brighter
than the 0.4$L^{*}$ cut at the relevant photo-$z$. Using the filter
combination relevant for the BCG, we use the Gaussian Mixture Model to fit the
distribution of the colors of all the galaxies selected above. To remove
possible overlap of two or more clusters along the line of sight, we consider
only galaxies within a photo-$z$ window of $\pm 0.25$ around the BCG. To
determine the appropriate number of Gaussian components for the fitting the
color distribution, we calculate the Akaike Information Criterion (AIC Akaike,
1974). Around a cluster, AIC normally chooses two Gaussian as best fit, one
narrow and one broad, and the former is chosen as the red sequence as it sits
red-ward. Using the fixed 0.5 Mpc aperture it is, however, possible that the
field of view is dominated by a large cluster and therefore the best fit to
the color distribution is a single Gaussian representing the red-sequence.
This highlights the need for a scaled aperture (in this case, enlarged) which
would include more background galaxies and push the fitting towards two color
components.
Next, for the two mixture case, we need to determine to which Gaussian
component the candidate BCG belongs. We compare its corresponding likelihoods
of the candidate BCG’s color belonging to each of the two Gaussian components
and assign the most likely Gaussian component to the candidate BCG. If this
Gaussian component is wider than than the other Gaussian component, we flag
this candidate BCG as a field galaxy and remove it from the searching list for
next steps. For the case where there is only one Gaussian component, we impose
a threshold on its width, beyond which we do not deem it suggestive of a red
sequence and remove the corresponding candidate BCG from consideration.
Extensive testing on rich clusters in the SDSS sets a color width of 0.16
(about twice the intrinsic width) as an appropriate threshold in both the
$g-r$ and $r-i$ colors.
Following this process, we consider only the candidate BCGs with an
appropriate red sequence measured. All the galaxies whose colors are within
$\pm$ 2 standard deviations of the mean of the corresponding Gaussian
component are flagged as members. The number of member galaxies selected this
way is denoted as $N_{gals}^{0.5Mpc}$. The $\pm\texttt{2}\sigma$ cut
corresponds roughly the level where the background likelihood dominates over
cluster likelihood. It is shown elsewhere that indeed the two component
Gaussian Mixture Model can reliably pick up the correct peak in color space as
verified by simulations (Hao et al., 2009).
#### 3.4.3 Scaled Aperture Size and Richness
Scaled apertures are required to measure clusters of different sizes. To
select the appropriate aperture, we assume there is a scaling relation between
the aperture and the richness we measured with 0.5 Mpc aperture, as motivated
by Hansen et al. (2007).
$R_{scale}=N(N_{gals}^{0.5Mpc})^{P}$ (1)
where $N$ and $P$ are the normalization and power respectively, which need to
be set so that the resulting $R_{scale}$ corresponds roughly to the relevant
value of $R_{200}$. To determine the scaling relation, we measure the
$N_{gals}^{0.5Mpc}$ for maxBCG clusters (Koester et al., 2007a). For every
maxBCG cluster, there is a $R_{200}^{lens}$ measured, interior to which the
mean mass density of the cluster is 200 times of the critical energy density.
This $R_{200}^{lens}$ is measured based on an exhaustive weak lensing analysis
(Johnston et al., 2007; Hansen et al., 2007). We find that $N_{gals}^{0.5Mpc}$
and the corresponding $R_{200}^{lens}$ follow a simple relation,
$R_{scale}=0.133\times(N_{gals}^{0.5Mpc})^{0.539},$ (2)
where $R_{scale}$, measured in Mpc, plays the role of the $R_{200}^{lens}$ in
Johnston et al. (2007) and Hansen et al. (2007). Once we have the scaled
aperture, we repeat the procedure for the fixed aperture richness measurement,
substituting the corresponding scaled aperture for 0.5 Mpc. The corresponding
richness is denoted as $N_{gal}^{scaled}$, and is used as the primary estimate
of richness for the cluster catalog.
#### 3.4.4 GMM vs ECGMM and Weighted Richness
In this prescription for cluster member selection, we rely on the detection of
the red sequence as well as the measurement of its width. The Gaussian Mixture
Model (GMM) and its generalization with error correction (ECGMM) are well-
suited to detecting the red sequence in a cluster. An unbiased measurement of
the evolution of the red sequence and its width requires the ECGMM (Hao et
al., 2009). However, as the measurement errors increase, we cannot simply
select member galaxies using ECGMM with a 2$\sigma$ ($\sigma$ is the standard
deviation of the Gaussian component corresponding to the red sequence) cut in
a consistent way. GMM does give consistent membership selection. This is
mainly due to the fact that the ECGMM measures “true” ridgeline width while
our cuts are made in terms of the observed colors. However, as the measurement
error increases (e.g. at higher redshift), GMM struggles to discern the
correct number of Gaussian components, as the measurement errors “blur” the
color distribution. In this case, GMM will more likely favor a single Gaussian
component over two based on AIC, but ECGMM more accurately recovers the
correct number of mixtures because it properly models the measurement errors.
On the other hand, we can also measure a weighted richness. When we apply GMM
(ECGMM) to fit the color distribution, each Gaussian component has a weight
from the fitting. This weight quantifies how much of the total population is
from the corresponding Gaussian component. By multiplying the relative weight
of the cluster component to the total number of galaxies in the field, we
measure the weighted richness. It turns out that this weighted richness
correlates better with the true richness of the cluster when they are well
measured202020Note if there is only one Gaussian component, this weighted
richness does not make sense.. To demonstrate this, we performed some Monte
Carlo tests. First, we generate the mock colors from two Gaussian
distribution, one corresponds to the background and another corresponds to the
cluster. We fix the number of galaxies in the background component as 40 while
vary the number of galaxies in cluster component from 10 to 70 with increment
of 5. Then, we generate the measurement errors from a uniform distribution
scaled by a noise level (0.1 and 0.2 respectively in our case). The mock color
will be updated by adding realizations from a Gaussian distribution with the
width specified by the measurement errors.
For each given mock cluster richness, we repeat the above procedure 100 times
and obtain a richness and weighted richness measurements using GMM/ECGMM each
time. In Figure 4, we plot our measured mean richness (NFound) and mean
weighted richness (Weighted NFound) vs the true richness (NTrue) at different
measurement noise level.
Figure 4: Reconstruction of richness using GMM and ECGMM at noise level 0.1
and 0.2. The noise on the plot indicate the scale we used to generate the mock
measurement errors. GMM results in better number counts reconstruction, while
ECGMM gives better weighted richness as measurement noise varies.
Based on these analyses, we conclude that GMM can give better richness counts
while ECGMM can give better weighted richness. Therefore, in practice, we will
use a hybrid of both GMM and ECGMM. We firstly detect the red sequence using
ECGMM and measure the weighted richness, and then we use GMM with fixed number
of mixtures (according to the results of ECGMM) to do a follow-up measurement
and select the red sequence members.
### 3.5 Clustering Strength
We now have sufficient machinery to detect red sequence around a given BCG
candidate. If there is red sequence detected, it is still possible that the
candidate BCG is, e.g., a bright foreground galaxy, and does not belong to the
red sequence. Criteria must be chosen to determine the association of a
candidate BCG with the identified red sequence. We thus consider it to be
“associated” with the red sequence if its color lies within 3 standard
deviations of the peak of the identified red sequence Gaussian.
Next, we quantify the strength of spatial clustering in the ra/dec plane by
convolving the selected members with a projected NFW (Bartelmann, 1996;
Navarro et al., 1997; Koester et al., 2007b) radial kernel. It is worth noting
that the type of kernel used is not as important as its scale, which has been
revealed by statistical kernel density analyses (Silverman, 1986; Scott,
1992). Therefore, the specific kernel does not significantly bias the
detection of clusters that deviate from the kernel shape. We introduce the
clustering strength as
$S_{cluster}=\sum_{k=1}^{N_{g}}\Sigma(x_{k})$ (3)
where $N_{g}$ is the total number of member galaxies and
$\Sigma(x)=\frac{2\rho_{s}r_{s}}{x^{2}-1}f(x),$ (4)
$r_{s}=r_{200}/c$ is the the scale radius, $\rho_{s}$ is the projected
critical density, $x=r/r_{s}$ and
$f(x)=\cases{1-{2\over\sqrt{x^{2}-1}}\ \mbox{tan}^{-1}\sqrt{{x-1\over
x+1}}&$x>1$\cr 1-{2\over\sqrt{1-x^{2}}}\ \mbox{tanh}^{-1}\sqrt{{1-x\over
x+1}}&$x<1$\cr 0&$x=1$\cr 0&$x>20$.}$ (5)
Similar to Koester et al. (2007b), we choose $r_{s}=150$ kpc, regardless of
richness. The clustering strength parameter $S_{cluster}$ is essentially the
height of the peak of the smoothed red sequence density field at the position
of the BCG.
### 3.6 Luminosity Weighted Clustering Strength
In addition to the clustering strength parameter introduced in the preceding
section, we also measure another luminosity weighted clustering strength
$S_{cluster}^{lum}$. The measurement is similar to $S_{cluster}^{strength}$
except that a luminosity weight ($W_{lum}$) is attached to each galaxy. The
luminosity weight is simply defined as the ratio of each galaxy’s $i$-band
magnitude to the $i$-band magnitude corresponding to 0.4$L^{*}$ at the
candidate cluster BCG’s redshift.
$S_{cluster}^{lum}=\sum_{k=1}^{N_{g}}\Sigma(x_{k})\times W_{lum}(k)$ (6)
The advantage of introducing such a measure is that its ratio to the non-
luminosity weighted $S_{cluster}$ is a good indicator of whether the candidate
BCG is a contaminating bright star. This forms an important double check of
the star/galaxy separation of the input catalog, which is a minor, but non-
negligible source of contamination.
### 3.7 Implementation of the Algorithm
With all the quantities calculated from the above definitions, the
implementation of the cluster selection is straightforward. There are
basically three steps:
1. 1.
For every galaxy in the catalog, evaluate the clustering strength
$S_{cluster}$ inside a 0.5 Mpc searching aperture. This $S_{cluster}$ is
calculated using galaxies fainter than the candidate BCG and belonging to the
identified red sequence.
2. 2.
Percolation procedure: rank the candidate BCGs by their clustering strength
and remove candidates from the BCG list if they are identified as “members” of
another candidate BCG with higher clustering strength. Figure 6 illustrates
the distribution of clustering strength around a candidate BCG.
3. 3.
Repeat the above process and finally obtain a list of BCGs and their cluster
members. Based on the richness measured in 0.5 Mpc, one calculates a scaling
$R_{scaled}$ for every BCG. Then processes 1) – 2) are repeated by changing
the searching aperture to $R_{scaled}$ from 0.5 Mpc. This concludes the search
and completes the final list of BCG members and BCGs with scaled richness
$N_{gals}^{scale}$.
The procedures are summarized as a flowchart in Figure 5.
Figure 5: Flowchart for the implementation of the GMBCG algorithm
### 3.8 Post Percolation Procedure
The above process is essentially a process of detecting the peaks of the
smoothed density field, where the height of the peaks is measured by
$S_{cluster}$. In Figure 6, we show the $S_{cluster}$ measured around Abell
1689.
In this cluster finding process, the center of the cluster is assumed to be
the brightest cluster galaxy. Therefore, it is possible that several higher
peaks (quantified by $S_{cluster}$) are identified in the field of a brightest
cluster galaxy and survive the previous percolation procedure. Multiple peaks
must be identified and merged into one cluster using some criteria. This
process is deemed “post percolation”, in contrast to the previous percolation
procedure. The major motivation for not directly blending the peaks during the
cluster finding process is the need for additional flexibility in both merging
the peaks and avoiding “over-percolation” the true BCGs by some bright stars.
Perhaps most importantly, the sub-peaks are indicators of potential cluster
sub-structure, and probe the internal structures of clusters.
Figure 6: Left panel shows the clustering strength distribution around a
galaxy cluster (Abell 1689). In this case, the BCG is the highest peak. Right
panel: SDSS image of A1689.
We settle on the following post-percolation prescription: for a given
candidate BCG (denoted as A), we identify a cylindrical region in the ra/dec
plane and redshift space around the BCG (A). The radius of the cylinder is
$R_{scale}$ of BCG (A), and the height is specified by the BCG (A)’s photo-$z$
$\pm 0.05$. Then, if another candidate BCG (denoted as B) falls inside this
cylinder and BCG (B) is fainter than BCG (A) but BCG (B)’s clustering strength
is not more than 4 times of that of BCG (A), we will merge BCG (B) into BCG
(A). Setting the clustering strength threshold of BCG (B) at a level of 4
times more than that of BCG (A) avoids merging a true BCG into a very bright
galaxy. The value 4 is obtained explicitly by testing in known situations in
the SDSS, where bright foreground objects (e.g. stars) confuse identification.
### 3.9 Comparison with MaxBCG Algorithm
It is interesting to explore the major differences between the GMBCG and
maxBCG algorithms (Koester et al., 2007b). maxBCG is a matched filter based
algorithm with an additional filter from the red sequence colors. Using this
algorithm, a large optical cluster catalog has been created (Koester et al.,
2007a), which has high purity and completeness based on tests on both a Monte
Carlo catalog and a N-body mock catalog.
The difference between GMBCG and maxBCG can be summarized in three major
respects:
1. 1.
maxBCG is a generalized matched filter algorithm with the inclusion of a color
filter in addition to radial and luminosity filters. It varies the filter at a
grid of testing redshifts to maximize the match to a model filter. The
redshift at which the model filter maximizes the match with data is selected
as the redshift of the cluster. GMBCG does not maximize the match for a
redshift dependent filter. It uses a statistically well-motivated mixture
model to identify the red sequence plus BCG feature. The
radial NFW kernel serves as a smoothing kernel rather than a model filter.
Therefore, GMBCG will be less biased against clusters that do not follow the
assumed model filter in maxBCG.
2. 2.
maxBCG assumes an average ridgeline redshift model for all clusters while
GMBCG does not assume any model as a priori. It uses the Gaussian Mixture
Model to detect the red sequence and background in a cluster by cluster way.
The advantage is that it automatically adjusts the cluster and background
parameters across a wide redshift range.
3. 3.
In the maxBCG algorithm, the photo-$z$s of the clusters are estimated as a
part of the execution of the algorithm. In GMBCG, photo-$z$s are obtained from
other methods such as neural networks, nearest neighbour polynomial, etc. A
photo-$z$ can also be estimated based on the measured red sequence colors as a
by product.
For these reasons, GMBCG is more easily extendible to a wide redshift range
and less biased against atypical clusters.
## 4 GMBCG catalog For SDSS DR7
In this section, we apply the GMBCG algorithm to the Data Release 7 of the
Sloan Digital Sky Survey (SDSS DR7), and construct an optical cluster catalog
of more than 55,000 rich clusters across $0.1<z<0.55$. To check the quality of
the cluster catalog, we cross-match the GMBCG clusters to X-ray clusters and
maxBCG clusters. We also create a mock catalog based on DR7 data to test the
completeness and purity of the catalog. The details of the catalog
construction are covered in the following section.
### 4.1 Input catalog
The Sloan Digital Sky Survey (SDSS) (York et al., 2000) is a multi-color
digitized CCD imaging and spectroscopic sky survey, utilizing a dedicated
2.5-meter telescope at Apache Point Observatory, New Mexico. It has recently
completed mapping over one quarter of the whole sky in $ugriz$ filters. DR7 is
a mark of the completion of the original goals of the SDSS and the end of the
phase known as SDSS-II (Abazajian & Sloan Digital Sky Survey, 2008). It
includes a total imaging area of 11663 square degrees with 357 million unique
objects identified.
In this paper, we will mainly detect clusters on the so called Legacy Survey
area, which “provided a uniform, well-calibrated map in $ugriz$ of more than
7,500 square degrees of the North Galactic Cap, and three stripes in the South
Galactic Cap totaling 740 square degrees” (Abazajian & Sloan Digital Sky
Survey, 2008). We construct the input galaxy catalog from the CasJobs
(http://casjobs.sdss.org/CasJobs/) PhotoPrimary view of the SDSS Catalog
Archive Server with type set to 3 (galaxy) and $i$-band magnitude less than
21.0. In addition, we also apply the following flags to keep the catalog
clean: SATURATED, SATUR_CENTER, BRIGHT, AMOMENT_MAXITER, AMOMENT_SHIFT and
AMOMENT_FAINT. We download the photo-$z$ table and cross match the objects to
the galaxy catalog to attach photo-$z$s to each galaxy we selected. In DR7,
the photo-$z$s in the photo-$z$ table are calculated based on a nearest
neighbor polynomial algorithm (Abazajian & Sloan Digital Sky Survey, 2008).
In addition to the above selection requirements, we also throw away those
galaxies with bad measurements (photometric errors in $g$ and $r$ band greater
than 10 percent). In principle, we should search all galaxies as candidate
BCGs. However, as BCG are well-known and form a subset of the total galaxy
population, the list (and computational time) can be reduced. Based on Figure
3, we make cuts in color space as shown in the red regions of Figure 7.
Additionally, each galaxy has a well-measured ellipticity through the SDSS
data processing pipeline based on adaptive moments (Bernstein & Jarvis, 2002).
We require the ellipticity in the $r$-band to be less than 0.7 for candidate
BCGs. This ellipticity cut helps to remove edge on spiral galaxies which, when
reddened by dust, often take on the colors of much higher redshift red
sequence galaxies, and hence can appear as false projected BCGs. All these
cuts keep $\sim$ 70% of the total galaxies in our candidate BCG search list,
effectively eliminating only those with quite atypical colors and
morphologies.
Figure 7: BCG preselection in color - color space for the SDSS DR7 data. Red
regions indicate the area of $g-r$ vs. $r-i$ (left panel) and $r-i$ vs. $i-z$
(right panel) color-color space in which we preselect BCGs. This preselection
keeps $\sim$ 70% of the total galaxies.
After the above procedures, we prepare an input catalog for our cluster
finder. It is worth noting that we did not apply any star/galaxy separation
procedures other than the ones generated by the standard DR7 pipeline. This is
a relatively tolerant selection that may be contaminated by occasional bright
stars that are not well separated from galaxies. As described earlier, we
handle these stragglers by comparing the measured luminosity weighted
clustering strength ($S_{cluster}^{lum}$) with the non-luminosity weighted
clustering strength ($S_{cluster}$) to reject those bright stars.
### 4.2 Richness Re-scaling
In the redshift range $0.1\sim 0.55$, only the $g-r$ or $r-i$ ridgeline colors
are used, and the switch between them is determined by the photo-$z$ of the
candidate BCG. Since we measure the richness by counting the number of
galaxies falling within $2\sigma$ of the ridgeline, the resulting richness
from $g-r$ or $r-i$ are not directly comparable. In part this is due to a
changing degree of background contamination as the ridgeline moves through
color space (see Figure 14). Generally, the richness measured from $r-i$ is
higher than that measured from $g-r$. To make the richnesses more consistent
across the whole redshift range, we rescale those measured from $r-i$ color.
Clearly, mass is the only true parameter with which we should relate the two
different richness. Therefore, a complete resolution of this problem requires
a carefully mapping of the mass-richness relation for richness in both
redshift ranges. However, for the moment, we settle for the simpler first
order approach. That is, we require the statistical distribution of richness
measured from $g-r$ color at redshift range [0.41 - 0.43] and richness
measured from $r-i$ color at redshift range [0.43, 0.45] to be the same since
the true richness of the clusters in these narrow redshift ranges should vary
only mildly. The scaling relation that matches the two distributions is not
necessarily linear. To ensure the distribution to be the same, we match the
richness at different percentile bins of the two distributions and re-scale
them linearly in each bin. Then, we fit a polynomial to the scaling relation
across all the bins to derive a “continuous” scaling relation. The richness
from the $r-i$ color will be accordingly re-scaled by this relation. In Figure
8, we show the richness distribution before and after the re-scaling. Since
the scaling relation is monotonously increasing, the scaled richness will not
alter the cluster ranking based on the original richness in the $r-i$ region
(it will affect the global ranking for sure). In a similar fashion, we also
re-scale the weighted richness and the clustering strength. In the following,
unless noted otherwise, the richness and clustering strength all refer to the
rescaled values.
Figure 8: Richness ($N_{gal}^{scaled}$) before and after the re-scaling. This
demonstrates that rescaling removes much of the difference in richness
measurements between the g-r and r-i bands.
#### 4.2.1 Catalog Cleaning and Masking
We apply the GMBCG algorithm to the input catalog and generate a full catalog
of galaxy clusters for the SDSS DR7. We search clusters from redshift
$0.05<z<0.60$, but only include in the final catalog the redshift range
$0.1<z<0.55$ to reduce redshift range edge effects. The luminosity weighted
and non-luminosity weighted clustering strength (see above) are employed. For
stars, the luminosity weighted clustering strength is much greater than its
non-luminosity weighted counterpart. By hand scanning the corresponding
images, we found the cuts as shown in Figure 9 are good for removing the
contaminated stars.
Figure 9: Density contour of BCGs in the space of luminosity weighted and non-
luminosity weighted clustering strength. Blue contours show the results for
all candidate BCGs. The green region shows cuts applied to candidate BCGs, as
described in §4.2.1, which removes bright stars that pass the star/galaxy
separation in the SDSS data processing pipeline.
In addition to the above cuts, we also mask out those clusters that are close
to the brightest stars. We apply the bright star mask from the NYU VAGC
(valued added galaxy catalog) release for SDSS DR7 (Blanton et al., 2005) and
mask out all clusters that fall inside the bright star mask polygons.
#### 4.2.2 Catalog Facts
Cleaning and masking trims the final catalog down to 380,000 clusters, which
we will refer as full catalog. When we apply a richness cut
$N_{gals}^{scaled}\geq 8$, we are left with about 55,000 rich clusters, which
we release with this paper. We refer this as the “public catalog” and its sky
coverage is shown in Figure 10. In Table 3, we list the tags in the public
cluster catalog and their corresponding definitions. The redshift and richness
distributions of the clusters in the public catalog are shown in Figure 11.
Images of example clusters at different redshifts are shown in Figure 12.
Figure 10: Sky coverage in the GMBCG public catalog based on SDSS DR7. Each point shows the position of one cluster on the sky. Table 3: Tags in the cluster catalog Tag Name in catalog | Definition
---|---
OBJID | Unique ID of each galaxy in SDSS DR7
RA | Right Ascention
DEC | Declination
PHOTOZ | photo-$z$ from the photo-$z$ table in DR7
PHOTOZ_ERR | Errors of photo-$z$
SPZ | Spectroscopic redshift
GMR | $g-r$ color212121All colors are calculated using model magnitude
GMR_ERR | Error of $g-r$ color
RMI | $r-i$ color
RMI_ERR | Error of $r-i$ color
MODEL_MAG | Dust extinction corrected model magnitude222222For details, see http://www.sdss.org/DR7/algorithms/photometry.html
MODEL_MAG_ERR | Error of model magnitude
S_CLUSTER | Clustering strength, $S_{cluster}$
GM_SCALED_NGALS | Number of member galaxies inside GM_SCALEDR from BCG
GM_NGALS_WEIGHTED | Weighted richness.
WEIGHTOK | If it is set to 1, we recommend the use of weighted richness for this cluster
Figure 11: Redshift and richness distribution of GMBCG clusters in the public
catalog. Left panel shows the redshift distribution of clusters, cut at
$0.1<photo-$z$<0.55$. Right panel shows the scaled richness distribution,
GM_scaled_Ngals, for clusters with GM_scaled_Ngals $>8$.
Figure 12: Sample cluster images from SDSS DR7 cluster catalog. The BCG
spectroscopic redshift is given in green.
An inherent assumption in GMBCG is that the BCG’s photo-$z$ should be
determined much better than the rest of galaxies. We now test that assumption.
In the public catalog, about 20,000 BCGs have spectroscopic redshift. In
Figure 13, we show the performance of photo-$z$ for BCGs. The rms of the
difference between BCG and photo-$z$ is $\sim 0.015$, which is almost the same
as the photo-$z$s from maxBCG clusters (Koester et al., 2007a), an indication
that the assumption is secure.
Figure 13: The difference between photo-$z$ and spec-z for the BCGs in the
public catalog.
### 4.3 Bimodality in color Space
As we have shown in previous sections, the apparent color distribution around
a cluster generally shows bi-modality. However, there are situations where the
cluster is so big that its members completely dominate the field within the
aperture we impose; in this case, the color distribution may be uni-modal. In
our implementation of the GMBCG algorithm, we also consider this situation as
a potential cluster as long as the width of the dominant uni-modal
distribution is narrow enough (width $<0.16$).
In the case of a bimodal color distribution, the separation between the two
Gaussian components will vary as redshift changes, leading to different
degrees of overlap. This overlap of the two Gaussian components measures the
fraction of projected galaxies when we impose the color cuts on the red
sequence galaxies. Therefore, the richness for the clusters should be
appropriately weighted to account for the projection. In Figure 14, we show
the color distribution of clusters at different redshifts. From the plot, the
$2\sigma$ ($\sigma$ is the standard deviation of the Gaussian component
corresponding to red sequence) cut we imposed for selecting red sequence
members coincides with point at which the likelihood of red sequence galaxy
becomes equal to that of background/blue galaxies.
Figure 14: The bimodal distribution of red sequence galaxy colors and
background/blue galaxies. The results are based on the average results of
clusters falling in each redshift bin as indicated in the plots. The green
vertical lines are the $2\sigma$ clip of the red sequence peak.
This information is important for getting consistent richness estimates across
the redshift range. The 2 $\sigma$ clip we use to select member galaxies will
lead to different levels of background galaxy contamination at different
redshifts. The weighted richness introduced in § 3.4.4 takes this overlap into
account automatically and thus is a better richness estimator than the direct
cluster member counts based on the top-hat $2\sigma$ color cuts. However, the
weighted richness is not always better than the direct number counts. There
are two cases that demand caution when the weighted richness is used. In first
case there is only one Gaussian component, which does not permit a weighted
richness. The second case is that there are situations where the relative
weight estimates from the ECGMM is not reliable, e.g. very small, leading to a
very small weighted richness. In this case, we recommend the direct richness
counts, i.e. $N_{gal}^{scale}$. To make this more clear in the public catalog,
we have a tag “WEIGHTOK”. The weighted richness is recommended if “WEIGHTOK”
is set to one.
## 5 Evaluating the catalog
Any cluster finding algorithm can be evaluated by two simple criteria:
completeness and purity. Completeness quantifies whether the cluster finder
can find all true clusters, while purity quantifies whether the clusters found
by the cluster finder are real clusters. However, calculating the completeness
and purity requires that we know in advance what is a true cluster. Ideally,
the true cluster here should correspond to a dark matter halo. This issue can
only be completely resolved when we have a high resolution simulation that can
properly reflect the galaxies’ colors as well as their interaction with dark
matter halos. However, creating a realistic galaxy catalog from the N-body
simulation has proven to be very challenging, complicated by various factors
such as unknown physics processes, limited resolution of simulation, unknown
behaviour of galaxies at high redshift, and other complications that affect
the evolution of galaxy colors and distribution. Therefore, in practice, we
need to slightly change the definition of true cluster to certain model
clusters we defined in terms of observational features.
In this section, we introduce a simple but realistic mock catalog to test our
cluster finder. The result can tell us the purity and completeness of our
cluster catalog with respect to the model clusters we put in. In addition, as
a check of completeness of the cluster catalog, we also cross match our
clusters to X-ray clusters and clusters from maxBCG catalog. Considering
uncertainties in cluster richness measurement, we will use the full catalog in
this section to accommodate the richness variances.
### 5.1 Mock catalog
Inserting model clusters into a realistic background is a widely used method
to create mock catalogs for evaluating cluster finding algorithms (Diaferio et
al., 1999; Adami et al., 2000; Postman et al., 2002; Kim et al., 2002; Goto et
al., 2002; Koester et al., 2007a). In practice, there are different schemes to
make the mock catalog as realistic as possible. In this paper, we develop a
Monte Carlo scheme that is similar to those used in (Goto et al., 2002;
Koester et al., 2007a), but with additional features. We construct mock
catalogs in four steps:
1. 1.
_The background galaxy distribution_ : To make a realistic background, we
consider 25 stripes from our input galaxy catalogs from SDSS DR7. We remove
the rich clusters (richness greater than 20 in our cluster catalog, about 4%
of the total galaxy in the input catalog) and shuffle the remaining galaxies’
positions (ra/dec), while keeping their colors and other properties unchanged,
creating a ’base’ catalog.
2. 2.
_Model cluster selection_ : We select 49 rich clusters whose redshift ranges
from 0.1 to 0.55 from our cluster catalog. About 60% of these clusters have a
match with known x-ray clusters (see §5.6) and all of them have been visually
checked to be very rich. Each cluster has a BCG and about 30-100 member
galaxies brighter than $0.4L^{*}$.
3. 3.
_Model mock clusters re-sampling_ : Pick a BCG randomly from the 49 model
clusters and then select a fixed number of member galaxies from the
corresponding model cluster’s members. The fixed number is randomly chosen
from [10, 15, 20, 25, 30, 35, 40, 45, 50]. The relative positions, colors and
luminosities of these galaxies all remain unchanged with respect to BCG. In
this way, we can generate a re-sampled model cluster of a given richness.
4. 4.
_Putting re-sampled model clusters into base catalog:_ For every stripe of the
base catalog, we select 500 re-sampled model clusters (roughly the number of
clusters removed in step 1) and put them into the background galaxy catalog so
that their corresponding BCGs replace 500 randomly chosen galaxies in the base
catalog. Then, we will have a Monte Carlo catalog that are based on the real
photometry of the SDSS DR7 data.
By construction, the Monte Carlo catalog is based on actual SDSS photometry,
and produces a mock catalog with reasonably realistic background galaxies.
### 5.2 Completeness and Purity
To test the completeness and purity of our cluster finder, we run it on the
mock catalog created above. Then, we cross match the detected clusters and the
model clusters using a simple cylinder matching, i.e. searching in a cylinder
of $R_{scale}$ in radius and $\pm 0.05$ in redshift. When we test the
completeness, we firstly sort the model cluster list by the cluster richness
and then match the detected clusters to them through the cylinder match. While
we test the purity, we sort the detected cluster list by their richness and
then match the model clusters to them via the cylinder match. In both cases,
we will consider only those unique and exclusive matches, meaning that a model
cluster will not be used any more once it is matched to a detected cluster for
purity test and a detected cluster will not be used any more once it is
matched to a model cluster for the completeness test. If more than one cluster
falls in the cylinder, we choose the richest one. After doing the matching, at
a given redshift bin and above a given $N_{gal}$, the completeness and purity
can then be defined as
${\rm completeness}=\frac{N_{model}^{match}(z,N_{gal})}{N_{model}(z,N_{gal})}$
(7) ${\rm
purity}=\frac{N_{found}^{match}(z,N_{gal}^{scaled})}{N_{found}(z,N_{gal}^{scaled})}$
(8)
where $N_{model}^{match}(z,N_{gal})$ denote the number of model clusters that
are matched to the found clusters by , $N_{model}(z,N_{gal})$ is the total
number of model clusters , $N_{found}^{match}(z,N_{gal}^{scaled})$ is the
number of found clusters that are matched to model clusters and
$N_{found}(z,N_{gal}^{scaled})$ is the total number of found clusters. The
results of the completeness and purity are plotted in Figure 15. The plot show
that the GMBCG algorithm can yield a highly complete and pure cluster catalog.
Figure 15: The completeness and purity of the GMBCG catalog based on the Monte
Carlo catalog. In the completeness plot, “Richness” is the number of member
galaxies of our input model clusters. In the purity plot, “Richness” is the
number of member galaxies measured by the cluster finder.
### 5.3 Richness Recovery
In addition the the completeness and purity, it is also important to compare
the richness estimated from GMBCG and the input richness of the mock clusters.
Note that when we create the mock clusters, the mock cluster richness is
randomly sampled from [10, 15, 20, 25, 30, 35, 40, 45, 50]. That is, the
possible input richness of the mock clusters are only the above numbers.
Therefore, for a given input richness, we look at the distribution of the
recovered richness from GMBCG. The results are shown in Figure 16. From the
plot, GM_Scaled_Ngals well recover the input N_true, though systematically
underestimate the true richness for low richness clusters. The reason for this
underestimation is primarily due to an artifact of our mock cluster creation.
Our low richness mock clusters are essentially re-sampled from real rich
clusters but their relative positions are retained. Therefore, some low
richness cluster may have members locate outside of the $R_{scaled}$ based on
the low richness, leading to an underestimated richness from the cluster
finder.
Figure 16: The recovered richness (GM_Scaled_Ngals) from GMBCG vs. the input
richness (N_true). The error bars are the scatters of the recovered richness
for the given input richnesses (N_true). The red dotted line is the 45 degree
line.
### 5.4 Cross-Matching of GMBCG clusters to MaxBCG Clusters
As a further test of the completeness of the GMBCG catalog, we make a
comparison to the maxBCG catalog (Koester et al., 2007a). The maxBCG catalog
consists of 13,823 clusters in the redshift range $0.1<z<0.3$ with a threshold
on richness set at $N_{200}=10$. It is derived from DR5 of the Sloan Digital
Sky Survey and covers a slightly smaller area than the new GMBCG catalog.
Several complications arise in the process of performing cluster-to-cluster
matches between catalogs, namely redshift uncertainties, centring differences
between the two algorithms, and scatter in the richness measurements. Although
many similarities exist between the maxBCG and GMBCG algorithms, it is not
always the case that they choose the same central galaxy for a given cluster.
When matching clusters, a careful cut must be made in the two-dimensional
physical separation in order to allow for this centering ambiguity, while at
the same time minimizing matches due to random projection. Uncertainty in the
photometric redshifts can yield a similar problem along the line of sight; a
cut in $\Delta z=|z_{maxBCG}-z_{GMBCG}|$ must be made to accommodate these
errors. Finally, the richness measurements themselves have large scatter, i.e.
clusters that appear in one catalog may have richness values below the
richness threshold of the counterpart catalog, rendering them unavailable to
match. These problems ultimately will determine a reasonable matching scheme
to that can be used to quantify the agreement between the GMBCG catalog and
the maxBCG catalog. We now consider these effects.
The uncertainty in redshift estimates for maxBCG clusters is $\sigma_{z}\sim
0.015$ (Koester et al., 2007a). In the GMBCG catalog, the uncertainty of the
photo-$z$s at redshift below 0.3 is $\sim 0.016$ (Figure13). Therefore, a
redshift difference of $\sim 0.05$ between the two catalogs is an appropriate
selection window for matching. As for the radial separation, given the fact
that the maxBCG clusters are percolated within a separation of $R200\sim
1.0-2.0$ Mpc (Koester et al., 2007b), a radial separation of $\sim 2.0$ Mpc is
appropriate for our matching search. Generally speaking, the smaller the
matching separation, the higher the probability of real matches. Also, the
lower the richness of the maxBCG cluster, the less likely they are true
clusters. Therefore, we will represent our matching with respect to both the
separation and the richness of maxBCG clusters.
We hold the maxBCG clusters as target and match the clusters from our full
GMBCG catalogs to them. In other words, it is essentially a completeness test
of the GMBCG catalog. We then execute the cylindrical matching algorithm
described above. The matching yields that 13,374 out of 13,823 ($\sim 96.8\%$)
clusters in maxBCG catalog have a match in the GMBCG catalog. Those non-
matched clusters are mostly at low richness end, which is mainly due to the
low end cuts placed on the catalog. There are also 8,818 of the 13,374 matched
clusters ($\sim 65.9\%$) that have identical BCGs in both catalogs. In the
left panel of Figure 17, we show the matching fraction of the GMBCG clusters
to maxBCG clusters at different maxBCG richness and separation. As a
comparison, we create a control catalog of the same size as the GMBCG catalog,
but with the ra and dec randomized. The matching results of this control
catalog to maxBCG clusters are shown in the right panel of Figure 17.
Figure 17: Left panel is the contour of matching fraction of the maxBCG
clusters to the GMBCG clusters as a function of richness (Ngals_R200) and
separations. From the plot, we can read that for clusters with richness above
20 in maxBCG catalog, 90% of them can be matched to GMBCG clusters with
separations less than 0.5 Mpc. In the right panel, we show the matching
results from the control catalog of random positions.
On the other hand, it is interesting to compare the richness estimate for the
cross matched clusters. Note that in maxBCG, the cluster members are counted
from the BCG color while in GMBCG, members are selected from the ridgeline.
This will lead to large scatters among the two richness estimates. However, in
Rozo et al. (2008a), a new richness estimator, Lambda, was proposed and it
out-performs the original richness estimator (Ngals_R200) in Koester et al.
(2007a). So, the richness estimate from the GMBCG catalog should have tighter
correlation with the Lambda than with the Ngals_R200. In Figure 18, we plot
the richness comparisons.
Figure 18: Left panel is the comparison of richness measured in maxBCG
(Ngals_R200) and in GMBCG (GM_Scaled_Ngals) for those matched clusters. In the
right panel, we show the comparison of GM_Scaled_Ngals and Lambda for the same
sample of matched clusters.
### 5.5 Matching with WHL clusters
In addition to maxBCG clusters, we also match the GMBCG clusters with clusters
from the catalog by Wen et al. (2009) (WHL catalog hereafter), which is built
based on SDSS DR6 and ranges from 0.05 to 0.6 in redshift. We apply the same
matching codes we used for maxBCG catalog to the WHL catalog. There are about
22,000 clusters in WHL catalog can find matched clusters in GMBCG catalog by
cylindrical matching (2 Mpc, 0.05 redshift uncertainty). Among these matches,
13,531 of them have identical BCGs detected in both catalogs.
In Figure 19, we show the matching results. The richness scales reasonably in
the two catalogs.
Figure 19: Left panel is the distribution of GMBCG clusters and the matched
clusters with WHL catalog. Middle panel is the same for cluster richness
distribution. The red dashed lines in both panel denotes the distribution of
those cross matched clusters. Right panel is the comparison of richness
measured in WHL catalog (R) and in GMBCG catalog (GM_Scaled_Ngals) for those
matched clusters.
### 5.6 Cross-Matching of GMBCG to ROSAT X-ray Clusters
Optical identification of peaks in the galaxy distribution represents only one
of many methods used to find clusters. Other observables employed in cluster
detection include thermal emission of x-rays from the hot intra-cluster
medium, weak-lensing distortion of background sources, and the Sunyaev-
Zeldovich effect of hot gas on the cosmic microwave background. Each method
has certain advantages and disadvantages. Each also provides a distinct proxy
for the mass of a cluster, which can be used to probe cosmological
constraints. It is important that our cluster finding algorithm be able to
detect those clusters found by alternative means. X-ray cluster catalogs are
the most appealing candidate for exploring this question. Numerous x-ray
catalogs exist with large sky coverage overlapping the DR7 survey area. Follow
up optical examination is frequently performed on these catalogs to confirm
their identity as clusters and is required to obtain redshifts.
Matching complications likewise arise when comparing to X-ray catalogs. It is
not always the case that the BCG lies exactly on the X-ray peak. There also
exists significant scatter in the x-ray luminosity-richness relation (Rykoff
et al., 2008). Furthermore, the DR7 catalog contains clusters down to a
richness threshold much lower then current x-ray catalogs can detect. The main
goal of this subsection is to test the extent to which our algorithm is able
to identify the most luminous x-ray clusters.
We compare the DR7 catalog to three x-ray identified cluster catalogs: NORAS
(Böhringer et al., 2000), REFLEX (Böhringer et al., 2004) and 400 deg2
(Burenin et al., 2007). NORAS and REFLEX consist of clusters identified from
extended sources on the ROSAT all-sky survey x-ray maps. Together they cover
the northern and southern galactic caps and are flux limited at $3\times
10^{-12}$ ergs s-1cm-2 in the 0.1 - 2.4 KeV energy band. The 400 deg2 catalog
is composed of serendipitous clusters found in the high galactic latitude
ROSAT pointings. It is flux limited at 1.4 s-1cm-2 in the 0.5 - 2.0 KeV energy
band. Sources from all three catalogs have been confirmed as clusters through
follow up optical identification. Combining these catalogs yields 229 unique
clusters in the survey area spanned by DR7.
A cylindrical search is performed on the combined x-ray catalogs in order to
determine if these clusters were found by the GMBCG algorithm, effectively a
completeness test of GMBCG. We consider two clusters a match if they have a
physical separation in the projected plane $sep<2.0$ Mpc and a redshift
difference $|z_{xray}-z_{photo}|<0.05$. By this criteria, 227 out of 229 X-ray
clusters are matched with at least one GMBCG cluster. In Figure 20, we show
the images of the two non-matched X-Ray clusters. In Figure 21, we show the
scatter plot of the richness and X-Ray luminosity as well as the matching
separation vs. X-Ray luminosity for those matched clusters. The results show
that we can reliably recover about 90% of the X-Ray clusters with separation
less than 0.6 Mpc.
Figure 20: Two non-matched X-Ray clusters. The cluster on right panel actually
has a BCG identified in the GMBCG catalog, but it is not recorded as a match
because of the photo-$z$ of the BCG is assigned as 0.549, falling outside of
our redshift matching envelope.
Figure 21: Matched ROSAT clusters in GMBCG catalog. Top panel shows the
location of matched clusters in the phase plane of X-Ray luminosity and
matching separation. 90% of the matched X-Ray clusters are within a matching
separation less than 0.6 Mpc. Bottom panel shows the scatter plot of cluster
richness vs. X-Ray luminosity for those matched clusters. The over-plotted red
dots and error bars are the median relation and scatter in each richness bin
of size 10.
## 6 Discussion
In this paper, we present a new cluster finding algorithm, GMBCG, and publish
the largest ever optical cluster catalog, with more than 55,000 rich clusters.
Compared to the public maxBCG cluster catalog that goes from redshift 0.1 to
0.3, the current GMBCG catalog covers a wider redshift range from 0.1 to 0.55.
GMBCG identifies galaxy clusters by detecting the BCG plus red sequence
feature that exists only in galaxy clusters and is not possessed by field
galaxies. This feature provides a powerful means for detection of galaxy
clusters with minimal line-of-sight projection contamination. The
effectiveness of this algorithm is based on the assumption that a BCG plus red
sequence feature is “universal” among galaxy clusters. Though this feature is
preserved in almost all clusters known to us, we cannot exclude the
possibility that there are some clusters that do not have this feature. In
particular, this is more likely at very high redshift where clusters are
forming. However, even if such “blue” clusters exist, it will be very
challenging to detect them using photometric data in optical bands in a
consistent way across a range of richness unless they also exhibit a tight
blue-sequence. But it is not very likely for the blue galaxy to be tightly
clustered in color since their spectra are not as regular as red galaxy and
the effect of 4000 Åbreak in their color normally shows large scatters.
The GMBCG algorithm uses the BCG’s photo-$z$ to determine the metric aperture
size, and uses the red sequence color to select member galaxies. It separates
the process of getting photo-$z$s and detecting clusters. This differentiates
it from matched filter algorithms (including maxBCG). For the SDSS data the
precision of the photo-$z$s for the BCGs from the machine learning algorithms
are within a factor of 2 of the photo-$z$’s from maxBCG, which means there is
not a serious disadvantage in this choice. Using existing photo-$z$s
significantly boosts the computation efficiency. GMBCG can produce a cluster
catalog for the full SDSS DR7 within 23 hours on a DELL computer with a single
quad core CPU and 8G RAM. As long as the photo-$z$ is not catastrophically
bad, GMBCG can detect the BCG plus red sequence feature of clusters; though
richness measurements may be affected by imprecise redshift estimates.
It is worth noting that for cosmological application, we generally want to
know the best mass proxy. Recent work has shown that weighted richnesses are
among the best optical mass proxies, rather than the direct counts of member
galaxies (Rozo et al., 2008b). However, this does not mean that we should
abandon the direct member galaxy count and identification. On the contrary, it
will be very interesting to have the member galaxies explicitly determined for
cluster science, i.e., the formation and evolution of clusters.
Though GMBCG works very well for the current SDSS DR7 data, there is still
room for improvement, especially for deeper data. For example, GMBCG does not
work well for very low richness clusters, say richness less than 4 for SDSS
DR7 data. This is mainly because GMM/ECGMM will not reliably detect the red
sequence at such low richness. GMBCG relies on the good photo-$z$s for BCGs,
which may be risky at very high redshift where photo-$z$ precision is not
guaranteed. The current GMBCG implementation relies on the photo-$z$ to decide
the color to search for the red sequence. This is not a serious issue for the
current data set, but will be preferable to perform a more comprehensive
analysis on color space spanned by all colors. These are beyond the scope of
this paper and additional improvements are left to future work on deeper data,
such as SDSS co-added data and the incoming Dark Energy Survey data (The Dark
Energy Survey Collaboration, 2005).
## References
* Abazajian & Sloan Digital Sky Survey (2008) Abazajian, K., & Sloan Digital Sky Survey, f. t. 2008, ArXiv e-prints
* Abell (1957) Abell, G. O. 1957, The distribution of rich clusters of galaxies. A catalogue of 2712 rich clusters found on the National Geographic Society Palomar Observatory Sky Survey (Chicago: Univiersity of Chicago Press, 1957)
* Adami et al. (2000) Adami, C., Ulmer, M. P., Romer, A. K., Nichol, R. C., Holden, B. P., & Pildis, R. A. 2000, ApJS, 131, 391
* Akaike (1974) Akaike, H. 1974, Automatic Control, IEEE Transactions on, 19, 716
* Annis et al. (1999) Annis, J., et al. 1999, in Bulletin of the American Astronomical Society, Vol. 31, Bulletin of the American Astronomical Society, 1391–+
* Barger et al. (1998) Barger, A. J., et al. 1998, ApJ, 501, 522
* Barrientos (1999) Barrientos, L. F. 1999, PhD thesis, AA(UNIVERSITY OF TORONTO (CANADA))
* Bartelmann (1996) Bartelmann, M. 1996, A&A, 313, 697
* Berlind et al. (2006) Berlind, A. A., et al. 2006, ApJS, 167, 1
* Bernstein & Jarvis (2002) Bernstein, G. M., & Jarvis, M. 2002, AJ, 123, 583
* Blakeslee et al. (2003) Blakeslee, J. P., et al. 2003, ApJ, 596, L143
* Blakeslee et al. (2006) —. 2006, ApJ, 644, 30
* Blanton et al. (2005) Blanton, M. R., et al. 2005, AJ, 129, 2562
* Böhringer et al. (2004) Böhringer, H., et al. 2004, A&A, 425, 367
* Böhringer et al. (2000) —. 2000, ApJS, 129, 435
* Botzler et al. (2004) Botzler, C. S., Snigula, J., Bender, R., & Hopp, U. 2004, MNRAS, 349, 425
* Bower et al. (1992) Bower, R. G., Lucey, J. R., & Ellis, R. S. 1992, MNRAS, 254, 601
* Burenin et al. (2007) Burenin, R. A., Vikhlinin, A., Hornstrup, A., Ebeling, H., Quintana, H., & Mescheryakov, A. 2007, ApJS, 172, 561
* Couch et al. (1991) Couch, W. J., Ellis, R. S., MacLaren, I., & Malin, D. F. 1991, MNRAS, 249, 606
* Davis et al. (1985) Davis, M., Efstathiou, G., Frenk, C. S., & White, S. D. M. 1985, ApJ, 292, 371
* De Lucia et al. (2007) De Lucia, G., et al. 2007, MNRAS, 374, 809
* Diaferio et al. (1999) Diaferio, A., Kauffmann, G., Colberg, J. M., & White, S. D. M. 1999, MNRAS, 307, 537
* Dong et al. (2008) Dong, F., Pierpaoli, E., Gunn, J. E., & Wechsler, R. H. 2008, ApJ, 676, 868
* Efstathiou et al. (1988) Efstathiou, G., Frenk, C. S., White, S. D. M., & Davis, M. 1988, MNRAS, 235, 715
* Eisenhardt et al. (2005) Eisenhardt, P. R., et al. 2005, in Bulletin of the American Astronomical Society, Vol. 37, Bulletin of the American Astronomical Society, 1344–+
* Eisenstein et al. (2005) Eisenstein, D. J., et al. 2005, ApJ, 633, 560
* Evrard (1989) Evrard, A. E. 1989, ApJ, 341, L71
* Gal (2006) Gal, R. R. 2006, ArXiv Astrophysics e-prints
* Gal et al. (2003) Gal, R. R., de Carvalho, R. R., Lopes, P. A. A., Djorgovski, S. G., Brunner, R. J., Mahabal, A., & Odewahn, S. C. 2003, AJ, 125, 2064
* Gal et al. (2000) Gal, R. R., de Carvalho, R. R., Odewahn, S. C., Djorgovski, S. G., & Margoniner, V. E. 2000, AJ, 119, 12
* Gerdes et al. (2009) Gerdes, D. W., Sypniewski, A. J., McKay, T. A., Hao, J., Weis, M. R., Wechsler, R. H., & Busha, M. T. 2009, ArXiv e-prints
* Gladders et al. (1998) Gladders, M. D., Lopez-Cruz, O., Yee, H. K. C., & Kodama, T. 1998, ApJ, 501, 571
* Gladders & Yee (2000) Gladders, M. D., & Yee, H. K. C. 2000, AJ, 120, 2148
* Gladders & Yee (2005a) —. 2005a, ApJS, 157, 1
* Gladders & Yee (2005b) —. 2005b, ApJS, 157, 1
* Gladders et al. (2007) Gladders, M. D., Yee, H. K. C., Majumdar, S., Barrientos, L. F., Hoekstra, H., Hall, P. B., & Infante, L. 2007, ApJ, 655, 128
* Goto et al. (2002) Goto, T., et al. 2002, AJ, 123, 1807
* Hansen et al. (2007) Hansen, S. M., Sheldon, E. S., Wechsler, R. H., & Koester, B. P. 2007, ArXiv e-prints, 710
* Hao (2009) Hao, J. 2009, PhD thesis, University of Michigan
* Hao et al. (2009) Hao, J., et al. 2009, ApJ, 702, 745
* Hao & Mckay (2009) Hao, J., & Mckay, T. 2009, in Bulletin of the American Astronomical Society, Vol. 41, Bulletin of the American Astronomical Society, 335–+
* Hao & Mckay (2008) Hao, J., & Mckay, T. A. 2008, in Bulletin of the American Astronomical Society, Vol. 40, Bulletin of the American Astronomical Society, 218–+
* Hu (2003) Hu, W. 2003, Phys. Rev. D, 67, 081304
* Huchra & Geller (1982) Huchra, J. P., & Geller, M. J. 1982, ApJ, 257, 423
* Johnston et al. (2007) Johnston, D. E., et al. 2007, ArXiv e-prints, 709
* Kepner et al. (1999) Kepner, J., Fan, X., Bahcall, N., Gunn, J., Lupton, R., & Xu, G. 1999, ApJ, 517, 78
* Kim et al. (2002) Kim, R. S. J., et al. 2002, AJ, 123, 20
* Koester et al. (2007a) Koester, B. P., et al. 2007a, ApJ, 660, 239
* Koester et al. (2007b) —. 2007b, ApJ, 660, 221
* Li & Yee (2008) Li, I. H., & Yee, H. K. C. 2008, AJ, 135, 809
* Lidman & Peterson (1996) Lidman, C. E., & Peterson, B. A. 1996, AJ, 112, 2454
* Lima & Hu (2004) Lima, M., & Hu, W. 2004, Phys. Rev. D, 70, 043504
* Lima & Hu (2005) —. 2005, Phys. Rev. D, 72, 043006
* Lopes et al. (2004) Lopes, P. A. A., de Carvalho, R. R., Gal, R. R., Djorgovski, S. G., Odewahn, S. C., Mahabal, A. A., & Brunner, R. J. 2004, AJ, 128, 1017
* López-Cruz et al. (2004) López-Cruz, O., Barkhouse, W. A., & Yee, H. K. C. 2004, ApJ, 614, 679
* Majumdar & Mohr (2004) Majumdar, S., & Mohr, J. J. 2004, ApJ, 613, 41
* Mei et al. (2009) Mei, S., et al. 2009, ApJ, 690, 42
* Miller et al. (2005) Miller, C. J., et al. 2005, AJ, 130, 968
* Mullis et al. (2005) Mullis, C. R., Rosati, P., Lamer, G., Böhringer, H., Schwope, A., Schuecker, P., & Fassbender, R. 2005, ApJ, 623, L85
* Navarro et al. (1997) Navarro, J. F., Frenk, C. S., & White, S. D. M. 1997, ApJ, 490, 493
* Oukbir & Blanchard (1992) Oukbir, J., & Blanchard, A. 1992, A&A, 262, L21
* Oyaizu et al. (2007) Oyaizu, H., Lima, M., Cunha, C. E., Lin, H., Frieman, J., & Sheldon, E. S. 2007, ArXiv e-prints, 708
* Papovich et al. (2010) Papovich, C., et al. 2010, ApJ, 716, 1503
* Perlmutter et al. (1999) Perlmutter, S., et al. 1999, Astrophys. J., 517, 565
* Postman et al. (2002) Postman, M., Lauer, T. R., Oegerle, W., & Donahue, M. 2002, ApJ, 579, 93
* Postman et al. (1996) Postman, M., Lubin, L. M., Gunn, J. E., Oke, J. B., Hoessel, J. G., Schneider, D. P., & Christensen, J. A. 1996, AJ, 111, 615
* Ramella et al. (2002) Ramella, M., Geller, M. J., Pisani, A., & da Costa, L. N. 2002, AJ, 123, 2976
* Riess et al. (1998) Riess, A. G., et al. 1998, Astron. J., 116, 1009
* Rozo et al. (2008a) Rozo, E., et al. 2008a, ArXiv e-prints
* Rozo et al. (2008b) —. 2008b, ArXiv e-prints
* Rozo et al. (2007a) Rozo, E., Wechsler, R. H., Koester, B. P., Evrard, A. E., & McKay, T. A. 2007a, ArXiv Astrophysics e-prints
* Rozo et al. (2007b) Rozo, E., et al. 2007b, ArXiv Astrophysics e-prints
* Rozo et al. (2010) —. 2010, ApJ, 708, 645
* Rykoff et al. (2008) Rykoff, E. S., et al. 2008, ApJ, 675, 1106
* Sandage et al. (1985) Sandage, A., Binggeli, B., & Tammann, G. A. 1985, AJ, 90, 1759
* Scott (1992) Scott, D. 1992, John Wiley
* Shectman (1985) Shectman, S. A. 1985, ApJS, 57, 77
* Sheldon et al. (2007) Sheldon, E. S., et al. 2007, ArXiv e-prints, 709
* Silverman (1986) Silverman, B. W. 1986, Chapman & Hall
* Smail et al. (1998) Smail, I., Edge, A. C., Ellis, R. S., & Blandford, R. D. 1998, MNRAS, 293, 124
* Spergel et al. (2003) Spergel, D. N., et al. 2003, Astrophys. J. Suppl., 148, 175
* Spergel et al. (2007) —. 2007, Astrophys. J. Suppl., 170, 377
* Stott et al. (2009) Stott, J. P., Pimbblet, K. A., Edge, A. C., Smith, G. P., & Wardlow, J. L. 2009, MNRAS, 394, 2098
* Tegmark et al. (2004) Tegmark, M., et al. 2004, Phys. Rev. D, 69, 103501
* The Dark Energy Survey Collaboration (2005) The Dark Energy Survey Collaboration. 2005, ArXiv Astrophysics e-prints
* Visvanathan & Sandage (1977) Visvanathan, N., & Sandage, A. 1977, ApJ, 216, 214
* Wen et al. (2009) Wen, Z. L., Han, J. L., & Liu, F. S. 2009, ApJS, 183, 197
* Wen et al. (2010) —. 2010, MNRAS, 407, 533
* Yang et al. (2007) Yang, X., Mo, H. J., van den Bosch, F. C., Pasquali, A., Li, C., & Barden, M. 2007, ApJ, 671, 153
* York et al. (2000) York, D. G., et al. 2000, AJ, 120, 1579
## Acknowledgments
JH and TM gratefully acknowledge support from NSF grant AST 0807304 and DoE
Grant DE-FG02-95ER40899. JH thanks Brian Nord, Jeffery Kubo, Marcelle Soares-
Santos and Heinz Andernach for helpful conversation. AEE acknowledges support
from NSF AST-0708150 and NASA NNX10AF61G. This work was supported in part by a
Department of Energy contract DE-AC02-76SF00515. This project was made
possible by workshops support from the Michigan Center for Theoretical
Physics.
Funding for the SDSS and SDSS-II has been provided by the Alfred P. Sloan
Foundation, the Participating Institutions, the National Science Foundation,
the U.S. Department of Energy, the National Aeronautics and Space
Administration, the Japanese Monbukagakusho, the Max Planck Society, and the
Higher Education Funding Council for England. The SDSS Web Site is
http://www.sdss.org/.
The SDSS is managed by the Astrophysical Research Consortium for the
Participating Institutions. The Participating Institutions are the American
Museum of Natural History, Astrophysical Institute Potsdam, University of
Basel, University of Cambridge, Case Western Reserve University, University of
Chicago, Drexel University, Fermilab, the Institute for Advanced Study, the
Japan Participation Group, Johns Hopkins University, the Joint Institute for
Nuclear Astrophysics, the Kavli Institute for Particle Astrophysics and
Cosmology, the Korean Scientist Group, the Chinese Academy of Sciences
(LAMOST), Los Alamos National Laboratory, the Max-Planck-Institute for
Astronomy (MPIA), the Max-Planck-Institute for Astrophysics (MPA), New Mexico
State University, Ohio State University, University of Pittsburgh, University
of Portsmouth, Princeton University, the United States Naval Observatory, and
the University of Washington.
|
arxiv-papers
| 2010-10-26T20:01:37 |
2024-09-04T02:49:14.247595
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jiangang Hao, Timothy A. McKay, Benjamin P. Koester, Eli S. Rykoff,\n Eduardo Rozo, James Annis, Risa H. Wechsler, August Evrard, Seth R. Siegel,\n Matthew Becker, Michael Busha, David Gerdes, David E. Johnston and Erin\n Sheldon",
"submitter": "Jiangang Hao",
"url": "https://arxiv.org/abs/1010.5503"
}
|
1010.5555
|
# Quantum mechanical photon-count formula derived by entangled state
representation
Li-yun Hu1, Z. S. Wang1, L. C. Kwek2, and Hong-yi Fan3 1College of Physics &
Communication Electronics, Jiangxi Normal University, Nanchang 330022, China
2Center for Quantum Technologies, National University of Singapore, Singapore
117543
3Department of Physics, Shanghai Jiao Tong University, Shanghai, 200030, China
###### Abstract
By introducing the thermo entangled state representation, we derived four new
photocount distribution formulas for a given density operator of light field.
It is shown that these new formulas, which is convenient to calculate the
photocount , can be expressed as such integrations over Laguree-Gaussian
function with characteristic function, Wigner function, Q-function, and
P-function, respectively.
In quantum optics photon counting is important for judging the nonclassical
features of light field, most measurements of the electromagnetic field are
based on the absorption of photons via the photoelectric effect. This is true
not only for used insofar as photodiodes, photomultipliers, etc., but also for
such homely devices as the photographic plate and the eye. So the problem of
photo-electric detection attracts an increasing attention of many physicists
and scientists. Expressions for the detection probability have been presented
in many works 1 ; 2 . The quantum mechanical photon counting distribution
formula was first derived by Kelley and Kleiner 3 . As shown in Refs. 3 ; 4 ;
5 for the single radiation mode, the probability distribution
$\mathfrak{p}\left(m,T\right)$ of registering $m$ photoelectrons in the time
interval $T$ is given by
$\mathfrak{p}\left(m,T\right)=\mathtt{Tr}\left\\{\mathbf{\rho\colon}\frac{\left(\zeta
a^{\dagger}a\right)^{m}}{m!}e^{-\zeta a^{\dagger}a}\colon\right\\},$ (1)
where $\zeta\propto T$ is called the quantum efficiency (a measure) of the
detector, and $\mathbf{\colon\colon}$ denotes normal ordering. $\mathbf{\rho}$
is a single-mode density operator of the light field concerned. The aim of
this Letter is to derive some other quantum mechanical photon-count formula by
introducing the thermal entangled state representation and convert the
calculations of Wigner function (WF) and the characteristic function of
density operator to an overlap between “two pure”states in a two-mode enlarged
Fock space, so that it is convenient to calculate the photocount when a light
field’s density operator is given. In addition, this new method seems concise
and easy to be accepted by readers.
Recall that the thermal entangled state representation (TESR) is constructed
in the doubled Fock space 6 ; 7 based on Umezawa-Takahash thermo field
dynamics (TFD) 8 ; 9 ; 10 , i.e.,
$\displaystyle\left|\eta\right\rangle$ $\displaystyle=$
$\displaystyle\exp\left[-\frac{1}{2}|\eta|^{2}+\eta
a^{\dagger}-\eta^{\ast}\tilde{a}^{\dagger}+a^{\dagger}\tilde{a}^{\dagger}\right]\left|0,\tilde{0}\right\rangle$
(2) $\displaystyle=$ $\displaystyle
D\left(\eta\right)\left|\eta=0\right\rangle,$
$\displaystyle\left|\xi\right\rangle$ $\displaystyle=$
$\displaystyle\exp\left[-\frac{1}{2}|\xi|^{2}+\xi
a^{\dagger}+\xi^{\ast}\tilde{a}^{\dagger}-a^{\dagger}\tilde{a}^{\dagger}\right]\left|0,\tilde{0}\right\rangle$
(3) $\displaystyle=$ $\displaystyle
D\left(\xi\right)\left|\xi=0\right\rangle,$
where the state vector $\left|\xi\right\rangle$ is conjugate to the state
$\left|\eta\right\rangle,$ $D\left(\eta\right)=e^{\eta
a^{\dagger}-\eta^{\ast}a}$ is a displacement operator, and
$\tilde{a}^{\dagger}$ is a fictitious mode accompanying the real photon
creation operator $a^{\dagger},$
$\left|0,\tilde{0}\right\rangle=\left|0\right\rangle\left|\tilde{0}\right\rangle,$
and $\left|\tilde{0}\right\rangle$ is annihilated by $\tilde{a}$ with the
relations $\left[\tilde{a},\tilde{a}^{\dagger}\right]=1$ and
$\left[a,\tilde{a}^{\dagger}\right]=0$. It is easily seen that
$\left|\eta=0\right\rangle$ and $\left|\xi=0\right\rangle$ have the
properties,
$\displaystyle\left|I\right\rangle$ $\displaystyle\equiv$
$\displaystyle\left|\eta=0\right\rangle=e^{a^{\dagger}\tilde{a}^{\dagger}}\left|0,\tilde{0}\right\rangle=\sum_{n=0}^{\infty}\left|n,\tilde{n}\right\rangle,$
(4) $\displaystyle\left|\xi=0\right\rangle$ $\displaystyle=$
$\displaystyle(-1)^{a^{{\dagger}}a}\left|\eta=0\right\rangle,$ (5)
where $\tilde{n}=n$, and $\tilde{n}$ denotes the number in the fictitious
Hilbert space.
According to the TFD and Eq.(4), we can reform the probability distribution
$\mathfrak{p}\left(m,T\right)$ as
$\displaystyle\mathfrak{p}\left(m,T\right)$ $\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}\left\langle
n\right|\mathbf{\rho\colon}\frac{\left(\zeta
a^{\dagger}a\right)^{m}}{m!}e^{-\zeta a^{\dagger}a}\colon\left|n\right\rangle$
(6) $\displaystyle=$ $\displaystyle\sum_{n,l=0}^{\infty}\left\langle
n,\tilde{n}\right|\mathbf{\rho\colon}\frac{\left(\zeta
a^{\dagger}a\right)^{m}}{m!}e^{-\zeta
a^{\dagger}a}\colon\left|l,\tilde{l}\right\rangle$ $\displaystyle=$
$\displaystyle\frac{\zeta^{m}}{m!}\left\langle\mathbf{\rho}\right|a^{\dagger
m}\left(1-\zeta\right)^{a^{\dagger}a}a^{m}\left|I\right\rangle,$
where in the last step, we have used the operator identity: $\exp\left(\lambda
a^{{\dagger}}a\right)=\colon\exp\left[\left(e^{\lambda}-1\right)a^{{\dagger}}a\right]\colon$.
Note that the density operators $\mathbf{\rho}$($a^{\dagger}$,$a)$ are defined
in the real space which are commutative with operators
($\tilde{a}^{\dagger}$,$\tilde{a})$ in the tilde space with
$\left|\rho\right\rangle\equiv\rho\left|I\right\rangle,$ as well as
$\left\langle\tilde{n}\right|\left.\tilde{l}\right\rangle=\delta_{n,l}$
($n=\tilde{n},l=\tilde{l}$). By using
$a^{m}\left|l\right\rangle=\sqrt{l!/(l-m)!}\left|l-m\right\rangle,a^{{\dagger}m}\left|l\right\rangle=\sqrt{(l+m)!/l!}\left|l+m\right\rangle,$
Eq.(6) becomes
$\displaystyle\mathfrak{p}\left(m,T\right)$ $\displaystyle=$
$\displaystyle\frac{\zeta^{m}}{m!}\left\langle\mathbf{\rho}\right|\sum_{l=0}^{\infty}\frac{\left(l+m\right)!}{l!}\left(1-\zeta\right)^{l}\left|l+m,\widetilde{l+m}\right\rangle$
(7) $\displaystyle=$
$\displaystyle\zeta^{m}\left\langle\mathbf{\rho}\right|\sum_{l=0}^{\infty}\frac{\left[\left(1-\zeta\right)a^{{\dagger}}\tilde{a}^{{\dagger}}\right]^{l}}{l!}\left|m,\tilde{m}\right\rangle$
$\displaystyle=$
$\displaystyle\zeta^{m}\left\langle\mathbf{\rho}\right|e^{\left(1-\zeta\right)a^{{\dagger}}\tilde{a}^{{\dagger}}}\left|m,\tilde{m}\right\rangle.$
In order to derive four new formulas for $\mathfrak{p}\left(m,T\right)$, we
first bridge the relation between the characteristic function (CF) and the
entangled state representation $\left\langle\eta\right|$. Similarly to
Eqs.(6), after using the TFD theory, the CF of density operator $\rho$,
$\chi_{S}\left(\lambda,\lambda^{\ast}\right)=\mathtt{tr}\left(\rho e^{\lambda
a^{{\dagger}}-\lambda^{\ast}a}\right),$ can be calculated as
$\displaystyle\chi_{S}\left(\lambda,\lambda^{\ast}\right)$ $\displaystyle=$
$\displaystyle\sum_{m,n}^{\infty}\left\langle n,\tilde{n}\right|\rho
e^{\lambda a^{{\dagger}}-\lambda^{\ast}a}\left|m,\tilde{m}\right\rangle$ (8)
$\displaystyle=$
$\displaystyle\left\langle\rho\right|D\left(\lambda\right)\left|\eta=0\right\rangle$
$\displaystyle=$
$\displaystyle\left\langle\rho\right|\left.\eta=\lambda\right\rangle,$
which is the CF formula in thermo entangled state representation, with which
the characteristic function of density operator is simplified as an overlap
between two “pure states” in enlarged Fock space, rather than using ensemble
average in the system-mode space. Thus we can then simplify the calculation of
$\chi_{S}\left(\lambda,\lambda^{\ast}\right)$ by virtue of some important
properties of the entangled state representation $\left\langle\eta\right|.$
Using the expression of $\left\langle\eta\right|$ in Fock space, i.e.,
$\left\langle\eta\right|=\left\langle
0,\tilde{0}\right|\sum_{m,n=0}^{\infty}i^{m+n}\frac{a^{m}\tilde{a}^{n}}{m!n!}H_{m,n}\left(-i\eta^{\ast},i\eta\right)e^{-\left|\eta\right|^{2}/2},$
(9)
where $H_{m,n}\left(\xi^{\ast},\xi\right)$ is the two-variable Hermite
polynomials 11 ; 12 , one finds
$\left\langle\eta\right|\left.m,\tilde{n}\right\rangle=i^{m+n}H_{m,n}(-i\eta^{\ast},i\eta)e^{-\left|\eta\right|^{2}/2}/\sqrt{m!n!},$
(10)
which leads to
$\displaystyle\left\langle\eta\right|e^{\left(1-\zeta\right)a^{{\dagger}}\tilde{a}^{{\dagger}}}\left|m,\tilde{m}\right\rangle$
(11) $\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}\frac{\left(1-\zeta\right)^{n}}{n!}\frac{\left(m+n\right)!}{m!}\left\langle\eta\right.\left|m+n,\widetilde{m+n}\right\rangle$
$\displaystyle=$
$\displaystyle\frac{\left(-1\right)^{m}e^{-\left|\eta\right|^{2}/2}}{m!}\sum_{n=0}^{\infty}\frac{\left(\zeta-1\right)^{n}}{n!}H_{m+n,m+n}(-i\eta^{\ast},i\eta)$
$\displaystyle=$
$\displaystyle\frac{1}{\zeta^{m+1}}e^{-\frac{2-\zeta}{2\zeta}\left|\eta\right|^{2}}L_{m}\left(\frac{1}{\zeta}\left|\eta\right|^{2}\right),$
where in the last step, we have used the formula 13 ,
$\displaystyle\sum_{l=0}^{\infty}\frac{\alpha^{l}}{l!}H_{m+l,n+l}\left(x,y\right)$
(12) $\displaystyle=$ $\displaystyle\frac{e^{\frac{\alpha\allowbreak
xy}{\alpha+1}}}{\left(\alpha+1\right)^{(m+n+2)/2}}H_{m,n}\left(\frac{x}{\sqrt{\alpha+1}},\frac{y}{\sqrt{\alpha+1}}\right),$
and the relation between two-variable Hermite polynomials and Laguree
polynomials,
$L_{m}\left(xy\right)=\frac{(-1)^{m}}{m!}H_{m,m}\left(x,y\right).$ (13)
Further inserting the completeness relation of $\left\langle\eta\right|,$i.e.,
$\int\frac{1}{\pi}\mathtt{d}^{2}\eta\left|\eta\right\rangle\left\langle\eta\right|=1$
(it can be proved by using the normally ordered form of vacuum projector
$\left|0,\tilde{0}\right\rangle\left\langle
0,\tilde{0}\right|=\colon\exp\left(-a^{\dagger}a-\tilde{a}^{\dagger}\tilde{a}\right)\colon$
and the technique of integration within an ordered product (IWOP) of operators
14 ; 15 ; 16 ), into Eq.(7), we can rewrite it as
$\mathfrak{p}\left(m,T\right)=\frac{1}{\zeta}\int\frac{\mathtt{d}^{2}\lambda}{\pi}e^{-\frac{2-\zeta}{2\zeta}\left|\lambda\right|^{2}}L_{m}\left(\frac{1}{\zeta}\left|\lambda\right|^{2}\right)\chi_{S}\left(\lambda,\lambda^{\ast}\right),$
(14)
which is just a new relation about the CF and the photon-count distribution.
When the characteristic function $\chi_{S}\left(\lambda,\lambda^{\ast}\right)$
of density operator for Wigner-Weyl form is known, the photocount distribution
can be calculated by using Eq.(14).
For instance, we first consider the single-mode coherent states
$\left|\beta\right\rangle$, whose CF reads
$\chi_{\text{coh}}\left(\lambda,\lambda^{\ast}\right)=\exp\left[-\frac{1}{2}\left|\lambda\right|^{2}+\lambda\beta^{\ast}-\lambda^{\ast}\beta\right],$
(15)
substituting it into Eq.(14) yields
$\displaystyle\mathfrak{p}\left(m,T\right)$ $\displaystyle=$
$\displaystyle\int\frac{\mathtt{d}^{2}\lambda}{\pi\zeta}e^{-\frac{1}{\zeta}\left|\lambda\right|^{2}+\lambda\beta^{\ast}-\lambda^{\ast}\beta}L_{m}\left(\frac{1}{\zeta}\left|\lambda\right|^{2}\right)$
(16) $\displaystyle=$
$\displaystyle\frac{\left(\zeta\bar{n}\right)^{m}}{m!}e^{-\zeta\bar{n}},\text{\
}(\bar{n}=\left\langle\beta\right|a^{\dagger}a\left|\beta\right\rangle=\left|\beta\right|^{2}),$
where we use the limiting expression $\lim_{x\rightarrow
0}x^{m}L_{m}(-\left|\alpha\right|^{2}/x)=\frac{1}{m!}\left|\alpha\right|^{2m}$
and the following integrational formula (see Appendix B),
$\displaystyle\int\frac{d^{2}\alpha}{\pi}e^{-\allowbreak
B\left|\alpha\right|^{2}+C\alpha-C^{\ast}\alpha^{\ast}}L_{m}\left\\{A\left|\alpha\right|^{2}\right\\}$
(17) $\displaystyle=$
$\displaystyle\frac{\left(B-A\right)^{m}}{B^{m+1}}e^{\frac{-CC^{\ast}}{B}}L_{m}\left(\frac{ACC^{\ast}/B}{A-B}\right).$
Eq.(16) is the Poisson distribution coinciding with the result in Refs. 4 ; 5
.
As another example, we consider the single-mode squeezed vacuum state,
$\exp\left[r\left(a^{{\dagger}2}-a^{2}\right)/2\right]\left|0\right\rangle,$
whose CF reads
$\chi_{sq}\left(\lambda,\lambda^{\ast}\right)=\exp\left[-\frac{1}{2}\left|\lambda\right|^{2}\cosh
2r+\frac{1}{4}\left(\lambda^{2}+\lambda^{\ast 2}\right)\sinh 2r\right],$ (18)
substituting Eq.(18) into (14), we have (Appendix C)
$\displaystyle\mathfrak{p}\left(m,T\right)$ $\displaystyle=$
$\displaystyle\frac{\xi^{m}\text{sech}r\tanh^{m}r}{\left(V^{2}-1\right)^{m/2}\left(1-V^{2}\right)^{1/2}}P_{m}\left(\frac{V}{\sqrt{V^{2}-1}}\right),$
(19) $\displaystyle(V$ $\displaystyle=$ $\displaystyle\left(1-\xi\right)\tanh
r),$
which $P_{m}\left(x\right)$ is the Legendre polynomial and Eq.(19) is a new
result.
Next, we derive other three new formula. Notice that the characteristic
function $\chi_{S}\left(\lambda,\lambda^{\ast}\right)$ is related to the
Wigner function, Q-function and P-representation by the following Fourier
transforms,
$\displaystyle\chi_{S}\left(\lambda,\lambda^{\ast}\right)$ $\displaystyle=$
$\displaystyle\int
e^{\lambda\alpha^{\ast}-\lambda^{\ast}\alpha}W\left(\alpha\right)d^{2}\alpha,$
(20) $\displaystyle\chi_{S}\left(\lambda,\lambda^{\ast}\right)$
$\displaystyle=$ $\displaystyle e^{\frac{\left|\lambda\right|^{2}}{2}}\int
e^{\lambda\alpha^{\ast}-\lambda^{\ast}\alpha}Q\left(\alpha\right)d^{2}\alpha,$
(21) $\displaystyle\chi_{S}\left(\lambda,\lambda^{\ast}\right)$
$\displaystyle=$ $\displaystyle e^{-\frac{\left|\lambda\right|^{2}}{2}}\int
e^{\lambda\alpha^{\ast}-\lambda^{\ast}\alpha}P\left(\alpha\right)d^{2}\alpha,$
(22)
respectively, thus substituting Eqs.(20)-(22) into (14) we can directly obtain
$\displaystyle\mathfrak{p}\left(m,T\right)$ $\displaystyle=$
$\displaystyle\frac{2\left(-\zeta\right)^{m}}{\left(2-\zeta\right)^{m+1}}\int
d^{2}\alpha
e^{-\frac{2\zeta\left|\alpha\right|^{2}}{2-\zeta}}L_{m}\left\\{\frac{4\left|\alpha\right|^{2}}{2-\zeta}\right\\}W\left(\alpha\right),$
(23) $\displaystyle\mathfrak{p}\left(m,T\right)$ $\displaystyle=$
$\displaystyle\frac{\left(-\zeta\right)^{m}}{\left(1-\zeta\right)^{m+1}}\int
d^{2}\alpha
e^{\frac{-\zeta\left|\alpha\right|^{2}}{1-\zeta}}L_{m}\left\\{\frac{\left|\alpha\right|^{2}}{1-\zeta}\right\\}Q\left(\alpha\right),$
(24) $\displaystyle\mathfrak{p}\left(m,T\right)$ $\displaystyle=$
$\displaystyle\frac{\zeta^{m}}{m!}\int
d^{2}\alpha\left|\alpha\right|^{2m}e^{-\zeta\left|\alpha\right|^{2}}P\left(\alpha\right),$
(25)
where
$W\left(\alpha\right)=2\mathtt{tr}\left(\rho\Delta\left(\alpha,\alpha^{\ast}\right)\right),Q\left(\alpha\right)=\frac{1}{\pi}\left\langle\alpha\right|\rho\left|\alpha\right\rangle$,
and the integrational formula (17) is used. Eqs.(23)-(25) are the new formula
for evaluating photon count distribution. Therefore, once one of these
distributions of $\mathbf{\rho}$ is known, the photocount distribution can be
calculated by using Eq.(23)-(25), which involve the Wigner function,
Q-function, and P-representation of $\rho$, respectively. To confirm their
correctness, we still consider the coherent light field
$\left|\beta\right\rangle\left\langle\beta\right|,$ its Wigner function,
Q-function and P-function are given by
$W\left(\alpha\right)=\frac{2}{\pi}e^{-2\left|\beta-\alpha\right|^{2}},$
$P\left(\alpha\right)=\delta^{(2)}\left(\beta-\alpha\right)$, and
$Q\left(\alpha\right)=\frac{1}{\pi}e^{-\left|\beta-\alpha\right|^{2}}$,
respectively, then according to (23)-(25) and using (17) and the above
limiting expression $\lim_{x\rightarrow
0}x^{m}L_{m}(-\left|\alpha\right|^{2}/x)=\frac{1}{m!}\left|\alpha\right|^{2m}$,
one can draw the same result as Eq.(16).
At last, we should mention that using Eqs. (2)-(5) it is shown that the Wigner
function of a mixed state $\rho$,
$W_{\rho}\left(\alpha,\alpha^{\ast}\right)\equiv
2\mathtt{tr}\left(\Delta\left(\alpha,\alpha^{\ast}\right)\rho\right),$ where
$\Delta\left(\alpha,\alpha^{\ast}\right)$ is the single-mode Wigner operator
17 ; 18 , whose explicit normally ordered form is 19
$\Delta\left(\alpha,\alpha^{\ast}\right)=\frac{1}{\pi}\colon
e^{-2\left(a^{\dagger}-\alpha^{\ast}\right)\left(a-\alpha\right)}\colon=\frac{1}{\pi}D\left(2\alpha\right)(-1)^{a^{\dagger}a},$
(26)
which can also be converted to a overlap between two “pure state”in the
enlarged Fock space,
$\displaystyle W_{\rho}\left(\alpha,\alpha^{\ast}\right)$
$\displaystyle=\sum_{m,n}^{\infty}\left\langle
n,\tilde{n}\right|\Delta\left(\alpha,\alpha^{\ast}\right)\rho\left|m,\tilde{m}\right\rangle$
$\displaystyle=\frac{1}{\pi}\left\langle\eta=0\right|D\left(2\alpha\right)(-1)^{a^{\dagger}a}\left|\rho\right\rangle$
$\displaystyle=\frac{1}{\pi}\left\langle\eta=-2\alpha\right|(-1)^{a^{\dagger}a}\left|\rho\right\rangle$
$\displaystyle=\frac{1}{\pi}\left\langle\xi=2\alpha\right|\left.\rho\right\rangle,$
(27)
which is the Wigner function formula in thermo entangled state representation,
with which the Wigner function of density operator is simplified as an overlap
between two “pure states” in enlarged Fock space. Employing its completeness,
i.e.,
$\int\frac{\mathtt{d}^{2}\xi}{\pi}\left|\xi\right\rangle\left\langle\xi\right|=1,$
one can derive these above new formula. In addition, the expression in Eq.(27)
can also examine the evolution of Wigner function of density operator
interacting with the environments 20 .
In summary, based on Umezawa-Takahash thermo field dynamics theory, after
introducing the thermo entangled state representation, we converted the
calculation of CF to an overlap between two “pure states” in enlarged Fock
space. Then we bridge the relation between the characteristic function and the
photo-count distribution. Once the CF of density operator for Wigner-Weyl form
is known, the photocount distribution can be calculated conveniently. Using
the Fourier transform relation between the CF and the distribution functions,
we further derive other three new formula so as to be convenient for
calculating photo-count distribution by using these formulas.
Acknowledgements: Work supported by a grant from the Key Programs Foundation
of Ministry of Education of China (No. 210115) and the Research Foundation of
the Education Department of Jiangxi Province of China (No. GJJ10097).
Appendix A: Derivation of sum-formula in Eq.(12)
Using the integration of two-variable Hermite polynomials,
$H_{m,n}\left(\xi,\eta\right)=(-1)^{n}e^{\xi\eta}\int\frac{d^{2}z}{\pi}z^{n}z^{\ast
m}e^{-\left|z\right|^{2}+\xi z-\eta z^{\ast}},$ (A1)
we have
$\displaystyle\sum_{l=0}^{\infty}\frac{\alpha^{l}}{l!}H_{m+l,n+l}\left(x,y\right)$
$\displaystyle=\sum_{l=0}^{\infty}\frac{\alpha^{l}}{l!}(-1)^{n+l}e^{xy}\int\frac{d^{2}z}{\pi}z^{n+l}z^{\ast
m+l}e^{-\left|z\right|^{2}+xz-yz^{\ast}}$
$\displaystyle=e^{xy}(-1)^{n}\int\frac{d^{2}z}{\pi}z^{n}z^{\ast
m}e^{-\left(\alpha+1\right)\left|z\right|^{2}+xz-yz^{\ast}}.$ (A2)
Then making scale transform and using Eq.(A1) again, Eq.(A2) can be put into
the following form
$\displaystyle\sum_{l=0}^{\infty}\frac{\alpha^{l}}{l!}H_{m+l,n+l}\left(x,y\right)$
$\displaystyle=\frac{(-1)^{n}e^{xy}}{\left(\alpha+1\right)^{(m+n+2)/2}}\int\frac{d^{2}z}{\pi}z^{n}z^{\ast
m}e^{-\left|z\right|^{2}+\frac{xz}{\sqrt{\alpha+1}}-\frac{yz^{\ast}}{\sqrt{\alpha+1}}}$
$\displaystyle=\text{Right hand side of Eq.(\ref{p14}).}$ (A3)
Appendix B: Derivation of integration-formula in Eq.(17)
Using Eq.(13) and the generating function of the two-variable Hermite
polynomials,
$\left.\frac{\partial^{m+n}}{\partial\tau^{m}\partial\upsilon^{n}}e^{-A\tau\upsilon+B\tau+C\upsilon}\right|_{\tau=\upsilon=0}=\left(\sqrt{A}\right)^{m+n}H_{m,n}\left(\frac{B}{\sqrt{A}},\frac{C}{\sqrt{A}}\right),$
(B1)
we find
$\displaystyle\int\frac{d^{2}\alpha}{\pi}L_{m}\left\\{A^{2}\left|\alpha\right|^{2}\right\\}e^{-\allowbreak
B^{2}\left|\alpha\right|^{2}+C\alpha+C^{\ast}\alpha^{\ast}}$
$\displaystyle=\int\frac{d^{2}\alpha}{\pi}\frac{(-1)^{m}}{m!}H_{m,m}\left\\{A\alpha,A\alpha^{\ast}\right\\}e^{-\allowbreak
B^{2}\left|\alpha\right|^{2}+C\alpha+C^{\ast}\alpha^{\ast}}$
$\displaystyle=\frac{(-1)^{m}}{m!}\frac{\partial^{2m}}{\partial t^{m}\partial
t^{\prime m}}e^{-tt^{\prime}}\int\frac{d^{2}\alpha}{\pi}\left.e^{-\allowbreak
B^{2}\left|\alpha\right|^{2}+\left(C+At\right)\alpha+\left(C^{\ast}+At^{\prime}\right)\alpha^{\ast}}\right|_{t=t^{\prime}=0}$
$\displaystyle=\frac{(-1)^{m}}{m!}\frac{\left(B^{2}-A^{2}\right)^{m}}{B^{2\left(m+1\right)}e^{-CC^{\ast}/B^{2}}}\frac{\partial^{2m}}{\partial
t^{m}\partial\tau^{m}}\left.e^{-t\tau+\frac{AC/B}{\sqrt{B^{2}-A^{2}}}\tau+\frac{AC^{\ast}/B}{\sqrt{B^{2}-A^{2}}}t\allowbreak}\right|_{t=\tau=0},$
(B2)
where we have used the formula
$\int\frac{d^{2}z}{\pi}e^{\zeta\left|z\right|^{2}+\xi z+\eta
z^{\ast}}=-\frac{1}{\zeta}e^{-\frac{\xi\eta}{\zeta}},\text{Re}\left(\zeta\right)<0$
(B3)
Using Eqs.(B1) and (13) again, one can get the integration-formula in Eq.(17).
Appendix C: Derivation of the result in Eq.(19)
In order to obtain Eq.(19), we first derive a new integral formula,
$I\equiv\int\frac{d^{2}\lambda}{\pi}L_{m}\left\\{A\left|\lambda\right|^{2}\right\\}e^{-B\left|\lambda\right|^{2}+C\lambda^{2}+C\lambda^{\ast
2}}.$ (C1)
Using Eqs.(13) and (B1), Eq.(C1) can be put into the form
$\displaystyle I$
$\displaystyle=\frac{(-1)^{m}}{m!}\int\frac{d^{2}\lambda}{\pi}H_{m,m}\left(\sqrt{A}\lambda,\sqrt{A}\lambda^{\ast}\right)e^{-B\left|\lambda\right|^{2}+C\lambda^{2}+C\lambda^{\ast
2}}$ $\displaystyle=\frac{(-1)^{m}}{m!}\frac{\partial^{2m}}{\partial
t^{m}\partial\tau^{m}}e^{-\tau
t}\int\frac{d^{2}\lambda}{\pi}\left.e^{-B\left|\lambda\right|^{2}+t\sqrt{A}\lambda+\tau\sqrt{A}\lambda^{\ast}+C\lambda^{2}+C\lambda^{\ast
2}}\right|_{t=\tau=0}$
$\displaystyle=\frac{(-1)^{m}}{m!\sqrt{B^{2}-4C^{2}}}\frac{\partial^{2m}}{\partial
t^{m}\partial\tau^{m}}\exp\left[-\frac{B^{2}-4C^{2}-BA}{B^{2}-4C^{2}}\tau
t+\frac{CA\left(\tau^{2}+t^{2}\right)}{B^{2}-4C^{2}}\right]_{t=\tau=0},$ (C2)
where in the last step, we used the formula 21
$\displaystyle\int\frac{d^{2}z}{\pi}\exp\left(\zeta\left|z\right|^{2}+\xi
z+\eta z^{\ast}+fz^{2}+gz^{\ast 2}\right)$
$\displaystyle=\frac{1}{\sqrt{\zeta^{2}-4fg}}\exp\left[\frac{-\zeta\xi\eta+\xi^{2}g+\eta^{2}f}{\zeta^{2}-4fg}\right],$
(C3)
whose convergent condition is Re($\zeta\pm f\pm g)<0,\
$Re[$(\zeta^{2}-4fg)/(\zeta\pm f\pm g)]<0$.
Expanding the exponential item involved in Eq.(C2), we see
$\displaystyle I$
$\displaystyle=\frac{(-1)^{m}}{m!\sqrt{B^{2}-4C^{2}}}\sum_{n,l,k=0}^{\infty}\frac{\left(-1\right)^{k}}{n!l!k!}\left.\frac{\left(B^{2}-4C^{2}-BA\right)^{k}}{\left(B^{2}-4C^{2}\right)^{k+n+l}/\left(CA\right)^{n+l}}\frac{\partial^{2m}}{\partial
t^{m}\partial\tau^{m}}\tau^{2n+k}t^{2l+k}\right|_{t=\tau=0}$
$\displaystyle=\frac{\left(B^{2}-4C^{2}-BA\right)^{m}}{\left(B^{2}-4C^{2}\right)^{m+1/2}}\sum_{l=0}^{[m/2]}\frac{m!}{2^{2l}l!l!\left(m-2l\right)!}\left(\frac{1}{y}\right)^{2l},$
(C4)
where
$y=\frac{B^{2}-4C^{2}-AB}{2AC}.$ (C5)
Recalling that newly found expression of Lagendre polynomial (it is
equivalence to the well-known Legendre polynomial’s expression 22 ),
$x^{m}\sum_{l=0}^{[m/2]}\frac{m!}{2^{2l}l!l!\left(m-2l\right)!}\left(1-\frac{1}{x^{2}}\right)^{l}=P_{m}\left(x\right),$
(C6)
the compact form for $I$ is written as
$I=\frac{\left(\left(A-B\right)^{2}-4C^{2}\right)^{m/2}}{\left(B^{2}-4C^{2}\right)^{(m+1)/2}}P_{m}\left(\frac{y}{\sqrt{y^{2}-1}}\right),$
(C7)
which is a new integration formula.
Substituting Eq.(18) into (14) we have
$\mathfrak{p}\left(m,T\right)=\frac{1}{\zeta}I^{\prime},$ (C8)
where $I^{\prime}$ shown in Eq.(C7) characteristic of
$A=\frac{1}{\zeta},B=\frac{1}{\zeta}+\sinh^{2}r,C=\frac{1}{4}\sinh 2r,$ (C9)
which leads to
$y=\left(1-\zeta\right)\tanh r,$ (C10)
$A-B=\left(A-B\right)^{2}-4C^{2}=-\sinh^{2}r,$ (C11)
$B^{2}-4C^{2}=\frac{1}{\zeta^{2}}\left[\left(2-\zeta\right)\zeta\sinh^{2}r+1\right],$
(C12)
and
$\displaystyle\frac{\left(\left(A-B\right)^{2}-4C^{2}\right)^{m/2}}{\left(B^{2}-4C^{2}\right)^{(m+1)/2}}$
$\displaystyle=\frac{\zeta^{m+1}\text{sech}r\left(-\tanh^{2}r\right)^{m/2}}{\left(\left(2-\zeta\right)\zeta\tanh^{2}r+\text{sech}^{2}r\right)^{(m+1)/2}}$
$\displaystyle=\frac{\zeta^{m+1}\text{sech}r\left(-\tanh^{2}r\right)^{m/2}}{\left(1-y^{2}\right)^{(m+1)/2}},$
(C13)
so
$\mathfrak{p}\left(m,T\right)=\frac{\zeta^{m}\text{sech}r\tanh^{m}r}{\left(y^{2}-1\right)^{m/2}\left(1-y^{2}\right)^{1/2}}P_{m}\left(\frac{y}{\sqrt{y^{2}-1}}\right),$
(C14)
which is the photon-count distribution of squeezed vacuum state.
## References
* (1) L. Mandel, E. Wolf, Optical Coherence and Quantum Optics, (Cambridge University Press, Cambridge, England, 1995) pp. 623.
* (2) R. J. Glauber, Phys. Rev. 130, 2529 (1963).
* (3) P. L. Kelley and W. H. Kleiner, Phys. Rev. 136, 316 (1964).
* (4) M. O. Scully and W. E. Lamb, Phys. Rev. 179, 368 (1969).
* (5) B. R. Mollow, Phys. Rev. 168, 1896 (1968).
* (6) Hong-yi Fan and J. R. Klauder, Phys. Rev. A 49, 704 (1994).
* (7) Hong-yi Fan and Yue Fan, J. Phys. A 35, 6873 (2002).
* (8) Memorial Issue for H. Umezawa, Int. J. Mod. Phys. B 10, 1695 (1996) memorial issue and references therein.
* (9) H. Umezawa, Advanced Field Theory – Micro, Macro, and Thermal Physics (AIP 1993).
* (10) Y. Takahashi and H. Umezawa, Collecive Phenomena 2, 55 (1975).
* (11) A. Wünsche, J. Computational and Appl. Math. 133, 665 (2001).
* (12) A. Wünsche, J. Phys. A: Math. and Gen. 33, 1603 (2000).
* (13) Li-yun Hu, Zheng-lu Duan, Xue-xiang Xu, and Zi-sheng Wang, arXiv:1010.0584 [quant-ph].
* (14) Hong-yi Fan, Hai-liang Lu and Yue Fan, Ann. Phys _._ 321, 480 (2006).
* (15) Hong-yi Fan, H. R. Zaidi and J. R. Klauder, Phys. Rev. D 35, 1831 (1987).
* (16) A. Wünsche, J. Opt. B: Quantum Semiclass. Opt. 1, R11 (1999).
* (17) E. Wigner, Phys. Rev. 40, 749 (1932).
* (18) Hong-yi Fan, Representation and Transformation Theory in Quantum Mechanics, (Shanghai Scientific & Technical, Shanghai, 1997) (in Chinese).
* (19) Hong-yi Fan and H. R. Zaidi, Phys. Lett. A 124, 303 (1987).
* (20) Li-yun Hu and Hong-yi Fan, Opt. Commun. 282, 4379 (2009).
* (21) R. R. Puri, Mathematical Method of Quantum Optics (Springer-Verlag, 2001), Appendix A.
* (22) Li-yun Hu and Hong-yi Fan, J. Opt. Soc. Am. B, 25, 1955 (2008).
|
arxiv-papers
| 2010-10-27T02:44:26 |
2024-09-04T02:49:14.262544
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Li-yun Hu, Z. S. Wang, L. C. Kwek, and Hong-yi Fan",
"submitter": "Liyun Hu",
"url": "https://arxiv.org/abs/1010.5555"
}
|
1010.5585
|
2010 Vol.XX No.X, 000–000
11institutetext: Central Department of Physics, Tribhuvan University,
Kirtipur, Nepal binil.aryal@uibk.ac.at
22institutetext: Institut of Astro- and Particle Physics, Innsbruck
University, Technikerstrasse 25, A-6020 Innsbruck, Austria
Received [2010] [January] [14]; accepted [2010] [October] [23]
# Winding sense of galaxies around the Local Supercluster
B. Aryal 1122
###### Abstract
We present an analysis of the winding sense (S and Z-shaped) of 1 621 field
galaxies that have radial velocity between 3 000 km s-1 to 5 000 km s-1. The
preferred alignments of S- and Z-shaped galaxies are studied using chi-square,
auto-correlation and the Fourier tests. We classified total galaxies into 32
subsamples and noticed a good agreement between the position angle (PA)
distribution of S- and Z-shaped galaxies. The homogeneous distribution of the
S- and Z-shaped galaxies is noticed for the late-type spirals (Sc, Scd, Sd and
Sm) than that of the early-types (Sa, Sab, Sb and Sbc). A significant
dominance of S-mode galaxies is noticed in the barred spirals. A random
alignment is noticed in the PA-distribution of Z- and S-mode spirals. In
addition, homogeneous distribution of the S- and Z-shaped galaxies is found to
be invariant under the global expansion. The PA-distribution of the total
S-mode galaxies is found to be random, whereas preferred alignment is noticed
for the total Z-mode galaxies. It is found that the galactic planes of Z-mode
galaxies tend to lie in the equatorial plane.
###### keywords:
spiral galaxies – clusters: individual (Local Supercluster)
## 1 Introduction
Differential rotation in a galaxy’s disc generate density waves in the disc,
leading to spiral arms. According to gravitational theory, the spiral arms
born as leading and subsequently transform to trailing modes. With the passage
of time, the spiral pattern deteriorates gradually by the differential
rotation of the equatorial plane of the galaxy, but the bar structure persists
for a long time (Oort 1970a). This structure can again regenerate spiral
pattern in the outer region. Thus, a close relation between the origin of the
arms in the spirals and barred spirals can not be denied (Oort 1970b).
Land et al. (2008) studied the distribution of projected spin vectors of
$\sim$ 37 000 spiral galaxies taken from Solon Dizital Sky Survey. They did
not notice any evidence for overall preferred handedness of Universe. In a
similar study, Longo (2007) found evidence for a preferred axis. Sugai & Iye
(1995) used statistics and studied the winding sense of galaxies (S- and
Z-patterns) in 9 825 spirals. No significant dominance from a random
distribution is noticed. Aryal & Saurer (2005) studied the spatial
orientations of spin vectors of 4 073 galaxies in the Local Supercluster. No
preferred alignment is noticed for the total sample. These results hint that
the distribution of angular momentum of galaxies is entirely random in two-
(S- and Z-shaped) and three-dimensional (spin vector) analysis provided the
database is rich.
In order to understand true structural modes (leading or trailing) of spiral
galaxies, we need to know the direction of the spiral pattern (S- or
Z-patterns), the approaching and receding sides and the near and far parts,
since galaxies are commonly inclined in space to the line of the sight. The S
and Z-patterns can be determined from the image of the galaxy. Similarly, the
approaching and receding sides can be defined if spectroscopy data on rotation
is available. The third one is fairly hard to established. For this, Pasha
(1985) used ‘tilt’ criteria and studied the sense of winding of the arms in
132 spirals. He found 107 spirals to have trailing arm. Thomasson et al.
(1989) studied theoretically and performed $N$-body simulations in order to
understand the formation of spiral structures in retrograde galaxy encounters.
Interestingly, they noticed the importance of halo mass. They concluded that
the spirals having halos with masses larger than the disk mass exhibit leading
pattern. Thus, the makeup of galactic haloes is important to cosmology in
order to understand the evolution of galaxies.
By considering the group of transformations acting on the configuration space,
Capozziello & Lattanzi (2006) predicted that the progressive loss of
inhomogeneity in the S- and Z-shaped galaxies might have some connection with
the rotationally-supported (spirals, barred spirals) and randomized stellar
systems (lenticulars, ellipticals).
The preferred alignments of galaxies can be an indicator of initial conditions
when galaxies and clusters formed provided the angular momenta of galaxies
have not been altered too much since their formation. A useful property of
galaxies in clusters for which theories make different predictions is the
angular momentum distribution. The ‘Pancake model’ by Doroshkevich
Doroshkevich (1973), the ‘Hierarchy model’ by Peebles Peebles (1969) and, the
‘Primordial vorticity model’ by Ozernoy Ozernoy (1978) predict different
scenarios concerning the formation of large-scale structure. Thus, the study
of galaxy orientation has the potential to yield important information
regarding the formation and evolution of cosmic structures. In this work, we
present an analysis of winding sense and preferred alignments of galaxies that
have radial velocity (RV) 3 000 km s-1 to 5 000 km s-1. These are field
galaxies. We intend to study the importance of winding sense in order to
understand the true structural modes (i.e., leading and trailing arm) of the
galaxy. We expect to study the following: (1) Are the distribution of S- and
Z-shaped galaxies homogeneous in the field? (2) Is there any correlation
between the preferred alignment and the winding sense of galaxies? (3) Does
radial velocity dependence exist concerning winding sense of galaxies? and
finally, (4) What can we say about the distribution of true structural modes
(i.e., leading or trailing arm) of galaxies in the large scale structure?
This paper is organized as follows: in Sect. 2 we describe the method of data
reduction. In Sect. 3 we give a brief account of the methods and the
statistics used. Finally, a discussion of the statistical results and the
conclusions are presented in Sects. 4 and 5.
## 2 The sample: data reduction
Figure 1: (a) A sketch representing the winding sense (S or Z) of the galaxy.
(b) All-sky distribution of Z-mode ($\triangle$) and S-mode ($\circ$) galaxies
that have RVs in the range 3 000 km s-1 to 5 000 km s-1. The morphology (c),
radial velocity (d), axial ratio (e) and the magnitude (f) distribution of Z
and S-mode galaxies in our database. The statistical $\pm$1$\sigma$ error bars
are shown for the S-mode ($\bullet$) subsample. The dashed line (e) represents
the expected distribution.
Eighteen catalogues were used for the data compilation. A list of the
catalogues and their references are given in Table 1. The abbreviations given
in the first column of Table 1 are as follows: NGC - New General Catalogue,
UGC - Uppsala General Catalogue of Galaxies, ESO - ESO/Uppsala Survey of the
ESO (B) Atlas, IC - Index Catalogue, MCG - Morphological Galaxy Catalogue,
UGCA - Uppsala obs. General Catalogue, Addendum, CGCG - Catalogue of Galaxies
and Clusters of Galaxies, KUG - Kiso Ultraviolet Galaxy Catalogue, MRK -
Markarian Galaxy Catalogue, MESSIER - Catalogue des nebuleuses et des amas
d’etoiles, BCG - Brandner+Grebel+Chu Catalogue, LSBG - Low Surface Brightness
Galaxies, SBS - Second Byurakan Survey, LCRS - Las Companas Red Shift Survey,
DDO - David Dunlap Observatory Publications, IRAS - Infrared Astronomical
Satellite, SGC - Southern Galaxy Catalogue and UM - University of Michigan:
Curtis Schmidt-thin prism survey for extragalactic emission-line objects: List
I-V.
The NASA/IPAC extragalactic database (NED, http://nedwww.ipac.caltech.
edu/) was used to compile these catalogues. The main editing process was as
follows: first, galaxies having RVs in the range 3 000 km s-1 to 5 000 km s-1
were collected. We downloaded the image of all these galaxies from NED in FITS
format. The second step was to compile the morphology of these galaxies from
the catalog. A galaxy with doubtful morphology (eg., ‘S?’, ‘S0’ or ‘Sa’) is
omitted. Finally, the position angles (PAs) of galaxies were added from the
UGC, ESO, and Third Reference Catalogue of Bright Galaxies (de Vaucouleurs et
al. 1991).
Table 1: The list of catalogues used for the data compilation. The first
column lists the abbreviation of the catalogue. The second column gives the
total number of galaxies. The references are listed in the last column.
$\begin{array}[]{p{0.12\linewidth}rll}\hline\cr\vskip 3.0pt plus 1.0pt minus
1.0pt\cr Catalogue&$N$&$References$\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\cr\hline\cr\vskip 3.0pt plus 1.0pt minus 1.0pt\cr NGC&623&$Dreyer (1895,
1908) $\\\ UGC&276&$Nilson (1973)$\\\ ESO&123&$Lauberts (1982)$\\\
IC&93&$Dreyer (1895, 1908) $\\\ MCG&88&$Vorontsov-Vel'Yaminov et al.$\\\
&&$(1962-74)$\\\ UGCA&84&$Nilson (1974)$\\\ CGCG&75&$Zwicky et al.
(1961-68)$\\\ KUG&44&$Takase (1980-2000)$\\\ MRK&40&$Markarian (1967)$\\\
MESSIER&41&$Messier (1784)$\\\ BCG&28&$Brandner et al. (2000)$\\\
LSBG&23&$Impey et al. (1996)$\\\ SBS&17&$Markarian et al. (1983)$\\\
LCRS&16&$Shectman et al. (1996)$\\\ DDO&15&$Bergh (1959, 1966)$\\\
IRAS&15&$Infrared Astronomical Satellite$\\\ &&$(1983)$\\\ SGC&10&$Corwin et
al. (1985)$\\\ UM&10&$MacAlpine et al.$\\\ &&$(1977a,b,c; 1978; 1981)$\\\
\vskip 3.0pt plus 1.0pt minus 1.0pt\cr\hline\cr\end{array}$
There were two clusters Abell 0426 ($\alpha$(J2000) = 03h18m 36.4s,
$\delta$(J2000) = +41∘30’54”) and Abell 3627 ($\alpha$(J2000) = 16h15m32.8s,
$\delta$(J2000) = –60∘54’30”) in our region. These clusters have mean RVs 5
366 km s-1 (75 $\pm$ 5 Mpc) and 4 881 km s-1 (63 $\pm$ 4 Mpc), respectively
(Abell, Corwin & Olowin 1989, Struble & Rodd 1999). We removed the galaxies
belong to the cluster Abell 0426 using the catalog established by Brunzendorf
& Meusinger (1999). For the cluster Abell 3627 galaxies, we used Photometric
Atlas of Northern Bright Galaxies (Kodaira, Okamura & Ichikawa 1990) and
Uppsala Galaxy Catalogue (Nilson 1973). There were 174 galaxies belongs to
these clusters in our database. We remove these galaxies.
The RVs were compiled from Las Campanas Redshift Survey (Shectman 1996). The
PAs and the diameters of galaxies were added from the Uppsala Galaxy Catalogue
(Nilson 1973), Uppsala obs. General Catalogue, Addendum (Nilson 1974),
Photometric Atlas of Northern Bright Galaxies (Kodaira, Okamura, & Ichikawa
1990), ESO/Uppsala Survey of the European Southern Observatory (Lauberts
1982), Southern Galaxy Catalogue (Corwin et al. 1985) and Third Reference
Catalogue of Bright Galaxies (de Vaucouleurs et al. 1991).
In the NED, 6 493 galaxies having RVs 3 000 km s-1 to 5 000 km s-1 were listed
until the cutoff date. Morphological information was given in the catalogues
for 3 276 (50%) galaxies. We visually inspected all these galaxies using
ALADIN2.5 software.
The arm patterns (S- or Z-type) of the galaxies were studied visually by the
author in order to maintain homogeneity. The contour maps of the galaxies were
studied in order to identify their structural modes. For this, we used
ALADIN2.5 software. The Z-mode is one whose outer tip points towards the anti-
clockwise direction (see Fig. 1a). Similarly, the outer tip of S-mode directs
in the clockwise direction. These two patterns are obviously the two
dimensional projections of three dimensional galaxy. The re-examination of the
S- and Z-modes using MIDAS software resulted the rejection of more than 17% of
the objects. These rejected galaxies were nearly edge-on spiral and barred
spiral galaxies. As expected, it was relatively easier to identify the
structural modes of nearly face-on than that of nearly edge-on galaxies.
In this way, we compiled a database of 1 621 galaxies showing either S- or
Z-structural mode. There were 807 Z-mode and 821 S-mode galaxies in our
database. All sky distribution of Z- and S-mode galaxies is shown in Fig. 1b.
The symbols “$\circ$” and “$\triangle$” represent the positions of the S-mode
and Z-mode galaxies, respectively. Several groups and aggregations of the
galaxies can be seen in the figure.
The morphology, radial velocity, axial ratio and the magnitude distributions
of S- and Z-mode galaxies are shown in Figs. 1c,d,e,f. The spirals (47%)
dominate our database (Fig. 1c). However, a significant dominance of S-modes
are noticed in the barred spirals whereas a weak dominance of Z-modes are
found in the spirals. The population of galaxies in the RV distribution
($\Delta$RV = 5 00 km s-1) were nearly equal (Fig. 1d). The axial ratio
distribution shows a good agreement with the expected cosine curve in the
limit 0.2 $<$ $b$/$a$ $\leq$ 0.9 (Fig. 1e). The values of absolute magnitude
lie between 13.0 and 16.0 for 82% galaxies in our database (Fig. 1f).
We classified the database into 32 subsamples for both the S- and Z-modes on
the basis of the morphology, radial velocity, area and the group of the
galaxies. The galaxies with doubtful morphology are omitted in the spiral and
barred spiral subsamples. The total number of early- and late-type spirals or
barred spirals is much less than that of the total spiral or total barred
spirals. It is because of the fact that the galaxy with incomplete morphology,
say, simply ‘S’ or ‘SB’ can not be included in the subsamples. In other words,
the galaxy with morphology Sa, Sab, Sb, Sbc are included in the early spirals
whereas the galaxies with morphology Sc, Scd, Sd and Sdm are classified as
late type spirals. The galaxies having morphology other than Sa, Sab, Sb, Sbc,
Sc, Scd, Sd and Sm can not be included in the early and late subsamples. A
statistical study of these subsamples are given in Table 1 and discussed in
Sect. 4.1.
## 3 Method of analysis
Basic statistics is used to study the dominance of Z- or S-mode galaxies. At
first, morphology and RV dependence are studied. Secondly, sky is divided into
16 equal parts in order to observe deviation from the homogeneity. Several
galaxy groups are identified in the all-sky map where the structural dominance
are noticed. Finally, we study the dominance of Z- or S-mode galaxies in these
groups.
We assume isotropic distribution as a theoretical reference and studied the
equatorial PA-distribution in the total sample and subsamples. In order to
measure the deviation from isotropic distribution we have carried out three
statistical tests: chi-square, auto-correlation and the Fourier.
We set the chi-square probability P($>\chi^{2}$) = 0.050 as the critical value
to discriminate isotropy from anisotropy, this corresponds to a deviation from
isotropy at the 2$\sigma$ level (Godlowski 1993). Auto correlation test takes
account the correlation between the number of galaxies in adjoining angular
bins. We expect, auto correlation coefficient C$\rightarrow$0 for an isotropic
distribution. The critical limit is the standard deviation of the correlation
coefficient C.
If the deviation from isotropy is only slowly varying with angles (in our
case: PA) the Fourier test can be applied (Godlowski 1993). A method of
expanding a function by expressing it as an infinite series of periodic
functions (sine and cosine) is called Fourier series. Let $N$ denote the total
number of solutions for galaxies in the sample, $N$k the number of solutions
in the kth bin, $N$0 the mean number of solutions per bin, and $N$0k the
expected number of solutions in the kth bin. Then the Fourier series is given
by (taking first order Fourier mode),
$\begin{array}[]{l}N_{k}=N_{k}(1+\Delta_{11}\cos 2\beta_{k}+\Delta_{21}\sin
2\beta_{k}+......)\\\ \end{array}$ (1)
Here the angle $\beta$k represents the polar angle in the kth bin. The Fourier
coefficients $\Delta_{11}$ and $\Delta_{21}$ are the parameters of the
distributions. We obtain the following expressions for the Fourier
coefficients $\Delta_{11}$ and $\Delta_{21}$,
$\begin{array}[]{l}\Delta_{11}=\sum(N_{k}-N_{0k})\cos 2\beta_{k}/\sum
N_{0k}\cos^{2}2\beta_{k}\\\ \end{array}$ (2)
$\begin{array}[]{l}\Delta_{21}=\sum(N_{k}-N_{0k})\sin 2\beta_{k}/\sum
N_{0k}\sin^{2}2\beta_{k}\\\ \end{array}$ (3)
The standard deviations ($\sigma$($\Delta_{11}$)) and
($\sigma$($\Delta_{21}$)) can be estimated using the expressions,
$\begin{array}[]{l}\sigma(\Delta_{11})=(\sum
N_{0k}\cos^{2}2\beta_{k})^{-1/2}\\\ \end{array}$ (4)
$\begin{array}[]{l}\sigma(\Delta_{21})=(\sum
N_{0k}\sin^{2}2\beta_{k})^{-1/2}\\\ \end{array}$ (5)
The probability that the amplitude
$\begin{array}[]{l}\Delta_{1}=(\Delta_{11}^{2}+\Delta_{21}^{2})^{1/2}\\\
\end{array}$ (6)
greater than a certain chosen value is given by the formula
$\begin{array}[]{l}P(>\Delta_{1})=\exp(-nN_{0}\Delta_{1}^{2}/4)\\\
\end{array}$ (7)
with standard deviation
$\begin{array}[]{l}\sigma(\Delta_{1})=(2/nN_{0})^{1/2}\\\ \end{array}$ (8)
The Fourier coefficient $\Delta_{11}$ gives the direction of departure from
isotropy. The first order Fourier probability function $P$($>$$\Delta_{1}$)
estimates whether (smaller value of $P$($>$$\Delta_{1}$) or not (higher value
of $P$($>$$\Delta_{1}$) a pronounced preferred orientation occurs in the
sample.
## 4 Results
First we present the statistical result concerning the distribution of Z- and
S-mode galaxies in the total sample and subsamples. Second, we study the
distribution of Z- and S-mode galaxies in the unit area of the sky and in the
groups. Then, the equatorial PA-distribution of galaxies in the total sample
and subsamples are discussed. At the end, a general discussion and a
comparison with the previous results will be presented.
### 4.1 Distribution of Z and S mode galaxies
A statistical comparison between the total sample and subsamples of the Z- and
S-modes of galaxies is given in Table 2. Fig. 2 shows this comparison in the
histogram. The $\Delta$(%) in Table 1 and Fig. 2 represent the percentage
difference between the number Z- and S-mode galaxies. We studied the standard
deviation of the major diameters ($a$) of galaxies in the total sample and
subsamples for both the Z- and S-modes. In Table 2, $\Delta(a\,$sde$)$
represents the difference between the standard deviation of the major
diameters of Z- and S-mode galaxies.
An insignificant difference (0.4% $\pm$ 0.2%) between the total number of Z-
and S-mode galaxies are found (Table 2). The difference between the standard
deviation of the major diameters ($\Delta(a\,$sde$)$) of the Z- and S-mode
galaxies is found less than 0.019 (eighth column, Table 2). Interestingly, the
sum of the major diameters of total Z- and S-mode galaxies coincide. This
result suggests the homogeneous distribution of Z- and S-mode field galaxies
that have RV in the range 3 000 km s-1 to 4 000 km s-1.
In Fig. 2, the slanting-line (grey-shaded) region corresponds to the region
showing $\leq$ 10% (5%) $\Delta$ value. Almost all subsamples lie in this
region, suggesting the homogeneous distribution of Z- and S-mode galaxies
within 10% error limit. Now, we present the distribution Z- and S-mode
galaxies in the subsamples classified according as their morphology, RVs, area
and the groups below.
#### 4.1.1 Morphology
In the spirals, Z-mode galaxies are found 3.7% ($\pm$1.8%) more than that of
S-mode. The homogeneous distribution of Z- and S-modes is found for the late-
type spirals (Sc, Scd, Sd and Sm) than that of early-type (Sa, Sab, Sb and
Sbc): $\Delta$ value turned out to be 9.5% ($\pm$4.8%) and 1.8% ($\pm$1.0%)
for early- and late-types (Table 2). Thus, no preferred winding pattern is
noticed in the late-type spirals than that of early-types.
Figure 2: The basic statistics of the Z and S-mode of galaxies in the total
sample and subsamples. The full form of the abbreviations (X-axis) are given
in Table 2 (first column). $\Delta$(%)=$S$$-$$Z$, where $S$ and $Z$ represent
the number of S- and Z-mode galaxies, respectively. The statistical error bars
$\sigma$(%) shown in the figure are calculated as: $\sigma$(%) =
$\sigma$/($\sqrt{S}$+$\sqrt{Z}$)$\times$100, where $\sigma$ =
($\sqrt{S}$-$\sqrt{Z}$). The grey-shaded and the slanting-line region
represent the $\leq$$\pm$5% and $\leq$$\pm$10% $\Delta$ value, respectively.
Table 2: Statistics of leading (column 3) and trailing arm (column 4) galaxies
in the total sample and subsamples. The fifth and sixth column give the
numeral and percentage difference ($\Delta$ = $S$–$Z$) between the S- ($S$)
and the Z- ($Z$) modes. The next two columns give the error: $\sigma$ =
($\sqrt{S}$–$\sqrt{Z}$) and $\sigma$(%) =
$\sigma$/($\sqrt{S}$+$\sqrt{Z}$)$\times$100\. The eighth column gives the
difference between the standard deviation (in arcmin) of the major diameters
($a$) of the S- and Z-modes galaxies ($\Delta$($a\,sde)$). The difference
between the sum of the major diameters ($\Delta$($a$)%) are listed in the last
column. The sample/subsample and their abbreviations are given in first two
columns. $\begin{array}[]{p{0.30\linewidth}rccrrrrrr}\hline\cr\hline\cr\vskip
3.0pt plus 1.0pt minus 1.0pt\cr
sample/subsample&$symbol$&$Z$&$S$&$$\Delta$$&$$\Delta$(\%)$&$$\sigma$(\%)$&$$\Delta$($a$\,sde)$&$$\Delta$($a$)(\%)$\\\
\hline\cr\hline\cr\vskip 3.0pt plus 1.0pt minus 1.0pt\cr
Total&$T$&814&807&-7&-0.4&-0.2&0.019&0.0\\\
Spiral&$S$&395&367&-28&-3.7&-1.8&0.031&3.0\\\ Spiral (early-
type)&$SE$&150&124&-26&-9.5&-4.8&0.058&8.1\\\ Spiral (late-
type)&$SL$&131&126&-5&-1.9&-1.0&0.031&0.3\\\ Barred
Spiral&$SB$&191&269&78&17.0&8.5&0.062&15.2\\\ Barred Spiral (early-
type)&$SBE$&97&140&43&18.1&9.1&0.091&14.6\\\ Barred Spiral (late-
type)&$SBL$&68&83&15&9.9&5.0&0.066&9.2\\\ 3\,000$<$RV (km
s${}^{-1}$)$\leq$3\,500&$RV1$&208&194&-14&-3.5&-1.7&0.046&2.6\\\ 3\,500$<$RV
(km s${}^{-1}$)$\leq$4\,000&$RV2$&201&201&0&0.0&0.0&0.031&3.1\\\ 4\,000$<$RV
(km s${}^{-1}$)$\leq$4\,500&$RV3$&172&182&10&2.8&1.4&0.034&0.6\\\ 4\,500$<$RV
(km s${}^{-1}$)$\leq$5\,000&$RV4$&226&237&11&2.4&1.2&0.041&1.0\\\ Grid
1&$G1$&20&21&1&2.4&1.2&0.068&6.8\\\ Grid
2&$G2$&121&116&-5&-2.1&-1.1&0.014&1.6\\\ Grid
3&$G3$&88&112&24&12.0&6.0&0.076&9.3\\\ Grid
4&$G4$&12&14&2&7.7&3.9&0.647&9.7\\\ Grid
5&$G5$&11&8&-3&-15.8&-7.9&0.042&22.3\\\ Grid
6&$G6$&75&80&5&3.2&1.6&0.081&4.5\\\ Grid
7&$G7$&62&56&-6&-5.1&-2.5&0.095&8.9\\\ Grid
8&$G8$&22&33&11&20.0&10.1&0.073&12.9\\\ Grid
9&$G9$&20&31&11&21.6&10.9&0.028&18.1\\\ Grid
10&$G10$&124&108&-16&-6.9&-3.5&0.004&5.4\\\ Grid
11&$G11$&61&52&-9&-8.0&-4.0&0.025&6.2\\\ Grid
12&$G12$&20&20&0&0.0&0.0&0.409&7.4\\\ Grid
13&$G13$&78&66&-12&-8.3&-4.2&0.039&2.9\\\ Grid
14&$G14$&37&44&7&8.6&4.3&0.356&10.3\\\ Grid
15&$G15$&47&44&-3&-3.3&-1.6&0.050&5.2\\\ Grid
16&$G16$&9&9&0&0.0&0.0&0.191&8.6\\\ Group
1&$Gr1$&37&30&-7&-10.4&-5.2&0.032&7.1\\\ Group
2&$Gr2$&48&70&22&18.6&9.4&0.097&12.6\\\ Group
3&$Gr3$&31&37&6&8.8&4.4&0.027&4.3\\\ Group
4&$Gr4$&34&40&6&8.1&4.1&0.031&3.9\\\ Group
5&$Gr5$&107&85&-22&-11.5&-5.7&0.089&11.2\\\ Group
6&$Gr6$&42&45&3&3.4&1.7&0.024&1.6\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\cr\hline\cr\hline\cr\end{array}$
A significant dominance of S-mode galaxies are noticed (17%$\pm$8.5%) in
spiral barred galaxies. The $\Delta$ value is found $>$ 9% for both early-
(SBa, SBab, SBb and SBbc) and late-type (SBc, SBcd, SBd, SBm) barred spirals.
Similar result (i.e., $\Delta$ $>$ 8%) is found for the irregulars and the
morphologically unidentified galaxies.
A similarity is noticed between the late-type spirals and barred spirals: the
$\Delta$ value for both the late-types are found to be less than that of
early-types (see Table 2).
The difference between the standard deviation of the major diameters
($\Delta(a\,$sde$)$) for S- and Z-mode galaxies is found less than 0.050 arc
minute for the total sample, spirals and the late-type spirals (eighth column,
Table 1). These samples showed $\Delta$ value $<$ 5% (grey-shaded region, Fig.
2a). Thus, we noticed a good correlation between the $\Delta$(%) and
$\Delta(a\,$sde$)$ value.
The difference between the sum of the major diameters (in percentage) are
found greater than 10% for the barred spirals and early-type barred spirals.
Interestingly, these two subsamples showed $\Delta$ value greater than 15%
(Fig. 2a). Thus, inhomogeniety in the distribution of S- and Z-mode galaxies
is noticed for barred spirals.
#### 4.1.2 Radial velocity
A very good correlation between the number of S- and Z-mode can be seen in the
RV classifications (Fig. 1d). All 4 subsamples show the $\Delta$ and
$\Delta(a\,$sde$)$ values less than 5% and 0.050, respectively (Table 2). In
addition, $\Delta$($a$) is found to be $<$ 5%. This result is important in the
sense that the statistics in these subsamples is rich (number of galaxies $>$
170) enough. Thus, we could not observe preference structural modes (S or Z)
in the low and high RV galaxies in our database.
A difference is noticed: dominance of Z- and S- modes, respectively in low
(RV1) and high (RV3, RV4) RV subsamples. However, this dominance is not
significant (i.e., $\Delta$ $<$ 5%). An equal number of S- and Z-mode galaxies
are found in the subsample RV2 (3 500 $<$ RV (km s-1) $\leq$ 4 000) (Table 1).
In order to check the binning effect, we further classify the total galaxies
in 6 ($\Delta$RV = 333 km s-1) and 8 bins ($\Delta$RV = 250 km s-1) and study
the statistics. No significant dominance of either S- and Z-modes are noticed.
Thus, it is found that the homogeneous distribution of S- and Z-mode galaxies
remain invariant with the global expansion (i.e, expansion of the Universe).
We further discuss this result below.
#### 4.1.3 Area
We study the distribution of S- and Z-mode galaxies by dividing the sky into
16 equal parts (Fig. 3a). The area of the grid (G) is 90∘ $\times$ 45∘ (RA
$\times$ Dec). The area distribution of S- and Z-mode galaxies are plotted,
that can be seen in Fig. 3a’. The statistical parameters are given in Table 2.
Figure 3: (a) All sky distribution of Z-mode (hollow circle) and S-mode
(hollow triangle) galaxies in 16 area grids. (a’) The histogram showing the
distribution of the Z- and S-mode galaxies in the grids G1 to G16. (b) Six
groups of the galaxies, represented by the grey-shaded region. (b’) The
distribution of Z- and S-modes in 6 groups. The statistical error bar
$\pm$1$\sigma$ is shown. The positions of the clusters Abell 0426 and Abell
3627 are shown by the symbol “$\times$” (a,b).
A significant dominance ($>$2$\sigma$) of S-mode is noticed in grid 3 (RA: 0∘
to 90∘, Dec: 0∘ to 45∘ (J2000)) (Fig. 3a,a’). An elongated group of galaxies
can be seen in this grid. In this grid, $\Delta$, $\Delta(a\,$sde$)$ and
$\Delta(a)\%$ are found to be 12% $\pm$ 6%, 0.076 and 9.3%, respectively.
These figures suggest that the distribution of S- and Z-mode galaxies in G3 is
not homogeneous. Probably, this is due to the apparent subgroupings or
subclusterings of the galaxies.
The S-mode galaxies dominate in the grids G8 and G9 (Fig. 3a’). However, the
statistics is poor ($<$ 40) in these grids (Table 2). In addition, no
groupings or subclustering are noticed.
A dominance ($\sim$1.5$\sigma$) of Z-mode is noticed in G10 (RA: 180∘ to 270∘,
Dec: –45∘ to 0∘ (J2000)) and G13 (RA: 270∘ to 360∘, Dec: –90∘ to –45∘ (J2000))
(Fig. 3a,a’). In both the grids, a large aggregation of the galaxies can be
seen. A subcluster-like aggregation can be seen in G10. An elongated structure
can be seen in G13. In both the grids, $\Delta$ value is found to be greater
than 5% (Table 2).
No dominance of either S- and Z-mode galaxies is noticed in the groups G1, G2,
G4, G5, G6, G7, G11, G12, G14, G15 and G16. Thus, homogeneous distribution of
S- and Z-mode galaxies is found intact in $\sim$ 80% area of the sky. We
suspect that the groupings or subclusterings of the galaxies lead the
preference structural modes (S or Z) in G3, G10 and G13.
#### 4.1.4 Galaxy groups
In all-sky map, several groups of galaxies can be seen (Fig. 3a). It is
interesting to study the distribution of structural modes (S or Z) of galaxies
in these groups. For this, we systematically searched for the groups
fulfilling following selection criteria: (a) major diameter $>$ 30∘, (b)
cutoff diameter $<$ 2 times the background galaxies, (c) number of galaxies
$>$ 50\. We found 6 groups fulfilling these criteria (Fig. 3b). All 6 groups
(Gr) are inspected carefully. In 3 groups (Gr2, Gr5 and Gr6), subgroups can be
seen. The number of galaxies in the groups Gr2 and Gr5 are found more than
100.
The clusters Abell 0426 and Abell 3627 are located close to the groups Gr2 and
Gr6. The symbol “$\times$” represents the position of the cluster center in
Fig. 3b. The mean radial velocities of these clusters are 5 366 km s-1 and 4
881 km s-1, respectively. However, we have removed the member galaxies of
these clusters from our database.
A significant dominance ($>$2$\sigma$) of S-mode galaxies is noticed in the
group Gr2 (Fig. 3b,b’). The $\Delta$, $\Delta(a\,$sde$)$ and $\Delta(a)\%$
values are found to be 18.6% ($\pm$9.4%), 0.097 and 12.6%, respectively (Table
2). We suspect that the galaxies in this group is under the influence of the
cluster Abell 0426, due to which apparent subclustering of the galaxies is
seen.
The galaxies in Gr5 shows an opposite preference: a significant dominance of
the Z-mode galaxies ($>$2$\sigma$) (Fig. 3b,b’). In this group, $\Delta$,
$\Delta(a\,sde)$ and $\Delta(a)\%$ are found to be 11.5% $\pm$ 5.7%, 0.089 and
11.2%, suggesting inhomogeneous distribution of structural modes (Table 2).
No humps or dips can be seen in the groups Gr1, Gr3, Gr4 and Gr6 (Fig.
3.2b,b’). Thus, the distribution of S- and Z-mode galaxies in these groups are
found to be homogeneous. The number of galaxies in these groups are less than
100.
In the group 6, we could not notice the influence of the cluster Abell 3627.
This might be due to the off location of the cluster center from the group
center.
### 4.2 Anisotropy in the position angle distribution
Figure 4: The equatorial position angle (PA) distribution of total Z- and
S-mode galaxies plotted in 9 (a) and 18 (b) bins. The solid and the dashed
line represent the expected isotropic distribution for S- and Z-mode galaxies,
respectively. The observed counts with statistical $\pm$1$\sigma$ error bars
are shown. PA = 90∘$\pm$45∘ (grey-shaded region) corresponds to the galactic
rotation axes tend to be oriented perpendicular with respect to the equatorial
plane.
We study the equatorial position angle (PA) distribution of S- and Z-mode
galaxies in the total sample and subsamples. A spatially isotropic
distribution is assumed in order to examine non-random effects in the PA-
distribution. In order to discriminate the deviation from the randomness, we
use three statistical tests: chi-square, auto correlation and the Fourier. The
bin size was chosen to be 20∘ (9 bins) in all these tests. The statistically
poor bins (number of solution $<$ 5) are omitted in the analysis. The
conditions for anisotropy are the following: the chi-square probability
P($>\chi^{2}$) $<$ 0.050, correlation coefficient $C$/$\sigma(C)$ $>$ 1, first
order Fourier coefficient $\Delta_{11}$/$\sigma(\Delta_{11}$) $>$ 1 and the
first order Fourier probability P($>\Delta_{1}$)$<$0.150 as used by Godlowski
(1993). Table 3 lists the statistical parameters for the total samples and
subsamples.
In the Fourier test, $\Delta_{11}$ $<$ 0 (i.e., negative) indicates an excess
of galaxies with the galactic planes parallel to the equatorial plane. In
other words, a negative $\Delta_{11}$ suggests that the rotation axis of
galaxies tend to be oriented perpendicular with respect to the equatorial
plane. Similarly, $\Delta_{11}$ $>$ 0 (i.e., positive) indicates that the
rotation axis of galaxies tend to lie in the equatorial plane.
In the histograms (see Figs. 4-7), a hump at 90∘$\pm$45∘ (grey-shaded region)
suggests that the galactic planes of galaxies tend to lie in the equatorial
plane. In other words, the rotation axes of galaxies tend to be oriented
perpendicular with respect to the equatorial plane when there is excess number
of solutions in the grey-shaded region in the histogram.
All three statistical tests show isotropy in the total S-mode galaxies. Thus,
no preferred alignment is noticed for the total S-mode galaxies (solid circles
in Fig. 4a). Interestingly, all three statistical tests show anisotropy in the
total Z-mode galaxies. The chi-square and Fourier probabilities
(P$(>\chi^{2})$, P($>\Delta_{1}$)) are found 1.5% ($<$ 5% limit) and 8.5% ($<$
15% limit), respectively (Table 2). The auto correlation coefficient
(C/C($\sigma$)) turned –3.2 ($>>$1). The $\Delta_{11}$/$\sigma(\Delta_{11}$)
value is found to be negative at $\sim$ 2$\sigma$ level, suggesting that the
rotation axes of Z-mode galaxies tend to be oriented perpendicular the
equatorial plane. Three humps at 50∘ ($>$1.5$\sigma$), 90∘ ($>$2$\sigma$) and
130∘ (1.5$\sigma$) support this result (Fig. 4a). We checked the biasness in
the results due to bin size by increasing and decreasing the number of bins. A
similar statistical result is found for both structural modes. Fig. 4b shows
the PA-distribution histogram for the total sample in 18 bins. The Z-mode
galaxies show three significant humps in the grey-shaded region, supporting
the results mentioned above.
Thus, we conclude isotropy for S-mode whereas anisotropy for Z-mode galaxies
in the total sample.
Table 3: Statistics of the PA-distribution of galaxies in the total sample and
subsamples (first column). The second, third, fourth and fifth columns give
the chi-square probability (P$(>\chi^{2})$), correlation coefficient
(C/C($\sigma$)), first order Fourier coefficient
($\Delta_{11}$/$\sigma$($\Delta_{11}$)), and first order Fourier probability
P($>\Delta_{1}$), respectively. The last four columns repeats the previous
columns. $\begin{array}[]{p{0.1\linewidth}ccccccccc}\hline\cr\vskip 3.0pt plus
1.0pt minus 1.0pt\cr sample&&$S-mode$&&&&$Z-mode$&&\\\ \vskip 3.0pt plus 1.0pt
minus
1.0pt\cr&$P$(>\chi^{2})$$&$C/C($\sigma$)$&$$\Delta_{11}$/${\sigma}$($\Delta_{11}$)$&$P(${>}\Delta_{1}$)$&$P$(>\chi^{2})$$&$C/C($\sigma$)$&$$\Delta_{11}$/${\sigma}$($\Delta_{11}$)$&$P(${>}\Delta_{1}$)$\\\
\hline\cr\vskip 3.0pt plus 1.0pt minus 1.0pt\cr
total&0.666&+0.0&-0.9&0.381&0.015&-3.2&-1.9&0.085\\\
S&0.511&-0.7&-1.2&0.434&0.225&+0.4&+0.8&0.383\\\
SE&0.973&+0.1&-0.9&0.569&0.031&+2.0&+2.8&0.015\\\
SL&0.234&+0.5&+0.8&0.209&0.460&-0.1&-0.5&0.345\\\
SB&0.729&+0.3&+1.0&0.454&0.285&-1.0&-0.2&0.497\\\
SBE&0.739&+0.1&-0.5&0.566&0.230&-0.7&+0.1&0.521\\\
SBL&0.043&+1.8&+1.7&0.046&0.620&-0.9&-0.2&0.872\\\
RV1&0.910&+0.3&+0.8&0.362&0.369&-0.9&-0.6&0.285\\\
RV2&0.790&+0.3&-1.0&0.496&0.925&-0.4&-0.2&0.887\\\
RV3&0.050&+1.6&-1.5&0.083&0.033&-1.8&-2.3&0.046\\\
RV4&0.043&-2.3&-1.5&0.116&0.636&+0.2&-0.7&0.692\\\
Gr2&0.455&+0.6&+0.8&0.861&0.033&-1.8&+1.7&0.116\\\
Gr5&0.033&-1.4&-2.0&0.085&0.516&+0.4&-0.4&0.548\\\ \vskip 3.0pt plus 1.0pt
minus 1.0pt\cr\hline\cr\end{array}$
#### 4.2.1 Morphology
In the spirals, the chi-square and auto correlation tests show isotropy for
both the S- and Z-modes. The first order Fourier probability is found greater
than 35%, suggesting no preferred alignment. However, the $\Delta_{11}$ value
exceeds 1$\sigma$ limit (–1.2$\sigma$) in the S-mode spirals. A hump at 90∘ is
not enough to turn the $\Delta_{11}$/$\sigma(\Delta_{11}$) $>$ 1.5 (Fig. 5a).
Similarly, a hump at 150∘ is not enough to make the
$\Delta_{11}$/$\sigma(\Delta_{11}$) $>$ 1.5 in the Z-mode spirals. Hence, the
preferred alignment is not profounded in both the S- and Z-mode spirals. Thus,
we conclude a random orientation of S- and Z-mode spirals.
Figure 5: The equatorial PA-distribution of Z- and S-mode galaxies in the
spirals (a), early-type spirals (b), late-type spirals (c), barred spirals
(d), early-type barred spirals (e) and late-type barred spirals (f). The
symbols, error bars, dashed lines and the explanations are analogous to Fig.
4.
Early- and late-type S-mode spirals show isotropy in all three statistical
tests (Table 3). No humps and the dips are seen in the histograms (solid
circles in Fig. 5b,c). Thus, the S-mode spirals show a random alignment in the
PA-distribution. In the subsample SE, all three statistical tests show
anisotropy (Table 2). Two significant humps at $>$ 150∘ cause the first order
Fourier coefficient ($\Delta_{11}$) $>$ +2.5$\sigma$ (hollow circle in Fig.
5b). Thus, a preferred alignment is noticed in the early-type Z-mode spirals:
the galactic rotation axes tend to lie in the equatorial plane. The late-type
Z-mode spirals show a random alignment.
The spiral barred galaxies show a random alignment in both the S- and Z-modes.
In Fig. 5d, no deviation from the expected distribution can be seen. All three
statistical tests support this result (Table 2). A similar result is found for
the early-type SB galaxies in both structural modes (Table 3, Fig. 5e).
The P$(>\chi^{2})$ and P($>\Delta_{1}$) are found less than 5%, suggesting a
preferred alignment for the late-type S-mode SB galaxies (Table 3). The auto
correlation coefficient (C/C($\sigma$)) and the hump at $>$ 150∘ support this
result (Fig. 5f). The $\Delta_{11}$/$\sigma(\Delta_{11}$) is found to be
positive at 1.7$\sigma$ level, suggesting that the S-mode SBL galaxies tend to
lie in the equatorial plane. Thus, the late-type S- and Z-mode SB galaxies
show preferred and random alignments, respectively.
#### 4.2.2 Radial velocity
The subsamples RV1 and RV2 show isotropy in all three statistical tests (Table
2). No humps or dips can be seen in Figs. 6a,b. Thus, the galaxies having
radial velocity in the range 3 000 km s-1 to 4 000 km s-1 show a random
alignment for both the S- and Z-mode galaxies.
The humps at 90∘ ($>$2$\sigma$) and 110∘ ($>$2$\sigma$) are found in the Z-
and S-mode RV3 galaxies, respectively (Fig. 6c). These two significant humps
lead the subsample to show anisotropy in the statistical tests (Table 2). The
$\Delta_{11}$ values are found negative at $\geq$1.5 level, suggesting a
similar preferred alignment for both modes: the galaxy rotation axes tend to
be directed perpendicular to the equatorial plane.
Figure 6: The equatorial PA-distribution of Z- and S-mode galaxies in RV1 (a),
RV2 (b), RV3 (c) and RV4 (d). The abbreviations are listed in Table 1. The
symbols, error bars, dashed lines and the explanations are analogous to Fig.
4.
A hump at 70∘ ($>$1.5$\sigma$) and a dip at 150∘ ($\sim$2$\sigma$) cause the
S-mode RV4 galaxies to show anisotropy in all three statistical tests (Fig.
6d). Thus, the S-mode galaxies having radial velocity in the range 4 500 km
s-1 to 5 000 km s-1 show a similar preferred alignments as shown by the
subsample RV3: galactic planes of galaxies tend to lie in the equatorial
plane. The leading arm galaxies in the subsample RV4 show a random alignment
(Table 3, Fig. 6d).
#### 4.2.3 Groups
We do not study PA-distribution of S- and Z-mode galaxies in the groups Gr1,
Gr3, Gr4 and Gr6 because of poor statistics (number $<$ 50).
We study the PA-distribution of S- and Z-mode galaxies in the groups Gr2 and
Gr5, where the dominance of either Z- or S-mode is noticed. In addition, the
statistics is relatively better in these groups.
Figure 7: The equatorial PA-distribution of Z- and S-mode galaxies in the
groups Gr2 and Gr5. The abbreviations are listed in Table 1. The symbols,
error bars, dashed lines and the explanations are analogous to Fig. 4.
In the group Gr2, Z-mode galaxies dominate the S-mode galaxies. In this group,
the Z-mode galaxies show a preferred alignment whereas S-mode galaxies show a
random alignment in the PA-distribution. All three statistical tests suggest
anisotropy in the Z-mode galaxies (Table 3). The humps at $>$ 150∘ cause the
$\Delta_{11}$ value to be positive at $>$ 1.5$\sigma$ level (Fig. 7a),
suggesting that the rotation axes of Z-mode galaxies in Gr2 tend to be
oriented parallel the equatorial plane.
The S-mode galaxies dominate in the group Gr5. Interestingly, a preferred
alignment of S-mode galaxies is noticed in the PA-distribution. In Fig. 7b,
two significant humps at 90∘ ($\sim$2$\sigma$) and 110∘ ($>$2$\sigma$) can be
seen. These humps lead the subsample (S-mode Gr5) to show anisotropy in the
statistical tests (Table 3). No preferred alignment is noticed in the Z-mode
galaxies of this group.
Thus, the dominating structural modes (Z or S) show a preferred alignment in
the PA-distribution. This is noticeable result.
### 4.3 Discussion
Fig. 8 shows a comparison between the number ($\Delta$) and position angle
($\Delta_{11}$/$\sigma(\Delta_{11}$)) distribution of S- and Z-mode galaxies
in the total sample and subsamples. This plot deals the correlation between
the homogeneity in the structural modes and the random alignment in the
subsamples. The grey-shaded region represents the region of isotropy and
homogeneity for the $\Delta_{11}$/$\sigma(\Delta_{11}$) and $\Delta$(%),
respectively.
Figure 8: A comparison between the number ($\Delta$%) and the position angle
($\Delta_{11}$/$\sigma(\Delta_{11}$)) distribution of Z- and S-mode galaxies
in the total sample and subsamples.
Twenty five (out of 39, 64%) subsamples lie in the grey-shaded region (Fig.
8a), suggesting a good agreement between the homogeneous distribution of S-
and Z-mode galaxies and the random alignment of the rotation axes of galaxies.
In four subsamples (SE, SBL, Gr2 and Gr5), a good correlation between the
preferred alignment and the dominance of either S- or Z-mode galaxies is
noticed (Fig. 8a). Thus, it is noticed that the random alignment of the PAs of
galaxies hint the existence of inhomogeneity in the structural modes.
Aryal & Saurer (2006) and Aryal, Paudel & Saurer (2007) studied the spatial
orientation of galaxies in 32 Abell clusters of BM type I, II, II-III and III
and found a significant preferred alignment in the late-type cluster (BM type
II-III, III). They concluded that the randomness decreases systematically in
galaxy alignments from early-type (BM type I, II) to late-type (BM type II-
III, III) clusters.
We noticed a very good correlation between the random alignments and the
homogeneity in the structural modes. Probably, this result reveals the fact
that the progressive loss of homogeneity in the structural modes might have
some connection with the rotationally supported (spirals, barred spirals) to
the randomized (lenticulars, ellipticals) system. Thus, we suspect that the
dynamical processes in the cluster evolution (such as late-type clusters) give
rise to a dynamical loss of homogeneity in the structural modes. It would be
interesting to test this prediction by studying the S- and Z-type spirals in
the late-type clusters in the future.
As 60% of galaxies in the nearby universe are rotationally supported discs,
understanding angular-momentum acquisition is obviously a crucial part of
understanding galaxy evolution. The winding sense of spiral arm patterns
(morphological feature) allows us to infer the orientation of the angular-
momentum vector of the disc galaxy. The expected distribution of spin vectors
of galaxies shows markedly different trends according to the galaxy formation
scenarios. One can suspect the possibility that the actual distribution of
galaxy spin shows a dipole or a quadrupole component depending on the
scenarios of galaxy formation. If galaxy spins were generated according to the
primordial whirl scenario, a strong bias in either the S or Z patterns would
be seen in a face-on sample of galaxies. We did not find this trend in our
sample. A quadrupole distribution of S/Z might be observed if the primary
process was the generation of spins due to the pancake shock scenario or the
explosion scenario. On the other hand, if the galaxy spins were produced by
the tidal spin-up process, there would be no global anisotropy as we noticed
in many cases, unless galaxy-cluster tidal interaction rather than galaxy-
galaxy tidal interaction were the primary process. No significant correlation,
however, was identified in any ensemble.
## 5 Conclusions
We studied the winding sense of 1 621 field galaxies around the Local
Supercluster. These galaxies have radial velocity (RV) in the range 3 000 km
s-1 to 5 000 km s-1. The distribution of Z- and S-mode galaxies is studied in
the total sample and 32 subsamples. To examine non-random effects, the
equatorial position angle (PA) distribution of galaxies in the total sample
and subsamples are studied. In order to discriminate anisotropy from the
isotropy we have performed three statistical tests: chi-square, auto-
correlation and the Fourier. Our results are as follows:
1. 1.
The homogeneous distribution of the total Z- and S-mode galaxies is found,
suggesting the homogeneous distribution of winding sense (S or Z) of galaxies
having RVs 3 000 km s-1 to 5 000 km s-1. The PA-distribution of S-mode
galaxies is found to be random, whereas preferred alignment is noticed for
Z-mode galaxies. It is found that the galactic rotation axes of Z-mode
galaxies tend to be oriented perpendicular the equatorial plane.
2. 2.
Z-mode are found 3.7% ($\pm$1.8%) more than that of the S-mode in the spirals
whereas a significant dominance (17% $\pm$ 8.5%) of S-mode is noticed in the
barred spirals. This difference is found $>$ 8% for the irregulars and the
morphologically unidentified galaxies. A random alignment is noticed in the
PA-distribution of Z- and S-mode spirals. Thus, it is noticed that the random
alignment of the PAs of galaxies lead the existence of inhomogeneity in the
structural modes of galaxies.
3. 3.
The inhomogeneity in the structural modes is found stronger for the late-type
spirals (Sc, Scd, Sd and Sm) than that of early-type (Sa, Sab, Sb and Sbc).
Similar result is found for the late-type barred spirals.
4. 4.
A very good correlation between the number of Z- and S-mode galaxies are found
in the RV subsamples. All 4 subsamples show the $\Delta$ value less than 5%.
Thus, we conclude that the homogeneous distribution of structural modes of
field galaxies remain invariant with the global expansion.
5. 5.
The galaxies having RVs 3 000 km s-1 to 4 000 km s-1 show a random alignment
for both the Z- and S-modes. The rotation axes of Z- and S-mode galaxies
having 4 000 $<$ RV (km s-1) $\leq$ 4 500 tend to be oriented perpendicular
the equatorial plane.
6. 6.
The distribution of the winding sense of galaxies is found homogeneous in
$\sim$ 80% area of the sky. This property is found to be violated in few
groups of galaxies. Two such groups (Gr2 and Gr8) are identified. In these
groups, the structural dominance and the preferred alignments of galaxies are
found to oppose each other.
The true structural mode of a galaxy must involve a determination of which
side of the galaxy is closer to the observer (Binney and Tremaine 1987).
Three-dimensional determination of the leading and the trailing arm patterns
in the galaxies is a very important problem. We intend to address this problem
in the future.
###### Acknowledgements.
We are indebted to the referee for his/her constructive criticism and useful
comments. I acknowledge Profs. R. Weinberger and W. Saurer of Innsbruck
University, Austria for insightful discussions. I am thankful to Tribhuvan
University, Nepal and Innsbruck University, Austria for providing financial
assistance to visit Innsbruck University during Jan-March 2009.
## References
* (1) Abell, G.O., Corwin, H.G., Olowin, R.P. 1989, ApJS, 70, 1
* (2) Aryal, B., Paudel, S., Saurer, W. 2007, MNRAS, 379, 1011
* (3) Aryal, B., Saurer, W. 2006, MNRAS, 366, 438
* (4) Aryal, B., Saurer, W. 2005, A&A, 432, 841
* (5) Bergh, V.D. 1959, Publ. David Dunlop Obs. 2, 145
* (6) Bergh, V.D. 1966, AJ, 71, 922
* (7) Binney J., Tremaine, S. 1987, Galactic Dynamics, Princeton Univ. press, Princeton, New Jersey
* (8) Brandner, W., Grebel, E.K., Chu, Y.-H. et al. 2000, AJ, 119, 292
* (9) Brunzendorf, J., Meusinger H. 1999, A&AS, 139, 141
* (10) Capozziello, S., Lattanzi, A. 2006, Ap&SS, 301, 1-4, 189
* (11) Corwin, H.G., de Vaucouleurs, A., de Vaucouleurs, G. 1985, Univ. Texas Monogr. Astron., 4, 1
* (12) Dreyer, J.L.E. 1895, MNRAS, 51, 185
* (13) Dreyer, J.L.E. 1908, MNRAS, 59, 105
* (14) Doroshkevich, A.G. 1973, ApJ, 14, L11
* (15) de Vaucouleurs, G., de Vaucouleurs, A., Corwin, et al. 1991, Third Reference Catalogue of Bright Galaxies, Springer-Verlag, New York
* (16) Godlowski, W. 1993, MNRAS, 265, 874
* (17) Impey, C.D., Sprayberry, D., Irwin, M.J., Bothun, G.D. 1996, Astrophys. J. Supp., 105, 209
* (18) Kodaira, K., Okamura, S., Ichikawa, S. 1990, Photometric Atlas of Northern Bright Galaxies, Univ. of Tokyo Press, Tokyo
* (19) Lauberts, A. 1982, ESO/Uppsala Survey of the ESO B Atlas, Garching bei Muenchen
* (20) Longo, M. J. 2007, preprint (astro-ph/0703325)
* (21) Land K., Slosar A., Lintott C. et al. 2008, MNRAS, 388, 1686
* (22) MacAlpine, G.M., Smith, S.B., Lewis, D.W. 1977a, ApJS, 34, 95
* (23) MacAlpine, G.M., Smith, S.B., Lewis, D.W. 1977b, ApJS, 35, 197
* (24) MacAlpine, G.M., Lewis, D.W., Smith, S.B. 1977c, ApJS, 35, 203
* (25) MacAlpine, G.M., Lewis, D.W. 1978, ApJS, 36, 587
* (26) MacAlpine, G.M., Williams, G.A. 1981, ApJS, 45, 113
* (27) Markarian, B.E. 1967, Astrofizika, 3, 24
* (28) Markarian, B.E., Lipovetskii, V.A., Stepanian, D.A. 1983, Astrofizika, 19, 29
* (29) Messier, C. 1784, Catalogue des nebuleuses et des amas d’etoiles, Connaissance des Temps
* (30) Nilson, P. 1973, Uppsala General Catalogue of Galaxies, Nova Acta Uppsala University, Ser. V:A, Vol.1
* (31) Nilson, P. 1974, Upps. Astron. Obs. Rep., 5
* (32) Oort, J.H. 1970a, Science, 170, 1363
* (33) Oort, J.H. 1970b, A&A, 7, 405
* (34) Ozernoy, L.M. 1978, in: Longair M.S., Einasto J., eds., The Large Scale Structure of the Universe, Proc. IAU Symp. No. 79. Reidel, Dordrecht, p. 427
* (35) Peebles, P.J.E. 1969, ApJ, 155, 393
* (36) Pasha, I.I. 1985, Sov. Astron. Lett., 11, 1
* (37) Shectman, S.A., Landy, S.D., Oemler, A. et al. 1996, ApJ, 470, 172
* (38) Struble, M.F., Rodd, H.J. 1999, ApJS, 125, 355
* (39) Sugai, H. & Iye, M. 1995, MNRAS, 276, 327
* 1980- (2000) Takase, et al. 1980-2000, Kiso Ultraviolet Galaxy Catalog, Publ. Nat. Astron. Obs. Japan
* (41) Thomasson, M., Donner, K. J., Sundelius, B. et al. 1989, A&A, 211, 25
* 1962- (74) Vorontsov-Vel, Y.B.A., Krasnogorskaya, A.A., Arkhipova, V.P. 1962-74, Morphological Catalogue of Galaxies (Part I-V), Trudy Gosud. Astron. Inst. Shternberga
* 1961- (68) Zwicky, F., Wild, P., Karpowicz, M., Kowal, C.T. 1961-68, The Catalogue of Galaxies and Clusters of Galaxies, California Inst. of Tech., Pasadena
|
arxiv-papers
| 2010-10-27T06:44:56 |
2024-09-04T02:49:14.270039
|
{
"license": "Public Domain",
"authors": "B. Aryal",
"submitter": "Binil Aryal",
"url": "https://arxiv.org/abs/1010.5585"
}
|
1010.5595
|
# The Role of Monotonicity in the Epistemic Analysis of Strategic Games
Krzysztof R. Apt ${}^{\safe@setref t1thankst1{\@nil},\safe@setref
t1thanks\@nil\@@t1{,\safe@setref t1thanks}$}\hbox{${}^{\safe@setref
t2thankst2{\@nil},\safe@setref t2thanks\@nil\@@t2{,\safe@setref
t2thanks}$}andJonathanA.Zvesper\hbox{${}^{\safe@setref
t3thankst3{\@nil},\safe@setref t3thanks\@nil\@@t3{,\safe@setref t3thanks}$}}}$
###### Abstract
It is well-known that in finite strategic games true common belief (or common
knowledge) of rationality implies that the players will choose only strategies
that survive the iterated elimination of strictly dominated strategies. We
establish a general theorem that deals with monotonic rationality notions and
arbitrary strategic games and allows to strengthen the above result to
arbitrary games, other rationality notions, and transfinite iterations of the
elimination process. We also clarify what conclusions one can draw for the
customary dominance notions that are not monotonic. The main tool is Tarski’s
Fixpoint Theorem.
## 1 Introduction
### 1.1 Contributions
In this paper we provide an epistemic analysis of arbitrary strategic games
based on possibility correspondences. We prove a general result that is
concerned with monotonic program properties111The concept of a monotonic
property is introduced in Section 2. used by the players to select optimal
strategies.
More specifically, given a belief model for the initial strategic game, denote
by $\textbf{RAT}(\phi)$ the property that each player $i$ uses a property
$\phi_{i}$ to select his strategy (‘each player $i$ is $\phi_{i}$-rational’).
We establish in Section 3 the following general result:
Assume that each property $\phi_{i}$ is monotonic. The set of joint strategies
that the players choose in the states in which $\textbf{RAT}(\phi)$ is a true
common belief is included in the set of joint strategies that remain after the
iterated elimination of the strategies that for player $i$ are not
$\phi_{i}$-optimal.
In general, transfinite iterations of the strategy elimination are possible.
For some belief models the inclusion can be reversed.
This general result covers the usual notion of rationalizability in finite
games and a ‘global’ version of the iterated elimination of strictly dominated
strategies used in [17] and studied for arbitrary games in [11]. It does not
hold for the ‘global’ version of the iterated elimination of weakly dominated
strategies. For the customary, ‘local’ version of the iterated elimination of
strictly dominated strategies we justify in Section 4 the statement
> _true common belief (or common knowledge) of rationality implies that the
> players will choose only strategies that survive the iterated elimination of
> strictly dominated strategies_
for arbitrary games and transfinite iterations of the elimination process.
Rationality refers here to the concept studied in [6]. We also show that the
above general result yields a simple proof of the well-known version of the
above result for finite games and strict dominance by a mixed strategy.
The customary, local, version of strict dominance is non-monotonic, so the use
of monotonic properties has allowed us to provide epistemic foundations for a
non-monotonic property. However, weak dominance, another non-monotonic
property, remains beyond the reach of this approach. In fact, we show that in
the above statement we cannot replace strict dominance by weak dominance. A
mathematical reason is that its global version is also non-monotonic, in
contrast to strict dominance, the global version of which is monotonic. To
provide epistemic foundations of weak dominance the only currently known
approaches are [10] based on lexicographic probability systems and [12] based
on a version of the ‘all I know’ modality.
### 1.2 Connections
The relevance of monotonicity in the context of epistemic analysis of finite
strategic games has already been pointed out in [23]. The distinction between
local and global properties is from [2] and [3].
To show that for some belief models an equality holds between the set of joint
strategies chosen in the states in which $\textbf{RAT}({\phi})$ is true common
belief and the set of joint strategies that remain after the iterated
elimination of the strategies that for player $i$ are not $\phi_{i}$-rational
requires use of transfinite ordinals. This complements the findings of [14] in
which transfinite ordinals are used in a study of limited rationality, and
[15], where a two-player game is constructed for which the $\omega_{0}$ (the
first infinite ordinal) and $\omega_{0}+1$ iterations of the rationalizability
operator of [6] differ.
In turn, [13] show that arbitrary ordinals are necessary in the epistemic
analysis of arbitrary strategic games based on partition spaces. Further, as
shown in [11], the global version of the iterated elimination of strictly
dominated strategies, when used for arbitrary games, also requires transfinite
iterations of the underlying operator.
Finally, [16] invokes Tarski’s Fixpoint Theorem, in the context of what the
author calls “general systems”, and uses this to prove that the set of
rationalizable strategies in a finite non-cooperative game is the largest
fixpoint of a certain operator. That operator coincides with the global
version of the elimination of never-best-responses.
Some of the results presented here were initially reported in a different
presentation, in [1].
## 2 Preliminaries
### 2.1 Strategic Games
Given $n$ players ($n>1$) by a _strategic game_ (in short, a _game_) we mean a
sequence $(S_{1},\mbox{$\ldots$},S_{n},p_{1},\mbox{$\ldots$},p_{n}),$ where
for all $i\in\\{1,\mbox{$\ldots$},n\\}$
* •
$S_{i}$ is the non-empty set of _strategies_ available to player $i$,
* •
$p_{i}$ is the _payoff function_ for the player $i$, so
$p_{i}:S_{1}\times\mbox{$\ldots$}\times S_{n}\mbox{$\>\rightarrow\>$}\cal{R},$
where $\cal{R}$ is the set of real numbers.
We denote the strategies of player $i$ by $s_{i}$, possibly with some
superscripts. We call the elements of $S_{1}\times\mbox{$\ldots$}\times S_{n}$
_joint strategies_. Given a joint strategy $s$ we denote the $i$th element of
$s$ by $s_{i}$, write sometimes $s$ as $(s_{i},s_{-i})$, and use the following
standard notation:
* •
$s_{-i}:=(s_{1},\mbox{$\ldots$},s_{i-1},s_{i+1},\mbox{$\ldots$},s_{n})$,
* •
$S_{-i}:=S_{1}\times\mbox{$\ldots$}\times S_{i-1}\times
S_{i+1}\times\mbox{$\ldots$}\times S_{n}$.
Given a finite non-empty set $A$ we denote by $\Delta A$ the set of
probability distributions over $A$ and call any element of $\Delta S_{i}$ a
_mixed strategy_ of player $i$.
In the remainder of the paper we assume an initial strategic game
$H:=(H_{1},\mbox{$\ldots$},H_{n},p_{1},\mbox{$\ldots$},p_{n})$
A _restriction_ of $H$ is a sequence $(G_{1},\mbox{$\ldots$},G_{n})$ such that
$G_{i}\mbox{$\>\subseteq\>$}H_{i}$ for all $i\in\\{1,\mbox{$\ldots$},n\\}$.
Some of $G_{i}$s can be the empty set. We identify the restriction
$(H_{1},\mbox{$\ldots$},H_{n})$ with $H$. We shall focus on the complete
lattice that consists of the set of all restrictions of the game $H$ ordered
by the componentwise set inclusion:
$(G_{1},\mbox{$\ldots$},G_{n})\mbox{$\>\subseteq\>$}(G^{\prime}_{1},\mbox{$\ldots$},G^{\prime}_{n})$
iff $G_{i}\mbox{$\>\subseteq\>$}G^{\prime}_{i}$ for all
$i\in\\{1,\mbox{$\ldots$},n\\}$
So in this lattice $H$ is the largest element in this lattice.
### 2.2 Possibility Correspondences
In this and the next subsection we essentially follow the survey of [5]. Fix a
non-empty set $\Omega$ of _states_. By an _event_ we mean a subset of
$\Omega$.
A _possibility correspondence_ is a mapping from $\Omega$ to the powerset
${\cal P}(\Omega)$ of $\Omega$. We consider three properties of a possibility
correspondence $P$:
1. (i)
for all $\omega$, $P(\omega)\neq\mbox{$\emptyset$}$,
2. (ii)
for all $\omega$ and $\omega^{\prime}$, $\omega^{\prime}\in P(\omega)$ implies
$P(\omega^{\prime})=P(\omega)$,
3. (iii)
for all $\omega$, $\omega\in P(\omega)$.
If the possibility correspondence satisfies properties (i) and (ii), we call
it a _belief correspondence_ and if it satisfies properties (i)–(iii), we call
it a _knowledge correspondence_.222Note that the notion of a belief has two
meanings in the literature on epistemic analysis of strategic games, so also
in this paper. From the context it is always clear which notion is used. In
the modal logic terminology a belief correspondence is a frame for the modal
logic KD45 and a knowledge correspondence is a frame for the modal logic S5,
see, e.g. [7]. Note that each knowledge correspondence $P$ yields a partition
$\\{P(\omega)\mid\omega\in\Omega\\}$ of $\Omega$.
Assume now that each player $i$ has at its disposal a possibility
correspondence $P_{i}$. Fix an event $E$. We define
$\square
E:=\square^{1}E:=\\{\omega\in\Omega\mid\mbox{$\forall$}i\in\\{1,\mbox{$\ldots$},n\\}\>P_{i}(\omega)\mbox{$\>\subseteq\>$}E\\}$
by induction on $k\geq 1$
$\square^{k+1}E:=\square\square^{k}E$
and finally
$\square^{*}E:=\bigcap_{k=1}^{\infty}\square^{k}E$
If all $P_{i}$s are belief correspondences, we usually write $B$ instead of
$\square$ and if all $P_{i}$s are knowledge correspondences, we usually write
$K$ instead of $\square$. When $\omega\in B^{*}E$, we say that the event $E$
is _common belief in the state $\omega$_ and when $\omega\in K^{*}E$, we say
that the event $E$ is _common knowledge in the state $\omega$_.
An event $F$ is called _evident_ if $F\mbox{$\>\subseteq\>$}\square F$. That
is, $F$ is evident if for all $\omega\in F$ we have
$P_{i}(\omega)\mbox{$\>\subseteq\>$}F$ for all
$i\in\\{1,\mbox{$\ldots$},n\\}$. In what follows we shall use the following
alternative characterizations of common belief and common knowledge based on
evident events:
$\begin{array}[]{l}\mbox{$\omega\in\square^{*}E$ iff for some evident event
$F$ we have $\omega\in F\mbox{$\>\subseteq\>$}\square E$}\end{array}$ (1)
where $\square=B$ or $\square=K$ (see [18], respectively Proposition 4 on page
180 and Proposition on page 174), and
$\omega\in K^{*}E$ iff for some evident event $F$ we have $\omega\in
F\mbox{$\>\subseteq\>$}E$ (2)
([4], page 1237).
### 2.3 Models for Games
We now relate these considerations to strategic games. Given a restriction
$G:=(G_{1},\mbox{$\ldots$},G_{n})$ of the initial game $H$, by a _model_ for
$G$ we mean a set of states $\Omega$ together with a sequence of functions
$\overline{s_{i}}:\Omega\mbox{$\>\rightarrow\>$}G_{i}$, where
$i\in\\{1,\mbox{$\ldots$},n\\}$. We denote it by
$(\Omega,\overline{s_{1}},\mbox{$\ldots$},\overline{s_{n}})$.
In what follows, given a function $f$ and a subset $E$ of its domain, we
denote by $f(E)$ the range of $f$ on $E$ and by $f\\!\mid\\!{E}$ the
restriction of $f$ to $E$.
By the _standard model_ ${\cal M}$ for $G$ we mean the model in which
* •
$\Omega:=G_{1}\times\mbox{$\ldots$}\times G_{n}$
* •
$\overline{s_{i}}(\omega):=\omega_{i}$, where
$\omega=(\omega_{1},\mbox{$\ldots$},\omega_{n})$
So the states of the standard model for $G$ are exactly the joint strategies
in $G$, and each $\overline{s_{i}}$ is a projection function. Since the
initial game $H$ is given, we know the payoff functions
$p_{1},\mbox{$\ldots$},p_{n}$. So in the context of $H$ the standard model is
an alternative way of representing a restriction of $H$.
Given a (not necessarily standard) model ${\cal
M}:=(\Omega,\overline{s_{1}},\mbox{$\ldots$},\overline{s_{n}})$ for a
restriction $G$ and a sequence of events
$\overline{E}=(E_{1},\mbox{$\ldots$},E_{n})$ in ${\cal M}$ (_i.e._ , of
subsets of $\Omega$) we define
$G_{\overline{E}}:=(\overline{s_{1}}(E_{1}),\mbox{$\ldots$},\overline{s_{n}}(E_{n}))$
and call it the _restriction of $G$ to $\overline{E}$_. When each $E_{i}$
equals $E$ we write $G_{E}$ instead of $G_{\overline{E}}$.
Finally, we extend the notion of a model for a restriction $G$ to a _belief
model_ for $G$ by assuming that each player $i$ has a belief correspondence
$P_{i}$ on $\Omega$. If each $P_{i}$ is a knowledge correspondence, we refer
then to a _knowledge model_. We write each belief model as
$(\Omega,\overline{s_{1}},\mbox{$\ldots$},\overline{s_{n}},P_{1},\mbox{$\ldots$},P_{n})$
### 2.4 Operators
Consider a fixed complete lattice $(D,\mbox{$\>\subseteq\>$})$ with the
largest element $\top$. In what follows we use ordinals and denote them by
$\alpha,\beta,\gamma$. Given a, possibly transfinite, sequence
$(G_{\alpha})_{\alpha<\gamma}$ of elements of $D$ we denote their join and
meet respectively by $\bigcup_{\alpha<\gamma}G_{\alpha}$ and
$\bigcap_{\alpha<\gamma}G_{\alpha}$.
Let $T$ be an operator on $(D,\mbox{$\>\subseteq\>$})$, _i.e._ ,
$T:D\mbox{$\>\rightarrow\>$}D$.
* •
We call $T$ _monotonic_ if for all $G,G^{\prime}$,
$G\mbox{$\>\subseteq\>$}G^{\prime}$ implies
$T(G)\mbox{$\>\subseteq\>$}T(G^{\prime})$, and _contracting_ if for all $G$,
$T(G)\mbox{$\>\subseteq\>$}G$.
* •
We say that an element $G$ is a _fixpoint_ of $T$ if $G=T(G)$ and a _post-
fixpoint_ of $T$ if $G\mbox{$\>\subseteq\>$}T(G)$.
* •
We define by transfinite induction a sequence of elements $T^{\alpha}$ of $D$,
where $\alpha$ is an ordinal, as follows:
* –
$T^{0}:=\top$,
* –
$T^{\alpha+1}:=T(T^{\alpha})$,
* –
for all limit ordinals $\beta$, $T^{\beta}:=\bigcap_{\alpha<\beta}T^{\alpha}$.
* •
We call the least $\alpha$ such that $T^{\alpha+1}=T^{\alpha}$ the _closure
ordinal_ of $T$ and denote it by $\alpha_{T}$. We call then $T^{\alpha_{T}}$
the _outcome of_ (iterating) $T$ and write it alternatively as $T^{\infty}$.
So an outcome is a fixpoint reached by a transfinite iteration that starts
with the largest element. In general, the outcome of an operator does not need
to exist but we have the following classic result due to [22].333We use here
its ‘dual’ version in which the iterations start at the largest and not at the
least element of a complete lattice.
Tarski’s Fixpoint Theorem Every monotonic operator $T$ on
$(D,\mbox{$\>\subseteq\>$})$ has an outcome, _i.e._ , $T^{\infty}$ is well-
defined. Moreover,
$T^{\infty}=\nu T=\cup\\{G\mid G\mbox{$\>\subseteq\>$}T(G)\\}$
where $\nu T$ is the largest fixpoint of $T$.
In contrast, a contracting operator does not need to have a largest fixpoint.
But we have the following obvious observation.
###### Note 1
Every contracting operator $T$ on $(D,\mbox{$\>\subseteq\>$})$ has an outcome,
_i.e._ , $T^{\infty}$ is well-defined. $\Box$
In Section 4 we shall need the following lemma, that modifies the
corresponding lemma from [3] from finite to arbitrary complete lattices.
###### Lemma 1
Consider two operators $T_{1}$ and $T_{2}$ on $(D,\mbox{$\>\subseteq\>$})$
such that
* •
for all $G$, $T_{1}(G)\mbox{$\>\subseteq\>$}T_{2}(G)$,
* •
$T_{1}$ is monotonic,
* •
$T_{2}$ is contracting.
Then $T_{1}^{\infty}\mbox{$\>\subseteq\>$}T_{2}^{\infty}$.
Proof. We first prove by transfinite induction that for all $\alpha$
$T_{1}^{\alpha}\mbox{$\>\subseteq\>$}T_{2}^{\alpha}$ (3)
By the definition of the iterations we only need to consider the induction
step for a successor ordinal. So suppose the claim holds for some $\alpha$.
Then by the first two assumptions and the induction hypothesis we have the
following string of inclusions and equalities:
$T_{1}^{\alpha+1}=T_{1}(T_{1}^{\alpha})\mbox{$\>\subseteq\>$}T_{1}(T_{2}^{\alpha})\mbox{$\>\subseteq\>$}T_{2}(T_{2}^{\alpha})=T_{2}^{\alpha+1}$
This shows that for all $\alpha$ (3) holds. By Tarski’s Fixpoint Theorem and
Note 1 the outcomes of $T_{1}$ and $T_{2}$ exist, which implies the claim.
$\Box$
### 2.5 Iterated Elimination of Non-Rational Strategies
In this paper we are interested in analyzing situations in which each player
pursues his own notion of rationality and this information is common knowledge
or true common belief. As a special case we cover then the usually analyzed
situation in which all players use the same notion of rationality.
Given player $i$ in the initial strategic game
$H:=(H_{1},\mbox{$\ldots$},H_{n},p_{1},\mbox{$\ldots$},p_{n})$ we formalize
his notion of rationality using an _optimality property_
$\phi(s_{i},G_{i},G_{-i})$ that holds between a strategy $s_{i}\in H_{i}$, a
set $G_{i}$ of strategies of player $i$ and a set $G_{-i}$ of joint strategies
of his opponents. Intuitively, $\phi_{i}(s_{i},G_{i},G_{-i})$ holds if $s_{i}$
is an ‘optimal’ strategy for player $i$ within the restriction
$G:=(G_{i},G_{-i})$, assuming that he uses the property $\phi_{i}$ to select
optimal strategies. In Section 4 we shall provide several natural examples of
such properties.
We say that the property $\phi_{i}$ used by player $i$ is _monotonic_ if for
all $G_{-i},G^{\prime}_{-i}\mbox{$\>\subseteq\>$}H_{-i}$ and $s_{i}\in H_{i}$
$G_{-i}\mbox{$\>\subseteq\>$}G^{\prime}_{-i}$ and $\phi(s_{i},H_{i},G_{-i})$
imply $\phi(s_{i},H_{i},G^{\prime}_{-i})$
So monotonicity refers to the situation in which the set of strategies of
player $i$ is set to $H_{i}$ and the set of joint strategies of player $i$’s
opponents is increased.
Each sequence of properties $\phi:=(\phi_{1},\mbox{$\ldots$},\phi_{n})$
determines an operator $T_{\phi}$ on the restrictions of $H$ defined by
$T_{\phi}(G):=G^{\prime}$
where $G:=(G_{1},\mbox{$\ldots$},G_{n})$,
$G^{\prime}:=(G^{\prime}_{1},\mbox{$\ldots$},G^{\prime}_{n})$, and for all
$i\in\\{1,\mbox{$\ldots$},n\\}$
$G^{\prime}_{i}:=\\{s_{i}\in G_{i}\mid\phi_{i}(s_{i},H_{i},G_{-i})\\}$
Note that in defining the set of strategies $G^{\prime}_{i}$ we use in the
second argument of $\phi_{i}$ the set $H_{i}$ of player’s $i$ strategies in
the _initial_ game $H$ and not in the _current_ restriction $G$. This captures
the idea that at every stage of the elimination process player $i$ analyzes
the status of each strategy in the context of his initial set of strategies.
Since $T_{\phi}$ is contracting, by Note 1 it has an outcome, _i.e._ ,
$T_{\phi}^{\infty}$ is well-defined. Moreover, if each $\phi_{i}$ is
monotonic, then $T_{\phi}$ is monotonic and by Tarski’s Fixpoint Theorem its
largest fixpoint $\nu T_{\phi}$ exists and equals $T_{\phi}^{\infty}$.
Finally, $G$ is a fixpoint of $T_{\phi}$ iff for all
$i\in\\{1,\mbox{$\ldots$},n\\}$ and all $s_{i}\in G_{i}$,
$\phi_{i}(s_{i},H_{i},G_{-i})$ holds.
Intuitively, $T_{\phi}(G)$ is the result of removing from $G$ all strategies
that are not $\phi_{i}$-rational. So the outcome of $T_{\phi}$ is the result
of the iterated elimination of strategies that for player $i$ are not
$\phi_{i}$-rational.
## 3 Two Theorems
We now assume that each player $i$ employs some property $\phi_{i}$ to select
his strategies, and we analyze the situation in which this information is true
common belief or common knowledge. To determine which strategies are then
selected by the players we shall use the $T_{\phi}$ operator.
We begin by fixing a belief model
$(\Omega,\overline{s_{1}},\mbox{$\ldots$},\overline{s_{n}},P_{1},\mbox{$\ldots$},P_{n})$
for the initial game $H$. Given an optimality property $\phi_{i}$ of player
$i$ we say that player $i$ is $\phi_{i}$-_rational in the state_ $\omega$ if
$\phi_{i}(\overline{s_{i}}(\omega),H_{i},(G_{P_{i}(\omega)})_{-i})$ holds.
Note that when player $i$ believes (respectively, knows) that the state is in
$P_{i}(\omega)$, the set $(G_{P_{i}(\omega)})_{-i}$ represents his belief
(respectively, his knowledge) about other players’ strategies. That is,
$(H_{i},(G_{P_{i}(\omega)})_{-i})$ is the restriction he believes
(respectively, knows) to be relevant to his choice.
Hence $\phi_{i}(\overline{s_{i}}(\omega),H_{i},(G_{P_{i}(\omega)})_{-i})$
captures the idea that if player $i$ uses $\phi_{i}$ to select his strategy in
the game he considers relevant, then in the state $\omega$ he indeed acts
‘rationally’.
To reason about common knowledge and true common belief we introduce the event
$\textbf{RAT}({\phi}):=\\{\omega\in\Omega\mid$ each player $i$ is
$\phi_{i}$-rational in $\omega$}
and consider the following two events constructed out of it:
$K^{*}\textbf{RAT}({\phi})$ and $\textbf{RAT}({\phi})\cap
B^{*}\textbf{RAT}({\phi})$. We then focus on the corresponding restrictions
$G_{K^{*}\textbf{RAT}({\phi})}$ and $G_{\textbf{RAT}({\phi})\cap
B^{*}\textbf{RAT}({\phi})}$.
So strategy $s_{i}$ is an element of the $i$th component of
$G_{K^{*}\textbf{RAT}({\phi})}$ if $s_{i}=\overline{s_{i}}(\omega)$ for some
$\omega\in K^{*}\textbf{RAT}({\phi})$. That is, $s_{i}$ is a strategy that
player $i$ chooses in a state in which it is common knowledge that each player
$j$ is $\phi_{j}$-rational, and similarly for $G_{\textbf{RAT}({\phi})\cap
B^{*}\textbf{RAT}({\phi})}$.
The following result then relates for arbitrary strategic games the
restrictions $G_{\textbf{RAT}({\phi})\cap B^{*}\textbf{RAT}({\phi})}$ and
$G_{K^{*}\textbf{RAT}({\phi})}$ to the outcome of the iteration of the
operator $T_{\phi}$.
###### Theorem 1
1. (i)
Suppose that each property $\phi_{i}$ is monotonic. Then for all belief models
for $H$
$G_{\textbf{RAT}({\phi})\cap
B^{*}\textbf{RAT}({\phi})}\mbox{$\>\subseteq\>$}T_{\phi}^{\infty}$
2. (ii)
Suppose that each property $\phi_{i}$ is monotonic. Then for all knowledge
models for $H$
$G_{K^{*}\textbf{RAT}({\phi})}\mbox{$\>\subseteq\>$}T_{\phi}^{\infty}$
3. (iii)
For some standard knowledge model for $H$
$T_{\phi}^{\infty}\mbox{$\>\subseteq\>$}G_{K^{*}\textbf{RAT}({\phi})}$
So part $(i)$ (respectively, $(ii)$) states that true common belief
(respectively, common knowledge) of $\phi_{i}$-rationality of each player $i$
implies that the players will choose only strategies that survive the iterated
elimination of non-$\phi$-rational strategies.
Proof.
$(i)$ Fix a belief model
$(\Omega,\overline{s_{1}},\mbox{$\ldots$},\overline{s_{n}},P_{1},\mbox{$\ldots$},P_{n})$
for $H$. Take a strategy $s_{i}$ that is an element of the $i$th component of
$G_{\textbf{RAT}({\phi})\cap B^{*}\textbf{RAT}({\phi})}$. Thus we have
$s_{i}=\overline{s_{i}}(\omega)$ for some state $\omega$ such that
$\omega\in\textbf{RAT}({\phi})$ and $\omega\in B^{*}\textbf{RAT}({\phi})$. The
latter implies by (1) that for some evident event $F$
$\omega\in
F\mbox{$\>\subseteq\>$}\\{\omega^{\prime}\in\Omega\mid\mbox{$\forall$}i\in\\{1,\mbox{$\ldots$},n\\}\>P_{i}(\omega^{\prime})\mbox{$\>\subseteq\>$}\textbf{RAT}({\phi})\\}$
(4)
Take now an arbitrary $\omega^{\prime}\in F\cap\textbf{RAT}({\phi})$ and
$i\in\\{1,\mbox{$\ldots$},n\\}$. Since
$\omega^{\prime}\in\textbf{RAT}({\phi})$, it holds that player $i$ is
$\phi_{i}$-rational in $\omega^{\prime}$, _i.e._ ,
$\phi_{i}(\overline{s_{i}}(\omega^{\prime}),H_{i},(G_{P_{i}(\omega^{\prime})})_{-i})$
holds. But $F$ is evident, so $P_{i}(\omega^{\prime})\mbox{$\>\subseteq\>$}F$.
Moreover by (4)
$P_{i}(\omega^{\prime})\mbox{$\>\subseteq\>$}\textbf{RAT}({\phi})$, so
$P_{i}(\omega^{\prime})\mbox{$\>\subseteq\>$}F\cap\textbf{RAT}({\phi})$. Hence
$(G_{P_{i}(\omega^{\prime})})_{-i}\mbox{$\>\subseteq\>$}(G_{F\cap\textbf{RAT}({\phi})})_{-i}$
and by the monotonicity of $\phi_{i}$ we conclude that
$\phi_{i}(\overline{s_{i}}(\omega^{\prime}),H_{i},(G_{F\cap\textbf{RAT}({\phi})})_{-i})$
holds.
By the definition of $T_{\phi}$ this means that
$G_{F\cap\textbf{RAT}({\phi})}\mbox{$\>\subseteq\>$}T_{\phi}(G_{F\cap\textbf{RAT}({\phi})})$,
_i.e._ $G_{F\cap\textbf{RAT}({\phi})}$ is a post-fixpoint of $T_{\phi}$. But
$T_{\phi}$ is monotonic since each property $\phi_{i}$ is. Hence by Tarski’s
Fixpoint Theorem
$G_{F\cap\textbf{RAT}({\phi})}\mbox{$\>\subseteq\>$}T_{\phi}^{\infty}$. But
$s_{i}=\overline{s_{i}}(\omega)$ and $\omega\in F\cap{\textbf{RAT}({\phi})}$,
so we conclude by the above inclusion that $s_{i}$ is an element of the $i$th
component of $T_{\phi}^{\infty}$. This proves the claim.
$(ii)$ By the definition of common knowledge for all events $E$ we have
$K^{*}E\mbox{$\>\subseteq\>$}E$. Hence for all $\phi$ we have
$K^{*}\textbf{RAT}({\phi})\mbox{$\>\subseteq\>$}\textbf{RAT}({\phi})\cap
K^{*}\textbf{RAT}({\phi})$ and consequently
$G_{K^{*}\textbf{RAT}({\phi})}\mbox{$\>\subseteq\>$}G_{\textbf{RAT}({\phi})\cap
K^{*}\textbf{RAT}({\phi})}$.
So part (ii) follows from part (i).
$(iii)$ Suppose $T^{\infty}_{\phi}=(G_{1},\mbox{$\ldots$},G_{n})$. Consider
the event $F:=G_{1}\times\mbox{$\ldots$}\times G_{n}$ in the standard model
for $H$. Then $G_{F}=T^{\infty}_{\phi}$. Define each possibility
correspondence $P_{i}$ by
$P_{i}(\omega):=\left\\{\begin{array}[]{l@{\extracolsep{3mm}}l}F&\mathrm{if}\
\omega\in F\\\ \Omega\setminus F&\mathrm{otherwise}\end{array}\right.$
Each $P_{i}$ is a knowledge correspondence (also when $F=\mbox{$\emptyset$}$
or $F=\Omega$) and clearly $F$ is an evident event.
Take now an arbitrary $i\in\\{1,\mbox{$\ldots$},n\\}$ and an arbitrary state
$\omega\in F$. Since $T^{\infty}_{\phi}$ is a fixpoint of $T_{\phi}$ and
$\overline{s_{i}}(\omega)\in G_{i}$ we have
$\phi_{i}(\overline{s_{i}}(\omega),H_{i},(T^{\infty}_{\phi})_{-i})$, so by the
definition of $P_{i}$ we have
$\phi_{i}(\overline{s_{i}}(\omega),H_{i},(G_{P_{i}(\omega)})_{-i})$. This
shows that each player $i$ is $\phi_{i}$-rational in each state $\omega\in F$,
_i.e._ , $F\mbox{$\>\subseteq\>$}\textbf{RAT}(\phi)$.
Since $F$ is evident, we conclude by (2) that in each state $\omega\in F$ it
is common knowledge that each player $i$ is $\phi_{i}$-rational, _i.e._ ,
$F\mbox{$\>\subseteq\>$}K^{*}\textbf{RAT}(\phi)$. Consequently
$T_{\phi}^{\infty}=G_{F}\mbox{$\>\subseteq\>$}G_{K^{*}\textbf{RAT}(\phi)}$
$\Box$
Items $(i)$ and $(ii)$ show that when each property $\phi_{i}$ is monotonic,
for all belief models of $H$ it holds that the joint strategies that the
players choose in the states in which each player $i$ is $\phi_{i}$-rational
and it is common belief that each player $i$ is $\phi_{i}$-rational (or in
which it is common knowledge that each player $i$ is $\phi_{i}$-rational) are
included in those that remain after the iterated elimination of the strategies
that are not $\phi_{i}$-rational.
Note that monotonicity of the $\phi_{i}$ properties was not needed to
establish item $(iii)$.
By instantiating the $\phi_{i}$’s with specific properties we get instances of
the above result that refer to specific definitions of rationality. This will
allow us to relate the above result to the ones established in the literature.
Before we do this we establish a result that identifies a large class of
properties $\phi_{i}$ for which Theorem 1 does not apply.
###### Theorem 2
Suppose that a joint strategy $s\not\in T_{\phi}^{\infty}$ exists such that
$\phi_{i}(s_{i},H_{i},(\\{s_{j}\\}_{j\neq i}))$
holds all $i\in\\{1,\mbox{$\ldots$},n\\}$. Then for some knowledge model for
$H$ the inclusion
$G_{K^{*}\textbf{RAT}({\phi})}\mbox{$\>\subseteq\>$}T_{\phi}^{\infty}$
does not hold.
Proof. We extend the standard model for $H$ by the knowledge correspondences
$P_{1},\mbox{$\ldots$},P_{n}$ where for all $i\in\\{1,\mbox{$\ldots$},n\\}$,
$P_{i}(\omega)=\mbox{$\\{{\omega}\\}$}$. Then for all $\omega$ and all
$i\in\\{1,\mbox{$\ldots$},n\\}$
$G_{P_{i}(\omega)}=(\mbox{$\\{{\overline{s_{1}}(\omega)}\\}$},\mbox{$\ldots$},\mbox{$\\{{\overline{s_{n}}(\omega)}\\}$})$
Let $\omega^{\prime}:=s$. Then for all $i\in\\{1,\mbox{$\ldots$},n\\}$,
$G_{P_{i}(\omega^{\prime})}=(\mbox{$\\{{s_{1}}\\}$},\mbox{$\ldots$},\mbox{$\\{{s_{n}}\\}$})$,
so by the assumption each player $i$ is $\phi_{i}$-rational in
$\omega^{\prime}$, _i.e._ , $\omega^{\prime}\in\textbf{RAT}(\phi)$. By the
definition of $P_{i}$s the event $\\{{\omega^{\prime}}\\}$ is evident and
$\omega^{\prime}\in K\textbf{RAT}(\phi)$. So by (1) $\omega^{\prime}\in
K^{*}\textbf{RAT}(\phi)$. Consequently
$s=(\overline{s_{1}}(\omega^{\prime}),\mbox{$\ldots$},\overline{s_{n}}(\omega^{\prime}))\in
G_{K^{*}\textbf{RAT}(\phi)}$.
This yields the desired conclusion by the choice of $s$. $\Box$
## 4 Applications
We now analyze to what customary game-theoretic properties the above two
results apply. By a _belief_ of player $i$ about the strategies his opponents
play given the set $G_{-i}$ of their joint strategies we mean one of the
following possibilities:
* •
a joint strategy of the opponents of player $i$, _i.e._ , $s_{-i}\in G_{-i}$,
called a _point belief_ ,
* •
or, in the case the game is finite, a joint mixed strategy of the opponents of
player $i$ (_i.e._ ,
$(m_{1},\mbox{$\ldots$},m_{i-1},m_{i+1},\mbox{$\ldots$},m_{n})$, where
$m_{j}\in\Delta G_{j}$ for all $j\neq i$), called an _independent belief_ ,
* •
or, in the case the game is finite, an element of $\Delta G_{-i}$, called a
_correlated belief_.
In the second and third case the payoff function $p_{i}$ can be lifted in the
standard way to an _expected payoff_ function $p_{i}:H_{i}\times{\cal
B}_{i}(G_{-i})\mbox{$\>\rightarrow\>$}\cal{R}$, where ${\cal B}_{i}(G_{-i})$
is the corresponding set of beliefs of player $i$ held given $G_{-i}$.
We use below the following abbreviations, where $s_{i},s^{\prime}_{i}\in
H_{i}$ and $G_{-i}$ is a set of the strategies of the opponents of player $i$:
* •
(_strict dominance_) $s^{\prime}_{i}\succ_{G_{-i}}s_{i}$ for
$\mbox{$\forall$}s_{-i}\in
G_{-i}\>p_{i}(s^{\prime}_{i},s_{-i})>p_{i}(s_{i},s_{-i})$
* •
(_weak dominance_) $s^{\prime}_{i}\succ^{w}_{G_{-i}}s_{i}$ for
$\mbox{$\forall$}s_{-i}\in G_{-i}\>p_{i}(s^{\prime}_{i},s_{-i})\geq
p_{i}(s_{i},s_{-i})\mbox{$\ \wedge\ $}\mbox{$\exists$}s_{-i}\in
G_{-i}\>p_{i}(s^{\prime}_{i},s_{-i})>p_{i}(s_{i},s_{-i})$
In the case of finite games the relations $\succ_{G_{-i}}$ and
$\succ^{w}_{G_{-i}}$ between a mixed strategy and a pure strategy are defined
in the same way.
We now introduce natural examples of the optimality notion.
* •
$sd_{i}(s_{i},G_{i},G_{-i})\equiv\neg\mbox{$\exists$}s^{\prime}_{i}\in
G_{i}\>s^{\prime}_{i}\succ_{G_{-i}}s_{i}$
* •
(assuming $H$ is finite)
$msd_{i}(s_{i},G_{i},G_{-i})\equiv\neg\mbox{$\exists$}m^{\prime}_{i}\in\Delta
G_{i}\>m^{\prime}_{i}\succ_{G_{-i}}s_{i}$
* •
$wd_{i}(s_{i},G_{i},G_{-i})\equiv\neg\mbox{$\exists$}s^{\prime}_{i}\in
G_{i}\>s^{\prime}_{i}\succ^{w}_{G_{-i}}s_{i}$
* •
(assuming $H$ is finite)
$mwd_{i}(s_{i},G_{i},G_{-i})\equiv\neg\mbox{$\exists$}m^{\prime}_{i}\in\Delta
G_{i}\>m^{\prime}_{i}\succ^{w}_{G_{-i}}s_{i}$
* •
$br_{i}(s_{i},G_{i},G_{-i})\equiv\mbox{$\exists$}\mu_{i}\in{\cal
B}_{i}(G_{-i})\>\mbox{$\forall$}s^{\prime}_{i}\in
G_{i}\>p_{i}(s_{i},\mu_{i})\geq p_{i}(s^{\prime}_{i},\mu_{i})$
So $sd_{i}$ and $wd_{i}$ are the customary notions of strict and weak
dominance and $msd_{i}$ and $mwd_{i}$ are their counterparts for the case of
dominance by a mixed strategy. Note that the notion $br_{i}$ of best response,
comes in three ‘flavours’ depending on the choice of the set ${\cal
B}_{i}(G_{-i})$ of beliefs.
Consider now the iterated elimination of strategies as defined in Subsection
2.5, so _with_ the repeated reference by player $i$ to the strategy set
$H_{i}$. For the optimality notion $sd_{i}$ such a version of iterated
elimination was studied in [11], for $mwd_{i}$ it was used in [10], while for
$br_{i}$ it corresponds to the rationalizability notion of [6].
In [15], [11] and [2] examples are provided showing that for the properties
$sd_{i}$ and $br_{i}$ in general transfinite iterations (_i.e._ , iterations
beyond $\omega_{0}$) of the corresponding operator are necessary to reach the
outcome. So to establish for them part $(iii)$ of Theorem 1 transfinite
iterations of the $T_{\phi}$ operator are necessary.
The following lemma holds.
###### Lemma 2
The properties $sd_{i},\ msd_{i}$ and $br_{i}$ are monotonic.
Proof. Straightforward. $\Box$
So Theorem 1 applies to the above three properties. In contrast, Theorem 1
does not apply to the remaining two properties $wd_{i}$ and $mwd_{i}$, since,
as indicated in [3], the corresponding operators $T_{wd}$ and $T_{mwd}$ are
not monotonic, and hence the properties $wd_{i}$ and $mwd_{i}$ are not
monotonic.
In fact, the desired inclusion does not hold and Theorem 2 applies to these
two optimality properties. Indeed, consider the following game:
${{\begin{array}[c]{@{}r|*{2}{c|}}\hfil\hbox{}\hfil\hbox{\vrule
height=9.41666pt,depth=2.58334pt,width=0.0pt}\hfil\hbox{\color[rgb]{0,0,0}\ignorespaces$L$
}\hfil\hbox{\vrule
height=9.41666pt,depth=2.58334pt,width=0.0pt}\hfil\hbox{\color[rgb]{0,0,0}\ignorespaces$R$}\hfil\hbox{\vrule
height=9.41666pt,depth=2.58334pt,width=0.0pt}\cr{\cline{2-}}{}{}{}\hfil\hbox{\ignorespaces$U$
}{}\hbox{\vrule
height=9.41666pt,depth=2.58334pt,width=0.0pt}\hfil\hbox{\ignorespaces\ignorespaces$1,1$
}\hfil\hbox{\vrule
height=9.41666pt,depth=2.58334pt,width=0.0pt}\hfil\hbox{\ignorespaces\ignorespaces$0,1$}\hfil\hbox{\vrule
height=9.41666pt,depth=2.58334pt,width=0.0pt}\cr{\cline{2-}}{}{}{}\hfil\hbox{\ignorespaces$D$
}{}\hbox{\vrule
height=9.41666pt,depth=2.58334pt,width=0.0pt}\hfil\hbox{\ignorespaces\ignorespaces$1,0$
}\hfil\hbox{\vrule
height=9.41666pt,depth=2.58334pt,width=0.0pt}\hfil\hbox{\ignorespaces\ignorespaces$1,1$
\color[rgb]{0,0,0}}\hfil\hbox{\vrule
height=9.41666pt,depth=2.58334pt,width=0.0pt}\cr{\cline{2-}}\hskip 6.0pt\hbox
to17.77777pt{\hfil}\hskip 6.0pt\hskip 6.0pt\hbox to17.77777pt{\hfil}\hskip
6.0pt\crcr}}\end{array}$
Then the outcome of iterated elimination for both $wd_{i}$ and $mwd_{i}$
yields $G:=(\mbox{$\\{{D}\\}$},\mbox{$\\{{R}\\}$})$. Further, we have
$wd_{1}(U,\\{U,D\\},\\{L\\})$ and $wd_{2}(L,\\{L,R\\},\\{U\\})$, and
analogously for $mwd_{1}$ and $mwd_{2}$.
So the joint strategy $(U,L)$ satisfies the conditions of Theorem 2 for both
$wd_{i}$ and $mwd_{i}$. Note that this game also furnishes an example for non-
monotonicity of $wd_{i}$ since $wd_{1}(U,\\{U,D\\},\\{L,R\\})$ does not hold.
This shows that the optimality notions $wd_{i}$ and $mwd_{i}$ cannot be
justified in the used epistemic framework as ‘stand alone’ concepts of
rationality.
## 5 Consequences of Common Knowledge of Rationality
In this section we show that common knowledge of rationality is sufficient to
entail the customary iterated elimination of strictly dominated strategies. We
also show that weak dominance is not amenable to such a treatment.
Given a sequence of properties $\phi:=(\phi_{1},\mbox{$\ldots$},\phi_{n})$, we
introduce an operator $U_{\phi}$ on the restrictions of $H$ defined by
$U_{\phi}(G):=G^{\prime},$
where $G:=(G_{1},\mbox{$\ldots$},G_{n})$,
$G^{\prime}:=(G^{\prime}_{1},\mbox{$\ldots$},G^{\prime}_{n})$, and for all
$i\in\\{1,\mbox{$\ldots$},n\\}$
$G^{\prime}_{i}:=\\{s_{i}\in G_{i}\mid\phi_{i}(s_{i},G_{i},G_{-i})\\}.$
So when defining the set of strategies $G^{\prime}_{i}$ we use in the second
argument of $\phi_{i}$ the set $G_{i}$ of player’s $i$ strategies in the
_current_ restriction $G$. That is, $U_{\phi}(G)$ determines the ‘locally’
$\phi$-optimal strategies in $G$. In contrast, $T_{\phi}(G)$ determines the
‘globally’ $\phi$-optimal strategies in $G$, in that each player $i$ must
consider all of his strategies $s^{\prime}_{i}$ that occur in his strategy set
$H_{i}$ in the _initial game_ $H$.
So the ‘global’ form of optimality coincides with rationality, as introduced
in Subsection 2.5, while the customary definition of iterated elimination of
strictly (or weakly) dominated strategies refers to the iterations of the
appropriate instantiation of the ‘local’ $U_{\phi}$ operator.
Note that the $U_{\phi}$ operator is non-monotonic for all non-trivial
optimality notions $\phi_{i}$ such that
$\phi_{i}(s_{i},\\{s_{i}\\},(\\{s_{j}\\}_{j\neq i}))$ for all joint strategies
$s$, so in particular for $br_{i},sd_{i},msd_{i},wd_{i}$ and $mwd_{i}$.
Indeed, given $s$ let $G_{s}$ denote the corresponding restriction in which
each player $i$ has a single strategy $s_{i}$. Each restriction $G_{s}$ is a
fixpoint of $U_{\phi}$. By non-triviality of $\phi_{i}$s we have
$U_{\phi}(H)\neq H$, so for each restriction $G_{s}$ with $s$ including an
eliminated strategy the inclusion
$U_{\phi}(G_{s})\mbox{$\>\subseteq\>$}U_{\phi}(H)$ does not hold, even though
$G_{s}\mbox{$\>\subseteq\>$}H$. In contrast, as we saw, by virtue of Lemma 2
the $T_{\phi}$ operator is monotonic for $br_{i},sd_{i}$ and $msd_{i}$.
First we establish the following consequence of Theorem 1. When each property
$\phi_{i}$ equals $\textit{br}_{i}$, we write here
$\textbf{RAT}({\textit{br}})$ and similarly with $U_{sd}$.
###### Corollary 1
1. (i)
For all belief models
$G_{\textbf{RAT}({\textit{br}})\cap
B^{*}\textbf{RAT}({\textit{br}})}\mbox{$\>\subseteq\>$}U^{\infty}_{\textit{sd}}$
2. (ii)
for all knowledge models
$G_{K^{*}\textbf{RAT}({\textit{br}})}\mbox{$\>\subseteq\>$}U^{\infty}_{\textit{sd}}$
where in both situations we use in $br_{i}$ the set of poinr beliefs.
Proof.
$(i)$ By Lemma 2 and Theorem 1$(i)$ $G_{\textbf{RAT}(\textit{br})\cap
B^{*}\textbf{RAT}(\textit{br})}\mbox{$\>\subseteq\>$}T^{\infty}_{\textit{br}}$
Each best response to a joint strategy of the opponents is not strictly
dominated, so for all restrictions $G$
$T_{\textit{br}}(G)\mbox{$\>\subseteq\>$}T_{\textit{sd}}(G)$
Also, for all restrictions $G$,
$T_{\textit{sd}}(G)\mbox{$\>\subseteq\>$}U_{\textit{sd}}(G)$. So by Lemma 1
$T^{\infty}_{\textit{br}}\mbox{$\>\subseteq\>$}U^{\infty}_{\textit{sd}}$,
which concludes the proof.
$(ii)$ By part $(i)$ and the fact that
$K^{*}\textbf{RAT}({\textit{br}})\mbox{$\>\subseteq\>$}\textbf{RAT}({\textit{br}})$.
$\Box$
Part $(ii)$ formalizes and justifies in the epistemic framework used here the
often used statement:
> common knowledge of rationality implies that the players will choose only
> strategies that survive the iterated elimination of strictly dominated
> strategies
for games with _arbitrary strategy sets_ and _transfinite iterations_ of the
elimination process, and where best response means best response to a point
belief.
In the case of finite games Theorem 1 implies the following result. For the
case of independent beliefs it is implicitly stated in [8], explicitly
formulated in [21] (see [5, page 181]) and proved using Harsanyi type spaces
in [9].
###### Corollary 2
Assume the initial game $H$ is finite.
1. (i)
For all belief models for $H$
$G_{\textbf{RAT}({\textit{br}})\cap
B^{*}\textbf{RAT}({\textit{br}})}\mbox{$\>\subseteq\>$}U^{\infty}_{\textit{msd}},$
2. (ii)
for all knowledge models for $H$
$G_{K^{*}\textbf{RAT}({\textit{br}})}\mbox{$\>\subseteq\>$}U^{\infty}_{\textit{msd}},$
where in both situations we use in $br_{i}$ either the set of point beliefs or
the set of independent beliefs or the set of correlated beliefs.
Proof. The argument is analogous as in the previous proof but relies on a
subsidiary result and runs as follows.
$(i)$ Denote respectively by $brp_{i}$, $bri_{i}$ and $brc_{i}$ the best
response property w.r.t. _point_ , _independent_ and _correlated_ beliefs of
the opponents. Below $\phi$ stands for either $brp$, $bri$ or $brc$.
By Lemma 2 and Theorem 1 $G_{\textbf{RAT}(\phi)\cap
B^{*}\textbf{RAT}(\phi)}\mbox{$\>\subseteq\>$}T^{\infty}_{\phi}$. Further, for
all restrictions $G$ we have both
$T_{\phi}(G)\mbox{$\>\subseteq\>$}U_{\phi}(G)$ and
$U_{\textit{br}}(G)\mbox{$\>\subseteq\>$}U_{\textit{bri}}(G)\mbox{$\>\subseteq\>$}U_{\textit{brc}}(G).$
So by Lemma 1
$T^{\infty}_{\phi}\mbox{$\>\subseteq\>$}U^{\infty}_{\textit{brc}}$. But by the
result of [19], (page 60) (that is a modification of the original result of
[20]), for all restrictions $G$ we have
$U_{\textit{brc}}(G)=U_{\textit{msd}}(G)$, so
$U^{\infty}_{\textit{brc}}=U^{\infty}_{\textit{msd}}$, which yields the
conclusion.
$(ii)$ By $(i)$ and the fact that
$K^{*}\textbf{RAT}({\textit{br}})\mbox{$\>\subseteq\>$}\textbf{RAT}({\textit{br}})$.
$\Box$
Finally, let us clarify the situation for the remaining two optimality
notions, $wd_{i}$ and $mwd_{i}$. For them the inclusions of Corollaries 1 and
2 do not hold. Indeed, it suffices to consider the following initial game $H$:
${{\begin{array}[c]{@{}r|*{2}{c|}}\hfil\hbox{}\hfil\hbox{\vrule
height=9.41666pt,depth=2.58334pt,width=0.0pt}\hfil\hbox{\color[rgb]{0,0,0}\ignorespaces$L$
}\hfil\hbox{\vrule
height=9.41666pt,depth=2.58334pt,width=0.0pt}\hfil\hbox{\color[rgb]{0,0,0}\ignorespaces$R$}\hfil\hbox{\vrule
height=9.41666pt,depth=2.58334pt,width=0.0pt}\cr{\cline{2-}}{}{}{}\hfil\hbox{\ignorespaces$U$
}{}\hbox{\vrule
height=9.41666pt,depth=2.58334pt,width=0.0pt}\hfil\hbox{\ignorespaces\ignorespaces$1,0$
}\hfil\hbox{\vrule
height=9.41666pt,depth=2.58334pt,width=0.0pt}\hfil\hbox{\ignorespaces\ignorespaces$1,0$}\hfil\hbox{\vrule
height=9.41666pt,depth=2.58334pt,width=0.0pt}\cr{\cline{2-}}{}{}{}\hfil\hbox{\ignorespaces$D$
}{}\hbox{\vrule
height=9.41666pt,depth=2.58334pt,width=0.0pt}\hfil\hbox{\ignorespaces\ignorespaces$1,0$
}\hfil\hbox{\vrule
height=9.41666pt,depth=2.58334pt,width=0.0pt}\hfil\hbox{\ignorespaces\ignorespaces$0,0$
\color[rgb]{0,0,0}}\hfil\hbox{\vrule
height=9.41666pt,depth=2.58334pt,width=0.0pt}\cr{\cline{2-}}\hskip 6.0pt\hbox
to17.77777pt{\hfil}\hskip 6.0pt\hskip 6.0pt\hbox to17.77777pt{\hfil}\hskip
6.0pt\crcr}}\end{array}$
Here every strategy is a best response but $D$ is weakly dominated by $U$. So
both $U^{\infty}_{\textit{wd}}$ and $U^{\infty}_{\textit{mwd}}$ are proper
subsets of $T^{\infty}_{\textit{br}}$. On the other hand by Theorem 1$(iii)$
for some standard knowledge model for $H$ we have
$G_{K^{*}\textbf{RAT}({\textit{br}})}=T^{\infty}_{\textit{br}}$. So for this
knowledge model neither
$G_{K^{*}\textbf{RAT}({\textit{br}})}\mbox{$\>\subseteq\>$}U^{\infty}_{\textit{wd}}$
nor
$G_{K^{*}\textbf{RAT}({\textit{br}})}\mbox{$\>\subseteq\>$}U^{\infty}_{\textit{mwd}}$
holds.
## Acknowledgements
We thank one of the referees for useful comments. We acknowledge helpful
discussions with Adam Brandenburger, who suggested Corollaries 1 and 2, and
with Giacomo Bonanno who, together with a referee of [2], suggested to
incorporate common beliefs in the analysis. Joe Halpern pointed us to [18].
This paper was previously sent for consideration to another major game theory
journal, but ultimately withdrawn because of different opinions with the
referee. We would like to thank the referee and associate editor of that
journal for their comments and help provided.
## References
* [1] K. R. Apt. Epistemic analysis of strategic games with arbitrary strategy sets. In Proceedings 11th Conference on Theoretical Aspects of Reasoning about Knowledge (TARK07), pages 22–38. The ACM Digital Library, 2007\. Available from http://portal.acm.org.
* [2] K. R. Apt. The many faces of rationalizability. The B.E. Journal of Theoretical Economics, 7(1), 2007. (Topics), Article 18, 39 pages. Available from http://arxiv.org/abs/cs.GT/0608011.
* [3] K. R. Apt. Relative strength of strategy elimination procedures. Economics Bulletin, 3(21):1–9, 2007. Available from http://www.economicsbulletin.com/.
* [4] R. Aumann. Agreeing to disagree. The Annals of Statistics, 4(6):1236–1239, 1976.
* [5] P. Battigalli and G. Bonanno. Recent results on belief, knowledge and the epistemic foundations of game theory. Research in Economics, 53(2):149–225, June 1999.
* [6] B. D. Bernheim. Rationalizable strategic behavior. Econometrica, 52(4):1007–1028, 1984.
* [7] P. Blackburn, M. de Rijke, and Y. Venema. Modal Logic. Cambridge University Press, 2001.
* [8] A. Brandenburger and E. Dekel. Rationalizability and correlated equilibria. Econometrica, 55(6):1391–1402, 1987.
* [9] A. Brandenburger and A. Friedenberg. Intrinsic correlation in games. Journal of Economic Theory, 141:28–67, 2008.
* [10] A. Brandenburger, A. Friedenberg, and H. Keisler. Admissibility in games. Econometrica, 76(2):307 –352, 2008.
* [11] Y.-C. Chen, N. V. Long, and X. Luo. Iterated strict dominance in general games. Games and Economic Behavior, 61(2):299 – 315, 2007.
* [12] J. Halpern and R. Pass. A logical characterization of iterated admissibility. In Proceedings of the 12th Conference on Theoretical Aspects of Rationality and Knowledge (TARK XII), pages 146–155. The ACM Digital Library, 2009.
* [13] A. Heifetz and D. Samet. Knowledge spaces with arbitrarily high rank. Games and Economic Behavior, 22:260–273, 1998.
* [14] B. L. Lipman. How to decide how to decide how to $\dots$: Modeling limited rationality. Econometrica, 59(4):1105–1125, 1991.
* [15] B. L. Lipman. A note on the implications of common knowledge of rationality. Games and Economic Behavior, 6:114–129, 1994.
* [16] X. Luo. General systems and $\phi$-stable sets—a formal analysis of socioeconomic environments. Journal of Mathematical Economics, 36:95–109, 2001.
* [17] P. Milgrom and J. Roberts. Rationalizability, learning, and equilibrium in games with strategic complementarities. Econometrica, 58:1255–1278, 1990.
* [18] D. Monderer and D. Samet. Approximating common knowledge with common beliefs. Games and Economic Behaviour, 1:170–190, 1989.
* [19] M. J. Osborne and A. Rubinstein. A Course in Game Theory. The MIT Press, Cambridge, Massachusetts, 1994.
* [20] D. G. Pearce. Rationalizable strategic behavior and the problem of perfection. Econometrica, 52(4):1029–1050, 1984.
* [21] R. Stalnaker. On the evaluation of solution concepts. Theory and Decision, 37(1):49–73, 1994.
* [22] A. Tarski. A lattice-theoretic fixpoint theorem and its applications. Pacific J. Math, 5:285–309, 1955.
* [23] J. van Benthem. Rational dynamics and epistemic logic in games. International Game Theory Review, 9(1):13–45, 2007.
|
arxiv-papers
| 2010-10-27T07:54:46 |
2024-09-04T02:49:14.280851
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Krzysztof R. Apt and Jonathan A. Zvesper",
"submitter": "Krzysztof R. Apt",
"url": "https://arxiv.org/abs/1010.5595"
}
|
1010.5829
|
# Robustness of a Network of Networks
Jianxi Gao,1,2 Sergey V. Buldyrev,3 Shlomo Havlin,4 and H. Eugene Stanley1
1Center for Polymer Studies and Department of Physics, Boston University,
Boston, MA 02215 USA
2Department of Automation, Shanghai Jiao Tong University, 800 Dongchuan Road,
Shanghai 200240, PR China
3Department of Physics, Yeshiva University, New York, NY 10033 USA
4Minerva Center and Department of Physics, Bar-Ilan University, 52900 Ramat-
Gan, Israel
(25 October 2010 — gbhs25oct.tex)
###### Abstract
Almost all network research has been focused on the properties of a single
network that does not interact and depends on other networks. In reality, many
real-world networks interact with other networks. Here we develop an
analytical framework for studying interacting networks and present an exact
percolation law for a network of $n$ interdependent networks. In particular,
we find that for $n$ Erdős-Rényi networks each of average degree $k$, the
giant component, $P_{\infty}$, is given by
$P_{\infty}=p[1-\exp(-kP_{\infty})]^{n}$ where $1-p$ is the initial fraction
of removed nodes. Our general result coincides for $n=1$ with the known Erdős-
Rényi second-order phase transition for a single network. For any $n\geq 2$
cascading failures occur and the transition becomes a first-order percolation
transition. The new law for $P_{\infty}$ shows that percolation theory that is
extensively studied in physics and mathematics is a limiting case ($n=1$) of a
more general general and different percolation law for interdependent
networks.
In recent years dramatic advances in the field of complex networks have
occurred Strogatz1998 ; bara2000 ; Callaway2000 ; Albert2002 ; Cohen2000 ;
Newman2003 ; Dorogovtsev2003 ; song2005 ; Pastor2006 ; Caldarelli2007 ;
Barrat2008 ; Shlomo2010 ; Neman2010 . The internet, airline routes, and
electric power grids are all examples of networks whose function relies
crucially on the connectivity between the network components. An important
property of such systems is their robustness to node failures. Almost all
research has been concentrated on the case of a single or isolated network
which does not interact with other networks. Recently, based on the motivation
that modern infrastructures are becoming significantly more dependent on each
other, a system of two coupled interdependent networks has been studied
Sergey2010 . A fundamental property of interdependent networks is that when
nodes in one network fail, they may lead to the failure of dependent nodes in
other networks which may cause further damage in the first network and so on,
leading to a global cascade of failures. Buldyrev et al. Sergey2010 developed
a framework for analyzing robustness of two interacting networks subject to
such cascading failures. They found that interdependent networks become
significantly more vulnerable compared to their noninteracting counterparts.
For many important examples, more than two networks depend on each other. For
example, diverse infrastructures are coupled together, such as water and food
supply, communications, fuel, financial transactions, and power stations
Peerenboom2001 ; Rinaldi2001 ; Rosato2008 ; Alessandro2010 . For further
examples see Section i@ in the Supplementary Information (SI). Understanding
the robustness due to such interdependencies is one of the major challenges
for designing resilient infrastructures.
We introduce here a model system, comprising a network of $n$ coupled
networks, where each network consists of $N$ nodes (See Fig. 1). The $N$ nodes
in each network are connected to nodes in neighboring networks by
bidirectional dependency links, thereby establishing a one-to-one
correspondence as illustrated in Fig. 2 in SI. We develop a mathematical
framework to study the robustness of this “network of networks” (NON). We find
an exact analytical law for percolation of a NON system composed of $n$
coupled randomly connected networks. Our result generalizes the known Erdős-
Rényi (ER) ER1959 ; ER1960 ; Bollob1985 result for the giant component of a
single network and the $n=2$ result found recently Sergey2010 , and shows that
while for $n=1$ the percolation transition is a second order transition, for
$n>1$ cascading failures occur and the transition becomes a first order
transition. Our results for $n$ interdependent networks suggest that the
classical percolation theory extensively studied in physics and mathematics is
a limiting case of a general theory of percolation in NONs, or networks with
multiple types of connectivity links. This general theory has many novel
features which are not present in classical percolation.
Additionally, we find:
(i) the robustness of NON significantly decreases with $n$, and
(ii) for a network of $n$ ER networks all with the same average degree $k$,
there exists a minimum degree $k_{\min}(n)$ increasing with $n$, below which
$p_{c}=1$, i.e., for $k<k_{\min}$ the NON will collapse once any finite number
of nodes fail. We find an analytical expression for $k_{min}(n)$, which
generalizes the known result $k_{\min}(1)=1$ for ER below which the network
collapses. We also discuss the critical effect of loops in the NON structure.
Real-world interacting networks (See SI for more details) are characterized by
complex correlations and a variety of organizational principles governing
their internal structures and interdependencies. Once these correlations are
quantified from the statistical analysis of actual data bases and the
organizational principles are specified from the engineering literature, real
world networks can be studied by computer simulations. These simulations will
have many parameters and therefore their outcome will also require complex
interpretation. It is therefore very important to develop simple analytically
tractable models for the robustness of interdependent networks against which
such simulations can be tested. Well-known examples of simplified models that
both demonstrate a fundamental phenomenon and significantly advance our
knowledge are the Ising model in statistical mechanics and the Erdős-Rényi
model in graph theory. This paper presents a simple model that can serve as a
benchmark for further studies of NONs.
We assume that a network $m$ $(m=1,2,...,n)$ in the NON is a randomly
connected network with a degree distribution $P_{m}(k)$. We call a pair of
networks A and B a fully interdependent pair if it satisfies the following
condition: each node $A_{i}$ of network A is connected to one and only one
node $B_{i}$ in network B by a bidirectional dependency link such that if node
$A_{i}$ fails, $B_{i}$ also fails and vice versa. Since the number of nodes in
each network is the same, these bidirectional links establish a one-to-one
correspondence between the nodes in the networks belonging to an
interdependent pair. Each node of the NON represents a network and each edge
represents a fully interdependent pair of networks. First, we will discuss the
case when the NON is a loopless tree of $n$ networks (Fig. 1). The dependency
edges in such a NON establish a unique one-to-one correspondence between the
nodes of any two networks not necessarily belonging to the same fully
interdependent pair. This one-to-one correspondence established by the
interdependency links between the nodes of different networks in the loopless
NON uniquely maps any set of nodes in one of the networks to a set of nodes
(which we will call an image of the original set) in any other network of the
NON (See SI for more details). In principle, the assumption of full
interdependence allows one to collapse all the networks of the NON onto a
single network with multiple types of links.
We assume that in order to remain functional a node must belong to a
sufficiently large mutually connected cluster Sergey2010 (See detailed
definition in SI). We will show that a large mutually connected cluster which
includes a finite fraction of the nodes in each network exists only in
networks of sufficiently high mean degree. We call this large mutually
connected cluster a mutual giant component, and we postulate that only nodes
in the mutual giant component remain functional.
We assume that due to an attack or random failure only a fraction of nodes $p$
in one particular network which we will call the root of the NON. We can now
observe a cascade of failures caused by the failure of the dependent nodes in
the networks connected directly to the root by the edges of the NON. The
damage will further spread to more distant networks. Moreover, fragmentation
of each network caused by the removal of certain nodes will cause malfunction
of other nodes which will now belong to small isolated clusters. This
malfunction will cause dependent nodes in neighboring networks to malfunction
as well. Depending on the time scales of these processes, the damage can
spread across the NON back and forth, which we can visualize as cascades of
failures, as shown in Fig. 3 of the SI section. At the end of this process
only the mutual giant component of the NON, if it exists, will remain
functional.
We now introduce generating functions Sergey2010 ; Newman2001 ; Newman2002PRE
; Shao2008 ; Shao2009 of each network, $G_{m0}(z)=\sum P_{m}(k)z^{k}$, and
the generating function of the associated branching process,
$G_{m1}(z)=G^{\prime}_{m0}(z)/G^{\prime}_{m0}(1)$. It is known Newman2001 ;
Newman2002PRE that the generating functions of a randomly connected network
once a fraction $1-p$ of nodes has been randomly removed are the same as the
generating functions of the original network with the new argument $1-p(1-z)$.
Furthermore it is known Shao2008 ; Shao2009 that the fraction of nodes in the
giant component of a single randomly connected network is
$\mu_{\infty,1}=pg_{m}(p)$, where $g_{m}(p)=1-G_{m0}(1-p(1-f_{m}))$ and
$f_{m}$ satisfies a transcendental equation
$f_{m}(p)=G_{m1}(1-p(1-f_{m}(p)))$.
We next prove that the fraction of nodes, $\mu_{\infty,n}$, in the mutual
giant component of a NON composed of $n$ networks is the product:
$\mu_{\infty,n}=p\prod_{m=1}^{n}g_{m}(x_{m}),$ (1)
where each $x_{m}$ satisfies the equation
$x_{m}=\mu_{\infty,n}/g_{m}(x_{m}).$ (2)
The system of $n+1$ equations (1) and (2) defines $n+1$ unknowns:
$\mu_{\infty,n},x_{1},x_{2},...,x_{n}$ as functions of $p$ and the degree
distributions $P_{m}(k)$.
We derive Eqs. (1) and (2) by mathematical induction. (An alternative proof is
given in the SI). Indeed, for $n=1$, Eqs. (1) and (2) follow directly from the
definition of $\mu_{\infty,1}$. Assuming that the formulas are valid for a NON
of $n-1$ networks we will prove that they are valid also for a NON of $n$
networks. A loopless NON of $n$ networks can be always represented as one of
its networks connected by a single edge to the other $n-1$ networks in the
NON. All the nodes in the $n$-th network, which do not belong to the image of
the mutual giant component $\mu_{\infty,n-1}$ of the NON of $n-1$ networks
will stop to function. The fraction of the nodes in the image of this mutual
giant component onto the $n$-th network satisfies the equation
$x_{1,n}=\mu_{\infty,n-1}(p)$. The fraction of nodes belonging to the giant
component of this dependency image is $\mu_{1,n}=x_{1,n}g_{n}(x_{1,n})$. Only
the nodes in the NON of $n-1$ networks which belong to the dependency image of
the giant component of the $n$-th network will remain functional. Due to the
randomness of the connectivity links in different networks, this dependency
image can be represented as a random selection of the fraction
$g_{n}(x_{1,n})$ out of the originally survived nodes, or as random selection
of $p_{1}=pg_{n}(x_{1,n})$ fraction of nodes in one of the networks comprising
the NON of $n-1$ networks. The fraction of nodes in the new mutual giant
component of the NON of $n-1$ networks corresponding to this new random
selection will be $\mu_{\infty,n-1}(p_{1})$. The image of this mutual giant
component in the $n$-th network is equivalent to a random selection of
$x_{2,n}=\mu_{\infty,n-1}(p_{1})/g_{n}(x_{1,n})$ fraction of nodes out of the
entire $n$-th network. As we continue this process, the sequence of giant
components $\mu_{j,n}$ in the $n$-th network, randomly selected sets $x_{n}$
in the $n$-th network and randomly selected sets $p_{j}$ in the NON of $n-1$
networks will satisfy the recursion relations
$x_{j+1,n}=\mu_{\infty,n-1}(p_{j})/g_{n}(x_{j,n})$,
$\mu_{j+1,n}=x_{j+1,n}g_{n}(x_{j+1,n})$, $p_{j+1}=pg_{n}(x_{j+1,n})$.
In the limit $j\to\infty$, this process will converge, i.e. all the parameters
in the two successive steps will coincide: $x_{j+1,n}\to x_{j,n}\equiv x_{n}$,
$p_{j}\to pg_{n}(x_{n})$ and $\mu_{\infty,n-1}(p_{j})\to\mu_{\infty,n}$. Then
$x_{n}=\mu_{\infty,n}/g_{n}(x_{n})$ which is identical to the last equation in
Eqs. (2) and $\mu_{\infty,n-1}(p_{j})\to
pg_{n}(x_{n})\prod_{m=1}^{n-1}g_{m}(x_{m})=p\prod_{m=1}^{n}g_{m}(x_{m})\equiv\mu_{\infty,n}$
which is identical to Eq. (1). By the assumption of induction
$x_{m}=\mu_{\infty,n-1}(p_{j})/g_{m}(x_{m})=\mu_{\infty,n}/g_{m}(x_{m})$ which
completes the set of Eqs. (2). Finally $\mu_{j+1,n}\to
x_{n}g_{n}(x_{n})=\mu_{\infty,n}/g_{n}(x_{n})$ and thus the fraction of nodes
in the giant $n$-th network coincides with the mutual giant component in the
NON of $n-1$ networks. The SI presents an alternative analytical derivation of
Eqs. (1) and (2), which represent a certain type of cascading failures. The SI
also presents simulation results which agree well with the theory (Figs. 5 and
6 in SI).
For the case of a NON with loops, the closed path of fully interdependent
pairs starting form a network A and ending at the same network A will
establish a dependence of nodes $A_{i}$ on node $A_{j_{i}}$, where $j_{i}$ is
a transposition of $i$. Then the failure of single node $i$ will cause an
entire cycle in the transposition to fail. The average size of a cycle in the
transposition of $N$ elements grows as $N/\ln N$, so the initial failure of
$\ln N$ nodes will cause almost all the nodes of the NON to fail without
taking into account any connectivity links which will cause additional damage.
So the NON with loops is unstable against removal of an infinitely small
fraction of nodes unless the transposition $j_{i}$ is not random. In case when
the transposition $j_{i}$ is trivial, $j_{i}=i$, we have the same one-to-one
correspondence between the nodes as in the loopless NON and then Eq. (1) and
(2) are valid. This is since in our proof we did not use any other property of
a NON except the unique one-to-one correspondence of the nodes in different
networks.
The case of NON of $n$ Erdős-Rényi (ER) ER1959 ; ER1960 ; Bollob1985 networks
with average degrees $k_{1},k_{2},...k_{i},...,k_{n}$ can be solved
explicitly. In this case, we have $G_{1,i}(x)=G_{0,i}(x)=\exp[k_{i}(x-1)]$
Newman2002PRE . Accordingly $g_{i}(x_{i})=1-\exp[k_{i}x_{i}(f_{i}-1)]$, where
$f_{i}=\exp[k_{i}x_{i}(f_{i}-1)]$ and thus $g_{i}(x_{i})=1-f_{i}$. Using Eq.
(2) for $x_{i}$ we get
$f_{i}=\exp[-pk_{i}\prod_{j=1}^{n}(1-f_{j})],i=1,2,...,n.$ (3)
These equations can be solved analytically, as shown in detail in the SI
section. They have only a trivial solution ($f_{i}=1$) if $p<p_{c}$, where
$p_{c}$ is the mutual percolation threshold. When the $n$ networks have the
same average degree $k$, $k_{i}=k$ ($i=1,2,...,n$), we obtain from Eq. (3)
that $f_{c}\equiv f_{i}(p_{c})$ satisfies
$f_{c}=e^{\frac{f_{c}-1}{nf_{c}}}.$ (4)
where the solution can be expressed in term of the Lambert function $W(x)$
Lambert1758 ; Corless1996 , $f_{c}=-[nW(-\frac{1}{n}e^{-\frac{1}{n}})]^{-1}$.
Once $f_{c}$ is known, we obtain $p_{c}$ and $\mu_{\infty,n}\equiv P_{\infty}$
by substituting $k_{i}=k$ into Eq. (S10) of the SI section
$\begin{array}[]{lcl}p_{c}=[nkf_{c}(1-f_{c})^{(n-1)}]^{-1},&\mbox{}&\\\
P_{\infty}=\frac{1-f_{c}}{nkf_{c}}.&\mbox{}&\\\ \end{array}.$ (5)
For $n=1$ we obtain the known result $p_{c}=1/k$ of Erdős-Rényi ER1959 ;
ER1960 ; Bollob1985 . Substituting $n=2$ in Eqs. (4) and (5) one obtains the
exact results of Sergey2010 .
To analyze $p_{c}$ as a function of $n$, we find $f_{c}$ from Eq. (4) and
substitute it into Eq. (5), and we obtain $p_{c}$ as a function of $n$ for
different $k$ values, as shown in Fig. 2(a). It is seen that the NON becomes
more vulnerable with increasing $n$ or decreasing $k$ ($p_{c}$ increases when
$n$ increases or $k$ decreases). Furthermore, for a fixed $n$, when $k$ is
smaller than a critical number $k_{min}(n)$, $p_{c}\geq 1$ meaning that for
$k<k_{min}(n)$, the NON will collapse even if a single node fails. Fig. 2 (b)
shows the minimum average degree $k_{\min}$ as a function of the number of
networks $n$. From Eq. (5) we get the minimum of $k$ as a function of $n$
$k_{\min}(n)=[nf_{c}(1-f_{c})^{(n-1)}]^{-1},$ (6)
Note that Eq. (6) together with Eq. (4) yield the value of $k_{\min}(1)=1$,
reproducing the known ER result, that $\langle k\rangle=1$ is the minimum
average degree needed to have a giant component. For $n=2$, Eq. (6) yields the
result obtained in Sergey2010 , i.e., $k_{\min}=2.4554$.
From Eqs. (1)-(3) we obtain the percolation law for the order parameter, the
size of the mutual giant component for all $p$ values and for all $k$ and $n$,
$\mu_{\infty,n}\equiv P_{\infty}=p[1-\exp(-kP_{\infty})]^{n}.$ (7)
The solutions of equation (7) are shown in Fig. 3 for several values of $n$.
Results are in excellent agreement with simulations. The special case $n=1$ is
the known ER percolation law for a single network ER1959 ; ER1960 ; Bollob1985
. Note that Eqs. (4)–(7) are based on the assumption that all $n$ networks
have the same average degree $k$.
In summary, we have developed a framework, Eqs. (1) and (2), for studying
percolation of NON from which we derived an exact analytical law, Eq. (7), for
percolation in the case of a network of $n$ coupled ER networks. Equation (7)
represents a bound for the case of partially dependent networks parshani2010 ,
which will be more robust. In particular for any $n\geq 2$, cascades of
failures naturally appear and the phase transition becomes first order
transition compared to a second order transition in the classical percolation
of a single network. These findings show that the percolation theory of a
single network is a limiting case of a more general case of percolation of
interdependent networks. Due to cascading failures which increase with $n$,
vulnerability significantly increases with $n$. We also find that for any
loopless network of networks the critical percolation threshold and the mutual
giant component depend only on the number of networks and not on the topology
(see Fig. 1(a)). When the NON includes loops, and dependency links are random,
$p_{c}=1$ and no mutual giant component exists.
## References
* (1) Watts D. J. & Strogatz S. H. Nature 393, 440-442 (1998).
* (2) Albert R., Jeong H. & Barabási A. L. Nature 406, 378-382 (2000).
* (3) Cohen R. et al. Phys. Rev. Lett. 85, 4626–4628 (2000).
* (4) Callaway D. S. et al. Phys. Rev. Lett. 85, 5468-5471 (2000).
* (5) Albert R. & Barabási A. L. Rev. Mod. Phys. 74, 47-97 (2002).
* (6) Newman M. E. J. SIAM Review 45, 167-256 (2003).
* (7) Dorogovtsev S. N. & Mendes J. F. F. Evolution of Networks: From Biological Nets to the Internet and WWW (Physics) (Oxford Univ. Press, New York, 2003).
* (8) Song C. et al. Nature 433, 392-395 (2005).
* (9) Satorras R. P. & Vespignani A. Evolution and Structure of the Internet: A Statistical Physics Approach (Cambridge Univ. Press, England, 2006).
* (10) Caldarelli G. & Vespignani A. Large scale Structure and Dynamics of Complex Webs (World Scientific, 2007).
* (11) Barrát A., Barthélemy M. & Vespignani A. Dynamical Processes on Complex Networks (Cambridge Univ. Press, England, 2008).
* (12) Havlin S. & Cohen R. Complex Networks: Structure, Robustness and Function (Cambridge Univ. Press, England, 2010).
* (13) Newman M. E. J. Networks: An Introduction (Oxford Univ. Press, New York, 2010).
* (14) Buldyrev S. V. et al. Nature 464, 1025-1028 (2010).
* (15) Peerenboom J., Fischer R. & Whitfield R. in Pro. CRIS/DRM/IIIT/NSF Workshop Mitigat. Vulnerab. Crit. Infrastruct. Catastr. Failures (2001).
* (16) Rinaldi S., Peerenboom J. & Kelly T. IEEE Contr. Syst. Mag. 21, 11-25 (2001).
* (17) Rosato V. et al. Int. J. Crit. Infrastruct. 4, 63-79 (2008).
* (18) Vespignani A. Nature 464, 984-985 (2010).
* (19) Erdős P. & Rényi A. I. Publ. Math. 6, 290-297 (1959).
* (20) Erdős P. & Rényi A. Publ. Math. Inst. Hung. Acad. Sci. 5, 17-61 (1960).
* (21) Bollobás B. Random Graphs (Academic, London, 1985).
* (22) Newman M. E. J. Strogatz S. H. & Watts D. J., Phys. Rev. E 64, 026118 (2001).
* (23) Newman M. E. J. Phys. Rev. E 66, 016128 (2002).
* (24) Shao J. et al. Europhys. Lett. 84, 48004 (2008).
* (25) Shao J. et al. Phys. Rev. E 80, 036105 (2009).
* (26) Lambert J. H. Acta Helveticae physico mathematico anatomico botanico medica, Band III, 128-168, (1758).
* (27) Corless R. M. et al. Adv. Computational Maths. 5, 329-359 (1996).
* (28) Parshani R. et al. Phys. Rev. Lett. 105, 048701 (2010).
Figure 1: (color online) Three types of loopless networks of networks composed
of five coupled networks. All have same percolation threshold and same giant
component. The darker green node is the origin network on which failures
occur.
Figure 2: (a) The critical fraction $p_{c}$ for different $k$ and $n$ and (b)
minimum average degree $k_{\min}$ as a function of the number of networks $n$.
The results of (a) and (b) are obtained using Eqs. (5) and (6) respectively
and are in good agreement with simulations. In simulations $p_{c}$ was
calculated from the number of cascading failures which diverge at $p_{c}$
parshani2010 (see also Fig. 7 in SI). Figure 3: For loopless NON,
$P_{\infty}$ as a function of $p$ for $k=5$ and several values of $n$. The
results obtained using Eq. (7) agree well with simulations.
|
arxiv-papers
| 2010-10-28T00:11:40 |
2024-09-04T02:49:14.295468
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jianxi Gao, Sergey V. Buldyrev, Shlomo Havlin, and H. Eugene Stanley",
"submitter": "Jianxi Gao",
"url": "https://arxiv.org/abs/1010.5829"
}
|
1010.5848
|
11institutetext: School of Physics, and State Key Laboratory of Nuclear
Physics & Technology, Peking University, China 22institutetext: Milano-Bicocca
University, Italy 33institutetext: INFN Milano-Bicocca, Italy 44institutetext:
European Organization for Nuclear Research 55institutetext: INFN Torino, Italy
66institutetext: Fermi National Accelerator Laboratory, Batavia, IL, USA
# Same Sign WW Scattering Process as a Probe of Higgs Boson in pp Collision at
$\sqrt{s}$ = 10 TeV
Bo Zhu 11 Pietro Govoni 223344 Yajun Mao 11 Chiara Mariotti 55 Weimin Wu
Supported by the National Natural Science Foundation of China (10099630),
Ministry of Science and Technology of China(2007CB816101) and China
Scholarship Council.66
(Received: date / Revised version: date)
###### Abstract
WW scattering is an important process to study electroweak symmetry breaking
in the Standard Model at the LHC, in which the Higgs mechanism or other new
physics processes must intervene to preserve the unitarity of the process
below 1 TeV. This channel is expected to be one of the most sensitive to
determine whether the Higgs boson exists. In this paper, the final state with
two same sign Ws is studied, with a simulated sample corresponding to the
integrated luminosity of 60 fb-1 in pp collision at $\sqrt{s}=$10 TeV. Two
observables, the invariant mass of $\mu\mu$ from W decays and the azimuthal
angle difference between the two $\mu$s, are utilized to distinguish the Higgs
boson existence scenario from the Higgs boson absence scenario. A good signal
significance for the two cases can be achieved. If we define the separation
power of the analysis as the distance, in the log-likelihood plane, of pseudo-
experiments outcomes in the two cases, with the total statistics expected from
the ATLAS and CMS experiments at the nominal centre-of-mass energy of 14 TeV,
the separation power will be at the level of 4 $\sigma$.
###### pacs:
14.80.Bnstandard model Higgs Bosons and 14.70.FmW bosons
## 1 Introduction
It is predicted by the Standard Model(SM) that perturbative unitarity is
violated in vector boson scattering process at high energy if the Higgs
particle is absentEWSB . This implies that the existence of a Higgs boson or
new physics must intervene below 1 TeV. If the Higgs boson does exist, a
resonance could be observed in the VV (WW or ZZ) invariant mass spectrum. On
the other hand, new physics may appear in the form of vector boson pair
resonances, as predicted by Little Higgs, Dynamical symmetry breaking, or
Higgsless modelsEWSB . Therefore, a measurement of WW scattering processes is
a model independent approach to probe the existence or absence of a Higgs
boson.
Figure 1: Same Sign WW Scattering Diagram.
The same sign WW scattering with W decaying to $\mu\nu$ is expected to be a
very clean process to study the difference between the standard model and new
physics scenarios accomando . It has the best separation power between the two
scenarios with respect to the other final states (WW, ZZ, WZ) as shown in
accomando . It will help clarify the electroweak breaking mechanism in case a
Higgs boson like resonance will not be observed or to finally test the
unitarity of the theory. A characteristic signature of the same sign WW
scattering is the presence of two forward jets (tag jets) with high energy
(see Fig.1) which can thus be efficiently extracted from most backgrounds. The
other signature, namely the presence of a same sign isolated muons pair, can
help in suppressing other backgrounds. In this work, we take into account all
the possible backgrounds, including that due to the mis-identification of
leptons (which is usually neglected in other same sign WW scattering studies).
We will show that we can get an almost background free result with the help of
isolation techniques. The final state with 2 electrons or 1 electron and 1
muons have been studied, but the background subtraction result is much less
effective, due to the high rate of mis-identified electrons.
Two same sign WW are produced only via t-channel process, thus no resonances
are expected in the $m_{WW}$ spectrum. The invariant mass of the WW is shown
in Fig.2 at parton level for two different values of the Higgs boson mass and
for the case of no-Higgs. Because of the Parton Distribution Functions, the
expected rise at large $m_{WW}$ values is dramatically suppressed, but still a
substantial difference between the two scenarios (Higgs and “no-Higgs”) can
clearly be observed.
Figure 2: $m_{WW}$ distribution for $m_{H}$ = 200 GeV$/c^{2}$, $m_{H}$ = 500
GeV$/c^{2}$ and no-Higgs Scenarios. The distribution is normalized to 1.
## 2 Monte Carlo Samples
The PHANTOM events generator phantom is used to generate $qq\rightarrow
qq\mu^{\pm}\nu\mu^{\pm}\nu$ processes at $\mathcal{O}(\alpha_{EW}^{6})$ ,
since it performs the full calculations at
$\mathcal{O}(\alpha_{EW}^{6}+\alpha_{EW}^{4}\alpha_{S}^{2})$ order. This is
necessary, since the study aims at comparing the WW scattering spectra under
two different Higgs boson hypotheses: thus it is of crucial importance to
correctly calculate the cross sections, by considering the interferences
between the various tree-level diagrams present in the WW scattering process
calculation.
Different Higgs boson hypotheses samples are generated for the signal: $m_{H}$
= 200 GeV$/c^{2}$, $m_{H}$ = 500 GeV$/c^{2}$ and no-Higgs scenarios. Out of
all the possible diagrams calculated by PHANTOM, the WW scattering process is
isolated by means of the following cuts at parton level: the invariant mass
constraint $|m_{\mu\nu}-m_{W}|$ $<$ 10 GeV$/c^{2}$, the pseudo-rapidity
difference of the final state quarks $\Delta\eta_{qq}$ $>$ 2.0, the invariant
mass of the quarks $m_{qq}$ $>$ 300 GeV$/c^{2}$, the minimal angle between the
final state muon and quark $\Delta R(\mu q)^{min}$ $>$ 1.2. After these
selections surviving events are considered as signal events, the remaining
events are studied as irreducible background.
Besides the irreducible background, some other processes at
$\mathcal{O}(\alpha_{EW}^{4}\alpha_{s}^{2})$ phantom with the same final
states particles are also produced by PHANTOM. These processes are denoted as
“QCD background” in the following.
The $t\bar{t}\rightarrow W^{+}bW^{-}b$ production is another very important
background, in which one hard muon comes from W, the other same sign muon is
from a b-hadron leptonic decay. Single top quark in association with W process
is also considered because of the same reason.
The production of single W along with jets, in which the W decays into
$\mu\nu$ is another dangerous background, because charged long lived hadrons
$(k^{\pm}$,$\pi^{\pm}$,$p^{\pm}$) may be wrongly identified as muons, and the
large cross section compensates for the low probability of the mis-
identification. We assume the probability of mis-identification to be $5\times
10^{-4}$ cmsptdr . In addition to the dominant backgrounds discussed above,
single top, $t\bar{t}$W and di-boson backgrounds (WW, WZ and ZZ) are studied
as well.
QCD and irreducible background samples are produced with PHANTOM, $t\bar{t}$,
W+jet and $t\bar{t}$W backgrounds are generated with Madgraphmadgraph and the
other backgrounds are simulated with PYTHIA at a collision energy of
$\sqrt{s}=$10 TeV. The cross sections of the samples which are produced by
PHANTOM are calculated at the Leading Order (LO), the cross sections of the
other samples are calculated at the Next-to-Leading Order (NLO). The cross
section will be roughly doubled if the collision energy is raised from 10 TeV
to 14 TeV. In all cases including signal and background samples, the parton
showering and hadronization are performed with PYTHIA, and the jet
reconstruction algorithm is also provided by PYTHIA. To include the detector
effect, the muons and jets momenta are smeared by a gaussian distribution with
the resolution based on the following $p_{T}$ resolution
parameterizationresolution ,
for muons:
$\displaystyle\frac{\sigma(p_{T})}{p_{T}}=e^{-4+0.0014\times p_{T}};$ (1)
for jets:
$\displaystyle\frac{\sigma(p_{T})}{p_{T}}=\sqrt{\frac{0.813^{2}}{p_{T}}+\frac{3.9^{2}}{p^{2}_{T}}+0.017^{2}}.$
(2)
## 3 Event Selection
The aim of the selection strategy is to achieve a reasonable level of signal
over background ratio. We concentrate on a cut-based selection strategy. The
selection chain includes two main parts: muon selection and jet selection.
A pair of same sign isolated hard muons is one of the most significant
characteristics of the signal process. Most standard model background events,
such as W+jet, $t\bar{t}$, single top and di-boson, comprise only one muon or
two opposite charged muons in the final state. If there are two same sign
muons in these events, one muon should come from b-hadron decay or muon mis-
identification from other backgrounds. Most of the non-top background events
contain at least one fake muon mostly in the low $p_{T}$ region. A $p_{T}$
threshold of 15 GeV$/c$ is required to suppress these kinds of background,
especially the W+jet events.
The muon isolation criteria are applied to all the tracks of charged
particles, which can be well reconstructed with an efficiency of almost
$100\%$ when $p_{T}$ $>$ 0.5 GeV$/c$ resolution . The isolation parameter is
defined as the sum of the $p_{T}$ of charged particles in an isolation cone of
0.3 rad centered around the muon at the primary vertex, in the ($\eta$,$\phi$)
plane. The footprint of the muon itself is removed by an inner veto cone of
0.01 rad:
$\displaystyle\beta={\Sigma p_{T}(0.01<\Delta R<0.3)}.$ (3)
As the top background is the most important one, the following isolation cuts
are tuned to reduce this contribution: $\beta<$1 GeV$/c$ and
$\beta/p_{T}(\mu)$ $<$ 0.05.
The vector boson scattering signature is exploited as well to further reduce
the backgrounds contribution. The tag jets are identified as the ones with
highest $p_{T}$ in the event. There will be very high fake rate for low
$p_{T}$ jets, so the $p_{T}$ threshold of the tag jets is 30 GeV$/c$. A number
of different strategies to implement tag jets selection were compared, and the
best rejection factor for a given efficiency is obtained by requiring the tag
jets with the opposite sign of pseudo-rapidity ($\eta$), to satisfy the $\eta$
difference $\Delta\eta_{jj}$ $>$ 4 and tag jets invariant mass $m_{jj}$ $>$
600 GeV$/c^{2}$.
The event number after the cut-based selection for signal and background are
shown in Table 1. The results are normalized to an integrated luminosity of 60
fb-1. For $t\bar{t}$, W+jet, single top and di-boson backgrounds, Monte Carlo
samples corresponding to 60 fb-1 are too large to be simulated, due to the
very large cross section. Only few events survive after the selection chain
with high statistics error. The expected number of events therefore will be
estimated with the efficiency factorization as discussed below.
$m_{H}$ = 200 GeV$/c^{2}$ | no-Higgs | Backgrounds
---|---|---
12.2 | 13.7 | 5.9
Table 1: Number of surviving events for signal and background after muon and
jet selection with an integrated luminosity of 60 fb-1
## 4 Higgs versus no-Higgs scenario
To distinguish the scenario where the Higgs boson is existing from the one
where the Higgs boson is absent, two possible additional selections have been
investigated. We choose the following relative separation definition to
optimize the selections:
$\displaystyle\alpha=\frac{N_{NoH}-N_{m_{H}(200)}}{\sqrt{N_{m_{H}(200)}+N_{Bkg}}}.$
(4)
where $N_{m_{H}(200)}$, $N_{NoH}$ and $N_{Bkg}$ are the number of events for
the two cases and for the backgrounds respectively. For this study the value
of the Higgs boson mass is not relevant, as explained in detailed in
ref.accomando
The region of high values of invariant mass of W bosons $(m_{WW})$ should be
sensitive to the presence of a Higgs particle (Fig.2). Unfortunately, because
of the presence of neutrinos, it is impossible to reconstruct the invariant
mass of the W bosons. Therefore, the invariant mass of the two muons system is
used to replace $m_{WW}$ and the events count for Equation 4 is performed
after a cut on the $m_{\mu\mu}$ value.
Fig.3 (a) shows the $m_{\mu\mu}$ distribution for the two scenarios
($m_{Higgs}$ = 200 GeV$/c^{2}$ and no-Higgs). Fig.3 (b) shows the number of
surviving signal events as a function of the $m_{\mu\mu}$ cut. Fig.3 (c) shows
the distribution of the relative separation (as defined in Equation 4) $vs.$
the cut on $m_{\mu\mu}$. To obtain a better separation between the two cases,
we require the muon to be in the central region: $|\eta_{\mu}|$ $<$ 2\. By
asking $m_{\mu\mu}$ $>$ 200 GeV$/c^{2}$, we can achieve good signal
significance and background control. However, the request is too tight, since
it eliminates about 80$\%$ of signal events (Fig.3 (b)).
a b c
Figure 3: Invariant mass distribution of the two muons ($m_{\mu\mu}$) (a) ,
the number of surviving events as a function of the cut on $m_{\mu\mu}$ (b),
relative separation $\alpha$ $vs.$ the $m_{\mu\mu}$ cut value (c). Results are
normalized to 60 fb-1.
Alternatively, a selection on the azimuthal angle between muons is
investigated, as the vector bosons tend to be back to back in a scattering
topology. Fig.4 (a) shows the $\Delta\phi$ distribution between the two muons
for the two cases. Fig.4 (b) shows the number of surviving events as a
function of a minimum $\Delta\phi_{\mu\mu}$ cut. Fig.4 (c) is the distribution
of the relative separation as defined in Equation 4 $vs.$ different
$\Delta\phi_{\mu\mu}$ cuts. With the cut $\Delta\phi_{\mu\mu}$ $>$ 2, the
highest separation is obtained with a loss of about 50$\%$ of signal events.
Only QCD and irreducible backgrounds are considered in Fig.3 (c) and Fig.4
(c).
a b c
Figure 4: $\Delta\phi_{\mu\mu}$ distribution (a), the number of surviving
events as a function of the $\Delta\phi_{\mu\mu}$ cut (b), relative separation
$\alpha$ $vs.$ the $\Delta\phi_{\mu\mu}$ cut value (c). Results are normalized
to 60 fb-1.
## 5 Background estimation
The main uncertainty comes from the simulated background samples statistical
error. Because of the limited statistics available, no event remains for
W+jet, top and di-boson samples. However, we cannot ignore those backgrounds
because of their very large cross sections.
Assuming there is no correlation among the single selections, we estimate the
number of surviving events by multiplying the single efficiencies:
$\displaystyle N=\sigma\times
L(60fb^{-1})\times\xi_{cut1}\times\xi_{cut2}...\times\xi_{cuti},$ (5)
where the $\xi_{cuti}$ is the efficiency for the i-th selection alone on each
sample. There is a very low level of correlation between the two main
selections, namely the jet selections and muon selections. The expected number
of background events for each sample using two different discriminators are
summarized in Table 2 with an integrated luminosity of 60 fb-1 .
Discriminator | top | W+jet | di-boson
---|---|---|---
$m_{\mu\mu}$ | 0.65 | 0.05 | 0.02
$\Delta\phi_{\mu\mu}$ | 2.6 | 0.2 | 0.1
Table 2: Estimated number of events of backgrounds
The signal significance is determined using the likelihood ratio method, with
poissonian probability density distributions, for both the $m_{\mu\mu}$ and
$\Delta\phi_{\mu\mu}$ selections with the background estimates in Table 2.
Results are listed in Table 3. The number of signal and background events are
shown after the selection. We make the hypothesis that the correlation between
the cuts will give 100$\%$ uncertainty for W+jet, top and di-boson
backgrounds. For the other samples, only the statistical error is considered.
Discriminator | No H | $m_{H}(200)$ | ${NoH}/{m_{H}(200)}$ | Background | Relative Separation | $S_{(m_{H}(200))}$ | $S_{NoH}$
---|---|---|---|---|---|---|---
$m_{\mu\mu}$ | 2.4$\pm$0.1 | 1.8$\pm$0.1 | 1.3$\pm$0.1 | 1.0$\pm$0.8 | 0.35 | 1.5 | 1.9
$\Delta\phi_{\mu\mu}$ | 6.5$\pm$0.1 | 5.4$\pm$0.1 | 1.2$\pm$0.1 | 3.5$\pm$2.9 | 0.36 | 2.4 | 2.8
Table 3: Signal significances, ratio and separation power between Higgs case
and no-Higgs case with an integrated luminosity of 60 fb-1, at 10 TeV centre-
of-mass energy.
## 6 Summary and Discussion
Figure 5: Normalized Likelihood Ratio with $m_{\mu\mu}$ cut, H0 hypothesis is
$m_{H}$ = 200GeV, H1 is no higgs. Result is corresponding to an inverse
luminosity of 6 ab-1 at $\sqrt{s}$ = 14 TeV
Assuming a poissonian pdf of the measurements in the Higgs boson existing
scenario and Higgs boson absence scenario, a likelihood-ratio is built to
distinguish the two hypotheses, giving the number of measured events. To
assess the separation power of the analysis, a set of toy-montecarlo
experiments have been generated for each of the cases, and the distributions
of the corresponding likelihood-ratios have been compared. To evaluate the
separation between the two curves, the distance between their maxima,
normalized to their sigma, is calculated (the sigma is taken as the half-width
of the narrowest interval containing 68% of the distribution):
$\delta~{}=~{}\frac{|\max(LLR)_{H0}-\max(LLR)_{H1}|}{\sigma_{H0}~{}\oplus~{}\sigma_{H1}}$
(6)
where $H0$ and $H1$ represent the Higgs boson and no-Higgs boson hypotheses
respectively. Fig.5 shows the distributions when we scale the results by the
total expected statistics collected by ATLAS and CMS (corresponding to an
inverse luminosity of 6 ab-1 at 14 TeV of LHC centre-of-mass energy). A
4$\sigma$ separation for the two hypotheses can be achieved.
We present an exploratory study of the same sign W scattering process with W
decay into $\mu\nu$ as probe of Higgs boson existence in pp collisions at
$\sqrt{s}$ = 10 TeV. All the standard model backgrounds are considered, with
detector effects parameterized, including muon mis-identification effect. It
is a clean channel compared with the other VV scattering processes accomando
vvscat because of the two main signatures, which are the same sign isolated
muons pair and energetic forward jets. $m_{\mu\mu}$ and $\Delta\phi_{\mu\mu}$
are both good discriminants to distinguish a Higgs scenario from the no-Higgs
one.
Although the cross section is not as large as searching for Higgs Boson via
di-boson resonances directly, it is a model independent channel to determine
if the Higgs boson exists, whatever the value of its mass, and to verify the
unitarity of the theory.
With the total statistics expected from the ATLAS and CMS experiments at 14
TeV, the separation power between Higgs boson and no-Higgs boson scenarios
will be at the level of 4 $\sigma$.
## References
* (1) M.J.G. Veltman, CERN-97-05; M.S. Chanowitz, [hep-ph/9812215]; S. Dawson, [hep-ph/9901280]; Chris Quigg, Acta Phys.Polon. B30 (1999) 2145\. [hep-ph/9905369]; S. Dawson Int.J.Mod.Phys. A21 (2006) 1629. [hep-ph/0510385]; R. Rattazzi PoS HEP2005 (2006) 399.[hep-ph/0607058]
* (2) E. Accomando et al., JHEP 0603 (2006) 093;
* (3) J. Bagger et al Phys.Rev.D52:3878-3889,1995.
* (4) A.Ballestrero, A.Belhouari, G.Bevilacqua etc. Computer Physics Communications 180 (2009) 401 C417
* (5) T.Sjostrand, S.Mrenna and P.Skands, JHEP 0605 (2006) 026
* (6) The CMS Collaboration, CERN-LHCC-2006-001
* (7) F. Maltoni and T. Stelzer, JHEP 0302 (2003) 027 [arXiv:hep-ph/0208156]; T. Stelzer and W. F. Long, Comput. Phys. Commun. 81 (1994) 357 [arXiv:hep-ph/9401258].
* (8) The ATLAS Collaboration, CERN-LHCC-1999-14 vol 1
* (9) A.Sznajder for CMS Collaboration, [arXiv:0810.3604v2]
|
arxiv-papers
| 2010-10-28T03:33:12 |
2024-09-04T02:49:14.303610
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Bo Zhu and Pietro Govoni and Yajun Mao and Chiara Mariotti and Weimin\n Wu",
"submitter": "Bo Zhu",
"url": "https://arxiv.org/abs/1010.5848"
}
|
1010.5861
|
# Lattice QCD analysis for instantaneous interquark potential in generalized
Landau gauge
Kyoto University
E-mail Hideo Suganuma
Kyoto University
E-mail suganuma@ruby.scphys.kyoto-u.ac.jp
###### Abstract:
Using generalized Landau gauge, we study the continuous change of gluon
properties from the Landau gauge toward the Coulomb gauge in SU(3) lattice
QCD. We investigate “instantaneous interquark potential”, which is defined by
the spatial correlation of the temporal link-variable $U_{4}$ and is an
interesting gauge-dependent concept. In the Coulomb gauge, the instantaneous
potential is expressed by the Coulomb plus linear potential, where the slope
is, however, 2-3 times as large as the physical string tension. In the Landau
gauge, the instantaneous potential has no linear part. We find that the linear
part is continuously growing by varying gauge from the Landau gauge toward the
Coulomb gauge. We also find that the instantaneous potential approximately
reproduces the physical interquark potential in a specific intermediate gauge,
$\lambda_{C}$-gauge. This $\lambda_{C}$-gauge is expected to be a useful gauge
for modeling effective theories such as the quark potential model.
## 1 Introduction
Quantum Chromodynamics (QCD) is the fundamental theory of the strong
interaction, and color SU(3) gauge symmetry is one of the guiding principles
to construct the theory. In fact, QCD is formulated to satisfy the SU(3) gauge
symmetry. In actual calculations, however, one needs to choose some gauge in
order to remove redundant gauge degrees of freedom. According to the choice of
the gauge, different physical pictures could be obtained, while physical
quantities and phenomena do not depend on the gauges.
On the confinement, the most intuitive picture would be the dual-
superconductor effect, which was proposed by Nambu, Mandelstam and ’t Hooft
[1]. In this scenario, the linear confinement potential between quarks is
derived from the one-dimensional formation of color-electric flux tube in the
dual-superconductor, and is mainly discussed in the maximally Abelian gauge.
One of the other famous confinement scenarios is Kugo-Ojima criterion [2]. In
this scenario, the confinement is mathematically analyzed in terms of the BRST
charge, which is formulated in a covariant and globally SU($N_{c}$) symmetric
gauge such as the Landau gauge.
In the Coulomb gauge, Gribov and Zwanziger proposed that the confinement force
is closely related to “instantaneous color Coulomb interaction” between quarks
[3, 4], which is known as Gribov-Zwanziger scenario. Greensite et al. showed
that the instantaneous interaction actually produces a linear interquark
potential from lattice QCD calculation [5]. However, the slope of the
instantaneous potential is 2-3 times larger than the actual value of the
physical string tension.
In this paper, we investigate the change of gluon properties and physical
picture by varying gauge continuously from the Landau gauge toward the Coulomb
gauge. In particular, we focus on behavior of the instantaneous interquark
potential, which is a key concept in Gribov-Zwanziger scenario. We also
discuss about the linkage between QCD and the quark potential model, which is
one of the most successful effective models to describe hadron properties.
## 2 Formalism
### 2.1 Generalization of the Landau gauge
The Landau gauge is one of the most popular gauges in QCD, and its gauge
fixing is given by
$\partial_{\mu}A_{\mu}=0,$ (1)
where $A_{\mu}$ are $\mathrm{SU}(N_{c})$ gauge fields. The Landau gauge keeps
the Lorentz covariance and global $\mathrm{SU}(N_{c})$ color symmetry. In the
Euclidean space-time, the Landau gauge is also defined as the global condition
to minimize the quantity $R_{\mathrm{Landau}}\equiv\int d^{4}x\
\mathrm{Tr}\left\\{A_{\mu}(x)A_{\mu}(x)\right\\}$ by gauge transformation [6].
The Coulomb gauge is also one of the most popular gauges, and is defined as
$\partial_{i}A_{i}=0.$ (2)
This condition resembles the Landau gauge, but there are no constraints on
$A_{4}$. In the Coulomb gauge, the Lorentz covariance is partially broken, and
gauge field components are completely decoupled into $\vec{A}$ and $A_{4}$:
$\vec{A}$ behave as canonical variables and $A_{4}$ becomes an instantaneous
potential.
From Eqs.(1) and (2), we consider generalization of the Landau gauge, which is
defined as
$\partial_{i}A_{i}+\lambda\partial_{4}A_{4}=0.$ (3)
This generalized Landau gauge is called as “$\lambda$-gauge” [7]. By varying
$\lambda$-parameter from $1$ to $0$, we can change the gauge continuously from
the Landau gauge toward the Coulomb gauge.
The lattice QCD action is constructed from link-variables
$U_{\mu}(x)\in\mathrm{SU}(N_{c})$ instead of the gauge fields
$A_{\mu}(x)\in\mathfrak{su}(N_{c})$, and gauge fixing condition is also
expressed in terms of $U_{\mu}(x)$. On the lattice, $\lambda$-gauge fixing is
defined as the maximization of
$R_{\lambda}[U]\equiv\sum_{x}\large\\{\sum_{i=1}^{3}\mathrm{Re}\ \mathrm{Tr}\
U_{i}(x)+\lambda\mathrm{Re}\ \mathrm{Tr}\ U_{4}(x)\large\\}$ (4)
by gauge transformation of link-variables, $U_{\mu}(x)\rightarrow
U^{\prime}_{\mu}(x)=\Omega(x)U_{\mu}(x)\Omega^{\dagger}(x+\hat{\mu}),\
\Omega\in\mathrm{SU}(N_{c})$.
### 2.2 Terminated Polyakov line and instantaneous potential
After generalized Landau gauge fixing, we calculate “$T$-length terminated
Polyakov line” $L(\vec{x},T)$, which is defined as
$L(\vec{x},T)\equiv U_{4}(\vec{x},1)U_{4}(\vec{x},2)\cdots
U_{4}(\vec{x},T),\quad T=1,2,\dots,N_{t}$ (5)
on $N_{s}^{3}\times N_{t}$ lattice. Here, we use the lattice unit of $a=1$. We
note that $L(\vec{x},T)$ is a gauge-dependent quantity. For $T=N_{t}$, the
trace of $T$-length Polyakov line, $\mathrm{Tr}\ L(\vec{x},N_{t})$, results in
the Polyakov loop.
Using $T$-length Polyakov line, we define “finite-time potential”
$V_{\lambda}(R,T)$ in $\lambda$-gauge as
$V_{\lambda}(R,T)\equiv-\frac{1}{T}\ln\langle\mathrm{Tr}[L^{\dagger}(\vec{x},T)L(\vec{y},T)]\rangle,\quad
R=|\vec{x}-\vec{y}|.$ (6)
$V_{\lambda}(R,T)$ gives the energy between two color sources, which are
created at $t=0$ and annihilated at $t=T$.
Especially for $T=1$, we call $V_{\lambda}(R,1)$ as “instantaneous potential”
$V_{\lambda}(R)\equiv
V_{\lambda}(R,1)=-\ln\langle\mathrm{Tr}[U_{4}^{\dagger}(\vec{x},1)U_{4}(\vec{y},1)]\rangle,\qquad
R=|\vec{x}-\vec{y}|.$ (7)
In the Coulomb gauge, the instantaneous potential is expressed by the Coulomb
plus linear potential [5], while no linear part appears in this potential in
the Landau gauge [6, 8].
## 3 Lattice QCD calculation
We perform SU(3) lattice QCD Monte Carlo calculations on $16^{4}$ with
lattice-parameter $\beta=5.8$ at the quenched level. The lattice spacing $a$
is $0.152$fm, which is determined so as to reproduce the string tension as
$\sqrt{\sigma}_{\rm phys}=427$MeV [9]. We investigate the Landau gauge, the
Coulomb gauge, and their intermediate gauges, i.e., $\lambda$-gauge with
$\lambda=0.75,0.50,0.25,0.10,0.05,0.04,0.03$, $0.02,0.01$. The number of gauge
configurations is 50 for each $\lambda$. The statistical error is estimated by
the jackknife method.
### 3.1 Instantaneous potential
We investigate the instantaneous potential $V_{\lambda}(R)$ in generalized
Landau gauge. Figure 2 shows gauge dependence of $V_{\lambda}(R)$. In this
figure, the statistic error is small and hidden in the symbols.
In the Coulomb gauge ($\lambda=0$), the instantaneous potential shows linear
behavior, while there is no linear part at all in the Landau gauge
($\lambda=1$). Thus, there is a large gap between these gauges in terms of the
instantaneous potential. By varying gauge from the Landau gauge toward the
Coulomb gauge, the potential $V_{\lambda}(R)$ grows monotonically, and these
two gauges are connected continuously. (See Fig.2.)
To analyze the instantaneous potential quantitatively, we fit the lattice QCD
results using Coulomb plus linear functional form as
$V_{\lambda}(R)=-\frac{A_{\lambda}}{R}+\sigma_{\lambda}R+C_{\lambda},$ (8)
where $A_{\lambda}$ is Coulomb coefficient, $\sigma_{\lambda}$ slope of the
potential (string tension), and $C_{\lambda}$ a constant. Here, besides the
Coulomb plus linear Ansatz, we try several candidates of the functional form,
$-A/R+\sigma(1-e^{-\varepsilon R})/\varepsilon$, $-A\exp(-mR)/R$, $-A/R+\sigma
R^{d}$, and $-A/R^{d}$, but they are less workable. The curves in Fig. 2 are
the best-fit results using Eq.(8). The Coulomb plus linear Ansatz works well
at least for $R\lesssim 0.8$fm, which is relevant region for hadron physics.
In the deep IR limit, $R\rightarrow\infty$, $V_{\lambda}(R)$ goes to a
saturated value, except for $\lambda=0$.
Figure 1: “Instantaneous potential” $V_{\lambda}(R)$ in generalized Landau
gauge for typical values of $\lambda$. Symbols are lattice QCD results, and
curves are fit results using Coulomb plus linear Ansatz.
Figure 2: “Instantaneous string tension” $\sigma_{\lambda}$ in generalized
Landau gauge. $\sigma_{\lambda}$ changes continuously from the Landau gauge to
the Coulomb gauge. $\sigma_{\lambda}$ coincides with physical value
$\sigma_{\rm phys}$ at $\lambda_{C}\sim 0.02$.
We focus on the gauge dependence of the linear slope $\sigma_{\lambda}$, which
we call “instantaneous string tension”. (See Fig.2.) For $\lambda\gtrsim 0.1$,
$\sigma_{\lambda}$ is almost zero, so that this region can be regarded as
“Landau-like.” For $\lambda\lesssim 0.1$, $V_{\lambda}(R)$ is drastically
changed near the Coulomb gauge, and $\sigma_{\lambda}$ grows rapidly in this
small region. Finally, in the Coulomb gauge, one finds $\sigma_{\lambda}\simeq
2.6\sigma_{\rm phys}$ ($\sigma_{\rm phys}=0.89$GeV/fm). Thus, the slope of the
potential grows continuously from the Landau gauge ($\sigma_{\lambda}\simeq
0$) towards the Coulomb gauge ($\sigma_{\lambda}\simeq 2.6\sigma_{\rm phys}$),
and therefore there exists some specific $\lambda$-parameter of $\lambda_{C}$
where the slope of the instantaneous potential coincides with the physical
string tension.
From Fig.2, the value of $\lambda_{C}$ is estimated to be about 0.02. In this
$\lambda_{C}$-gauge, the physical static interquark potential $V_{\rm
phys}(R)$ is approximately reproduced by the instantaneous potential. (See
Fig.4.) While $V_{\rm phys}(R)$ is derived from large $T$ behavior of the
Wilson loop $W(R,T)$ as $V_{\rm
phys}(R)=-\lim_{T\rightarrow\infty}\frac{1}{T}\ln\langle W(R,T)\rangle$, only
instantaneous correlation of $U_{4}$ approximately reproduces the physical
static potential in $\lambda_{C}$-gauge. (See Fig.4.)
Figure 3: The instantaneous potential at $\lambda=0.02$ $(\sim\lambda_{C})$,
the solid line is physical interquark potential $V_{\rm phys}(R)=-A_{\rm
phys}/R+\sigma_{\rm phys}R$ with $A_{\rm phys}=0.27$, and $\sigma_{\rm
phys}=0.89$GeV/fm.
Figure 4: Schematic picture of physical interquark potential and
instantaneous potential. In $\lambda_{C}$-gauge, instantaneous potential
$V_{\rm inst}$ approximately reproduces the physical potential $V_{\rm phys}$.
### 3.2 Finite-time potential
Next, we analyze finite-time potential $V_{\lambda}(R,T)$ defined by Eq.(6),
which is a generalization of instantaneous potential.
First, we consider the Coulomb gauge. Figure 6 shows the lattice QCD result
for $V_{\lambda}(R,T)$ in the Coulomb gauge. Similar to the instantaneous
potential, $V_{\lambda}(R,T)$ is well reproduced by the Coulomb plus linear
form. However, the parameter values are changed according to $T$-length. In
particular, the slope of the potential becomes smaller as $T$ becomes larger,
which shows an “instability” of $V_{\lambda}(R,T)$ in terms of $T$ in the
Coulomb gauge.
For general $\lambda$, finite-time potential $V_{\lambda}(R,T)$ is found to be
reproduced by the Coulomb plus linear form as
$V_{\lambda}(R,T)=-\frac{A_{\lambda}(T)}{R}+\sigma_{\lambda}(T)R+C_{\lambda}(T),$
(9)
at least for $R\lesssim 0.8$fm, similarly for the instantaneous potential.
We focus on $T$-length dependence of the slope $\sigma_{\lambda}(T)$ of
$V_{\lambda}(R,T)$ at each $\lambda$. (See Fig. 6.) In the Coulomb gauge
($\lambda=0$), $\sigma_{\lambda}(T)$ is a decreasing function: starting from
2-3 times larger value, it approaches to the physical string tension
$\sigma_{\rm phys}$, as $T$ increases. Around $\lambda_{C}$-gauge, i.e., for
$\lambda\sim\lambda_{C}(\simeq 0.02)$, $T$-dependence is relatively weak, and
$\sigma_{\lambda}(T)$ seems to converge on the same value of about
$1.3\sigma_{\rm phys}$ around $T\sim 1$fm. For $\lambda\gtrsim 0.1$ (Landau-
like), $\sigma_{\lambda}(T)$ is an increasing function of $T$: starting from
zero at $T=1$, the linear part of $V_{\lambda}(R,T)$ appears and grows, as $T$
increases. At each $\lambda$, $\sigma_{\lambda}(T)$ seems to approach to the
physical string tension $\sigma_{\rm phys}$ for sufficiently large $T$.
Figure 5: “Finite-time potential” $V_{\lambda}(R,T)$ in the Coulomb gauge
$(\lambda=0)$. An irrelevant constant is shifted. The curves are the fit
results using Coulomb plus linear function.
Figure 6: $T$-length dependence of the slope $\sigma_{\lambda}(T)$ of finite-
time potential $V_{\lambda}(R,T)$ in generalized Landau gauge for several
typical $\lambda$-values.
## 4 Summary and Discussion
In this paper, aiming to grasp the gauge dependence of gluon properties, we
have investigated generalized Landau gauge and applied it to instantaneous
interquark potential in SU(3) lattice QCD at $\beta$=5.8. In the Coulomb
gauge, the instantaneous potential is expressed by the sum of Coulomb
potential and linear potential with 2-3 times larger string tension. In
contrast, the instantaneous potential has no linear part in the Landau gauge.
Thus, there is a large gap between these two gauges. Using generalized Landau
gauge, we have found that the instantaneous potential $V_{\lambda}(R)$ is
connected continuously from the Landau gauge towards the Coulomb gauge, and
the linear part in $V_{\lambda}(R)$ grows rapidly in the neighborhood of the
Coulomb gauge.
Since the slope $\sigma_{\lambda}$ of the instantaneous potential
$V_{\lambda}(R)$ grows continuously from 0 to 2-3$\sigma_{\rm phys}$, there
must exist some specific intermediate gauge where the slope $\sigma_{\lambda}$
coincides with the physical string tension $\sigma_{\rm phys}$. From the
lattice QCD calculation, the specific $\lambda$-parameter, $\lambda_{C}$, is
estimated to be about $0.02$. In this $\lambda_{C}$-gauge, the physical static
interquark potential $V_{\rm phys}(R)$ is approximately reproduced by the
instantaneous potential $V_{\lambda}(R)$. (See Fig.4.)
We have also investigated finite-time potential $V_{\lambda}(R,T)$, which is
defined from $T$-length terminated Polyakov line and a generalization of the
instantaneous potential. The behavior of the slope $\sigma_{\lambda}(T)$ of
finite-time potential is classified into three groups: the Coulomb-like gauge
($\lambda\lesssim 0.01$), the Landau-like gauge ($\lambda\gtrsim 0.1$), and
neighborhood of $\lambda_{C}$-gauge ($\lambda\sim\lambda_{C}$). In the
Coulomb-like gauge, the slope $\sigma_{\lambda}(T)$ is a decreasing function
of $T$, and seems to approach to physical string tension $\sigma_{\rm phys}$
for large $T$. In the Landau-like gauge, $\sigma_{\lambda}(T)$ is an
increasing function. Around the $\lambda_{C}$-gauge, $\sigma_{\lambda}(T)$ has
a weak $T$-length dependence.
Finally, we consider a possible gauge of QCD to describe the quark potential
model from the viewpoint of instantaneous potential. The quark potential model
is a successful nonrelativistic framework with a potential instantaneously
acting among quarks, and describes many hadron properties in terms of quark
degrees of freedom. In this model, there are no dynamical gluons, and gluonic
effects indirectly appear as the instantaneous interquark potential.
As for the Coulomb gauge, the instantaneous potential has too large linear
part, which gives an upper bound on the static potential [10]. It has been
suggested by Greensite et al. that the energy of the overconfining state is
lowered by inserting dynamical gluons between (anti-)quarks, which is called
“gluon-chain picture”. This gluon-chain state is considered as the ground
state in the Coulomb gauge [5, 11]. Therefore, dynamical gluon degrees of
freedom must be also important to describe hadron states in the Coulomb gauge.
For $\lambda_{C}$-gauge, the physical interquark potential $V_{\rm phys}(R)$
is approximately reproduced by the instantaneous potential
$V_{\lambda_{C}}(R)$. This physically means that all other complicated effects
including dynamical gluons and ghosts are approximately cancelled in the
$\lambda_{C}$-gauge, and therefore we do not need to introduce any redundant
gluonic degrees of freedom. The absence of dynamical gluon degrees of freedom
would be a desired property for the quark model picture. The weak $T$-length
dependence of $\sigma_{\lambda}(T)$ around the $\lambda_{C}$-gauge ($T$-length
stability) is also a suitable feature for the potential model. In this way, as
an interesting possibility, the $\lambda_{C}$-gauge is expected to be a useful
gauge in considering the linkage from QCD to the quark potential model.
## Acknowledgements
This work is supported by the Global COE Program, “The Next Generation of
Physics, Spun from Universality and Emergence” at Kyoto University. H.S. is
supported in part by the Grant for Scientific Research [(C) No. 19540287,
Priority Areas “New Hadrons” (E01:21105006)] from the Ministry of Education,
Culture, Science, and Technology (MEXT) of Japan. The lattice QCD calculations
have been done on NEC-SX8 at Osaka University.
## References
* [1] Y. Nambu, Phys. Rev. D10, 4262 (1974); S. Mandelstam, Phys. Rept. 23, 245 (1976);
G. ’t Hooft, Nucl. Phys. B190, 455 (1981).
* [2] T. Kugo and I. Ojima, Suppl. Prog. Theor. Phys. 66, 1-130 (1979);
T. Kugo, Proc. of Int. Symp. on “BRS Symmetry on the Occasion of Its 20th
Anniversary”, 107 [arXiv:hep-th/9511033].
* [3] V. Gribov, Nucl. Phys. B139, 1 (1978).
* [4] D. Zwanziger, Nucl. Phys. B518, 237 (1998).
* [5] J. Greensite and S. Olejník, Phys. Rev. D 67, 094503 (2003);
J. Greensite, S. Olejník, and D. Zwanziger, Phys. Rev. D 69, 074506 (2004);
J. Greensite, Prog. Part. Nucl. Phys. 51, 1 (2003).
* [6] T. Iritani, H. Suganuma, and H. Iida, Phys. Rev. D 80, 114505 (2009);
H. Suganuma, T. Iritani, A. Yamamoto, and H. Iida, PoS (QCD-TNT09), 044 (2009)
[arXiv:hep-lat/0912.0437].
* [7] C. Bernard, D. Murphy, A. Soni, and K. Yee, Nucl. Phys. B. (Proc. Suppl.) 17, 593 (1990);
C. Bernard, D. Murphy, and A. Soni, Nucl. Phys. B. (Proc. Suppl.) 20, 410
(1991).
* [8] A. Nakamura and T. Saito, Prog. Theor. Phys. 115, 189 (2006).
* [9] H. Suganuma, T.T. Takahashi, and H. Ichie, Color Confinement and Hadrons in Quantum Chromodynamics (World Scientific, Singapore, 2004), p.249;
T.T. Takahashi et al., Phys. Rev. D 65, 114509 (2002); Phys. Rev. Lett. 86, 18
(2001).
* [10] D. Zwanziger, Phys. Rev. Lett. 90, 102001 (2003).
* [11] J. Greensite and C.B. Thorn, JHEP 02, 014 (2002).
|
arxiv-papers
| 2010-10-28T04:56:28 |
2024-09-04T02:49:14.314560
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Takumi Iritani, Hideo Suganuma",
"submitter": "Takumi Iritani",
"url": "https://arxiv.org/abs/1010.5861"
}
|
1010.5915
|
# hypercyclic abelian semigroups of matrices on $\mathbb{R}^{n}$
Adlene Ayadi1 and Habib Marzougui2 Adlene Ayadi1, University of Gafsa, Faculty
of Science of Gafsa, Department of Mathematics, Tunisia; Habib Marzougui2,
University of 7th November at Carthage, Faculty of Science of Bizerte,
Department of Mathematics, Zarzouna. 7021. habib.marzouki@fsb.rnu.tn;
adleneso@yahoo.fr
###### Abstract.
We give a complete characterization of existence of dense orbit for any
abelian semigroup of matrices on $\mathbb{R}^{n}$. For finitely generated
semigroups, this characterization is explicit and it is used to determine the
minimal number of matrices in normal form over $\mathbb{R}$ which form a
hypercyclic abelian semigroup on $\mathbb{R}^{n}$. In particular, we show that
no abelian semigroup generated by $\left[\frac{n+1}{2}\right]$ matrices on
$\mathbb{R}^{n}$ can be hypercyclic. ($[\ ]$ denotes the integer part).
###### Key words and phrases:
Hypercyclic, matrices, dense orbit, locally dense, semigroup, abelian subgroup
###### 2000 Mathematics Subject Classification:
37C85, 47A16
This work is supported by the research unit: systèmes dynamiques et
combinatoire: 99UR15-15
## 1\. Introduction
Let $M_{n}(\mathbb{R})$ be the set of all square matrices over $\mathbb{R}$ of
order $n\geq 1$ and by GL($n,\mathbb{R})$ the group of invertible matrices of
$M_{n}(\mathbb{R})$. Let $G$ be an abelian sub-semigroup of
$M_{n}(\mathbb{R})$. For a vector $v\in\mathbb{R}^{n}$, we consider the orbit
of $G$ through $v$: $G(v)=\\{Av:\ A\in G\\}\subset\mathbb{\mathbb{R}}^{n}$.
The orbit $G(v)\subset\mathbb{R}^{n}$ is dense (resp. locally dense) in
${\mathbb{R}}^{n}$ if $\overline{G(v)}={\mathbb{R}}^{n}$ (resp.
$\mathring{\overline{G(v)}}\neq\emptyset$), where $\overline{E}$ (resp.
$\overset{\circ}{E}$ ) denotes the closure (resp. the interior) of a subset
$E\subset\mathbb{R}^{n}$. We say that $G$ is hypercyclic (resp. locally
hypercyclic) if there exists a vector $v\in{\mathbb{R}}^{n}$ such that $G(v)$
is dense (resp. locally dense) in ${\mathbb{R}}^{n}$. Hypercyclic is also
called topologically transitive. We refer the reader to the recent book [5]
and [10] for a thoroughly account on hypercyclicity.
So, the question to investigate is the following: When an abelian sub-
semigroup of $M_{n}(\mathbb{R})$ can be hypercyclic (resp. locally
hypercyclic)?
The main purpose of this paper is twofold: firstly, we give a general result
answering the above question for any abelian sub-semigroup of
$M_{n}(\mathbb{R})$. Notice that in [2] (resp. [4]), the authors answer this
question for any abelian subgroup (resp. sub-semigroup) of
$\mathrm{GL}(n;\mathbb{R})$ (resp. $\mathrm{GL}(n;\mathbb{C})$), so this paper
can be viewed as a continuation of these works. Secondly, we prove that the
minimal number of matrices in normal form in
$\mathcal{K}_{\eta,r,s}(\mathbb{R})$ (see definition below) required to form a
hypercyclic abelian semigroup in $\mathbb{R}^{n}$ is $n-s+1$ (Corollary 1.6).
In particular, $\left[\frac{n+1}{2}\right]+1$ is the minimal number of
matrices on $\mathbb{R}^{n}$ required to form a hypercyclic abelian semigroup
on $\mathbb{R}^{n}$, this answer a question raised by Feldman in ([8], Section
6). Notice that in [8], Feldman showed that no semigroup generated by n-tuple
of diagonalizable matrices on $\mathbb{R}^{n}$ can be hypercyclic. If one
remove the diagonalizability condition, Costakis and al. proved in [7] that
there exist $n$-tuple of non diagonalizable matrices on $\mathbb{R}^{n}$ which
is hypercyclic. Recently, Costakis and Parissis proved in [8] that the minimal
number of matrices in Jordan form on $\mathbb{R}^{n}$ which form a hypercyclic
tuple is $n+1$.
I learned that Shkarin [12], Abels and Manoussos [1] have, independently
proved, similar results to Corollaries 1.8 and 1.9. The methods of proof in
[12] and in this paper are quite different and have different consequences.
To state our main results, we need to introduce the following notations and
definitions for the sequel.
Write $\mathbb{N}_{0}=\mathbb{N}\backslash\\{0\\}$. Let $n\in\mathbb{N}_{0}$
fixed. For each $m=1,2,\dots,n,$ denote by:
$\bullet$ $\mathbb{T}_{n}(\mathbb{R})$ the set of matrices over $\mathbb{R}$
of the form:
$\left[\begin{array}[]{cccc}\mu&&&0\\\ a_{2,1}&\ddots&&\\\
\vdots&\ddots&\ddots&\\\ a_{m,1}&\dots&a_{m,m-1}&\mu\end{array}\right]\ \ \ \
(1)$
$\bullet$
$\mathbb{T}_{n}^{*}(\mathbb{R})=\mathbb{T}_{n}(\mathbb{R})\cap\textrm{GL}(n,\mathbb{R})$
the group of matrices of the form (1) with $\mu\neq 0$.
• $\mathbb{T}_{n}^{+}(\mathbb{R})$ the group of matrices over $\mathbb{R}$ of
the form $(1)$ with $\mu>0$.
For each $1\leq m\leq\frac{n}{2}$, denote by
$\bullet$ $\mathbb{B}_{m}(\mathbb{R})$ the set of matrices of
$M_{2m}(\mathbb{R})$ of the form
$\begin{bmatrix}C&&&0\\\ C_{2,1}&C&&\\\ \vdots&\ddots&\ddots&\\\
C_{m,1}&\dots&C_{m,m-1}&C\end{bmatrix}:\ C,\ C_{i,j}\in\mathbb{S},\ 2\leq
i\leq m,1\leq j\leq m-1\ \qquad(2)$
where $\mathbb{S}$ is the set of matrices over $\mathbb{R}$ of the form
$\begin{bmatrix}\alpha&\beta\\\ -\beta&\alpha\\\ \end{bmatrix}.$
$\bullet$
$\mathbb{B}^{*}_{m}(\mathbb{R}):=\mathbb{B}_{m}(\mathbb{R})\cap\textrm{GL}(2m,\mathbb{R})$
the group of matrices over $\mathbb{R}$ of the form (2) with $C$ invertible.
Let $r,\ s\in\mathbb{N}$ and $\eta=\begin{cases}(n_{1},\dots,n_{r};\
m_{1},\dots,m_{s})&\textrm{ if }rs\neq 0,\\\ (m_{1},\dots,m_{s})&\textrm{ if
}r=0,\\\ (n_{1},\dots,n_{r})&\textrm{ if }s=0\end{cases}$
be a sequence of positive integers such that
$(n_{1}+\dots+n_{r})+2(m_{1}+\dots+m_{s})=n.\qquad(3)$
In particular, we have $r+2s\leq n$. Denote by
•
$\mathcal{K}_{\eta,r,s}(\mathbb{R}):=\mathbb{T}_{n_{1}}(\mathbb{R})\oplus\dots\oplus\mathbb{T}_{n_{r}}(\mathbb{R})\oplus\mathbb{B}_{m_{1}}(\mathbb{R})\oplus\dots\oplus\mathbb{B}_{m_{s}}(\mathbb{R}).$
In particular:
\- If $r=1,\ s=0$ then
$\mathcal{K}_{\eta,1,0}(\mathbb{R})=\mathbb{T}_{n}(\mathbb{R})$ and
$\eta=(n)$.
\- If $r=0,\ s=1$ then
$\mathcal{K}_{\eta,0,1}(\mathbb{R})=\mathbb{B}_{m}(\mathbb{R})$ and
$\eta=(m)$, $n=2m$.
\- If $r=0,\ s>1$ then
$\mathcal{K}_{\eta,0,s}(\mathbb{R})=\mathbb{B}_{m_{1}}(\mathbb{R})\oplus\dots\oplus\mathbb{B}_{m_{s}}(\mathbb{R})$
and $\eta=(m_{1},\dots,m_{s})$.
•
$\mathcal{K}^{*}_{\eta,r,s}(\mathbb{R}):=\mathcal{K}_{\eta,r,s}(\mathbb{R})\cap\textrm{GL}(n,\
\mathbb{R})$.
•
$\mathcal{K}^{+}_{\eta,r,s}(\mathbb{R}):=\mathbb{T}^{+}_{n_{1}}(\mathbb{R})\oplus\dots\oplus\mathbb{T}^{+}_{n_{r}}(\mathbb{R})\oplus\mathbb{B}^{*}_{m_{1}}(\mathbb{R})\oplus\dots\oplus\mathbb{B}^{*}_{m_{s}}(\mathbb{R}).$
Consider the matrix exponential map
$\mathrm{exp}:M_{n}(\mathbb{R})\longrightarrow GL(n,\mathbb{R})$, set
$\mathrm{exp}(M)=e^{M}$.
Let $G$ be a sub-semigroup of $M_{n}(\mathbb{R})$. It is proved (see
Proposition 2.2) that there exists a $P\in\textrm{GL}(n,\mathbb{R})$ such that
$P^{-1}GP$ is an abelian sub-semigroup of
$\mathcal{K}_{\eta,r,s}(\mathbb{R})$, for some $1\leq r,s\leq n$, where
$\eta=(n_{1},\dots,n_{r};\ m_{1},\dots,m_{s})\in\mathbb{N}_{0}^{r+s}$ is a
partition of $n$. We call $P^{-1}GP$ the normal form of $G$. For such a choice
of matrix $P$, we let:
$\bullet$
$\mathrm{g}:=\textrm{exp}^{-1}(G)\cap\left[P(\mathcal{K}_{\eta,r,s}(\mathbb{R}))P^{-1}\right]$.
$\bullet$ $\mathrm{g}_{u}:=\\{Bu:\ B\in\mathrm{g}\\},\ u\in\mathbb{R}^{n}.$
$\bullet$ $G^{2}=\\{A^{2}:A\in G\\}$
$\bullet$
$\mathrm{g}^{2}=\textrm{exp}^{-1}(G^{2})\cap\left[P(\mathcal{K}_{\eta,r,s}(\mathbb{R}))P^{-1}\right]$
$\bullet$ $G^{*}=G\cap\textrm{GL}(n,\mathbb{R})$, it is a sub-semigroup of
$\textrm{GL}(n,\mathbb{R})$.
$\bullet$ $G^{*2}=\\{A^{2}:\ A\in G^{*}\\}$
In particular when $G\subset\mathcal{K}_{\eta,r,s}(\mathbb{R})$,
$\mathrm{g}=\textrm{exp}^{-1}(G)\cap\mathcal{K}_{\eta,r,s}(\mathbb{R})$.
• For every $M\in G^{*}$, one can write
$\widetilde{M}:=P^{-1}MP=\mathrm{diag}(M_{1},\dots,M_{r}$;
$\widetilde{M}_{1},\dots,\widetilde{M}_{s})\in\mathcal{K}^{*}_{\eta,r,s}(\mathbb{R})$.
Set $\widetilde{G^{*}}=P^{-1}G^{*}P$. Let $\mu_{k}$ be the eigenvalue of
$M_{k}$, $k=1,\dots,r$, and define the index $\textrm{ind}(\widetilde{G^{*}})$
of $\widetilde{G^{*}}$ to be
$\textrm{ind}(\widetilde{G^{*}}):=\begin{cases}0,\ if\ r=0\\\ \\\
\left\\{\begin{array}[]{c}1,\ \ \mathrm{if}\ \ \ \mathrm{\exists}\
\widetilde{M}\in\widetilde{G^{*}}\ \ \mathrm{with}\ \mu_{1}<0\\\ 0,\ \
\mathrm{otherwise}\end{array}\right.,\ if\ \ r=1\\\ \\\
\textrm{card}\left\\{k\in\\{1,\dots,r\\}:\
\exists\widetilde{M}\in\widetilde{G^{*}}\ \textrm{ with }\ \mu_{k}<0,\
\mu_{i}>0,\ \forall\ i\neq k\right\\},\ if\ r\notin\\{0,\ 1\\}.\end{cases}$
($\textrm{card}(E)$ denotes the number of elements of a subset $E$ of
$\mathbb{N}$).
In particular,
\- If $\widetilde{G^{*}}\subset\mathcal{K}^{+}_{\eta,r,s}(\mathbb{R})$ with
$r\neq 0$ then $\textrm{ind}(\widetilde{G})=0$.
\- If $\widetilde{G^{*}}\subset\mathbb{B}^{*}_{m}(\mathbb{R})$, then
$\textrm{ind}(\widetilde{G})=0$ (since $r=0$).
We define the index of $G$ to be
$\textrm{ind}(G):=\textrm{ind}(\widetilde{G^{*}})$. It is plain that this
definition does not depend on $P$.
Throughout the paper, denote by
$\bullet$ $v^{T}$ the transpose of a vector $v\in\mathbb{R}^{n}$.
$\bullet$ $\mathcal{B}_{0}=(e_{1},\dots,e_{n})$ the canonical basis of
$\mathbb{R}^{n}$ and $I_{n}$ the identity matrix on $\mathbb{R}^{n}$.
$\bullet$
$u_{0}=[e_{1,1},\dots,e_{r,1};f_{1,1},\dots,f_{s,1}]^{T}\in\mathbb{R}^{n}$
where $e_{k,1}=[1,0,\dots,0]^{T}\in\mathbb{R}^{n_{k}}$,
$f_{l,1}=[1,0,\dots,0]^{T}\in\mathbb{R}^{2m_{l}}$, $k=1,\dots,r;\
l=1,\dots,s$.
$\bullet$ $v_{0}=Pu_{0}$.
$\bullet$
$f^{(l)}=[0,\dots,0,f^{(l)}_{1},\dots,f^{(l)}_{s}]^{T}\in\mathbb{R}^{n}$
where for $i=1,\dots,r$, $j=1,\dots,s$:
$f^{(l)}_{j}=\begin{cases}0\in\mathbb{R}^{2m_{j}}&\mathrm{if}\ \ j\neq l,\\\
[0,1,0,\dots,0]^{T}\in\mathbb{R}^{2m_{l}}&\mathrm{if}\ \ j=l.\end{cases}$
An equivalent formulation is $f^{(1)}=e_{t_{1}}$, $\dots,f^{(l)}=e_{t_{l}}$
where $t_{1}=\underset{j=1}{\overset{r}{\sum}}n_{j}+2,$
$t_{l}=\underset{j=1}{\overset{r}{\sum}}n_{j}+2\underset{j=1}{\overset{l-1}{\sum}}m_{j}+2,$
$l=2,\dots,s$.
Our principal results can now be stated as follows:
###### Theorem 1.1.
Let $G$ be an abelian sub-semigroup of $M_{n}(\mathbb{R})$. The following
properties are equivalent:
* (i)
$G$ is locally hypercyclic.
* (ii)
$G(v_{0})$ is locally dense in $\mathbb{R}^{n}$.
* (iii)
$\mathrm{g}_{v_{0}}$ is an additif sub-semigroup, dense in $\mathbb{R}^{n}$
###### Corollary 1.2.
Let $G$ be an abelian sub-semigroup of $M_{n}(\mathbb{R})$ and
$P\in\textrm{GL}(n,\mathbb{R})$ such that
$P^{-1}GP\subset\mathcal{K}_{\eta,r,s}(\mathbb{R})$ for some $0\leq r,s\leq
n$. The following properties are equivalent:
* (i)
$G$ is hypercyclic.
* (ii)
$G(v_{0})$ is dense in $\mathbb{R}^{n}$.
* (iii)
$\mathrm{g}_{v_{0}}$ is an additif sub-semigroup dense in $\mathbb{R}^{n}$ and
$\mathrm{ind}(G)=r$.
Remark 1. If all matrices of $G\backslash I_{n}$ are non invertible (i.e.
$G^{*}=\\{I_{n}\\}$) then $G$ is not hypercyclic (Proposition 4.1).
###### Theorem 1.3.
Let $G$ be an abelian sub-semigroup of $M_{n}(\mathbb{R})$ and
$P\in\textrm{GL}(n,\mathbb{R})$ so that
$P^{-1}GP\subset\mathcal{K}_{\eta,r,s}(\mathbb{R})$, for some $0\leq r,s\leq
n$. Let $A_{1},\dots,A_{p}$ generate $G$ and let
$B_{1},\dots,B_{p}\in\mathrm{g}$ such that
$A_{1}^{2}=e^{B_{1}},\dots,A_{p}^{2}=e^{B_{p}}$. The following properties are
equivalent:
* (i)
$G$ is locally hypercyclic.
* (ii)
$G(v_{0})$ is locally dense in $\mathbb{R}^{n}$.
* (iii)
$\mathrm{g}^{2}_{v_{0}}=\underset{k=1}{\overset{p}{\sum}}\mathbb{N}B_{k}v_{0}+\underset{k=1}{\overset{s}{\sum}}2\pi\mathbb{Z}Pf^{(l)}$
is an additive sub-semigroup dense in $\mathbb{R}^{n}$.
###### Corollary 1.4.
Under the hypothesis of Theorem 1.3, the following properties are equivalent:
* (i)
$G$ is hypercyclic.
* (ii)
$G(v_{0})$ is dense in $\mathbb{R}^{n}$.
* (iii)
$\mathrm{g}^{2}_{v_{0}}=\underset{k=1}{\overset{p}{\sum}}\mathbb{N}B_{k}v_{0}+\underset{k=1}{\overset{s}{\sum}}2\pi\mathbb{Z}Pf^{(l)}$
is an additive sub-semigroup dense in $\mathbb{R}^{n}$ and
$\mathrm{ind}(G)=r$.
###### Corollary 1.5.
If $G$ is an abelian semigroup $($with normal form in
$\mathcal{K}_{\eta,r,s}(\mathbb{R}))$ generated by $(n-s)$ matrices of
$M_{n}(\mathbb{R})$, it has nowhere dense orbit.
###### Corollary 1.6.
If $G$ is an abelian semigroup generated by $[\frac{n+1}{2}]$ matrices of
$M_{n}(\mathbb{R})$, it has nowhere dense orbit.
###### Theorem 1.7.
For every $n\in\mathbb{N}_{0}$, $r=1,\dots,n$, and $\eta=(n_{1},\dots,n_{r};\
m_{1},\dots,m_{s})\in\mathbb{N}_{0}^{r+s}$, there exist $(n-s+1)$ matrices in
$\mathcal{K}_{\eta,r,s}^{*}(\mathbb{R})$ that generate an hypercyclic abelian
semigroup.
As a consequence, from Theorem 1.7 and Corollary 1.5, we obtain the following
Corollary.
###### Corollary 1.8.
For every $n\in\mathbb{N}_{0}$, $r=1,\dots,n$, and $\eta=(n_{1},\dots,n_{r};\
m_{1},\dots,m_{s})\in\mathbb{N}_{0}^{r+s}$, the minimum number of matrices of
$M_{n}(\mathbb{R})$ with normal form in $\mathcal{K}_{\eta,r,s}(\mathbb{R})$,
that generate an hypercyclic abelian semigroup is $(n-s+1)$.
###### Corollary 1.9.
The minimum number of matrices of $M_{n}(\mathbb{R})$, that generate an
hypercyclic abelian semigroup is $\left[\frac{n+1}{2}\right]+1$.
In particular:
For $r=n$, we obtain Feldman’s Theorem:
###### Corollary 1.10.
$($[9]$)$ The minimum number of diagonalizable matrices of $M_{n}(\mathbb{R})$
that generate an abelian hypercyclic semigroup is $n+1$.
For $r=1$ and $s=0$, we obtain the following Corollary:
###### Corollary 1.11.
The minimum number of matrices of $\mathbb{T}_{n}(\mathbb{R})$ that generate
an hypercyclic abelian semigroup is $n+1$.
This paper is organized as follows: In Section 2, we introduce the normal form
of an abelian sub-semigroup of $M_{n}(\mathbb{R})$. Section 3 is devoted to
the characterization of abelian sub-semigroups of
$\mathcal{K}^{*}_{\eta,r,s}(\mathbb{R})$ with a locally dense orbit. The proof
of Theorem 1.1 and Corollary 1.2 are done in Section 4. In Section 5, we prove
Theorem 1.3, Corollaries 1.4, 1.5 and 1.6. Theorem 1.7 and Corollary 1.9 are
proved in Section 6. In Section 7, we give an example for n = 2.
## 2\. Normal form of abelian sub-semigroups of $M_{n}(\mathbb{R})$
Recall first the following proposition.
###### Proposition 2.1.
$($[3], Proposition 2.6$)$ Let $G$ be an abelian subgroup of
$\textrm{GL}(n,\mathbb{R})$. Then there exists a
$P\in\textrm{GL}(n,\mathbb{R})$ such that $P^{-1}GP$ is an abelian subgroup of
$\mathcal{K}_{\eta,r,s}^{*}(\mathbb{R})$, for some
$\eta\in\mathbb{N}_{0}^{r+s}$ and $r,s\in\\{1,\dots,n\\}$.
The analogous of Proposition 2.1 for sub-semigroup is the following:
###### Proposition 2.2.
Let $G$ be an abelian sub-semigroup of $M_{n}(\mathbb{R})$. Then there exists
a $P\in GL(n,\mathbb{R})$ such that $P^{-1}GP$ is an abelian sub-semigroup of
$\mathcal{K}_{\eta,r,s}(\mathbb{R})$, for some $\eta\in\mathbb{N}_{0}^{r+s}$
and $r,s\in\\{1,\dots,n\\}$.
###### Proof.
For every $A\in G$, there exists $\lambda_{A}\in\mathbb{R}$ such that
$(A-\lambda_{A}I_{n})\in\textrm{GL}(n,\mathbb{R})$ (it suffices to take
$\lambda_{A}$ not an eigenvalue of $A$). We let $\widehat{L}$ be the group
generated by $L:=\left\\{A-\lambda_{A}I_{n}:\ A\in G\right\\}$. Then
$\widehat{L}$ is an abelian subgroup of $\textrm{GL}(n,\mathbb{R})$ and by
Proposition 2.1, there exists a $P\in\textrm{GL}(n,\mathbb{R})$ such that
$P^{-1}\widehat{L}P\subset\mathcal{K}_{\eta,r,s}^{*}(\mathbb{R})$, for some
$\eta\in\mathbb{N}_{0}^{r}$. As
$P^{-1}LP=\left\\{P^{-1}AP-\lambda_{A}I_{n}:\ A\in G\right\\}$
then $P^{-1}GP\subset\mathcal{K}_{\eta,r,s}(\mathbb{R})$, this proves the
proposition. ∎
## 3\. Abelian sub-semigroup of $\mathcal{K}^{*}_{\eta,r,r}(\mathbb{R})$
with a locally dense orbit
The aim of this section is to give results for abelian sub-semigroups of
$\mathcal{K}_{\eta,r,s}^{*}(\mathbb{R})$ analogous to those for abelian groups
in ([3], Sections 3, 4 and 7).
Let $G$ be an abelian sub-semigroup of
$\mathcal{K}^{*}_{\eta,r,s}(\mathbb{R})$. Let us first recall that a subset
$E\subset\mathbb{R}^{n}$ is called _$G$ -invariant_ if $A(E)\subset E$ for any
$A\in G$. One can check that if $E$ is $G$-invariant then so is
$\overline{E}$, $\mathring{E}$ and $\mathbb{R}^{n}\backslash E$.
Denote by
$\bullet$ $\mathcal{C}(G):=\\{A\in\mathcal{K}_{\eta,r,s}(\mathbb{R}):\ AB=BA,\
\forall\ B\in G\\}.$ Since $G$ is abelian, $G\subset\mathcal{C}(G)$.
$\bullet$ $G^{+}:=G\cap\mathcal{K}^{+}_{\eta,r,s}(\mathbb{R})$.
$\bullet$ $\widehat{G}$ the group generated by $G$.
$\bullet$
$\widehat{\mathrm{g}}:=\mathrm{exp}^{-1}(\widehat{G})\cap\mathcal{K}_{\eta,r,s}(\mathbb{R})$.
Let now recall the following results from [3].
###### Lemma 3.1.
$($[3], Proposition 3.2$)$.
$\mathrm{exp}(\mathcal{K}_{\eta,r,s}(\mathbb{R}))=\mathcal{K}^{+}_{\eta,r,s}(\mathbb{R})$.
###### Lemma 3.2.
$($[3], Proposition 3.3$)$. Let $A,\ B\in\mathcal{K}_{\eta,r,s}(\mathbb{R})$.
If $e^{A}e^{B}=e^{B}e^{A}$ then $AB=BA$.
###### Lemma 3.3.
Let $G$ be an abelian sub-semigroup of
$\mathcal{K}^{*}_{\eta,r,s}(\mathbb{R})$. Then:
* (i)
$\mathcal{C}(\widehat{G})=\mathcal{C}(G)$.
* (ii)
$\mathrm{g}\subset\mathcal{C}(G)$ and all matrix of $\mathrm{g}$ commute. In
particular, $\mathrm{g}\subset\mathcal{C}(\mathrm{g})$.
* (iii)
$\mathrm{exp}(\mathrm{g})=G^{+}$.
###### Proof.
(i) If $B\in\mathcal{C}(G)$ and $A\in G$ then $A^{-1}B=BA^{-1}$ (since
$AB=BA$). We conclude that $B\in\mathcal{C}(\widehat{G})$.
(ii) Since $\widehat{\mathrm{g}}\subset\mathcal{C}(\widehat{G})$ $($[3], Lemma
3.10, (iv)$)$ and $\mathrm{g}\subset\widehat{\mathrm{g}}$, it follows that
$\mathrm{g}\subset\mathcal{C}(\widehat{G})$ and by (i),
$\mathrm{g}\subset\mathcal{C}(G)$. By Lemma 3.2, all element of $\mathrm{g}$
commute, hence $\mathrm{g}\subset\mathcal{C}(\mathrm{g})$.
(iii) By Lemma 3.1,
$\textrm{exp}(\mathrm{g})\subset\mathcal{K}^{+}_{\eta,r,s}(\mathbb{R})$, hence
$\textrm{exp}(\mathrm{g})\subset G^{+}$.
Conversely, let $A\in G^{+}$. There exists
$B\in\mathcal{K}_{\eta,r,s}(\mathbb{R})$ so that $e^{B}=A$ (Lemma 3.1). Hence
$B\in\textrm{exp}^{-1}(G)\cap\mathcal{K}_{\eta,r,s}(\mathbb{R})=\mathrm{g}$,
and then $A\in\textrm{exp}(\mathrm{g})$. So
$G^{+}\subset\textrm{exp}(\mathrm{g})$, this proves (iii). ∎
###### Lemma 3.4.
Let $G$ be an abelian sub-semigroup of $\mathcal{K}_{\eta,r,s}(\mathbb{R})$
and
$\mathrm{g}^{*}=\mathrm{exp}^{-1}(G^{*})\cap\mathcal{K}_{\eta,r,s}(\mathbb{R})$.
Then $\mathrm{g}=\mathrm{g}^{*}$.
###### Proof.
Since $G^{*}\subset G$, we see that $\mathrm{g}^{*}\subset\mathrm{g}$.
Conversely, if $B\in\mathrm{g}$ then $e^{B}\in G\cap GL(n,\mathbb{R})=G^{*}$,
so
$B\in\mathrm{exp}^{-1}(G^{*})\cap\mathcal{K}_{\eta,r,s}(\mathbb{R})=\mathrm{g}^{*}$,
hence $\mathrm{g}\subset\mathrm{g}^{*}$. It follows that
$\mathrm{g}=\mathrm{g}^{*}$. ∎
Denote by
$\bullet$
$U:=\begin{cases}\underset{k=1}{\overset{r}{\prod}}(\mathbb{R}^{*}\times\mathbb{R}^{n_{k}-1})\times\underset{l=1}{\overset{s}{\prod}}(\mathbb{R}^{2}\backslash\\{(0,0)\\})\times\mathbb{R}^{2m_{l}-2},&\
if\ r,s\geq 1\\\
\underset{l=1}{\overset{s}{\prod}}(\mathbb{R}^{2}\backslash\\{(0,0)\\})\times\mathbb{R}^{2m_{l}-2},&\
if\ r=0\\\
\underset{k=1}{\overset{r}{\prod}}\mathbb{R}^{*}\times\mathbb{R}^{n_{k}-1},&\
if\ s=0\end{cases}$
$\bullet$
$C_{u_{0}}:=\begin{cases}\underset{k=1}{\overset{r}{\prod}}(\mathbb{R}^{*}_{+}\times\mathbb{R}^{n_{k}-1})\times\underset{l=1}{\overset{s}{\prod}}(\mathbb{R}^{2}\backslash\\{(0,0)\\})\times\mathbb{R}^{2m_{l}-2},&\
if\ r,s\geq 1\\\
\underset{l=1}{\overset{s}{\prod}}(\mathbb{R}^{2}\backslash\\{(0,0)\\})\times\mathbb{R}^{2m_{l}-2},&\
if\ r=0\\\
\underset{k=1}{\overset{r}{\prod}}\mathbb{R}^{*}_{+}\times\mathbb{R}^{n_{k}-1},&\
if\ s=0\end{cases}$
Note that $C_{u_{0}}$ is the connected component of $U$ containing $u_{0}$.
Moreover, $U$ is a dense open subset in $\mathbb{R}^{n}$ and a simple
calculation from the definition yields that $U$ is a $G$-invariant.
$\bullet$ $\Gamma$ the subgroup of $\mathcal{K}^{*}_{\eta,r,s}(\mathbb{R})$
generated by $(S_{k})_{1\leq k\leq r}$ where
$S_{k}:=\mathrm{diag}\left(\varepsilon_{1,k}I_{n_{1}},\dots,\varepsilon_{r,k}I_{n_{r}};\
I_{2m_{1}},\dots,I_{2m_{s}}\right)\in\mathcal{K}^{*}_{\eta,r,s}(\mathbb{R}),$
and
$\varepsilon_{i,k}:=\begin{cases}-1,&\ if\ {i=k}\\\ 1,&\ if\ {i\neq
k},\end{cases},\ 1\leq i,\ k\leq r$
###### Lemma 3.5.
Let $G$ be an abelian sub-semigroup of
$\mathcal{K}^{*}_{\eta,r,s}(\mathbb{R})$. Then:
* (i)
$U=\underset{S\in\Gamma}{\bigcup}S(C_{u_{0}})$.
* (ii)
$S_{k}M=MS_{k}$, for every $M\in\mathcal{K}^{*}_{\eta,r,s}(\mathbb{R})$,
$k=1,\dots,r$.
* (iii)
$G^{+}(u_{0}):=G(u_{0})\cap C_{u_{0}}$.
* (iv)
if $\mathrm{ind}(G)=r$ then $G(u_{0})\cap S(C_{u_{0}})\neq\emptyset$ for every
$S\in\Gamma$.
###### Proof.
The proof is analogous to that of ([3], Lemma 4.6). ∎
For an abelian subgroup $G$ of $\textrm{GL}(n,\mathbb{R})$, denote by:
* -
$E(u):=\textrm{vect}(G(u))$ the vector subspace of $\mathbb{R}^{n}$ generated
by $G(u)$, $u\in\mathbb{R}^{n}$.
* -
vect($G)$ the vector subspace of $M_{n}(\mathbb{R})$ generated by $G$.
One can easily check that vect($G)$ is the algebra generated by $G$. In
particular, if $G$ is an abelian subgroup of
$\mathcal{K}_{\eta,r,s}(\mathbb{R})$ then
vect($G)\subset\mathcal{C}(G)\subset\mathcal{K}_{\eta,r,s}(\mathbb{R})$.
###### Lemma 3.6.
$($[6], Proposition 3.1$)$ Let $G$ be an abelian subgroup of
$\textrm{GL}(n,\mathbb{R})$. If $u\in\mathbb{R}^{n}$ and $v\in E(u)$, then
there exists $B\in\mathrm{vect}(G)$ such that $Bu=v$.
###### Proposition 3.7.
Let $G$ be an abelian sub-semigroup of
$\mathcal{K}^{*}_{\eta,r,s}(\mathbb{R})$.
* (i)
if $\overset{\circ}{\overline{G(u)}}\neq\emptyset$, for some
$u\in\mathbb{R}^{n}$, then for every $v\in U$, there exists
$B\in\mathrm{vect}(G)\cap\textrm{GL}(n,\mathbb{R})$ such that $Bu=v$ and
$B(G(u))=G(v)$.
* (ii)
$G$ has a dense $($resp. locally dense$)$ orbit in $\mathbb{R}^{n}$ if and
only if $G(u_{0})$ is dense $($resp. locally dense$)$ in $\mathbb{R}^{n}$.
###### Proof.
(i): Let $\widehat{G}\subset\mathcal{K}^{*}_{\eta,r,s}(\mathbb{R})$ be the
group generated by $G$. Then
$\overset{\circ}{\overline{\widehat{G}(u)}}\neq\emptyset$ and thus
$E(u):=\mathrm{vect}(\widehat{G}(u))=\mathbb{R}^{n}$. As $U$ is a
$\widehat{G}$-invariant and dense open subset of $\mathbb{R}^{n}$, hence $u\in
U$. Write $u=[u_{1},\dots,u_{r};\
\widetilde{u}_{1},\dots,\widetilde{u}_{s}]^{T}$ with
$u_{k}=[x_{k,1},\dots,x_{k,n_{k}}]^{T}\in\mathbb{R}^{*}\times\mathbb{R}^{n_{k}-1}$
and
$\widetilde{u}_{l}=[y_{l,1},y^{\prime}_{l,1},\dots,y_{l,m_{l}},y^{\prime}_{l,m_{l}}]^{T}\in(\mathbb{R}^{2}\backslash\\{(0,0)\\})\times\mathbb{R}^{2m_{l}-2}$,
$k=1,\dots,r$, $l=1,\dots,s$. Applying Lemma 3.6 to $\widehat{G}$, then there
exists $B\in\mathrm{vect}(\widehat{G})$ such that $Bu=v$. Let’s prove that
$B\in\textrm{GL}(n,\mathbb{R})$:
Since $\mathcal{K}_{\eta,r,s}(\mathbb{R})$ is a vector space,
vect$(\widehat{G})\subset\mathcal{K}_{\eta,r,s}(\mathbb{R})$ and one can write
$B=\mathrm{diag}(B_{1},\dots,B_{r};\
\widetilde{B}_{1},\dots,\widetilde{B}_{s})$ with
$B_{k}\in\mathbb{T}_{n_{k}}(\mathbb{R})$, $k=1,\dots,r$ and
$\widetilde{B}_{l}\in\mathbb{B}_{m_{l}}(\mathbb{R})$, $l=1,\dots,s$. Let
$\mu_{k}$ be the real eigenvalue of $B_{k}$ and
$\lambda_{l}=\alpha_{l}+i\beta_{l}$ the complex eigenvalue of
$\widetilde{B}_{l}$ such that
$\widetilde{B}_{l}=\left[\begin{array}[]{cccc }C^{(l)}&&&0\\\
C^{(l)}_{2,1}&\ddots&&\\\ \vdots&\ddots&\ddots&\\\
C^{(l)}_{m_{l},1}&\dots&C^{(l)}_{m_{l},m_{l}-1}&C^{(l)}\end{array}\right]\in
M_{2m_{l}}(\mathbb{R}),\ \ \mathrm{with}\ \
C^{(l)}=\left[\begin{array}[]{cc}\alpha_{l}&\beta_{l}\\\
-\beta_{l}&\alpha_{l}\end{array}\right].$
Write $v=[v_{1},\dots,v_{r};\ \widetilde{v}_{1},\dots,\widetilde{v}_{s}]^{T}$
with
$v_{k}=[z_{k,1},\dots,z_{k,n_{k}}]^{T}\in\mathbb{R}^{*}\times\mathbb{R}^{n_{k}-1}$
and
$\widetilde{v}_{l}=[a_{l,1},a^{\prime}_{l,1},\dots,a_{l,m_{l}},a^{\prime}_{l,m_{l}}]^{T}\in(\mathbb{R}^{2}\backslash\\{(0,0)\\})\times\mathbb{R}^{2m_{l}-2}$,
$k=1,\dots,r$, $l=1,\dots,s$. From $Bu=v$, we see that $B_{k}u_{k}=v_{k}$ and
$\widetilde{B}_{l}\widetilde{u}_{l}=\widetilde{v}_{l}$ for all $k=1,\dots,r$;
$l=1,\dots,s$. It follows that $\mu_{k}x_{k,1}=z_{k,1}$ and
$C^{(l)}[y_{l,1},y^{\prime}_{l,1}]^{T}=[a_{l,1},a^{\prime}_{l,1}]^{T}$. Since
$z_{k,1}\neq 0$ and $[a_{l,1},a^{\prime}_{l,1}]^{T}\neq[0,0]^{T}$, thus
$\mu_{k}\neq 0$ and $\lambda_{l}\neq 0$ for all $k=1,\dots,r$; $l=1,\dots,s$.
It follows that $B\in\textrm{GL}(n,\mathbb{R})$. As
vect$(\widehat{G})\subset\mathcal{C}(\widehat{G})=\mathcal{C}(G)$ (Lemma
3.3,(i)), then $B\in\mathcal{C}(G)$ and so $B(G(u))=G(v)$.
(ii): Suppose that $\overline{G(u)}=\mathbb{R}^{n}$ $($resp.
$\overset{\circ}{\overline{G(u)}}\neq\emptyset)$ for some
$u\in\mathbb{R}^{n}$. Then applying (i) for $v=u_{0}\in U$; there exists
$B\in\mathrm{vect}(G)\cap\textrm{GL}(n,\mathbb{R})$ such that $Bu=u_{0}$ and
$B(G(u))=G(u_{0})$. Therefore $\overline{G(u_{0})}=\mathbb{R}^{n}$ $($resp.
$\overset{\circ}{\overline{G(u_{0})}}\neq\emptyset)$. ∎
###### Lemma 3.8.
Let $G$ be an abelian sub-semigroup of $\mathcal{K}_{\eta,r,s}(\mathbb{R})$.
If $\overset{\circ}{\overline{G(u_{0})}}\neq\emptyset$ then for any $v\in
C_{u_{0}}$, we have $\overline{G(v)}\cap C_{u_{0}}=C_{u_{0}}$.
###### Proof.
By Proposition 3.7,(i), there exists
$B\in\mathrm{vect}(G)\cap\textrm{GL}(n,\mathbb{R})$ such that $Bu_{0}=v$ and
$B(G(u_{0}))=G(v)$. Hence $\overset{\circ}{\overline{G(v)}}\neq\emptyset$ for
any $v\in C_{u_{0}}$. Let’s prove that $\overline{G(v)}\cap
C_{u_{0}}=\overset{\circ}{\overline{G(v)}}\cap C_{u_{0}}$: if there exists
$w\in(\overline{G(v)}\backslash\overset{\circ}{\overline{G(v)}})\cap
C_{u_{0}}$, then
$\emptyset\neq\overset{\circ}{\overline{G(w)}}\subset(\overline{G(v)}\backslash\overset{\circ}{\overline{G(v)}})\cap
C_{u_{0}}$. Hence
$\overset{\circ}{\overline{G(w)}}\subset\overset{\circ}{\overline{G(v)}}$, a
contradiction. Since $C_{u_{0}}$ is connected, it follows that
$\overline{G(v)}\cap C_{u_{0}}=C_{u_{0}}$, this completes the proof. ∎
###### Lemma 3.9.
Let $G$ be an abelian sub-semigroup of
$\mathcal{K}^{*}_{\eta,r,s}(\mathbb{R})$. Then
$\overset{\circ}{\overline{G(u_{0})}}\neq\emptyset$ if and only if
$\overset{\circ}{\overline{G^{+}(u_{0})}}\neq\emptyset.$
###### Proof.
Suppose that $\overset{\circ}{\overline{G^{+}(u_{0})}}\neq\emptyset$. It is
plain that $\overset{\circ}{\overline{G(u_{0})}}\neq\emptyset$ since
$G^{+}(u_{0})\subset G(u_{0})$. Conversely, suppose that
$\overset{\circ}{\overline{G(u_{0})}}\neq\emptyset$. Then by Lemma 3.8,
$\overline{G(u_{0})}\cap C_{u_{0}}=C_{u_{0}}$ and by Lemma 3.5,(iii),
$G^{+}(u_{0})=G(u_{0})\cap C_{u_{0}}$. It follows that
$\overset{\circ}{\overline{G^{+}(u_{0})}}=\overset{\circ}{\overline{C_{u_{0}}}}\supset
C_{u_{0}}$. ∎
###### Proposition 3.10.
Let $G$ be an abelian sub-semigroup of
$\mathcal{K}^{*}_{\eta,r,s}(\mathbb{R})$. Then the following properties are
equivalent:
* (i)
$\overline{G(u_{0})}=\mathbb{R}^{n}$
* (ii)
$\overset{\circ}{\overline{G(u_{0})}}\neq\emptyset$ and $\textrm{ ind}(G)=r$.
###### Proof.
$(i)\Longrightarrow(ii)$: Suppose that $\textrm{ind}(G)<r$. Then there exists
$1\leq k_{0}\leq r$ such that for every $B=\mathrm{diag}(B_{1},\dots,B_{r};\
\widetilde{B}_{1},\dots,\widetilde{B}_{s})\in G$ with
$B_{k}\in\mathbb{T}_{n_{k}}(\mathbb{R})$, $k=1,\dots,r$ having an eigenvalue
$\mu_{k}$ and $\widetilde{B}_{l}\in\mathbb{B}_{m_{l}}(\mathbb{R})$,
$l=1,\dots,s$, we have $\mu_{k_{0}}>0$ or $\mu_{i}<0$ for some $i\neq k_{0}$.
Therefore
$G(u_{0})\subset\mathbb{R}^{n}\backslash\mathcal{C}^{\prime}_{u_{0},k_{0}}$
where
$\mathcal{C}^{\prime}_{u_{0},k_{0}}:=\left(\underset{i=1}{\overset{k_{0}-1}{\prod}}\mathbb{R}_{+}^{*}\times\mathbb{R}^{n_{i}-1}\right)\times\left(\mathbb{R}^{*}_{-}\times\mathbb{R}^{n_{k_{0}}-1}\right)\times\left(\underset{i=k_{0}+1}{\overset{r}{\prod}}\mathbb{R}_{+}^{*}\times\mathbb{R}^{n_{i}-1}\right)\times\left(\underset{l=1}{\overset{s}{\prod}}(\mathbb{R}^{2}\backslash\\{(0,0)\\})\times\mathbb{R}^{2m_{l}-2}\right)$
and thus
$\overline{\mathbb{R}^{n}\backslash\mathcal{C}^{\prime}_{u_{0},k_{0}}}=\mathbb{R}^{n}$,
that is $\mathcal{C}^{\prime}_{u_{0},k_{0}}=\emptyset$, a contradiction.
$(ii)\Longrightarrow(i)$: By Lemma 3.5,(iv), $G(u_{0})\cap
S(C_{u_{0}})\neq\emptyset,$ for every $S\in\Gamma$. So let $v\in G(u_{0})\cap
S(C_{u_{0}})$ and write $w=S^{-1}(v)\in C_{u_{0}}$. By Lemma 3.8,
$\overline{G(w)}\cap C_{u_{0}}=C_{u_{0}}.$ By Lemma 3.5, (ii),
$G(v)=G(Sw)=S(G(w))$. Hence we have
$\overline{G(v)}\cap S(C_{u_{0}})=S\left(\overline{G(w)}\cap
C_{u_{0}}\right)=S(C_{u_{0}}),$
and hence $S(C_{u_{0}})\subset\overline{G(u_{0})}$. As
$U=\underset{S\in\Gamma}{\bigcup}S(C_{u_{0}})$ (Lemma 3.5, (i)), then
$U\subset\overline{G(v)}$ and therefore $\overline{G(v)}=\mathbb{R}^{n}$ since
$\overline{U}=\mathbb{R}^{n}$. It follows that
$\overline{G(u_{0})}=\mathbb{R}^{n}$ since $v\in G(u_{0})$. ∎
Analogous to ([3], Theorem 7.2) for semigroup is the following:
###### Proposition 3.11.
Let $G$ be an abelian $\mathrm{sub}$-$\mathrm{semigroup}$ of
$\mathcal{K}^{*}_{\eta,r,s}(\mathbb{R})$.
* (1)
If $\overset{\circ}{\overline{G(u_{0})}}\neq\emptyset$ then there exists a map
$f:\ \mathbb{R}^{n}\ \longrightarrow\ \mathbb{R}^{n}$ satisfying
* (i)
$f$ is continuous and open
* (ii)
$f(Bu_{0})=e^{B}u_{0}$ for every $B\in\mathcal{C}(G)$.
* (iii)
$f^{-1}(G^{+}(u_{0}))=\mathrm{g}_{u_{0}}$.
* (iv)
$f(\mathbb{R}^{n})=C_{u_{0}}$.
* (2)
If $\overset{\circ}{\overline{\mathrm{g}_{u_{0}}}}\neq\emptyset$ then there
exists a map $h:\ \mathbb{R}^{n}\ \longrightarrow\ \mathbb{R}^{n}$ satisfying
* (i)
$h$ is continuous and open
* (ii)
$h(Bu_{0})=e^{B}u_{0}$ for every $B\in\mathcal{C}(\mathrm{g})$. In particular,
$h(\mathrm{g}_{u_{0}})=G^{+}(u_{0})$.
* (iii)
$h(\mathbb{R}^{n})=C_{u_{0}}$.
###### Proof.
(2): Suppose that
$\overset{\circ}{\overline{\mathrm{g}_{u_{0}}}}\neq\emptyset$. Then
$\overset{\circ}{\overline{\widehat{\mathrm{g}}_{u_{0}}}}\neq\emptyset$.
Applying ([3], Theorem 7.2) to $\widehat{G}$, so there exists a continuous
open map $h:\mathbb{R}^{n}\longrightarrow\mathbb{R}^{n}$ satisfying
$h(\mathbb{R}^{n})=C_{u_{0}}$ and $h(Bu_{0})=e^{B}u_{0}$, for every
$B\in\mathcal{C}(\widehat{\mathrm{g}})$. By Lemma 3.3, (ii),
$\mathrm{g}\subset\widehat{\mathrm{g}}\subset\mathcal{C}(\widehat{\mathrm{g}})$,
hence $h(Bu_{0})=e^{B}u_{0}$, for every $B\in\mathrm{g}$ and so
$h(\mathrm{g}_{u_{0}})=G^{+}(u_{0})$ (Lemma 3.3, (iii)).
(1): Similar arguments as before apply to the proof of (1) using ([3], Theorem
7.2). ∎
###### Proposition 3.12.
Let $G$ be an abelian sub-semigroup of
$\mathcal{K}_{\eta,r,s}^{*}(\mathbb{R})$. The following assertions are
equivalent:
* (i)
$G(u_{0})$ is locally dense in $\mathbb{R}^{n}$
* (ii)
$\mathrm{g}_{u_{0}}$ is dense in $\mathbb{R}^{n}$.
###### Proof.
$(ii)\Longrightarrow(i):$ By Proposition 3.11, (2), there exists a continuous
open map $h:\mathbb{R}^{n}\longrightarrow\mathbb{R}^{n}$ such that
$h(\mathrm{g}_{u_{0}})=G^{+}(u_{0})$. Hence,
$C_{u_{0}}=h(\mathbb{R}^{n})=h(\overline{\mathrm{g}_{u_{0}}})\subset\overline{G(u_{0})}$
and therefore $\overset{\circ}{\overline{G(u_{0})}}\neq\emptyset$.
$(i)\Longrightarrow(ii)$ : By Proposition 3.11,(1),(iii) and (iv), there
exists a continuous open map $f:\mathbb{R}^{n}\longrightarrow\mathbb{R}^{n}$
such that $f(\mathbb{R}^{n})=C_{u_{0}}$ and
$f^{-1}(G^{+}(u_{0}))=\mathrm{g}_{u_{0}}$. As $\overline{G(u_{0})}\cap
C_{u_{0}}=C_{u_{0}}$ (Lemma 3.8) and $G^{+}(u_{0})=G(u_{0})\cap C_{u_{0}}$
(Lemma 3.5,(iii)), then
$C_{u_{0}}\subset\mathring{\overline{C_{u_{0}}}}=\overset{\circ}{\overline{G(u_{0})\cap
C_{u_{0}}}}=\overset{\circ}{\overline{G^{+}(u_{0})}}.$
It follows that
$\mathbb{R}^{n}=f^{-1}(C_{u_{0}})\subset\overline{f^{-1}(G^{+}(u_{0}))}=\overline{\mathrm{g}_{u_{0}}}$
and therefore $\overline{\mathrm{g}_{u_{0}}}=\mathbb{R}^{n}$. ∎
## 4\. Proof of Theorem 1.1 and Corollary 1.2
###### Proposition 4.1.
Let $G$ be an abelian sub-semigroup of $M_{n}(\mathbb{R})$ and
$u\in\mathbb{R}^{n}$. Then $G(u)$ is locally dense $($resp. dense$)$ if and
only if so is $G^{*}(u)$.
###### Proof.
Suppose that $\overset{\circ}{\overline{G^{*}(u)}}\neq\emptyset$ (resp.
$\overline{G^{*}(u)}=\mathbb{R}^{n}$). It is obvious that
$\overset{\circ}{\overline{G(u)}}\neq\emptyset$ (resp.
$\overline{G(u_{0})}=\mathbb{R}^{n}$) since $G^{*}(u)\subset G(u)$.
Conversely, suppose that $\overset{\circ}{\overline{G(u)}}\neq\emptyset$
(resp. $\overline{G(u_{0})}=\mathbb{R}^{n}$). We can assume, by Proposition
2.2, that $G\subset\mathcal{K}_{\eta,r,s}(\mathbb{R})$. Write
$G^{\prime}:=G\backslash G^{*}$.
\- If $G^{\prime}=\emptyset$, then $G=G^{*}$ is locally hypercyclic (resp.
hypercyclic).
\- If $G^{\prime}\neq\emptyset$, then
$G(u)\subset\left(\underset{A\in G^{\prime}}{\bigcup}\textrm{Im}(A)\right)\cup
G^{*}(u).$
Since any $A\in G^{\prime}$ is non invertible,
$\textrm{Im}(A)\subset\underset{k=1}{\overset{r}{\bigcup}}H_{k}\cup\underset{l=1}{\overset{s}{\bigcup}}F_{l}$
where
$H_{k}:=\left\\{u=[u_{1},\dots,u_{r};\
\widetilde{u}_{1},\dots,\widetilde{u}_{s}]^{T}\in\mathbb{R}^{n},\
\begin{array}[]{c}u_{j}\in\mathbb{R}^{n_{j}},\\\
u_{k}\in\\{0\\}\times\mathbb{R}^{n_{k}-1},\\\
\widetilde{u}_{l}\in\mathbb{R}^{2m_{l}},\end{array}\begin{array}[]{c}1\leq
j\leq r,\\\ j\neq k\\\ 1\leq l\leq s,\end{array}\right\\}$
and
$F_{l}:=\left\\{u=[u_{1},\dots,u_{r};\
\widetilde{u}_{1},\dots,\widetilde{u}_{s}]^{T}\in\mathbb{R}^{n},\
\begin{array}[]{c}u_{k}\in\mathbb{R}^{n_{k}},\\\
\widetilde{u}_{j}\in\mathbb{R}^{2m_{j}},\\\
\widetilde{u}_{l}\in\\{(0,0)\\}\times\mathbb{R}^{2m_{l}-2},\\\
\end{array}\begin{array}[]{c}1\leq k\leq r,\\\ 1\leq j\leq s,\\\ j\neq l\\\
\end{array}\right\\}.$
It follows that
$G(u)\subset\left(\underset{k=1}{\overset{r}{\bigcup}}H_{k}\cup\underset{l=1}{\overset{s}{\bigcup}}F_{l}\right)\cup
G^{*}(u)$
and so
$\overline{G(u)}\subset\left(\underset{k=1}{\overset{r}{\bigcup}}H_{k}\cup\underset{l=1}{\overset{s}{\bigcup}}F_{l}\right)\cup\overline{G^{*}(u)}.$
Since $H_{k}$ (resp. $F_{l}$) has dimension $n-1$ (resp. $n-2$), so
$\overset{\circ}{H_{k}}=\overset{\circ}{F_{l}}=\emptyset$, for every $1\leq
k\leq r$, $1\leq l\leq s$. We conclude that
$\overset{\circ}{\overline{G^{*}(u)}}\neq\emptyset$ (resp.
$\overline{G^{*}(u)}=\mathbb{R}^{n}$). ∎
_Proof of Theorem 1.1._ One can assume by Proposition 2.2 that $G$ is an
abelian sub-semigroup of $\mathcal{K}_{\eta,r,s}(\mathbb{R})$, so
$v_{0}=u_{0}$. $(iii)\Longrightarrow(ii)$: By Lemma 3.4,
$\overline{\mathrm{g}^{*}_{u_{0}}}=\mathbb{R}^{n}$ where
$\mathrm{g}^{*}=\mathrm{exp}^{-1}(G^{*})\cap\mathcal{K}_{\eta,r,s}(\mathbb{R})$
and $\mathrm{g}^{*}_{u_{0}}=\\{Bu_{0}:\ B\in\mathrm{g}^{*}\\}$. Applying
Proposition 3.12 to $G^{*}$, then we have
$\overset{\circ}{\overline{G^{*}(u_{0})}}\neq\emptyset$ and therefore
$\overset{\circ}{\overline{G(u_{0})}}\neq\emptyset$.
$(ii)\Longrightarrow(iii)$: By Proposition 4.1,
$\overset{\circ}{\overline{G^{*}(u_{0})}}\neq\emptyset$. By applying
Proposition 3.12 to $G^{*}$, we have
$\overline{\mathrm{g}^{*}_{u_{0}}}=\mathbb{R}^{n}$. Since
$\mathrm{g}^{*}_{u_{0}}=\mathrm{g}_{u_{0}}$ (Lemma 3.4), it follows that
$\overline{\mathrm{g}_{u_{0}}}=\mathbb{R}^{n}$.
The equivalence $(i)\Longleftrightarrow(ii)$ results directly from
Propositions 4.1 and 3.7. ∎
_Proof of Corollary 1.2._ One can assume by Proposition 2.2, that
$G\subset\mathcal{K}_{\eta,r,s}(\mathbb{R})$, in this case $v_{0}=u_{0}$.
$(i)\Longleftrightarrow(ii)$ follows directly from Propositions 4.1 and 3.7,
(ii).
$(iii)\Longrightarrow(ii)$: As by definition
$\textrm{ind}(G)=\mathrm{ind}(G^{*})=r$ and
$\mathrm{g}_{u_{0}}=\mathrm{g}^{*}_{u_{0}}$ (Lemma 3.4) then by Proposition
3.12, $\overline{G^{*}(u_{0})}\neq\emptyset$. It follows by Proposition 3.10,
that $\overline{G^{*}(u_{0})}=\mathbb{R}^{n}$ and hence
$\overline{G(u_{0})}=\mathbb{R}^{n}$.
$(ii)\Longrightarrow(iii):$ By Proposition 4.1, one can suppose that
$G\subset\mathcal{K}^{*}_{\eta,r,s}(\mathbb{R})$. Then by Proposition 3.10,
$\textrm{ind}(G)=r$ and by Proposition 3.12,
$\overline{\mathrm{g}_{u_{0}}}=\mathbb{R}^{n}$.
## 5\. Proof of Theorem 1.3, Corollaries 1.4, 1.5 and 1.6
###### Lemma 5.1.
$($[3], Proposition 3.6$)$
* (i)
Let $A,B\in\mathbb{T}_{n}(\mathbb{R})$ such that $AB=BA$. If $e^{A}=e^{B}$,
then $A=B$.
* (ii)
Let $A$, $B\in\mathbb{B}_{m}(\mathbb{R})$ such that $AB=BA$. If $e^{A}=e^{B}$
then $A=B+2k\pi J_{m}$, for some $k\in\mathbb{Z}$
$where\ \ \ J_{m}=\mathrm{diag}(J_{2},\dots,J_{2})\in\textrm{GL}(2m,\
\mathbb{R})\ \ \mathrm{with}\ \ J_{2}=\left[\begin{array}[]{cc}0&-1\\\ 1&0\\\
\end{array}\right].$
###### Proposition 5.2.
Let $G$ be an abelian sub-semigroup of
$\mathcal{K}^{+}_{\eta,r,s}(\mathbb{R})$ and let
$B_{1},\dots,B_{p}\in\mathrm{g}$ $(p\in\mathbb{N}_{0})$ be such that
$e^{B_{1}},\dots,e^{B_{p}}$ generate $G$. We have
$\mathrm{g}_{u_{0}}=\underset{k=1}{\overset{p}{\sum}}\mathbb{N}B_{k}u_{0}+\underset{l=1}{\overset{s}{\sum}}2\pi\mathbb{Z}f^{(l)}.$
###### Proof.
$\bullet$ First we determine $\mathrm{g}$. Let $C\in\mathrm{g}$. Then
$C=\mathrm{diag}(C_{1},\dots,C_{r};\
\widetilde{C}_{1},\dots,\widetilde{C}_{s})\in\mathcal{K}_{n,r,s}(\mathbb{R})$
and $e^{C}\in G$. So $e^{C}=\mathrm{diag}(e^{C_{1}},\dots,e^{C_{r}};\
e^{\widetilde{C}_{1}},\dots,e^{\widetilde{C}_{s}})=e^{m_{1}B_{1}}\dots
e^{m_{p}B_{p}}$ for some $m_{1},\dots,m_{p}\in\mathbb{N}$. Since
$B_{1},\dots,B_{p}\in\mathrm{g}$, they pairwise commute (Lemma 3.3,(ii)).
Therefore, $e^{C}=e^{m_{1}B_{1}+\dots+m_{p}B_{p}}$. Write
$B_{j}=\mathrm{diag}(B_{j,1},\dots,B_{j,r};\
\widetilde{B}_{j,1},\dots,\widetilde{B}_{j,s}),$
then $e^{C_{k}}=e^{m_{1}B_{1,k}+\dots+m_{p}B_{p,k}}$, $k=1,\dots,r$ and
$e^{\widetilde{C}_{l}}=e^{m_{1}\widetilde{B}_{1,l}+\dots+m_{p}\widetilde{B}_{p,l}}$,
$l=1,\dots,s$. As $CB_{j}=B_{j}C$ then $C_{k}B_{j,k}=B_{j,k}C_{k}$ and
$\widetilde{C}_{l}\widetilde{B}_{j,l}=\widetilde{B}_{j,l}\widetilde{C}_{l},$
$j=1,\dots,p$. From Lemma 5.1, it follows that:
$C_{k}=m_{1}B_{1,k}+\dots+m_{p}B_{p,k}$ and
$\widetilde{C}_{l}=m_{1}\widetilde{B}_{1,l}+\dots+m_{p}\widetilde{B}_{p,l}+2\pi
t_{l}J_{m_{l}}$ for some $t_{l}\in\mathbb{Z}$ where
$J_{m_{l}}=\mathrm{diag}(J_{2},\dots,J_{2})\in\textrm{GL}(2m_{l},\
\mathbb{R})\ \ \mathrm{with}\ \ J_{2}=\left[\begin{array}[]{cc}0&-1\\\ 1&0\\\
\end{array}\right].$
Therefore
$\displaystyle C$
$\displaystyle=\mathrm{diag}\left(\underset{j=1}{\overset{p}{\sum}}m_{j}B_{j,1},\dots,\underset{j=1}{\overset{p}{\sum}}m_{j}B_{j,r};\underset{j=1}{\overset{p}{\sum}}m_{j}\widetilde{B}_{j,1}+2\pi
t_{1}J_{m_{1}},\
\dots,\underset{j=1}{\overset{p}{\sum}}m_{j}\widetilde{B}_{j,s}+2\pi
t_{s}J_{m_{s}}\right)$
$\displaystyle=\underset{j=1}{\overset{p}{\sum}}m_{j}B_{j}+\mathrm{diag}\left(0,\dots,0;\
2\pi t_{1}J_{m_{1}},\dots,2\pi t_{s}J_{m_{s}}\right)$
Set
$L_{l}:=\mathrm{diag}(0,\dots,0;\
\widetilde{L}_{l,1},\dots,\widetilde{L}_{l,s})$
where
$\widetilde{L}_{l,i}=\left\\{\begin{array}[]{c}0\in B_{m_{i}}(\mathbb{R})\ \ \
if\ \ i\neq l\\\ J_{m_{l}}\ \ \ \ \ \ \ \ \ \ \ \ \ if\ \
i=l\end{array}\right.$
Then we have $\mathrm{diag}(0,\dots,0,\ 2\pi t_{1}J_{m_{1}},\dots,2\pi
t_{s}J_{m_{s}})=\underset{l=1}{\overset{s}{\sum}}2\pi t_{l}L_{l}$ and
therefore
$C=\underset{j=1}{\overset{p}{\sum}}m_{j}B_{j}+\underset{l=1}{\overset{s}{\sum}}2\pi
t_{l}L_{l}$. We conclude that
$\mathrm{g}=\underset{j=1}{\overset{p}{\sum}}\mathbb{N}B_{j}+\underset{l=1}{\overset{s}{\sum}}2\pi\mathbb{Z}L_{l}$.
$\bullet$ Second we determine $\mathrm{g}_{u_{0}}$. Let $B\in\mathrm{g}$. We
have
$B=\underset{j=1}{\overset{p}{\sum}}m_{j}B_{j}+\underset{l=1}{\overset{s}{\sum}}2\pi
t_{l}L_{l}$ for some $m_{1},\dots,m_{p}\in\mathbb{N}$, and
$t_{1},\dots,t_{s}\in\mathbb{Z}$. As $\widetilde{L}_{l,i}f_{i,1}=f^{(l)}_{i}$,
$i=1,\dots,s$ then
$\displaystyle L_{l}u_{0}$
$\displaystyle=\mathrm{diag}(0,\dots,\widetilde{L}_{l,1},\dots,\widetilde{L}_{l,s})[e_{1,1},\dots,e_{r,1};\
f_{1,1},\dots,f_{s,1}]^{T}$ $\displaystyle=[0,\dots,0;\
f^{(l)}_{1},\dots,f^{(l)}_{s}]^{T}$ $\displaystyle=f^{(l)}.$
Hence
$Bu_{0}=\underset{j=1}{\overset{p}{\sum}}m_{j}B_{j}u_{0}+\underset{l=1}{\overset{s}{\sum}}2\pi
t_{l}f^{(l)}$ and therefore
$\mathrm{g}_{u_{0}}=\underset{j=1}{\overset{p}{\sum}}\mathbb{N}B_{j}u_{0}+\underset{l=1}{\overset{s}{\sum}}2\pi\mathbb{Z}f^{(l)}$.
This proves the proposition. ∎
###### Lemma 5.3.
Let $G$ be an abelian sub-semigroup of
$\mathcal{K}_{\eta,r,s}^{*}(\mathbb{R})$. Then:
$\overset{\circ}{\overline{G(u_{0})}}\neq\emptyset\ \ \ \textrm{ if and only
if }\ \ \overset{\circ}{\overline{G^{2}(u_{0})}}\neq\emptyset.$
###### Proof.
Suppose that $\overset{\circ}{\overline{G^{2}(u_{0})}}\neq\emptyset$. It is
obvious that $\overset{\circ}{\overline{G(u_{0})}}\neq\emptyset$ since
$G^{2}(u_{0})\subset G(u_{0})$. Conversely, suppose that
$\overset{\circ}{\overline{G(u_{0})}}\neq\emptyset$. Then by Theorem 1.1,
$\overline{\mathrm{g}_{u_{0}}}=\mathbb{R}^{n}$. As
$\mathrm{g}\subset\frac{1}{2}\mathrm{g}^{2}$ (since if $B\in\mathrm{g}$, we
have $e^{2B}=(e^{B})^{2}\in G^{2}$), then
$\overline{\frac{1}{2}\mathrm{g}^{2}_{u_{0}}}=\mathbb{R}^{n}$ and so
$\overline{\mathrm{g}^{2}_{u_{0}}}=\mathbb{R}^{n}$. Applying Theorem 1.1 to
the abelian sub-semigroup $G^{2}$, it follows that
$\overset{\circ}{\overline{G^{2}(u_{0})}}\neq\emptyset$. ∎
###### Corollary 5.4.
Let $G$ be an abelian sub-semigroup of
$\mathcal{K}^{*}_{\eta,r,s}(\mathbb{R})$. Then $G$ has a locally dense orbit
if and only if so does $G^{2}$.
###### Proof.
This is a consequence from Proposition 3.7,(ii) and Lemma 5.3. ∎
###### Proof of Theorem 1.3.
One can assume by Propositions 2.2 and 4.1 that $G$ is an abelian sub-
semigroup of $\mathcal{K}_{\eta,r,s}^{*}(\mathbb{R})$. Applying Theorem 1.1 to
the sub-semigroup $G^{2}$ of $\mathcal{K}^{+}_{\eta,r,s}(\mathbb{R})$, then
$(ii)\Leftrightarrow(iii)$ follows from Proposition 5.2 and Lemma 5.3.
$(i)\Leftrightarrow(ii)$ follows from Theorem 1.1. ∎
Proof of Corollary 1.4. This follows from Corollary 1.2 and Theorem 1.3.∎
###### Proposition 5.5.
$($[12], Lemma 2.1$)$. Let $H=\mathbb{Z}u_{1}+\dots+\mathbb{Z}u_{m}$ with
$u_{k}\in\mathbb{R}^{n}$, $k=1,\dots,m$. If $m\leq n$ then $H$ is nowhere
dense in $\mathbb{R}^{n}$.
Proof of Corollary 1.5. Let $P\in\textrm{GL}(n,\mathbb{R})$ so that
$P^{-1}GP\subset\mathcal{K}_{\eta,r,s}(\mathbb{R})$. Let $A_{1},\dots,A_{p}$
generate $G$ and let $B_{1},\dots,B_{p}\in\mathrm{g}$ so that
$A_{1}^{2}=e^{B_{1}},\dots,A_{p}^{2}=e^{B_{p}}$. If $p\leq n-s$ then
$\mathrm{g}^{2}_{v_{0}}=\underset{k=1}{\overset{p}{\sum}}\mathbb{Z}(B_{k}v_{0})+\underset{l=1}{\overset{s}{\sum}}2\pi\mathbb{Z}Pf^{(l)}$
is nowhere dense in $\mathbb{R}^{n}$ (Proposition 5.5) and in particular, for
$\mathrm{g}_{v_{0}}=\underset{k=1}{\overset{p}{\sum}}\mathbb{N}B_{k}v_{0}+\underset{k=1}{\overset{s}{\sum}}2\pi\mathbb{Z}Pf^{(l)}$.
By Theorem 1.3, $G(v_{0})$ is nowhere dense in $\mathbb{R}^{n}$. ∎
Proof of Corollary 1.6. Let $P\in\textrm{GL}(n,\mathbb{R})$ so that
$P^{-1}GP\subset\mathcal{K}_{\eta,r,s}(\mathbb{R})$. Let $A_{1},\dots,A_{p}$
generate $G$ with $p\leq\left[\frac{n+1}{2}\right]$. Since $r+2s\leq n$, it
follows that $p+s\leq\left[\frac{n+1}{2}\right]+\frac{n-r}{2}\leq
n+\frac{1-r}{2}\leq n+\frac{1}{2}$. Hence, $p+s\leq n$ and therefore Corollary
1.6 follows from Corollary 1.5. ∎
## 6\. Proof of Theorem 1.7 and Corollary 1.9
We will construct for any any $n\in\mathbb{N}_{0}$ and any $r,s=1,\dots,n$,
and for any partition $\eta\in\mathbb{N}_{0}^{r+s}$ of $n$, ($n-s+1$) matrices
$A_{1},\dots,A_{n-s+1}\in\mathcal{K}_{\eta,r,s}^{*}(\mathbb{R})$ that they
generate an hypercyclic abelian semigroup. For this we need the following
propositions and lemmas:
We used repeatedly the following multidimensional version of Kronecker’s
Theorem stated below. (See for example [5], Theorem 442):
Kronecker’s Theorem. Let $\alpha_{1},\dots,\alpha_{n}$ be negative real
numbers such that the numbers $1,\alpha_{1},\dots,\alpha_{n}$ are linearly
independent over $\mathbb{Q}$. Then the set
$\mathbb{N}^{n}+\mathbb{N}[\alpha_{1},\dots,\alpha_{n}]^{T}:=\left\\{[s_{1},\dots,s_{n}]^{T}+k[\alpha_{1},\dots,\alpha_{n}]^{T}:\
k,s_{1},\dots,s_{n}\in\mathbb{N}\right\\}$ is dense in $\mathbb{R}^{n}$.
###### Proposition 6.1.
Let $n\in\mathbb{N}_{0}$ and $s=1,\dots,n$. Then there exist $n-s+1$ vectors
$u_{1},\dots,u_{n-s+1}$ of $\mathbb{R}^{n}$ such that
$H:=\underset{k=1}{\overset{n-s+1}{\sum}}\mathbb{N}u_{k}+\underset{l=1}{\overset{s}{\sum}}2\pi\mathbb{Z}f^{(l)}$
is dense in $\mathbb{R}^{n}$.
###### Proof.
Let $\alpha_{1},\dots,\alpha_{n}$ be negative real numbers such that the
numbers $1,\alpha_{1},\dots,\alpha_{n}$ are linearly independent over
$\mathbb{Q}$. Recall that $f^{(l)}=e_{t_{l}}$ where
$t_{1}=\underset{j=1}{\overset{r}{\sum}}n_{j}+2$,
$t_{l}=\underset{j=1}{\overset{r}{\sum}}n_{j}+2\underset{j=1}{\overset{l-1}{\sum}}m_{l}+2$,
$l=2,\dots,s$. Denote by
$\mathcal{B}_{0}\backslash(e_{t_{1}},\dots,e_{t_{s}}):=(e_{i_{s+1}},\dots,e_{i_{n}})$
and let $S$ denote the matrix defined by
$Se_{k}=\begin{cases}2\pi f^{(k)},&\ \mathrm{if}\ 1\leq k\leq s\\\
e_{i_{k}},&\ \mathrm{if}\ s+1\leq k\leq n\end{cases}$
We see that $S\in GL(n;\mathbb{R})$. Write
$u=[\alpha_{1},\dots,\alpha_{n}]^{T},$ $u_{k}=\begin{cases}Se_{s+k},&\
\mathrm{if}\ \ 1\leq k\leq n-s\\\ Su,&\ \mathrm{if}\ \ k=n-s+1\\\ \end{cases}$
and
$H^{\prime}:=\underset{k=1}{\overset{n-s}{\sum}}\mathbb{N}e_{s+k}+\mathbb{N}u+\underset{l=1}{\overset{s}{\sum}}\mathbb{Z}e_{l}.$
Then we have
$\displaystyle S(H^{\prime})$
$\displaystyle=\underset{k=1}{\overset{n-s}{\sum}}\mathbb{N}Se_{s+k}+\mathbb{N}Su+\underset{l=1}{\overset{s}{\sum}}\mathbb{Z}Se_{l}$
$\displaystyle=\underset{k=1}{\overset{n-s}{\sum}}\mathbb{N}u_{k}+\mathbb{N}u_{n-s+1}+\underset{l=1}{\overset{s}{\sum}}2\pi\mathbb{Z}f^{(l)}$
$\displaystyle=\underset{k=1}{\overset{n-s+1}{\sum}}\mathbb{N}u_{k}+\underset{l=1}{\overset{s}{\sum}}2\pi\mathbb{Z}f^{(l)}$
$\displaystyle=H$
Since $\mathbb{N}^{n}+\mathbb{N}u\subset H^{\prime}$, it follows by by
Kronecker’s theorem that $H^{\prime}$ is dense in $\mathbb{R}^{n}$ and thus so
is $H$. This proves the proposition. ∎
###### Proof of Theorem 1.7.
Let $u_{1},\dots,u_{n-s+1}\in\mathbb{R}^{n}$ so that
$H:=\underset{j=1}{\overset{n-s+1}{\sum}}\mathbb{N}u_{j}+\underset{l=1}{\overset{s}{\sum}}2\pi\mathbb{Z}f^{(l)}$
is dense in $\mathbb{R}^{n}$ (Proposition 6.1). Write
$u_{j}=[u_{j,1},\dots,u_{j,r};\widetilde{u}_{j,1},\dots,\widetilde{u}_{j,s}]^{T}$
with $u_{j,k}=[x^{(j)}_{k,1},\dots,x^{(j)}_{k,n_{k}}]^{T}$ and
$\widetilde{u}_{j,l}=[y^{(j)}_{l,1},y^{\prime(j)}_{l,1},\dots,y^{(j)}_{l,m_{l}},y^{\prime(j)}_{l,m_{l}}]^{T}$,
$1\leq j\leq n-s+1$, $1\leq k\leq r$, $1\leq l\leq s$. For every $1\leq j\leq
n-s+1$, define $B_{j}=\mathrm{diag}(B_{j,1},\dots,B_{j,r};\
\widetilde{B}_{j,1},\dots,\widetilde{B}_{j,s})$ where
$B_{j,k}=\left[\begin{array}[]{ccccc}x^{(j)}_{k,1}&&&&0\\\ \vdots&\ddots&&&\\\
\vdots&0&\ddots&&\\\ \vdots&\vdots&\ddots&\ddots&\\\
x^{(j)}_{k,n_{k}}&0&\dots&0&x^{(j)}_{k,1}\end{array}\right]\ \ \ \
\mathrm{and}\ \ \ \
\widetilde{B}_{j,l}=\left[\begin{array}[]{ccccc}C^{(j)}_{l,1}&&&&0\\\
\vdots&\ddots&&&\\\ \vdots&0&\ddots&&\\\ \vdots&\vdots&\ddots&\ddots&\\\
C^{(j)}_{l,2m_{l}}&0&\dots&0&C^{(j)}_{l,1}\end{array}\right],$
$\mathrm{where}\ \
C^{(j)}_{l,i}=\left[\begin{array}[]{cc}y^{(j)}_{l,i}&y^{\prime(j)}_{l,i}\\\
-y^{\prime(j)}_{l,i}&y^{(j)}_{l,i}\end{array}\right],$ $1\leq k\leq r,\ \
1\leq l\leq s,\ 1\leq i\leq m_{l}.$
Then we have $B_{j}u_{0}=u_{j}$. Write
$A_{j}=\mathrm{diag}(A_{j,1},\dots,A_{j,r};\widetilde{A}_{j,1},\dots,\widetilde{A}_{j,s})$
where
$\widetilde{A_{j,l}}=e^{\frac{1}{2}\widetilde{B_{j,l}}},\ \ l=1,\dots,s$
and
$A_{j,k}=\begin{cases}e^{\frac{1}{2}B_{j,k}},\ &\mathrm{if}\ 1\leq k\neq j\leq
r\\\ -e^{\frac{1}{2}B_{j,j}},\ &\mathrm{if}\ k=j\end{cases}$
Let $G$ be the sub-semigroup of $\mathcal{K}_{\eta,r,s}^{*}(\mathbb{R})$
generated by $A_{1},\dots,A_{n-s+1}$. Note that
\- $A^{2}_{j}=e^{B_{j}}$, $j=1,\dots,n-s+1$,
\- $r\leq n-s+1$ since $r+2s\leq n$.
\- $A_{j,j}$ has a negative eigenvalue: $-e^{x^{(j)}_{j,1}}$, for every $1\leq
j\leq r$,
Therefore $\mathrm{ind}(G)=r$.
Moreover, $G$ is abelian: For this, it suffices to show that
$A_{i}A_{j}=A_{j}A_{i}$ for every $1\leq i,j\leq n-s+1$, which is equivalent
to show that $B_{j}B_{j^{\prime}}=B_{j^{\prime}}B_{j}$ and
$\widetilde{B}_{j}\widetilde{B}_{j^{\prime}}=\widetilde{B}_{j^{\prime}}\widetilde{B}_{j}$
for every $j,\ j^{\prime}=1,\dots,n-s+1$:
Write $B_{j,k}=N_{j,k}+x^{(j)}_{k,1}I_{n_{k}},\
\widetilde{B}_{j,l}=\widetilde{N}_{j,l}+\widetilde{D}^{(j)}_{l,1}$
where
$N_{j,k}=\left[\begin{array}[]{cc}0&0\\\
T_{j,k}&0\end{array}\right]\in\mathbb{T}_{n_{k}}(\mathbb{R}),\
\widetilde{D}^{(j)}_{l,1}=\mathrm{diag}(C^{(j)}_{l,1},\dots,C^{(j)}_{l,1}),$
with
$T_{j,k}=\left[x^{(j)}_{k,2},\dots,x^{(j)}_{k,n_{k}}\right]^{T},\
k=1,\dots,r.$
and
$\widetilde{N}_{j,l}=\left[\begin{array}[]{cc}0&0\\\
\widetilde{T}_{j,l}&0\end{array}\right]\in\mathbb{B}_{m_{l}}(\mathbb{R}),$
with
$\widetilde{T}_{j,l}=\left[C^{(j)}_{l,2},\dots,C^{(j)}_{l,m_{l}}\right]^{T},\
l=1,\dots,s.$
We see that $N_{j,k}N_{j^{\prime},k}=N_{j^{\prime},k}N_{j,k}=0$, for every
$j,\ j^{\prime},\ k=1,\dots,s$. Hence
$B_{j,k}B_{j^{\prime},k}=B_{j^{\prime},k}B_{j,k}$. In the same way,
$\widetilde{N}_{j,k}\widetilde{N}_{j,k^{\prime}}=\widetilde{N}_{j,k^{\prime}}\widetilde{N}_{j,k}=0$,
$\widetilde{N}_{j,k}\widetilde{D}_{j,k^{\prime}}=\widetilde{D}_{j,k^{\prime}}\widetilde{N}_{j,k}$
and
$\widetilde{N}_{j^{\prime},k}\widetilde{D}_{j,k}=\widetilde{D}_{j,k}\widetilde{N}_{j^{\prime},k}$,
for every $k=1,\dots,s$. Therefore
$\widetilde{B}_{j,k}\widetilde{B}_{j^{\prime},k}=\widetilde{B}_{j^{\prime},k}\widetilde{B}_{j,k}$
and thus $B_{j}B_{j^{\prime}}=B_{j^{\prime}}B_{j}$.
Now by Proposition 5.2, we have
$\displaystyle\mathrm{g}^{2}_{u_{0}}$
$\displaystyle=\underset{k=1}{\overset{n-s+1}{\sum}}\mathbb{N}B_{k}u_{0}+\underset{l=1}{\overset{s}{\sum}}2\pi\mathbb{Z}f^{(l)}$
$\displaystyle=\underset{k=1}{\overset{n-s+1}{\sum}}\mathbb{N}u_{k}+\underset{l=1}{\overset{s}{\sum}}2\pi\mathbb{Z}f^{(l)}$
$\displaystyle=H$
By proposition 6.1, $\overline{\mathrm{g}^{2}_{u_{0}}}=\mathbb{R}^{n}$ and
since $\mathrm{ind}(G)=r$, it follows by Corollary 1.4, that
$\overline{G(u_{0})}=\mathbb{R}^{n}$. ∎
.
Proof of Corollary 1.9. Take $s=n-\left[\frac{n+1}{2}\right]$. Then
$s\in\mathbb{N}$, $0\leq s\leq\frac{n}{2}$ and so
$n-s+1\geq\left[\frac{n+1}{2}\right]$. By Theorem 1.7, there exist
$(\left[\frac{n+1}{2}\right]+1)$ matrices in
$\mathcal{K}_{\eta;r,s}^{*}(\mathbb{R})$ that they generate an hypercyclic
abelian semigroup. By Corollary 1.6, $(\left[\frac{n+1}{2}\right]+1)$ is the
minimum number of matrices having such property. The proof is complete.∎
## 7\. Example
###### Example 7.1.
Let $G$ be the semigroup generated by $A_{1}=\mathrm{diag}(e^{\pi},e^{\pi})$,
$A_{2}=\left[\begin{array}[]{cc}-1&0\\\ \pi&-1\end{array}\right]$ and
$A_{3}=e^{-\pi\sqrt{2}}\left[\begin{array}[]{cc}1&0\\\
-\pi\sqrt{3}&1\end{array}\right].$
Then $G$ is abelian and hypercyclic.
###### Proof.
By construction, $G$ is an abelian sub-semigroup of
$\mathbb{T}^{*}_{2}(\mathbb{R})$, $\mathrm{ind}(G)=1$. Moreover, $u_{0}=e_{1}$
and $A^{2}_{k}=e^{B_{k}}$, $k=1,2,3$ where $B_{1}=\mathrm{diag}(2\pi;2\pi)$,
$B_{2}=\left[\begin{array}[]{cc}0&0\\\ 2\pi&0\end{array}\right]$ and
$B_{3}=\left[\begin{array}[]{cc}-2\pi\sqrt{2}&0\\\
-2\pi\sqrt{3}&-2\pi\sqrt{2}\end{array}\right].$
By Proposition 5.2,
$\displaystyle\mathrm{g}^{2}_{e_{1}}$
$\displaystyle=\underset{k=1}{\overset{3}{\sum}}\mathbb{N}B_{k}e_{1}$
$\displaystyle=2\pi H$
where
$H:=\mathbb{N}e_{1}+\mathbb{N}e_{2}+\mathbb{N}[-\sqrt{2},-\sqrt{3}]^{T}=\mathbb{N}^{2}+\mathbb{N}[-\sqrt{2},-\sqrt{3}]^{T}$.
By Kronecker’s Theorem, $\overline{H}=\mathbb{R}^{2}$ since $1,-\sqrt{2}$ and
$-\sqrt{3}$ are linearly independent over $\mathbb{Q}$. We conclude, by
Corollary 1.4, that $\overline{G(e_{1})}=\mathbb{R}^{2}$. ∎
## References
* [1] H. Abels and A. Manoussos, _Groups generators and hypercyclic tuples of matrices_ , preprint.
* [2] A. Ayadi and H. Marzougui, _Dense orbits for abelian subgroups of GL(n, $\mathbb{C}$)_, Foliations 2005: World Scientific, Hackensack, NJ, (2006), 47-69.
* [3] A. Ayadi, H. Marzougui and E. Salhi, _Topological transitive abelian subgroups of GL(n, $\mathbb{R})$_, Preprint Hal: 00519070, version 2 (2010).
* [4] A. Ayadi and H. Marzougui, _Hypercyclic abelian semigroups of matrices on $\mathbb{C}^{n}$_, Preprint ICTP, IC/2010/059.
* [5] F. Bayart, E. Matheron, Dynamics of Linear Operators, Cambridge Tracts in Math., 179, Cambridge University Press, (2009).
* [6] S.Chihi, _On the minimal orbits of an abelian linear action_ , Differential Geometry-Dynamical System, 12, (2010), 61-72.
* [7] G. Costakis, D. Hadjiloucas and A. Manoussos, _On the minimal number of matrices which form a locally hypercyclic, non-hypercyclic tuple_ , J. Math. Anal. Appl. 365 (2010), 229–237.
* [8] G. Costakis and I. Parissis, _Dynamics of tuple of matrices in Jordan form_ , preprint, arXiv:1003.5321v2 (2010).
* [9] N.S. Feldman, _Hypercyclic tuples of operators and somewhere dense orbits_ , J. Math. Anal. Appl. 346 (2008), 82–98.
* [10] K.G. Grosse-Herdmann and A. Peris, Linear Chaos, Universitext, Springer, to appear.
* [11] M.S. Kulikov, _Schottky-type and minimal sets of horocycle and geodesic flows_ , Sbornik Mathematics 195: 1, (2004), 35-64.
* [12] S. Shkarin, _Hypercyclic tuples of operator on ${\mathbb{C}}^{n}$ and ${\mathbb{R}}^{n}$_, Preprint arXiv: 1008.3483v1 (2010).
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arxiv-papers
| 2010-10-28T10:49:18 |
2024-09-04T02:49:14.322833
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Adlene Ayadi and Habib Marzougui",
"submitter": "Habib Marzougui",
"url": "https://arxiv.org/abs/1010.5915"
}
|
1010.5969
|
# The event generator for the two-photon process
$e^{+}e^{-}\to e^{+}e^{-}R$ $(J^{PC}=0^{-+})$ in the single-tag mode
V. P. Druzhinin Budker Institute of Nuclear Physics, Novosibirsk 630090,
Russia Novosibirsk State University, Novosibirsk 630090, Russia L. A.
Kardapoltsev Novosibirsk State University, Novosibirsk 630090, Russia Budker
Institute of Nuclear Physics, Novosibirsk 630090, Russia V. A. Tayursky
tayursky@inp.nsk.su Budker Institute of Nuclear Physics, Novosibirsk 630090,
Russia
###### Abstract
The Monte Carlo event generator GGRESRC is described. The generator is
developed for simulation of events of the two-photon process $e^{+}e^{-}\to
e^{+}e^{-}R$, where R is a pseudoscalar resonance, $\pi^{0}$, $\eta$,
$\eta^{\prime}$, $\eta_{c}$, or $\eta_{b}$. The program is optimized for
generation of two-photon events in the single-tag mode. For single-tag events,
radiative correction simulation is implemented in the generator including
photon emission from the initial and final states.
## 1 Introduction
The purpose of this work is to develop an efficient event generator for the
process of the two-photon resonance production $e^{+}e^{-}\to e^{+}e^{-}R$ in
the so-called single-tag mode, when one of the final electrons111Unless
otherwise specified, we use the term ”electron” for either an electron or a
positron. is scattered at a large angle and detected. Such generator is needed
for simulation of experiments on the measurement of the meson-photon
transition form factors. The generator GGRESRC described in this work was used
for the measurement of the transition form factors for the $\pi^{0}$, $\eta$,
$\eta^{\prime}$, and $\eta_{c}$ mesons with the BABAR detector. To achieve
required accuracy ($\sim 1\%$), the radiative corrections to the Born cross
section are taken into account. In particular, extra photon emission from the
initial and final states are simulated.
In the two-photon process $e^{+}e^{-}\to e^{+}e^{-}R$, the virtual photons,
radiated by the colliding electrons, form a $C$-even resonance with the four-
momentum ${k}={k}_{1}+{k}_{2}$ (see Fig. 1).
Figure 1: The diagram of the two-photon process $e^{+}e^{-}\to e^{+}e^{-}+R$.
Let $Q_{2}^{2}$ be the absolute value of the four-momentum squared, carried by
the space-like photon connected with the tagged (detected) electron , while
$Q_{1}^{2}$ be the same parameter for the untagged (undetected) electron
($Q_{1}^{2}\approx 0$). The transition form factor is determined from the
measured differential cross section $({\rm d}\sigma/{\rm d}Q_{2}^{2})_{\rm
data}$ and the MC calculated cross section $({\rm d}\sigma/{\rm
d}Q_{2}^{2})_{\rm MC}$:
$|{F}^{\rm data}_{\gamma^{*}\gamma R}(Q_{2}^{2})|^{2}=\frac{({\rm
d}\sigma/{\rm d}Q_{2}^{2})_{\rm data}}{({\rm d}\sigma/{\rm d}Q_{2}^{2})_{\rm
MC}}|{F}^{\rm MC}_{\gamma^{*}\gamma R}(Q_{2}^{2})|^{2},$ (1)
where $|{F}^{MC}_{\gamma^{*}\gamma R}(Q_{2}^{2})|^{2}$ is the transition form
factor used in MC simulation.
## 2 Born cross section
To describe the process $e^{+}e^{-}\to e^{+}e^{-}R$ we use the notations
defined in Fig. 1, and the following six invariants:
$\displaystyle t_{1}=-Q_{1}^{2}={k}_{1}^{2},\quad
t_{2}=-Q_{2}^{2}={k}_{2}^{2},$ $\displaystyle
s_{1}=({p}_{1}^{\prime}+{k})^{2},\quad s_{2}=({p}_{2}^{\prime}+{k})^{2},$ (2)
$\displaystyle s=({p}_{1}+{p}_{2})^{2},\quad
W^{2}={k}^{2}=({k}_{1}+{k}_{2})^{2}.$
The differential cross-section for this process in the lowest QED order is
given by BGMS :
${\rm
d}\sigma=\frac{\alpha^{2}}{16\pi^{4}t_{1}t_{2}}\sqrt{\frac{({k}_{1}{k}_{2})^{2}-t_{1}t_{2}}{({p}_{1}{p}_{2})^{2}-m_{e}^{4}}}\Sigma\frac{{\rm
d}^{3}\vec{p^{\prime}}_{1}}{E^{\prime}_{1}}\frac{{\rm
d}^{3}\vec{p^{\prime}}_{2}}{E^{\prime}_{2}},$ (3)
where $\alpha$ is the fine structure constant, $m_{e}$ is the electron mass,
$E^{\prime}_{i}$ ($i$=1,2) are the energies of the scattered electrons and
$\displaystyle\Sigma$ $\displaystyle=$ $\displaystyle
4\rho^{++}_{1}\rho^{++}_{2}\sigma_{TT}+2\rho^{++}_{1}\rho^{00}_{2}\sigma_{TS}+2\rho^{00}_{1}\rho^{++}_{2}\sigma_{ST}+\rho^{00}_{1}\rho^{00}_{2}\sigma_{SS}$
$\displaystyle+$ $\displaystyle
2|\rho^{+-}_{1}\rho^{+-}_{2}|\tau_{TT}\cos\,2\tilde{\phi}-8|\rho^{+0}_{1}\rho^{+0}_{2}|\tau_{TS}\cos\,\tilde{\phi}.$
Here $\tilde{\phi}$ is the angle between the electron and positron scattering
planes in the center-of-mass (c.m.) frame of the virtual photons,
$\sigma_{ab}$ are the $\gamma^{\ast}\gamma^{\ast}\to R$ cross sections for
unpolarized transverse ($a,b=T$) and scalar ($a,b=S$) photons. The
interference terms containing the functions $\tau_{ab}$ arise due to virtual
photon polarization. The function $\tau_{TT}$ is the difference between cross
sections for transverse photons with the parallel and orthogonal linear
polarizations: $\tau_{TT}=\sigma_{\parallel}-\sigma_{\perp}$, while the cross
section for unpolarized photons is
$\sigma_{TT}=(\sigma_{\parallel}+\sigma_{\perp})/2$.
The effects of the strong interaction are completely contained in the
functions $\sigma_{ab}$ and $\tau_{ab}$. All other functions entering in Eq.
(2) are calculable with QED. The expressions for the virtual photon density
matrices $\rho^{++}_{i}$, $\rho^{+-}_{i}$, $\rho^{+0}_{i}$, $\rho^{00}_{i}$
($i=1,2$) can be found in Ref. BGMS .
In the case of the pseudoscalar meson production, only the functions
$\sigma_{TT}$ and $\tau_{TT}$ are non-zero, and $\tau_{TT}=-2\sigma_{TT}$
poppe . The cross section $\sigma_{TT}$ for a narrow pseudoscalar meson with
the mass $M_{R}$ can be written in term of the transition form factor:
$\sigma_{TT}(W,Q_{1}^{2},Q_{2}^{2})=8\pi\frac{\Gamma_{\gamma\gamma}}{M_{R}}\left|\frac{{F}(Q_{1}^{2},Q_{2}^{2})}{{F}(0,0)}\right|^{2},\,\,|{F}(0,0)|^{2}=\frac{4\Gamma_{\gamma\gamma}}{\pi\alpha^{2}M_{R}^{3}},$
(5)
where $\Gamma_{\gamma\gamma}$ is the meson two-photon width. It should be
noted that some two-photon event generators neglect the term with $\tau_{TT}$.
This approach may be reasonable only for study of two-photon processes in the
no-tag mode, when both the electrons are scattered at small angles. The
$\tau_{TT}$ term gives a sizable contribution to the differential cross
section ${\rm d}\sigma/{\rm d}Q_{2}^{2}$ at large $Q_{2}^{2}$ and should be
taken into account in simulation of single-tag experiments.
In the GGRESRC events generator we perform integration of the differential
cross section using invariant variables (2). For a narrow pseudoscalar
resonance, Eq. (3) can be rewritten:
${\rm d}\sigma=\frac{4\alpha^{2}\Gamma_{\gamma\gamma}}{\pi
s^{2}t_{1}^{2}t_{2}^{2}M_{R}^{3}}\left|\frac{F(t_{1},t_{2})}{F(0,0)}\right|^{2}B\frac{{\rm
d}t_{2}{\rm d}t_{1}{\rm d}s_{1}{\rm d}s_{2}}{\sqrt{-\Delta_{4}}},$ (6)
where $\Delta_{4}(s,s_{1},s_{2},t_{1},t_{2},M_{R}^{2},m_{e}^{2})$ is the Gram
determinant BK . The physical region in the variables $s_{1}$, $s_{2}$,
$t_{1}$, $t_{2}$ is defined by the condition $\Delta_{4}\leq 0$. The function
$B$ coincides, up to a factor, with the function $\Sigma$ (Eq. (2)) for
pseudoscalar mesons. It was calculated in Ref. BKT and is given by
$B=\frac{1}{4}t_{1}t_{2}B_{1}-4B_{2}^{2}+m_{e}^{2}B_{3},$ (7)
where
$\displaystyle B_{1}$ $\displaystyle=$
$\displaystyle(4{p_{1}}{p_{2}}-2{p_{1}}{k_{2}}-2{p_{2}}{k_{1}}+{k_{1}}{k_{2}})^{2}+({k_{1}}{k_{2}})^{2}-16t_{1}t_{2}-16m_{e}^{4},$
$\displaystyle B_{2}$ $\displaystyle=$
$\displaystyle({p_{1}}{p_{2}})({k_{1}}{k_{2}})-({p_{1}}{k_{2}})({p_{2}}{k_{1}}),$
(8) $\displaystyle B_{3}$ $\displaystyle=$ $\displaystyle
t_{1}(2{p_{1}}{k_{2}}-{k_{1}}{k_{2}})^{2}+t_{2}(2{p_{2}}{k_{1}}-{k_{1}}{k_{2}})^{2}+4m_{e}^{2}({k_{1}}{k_{2}})^{2},$
To describe the $Q_{1}^{2}$ and $Q_{2}^{2}$ dependencies of the transition
form factor $F(Q_{1}^{2},Q_{2}^{2})$, the two options are implemented in the
generator: $F(Q_{1}^{2},Q_{2}^{2})=F(0,0)$, and the vector-dominance model
(VDM)
$|F|^{2}=\frac{1}{(1+Q^{2}_{1}/\Lambda^{2})^{2}(1+Q^{2}_{2}/\Lambda^{2})^{2}},$
(9)
where $\Lambda=m_{\rho}$ for $\pi^{0}$, $\eta$, $\eta^{\prime}$ production,
$\Lambda=m_{J/\psi}$ for $\eta_{c}$, and $\Lambda=m_{\Upsilon}$ for
$\eta_{b}$. The $Q_{2}^{2}$ dependence of the $|F|^{2}$ calculated with Eq.
(9) at $Q_{1}^{2}=0$ for $\Lambda=m_{\rho}$, is shown in Fig. 2.
Figure 2: The $Q^{2}$ dependence of the form factor $|F|^{2}$ at
$Q_{1}^{2}=0$, $\Lambda=m_{\rho}$=0.7755 GeV.
Four-dimensional Monte-Carlo integration of Eq. (6) is performed using the
method developed for the GALUGA two-photon event generator S . In this method,
in particular, the invariant variables are generated in the order $t_{2}$,
$t_{1}$, $s_{1}$, $s_{2}$. This allows to set a restriction on $Q_{2}^{2}$ at
the beginning of the event generation and significantly increase the
generation efficiency for single-tag events. The values of the generated
invariants, are then used together with a random azimuthal angle of the system
of the final particles to calculate the 4-momenta of the scattered electron,
positron, and produced resonance. The formulae to do this can be found in Ref.
T . The main decay modes for $\pi^{0}$, $\eta$, and $\eta^{\prime}$ are also
simulated according to Ref. T .
The total widths of the $\eta_{c}$ and $\eta_{b}$ resonances are comparable or
even larger than the mass resolution of modern detectors. Therefore, the mass
distributions for these resonances are generated using Breit-Wigner
distributions.
## 3 Radiative correction
In the no-tag mode, when both the electron and the positron are scattered
predominantly at small angles, the radiative correction to the Born cross
section is expected to be small, less than 1% RC . The situation changes
drastically in the single-tag mode, at a large electron scattering angle. At
large $Q^{2}$ the correction due to extra photon emission from the initial
state may reach several percents and should be taken into account in
simulation.
The process-independent formula for the radiative correction in the next-to-
leading order for two-photon processes in the single-tag mode was obtained in
Ref. OK . The main contribution to the correction comes from the vertex of the
tagged electron. The corresponding contribution of the untagged-electron
vertex is expected to be smaller than 0.5% and neglected. Fig. 3 shows the
diagrams taken into account in Ref. OK . They substitutes for the left-hand
vertex in Fig. 1.
Figure 3: Diagrams used for calculation of the radiative correction.
The cross section for a single-tag experiment is given by:
${\rm d}\sigma={\rm d}\sigma_{B}(1+\delta)={\rm
d}\sigma_{B}(1+\delta^{\prime}+\delta_{VP}),$ (10)
where ${\rm d}\sigma_{B}$ is the lowest-order cross section for the two-photon
process given, for example, by Eq. (6). The total radiative correction is
separated into two parts:
1. i.
$\delta^{\prime}$, which includes the virtual correction due to the
interference between the diagrams (a) and (c), soft-photon part of diagrams
(d)+(e), and the corrections due to real photon emission from the initial
(diagram (e)) and final (diagram (d)) states,
2. ii.
$\delta_{VP}$, the vacuum polarization correction due to the interference
between the diagrams (a) and (b).
To obtain $\delta^{\prime}$ we have used the result of Ref. OK for the total
radiative correction, removing from it the contribution of the vacuum
polarization diagram, $\delta_{e}$ (in Ref. OK only electron contribution was
taken into account). The resulting $\delta^{\prime}$ is given by
$\delta^{\prime}=-\frac{\alpha}{\pi}\Biggl{\\{}\biggl{[}\ln\frac{1}{r_{max}}-\frac{3}{4}\biggr{]}(L-1)+\frac{1}{4}\Biggr{\\}}.$
(11)
where $r_{max}$ ($\ll 1$) is the maximum energy of the photon emitted from the
initial state in units of the beam energy $E_{b}$, $L=\ln{(Q^{2}/m_{e}^{2})}$,
and $Q^{2}$ is the absolute value of the momentum transfer squared to the
electron. The formula does not contain any restriction on the energy of the
photon emitted from the final state, i.e. the cross section given by Eq. (10)
is calculated for the case when the tagged electron is allowed to radiate a
photon of any possible energy. The values of the correction $\delta^{\prime}$
for nine representative sets of $Q^{2}$ and ${r_{max}}$ are listed in Table 1.
Table 1: The correction $\delta^{\prime}$ (%) for the various values of $r_{max}$ and $Q^{2}$. $Q^{2}$ (GeV2) | $r_{max}$=0.03 | $r_{max}$=0.05 | $r_{max}$=0.1
---|---|---|---
1 | -9.1 | -7.4 | -5.2
10 | -10.6 | -8.6 | -6.0
100 | -12.1 | -9.8 | -6.8
In the $Q^{2}$ region from 1 to 100 GeV2 available for experiments at
$B$-factories, the correction reach 5–7% even with the relatively loose
restriction ($r_{max}=0.1$) on the scaled energy of the undetected photon
emitted from the initial state.
The correction $\delta^{\prime}$ is partly compensated by the vacuum
polarization correction $\delta_{VP}$, for which we use the results of Ref.
CMD-2 , which includes the contributions from the $e$, $\mu$, $\tau$ leptons,
and hadrons. The $Q^{2}$ dependence of $\delta_{VP}$ is shown in Fig. 4 in
comparison with $\delta_{e}$.
Figure 4: The vacuum polarization correction as a function of $Q^{2}$. The
curve ”All” shows $\delta_{VP}$ calculated in Ref. CMD-2 with account of
contributions from $e$, $\mu$, $\tau$, and hadrons. The curve ”Electrons” –
represents the contribution only from electrons, $\delta_{e}$.
The values of the total correction $\delta=\delta^{\prime}+\delta_{VP}$
calculated for for nine representative sets of $Q^{2}$ and ${r_{max}}$ are
listed in Table 2.
Table 2: Total radiative correction $\delta=\delta^{\prime}+\delta_{VP}$. $Q^{2}$ (GeV 2) | $r_{max}$=0.03 | $r_{max}$=0.05 | $r_{max}$=0.1
---|---|---|---
1 | -5.9 | -4.3 | -2.0
10 | -5.6 | -3.7 | -1.0
100 | -4.8 | -2.6 | +0.4
The emission of the hard photon by the electron distorts the kinematics of
two-photon event. To model how this effect influences the detection
efficiency, the event generator includes generation of extra photons emitted
from the initial and final states.
### 3.1 Simulation of initial state radiation
For simulation of the initial state radiation (ISR), it is convenient to
represent the radiative correction in the form
$1+\delta^{\prime}\approx\left[1+\frac{\alpha}{\pi}\left(\frac{3}{4}L-1\right)\right]\int_{0}^{r_{max}}\frac{\beta
dr}{r^{1-\beta}},$ (12)
where $\beta=(\alpha/\pi)(L-1)$, $r=E_{\gamma}/E_{b}$, and $E_{\gamma}$ is the
energy of the ISR photon.
The function under the integral can be interpreted as the energy spectrum for
photons radiated from the initial state. Indeed, at $Q^{2}=1\div 100$ GeV2 the
parameter $\beta$ is small ($\beta=0.033\div 0.044$), and this function
coincides approximately with the energy spectrum for hard photons, radiated
from the initial state OK :
$\frac{{\rm d}N}{{\rm d}r}=\frac{\alpha(L-1)}{\pi r}.$ (13)
For simulation of the extra photon emission, we replace the four-dimensional
integration in Eq. (6) to five-dimensional one with $r$ as the outermost
integration variable
${\rm
d}\sigma=\left[1+\frac{\alpha}{\pi}\left(\frac{3}{4}L-1\right)\right]\frac{\beta}{r^{1-\beta}}{\rm
d}\sigma_{B}{\rm d}r$ (14)
The vacuum polarization correction is included by the substitution
$\alpha^{2}\to\alpha^{2}(1+\delta_{VP}(Q_{1}^{2}))(1+\delta_{VP}(Q_{2}^{2}))$
(15)
in the Born cross section ${\rm d}\sigma_{B}$.
In simulation of the initial state radiation, the approximation is used that
the photon is emitted strictly along the initial direction of the radiating
electron. Since the energy of the photon is restricted by the condition
$r<r_{max}$, we expect that this approximation does not lead to a significant
systematics in determination of the detection efficiency. Note that selection
criteria used in data analysis should provide the fulfillment of the condition
$r<r_{max}$ for both experimental and simulated events.
To increase simulation efficiency, the variable $r$ is initially generated
according to the $\beta_{0}/r^{1-\beta_{0}}$ distribution with
$\beta_{0}=\beta(Q^{2}_{min})$, where $Q^{2}_{min}$ is a lower bound on the
tagged-electron $Q^{2}$ for simulated single-tag event. If the generated value
of $r$ is higher than a threshold $r_{min}$, the photon is added to the list
of final particles in an event. The scattered $e^{+}$ and $e^{-}$, and the
pseudoscalar meson are then generated in the frame with the shifted c.m.
energy of $2E_{b}\sqrt{1-r}$. If $r<r_{min}$, the photon is not generated, and
the c.m. energy is not shifted, but the radiative correction factor in the
cross section (see Eq. (14)) is calculated.
### 3.2 Simulation of final state radiation
The final state radiation (FSR) is simulated after the generation of the two-
photon event. The final electron scattered at a large angle is “decayed” to
$e+\gamma$ with some probability. The final-meson four-momentum is then
modified to provide the energy and momentum balance. The probability of the
emission of the photon with the energy greater than $E_{\gamma,min}$ equals
$P(Q^{2},x_{min})=\frac{\alpha}{\pi(1+\delta^{\prime})}\biggl{[}(L-1)\ln\frac{1}{x_{min}}-\frac{3}{4}L+1\biggr{]},$
(16)
where $x_{min}=E_{\gamma,min}/E$, and $E$ is the electron energy before FSR
simulation. This formula is obtained by integration of the FSR photon spectrum
given by Eq. (23) of Ref. OCK . The $Q^{2}$ dependence of the FSR probability
calculated for $x_{min}=0.1$, 0.01 and 0.001 is shown in Fig. 5.
Figure 5: The $Q^{2}$ dependence of the final state radiation probability.
The photon energy $E_{\gamma}$ and angle $\theta_{\gamma}$ with respect to the
electron direction before radiation are generated according to the following
distribution function OCK :
$\frac{{\rm d}N}{{\rm d}x{\rm d}\cos\theta_{\gamma}}=\frac{\alpha}{\pi
x}\frac{1-x+x^{2}/2}{1-\beta\cos\theta_{\gamma}},\quad$ (17)
where $x=E_{\gamma}/E$, $\beta=\sqrt{1-m_{e}^{2}/E^{\prime 2}}$, and
$E^{\prime}$ is the electron energy after the photon emission.
## 4 Comparison with other generators
The comparison of the total cross sections in the no-tag mode obtained with
GGRESRC and the two other generators of two-photon events, GGRESPS T and
TWOGAM TWOGAM , was performed. The results of Monte-Carlo calculations are
identical for all the three generators, if the mass of the meson, its two-
photon width, and $Q^{2}$-dependence of the form factor are set to be the same
in the generators. The GGRESRC and GGRESPS use the same formula (Eq. (6)), but
different orders of integration over the invariant variables. The TWOGAM
generator was developed for the CLEO measurements of the meson-photon
transition form factors CLEO-2 . It is based on the BGMS formalism BGMS (see
Eq. (3)) and uses the completely different integration variables, the momenta
of the final electrons.
For GGRESRC in the regime without radiative corrections and TWOGAM, the
comparison of the $Q^{2}$ spectra, obtained for the process of the $\pi^{0}$
production in the single-tag mode, was performed. The spectra was found to be
in agreement within the Monte-Carlo statistical errors.
## 5 Generator parameters
The parameters of the event generator are listed in Table 3. The recommended
values for the parameters Rmax, Rmin, and Kmin are given in brackets. To
simulate no-tag events, the parameters Q1Smin, Q2Smin, and IRad should be set
to zero. The regime with radiative correction (IRad=1) is used only in the
single-tag mode.
Table 3: Parameters of the generator GGRESRC. Name | Description
---|---
Eb | beam energy (GeV)
IR | produced meson: $\pi^{0}$ ($=1$), $\eta$ ($=2$), $\eta^{\prime}$ ($=3$), $\eta_{c}$ ($=4$), $\eta_{b}$ ($=5$)
IMode | meson decay mode (see Table 4)
KVMDM | form factor model: constant ($=0$), VDM ($=1$)
Itag | tagged particle: positron ($=1$), electron ($=2$), mix ($=3$)
IRad | simulation with/without radiative correction ($=1/0$)
Rmax | maximal energy of ISR photon in units of Eb ($0.1$)
Rmin | minimal energy of ISR photon in units of Eb ($10^{-4}$)
Kmin | minimal energy of the FSR photon ( 0.001 GeV )
Q1Smin | minimal momentum transfer squared to the untagged electron
Q1Smax | maximal momentum transfer squared to the untagged electron
Q2Smin | minimal momentum transfer squared to the tagged electron
Q2Smax | maximal momentum transfer squared to the tagged electron
Fmax | maximum weight of events
The resonances decay modes implemented in the generator are listed in Table 4.
The decay models used are described in Ref. T . If parameter IMode equals 0,
the meson decay is not simulated.
Table 4: The meson decay modes in GGRESRC. If IMode=0, the meson decay is not simulated. Meson | IMode | Decay channel | Branching
---|---|---|---
| | | fraction PDG (%)
$\pi^{0}$ | 1 | $\pi^{0}\to 2\gamma$ | 98.798
| 2 | $\pi^{0}\to e^{+}e^{-}\gamma$ | 1.198
| 1 | $\eta\to 2\gamma$ | 39.31
$\eta$ | 2 | $\eta\to 3\pi^{0},\quad\pi^{0}\to 2\gamma$ | 31.4
| 3 | $\eta\to\pi^{+}\pi^{-}\pi^{0},\quad\pi^{0}\to 2\gamma$ | 22.457
| 4 | $\eta\to\pi^{+}\pi^{-}\gamma$ | 4.6
| 1 | $\eta^{\prime}\to 2\gamma$ | 2.1
$\eta^{\prime}$ | 2 | $\eta^{\prime}\to\pi^{+}\pi^{-}\eta,\quad\eta\to 2\gamma$ | 17.532
| 3 | $\eta^{\prime}\to\pi^{+}\pi^{-}\eta,\quad\eta\to 3\pi^{0},\quad\pi^{0}\to 2\gamma$ | 14.004
| 4 | $\eta^{\prime}\to\pi^{+}\pi^{-}\eta,\quad\eta\to\pi^{+}\pi^{-}\pi^{0},\quad\pi^{0}\to 2\gamma$ | 10.016
| 5 | $\eta^{\prime}\to\pi^{+}\pi^{-}\eta,\quad\eta\to\pi^{+}\pi^{-}\gamma$ | 2.0516
| 6 | $\eta^{\prime}\to 2\pi^{0}\eta,\quad\eta\to 2\gamma,\quad\pi^{0}\to 2\gamma$ | 7.943
| 7 | $\eta^{\prime}\to 2\pi^{0}\eta,\quad\eta\to 3\pi^{0},\quad\pi^{0}\to 2\gamma$ | 6.3445
| 8 | $\eta^{\prime}\to 2\pi^{0}\eta,\quad\eta\to\pi^{+}\pi^{-}\pi^{0},\quad\pi^{0}\to 2\gamma$ | 4.5375
| 9 | $\eta^{\prime}\to 2\pi^{0}\eta,\quad\eta\to\pi^{+}\pi^{-}\gamma,\quad\pi^{0}\to 2\gamma$ | 0.9294
| 10 | $\eta^{\prime}\to\rho^{0}\gamma,\quad\rho^{0}\to\pi^{+}\pi^{-}$ | 29.4
$\eta_{c}$ | 1 | $\eta_{c}\to K_{S}K^{+}\pi^{-}$ \+ c.c. | 2.33
| 2 | $\eta_{c}\to 2\gamma$ | 0.024
$\eta_{b}$ | 1 | $\eta_{b}\to 2\gamma$ | -
In general terms the GGRESRC simulation algorithm is the following:
* •
The electron and positron collide in the c.m. frame ($S_{0}$). In this frame
the positive $z$-axis is defined to coincide with the $e^{-}$ beam direction.
* •
The emission of a hard photon from the initial state is simulated. The photon
is emitted along the collision axis. If the ISR photon energy is greater than
$r_{min}E_{b}$, the photon momentum is stored in the list of final particles.
* •
The scattered electrons and the resonance are generated in the new c.m. frame
($S_{1}$) with the c.m. energy $2E_{b}\sqrt{1-E_{\gamma}/E_{b}}$;
$S_{1}=S_{0}$ if $E_{\gamma}/E_{b}<r_{min}$.
* •
The final state radiation is simulated. If the photon energy is greater than
$k_{min}$, its parameters are stored in the list of final particles. The
momenta of the tagged electron and the produced meson are modified.
* •
The meson decay is simulated.
* •
The momenta of the final particles are transformed from $S_{1}$ to $S_{0}$
frame.
When required statistics is collected, the total cross section for the two-
photon process with radiative corrections is calculated and printed.
Table 5: The simulation parameters used for calculation of the distributions shown in Figs. 6– 13. Parameter | Value | Comment
---|---|---
Eb | 5.29 | beam energy (GeV)
IR | 1 | meson: $\pi^{0}$
IMode | 1 | decay mode: $\pi^{0}\to 2\gamma$
KVMDM | 1 | VDM form factor (Eq. (9)) is used
Itag | 1 | the final positron is tagged
IRad | 1 | radiative corrections are simulated
Rmax | 0.1 | maximal energy of ISR photons in units of $E_{b}$
Rmin | 10-4 | minimal energy of ISR photons in units of $E_{b}$
Kmin | 0.001 | minimal energy of FSR photons (GeV)
Q1Smin | 0 | $Q^{2}_{min}$ for $e^{-}$ (GeV2)
Q1Smax | 1.5 | $Q^{2}_{max}$ for $e^{-}$ (GeV2)
Q2Smin | 1.5 | $Q^{2}_{min}$ for $e^{+}$ (GeV2)
Q2Smax | 9 | $Q^{2}_{max}$ for $e^{+}$ (GeV2)
## 6 Example of $e^{+}e^{-}\to e^{+}e^{-}\pi^{0}$ simulation
In this section, some distributions for the process $e^{+}e^{-}\to
e^{+}e^{-}+\pi^{0}$, $(\pi^{0}\to 2\gamma)$ obtained with the generator
GGRESRC, are presented. In Table 5 the parameters of the generator used in the
simulation are listed.
At these parameter values, 57% of events do not contain extra photons, 22%,
28%, and 6% of events contain ISR photon, FSR photon, and both ISR and FSR
photons, respectively. The obtained cross section of the process and average
radiative correction are: $\sigma=$0.99 pb, $\delta=$-0.6%.
The calculated cross section as a function of the restriction on $Q_{1}^{2}$
(the values of the other parameter are equal to those in Table 5) is shown in
Fig. 6. One can see that at $Q_{1max}^{2}\approx$ 1.5 GeV2 the cross section
reaches an asymptotic value.
Figure 6: The $e^{+}e^{-}\to e^{+}e^{-}\pi^{0}$ cross section as a function
of the limit on $Q_{1}^{2}$.
The energy spectra of tagged electrons obtained with and without radiative-
correction simulation are shown in Fig. 7. It is seen that emission of extra
photons significantly changes the shape of this spectrum.
Figure 7: The energy spectra of tagged electrons from the process
$e^{+}e^{-}\to e^{+}e^{-}\pi^{0}$ calculated without (left panel) and with
(right panel) radiative correction simulation.
Fig. 8 shows the energy spectra of the photons from the $\pi^{0}\to 2\gamma$
decay in the process $e^{+}e^{-}\to e^{+}e^{-}\pi^{0}$. The energy spectra of
photons emitted by the initial and final electrons are presented in Fig. 9.
Figure 8: The energy spectra of photons from the $\pi^{0}\to 2\gamma$ decay
in the process $e^{+}e^{-}\to e^{+}e^{-}\pi^{0}$ calculated without (left
panel) and with (right panel) radiative correction simulation.
Figure 9: The energy spectrum of ISR (left panel) and FSR (right panel)
photons in the simulation of the process $e^{+}e^{-}\to e^{+}e^{-}\pi^{0}$.
The polar-angle distributions of tagged electrons is shown in Figs. 10. It is
seen that at $E_{b}=5.29$ GeV the cut $Q^{2}_{min}>1.5$ GeV2 corresponds to
the minimum scattering angle of about $13^{\circ}$. This is in agreement with
an estimate for small scattering angles $\theta\approx Q/(E_{b}\cdot
E^{\prime})^{1}/2$, where energy of the scattered electron $E^{\prime}\approx
E_{b}$.
Figure 10: The polar-angle distributions of tagged electrons from the process
$e^{+}e^{-}\to e^{+}e^{-}\pi^{0}$ obtained without (left panel) and with
(right panel) radiative correction simulation.
The polar angle distribution of FSR photons is shown in Fig. 11. Since the FSR
photon is emitted predominantly along the tagged-electron direction, the
photon angular distribution is very close to that for the electron.
Figure 11: The polar-angle distributions of FSR photons from the process
$e^{+}e^{-}\to e^{+}e^{-}\pi^{0}$.
The polar-angle distribution of photons from the $\pi^{0}\to 2\gamma$ decay is
shown in Fig. 12. Photons have wide distribution, which becomes more uniform
with account of radiative corrections.
Figure 12: The polar-angle distributions of photons from the $\pi^{0}\to
2\gamma$ decay in the process $e^{+}e^{-}\to e^{+}e^{-}\pi^{0}$ obtained
without (left panel) and with (right panel) radiative correction simulation.
In Fig. 13 the distribution of the missing mass in the process $e^{+}e^{-}\to
e^{+}e^{-}+\pi^{0}$ is shown. The missing mass is calculated as
$\sqrt{(p_{1}+p_{2}-k-p_{2}^{\prime})^{2}}$, i.e. we assume that only the
tagged electron and the two photon from $\pi^{0}$ decay are detected. The
narrow peak at zero mass contains events (57% of the total number of events),
which do not have extra ISR or FSR photons. It is seen that emission of the
extra photons leads to significant widening of the missing mass distribution.
Figure 13: The missing mass distribution for the simulated $e^{+}e^{-}\to
e^{+}e^{-}\pi^{0}$ single-tag events.
## 7 Program components
### 7.1 Common blocks
COMMON /GGRSTA/Sum,Es,Sum1,Sum2,Fm,Fm1,NOBR,Nact,Ngt
Purpose: to collect simulation statistics.
`Sum Real*8 ` used for calculation of the total cross section
`Es Real*8 ` used for calculation of the cross-section error
`Sum1 Real*8 ` used for calculation of the average form factor
`Sum2 Real*8 ` used for calculation of the average radiative correction factor
`Fm Real*8 ` maximum weight of event
`Fm1 Real*8 ` maximum weight of event in FSR simulation
`NOBR Integer*4 ` number of calls of the generator
`Nact Integer*4 ` number of the generated events
`Ngt Integer*4 ` number of events with the weight greater than Fmax (see
`/GGRPAR/`)
COMMON /GGRPAR/Eb,Rmas,Rwid,Rg,Rm,Fmax,Rmax,Rmin,t1imin,t1imax,
t2imin,t2imax,Kmin,Fmax1,IR,IMode,KVMDM,ITag,IRad
Purpose: simulation parameters.
`Eb Real*8 ` beam energy (GeV)
`Rmas Real*8 ` meson mass (GeV)
`Rwid Real*8 ` meson total width (GeV)
`Rg Real*8 ` meson two-photon width (keV)
`Rm Real*8 ` meson mass in the current event (GeV)
`Fmax Real*8 ` expected maximum weight of event
`Rmax Real*8 ` maximal energy of ISR photon in `Eb` units
`Rmin Real*8 ` minimal energy of ISR photon in `Eb` units
`t1imin Real*8 ` minimal value of $t_{1}$ (GeV2)
`t1imax Real*8 ` maximal value of $t_{1}$ (GeV2)
`t2imin Real*8 ` minimal value of $t_{2}$ (GeV2)
`t2imax Real*8 ` maximal value of $t_{2}$ (GeV2)
`Kmin Real*8 ` minimal energy of the FSR photon (GeV)
`Fmax1 Real*8 ` maximum expected weight for FSR simulation
`IR Integer*4 ` meson type
`Imode Integer*4 ` meson decay mode
`KVMDM Integer*4 ` form factor model
`ITag Integer*4 ` tagged particle
`IRad Integer*4 ` switch for radiative correction calculation
COMMON /GGRCON/Alpha,PI,EM,mPi0,mPi,mEta,mEtap,mKs,mKc,mRho,mJpsi,
mUps,BrPi0(2),BrEta(4),BrEtaPrim(4),BrRho,BrTot
Purpose: constants.
`Alpha Real*8 ` fine structure constant (1/137.03604)
`Pi Real*8 ` $\pi$ (3.14159265)
`Em Real*8 ` electron mass (0.00051099891 GeV)
`mPi0 Real*8 ` $\pi^{0}$ mass (0.1349766 GeV)
`mPi Real*8 ` $\pi^{\pm}$ mass (0.13957018 GeV)
`mEta Real*8 ` $\eta$ mass (0.547853 GeV)
`mEtap Real*8 ` $\eta^{\prime}$ mass (0.95766 GeV)
`mKs Real*8 ` $K_{S}$ mass (0.497614 GeV)
`mKc Real*8 ` $K^{\pm}$ mass (0.493677 GeV)
`mRho Real*8 ` $\rho^{0}$ mass (0.77549 GeV)
`mJpsi Real*8 ` $J/\psi$ mass (3.096916 GeV)
`mUps Real*8 ` $\Upsilon$ mass (9.4603 GeV)
`BrPi0(2) Real*8 ` $\pi^{0}$ decay branching fractions
`BrEta(4) Real*8 ` $\eta$ decay branching fractions
`BrEtaPrim(4) Real*8 ` $\eta^{\prime}$ decay branching fractions
`BrRho Real*8 ` branching fraction of the decay $\rho^{0}\to\pi^{+}\pi^{-}$
`BrTot Real*8 ` total probability of the decay chain
COMMON /GGREV/pPart(4,25),mPart(25),Type(25),Mother(25),Npart
Purpose: final particle parameters (up to 25 particles).
`pPart(1-3,i) Real*8 ` momentum of i-th particle (GeV)
`pPart(4,i) Real*8 ` energy of i-th particle (GeV)
`mPart(i) Integer*4 ` mass of i-th particle (GeV)
`Type(i) Integer*4 ` type of i-th particle
`Mother(i) Integer*4 ` index of parent of i-th particle in `/GGREV/`
`Npart Integer*4 ` total number of particles in `/GGREV/`
In the common block /GGREV/: 1-st and 2-nd particles are the scattered
electrons, 3-rd particle is the produced resonance, 4-th e.t.c. particles are
the ISR photon (if exists), the FSR photon (if exists), resonance decay
products.
COMMON /GGRPOL/SETS(7330),SETPOL(7330)
Purpose: vacuum polarization correction.
`SETS Real*8 ` momentum transfer squared (GeV2)
`SETPOL Real*8 ` value of the vacuum polarization correction
Common blocks for internal use: `/GGRARIP/`, `/GGRFUC/`.
### 7.2 Subroutines of the generator
`GGRESRC ` the main subroutine
`GGRDEC2G ` simulation of resonance decay to 2$\gamma$
`GGRESEND ` print out of simulation results
`GGRESINI ` initialization
`GGRETCD ` simulation of $\eta_{c}$ decays
`GGRETD ` simulation of $\eta$ decays
`GGRET1D ` simulation of $\eta^{\prime}$ decays
`GGRET1D1 ` simulation of the decays $\eta^{\prime}\to\pi^{+}\pi^{-}\eta$ and
$\pi^{0}\pi^{0}\eta$
`GGRET1D2 ` simulation of the decay $\eta^{\prime}\to\rho^{0}\gamma$
`GGRFSR ` FSR simulation
`GGRFVP ` filling the common block /GGRPOL/
`GGRINV ` simulation of the invariants $t_{2}$, $t_{1}$, $s_{1}$, $s_{2}$
`GGRLMOM ` calculation of the laboratory momenta of the final electrons and
meson
`GGRLOR ` Lorentz transformation
`GGRPI0D ` simulation of $\pi^{0}$ decays
`GGRPI0D1 ` simulation of the $\pi^{0}\to e^{+}e^{-}\gamma$ decay
`GGRPREV ` print out of one event
`GGRRNDM ` wrapper of a pseudo-random numbers generator
`GGRSPC3 ` simulation of the three particle phase space
### 7.3 Double-precision functions
`GGRPOLAR ` calculation of the vacuum polarization correction
`GGRFU ` function used by the subroutine GGRFSR
`GGRFVMDM ` calculation of the form factor in the vector dominance model
### 7.4 Library subroutines
In the generator we use following functions from the CERN program library:
`RANLUX ` generation of pseudo-random numbers uniformly distributed in the
interval (0,1);
`DZEROX ` computing a zero of a real-valued function $f(x)$ in the given
interval [a, b].
## 8 Summary
The event generator GGRESRC for simulation of the two-photon process
$e^{+}e^{-}\to e^{+}e^{-}R$, where $R$ is a pseudoscalar meson, has been
developed. The generator allows to efficiently generate two-photon events in
the single-tag mode, when one of the final electrons is scattered at a large
angle and may be detected. In this mode simulation of radiative corrections
has been implemented in the generator including extra photon emission from the
initial and final states.
The generator is used for simulation of experiments with the BABAR detector on
measurements of the photon-meson transition form factors (see, for example,
Refs. Bab_pi0 ; Bab_etac ), and for simulation of two-photon experiments with
the KEDR detector at VEPP-4M collider.
The work is partially supported by the RF Presidential Grant for Sc. Sch.
NSh-6943.2010.2.
## References
* (1) V. M. Budnev, I. F. Ginzburg, G. V. Meledin and V. G. Serbo, Phys. Rep. 15, 181 (1975).
* (2) M. Poppe, Int. J. Mod. Phys. A 1, 545 (1986).
* (3) E. Byckling, K. Kajante, Particle Kinematics (John Wiley & Sons Ltd., New York, 1973).
* (4) S. J. Brodsky, T. Kinoshita, H. Terazawa, Phys. Rev. D 4 (1971) 1532.
* (5) G. A. Schuler, Comput. Phys. Commun. 108, 279 (1998).
* (6) V.A.Tayursky, Preprint INP 2001-61. Novosibirsk 2001 (in Russian).
* (7) M. Defrise, S. Ong, J. Silva and C. Carimalo, Phys. Rev. D 23, 663 (1981); W. L. van Neerven and J. A. M. Vermaseren, Nucl. Phys. B 238, 73 (1984).
* (8) S. Ong and P. Kessler, Phys. Rev. D 38, 2280 (1988).
* (9) S. Ong, C. Carimalo and P. Kessler, Phys. Lett. B 142, 429 (1984).
* (10) F. V. Ignatov, PHD thesis, Budker INP 2008 (in Russian).
* (11) TWOGAM, The Two-Photon Monte Carlo Simulation Program, written by D. M. Coffman (unpublished).
* (12) J. Gronberg et al. [CLEO Collaboration], Phys. Rev. D 57, 33 (1998).
* (13) Particle Data Group, Phys. Lett. B 667, 1 (2008).
* (14) B. Aubert et al. [BABAR Collaboration], Phys. Rev. D 80, 052002 (2009).
* (15) J. P. Lees et al. [BABAR Collaboration], Phys. Rev. D 81, 052010 (2010).
|
arxiv-papers
| 2010-10-28T13:55:45 |
2024-09-04T02:49:14.334908
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "V.P. Druzhinin and L.A. Kardapoltsev and V.A. Tayursky",
"submitter": "Evgueni Baldin",
"url": "https://arxiv.org/abs/1010.5969"
}
|
1010.5976
|
¡html¿ ¡head¿ ¡title¿CERN-2010-001¡/title¿ ¡/head¿ ¡body¿ ¡h1¿¡a
href=”http://physicschool.web.cern.ch/PhysicSchool/LatAmSchool/2009/Welcome.html”¿
2009 CERN‒Latin-American School of High-Energy Physics¡/a¿¡/h1¿
¡h2¿Recinto Quirama, Colombia, 15 - 28 March 2009¡/h2¿ ¡h2¿Proceedings - CERN
Yellow Report ¡a
href=”http://cdsweb.cern.ch/record/1119305?ln=en”¿CERN-2010-001¡/a¿¡/h2¿
¡h3¿editors: C. Grojean and M. Spiropulu ¡/h3¿
The CERN-Latin-American School of High-Energy Physics is intended to give
young physicists an introduction to the theoretical aspects of recent advances
in elementary particle physics. These proceedings contain lectures on quantum
field theory, quantum chromodynamics, physics beyond the Standard Model,
neutrino physics, flavour physics and CP violation, particle cosmology, high-
energy astro-particle physics, and heavy-ion physics, as well as trigger and
data acquisition, and commissioning and early physics analysis of the ATLAS
and CMS experiments. Also included are write-ups of short review projects
performed by the student discussions groups.
¡h2¿Lectures¡/h2¿
¡!– Introductory lectures on quantum field theory –¿ LIST:hep-th/0510040 ¡br¿
¡!– Quantum ChromoDynamics –¿ LIST:hep-ph/0505192 ¡br¿
¡!– Beyond the Standard Model for Montañeros –¿ LIST:arXiv:0911.4409 ¡br¿
¡!– Neutrino physics –¿ LIST:arXiv:1010.4131 ¡br¿
¡!– Flavour physics and CP violation –¿ LIST:arXiv:1010.2666 ¡br¿
¡!– Particle cosmology –¿ LIST:arXiv:1010.2642 ¡br¿
¡!– High-energy astroparticle physics –¿ LIST:arXiv:1010.2647 ¡br¿
¡!– Relativistic heavy-ion physics –¿ LIST:arXiv:1010.3164 ¡br¿
¡!– Trigger and data acquisition –¿ LIST:arXiv:1010.2942 ¡br¿
¡!– Commissioning and early physics analysis with the ATLAS and CMS
experiments –¿ LIST:arXiv:1002.2891 ¡br¿
¡h3¿Student project write-ups¡/h3¿ Group 1: ¡a
href=”http://cdsweb.cern.ch/record/1249755?ln=en”¿ High-energy cosmic-ray
acceleration¡/a¿ ¡br¿ Group 2: ¡a
href=”http://cdsweb.cern.ch/record/1249756?ln=en”¿ The inert doublet model¡/a¿
¡br¿ Group 3: ¡a href=”http://cdsweb.cern.ch/record/1249757?ln=en”¿ Searching
for new physics in two-body decays: Ideas and pitfalls¡/a¿ ¡br¿ Group 4: ¡a
href=”http://cdsweb.cern.ch/record/1249758?ln=en”¿ The accelerating
universe¡/a¿¡br¿
¡/body¿ ¡/html¿
|
arxiv-papers
| 2010-10-28T14:09:42 |
2024-09-04T02:49:14.342796
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "C. Grojean and M. Spiropulu",
"submitter": "Scientific Information Service Cern",
"url": "https://arxiv.org/abs/1010.5976"
}
|
1011.0155
|
# Linearized Tensor Renormalization Group Algorithm for Thermodynamics of
Quantum Lattice Models
Wei Li1, Shi-Ju Ran1, Shou-Shu Gong1, Yang Zhao1, Bin Xi1, Fei Ye2, and Gang
Su1 Email: gsu@gucas.ac.cn 1College of Physical Sciences, Graduate University
of Chinese Academy of Sciences, P. O. Box 4588, Beijing 100049, China
2College of Materials Science and Opto-Electronic Technology, Graduate
University of Chinese Academy of Sciences, P. O. Box 4588, Beijing 100049,
China
###### Abstract
A linearized tensor renormalization group (LTRG) algorithm is developed to
calculate the thermodynamic properties of low-dimensional quantum lattice
models. This new approach employs the infinite time-evolving block decimation
technique, and allows for treating directly the transfer-matrix tensor network
that makes it more scalable. To illustrate the performance, the thermodynamic
quantities of the quantum XY spin chain as well as the Heisenberg
antiferromagnet on a honeycomb lattice are calculated by the LTRG method,
showing the pronounced precision and high efficiency.
###### pacs:
75.10.Jm, 75.40.Mg, 05.30.-d, 02.70.-c
Since the appearance of White’s density-matrix renormalization group (DMRG)
theory White , the numerical renormalization group (RG) approaches have
achieved great success in studying low-dimensional strongly correlated lattice
models Schollwoek . In the past few years, a number of RG-based methods, e.g.,
the coarse-graining tensor renormalization group (TRG) Levin ; Jiang ; Gu ,
projected entangled pair states Cirac , entanglement renormalization Vidal ,
the infinite time-evolving block decimation (iTEBD) G. Vidal , finite-
temperature DMRG Verstraete ; White2 , etc., _have been proposed inspired by
the quantum information theory_. In spite of the great success in one- and
two-dimensional (1D and 2D) lattice models, it is still quite necessary to
develop new algorithms to improve the accuracy and efficiency of numerical
calculations for strongly correlated systems.
In this Letter, we propose a new algorithm to simulate the thermodynamics of
low-dimensional quantum lattice models. Our strategy is first to transform the
$D$ dimensional quantum lattice model to a $D+1$ dimensional classical tensor
network by means of the Trotter-Suzuki decomposition Trotter , and then to
decimate linearly the tensors following the lines developed in the iTEBD
scheme to obtain the thermodynamics of the original quantum many-body system.
This algorithm is so dubbed as the linearized TRG (LTRG). As is known, the
previous real space TRG approach deals with the 2D tensor network with
exponential decimation in the coarse-graining procedure, which was shown
effective for both 2D classical and quantum lattice models Chang ; Li1 ; Jiang
; Gu ; Chen ; Li2 . For the best illustration of the algorithm and performance
of the LTRG approach, we take the exactly solvable 1D quantum XY spin chain as
a prototype. The results show that the precision of the LTRG method is
comparable with that of the transfer-matrix renormalization group (TMRG) Xiang
, the method that is quite powerful for simulating the 1D quantum lattice
models at finite temperatures (e.g. Refs. gucas ; sirker ). To demonstrate its
scalability, a LTRG result with remarkable precision for a 2D Heisenberg
antiferromagnet on a honeycomb lattice is also included.
Figure 1: (Color online) (a) A transfer-matrix tensor network, where each bond
denotes the $\sigma$ index in Eqs. (2) and (3). (b) A local transformation of
a fourth-order tensor into two third-order tensors through a singular value
decomposition (SVD). (c) Transform the transfer-matrix tensor network to a
hexagonal one. (d) By contracting the intermediate bonds marked by dashed
ovals in (c), one gets a brick wall structure with the 4th-order tensors in
the bottom line.
Let us start with the Hamiltonian of a 1D quantum many-body model given by
$\displaystyle H$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{N}h_{i,i+1}=H_{1}+H_{2},$ $\displaystyle H_{1}$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{N/2}h_{2i-1,2i},\,\,H_{2}=\sum_{i=1}^{N/2}h_{2i,2i+1},$
(1)
where $N$ (even) is the number of sites. By inserting $2K$ (large $K$)
complete sets of states
$\\{|\sigma_{i}^{j}\rangle\\}(\sigma_{i}^{j}=1,\cdots,D)$ with $i$ the site
index and $j$ the Trotter index, the partition function of this model can be
represented as
$\displaystyle Z_{N}$ $\displaystyle\simeq$ $\displaystyle Tr[e^{-\beta
H_{1}/K}e^{-\beta H_{2}/K}]^{K}$ (2) $\displaystyle=$
$\displaystyle\sum_{\\{\sigma_{i}^{j}\\}}\prod_{j=1}^{K}\langle\sigma_{1}^{2j-1}...\sigma_{N}^{2j-1}|e^{-\beta
H_{1}/K}|\sigma_{1}^{2j}...\sigma_{N}^{2j}\rangle$ $\displaystyle\times$
$\displaystyle\langle\sigma_{1}^{2j}...\sigma_{N}^{2j}|e^{-\beta
H_{2}/K}|\sigma_{1}^{2j+1}...\sigma_{N}^{2j+1}\rangle,$
where the periodic boundary conditions along both spatial and temporal
directions are assumed, i.e., $\sigma_{i}^{1}=\sigma_{i}^{2K+1}$ and
$\sigma_{1}^{j}=\sigma_{N+1}^{j}$. Since the terms within $H_{1}$(and $H_{2}$)
mutually commute, Eq. (2) can be further decomposed as
$Z_{N}\simeq\sum_{\\{\sigma_{i}^{j}\\}}\prod_{i=1}^{N/2}\prod_{j=1}^{K}v_{\sigma_{2i-1}^{2j-1}\sigma_{2i}^{2j-1},\sigma_{2i-1}^{2j}\sigma_{2i}^{2j}}\,v_{\sigma_{2i}^{2j}\sigma_{2i+1}^{2j},\sigma_{2i}^{2j+1}\sigma_{2i+1}^{2j+1}},$
(3)
where the transfer matrix,
$v_{\sigma_{1}\sigma_{4},\sigma_{2}\sigma_{3}}\equiv\langle\sigma_{1}\sigma_{4}|\exp(-\beta
h_{i,i+1}/K)|\sigma_{2}\sigma_{3}\rangle$, is a 4th-order tensor. Obviously,
the partition function, Eq.(3), can be viewed as a classical transfer-matrix
tensor network, as illustrated in Fig. 1(a).
Figure 2: (Color online) A local evolution of the tensors by contraction and
SVD. (a) Contract the intermediate bonds; (b) obtain a 6th-order tensor $O$;
(c) calculate the singular value decomposition (SVD) of $O$, and update the
tensors $M_{a,b}$ and $\lambda$. The above manipulation has a computational
cost that scales as $O(D^{6}D_{c}^{3})$. Figure 3: (Color online) An
successive projection of each row of tensors onto the MPO in the bottom line
[(a)-(c)]. After the projection along the Trotter direction, by tracing out
the physical indices $t$ and $b$ of the MPO, one may get a 1D matrix product,
of which the trace can be obtained by a matrix RG procedure [(d)-(g)].
The partition function can be obtained by summing over all the intermediate
states $|\sigma_{i}^{j}\rangle$, namely, contracting all the bonds $\sigma$ in
the tensor network. This procedure is accomplished by first making a singular
value decomposition (SVD) of $\nu$-tensors in the following way
$\displaystyle\nu_{\sigma_{1}\sigma_{2},\sigma_{3}\sigma_{4}}$
$\displaystyle=$
$\displaystyle\sum_{x=1}^{D^{2}}U_{\sigma_{1}\sigma_{2},x}\lambda_{x}V^{\top}_{x,\sigma_{3}\sigma_{4}}$
(4) $\displaystyle\equiv$
$\displaystyle\sum_{x=1}^{D^{2}}(T_{a})_{x,\sigma_{1},\sigma_{2}}(T_{b})_{x,\sigma_{3},\sigma_{4}},$
where the diagonal matrix $\lambda$ collects $D^{2}$ singular values, and two
auxiliary tensors $(T_{a})_{x,\sigma_{1},\sigma_{2}}\equiv
U_{\sigma_{1}\sigma_{2},x}\sqrt{\lambda_{x}}$ and
$(T_{b})_{x,\sigma_{3},\sigma_{4}}\equiv
V_{\sigma_{3}\sigma_{4},x}\sqrt{\lambda_{x}}$ are introduced for convenience.
After this transformation, the square tensor network becomes a hexagonal one
with two 3rd-order tensors $T_{a}$ and $T_{b}$, as depicted in Fig. 1(b).
Then, one contracts the $\sigma$-bonds encircled by the dashed oval lines
between the last two rows in Fig. 1(c), which leads to the two 4th-order
tensors
$\displaystyle(M_{a})_{\alpha,t_{1},\beta,b_{1}}$ $\displaystyle=$
$\displaystyle\sum_{y=1}^{D}(T_{a})_{\beta,b_{1},y}(T_{b})_{\alpha,t_{1},y},$
$\displaystyle(M_{b})_{\beta,t_{2},\gamma,b_{2}}$ $\displaystyle=$
$\displaystyle\sum_{z=1}^{D}(T_{a})_{\gamma,z,t_{2}}(T_{b})_{\beta,z,b_{2}},$
(5)
which form a matrix product operator (MPO) lying in the bottom line of the
whole tensor network, that can also be viewed as a “superket” in the operator
Hilbert space Vidal2 . Each horizontal bond between $M_{a}$ and $M_{b}$ is
assigned with a diagonal matrix $\lambda_{1,2}$. Finally, we obtain a tensor
network with brick wall structure as shown in Fig. 1 (d).
Next, one can project the tensors $T_{a,b}$ onto $M_{a,b}$ successively. At
each time, we project one row of tensors $T_{a}$ and $T_{b}$ followed by
updating $M_{a,b}$ and $\lambda_{1,2}$. After two projections, the system
evolves one Trotter step forward. This procedure is illustrated in Fig. 2. One
first contracts the $\sigma$-bonds between $M$-tensors and $T$-tensors in Fig.
2 (a) to obtain a 6th-order tensor in Fig. 2 (b)
$\displaystyle O_{y,\alpha,b_{1},z,\gamma,b_{2}}$ $\displaystyle=$
$\displaystyle\sum_{x,t_{1},t_{2},\beta}(\lambda_{1})_{\alpha}\,(M_{a})_{\alpha,t_{1},\beta,b_{1}}\,(\lambda_{2})_{\beta}\,(M_{b})_{\beta,t_{2},\gamma,b_{2}}\,(\lambda_{1})_{\gamma}$
(6)
$\displaystyle\,\,\,\,\,\,\,\,\,\,\,\,(T_{a})_{x,t_{1},y}\,(T_{b})_{x,z,t_{2}},$
and then, takes a SVD of the $O$-tensors (after matricization). $O_{y\alpha
b_{1},z\gamma b_{2}}\simeq\sum_{\beta}^{D_{c}}U_{y\alpha
b_{1},\beta}(\lambda^{\prime}_{2})_{\beta}V^{\top}_{\beta,z\gamma b_{2}},$
while keeps only the largest $D_{c}$ singular values of
$\lambda^{\prime}_{2}$. One can define new $M$-tensors
$(M_{a}^{\prime})_{\alpha,y,\beta,b_{1}}=U_{y\alpha
b_{1},\beta}/(\lambda_{1})_{\alpha}$ and
$(M_{b}^{\prime})_{\beta,z,\gamma,b_{2}}=V_{z\gamma
b_{2},\beta}/(\lambda_{1})_{\gamma}$, and update the horizontal bonds with
$\lambda^{\prime}_{2}$. After these operations, the last row of the tensor
network is half updated as shown in Fig. 2 (c). To project the next row of
tensors, one can simply exchange $M_{a}$ and $M_{b}$ as well as $\lambda_{1}$
and $\lambda_{2}$ in Eq. (6). These two successive projections make up of a
full Trotter step $\tau$, as illustrated from Fig. 3(a) to 3(c). In each
Trotter step, the transfer-matrix tensor network is decimated linearly with
only $O(D_{c})$ singular values discarded, which improves greatly the
efficiency compared with the original TRG approach where $O(D_{c}^{n})$ ($n=2$
for honeycomb network) ones are discarded in the coarse-graining procedure
expla .
In order to avoid the divergence in the imaginary time evolution, one has to
normalize all the singular values in $\lambda$ with its largest one $n_{i}$ in
$i$-th step. After projecting all the $T$-tensors at inverse temperature
$\beta$, one is left with the matrix product density operator of the present
system. It consists of 4th-order $M$-tensors [see Fig. 3 (c)], each of which
has two legs with physical indices $t$ and $b$ in the Trotter direction, that
can be further traced out due to the periodic boundary condition. Thus, we
obtain a 1D matrix product (MP) extended in the spatial direction, where the
matrices are labeled as $cM_{a,b}$ as shown in Fig. 3 (d). It is convenient to
assume the number of matrices is $2^{p}$. To get the trace of the product of
these $2^{p}$ matrices, one can contract the neighboring matrices pairwise to
obtain a new product of $2^{p-1}$ matrices, each of which should be normalized
by the absolute value of its largest elements to avoid divergence. This
contraction procedure is represented in Figs. 3(d)-3(g). After $p$ steps, the
$2^{p}$ matrices shrink to a single one, of which the trace can be easily
calculated. In each coarse graining step, all the normalization factors
denoted by $m_{j}$ with $j=1,\cdots,p$ need to be collected for the
calculation of physical quantities, e.g., the free energy per site $f$ at
inverse temperature $\beta=K\tau$ can then be determined by the normalization
factors $n_{j}$’s and $m_{j}$’s
$\displaystyle f$ $\displaystyle=$ $\displaystyle-\frac{1}{\beta
L}\ln[\prod_{i=1}^{2K-2}(n_{i})^{\frac{L}{2}}\prod_{j=1}^{p}(m_{j})^{\frac{L}{2^{j}}}]$
(7) $\displaystyle=$
$\displaystyle-\frac{1}{K\tau}(\sum_{i=1}^{2K-2}\frac{\ln{n_{i}}}{2}+\sum_{j=1}^{p}\frac{\ln{m_{j}}}{2^{j}}).$
Figure 4: (Color online) The relative error of the free energy per site,
$\delta f$, of the quantum XY spin chain at high temperatures. $\delta f$
converges rapidly with $D_{c}$, and the lines with $D_{c}=100$ and $150$
coincide with each other ($\tau=0.05,0.02$). In addition, the TMRG results
($\tau=0.1,0.05$) are also presented for a comparison.
In the above descriptions, we illustrate the LTRG algorithm by first
decimating the tensors along the Trotter direction, and then contracting the
matrices in the spatial direction. Alternatively, one can also perform the
decimation first in the spatial direction, and then do the matrix contraction
in the Trotter direction.
As an example, we are going to demonstrate the efficiency of the LTRG
algorithm by computing the free energy and other thermodynamic quantities of
the quantum XY spin-1/2 chain with a local Hamiltonian
$h_{i,i+1}=-J(S_{i}^{x}S_{i+1}^{x}+S_{i}^{y}S_{i+1}^{y})$ in Eq. (1) with
$J=1$. We take the chain length to be $2^{100}$, which definitely reaches the
thermodynamic limit.
In Fig. 4, we show the relative error of the free energy $f$ with respect to
the exact solution, i.e., $\delta f=|(f-f_{exact})/f_{exact}|$, for different
Trotter steps $\tau=0.1,0.05,0.02,0.01$. We observe that the accuracy is
enhanced with decreasing $\tau$, as well as increasing $D_{c}$. Owing to the
close relation between iTEBD and DMRG, the truncation parameter $D_{c}$ plays
a role similar to the number of states kept $M$ in the TMRG method. As shown
in Fig. 4, we compare the LTRG results to those of TMRG, both of which show
the same accuracy for $\tau=0.1$ and $0.05$. It is also noticed that the
relative errors saturate rapidly with increasing $D_{c}$, implying that the
errors at high temperatures (_e.g._ $T>0.2J$) mainly originate from the
Trotter-Suzuki decomposition. In order to check the truncation error, the LTRG
algorithm is also tested at very low temperatures. In Fig. 5, the temperature
is down to $T=J/120$ with a Trotter step $\tau=0.05$. As shown in Fig. 5 (a),
the accuracy of low $T$ results is remarkably improved by increasing $D_{c}$,
and the relative error $\delta f\simeq 7\times 10^{-6}$ at $\beta=120$ for
$D_{c}=150$.
Besides the free energy, other thermodynamic quantities, such as the internal
energy, can also be obtained. There are at least two ways to get them, one can
either introduce some impurity tensors in the tensor network (see, for
instance, Ref. G. Vidal, ), or do a numerical differentiation of free energy
with respect to temperature. Both ways are found to have a similar accuracy.
In Fig. 5 (b), the energy per site, $e$, is presented. We apply the LTRG
algorithm to approach the ground state energy $e_{0}$, and find the difference
$(e-e_{0})/e_{0}$ is about $10^{-4}$ at $\beta=120$ for $D_{c}$=150,
suggesting that the LTRG result is very close to the exact solution. The TMRG
results with various $M$ (up to $M=200$) are also included in Fig. 5 for a
comparison. The relative errors for the free energy and internal energy are
found to be of the same order down to $\beta=120$ for both approaches.
Figure 5: (Color online) LTRG and TMRG results of the quantum XY spin chain.
(a) Relative error of the free energy per site $\delta f$. (b) The energy per
site $e$. The inset shows the variation of $(e-e_{0})/e_{0}$ with inverse
temperature $\beta$ for various $D_{c}$.
The specific heat of the quantum XY spin chain is also calculated, as shown in
Fig. 6(a). The LTRG results agree quite well with the exact solution both at
high and low temperatures. As indicated in the inset, the accuracy will be
enhanced by increasing $D_{c}$. For $D_{c}=150$, the LTRG results coincide
with the exact solution down to very low temperature ($T/J\simeq 0.008$). The
TMRG results with states $M=200$ are also included, showing that both
numerical methods have the comparable accuracy.
To examine the scalability of the LTRG algorithm, we also calculate the energy
per site of a 2D spin-1/2 Heisenberg antiferromagnetic model on a honeycomb
lattice, whose Hamiltonian is
$H=J\sum_{<i,j>}\vec{S}_{i}\cdot\vec{S}_{j}+h_{s}\sum_{i}(-1)^{|i|}S_{i}^{z}$,
where $(-1)^{|i|}$ denotes the parity of the lattice and $h_{s}$ is a
staggered magnetic field, as shown in Fig. 6 (b). A pronounced agreement
between LTRG and quantum Monte Carlo (QMC) results is clearly seen.
Figure 6: (Color online) (a) Specific heat as a function of temperature
($T=1/\beta$) of the quantum XY spin chain. The inset shows the low
temperature results for $D_{c}=100$ and $200$, along with the TMRG data
($M=200$) for a comparison. (b) Energy per site of the 2D spin-1/2 Heisenberg
antiferromagnet on a honeycomb lattice for different staggered magnetic
fields. The QMC results are obtained by using the ALPS library AF .
In summary, we have proposed a linearized TRG algorithm to calculate the
thermodynamic properties of low dimensional quantum lattice models, and
obtained very accurate results. The LTRG algorithm can be readily generalized
to fermion and boson models, and also provides a quite promising way to
simulate the 2D quantum lattice models without involving the negative sign
problem.
We are indebted to Q. N. Chen, J. W. Cai, J. Sirker, T. Xiang, Z. Y. Xie, and
H. H. Zhao for stimulating discussions, and Z. Y. Chen, S. J. Hu, Y. T. Hu, G.
H. Liu, X. L. Sheng, Y. H. Su, Q. B. Yan, and Q. R. Zheng for helpful
assistance. This work is supported in part by the NSFC (Grants No. 10625419,
No. 10934008, No. 10904081, No. 90922033) and the Chinese Academy of Sciences.
## References
* (1) S. R. White, Phys. Rev. Lett. 69, 2863 (1992); Phys. Rev. B 48, 10345 (1993).
* (2) U. Schollwoeck, Rev. Mod. Phys. 77, 259 (2005); Ann. Phys. 326, 96 (2011).
* (3) M. Levin and C. P. Nave, Phys. Rev. Lett. 99, 120601 (2007).
* (4) H.C. Jiang, Z.Y. Weng, and T. Xiang, Phys. Rev. Lett. 101 090603 (2008); Z.Y. Xie _et al._ , Phys. Rev. Lett. 103, 160601 (2009); H. H. Zhao _et al._ , Phys. Rev. B 81, 174411 (2010).
* (5) Z.C. Gu, M. Levin, and X.G. Wen, Phys. Rev. B 78, 205116 (2008); Z.C. Gu and X.G. Wen, Phys. Rev. B 80, 155131 (2009).
* (6) F. Verstraete and J. I. Cirac, arXiv:0407066 (2004).
* (7) G. Vidal, Phys. Rev. Lett. 99, 220405 (2007); Phys. Rev. Lett. 101, 110501 (2008).
* (8) G. Vidal, Phys. Rev. Lett. 98, 070201 (2007); R. Orús and G. Vidal, Phys. Rev. B 78, 155117 (2008).
* (9) F. Verstraete _et al._ , Phys. Rev. Lett. 93, 207204 (2004).
* (10) A.E. Feiguin and S.R. White, Phys. Rev. B 72, 220401 (2005); S.R. White, Phys. Rev. Lett. 102, 190601 (2009); E.M. Stoudenmire and S.R. White, New J. Phys. 12, 055026 (2010).
* (11) M. Suzuki and M. Inoue, Prog. Theor. Phys. 78, 787 (1987); M. Inoue and M. Suzuki, Prog. Theor. Phys. 79, 645 (1988).
* (12) M.-C. Chang, M.-F. Yang, Phys. Rev. B 79, 104411 (2009).
* (13) W. Li _et al._ , Phys. Rev. B 82, 134434 (2010).
* (14) P. Chen, C.Y. Lai, and M.F. Yang, J. Stat. Mech. P10001 (2009).
* (15) W. Li _et al._ , Phys. Rev. B 81, 184427 (2010).
* (16) R. J. Bursill, T. Xiang, and G. A. Gehring, J. Phys: Condens. Matter 8, L583 (1996); Xiaoqun Wang, Tao Xiang, Phys. Rev. B 56, 5061 (1997); Tao Xiang, Phys. Rev. B 58, 9142 (1998).
* (17) B. Gu, G. Su and S. Gao, Phys. Rev. B 73, 134427 (2006); B. Gu and G. Su, Phys. Rev. Lett. 97, 089701 (2006); B. Gu and G. Su, Phys. Rev. B 75, 174437 (2007); S.-S. Gong, S. Gao, and G. Su, Phys. Rev. B 80, 014413 (2009); S.-S. Gong _et al._ , Phys. Rev. B 81, 214431 (2010).
* (18) J. Sirker, Phys. Rev. B 73, 224424 (2006); J. Sirker _et al._ , Phys. Rev. B 78, 235125 (2008); J. Sirker, Phys. Rev. B 81, 014419 (2010); J. Sirker, Phys. Rev. Lett. 105, 117203 (2010).
* (19) M. Zwolak and G. Vidal, Phys. Rev. Lett. 93, 207205 (2004).
* (20) We note that the present LTRG algorithm can also be applied to evaluate the thermodynamics of 2D classical models, achieving more accurate results than the coarse-graining TRG algorithm.
* (21) A. F. Albuquerque _et al._ , J. Magn. Magn. Mat. 310, 1187 (2007).
|
arxiv-papers
| 2010-10-31T13:37:22 |
2024-09-04T02:49:14.362066
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Wei Li, Shi-Ju Ran, Shou-Shu Gong, Yang Zhao, Bin Xi, Fei Ye, and Gang\n Su",
"submitter": "Wei Li",
"url": "https://arxiv.org/abs/1011.0155"
}
|
1011.0207
|
1310
# Geometry of Hermitian manifolds
Kefeng Liu, Xiaokui Yang
###### Abstract
On Hermitian manifolds, the second Ricci curvature tensors of various metric
connections are closely related to the geometry of Hermitian manifolds. By
refining the Bochner formulas for any Hermitian complex vector bundle
(Riemannain real vector bundle) with an arbitrary metric connection over a
compact Hermitian manifold, we can derive various vanishing theorems for
Hermitian manifolds and complex vector bundles by the second Ricci curvature
tensors. We will also introduce a natural geometric flow on Hermitian
manifolds by using the second Ricci curvature tensor.
## 1 Introduction
It is well-known([5]) that on a compact Kähler manifold, if the Ricci
curvature is positive, then the first Betti number is zero; if the Ricci
curvature is negative, then there is no holomorphic vector field. The key
ingredient for the proofs of such results is the Kähler symmetry. On the other
hand, on an Hermitian manifold, we don’t have such symmetry and there are
several different Ricci curvatures. While on a Kähler manifold, all these
Ricci curvatures coincide, since the Chern curvature on a Kähler manifold
coincides with the curvature of the (complexified) Levi-Civita connection. We
can see this more clearly on an abstract Hermitian holomorphic bundle $E$. The
Chern connection $\nabla^{CH}$ on $E$ is the unique connection which is
compatible with the holomorphic structure and the Hermitian metric on $E$.
Hence, the Chern curvature $\Theta^{E}\in\Gamma(M,\Lambda^{1,1}T^{*}M\otimes
E^{*}\otimes E)$. There are two methods to take trace of $\Theta^{E}$. If we
take trace of $\Theta^{E}$ on the part $End(E)=E^{*}\otimes E$, we get a
$(1,1)$-form on $M$ which it is called the first Ricci curvature. It is well
known that the first Ricci curvature represents the first Chern class of the
bundle. On the other hand, if we take trace on the $(1,1)$-part using the
metric of the manifold, we obtain an endomorphism of $E$,
$Tr_{\omega}\Theta^{E}\in\Gamma(M,E^{*}\otimes E)$. It is called the second
Ricci curvature of $\Theta^{E}$. The first and second Ricci curvatures have
different geometric meanings, which were not clearly studied in some earlier
literatures. We should point out that the nonexistence of holomorphic sections
is characterized by the second Ricci curvature. Let $E$ be the holomorphic
tangent bundle $T^{1,0}M$. If $M$ is Kähler, the first and second Ricci
curvature are the same by the Kähler symmetry. Unfortunately, on an Hermitian
manifold, the Chern curvature is not symmetric, i.e., the first and second
Ricci curvatures are different. Moreover, in general they can not be compared.
An interesting example is the Hopf manifold
${\mathbb{S}}^{2n+1}\times{\mathbb{S}}^{1}$. The canonical metric on it has
strictly positive second Ricci curvature!
In this paper, we study the nonexistence of holomorphic and harmonic sections
of an abstract vector bundle over a compact Hermitian manifold. Let $E$ be a
holomorphic vector bundle over a compact Hermitian manifold $(M,\omega)$.
Since the holomorphic section space $H^{0}(M,E)$ is independent on the
connections of $E$, we can choose any connection on it. As we mentioned above,
the key part, is the second Ricci curvature of the connection. For example, on
the holomorphic tangent bundle $T^{1,0}M$ of an Hermitian manifold $M$, there
are three common connections
(1)
the complexified Levi-Civita connection $\nabla$ on $T^{1,0}M$;
(2)
the Chern connection $\nabla^{CH}$ on $T^{1,0}M$;
(3)
the Bismut connection $\nabla^{B}$ on $T^{1,0}M$.
It is well-known that if $M$ is Kähler, all three connections are the same.
However, in general, the relations among them are somewhat mysterious. In this
paper, we derive certain relations about their curvatures on certain Hermitian
manifolds.
Let $E$ be an Hermitian _complex_ (possibly _non-holomorphic_) vector bundle
or a Riemannian _real_ vector bundle over a compact Hermitian manifold
$(M,\omega)$. Let $\partial_{E},\overline{\partial}_{E}$ be the $(1,0),(0,1)$
part of $\nabla^{E}$ respectively. The $(1,1)$-curvature of $\nabla^{E}$ is
denoted by $R^{E}\in\Gamma(M,\Lambda^{1,1}T^{*}M\otimes E^{*}\otimes E)$. It
can be viewed as a representation of the operator
$\partial_{E}\overline{\partial}_{E}+\overline{\partial}_{E}\partial_{E}$. We
can define harmonic section spaces associated to $(E,\nabla^{E})$ by
${\mathcal{H}}^{p,q}_{\overline{\partial}_{E}}(M,E)=\\{\varphi\in\Omega^{p,q}(M,E)\
|\ \overline{\partial}_{E}\varphi=\overline{\partial}_{E}^{*}\varphi=0\\}$
(1.1)
In general, on a complex vector bundle $E$, there is no such terminology like
“holomorphic section of $E$”. However, if the vector bundle $E$ is holomorphic
and $\nabla^{E}$ is the Chern connection on $E$ i.e.
$\overline{\partial}_{E}=\overline{\partial}$, then
${\mathcal{H}}^{p,q}_{\overline{\partial}_{E}}(M,E)$ is isomorphic to the
Dolbeault cohomology group $H_{\overline{\partial}}^{p,q}(M,E)$ and
$H^{0}_{\overline{\partial}}(M,E)$ is the holomorphic section space
$H^{0}(M,E)$ of $E$.
###### Theorem 1.1.
Let $E$ be an Hermitian complex vector bundle or a Riemannian real vector
bundle over a compact Hermitian manifold $(M,\omega)$ and $\nabla^{E}$ be any
metric connection on $E$.
(1)
If the second Hermitian-Ricci curvature $Tr_{\omega}R^{E}$ is nonpositive
everywhere, then every $\overline{\partial}_{E}$-closed section of $E$ is
parallel, i.e. $\nabla^{E}s=0$;
(2)
If the second Hermitian-Ricci curvature $Tr_{\omega}R^{E}$ is nonpositive
everywhere and negative at some point, then
${\mathcal{H}}^{0}_{\overline{\partial}_{E}}(M,E)=0$;
(3)
If the second Hermitian-Ricci curvature $Tr_{\omega}R^{E}$ is $p$-nonpositive
everywhere and $p$-negative at some point, then
${\mathcal{H}}^{0}_{\overline{\partial}_{E}}(M,\Lambda^{q}E)=0$ for any $p\leq
q\leq rank(E)$.
The proof of this theorem is based on generalized Bochner-Kodaira identities
on vector bundles over Hermitian manifolds (Theorem 4.5). We prove that
(Theorem 4.8) the torsion integral of the Hermitian manifold can be killed if
the background Hermitian metric $\omega$ is Gauduchon, i.e.
$\partial\overline{\partial}\omega^{n-1}=0$. On the other hand, in the
conformal class of any Hermitian metric, the Gauduchon metric always exists
([21]). So we can change the background metric in the conformal way and the
positivity of the second Hermitian-Ricci curvature is preserved. This method
is very useful on Hermitian manifolds. Kobayashi-Wu([31]) and Gauduchon([19])
obtained similar result in the special case when $\nabla^{E}$ is the Chern
connection of the Hermitian _holomorphic_ vector bundle $E$. Now we go back to
the Hermitian manifold $(M,\omega)$.
###### Corollary 1.2.
Let $(M,\omega)$ be a compact Hermitian manifold
(1)
if the second Ricci-Chern curvature $Tr_{\omega}\Theta$ is nonnegative
everywhere and positive at some point, then
$H^{p,0}_{\overline{\partial}}(M)=0$ for any $1\leq p\leq n$. In particular,
the arithmetic genus $\chi(M,{\mathcal{O}})=1$;
(2)
if the second Ricci-Chern curvature $Tr_{\omega}\Theta$ is nonpositive
everywhere and negative at some point, then the holomorphic vector bundle
$\Lambda^{p}T^{1,0}M$ has no holomorphic vector field for any $1\leq p\leq n$.
Since the first Ricci-Chern curvature and the second Ricci-Chern curvature of
an Hermitian manifold can not be compared, we can not derive that the manifold
$M$ is Kähler, even if the second Ricci-Chern curvature is positive
everywhere. In general, the first Ricci-Chern curvature is $d$-closed but the
second Ricci-Chern curvature is not $d$-closed and so they are in the
different $(d,\overline{\partial},\partial)$-cohomology classes. For example,
the Hopf manifold ${\mathbb{S}}^{2n+1}\times{\mathbb{S}}^{1}$ with standard
Hermitian metric has strictly positive second Ricci-Chern curvature and
nonnegative first Ricci-Chern curvature, but it is non-Kähler. For more
details, see Proposition 6.4.
Now we consider several special Hermitian manifolds. An interesting class of
Hermitian manifolds is the balanced Hermitian manifolds, i.e., Hermitian
manifolds with coclosed Kähler forms. It is well-known that every Kähler
manifold is balanced. In some literatures, they are also called semi-Kähler
manifolds. In complex dimension $1$ and $2$, every balanced Hermitian manifold
is Kähler. However, in higher dimensions, there exist non-Kähler manifolds
which admit balanced Hermitian metrics. Such examples were constructed by E.
Calabi([7]), see also [23] and [36]. There are also some other important
classes of non-Kähler balanced manifolds, such as: complex solvmanifolds,
1-dimensional families of Kähler manifolds (see [36]) and compact complex
parallelizable manifolds (except complex torus) (see [46]). On the other hand,
Alessandrini- Bassaneli( [2]) proved that every Moishezon manifold is balanced
and so balanced manifolds can be constructed from Kähler manifolds by
modification. For more examples, see [3], [36], [16] and [17].
Every balanced metric $\omega$ is a Gauduchon metric. In fact, $d^{*}\omega=0$
is equivalent to $d\omega^{n-1}=0$ and so
$\partial\overline{\partial}\omega^{n-1}=0$. By [21], every Hermitian manifold
has a Gauduchon metric. However, there are many manifolds which can not
support balanced metrics. For example, the Hopf surface
${\mathbb{S}}^{3}\times{\mathbb{S}}^{1}$ is non-Kähler, so it has no balanced
metric. For more discussion , one can see [7], [36],[42], [2] and [3].
On a compact balanced Hermitian manifold $M$, we can detect the holomorphic
section spaces $H^{p,0}_{\overline{\partial}}(M)$ by Levi-Civita connection.
Let $\nabla$ be the complexified Levi-Civita connection and $\nabla^{\prime}$,
$\nabla^{\prime\prime}$ the $(1,0)$ and $(0,1)$ components of $\nabla$
respectively. In general, holomorphic $p$-forms are not
$\nabla^{\prime\prime}$-closed. The Ricci curvatures related to the Levi-
Civita connection are defined in 2.11 and 2.29.
###### Theorem 1.3.
Let $(M,\omega)$ be a compact balanced Hermitian manifold. If the Hermitian-
Ricci curvature $(R_{i\overline{j}})$ of $M$ is nonnegative everywhere, then
(1)
If $\varphi$ is a holomorphic $p$-form, then $\Delta_{\partial}\varphi=0$ and
so $h^{p,0}(M)\leq h^{0,p}(M)$ for any $1\leq p\leq n$;
(2)
If the Hermitian-Ricci curvature $(R_{i\overline{j}})$ is positive at some
point, then $H^{p,0}_{\overline{\partial}}(M)=0$ for any $1\leq p\leq n$. In
particular, the arithmetic genus $\chi(M,{\mathcal{O}})=1$.
The dual of Theorem 1.3 is
###### Theorem 1.4.
Let $(M,\omega)$ be a compact balanced Hermitian manifold. If
$2\widehat{R}^{(2)}_{i\overline{j}}-R_{i\overline{j}}$ is nonpositive
everywhere and negative at some point, there is no holomorphic vector field on
M.
###### Remark 1.5.
It is easy to see that the Hermitian-Ricci curvature tensor
$(R_{i\overline{j}})$ and second Ricci-Chern curvature tensor
$\Theta^{(2)}:=Tr_{\omega}\Theta$ can not be compared. Therefore, Theorem 1.3
and Corollary 1.2 are independent of each other. For the same reason, Theorem
1.4 and Corollary 1.2 are independent. Balanced Hermitian manifolds with
nonnegative Hermitian-Ricci curvatures are discussed in Proposition 3.5.
As we discuss in the above, on Hermitian manifolds, the second Ricci curvature
tensors of various metric connections are closely related to the geometry of
Hermitian manifolds. A natural idea is to define a flow by using second Ricci
curvature tensors of various metric connections. For example,
$\frac{\partial h}{\partial t}=-\Theta^{(2)}+\mu h,\ \ \ \mu\in{\mathbb{R}}$
(1.2)
on a general Hermitian manifold $(M,h)$ by using the second Ricci-Chern
curvature. This flow preserves the Kähler and the Hermitian structure and has
short time solution on any compact Hermitian manifold. It is very similar to
and closely related to the Hermitian Yang-Mills flow, the Kähler-Ricci flow
and the harmonic map heat flow. It may be a bridge to connect them. In this
paper we only briefly discuss its basic properties. In a subsequent
paper([33]) we will study its geometric and analytic property in detail.
We would like to thank Yi Li, Jeffrey Streets, Valetino Tosatti for their
useful comments on an earlier version of this paper.
## 2 Various connections and curvatures on Hermitian manifolds
### 2.1 Complexified Riemannian curvature
Let $(M,g)$ be a Riemannian manifold with Levi-Civita connection $\nabla$, the
curvature $R$ of $(M,g,\nabla)$ is defined as
$R(X,Y,Z,W)=g\left(\left(\nabla_{X}\nabla_{Y}-\nabla_{Y}\nabla_{X}-\nabla_{[X,Y]}\right)Z,W\right)$
(2.1)
On an Hermitian manifold $(M,h)$, let $\nabla$ be the complexified Levi-Civita
connection and $g$ the background Riemannian metric. Two metrics are related
by
$ds^{2}_{h}=ds^{2}_{g}-\sqrt{-1}\omega_{h}$ (2.2)
where $\omega_{h}$ is the fundamental $(1,1)$-form (or Kähler form) associated
to $h$. For any two holomorphic vector fields $X,Y\in\Gamma(M,T^{1,0}M)$,
$h(X,Y)=2g(X,\overline{Y})$ (2.3)
_This formula will be used in several definitions_. In the local holomorphic
coordinates $\\{z^{1},\cdots,z^{n}\\}$ on $M$, the complexified Christoffel
symbols are given by
$\Gamma_{AB}^{C}=\sum_{E}\frac{1}{2}g^{CE}\big{(}\frac{\partial
g_{AE}}{\partial z^{B}}+\frac{\partial g_{BE}}{\partial z^{A}}-\frac{\partial
g_{AB}}{\partial z^{E}}\big{)}=\sum_{E}\frac{1}{2}h^{CE}\big{(}\frac{\partial
h_{AE}}{\partial z^{B}}+\frac{\partial h_{BE}}{\partial z^{A}}-\frac{\partial
h_{AB}}{\partial z^{E}}\big{)}$ (2.4)
where $A,B,C,E\in\\{1,\cdots,n,\overline{1},\cdots,\overline{n}\\}$ and
$z^{A}=z^{i}$ if $A=i$, $z^{A}=\overline{z}^{i}$ if $A=\overline{i}$. For
example
$\Gamma_{ij}^{k}=\frac{1}{2}h^{k\overline{\ell}}\left(\frac{\partial
h_{j\overline{\ell}}}{\partial z^{i}}+\frac{\partial
h_{i\overline{\ell}}}{\partial z^{j}}\right),\
\Gamma_{\overline{i}j}^{k}=\frac{1}{2}h^{k\overline{\ell}}\left(\frac{\partial
h_{j\overline{\ell}}}{\partial\overline{z}^{i}}-\frac{\partial
h_{j\overline{i}}}{\partial\overline{z}^{\ell}}\right)$ (2.5)
The complexified curvature components are
$\displaystyle R_{ABCD}:$ $\displaystyle=$ $\displaystyle
2\textbf{g}\left(\left(\nabla_{\frac{\partial}{\partial
z^{A}}}\nabla_{\frac{\partial}{\partial
z^{B}}}-\nabla_{\frac{\partial}{\partial
z^{B}}}\nabla_{\frac{\partial}{\partial z^{A}}}\right)\frac{\partial}{\partial
z^{C}},\frac{\partial}{\partial z^{D}}\right)$ $\displaystyle=$
$\displaystyle\textbf{h}\left(\left(\nabla_{\frac{\partial}{\partial
z^{A}}}\nabla_{\frac{\partial}{\partial
z^{B}}}-\nabla_{\frac{\partial}{\partial
z^{B}}}\nabla_{\frac{\partial}{\partial z^{A}}}\right)\frac{\partial}{\partial
z^{C}},\frac{\partial}{\partial z^{\overline{D}}}\right)$
Hence
$R_{ABC}^{D}=\sum_{E}R_{ABCE}h^{ED}=-\left(\frac{\partial\Gamma_{AC}^{D}}{\partial
z^{B}}-\frac{\partial\Gamma_{BC}^{D}}{\partial
z^{A}}+\Gamma_{AC}^{F}\Gamma_{FB}^{D}-\Gamma_{BC}^{F}\Gamma_{AF}^{D}\right)$
(2.6)
By the Hermitian property, we have, for example
$R_{i\overline{j}k}^{l}=-\left(\frac{\partial\Gamma^{l}_{ik}}{\partial\overline{z}^{j}}-\frac{\partial\Gamma^{l}_{\overline{j}k}}{\partial
z^{i}}+\Gamma_{ik}^{s}\Gamma^{l}_{\overline{j}s}-\Gamma_{\overline{j}k}^{s}\Gamma^{l}_{is}-{\Gamma_{\overline{j}k}^{\overline{s}}\Gamma_{i\overline{s}}^{l}}\right)$
(2.7)
###### Remark 2.1.
We have $R_{ABCD}=R_{CDAB}$. In particular,
$R_{i\overline{j}k\overline{\ell}}=R_{k\overline{\ell}i\overline{j}}$ (2.8)
Unlike the Kähler case, we can define several Ricci curvatures:
###### Definition 2.2.
(1)
The _complexified Ricci curvature_ on $(M,h)$ is defined by
$\mathscr{R}_{k\overline{\ell}}:=h^{i\overline{j}}\left(R_{k\overline{j}i\overline{\ell}}+R_{ki\overline{j}\overline{\ell}}\right)$
(2.9)
The _complexified scalar curvature_ of $h$ is defined as
$s_{h}:=h^{k\overline{\ell}}\mathscr{R}_{k\overline{\ell}}$ (2.10)
(2)
The _Hermitian-Ricci curvature_ is
$R_{k\overline{\ell}}:=h^{i\overline{j}}R_{i\overline{j}k\overline{\ell}}$
(2.11)
The _Hermitian-scalar curvature_ of $h$ is given by
$S:=h^{k\overline{\ell}}R_{k\overline{\ell}}$ (2.12)
###### Lemma 2.3.
On an Hermitian manifold,
$\overline{R_{ABCD}}=R_{\overline{A}\overline{B}\overline{C}\overline{D}},\ \
\overline{\mathscr{R}_{k\overline{\ell}}}=\mathscr{R}_{\ell\overline{k}},\ \ \
\overline{R_{k\overline{\ell}}}=R_{\ell\overline{k}}$ (2.13)
and
$\mathscr{R}_{k\overline{\ell}}=h^{i\overline{j}}\left(2R_{k\overline{j}i\overline{\ell}}-R_{k\overline{\ell}i\overline{j}}\right)$
(2.14)
###### Proof.
The Hermitian property of curvature tensors is obvious. By first Bianchi
identity, we have
$R_{ki\overline{j}\overline{\ell}}+R_{k\overline{j}\overline{\ell}i}+R_{k\overline{\ell}i\overline{j}}=0$
That is
$R_{ki\overline{j}\overline{\ell}}=R_{k\overline{j}i\overline{\ell}}-R_{k\overline{\ell}i\overline{j}}$.
The curvature formula 2.9 turns to be
$\mathscr{R}_{k\overline{\ell}}=h^{i\overline{j}}\left(2R_{k\overline{j}i\overline{\ell}}-R_{k\overline{\ell}i\overline{j}}\right)$
(2.15)
∎
###### Definition 2.4.
The Ricci curvatures are called _positive_ ( resp. _nonnegative, negative,
non-positive_) if the corresponding Hermitian matrices are positive ( resp.
nonnegative, negative, non-positive).
The following three formulas are used frequently in the sequel.
###### Lemma 2.5.
Assume $h_{i\overline{j}}=\delta_{ij}$ at a fixed point $p\in M$, we have the
following formula
$\displaystyle R_{i\overline{j}k\overline{\ell}}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\left(\frac{\partial^{2}h_{i\overline{\ell}}}{\partial
z^{k}\partial\overline{z}^{j}}+\frac{\partial^{2}h_{k\overline{j}}}{\partial
z^{i}\partial\overline{z}^{\ell}}\right)$
$\displaystyle+\frac{1}{4}\left(\frac{\partial h_{k\overline{q}}}{\partial
z^{i}}\frac{\partial
h_{q\overline{\ell}}}{\partial\overline{z}^{j}}+\frac{\partial
h_{i\overline{q}}}{\partial z^{k}}\frac{\partial
h_{q\overline{j}}}{\partial\overline{z}^{\ell}}\right)+\frac{1}{4}\left(\frac{\partial
h_{i\overline{q}}}{\partial z^{k}}\frac{\partial
h_{q\overline{\ell}}}{\partial\overline{z}^{j}}+\frac{\partial
h_{k\overline{q}}}{\partial z^{i}}\frac{\partial
h_{q\overline{j}}}{\partial\overline{z}^{\ell}}\right)$
$\displaystyle+\frac{1}{4}\left(\frac{\partial h_{q\overline{\ell}}}{\partial
z^{i}}\frac{\partial
h_{k\overline{j}}}{\partial\overline{z}^{q}}+\frac{\partial
h_{q\overline{j}}}{\partial z^{k}}\frac{\partial
h_{i\overline{\ell}}}{\partial\overline{z}^{q}}\right)+\frac{1}{4}\left(\frac{\partial
h_{i\overline{\ell}}}{\partial z^{q}}\frac{\partial
h_{k\overline{q}}}{\partial\overline{z}^{j}}+\frac{\partial
h_{k\overline{j}}}{\partial z^{q}}\frac{\partial
h_{i\overline{q}}}{\partial\overline{z}^{\ell}}\right)$
$\displaystyle-\frac{1}{4}\left(\frac{\partial h_{q\overline{\ell}}}{\partial
z^{i}}\frac{\partial
h_{k\overline{q}}}{\partial\overline{z}^{j}}+\frac{\partial
h_{q\overline{j}}}{\partial z^{k}}\frac{\partial
h_{i\overline{q}}}{\partial\overline{z}^{\ell}}\right)-\frac{1}{4}\left(\frac{\partial
h_{i\overline{\ell}}}{\partial z^{q}}\frac{\partial
h_{k\overline{j}}}{\partial\overline{z}^{q}}+\frac{\partial
h_{k\overline{j}}}{\partial z^{q}}\frac{\partial
h_{i\overline{\ell}}}{\partial\overline{z}^{q}}\right)$
By a linear transformation on the local holomorphic coordinates, one can get
the following Lemma. For more details, we refer the reader to [44].
###### Lemma 2.6.
Let $(M,h,\omega)$ be an Hermitian manifold. For any $p\in M$, there exist
local holomorphic coordinates $\\{z^{i}\\}$ centered at a point $p$ such that
$h_{i\overline{j}}(p)=\delta_{ij}\quad\mbox{and}\quad\Gamma_{ij}^{k}(p)=0$
(2.17)
By Lemma 2.6, we have a simplified version of curvatures:
###### Lemma 2.7.
Assume $h_{i\overline{j}}(p)=\delta_{ij}$ and $\Gamma_{ij}^{k}(p)=0$ at a
fixed point $p\in M$,
$R_{i\overline{j}k\overline{\ell}}=-\frac{1}{2}\left(\frac{\partial^{2}h_{i\overline{\ell}}}{\partial
z^{k}\partial\overline{z}^{j}}+\frac{\partial^{2}h_{k\overline{j}}}{\partial
z^{i}\partial\overline{z}^{\ell}}\right)-\sum_{q}\left(\frac{\partial
h_{q\overline{\ell}}}{\partial z^{i}}\frac{\partial
h_{k\overline{q}}}{\partial\overline{z}^{j}}+\frac{\partial
h_{q\overline{j}}}{\partial z^{k}}\frac{\partial
h_{i\overline{q}}}{\partial\overline{z}^{\ell}}\right)$ (2.18)
For Hermitian-Ricci curvatures
$R_{k\overline{\ell}}=h^{i\overline{j}}R_{i\overline{j}k\overline{\ell}}=-\frac{1}{2}\sum_{s}\left(\frac{\partial^{2}h_{s\overline{\ell}}}{\partial
z^{k}\partial\overline{z}^{s}}+\frac{\partial^{2}h_{k\overline{s}}}{\partial
z^{s}\partial\overline{z}^{\ell}}\right)-\sum_{q,s}\left(\frac{\partial
h_{q\overline{\ell}}}{\partial z^{s}}\frac{\partial
h_{k\overline{q}}}{\partial\overline{z}^{s}}+\frac{\partial
h_{k\overline{q}}}{\partial z^{s}}\frac{\partial
h_{q\overline{\ell}}}{\partial\overline{z}^{s}}\right)$ (2.19)
and
$h^{i\overline{j}}R_{k\overline{j}i\overline{\ell}}=h^{i\overline{j}}R_{i\overline{\ell}k\overline{j}}=-\frac{1}{2}\sum_{s}\left(\frac{\partial^{2}h_{k\overline{\ell}}}{\partial
z^{s}\partial\overline{z}^{s}}+\frac{\partial^{2}h_{s\overline{s}}}{\partial
z^{k}\partial\overline{z}^{\ell}}\right)-\sum_{q,s}\left(\frac{\partial
h_{q\overline{\ell}}}{\partial z^{k}}\frac{\partial
h_{s\overline{q}}}{\partial\overline{z}^{s}}+\frac{\partial
h_{q\overline{s}}}{\partial z^{s}}\frac{\partial
h_{k\overline{q}}}{\partial\overline{z}^{\ell}}\right)$ (2.20)
For complexified Ricci curvature,
$\displaystyle\mathscr{R}_{k\overline{\ell}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{s}\left(\frac{\partial^{2}h_{s\overline{\ell}}}{\partial
z^{k}\partial\overline{z}^{s}}+\frac{\partial^{2}h_{k\overline{s}}}{\partial
z^{s}\partial\overline{z}^{\ell}}\right)-\sum_{s}\left(\frac{\partial^{2}h_{k\overline{\ell}}}{\partial
z^{s}\partial\overline{z}^{s}}+\frac{\partial^{2}h_{s\overline{s}}}{\partial
z^{k}\partial\overline{z}^{\ell}}\right)$ (2.21) $\displaystyle+$
$\displaystyle\sum_{q,s}\left(\frac{\partial h_{q\overline{\ell}}}{\partial
z^{s}}\frac{\partial
h_{k\overline{q}}}{\partial\overline{z}^{s}}+\frac{\partial
h_{k\overline{q}}}{\partial z^{s}}\frac{\partial
h_{q\overline{\ell}}}{\partial\overline{z}^{s}}\right)-2\sum_{q,s}\left(\frac{\partial
h_{q\overline{\ell}}}{\partial z^{k}}\frac{\partial
h_{s\overline{q}}}{\partial\overline{z}^{s}}+\frac{\partial
h_{q\overline{s}}}{\partial z^{s}}\frac{\partial
h_{k\overline{q}}}{\partial\overline{z}^{\ell}}\right)$
### 2.2 Curvature of complexified Levi-Civita connection on $T^{1,0}M$
Since $T^{1,0}M$ is a subbundle of $T_{{\mathbb{C}}}M$, there is an induced
connection $\widehat{\nabla}$ on $T^{1,0}M$ given by
$\widehat{\nabla}=\pi\circ\nabla:T^{1,0}M\stackrel{{\scriptstyle\nabla}}{{\rightarrow}}\Gamma(M,T_{{\mathbb{C}}}M\otimes
T_{{\mathbb{C}}}M)\stackrel{{\scriptstyle\pi}}{{\rightarrow}}\Gamma(M,T_{{\mathbb{C}}}M\otimes
T^{1,0}M)$ (2.22)
The curvature $\widehat{R}\in\Gamma(M,\Lambda^{2}T_{{\mathbb{C}}}M\otimes
T^{*1,0}M\otimes T^{1,0}M)$ of $\widehat{\nabla}$ is given by
$\widehat{R}(X,Y)s=\widehat{\nabla}_{X}\widehat{\nabla}_{Y}s-\widehat{\nabla}_{Y}\widehat{\nabla}_{X}s-\widehat{\nabla}_{[X,Y]}s$
(2.23)
for any $X,Y\in T_{{\mathbb{C}}}M$ and $s\in T^{1,0}M$. It has components
$\widehat{R}_{ABk}^{l}=\frac{\partial\Gamma_{Bk}^{l}}{\partial
z^{A}}-\frac{\partial\Gamma_{Ak}^{l}}{\partial
z^{B}}-\Gamma_{Ak}^{s}\Gamma_{Bs}^{l}+\Gamma_{Bk}^{s}\Gamma_{As}^{l}$ (2.24)
where
$\widehat{R}\left(\frac{\partial}{\partial z^{A}},\frac{\partial}{\partial
z^{B}}\right)\frac{\partial}{\partial
z^{k}}=\sum_{l}\widehat{R}_{ABk}^{l}\frac{\partial}{\partial z^{\ell}}$ (2.25)
For example,
$\widehat{R}_{i\overline{j}k}^{l}=-\left(\frac{\partial\Gamma^{l}_{ik}}{\partial\overline{z}^{j}}-\frac{\partial\Gamma^{l}_{\overline{j}k}}{\partial
z^{i}}+\Gamma_{ik}^{s}\Gamma^{l}_{\overline{j}s}-\Gamma_{\overline{j}k}^{s}\Gamma^{l}_{si}\right)$
(2.26)
With respect to the Hermitian metric $h$ on $T^{1,0}M$, we can define
$\widehat{R}_{ABk\overline{l}}=\sum_{s=1}^{n}\widehat{R}_{ABk}^{s}h_{s\overline{\ell}}$
(2.27)
###### Definition 2.8.
The _first Ricci curvature_ of the Hermitian vector bundle
$\left(T^{1,0}M,\widehat{\nabla}\right)$ is defined by
$\widehat{R}^{(1)}_{i\overline{j}}=h^{k\overline{\ell}}\widehat{R}_{i\overline{j}k\overline{\ell}}$
(2.28)
The _second Ricci curvature_ of it is
$\widehat{R}^{(2)}_{k\overline{\ell}}=h^{i\overline{j}}\widehat{R}_{i\overline{j}k\overline{\ell}}$
(2.29)
The _scalar curvature_ of $\widehat{\nabla}$ on $T^{1,0}M$ is denoted by
$S^{LC}=h^{i\overline{j}}h^{k\overline{\ell}}\widehat{R}_{i\overline{j}k\overline{\ell}}$
(2.30)
By Lemma 2.6, we have the following formulas
###### Lemma 2.9.
On an Hermitian manifold $(M,h)$, on a point $p$ with
$h_{i\overline{j}}(p)=\delta_{ij}$ and $\Gamma_{ij}^{k}(p)=0$,
$\displaystyle\widehat{R}_{i\overline{j}k\overline{\ell}}=-\frac{1}{2}\left(\frac{\partial^{2}h_{i\overline{\ell}}}{\partial
z^{k}\partial\overline{z}^{j}}+\frac{\partial^{2}h_{k\overline{j}}}{\partial
z^{i}\partial\overline{z}^{\ell}}\right)-\sum_{q}\frac{\partial
h_{q\overline{\ell}}}{\partial z^{i}}\frac{\partial
h_{k\overline{q}}}{\partial\overline{z}^{j}}$ (2.31)
For the Ricci curvatures,
$\widehat{R}^{(1)}_{i\overline{j}}=-\frac{1}{2}\sum_{k}\left(\frac{\partial^{2}h_{i\overline{k}}}{\partial
z^{k}\partial\overline{z}^{j}}+\frac{\partial^{2}h_{k\overline{j}}}{\partial
z^{i}\partial\overline{z}^{k}}\right)-\sum_{k,q}\frac{\partial
h_{q\overline{k}}}{\partial z^{i}}\frac{\partial
h_{k\overline{q}}}{\partial\overline{z}^{j}}$ (2.32)
and
$\widehat{R}^{(2)}_{i\overline{j}}=-\frac{1}{2}\sum_{k}\left(\frac{\partial^{2}h_{i\overline{k}}}{\partial
z^{k}\partial\overline{z}^{j}}+\frac{\partial^{2}h_{k\overline{j}}}{\partial
z^{i}\partial\overline{z}^{k}}\right)-\sum_{k,q}\frac{\partial
h_{i\overline{q}}}{\partial\overline{z}^{k}}\frac{\partial
h_{q\overline{j}}}{\partial z^{k}}$ (2.33)
Moreover,
$\widehat{R}^{(1)}_{i\overline{j}}-\widehat{R}_{i\overline{j}}^{(2)}=h_{m\overline{j}}h^{\ell\overline{k}}\Gamma_{\overline{k}i}^{\overline{q}}\Gamma_{\ell\overline{q}}^{m}-\Gamma_{k\overline{j}}^{\overline{q}}\Gamma_{i\overline{q}}^{k}=\sum_{k,q}\left(\frac{\partial
h_{i\overline{q}}}{\partial\overline{z}^{k}}\frac{\partial
h_{q\overline{j}}}{\partial z^{k}}-\frac{\partial h_{i\overline{q}}}{\partial
z^{k}}\frac{\partial h_{q\overline{j}}}{\partial\overline{z}^{k}}\right)$
(2.34)
### 2.3 Curvature of Chern connection on $T^{1,0}M$
On the Hermitian holomorphic vector bundle $(T^{1,0}M,h)$, the Chern
connection $\nabla^{CH}$ is the unique connection which is compatible with the
complex structure and the Hermitian metric. Its curvature components are
$\Theta_{i\overline{j}k\overline{\ell}}=-\frac{\partial^{2}h_{k\overline{\ell}}}{\partial
z^{i}\partial\overline{z}^{j}}+h^{p\overline{q}}\frac{\partial
h_{p\overline{\ell}}}{\partial\overline{z}^{j}}\frac{\partial
h_{k\overline{q}}}{\partial z^{i}}$ (2.35)
It is well-known that the _first Ricci-Chern curvature_
$\Theta^{(1)}:=\frac{\sqrt{-1}}{2\pi}\Theta^{(1)}_{i\overline{j}}dz^{i}\wedge
d\overline{z}^{j}$ (2.36)
represents the first Chern class of $M$ where
$\Theta^{(1)}_{i\overline{j}}=h^{k\overline{\ell}}\Theta_{i\overline{j}k\overline{\ell}}=-\frac{\partial^{2}\log\det(h_{k\overline{\ell}})}{\partial
z^{i}\partial\overline{z}^{j}}$ (2.37)
The _second Ricci-Chern curvature_ components are
$\Theta^{(2)}_{i\overline{j}}=h^{k\overline{\ell}}\Theta_{k\overline{\ell}i\overline{j}}$
(2.38)
The _scalar curvature_ of the Chern connection is defined by
$S^{CH}=h^{i\overline{j}}h^{k\overline{\ell}}\Theta_{i\overline{j}k\overline{\ell}}$
(2.39)
### 2.4 Curvature of Bismut connection on $T^{1,0}M$
In [4], Bismut defined a class of connections on Hermitian manifolds. In this
subsection, we choose one of them (see [35], p. $21$). The _Bismut connection_
$\nabla^{B}$ on the holomorphic tangent bundle $(T^{1,0}M,h)$ is characterized
by
$\nabla^{B}=\nabla+S^{B}$ (2.40)
where $S^{B}$ is $1$-form with values in $End(T^{1,0}M)$
$\textbf{h}(S^{B}(X)Y,Z)=2\textbf{g}(S^{B}(X)Y,\overline{Z})=\sqrt{-1}(\partial-\overline{\partial})\omega_{h}(X,Y,\overline{Z})$
(2.41)
for any $Y,Z\in T^{1,0}M$ and $X\in T_{{\mathbb{C}}}M$. Let
$\widetilde{\Gamma}_{i\alpha}^{\beta}$ and
$\widetilde{\Gamma}_{\overline{j}\alpha}^{\beta}$ be the Christoffel symbols
of the Bismut connection where $i,j,\alpha,\beta\in\\{1,\cdots,n\\}$. We use
different types of letters since the Bismut connection is not torsion free.
###### Lemma 2.10.
We have the following relations between $\widetilde{\Gamma}$ and $\Gamma$,
$\widetilde{\Gamma}_{i\alpha\overline{\beta}}:=h_{\beta\overline{\gamma}}\Gamma_{i\alpha}^{\overline{\gamma}}=\Gamma_{i\alpha\overline{\beta}}+\Gamma_{\alpha\overline{\beta}i}=\frac{\partial
h_{i\overline{\beta}}}{\partial z^{\alpha}},\ \ \ \
\widetilde{\Gamma}_{\overline{j}\alpha\overline{\beta}}=2\Gamma_{\overline{j}\alpha\overline{\beta}}$
(2.42)
###### Proof.
Let $X=\frac{\partial}{\partial z^{i}},Y=\frac{\partial}{\partial
z^{j}},Z=\frac{\partial}{\partial z^{k}}$. Since
$\omega_{h}=\frac{\sqrt{-1}}{2}h_{m\overline{n}}dz^{m}\wedge
d\overline{z}^{n}$, we obtain
$\displaystyle\sqrt{-1}(\partial-\overline{\partial})\omega_{h}(X,Y,\overline{Z})$
$\displaystyle=$ $\displaystyle-\frac{1}{2}\frac{\partial
h_{m\overline{n}}}{\partial
z^{p}}dz^{p}dz^{m}d\overline{z}^{n}\left(\frac{\partial}{\partial
z^{i}},\frac{\partial}{\partial
z^{j}},\frac{\partial}{\partial\overline{z}^{k}}\right)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\frac{\partial h_{i\overline{k}}}{\partial
z^{j}}-\frac{\partial h_{j\overline{k}}}{\partial z^{i}}\right)$
$\displaystyle=$
$\displaystyle\Gamma_{j\overline{k}}^{\overline{s}}h_{i\overline{s}}=\Gamma_{j\overline{k}i}$
On the other hand
$h\left(\nabla^{B}_{\frac{\partial}{\partial z^{i}}}\frac{\partial}{\partial
z^{j}},\frac{\partial}{\partial
z^{k}}\right)=\widetilde{\Gamma}_{ij\overline{k}}$ (2.43)
Using the definition of Bismut connection, we get
$\widetilde{\Gamma}_{i\alpha\overline{\beta}}=\Gamma_{i\alpha\overline{\beta}}+\Gamma_{\alpha\overline{\beta}i}=\frac{\partial
h_{i\overline{\beta}}}{\partial z^{\alpha}}$ (2.44)
The proof of the other one is similar. ∎
The Bismut curvature $B\in\Gamma\left(M,\Lambda^{1,1}T^{*}M\otimes
End(T^{1,0}M)\right)$ is given by
$B_{i\overline{j}\alpha}^{\beta}=-\frac{\partial\widetilde{\Gamma}_{i\alpha}^{\beta}}{\partial\overline{z}^{j}}+\frac{\partial\widetilde{\Gamma}_{\overline{j}\alpha}^{\beta}}{\partial
z^{i}}-\widetilde{\Gamma}_{i\alpha}^{\gamma}\widetilde{\Gamma}_{\overline{j}\gamma}^{\beta}+\widetilde{\Gamma}_{\overline{j}\alpha}^{\gamma}\widetilde{\Gamma}_{i\gamma}^{\beta}$
(2.45)
###### Lemma 2.11.
Assume $h_{i\overline{j}}(p)=\delta_{ij}$ and $\Gamma_{ij}^{k}(p)=0$ at a
fixed point $p\in M$,
$B_{i\overline{j}\alpha\overline{\beta}}=-\left(\frac{\partial^{2}h_{i\overline{\beta}}}{\partial\overline{z}^{j}\partial
z^{\alpha}}+\frac{\partial^{2}h_{\alpha\overline{j}}}{\partial
z^{i}\partial\overline{z}^{\beta}}-\frac{\partial^{2}h_{\alpha\overline{\beta}}}{\partial
z^{i}\partial\overline{z}^{j}}\right)+\sum_{\gamma}\frac{\partial
h_{\alpha\overline{\gamma}}}{\partial z^{i}}\frac{\partial
h_{\gamma\overline{\beta}}}{\partial\overline{z}^{j}}-4\sum_{\gamma}\frac{\partial
h_{\alpha\overline{\gamma}}}{\partial\overline{z}^{j}}\frac{\partial
h_{\gamma\overline{\beta}}}{\partial z^{i}}$ (2.46)
###### Proof.
It follows by 2.42 and 2.45.∎
We can define the first Ricci-Bismut curvature $B^{(1)}_{i\overline{j}}$, the
second Ricci-Bismut curvature $B^{(2)}_{i\overline{j}}$ and scalar curvature
$S^{BM}$ similarly.
### 2.5 Relations among the four curvatures on Hermitian manifolds
###### Proposition 2.12.
On an Hermitian manifold $(M,h)$, we have
$R_{ijk\overline{l}}=\widehat{R}_{ijk\overline{\ell}},\ \ \
R_{\overline{i}\overline{j}k\overline{\ell}}=\widehat{R}_{\overline{i}\overline{j}k\overline{\ell}}$
(2.47)
and for any $u,v\in{\mathbb{C}}^{n}$,
$\left(R_{i\overline{j}k\overline{\ell}}-\widehat{R}_{i\overline{j}k\overline{\ell}}\right)u^{i}\overline{u}^{j}v^{k}\overline{v}^{\ell}\leq
0$ (2.48)
In particular, $R_{i\overline{j}}\leq\widehat{R}^{(1)}_{i\overline{j}}$ and
$R_{i\overline{j}}\leq\widehat{R}^{(2)}_{i\overline{j}}$ in the sense of
Hermitian matrices.
###### Proof.
Let
$T_{i\overline{j}k\overline{\ell}}=R_{i\overline{j}k\overline{\ell}}-\widehat{R}_{i\overline{j}k\overline{\ell}}=\Gamma_{\overline{j}k}^{\overline{s}}\Gamma_{i\overline{s}}^{t}h_{t\overline{\ell}}$
(2.49)
Without loss generality, we assume $h_{i\overline{j}}=\delta_{ij}$ at a fixed
point, then
$T_{i\overline{j}k\overline{\ell}}=\sum_{s}\Gamma_{\overline{j}ks}\Gamma_{i\overline{s}\overline{\ell}}=-\sum_{s}\Gamma_{i\overline{s}\overline{\ell}}\overline{\Gamma_{j\overline{s}\overline{k}}}$
(2.50)
where
$\Gamma_{i\overline{s}\overline{\ell}}=\frac{1}{2}\left(\frac{\partial
h_{i\overline{\ell}}}{\partial\overline{z}^{s}}-\frac{\partial
h_{i\overline{s}}}{\partial\overline{z}^{\ell}}\right)=-\Gamma_{i\overline{\ell}\overline{s}}$
(2.51)
and so
$T_{i\overline{j}k\overline{\ell}}u^{i}\overline{u}^{j}v^{k}\overline{v}^{\ell}\leq
0$. ∎
###### Remark 2.13.
(1)
Because of the second order terms in $R$, $\widehat{R}$, $\Theta$ and $B$, we
can not compare $R,\widehat{R}$ with $\Theta$, $B$.
(2)
Since the third order terms of $\partial\Theta^{(2)}$ are not zero in general.
Therefore it is possible that $\Theta^{(1)}$ and $\Theta^{(2)}$ are not in the
same $(d,\partial,\overline{\partial})$-cohomology class. For the same reason
$B^{(1)}$ and $B^{(2)}$ are not in the same
$(d,\partial,\overline{\partial})$-cohomology class.
(3)
If the manifold $(M,h)$ is Kähler, then all curvatures are the same.
## 3 Curvature relations on special Hermitian manifolds
### 3.1 Curvatures relations on balanced Hermitian manifolds
The following lemma is well-known( for example [18]), and we include a proof
here in our setting.
###### Lemma 3.1.
Let $(M,\omega)$ be a compact Hermitian manifold. The following conditions are
equivalent:
(1)
$d^{*}\omega=0$;
(2)
$d\omega^{n-1}=0$;
(3)
For any smooth function $f\in C^{\infty}(M)$,
$\frac{1}{2}\Delta_{d}f=\Delta_{\overline{\partial}}f=\Delta_{\partial}f=-h^{i\overline{j}}\frac{\partial^{2}f}{\partial
z^{i}\partial\overline{z}^{j}}$ (3.1)
(4)
$\Gamma_{\overline{i}\ell}^{\ell}=0$ for any $1\leq i\leq n$.
###### Proof.
On a compact Hermitian manifold, $d^{*}\omega=-*d*\omega=-c_{n}*d\omega^{n-1}$
where $c_{n}$ is a constant depending only on the complex dimension $n$ of
$M$. On the other hand, the Hodge $*$ is an isomorphism, and so $(1)$ and
$(2)$ are equivalent. If $f$ is a smooth function on $M$,
$\begin{cases}\Delta_{\overline{\partial}}f=-h^{i\overline{j}}\frac{\partial^{2}f}{\partial
z^{i}\partial\overline{z}^{j}}+2h^{i\overline{j}}\Gamma_{i\overline{j}}^{\overline{\ell}}\frac{\partial
f}{\partial\overline{z}^{\ell}}\\\
\Delta_{\partial}f=-h^{i\overline{j}}\frac{\partial^{2}f}{\partial
z^{i}\partial\overline{z}^{j}}+2h^{i\overline{j}}\Gamma_{\overline{j}i}^{k}\frac{\partial
f}{\partial z^{k}}\end{cases}$ (3.2)
On the other hand,
$h^{i\overline{j}}\Gamma_{i\overline{j}}^{\overline{\ell}}=-\Gamma_{k\overline{j}}^{\overline{j}}h^{k\overline{\ell}}\quad\mbox{and}\quad
h^{i\overline{j}}\Gamma_{\overline{j}i}^{k}=-\Gamma_{\overline{\ell}i}^{i}h^{k\overline{\ell}}$
(3.3)
Therefore $(3)$ and $(4)$ are equivalent. For the equivalence of $(1)$ and
$(4)$, see Lemma 8.8. ∎
###### Definition 3.2.
An Hermitian manifold $(M,\omega)$ is called _balanced_ if it satisfies one of
the conditions in Lemma 3.1.
On a balanced Hermitian manifold, there are more symmetries on the second
derivatives of the metric.
###### Lemma 3.3.
Let $(M,h)$ be a balanced Hermitian manifold. On a point $p$ with
$h_{i\overline{j}}(p)=\delta_{ij}$ and $\Gamma_{ij}^{k}(p)=0$,
$\sum_{s}\frac{\partial
h_{s\overline{i}}}{\partial\overline{z}^{s}}=\sum_{s}\frac{\partial
h_{s\overline{s}}}{\partial\overline{z}^{i}}=0$ (3.4)
and
$\sum_{i}\frac{\partial^{2}h_{i\overline{\ell}}}{\partial
z^{k}\partial\overline{z}^{i}}=\sum_{i}\frac{\partial^{2}h_{k\overline{i}}}{\partial
z^{i}\partial\overline{z}^{\ell}}=\sum_{i}\frac{\partial^{2}h_{i\overline{i}}}{\partial
z^{k}\partial\overline{z}^{\ell}}-2\sum_{i,q}\frac{\partial
h_{q\overline{\ell}}}{\partial\overline{z}^{i}}\frac{\partial
h_{k\overline{q}}}{\partial z^{i}}$ (3.5)
###### Proof.
At a fixed point $p$, if $h_{i\overline{j}}=0$ and $\Gamma_{ij}^{k}=0$, then
$\frac{\partial h_{i\overline{j}}}{\partial\overline{z}^{k}}=-\frac{\partial
h_{i\overline{k}}}{\partial\overline{z}^{j}}$ (3.6)
The balanced condition $\sum_{s}\Gamma_{\overline{i}s}^{s}=0$ is reduced to
$\sum_{s}\frac{\partial
h_{s\overline{s}}}{\partial\overline{z}^{i}}=\sum_{s}\frac{\partial
h_{s\overline{i}}}{\partial\overline{z}^{s}}=0$ (3.7)
by formula 3.6. By the balanced condition
$\displaystyle 0=\frac{\partial\Gamma_{\overline{\ell}i}^{i}}{\partial z^{k}}$
$\displaystyle=$ $\displaystyle\frac{\partial}{\partial
z^{k}}\left(\frac{1}{2}h^{i\overline{q}}\left(\frac{\partial
h_{i\overline{q}}}{\partial\overline{z}^{\ell}}-\frac{\partial
h_{i\overline{\ell}}}{\partial\overline{z}^{q}}\right)\right)$
$\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{i}\left(\frac{\partial^{2}h_{i\overline{i}}}{\partial
z^{k}\partial\overline{z}^{\ell}}-\frac{\partial^{2}h_{i\overline{\ell}}}{\partial
z^{k}\partial\overline{z}^{i}}\right)-\sum_{i,q}\frac{\partial
h_{q\overline{\ell}}}{\partial\overline{z}^{i}}\frac{\partial
h_{k\overline{q}}}{\partial z^{i}}$
Hence, we obtain formula 3.5. ∎
###### Proposition 3.4.
Let $(M,h)$ be a balanced Hermitian manifold. At a point $p$ with
$h_{i\overline{j}}(p)=\delta_{ij}$ and $\Gamma_{ij}^{k}(p)=0$, we have
following formulas about various Ricci curvatures:
$\displaystyle\Theta^{(1)}_{k\overline{\ell}}$ $\displaystyle=$
$\displaystyle\widehat{R}^{(1)}_{k\overline{\ell}}=B^{(1)}_{k\overline{\ell}}=-\sum_{i}\frac{\partial^{2}h_{i\overline{i}}}{\partial
z^{k}\partial\overline{z}^{\ell}}+\sum_{q,i}\frac{\partial
h_{q\overline{\ell}}}{\partial\overline{z}^{i}}\frac{\partial
h_{k\overline{q}}}{\partial z^{i}}$ (3.8)
$\displaystyle\Theta^{(2)}_{k\overline{\ell}}$ $\displaystyle=$
$\displaystyle-\sum_{i}\frac{\partial^{2}h_{k\overline{\ell}}}{\partial
z^{i}\partial\overline{z}^{i}}+\sum_{i,q}\frac{\partial
h_{q\overline{\ell}}}{\partial\overline{z}^{i}}\frac{\partial
h_{k\overline{q}}}{\partial z^{i}}$ (3.9)
$\displaystyle\widehat{R}^{(2)}_{k\overline{\ell}}$ $\displaystyle=$
$\displaystyle-\sum_{i}\frac{\partial^{2}h_{i\overline{i}}}{\partial
z^{k}\partial\overline{z}^{\ell}}+\sum_{i,q}\left(2\frac{\partial
h_{q\overline{\ell}}}{\partial\overline{z}^{i}}\frac{\partial
h_{k\overline{q}}}{\partial z^{i}}-\frac{\partial
h_{k\overline{q}}}{\partial\overline{z}^{i}}\frac{\partial
h_{q\overline{\ell}}}{\partial z^{i}}\right)$ (3.10) $\displaystyle
B_{k\overline{\ell}}^{(2)}$ $\displaystyle=$
$\displaystyle-\sum_{i}\frac{\partial^{2}h_{i\overline{i}}}{\partial
z^{k}\partial\overline{z}^{\ell}}+\sum_{i,q}\left(5\frac{\partial
h_{q\overline{\ell}}}{\partial\overline{z}^{i}}\frac{\partial
h_{k\overline{q}}}{\partial z^{i}}-4\frac{\partial
h_{k\overline{q}}}{\partial\overline{z}^{i}}\frac{\partial
h_{q\overline{\ell}}}{\partial z^{i}}\right)$ (3.11) $\displaystyle
R_{k\overline{\ell}}$ $\displaystyle=$
$\displaystyle-\sum_{i}\frac{\partial^{2}h_{i\overline{i}}}{\partial
z^{k}\partial\overline{z}^{\ell}}+\sum_{i,q}\left(\frac{\partial
h_{q\overline{\ell}}}{\partial\overline{z}^{i}}\frac{\partial
h_{k\overline{q}}}{\partial z^{i}}-\frac{\partial
h_{k\overline{q}}}{\partial\overline{z}^{i}}\frac{\partial
h_{q\overline{\ell}}}{\partial z^{i}}\right)$ (3.12)
$\displaystyle\mathscr{R}_{k\overline{\ell}}$ $\displaystyle=$
$\displaystyle-\sum_{i}\frac{\partial^{2}h_{k\overline{\ell}}}{\partial
z^{i}\partial\overline{z}^{i}}-\sum_{i,q}\left(\frac{\partial
h_{q\overline{\ell}}}{\partial\overline{z}^{i}}\frac{\partial
h_{k\overline{q}}}{\partial z^{i}}-\frac{\partial
h_{k\overline{q}}}{\partial\overline{z}^{i}}\frac{\partial
h_{q\overline{\ell}}}{\partial z^{i}}\right)$ (3.13)
###### Proof.
In 2.32, 2.33, 2.37,2.38, 2.19, 2.21, we get expressions for all Ricci
curvatures on Hermitian manifolds. By balanced relations 3.4 and 3.5, we get
simplified versions of all Ricci curvatures. ∎
###### Proposition 3.5.
(1)
A balanced Hermitian manifold with positive Hermitian-Ricci curvature
$R_{i\overline{j}}$ is Kähler.
(2)
Let $(M,h)$ be a compact balanced Hermitian manifold. If the Hermitian-Ricci
curvature is nonnegative everywhere and positive at some point, then $M$ is
Moishezon.
###### Proof.
(1) On a balanced Hermitian manifold
$\Theta^{(1)}_{i\overline{j}}=\widehat{R}^{(1)}_{i\overline{j}}\geq
R_{i\overline{j}}$ (3.14)
If $R_{i\overline{j}}$ is Hermitian positive, then
$\Theta^{(1)}_{i\overline{j}}$ is Hermitian positive, and so
$\Omega=-\frac{\sqrt{-1}}{2\pi}\partial\overline{\partial}\log\det(h_{k\overline{\ell}})$
(3.15)
is a Kähler metric.
(2) If the Hermitian-Ricci curvature is nonnegative everywhere and positive at
some point, so is $\Theta^{(1)}_{i\overline{j}}$. The Hermitian line bundle
$L=\det(T^{1,0}M)$ satisfies
$\int_{M}c_{1}(L)^{n}>0$ (3.16)
By Siu-Demailly’s solution of Grauert-Riemenschneider conjecture ([39] [9]),
$M$ is Moishezon. ∎
### 3.2 Curvature relations on Hermitian manifolds with
$\Lambda(\partial\overline{\partial}\omega)=0$
Now we consider a compact Hermitian manifold $(M,\omega)$ with
$\Lambda(\partial\overline{\partial}\omega)=0$. The condition
$\Lambda(\partial\overline{\partial}\omega)=0$ is equivalent to
$\sum_{k}\left(\frac{\partial h_{i\overline{j}}}{\partial
z^{k}\partial\overline{z}^{k}}+\frac{\partial h_{k\overline{k}}}{\partial
z^{i}\partial\overline{z}^{j}}\right)=\sum_{k}\left(\frac{\partial
h_{i\overline{k}}}{\partial
z^{k}\partial\overline{z}^{j}}+\frac{\partial^{2}h_{k\overline{j}}}{\partial
z^{i}\partial\overline{z}^{k}}\right)$ (3.17)
for any $i,j$. We can use 3.17 to simplify Ricci curvatures and get relations
among them.
###### Proposition 3.6.
Let $(M,h)$ be a compact Hermitian manifold with
$\Lambda(\partial\overline{\partial}\omega)=0$. At a point $p$ with
$h_{i\overline{j}}(p)=\delta_{ij}$ and $\Gamma_{ij}^{k}(p)=0$, the following
identities about Ricci curvatures hold:
$\displaystyle\Theta^{(1)}_{k\overline{\ell}}$ $\displaystyle=$
$\displaystyle-\sum_{i}\frac{\partial^{2}h_{i\overline{i}}}{\partial
z^{k}\partial\overline{z}^{\ell}}+\sum_{q,i}\frac{\partial
h_{q\overline{\ell}}}{\partial\overline{z}^{i}}\frac{\partial
h_{k\overline{q}}}{\partial z^{i}}$ (3.18)
$\displaystyle\Theta^{(2)}_{k\overline{\ell}}$ $\displaystyle=$
$\displaystyle-\sum_{i}\frac{\partial^{2}h_{k\overline{\ell}}}{\partial
z^{i}\partial\overline{z}^{i}}+\sum_{i,q}\frac{\partial
h_{q\overline{\ell}}}{\partial\overline{z}^{i}}\frac{\partial
h_{k\overline{q}}}{\partial z^{i}}$ (3.19)
$\displaystyle\widehat{R}^{(1)}_{k\overline{\ell}}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\sum_{i}\left(\frac{\partial^{2}h_{k\overline{\ell}}}{\partial
z^{i}\partial\overline{z}^{i}}+\frac{\partial^{2}h_{i\overline{i}}}{\partial
z^{k}\partial\overline{z}^{\ell}}\right)-\sum_{i,q}\frac{\partial
h_{q\overline{\ell}}}{\partial\overline{z}^{i}}\frac{\partial
h_{k\overline{q}}}{\partial z^{i}}$ (3.20)
$\displaystyle\widehat{R}^{(2)}_{k\overline{\ell}}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\sum_{i}\left(\frac{\partial^{2}h_{k\overline{\ell}}}{\partial
z^{i}\partial\overline{z}^{i}}+\frac{\partial^{2}h_{i\overline{i}}}{\partial
z^{k}\partial\overline{z}^{\ell}}\right)-\sum_{i,q}\frac{\partial
h_{k\overline{q}}}{\partial\overline{z}^{i}}\frac{\partial
h_{q\overline{\ell}}}{\partial z^{i}}$ (3.21) $\displaystyle
B^{(1)}_{k\overline{\ell}}$ $\displaystyle=$
$\displaystyle-\sum_{i}\frac{\partial^{2}h_{k\overline{\ell}}}{\partial
z^{i}\partial\overline{z}^{i}}+\sum_{i,q}\left(\frac{\partial
h_{q\overline{\ell}}}{\partial\overline{z}^{i}}\frac{\partial
h_{k\overline{q}}}{\partial z^{i}}-4\frac{\partial
h_{q\overline{\ell}}}{\partial z^{i}}\frac{\partial
h_{k\overline{q}}}{\partial\overline{z}^{i}}\right)$ (3.22) $\displaystyle
B_{k\overline{\ell}}^{(2)}$ $\displaystyle=$
$\displaystyle-\sum_{i}\frac{\partial^{2}h_{i\overline{i}}}{\partial
z^{k}\partial\overline{z}^{\ell}}+\sum_{i,q}\left(\frac{\partial
h_{q\overline{\ell}}}{\partial\overline{z}^{i}}\frac{\partial
h_{k\overline{q}}}{\partial z^{i}}-4\frac{\partial
h_{k\overline{q}}}{\partial\overline{z}^{i}}\frac{\partial
h_{q\overline{\ell}}}{\partial z^{i}}\right)$ (3.23) $\displaystyle
R_{k\overline{\ell}}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\sum_{i}\left(\frac{\partial^{2}h_{k\overline{\ell}}}{\partial
z^{i}\partial\overline{z}^{i}}+\frac{\partial^{2}h_{i\overline{i}}}{\partial
z^{k}\partial\overline{z}^{\ell}}\right)-\sum_{i,q}\left(\frac{\partial
h_{q\overline{\ell}}}{\partial\overline{z}^{i}}\frac{\partial
h_{k\overline{q}}}{\partial z^{i}}+\frac{\partial
h_{k\overline{q}}}{\partial\overline{z}^{i}}\frac{\partial
h_{q\overline{\ell}}}{\partial z^{i}}\right)$ (3.24)
$\displaystyle\mathscr{R}_{k\overline{\ell}}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\sum_{i}\left(\frac{\partial^{2}h_{k\overline{\ell}}}{\partial
z^{i}\partial\overline{z}^{i}}+\frac{\partial^{2}h_{i\overline{i}}}{\partial
z^{k}\partial\overline{z}^{\ell}}\right)+\sum_{i,q}\left(\frac{\partial
h_{q\overline{\ell}}}{\partial\overline{z}^{i}}\frac{\partial
h_{k\overline{q}}}{\partial z^{i}}+\frac{\partial
h_{k\overline{q}}}{\partial\overline{z}^{i}}\frac{\partial
h_{q\overline{\ell}}}{\partial z^{i}}\right)$
$\displaystyle-2\sum_{q,i}\left(\frac{\partial h_{q\overline{\ell}}}{\partial
z^{k}}\frac{\partial
h_{i\overline{q}}}{\partial\overline{z}^{i}}+\frac{\partial
h_{q\overline{i}}}{\partial z^{i}}\frac{\partial
h_{k\overline{q}}}{\partial\overline{z}^{\ell}}\right)$
###### Proposition 3.7.
If $(M,\omega)$ is a compact Hermitian manifold with
$\Lambda(\partial\overline{\partial}\omega)=0$, then
$B^{(2)}\leq\Theta^{(1)}\quad\mbox{and}\quad B^{(1)}\leq\Theta^{(2)}$ (3.26)
in the sense of Hermitian matrices and identities hold if and only if
$(M,\omega)$ is Kähler. Moreover,
$\Theta^{(2)}+B^{(2)}=\Theta^{(1)}+R^{(1)}$ (3.27)
Finally, we would like to discuss the relations of balanced manifolds and
strong Kähler manifolds with torsion. By [2], every Moishezon manifold is
balanced, i.e. there exists a smooth Hermitian metric $\omega$ such that
$d^{*}\omega=0$. On the other hand, by Demailly-Paun [10]( see also [27]), on
each Moishezon manifold, there exists a singular Hermitian metric $\omega$
such that $\partial\overline{\partial}\omega=0$ in the sense of current.
However, these two conditions can not be satisfied simultaneously in the
smooth sense on an Hermitian non-Kähler manifold. It is known in [1] and also
[15], but merits a proof in our setting.
###### Proposition 3.8.
Let $(M,\omega)$ be a compact Hermitian manifold. If $d^{*}\omega=0$ and
$\Lambda(\partial\overline{\partial}\omega)=0$, then $d\omega=0$, i.e.
$(M,\omega)$ is Kähler. In particular, if a compact Hermitian manifold admits
a smooth metric $\omega$ such that $d^{*}\omega=0$ and
$\partial\overline{\partial}\omega=0$, then it is Kähler.
###### Proof.
Let $(M,\omega)$ be a balanced Hermitian manifold with
$\Lambda(\partial\overline{\partial}\omega)=0$. The condition
$\Lambda(\partial\overline{\partial}\omega)=0$ is equivalent to
$\sum_{i}\frac{\partial h_{i\overline{i}}}{\partial
z^{k}\partial\overline{z}^{\ell}}+\sum_{i}\frac{\partial^{2}h_{k\overline{\ell}}}{\partial
z^{i}\partial\overline{z}^{i}}=\sum_{i}\frac{\partial
h_{i\overline{\ell}}}{\partial
z^{k}\partial\overline{z}^{i}}+\sum_{i}\frac{\partial^{2}h_{k\overline{i}}}{\partial
z^{i}\partial\overline{z}^{\ell}}$ (3.28)
By formula 3.5, at a point $p$ with $h_{i\overline{j}}=\delta_{ij}$ and
$\Gamma_{ij}^{k}(p)=0$, we have
$\displaystyle\sum_{i}\frac{\partial h_{i\overline{i}}}{\partial
z^{k}\partial\overline{z}^{\ell}}+\sum_{i}\frac{\partial^{2}h_{k\overline{\ell}}}{\partial
z^{i}\partial\overline{z}^{i}}$ $\displaystyle=$
$\displaystyle\sum_{i}\frac{\partial h_{i\overline{\ell}}}{\partial
z^{k}\partial\overline{z}^{i}}+\sum_{i}\frac{\partial^{2}h_{k\overline{i}}}{\partial
z^{i}\partial\overline{z}^{\ell}}$ $\displaystyle=$ $\displaystyle
2\sum_{i}\frac{\partial h_{i\overline{i}}}{\partial
z^{k}\partial\overline{z}^{\ell}}-4\sum_{q,i}\frac{\partial
h_{q\overline{\ell}}}{\partial\overline{z}^{i}}\frac{\partial
h_{k\overline{q}}}{\partial z^{i}}$
That is
$\sum_{i}\frac{\partial h_{i\overline{i}}}{\partial
z^{k}\partial\overline{z}^{\ell}}=\sum_{i}\frac{\partial^{2}h_{k\overline{\ell}}}{\partial
z^{i}\partial\overline{z}^{i}}+4\sum_{q,i}\frac{\partial
h_{q\overline{\ell}}}{\partial\overline{z}^{i}}\frac{\partial
h_{k\overline{q}}}{\partial z^{i}}$ (3.29)
Taking trace of it, we obtain
$4\sum_{q,i,k}\frac{\partial
h_{q\overline{k}}}{\partial\overline{z}^{i}}\frac{\partial
h_{k\overline{q}}}{\partial z^{i}}=0\Longleftrightarrow\frac{\partial
h_{k\overline{q}}}{\partial z^{i}}=0$ (3.30)
at point $p$. Since $p$ is arbitrary, we have $d\omega\equiv 0$, therefore,
$(M,\omega)$ is Kähler. ∎
## 4 Bochner formulas on Hermitian complex and Riemannian real vector bundles
over compact Hermitian manifolds
Let $(M,h,\omega)$ be a compact Hermitian manifold. The complexified Levi-
Civita connection $\nabla$ on $T_{{\mathbb{C}}}M$ induces a linear connection
on $\Omega^{p,q}(M)$:
$\nabla:\Omega^{p,q}(M)\mathrel{\mathop{\hbox
to16.11119pt{\rightarrowfill}}\limits}\Omega^{1}(M)\otimes\left(\Omega^{p,q}(M)\oplus\Omega^{p-1,q+1}(M)\oplus\Omega^{p+1,q-1}(M)\right)$
(4.1)
We consider the following two canonical components of $\nabla$,
$\begin{cases}\nabla^{\prime}:\Omega^{p,q}(M)\rightarrow\Omega^{1,0}(M)\otimes\Omega^{p,q}(M)\\\
\nabla^{\prime\prime}:\Omega^{p,q}(M)\rightarrow\Omega^{0,1}(M)\otimes\Omega^{p,q}(M)\end{cases}$
(4.2)
Note that $\nabla\neq\nabla^{\prime}+\nabla^{\prime\prime}$ if $(M,h,\omega)$
is not Kähler. The following calculation rule follows immediately
$\nabla^{\prime}(\varphi\wedge\psi)=\left(\nabla^{\prime}\varphi\right)\wedge\psi+\varphi\wedge\nabla^{\prime}\psi$
(4.3)
for any $\varphi,\psi\in\Omega^{\bullet}(M)$.
###### Lemma 4.1.
On an Hermitian manifold $(M,h)$, we have
$\begin{cases}\partial
h(\varphi,\psi)=h(\nabla^{\prime}\varphi,\psi)+h(\varphi,\nabla^{\prime\prime}\psi)\\\
\overline{\partial}h(\varphi,\psi)=h(\nabla^{\prime\prime}\varphi,\psi)+h(\varphi,\nabla^{\prime}\psi)\end{cases}\qquad\Longleftrightarrow\begin{cases}\frac{\partial}{\partial
z^{i}}h(\varphi,\psi)=h(\nabla^{\prime}_{i}\varphi,\psi)+h(\varphi,\nabla^{\prime\prime}_{\overline{i}}\psi)\\\
\frac{\partial}{\partial\overline{z}^{j}}h(\varphi,\psi)=h(\nabla^{\prime\prime}_{\overline{j}}\varphi,\psi)+h(\varphi,\nabla^{\prime}_{j}\psi)\end{cases}$
(4.4)
for any $\varphi,\psi\in\Omega^{p,q}(M)$.
###### Remark 4.2.
(1)
Here we use the compact notations
$\nabla^{\prime}_{i}=\nabla^{\prime}_{\frac{\partial}{\partial z^{i}}},\ \
\nabla^{\prime\prime}_{\overline{j}}=\nabla^{\prime\prime}_{\frac{\partial}{\partial\overline{z}^{j}}}$
Note that $\nabla^{\prime}_{\overline{j}}=\nabla^{\prime\prime}_{i}=0$ and
$\nabla_{i}\neq\nabla^{\prime}_{i}$,
$\nabla_{\overline{j}}\neq\nabla^{\prime}_{\overline{j}}$.
(2)
If we regard $\Lambda^{p,q}T^{*}M$ as an abstract vector bundle $E$, the above
lemma says that $\nabla^{\prime}$ and $\nabla^{\prime\prime}$ are compatible
with the Hermitian metric on $E$.
Now we go to an abstract setting. Let $E$ be an Hermitian _complex_ (possibly
_non-holomorphic_) vector bundle or a _Riemannian_ real vector bundle over a
compact Hermitian manifold $(M,\omega)$. There is a natural decomposition
$\nabla=\nabla^{{}^{\prime}E}+\nabla{{}^{\prime\prime E}}$ (4.5)
where
$\begin{cases}\nabla^{{}^{\prime}E}:\Gamma(M,E)\mathrel{\mathop{\hbox
to16.11119pt{\rightarrowfill}}\limits}\Omega^{1,0}(M,E)\\\
\nabla^{{}^{\prime\prime}E}:\Gamma(M,E)\mathrel{\mathop{\hbox
to16.11119pt{\rightarrowfill}}\limits}\Omega^{0,1}(M,E)\end{cases}$ (4.6)
$\nabla^{{}^{\prime}E}$ and $\nabla^{{}^{\prime\prime}E}$ induce two
differential operators. The first one is
$\partial_{E}:\Omega^{p,q}(M,E)\mathrel{\mathop{\hbox
to16.11119pt{\rightarrowfill}}\limits}\Omega^{p+1,q}(M,E)$ defined by
$\partial_{E}(\varphi\otimes s)=\left(\partial\varphi\right)\otimes
s+(-1)^{p+q}\varphi\wedge\nabla^{{}^{\prime}E}s$ (4.7)
for any $\varphi\in\Omega^{p,q}(M)$ and $s\in\Gamma(M,E)$. The other one is
$\overline{\partial}_{E}:\Omega^{p,q}(M,E)\mathrel{\mathop{\hbox
to16.11119pt{\rightarrowfill}}\limits}\Omega^{p+1,q}(M,E)$ defined by
$\overline{\partial}_{E}(\varphi\otimes
s)=\left(\overline{\partial}\varphi\right)\otimes
s+(-1)^{p+q}\varphi\wedge\nabla^{{}^{\prime\prime}E}s$ (4.8)
for any $\varphi\in\Omega^{p,q}(M)$ and $s\in\Gamma(M,E)$. The following
formula is well-known
$\left(\partial_{E}\overline{\partial}_{E}+\overline{\partial}_{E}\partial_{E}\right)(\varphi\otimes
s)=\varphi\wedge\left(\partial_{E}\overline{\partial}_{E}+\overline{\partial}_{E}\partial_{E}\right)s$
(4.9)
for any $\varphi\in\Omega^{p,q}(M)$ and $s\in\Gamma(M,E)$. The operator
$\partial_{E}\overline{\partial}_{E}+\overline{\partial}_{E}\partial_{E}$ is
represented by its $(1,1)$ curvature tensor
$R^{E}\in\Gamma(M,\Lambda^{1,1}T^{*}M\otimes E)$. For any
$\varphi,\psi\in\Omega^{\bullet,\bullet}(M,E)$, there is a _sesquilinear
pairing_
$\left\\{\varphi,\psi\right\\}=\varphi^{\alpha}\wedge\overline{\psi^{\beta}}\langle
e_{\alpha},e_{\beta}\rangle$ (4.10)
if $\varphi=\varphi^{\alpha}e_{\alpha}$ and $\psi=\psi^{\beta}e_{\beta}$ in
the local frames $\\{e_{\alpha}\\}$ on $E$. By the metric compatible property
of $\nabla^{E}$,
$\partial\\{\varphi,\psi\\}=\\{\partial_{E}\varphi,\psi\\}+(-1)^{p+q}\\{\varphi,\overline{\partial}_{E}\psi\\}$
(4.11)
if $\varphi\in\Omega^{p,q}(M,E)$.
Let $\omega$ be the Kähler form of the Hermitian metric $h$, i.e.,
$\omega=\frac{\sqrt{-1}}{2}h_{i\overline{j}}dz^{i}\wedge d\overline{z}^{j}$
(4.12)
On the Hermitian manifold $(M,h,\omega)$, the norm on $\Omega^{p,q}(M)$ is
defined by
$(\varphi,\psi)=\int_{M}\langle\varphi,\psi\rangle\frac{\omega^{n}}{n!}=\frac{2^{n}}{(p+q)!}\int_{M}h(\varphi,\psi)\frac{\omega^{n}}{n!}=\int_{M}\varphi\wedge*\overline{\psi}$
(4.13)
The norm on $\Omega^{p,q}(M,E)$ is defined by
$(\varphi,\psi)=\int_{M}\\{\varphi,*\psi\\}=\int_{M}\left(\varphi^{\alpha}\wedge*\overline{\psi^{\beta}}\right)\langle
e_{\alpha},e_{\beta}\rangle$ (4.14)
for $\varphi,\psi\in\Omega^{p,q}(M,E)$. The dual operators of
$\partial,\overline{\partial},\partial_{E}$ and $\overline{\partial}_{E}^{*}$
are denoted by $\partial^{*},\overline{\partial}^{*},\partial^{*}_{E}$ and
$\overline{\partial}_{E}^{*}$ respectively.
The following lemma was firstly shown by Demailly using Taylor expansion
method( e.g. [8]). For the convenience of the reader, we will take another
approach which seems to be useful in local computations.
###### Lemma 4.3.
Let $(M,h,\omega)$ be a compact Hermitian manifold. If $\tau$ is the operator
of type $(1,0)$ defined by $\tau=[\Lambda,2\partial\omega]$ on
$\Omega^{\bullet}(M,E)$,
$\begin{cases}\left[\Lambda,\partial\right]=\sqrt{-1}\left(\overline{\partial}^{*}+\overline{\tau}^{*}\right)\\\
\left[\Lambda,\overline{\partial}\right]=-\sqrt{-1}(\partial^{*}+\tau^{*})\end{cases}$
(4.15)
For the dual equation, it is
$\begin{cases}[\overline{\partial}^{*},L]=\sqrt{-1}(\partial+\tau)\\\
[\partial^{*},L]=-\sqrt{-1}(\overline{\partial}+\overline{\tau})\end{cases}$
(4.16)
where $L$ is the operator $L\varphi=2\omega\wedge\varphi$ and $\Lambda$ is the
adjoint operator of $L$.
###### Proof.
See Appendix Lemma 8.7. ∎
In the rest of this section $E$ is assumed to be an Hermitian complex vector
bundles or a Riemannian real vector bundle over a compact Hermitian manifold
$M$.
###### Lemma 4.4.
Let $\nabla^{E}$ be a metric connection on $E$ over a compact Hermitian
manifold $(M,\omega)$. If $\tau$ is the operator of type $(1,0)$ defined by
$\tau=[\Lambda,2\partial\omega]$ on $\Omega^{\bullet}(M,E)$, then
(1)
$[\overline{\partial}_{E}^{*},L]=\sqrt{-1}(\partial_{E}+\tau)$;
(2)
$[\partial^{*}_{E},L]=-\sqrt{-1}(\overline{\partial}_{E}+\overline{\tau})$;
(3)
$[\Lambda,\partial_{E}]=\sqrt{-1}(\overline{\partial}_{E}^{*}+\overline{\tau}^{*})$
;
(4)
$[\Lambda,\overline{\partial}_{E}]=-\sqrt{-1}(\partial_{E}^{*}+\tau^{*})$.
###### Proof.
See Appendix Lemma 8.10. ∎
###### Theorem 4.5.
Let $\nabla^{E}$ be a metric connection $E$ over a compact Hermitian manifold
$(M,\omega)$.
$\Delta_{\overline{\partial}_{E}}=\Delta_{\partial_{E}}+\sqrt{-1}\left[\partial_{E}\overline{\partial}_{E}+\overline{\partial}_{E}\partial_{E},\Lambda\right]+(\partial_{E}\tau^{*}+\tau^{*}\partial_{E})-(\overline{\partial}_{E}\overline{\tau}^{*}+\overline{\tau}^{*}\overline{\partial}_{E})$
(4.17)
where
$\begin{cases}\Delta_{\overline{\partial}_{E}}=\overline{\partial}_{E}\overline{\partial}_{E}^{*}+\overline{\partial}_{E}^{*}\overline{\partial}_{E}\\\
\Delta_{\partial_{E}}=\partial_{E}\partial_{E}^{*}+\partial_{E}^{*}\partial_{E}\end{cases}$
(4.18)
###### Proof.
It follows from Lemma 4.4. ∎
We make a useful observation on the torsion $\tau$:
###### Lemma 4.6.
For any $s\in\Gamma(M,E)$, we have
$\tau(s)=-2\sqrt{-1}\left(\overline{\partial}^{*}\omega\right)\cdot s,\ \ \ \
\ \ \overline{\tau}(s)=2\sqrt{-1}\left(\partial^{*}\omega\right)\cdot s$
(4.19)
###### Proof.
By definition
$\displaystyle\left([\Lambda,2\partial\omega]\right)s$ $\displaystyle=$
$\displaystyle 2\Lambda\left((\partial\omega)\cdot s\right)$ $\displaystyle=$
$\displaystyle 2\left(\Lambda(\partial\omega)\right)\cdot s$ $\displaystyle=$
$\displaystyle-2\sqrt{-1}\left(\overline{\partial}^{*}\omega\right)\cdot s$
Here we use the identity
$\overline{\partial}^{*}\omega=\sqrt{-1}\Lambda(\partial\omega)$ (4.20)
where the proof of it is contained in Lemma 8.8 of the Appendix. ∎
###### Corollary 4.7.
If $(M,\omega)$ is a compact balanced Hermitian manifold, and $\nabla^{E}$ a
metric connection on $E$ over $M$, then
$\|\overline{\partial}_{E}s\|^{2}=\|\partial_{E}s\|^{2}+\left(\sqrt{-1}\left[\partial_{E}\overline{\partial}_{E}+\overline{\partial}_{E}\partial_{E},\Lambda\right]s,s\right)$
(4.21)
for any $s\in\Gamma(M,E)$.
###### Proof.
Since for any $s\in\Gamma(M,E)$, $\tau s=\overline{\tau}s=0$ and
$\tau^{*}s=\overline{\tau}^{*}s=0$ on a balanced Hermitian manifold. ∎
###### Theorem 4.8.
Let $(M,\omega)$ be an Hermitian manifold with
$\partial\overline{\partial}\omega^{n-1}=0$. If $\nabla^{E}$ is a metric
connection on $E$ over $M$, then
$0=\|\overline{\partial}_{E}s\|^{2}=\|\partial_{E}s\|^{2}+\left(\sqrt{-1}\left[\partial_{E}\overline{\partial}_{E}+\overline{\partial}_{E}\partial_{E},\Lambda\right]s,s\right)$
(4.22)
for any $s\in\Gamma(M,E)$ with $\overline{\partial}_{E}s=0$.
###### Proof.
We only have to prove that
$\left((\partial_{E}\tau^{*}+\tau^{*}\partial_{E})s-(\overline{\partial}_{E}\overline{\tau}^{*}+\overline{\tau}^{*}\overline{\partial}_{E})s,s\right)=0$
(4.23)
which is equivalent to $\left(\partial_{E}s,\tau s\right)=0$ since
$\tau^{*}s=\overline{\tau}^{*}s=\overline{\partial}_{E}s=0$. By formula 4.19
and Stokes’ Theorem,
$\displaystyle\left(\tau^{*}\partial_{E}s,s\right)$ $\displaystyle=$
$\displaystyle\left(\partial_{E}s,\tau
s\right)=\int_{M}\left\\{\partial_{E}s,*(\tau s)\right\\}$ $\displaystyle=$
$\displaystyle
2\sqrt{-1}\int_{M}\left\\{\partial_{E}s,*\left(\overline{\partial}^{*}\omega\cdot
s\right)\right\\}$ $\displaystyle=$ $\displaystyle
2\sqrt{-1}\int_{M}\left\\{\partial_{E}s,\left(*\overline{\partial}^{*}\omega\right)\cdot
s\right\\}$ $\displaystyle=$
$\displaystyle-2\sqrt{-1}\int_{M}\left\\{s,\overline{\partial}_{E}\left(\left(*\overline{\partial}^{*}\omega\right)\cdot
s\right)\right\\}$ $\displaystyle=$
$\displaystyle-2\sqrt{-1}\int_{M}\left\\{s,\left(\overline{\partial}*\overline{\partial}^{*}\omega\right)\cdot
s-\left(*\overline{\partial}^{*}\omega\right)\wedge\overline{\partial}_{E}s\right\\}$
It is easy to see that
$\overline{\partial}*\overline{\partial}^{*}\omega=-\overline{\partial}**\partial*\omega=c_{n}\overline{\partial}\partial\omega^{n-1}=0$
(4.24)
since $*\omega=c_{n}\omega^{n-1}$ where $c_{n}$ is a constant depending only
on the complex dimension of $M$. Hence
$(\partial_{E}s,\tau
s)=2\sqrt{-1}\int_{M}\left\\{s,\left(*\overline{\partial}^{*}\omega\right)\wedge\overline{\partial}_{E}s\right\\}=0$
(4.25)
since $\overline{\partial}_{E}s=0$. ∎
###### Remark 4.9.
By these formula, we can obtain classical vanishing theorems on Kähler
manifolds and rigidity of harmonic maps between Hermitian and Riemannian
manifolds.
## 5 Vanishing theorems on Hermitian manifolds
### 5.1 Vanishing theorems on compact Hermitian manifolds
Let $E$ be an Hermitian _complex_ (possibly _non-holomorphic_) vector bundle
or a Riemannian _real_ vector bundle over a compact Hermitian manifold
$(M,\omega)$. Let $\partial_{E},\overline{\partial}_{E}$ be the $(1,0),(0,1)$
part of $\nabla^{E}$ respectively. The $(1,1)$-curvature of $\nabla^{E}$ is
denoted by $R^{E}\in\Gamma(M,\Lambda^{1,1}T^{*}M\otimes E^{*}\otimes E)$. It
is a representation of the operator
$\partial_{E}\overline{\partial}_{E}+\overline{\partial}_{E}\partial_{E}$. We
can define harmonic section spaces associated to $(E,\nabla^{E})$ by
${\mathcal{H}}^{p,q}_{\overline{\partial}_{E}}(M,E)=\\{\varphi\in\Omega^{p,q}(M,E)\
|\ \overline{\partial}_{E}\varphi=\overline{\partial}_{E}^{*}\varphi=0\\}$
(5.1)
In general, on a complex vector bundle $E$, there is no such terminology like
“holomorphic section of $E$”. However, if the vector bundle $E$ is holomorphic
and $\nabla^{E}$ is the Chern connection of $E$ i.e.
$\overline{\partial}_{E}=\overline{\partial}$, then
${\mathcal{H}}^{p,q}_{\overline{\partial}_{E}}(M,E)$ is isomorphic to the
Dolbeault cohomology group $H_{\overline{\partial}}^{p,q}(M,E)$ and
$H^{0}_{\overline{\partial}}(M,E)$ is the holomorphic section spaces
$H^{0}(M,E)$ of $E$.
###### Definition 5.1.
Let $A$ be an $r\times r$ Hermitian matrix and
$\lambda_{1}\leq\cdots\leq\lambda_{r}$ be eigenvalues of $A$. $A$ is said to
be _$p$ -nonnegative_ (resp. _positive, negative, nonpositive_) for $1\leq
p\leq r$ if
$\lambda_{i_{1}}+\cdots+\lambda_{i_{p}}\geq 0(\quad\mbox{resp.}\quad>0,<0,\leq
0)\quad\mbox{for any}\quad 1\leq i_{1}<i_{2}<\cdots<i_{p}\leq n$ (5.2)
###### Theorem 5.2.
Let $\nabla^{E}$ be any metric connection of an Hermitian complex vector
bundle or a Riemannian real vector bundle $E$ over a compact Hermitian
manifold $(M,h,\omega)$.
(1)
If the second Hermitian-Ricci curvature $Tr_{\omega}R^{E}$ is nonpositive
everywhere, then every $\overline{\partial}_{E}$-closed section of $E$ is
parallel, i.e. $\nabla^{E}s=0$;
(2)
If the second Hermitian-Ricci curvature $Tr_{\omega}R^{E}$ is nonpositive
everywhere and negative at some point, then
${\mathcal{H}}^{0}_{\overline{\partial}_{E}}(M,E)=0$;
(3)
If the second Hermitian-Ricci curvature $Tr_{\omega}R^{E}$ is $p$-nonpositive
everywhere and $p$-negative at some point, then
${\mathcal{H}}^{0}_{\overline{\partial}_{E}}(M,\Lambda^{q}E)=0$ for any $p\leq
q\leq rank(E)$.
###### Proof.
By [21], there exists a smooth function $u:M\mathrel{\mathop{\hbox
to16.11119pt{\rightarrowfill}}\limits}{\mathbb{R}}$ such that
$\omega_{G}=e^{u}\omega$ is a Gauduchon metric, i.e.
$\partial\overline{\partial}\omega^{n-1}_{G}=0$. Now we replace the metric
$\omega$ on $M$ by the Gauduchon metric $\omega_{G}$. By the relation
$\omega_{G}=e^{u}\omega$, we get
$Tr_{\omega_{G}}R^{E}=e^{-u}Tr_{\omega}R^{E}$ (5.3)
Therefore, the positivity conditions in the Theorem are preserved. Let
$s\in\Gamma(M,E)$ with $\overline{\partial}_{E}s=0$, by formula 4.22, we
obtain
$0=\|\partial_{E}s\|^{2}+\left(\sqrt{-1}\left[\partial_{E}\overline{\partial}_{E}+\overline{\partial}_{E}\partial_{E},\Lambda_{G}\right]s,s\right)=\|\partial_{E}s\|^{2}-\left(Tr_{\omega_{G}}R^{E}s,s\right)$
(5.4)
where
$R^{E}=\partial_{E}\overline{\partial}_{E}+\overline{\partial}_{E}\partial_{E}=R_{i\overline{j}\alpha}^{\beta}dz^{i}\wedge
d\overline{z}^{j}\otimes e^{\alpha}\otimes e_{\beta}$ (5.5)
Since the second Hermitian-Ricci curvature $Tr_{\omega_{G}}R^{E}$ has
components
$R_{\alpha\overline{\beta}}=h^{i\overline{j}}_{G}R_{i\overline{j}\alpha\overline{\beta}}$
(5.6)
formula 5.4 can be written as
$0=\|\partial_{E}s\|^{2}-\int_{M}R_{\alpha\overline{\beta}}s^{\alpha}\overline{s}^{\beta}$
(5.7)
Now (1) and (2) follow by identity 5.7 with the curvature conditions
immediately. For (3), we set $F=\Lambda^{q}E$ with $p\leq q\leq r=rank(E)$.
Let $\lambda_{1}\leq\cdots\leq\lambda_{r}$ be the eigenvalues of
$Tr_{\omega_{G}}R^{E}$, then we know
$\lambda_{1}+\cdots+\lambda_{p}\geq 0$ (5.8)
and it is strictly positive at some point. If $p\leq q\leq r$, the smallest
eigenvalue of $Tr_{\omega_{G}}R^{F}$ is $\lambda_{1}+\cdots+\lambda_{q}\geq 0$
and it is strictly positive at some point. By (2), we know
${\mathcal{H}}^{0}_{\overline{\partial}_{E}}(M,F)=0$. ∎
If $\nabla^{E}$ is the Chern connection of the Hermitian holomorphic vector
bundle $E$, we know
${\mathcal{H}}_{\overline{\partial}_{E}}^{0}(M,E)\cong H^{0}(M,E)$
since
$\overline{\partial}_{E}=\nabla^{{}^{\prime\prime}E}=\overline{\partial}$ for
the Chern connection.
###### Corollary 5.3 (Kobayashi-Wu[31], Gauduchon [19]).
Let $\nabla^{E}$ be the Chern connection of an Hermitian holomorphic vector
bundle $E$ over a compact Hermitian manifold $(M,h,\omega)$.
(1)
If the second Ricci-Chern curvature $Tr_{\omega}R^{E}$ is nonpositive
everywhere, then every holomorphic section of $E$ is parallel, i.e.
$\nabla^{E}s=0$;
(2)
If the second Ricci-Chern curvature $Tr_{\omega}R^{E}$ is nonpositive
everywhere and negative at some point, then $E$ has no holomorphic section,
i.e. $H^{0}(M,E)=0$;
(3)
If the second Ricci-Chern curvature $Tr_{\omega}R^{E}$ is $p$-nonpositive
everywhere and $p$-negative at some point, then $\Lambda^{q}E$ has no
holomorphic section for any $p\leq p\leq rank(E)$.
Now we can apply it to the tangent and cotangent bundles of compact Hermitian
manifolds.
###### Corollary 5.4.
Let $(M,\omega)$ be a compact Hermitian manifold and $\Theta$ is the Chern
curvature of the Chern connection $\nabla^{CH}$ on the holomorphic tangent
bundle $T^{1,0}M$.
(1)
If the second Ricci-Chern curvature $\Theta^{(2)}$ is nonpositive everywhere
and negative at some point, then $M$ has no holomorphic vector field, i.e.
$H^{0}(M,T^{1,0}M)=0$;
(2)
If the second Ricci-Chern curvature $\Theta^{(2)}$ is nonnegative everywhere
and positive at some point, then $M$ has no holomorphic $p$-form for any
$1\leq p\leq n$, i.e. $H^{p,0}_{\overline{\partial}}(M)=0$; In particular, the
arithmetic genus
$\chi(M,{\mathcal{O}})=\sum(-1)^{p}h^{p,0}(M)=1$ (5.9)
(3)
If the second Ricci-Chern curvature $\Theta^{(2)}$ is $p$-nonnegative
everywhere and $p$-positive at some point, then $M$ has no holomorphic
$q$-form for any $p\leq q\leq n$, i.e. $H^{q,0}_{\overline{\partial}}(M)=0$.
In particular, if the scalar curvature $S^{CH}$ is nonnegative everywhere and
positive at some point, then $H^{0}(M,mK_{M})=0$ for all $m\geq 1$ where
$K_{M}$ is the canonical line bundle of $M$.
###### Proof.
Let $E=T^{1,0}M$ and $h$ be an Hermitian metric on $E$ such that the second
Ricci-Chern curvature $Tr_{\omega_{h}}\Theta$ of $(E,h)$ satisfies the
assumption. It is obvious that all section spaces in consideration are
independent of the choice of the metrics and connections.
The metric on the vector bundle $E$ is fixed. Now we choose a Gauduchon metric
$\omega_{G}=e^{u}\omega_{h}$ on $M$. Then the second Ricci-Chern curvature
$\widetilde{\Theta}^{(2)}=Tr_{\omega_{G}}\Theta=e^{-u}Tr_{\omega_{h}}\Theta$
shares the semi-definite property with $\Theta^{(2)}=Tr_{\omega_{h}}\Theta$.
For the safety, we repeat the arguments in Theorem 5.2 briefly. If $s$ is a
holomorphic section of $E$, i.e.,
$\overline{\partial}_{E}s=\overline{\partial}s=0$, by formula 4.22, we obtain
$0=\|\partial_{E}s\|^{2}+\left(\sqrt{-1}\left[\partial_{E}\overline{\partial}_{E}+\overline{\partial}_{E}\partial_{E},\Lambda_{G}\right]s,s\right)=\|\partial_{E}s\|^{2}-\left(Tr_{\omega_{G}}\Theta
s,s\right)$ (5.10)
If $Tr_{\omega}\Theta$ is nonpositive everywhere, then $\partial_{E}s=0$ and
so $\nabla^{E}s=0$. If $Tr_{\omega}\Theta$ is nonpositive everywhere and
negative at some point, we get $s=0$, therefore $H^{0}(M,T^{1,0}M)=0$. The
proofs of the other parts are similar. ∎
###### Remark 5.5.
It is well-known that the first Ricci-Chern curvature $\Theta^{(1)}$
represents the first Chern class of $M$. But on an Hermitian manifold, it is
possible that the second Ricci-Chern curvature $\Theta^{(2)}$ is not in the
same $(d,\partial,\overline{\partial})$-cohomology class as $\Theta^{(1)}$.
For example, ${\mathbb{S}}^{3}\times{\mathbb{S}}^{1}$ with canonical metric
has strictly positive second Ricci-Chern curvature but it is well-known that
it has vanishing first Chern number $c_{1}^{2}$. For more details see
Proposition 6.4. Therefore, $\Theta^{(2)}$ in Proposition 5.4 can not be
replaced by $\Theta^{(1)}$. It seems to be an interesting question: if
$(M,\omega)$ is a compact Hermitian manifold and its first Ricci-Chern
curvature is nonnegative everywhere and positive at some point, is the first
Betti number of $M$ zero? In particular, is it Kähler in dimension $2$?
As special cases of our results, the following results for Kähler manifolds
are well-known, and we list them here for the convenience of the reader. Let
$(M,h,\omega)$ be a compact Kähler manifold.
(1)
If the Ricci curvature is nonnegative everywhere, then any holomorphic $(p,0)$
form is parallel;
(2)
If the Ricci curvature is nonnegative everywhere and positive at some point,
then $h^{p,0}=0$ for $p=1,\cdots,n$. In particular, the arithmetic genus
$\chi(M,{\mathcal{O}})=1$ and $b_{1}(M)=0$;
(3)
If the scalar curvature is nonnegative everywhere and positive at some point,
then $h^{n,0}=0$.
(A)
If the Ricci curvature is nonpositive everywhere, then any holomorphic vector
field is parallel;
(B)
If the Ricci curvature is nonpositive everywhere and negative at some point,
there is no holomorphic vector field.
### 5.2 Vanishing theorems on special Hermitian manifolds
Let $(M,h,\omega)$ be a compact Hermitian manifold and $\nabla$ be the Levi-
Civita connection.
###### Lemma 5.6.
Let $(M,\omega)$ be a compact balanced Hermitian manifold. For any
$(p,0)$-form $\varphi$ on $M$,
(1)
If $\varphi$ is holomorphic, then $\partial^{*}\varphi=0$;
(2)
If $\nabla^{\prime}\varphi=0$, then $\partial\varphi=0$.
###### Proof.
For simplicity, we assume $p=1$. For the general case, the proof is the same.
By Lemma 8.5, we know, for any $(1,0)$-form $\varphi=\varphi_{i}dz^{i}$,
$\partial^{*}\varphi=-h^{i\overline{j}}\frac{\partial\varphi_{i}}{\partial\overline{z}^{j}}$
(5.11)
where we use the balanced condition
$h^{i\overline{j}}\Gamma_{i\overline{j}}^{s}=0$. If $\varphi$ is holomorphic,
then $\frac{\partial\varphi_{i}}{\partial\overline{z}^{j}}=0$, hence
$\partial^{*}\varphi=0$. On the other hand,
$\nabla^{\prime}\varphi=\left(\frac{\partial\varphi_{i}}{\partial
z^{j}}-\Gamma_{ji}^{m}\varphi_{m}\right)dz^{j}\otimes dz^{i}$ (5.12)
If $\nabla^{\prime}\varphi=0$, we obtain
$\partial\varphi=\frac{\partial\varphi_{i}}{\partial z^{j}}dz^{j}\wedge
dz^{i}=\Gamma_{ji}^{m}\varphi_{m}dz^{j}\wedge dz^{i}=0$ (5.13)
∎
###### Theorem 5.7.
Let $(M,\omega)$ be a compact balanced Hermitian manifold with Levi-Civita
connection $\nabla$.
(1)
If the Hermitian-Ricci curvature $(R_{i\overline{j}})$ is $p$-nonnegative
everywhere, then any holomorphic $(q,0)$-form ($p\leq q\leq n$) is
$\partial$-harmonic; in particular, $h^{q,0}(M)\leq h^{0,q}(M)$ for any $p\leq
q\leq n$;
(2)
If the Hermitian-Ricci curvature $(R_{i\overline{j}})$ is $p$-nonnegative
everywhere and $p$-positive at some point,
$H^{q,0}_{\overline{\partial}}(M)=0$ for any $p\leq q\leq n$;
In particular,
(3)
if the Hermitian-Ricci curvature $(R_{i\overline{j}})$ is nonnegative
everywhere and positive at some point, then
$H^{p,0}_{\overline{\partial}}(M)=0$, for $p=1,\cdots,n$ and so the arithmetic
genus $\chi(M,{\mathcal{O}})=1$ and $b_{1}(M)\leq h^{0,1}(M)$.
(4)
if the Hermitian-scalar curvature $S$ is nonnegative everywhere and positive
at some point, then
$H^{0}(M,mK_{M})=0\quad\mbox{for any}\quad m\geq 1$
where $K_{M}=\det T^{*1,0}M$.
###### Proof.
At first, we assume $p=1$ for (1) and (2). Now we consider $E=T^{*1,0}M$ with
the induced metric connection $\nabla^{E}=\widehat{\nabla}$ for $h$ (see
2.22). By formula 4.7, we have
$\|\overline{\partial}_{E}s\|^{2}=\|\partial_{E}s\|^{2}+\sqrt{-1}\left(\left[R^{E},\Lambda\right]s,s\right)$
(5.14)
where $R^{E}$ is the $(1,1)$-part curvature of $E$ with respect to the
connection $\nabla^{E}$. More precisely,
$R^{E}=\partial_{E}\overline{\partial}_{E}+\overline{\partial}_{E}\partial_{E}=-\widehat{R}_{i\overline{j}k}^{\ell}dz^{i}\wedge
d\overline{z}^{j}\otimes\frac{\partial}{\partial z^{\ell}}\otimes dz^{k}$
(5.15)
since $E$ is the dual vector bundle of $T^{1,0}M$ and the $(1,1)$-part of the
curvature of $T^{1,0}M$ is
$\widehat{R}_{i\overline{j}k}^{\ell}dz^{i}\wedge d\overline{z}^{j}\otimes
dz^{k}\otimes\frac{\partial}{\partial z^{\ell}}$ (5.16)
If $s=f_{i}dz^{i}$ is a holomorphic $1$-form, i.e.
$\overline{\partial}s=\frac{\partial
f_{i}}{\partial\overline{z}^{j}}d\overline{z}^{j}\wedge dz^{i}=0$ (5.17)
then
$\overline{\partial}_{E}s=\left(\frac{\partial
f_{i}}{\partial\overline{z}^{j}}-f_{k}\Gamma_{\overline{j}i}^{k}\right)d\overline{z}^{j}\otimes
dz^{i}=-f_{k}\Gamma_{\overline{j}i}^{k}d\overline{z}^{j}\otimes dz^{i}$ (5.18)
Without loss of generality, we assume $h_{i\overline{j}}=\delta_{ij}$ at a
given point. By Proposition 2.12, the quantity
$|\overline{\partial}_{E}s|^{2}=\sum_{i,j,t,n}f_{i}\overline{f}_{n}\Gamma_{\overline{j}t\overline{i}}\overline{\Gamma_{\overline{j}t\overline{n}}}=\sum_{i,n}\left(\widehat{R}^{(2)}_{n\overline{i}}-R_{n\overline{i}}\right)f_{i}\overline{f}_{n}$
(5.19)
On the other hand
$\sqrt{-1}\left\langle\left[R^{E},\Lambda\right]s,s\right\rangle=\sum_{i,n}\widehat{R}^{(2)}_{n\overline{i}}f_{i}\overline{f}_{n}$
(5.20)
That is
$|\overline{\partial}_{E}s|^{2}-\sqrt{-1}\left\langle\left[R^{E},\Lambda\right]s,s\right\rangle=-\sum_{i,n}R_{n\overline{i}}f_{i}\overline{f}_{n}\leq
0$ (5.21)
if the Hermitian-Ricci curvature $(R_{n\overline{i}})$ of $(M,h,\omega)$ is
nonnegative everywhere. Then we get
$0\leq\|\partial_{E}s\|^{2}=\|\overline{\partial}_{E}s\|^{2}-\sqrt{-1}\left(\left[R^{E},\Lambda\right]s,s\right)\leq
0$ (5.22)
That is $\partial_{E}s=0$. Since
$\partial_{E}s=\nabla^{{}^{\prime}E}s=\widehat{\nabla}^{\prime}s=\nabla^{\prime}s=\left(\frac{\partial
f_{i}}{\partial z^{j}}-f_{\ell}\Gamma_{ij}^{\ell}\right)dz^{j}\otimes dz^{i}$
we obtain $\nabla^{\prime}s=0$. By Lemma 5.6, we know $\Delta_{\partial}s=0$.
In summary, we get
$H^{1,0}_{\overline{\partial}}(M)\subset H^{1,0}_{\partial}(M)\cong
H^{0,1}_{\overline{\partial}}(M)$ (5.23)
If the Hermitian-Ricci curvature $(R_{n\overline{i}})$ is nonnegative
everywhere and positive at some point, then $f_{i}=0$ for each $i$, that is
$s=0$. So we proved $H^{1,0}_{\overline{\partial}}(M)=0$. The general cases
follow by the same arguments as Theorem 5.2 and Theorem 5.4. ∎
The dual of Theorem 5.7 is
###### Theorem 5.8.
Let $(M,h,\omega)$ be a compact balanced Hermitian manifold.
(1)
If $2\widehat{R}^{(2)}_{i\overline{j}}-R_{i\overline{j}}$ is nonpositive
everywhere, then any holomorphic vector field is $\nabla^{\prime}$-closed;
(2)
If $2\widehat{R}^{(2)}_{i\overline{j}}-R_{i\overline{j}}$ is nonpositive
everywhere and negative at some point, there is no holomorphic vector field.
###### Proof.
Let $E=T^{1,0}M$ and $\widehat{\nabla}$ the induced connection on it. If
$s=f^{i}\frac{\partial}{\partial z^{i}}$ is a holomorphic section, then
$\overline{\partial}_{E}s=f^{i}\Gamma_{\overline{j}i}^{\ell}d\overline{z}^{j}\otimes\frac{\partial}{\partial
z^{\ell}}$ (5.24)
Without loss generality, we assume $h_{i\overline{j}}=\delta_{ij}$ at a given
point. By Proposition 2.12,
$\displaystyle|\overline{\partial}_{E}s|^{2}-\sqrt{-1}\left\langle\left[\widehat{R}^{1,1},\Lambda\right]s,s\right\rangle$
$\displaystyle=$
$\displaystyle\left(\widehat{R}^{(2)}_{i\overline{j}}-R_{i\overline{j}}\right)f^{i}\overline{f}^{j}+\widehat{R}_{i\overline{j}}^{(2)}f^{i}\overline{f}^{j}$
$\displaystyle=$
$\displaystyle\left(2\widehat{R}^{(2)}_{i\overline{j}}-R_{i\overline{j}}\right)f^{i}\overline{f}^{j}$
By formula 4.17,
$0\leq\|\partial_{E}s\|^{2}=\|\overline{\partial}_{E}s\|^{2}-\sqrt{-1}\left(\left[\widehat{R}^{1,1},\Lambda\right]s,s\right)$
(5.25)
So if $2\widehat{R}^{(2)}_{i\overline{j}}-R_{i\overline{j}}$ is nonpositive
everywhere, $\partial_{E}s=\nabla^{\prime}s=0$. If
$2\widehat{R}^{(2)}_{i\overline{j}}-R_{i\overline{j}}$ is nonpositive
everywhere and negative at some point, there is no holomorphic vector field. ∎
###### Remark 5.9.
(1)
It is obvious that the second Ricci-Chern curvature
$\Theta^{(2)}_{k\overline{\ell}}$ and Hermitian-Ricci curvature
$R_{k\overline{\ell}}$ can not be compared. Therefore, Theorem 5.4 and Theorem
5.7 are independent of each other. For the same reason, Theorem 5.4 and
Theorem 5.8 are independent.
(2)
For a special case in Theorem 5.7, if the Hermitian-Ricci curvature
$R_{k\overline{\ell}}$ is nonnegative everywhere and positive at some point,
by Proposition 3.5, the manifold $(M,\omega)$ is Moishezon. It is well-known
that every $2$-dimensional Moishezon/balanced manifold is Kähler, but there
are many Moishezon non-Kähler manifolds in higher dimension( See [36]).
The following result was firstly obtained in [25]:
###### Corollary 5.10.
Let $(M,\omega)$ be a compact Hermitian manifold with
$\Lambda(\partial\overline{\partial}\omega)=0$. Let $\nabla^{B}$ be the Bismut
connection on $T^{1,0}M$.
(1)
If the first Ricci-Bismut curvature $B^{(1)}$ is nonnegative everywhere, then
every holomorphic $(p,0)$-form is parallel with respect to the Chern
connection $\nabla^{CH}$;
(2)
If the first Ricci-Bismut curvature $B^{(1)}$ is nonnegative everywhere and
positive at some point, then $M$ has no holomorphic $(p,0)$-form for any
$1\leq p\leq n$, i.e. $H^{p,0}_{\overline{\partial}}(M)=0$; in particular, the
arithmetic genus $\chi(M,{\mathcal{O}})=1$.
(3)
If the first Ricci-Bismut curvature $B^{(1)}$ is $p$-nonnegative everywhere
and $p$-positive at some point then $M$ has no holomorphic $(q,0)$-form for
any $p\leq q\leq n$, i.e. $H^{q,0}_{\overline{\partial}}(M)=0$. In particular,
if the scalar curvature $S^{BM}$ of the Bismut connection is nonnegative
everywhere and positive at some point, then $H^{0}(M,mK_{M})=0$ for any $m\geq
1$.
###### Proof.
By Proposition 3.7, if $\Lambda(\partial\overline{\partial}\omega)=0$, then
$B^{(1)}\leq\Theta^{(2)}$ (5.26)
Now we can apply Corollary 5.4 to get $(1)$, $(2)$ and $(3)$. ∎
###### Remark 5.11.
For more vanishing theorems on special Hermitian manifolds, one can see [1],
[25], [16], [17] and references therein.
## 6 Examples of non-Kähler manifolds with nonnegative curvatures
Let $M={\mathbb{S}}^{2n-1}\times{\mathbb{S}}^{1}$ be the standard
$n$-dimensional ($n\geq 2$) Hopf manifold. It is diffeomorphic to
${\mathbb{C}}^{n}-\\{0\\}/G$ where $G$ is cyclic group generated by the
transformation $z\rightarrow\frac{1}{2}z$. It has an induced complex structure
of ${\mathbb{C}}^{n}-\\{0\\}$. For more details about such manifolds, we refer
the reader to [30]. On $M$, there is a natural metric
$h=\sum_{i=1}^{n}\frac{4}{|z|^{2}}dz^{i}\otimes d\overline{z}^{i}$ (6.1)
The following identities follow immediately
$\frac{\partial h_{k\overline{\ell}}}{\partial
z^{i}}=-\frac{4\delta_{k\ell}\overline{z}^{i}}{|z|^{4}},\ \ \ \ \frac{\partial
h_{k\overline{\ell}}}{\partial\overline{z}^{j}}=-\frac{4\delta_{k\ell}z^{j}}{|z|^{4}}$
(6.2)
and
$\frac{\partial^{2}h_{k\overline{\ell}}}{\partial
z^{i}\partial\overline{z}^{j}}=-4\delta_{k\ell}\frac{\delta_{i\overline{j}}|z|^{2}-2\overline{z}^{i}z^{j}}{|z|^{6}}$
(6.3)
###### Example 6.1 (Curvatures of Chern connection).
Direct computation shows that, the Chen curvature components are
$\Theta_{i\overline{j}k\overline{\ell}}=-\frac{\partial^{2}h_{k\overline{\ell}}}{\partial
z^{i}\partial\overline{z}^{j}}+h^{p\overline{q}}\frac{\partial
h_{{k\overline{q}}}}{\partial z^{i}}\frac{\partial
h_{p\overline{\ell}}}{\partial\overline{z}^{j}}=\frac{4\delta_{kl}(\delta_{ij}|z|^{2}-z^{j}\overline{z}^{i})}{|z|^{6}}$
(6.4)
and the first and second Ricci-Chern curvatures are
$\Theta^{(1)}_{k\overline{\ell}}=\frac{n\left(\delta_{k\ell}|z|^{2}-z^{\ell}\overline{z}^{k}\right)}{|z|^{4}},\
\ \ \ \Theta^{(2)}_{k\overline{\ell}}=\frac{(n-1)\delta_{k\ell}}{|z|^{2}}$
(6.5)
It is easy to see that the eigenvalues of $\Theta^{(1)}$ are
$\lambda_{1}=0,\lambda_{2}=\cdots=\lambda_{n}=\frac{n}{|z|^{2}}$ (6.6)
Hence, $\Theta^{(1)}$ is nonnegative and $2$-positive everywhere.
###### Example 6.2 (Curvatures of Levi-Civita connection).
Similarly, we have
$\Gamma_{ik}^{\ell}=-\frac{\delta_{i\ell}\overline{z}^{k}+\delta_{k\ell}\overline{z}^{i}}{2|z|^{2}},\
\ \ \
\Gamma_{\overline{j}k}^{\ell}=\frac{\delta_{jk}z^{\ell}-\delta_{k\ell}z^{j}}{2|z|^{2}}$
(6.7)
and
$\frac{\partial\Gamma_{ik}^{\ell}}{\partial\overline{z}^{j}}=-\frac{\delta_{{k\ell}}\delta_{ij}+\delta_{i\ell}\delta_{jk}}{2|z|^{2}}+\frac{\delta_{i\ell}z^{j}\overline{z}^{k}+\delta_{k\ell}z^{j}\overline{z}^{i}}{2|z|^{4}}$
(6.8)
$\frac{\partial\Gamma_{\overline{j}k}^{\ell}}{\partial
z^{i}}=\frac{\delta_{jk}\delta_{i\ell}-\delta_{k\ell}\delta_{ij}}{2|z|^{2}}-\frac{(\delta_{jk}z^{\ell}-\delta_{k\ell}z^{j})\overline{z}^{i}}{2|z|^{4}}$
(6.9)
The complexified Riemannian curvature components are
$R_{i\overline{j}k}^{\ell}=-\left(\frac{\partial\Gamma^{\ell}_{ik}}{\partial\overline{z}^{j}}-\frac{\partial\Gamma^{\ell}_{\overline{j}k}}{\partial
z^{i}}+\Gamma_{ik}^{s}\Gamma^{\ell}_{\overline{j}s}-\Gamma_{\overline{j}k}^{s}\Gamma^{\ell}_{is}-{\Gamma_{\overline{j}k}^{\overline{s}}\Gamma_{i\overline{s}}^{\ell}}\right)=\frac{\delta_{i\ell}\delta_{jk}}{2|z|^{2}}-\frac{\delta_{i\ell}z^{j}\overline{z}^{k}+\delta_{jk}z^{\ell}\overline{z}^{i}}{4|z|^{4}}$
(6.10)
and
$R_{i\overline{j}k\overline{\ell}}=\frac{2\delta_{i\ell}\delta_{jk}}{|z|^{4}}-\frac{\delta_{i\ell}z^{j}\overline{z}^{k}+\delta_{jk}z^{\ell}\overline{z}^{i}}{|z|^{6}},\
\ \ \
R_{k\overline{\ell}}=\frac{\delta_{k\ell}|z|^{2}-z^{\ell}\overline{z}^{k}}{2|z|^{4}}$
(6.11)
###### Example 6.3 ( Curvatures of Bismut connection).
By definition 2.45 and Lemma 2.10, we obtain
$B_{i\overline{j}k}^{\ell}=\frac{\delta_{jk}\delta_{i\ell}-\delta_{k\ell}\delta_{ij}}{|z|^{2}}+\frac{\delta_{ij}\overline{z}^{k}z^{\ell}+\delta_{k\ell}\overline{z}^{i}z^{j}-\delta_{i\ell}\overline{z}^{k}z^{j}-\delta_{jk}\overline{z}^{i}z^{\ell}}{|z|^{4}}$
(6.12)
Two Ricci curvatures are
$B^{(1)}_{i\overline{j}}=B^{(2)}_{i\overline{j}}=\frac{(2-n)(\delta_{ij}|z|^{2}-\overline{z}^{i}z^{j})}{4|z|^{2}}$
(6.13)
On the other hand, by 6.3, it is easy to see
$\partial\overline{\partial}\omega=0$ and $B^{(1)}=0$ for $n=2$.
###### Proposition 6.4.
Let $M={\mathbb{S}}^{2n-1}\times{\mathbb{S}}^{1}$ be the standard
$n$-dimensional ($n\geq 2$) Hopf manifold with canonical metric $h$,
(1)
$(M,h)$ has positive second Ricci-Chern curvature $\Theta^{(2)}$;
(2)
$(M,h)$ has nonnegative first Ricci-Chern curvature $\Theta^{(1)}$, i.e.,
$c_{1}(M)\geq 0$. Moreover,
$\int_{M}c_{1}^{n}(M)=0$ (6.14)
(3)
$(M,h)$ is semi-positive in the sense of Griffiths, i.e.
$\Theta_{i\overline{j}k\overline{\ell}}u^{i}\overline{u}^{j}v^{k}\overline{v}^{\ell}\geq
0$ (6.15)
for any $u,v\in{\mathbb{C}}^{n}$;
(4)
$R_{k\overline{\ell}}$ is nonnegative and $2$-positive everywhere;
(5)
$(M,h)$ has nonpositive and $2$-negative first Ricci-Bismut curvature. In
particular, $({\mathbb{S}}^{3}\times{\mathbb{S}}^{1},\omega)$ satisfies
$\partial\overline{\partial}\omega=0$ and has vanishing first Ricci-Bismut
curvature $B^{(1)}$.
Although we know all Betti numbers of Hopf manifold
${\mathbb{S}}^{2n-1}\times{\mathbb{S}}^{1}$, $h^{p,0}$ is not so obvious.
###### Corollary 6.5.
Let $(M,h)$ be $n$-dimensional Hopf manifold with $n\geq 2$,
(1)
$h^{p,0}(M)=0$ for $p\geq 1$ and $\chi(M,{\mathcal{O}})=1$. In particular,
$h^{0,1}(M)\geq 1$.
(2)
$\dim_{\mathbb{C}}H^{0}(M,mK)=0$ for any $m\geq 1$ where $K=\det(T^{*1,0}M)$.
## 7 A natural geometric flow on Hermitian manifolds
As we discussed in the above sections, on Hermitian manifolds, the second
Ricci curvature tensors of various metric connections are closely related to
the geometry of Hermitian manifolds. A natural idea is to define a flow by
using second Ricci curvature tensors of various metric connections. We
describe it in the following.
Let $(M,h)$ be a compact Hermitian manifold. Let $\nabla$ be an _arbitrary
metric connection_ on the holomorphic tangent bundle $(E,h)=(T^{1,0}M,h)$.
$\nabla:E\mathrel{\mathop{\hbox
to16.11119pt{\rightarrowfill}}\limits}\Omega^{1}(E)$ (7.1)
It has two components $\nabla^{{}^{\prime}}$ and $\nabla^{{}^{\prime\prime}}$,
$\nabla=\nabla^{{}^{\prime}}+\nabla^{{}^{\prime\prime}}$ (7.2)
$\nabla^{{}^{\prime}}$ and $\nabla^{{}^{\prime\prime}}$ induce two
differential operators
$\partial_{E}:\Omega^{p,q}(E)\mathrel{\mathop{\hbox
to16.11119pt{\rightarrowfill}}\limits}\Omega^{p+1,q}(E)$ (7.3)
$\overline{\partial}_{E}:\Omega^{p,q}(E)\mathrel{\mathop{\hbox
to16.11119pt{\rightarrowfill}}\limits}\Omega^{p,q+1}(E)$ (7.4)
Let $R^{E}$ be the $(1,1)$ curvature of the metric connection $\nabla$. More
precisely $R^{E}$ is a representation of
$\partial_{E}\overline{\partial}_{E}+\overline{\partial}_{E}\partial_{E}$. It
is easy to see that
$R^{E}\in\Gamma(M,\Lambda^{1,1}T^{*}M\otimes End(E))$ (7.5)
and locally, we can write it as
$R^{E}=R_{i\overline{j}A}^{B}dz^{i}\wedge dz^{j}\otimes e^{A}\otimes e_{B}$
(7.6)
Here we set $e_{A}=\frac{\partial}{\partial z^{A}},e^{B}=dz^{B}$ where
$A,B=1,\cdots,n$, since the geometric meanings of $j$ and $A$ are different.
It is well-known that a metric connection $\nabla$ is determined by its
Christoffel symbols
$\nabla_{\frac{\partial}{\partial z^{i}}}e_{A}=\Gamma_{iA}^{B}e_{B},\ \ \ \
\nabla_{\frac{\partial}{\partial\overline{z}^{j}}}e_{A}=\Gamma_{\overline{j}A}^{B}e_{B}$
(7.7)
In particular, we don’t have notations such as $\Gamma_{Ai}^{B}$. It is
obvious that
$R_{i\overline{j}B}^{A}=-\frac{\partial\Gamma_{iA}^{B}}{\partial\overline{z}^{j}}+\frac{\partial\Gamma_{\overline{j}A}^{B}}{\partial
z^{i}}-\Gamma_{iA}^{C}\Gamma_{\overline{j}C}^{B}+\Gamma_{\overline{j}A}^{C}\Gamma_{iC}^{B}$
(7.8)
We set the second Hermitian-Ricci curvature tensor of $(\nabla,h)$ as
$R^{(2)}=h^{i\overline{j}}R_{i\overline{j}A\overline{B}}e^{A}\otimes\overline{e}^{B}\in\Gamma(M,E^{*}\otimes\overline{E}^{*})$
(7.9)
In general we can study a new class of flows on Hermitian manifolds
$\begin{cases}\frac{\partial h}{\partial t}={\mathcal{F}}(h)+\mu h\\\
h(0)=h_{0}\end{cases}$ (7.10)
where ${\mathcal{F}}$ can be a linear combination of the first and the second
Hermitian-Ricci curvature tensors of different metric connections on
$(T^{1,0}M,h)$. For examples, ${\mathcal{F}}(h)=-\Theta^{(2)}$, the second
Ricci-Chern curvature tensor of the Chern connection, and
${\mathcal{F}}(h)=-\widehat{R}^{(2)}$, the second Hermitian-Ricci curvature
tensor of the complexified Levi-Civita connection, or the second Ricci
curvature of any other Hermitian connection. Quite interesting is to take
${\mathcal{F}}(h)=s\Theta^{(1)}+(1-s)\Theta^{(2)}$ as the mixed Ricci-Chern
curvature, or ${\mathcal{F}}(h)=B^{(2)}-2\widehat{R}^{(2)}$ where $B^{(2)}$ is
the second Ricci curvature of the Bismut connection. More generally, we can
set ${\mathcal{F}}(h)$ to be certain suitable functions on the metric $h$. For
example, if ${\mathcal{F}}(h)=\left(\Delta_{h}S\right)h$, the above equation
will be the Hermitian Calabi flows.
The following result holds for quite general ${\mathcal{F}}(h)$, but here for
simplicity we will only take ${\mathcal{F}}(h)=-\Theta^{(2)}$ as an example.
$\begin{cases}\frac{\partial h}{\partial t}=-\Theta^{(2)}+\mu h\\\
h(0)=h_{0}\end{cases}$ (7.11)
where $\mu$ is a real parameter. By formula 2.38, the second Ricci-Chern
curvature tensor has components
$\Theta^{(2)}_{k\overline{\ell}}=h^{i\overline{j}}\Theta_{i\overline{j}k\overline{\ell}}=-h^{i\overline{j}}\frac{\partial^{2}h_{k\overline{\ell}}}{\partial
z^{i}\partial\overline{z}^{j}}+h^{i\overline{j}}h^{p\overline{q}}\frac{\partial
h_{k\overline{q}}}{\partial z^{i}}\frac{\partial
h_{p\overline{\ell}}}{\partial\overline{z}^{j}}$ (7.12)
###### Theorem 7.1.
Let $(M,h_{0})$ be a compact Hermitian manifold.
(1)
There exists small $\varepsilon$ such that, the solution of flow 7.11 exists
for $|t|<\varepsilon$, and it preserves the Hermitian structure;
(2)
The flow 7.11 preserves the Kähler structure, i.e., if the initial metric
$h_{0}$ is Kähler, then $h(t)$ are also Kähler.
###### Proof.
(1). Let $\Delta_{c}$ be the canonical Laplacian operator on the Hermitian
manifold $(M,h)$ defined by
$\Delta_{c}=h^{p\overline{q}}\frac{\partial^{2}}{\partial
z^{p}\partial\overline{z}^{q}}.$ (7.13)
Therefore, the second Ricci-Chern curvature $-\Theta^{(2)}_{i\overline{j}}$
has leading term $\Delta_{c}h_{i\overline{j}}$ which is strictly elliptic. The
local existence of the flow 7.11 follows by general theory of parabolic PDE,
and the solution is an Hermitian metric on $M$.
(2). The coefficients of the tensor $\partial\omega$ are given by
$f_{i\overline{j}k}=\frac{\partial h_{i\overline{j}}}{\partial
z^{k}}-\frac{\partial h_{k\overline{j}}}{\partial z^{i}}$ (7.14)
Under the flow 7.11, we have
$\begin{cases}\frac{\partial f_{i\overline{j}k}}{\partial
t}=\frac{\partial\Theta^{(2)}_{k\overline{j}}}{\partial
z^{i}}-\frac{\partial\Theta^{(2)}_{i\overline{j}}}{\partial z^{k}}+\mu
f_{i\overline{j}k}\\\ f_{i\overline{j}k}(0)=0\end{cases}$ (7.15)
At first, we observe that $f_{i\overline{j}k}(t)\equiv 0$ is a solution of
7.15. In fact, if $f_{i\overline{j}k}(t)\equiv 0$, then $h_{i\overline{j}}(t)$
are Kähler metrics, and so
$\Theta^{(2)}_{i\overline{j}}=\Theta^{(1)}_{i\overline{j}}=-\frac{\partial^{2}\log\det(h_{m\overline{n}})}{\partial
z^{i}\partial\overline{z}^{j}}$
Therefore,
$\frac{\partial\Theta^{(2)}_{k\overline{j}}}{\partial
z^{i}}-\frac{\partial\Theta^{(2)}_{i\overline{j}}}{\partial
z^{k}}=-\frac{\partial^{3}\log\det(h_{m\overline{n}})}{\partial z^{i}\partial
z^{k}\partial\overline{z}j}+\frac{\partial^{3}\log\det(h_{m\overline{n}})}{\partial
z^{i}\partial z^{k}\partial\overline{z}j}=0$ (7.16)
On the other hand,
$\frac{\partial\Theta^{(2)}_{k\overline{j}}}{\partial
z^{i}}-\frac{\partial\Theta^{(2)}_{i\overline{j}}}{\partial
z^{k}}=\Delta_{c}\left(f_{i\overline{j}k}\right)+\quad\mbox{lower order
terms}\quad$ (7.17)
Hence the solution of 7.15 is unique. ∎
###### Remark 7.2.
Theorem 7.1 holds also for quite general ${\mathcal{F}}(h)$ which we will
study in detail in a subsequent paper [33].
The flow 7.11 has close connections to several important geometric flows:
1. 1.
It is very similar to the Hermitian Yang-Mills flow on holomorphic vector
bundles. More precisely, if the flow 7.11 has long time solution and it
converges to an Hermitian metric $h_{\infty}$ such that
$\Theta^{(2)}_{i\overline{j}}=\mu h_{i\overline{j}}$ (7.18)
The Hermitian metric $h_{\infty}$ is Hermitian-Einstein. So, by [34], the
holomorphic tangent bundle $T^{1,0}M$ is stable. As shown in Example 6.1, the
Hopf manifold ${\mathbb{S}}^{2n+1}\times{\mathbb{S}}^{1}$ is stable for any
$n\geq 1$. In fact, in the definition of $\Theta^{(2)}_{i\overline{j}}$, if we
take trace by using the initial metric $h_{0}$, then we get the original
Hermitian-Yang-Mills flow equation.
2. 2.
If the initial metric is Kähler, then this flow is reduced to the usual
Kähler-Ricci flow([6]).
3. 3.
The flow 7.11 is similar to the harmonic map flow equation as shown in Theorem
7.1. It is strictly parabolic, and so the long time existence depends on
certain curvature condition of the target manifold as discussed in the
pioneering work of Eells-Sampson in [11]. The long time existence of this flow
and other geometric properties of our new flow will be studied in our
subsequent work.
Certain geometric flows and related results have been considered on Hermitian
manifolds recently, we refer the reader to [43], [44], [45] and [22].
## 8 Appendix: The proof of the refined Bochner formulas
###### Lemma 8.1.
On a compact Hermitian manifold $(M,h,\omega)$, we have
$[\Lambda,2\partial\omega]=A+B+C$ (8.1)
where
$\begin{cases}A=-h^{k\overline{\ell}}h_{i\overline{m}}\Gamma_{s\overline{\ell}}^{\overline{m}}dz^{s}\wedge
dz^{i}I_{k}\\\
\overline{A}^{*}=-h^{s\overline{t}}\Gamma_{s\overline{k}}^{\overline{i}}d\overline{z}^{k}I_{\overline{i}}I_{\overline{t}}\end{cases}$
(8.2)
$\begin{cases}B=-2\Gamma_{i\overline{j}}^{\overline{\ell}}dz^{i}\wedge
d\overline{z}^{j}I_{\overline{\ell}}\\\
\overline{B}^{*}=2h^{p\overline{j}}\Gamma_{\ell\overline{j}}^{\overline{s}}dz^{\ell}I_{p}I_{\overline{s}}\end{cases}$
(8.3)
$\begin{cases}C=\Lambda(2\partial\omega)=2\Gamma_{j\overline{\ell}}^{\overline{\ell}}dz^{j}\\\
\overline{C}^{*}=2h^{j\overline{\ell}}\Gamma_{j\overline{s}}^{\overline{s}}I_{\overline{\ell}}=-2h^{j\overline{i}}\Gamma_{j\overline{i}}^{\overline{\ell}}I_{\overline{\ell}}\end{cases}$
(8.4)
Moreover,
(1)
$[\Lambda,A]=-\sqrt{-1}\overline{B}^{*}$;
(2)
$[\Lambda,B]=-\sqrt{-1}(2\overline{A}^{*}+\overline{B}^{*}+\overline{C}^{*})$;
(3)
$[\Lambda,C]=-\sqrt{-1}\overline{C}^{*}$.
###### Proof.
All formulas follow by direct computation. ∎
###### Definition 8.2.
With respect to $\nabla^{\prime}$ and $\nabla^{\prime\prime}$, we define
$\begin{cases}D^{\prime}:=dz^{i}\wedge\nabla^{\prime}_{i}\\\
D^{\prime\prime}:=d\overline{z}^{j}\wedge\nabla^{\prime\prime}_{\overline{j}}\end{cases}$
(8.5)
The dual operators of
$\partial,\overline{\partial},D^{\prime},D^{\prime\prime}$ with respect to the
norm in 4.13 are denoted by
$\partial^{*},\overline{\partial}^{*},\delta^{\prime},\delta^{\prime\prime}$
and define
$\begin{cases}\delta_{0}^{\prime}:=-h^{i\overline{j}}I_{i}\nabla^{\prime\prime}_{\overline{j}}\\\
\delta_{0}^{\prime\prime}:=-h^{j\overline{i}}I_{\overline{i}}\nabla^{\prime}_{j}\end{cases}$
(8.6)
where $I$ the contraction operator and $I_{i}=I_{\frac{\partial}{\partial
z^{i}}}$ and $I_{\overline{i}}=I_{\frac{\partial}{\partial\overline{z}^{i}}}$.
###### Remark 8.3.
It is obvious that these first order differential operators
$D^{\prime},D^{\prime\prime},\delta_{0}^{\prime}$ and
$\delta_{0}^{\prime\prime}$ are well-defined and they don’t depend on the
choices of holomorphic frames. If $(M,h)$ is Kähler, $D^{\prime}=\partial$,
$D^{\prime\prime}=\overline{\partial}$,
$\delta_{0}^{\prime}=\delta^{\prime}=\partial^{*}$ and
$\delta_{0}^{\prime\prime}=\delta^{\prime\prime}=\overline{\partial}^{*}$.
###### Lemma 8.4.
In the local holomorphic coordinates,
$\partial=D^{\prime}-\frac{B}{2}\quad\mbox{and}\quad\overline{\partial}=D^{\prime\prime}-\frac{\overline{B}}{2}$
(8.7)
###### Proof.
We only have to check them on functions and $1$-forms. ∎
###### Lemma 8.5.
On a compact Hermitian manifold $(M,h)$, we have
$\begin{cases}\delta^{\prime\prime}=\delta_{0}^{\prime\prime}-\frac{\overline{C}^{*}}{2}\\\
\delta^{\prime}=\delta_{0}^{\prime}-\frac{C^{*}}{2}\end{cases}$ (8.8)
For $\partial$ and $\overline{\partial}$, we have
$\begin{cases}\partial^{*}=\delta_{0}^{\prime}-\frac{B^{*}+C^{*}}{2}\\\
\overline{\partial}^{*}=\delta^{\prime\prime}_{0}-\frac{\overline{B}^{*}+\overline{C}^{*}}{2}\end{cases}$
(8.9)
###### Proof.
For any $\varphi\in\Omega^{p,q-1}(M)$ and $\psi\in\Omega^{p,q}(M)$, by stokes’
theorem
$\displaystyle 0$ $\displaystyle=$
$\displaystyle\int_{M}\overline{\partial}(\varphi\wedge*\overline{\psi})$
$\displaystyle=$
$\displaystyle\int_{M}\frac{\partial}{\partial\overline{z}^{j}}\left(d\overline{z}^{j}\wedge\varphi\wedge*\overline{\psi}\right)$
$\displaystyle=$
$\displaystyle\int_{M}\frac{\partial}{\partial\overline{z}^{j}}\left(\langle
d\overline{z}^{j}\wedge\varphi,\psi\rangle\frac{\omega^{n}}{n!}\right)$
$\displaystyle=$
$\displaystyle\int_{M}\frac{\partial}{\partial\overline{z}^{j}}\left(\left\langle\varphi,h^{j\overline{i}}I_{\overline{i}}\psi\right\rangle\frac{\omega^{n}}{n!}\right)$
$\displaystyle=$
$\displaystyle\int_{M}\left(\left\langle\nabla^{\prime\prime}_{\overline{j}}\varphi,h^{j\overline{i}}I_{\overline{i}}\psi\right\rangle+\left\langle\varphi,\nabla^{\prime}_{j}h^{j\overline{i}}I_{\overline{i}}\psi\right\rangle+\left\langle\varphi,h^{j\overline{i}}I_{\overline{i}}\psi\right\rangle\frac{\partial\log\det(h_{m\overline{n}})}{\partial\overline{z}^{j}}\right)\frac{\omega^{n}}{n!}$
$\displaystyle=$ $\displaystyle\int_{M}\left(\left\langle
d\overline{z}^{j}\wedge\nabla^{\prime\prime}_{\overline{j}}\varphi,\psi\right\rangle+\left\langle\varphi,h^{j\overline{i}}\nabla^{\prime}_{j}I_{\overline{i}}\psi\right\rangle+\left\langle\varphi,\frac{\partial
h^{j\overline{i}}}{\partial
z^{j}}I_{\overline{i}}\psi\right\rangle+\left\langle\varphi,h^{j\overline{i}}I_{\overline{i}}\psi\right\rangle\frac{\partial\log\det(h_{m\overline{n}})}{\partial\overline{z}^{j}}\right)\frac{\omega^{n}}{n!}$
That is
$(D^{\prime\prime}\varphi,\psi)=\left(d\overline{z}^{j}\wedge\nabla^{\prime\prime}_{\overline{j}}\varphi,\psi\right)=-\left(\varphi,h^{j\overline{i}}\nabla^{\prime}_{j}I_{\overline{i}}\psi\right)-\left(\varphi,\left(\frac{\partial
h^{j\overline{i}}}{\partial
z^{j}}+h^{j\overline{i}}\frac{\partial\log\det(h_{m\overline{n}})}{\partial
z^{j}}\right)I_{\overline{i}}\psi\right)$ (8.10)
Now we will compute the second and third terms on the right hand side.
$\frac{\partial h^{j\overline{i}}}{\partial
z^{j}}+h^{j\overline{i}}\frac{\partial\log\det(h_{m\overline{n}})}{\partial
z^{j}}=h^{j\overline{i}}h^{s\overline{t}}\left(\frac{\partial
h_{s\overline{t}}}{\partial z^{j}}-\frac{\partial h_{j\overline{t}}}{\partial
z^{s}}\right)=2h^{j\overline{i}}\Gamma_{j\overline{t}}^{\overline{t}}=-2h^{j\overline{\ell}}\Gamma_{j\overline{\ell}}^{\overline{i}}$
(8.11)
On the other hand
$\displaystyle-h^{j\overline{i}}\nabla^{\prime}_{j}I_{\overline{i}}$
$\displaystyle=$
$\displaystyle-h^{j\overline{i}}I_{\overline{i}}\nabla^{\prime}_{j}-h^{j\overline{i}}I\left(\nabla^{\prime}_{j}\frac{\partial}{\partial\overline{z}^{i}}\right)$
(8.12) $\displaystyle=$
$\displaystyle\delta^{\prime\prime}_{0}-h^{j\overline{i}}\Gamma_{j\overline{i}}^{\overline{\ell}}I_{\overline{\ell}}$
In summary, by 8.10, 8.11 and 8.12, the adjoint operator
$\delta^{\prime\prime}$ of $D^{\prime\prime}$ is
$\delta^{\prime\prime}=\left(\delta^{\prime\prime}_{0}-h^{j\overline{i}}\Gamma_{j\overline{i}}^{\overline{\ell}}I_{\overline{\ell}}\right)+2h^{j\overline{i}}\Gamma_{j\overline{i}}^{\overline{\ell}}I_{\overline{\ell}}=\delta_{0}^{\prime\prime}-\frac{\overline{C}^{*}}{2}$
Since $\overline{\partial}=D^{\prime\prime}-\frac{\overline{B}}{2}$, we get
$\overline{\partial}^{*}=\delta^{\prime\prime}-\frac{\overline{B}^{*}}{2}=\delta_{0}^{\prime\prime}-\frac{\overline{B}^{*}+\overline{C}^{*}}{2}$
∎
###### Lemma 8.6.
On a compact Hermitian manifold $(M,h)$, we have
$\begin{cases}\left[\Lambda,D^{\prime}\right]=\sqrt{-1}\left(\delta^{\prime\prime}+\frac{\overline{C}^{*}}{2}\right)\\\
\left[\Lambda,D^{\prime\prime}\right]=-\sqrt{-1}(\delta^{\prime}+\frac{C^{*}}{2})\end{cases}\quad\mbox{and}\quad\begin{cases}[\delta^{\prime\prime},L]=\sqrt{-1}(D^{\prime}+\frac{C}{2})\\\
[\delta^{\prime},L]=-\sqrt{-1}(D^{\prime\prime}+\frac{\overline{C}}{2})\end{cases}$
(8.13)
###### Proof.
By definition
$\displaystyle(\Lambda D^{\prime})\varphi$ $\displaystyle=$
$\displaystyle\left(\sqrt{-1}h^{i\overline{j}}I_{i}I_{\overline{j}}\right)(dz^{k}\wedge\nabla^{\prime}_{k}\varphi)$
$\displaystyle=$
$\displaystyle-\sqrt{-1}h^{i\overline{j}}I_{i}\left(dz^{k}\wedge
I_{\overline{j}}\nabla^{\prime}_{k}\varphi\right)$ $\displaystyle=$
$\displaystyle-\sqrt{-1}h^{i\overline{j}}I_{\overline{j}}\nabla^{\prime}_{i}\varphi+\sqrt{-1}h^{i\overline{j}}dz^{k}I_{i}I_{\overline{j}}\nabla^{\prime}_{k}\varphi$
$\displaystyle=$
$\displaystyle\sqrt{-1}\delta_{0}^{\prime\prime}+dz^{k}\wedge\nabla^{\prime}_{k}\left(\sqrt{-1}h^{i\overline{j}}I_{i}I_{\overline{j}}\varphi\right)$
$\displaystyle=$
$\displaystyle\sqrt{-1}\delta_{0}^{\prime\prime}+D^{\prime}\Lambda\varphi$
where we use the metric compatible condition
$\nabla^{\prime}\omega=0\Longrightarrow\nabla_{k}^{\prime}(\Lambda\varphi)=\Lambda(\nabla_{k}^{\prime}\varphi)$
(8.14)
∎
###### Lemma 8.7.
On a compact Hermitian manifold $(M,h)$, we have
$\begin{cases}\left[\Lambda,\partial\right]=\sqrt{-1}\left(\overline{\partial}^{*}+\overline{\tau}^{*}\right)\\\
\left[\Lambda,\overline{\partial}\right]=-\sqrt{-1}(\partial^{*}+\tau^{*})\end{cases}$
(8.15)
For the dual case, it is
$\begin{cases}[\overline{\partial}^{*},L]=\sqrt{-1}(\partial+\tau)\\\
[\partial^{*},L]=-\sqrt{-1}(\overline{\partial}+\overline{\tau})\end{cases}$
(8.16)
###### Proof.
By Lemma 8.6, 8.4 and 8.1,
$\displaystyle[\Lambda,\partial]$ $\displaystyle=$
$\displaystyle[\Lambda,D^{\prime}]-\left[\Lambda,\frac{B}{2}\right]$
$\displaystyle=$
$\displaystyle\sqrt{-1}\left(\delta_{0}^{\prime\prime}+\frac{2\overline{A}^{*}+\overline{B}^{*}+\overline{C}^{*}}{2}\right)$
$\displaystyle=$
$\displaystyle\sqrt{-1}\left(\delta^{\prime\prime}+\frac{\overline{C}^{*}}{2}+\frac{2\overline{A}^{*}+\overline{B}^{*}+\overline{C}^{*}}{2}\right)$
$\displaystyle=$
$\displaystyle\sqrt{-1}(\overline{\partial}^{*}+\overline{\tau}^{*})$
The other relations follow by complex conjugate and adjoint operations. ∎
###### Lemma 8.8.
On an Hermitian manifold $(M,h,\omega)$,
$\overline{\partial}^{*}\omega=\sqrt{-1}\Lambda(\partial\omega)=\sqrt{-1}\Gamma_{\ell\overline{j}}^{\overline{j}}dz^{\ell}$
(8.17)
###### Proof.
We have
$\frac{C}{2}=\Lambda(\partial\omega)=\Gamma_{j\overline{\ell}}^{\overline{\ell}}dz^{j}$
On the other hand, by Lemma 8.5 and $\delta_{0}^{\prime\prime}\omega=0$
$\displaystyle\overline{\partial}^{*}\omega$ $\displaystyle=$
$\displaystyle\left(\delta^{\prime\prime}_{0}-\frac{\overline{B}^{*}+\overline{C}^{*}}{2}\right)\omega=-\frac{\overline{B}^{*}\omega}{2}-\frac{\overline{C}^{*}}{2}\omega$
$\displaystyle=$
$\displaystyle\left(h_{\ell\overline{k}}h^{p\overline{j}}h^{i\overline{s}}\Gamma_{i\overline{j}}^{\overline{k}}dz^{\ell}I_{p}I_{\overline{s}}\right)\left(\frac{\sqrt{-1}}{2}h_{m\overline{n}}dz^{m}\wedge
d\overline{z}^{n}\right)-\frac{\overline{C}^{*}}{2}\omega$ $\displaystyle=$
$\displaystyle-\frac{\sqrt{-1}}{2}h_{\ell\overline{k}}h^{i\overline{j}}\Gamma_{i\overline{j}}^{\overline{k}}dz^{\ell}-\frac{\overline{C}^{*}}{2}\omega$
$\displaystyle=$
$\displaystyle\frac{\sqrt{-1}}{2}\Gamma_{\ell\overline{j}}^{\overline{j}}dz^{\ell}-\frac{\overline{C}^{*}}{2}\omega$
$\displaystyle=$
$\displaystyle\sqrt{-1}\Gamma_{\ell\overline{j}}^{\overline{j}}dz^{\ell}$
$\displaystyle=$ $\displaystyle\sqrt{-1}\Lambda(\partial\omega)$
∎
Now we assume $E$ is an Hermitian complex vector bundle or a Riemannian vector
bundle over a compact Hermitian manifold $(M,h,\omega)$ and $\nabla^{E}$ is a
metric connection on $E$.
###### Lemma 8.9.
We have the following formula:
$\overline{\partial}_{E}^{*}(\varphi\otimes
s)=(\overline{\partial}^{*}\varphi)\otimes
s-h^{i\overline{j}}\left(I_{\overline{j}}\varphi\right)\wedge\nabla_{i}^{E}s$
(8.18)
for any $\varphi\in\Omega^{p,q}(M)$ and $s\in\Gamma(M,E)$.
###### Proof.
The proof of is the same as Lemma 8.5. ∎
###### Lemma 8.10.
If $\tau$ is the operator of type $(1,0)$ defined by
$\tau=[\Lambda,2\partial\omega]$ on $\Omega^{\bullet}(M,E)$, then
(1)
$[\overline{\partial}_{E}^{*},L]=\sqrt{-1}(\partial_{E}+\tau)$;
(2)
$[\partial^{*}_{E},L]=-\sqrt{-1}(\overline{\partial}_{E}+\overline{\tau})$;
(3)
$[\Lambda,\partial_{E}]=\sqrt{-1}(\overline{\partial}_{E}^{*}+\overline{\tau}^{*})$
;
(4)
$[\Lambda,\overline{\partial}_{E}]=-\sqrt{-1}(\partial_{E}^{*}+\tau^{*})$.
###### Proof.
We only have to prove (3). For any $\varphi\in\Omega^{\bullet}(M)$ and
$s\in\Gamma(M,E)$,
$\displaystyle(\Lambda\partial_{E})(\varphi\otimes s)$ $\displaystyle=$
$\displaystyle\Lambda\left(\partial\varphi\otimes
s+(-1)^{|\varphi|}\varphi\wedge\partial_{E}s\right)$ $\displaystyle=$
$\displaystyle(\Lambda\partial\varphi)\otimes
s+(-1)^{|\varphi|}\sqrt{-1}h^{k\overline{\ell}}I_{k}I_{\overline{\ell}}\left(\varphi\wedge\partial_{E}s\right)$
$\displaystyle=$ $\displaystyle(\Lambda\partial\varphi)\otimes
s+(-1)^{|\varphi|}\sqrt{-1}h^{k\overline{\ell}}I_{k}\left(\left(I_{\overline{\ell}}\varphi\right)\wedge\partial_{E}s\right)$
$\displaystyle=$ $\displaystyle(\Lambda\partial\varphi)\otimes
s+(-1)^{|\varphi|}\sqrt{-1}h^{k\overline{\ell}}\left(I_{k}\left(I_{\overline{\ell}}\varphi\right)\right)\wedge\partial_{E}s-\sqrt{-1}h^{k\overline{\ell}}I_{\overline{\ell}}(\varphi)\wedge
I_{k}\partial_{E}s$ $\displaystyle=$
$\displaystyle(\Lambda\partial\varphi)\otimes
s+(-1)^{|\varphi|}(\Lambda\varphi)\wedge\partial_{E}s-\sqrt{-1}h^{k\overline{\ell}}I_{\overline{\ell}}(\varphi)\wedge\nabla_{k}^{E}s$
On the other hand
$\displaystyle(\partial_{E}\Lambda)(\varphi\otimes s)$ $\displaystyle=$
$\displaystyle\partial_{E}\left((\Lambda\varphi)\otimes s\right)$
$\displaystyle=$ $\displaystyle(\partial\Lambda\varphi)\otimes
s+(-1)^{|\varphi|}(\Lambda\varphi)\wedge\partial_{E}s$
Therefore
$\displaystyle[\Lambda,\partial_{E}](\varphi\otimes s)$ $\displaystyle=$
$\displaystyle\left([\Lambda,\partial]\varphi\right)\otimes
s-\sqrt{-1}h^{k\overline{\ell}}I_{\overline{\ell}}(\varphi)\wedge\nabla_{k}^{E}s$
$\displaystyle=$
$\displaystyle\sqrt{-1}\left(\left(\overline{\partial}^{*}+\overline{\tau}^{*}\right)\varphi\right)\otimes
s-\sqrt{-1}h^{k\overline{\ell}}I_{\overline{\ell}}(\varphi)\wedge\nabla_{k}^{E}s$
$\displaystyle=$
$\displaystyle\sqrt{-1}\left(\overline{\partial}_{E}^{*}+\overline{\tau}^{*}\right)(\varphi\otimes
s)$
where the last step follows by 8.18.∎
## References
* [1] Alexandrov, B.; Ivanov, S. Vanishing theorems on Hermitian manifolds. Differential Geom. Appl. 14 (2001), no. 3, 251-265.
* [2] Alessandrini, L.; Bassaneli, G. Metric properties of manifolds bimeromorphic to compact Kaehler manifolds, J. Diff. Geometry 37 (1993), 95-121.
* [3] Alessandrini, L.; Bassanelli, G. A class of balanced manifolds. Proc. Japan Acad. Ser. A Math. Sci. 80 (2004), no. 1, 6-7.
* [4] Bismut, J.-M. A local index theorem for non Kähler manifolds. Math. Ann. 284 (1989), no. 4, 681-699.
* [5] Bochner, S. Vector fields and Ricci curvature, Bull. Amer. Math. Soc. 52, 776-797 (1946).
* [6] Cao, H.-D. Deformation of Kähler metrics to Kähler-Einstein metrics on compact Kähler manifolds. Invent. Math. 81 (1985), no. 2, 359-372.
* [7] Calabi, E.; Eckmann, B. A class of compact, complex manifolds which are not algebraic. Ann. of Math. (2) 58, (1953) 494-500.
* [8] Demailly, J-P. _Complex analytic and algebraic geometry_. book online http://www-fourier.ujf-grenoble.fr/ demailly/books.html.
* [9] Demailly, J-P. Une preuve simple de la conjecture de Grauert-Riemenschneider. Lecture Notes in Math., 1295, 24-47, Springer, Berlin, 1987.
* [10] Demailly, J.; Paun, M. Numerical characterization of the Kähler cone of a compact Kähler manifold. Ann. of Math. (2) 159 (2004), no. 3, 1247-1274.
* [11] Eells, J.; Sampson, J. Harmonic mappings of Riemannian manifolds. Amer. J. Math. 86(1964) 109-160.
* [12] Enrietti, N. Static SKT metrics on Lie groups. arXiv:1009.0620
* [13] Fino, A.; Grantcharov, G. Properties of manifolds with skew-symmetric torsion and special holonomy. Adv. Math. 189 (2004), no. 2, 439-450.
* [14] Fino, A.; Tomassini, A. Blow-ups and resolutions of strong Kähler with torsion metrics. Adv. Math. 221 (2009), no. 3, 914–935
* [15] Fino, A.; Parton, M.; Salamon, Simon Families of strong KT structures in six dimensions. Comment. Math. Helv. 79 (2004), no. 2, 317-340.
* [16] Ganchev, G.; Ivanov, S. Holomorphic and Killing vector fields on compact balanced Hermitian manifolds. Internat. J. Math. 11 (2000), no. 1, 15-28.
* [17] Ganchev, G.; Ivanov, S. Harmonic and holomorphic 1-forms on compact balanced Hermitian manifolds. Differential Geom. Appl. 14 (2001), no. 1, 79-93.
* [18] Gauduchon, P. Le theoreme de l’excentricité nulle, C. R. Acad. Sci. Paris Ser. A, 285 (1977), 387-390.
* [19] Gauduchon, P. Fibrés hermitiens à endomorphisme de Ricci non-négatif, Bull. Soc. Math. France 105 (1977), 113-140.
* [20] Gauduchon, P. Hermitian connections and Dirac operators, Bol. U. M. I. ser. VII, vol. XI-B, supl. 2 (1997), 257-289.
* [21] Gauduchon, P. La $1$-forme de torsion d’une variété hermitienne compacte. Math. Ann. 267 (1984), no. 4, 495-518.
* [22] Gill, M. Convergence of the parabolic complex Monge-Amp re equation on compact Hermitian manifolds. arXiv:1009.5756.
* [23] Gray, A. Some examples of almost Hermitian manifolds. Illinois J. Math. 10 1966, 353-366.
* [24] Hamilton, R. Three-manifolds with positive Ricci curvature. J. Differential Geom. 17 (1982), no. 2, 255-306.
* [25] Ivanov, S.; Papadopoulos, G. Vanishing theorems and string backgrounds. Classical Quantum Gravity 18 (2001), no. 6, 1089-1110.
* [26] Ji, S. Currents, metrics and Moishezon manifolds. Pacific J. Math. 158 (1993), no. 2, 335-351.
* [27] Ji, S.; Shiffman, B. Properties of compact complex manifolds carrying closed positive currents. J. Geom. Anal. 3 (1993), no. 1, 37-61.
* [28] Jost, J.; Yau, S.-T. A nonlinear elliptic system for maps from Hermitian to Riemannian manifolds and rigidity theorems in Hermitian geometry. Acta Math. 170 (1993), no. 2, 221-254.
* [29] Kobayashi, S. Differential Geometry of complex vector bundles, Iwanami Shoten, Publishers and Princeton Univ. Press. $1987$.
* [30] Kobayashi, S.; Nomizu, K. _Foundations of differential geometry. Vol. II._ Interscience Publishers John Wiley-Sons, Inc., New York-London-Sydney 1969\.
* [31] Kobayashy, S.; Wu, H. On holomorphic sections of certain Hermitian vector bundles, Math. Ann. 189 $(1970)$, 1-4.
* [32] Li, J.; Yau, S.-T. Hermitian-Yang-Mills connection on non-Kähler manifolds. _Mathematical aspects of string theory_ (San Diego, Calif., 1986), 560 C573, Adv. Ser. Math. Phys., 1, World Sci. Publishing, Singapore, 1987\.
* [33] Li,Y.; Liu, K.; Yang, X. Hermitian geometric flow I.
* [34] Liu, K.; Yang, X. Harmonic maps between compact Hermitian manifolds. Sci. China Ser. A 51 (2008), no. 12, 2149-2160.
* [35] Ma, X.; Marinescu, G. _Holomorphic Morse inequalities and Bergman kernels_. Progress in Mathematics, 254. Birkhäuser Verlag, Basel, 2007
* [36] Michelson, M.L. On the existence of special metrics in complex geometry, Acta Math. 143 (1983), 261-295.
* [37] Petersen, P. Riemannian geometry. Second edition. Graduate Texts in Mathematics, 171. Springer, New York, 2006.
* [38] Shiffman, B.; Sommese, A.J. _Vanishing theorems on complex manifolds_. Progress in Mathematics, 56. Birkhäuser 1985.
* [39] Siu, Y. A vanishing theorem for semipositive line bundles over non-Kähler manifolds. J. Differential Geom. 19 (1984), no. 2, 431-452.
* [40] Siu, Y-T.; Yau, S-T. Compact Kähler manifolds of positive bisectional curvature. Invent. Math. 59 (1980), no. 2, 189-204.
* [41] Tosatti, V.; Weinkove, B. The complex Monge-Amp re equation on compact Hermitian manifolds. J. Amer. Math. Soc. 23 (2010), no.4, 1187-1195
* [42] Tosatti, V.; Weinkove, B. Estimates for the complex Monge-Amp re equation on Hermitian and balanced manifolds. Asian J. Math. 14 (2010), no.1, 19-40.
* [43] Streets, J., Tian, G., Hermitian curvature flow, arXiv: 0804.4109.
* [44] Streets, J., Tian, G., A parabolic flow of pluriclosed metrics, arXiv: 0903.4418.
* [45] Streets, J., Tian, G., Regularity results for pluriclosed flow, 1008.2794.
* [46] Urakawa, H. Complex Laplacians on compact complex homogeneous spaces. J. Math. Soc. Japan 33 (1981), no. 4, 619-638
* [47] Wu, H. Bochner technique in Differential Geometry, Math. Reports, vol.$3$, part $2$, $1988$.
* [48] Yang, H-C. Complex parallelisable manifold, Proc. Amer. Math. Soc., 51(1954), 771-776.
* [49] Yano, K.; Bochner, S. Curvature and Betti numbers. Annals of Mathematics Studies, No. 32. Princeton University Press, Princeton, N. J., 1953
* [50] Yau, S.-T. On the Ricci curvature of a compact Kähler manifold and the complex Monge-Amp re equation. I. Comm. Pure Appl. Math. 31 (1978), no. 3, 339-411.
* [51] Zheng, F. _Complex differential geometry_. AMS/IP Studies in Advanced Mathematics, 18. American Mathematical Society, Providence, RI; International Press, Boston, MA, 2000 Department of Mathematics, University of California at Los Angeles,
Los Angeles, CA, 90095-1555
_E-mail Address_ : liu@math.ucla.edu; xkyang@math.ucla.edu
|
arxiv-papers
| 2010-10-31T20:52:39 |
2024-09-04T02:49:14.369782
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Kefeng Liu, Xiaokui Yang",
"submitter": "Xiaokui Yang",
"url": "https://arxiv.org/abs/1011.0207"
}
|
1011.0257
|
# Finite temperature QCD at fixed Q with overlap fermions
JLQCD Collaboration: a, Sinya Aokib, Shoji Hashimotoa,c, Takashi Kanekoa,c,
Hideo Matsufurua, Jun-ichi Noakia, Eigo Shintanid
aTheory Center, IPNS, High Energy Accelerator Research Organization (KEK),
Tsukuba 305-0801, Japan
bGraduate School of Pure and Applied Sciences, University of Tsukuba, Tsukuba
305-8571, Japan
cSchool of High Energy Accelerator Science, The Graduate University for
Advanced Studies (Sokendai), Tsukuba 305-0801, Japan
dRIKEN-BNL Research Center, Upton, NY 11973-5000, USA E-mail:
cossu@post.kek.jp
###### Abstract:
We present some preliminary results of the project on finite temperature QCD
with overlap fermions at KEK. We performed a series of simulations to assess
the effects of fixing the topological sector at finite temperature and we will
show the first calculations of topological susceptibility and meson masses for
quenched and full QCD.
## 1 Introduction and motivation
Among several features of QCD the chiral symmetry breaking is one of the most
interesting ones. At low temperature this symmetry is spontaneously broken and
the vacuum develops a quark anti-quark condensate, $\langle\bar{q}q\rangle\neq
0$. Massless Nambu-Goldstone (NG) bosons should appear in the spectrum of the
massless theory. In real QCD, of course, the light quark masses explicitly
break the chiral symmetry, giving a small mass to the NG bosons that have been
identified as the 8 lightest mesons (pions, kaons, $\eta$). Classically, the
pattern of chiral symmetry breaking is the following ($N_{f}$ being the number
of light quarks)111$U(1)_{V}$ gives the conserved baryon number and
$SU(N_{f})_{V}$ is only softly broken by the small quark mass difference.:
$SU(N_{f})_{V}\times SU(N_{f})_{A}\times U(1)_{V}\times U(1)_{A}\rightarrow
SU(N_{f})_{V}\times U(1)_{V}$ (1)
that should actually give 9 NG bosons, but the ninth particle is absent in the
spectrum. This apparent problem was solved noticing that the flavor-singlet
axial $U(1)_{A}$ rotation is no more a symmetry at the quantum level (even in
the massless limit); it is anomalous [1] due to the presence of instanton-like
configurations. A direct effect of the anomaly is the large splitting in the
mass of flavor-singlet and non-singlet pseudoscalar mesons (see Witten-
Veneziano formula [2, 3]).
While at zero temperature the physics is quite clear, at finite temperature
still there is no definite answer to the question if axial $U(1)_{A}$ symmetry
is restored or not. If it is restored, an interesting problem is to establish
if this happens at the same critical temperature of chiral symmetry
restoration. This would have relevant effects on the pattern of symmetry
breaking and so on the critical exponents of the phase transition [4].
By semiclassical calculations of dilute instanton gas at very high temperature
$T\gg T_{c}$, we expect a strong suppression but not an exact restoration of
the $U(1)_{A}$ symmetry. The most advanced lattice result in this context is
the one by Vranas [5], where, using domain wall fermions, he concluded that
just above chiral phase transition the $U(1)_{A}$ symmetry remains broken but
by a very small amount. It is an open question what is the effect that may
have to the order of the transition.
Our aim is to study the fate of chiral and $U(1)_{A}$ symmetry (and the mass
of particles) at finite temperature around and above the phase transition
using the fermionic action that retains the maximal amount of chirality on the
lattice, i.e. the overlap formulation [6]. The JLQCD and TWQCD collaborations
have performed large scale QCD simulations using the overlap action [7]. All
simulations were done at zero temperature and we investigated the chiral
behavior of spectra, low energy constants, chiral condensate and topological
susceptibility [8, 9, 10, 11]. A non-zero topological susceptibility,
indicating anomalous breaking of $U(1)_{A}$ symmetry, was clearly observed in
those simulations at $T=0$ and also the linear dependence with sea quark mass
was obtained, as predicted by chiral perturbation theory.
In order to simulate QCD using HMC with dynamical overlap fermions, fixing the
topological sector was crucial because allowing for topology change would be
extremely expensive [12]. To run a simulation at fixed topological charge $Q$
it was introduced an irrelevant term in the action [12], that suppresses the
occurrence of zero eigenvalues of the hermitian Wilson-Dirac operator that
have to be crossed in order to change topological sector. Fixing topology, of
course, creates a bias in physical results that must be corrected. A full
theory describing the effects of working at fixed $Q$ was developed [13]: the
effects at zero temperature are understood, under control and $O(1/V)$. We
will discuss with more details the subject in the following section.
In order to obtain reliable results at finite temperature we need to check
whether the same methods used at zero temperature to correct for fixed
topology effects work even in this case. So, we started with some exploratory
studies using quenched theory but fixing topology in order to compare with
previous results in the literature. We measured the topological susceptibility
and several meson correlators at finite temperature. We will report the
results of these simulations and the preliminary results in full QCD to
investigate $U(1)_{A}$ restoration.
## 2 Simulations and results
Before discussing the results, let us briefly describe the methods used to
measure correlators and the topological susceptibility at fixed topology.
Detailed derivation of equations can be found in [13].
By using a saddle point expansion of the QCD partition function in a finite
volume we can derive an expression for $Z_{Q}$, partition function at fixed
topology ($V$ is the 4-volume):
$Z_{Q}=\frac{1}{\sqrt{2\pi\chi_{t}V}}\exp\Bigl{[}-\frac{Q^{2}}{2\chi_{t}V}\Bigr{]}\Bigl{[}1-\frac{c_{4}}{8V\chi_{t}^{2}}+O\Bigl{(}\frac{1}{V^{2}}\Bigr{)}\Bigr{]},$
(2)
a gaussian distribution for topological charge that can be used to show that:
$\lim_{|x|\rightarrow\infty}\langle\rho(x)\rho(0)\rangle=\frac{1}{V}\Bigl{(}\frac{Q^{2}}{V}-\chi_{t}-\frac{c_{4}}{2\chi_{t}V}\Bigr{)}+O(V^{-3}).$
(3)
which implies that the topological susceptibility can be extracted from a long
range correlation of the topological charge density $\rho(x)$. At first order
in $1/V$ we can ignore the contribution of the $c_{4}$ term, and check later
the consistency of the assumption.
Using the overlap operator we can define an object that has the same
properties as the topological charge density:
$\rho_{m}(x)=m\,{\rm tr}[\gamma_{5}(D_{c}+m)_{x,x}^{-1}],$ (4)
where $(D_{c}+m)^{-1}$ is the valence quark propagator constructed using the
chirally symmetric overlap operator. An alternative way is to consider the
pseudoscalar isosinglet $\eta^{\prime}$ correlator, whose disconnected part is
equal to $\langle\rho_{m}(x)\rho_{m}(0)\rangle$ at large distances and couples
only to the fast decaying $\eta^{\prime}$ state, making it a better choice in
order to estimate the long distance limit.
We measured the connected and disconnected part of pseudoscalar correlators at
long distance to extract $\chi_{t}$ [8]. Since we are working at finite
temperature the correlators are measured and averaged over spatial directions.
We reconstructed the correlators by using the first 50 eigenvectors of the
overlap operator assuming low mode dominance. For example, the connected
scalar correlator is given by
$C(x,y)_{N}={\rm
Tr}\sum_{ij}^{N}\frac{\psi_{\lambda_{i}}(x)\psi^{\dagger}_{\lambda_{i}}(y)}{i\lambda_{i}+m}\frac{\psi_{\lambda_{j}}(y)\psi^{\dagger}_{\lambda_{j}}(x)}{i\lambda_{j}+m}$
(5)
and similar expression of the pseudoscalar (just
$i\lambda_{i}\rightarrow-i\lambda_{i}$). We checked that the saturation with
50 eigenmodes is sufficiently accurate for the infrared behavior.
### 2.1 Pure gauge simulations
By introducing the topology fixing term in pure gauge simulations we can check
if, even at finite temperature, we can reconstruct topological susceptibility
using the method described above by comparing with the literature.
The setup is the following: Iwasaki action + topology fixing term at
temperatures ranging from $[0.8,1.3]T_{c}$ on two different volumes
$16^{3}\times 6$ and $24^{3}\times 6$. The critical point was estimated to be
at $\beta_{c}=2.445$ by inspecting the Polyakov loop.
We first check whether the eigenvalue distribution behaves as expected. The
typical distribution is shown in Figure 1. We do not find any discrepancy with
previous results (e.g. [14]). The presence of a peak for small eigenvalues in
the high temperature side ($T>T_{c}$) was confirmed. In [14] these modes are
associated with the presence of dilute gas of instantons-anti instantons.
Figure 1: Spectral density of the overlap-Dirac operator at finite temperature
on the $24^{3}\times 6$ quenched lattice.
The most interesting result at this stage is the behavior of the topological
susceptibility at finite temperature in comparison with the results of
Gattringer et al. [15], shown in Figure 2. The asymptotic value for the
disconnected correlator (see equation (3) ) was estimated using a joint fit of
the connected and disconnected parts and assuming a double pole form for the
last one in the quenched theory. The decay is dominated by pionic states for
both of them. Then we use (3) to extract $\chi_{t}$ assuming that the $c_{4}$
term is negligible.
Figure 2: Comparison of topological susceptibility results with [15], lattice
$24^{3}\times 6$
In [15] the topological susceptibility was measured using the index theorem by
just counting the number of zero modes of an approximated version of the
overlap operator, which is a clean definition without the ambiguities due to
cooling. By also performing a simulation without the topology fixing term, we
checked their results also with our exact overlap operator, finding no
significant deviations within errors (cross symbol in Figure 2, temperature
$T/T_{c}=1.1$).
Below the transition temperature there is agreement between the two sets of
data, showing that the method works very well at least until $T_{c}$. Above
the critical temperature our results are systematically lower than the
reference ones. We are currently investigating the source of this discrepancy
which could be traced back in the assumptions leading to (2) or, the $c_{4}$
term could be non negligible in this regime. We are trying to estimate this
quantity but still we are getting too large error bars for the four-point
spatial correlators.
Another possibility to check the validity of (3) is to measure it on different
topological sectors. Simulations are on the way at the time of writing.
### 2.2 Full QCD simulations
In full QCD with two flavors of dynamical overlap fermions we concentrated on
the channels $\pi,\delta,\eta^{\prime},\sigma$ given by the correlators of the
operators
$\bar{\psi}\gamma_{5}\vec{\tau}\psi,\bar{\psi}\vec{\tau}\psi,\bar{\psi}\gamma_{5}\psi,\bar{\psi}\psi$,
respectively. Here, $\tau$ is the Pauli matrix in the flavor space.
If chiral symmetry is restored we expect that the pairs $(\sigma,\pi)$,
$(\eta^{\prime},\delta)$ become degenerate. Similarly, if the flavour-singlet
axial symmetry is restored too at high temperatures we would have that all the
channels become degenerate. The $\pi$ and the $\eta^{\prime}$ differ just by
the disconnected part, which is essentially given by the near-zero modes. This
observation implies that the $U(1)_{A}$ breaking is driven by near-zero modes.
By looking at the spatial correlators in those channels, we can check whether
it happens. The problem of establishing if the axial symmetry is restored at
the critical point is still open and has a relevance on the possible order of
the phase transition [4].
Some details on the simulation follow. The algorithm is HMC, using Iwasaki
action with topology fixing term and two flavors of sea quarks. The size of
the lattice is $16^{3}\times 8$. We choose $N_{t}=8$ to ensure that the
configurations are smooth enough. Masses start from $am=0.05$ down to
$am=0.01$ giving a pion mass of around 400 MeV at lowest temperature. The
$\beta$s were chosen to be in the temperature range $T=[171,243]$ MeV,
$T/T_{c}=[0.97,1.39]$ (assuming the critical temperature to be $T_{c}=175$
MeV). A comment is in order here: we couldn’t estimate directly the transition
temperature because it requires long history runs to measure susceptibilities.
Anyway we checked that, by looking at eigenvalues of the Dirac operator
(Figure 3), and the Polyakov loop, we simulated $\beta$s just below and above
the transition temperature. The topological sector is mainly $Q=0$ but we have
some simulations also at $Q=2$. All configurations were generated and analyzed
using the BlueGene/L installation at KEK.
Figure 3: Eigenvalues density in full QCD simulations, lattice $16^{3}\times
8$. Density around zero gives the chiral condensate by Banks-Casher relation.
Some preliminary results are shown in Figure 4. At this stage we cannot say
anything quantitative since the volume seems still too small to extract
masses. We can just check by inspection of the plots when the correlators
become equal. We observe that the correlators start to become almost
degenerate after the transition temperature and that as the sea quark mass is
decreased this degeneracy is improved. We are collecting more data points to
extrapolate this result toward the transition temperature.
Figure 4: Scalar and pseudoscalar spatial correlators at finite temperature.
The estimate for the $\sigma$, green area, correlator has still huge errors in
comparison to the others. We see some degeneracy for the correlators at
$\beta=2.30,T=208$ MeV, right panel.
## 3 Conclusions
We described our project on finite temperature QCD with overlap fermions.
Simulating overlap fermions forces to fix the topology in our case. A method
to extract physical results from fixed $Q$ simulations was previosly developed
for the zero temperature regime. To obtain reliable results we must check that
we can apply the same method at finite temperature or find the necessary
modifications. We started the investigation analyzing the behavior of
topological susceptibility in pure gauge theory at $Q=0$. We found that our
results differ at high temperature from the previous works. We are currently
investigating the source of this discrepancy. A deep understanding of this
problem is essential in the interpretation of full QCD results where we found
restoration of axial symmetry at least from temperatures above $1.1T_{c}$.
## Acknowledgements
Numerical simulations are performed on Hitachi SR11000 and IBM System Blue
Gene Solution at High Energy Accelerator Research Organization (KEK) under a
support of its Large Scale Simulation Program (No. 09/10-09). This work is
supported in part by the Grant-in-Aid of the Ministry of Education, Culture,
Sports, Science and Technology (No. 21674002, No. 20105005, No. 21684013, No.
220340047, No. 21105508) and by Grant-in-Aid for Scientific Research on
Innovative Areas (No. 20105001, No. 20105003).
## References
* [1] G. ’t Hooft, Phys. Rev. Lett. 37, 8-11 (1976).
* [2] E. Witten, Nucl. Phys. B156, 269 (1979).
* [3] G. Veneziano, Nucl. Phys. B159, 213-224 (1979).
* [4] A. Butti, A. Pelissetto, E. Vicari, JHEP 0308, 029 (2003). [hep-ph/0307036].
* [5] P. M. Vranas, Nucl. Phys. Proc. Suppl. 83, 414-416 (2000). [hep-lat/9911002].
* [6] M. Luscher, Phys. Lett. B428, 342-345 (1998).
* [7] T. Kaneko et al. [ JLQCD Collaboration ], PoS LAT2006, 054 (2006).
* [8] S. Aoki et al. [JLQCD and TWQCD Collaborations], Phys. Lett. B 665, 294 (2008) [arXiv:0710.1130 [hep-lat]].
* [9] J. Noaki et al. [JLQCD and TWQCD Collaborations], Phys. Rev. Lett. 101, 202004 (2008) [arXiv:0806.0894 [hep-lat]].
* [10] S. Aoki et al. [JLQCD Collaboration and TWQCD Collaboration], Phys. Rev. D 80, 034508 (2009) [arXiv:0905.2465 [hep-lat]].
* [11] H. Fukaya, S. Aoki, S. Hashimoto, T. Kaneko, J. Noaki, T. Onogi and N. Yamada [JLQCD collaboration], Phys. Rev. Lett. 104, 122002 (2010) [Erratum-ibid. 105, 159901 (2010)] [arXiv:0911.5555 [hep-lat]].
* [12] H. Fukaya et al. [ JLQCD Collaboration ], Phys. Rev. D74, 094505 (2006). [hep-lat/0607020].
* [13] S. Aoki, H. Fukaya, S. Hashimoto, T. Onogi, Phys. Rev. D76, 054508 (2007).
* [14] R. G. Edwards, U. M. Heller, J. E. Kiskis and R. Narayanan, Phys. Rev. D 61, 074504 (2000)
* [15] C. Gattringer, R. Hoffmann and S. Schaefer, Phys. Lett. B 535, 358 (2002)
|
arxiv-papers
| 2010-11-01T06:30:30 |
2024-09-04T02:49:14.384365
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "JLQCD Collaboration: Guido Cossu, Sinya Aoki, Shoji Hashimoto, Takashi\n Kaneko, Hideo Matsufuru, Jun-ichi Noaki, Eigo Shintani",
"submitter": "Guido Cossu",
"url": "https://arxiv.org/abs/1011.0257"
}
|
1011.0269
|
# Determination of the strong coupling $g_{B^{*}B\pi}$ from semi-leptonic
$B\to\pi\ell\nu$ decay
Xin-Qiang Li1,2, Fang Su3 and Ya-Dong Yang3,4
1Department of Physics, Henan Normal University, Xinxiang, Henan 453007, P. R.
China
2IFIC, Universitat de València-CSIC, Apt. Correus 22085, E-46071 València,
Spain
3Institute of Particle Physics, Huazhong Normal University, Wuhan, Hubei
430079, P. R. China
4Key Laboratory of Quark & Lepton Physics, Ministry of Education, Huazhong
Normal University
Wuhan, Hubei, 430079, P. R. China
###### Abstract
According to heavy-meson chiral perturbation theory, the vector form factor
$f_{+}(q^{2})$ of exclusive semi-leptonic decay $B\to\pi\ell\nu$ is closely
related, at least in the soft-pion region (i.e.,
$q^{2}\sim(m_{B}-m_{\pi})^{2}$), to the strong coupling $g_{B^{*}B\pi}$ or the
normalized coupling $\hat{g}$. Combining the precisely measured $q^{2}$
spectrum of $B\to\pi\ell\nu$ decay by the BaBar and Belle collaborations with
several parametrizations of the form factor $f_{+}(q^{2})$, we can extract
these couplings from the residue of the form factor at the $B^{*}$ pole, which
relies on an extrapolation of the form factor from the semi-leptonic region to
the unphysical point $q^{2}=m_{B^{*}}^{2}$. Comparing the extracted values
with the other experimental and theoretical estimates, we can test these
various form-factor parametrizations, which differ from each other by the
amount of physical information embedded in. It is found that the extracted
values based on the BK, BZ and BCL parametrizations are consistent with each
other and roughly in agreement with the other theoretical and lattice
estimates, while the BGL ansatz, featured by a spurious, unwanted pole at the
threshold of the cut, gives a neatly larger value.
## 1 Introduction
The most promising decay mode for a precise determination of the Cabibbo-
Kobayashi-Maskawa (CKM) [1] matrix element $|V_{ub}|$, both experimentally and
theoretically, is the exclusive semi-leptonic $B\to\pi\ell\nu$ decay [2], for
which a number of measurements have been made by various collaborations (CLEO
[3], BaBar [4, 5, 6] and Belle [7, 8]). A fit to the measured $q^{2}$
spectrum, on the other hand, allows for a precise extraction of the $q^{2}$
dependence of the vector form factor $f_{+}(q^{2})$, and thus provides a
stringent check on our understanding of the dynamics of hadrons governed by
QCD.
The heavy-to-light form factors are complicated nonperturbative objects, which
have attracted extensive investigations in the literature. Besides various
quark models (see, e.g., [9, 10]), which in many aspects help our
phenomenological understanding of the heavy-to-light transitions, there exist
two more quantitative predictions based on first principles of QCD, the
lattice QCD (LQCD) simulation (see, e.g., [11, 12, 13]) and the QCD sum rules
on the light-cone (LCSR) (see, e.g., [14, 15, 16, 17]). These two methods are
complementary to each other with respect to the momentum transfer $q^{2}$:
while the LQCD calculations are restricted to the high $q^{2}$ region,
reliable predictions of the LCSR method can only be made at the low $q^{2}$
region.
Due to our limited theoretical knowledge of the $q^{2}$ dependence of the
transition form factors, a variety of parametrizations have been proposed in
the literature, trying to capture as much information as possible on the
dynamics of the corresponding mesons. These include the two-parameter
Bećirević-Kaidalov (BK) ansatz [18], the three-parameter Ball-Zwicky (BZ)
ansatz [14, 19], the so-called Series Expansion (SE) ansatz [20, 21, 22, 23],
as well as the representation from the Omnes solution to the dispersive bounds
[24]. It turns out that most of them could fit the data equally well in the
semi-leptonic region [4, 7, 19]. A good review of these different
parametrizations could be found, for example, in Refs. [4, 19].
Most of the above parametrizations include the essential feature that the
vector form factor $f_{+}(q^{2})$ has a pole at $q^{2}=m_{B^{*}}^{2}$, where
$B^{*}(1^{-})$ is a narrow resonance with $m_{B^{*}}=5.325~{}{\rm
GeV}<m_{B}+m_{\pi}$. As the high-precision experimental data on
$B\to\pi\ell\nu$ decay is available only in the semi-leptonic region, $0\leq
q^{2}\leq(m_{B}-m_{\pi})^{2}$, in order to extract the pole residue we have to
extrapolate the form factor from this region to the unphysical point
$q^{2}=m_{B^{*}}^{2}$. Although lying outside the physical region, the pole
residue is of great phenomenological interest. It is related to the strong
coupling $g_{B^{*}B\pi}$, describing the low-energy interaction among the two
heavy B-mesons and a soft pion, or the normalized coupling $\hat{g}$, a
fundamental parameter in heavy-meson chiral perturbation theory (HMChPT) [25,
26]. Since the process $B^{*}\to B\pi$ is kinematically forbidden, the
coupling $g_{B^{*}B\pi}$ cannot be measured directly but should be fixed
phenomenologically. In this paper, exploiting the experimental knowledge on
the form factor $f_{+}(q^{2})$ extracted from the semi-leptonic
$B\to\pi\ell\nu$ decay, we determine the strong coupling $g_{B^{*}B\pi}$ and
$\hat{g}$ from the pole residue by extrapolating the form factor from the
physical region to the unphysical point $q^{2}=m_{B^{*}}^{2}$. By comparing
the extracted values with other theoretical and experimental estimates, we can
then test the various form-factor parametrizations.
Our paper is organized as follows. In Section 2, we provide the definition of
heavy-to-light form factors, their different parametrizations, and the pole
residue at $q^{2}=m_{B^{*}}^{2}$. In Section 3, after collecting the up-to-
date measured $B\to\pi$ form-factor shape parameters, we give our
determinations of the strong coupling $g_{B^{*}B\pi}$ and the corresponding
normalized coupling $\hat{g}$; some interesting phenomenological discussions
are also presented in this section. Our conclusions are made in Section 4.
## 2 Heavy-to-light form factor
### 2.1 Definition of the heavy-to-light form factor
In exclusive semi-leptonic $B\to\pi\ell\nu$ decay, the hadronic matrix element
is usually parameterized in terms of two form factors $f_{+}(q^{2})$ and
$f_{0}(q^{2})$ [27],
$\langle\pi(p_{\pi})|\bar{u}\gamma^{\mu}b|\bar{B}(p_{B})\rangle=f_{+}(q^{2})\left[(p_{B}+p_{\pi})^{\mu}-\frac{m_{B}^{2}-m_{\pi}^{2}}{q^{2}}\,q^{\mu}\right]+f_{0}(q^{2})\frac{m_{B}^{2}-m_{\pi}^{2}}{q^{2}}\,q^{\mu}\,,$
(1)
where $q\equiv p_{B}-p_{\pi}$ is the momentum transferred to the lepton pair,
with $p_{B}$ and $p_{\pi}$ the four-momenta of the parent B-meson and the
final-state pion, and $m_{B}$ and $m_{\pi}$ their masses. For massless
leptons, which is a good approximation for electrons and muons, the form
factor $f_{0}(q^{2})$ is absent and we are left with only a single form factor
$f_{+}(q^{2})$.
Precise knowledge of the heavy-to-light form factors is of primary importance
for flavour physics. It is needed for the determination of the CKM matrix
element $|V_{ub}|$ from exclusive semi-leptonic $B\to\pi\ell\nu$ decay. They
are also needed as ingredients in the analysis of hadronic B-meson decays,
such as $B\to\pi\pi$ and $B\to\pi K$, in the framework of QCD factorization
[28], again with the objective to provide precision determinations of the
quark flavour mixing parameters.
The two QCD methods, LQCD and LCSR, result in predictions for different
$q^{2}$ regions. The LCSR combines the idea of QCD sum rules with twist
expansions performed up to ${\cal O}(\alpha_{s})$, and provides estimates of
various form factors at low intermediate $q^{2}$ regions, $0<q^{2}<14~{}{\rm
GeV}^{2}$. The overall normalization is predicted at the zero momentum
transfer with typical uncertainties of $10-13\%$ [14, 15]. The LQCD simulation
can, on the other hand, potentially provide the heavy-to-light form factors in
the high-$q^{2}$ region from first principles of QCD. The unquenched lattice
calculations, in which quark-loop effects in the QCD vacuum and three
dynamical quark flavours (the mass-degenerate $u$ and $d$ quarks and a heavier
$s$ quark) are incorporated, are now available for $B\to\pi$ form factors [11,
12, 13]. Unfortunately, neither the LQCD nor the LCSR can predict the form
factors over the full $q^{2}$ range.
### 2.2 Form-factor parametrizations
While predictions of the exact form-factor shape are challenged for any
theoretical calculations, it is well established that the general properties
of analyticity, crossing symmetry and unitarity largely constrain the $q^{2}$
behavior of the form factor [21, 22, 23]. Specifically, it is expected to be
an analytic function everywhere in the complex $q^{2}$ plane outside of a cut
that extends along the positive $q^{2}$ axis from the mass of the lowest-lying
$b\bar{d}$ vector meson. This assumption leads to an un-subtracted dispersion
relation [21],
$f_{+}(q^{2})=\frac{f_{+}(0)/(1-\alpha)}{1-q^{2}/m^{2}_{B^{*}}}+\frac{1}{\pi}\int_{(m_{B}+m_{\pi})^{2}}^{\infty}dt{\frac{{\rm
Im}f_{+}(t)}{t-q^{2}-i\epsilon}}\,,$ (2)
which means that we have a pole residue at $q^{2}=m_{B^{*}}^{2}$ and a cut
from the $B\,\pi$ continuum, and the parameter $\alpha$ gives the relative
size of contribution to $f_{+}(0)$ from the $B^{*}$ pole.
The various parametrizations proposed in the literature make explicitly or
implicitly different simplifications in the treatment of the cut, and the
following four ones are widely used, with their respective salient features
sketched below:
1. 1.
Bećirević-Kaidalov (BK) ansatz [18]:
$f_{+}(q^{2})=\frac{f_{+}(0)}{(1-q^{2}/m_{B^{*}}^{2})(1-\alpha_{BK}\,q^{2}/m_{B^{*}}^{2})}\,,$
(3)
where $f_{+}(0)$ sets the normalization and $\alpha_{BK}$ defines the shape of
the form factor. It is mainly motivated by the scaling laws of the form
factors in the heavy quark limit, and provides an approximate representation
of the second term in Eq. (2) by an additional effective pole
$m_{B^{*}}^{2}/\alpha_{BK}$, with $\alpha_{BK}<1$ to be consistent with the
location of the cut.
2. 2.
Ball-Zwicky (BZ) ansatz [14, 19]:
$f_{+}(q^{2})=f_{+}(0)\left[\frac{1}{1-q^{2}/m_{B^{*}}^{2}}+\frac{r_{BZ}\,q^{2}/m^{2}_{B^{*}}}{(1-q^{2}/m_{B^{*}}^{2})\,(1-\alpha_{BZ}\,q^{2}/m_{B^{*}}^{2})}\right]\,,$
(4)
where $f_{+}(0)$ is the normalization, and $\alpha_{BZ}$ and $r_{BZ}$
determine the shape of the form factor. This is an extension of the BK ansatz,
related to each other by the simplification $\alpha_{BK}=\alpha_{BZ}=r_{BZ}$.
The BK and BZ parametrizations are featured by both being intuitive and having
fewer free parameters.
3. 3.
Boyd-Grinstein-Lebed (BGL) ansatz [21, 22]:
$f_{+}(q^{2})=\frac{1}{P(q^{2})\phi(q^{2},q^{2}_{0})}\,\sum_{k=0}^{k_{max}}a_{k}(q^{2}_{0})\big{[}z(q^{2},q^{2}_{0})\big{]}^{k}\,,$
(5)
with the conformal mapping variable defined by
$z(q^{2},q^{2}_{0})=\frac{\sqrt{t_{+}-q^{2}}-\sqrt{t_{+}-q^{2}_{0}}}{\sqrt{t_{+}-q^{2}}+\sqrt{t_{+}-q^{2}_{0}}}\,,$
(6)
where $t_{\pm}=(m_{B}\pm m_{\pi})^{2}$ and $q^{2}_{0}$ is a free parameter.
The so-called Blaschke factor $P(q^{2})=z(q^{2},m_{B^{*}}^{2})$ accounts for
the pole at $q^{2}=m_{B^{*}}^{2}$, and the outer function
$\phi(q^{2},q^{2}_{0})$ is an arbitrary analytic function, the choice of which
affects only the particular values of the series coefficients $a_{k}$. The
form-factor shape is determined by the values of $a_{k}$, with truncation at
$k_{max}=2$ or $3$. The expansion parameters $a_{k}$ are bounded by unitarity,
$\sum_{k}a_{k}^{2}\leq 1$. Becher and Hill [21] have pointed out that due to
the large $b$-quark mass, this bound is far from being saturated. For more
details we refer to Refs. [21, 22].
4. 4.
Bourrely-Caprini-Lellouch (BCL) ansatz [23]:
$f_{+}(q^{2})=\frac{1}{1-q^{2}/m_{B^{*}}^{2}}\sum_{k=0}^{k_{max}}b_{k}\left\\{[z(q^{2},q^{2}_{0})]^{k}-(-1)^{k-k_{max}-1}\frac{k}{k_{max}+1}[z(q^{2},q^{2}_{0})]^{k_{max}+1}\right\\}\,,$
(7)
where the variable $z(q^{2},q^{2}_{0})$ is defined by Eq. (6), and the free
parameter $q^{2}_{0}$ can be chosen to make the maximum value of $|z|$ as
small as possible in the semi-leptonic region [23]. In this ansatz, the form-
factor shape is determined by the values of $b_{k}$, with truncation at
$k_{max}=2$ or $3$.
Although the BK and the BZ parametrization are intuitive and have few free
parameters, the presence of poles near the semi-leptonic region creates doubt
on whether truncating all but the first one or two terms leaves an accurate
estimate of the true form-factor shape. The BGL and the BCL parametrization
are based on some fundamental theoretical concepts like analyticity and
unitarity, and avoid ad hoc assumptions about the number of poles and the pole
masses. Fits to the measured $q^{2}$ spectrum of $B\to\pi\ell\nu$ decay have,
on the other hand, shown that these different form-factor parametrizations
could describe the data equally well [4].
### 2.3 Pole residue at $q^{2}=m_{B^{*}}^{2}$ and the strong coupling
$g_{B^{*}B\pi}$
All the above four parametrizations have the essential feature that the vector
form factor $f_{+}(q^{2})$ has a pole at $q^{2}=m_{B^{*}}^{2}$. Although lying
outside the semi-leptonic region, the pole residue at $q^{2}=m_{B^{*}}^{2}$ is
phenomenologically very interesting. With the following standard definitions
[18],
$\langle
0|\bar{d}\gamma_{\mu}b|\bar{B}^{*0}(p,\epsilon)\rangle=f_{B^{*}}m_{B^{*}}\epsilon_{\mu},\qquad\langle
B^{-}(p)\pi^{+}(q)|\bar{B}^{*0}(p+q,\epsilon)\rangle=g_{B^{*}B\pi}(q\cdot\epsilon)\,,$
(8)
it is given by the product of the strong coupling $g_{B^{*}B\pi}$ and the
vector decay constant $f_{B^{*}}$ [14, 18],
$\displaystyle r_{1}$ $\displaystyle=$
$\displaystyle\lim_{q^{2}=m_{B^{*}}^{2}}(1-q^{2}/m_{B^{*}}^{2})\,f_{+}(q^{2})\,$
(9) $\displaystyle=$
$\displaystyle\frac{f_{B^{*}}\,g_{B^{*}B\pi}}{2m_{B^{*}}}\,.$
In fact, at the upper end of the physical region (i.e., at the zero recoil
point $q^{2}=(m_{B}-m_{\pi})^{2}$), the vector-meson dominance (VMD) of
$f_{+}(q^{2})$ is expected to be very effective [29, 30]. It has been argued
that, in the combined heavy quark and chiral limit, the VMD becomes even exact
[31]. Thus, the strong coupling $g_{B^{*}B\pi}$ determines the normalization
of the vector form factor $f_{+}(q^{2})$ near the zero recoil of pion. The
strong coupling $g_{B^{*}B\pi}$ also provides access to the normalized
coupling $\hat{g}$, which is, in the limit of exact chiral, heavy flavour and
spin symmetries, the single parameter for heavy-meson chiral perturbation
theory (HMChPT) [25, 26]. They are related to each other through [32]
$\hat{g}=\frac{g_{B^{*}B\pi}}{2\,\sqrt{m_{B}m_{B^{*}}}}\,f_{\pi}\,,$ (10)
where the convention $f_{\pi}\simeq 131~{}{\rm MeV}$ is used. Unlike the
$D^{*}D\pi$ coupling $g_{D^{*}D\pi}$, which could be extracted from the
available experimental data on the decay $D^{*}\to D\pi$ [33], there cannot be
a direct experimental indication on the coupling $g_{B^{*}B\pi}$, because
there is no phase space for the decay $B^{*}\to B\pi$. They could however be
related through the heavy quark symmetry [26].
As a result, a precise determination of the couplings $g_{B^{*}B\pi}$ and
$\hat{g}$ is of particular importance. During recent years a large number of
theoretical studies have been devoted to the calculation of these couplings in
various versions of quark models [29, 34] and QCD sum rules [35, 36]. However,
the variation of the obtained values, even within a single class of models,
turns out to be quite large [16, 26], for an overview see [16, 26]111Values
for the couplings obtained prior to 1995 with different approaches could be
found, for example, in [36] and references therein.. In addition, there have
been several LQCD simulations of these couplings in both quenched [37, 38] and
unquenched [39, 40] approximations. These strong couplings have also been
calculated using a framework based on QCD Dyson-Schwinger equations [41, 42].
Motivated by the precise experimental knowledge on the vector form factor
$f_{+}(q^{2})$, one can extract indirectly the values of $g_{B^{*}B\pi}$ via
Eq. (9) and $\hat{g}$ via Eq. (10), by an extrapolation of the form factor
from the physical region to the pole $m_{B^{*}}^{2}$, which will be detailed
in the next section.
## 3 Numerical results and discussions
### 3.1 The fitted $B\to\pi$ form-factor shape parameters
In order to extrapolate the vector form factor $f_{+}(q^{2})$ to the $B^{*}$
pole based on the various form-factor parametrizations, we first need to
determine their shape parameters from the current experimental data on
$B\to\pi\ell\nu$ decay reported by the BaBar [4, 5, 6] and Belle [7, 8]
collaborations. Although these measurements employ different experimental
techniques in treating the second B meson in the $B\bar{B}$ event, the
measured total and partial branching fractions agree well among each other.
For more details, we refer to these original references [4, 5, 6, 7, 8].
These experiments have also measured the $q^{2}$ spectrum of $B\to\pi\ell\nu$
decay, a fit to which allows for an extraction of the $q^{2}$ dependence of
the vector form factor $f_{+}(q^{2})$. It is generally observed that all the
four form-factor parametrizations introduced in section 2.2 could describe the
measured spectrum equally well [4, 7, 19]. A summary of the fitted form-factor
shape parameters based on various parametrizations is given in Table 1, where
both a linear (2 para., with $k_{max}=2$) and a quadratic (3 para., with
$k_{max}=3$) ansatz for the BGL and BCL parametrizations are considered in
[4], while a third-order polynomial fit (4 para., with $k_{max}=4$) is
perfermed in [7]. The value of the product $|V_{ub}|f_{+}(0)$ obtained from
the fit extrapolated to $q^{2}=0$, if available, are listed in the last
column.
Table 1: Summary of the form-factor shape parameters obtained by fitting to the BaBar [4] (top) and Belle [7] (bottom) measurements for the isospin-combined $B\to\pi\ell\nu$ decays, based on various parametrizations of the vector form factor $f_{+}(q^{2})$. Parametrization | Fit parameters | $|V_{ub}|f_{+}(0)~{}[10^{-3}]$
---|---|---
BK | $\alpha_{BK}=+0.310\pm 0.085$ | $1.052\pm 0.042$ [4]
BZ | $r_{BZ}=+0.170\pm 0.124$ | $1.079\pm 0.046$ [4]
| $\alpha_{BZ}=+0.761\pm 0.337$ |
BCL (2 par.) | $b_{1}/b_{0}=-0.67\pm 0.18$ | $1.065\pm 0.042$ [4]
BCL (3 par.) | $b_{1}/b_{0}=-0.90\pm 0.46$ | $1.086\pm 0.055$ [4]
| $b_{2}/b_{0}=+0.47\pm 1.49$ |
BGL (2 par.) | $a_{1}/a_{0}=-0.94\pm 0.20$ | $1.103\pm 0.042$ [4]
BGL (3 par.) | $a_{1}/a_{0}=-0.82\pm 0.29$ | $1.080\pm 0.056$ [4]
| $a_{2}/a_{0}=-1.14\pm 1.81$ |
BK | $\alpha_{BK}=+0.60\pm 0.04$ | $0.924\pm 0.028$ [7]
BGL (4 par.) | $a_{0}=+0.022\pm 0.002$ | $---$ [7]
| $a_{1}=-0.032\pm 0.004$ |
| $a_{2}=-0.080\pm 0.020$ |
| $a_{3}=+0.081\pm 0.066$ |
As concluded in Refs. [4, 19], all these form-factor parametrizations could
describe the experimental data equally well, and the central values of the
product $|V_{ub}|f_{+}(0)$ agree with each other. Thus, all the four form-
factor parametrizations are valid choices to describe the $q^{2}$ dependence
of the vector form factor $f_{+}(q^{2})$, at least in the physical region. To
further test these different form-factor parametrizations, more precise and
additional information is needed.
### 3.2 The relevant input parameters
Before presenting the results for the strong coupling $g_{B^{*}B\pi}$, we
would like to first fix the relevant input parameters, such as the decay
constants, the CKM matrix element $|V_{ub}|$, as well as the free parameter
$q^{2}_{0}$ in the BGL and BCL parametrizations.
The vector decay constant defined by Eq. (8) is not relevant from a
phenomenological point of view, since the meson $B^{*}$ will decay
predominantly through the electromagnetic interaction. It is, however, needed
in our case to extract the strong coupling $g_{B^{*}B\pi}$ from the pole
residue Eq. (9). To take into account the uncertainties induced by this
quantity, we shall use the following two inputs: one is taken from the UKQCD
collaboration [43],
$\tilde{f}_{B^{*}}=28(1)^{+3}_{-4}\,,$ (11)
which is related to the vector decay constant by
$\tilde{f}_{B^{*}}=m_{B^{*}}/f_{B^{*}}$, with the first error quoted
statistical and the second systematic, and hence we get $f_{B^{*}}=(190\pm
7_{stat.}{}{{}^{+32}_{-18}}_{syst.})~{}{\rm MeV}$; the other one is taken from
the quenched LQCD calculation [44],
$f_{B^{*}}=(177\pm 6_{stat.}\pm 17_{syst.})~{}{\rm MeV}\,.$ (12)
To extract the normalized form factor $f_{+}(0)$ from the fitted results of
the product $|V_{ub}|f_{+}(0)$, one needs to know the value of the CKM matrix
element $|V_{ub}|$. The two avenues for $|V_{ub}|$ determination through
inclusive and exclusive $b\to u\ell\nu$ decays have been reviewed in [45, 46].
How to reconcile the difference between the values for $|V_{ub}|$ obtained
from these two methods remains an intriguing puzzle. At the same time,
$|V_{ub}|$ can also be most precisely determined by a global fit of the
unitarity triangle (UT) that uses all available measurements [47, 48]. Since
the presence of New Physics (NP) might, in principle, affect the result of the
UT analysis, here we shall use the tree-level fit result performed by the
UTfit collaboration [48],
$|V_{ub}|=(3.76\pm 0.20)\times 10^{-3}\,,$ (13)
which is almost unchanged by the presence of NP.
In the BGL and BCL parametrizations, both the free parameter $q_{0}^{2}$ and
the outer function $\phi(q^{2},q_{0}^{2})$ have to be specified. Following the
BaBar collaboration [4] and references therein, we choose the values
$q_{0}^{2}=0.65t_{-}$ for the BGL, and
$q_{0}^{2}=(m_{B}+m_{\pi})(\sqrt{m_{B}}-\sqrt{m_{\pi}})^{2}$ for the BCL
parametrization. The outer function $\phi(q^{2},q_{0}^{2})$ in the BGL
parametrization is given explicitly as [20],
$\displaystyle\phi_{+}(q^{2},q_{0}^{2})$ $\displaystyle=$
$\displaystyle\sqrt{\frac{1}{32\pi\chi^{(0)}_{J}}}\big{(}\sqrt{t_{+}-q^{2}}+\sqrt{t_{+}-q_{0}^{2}}\big{)}\big{(}\sqrt{t_{+}-q^{2}}+\sqrt{t_{+}-t_{-}}\big{)}^{3/2}$
(14) $\displaystyle\times$
$\displaystyle\big{(}\sqrt{t_{+}-q^{2}}+\sqrt{t_{+}}\big{)}^{-5}\frac{(t_{+}-q^{2})}{(t_{+}-q_{0}^{2})^{1/4}}\,,$
where $\chi^{(0)}_{J}$ is a numerical factor that can be calculated via
operator product expansion [49]. At two loops in terms of the pole mass and
condensates and taking $\mu=m_{b}$, it is given as [20]
$\chi^{(0)}_{J}=\frac{3\big{[}1\\!+\\!1.140\,\alpha_{s}(m_{b})\big{]}}{32\pi^{2}m_{b}^{2}}\\!-\\!\frac{\overline{m}_{b}\>\langle\bar{u}u\rangle}{m_{b}^{6}}\\!-\\!\frac{\langle\alpha_{s}G^{2}\rangle}{12\pi
m_{b}^{6}}\,,$ (15)
with $m_{b}=4.88~{}{\rm GeV}$,
$\overline{m}_{b}\langle\bar{u}u\rangle\simeq-0.076\,{\rm GeV}^{4}$,
$\langle\alpha_{s}G^{2}\rangle\simeq 0.063{\rm GeV}^{4}$ [20]. Explicitly the
BaBar collaboration uses $\chi^{(0)}_{J}=6.889\times 10^{-4}$ [4].
For all the other input parameters, we list them in Table 2. Throughout the
paper, we use the isospin-averaged meson masses, for example,
$m_{\pi}=(m_{\pi^{+}}+m_{\pi^{0}})/2$.
Table 2: The relevant input parameters used in our calculation. All meson masses are taken directly from the Particle Data Group [46]. $m_{\pi^{+}}=139.6~{}{\rm MeV}$ | $m_{\pi^{0}}=135.0~{}{\rm MeV}$ | $f_{\pi}=130.41\pm 0.20~{}{\rm MeV}$ [46]
---|---|---
$m_{B^{+}}=5279.2~{}{\rm MeV}$ | $m_{B^{0}}=5279.5~{}{\rm MeV}$ | $m_{B^{\ast}}=5325.1~{}{\rm MeV}$
### 3.3 Numerical results for the couplings $g_{B^{*}B\pi}$ and $\hat{g}$
In this subsection, assuming a definite behavior of the $q^{2}$ dependence of
the vector form factor $f_{+}(q^{2})$ and using the fitted shape parameters
listed in Table 1, we shall extrapolate the form factor to the $B^{*}$ pole
and extract the strong couplings $g_{B^{*}B\pi}$ and $\hat{g}$ through Eqs.
(9) and (10).
#### 3.3.1 The coupling $g_{B^{*}B\pi}$
As mentioned already, the coupling $g_{B^{*}B\pi}$ is only poorly known
phenomenologically and the literature exhibits a wide spread of values [16,
26, 37, 38, 39, 40]. In this subsection, we first present in Table 3 the
extracted values of $g_{B^{*}B\pi}$ from the pole residue.
Table 3: The extracted values of the strong couplings $g_{B^{*}B\pi}$ and $\hat{g}$ using different form-factor parametrizations with the shape parameters given in Table 1. The columns Eq. (11) and Eq. (12) denote the results obtained with the corresponding input for $f_{B^{*}}$ given by these two equations. Parametrization | $f_{B^{*}}g_{B^{*}B\pi}~{}[{\rm GeV}]$ | $g_{B^{*}B\pi}$ | $\hat{g}$
---|---|---|---
Eq. (11) | Eq. (12) | Eq. (11) | Eq. (12)
BK | $4.32^{+0.68}_{-0.55}$ | $22.71^{+4.38}_{-4.42}$ | $24.40^{+4.72}_{-3.84}$ | $0.28^{+0.05}_{-0.05}$ | $0.30^{+0.06}_{-0.05}$ [4]
BZ | $5.23^{+1.63}_{-1.62}$ | $27.50^{+9.11}_{-9.45}$ | $29.55^{+9.79}_{-9.57}$ | $0.34^{+0.11}_{-0.12}$ | $0.36^{+0.12}_{-0.12}$ [4]
BCL (2 par.) | $4.82^{+0.74}_{-0.65}$ | $25.34^{+4.85}_{-5.07}$ | $27.23^{+5.21}_{-4.46}$ | $0.31^{+0.06}_{-0.06}$ | $0.33^{+0.06}_{-0.05}$ [4]
BCL (3 par.) | $5.78^{+2.11}_{-1.56}$ | $30.38^{+11.61}_{-9.32}$ | $32.64^{+12.48}_{-9.29}$ | $0.37^{+0.14}_{-0.11}$ | $0.40^{+0.15}_{-0.11}$ [4]
BGL (2 par.) | $10.57^{+1.60}_{-1.44}$ | $55.58^{+10.48}_{-11.16}$ | $59.72^{+11.28}_{-9.85}$ | $0.68^{+0.13}_{-0.14}$ | $0.73^{+0.14}_{-0.12}$ [4]
BGL (3 par.) | $7.76^{+3.44}_{-3.89}$ | $40.81^{+18.67}_{-21.30}$ | $43.85^{+20.06}_{-22.33}$ | $0.50^{+0.23}_{-0.26}$ | $0.54^{+0.25}_{-0.27}$ [4]
BK | $6.54^{+0.77}_{-0.66}$ | $34.40^{+5.61}_{-6.15}$ | $36.97^{+6.04}_{-5.07}$ | $0.42^{+0.07}_{-0.08}$ | $0.45^{+0.07}_{-0.06}$ [7]
BGL (4 par.) | $0.34^{+4.59}_{-4.59}$ | $1.78^{+24.12}_{-24.12}$ | $1.92^{+25.91}_{-25.91}$ | $0.02^{+0.30}_{-0.30}$ | $0.02^{+0.32}_{-0.32}$ [7]
Since the vector decay constant $f_{B^{*}}$ could not be measured directly and
the lattice calculation still has a large uncertainty [43, 44], we also give
the values of the product $f_{B^{*}}g_{B^{*}B\pi}$ in Table 3, which is free
of the uncertainty induced by $f_{B^{*}}$. Comparing the values listed in the
two columns Eq. (11) and Eq. (12), we can see that the extracted values of
$g_{B^{*}B\pi}$ and $\hat{g}$ are not so sensitive to the vector decay
constant, and are consistent with each other within their respective error
bars. Further reduction of the uncertainty on the vector decay constant
$f_{B^{*}}$ is welcome from the LQCD simulation.
As can be seen from the upper part in Table 3, the extracted results of the
parameters based on all the four parametrizations are roughly consistent with
each other with their respective uncertainties taken into account; the central
values obtained with the BGL parametrization, on the other hand, are neatly
larger than the ones with the other three parametrizations. As noted in Refs.
[23, 50], this is due to the spurious zero at $q^{2}=t_{+}$ in definition of
the outer function $\phi(q^{2},q^{0})$ in Eq. (14), implying that the BGL
parametrization includes a spurious, unwanted pole at the threshold of the
cut. Although being also a series-expansion-based ansatz, the BCL
parametrization could yield a value in good agreement with the BK and BZ ones,
which confirms the reason for generating such a larger value in the BGL ansatz
caused by the spurious zero in $\phi(q^{2},q^{0})$. In addition, comparing the
linear and the quadratic fits in the BGL and BCL parametrizations, we can see
that the errors increase with more expansion parameters added, leading to a
loss of predictive power. This means that the BGL and BCL parametrizations
with more fitting parameters could not be well constrained by the current data
of the semi-leptonic B decays.
From the lower part in Table 3, on the other hand, we can see that, while the
results of the BK parametrization are roughly consisitent with the ones using
the other ansatz based the BaBar data [4], the BGL parametrization performed
by the Belle collaboration [7] gives much smaller results, but with larger
uncertainties. This might be due to the fact that the Belle collaboration [7]
uses a different fitting strategy: rather than treating the model-independent
quantity $|V_{ub}|f_{+}(0)$ as a free parameter (as does the BaBar
collabortion [4]), they perform a simultaneous fit of the experimental [7] and
the FNAL/MILC [11] LQCD results, where the free parameters are the CKM matrix
element $|V_{ub}|$ and the series-expansion parameters $a_{i}$. In order to
compare directly with the BaBar results, a similar fit from the Belle
collaboration is necessarily needed.
To check the validity of the form-factor extrapolation, we would like to
compare the values of $f_{B^{*}}g_{B^{*}B\pi}$ given in Table 3 with the ones
existing in the literature,
$f_{B^{*}}g_{B^{*}B\pi}=\left\\{\begin{array}[]{l}(4.44\pm 0.97)~{}{\rm
GeV}~{}\cite[cite]{[\@@bibref{}{qcdsr2}{}{}]}\,,\\\
(7.77,7.88,8.20,10.01)~{}{\rm
GeV}\quad\mbox{for~{}sets~{}1~{}to~{}4}~{}\cite[cite]{[\@@bibref{}{Ball:2004ye}{}{}]}\,,\end{array}\right.$
(16)
from which we can see that our results are generally consistent with them. On
the other hand, it is observed that the result obtained in the LCSR method
[36] is smaller than the fits given in Ref. [14]; this might be due to the
failure of the simple quark-hadron duality used for the contribution of higher
resonances and the continuum to the sum rules [51]; the inclusion of a radial
excitation with negative residue in the hadronic parametrization of the
correlation function does increase the value [51]. With this fact taken into
account, our central values are a bit smaller than that given in Eq. (16).
#### 3.3.2 The normalized coupling $\hat{g}$
The normalized coupling $\hat{g}$ is the single constant in the limit of exact
chiral, heavy flavour and spin symmetries [25, 26]. However, being the
parameter of the effective theory, its value cannot be predicted but should be
fixed phenomenologically. Our results are given in last two columns in Table
3. As is the case for $g_{B^{*}B\pi}$, the central values based on the BK, BZ
and BCL parametrizations are consistent with each other, while the ones in the
BGL ansatz are larger.
As an improved determination of the $B^{*}B\pi$ coupling can reduce the
systematic uncertainty in most lattice calculations of B-meson quantities, it
has aroused a lot of precise determinations of the $B^{*}B\pi$ coupling in the
literature [37, 38, 39, 40]. The most recent lattice results are
$\hat{g}=\left\\{\begin{array}[]{l}0.42\pm 0.04_{stat}\pm
0.08_{syst}\qquad{\rm
for}N_{f}=0~{}\cite[cite]{[\@@bibref{}{deDivitiis:1998kj}{}{}]}\,,\\\ 0.58\pm
0.06_{stat}\pm 0.10_{syst}\qquad{\rm
for}N_{f}=0~{}\cite[cite]{[\@@bibref{}{Abada:2003un}{}{}]}\,,\\\ 0.44\pm
0.03_{stat}{}^{+0.07}_{-0.00}{}_{syst}\qquad{\rm
for}N_{f}=2~{}\cite[cite]{[\@@bibref{}{Becirevic:2009yb}{}{}]}\,,\\\ 0.516\pm
0.005_{stat}\pm 0.033_{chiral}\pm 0.028_{pert}\pm 0.028_{dics}\qquad{\rm
for}N_{f}=2~{}\cite[cite]{[\@@bibref{}{Ohki:2008py}{}{}]}\,,\end{array}\right.$
(17)
which have about $5\%$ and $15\%$ statistical errors for the quenched and
unquenched cases, respectively. With their respective uncertainties taken into
account, our extracted values are generally consistent with the above lattice
data.
Other estimates of the coupling $\hat{g}$ are derived using various versions
of quark models and QCD sum rules [16, 26]. The best estimate based on the
analyses of both QCD sum rules and relativistic quark model, quoted in the
review [26], is
$\hat{g}\simeq 0.38\,,$ (18)
with an uncertainty around $20\%$, which is also in agreement with our results
given in Table 3.
Both the strong couplings $g_{B^{*}B\pi}$ and $\hat{g}$ have also been
calculated using a framework based on QCD’s Dyson-Schwinger equations [41,
42]. By implementing a more realistic representation of heavy-light mesons,
the updated analysis based on this framework gives
$g_{B^{*}B\pi}=30.0^{+3.2}_{-1.4}$ and $\hat{g}=0.37^{+0.04}_{-0.02}$ [41],
both of which are also consistent with our extracted values from the semi-
leptonic $B\to\pi\ell\nu$ decays.
The coupling $\hat{g}$ is also related to the measured decay width
$\Gamma(D^{*}\to D\pi)$ [33]. From the width of the charged $D^{*}$-meson
measured by CLEO, $\Gamma^{\rm exp}(D^{*+})=(96\pm 22)~{}{\rm KeV}$ [33], and
by using the experimentally established branching fraction
${\mathcal{B}}(D^{*+}\to D^{+}\gamma)=(1.6\pm 0.4)\%$ [46], we can get
$\displaystyle\Gamma^{\rm exp}(D^{*+})\left[1-{\mathcal{B}}(D^{*+}\to
D^{+}\gamma)\right]$ $\displaystyle=$ $\displaystyle\Gamma(D^{*+}\to
D^{0}\pi^{+})+\Gamma(D^{*+}\to D^{+}\pi^{0})\,$ (19) $\displaystyle=$
$\displaystyle\frac{2\,m_{D^{0}}\,|\vec{k}_{\pi^{+}}|^{3}+m_{D^{+}}\,|\vec{k}_{\pi^{0}}|^{3}}{12\,\pi\,m_{D^{*+}}\,f_{\pi}^{2}}\,\hat{g}^{2}\,,$
where
$|\vec{k}_{\pi^{+}}|=\frac{\sqrt{[m_{D^{*}}^{2}-(m_{D}+m_{\pi})^{2}]\,[m_{D^{*}}^{2}-(m_{D}-m_{\pi})^{2}]}}{2\,m_{D^{*}}}$
is the three-momentum of pion in the rest frame of $D^{*}$ meson. Using the
inputs listed in Table 2, we get numerically
$\hat{g}=0.61\pm 0.07,$ (20)
which is a bit larger than both the LQCD simulation and our results. This
discrepancy might be due to the fact that the charm quark is not very heavy
and there are potentially large ${\cal O}(1/m_{c}^{n})$ corrections to the
relation Eq. (10) with $B$ replaced by $D$.
## 4 Conclusions
In this paper, motivated by the precisely measured $q^{2}$ spectrum of semi-
leptonic $B\to\pi\ell\nu$ decays by the BaBar [4, 5, 6] and Belle [7, 8]
collaborations, we have performed a phenomenological study of the strong
coupling $g_{B^{*}B\pi}$ and the normalized coupling $\hat{g}$ appearing in
the HMChPT, which is related to the pole residue of the vector form factor
$f_{+}(q^{2})$ at the unphysical point $q^{2}=m_{B^{*}}^{2}$.
Through a detailed analysis, we found that the extracted values based on the
BK, BZ and BCL parametrizations are consistent with each other and also
roughly in agreement with other theoretical and lattice estimates, while the
BGL ansatz gives much larger values, which is due to the spurious zero at
$q^{2}=t_{+}$ in definition of the outer function $\phi(q^{2},q^{0})$. It is
also found that the errors increase with more expansion parameters added in
the BGL and BCL parametrizations, leading to a loss of predictive power; the
BGL and BCL parametrizations with more fitting parameters could not be well
constrained by the current data in the physical region.
In order to gain further information about the $q^{2}$ behavior of heavy-to-
light transition form factors, much more precise experimental data on
exclusive semi-leptonic B-meson decays, as well as additional information on
the behavior of the vector form factor $f_{+}(q^{2})$ outside the physical
region are urgently needed.
## Acknowledgments
The work was supported in part by the National Natural Science Foundation
under contract Nos. 11075059, 10735080, 11005032 and 11047165. X. Q. Li was
also supported in part by MEC (Spain) under Grant FPA2007-60323 and by the
Spanish Consolider Ingenio 2010 Programme CPAN (CSD2007-00042).
## References
* [1] N. Cabibbo, Phys. Rev. Lett. 10 (1963) 531; M. Kobayashi and T. Maskawa, Prog. Theor. Phys. 49 (1973) 652.
* [2] M. Antonelli et al., Phys. Rept. 494 (2010) 197 [arXiv:0907.5386 [hep-ph]]; M. Artuso et al., Eur. Phys. J. C 57 (2008) 309 [arXiv:0801.1833 [hep-ph]]; J. D. Richman and P. R. Burchat, Rev. Mod. Phys. 67 (1995) 893 [arXiv:hep-ph/9508250].
* [3] S. B. Athar et al. [CLEO Collaboration], Phys. Rev. D 68 (2003) 072003 [arXiv:hep-ex/0304019]; N. E. Adam et al. [CLEO Collaboration], Phys. Rev. Lett. 99 (2007) 041802 [arXiv:hep-ex/0703041].
* [4] P. del Amo Sanchez et al. [BABAR Collaboration], arXiv:1005.3288 [hep-ex].
* [5] P. del Amo Sanchez et al. [BABAR Collaboration], arXiv:1010.0987 [hep-ex].
* [6] B. Aubert et al. [BABAR Collaboration], Phys. Rev. Lett. 98 (2007) 091801 [arXiv:hep-ex/0612020].
* [7] H. Ha et al., arXiv:1012.0090 [hep-ex].
* [8] T. Hokuue et al. [Belle Collaboration], Phys. Lett. B 648 (2007) 139 [arXiv:hep-ex/0604024].
* [9] N. Isgur, D. Scora, B. Grinstein and M. B. Wise, Phys. Rev. D 39 (1989) 799; D. Scora and N. Isgur, Phys. Rev. D 52 (1995) 2783 [arXiv:hep-ph/9503486].
* [10] J. M. Soares, Phys. Rev. D 54 (1996) 6837 [arXiv:hep-ph/9607284]; D. Melikhov, Phys. Rev. D 53 (1996) 2460 [arXiv:hep-ph/9509268]; D. Melikhov and B. Stech, Phys. Rev. D 62 (2000) 014006 [arXiv:hep-ph/0001113].
* [11] J. A. Bailey et al., Phys. Rev. D 79 (2009) 054507 [arXiv:0811.3640 [hep-lat]].
* [12] C. Bernard et al., Phys. Rev. D 80 (2009) 034026 [arXiv:0906.2498 [hep-lat]].
* [13] E. Dalgic, A. Gray, M. Wingate, C. T. H. Davies, G. P. Lepage and J. Shigemitsu, Phys. Rev. D 73 (2006) 074502 [Erratum-ibid. D 75 (2007) 119906] [arXiv:hep-lat/0601021].
* [14] P. Ball and R. Zwicky, Phys. Rev. D 71 (2005) 014015 [arXiv:hep-ph/0406232].
* [15] G. Duplancic, A. Khodjamirian, T. Mannel, B. Melic and N. Offen, JHEP 0804 (2008) 014 [arXiv:0801.1796 [hep-ph]]; A. Khodjamirian, T. Mannel and N. Offen, Phys. Rev. D 75 (2007) 054013 [arXiv:hep-ph/0611193].
* [16] A. Khodjamirian and R. Ruckl, Adv. Ser. Direct. High Energy Phys. 15, 345 (1998) [arXiv:hep-ph/9801443]; P. Colangelo and A. Khodjamirian, arXiv:hep-ph/0010175.
* [17] X. G. Wu and T. Huang, Phys. Rev. D 79 (2009) 034013 [arXiv:0901.2636 [hep-ph]]; X. G. Wu, T. Huang and Z. Y. Fang, Phys. Rev. D 77 (2008) 074001 [arXiv:0712.0237 [hep-ph]].
* [18] D. Becirevic and A. B. Kaidalov, Phys. Lett. B 478 (2000) 417 [arXiv:hep-ph/9904490].
* [19] P. Ball, Phys. Lett. B 644 (2007) 38 [arXiv:hep-ph/0611108].
* [20] M. C. Arnesen, B. Grinstein, I. Z. Rothstein and I. W. Stewart, Phys. Rev. Lett. 95 (2005) 071802 [arXiv:hep-ph/0504209].
* [21] T. Becher and R. J. Hill, Phys. Lett. B 633 (2006) 61 [arXiv:hep-ph/0509090]; R. J. Hill, Phys. Rev. D 73 (2006) 014012 [arXiv:hep-ph/0505129].
* [22] C. G. Boyd, B. Grinstein and R. F. Lebed, Phys. Rev. Lett. 74 (1995) 4603 [arXiv:hep-ph/9412324]; C. G. Boyd and M. J. Savage, Phys. Rev. D 56 (1997) 303 [arXiv:hep-ph/9702300].
* [23] C. Bourrely, I. Caprini and L. Lellouch, Phys. Rev. D 79 (2009) 013008 [Erratum-ibid. D 82 (2010) 099902] [arXiv:0807.2722 [hep-ph]].
* [24] J. M. Flynn and J. Nieves, Phys. Rev. D 76 (2007) 031302 [arXiv:0705.3553 [hep-ph]].
* [25] M. B. Wise, Phys. Rev. D 45 (1992) 2188; T. M. Yan, H. Y. Cheng, C. Y. Cheung, G. L. Lin, Y. C. Lin and H. L. Yu, Phys. Rev. D 46 (1992) 1148 [Erratum-ibid. D 55 (1997) 5851]; G. Burdman and J. F. Donoghue, Phys. Lett. B 280 (1992) 287.
* [26] R. Casalbuoni, A. Deandrea, N. Di Bartolomeo, R. Gatto, F. Feruglio and G. Nardulli, Phys. Rept. 281 (1997) 145 [arXiv:hep-ph/9605342].
* [27] M. Wirbel, B. Stech and M. Bauer, Z. Phys. C 29 (1985) 637.
* [28] M. Beneke, G. Buchalla, M. Neubert and C. T. Sachrajda, Phys. Rev. Lett. 83 (1999) 1914 [hep-ph/9905312]; M. Beneke, G. Buchalla, M. Neubert and C. T. Sachrajda, Nucl. Phys. B 591 (2000) 313 [hep-ph/0006124].
* [29] N. Isgur and M. B. Wise, Phys. Rev. D 41 (1990) 151.
* [30] G. Burdman, Z. Ligeti, M. Neubert and Y. Nir, Phys. Rev. D 49 (1994) 2331 [arXiv:hep-ph/9309272].
* [31] B. Grinstein and P. F. Mende, Nucl. Phys. B 425 (1994) 451 [arXiv:hep-ph/9401303].
* [32] A. Abada et al., Phys. Rev. D 66 (2002) 074504 [arXiv:hep-ph/0206237].
* [33] S. Ahmed et al. [CLEO Collaboration], Phys. Rev. Lett. 87 (2001) 251801 [arXiv:hep-ex/0108013]; A. Anastassov et al. [CLEO Collaboration], Phys. Rev. D 65 (2002) 032003 [arXiv:hep-ex/0108043].
* [34] D. Becirevic and A. L. Yaouanc, JHEP 9903 (1999) 021 [arXiv:hep-ph/9901431]; D. Melikhov and M. Beyer, Phys. Lett. B 452 (1999) 121 [arXiv:hep-ph/9901261]; P. Colangelo, F. De Fazio and G. Nardulli, Phys. Lett. B 334 (1994) 175 [arXiv:hep-ph/9406320]; P. J. O’Donnell and Q. P. Xu, Phys. Lett. B 336 (1994) 113 [arXiv:hep-ph/9406300].
* [35] P. Colangelo, G. Nardulli, A. Deandrea, N. Di Bartolomeo, R. Gatto and F. Feruglio, Phys. Lett. B 339 (1994) 151 [arXiv:hep-ph/9406295]; P. Colangelo and F. De Fazio, Eur. Phys. J. C 4 (1998) 503 [arXiv:hep-ph/9706271].
* [36] V. M. Belyaev, V. M. Braun, A. Khodjamirian and R. Ruckl, Phys. Rev. D 51 (1995) 6177 [arXiv:hep-ph/9410280]; A. Khodjamirian, R. Ruckl, S. Weinzierl and O. I. Yakovlev, Phys. Lett. B 457 (1999) 245 [arXiv:hep-ph/9903421].
* [37] G. M. de Divitiis, L. Del Debbio, M. Di Pierro, J. M. Flynn, C. Michael and J. Peisa [UKQCD Collaboration], JHEP 9810 (1998) 010 [arXiv:hep-lat/9807032].
* [38] A. Abada, D. Becirevic, P. Boucaud, G. Herdoiza, J. P. Leroy, A. Le Yaouanc and O. Pene, JHEP 0402 (2004) 016 [arXiv:hep-lat/0310050].
* [39] D. Becirevic, B. Blossier, E. Chang and B. Haas, Phys. Lett. B 679 (2009) 231 [arXiv:0905.3355 [hep-ph]].
* [40] H. Ohki, H. Matsufuru and T. Onogi, Phys. Rev. D 77 (2008) 094509 [arXiv:0802.1563 [hep-lat]]; S. Negishi, H. Matsufuru and T. Onogi, Prog. Theor. Phys. 117 (2007) 275 [arXiv:hep-lat/0612029].
* [41] B. El-Bennich, M. A. Ivanov and C. D. Roberts, arXiv:1012.5034 [nucl-th].
* [42] M. A. Ivanov, Yu. L. Kalinovsky and C. D. Roberts, Phys. Rev. D 60 (1999) 034018 [arXiv:nucl-th/9812063].
* [43] K. C. Bowler, L. Del Debbio, J. M. Flynn, G. N. Lacagnina, V. I. Lesk, C. M. Maynard and D. G. Richards [UKQCD Collaboration], Nucl. Phys. B 619 (2001) 507 [arXiv:hep-lat/0007020].
* [44] C. Bernard et al., Phys. Rev. D 65 (2002) 014510 [arXiv:hep-lat/0109015].
* [45] The Heavy Flavor Averaging Group et al., arXiv:1010.1589 [hep-ex]; and online update at http://www.slac.stanford.edu/xorg/hfag.
* [46] K. Nakamura et al. [Particle Data Group], J. Phys. G 37 (2010) 075021.
* [47] J. Charles et al. [CKMfitter Group], Eur. Phys. J. C 41, 1 (2005) [arXiv:hep-ph/0406184]; and online update at http://ckmfitter.in2p3.fr/.
* [48] M. Bona et al. [UTfit Collaboration], JHEP 0507 (2005) 028 [arXiv:hep-ph/0501199]; and online update at http://www.utfit.org/UTfit/.
* [49] L. Lellouch, Nucl. Phys. B 479 (1996) 353 [arXiv:hep-ph/9509358]; S. C. Generalis, J. Phys. G 16 (1990) 367; J. Phys. G 16 (1990) 785.
* [50] S. Descotes-Genon and A. Le Yaouanc, J. Phys. G 35 (2008) 115005 [arXiv:0804.0203 [hep-ph]].
* [51] D. Becirevic, J. Charles, A. LeYaouanc, L. Oliver, O. Pene and J. C. Raynal, JHEP 0301 (2003) 009 [arXiv:hep-ph/0212177].
|
arxiv-papers
| 2010-11-01T08:52:07 |
2024-09-04T02:49:14.389724
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xin-Qiang Li, Fang Su, Ya-Dong Yang",
"submitter": "Yadong Yang",
"url": "https://arxiv.org/abs/1011.0269"
}
|
1011.0390
|
# A Path Algebra for Multi-Relational Graphs
Marko A. Rodriguez1, Peter Neubauer2
$~{}^{1}$Graph System Architect, AT&T Interactive
Santa Fe, NM 87506 USA
marko@markorodriguez.com
$~{}^{2}$VP Product Development, Neo Technology
21119 Malmö, Sweden
peter.neubauer@neotechnology.com
###### Abstract
A multi-relational graph maintains two or more relations over a vertex set.
This article defines an algebra for traversing such graphs that is based on an
$n$-ary relational algebra, a concatenative single-relational path algebra,
and a tensor-based multi-relational algebra. The presented algebra provides a
monoid, automata, and formal language theoretic foundation for the
construction of a multi-relational graph traversal engine.
## I Introduction
The adjacency of vertex $i$ and vertex $j$ is defined by the edge $(i,j)$. A
structure of this form is called a graph and is usually defined as
$\ddot{G}=(\ddot{V},\ddot{E})$, where $i,j\in\ddot{V}$ are vertices and
$(i,j)\in\ddot{E}$ is the edge adjoining those vertices.111The “high dot”
notation denotes that $\ddot{G}\neq\dot{G}\neq G$, where $G$ is the main
definition used throughout the article. When the only distinguishing
characteristic between two edges is the vertices they join, the graph is
called single-relational. The reason for this is that there is only a single
type of relation in the graph—namely, the binary relation
$\ddot{E}\subseteq(\ddot{V}\times\ddot{V})$. Single-relational graphs have
been used widely to model various systems of homogenous elements related by a
single type of relation and as such, have numerous algorithms associated with
their analysis [1].
When the domain of discourse is variegated by a heterogeneous set of
relations, then the multi-relational graph becomes the more applicable
construct. A multi-relational graph can be defined as
$\dot{G}=(\dot{V},\dot{\mathbb{E}})$, where $\dot{\mathbb{E}}$ is a family of
edge sets and
$\dot{\mathbb{E}}=\\{\dot{E_{1}},\dot{E_{2}},\ldots,\dot{E_{m}}\subseteq(\dot{V}\times\dot{V})\\}$.
When $m>1$, then there are multiple relations between the vertices of
$\dot{V}$. Multi-relational graphs not only specify which vertices are
adjacent to one another, they also specify the way in which they are adjacent.
With respect to the formalisms of this article and without loss of generality,
a multi-relational graph can also be represented as $G=(V,E)$, where $E$ is
ternary relation, $E\subseteq(V\times\Omega\times V)$, and $\Omega$ is a set
of edge labels (i.e. relation types). Thus, in reference to the structure
$\dot{G}=(\dot{V},\dot{\mathbb{E}})$, $|\dot{\mathbb{E}}|=|\Omega|$ and
$\sum_{n=1}^{n\leq|\dot{\mathbb{E}}|}|\dot{E}_{n}|=|E|:\dot{E}_{n}\in\dot{\mathbb{E}}$.
The ternary relation model is the multi-relational graph structure used
throughput this article. The reason for the use of this particular $G$
definition will be explained in §II.
Given the growing use of multi-relational graphs in computing [2] and the lack
of graph techniques for such structures (relative to single-relational
graphs), an algebraic model for traversing multi-relational graphs is
presented. This article can be interpreted as a convergence of the $n$-ary
relational algebra of [3], the concatenative single-relational path algebra in
[4], and the multi-relational tensor algebra presented in [5]. However, unlike
[3], the presented algebra is tied specifically to path construction by means
of graph traversals as in [5] and [4]. Next, unlike the algebra in [4], which
is oriented primarily towards single-relational graphs, the presented algebra
conveniently supports multiple relations as in [3] and [5]. Finally, unlike
[5], the presented algebra is a concatenative, order-preserving variation of
the relational algebra in [3] and, as such, more aligned with [4].
The operations presented are summarized in the itemization below and are
provided here as a consolidated summary for ease of reference.
* •
$\|a\|$: the path length of path $a$.
* •
$\circ:E^{*}\times E^{*}\rightarrow E^{*}$ : the concatenation of two
paths.222The unary Kleene star operation ∗ forms the free monoid
$E^{*}=\bigcup_{n=0}^{\infty}E^{i}$, where $E^{0}=\\{\epsilon\\}$ and
$\epsilon$ is the empty/identity element.
* •
$\sigma:E^{*}\times\mathbb{N}^{+}\rightarrow E$: the projection of the
$n^{\text{th}}$ edge of a path.
* •
$\gamma^{-}:E^{*}\rightarrow V$: the projection of the tail (first element) of
a path.
* •
$\gamma^{+}:E^{*}\rightarrow V$: the projection of the head (last element) of
a path.
* •
$\omega:E\rightarrow\Omega$: the projection of the label of an edge.
* •
$\cup:\mathcal{P}(E^{*})\times\mathcal{P}(E^{*})\rightarrow\mathcal{P}(E^{*})$:
the union of two path sets.
* •
$\bowtie_{\circ}:\mathcal{P}(E^{*})\times\mathcal{P}(E^{*})\rightarrow\mathcal{P}(E^{*})$:
the concatenative join of two path sets.
* •
$\times_{\circ}:\mathcal{P}(E^{*})\times\mathcal{P}(E^{*})\rightarrow\mathcal{P}(E^{*})$:
the concatenative product of two path sets.
Definitions of these operations are provided in §II. The use of these
operations to represent basic traversal idioms is presented in §III. In §IV,
regular paths can be recognized and generated as demonstrated in §IV-A and
§IV-B, respectively. Making use of the algebra to evaluate single-relational
graph algorithms is presented in §IV-C. The algebra provides a set of core
operations for constructing a multi-relational graph traversal engine that is
founded on monoid, automata, and formal language theory.
## II Core Operations
Traversing a graph is the process of moving over the edges specified in $E$.
During a traversal, paths are derived and properties of those paths can be
extracted.
###### Definition 1 (Path)
A path $a$ in a multi-relational graph is a sequence, or string, where $a\in
E^{*}$ and $E\subseteq(V\times\Omega\times V)$. A path allows for repeated
edges. The path length is denoted $\|a\|$ and is equal to the number of edges
in $a$. Any edge in $E$ is a path with a path length of $1$ as $e\in E\subset
E^{*}$.
The binary operation $\circ:E^{*}\times E^{*}\rightarrow E^{*}$ is the
concatenation of two paths into a new path such that if $(i,\alpha,j)$ and
$(j,\beta,k)$ are two edges in $E$, then their concatenation is the path
$(i,\alpha,j,j,\beta,k)$, where $i,j,k\in V$ and $\alpha,\beta\in\Omega$.
Concatenation is associative (i.e. $(a\circ b)\circ c=a\circ(b\circ c)$), not
commutative (i.e. it is generally true that $a\circ b\neq b\circ a$), and
$\epsilon$ serves as an identity (i.e. $\epsilon\circ a=a=a\circ\epsilon$).
Operations exist to extract information out of a path. The operation
$\sigma:E^{*}\times\mathbb{N}^{+}\rightarrow E$ is a projection that maps a
path to the $n^{\text{th}}$ edge in that path. For example, if
$a=(i,\alpha,j,j,\beta,k)$, then $\sigma(a,1)=(i,\alpha,j)$ and
$\sigma(a,2)=(j,\beta,k)$. Next, for any path, $\gamma^{-}:E^{*}\rightarrow V$
projects the tail (first vertex) of the path such that
$\gamma^{-}((i,\alpha,j))=i$. Likewise, $\gamma^{+}:E^{*}\rightarrow V$, where
$\gamma^{+}((i,\alpha,j))=j$. Similarly, for edge labels,
$\omega:E\rightarrow\Omega$, where $\omega((i,\alpha,j))=\alpha$.333All
projection operations can be reduced to a single string indexing operation,
but for the sake of clarity in the following discussion, they are presented as
being atomic.
###### Definition 2 (Path Label)
The path label of path $a$ is defined as the edge labels contained in $a$.
Formally, if $a$ is a path, then the path label is constructed by
$\omega^{\prime}:E^{*}\rightarrow\Omega^{*}$, where, using concatenation,
$\omega^{\prime}(a)=\prod_{n=1}^{n\leq\|a\|}\omega\left(\sigma\left(a,n\right)\right).$
The path label of any single edge $e\in E$ is simply the edge’s label as
$\|e\|=1$ and $\omega^{\prime}(e)=\omega(\sigma(e,1))=\omega(e)$.
The binary operation
$\cup:\mathcal{P}(E^{*})\times\mathcal{P}(E^{*})\rightarrow\mathcal{P}(E^{*})$
is standard set union. The binary operation
$\bowtie_{\circ}:\mathcal{P}(E^{*})\times\mathcal{P}(E^{*})\rightarrow\mathcal{P}(E^{*})$
is the concatenative join of two sets of paths such that if
$A,B\in\mathcal{P}(E^{*})$, then
$\displaystyle A\bowtie_{\circ}B=$ $\displaystyle\;\\{a\circ b\;|\;a\in
A\;\wedge\;b\in B$
$\displaystyle\;\;\wedge\;\left(a=\epsilon\;\vee\;b=\epsilon\;\vee\;\gamma^{+}(a)=\gamma^{-}(b)\right)\\},$
where $\gamma^{+}(a)=\gamma^{-}(b)$ ensures that only joint (i.e. adjacent)
paths are concatenated.444The defined concatenative join is analogous to the
$\theta$-join in [3], where $\begin{array}[]{c}A\bowtie B\\\
\gamma^{+}(a)=\gamma^{-}(b)\end{array}$. In this form, its known as an
equijoin. A discussion relating concatenative join and the relational algebra
is found in [6]. For example, if
$A=\left\\{(i,\alpha,j),(j,\beta,k,k,\alpha,j)\right\\}$
and
$B=\left\\{(j,\beta,j),(j,\beta,i,i,\alpha,k),(i,\beta,k)\right\\},$
then
$\displaystyle A\bowtie_{\circ}B=$
$\displaystyle\;\\{(i,\alpha,j,j,\beta,j),(i,\alpha,j,j,\beta,i,i,\alpha,k),$
$\displaystyle\;\;(j,\beta,k,k,\alpha,j,j,\beta,j),$
$\displaystyle\;\;(j,\beta,k,k,\alpha,j,j,\beta,i,i,\alpha,k)\\},$
where $i,j,k\in V$, $\alpha,\beta\in\Omega$, and
$(i,\alpha,j),(j,\beta,k),(k,\alpha,j),\\\
(j,\beta,j),(j,\beta,i),(i,\alpha,k),(i,\beta,k)\in E$. Given that
$\bowtie_{\circ}$ is based on $\circ$, $\bowtie_{\circ}$ is associative, but
not commutative.
###### Definition 3 (Path Jointness)
A path is joint is it satisfies the characteristic function
$f:E^{*}\rightarrow\\{\top,\bot\\}$ with the function rule
$\displaystyle f(a)=\begin{cases}\top&\text{if }\|a\|=1,\\\ \top&\text{if
}\bigwedge_{n=1}^{n<\|a\|-1}\gamma^{+}(\sigma(a,n))=\gamma^{-}(\sigma(a,n+1)),\\\
\bot&\text{otherwise}.\end{cases}$
The function maps to $\top$ if the path is joint and $\bot$ if it is disjoint.
The binary operation $\bowtie_{\circ}$ constructs joint paths. It may be the
case that traversing disjoint paths is desirable.555For example, priors-based
algorithms require the concept of “teleportation” in order to make a disjoint
jump in the graph. The Cartesian product supports the concatenation of
potentially disjoint paths. As such,
$\times_{\circ}:\mathcal{P}(E^{*})\times\mathcal{P}(E^{*})\rightarrow\mathcal{P}(E^{*})$,
where $A\times_{\circ}B=\\{a\circ b\;|\;a\in A\;\wedge\;b\in B\\}$.
Finally, to conclude this section, the reason why the
$\dot{G}=(\dot{V},\dot{\mathbb{E}}=\\{\dot{E_{1}},\dot{E_{2}},\ldots,\dot{E_{m}}\subseteq(\dot{V}\times\dot{V})\\})$
definition of a multi-relational graph is not used is because when evaluating
concatenative joins over binary relations, the edge label information is lost
and thus, the path label can not be determined. In other words, if $e$ and $f$
are edges from two different binary relations, then $e\circ f$ would only
provide a sequence of vertices and as such would not specify from which
relations the join was constructed. This is a deficiency of the algebra in
[4], where binary relations are used and $\circ:V^{*}\times V^{*}\rightarrow
V^{*}$ as opposed to $\circ:E^{*}\times E^{*}\rightarrow E^{*}$, where
$E=(V\times\Omega\times V)$. While the algebra in [4] is applicable to multi-
relational graphs (as any two relations can be joined), it was specifically
intended for single-relational graphs, where problems involving path labels
are not considered. In contrast, the specification defined in this article
preserves path labels.
## III Basic Traversals
From the explicit adjacencies (edges) defined in the edge set $E$, there
exists implicit adjacencies (paths) defined by $e\circ f$, where $e,f\in E$
and $e\circ f\in E^{*}$. Given the previously defined operations, different
types of common traversal idioms can be affected.
### III-A Complete Traversal
All joint paths through a graph of length $n$ can be constructed using
$\underbrace{E\bowtie_{\circ}\ldots\bowtie_{\circ}E}_{n\text{ times}}$. This
type of traversal is called a complete traversal because there is no
discrimination when joining except that the join vertex (i.e. the head of the
first path and tail of the second) be equal. When it is desirable to limit the
set of paths derived by the traversal then the sets $A,B\subseteq E$ need to
be defined and joined.
### III-B Source Traversal
A source traversal emanates from a particular set of vertices. Such a
traversal is left restricting as it constructs paths whose tail vertex is an
element of $V_{s}\subseteq V$. The first concatenative join must, on its left
side, contain the set of all edges in $E$ that have their tail vertex in
$V_{s}$. Therefore, when
$A=\\{e\;|\;e\in E\;\wedge\;\gamma^{-}(e)\in V_{s}\\},$
$\underbrace{A\bowtie_{\circ}E\ldots\bowtie_{\circ}E}_{n\text{ times}}$ yields
all joint paths of length $n$ emanating from the vertices in $V_{s}$. When
$V_{s}=V$, a complete traversal is evaluated since $A=E$. For ease of
expression, the complement of the set $V_{s}$ can be used to denote where not
to start a traversal from. For example, $\overline{V_{s}}=V\setminus V_{s}$
states to start the traversal from all vertices in $V$ except those in
$V_{s}$.
### III-C Destination Traversal
A destination traversal is similar to a source traversal, except that it is
right restricting as it constructs all paths of length $n$ whose head, or
terminal, vertex is in $V_{d}\subseteq V$. In this way, when
$B=\\{e\;|\;e\in E\;\wedge\;\gamma^{+}(e)\in V_{d}\\},$
$\underbrace{E\bowtie_{\circ}\ldots E\bowtie_{\circ}B}_{n\text{ times}}$ is a
destination traversal. When $V_{d}=V$, a complete traversal is evaluated
because $B=E$ in such situations.
By combining a source and destination traversal, its possible to emanate from
particular vertices and arrive at particular vertices, where
$\underbrace{A\bowtie_{\circ}E\ldots E\bowtie_{\circ}B}_{n\text{ times}}$ is
the set of all joint paths that start from vertices in $V_{s}$, end at
vertices in $V_{d}$, and are of length $n$. Source and destination traversals
can also be used to ensure that each edge in the path goes through a
particular set of vertices by specifying, at some particular $\bowtie_{\circ}$
step, the source (or destination) vertex set as $V_{s}$ (or $V_{d}$) before
enacting the next concatenative join.
### III-D Labeled Traversal
A traversal can be constrained to particular path labels by defining an edge
set that is a function of its edge labels. For example, if
$\Omega_{e}\subseteq\Omega$, $\Omega_{f}\subseteq\Omega$,
$A=\\{e\;|\;e\in E\;\wedge\;\omega(e)\in\Omega_{e}\\},$
and
$B=\\{f\;|\;f\in E\;\wedge\;\omega(f)\in\Omega_{f}\\},$
then $A\bowtie_{\circ}B$ denotes all paths where
$\omega(\sigma(a,1))\in\Omega_{e}$ and $\omega(\sigma(a,2))\in\Omega_{f}$.
When $\Omega_{e}=\Omega_{f}=\Omega$, a complete traversal is enacted as, in
such situations, $A=B=E$. The labeled traversal is possible because the
relation type is represented in the edge definition
$E\subseteq(V\times\Omega\times V)$ and there exists the label projection
function $\omega:E\rightarrow\Omega$.
## IV Derivative Traversals
The basic traversals defined in §III can be mixed and matched to yield
different types of joint paths in $E^{*}$. This section will introduce some
typical applications of the presented multi-relational path algebra to
problems that are specific to multi-relational graphs—focusing primarily on
problems involving regular paths.666For the sake of simplicity, only regular
paths are discussed. However, with more machinery (e.g. memory structures),
more complex traversals can be expressed using the core operations presented
in §II.
### IV-A Regular Path Recognizer
The presented multi-relational path algebra has application to regular
expressions and their corresponding finite state automata. Before presenting
this application, an example-specific set-builder notation is introduced in
order to specify subsets of $E$ in a more concise, readable manner than
previously presented. A source edge set can be specified as
$[i,\\_,\\_]\equiv\bigcup_{\alpha\in\Omega}\bigcup_{j\in
V}(i,\alpha,j):(i,\alpha,j)\in E$ in order to denote the set of all edges that
emanate from vertex $i$. A destination edge set can be specified as
$[\\_,\\_,j]\equiv\bigcup_{i\in
V}\bigcup_{\alpha\in\Omega}(i,\alpha,j):(i,\alpha,j)\in E$ in order to denote
the set of all edges that terminate at vertex $j$. A labeled edge set can be
specified as $[\\_,\alpha,\\_]\equiv\bigcup_{i\in V}\bigcup_{j\in
V}(i,\alpha,j):(i,\alpha,j)\in E$ in order to denote the set of all edges that
have $\alpha$ as their label. Finally $[\\_,\\_,\\_]=E$.
If $E$ is the regular expression alphabet, then $\emptyset$, $\epsilon$, and
any $e\in E$ are regular expressions. If $R$ and $Q$ are regular expressions,
then $R\cup Q$, $R\bowtie_{\circ}Q$, and $R^{*}$ are regular expressions
[7].777The $\times_{\circ}$ operation can be used to recognize potentially
disjoint paths, but in practice, when only joint paths are being recognized
then $\bowtie_{\circ}$ is a more efficient use of resources as
$R\bowtie_{\circ}Q\subseteq R\times_{\circ}Q$. A regular expression over $E$,
and corresponding finite state automaton, recognize a set of joint paths in
$\mathcal{P}(E^{*})$.888The common operations $R^{+}$, $R?$, and $R^{n}$ used
in practice can be represented as $R\bowtie_{\circ}R^{*}$,
$R\cup\\{\epsilon\\}$, and
$\underbrace{R\bowtie_{\circ}\ldots\bowtie_{\circ}R}_{n\text{ times}}$,
respectively. For example,
$[i,\alpha,\\_]\bowtie_{\circ}[\\_,\beta,\\_]^{*}\left(\left([\\_,\alpha,j]\bowtie_{\circ}\\{(j,\alpha,i)\\}\right)\;\cup\;[\\_,\alpha,k]\right)$
recognizes all paths emanating from $i$, terminating at $i$ or $k$, with the
first and last label traversed being $\alpha$, and all intermediate edge
labels (zero or more) being $\beta$. The corresponding finite state automaton
is diagrammed in Figure 1, where the transition function is based on set
membership, not equality.999Given that set membership can be represented
element-wise as element equality under or, each element of the transition
label edge set can be individually denoted as a transition with the same tail
and head state. As such, the typical finite state automaton transition exists.
For diagram clarity, set membership is used instead of equality.
Figure 1: A finite state automaton to recognize and generate a set of paths
in $\mathcal{P}(E^{*})$. The left most state is the start state and the
double-circle states denote accepting states.
Regular paths in graphs are explored in depth in [8], where only paths with
particular path labels are considered for recognition. In other words, in [8],
a regular expression is defined for the alphabet $\Omega$, where above, its
defined for $E$.
### IV-B Regular Path Generator
By making use of a non-deterministic single-stack automaton with a stack
alphabet of $\mathcal{P}(E^{*})$, it is possible to generate all paths in $G$
that can be recognized by some regular expression. The non-deterministic
aspect of the automaton ensures that all branches in the state machine are
taken “in parallel.” The single-stack aspect refers to the fact that the
automaton (and thus, its cloned/branched automata) maintain a first-in/last-
out stack memory that can be pushed and popped.
Initially, the automaton’s stack contains the element $\\{\epsilon\\}$. The
automaton will halt whenever its stack element is $\emptyset$ or is in an
accepting state. For each state transition (which happens unless the automaton
has been halted), the path set defined on the transition label is joined on
the right with the path set popped off the stack. The result of the join is
then pushed back onto the stack. Whenever a branch in the automaton’s state
graph is approached, all branches are taken “in parallel.” Thus, given the
automaton diagrammed in Figure 1, the following joins are evaluated.
$\displaystyle\\{\epsilon\\}\bowtie_{\circ}[i,\alpha,\\_]\bowtie_{\circ}[\\_,\alpha,j]\bowtie_{\circ}\\{(j,\alpha,i)\\}$
$\displaystyle\\{\epsilon\\}\bowtie_{\circ}[i,\alpha,\\_]\bowtie_{\circ}[\\_,\alpha,k]$
$\displaystyle\\{\epsilon\\}\bowtie_{\circ}[i,\alpha,\\_]\bowtie_{\circ}[\\_,\beta,\\_]\ldots\bowtie_{\circ}[\\_,\alpha,j]\bowtie_{\circ}\\{(j,\alpha,i)\\}$
$\displaystyle\\{\epsilon\\}\bowtie_{\circ}[i,\alpha,\\_]\bowtie_{\circ}[\\_,\beta,\\_]\ldots\bowtie_{\circ}[\\_,\alpha,k]$
The union of the first (and only) element of all the stacks across all
branches of accept-state automaton forms the set of all paths in $G$ that
satisfy the regular expression.
### IV-C Constructing Semantically-Rich Single-Relational Graphs
Most of the graph algorithms in existence today have been developed for
single-relational graphs. Examples of such algorithms include the geodesics
(e.g. closeness centrality, betweenness centrality), spectral (e.g.
eigenvector centrality, spreading activation), and assortative (e.g. scalar
and discrete) algorithms (see [1] for a consolidate review and analysis of
many such algorithms). When applied to multi-relational graphs, these
algorithms have the potential drawback of losing their meaning and thus, their
applicability. To explicate this statement, it is important to consider the
way in which a single-relational graph algorithm can be formally applied to
multi-relational graphs. One method that can be employed is to simply ignore
edge labels and, potentially, repeated edges between the same two vertices.
However, when there are numerous ways in which one vertex can be related to
another vertex, what is the resulting semantics of, say, a centrality
algorithm? Another method is to extract a single edge relation, based on its
label, from the multi-relational graph. For example, its possible to construct
the binary edge set
$E_{\alpha}=\\{(\gamma^{-}(e),\gamma^{+}(e))\;|\;e\in
E\;\wedge\;\omega(e)=\alpha\\}$
and utilize that subgraph as the source of a single-relational graph
algorithm. However, with multiple ways in which vertices can be related, more
abstract relationships can be inferred through paths. Thus, in the final
method, single-relational graphs can be generated from the multi-relational
graph through the derivation of implicit edges defined through paths. Using a
simple example, if $\alpha,\beta\in\Omega$ are two edge labels, then all
$\alpha\beta$-paths can be constructed when $A=\\{e\;|\;e\in
E\;\wedge\;\omega(e)=\alpha\\}$, $B=\\{e\;|\;e\in
E\;\wedge\;\omega(e)=\beta\\}$ and $A\bowtie_{\circ}B$. The tail and head
vertices of these paths can then be projected to form a new binary edge set
$E_{\alpha\beta}=\bigcup_{a\in
A\bowtie_{\circ}B}\left(\gamma^{-}(a),\gamma^{+}(a)\right).$
Thus, $E_{\alpha\beta}\subseteq(V\times V)$ can be subjected to all known
single-relational graph algorithms. For regular paths, a regular path
generator can be used as in §IV-B. Mapping single-relational graph algorithms
over to the multi-relational domain is explored in depth in [5].
## V Conclusion
This article defined a path algebra for multi-relational graphs represented as
$G=(V,E\subseteq(V\times\Omega\times V)$. The core traversal types (complete,
source, destination, and labeled) allow for the expression of more expressive
traversals through the restriction of the join set $E$. Applications to
regular path recognizers (§IV-A), generators (§IV-B), and “semantically-rich”
single-relational graph construction (§IV-C) were presented. Generally, the
algebra has applicability to the construction of a multi-relational graph
traversal engine.
## References
* [1] U. Brandes and T. Erlebach, Eds., _Network Analysis: Methodolgical Foundations_. Berling, DE: Springer, 2005\.
* [2] M. A. Rodriguez and P. Neubauer, “Constructions from dots and lines,” _Bulletin of the American Society for Information Science and Technology_ , vol. 36, no. 6, pp. 35–41, August 2010.
* [3] E. F. Codd, “A relational model of data for large shared data banks,” _Communications of the ACM_ , vol. 13, no. 6, pp. 377–387, 1970.
* [4] M. Russling, “A general scheme for breadth-first graph traversal,” in _Mathematics of Program Construction_ , ser. Lecture Notes in Computer Science, M. Russling, Ed., vol. 947, no. 380–398. Springer-Verlag, 1995, pp. 380–398.
* [5] M. A. Rodriguez and J. Shinavier, “Exposing multi-relational networks to single-relational network analysis algorithms,” _Journal of Informetrics_ , vol. 4, no. 1, pp. 29–41, 2009. [Online]. Available: http://arxiv.org/abs/0806.2274
* [6] P. Pucheral and J.-M. Thévenin, “A graph based data structure for efficient implementation of main memory dbms,” in _Proceedings of the Sixth International Workshop on Database Machines_. London, UK: Springer-Verlag, 1989, pp. 73–96.
* [7] B. Moret, _The Theory of Computation_. Addison-Wesley, 1997.
* [8] A. O. Mendelzon and P. T. Wood, “Finding regular simple paths in graph databases,” in _Proceedings of the 15th International Conference on Very Large Data Bases_. San Francisco, CA, USA: Morgan Kaufmann Publishers Inc., 1989, pp. 185–193.
|
arxiv-papers
| 2010-11-01T17:33:46 |
2024-09-04T02:49:14.401641
|
{
"license": "Public Domain",
"authors": "Marko A. Rodriguez and Peter Neubauer",
"submitter": "Marko A. Rodriguez",
"url": "https://arxiv.org/abs/1011.0390"
}
|
1011.0405
|
# Unitarity in Dirichlet Higgs Model
Kenji Nishiwaki and Kin-ya Oda
∗Department of Physics, Kobe University, Kobe 657-8501, Japan
†Department of Physics, Osaka University, Osaka 560-0043, Japan
E-mail: nishiwaki@stu.kobe-u.ac.jp E-mail: odakin@phys.sci.osaka-u.ac.jp
###### Abstract
We show that a five-dimensional Universal Extra Dimension model, compactified
on a line segment, is consistently formulated even when the gauge symmetry is
broken solely by non-zero Dirichlet boundary conditions on a bulk Higgs field,
without any quartic interaction. We find that the longitudinal $W^{+}W^{-}$
elastic scattering amplitude, under the absence of the Higgs zero mode, is
unitarized by exchange of infinite towers of KK Higgs bosons. Resultant
amplitude scales linearly with the scattering energy $\propto\sqrt{s}$,
exhibiting five-dimensional nature. A tree-level partial-wave unitarity
condition is satisfied up to $6.7\,(5.7)\,\text{TeV}$ for the KK scale
$m_{\text{KK}}=430\,(500)\,\text{GeV}$, favored by the electroweak data within
90% CL.
CERN-PH-TH/2010-248
KOBE-TH-10-03
OU-HET-684/2010
## 1 Introduction
More than four decades after the birth of the Standard Model (SM) [1, 2, 3],
finally the CERN Large Hadron Collider (LHC) is accumulating data that will
eventually reveal whether or not there exists the last missing piece of the
SM, the Higgs boson, and if the Electro-Weak Symmetry Breaking (EWSB) is truly
caused by the Higgs mechanism [4, 5, 6, 7, 8, 9, 10], namely, if or not the SM
is ultimately the right description of nature at around the weak scale. The
EWSB sector is the key element of the SM which eventually supplies all the
masses for the elementary particles through the Yukawa couplings, but is the
least experimentally confirmed part.
Even if we find a particle that looks similar to the SM Higgs boson, it is not
the end of the story. It takes long way to establish whether the observed
particle is really the one in the SM; see e.g. Ref.[11, 12]. Indeed there are
many alternative EWSB mechanisms to the SM Higgs one that possess their own
virtues; see e.g. [13, 14] for brief overviews. Also for more reviews on
Higgs/EWSB in a particular model, see e.g. Refs. [15, 16] for gauge-Higgs
Unification models, Ref. [17, 18] for the Higgsless EWSB, Refs. [19, 20] for
the little Higgs models, Ref. [21] for the Minimal Supersymmetric Standard
Model, and Refs. [22, 23] for walking technicolor models.
In Refs. [24, 25], it has been proposed that the EWSB can be caused without
any Higgs potential if we put general non-zero Dirichlet boundary conditions
on a bulk Higgs field in five dimensions, compactified on a line segment,
where all the SM fields propagate in the bulk. This Dirichlet Higgs model,
which is essentially the same as the Universal Extra Dimension (UED) model
[26, 27] except for the Higgs sector, predicts that there are no zero modes
for the Higgs and its first Kaluza–Klein (KK) mode couples to the SM zero
modes (quarks, leptons, and gauge bosons) with its coupling universally
multiplied by $2\sqrt{2}/\pi\simeq 0.9$. In the first look, this Dirichlet
Higgs model might appear to be equivalent to the infinitely large quartic
coupling limit of the boundary-localized Higgs potential [28]. However, there
are no quartic coupling for the KK Higgs field in the former Dirichlet Higgs
model, in contrast to the latter large boundary coupling limit that gives
large quartic couplings for the KK Higgs fields. The first KK Higgs in the
Dirichlet Higgs model is a “Higgs impostor” which has no quartic coupling and
has couplings to SM sector that are always universally 10% smaller than those
in the SM.
In the Dirichlet Higgs model, the EWSB is caused by the seemingly explicit
breaking at the boundaries. As we will see in Section 2, the boundary
conditions on the Higgs leave no gauge symmetry even in the bulk at the
classical level. Therefore one might worry if the theory possesses a gauge
symmetry at all. Furthermore, the longitudinal SM gauge bosons (zero modes) do
not couple to KK gauge bosons, under the assumption that the boundary
conditions respect the KK parity, i.e., when the non-zero Dirichlet boundary
conditions take the same value at both boundaries.111 The zero mode gauge
bosons do not couple to a pair of KK gauge bosons nor to a single KK-even
gauge boson because of the accidental conservation of the KK number among the
(von Neumann) gauge fields. Therefore, the KK gauge bosons do not help to
unitarize the high energy behavior of the elastic scattering of longitudinal
gauge bosons $W^{+}W^{-}\to W^{+}W^{-}$, unlike the Higgsless models. (Recall
that there is no Higgs zero mode either.)
In this Letter, we answer above concerns. First we explain that the theory
possesses a nilpotent Becchi–Rouet–Stora–Tyutin (BRST) symmetry both in five
dimensions and also in a KK-expanded picture, under the non-zero Dirichlet
boundary condition on the bulk Higgs field. Therefore, the Dirichlet Higgs
model is fully gauge invariant as a path-integrated (or canonically quantized)
quantum field theory and is unitary in the sense that there appears no
unphysical degrees of freedom in external lines.
Then we study high energy behavior of the tree-level scattering amplitude of
the longitudinal SM gauge boson zero modes. We will show that the growth
$\propto s$ of the elastic scattering amplitude of longitudinal $W^{+}W^{-}$
zero modes is indeed canceled by the exchanges of infinite tower of Higgs KK
modes. Further, we will show that the first KK Higgs boson contributes most
since the overlap of the KK wave function to zero modes decreases for higher-
modes, which explains why the first KK Higgs has a coupling to all the
Standard Model (SM) zero modes very close to the SM value that is multiplied
by a factor $2\sqrt{2}/\pi\simeq 0.9$. We also examine the partial-wave
unitarity.
The organization of the paper is as follows. In Section 2, we present the
setup of our theory and show where arises a potential difficulty. Section 3
can be skipped by a reader who is not interested in formal consistency of the
theory. First we explain that the background gauge transformation is viewed as
a field redefinition and that the non-zero Dirichlet boundary condition can be
rotated into a simpler basis. We then briefly sketch how a nilpotent BRST
transformation is implemented in our model. In Section 4, we show the KK
expansion of the bulk gauge, Higgs, and ghost fields. Section 5 is the main
part of this Letter, where we show the high energy scattering of the
longitudinal components of the zero mode gauge fields $W_{L}^{\pm}$ to exhibit
the tree-level unitarity of the amplitude. In the last section, we summarize
our results.
## 2 Classical setup
We consider a UED model in a flat five-dimensional spacetime
$\displaystyle ds^{2}$ $\displaystyle=\eta_{\mu\nu}dx^{\mu}dx^{\nu}+dz^{2},$
(1)
compactified on a line segment $-L/2\leq z\leq L/2$, where indices
$\mu,\nu,\dots$ run for $0,\dots,3$ and the metric signature is
$\eta_{\mu\nu}=\operatorname{diag}\left(-1,+1,+1,+1\right)$. We also let
$M,N,\dots$ be five-dimensional indices running for $0,1,2,3,z$. The gauge
kinetic action is
$\displaystyle S_{g}$ $\displaystyle=\int
d^{4}x\int_{-L/2}^{L/2}dz\left[-{1\over
2}\operatorname{tr}\left(\mathcal{F}_{MN}\mathcal{F}^{MN}\right)-{1\over
4}(\mathcal{F}^{Y})_{MN}(\mathcal{F}^{Y})^{MN}\right],$ (2)
where
$\displaystyle\mathcal{F}_{MN}$
$\displaystyle:=\partial_{M}\mathcal{W}_{N}-\partial_{N}\mathcal{W}_{M}+ig[\mathcal{W}_{M},\mathcal{W}_{N}],$
$\displaystyle(\mathcal{F}^{Y})_{MN}$
$\displaystyle:=\partial_{M}\mathcal{B}_{N}-\partial_{N}\mathcal{B}_{M},$ (3)
$\mathcal{B}_{M}$ is the $U(1)_{Y}$ gauge field and
$\mathcal{W}_{M}:=\mathcal{W}^{a}_{M}T^{a}$, with $a,b,\dots$ running for
$SU(2)_{W}$ adjoint indices $1,2,3$ whose summation is being understood unless
otherwise stated, and $[T^{a},T^{b}]=i\epsilon^{abc}T^{c}$; We have normalized
to $\operatorname{tr}\left(T^{a}T^{b}\right)=1/2$, as usual. We also write
collectively
$\displaystyle\boldsymbol{\mathcal{W}}_{M}$
$\displaystyle:=\sum_{A}\mathcal{W}_{M}^{A}T^{A},$
$\displaystyle\boldsymbol{g}\boldsymbol{\mathcal{W}}_{M}$
$\displaystyle:=\sum_{A}g_{A}\mathcal{W}_{M}^{A}T^{A},$ (4)
where $A$ run for $1,2,3,Y$ with $g_{1}=g_{2}=g_{3}:=g$, and correspondingly
$\mathcal{W}^{Y}_{M}:=\mathcal{B}_{M}$ and $T^{Y}:=Y$.
The Higgs action is
$\displaystyle S_{\Phi}$ $\displaystyle=\int
d^{4}x\int_{-L/2}^{L/2}dz\left[-\left(D_{M}\Phi\right)^{\dagger}D^{M}\Phi-V(\Phi)\right],$
(5)
where
$\displaystyle D_{M}\Phi$
$\displaystyle:=\left(\partial_{M}+i\boldsymbol{g}\boldsymbol{\mathcal{W}}_{M}\right)\Phi.$
(6)
On $\Phi$, $Y=1/2$ and $T^{a}=\sigma^{a}/2$ with $\sigma^{a}$ being Pauli
matrices. In this Letter we set $V(\Phi)=0$ since we are interested in the
theoretical consistency of putting the general non-zero Dirichlet boundary
condition on the Higgs field [24, 25]. Essential features such as BRST
invariance and unitarization of longitudinal gauge boson scattering are not
altered by inclusion of $V(\Phi)$.
On all the gauge fields, we put the standard von Neumann and Dirichlet
boundary conditions on $\mathcal{A}_{\mu}(x,z)$ and $\mathcal{A}_{z}(x,z)$,
respectively, at both ends of the line segment:
$\displaystyle\partial_{z}\mathcal{W}^{A}_{\mu}(x,\pm L/2)$ $\displaystyle=0,$
$\displaystyle\mathcal{W}^{A}_{z}(x,\pm L/2)$ $\displaystyle=0.$ (7)
On the Higgs field $\Phi(x,z)$, we impose the most general non-zero Dirichlet
boundary condition [24, 25]:
$\displaystyle\Phi(x,\pm L/2)$ $\displaystyle=\begin{bmatrix}\phi_{D}^{1}\\\
\phi_{D}^{2}\end{bmatrix}=:\Phi_{D},$ (8)
where $\phi_{D}^{1}$ and $\phi_{D}^{2}$ are arbitrary complex constants and we
have assumed that the KK parity $z\to-z$ is preserved by the boundary
conditions, $\Phi(x,L/2)=\Phi(x,-L/2)$, for simplicity. Note that, without
loss of generality, we can perform a field redefinition to rotate the boundary
condition to become
$\displaystyle\Phi_{D}$
$\displaystyle\to\Phi_{D}^{\text{new}}=\begin{bmatrix}0\\\
v/\sqrt{2}\end{bmatrix},$ (9)
where $v$ is a real parameter, but we leave it general as in Eq. (8) for the
moment to see below how the background gauge invariance is implemented in the
Dirichlet Higgs model.
For our purpose, is it most convenient to employ the background field method,
see e.g. Ref. [29], in which we separate a field into a classical background
and a quantum fluctuation around it:
$\displaystyle\Phi$ $\displaystyle=\Phi^{{}^{\rm c}}+\Phi^{{}^{\prime}},$
$\displaystyle\boldsymbol{\mathcal{W}}_{M}$
$\displaystyle=\boldsymbol{\mathcal{W}}_{M}^{{}^{\rm
c}}+\boldsymbol{\mathcal{W}}_{M}^{{}^{\prime}}.$ (10)
Throughout this paper, ′ on a field does not denote derivative. The classical
equation of motion for $\Phi^{{}^{\rm c}}(x,z)$ is given by the variation in
the bulk as
$\displaystyle\left(\Box+\partial_{z}^{2}\right)\Phi^{{}^{\rm c}}(x,z)=0,$
(11)
where $\Box:=\partial_{\mu}\partial^{\mu}$. An obvious classical solution to
the e.o.m. (11) under the boundary condition (8) is the constant one
$\displaystyle\Phi^{{}^{\rm c}}(x,z)=\Phi_{D}.$ (12)
Around this vacuum expectation value (vev), the Higgs field is now expanded as
$\displaystyle\Phi(x,z)$ $\displaystyle=\Phi_{D}+\Phi^{{}^{\prime}}(x,z),$
(13)
Let us emphasize that the non-zero Dirichlet boundary condition (8) implies
that the boundary condition for the quantum fluctuation reduces to the
ordinary vanishing Dirichlet condition
$\displaystyle\Phi^{{}^{\prime}}(x,\pm L/2)$ $\displaystyle=0.$ (14)
We note that, at classical level (omitting c), a gauge transformation in five
dimensions,
$\displaystyle\Phi(x,z)$ $\displaystyle\to
e^{i\boldsymbol{g}\boldsymbol{\theta}(x,z)}\Phi(x,z),$ $\displaystyle
i\boldsymbol{g}\boldsymbol{\mathcal{W}}_{M}(x,z)$ $\displaystyle\to
e^{i\boldsymbol{g}\boldsymbol{\theta}(x,z)}\left(\partial_{M}+i\boldsymbol{g}\boldsymbol{\mathcal{W}}_{M}(x,z)\right)e^{-i\boldsymbol{g}\boldsymbol{\theta}(x,z)},$
(15)
where
$\boldsymbol{g}\boldsymbol{\theta}(x,z):=\sum_{A}g_{A}\theta^{A}(x,z)T^{A}$,
does not change the boundary conditions on the gauge fields (7) when and only
when all the gauge parameters satisfy the von Neumann condition:
$\displaystyle\partial_{z}\theta^{A}(x,\pm L/2)$ $\displaystyle=0.$ (16)
However, for a general non-zero Dirichlet boundary condition on Higgs (8), it
appears that the broken gauge parameter for $SU(2)_{W}/U(1)_{\text{EM}}$ must
also obey the (vanishing) Dirichlet condition $\theta^{A}(x,\pm L/2)=0$,
which, with Eq. (16), shows that $\theta^{A}(x,z)=0$ everywhere. It looks as
if the symmetry breaking by the conditions (7) and (8) were an explicit
breaking and there remained no $SU(2)_{W}$ symmetry even in the bulk of five-
dimensional space. By this classical argument, the theory looks pathetic. How
can we overcome this difficulty?
The key observation is that the Dirichlet boundary condition on the Higgs
field fluctuation (14) remains to be Dirichlet when multiplied by a function
$\theta(x,z)$ with von Neumann condition (16), that is, the condition (14) is
preserved by the von Neumann transformation $\theta(x,z)$. We will see how
this observation is implemented as the nilpotent BRST transformation in the
following.
## 3 Background and BRST transformations
In this section, we briefly outline how the theory is consistently defined. A
reader who is not interested in formal consistency may skip the entire
section.
### 3.1 Background-field $R_{\xi}$ gauge fixing
We employ the following gauge fixing, the background-field $R_{\xi}$ gauge:
$\displaystyle S_{\xi}$ $\displaystyle=\int
d^{4}x\int_{-L/2}^{L/2}dz\left[-{1\over 2\xi}f^{A}f^{A}\right],$ (17)
with $A$ running for $1,2,3,Y$ and the gauge fixing function being given by
$\displaystyle f^{A}$ $\displaystyle:=D_{\mu}^{{}^{\rm
c}}\mathcal{W}^{{{}^{\prime}}A\mu}+\xi D_{z}^{{}^{\rm
c}}\mathcal{W}^{{{}^{\prime}}Az}+ig_{A}\xi\left((\Phi^{{}^{\prime}})^{\dagger}T^{A}\Phi^{{}^{\rm
c}}-(\Phi^{{}^{\rm c}})^{\dagger}T^{A}\Phi^{{}^{\prime}}\right),$ (18)
where $g_{1}=g_{2}=g_{3}=:g$, $T^{Y}:=Y$,
$\mathcal{W}^{Y}_{M}:=\mathcal{B}_{M}$, $\xi$ is a dimensionless positive
constant, and we define the background covariant derivative on an arbitrary
$SU(2)_{W}$ adjoint field $\Phi_{\text{ad}}$ as $D^{{}^{\rm
c}}_{M}\Phi_{\text{ad}}:=\partial_{M}\Phi_{\text{ad}}+ig[\mathcal{W}^{{}^{\rm
c}}_{M},\Phi_{\text{ad}}]$. Note that $D^{{}^{\rm
c}}_{M}\mathcal{B}_{N}=\partial_{M}\mathcal{B}_{N}$.
The true gauge transformation that is fixed by the gauge choice (18) is, in
its infinitesimal form,
$\displaystyle\Delta^{\text{true}}_{\boldsymbol{\epsilon}}\mathcal{W}_{M}^{{{}^{\prime}}A}$
$\displaystyle=-D^{{}^{\rm
c}}_{M}\epsilon^{A}+i[\boldsymbol{g}{\boldsymbol{\epsilon}},\boldsymbol{\mathcal{W}}^{{}^{\prime}}_{M}]^{A},$
$\displaystyle\Delta^{\text{true}}_{\boldsymbol{\epsilon}}\mathcal{W}_{M}^{{{}^{\rm
c}}A}$ $\displaystyle=0,$
$\displaystyle\Delta^{\text{true}}_{\boldsymbol{\epsilon}}\Phi^{{}^{\prime}}$
$\displaystyle=i\boldsymbol{g}{\boldsymbol{\epsilon}}\,(\Phi^{{}^{\rm
c}}+\Phi^{{}^{\prime}}),$
$\displaystyle\Delta^{\text{true}}_{\boldsymbol{\epsilon}}\Phi^{{}^{\rm c}}$
$\displaystyle=0,$ (19)
with $\boldsymbol{g}{\boldsymbol{\epsilon}}:=\sum_{A}g_{A}\epsilon^{A}T^{A}$,
from which the ghost Lagrangian can be read off as
$\displaystyle\mathcal{L}_{\omega}$
$\displaystyle=-\bar{\omega}^{A}\Delta^{\text{true}}_{\boldsymbol{\omega}}f^{A}$
$\displaystyle=-\bar{\omega}^{A}D^{{{}^{\rm c}}\mu}\left(-D^{{}^{\rm
c}}_{\mu}\omega^{A}+i[\boldsymbol{g}\boldsymbol{\omega},\boldsymbol{\mathcal{W}}^{{}^{\prime}}_{\mu}]^{A}\right)-\xi\bar{\omega}^{A}D_{z}^{{}^{\rm
c}}\left(-D^{{}^{\rm
c}}_{z}\epsilon^{A}+i[\boldsymbol{g}\boldsymbol{\omega},\boldsymbol{\mathcal{W}}^{{}^{\prime}}_{z}]^{A}\right)$
$\displaystyle\quad-\xi\left(-(\Phi^{{}^{\rm
c}}+\Phi^{{}^{\prime}})^{\dagger}\left(\boldsymbol{g}\boldsymbol{\omega}\right)\left(\boldsymbol{g}\bar{\boldsymbol{\omega}}\right)\Phi^{{}^{\rm
c}}+(\Phi^{{}^{\rm
c}})^{\dagger}\left(\boldsymbol{g}\bar{\boldsymbol{\omega}}\right)\left(\boldsymbol{g}\boldsymbol{\omega}\right)(\Phi^{{}^{\rm
c}}+\Phi^{{}^{\prime}})\right),$ (20)
where $\boldsymbol{g}\boldsymbol{\omega}:=\sum_{A}g_{A}\omega^{A}T^{A}$ and
$\boldsymbol{g}\bar{\boldsymbol{\omega}}:=\sum_{A}g_{A}\bar{\omega}^{A}T^{A}$.
The background gauge transformation is given, with
$\boldsymbol{\omega}:=\sum_{A}\omega^{A}T^{A}$, by
$\displaystyle\delta\mathcal{W}_{M}^{{{}^{\prime}}A}$
$\displaystyle=i[\boldsymbol{g}{\boldsymbol{\epsilon}},\boldsymbol{\mathcal{W}}^{{}^{\prime}}_{M}]^{A},$
$\displaystyle\delta\mathcal{W}_{M}^{{{}^{\rm c}}A}$
$\displaystyle=-D^{{}^{\rm c}}_{M}\epsilon^{A},$
$\displaystyle\delta\Phi^{{}^{\prime}}$
$\displaystyle=i\boldsymbol{g}{\boldsymbol{\epsilon}}\Phi^{{}^{\prime}},$
$\displaystyle\delta\Phi^{{}^{\rm c}}$
$\displaystyle=i\boldsymbol{g}{\boldsymbol{\epsilon}}\Phi^{{}^{\rm c}},$
$\displaystyle\delta\omega^{{{}^{\prime}}A}$
$\displaystyle=i[\boldsymbol{g}{\boldsymbol{\epsilon}},\boldsymbol{\omega}^{{}^{\prime}}]^{A},$
$\displaystyle\delta\omega^{{{}^{\rm c}}A}$
$\displaystyle=i[\boldsymbol{g}{\boldsymbol{\epsilon}},\boldsymbol{\omega}^{{}^{\rm
c}}]^{A},$ $\displaystyle\delta\bar{\omega}^{{{}^{\prime}}A}$
$\displaystyle=i[\boldsymbol{g}{\boldsymbol{\epsilon}},\boldsymbol{\bar{\omega}}^{{}^{\prime}}]^{A},$
$\displaystyle\delta\bar{\omega}^{{{}^{\rm c}}A}$
$\displaystyle=i[\boldsymbol{g}{\boldsymbol{\epsilon}},\boldsymbol{\bar{\omega}}^{{}^{\rm
c}}]^{A},$ (21)
which transforms (anti-)ghost and the quantum fluctuation
$\mathcal{W}_{M}^{\prime}$ as adjoint and leaves the ghost Lagrangian (20)
manifestly invariant. Noting that the background transformation (21) varies
the gauge-fixing function as adjoint:
$\displaystyle\delta f^{A}$
$\displaystyle=i[\boldsymbol{g}{\boldsymbol{\epsilon}},\boldsymbol{f}]^{A},$
(22)
we find that the total action, i.e. the gauge fixing action (17) as well as
the original gauge (2) and Higgs (5) actions are invariant under the
background gauge transformation (21).
Note that the rotated field by the transformation (21) satisfies the following
boundary condition:
$\displaystyle\Phi^{{}^{\prime}}(x,\pm L/2)^{\text{new}}$ $\displaystyle=0,$
(23) $\displaystyle\Phi^{{}^{\rm c}}(x,\pm L/2)^{\text{new}}$
$\displaystyle=e^{i\boldsymbol{g}{\boldsymbol{\epsilon}}(x,\pm
L/2)}\Phi^{{}^{\rm c}}(x,\pm
L/2)=e^{i\boldsymbol{g}{\boldsymbol{\epsilon}}(x,\pm L/2)}\Phi_{D},$ (24)
that is, the quantum fluctuation does not change its boundary condition (b.c.)
by the background transformation though the vev does change its b.c. into
$\displaystyle\Phi_{D}^{\text{new}}(x,\pm L/2)$
$\displaystyle=e^{i\boldsymbol{g}{\boldsymbol{\epsilon}}(x,\pm L/2)}\Phi_{D}.$
(25)
This is natural since the background transformation (21) rotates the vevs
$\Phi^{{}^{\rm c}}$ and $\mathcal{A}_{M}^{{}^{\rm c}}$ and hence should be
regarded as a field redefinition, unlike the true gauge transformation (19).
The field redefinition certainly must change the b.c.
When we consider a background transformation (namely field redefinition) that
respects the KK parity $\epsilon^{A}(x,L/2)=\epsilon^{A}(x,-L/2)$, the rotated
boundary conditions remain to respect it too
$\Phi_{D}^{\text{new}}(x,L/2)=\Phi_{D}^{\text{new}}(x,-L/2)$. In particular,
by a global background transformation
$\displaystyle\Phi^{{}^{\prime}}(x,z)$ $\displaystyle\to
e^{i\boldsymbol{g}{\boldsymbol{\epsilon}}}\Phi^{{}^{\prime}}(x,z),$
$\displaystyle\Phi^{{}^{\rm c}}(x,z)$ $\displaystyle\to
e^{i\boldsymbol{g}{\boldsymbol{\epsilon}}}\Phi^{{}^{\rm c}}(x,z),$
$\displaystyle\boldsymbol{\mathcal{W}}^{{}^{\rm c}}_{M}(x,z)$
$\displaystyle\to
e^{i\boldsymbol{g}{\boldsymbol{\epsilon}}}\boldsymbol{\mathcal{W}}^{{}^{\rm
c}}_{M}(x,z)e^{-i\boldsymbol{g}{\boldsymbol{\epsilon}}},$
$\displaystyle\boldsymbol{\mathcal{W}}^{{}^{\prime}}_{M}(x,z)$
$\displaystyle\to
e^{i\boldsymbol{g}{\boldsymbol{\epsilon}}}\boldsymbol{\mathcal{W}}^{{}^{\prime}}_{M}(x,z)e^{-i\boldsymbol{g}{\boldsymbol{\epsilon}}},$
(26)
the boundary condition for Higgs can always be rotated to the form (9).
### 3.2 BRST invariance
The bulk BRST transformation can be introduced quite the same way as in the
four-dimensional (4D) gauge theory. On physical degrees of freedom, it is
defined as a true gauge transformation with its gauge parameter being replaced
by the ghost field:
$\displaystyle s\mathcal{W}_{M}^{{{}^{\prime}}A}$
$\displaystyle=-\partial_{M}\omega^{A}+i[\boldsymbol{g}\boldsymbol{\omega},\boldsymbol{\mathcal{W}}^{{}^{\prime}}_{M}]^{A},$
$\displaystyle s\mathcal{W}_{M}^{{{}^{\rm c}}A}$ $\displaystyle=0,$
$\displaystyle s\Phi^{{}^{\prime}}$
$\displaystyle=i\boldsymbol{g}\boldsymbol{\omega}\,(\Phi^{{}^{\rm
c}}+\Phi^{{}^{\prime}}),$ $\displaystyle s\Phi^{{}^{\rm c}}$
$\displaystyle=0.$ (27)
On unphysical fields, the BRST transformation reads
$\displaystyle s\omega^{A}$ $\displaystyle={i\over
2}[\boldsymbol{g}\boldsymbol{\omega},\boldsymbol{\omega}]^{A},$ $\displaystyle
s\bar{\omega}^{A}$ $\displaystyle=h^{A},$ $\displaystyle sh^{A}$
$\displaystyle=0,$ (28)
where we take $\omega^{{{}^{\rm c}}A}=\bar{\omega}^{{{}^{\rm c}}A}=h^{{{}^{\rm
c}}A}=0$ and drop ′ from the quantum fluctuations. We see that the action,
including the gauge fixing and ghost terms, is invariant under the BRST
transformation (27).
The only non-triviality here is the appearance of $\Phi^{{}^{\rm c}}$ in the
transformation of $\Phi^{{}^{\prime}}$ but it is still straightforward to show
the nilpotency of the BRST transformation on $\Phi^{{}^{\prime}}$. One might
worry that the flat configuration $\Phi^{{}^{\rm c}}$ is mixed with the
Dirichlet field $\Phi^{{}^{\prime}}$ after the transformation. To answer it,
we can KK-expand the transformation (27) and define it on the expanded fields.
More detailed explanation will be shown in a separate publication [30].222 In
[31], a higher-dimensional BRST symmetry is considered for orbifold gauge
theories. In [32], an orbifold GUT is studied with infinite number of 4D
gauge-fixing terms, where a BRST symmetry is proposed including the
corresponding infinite number of 4D ghosts, with its nilpotency being
untouched.
## 4 KK expansions
From now on, we choose the basis in which the b.c. becomes (9), which leads to
the vev
$\displaystyle\Phi^{{}^{\rm c}}(x,z)$ $\displaystyle=\begin{bmatrix}0\\\
{v/\sqrt{2}}\end{bmatrix},$ (29)
where
$v:=\sqrt{2}\left(|\varphi^{1}_{D}|^{2}+|\varphi^{2}_{D}|^{2}\right)^{1/2}$ in
terms of the original most general boundary condition (8). Let us rewrite the
Higgs fluctuation as:
$\displaystyle\Phi^{{}^{\prime}}(x,z)$
$\displaystyle=\begin{bmatrix}\chi^{+}(x,z)\\\
{\varphi(x,z)+i\chi(x,z)\over\sqrt{2}}\end{bmatrix},$ (30)
where we omit ′ from fluctuations $\varphi$, $\chi^{+}$, and $\chi$. The
boundary condition is now
$\displaystyle\chi^{\pm}(x,\pm L/2)=\varphi(x,\pm L/2)=\chi(x,\pm L/2)=0.$
(31)
On physical ground, we put $\mathcal{W}_{M}^{{{}^{\rm c}}A}=\omega^{{{}^{\rm
c}}A}=\bar{\omega}^{{{}^{\rm c}}A}=0$ hereafter (and drop ′ from the quantum
fluctuations unless otherwise stated).333 Since we are putting the (vanishing)
Dirichlet boundary condition on $\mathcal{W}^{A}_{z}$, we do not have
$\mathcal{W}^{{{}^{\rm c}}A}_{z}$ nor the Wilson line along the extra
dimension. Then gauge fields in the mass eigenbasis are, as usual,
$\displaystyle\mathcal{W}^{\pm}_{M}$
$\displaystyle:={1\over\sqrt{2}}\left(\mathcal{W}^{1}_{M}\mp
i\mathcal{W}^{2}_{M}\right),$ $\displaystyle\begin{bmatrix}\mathcal{Z}_{M}\\\
\mathcal{A}_{M}\end{bmatrix}$
$\displaystyle:=\begin{bmatrix}\cos\theta_{W}&-\sin\theta_{W}\\\
\sin\theta_{W}&\cos\theta_{W}\end{bmatrix}\begin{bmatrix}\mathcal{W}^{3}_{M}\\\
\mathcal{B}_{M}\end{bmatrix},$ (32)
where $\sin\theta_{W}:=g_{Y}/\sqrt{g^{2}+g_{Y}^{2}}$.
After some manipulations, all the von Neumann and Dirichlet fields $\Psi^{N}$
and $\Psi^{D}$, respectively, are KK-expanded as [30]444 In this notation, a
zero mode becomes canonically normalized in terms of a redefined field
$\psi^{N}_{n}(x)$, where the KK modes are normalized by
$\psi^{N}_{0}(x):=\Psi^{N}_{0}(x)/\sqrt{2}$ for $n=0$ and by
$\psi^{N}_{n}(x):=\Psi^{N}_{n}(x)$ for $n\neq 0$. We note that we are defining
the negative KK modes by $\Psi^{N}_{-n}(x)=\Psi^{N}_{n}(x)$ and
$\Psi^{D}_{-n}(x)=-\Psi^{D}_{n}(x)$, which is consistent with the choice of
the normalization $C_{-n}(z)=C_{n}(z)$ and $S_{-n}(z)=-S_{n}(z)$.
$\displaystyle\Psi^{N}(x,z)$
$\displaystyle=\sum_{n=-\infty}^{\infty}C_{n}(z)\Psi^{N}_{n}(x),$
$\displaystyle\Psi^{D}(x,z)$
$\displaystyle=\sum_{n=-\infty}^{\infty}S_{n}(z)\Psi^{D}_{n}(x),$ (33)
where
$\displaystyle C_{n}(z)$
$\displaystyle:={1\over\sqrt{2L}}\cos\\!\left[{n\pi\over L}\left(z+{L\over
2}\right)\right]={1\over\sqrt{2L}}\times\begin{cases}(-1)^{n\over 2}\cos{n\pi
z\over L}&\text{for $n$: even,}\\\ (-1)^{n+1\over 2}\sin{n\pi z\over
L}&\text{for $n$: odd,}\end{cases}$ $\displaystyle S_{n}(z)$
$\displaystyle:={1\over\sqrt{2L}}\sin\\!\left[{n\pi\over L}\left(z+{L\over
2}\right)\right]={1\over\sqrt{2L}}\times\begin{cases}(-1)^{n\over 2}\sin{n\pi
z\over L}&\text{for $n$: even,}\\\ (-1)^{n-1\over 2}\cos{n\pi z\over
L}&\text{for $n$: odd.}\end{cases}$ (34)
Concretely, the von Neumann boundary condition is satisfied by all the gauge
fields $\mathcal{W}^{\pm}_{\mu}$, $\mathcal{Z}_{\mu}$, $\mathcal{A}_{\mu}$ and
ghost fields (as well as all the quarks and leptons), whereas the (vanishing)
Dirichlet boundary condition is satisfied by all the Higgs fluctuations
$\varphi^{\pm}$, $\varphi$, $\chi$ and all the vector-scalars
$\mathcal{W}^{\pm}_{z}$, $\mathcal{Z}_{z}$, $\mathcal{A}_{z}$. A crucial point
is that fields with von Neumann and non-zero Dirichlet boundary conditions are
not necessarily orthogonal to each other though Dirichlet function is
orthogonal to Dirichlet ones and vice versa, as a line segment does not have
periodicity. This feature becomes important in the next section.
We find that the KK masses for physical degrees of freedom are [30]
$\displaystyle\mu^{2}_{W}$ $\displaystyle=m_{W}^{2}+{n^{2}\over R^{2}}$
$\displaystyle(n\geq 0),$ $\displaystyle\mu^{2}_{Z}$
$\displaystyle=m_{Z}^{2}+{n^{2}\over R^{2}}$ $\displaystyle(n\geq 0),$
$\displaystyle\mu^{2}_{\varphi}$ $\displaystyle={n^{2}\over R^{2}}$
$\displaystyle(n\geq 1).$ (35)
where $L=:\pi R$. Note that $S_{0}(z)=0$ and there are no zero mode for a
Dirichlet field. In particular, it is important that there is no zero mode for
the physical Higgs field $\varphi$ because it obeys the Dirichlet boundary
condition [24, 25]. Below we will see how the elastic scattering of
longitudinal $W^{+}W^{-}$ zero modes is unitarized in high energies in our
model where we do not have a Higgs zero mode.
## 5 Unitarity in elastic scattering
$A$,$Z$$W^{-}$$W^{+}$$W^{+}$$W^{-}$
$A$,$Z$$W^{-}$$W^{+}$$W^{+}$$W^{-}$
$W^{-}$$W^{+}$$W^{+}$$W^{-}$
Figure 1: SM gauge interactions involving only zero modes, where charges are
written as all incoming.
Let us consider the elastic scattering of longitudinal modes
$W^{+}_{L}W^{-}_{L}\to W^{+}_{L}W^{-}_{L}$. In the absence of the Higgs zero
mode, the SM contributions to the gauge boson scattering amplitude, shown in
Fig. 1, grows with energy as [33]
$\displaystyle\mathcal{M}^{\text{SM gauge only}}_{W_{L}^{+}W_{L}^{-}\to
W_{L}^{+}W_{L}^{-}}$ $\displaystyle={s\left(1+\cos\theta\right)\over
2v_{\text{EW}}^{2}}+\mathcal{O}(s^{0}),$ (36)
where $v_{\text{EW}}\simeq 246\,\text{GeV}$ is the electroweak scale, $\theta$
is the scattering angle in CM frame and $s$ is the Mandelstam variable. Note
that in our notation, $v_{\text{EW}}=v\sqrt{L}$.
In the Higgsless model, KK modes of the gauge fields served to unitarize this
high energy behavior. In our model with the KK parity respecting boundary
condition, no KK mode of gauge/vector-scalar fields can couple to the external
zero mode $W^{\pm}$ [30]. Then what can unitarize the $W^{+}_{L}W^{-}_{L}$
scattering in our model, where there are no zero mode Higgs? Hereafter, we
show that infinite tower of the Higgs KK modes $\varphi_{n}(x)$ do unitarize
the scattering of longitudinal modes.
### 5.1 KK Higgs exchange amplitude
In our model, the KK parity of the physical Higgs field becomes flipped from
that of a von Neumann field. Furthermore, as a result of non-orthogonality of
Dirichlet and von Neumann fields, the odd KK Higgs field can have a coupling
to the longitudinal $W^{\pm}$ zero mode:
$W^{-\nu}$$W^{+\mu}$$\varphi_{n}$ $\displaystyle={-2\sqrt{2}i\over
n\pi}g_{4}m_{W}\eta_{\mu\nu},$ (37)
where $n>0$ is a positive odd integer, $g_{4}:=g/\sqrt{L}$ is the four-
dimensional $SU(2)_{W}$ gauge coupling, and
$W^{\pm\mu}(x):=\mathcal{W}^{\pm\mu}_{0}(x)/\sqrt{2}$ is the canonically
normalized zero mode; see footnote 4. We note that the coupling of the $n$th
KK Higgs mode is multiplied by the factor $2\sqrt{2}/n\pi\simeq 0.9/n$. In
particular, the first KK Higgs mode coupling to all the zero mode SM fermions
and gauge bosons are multiplied by this factor $2\sqrt{2}/\pi\simeq 0.9$. We
will discuss below why this first KK Higgs behaves almost like the SM Higgs,
though it has no quartic coupling.
The $s$ and $t$-channel Higgs-exchange diagrams are shown in Fig. 2. In the
Feynman-’t Hooft gauge $\xi=1$, we can check that these are the only
additional diagrams and get
$\displaystyle\mathcal{M}_{W^{+}_{L}W^{-}_{L}\to W^{+}_{L}W^{-}_{L}}^{\text{KK
Higgs exchange}}$ $\displaystyle=-\sum_{\text{$n>0$,
odd}}{8g_{4}^{2}m_{W}^{2}\over n^{2}\pi^{2}}\left[{\left(1-{s\over
2m_{W}^{2}}\right)^{2}\over s-\left(n\over R\right)^{2}}+{\left(1+{2t\over
s-4m_{W}^{2}}{s\over 4m_{W}^{2}}\right)^{2}\over t-\left(n\over
R\right)^{2}}\right],$ (38)
where $t=-\left(s-4m_{W}^{2}\right)\left(1-\cos\theta\right)/2$. When we take
the hard scattering limit with large $s$ and fixed scattering angle $\theta$
for each contribution from the $n$th KK Higgs mode,
$\displaystyle\mathcal{M}_{W^{+}_{L}W^{-}_{L}\to W^{+}_{L}W^{-}_{L}}^{\text{KK
Higgs exchange}}$ $\displaystyle=-\sum_{\text{$n>0$, odd}}\left(2\sqrt{2}\over
n\pi\right)^{2}{s\left(1+\cos\theta\right)\over
2v_{\text{EW}}^{2}}+\mathcal{O}(s^{0}).$ (39)
As stated above, the first KK Higgs almost ($\simeq 81\%$) cancels the SM
gauge contribution (36) because the higher KK modes have smaller overlapping
with the von Neumann zero mode and the first one contributes most. This is why
the first KK Higgs behaves almost like the SM Higgs with all its coupling to
SM zero modes multiplied by ${2\sqrt{2}\over\pi}\simeq 0.9$. It almost
unitarizes the $WW$ scattering, hence it is almost a Higgs. Finally by
performing the summation, we get
$\displaystyle\mathcal{M}_{W^{+}_{L}W^{-}_{L}\to W^{+}_{L}W^{-}_{L}}^{\text{KK
Higgs exchange}}$ $\displaystyle=-{s\left(1+\cos\theta\right)\over
2v_{\text{EW}}^{2}}+\mathcal{O}(s^{0}),$ (40)
which exactly cancels and unitarizes the SM gauge contribution (36).
In general, an elastic scattering amplitude of massive gauge bosons is
expanded as
$\displaystyle\mathcal{M}$
$\displaystyle={s^{2}\over{v_{\text{EW}}^{4}}}\mathcal{M}^{(4)}+{s\over{v_{\text{EW}}^{2}}}\mathcal{M}^{(2)}+\mathcal{M}^{(0)}+\mathcal{O}\left({v_{\text{EW}}^{2}}\over
s\right).$ (41)
If the non-zero Dirichlet b.c. were not put on a Higgs, $v_{\text{EW}}$ would
be replaced by $m_{\text{KK}}:=1/R$ generally in Eq. (41). In such an
expansion, cancelation of $\mathcal{O}(s^{2})$ and $\mathcal{O}(s)$ terms has
been shown for gauge theories on $S^{1}/Z_{2}$ [34], for an electroweak
$SU(3)_{W}$ model555 The bulk $SU(3)_{W}$ is broken down to $SU(2)_{W}\times
U(1)_{Y}=:G_{\text{SM}}$, and the high energy scattering unitarity of KK gauge
bosons $W^{(1/2)}$, which belong to the broken non-SM sector
$SU(3)_{W}/G_{\text{SM}}$, is verified under the assumption that $W^{(1/2)}$
had the same interaction to $\gamma,Z$ as that of the SM $W^{\pm}$ living in
the unbroken $G_{\text{SM}}$ [35]. and an $SU(5)$ GUT model on the orbifold
$S^{1}/Z_{2}$ [35], and for Higgsless models on $S^{1}/Z_{2}$ [36] and on a
line segment [37], all of which are equivalent to taking the limit (39) before
summation. In our case, we have seen that the terms of $\mathcal{O}(s^{2})$
cancels within SM gauge amplitudes, while the sum over the terms of
$\mathcal{O}(s)$ from the SM gauge sector (36) is canceled by the infinite sum
over all the odd-$n$ KK Higgs modes (40). Actually, we can go one step further
from the analysis of Refs. [34, 35, 36, 37]. Let us see it below.
$\varphi_{n}$$W^{-}$$W^{+}$$W^{+}$$W^{-}$
$\varphi_{n}$$W^{-}$$W^{+}$$W^{+}$$W^{-}$
Figure 2: $s$ and $t$-channel KK Higgs exchange diagrams, where charge
convention is given as in Fig. 1. $n>0$ is odd.
One might still worry that the high energy limit $s\to\infty$ is taken before
the infinite summation. We can indeed exactly perform the infinite sum before
taking the limit, so as not to spoil five-dimensional symmetries:
$\displaystyle\mathcal{M}_{W^{+}_{L}W^{-}_{L}\to W^{+}_{L}W^{-}_{L}}^{\text{KK
Higgs exchange}}$ $\displaystyle=-{s\over
v_{\text{EW}}^{2}}\left(1-{2m_{W}^{2}\over s}\right)^{2}\left[1-{2\over\pi
R\sqrt{s}}\tan{\pi R\sqrt{s}\over 2}\right]$
$\displaystyle\quad+{\left|t\right|\over v_{\text{EW}}^{2}}\left({1\over
1-{4m_{W}^{2}\over
s}}-{2m_{W}^{2}\over\left|t\right|}\right)^{2}\left[1-{2\over\pi
R\sqrt{\left|t\right|}}\tanh{\pi R\sqrt{\left|t\right|}\over 2}\right],$ (42)
where $-s+4m_{W}^{2}\leq t\leq 0$. In the hard scattering limit $s\to\infty$
with fixed scattering angle $\theta$, the hyperbolic tangent goes to unity
exponentially:
$\displaystyle\tanh{{\pi R\sqrt{\left|t\right|}\over 2}}\to 1.$ (43)
How about the tangent: $\tan{\pi R\sqrt{s}\over 2}$? We see that there appear
poles at $\sqrt{s}=n/R=:m_{n}$ ($n=1,3,\dots$), which are nothing but the
remnant of the $s$-channel $\varphi_{n}$ resonance production. In string
theory, we know how to treat this kind of infinite number of poles. If we take
the higher loop corrections into account, these poles on the real axis of
complex $s$ plane will be shifted to
$\displaystyle{1\over s-m_{n}^{2}}\to{1\over s-m_{n}^{2}+im_{n}\Gamma_{n}},$
(44)
where $\Gamma_{n}$ is the decay rate of the $\varphi_{n}$ resonance. Under a
mild assumption that the decay rate increases with $m_{n}$ at least linearly,
effect of such a decay width can be taken into account by slightly shifting
the contour of the large $s$ limit: $s\to(1+i\epsilon)\infty$ where the
positive constant $\epsilon$ can be taken arbitrary small but must be kept
finite.666 We note that in our model, the decay rate of the resonance into
$W^{\pm}$ pair is indeed sizable already at the lowest KK Higgs mode [24]:
$\displaystyle\Gamma_{\varphi_{1}\to W^{+}W^{-}}$
$\displaystyle=\left(2\sqrt{2}\over\pi\right)^{2}{g_{4}^{2}m_{H}^{3}\over
64\pi m_{W}^{2}}\left(1-{2m_{W}^{2}\over
m_{H}^{2}}\right)^{2}\sqrt{1-{4m_{W}^{2}\over m_{H}^{2}}},$ where we note that
the mass of this first KK Higgs, the “Higgs impostor,” is exactly the KK
scale: $m_{H}=1/R$. This type of limit is taken when we get the Regge and
hard scattering limits from the tree-level string amplitude. See e.g. [38] for
more detailed discussion. By this prescription, we get the exponential limit:
$\displaystyle\tan{\pi R\sqrt{s}\over 2}\to-1,$ (45)
and finally
$\displaystyle\mathcal{M}_{W^{+}_{L}W^{-}_{L}\to W^{+}_{L}W^{-}_{L}}^{\text{KK
Higgs exchange}}$ $\displaystyle\to-{s\left(1+\cos\theta\right)\over
2v_{\text{EW}}^{2}}-{\sqrt{2s}\over v_{\text{EW}}^{2}\pi
R}\left(\sqrt{2}+\sqrt{1-\cos\theta}\right)+\mathcal{O}(s^{0}).$ (46)
That is, the total amplitude becomes
$\displaystyle\mathcal{M}_{W^{+}_{L}W^{-}_{L}\to W^{+}_{L}W^{-}_{L}}$
$\displaystyle\to-{\sqrt{2s}\over v_{\text{EW}}^{2}\pi
R}\left(\sqrt{2}+\sqrt{1-\cos\theta}\right)+\mathcal{O}(s^{0}).$ (47)
A differential cross section in CM frame is written, when all the masses for
incoming and outgoing four particles are equal, as
$\displaystyle{d\sigma\over d\Omega}$ $\displaystyle={1\over
64\pi^{2}s}\left|\mathcal{M}\right|^{2},$ (48)
and we get the elastic cross section that takes the dominant KK Higgs-exchange
contribution into account:
$\displaystyle\sigma_{W^{+}_{L}W^{-}_{L}\to
W^{+}_{L}W^{-}_{L}}=2\pi\int_{-1}^{1}d\cos\theta{d\sigma\over d\Omega}={1\over
32\pi s}\int_{-1}^{1}d\cos\theta\left|\mathcal{M}\right|^{2}\to{17\over
24\pi^{3}v_{\text{EW}}^{4}R^{2}}.$ (49)
We see that the tree-level elastic cross section remains constant in the high
energy limit and hence is marginally unitarized.
In the literature the question of the unitarity of $WW$ scattering is
typically addressed using the Nambu–Goldstone (NG) boson equivalence theorem.
Following [24, 25], one may speculate that the NG boson that is absorbed by
the gauge zero mode $W^{\mu}_{0}$ is an infinite sum:
$\chi_{\text{NG}}=\sum_{\text{$n$: odd}}{2\over n\pi}\widetilde{\chi}_{n}$,
with each $\widetilde{\chi}_{n}$ being a linear combination of $\chi_{n}$ and
$W^{z}_{n}$. To prove that, one has to compute an infinite number of KK-number
violating scattering amplitudes and sum them up correctly. In this paper we
have restricted ourselves to the simpler analysis as is presented above.
We have found the growing amplitude with energy $\mathcal{M}\propto\sqrt{s}$
after summing over infinite KK modes, though the original amplitude is
expanded as Eq. (41) and does not have such half power of $s$, where
$\mathcal{M}^{(4)}$ cancels within SM gauge amplitudes, while we have seen
that $\mathcal{M}^{(2)}$ cancels between the sum of SM gauge amplitudes (36)
and that of the KK Higgs amplitudes (40). This half power arises when one sums
over infinite KK modes and can be interpreted as follows.777 The half power
$\sqrt{s}$ resides within the terms proportional to (hyperbolic) tangent in
Eq. (42). The poles $\sqrt{s}=1/R,3/R,\dots$ in tangent correspond to the
resonances as is explained in Eq. (44). In KK picture, the half power could be
interpreted as the effect of taking into account the width. This behavior
should appear even when one considers scattering with Euclidean external
momenta. When we sum over infinite KK modes, we see a scattering within full
five-dimensional bulk. In five dimensions, the gauge coupling has mass
dimension $[g]=-1/2$ and hence from naive dimension counting, we expect
$\displaystyle\mathcal{M}^{\text{naive}}\sim g^{2}\sqrt{s}\sim{\sqrt{s}\over
v_{\text{EW}}^{2}R}{m_{W}^{2}\over m_{\text{KK}}^{2}},$ (50)
which is what we have found in Eq. (47), up to the extra factor
$m_{\text{KK}}^{2}/m_{W}^{2}$ to be multiplied.
### 5.2 Partial-wave unitarity
Let us expand the $W^{+}_{L}W^{-}_{L}$ scattering amplitude into partial-waves
$\displaystyle\mathcal{M}(s,\cos\theta)$
$\displaystyle=\sum_{J=0}^{\infty}\left(2J+1\right)\mathcal{M}_{J}(s)P_{J}(\cos\theta),$
(51)
where the $J$th partial amplitude is obtained inversely
$\displaystyle\mathcal{M}_{J}(s)$ $\displaystyle={1\over
2}\int_{-1}^{1}d\cos\theta\,P_{J}(\cos\theta)\mathcal{M}(s,\cos\theta).$ (52)
In the high energy limit (47), we get
$\displaystyle\mathcal{M}_{J}$ $\displaystyle=-{\sqrt{s}\over
v_{\text{EW}}^{2}\pi R}c_{J},$ (53)
where
$\displaystyle c_{J}$
$\displaystyle=\int_{-1}^{1}d\cos\theta\,P_{J}(\cos\theta)\left(1+\sqrt{1-\cos\theta\over
2}\right).$ (54)
Concretely, $c_{J}={10\over 3},-{4\over 15},-{4\over 105}\dots$ for
$J=0,1,2,\dots$, respectively.
The unitarity of the partial-wave amplitude reads
$\displaystyle\operatorname{Im}{\mathcal{M}_{J}}$
$\displaystyle\geq{\left|\boldsymbol{k}\right|\over
8\pi\sqrt{s}}\left|\mathcal{M}_{J}\right|^{2}\to{1\over
16\pi}\left|\mathcal{M}_{J}\right|^{2},$ (55)
where high energy limit $s\gg m_{W}^{2}$ is taken in the last step. At the
tree level, we do not have the imaginary part at all and it is customary to
use a corollary of the exact unitarity condition (55):
$\displaystyle 1$ $\displaystyle\geq{\left|\boldsymbol{k}\right|\over
8\pi\sqrt{s}}\left|\mathcal{M}_{J}\right|\to{1\over
16\pi}\left|\mathcal{M}_{J}\right|.$ (56)
This way, the tree-level partial-wave unitarity condition is, for the most
stringent $J=0$ partial-wave amplitude,
$\displaystyle\sqrt{s}<{24\pi^{2}v_{\text{EW}}^{2}\over
5m_{\text{KK}}}=:\Lambda,$ (57)
where $m_{\text{KK}}:=1/R$ is the first KK Higgs mass. Around the scale
$\Lambda$, higher loop corrections become important in the scattering, though
the gauge theory itself is still well defined as we can show that our theory
possesses a nilpotent BRST symmetry. When we require that there exists a weak
coupling region for, say, three KK modes: $\Lambda\gtrsim 3m_{\text{KK}}$, we
get
$\displaystyle m_{\text{KK}}\lesssim 980\,\text{GeV}.$ (58)
More concretely, for KK scales favored by the electroweak precision data
within 90% CL [24]: $m_{\text{KK}}=430$–$500\,\text{GeV}$, we get
$\displaystyle\Lambda=6.7\text{--}5.7\,\text{TeV},$ (59)
which are well beyond the corresponding KK scales, at least ten KK modes being
within tree-level unitarity range.
In this paper, we have concentrated on the elastic channels. In an analysis of
a Higgsless model [39], inclusion of the inelastic channels into KK $W$ bosons
leads to a lower 5D cutoff:
$\displaystyle\Lambda_{5}\sim\Lambda_{4}\sqrt{N_{\text{KK}}}$ (60)
than considering only the elastic ones in the Higgsless model, where
$N_{\text{KK}}$ is the number of KK modes below the 5D cutoff and
$\Lambda_{4}\simeq 2\,\text{TeV}$ is the cutoff of the four dimensional SM
without Higgs. (The relation (60) is consistent with the 5D Naive Dimensional
Analysis.) In Higgsless models, the second KK states must be much heavier than
twice the first KK mass to match the electroweak constraint. In contrast, our
model has the equal separation of the KK modes without contradicting to the
electroweak data: $N_{\text{KK}}=\Lambda_{5}/m_{\text{KK}}$. Therefore, the
relation (60) simply leads to $\Lambda_{5}\sim 8\,\text{TeV}$ for
$m_{\text{KK}}\simeq 500\,\text{GeV}$ in our case, which is the same order as
Eq. (59).
## 6 Summary
We have briefly sketched how the five-dimensional UED model, compactified on a
line segment, is consistently formulated when the EWSB is solely due to the
non-zero Dirichlet boundary conditions on the bulk Higgs field, in the limit
of vanishing bulk and boundary potentials. We have discussed how the elastic
scattering of the longitudinal $W^{+}W^{-}$ zero modes is unitarized, under
the absence of the Higgs zero mode, by showing that the sum over the
contribution of infinite tower of the KK Higgs modes exactly cancels the
$\mathcal{O}(s)$ contribution from the SM gauge sector. Further, we have
obtained the high energy limit taken _after_ summing over all the KK Higgs
modes, that exhibit the behavior $\mathcal{M}\propto\sqrt{s}$, which never
appears in four-dimensional level before summation and is genuinely five-
dimensional. Resultant tree-level partial-wave unitarity condition leads, for
a range favored by the electroweak precision data within 90% CL
$m_{\text{KK}}=430$–$500\,\text{GeV}$, to the strongly-coupled UV-cutoff scale
$\Lambda=6.7$–$5.7\,\text{TeV}$, which is well above the KK scale. Details of
our study and further discussions will be presented in a separate publication
[30].
### Acknowledgment
We are most grateful to Alex Pomarol for valuable comments. We appreciate
earlier discussions with Naoyuki Haba that brought attention to the unitarity
issues on the model. We also thank Tomohiro Abe, Arthur Hebecker, Victor Kim,
C.S. Lim, Hitoshi Murayama, Makoto Sakamoto, Marco Serone, and Ryo Takahashi
for useful discussions and Gianmassimo Tasinato, Yasuhiro Yamamoto and Ivonne
Zavala for helpful conversations. K.O. acknowledges the hospitality of the
particle theory group of Bonn University while this work is partly developed.
The stay of K.O. in Bonn University and CERN is financially supported in part
by the JSPS International Training Program of Osaka University. K.O. is
partially supported by Scientific Grant by Ministry of Education and Science
(Japan), Nos. 19740171 and 20244028.
## References
* [1] S. Weinberg, _A Model of Leptons_ , Phys. Rev. Lett. 19 (1967), 1264–1266.
* [2] A. Salam, in _Elementary Particle Theory_ (N. Svartholm, ed.), Almquist and Wiksells, Stockholm, 1969, p. 367.
* [3] S. L. Glashow, J. Iliopoulos, and L. Maiani, _Weak Interactions with Lepton-Hadron Symmetry_ , Phys. Rev. D2 (1970), 1285–1292.
* [4] P. W. Anderson, _PLASMONS, GAUGE INVARIANCE, AND MASS_ , Phys. Rev. 130 (1963), 439–442.
* [5] F. Englert and R. Brout, _BROKEN SYMMETRY AND THE MASS OF GAUGE VECTOR MESONS_ , Phys. Rev. Lett. 13 (1964), 321–322.
* [6] P. W. Higgs, _Broken symmetries, massless particles and gauge fields_ , Phys. Lett. 12 (1964), 132–133.
* [7] P. W. Higgs, _BROKEN SYMMETRIES AND THE MASSES OF GAUGE BOSONS_ , Phys. Rev. Lett. 13 (1964), 508–509.
* [8] G. S. Guralnik, C. R. Hagen, and T. W. B. Kibble, _GLOBAL CONSERVATION LAWS AND MASSLESS PARTICLES_ , Phys. Rev. Lett. 13 (1964), 585–587.
* [9] P. W. Higgs, _Spontaneous Symmetry Breakdown without Massless Bosons_ , Phys. Rev. 145 (1966), 1156–1163.
* [10] T. W. B. Kibble, _Symmetry breaking in non-Abelian gauge theories_ , Phys. Rev. 155 (1967), 1554–1561.
* [11] I. Low and J. Lykken, _Revealing the electroweak properties of a new scalar resonance_ , (2010), 1005.0872.
* [12] A. De Rujula, _To be or not to be: Higgs impostors at the LHC_ , (2010), 1005.2939.
* [13] H.-C. Cheng, _Little Higgs, Non-standard Higgs, No Higgs and All That_ , (2007), 0710.3407.
* [14] C. Grojean, _New approaches to electroweak symmetry breaking_ , Phys. Usp. 50 (2007), 1–35.
* [15] M. Serone, _The Higgs boson as a gauge field in extra dimensions_ , AIP Conf. Proc. 794 (2005), 139–142, hep-ph/0508019.
* [16] M. Serone, _Holographic Methods and Gauge-Higgs Unification in Flat Extra Dimensions_ , New J. Phys. 12 (2010), 075013, 0909.5619.
* [17] E. H. Simmons, R. S. Chivukula, H. J. He, M. Kurachi, and M. Tanabashi, _Higgsless models: Lessons from deconstruction_ , AIP Conf. Proc. 857 (2006), 34–45, hep-ph/0606019.
* [18] C. Csaki, J. Hubisz, and P. Meade, _Electroweak symmetry breaking from extra dimensions_ , (2005), hep-ph/0510275.
* [19] M. Schmaltz and D. Tucker-Smith, _Little Higgs Review_ , Ann. Rev. Nucl. Part. Sci. 55 (2005), 229–270, hep-ph/0502182.
* [20] M. Perelstein, _Little Higgs models and their phenomenology_ , Prog. Part. Nucl. Phys. 58 (2007), 247–291, hep-ph/0512128.
* [21] A. Djouadi, _The Anatomy of electro-weak symmetry breaking. II. The Higgs bosons in the minimal supersymmetric model_ , Phys. Rept. 459 (2008), 1–241, hep-ph/0503173.
* [22] F. Sannino, _Dynamical Stabilization of the Fermi Scale: Phase Diagram of Strongly Coupled Theories for (Minimal) Walking Technicolor and Unparticles_ , (2008), 0804.0182.
* [23] M. Piai, _Lectures on walking technicolor, holography and gauge/gravity dualities_ , (2010), 1004.0176.
* [24] N. Haba, K. Oda, and R. Takahashi, _Dirichlet Higgs in extra dimension, consistent with electroweak data_ , (2009), 0910.3356, v5.
* [25] N. Haba, K. Oda, and R. Takahashi, _Phenomenological Aspects of Invisibly Broad Higgs Model from Extra-Dimension_ , JHEP 07 (2010), 079, 1005.2306.
* [26] T. Appelquist, H.-C. Cheng, and B. A. Dobrescu, _Bounds on universal extra dimensions_ , Phys. Rev. D64 (2001), 035002, hep-ph/0012100.
* [27] T. Appelquist and H.-U. Yee, _Universal extra dimensions and the Higgs boson mass_ , Phys. Rev. D67 (2003), 055002, hep-ph/0211023.
* [28] N. Haba, K. Oda, and R. Takahashi, _Top Yukawa Deviation in Extra Dimension_ , Nucl. Phys. B821 (2009), 74–128, 0904.3813.
* [29] A. Denner, G. Weiglein, and S. Dittmaier, _Application of the background field method to the electroweak standard model_ , Nucl. Phys. B440 (1995), 95–128, hep-ph/9410338.
* [30] K. Nishiwaki and K. Oda, _Unitarity and BRST invariance in Dirichlet Higgs model_ , in preparation.
* [31] T. Ohl and C. Schwinn, _Unitarity, BRST symmetry and Ward identities in orbifold gauge theories_ , Phys. Rev. D70 (2004), 045019, hep-ph/0312263.
* [32] Y. Abe et al., _4D equivalence theorem and gauge symmetry on orbifold_ , Prog. Theor. Phys. 113 (2005), 199–213, hep-th/0402146.
* [33] B. W. Lee, C. Quigg, and H. B. Thacker, _Weak Interactions at Very High-Energies: The Role of the Higgs Boson Mass_ , Phys. Rev. D16 (1977), 1519.
* [34] R. S. Chivukula, D. A. Dicus, and H.-J. He, _Unitarity of compactified five-dimensional Yang-Mills theory_ , Phys. Lett. B525 (2002), 175–182, hep-ph/0111016.
* [35] Y. Abe, N. Haba, Y. Higashide, K. Kobayashi, and M. Matsunaga, _Unitarity in gauge symmetry breaking on orbifold_ , Prog. Theor. Phys. 109 (2003), 831–842, hep-th/0302115.
* [36] R. S. Chivukula, D. A. Dicus, H.-J. He, and S. Nandi, _Unitarity of the higher-dimensional standard model_ , Phys. Lett. B562 (2003), 109–117, hep-ph/0302263.
* [37] C. Csaki, C. Grojean, H. Murayama, L. Pilo, and J. Terning, _Gauge theories on an interval: Unitarity without a Higgs_ , Phys. Rev. D69 (2004), 055006, hep-ph/0305237.
* [38] T. Matsuo and K. Oda, _Geometric cross sections of rotating strings and black holes_ , Phys. Rev. D79 (2009), 026003, 0808.3645.
* [39] M. Papucci, _NDA and perturbativity in Higgsless models_ , (2004), hep-ph/0408058.
|
arxiv-papers
| 2010-11-01T18:24:41 |
2024-09-04T02:49:14.408118
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Kin-ya Oda and Kenji Nishiwaki",
"submitter": "Kin-ya Oda",
"url": "https://arxiv.org/abs/1011.0405"
}
|
1011.0620
|
# Rainbow Connection Number and Radius
Manu Basavaraju Department of Computer Science and Automation,
Indian Institute of Science,
Bangalore -560012, India.
{manu, sunil, deepakr, arunselvan}@csa.iisc.ernet.in L. Sunil Chandran
Department of Computer Science and Automation,
Indian Institute of Science,
Bangalore -560012, India.
{manu, sunil, deepakr, arunselvan}@csa.iisc.ernet.in Deepak Rajendraprasad
Partially supported by Microsoft Research India - PhD Fellowship. Department
of Computer Science and Automation,
Indian Institute of Science,
Bangalore -560012, India.
{manu, sunil, deepakr, arunselvan}@csa.iisc.ernet.in Arunselvan Ramaswamy
Department of Computer Science and Automation,
Indian Institute of Science,
Bangalore -560012, India.
{manu, sunil, deepakr, arunselvan}@csa.iisc.ernet.in
###### Abstract
The rainbow connection number, $rc(G)$, of a connected graph $G$ is the
minimum number of colours needed to colour its edges, so that every pair of
its vertices is connected by at least one path in which no two edges are
coloured the same. In this note we show that for every bridgeless graph $G$
with radius $r$, $rc(G)\leq r(r+2)$. We demonstrate that this bound is the
best possible for $rc(G)$ as a function of $r$, not just for bridgeless
graphs, but also for graphs of any stronger connectivity. It may be noted that
for a general $1$-connected graph $G$, $rc(G)$ can be arbitrarily larger than
its radius ($K_{1,n}$ for instance). We further show that for every bridgeless
graph $G$ with radius $r$ and chordality (size of a largest induced cycle)
$k$, $rc(G)\leq rk$. Hitherto, the only reported upper bound on the rainbow
connection number of bridgeless graphs is $4n/5-1$, where $n$ is order of the
graph [1].
It is known that computing $rc(G)$ is NP-Hard [2]. Here, we present a
$(r+3)$-factor approximation algorithm which runs in $O(nm)$ time and a
$(d+3)$-factor approximation algorithm which runs in $O(dm)$ time to rainbow
colour any connected graph $G$ on $n$ vertices, with $m$ edges, diameter $d$
and radius $r$.
Keywords: rainbow connectivity, rainbow colouring, radius, isometric cycle,
chordality, approximation algorithm.
## 1 Introduction
An edge colouring of a graph is a function from its edge set to the set of
natural numbers. A path in an edge coloured graph with no two edges sharing
the same colour is called a rainbow path. An edge coloured graph is said to be
rainbow connected if every pair of vertices is connected by at least one
rainbow path. Such a colouring is called a rainbow colouring of the graph. The
minimum number of colours required to rainbow colour a connected graph is
called its rainbow connection number, denoted by $rc(G)$. For example, the
rainbow connection number of a complete graph is $1$, that of a path is its
length, and that of a tree is its number of edges. For a basic introduction to
the topic, see Chapter $11$ in [5].
The concept of rainbow colouring was introduced in [4]. It was shown in [2]
that computing the rainbow connection number of a graph is NP-Hard. To rainbow
colour a graph, it is enough to ensure that every edge of some spanning tree
in the graph gets a distinct colour. Hence the order of the graph minus one is
an upper bound for its rainbow connection number. Many authors view rainbow
connectivity as one ‘quantifiable’ way of strengthening the connectivity
property of a graph [1, 2, 7]. Hence tighter upper bounds on the rainbow
connection number for graphs with higher connectivity have been a subject of
investigation. The following are the results in this direction reported in
literature: Let $G$ be a graph of order $n$. If $G$ is 2-edge-connected
(bridgeless), then $rc(G)\leq 4n/5-1$ and if $G$ is 2-vertex-connected, then
$rc(G)\leq\min\\{2n/3,n/2+O(\sqrt{n})\\}$ [1]. This was very recently improved
in [8], where it was shown that if $G$ is $2$-vertex-connected, then
$rc(G)\leq\lceil n/2\rceil$, which is the best possible upper bound for the
case. It also improved the previous best known upper bound of $3(n+1)/5$ for
$3$-vertex connected graphs [9]. It was shown in [7] that $rc(G)\leq
20n/\delta$ where $\delta$ is the minimum degree of $G$. The result was
improved in [3] where it was shown that $rc(G)\leq 3n/(\delta+1)+3$. Hence it
follows that $rc(G)\leq 3n/(\lambda+1)+3$ if $G$ is $\lambda$-edge-connected
and $rc(G)\leq 3n/(\kappa+1)+3$ if $G$ is $\kappa$-vertex-connected. It was
shown in [8] that the above bound in terms of edge connectivity is tight up to
additive constants and that the bound in terms of vertex connectivity can be
improved to $(2+\epsilon)n/\kappa+23/\epsilon^{2}$, for any $\epsilon>0$.
All the above upper bounds grow with $n$. The diameter of a graph, and hence
its radius, are obvious lower bounds for its rainbow connection number. Hence
it is interesting to see if there is an upper bound for rainbow connection
number which is a function of radius or diameter alone. Such upper bounds were
shown for some special graph classes in [3]. But, for a general graph, the
rainbow connection number cannot be upper bounded by a function of $r$ alone.
For instance, the star $K_{1,n}$ has radius $1$ but rainbow connection number
$n$. In fact, it is easy to see that the number of bridges in a graph is also
a lower bound on its rainbow connection number. Still, the question of whether
such an upper bound exists for graphs with higher connectivity remains. Here
we answer this question in the affirmative. In particular, we show that if $G$
is bridgeless, then $rc(G)\leq r(r+2)$ where $r$ is the radius of $G$
(Corollary 5). Moreover, we also demonstrate that the bound cannot be improved
even if we assume stronger connectivity (Example 6). The technique presented
in this paper of growing a connected multi-step dominating set was later
extended in [6] to show an upper bound for the rainbow connection number of a
general connected graph in terms of its radius and number of bridges.
Since the above bound is quadratic in $r$, we tried to see what additional
restriction would give an upper bound which is linear in $r$. To this end, we
show that if the size of isometric cycles or induced cycles in a graph is
bounded independently of $r$, then the rainbow connection number is linear in
$r$. In particular, we show that if $G$ is a bridgeless graph with radius $r$
and the size of a largest isometric cycle $\zeta$, then $rc(G)\leq r\zeta$
(Theorem 4). Since every isometric cycle is induced, it also follows that
$rc(G)\leq rk$ where $k$ is the chordality (size of a largest induced cycle)
of $G$ (Corollary 7).
Since computing $rc(G)$ is NP-Hard [2], it is natural to ask for approximation
algorithms for rainbow colouring a graph. Our proof for the $r(r+2)$ bound is
constructive and hence yields a $(r+2)$-factor approximation algorithm to
rainbow colour any bridgeless graph $G$ of radius $r$. Note that $r$ is a
lower bound on $rc(G)$ and hence the approximation factor. We show that this
algorithm runs in $O(nm)$ time, where $n$ and $m$ are the number of vertices
and edges of $G$ respectively. We also present an algorithm which has a
smaller running time of $O(dm)$ but with a slightly poorer approximation ratio
of $(d+2)$, where $d$ is the diameter of $G$. Both these algorithms are
described in Section 3.1. Bridges in a connected graph can be found in $O(m)$
time [10]. Contracting every bridge of a general connected graph gives a
bridgeless graph and its rainbow colouring can be extended to the original
graph by giving a new colour to every bridge. Using these ideas, we give a
$(r+3)$-factor approximation algorithm which runs in $O(nm)$ time and a
$(d+3)$-factor approximation algorithm which runs in $O(dm)$ time to rainbow
colour any connected graph $G$ on $n$ vertices, with $m$ edges, diameter $d$
and radius $r$ (Section 3.2).
### 1.1 Preliminaries
All the graphs considered in this article are finite, simple and undirected.
The length of a path $P$ is its number of edges and is denoted by $|P|$. An
edge in a connected graph is called a bridge, if its removal disconnects the
graph. A connected graph with no bridges is called a bridgeless (or $2$-edge-
connected) graph. If $S$ is a subset of vertices of a graph $G$, the subgraph
of $G$ induced by the vertices in $S$ is denoted by $G[S]$. The graph obtained
by contracting the set $S$ into a single vertex $v_{S}$ is denoted by $G/S$.
The vertex set and edge set of $G$ are denoted by $V(G)$ and $E(G)$
respectively.
###### Definition 1.
Let $G$ be a connected graph. The distance between two vertices $u$ and $v$ in
$G$, denoted by $d_{G}(u,v)$ is the length of a shortest path between them in
$G$. The eccentricity of a vertex $v$ is $ecc(v):=\max_{x\in
V(G)}{d_{G}(v,x)}$. The diameter of $G$ is $diam(G):=\max_{x\in
V(G)}{ecc(x)}$. The radius of $G$ is $rad(G):=\min_{x\in V(G)}{ecc(x)}$. The
distance between a vertex $v$ and a set $S\subseteq V(G)$ is
$d_{G}(v,S):=\min_{x\in S}{d_{G}(v,x)}$. The neighbourhood of $S$ is
$N(S):=\\{x\in V(G)|d_{G}(x,S)=1\\}$.
###### Definition 2.
Given a graph $G$, a set $D\subseteq V(G)$ is called a $k$-step dominating set
of $G$, if every vertex in $G$ is at a distance at most $k$ from $D$. Further
if $G[D]$ is connected, then $D$ is called a connected $k$-step dominating set
of $G$.
###### Definition 3.
A subgraph $H$ of a graph $G$ is called isometric if the distance between any
pair of vertices in $H$ is the same as their distance in $G$. The size of a
largest isometric cycle in $G$ is denoted by $iso(G)$.
###### Definition 4.
A graph is called chordal if it contains no induced cycles of length greater
than $3$. The chordality of a graph $G$ is the length of a largest induced
cycle in $G$.
Note that every isometric cycle is induced and hence $iso(G)$ is at most the
chordality of $G$. Also note that $3\leq iso(G)\leq 2\cdot diam(G)+1$ for
every bridgeless graph $G$.
## 2 Upper Bounds for Bridgeless Graphs
The most important idea in this note is captured in Lemma 3 and all the upper
bounds reported here will follow easily from it. The next important idea in
this note, which is used in the construction of all the tight examples, is
illustrated in Theorem 4. Before stating Lemma 3, we state and prove two small
lemmas which are used in its proof.
###### Lemma 1.
For every edge $e$ in a graph $G$, any shortest cycle containing $e$ is
isometric.
###### Proof.
Let $C$ be a shortest cycle containing $e$. For contradiction, assume that
there exists at least one pair $(x,y)\in V(C)\times V(C)$ such that
$d_{G}(x,y)<d_{C}(x,y)$. Choose $(x,y)$ to be one with minimum $d_{G}(x,y)$
among all such pairs. Let $P$ be a shortest $x\mbox{--}y$ path in $G$. First
we show that $P\cap C=\\{x,y\\}$. If $P\cap C$ contains some vertex
$z\notin\\{x,y\\}$, then $d_{G}(x,z)+d_{G}(z,y)=d_{G}(x,y)<d_{C}(x,y)\leq
d_{C}(x,z)+d_{C}(z,y)$. First equality follows since $P$ is a shortest
$x\mbox{--}y$ path, the strict inequality follows by assumption and the last
is triangle inequality. Therefore, either $d_{G}(x,z)<d_{C}(x,z)$ or
$d_{G}(y,z)<d_{C}(y,z)$. This contradicts the choice of $(x,y)$. Now it is
easy to see that $P$ together with the segment of $C$ between $x$ and $y$
containing $e$ will form a cycle of length strictly smaller than $C$ and
containing $e$. This contradicts the minimality of $C$. Hence $C$ is
isometric. ∎∎
###### Definition 5.
Given a graph $G$ and a set $D\subset V(G)$, a $D$-ear is a path
$P=(x_{0},x_{1},\ldots,x_{p})$ in $G$ such that $P\cap D=\\{x_{0},x_{p}\\}$.
$P$ may be a closed path, in which case $x_{0}=x_{p}$. Further, $P$ is called
an acceptable $D$-ear if either $P$ is a shortest $D$-ear containing
$(x_{0},x_{1})$ or $P$ is a shortest $D$-ear containing $(x_{p-1},x_{p})$.
###### Lemma 2.
If $P$ is an acceptable $D$-ear in a graph $G$ for some $D\subset V(G)$, then
$d_{G}(x,D)=d_{P}(x,D)$ for every $x\in P$.
###### Proof.
Without loss of generality, let $P=(x_{0},x_{1},\ldots,x_{p})$ be a shortest
$D$-ear containing $e=(x_{0},x_{1})$. Let $G^{\prime}=G/D$ be the graph
obtained by contracting $D$ into a single vertex $v_{D}$. It is easy to see
that $P^{\prime}=(v_{D},x_{1},x_{2},\ldots,x_{p-1},v_{D})$ is a shortest cycle
in $G^{\prime}$ containing $e=(v_{D},x_{1})$. Hence by Lemma 1, $P^{\prime}$
is isometric in $G^{\prime}$. Now the result follows since
$d_{G}(x,D)=d_{G^{\prime}}(x,v_{D})$ and $d_{P}(x,D)=d_{P^{\prime}}(x,v_{D})$.
∎∎
###### Lemma 3.
If $G$ is a bridgeless graph, then for every connected $k$-step dominating set
$D^{k}$ of $G$, $k\geq 1$, there exists a connected $(k-1)$-step dominating
set $D^{k-1}\supset D^{k}$ such that
$rc(G[D^{k-1}])\leq rc(G[D^{k}])+\min\\{2k+1,\zeta\\},$
where $\zeta=iso(G)$.
###### Proof.
Given $D^{k}$, we rainbow colour $G[D^{k}]$ with $rc(G[D^{k}])$ colours. Let
$m=\min\\{2k+1,\zeta\\}$ and let $\mathcal{A}=\\{a_{1},a_{2},\ldots\\}$ and
$\mathcal{B}=\\{b_{1},b_{2},\ldots\\}$ be two pools of colours, none of which
are used to colour $G[D^{k}]$. A $D^{k}$-ear $P=(x_{0},x_{1},\ldots,x_{p})$
will be called evenly coloured if its edges are coloured
$a_{1},a_{2},\ldots,a_{\lceil\frac{p}{2}\rceil},b_{\lfloor\frac{p}{2}\rfloor},\ldots,b_{2},b_{1}$
in that order. We prove the lemma by constructing a sequence of sets
$D^{k}=D_{0}\subset D_{1}\subset\cdots\subset D_{t}=D^{k-1}$ such that
$D_{i+1}=D_{i}\cup P$, where $P$ is an acceptable $D^{k}$-ear and then
colouring $G[D_{i+1}]$ in such a way that $P$ is evenly coloured using at most
$m$ colours from $\mathcal{A}\cup\mathcal{B}$. In particular, this ensures
that every $x\in D_{i}\backslash D^{k}$, $0\leq i\leq t$, lies in an evenly
coloured acceptable $D^{k}$-ear throughout the construction.
If $N(D^{k})\subset D_{i}$, then $D_{i}$ is a $(k-1)$-step dominating set and
we stop the procedure by setting $t=i$. Otherwise pick any edge
$e=(x_{0},x_{1})\in D^{k}\times(N(D^{k})\backslash D_{i})$ of $G$ and let
$Q=(x_{0},x_{1},\ldots,x_{q})$ be a shortest $D_{k}$-ear containing $e$. Such
an ear always exists since $G$ is bridgeless. Let $x_{l}$ be the first vertex
of $Q$ in $D_{i}$. If $x_{l}=x_{q}$, then evenly colour $Q$. Hence $P=Q$ is an
evenly coloured acceptable $D^{k}$-ear. Otherwise $x_{l}$ is on some evenly
coloured acceptable $D^{k}$-ear $P^{\prime}$ added in an earlier iteration. By
Lemma 2, $d_{P^{\prime}}(x_{l},D^{k})=d_{G}(x_{l},D^{k})$. Hence the shorter
segment $R$ of $P^{\prime}$ (from $x_{l}$ to $D^{k}$) together with
$L=(x_{0},x_{1},\ldots,x_{l})$ is also an acceptable $D^{k}$-ear, $P=L\cup R$
containing $e$. Colour the edges of $L$ so that $P$ is evenly coloured. This
is possible because (i) $R$ uses colours exclusively from one pool
($|R|\leq\lfloor|P^{\prime}|/2\rfloor$, since it is a shorter segment of
$P^{\prime}$) and (ii) $R$ forms a shorter segment of $P$ ($|L|\geq
d_{G}(x_{l},D^{k})=|R|$, by Lemma 2). Hence the colouring of $R$ can be evenly
extended to $L$. Set $D_{i+1}=D_{i}\cup P$.
Firstly, we claim that at most $m$ new colours are used in the above procedure
for constructing $D^{k-1}$ from $D^{k}$. Since $D^{k}$ is a $k$-step
dominating set and since the $D^{k}$-ear $P=(x_{0},x_{1},\ldots,x_{p})$ added
in each iteration is acceptable, it follows that $|P|\leq 2k+1$. Otherwise a
middle vertex $x_{\lfloor\frac{p}{2}\rfloor}$ of $P$ will be at a distance
more than $k$ from $D^{k}$ (Lemma 2). Let $C$ be a shortest cycle containing
$e=(x_{0},x_{1})$. $C$ exists since $G$ is bridgeless. By Lemma 1, $C$ is
isometric and hence $|C|\leq\zeta$. Further, $|P|\leq|C|$ since a sub-path of
$C$ is a $D^{k}$-ear containing $e$. Thus $|P|\leq m=\min\\{2k+1,\zeta\\}$ in
every iteration. Hence all the new colours used in the procedure are from
$\\{a_{1},\ldots,a_{\lceil\frac{m}{2}\rceil}\\}\cup\\{b_{1},\ldots
b_{\lfloor\frac{m}{2}\rfloor}\\}$, i.e., at most $m$ new colours are used.
Next, we claim that the $G[D^{k-1}]$ constructed this way is rainbow
connected. Any pair $(x,y)\in D^{k}\times D^{k}$, is rainbow connected in
$G[D^{k}]$. For any pair $(x,y)\in(D^{k-1}\backslash D^{k})\times D^{k}$, let
$P=(x_{0},x_{1},\ldots,x_{i}=x,\ldots,x_{p})$ be the evenly coloured
(acceptable) $D^{k}$-ear containing $x$. Joining
$(x=x_{i},x_{i+1},\ldots,x_{p})$ with a $x_{p}\mbox{--}y$ rainbow path in
$G[D^{k}]$ gives a $x\mbox{--}y$ rainbow path. For any pair
$(x,y)\in(D^{k-1}\backslash D^{k})\times(D^{k-1}\backslash D^{k})$, let
$P=(x_{0},x_{1},\ldots,x_{i}=x,\ldots,x_{p})$ and
$Q=(y_{0},y_{1},\ldots,y_{j}=y,\ldots,y_{q})$ be evenly coloured (acceptable)
$D^{k}$-ears containing $x$ and $y$ respectively. Recall that the vertices of
$P$ and $Q$ are ordered in such a way that their first halves get colours from
Pool $\mathcal{A}$. We consider the following $4$ cases. If
$i\leq\lfloor\frac{p}{2}\rfloor$ and $j>\lfloor\frac{q}{2}\rfloor$, then
joining $(y=y_{j},y_{j+1}\ldots,y_{q})$ (which is $\mathcal{B}$-coloured) to
the $y_{q}\mbox{--}x_{0}$ rainbow path in $G[D^{k}]$ followed by
$(x_{0},x_{1},\ldots,x_{i}=x)$ (which is $\mathcal{A}$-coloured) gives a
$x\mbox{--}y$ rainbow path. Case when $i>\lfloor\frac{p}{2}\rfloor$ and
$j\leq\lfloor\frac{q}{2}\rfloor$ is similar. When
$i\leq\lfloor\frac{p}{2}\rfloor$ and $j\leq\lfloor\frac{q}{2}\rfloor$ check if
$i\leq j$. If yes, join $(y=y_{j},y_{j+1},\ldots,y_{q})$ (which uses colours
from $\\{a_{l}\in\mathcal{A}:l\geq j+1\\}\cup\mathcal{B}$) to the
$y_{q}\mbox{--}x_{0}$ rainbow path in $G[D^{k}]$ followed by
$(x_{0},x_{1},\ldots,x_{i}=x)$ (which uses colours from
$\\{a_{l}\in\mathcal{A}:l\leq i\\}$) to get an $x\mbox{--}y$ rainbow path. If
$i>j$, then do the reverse. In the final case, when
$i>\lfloor\frac{p}{2}\rfloor$ and $j>\lfloor\frac{q}{2}\rfloor$ check if
$q-j\leq p-i$. If yes, join $(y=y_{j},y_{j+1},\ldots,y_{q})$ (which uses
colours from $\\{b_{l}\in\mathcal{B}:l\leq q-j\\}$ to the
$y_{q}\mbox{--}x_{0}$ rainbow path in $G[D^{k}]$ followed by
$(x_{0},x_{1},\ldots,x_{i}=x)$ (which uses colours from
$\mathcal{A}\cup\\{b_{l}\in\mathcal{B}:l\geq p-i+1\\}$) to get an
$x\mbox{--}y$ rainbow path. If $q-j>p-i$, then do the reverse. Any edge in
$G[D^{k-1}]$ left uncoloured by the procedure can be assigned any used colour
to complete the rainbow colouring. ∎∎
###### Theorem 4.
For every bridgeless graph $G$,
$rc(G)\leq\sum_{i=1}^{r}{\min\\{2i+1,\zeta\\}}\leq r\zeta,$
where $r$ is the radius of $G$ and $\zeta=iso(G)$.
Moreover, for every two integers $r\geq 1$, and $3\leq\zeta\leq 2r+1$, there
exists a bridgeless graph $G$ with radius $r$ and $iso(G)=\zeta$ such that
$rc(G)=\sum_{i=1}^{r}{\min\\{2i+1,\zeta\\}}$.
###### Proof.
If $u$ is a central vertex of $G$, i.e., $ecc(u)=r$, then $D^{r}=\\{u\\}$ is
an $r$-step dominating set in $G$ and $rc(G[D^{r}])=0$. The only $0$-step
dominating set in $G$ is $V(G)$. Hence, repeated application of Lemma 3 gives
the upper bound
To construct a tight example for a given $r\geq 1$ and $3\leq\zeta\leq 2r+1$,
consider the graph $H_{r,\zeta}$ in Figure 1. Note that (i) $H_{r,\zeta}$ is
bridgeless, (ii) the size of largest isometric cycle in $H_{r,\zeta}$ is
$\zeta$, and (iii) $ecc(u)=r$ for any $\zeta\leq 2r+1$.
$u=x_{r}$$P_{r}$$x_{r-1}$$x_{2}$$P_{2}$$x_{1}$$P_{1}$$x_{0}=v$ Figure 1: Graph
$H_{r,\zeta}$. Every $P_{i}$ is a $x_{i-1}$–$x_{i}$ path of length
$|P_{i}|=\min\\{2i,\zeta-1\\}$.
Let $m:=\sum_{i=1}^{r}{\min\\{2i+1,\zeta\\}}$. Construct a graph $G$ by taking
$m^{r}+1$ graphs $\\{H^{j}\\}_{j=0}^{m^{r}}$ where $V(H^{j})=\\{x^{j}:x\in
V(H_{r,\zeta})\\}$ and $E(H^{j})=\\{\\{x^{j},y^{j}\\}:\\{x,y\\}\in
E(H_{r,\zeta})\\}$. Identify the vertex $u^{j}$ as common in every copy
($u=u^{j},0\leq j\leq m^{r}$). It can be easily verified that (i) $G$ is
bridgeless (ii) $rad(G)=r$ and (ii) size of the largest isometric cycle in $G$
is $\zeta$. Hence, by first part of this theorem, $k:=rc(G)\leq m$. In any
edge colouring $c:E(G)\rightarrow\\{1,2,\ldots,k\\}$ of $G$, each $r$-length
$u\mbox{--}v^{j}$ path can be coloured in at most $k^{r}$ different ways. By
pigeonhole principle, there exist $p\neq q$, $0\leq p,q\leq m^{r}$ such that
$c(e_{i}^{p})=c(e_{i}^{q}),1\leq i\leq r$ where
$e_{i}^{j}=(x_{i-1}^{j},x_{i}^{j})$. Consider any rainbow path $R$ between
$v^{p}$ and $v^{q}$. For every $1\leq i\leq r$,
$|R\cap\\{e_{i}^{p},e_{i}^{q}\\}|\leq 1$ (since $c(e_{i}^{p})=c(e_{i}^{q})$)
and hence $P_{i}^{j}\subset R$ for some $j\in\\{p,q\\}$. Thus
$|R|\geq\sum_{i=1}^{r}{(1+|P_{i}|)}=m$. Hence $k\geq m$ and $G$ gives the
required tight example. ∎∎
###### Corollary 5.
For every bridgeless graph $G$ with radius $r$,
$rc(G)\leq r(r+2).$
Moreover, for every integer $r\geq 1$, there exists a bridgeless graph with
radius $r$ and $rc(G)=r(r+2)$.
###### Proof.
Noting that $\min\\{2i+1,\zeta\\}\leq 2i+1$, the upper bound follows from
Theorem 4. The tight examples are obtained by setting $\zeta=2r+1$ in the
tight examples for Theorem 4 ∎∎
A natural question at this stage is whether the upper bound of $r(r+2)$ can be
improved if we assume a stronger connectivity for $G$. But the following
example shows that it is not the case.
###### Example 6 (Construction of a $\kappa$-connected graph of radius $r$
whose rainbow connection number is $r(r+2)$ for any two given integers
$\kappa,r\geq 1$).
Let $s(0):=0$, $s(i):=2\sum_{j=r}^{r-i+1}j$ for $1\leq i\leq r$ and
$t:=s(r)=r(r+1)$. Let $V=V_{0}\uplus V_{1}\uplus\cdots\uplus V_{t}$ where
$V_{i}=\\{x_{i,0},x_{i,1},\ldots,x_{i,\kappa-1}\\}$ for $0\leq i\leq t-1$ and
$V_{t}=\\{x_{t,0}\\}$. Construct a graph $X_{r,\kappa}$ on $V$ by adding the
following edges.
$E(X)=\\{\\{x_{i,j},x_{i^{\prime},j^{\prime}}\\}:|i-i^{\prime}|\leq
1\\}\cup\\{\\{x_{s(i),0},x_{s(i+1),0}\\}:0\leq i\leq r-1\\}.$ Figure 2 depicts
$X_{3,2}$.
$V_{0}$$V_{1}$$V_{t}$$x_{0,0}$$x_{1,0}$$x_{2,0}$$x_{3,0}$$x_{3,0}$$x_{4,0}$$x_{4,0}$$x_{5,0}$$x_{6,0}$$x_{7,0}$$x_{8,0}$$x_{9,0}$$x_{10,0}$$x_{11,0}$$x_{12,0}$$x_{0,1}$$x_{1,1}$$x_{2,1}$$x_{3,1}$$x_{3,1}$$x_{4,1}$$x_{4,1}$$x_{5,1}$$x_{6,1}$$x_{7,1}$$x_{8,1}$$x_{9,1}$$x_{10,1}$$x_{11,1}$
Figure 2: Graph $X_{3,2}$. Note: (i) $X_{3,2}$ is $2$-connected and (ii)
$ecc(x_{0,0})=3$.
Let $m=r(r+2)$. Construct a new graph $G$ by taking $m^{r}+1$ copies of
$X_{r,k}$ and identifying the vertices in $V_{0}$ as common in every copy. It
is easily seen that $G$ is $\kappa$-connected and has a radius $r$ with
$x_{0,0}$ as the central vertex. By arguments similar to those in the tight
examples for Theorem 4, we can see that $rc(G)=m$.
###### Corollary 7.
For every bridgeless graph $G$ with radius $r$ and chordality $k$,
$rc(G)\leq\sum_{i=1}^{r}{\min\\{2i+1,k\\}}\leq rk.$
Moreover, for every two integers $r\geq 1$ and $3\leq k\leq 2r+1$, there
exists a bridgeless graph $G$ with radius $r$ and chordality $k$ such that
$rc(G)=\sum_{i=1}^{r}{\min\\{2i+1,k\\}}$.
###### Proof.
Since every isometric cycle is an induced cycle, the chordality of a graph is
at least the size of its largest isometric cycle. i.e, $k\geq\zeta$. Hence the
upper bound follows from that in Theorem 4. The tight example demonstrated in
Theorem 4 suffices here too. ∎∎
This generalises a result from [3] that the rainbow connection number of any
bridgeless chordal graph is at most three times its radius.
## 3 Approximation Algorithms
### 3.1 Bridgeless Graphs
Throughout this section, $G$ will be a bridgeless graph with $n$ vertices, $m$
edges, diameter $d$ and radius $r$. A set $S\subset V(G)$ will be called
rainbow coloured under a partial edge colouring of $G$ if every pair of
vertices in $S$ is connected by a rainbow path in $G[S]$.
#### 3.1.1 $O(nm)$ time $(r+2)$-factor Approximation Algorithm
Corollary 5 was proved by demonstrating a colouring procedure which assigns a
rainbow colouring to any bridgeless graph of radius $r$ using at most $r(r+2)$
colours. Since the proof is constructive, it automatically gives us an
algorithm for rainbow colouring $G$. Since $r$ is a lower bound on rainbow
connection number, this is a $(r+2)$-factor approximation algorithm. The
procedure starts by identifying a central vertex in the graph. This can be
done by computing the eccentricity of every vertex using a Breadth First
Search (BFS) rooted at it. Thus the time complexity for finding the central
vertex in any connected graph is $O(nm)$. The acceptable ears to be coloured
in each step can be found using a BFS rooted at the selected vertex in
$N(D^{k})$ on a subgraph of $G$ and hence takes $O(m)$ running time on any
connected graph. Since we do not start the BFS more than once from any vertex,
the total running time for finding all the acceptable ears that gets coloured
is $O(nm)$. The colouring of a selected acceptable ear takes a time
proportional to the number of uncoloured edges in that ear. Moreover, each
edge is coloured only once by the algorithm. Hence the total effect of colour
assignments on the algorithm’s running time is $O(m)$. Thus the total running
time for the algorithm is $O(nm)$.
Next we present an algorithm which has a smaller running time of $O(dm)$ but a
slightly poorer approximation ratio of $(d+2)$.
#### 3.1.2 $O(dm)$ time $(d+2)$-factor Approximation Algorithm
To the best of our knowledge, there is no known algorithm to find a central
vertex of a bridgeless graph in a time significantly smaller than
$\Theta(nm)$. Hence we start the procedure by picking any arbitrary vertex $v$
of $G$ ($O(1)$ time). Since $ecc(v)\leq d$, this is connected $d$-step
dominating set of $G$. Hence, by repeated application of Lemma 3, we can grow
the trivially rainbow coloured connected $d$-step dominating set
$D^{d}=\\{v\\}$ to a rainbow coloured connected $0$-step dominating set
$D^{0}=V(G)$ using at most $d(d+2)$ colours. So if we can grow a rainbow
coloured connected $k$-step dominating set $D^{k}$ to a rainbow coloured
connected $(k-1)$-step dominating set $D^{k-1}$ in $O(m)$ time, then we can
complete the rainbow colouring of $G$ using $d(d+2)$ colours in $O(dm)$ time.
Since $d$ is a lower bound on rainbow connection number this gives a
$(d+2)$-factor approximation algorithm.
In the proof of Lemma 3, given a rainbow coloured connected $k$-step
dominating set $D^{k}$, we pick any edge $e=(x_{0},x_{1})$ with $x_{0}\in
D^{k}$ and $x_{1}$ being an uncaptured vertex in $N(D^{k})$. Next, we find an
acceptable ear containing $e$ and evenly colour that ear. When every vertex in
$N(D^{k})$ is captured this way, we have a rainbow coloured connected
$(k-1)$-step dominating set $D^{k-1}$ in hand. It is easy to see that, once an
acceptable ear is found and the colours (if any) of its end edges are known,
it can be evenly coloured in a time proportional to number of uncoloured edges
in that ear. Since no edge is coloured more than once by the algorithm, the
total running time for the colouring subroutine (once the acceptable ears are
found) is only $O(m)$. Hence if we can capture every vertex in $N(D^{k})$
using acceptable ears in $O(m)$ time, we can construct the required $D^{k-1}$
from the given $D^{k}$ in $O(m)$ time. This is precisely what Algorithm 1
achieves.
Algorithm 1 accepts a partially edge coloured bridgeless graph $G$, a rainbow
coloured connected $k$-step dominating set $D^{k}$ in $G$ and two pools of
colours $\mathcal{A}=\\{a_{1},a_{2},\ldots,a_{k+1}\\}$ and
$\mathcal{B}=\\{b_{1},b_{2},\ldots,b_{k}\\}$ not used in colouring $G[D^{k}]$.
It returns a $(k-1)$-step dominating set $D^{k-1}$ of vertices and colours a
subset of $E(G[D^{k-1}])\setminus E(G[D^{k}])$ using colours from
$\mathcal{A}\cup\mathcal{B}$ such that $G[D^{k-1}]$ is rainbow coloured. It
achieves the same by running a single BFS on $G\setminus E(G[D^{k}])$ with the
BFS queue initialised with $D^{k}$ and maintaining enough side information to
detect meetings which result in acceptable ears. Once an acceptable ear is
found, that ear is evenly coloured using colours from pools $\mathcal{A}$ and
$\mathcal{B}$. The procedure terminates once every edge is examined and hence
runs in $O(m)$ time.
#### Side information associated with each vertex $v$ in Algorithm 1
$Parent$:
For each vertex $v$ visited by the BFS, $Parent(v)$ points to parent vertex of
$v$ in the BFS forest. It is initialised to $\emptyset$ for all vertices.
$ParentEdgeColour$:
For each new vertex $v$ captured by the algorithm ($v\in D^{k-1}\setminus
D^{k}$), $ParentEdgeColour(v)$ holds the colour assigned to the edge
$(v,Parent(v))$ by the algorithm. It is also initialised to $\emptyset$ for
all vertices. This information is updated for the vertices of an acceptable
ear when it gets evenly coloured during the algorithm. Note that it is only a
temporary and partial information of the colourings effected in one run of the
algorithm which is used to make an instant check of whether a vertex has been
already captured by an evenly coloured acceptable ear and to detect the colour
pool used. The colouring subroutine also encodes every colour assignment into
the adjacency list of $G$ and that is what is finally returned.
$Foot$:
For each vertex $v$ visited by the BFS, $Foot(v)$ is the ordered pair of last
two vertices in the BFS path from $v$ to $D^{k}$. It is set to $\emptyset$ for
all vertices in the initial queue $D^{k}$.
Algorithm 1 Construct and rainbow colour $D^{k-1}$ given rainbow coloured
$D^{k}$
0: $G$ is a partially edge coloured bridgeless graph. $D^{k}$ is a rainbow
coloured connected $k$-step dominating set in $G$.
$\mathcal{A}=\\{a_{1},a_{2},\ldots,a_{k+1}\\}$ and
$\mathcal{B}=\\{b_{1},b_{2},\ldots,b_{k}\\}$ are two pools of colours not used
to colour $G[D^{k}]$.
0: $D^{k-1}\supset D^{k}$ is a rainbow coloured connected $(k-1)$-step
dominating set in $G$ such that new colours used are from
$\mathcal{A}\cup\mathcal{B}$.
for each $u\in G$ do
$Parent(u)\leftarrow\emptyset$, $ParentEdgeColour(u)\leftarrow\emptyset$,
$Foot(u)\leftarrow\emptyset$
end for
Flush($\mathcal{Q}$), Enqueue($\mathcal{Q}$, $D^{k}$), $D^{k-1}\leftarrow
D^{k}$
repeat
$u\leftarrow Dequeue(\mathcal{Q})$
for each vertex $v\in N(u)\cap V(G\setminus D^{k})$ do
if $Foot(v)=\emptyset$ then $/\negthickspace/$ $v$ is an unvisited vertex
if $Foot(u)=\emptyset$ then $/\negthickspace/$ $u\in D^{k}$
$Foot(v)\leftarrow(v,u)$
else
$Foot(v)\leftarrow Foot(u)$
end if
$Parent(v)\leftarrow u$, $Enqueue(\mathcal{Q},v)$
else if $Foot(v)\neq Foot(u)$ then $/\negthickspace/$ we have found an
acceptable $D^{k}$-ear
if $Foot(u)=\emptyset$ then $/\negthickspace/$ $u\in D^{k}$
$u_{0}=u$, $c_{u}=\emptyset$
else $/\negthickspace/$ $u_{0}$ will hold the vertex of $Foot(u)$ in $D^{k}$
and $c_{u}$ will hold the colour of $Foot(u)$
$(u_{1},u_{0})\leftarrow Foot(u)$, $c_{u}\leftarrow ParentEdgeColour(u_{1})$
end if
$(v_{1},v_{0})\leftarrow Foot(v)$, $c_{v}\leftarrow ParentEdgeColour(v_{1})$
if $c_{u}=\emptyset$ or $c_{v}=\emptyset$ then $/\negthickspace/$ $u_{1}$ or
$v_{1}$ is an uncaptured vertex in $N(D^{k})$
$P\leftarrow u_{0}Tu\mbox{--}vTv_{0}$ where $xTy$ is the unique path from $x$
to $y$ in the BFS forest under construction. $/\negthickspace/$ Path $P$ is an
acceptable $D^{k}$ ear some of whose edges are still uncoloured
$p\leftarrow|P|$ $/\negthickspace/$ length of $P$
if $c_{u}=a_{1}$ or $c_{v}=b_{1}$ or $c_{u}=c_{v}=\emptyset$ then
The uncoloured edges of $P$ are coloured so that the edges of $P$ get the
colours
$a_{1},a_{2},\ldots,a_{\lceil\frac{p}{2}\rceil},b_{\lfloor\frac{p}{2}\rfloor},\ldots,b_{2},b_{1}$
in that order.
else
The uncoloured edges of $P$ are coloured so that the edges of $P$ get the
colours
$b_{1},b_{2},\ldots,b_{\lfloor\frac{p}{2}\rfloor},a_{\lceil\frac{p}{2}\rceil},\ldots,a_{2},a_{1}$
in that order.
end if
$D^{k-1}\leftarrow D^{k-1}\cup P$
end if
end if
end for
until $\mathcal{Q}$ is empty
### 3.2 General Connected Graphs
In this section, $G$ will be a connected graph with $n$ vertices, $m$ edges,
diameter $d$, radius $r$ and $b$ bridges. Let $G^{\prime}$ be the graph
obtained by contracting every bridge of $G$. The diameter (radius) of
$G^{\prime}$ is at most $d$ ($r$). We can extend a rainbow colouring of
$G^{\prime}$ to $G$ by giving a new colour to every bridge of $G$. Hence
$rc(G)\leq rc(G^{\prime})+b$. We can find all the bridges in a connected graph
in $O(m)$ time [10]. Now, using the algorithm in Section 3.1.1 to colour
$G^{\prime}$, we can colour $G$ using at most $r(r+2)+b$ colours in $O(nm)$
time. Since $r(r+2)+b\leq\max\\{r,b\\}(r+3)$ and since $\max\\{r,b\\}$ is a
lower bound on $rc(G)$, we immediately have a $(r+3)$-factor $O(nm)$
approximation algorithm to rainbow colour any connected graph.
Similarly by combining an $O(m)$ algorithm to find every bridge of $G$ with
the algorithm in Section 3.1.2 gives an $O(dm)$ algorithm to rainbow colour
$G$ using $d(d+2)+b$ colours. Since $d(d+2)+b\leq\max\\{d,b\\}(d+3)$ and since
$\max\\{d,b\\}$ is a lower bound on $rc(G)$, we immediately have a
$(d+3)$-factor $O(dm)$ approximation algorithm to rainbow colour any connected
graph.
## References
* [1] Yair Caro, Arie Lev, Yehuda Roditty, Zsolt Tuza, and Raphael Yuster. On rainbow connection. Electron. J. Combin., 15(1):Research paper 57, 13, 2008.
* [2] Sourav Chakraborty, Eldar Fischer, Arie Matsliah, and Raphael Yuster. Hardness and algorithms for rainbow connection. Journal of Combinatorial Optimization, pages 1–18, 2009.
* [3] L. Sunil Chandran, Anita Das, Deepak Rajendraprasad, and Nithin M. Varma. Rainbow connection number and connected dominating sets. Journal of Graph Theory, 2011.
* [4] Gary Chartrand, Garry L. Johns, Kathleen A. McKeon, and Ping Zhang. Rainbow connection in graphs. Math. Bohem., 133(1):85–98, 2008.
* [5] Gary Chartrand and Ping Zhang. Chromatic Graph Theory. Chapman & Hall, 2008.
* [6] Jiuying Dong and Xueliang Li. Rainbow connection number, bridges and radius. Preprint arXiv:1105.0790v1 [math.CO], 2011.
* [7] Michael Krivelevich and Raphael Yuster. The rainbow connection of a graph is (at most) reciprocal to its minimum degree. J. Graph Theory, 63(3):185–191, 2010.
* [8] Xueliang Li, Sujuan Liu, L. Sunil Chandran, Rogers Mathew, and Deepak Rajendraprasad. Rainbow connection number and connectivity. The Electronic Journal of Combinatorics, 19(1):P20, 2012.
* [9] Xueliang Li and Yongtang Shi. Rainbow connection in 3-connected graphs. Graphs and Combinatorics, pages 1–5, 2012. 10.1007/s00373-012-1204-9.
* [10] Robert Endre Tarjan. A note on finding the bridges of a graph. Information Processing Letters, 2(6):160–161, 1974.
|
arxiv-papers
| 2010-11-02T13:56:26 |
2024-09-04T02:49:14.423511
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Manu Basavaraju, L. Sunil Chandran, Deepak Rajendraprasad, and\n Arunselvan Ramaswamy",
"submitter": "Deepak Rajendraprasad",
"url": "https://arxiv.org/abs/1011.0620"
}
|
1011.0676
|
# Power-Law Entropic Corrections to Newton’s Law and Friedmann Equations
A. Sheykhi 1,2 and S. H. Hendi 2,3 email address: hendi@mail.yu.ac.iremail
address: sheykhi@mail.uk.ac.ir 1 Department of Physics, Shahid Bahonar
University, P.O. Box 76175, Kerman, Iran
2 Research Institute for Astronomy and Astrophysics of Maragha (RIAAM),
Maragha, Iran
3 Physics Department, College of Sciences, Yasouj University, Yasouj 75914,
Iran
###### Abstract
A possible source of black hole entropy could be the entanglement of quantum
fields in and out the horizon. The entanglement entropy of the ground state
obeys the area law. However, a correction term proportional to a fractional
power of area results when the field is in a superposition of ground and
excited states. Inspired by the power-law corrections to entropy and adopting
the viewpoint that gravity emerges as an entropic force, we derive modified
Newton’s law of gravitation as well as the corrections to Friedmann equations.
In a different approach, we obtained power-law corrected Friedmann equation by
starting from the first law of thermodynamics at apparent horizon of a FRW
universe, and assuming that the associated entropy with apparent horizon has a
power-law corrected relation. Our study shows a consistency between the
obtained results of these two approaches. We also examine the time evolution
of the total entropy including the power-law corrected entropy associated with
the apparent horizon together with the matter field entropy inside the
apparent horizon and show that the generalized second law of thermodynamics is
fulfilled in a region enclosed by the apparent horizon.
## I Introduction
Recently, Verlinde Verlinde demonstrated that gravity can be interpreted as
an entropic force caused by the changes in the information associated with the
positions of material bodies. In his new proposal, Verlinde obtained
successfully the Newton’s law of gravitation, the Poisson’s equation and
Einstein field equations by employing the holographic principle together with
the equipartition law of energy. As soon as Verlinde presented his idea, many
relevant works about entropic force appeared. For example, Friedmann equations
from entropic force have been derived in Refs. Shu ; Cai1 . The Newtonian
gravity Smolin , the holographic dark energy Li and thermodynamics of black
holes Tian have been investigated by using the entropic force approach. It
has been shown that uncertainty principle may arise in the entropic force
paradigm Vancea . Other studies on the entropic force, which raised a lot of
attention recently, have been carried out in other .
On the other hand, string theory, as well as the string inspired braneworld
scenarios such as RSII model, suggest a modification of Newton’s law of
gravitation at small distance scales Polchinski ; Randall . In addition, there
have been considerable works on quantum corrections to some basic physical
laws. The loop quantum corrections to the Newton and Coulomb potential have
been investigated in some references (see Donoghue and references therein).
Also, corrections to Friedmann equations from loop quantum gravity has been
studied in Taveras .
Inspired by Verlinde’s argument and considering the quantum corrections to the
area law of the black hole entropy, one is able to derive some physical
equations with correction terms. For example, modified Newton’s law of
gravitation has been studied in Modesto , while, modified Friedmann equations
have been constructed in Sheykhi1 ; BLi . In all these cases Modesto ;
Sheykhi1 ; BLi the corrected entropy has the logarithmic term which arises
from the inclusion of quantum effects, motivated from the loop quantum gravity
and is due to the thermal equilibrium fluctuations and quantum fluctuations
Rovelli . In addition, entropic corrections to Coulomb’s law have also been
investigated in modifiedNC . Very recently, by considering the quantum
corrections to the area law of black hole entropy, the modified forms of
Poisson’s equation of gravity, MOND theory of gravitation and Einstein field
equations were derived using the entropic force interpretation of gravity
hendisheykhi .
In this paper we would like to consider the effects of the power-law
correction terms to the entropy on the Newton’s law and Friedmann equation.
The power-law corrections to entropy appear in dealing with the entanglement
of quantum fields in and out the horizon Sau . Indeed, it has been shown that
the origin of black hole entropy may lie in the entanglement of quantum fields
between inside and outside of the horizon Sau . Since the modes of
gravitational fluctuations in a black hole background behave as scalar fields,
one is able to compute the entanglement entropy of such a field, by tracing
over its degrees of freedom inside a sphere. In this way the authors of Sau
showed that the black hole entropy is proportional to the area of the sphere
when the field is in its ground state, but a correction term proportional to a
fractional power of area results when the field is in a superposition of
ground and excited states. For large horizon areas, these corrections are
relatively small and the area law is recovered. Applying this power-law
corrected entropy, we obtain the corrections to Newton’s law as well as
modified Friedmann equation by adopting the viewpoint that gravity emerges as
an entropic force.
The outline of our paper is as follows. In the next section, we use Verlinde
approach to derive Newton’s law of gravitation with a correction term
resulting from the entanglement of quantum fields in and out the horizon. In
section III, we derive the power-law entropy-corrected Friedmann equation of
FRW universe by considering gravity as an entropic force. Then, in section IV,
we obtain modified Friedmann equation by applying the first law of
thermodynamics at apparent horizon of a FRW universe. In section V we examine
to see whether the power-law entropy-area relation together with the matter
field entropy inside the apparent horizon will satisfy the generalized second
law of thermodynamics. The last section is devoted to conclusions and
discussions.
## II Entropic correction to Newton’s law
According to Verlinde’s argument, when a test particle moves apart from the
holographic screen, the magnitude of the entropic force on this body has the
form
$F\triangle x=T\triangle S,$ (1)
where $\triangle x$ is the displacement of the particle from the holographic
screen, while $T$ and $\triangle S$ are the temperature and the entropy change
on the screen, respectively.
In Verlinde’s discussion, the black hole entropy $S$ plays a significant role.
Indeed, the derivation of Newton’s law of gravity depends on the entropy-area
relationship $S=k_{B}A/4\ell_{p}^{2}$ of black holes in Einstein s gravity,
where $A=4\pi R^{2}$ represents the area of the horizon and
$\ell_{p}=\sqrt{G\hbar/c^{3}}$ is the Planck length. However, the area law of
black hole entropy can be modified Sau . The corrected entropy takes the form
pavon1
$S=\frac{k_{B}A}{4\ell_{p}^{2}}\left[1-K_{\alpha}A^{1-\alpha/2}\right],$ (2)
where $\alpha$ is a dimensionless constant whose value is currently under
debate, $k_{B}$ stands for the Boltzmann constant and
$K_{\alpha}=\frac{\alpha(4\pi)^{\alpha/2-1}}{(4-\alpha)r_{c}^{2-\alpha}},$ (3)
where $r_{c}$ is the crossover scale. The second term in the above Eq. (2) may
be regarded as a power law correction to the area law, resulting from
entanglement, when the wave-function of the field is chosen to be a
superposition of ground state and exited states.
Considering the power-law correction to entropy, we show that Newton’s law of
gravitation as well as Friedman equations will be modified accordingly. First
of all, we rewrite Eq. (2) in the following form
$S=k_{B}\left[\frac{A}{4\ell_{p}^{2}}+{s}(A)\right],$ (4)
where $s(A)$ stands for the correction term in the entropy expression. Suppose
we have two masses one a test mass and the other considered as the source with
respective masses $m$ and $M$. Centered around the source mass $M$, is a
spherically symmetric surface $\mathcal{S}$ which will be defined with certain
properties that will be made explicit later. To derive the entropic law, the
surface $\mathcal{S}$ is between the test mass and the source mass, but the
test mass is assumed to be very close to the surface as compared to its
reduced Compton wavelength $\lambda_{m}=\frac{\hbar}{mc}$. When a test mass
$m$ is a distance $\triangle x=\eta\lambda_{m}$ away from the surface
$\mathcal{S}$, the entropy of the surface changes by one fundamental unit
$\triangle S$ fixed by the discrete spectrum of the area of the surface via
the relation
$\triangle S=\frac{\partial S}{\partial A}\triangle
A=k_{B}\left(\frac{1}{4\ell_{p}^{2}}+\frac{\partial{s}(A)}{\partial
A}\right)\triangle A.$ (5)
The energy of the surface $\mathcal{S}$ is identified with the relativistic
rest mass of the source mass:
$E=Mc^{2}.$ (6)
On the surface $\mathcal{S}$, there live a set of “bytes” of information that
scale proportional to the area of the surface so that
$A=QN,$ (7)
where $N$ represents the number of bytes and $Q$ is a fundamental constant
which should be specified later. Assuming the temperature on the surface is
$T$, and then according to the equipartition law of energy Pad1 , the total
energy on the surface is
$E=\frac{1}{2}Nk_{B}T.$ (8)
Finally, we assume that the force on the particle follows from the generic
form of the entropic force governed by the thermodynamic equation
$F=T\frac{\triangle S}{\triangle x},$ (9)
where $\triangle S$ is one fundamental unit of entropy when $|\triangle
x|=\eta\lambda_{m}$, and the entropy gradient points radially from the outside
of the surface to inside. Note that $N$ is the number of bytes and thus we set
$\triangle N=1$; hence from (7) we have $\triangle A=Q$. Combining Eqs. (5)-
(9), we find
$F=-\frac{Mm}{R^{2}}\left(\frac{Q^{2}c^{3}}{8\pi\hbar\eta\ell_{p}^{2}}\right)\left[1+4\ell_{p}^{2}\frac{\partial{s(A)}}{\partial
A}\right]_{A=4\pi R^{2}}.$ (10)
This is nothing but the Newton’s law of gravitation to the first order
provided we define $Q^{2}=8\pi\eta\ell_{p}^{4}$. Thus we reach
$F=-\frac{GMm}{R^{2}}\left[1+4\ell_{p}^{2}\frac{\partial{s}}{\partial
A}\right]_{A=4\pi R^{2}}.$ (11)
Using Eq. (2) we obtain
$\left(\frac{\partial{s}}{\partial A}\right)_{A=4\pi
R^{2}}=-\frac{K_{\alpha}(4-\alpha)}{8\ell_{p}^{2}}\left(4\pi
R^{2}\right)^{1-\alpha/2}$ (12)
Substituting Eq. (12) in Eq. (11) we obtain
$F=-\frac{GMm}{R^{2}}\left[1-\frac{K_{\alpha}}{2}(4-\alpha)\left(4\pi
R^{2}\right)^{1-\alpha/2}\right],$ (13)
Using Eq. (3) the above relation can be rewritten as
$F=-\frac{GMm}{R^{2}}\left[1-\frac{\alpha}{2}\left(\frac{r_{c}}{R}\right)^{\alpha-2}\right],$
(14)
This is the power-law correction to the Newton’s law of gravitation. When
$\alpha=0$, one recovers the usual Newton’s law. Since gravity is an
attractive force we should have $F<0$. This requires
$1-\frac{\alpha}{2}\left(\frac{r_{c}}{R}\right)^{\alpha-2}>0,$ (15)
which can also be rewritten as
$\alpha<2\left(\frac{R}{r_{c}}\right)^{\alpha-2},$ (16)
As we will see in section V, this condition is also necessary for satisfaction
of the generalized second law of thermodynamics for the universe with the
power-law corrected entropy.
## III Entropic Corrections to Friedmann Equations
Next, we extend our discussion to the cosmological setup. Assuming the
background spacetime to be spatially homogeneous and isotropic which is
described by the line element
$ds^{2}={h}_{\mu\nu}dx^{\mu}dx^{\nu}+R^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2}),$ (17)
where $R=a(t)r$, $x^{0}=t,x^{1}=r$, the two dimensional metric
$h_{\mu\nu}$=diag $(-1,a^{2}/(1-kr^{2}))$. Here $k$ denotes the curvature of
space with $k=0,1,-1$ corresponding to open, flat, and closed universes,
respectively. The dynamical apparent horizon, a marginally trapped surface
with vanishing expansion, is determined by the relation
$h^{\mu\nu}\partial_{\mu}R\partial_{\nu}R=0$. A simple calculation gives the
apparent horizon radius for the Friedmann-Robertson-Walker (FRW) universe
$R=ar=\frac{1}{\sqrt{H^{2}+k/a^{2}}},$ (18)
where $H=\dot{a}/a$ is the Hubble parameter. We also assume the matter source
in the FRW universe is a perfect fluid of mass density $\rho$ and pressure $p$
with stress-energy tensor
$T_{\mu\nu}=(\rho+p)u_{\mu}u_{\nu}+pg_{\mu\nu}.$ (19)
Due to the pressure, the total mass $M=\rho V$ in the region enclosed by the
boundary $\mathcal{S}$ is no longer conserved, the change in the total mass is
equal to the work made by the pressure $dM=-pdV$ , which leads to the well-
known continuity equation
$\dot{\rho}+3H(\rho+p)=0,$ (20)
It is instructive to first derive the dynamical equation for Newtonian
cosmology. Consider a compact spatial region $V$ with a compact boundary
$\mathcal{S}$, which is a sphere with physical radius $R=a(t)r$. Note that
here $r$ is a dimensionless quantity which remains constant for any
cosmological object partaking in free cosmic expansion. Combining the second
law of Newton for the test particle $m$ near the surface with gravitational
force (14) we get
$F=m\ddot{R}=m\ddot{a}r=-\frac{GMm}{R^{2}}\left[1-\frac{\alpha}{2}\left(\frac{r_{c}}{R}\right)^{\alpha-2}\right],$
(21)
We also assume $\rho=M/V$ is the energy density of the matter inside the the
volume $V=\frac{4}{3}\pi a^{3}r^{3}$. Thus, Eq. (21) can be rewritten as
$\frac{\ddot{a}}{a}=-\frac{4\pi
G}{3}\rho\left[1-\frac{\alpha}{2}\left(\frac{r_{c}}{R}\right)^{\alpha-2}\right],$
(22)
This is nothing but the power-law entropy-corrected dynamical equation for
Newtonian cosmology. The main difference between this equation and the
standard dynamical equation for Newtonian cosmology is that the correction
terms now depends explicitly on the radius $R$. However, we can remove this
confusion. Assuming that for Newtonian cosmology the spacetime is Minkowskian
with $k=0$, then we get $R=1/H$, and we can rewrite Eq. (22) in the form
$\frac{\ddot{a}}{a}=-\frac{4\pi
G}{3}\rho\left[1-\frac{\alpha}{2}{r_{c}}^{\alpha-2}\left(\frac{\dot{a}}{a}\right)^{\alpha-2}\right].$
(23)
It was argued in Cai4 that for deriving the Friedmann equations of FRW
universe in general relativity, the quantity that produces the acceleration is
the active gravitational mass $\mathcal{M}$ Pad2 , rather than the total mass
$M$ in the spatial region $V$. With the entropic correction term, the active
gravitational mass $\mathcal{M}$ will also modified as well. On one side, from
Eq. (22) with replacing $M$ with $\mathcal{M}$ we have
$\mathcal{M}=-\frac{\ddot{a}a^{2}}{G}r^{3}\left[1-\frac{\alpha}{2}\left(\frac{r_{c}}{R}\right)^{\alpha-2}\right]^{-1}.$
(24)
On the other side, the active gravitational mass is defined as Cai4
$\mathcal{M}=2\int_{V}{dV\left(T_{\mu\nu}-\frac{1}{2}Tg_{\mu\nu}\right)u^{\mu}u^{\nu}}.$
(25)
A simple calculation leads
$\mathcal{M}=(\rho+3p)\frac{4\pi}{3}a^{3}r^{3}.$ (26)
Equating Eqs. (24) and (26), we find
$\frac{\ddot{a}}{a}=-\frac{4\pi
G}{3}(\rho+3p)\left[1-\frac{\alpha}{2}\left(\frac{r_{c}}{R}\right)^{\alpha-2}\right].$
(27)
Multiplying $\dot{a}a$ on both sides of Eq. (27), and using the continuity
equation (20) we reach
$\frac{d}{dt}(\dot{a}^{2})=\frac{8\pi G}{3}\frac{d}{dt}(\rho
a^{2})\left[1-\frac{\alpha}{2}\left(\frac{r_{c}}{R}\right)^{\alpha-2}\right].$
(28)
Integrating of Eq. (28), we find
$H^{2}+\frac{k}{a^{2}}=\frac{8\pi G}{3}\rho\left[1-\frac{\alpha}{2\rho
a^{2}}\left(\frac{r_{c}}{r}\right)^{\alpha-2}\int{\frac{d(\rho
a^{2})}{a^{\alpha-2}}}\right],$ (29)
where $k$ is a constant of integration. Now, in order to calculate the
integral we need to find $\rho=\rho(a)$. Assuming the equation of state
parameter $w=p/\rho$ is a constant, the continuity equation (20) can be
integrated immediately to give
$\rho=\rho_{0}a^{-3(1+w)},$ (30)
where $\rho_{0}$ is the present value of the energy density. Inserting
relation (30) in Eq. (29), after integration, we obtain
$H^{2}+\frac{k}{a^{2}}=\frac{8\pi
G}{3}\rho\left[1-\beta\left(\frac{r_{c}}{R}\right)^{\alpha-2}\right],$ (31)
where we have defined
$\beta=\frac{\alpha}{2}\frac{(3w+1)}{(3w+\alpha-1)}.$ (32)
Using Eq. (18), we can rewrite Eq. (31) as
$\displaystyle H^{2}+\frac{k}{a^{2}}=\frac{8\pi G}{3}\rho$ (33)
$\displaystyle\times\left[1-\beta{r_{c}}^{\alpha-2}\left(H^{2}+\frac{k}{a^{2}}\right)^{\alpha/2-1}\right]$
In the absence of the correction terms $(\alpha=0=\beta)$, one recovers the
well-known Friedmann equation in standard cosmology. Let us note that the left
hand side of Eq. (33) is always positive thus the right hand side is also
positive. This is due to the fact that the right hand side of the usual
Friedmann equation is always positive ($\rho>0$ and $G>0$) so $H^{2}+k/a^{2}$
should be positive in our case to have a correct $\beta=0=\alpha$ limit. This
leads to the following condition
$\beta{r_{c}}^{\alpha-2}\left(H^{2}+\frac{k}{a^{2}}\right)^{\alpha/2-1}<1.$
(34)
Eq. (33) can also be written as
$\displaystyle\left(H^{2}+\frac{k}{a^{2}}\right)\left[1-\beta{r_{c}}^{\alpha-2}\left(H^{2}+\frac{k}{a^{2}}\right)^{\alpha/2-1}\right]^{-1}$
(35) $\displaystyle=\frac{8\pi G}{3}\rho.$
Taking into account condition (34) we can expand the above equation up to the
linear order of $\beta$. The result is
$\displaystyle\left(H^{2}+\frac{k}{a^{2}}\right)+\beta{r_{c}}^{\alpha-2}\left(H^{2}+\frac{k}{a^{2}}\right)^{\alpha/2}=\frac{8\pi
G}{3}\rho,$ (36)
where we have neglected $O(\beta^{2})$ terms and higher powers of $\beta$.
This is due to the fact that at the present time $R\gg 1$ and hence
$H^{2}+k/a^{2}\ll 1$ (see the right hand side of standard Friedman equation
where $G\sim 10^{-11}$ and $\rho\ll 1$). Indeed for the present time where the
apparent horizon area (radius) becomes large, the power-law correction terms
to entropy Sau and hence to Friedman equation are relatively small and the
usual Friedman equation is recovered. Thus, the corrections make sense only at
the early stage of the universe where $a\rightarrow 0$. When $a\rightarrow 0$,
even the higher powers of $\beta$ should be considered. These correction terms
at the early stage of the universe may affect on the number of e-folding
during the inflation. However this issue should be examined carefully
elsewhere. With expansion of the universe, the power-law entropy-corrected
Friedmann equation reduces to the usual Friedman equation.
## IV Modified Friedmann equations from the first law
In this section, we adopt another approach to derive the entropy-corrected
Friedmann equation. Indeed, we are able to derive modified Friedmann equation
by applying the first law of thermodynamics at apparent horizon of a FRW
universe, with the assumption that the associated entropy with apparent
horizon has the power-law corrected form (2). It was already shown that the
differential form of the Friedmann equation in the FRW universe can be written
in the form of the first law of thermodynamics on the apparent horizon
Sheykhi2 . We follow the method developed in Sheykhi3 . Throughout this
section we set $\hbar=c=k_{B}=1$ for simplicity. The associated temperature
with the apparent horizon can be defined as Cai5
$T=\frac{\kappa}{2\pi}=-\frac{1}{2\pi R}\left(1-\frac{\dot{R}}{2HR}\right).$
(37)
where $\kappa$ is the surface gravity. When $\dot{R}\leq 2HR$, the temperature
becomes negative $T\leq 0$. Physically it is not easy to accept the negative
temperature. In this case the temperature on the apparent horizon should be
defined as $T=|\kappa|/2\pi$. The work density is obtained as Hay2
$W=\frac{1}{2}(\rho-p).$ (38)
The work density term is regarded as the work done by the change of the
apparent horizon. We also assume the first law of thermodynamics on the
apparent horizon is satisfied and has the form
$dE=T_{h}dS_{h}+WdV,$ (39)
where $S_{h}$ is the power-law corrected entropy associated with the apparent
horizon which has the form (2). Suppose $E=\rho V$ is the total energy content
of the universe inside a $3$-sphere of radius $R$, where
$V=\frac{4\pi}{3}R^{3}$ is the volume enveloped by 3-dimensional sphere with
the area of apparent horizon $A=4\pi R^{2}$. Taking differential form of the
relation $E=\frac{4\pi}{3}\rho R^{3}$ for the total matter and energy inside
the apparent horizon, and using the continuity equation (20), we get
$dE=4\pi\rho R^{2}dR-4\pi HR^{3}(\rho+p)dt.$ (40)
Taking differential form of the corrected entropy (2), we have
$dS_{h}=\frac{2\pi
R}{G}\left[1-\frac{\alpha}{2}\left(\frac{r_{c}}{R}\right)^{\alpha-2}\right]dR.$
(41)
Inserting Eqs. (37), (38), (40) and (41) in the first law (39), we can get the
differential form of the modified Friedmann equation
$\frac{1}{4\pi
G}\frac{dR}{R^{3}}\left[1-\frac{\alpha}{2}\left(\frac{r_{c}}{R}\right)^{\alpha-2}\right]=H(\rho+p)dt.$
(42)
Using the continuity equation (20), we can rewrite it as
$-\frac{2}{R^{3}}\left[1-\frac{\alpha}{2}\left(\frac{r_{c}}{R}\right)^{\alpha-2}\right]dR=\frac{8\pi
G}{3}d\rho.$ (43)
Integrating (43) yields
$\frac{1}{R^{2}}-\frac{r_{c}^{\alpha-2}}{R^{\alpha}}=\frac{8\pi G}{3}\rho+C,$
(44)
where $C$ is an integration constant to be determined later. Substituting $R$
from Eq. (18) we obtain entropy-corrected Friedmann equation
$H^{2}+\frac{k}{a^{2}}-r_{c}^{\alpha-2}\left(H^{2}+\frac{k}{a^{2}}\right)^{\alpha/2}=\frac{8\pi
G}{3}\rho+C.$ (45)
The constant $C$ can be determined by taking the $\alpha\rightarrow 0$ limit
of the above expression. In this limit Eq. (45) reduces to the usual Friedmann
equation provided $C=-r_{c}^{-2}$. Thus we reach
$H^{2}+\frac{k}{a^{2}}-r_{c}^{-2}\left[r_{c}^{\alpha}\left(H^{2}+\frac{k}{a^{2}}\right)^{\alpha/2}-1\right]=\frac{8\pi
G}{3}\rho.$ (46)
This is the power-law entropy corrected Friedmann equation derived using the
first law on the apparent horizon. To show the consistency between the result
of this section with Eq. (36) derived in the previous section, let us note
that Eq. (36) in the previous section was derived for the late time where the
term $O(\beta^{2})$ can be neglected and thus we do not expect to be exactly
the same as the result obtained in this section which is valid for all epoch
of the universe. However, if one absorbs, in Eq. (45), the constant $C$ in
$\rho$, then one can rewrite Eq. (45) as
$H^{2}+\frac{k}{a^{2}}-r_{c}^{\alpha-2}\left(H^{2}+\frac{k}{a^{2}}\right)^{\alpha/2}=\frac{8\pi
G}{3}\rho,$ (47)
which is consistent with Eq. (36) derived using the entropic force approach in
the previous section provided one takes $\beta=-1$, which can be translated
into
$w=\frac{2-3\alpha}{3\alpha+6}.$ (48)
For $\alpha>2$, the above relation leads to $w<-1/3$. Two points should be
considered here carefully. First, relation (48) was derived for $\beta=-1$,
thus it does not have $\alpha=0$ limit, since in this case ($\alpha=0$), from
definition (32) we have $\beta=0$, which is in contradiction with condition
$\beta=-1$. Second, relation (48) appears when we want to show the consistency
between modified Friedman equation derived from two methods. In the absence of
correction terms $(\alpha=0=\beta)$ the obtained Friedman equations from two
different methods, namely Eqs. (36) and (46) exactly coincide regardless of
the value of $w$. This indicates that for usual Friedmann equation the
condition (48) is relaxed and hence $w$ can have any arbitrary value in
standard cosmology.
It is also notable to mention that Eq. (46) is consistent with the result
obtained in Karami . However, our derivation is quite different from Karami .
Let us stress the difference between our derivation in this section and Karami
. First of all, the authors of Karami have derived modified Friedmann
equations by applying the first law of thermodynamics, $TdS=-dE$, to the
apparent horizon of a FRW universe with the assumption that the apparent
horizon has corrected-entropy like (2). It is worthy to note that the notation
$dE$ in Karami is quite different from the same we used in this section. In
Karami , $-dE$ is actually just the heat flux crossing the apparent horizon
within an infinitesimal internal of time $dt$. But, here $dE$ is change in the
the matter energy inside the apparent horizon. Besides, in Karami the
apparent horizon radius $R$ has been assumed to be fixed. But, here, the
apparent horizon radius changes with time. This is the reason why we have
included the term $WdV$ in the first law (39). Indeed, the term $4\pi
R^{2}\rho dR$ in Eq. (40) contributes to the work term, while this term is
absent in $dE$ of Karami . This is consistent with the fact that in
thermodynamics the work is done when the volume of the system is changed.
## V Generalized Second law of thermodynamics
Finally, we investigate the validity of the generalized second law of
thermodynamics for the power-law entropy corrected Friedmann equations in a
region enclosed by the apparent horizon. Our method here differs from that of
Ref. pavon1 , in that they studied the generalized second law along with
either Clausius relation or the equipartition law of energy, while we apply
the first law of thermodynamics (39). The difference between our method and
Ref. Karami was also explained in the last paragraph of the previous section.
Substituting relation (18) in modified Friedmann Eq. (46) we find
$\frac{1}{R^{2}}-\frac{r_{c}^{\alpha-2}}{R^{\alpha}}+r_{c}^{\alpha-2}=\frac{8\pi
G}{3}\rho$ (49)
Differentiating Eq. (49) with respect to the cosmic time, after using the
continuity Eq. (20), we get
$\dot{R}=4\pi
GHR^{3}(\rho+p)\left[1-\frac{\alpha}{2}\left(\frac{r_{c}}{R}\right)^{\alpha-2}\right]^{-1}.$
(50)
Next, we calculate $T_{h}\dot{S_{h}}$. Using Eq. (41) we find
$T_{h}\dot{S_{h}}=\frac{1}{2\pi
R}\left(1-\frac{\dot{R}}{2HR}\right)\times\frac{2\pi
R}{G}\left[1-\frac{\alpha}{2}\left(\frac{r_{c}}{R}\right)^{\alpha-2}\right]\dot{R}$
(51)
After some simplification and using Eq. (50) we obtain
$T_{h}\dot{S_{h}}=4\pi HR^{3}(\rho+p)\left(1-\frac{\dot{R}}{2HR}\right).$ (52)
In the accelerating universe the dominant energy condition may violate,
$\rho+p<0$, indicating that the second law of thermodynamics ,$\dot{S_{h}}\geq
0$, does not hold. However, as we will see below the generalized second law of
thermodynamics, $\dot{S_{h}}+\dot{S_{m}}\geq 0$, is still fulfilled throughout
the history of the universe. From the Gibbs equation we have Pavon2
$T_{m}dS_{m}=d(\rho V)+pdV=Vd\rho+(\rho+p)dV,$ (53)
where $T_{m}$ and $S_{m}$ are, respectively, the temperature and the entropy
of the matter fields inside the apparent horizon. We limit ourselves to the
assumption that the thermal system bounded by the apparent horizon remains in
equilibrium so that the temperature of the system must be uniform and the same
as the temperature of its boundary. This requires that the temperature $T_{m}$
of the energy inside the apparent horizon should be in equilibrium with the
temperature $T_{h}$ associated with the apparent horizon, so we have
$T_{m}=T_{h}$ Pavon2 . This expression holds in the local equilibrium
hypothesis. If the temperature of the fluid differs much from that of the
horizon, there will be spontaneous heat flow between the horizon and the fluid
and the local equilibrium hypothesis will no longer hold. Therefore from the
Gibbs equation (53) we can obtain
$T_{h}\dot{S_{m}}=4\pi R^{2}\dot{R}(\rho+p)-4\pi R^{3}H(\rho+p).$ (54)
To check the generalized second law of thermodynamics, we have to examine the
evolution of the total entropy $S_{h}+S_{m}$. Adding equations (52) and (54),
we get
$T_{h}(\dot{S_{h}}+\dot{S_{m}})=2\pi
R^{2}(\rho+p)\dot{R}=\frac{A}{2}(\rho+p)\dot{R}.$ (55)
where $A=4\pi R^{2}$ is the apparent horizon area. Substituting $\dot{R}$ from
Eq. (50) into (55) we find
$T_{h}(\dot{S_{h}}+\dot{S_{m}})=2\pi
GAHR^{3}(\rho+p)^{2}\left[1-\frac{\alpha}{2}\left(\frac{r_{c}}{R}\right)^{\alpha-2}\right]^{-1}.$
(56)
As we argued after Eq. (14) the expression in the bracket of Eq. (56) is
always positive i.e.,
$\left[1-\frac{\alpha}{2}\left(\frac{r_{c}}{R}\right)^{\alpha-2}\right]>0.$
(57)
Thus the right hand side of Eq. (56) cannot be negative throughout the history
of the universe, which means that $\dot{S_{h}}+\dot{S_{m}}\geq 0$ always
holds. This implies that for a universe with power-law entropy corrected
relation the generalized second law of thermodynamics is fulfilled in a region
enclosed by the apparent horizon. Note that if we identify the crossover scale
$r_{c}$ with the present value of the apparent horizon, i.e., $r_{c}=R$, then
the condition (57) reduces to $\alpha<2$, which is consistent with the result
obtained in pavon1 ; Karami .
## VI Conclusions and discussions
It was argued that a possible source of black hole entropy could be the
entanglement of quantum fields in and out the horizon Sau . The entanglement
entropy of the ground state of field obeys the well-known area law. However,
the power-law correction to the area law appears when the wave-function of the
quantum field is chosen to be a superposition of ground state and exited state
Sau . Indeed, the excited states contribute to the correction, and more
excitations produce more deviation from the area law sau1 ; sau2 . Therefore,
the correction terms are more significant for higher excitations.
Motivated by the power-law corrected entropy and adopting the viewpoint that
gravity emerges as an entropic force, we derived modified Newton’s law of
gravitation as well as power-law correction to Friedmann equations. We found
that the correction term for Friedmann equation falls off rapidly with
apparent horizon radius and can be comparable to the first term only when the
scale factor $a$ is very small. Thus the corrections make sense only at early
stage of the universe. When the universe becomes large, the power-law entropy-
corrected Friedmann equation reduces to the standard Friedman equation. This
can be understood easily. At late time where $a$ is large, i.e., at low
energies, it is difficult to excite the modes and hence, the ground state
modes contribute to most of the entanglement entropy. However, at the early
stage, i.e., at high energies, a large number of field modes can be excited
and contribute significantly to the correction causing deviation from the area
law and hence deviation from the standard Friedmann equation.
We also derived modified Friedmann equation from different approach. Starting
from the first law of thermodynamics at apparent horizon of a FRW universe,
and assuming that the associated entropy with apparent horizon has power-law
corrected form (2), we obtained modified Friedmann equation. We find out that
the derived modified equations from these two different approaches (entropic
force approach and first law approach) can be consistent provided the equation
of state parameter satisfies in condition (48). However, in the absence of the
correction terms $(\alpha=0=\beta)$ the obtained Friedman equations from two
different methods, namely Eqs. (36) and (46) exactly coincide regradless of
the value of $w$. This indicates that for usual Friedmann equation the
condition (48) is relaxed and hence $w$ can have any arbitrary value in
standard cosmology.
Finally, we investigated the validity of the generalized second law of
thermodynamics for the FRW universe with any spatial curvature. We have shown
that, when thermal system bounded by the apparent horizon remains in
equilibrium with its boundary such that $T_{m}=T_{h}$, the generalized second
law of thermodynamics is fulfilled in a region enclosed by the apparent
horizon. The results obtained here for power-law corrected entropy area
relation further supports the thermodynamical interpretation of gravity and
provides more confidence on the profound connection between gravity and
thermodynamics.
###### Acknowledgements.
We thank the anonymous referee for constructive and valuable comments. This
work has been supported by Research Institute for Astronomy and Astrophysics
of Maragha.
## References
* (1) E. P. Verlinde, arXiv:1001.0785.
* (2) F. W. Shu and Y. G. Gong, arXiv:1001.3237.
* (3) R. G. Cai, L. M. Cao and N. Ohta, Phys. Rev. D 81, 061501 (2010).
* (4) L. Smolin, arXiv:1001.3668.
* (5) M. Li and Y. Wang, Phys. Lett. B 687, 243 (2010).
* (6) Y. Tian and X. Wu, Phys. Rev. D 81, 104013 (2010); Y. S. Myung, arXiv:1002.0871.
* (7) I. V. Vancea and M. A. Santos, arXiv:1002.2454.
* (8) Y. X. Liu, Y. Q. Wang and S.W. Wei, Class. Quantum Grav. 27, 185002 (2010);
V. V. Kiselev and S. A. Timofeev, Mod. Phys. Lett. A 25, 2223 (2010);
R. A. Konoplya, Eur. Phys. J. C 69, 555 (2010);
R. Banerjee and B. R. Majhi. Phys. Rev. D 81, 124006 (2010);
P. Nicolini, Phys. Rev. D 82, 044030 (2010);
C. Gao, Phys. Rev. D 81, 087306 (2010);
Y. S. Myung and Y. W. Kim, Phys. Rev. D 81, 105012 (2010);
H. Wei, Phys. Lett. B 692, 167 (2010);
Y. Ling and J.P. Wu, JCAP 1008, 017 (2010);
D. A. Easson, P. H. Frampton and G. F. Smoot, arXiv:1002.4278;
D. A. Easson, P. H. Frampton and G. F. Smoot, arXiv:1003.1528;
S. W. Wei, Y. X. Liu and Y. Q. Wang, arXiv:1001.5238.
* (9) J. Polchinski, String Theory Vol I and II, Cambridge Unversity Press, (1998).
* (10) L. Randall and R. Sundrum, Phys. Rev. Lett. 83, 4690 (1999); Shiromizu et al, Phys. Rev. D 62, 024012 (2000); D. K. Park and S. Tamaryan, Phys. Lett. B 554, 92 (2003); M. Azam et al, Phys. Rev. D 77, 101101 (2008).
* (11) J. F. Donoghue, Phys. Rev. D 50(6), 3874 (1994); J. F. Donoghue, Phys. Rev. Lett., 72(19), 2996 (1994); A. Akhundov et al, Phys. Lett. B 395, 16 (1997); B. F. L. Ward, Mod. Phys. Lett. A 17, 2371 (2002); B. F. L. Ward, Int. J. Mod. Phys. A 20, 3502 (2005); G. G. Kirilin and I. B. Khriplovich, J. Exp. Theor. Phys. 95, 981 (2002); S. Faller, Phys. Rev. D 77, 124039 (2008).
* (12) V. Taveras, Phys. Rev. D 78, 064072 (2008).
* (13) L. Modesto and A. Randono, arXiv:1003.1998.
* (14) A. Sheykhi, Phys. Rev. D 81, 104011 (2010).
* (15) B. Liu, Y. C. Dai, X. R. Hu and J. B. Deng, Mod. Phys. Lett. A 26, 489 (2011).
* (16) A. Sheykhi and S. H. Hendi, arXiv:1009.5561.
* (17) C. Rovelli, Phys. Rev. Lett. 77, 3288 (1996);
A. Ashtekar, J. Baez, A. Corichi and K. Krasnov, Phys. Rev. Lett. 80, 904
(1998).
* (18) S. H. Hendi and A. Sheykhi, Phys. Rev. D 83 (2011) 084012\.
* (19) Saurya Das, S. Shankaranarayanan and Sourav Sur, Phys. Rev. D, 77, 064013 (2008).
* (20) N. Radicella and D. Pavon, Phys. Lett. B 691, 121 (2010).
* (21) T. Padmanabhan, Mod. Phys. Lett. A 25, 1129 (2010);
T. Padmanabhan, Phys. Rev. D 81, 124040 (2010).
* (22) R.G. Cai, L. M. Cao and N. Ohta, Phys. Rev. D 81, 061501(R) (2010).
* (23) T. Padmanabhan, Class. Quantum Grav. 21, 4485 (2004).
* (24) R.G. Cai, S.P. Kim, JHEP 02, 050 (2005);
A. Sheykhi, B. Wang and R. G. Cai, Nucl. Phys. B779, 1 (2007);
A. Sheykhi, B. Wang and R. G. Cai, Phys. Rev. D 76, 023515 (2007);
R. G. Cai and L. M. Cao, Nucl. Phys. B 785, 135 (2007);
A. Sheykhi, JCAP 05, 019 (2009).
* (25) A. Sheykhi, Eur. Phys. J. C 69, 265 (2010).
* (26) A. Sheykhi, B. Wang, Phys. Lett. B 678, 434 (2009);
A. Sheykhi, B. Wang, Mod. Phys. Lett. A 25, 1199 (2010).
* (27) S. A. Hayward, S.Mukohyana, andM. C. Ashworth, Phys. Lett. A 256, 347 (1999); S. A. Hayward, Class. Quant. Grav. 15, 3147 (1998).
* (28) K. Karami, N. Sahraei and S. Ghaffari, arXiv:1009.3833.
* (29) S. Das, S. Shankaranarayanan and S. Sur, arXiv:1002.1129.
* (30) S. Das, S. Shankaranarayanan and S. Sur, arXiv:0806.0402.
* (31) G. Izquierdo and D. Pavon, Phys. Lett. B 633 (2006) 420\.
|
arxiv-papers
| 2010-11-02T16:54:07 |
2024-09-04T02:49:14.431841
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ahmad Sheykhi and Seyed Hossein Hendi",
"submitter": "Ahmad Sheykhi",
"url": "https://arxiv.org/abs/1011.0676"
}
|
1011.0706
|
UNIFORM MODEL OF GEOMETRIC SPACES
Alexandru Popa
Abstract. Full classification of geometric spaces was proposed by Isaak Yaglom
in [2]. By defining the elliptic, parabolic (or linear) and hyperbolic kinds
of measure and applying them to distance, plane and dihedral angle of
different dimensions we get $3^{n}$ geometric spaces of dimension $n$. In his
work [3] Yaglom says that ”finding a general description of all geometric
systems [was] considered by mathematicians the central question of the day.”
A. B. Khachaturean resumed Yaglom’s work in [8].
Author developed a uniform model for all these spaces where distance and angle
measure kinds are parameters. This model is calculus centric, but can also be
used in theoretical research. It is useful in the following domains:
* •
deduction of uniform equations among geometric spaces;
* •
uniform model applied to any space, which provides an easy way to calculate
distances, plane and dihedral angles of any dimension, areas and volumes as
well as parallel (where applied) and orthogonal property detection;
* •
study of not yet described spaces and more.
2000 Mathematics Subject Classification: 51N25, 51N15.
1\. Definitions
As was shown by Yaglom in [2], some $n$-dimensional geometric space can be
defined specifying its $n$ characteristics, or measure kinds. We will use
numbers 1 for elliptic characteristic, 0 for parabolic (or linear) one and
$-1$ for hyperbolic one. So, full space specification is a set of $n$
characteristics $k_{1},...,k_{n}\in\\{-1,0,1\\}$, which can be detected by a
simple algorithm.
Define
$K_{i}=\prod_{j=1}^{i}k_{j},\,\forall i=\overline{0,n}.$ (1)
For two vectors $x,y\in\mathbb{PR}^{n}$, $x=\left<x_{0}:...:x_{n}\right>$,
$y=\left<y_{0}:...:y_{n}\right>$ define a dot product in respect of
characteristics $k_{1}...k_{n}$ as
$x\odot y=\sum_{i=0}^{n}K_{i}x_{i}y_{i}.$ (2)
and cross product in respect of $k_{1}...k_{n}$ so that
$(x\odot y)^{2}+k_{1}(x\otimes y)^{2}=(x\odot x)(y\odot y),\,\forall
x,y\in\mathbb{PR}^{n}.$
It can be checked that111Here and further we will consider for simplicity that
$k^{0}=1$ for $k=0$ too. We will say $x$ divide $k^{i}$, $k=0$ if in
expression $x/k^{i}$ the exponent of $k$ in numerator is greater then or
equals to $i$.
$x\otimes
y=\sqrt{\frac{1}{k_{1}}\sum_{i<j=0}^{n}K_{i}K_{j}(x_{i}y_{j}-x_{j}y_{i})^{2}}.$
(3)
These products were considered by Klein in [1] for elliptic and hyperbolic
spaces.
A $(n+1)\times(n+1)$ matrix is generalized orthogonal in respect of
$k_{1}...k_{n}$ if for all columns $c_{i},c_{j}$ ($i,j=\overline{0,n}$)
$\frac{1}{K_{min(i,j)}}c_{i}\odot c_{j}=\begin{cases}1,i=j,\\\ 0,i\neq
j.\end{cases}$ (4)
Having characteristics $k\in\\{-1,0,1\\}$ consider functions
$C,S,T:\mathbb{R}\to\mathbb{R}$:
$\displaystyle C(x)=C(k,x)$ $\displaystyle=$
$\displaystyle\sum_{i=0}^{\infty}(-k)^{i}\frac{x^{2i}}{(2i)!},$ (5)
$\displaystyle S(x)=S(k,x)$ $\displaystyle=$
$\displaystyle\sum_{i=0}^{\infty}(-k)^{i}\frac{x^{2i+1}}{(2i+1)!},$ (6)
$\displaystyle T(x)=T(k,x)$ $\displaystyle=$
$\displaystyle\frac{S(k,x)}{C(k,x)}.$ (7)
It’s easy to see, that
$\displaystyle C(x)=\begin{cases}\cos x,&k=1,\\\ 1,&k=0,\\\ \cosh
x,&k=-1\end{cases}$ $\displaystyle S(s)=\begin{cases}\sin x,&k=1,\\\
x,&k=0,\\\ \sinh x,&k=-1\end{cases}$ $\displaystyle T(x)=\begin{cases}\tan
x,&k=1,\\\ x,&k=0,\\\ \tanh x,&k=-1\end{cases}$
Define a geometric space with characteristics $k_{1}...k_{n}$ as ”unit ball”
in projective space: $\mathbb{B}^{n}=\\{x\in\mathbb{PR}^{n}\,|\,x\odot
x=1\\}$. Consider ”points” $X\in\mathbb{B}^{n}$ corresponding vectors
$x\in\mathbb{PR}^{n}$. Consider ”space transformation” all linear mappings of
$\mathbb{PR}^{n}$ whose matrices are generalized orthogonal. They are also
transformations of $\mathbb{B}^{n}$ as they preserve it. Consider
$m$-dimensional planes images of $\mathbb{B}^{m}\subset\mathbb{B}^{n}$ on some
transformation. All $m$-dimensional planes are (restricted to
$\mathbb{B}^{n}$) linear combination of first $m+1$ columns of some
generalized orthogonal matrix. So, we can identify $m$-dimensional planes,
$m<n$ with such $(n+1)\times(m+1)$ matrices.
For two $m$-dimensional planes $X,Y$ define dot product in respect of
$k_{1}...k_{n}$ as
$X\odot
Y=\sum_{i_{0}<...<i_{m}=0}^{n}X_{i_{0}...i_{m}}Y_{i_{0}...i_{m}}\prod_{p=1}^{m}\frac{K_{i_{p}}}{K_{p}},$
(8)
where
$M_{l_{0}...l_{m}}=\begin{vmatrix}m_{l_{0}0}&\ldots&m_{l_{0}m}\\\
\vdots&\ddots&\vdots\\\ m_{l_{m}0}&\ldots&m_{l_{m}m}\\\ \end{vmatrix}$
and cross product so that
$(X\odot Y)^{2}+k_{m+1}(X\otimes Y)^{2}=(X\odot X)(Y\odot Y)$
It can be checked that
$X\otimes
Y=\sqrt{\frac{1}{k_{m+1}}\sum_{\begin{subarray}{c}i_{0}<...<i_{m}=0\\\
j_{0}<...<j_{m}=0\\\
i_{0}...i_{m}<j_{0}...j_{m}\end{subarray}}^{n}(X_{i_{0}...i_{m}}Y_{j_{0}...j_{m}}-X_{j_{0}...j_{m}}Y_{i_{0}...i_{m}})^{2}\prod_{p=1}^{m}\frac{K_{i_{p}}K_{j_{p}}}{K^{2}_{p}}}.$
(9)
This model generalizes spherical model of elliptic space, hyperboloid model of
hyperbolic space [6], projective euclidean space model [7] and describes many
new spaces.
2\. Calculus in uniform model
Author shows that dot and cross products of points and planes is invariant in
respect of space transformation. Moreover, it can be used for distance and
angle calculus based on equalities ($m<n$).
$\displaystyle X\odot Y$ $\displaystyle=$ $\displaystyle C_{m+1}(\phi),$ (10)
$\displaystyle X\otimes Y$ $\displaystyle=$ $\displaystyle S_{m+1}(\phi),$
(11)
where $X$ and $Y$ are two points (if $m=0$) and $\phi$ is distance between
them or $X$ and $Y$ are $m$-dimensional planes (if $m>0$) and $\phi$ is angle
between them and functions $C_{m+1}(x)=C(k_{m+1},x),S_{m+1}(x)=S(k_{m+1},x)$.
For some figure $F\subset\mathbb{B}^{n}$ volume can be calculated using the
following equation
$V_{\mathbb{R}}(F)=\frac{1}{n+1}V_{\mathbb{B}}(C_{F})$ (12)
where $C_{F}\subset\mathbb{R}^{n+1}$ is cone having origin
$O=\\{0,...,0\\}\notin\mathbb{B}^{n}$ as vertex and figure $F$ as base,
$V_{\mathbb{B}}$ is native volume in $\mathbb{B}^{n}$ and $V_{\mathbb{R}}$ is
volume in sense of $\mathbb{R}^{n+1}$. The advantage of this approach is the
fact $V_{\mathbb{R}}$ is volume in a linear vector space which is usually
easily to find.
Based on this unified model we can deduce common equation among all spaces.
For example, consider $\mathbb{B}^{2}$ with characteristics $k_{1}$ and
$k_{2}$ and triangle $ABC\in\mathbb{B}^{2}$ with edges $a$, $b$ and $c$,
interior angles $\alpha$, $\gamma$ and exterior angle $\beta^{\prime}$
(interior angle $\beta$ may not exist). Then sine and cosine I and II lows
have identical form in all 9 2-dimensional spaces:
$\frac{S_{1}(a)}{S_{2}(\alpha)}=\frac{S_{1}(b)}{S_{2}(\beta^{\prime})}=\frac{S_{1}(c)}{S_{2}(\gamma)},$
(13)
and
$\displaystyle C_{1}(a)$ $\displaystyle=$ $\displaystyle
C_{1}(b)C_{1}(c)+k_{1}S_{1}(b)S_{1}(c)C_{2}(\alpha),$ (14) $\displaystyle
C_{1}(b)$ $\displaystyle=$ $\displaystyle
C_{1}(a)C_{1}(c)-k_{1}S_{1}(a)S_{1}(c)C_{2}(\beta^{\prime}),$ (15)
$\displaystyle C_{1}(c)$ $\displaystyle=$ $\displaystyle
C_{1}(a)C_{1}(b)+k_{1}S_{1}(a)S_{1}(b)C_{2}(\gamma),$ (16) $\displaystyle
C_{2}(\alpha)$ $\displaystyle=$ $\displaystyle
C_{2}(\beta^{\prime})C_{2}(\gamma)+k_{2}S_{2}(\beta^{\prime})S_{2}(\gamma)C_{1}(a),$
(17) $\displaystyle C_{2}(\beta^{\prime})$ $\displaystyle=$ $\displaystyle
C_{2}(\alpha)C_{2}(\gamma)-k_{2}S_{2}(\alpha)S_{2}(\gamma)C_{1}(b),$ (18)
$\displaystyle C_{2}(\gamma)$ $\displaystyle=$ $\displaystyle
C_{2}(\alpha)C_{2}(\beta^{\prime})+k_{2}S_{2}(\alpha)S_{2}(\beta^{\prime})C_{1}(a),$
(19)
or
$\displaystyle
T_{1}^{2}(a)=\frac{T_{1}^{2}(b)+T_{1}^{2}(c)-2T_{1}(b)T_{1}(c)C_{2}(\alpha)+k_{1}k_{2}T_{1}^{2}(b)T_{1}^{2}(c)S_{1}^{2}(\alpha)}{(1+k_{1}T_{1}(b)T_{1}(c)C_{2}(\alpha))^{2}},$
(20) $\displaystyle
T_{1}^{2}(b)=\frac{T_{1}^{2}(a)+T_{1}^{2}(c)+2T_{1}(a)T_{1}(c)C_{2}(\beta^{\prime})+k_{1}k_{2}T_{1}^{2}(a)T_{1}^{2}(c)S_{1}^{2}(\beta^{\prime})}{(1-k_{1}T_{1}(a)T_{1}(c)C_{2}(\beta^{\prime}))^{2}},$
(21) $\displaystyle
T_{1}^{2}(c)=\frac{T_{1}^{2}(a)+T_{1}^{2}(b)-2T_{1}(a)T_{1}(b)C_{2}(\gamma)+k_{1}k_{2}T_{1}^{2}(a)T_{1}^{2}(b)S_{1}^{2}(\gamma)}{(1+k_{1}T_{1}(a)T_{1}(b)C_{2}(\gamma))^{2}},$
(22) $\displaystyle
T_{2}^{2}(\alpha)=\frac{T_{2}^{2}(\beta^{\prime})+T_{2}^{2}(\gamma)-2T_{2}(\beta^{\prime})T_{2}(\gamma)C_{1}(a)+k_{1}k_{2}T_{2}^{2}(\beta^{\prime})T_{2}^{2}(\gamma)S_{1}^{2}(a)}{(1+k_{2}T_{2}(\beta^{\prime})T_{2}(\gamma)C_{1}(a))^{2}},$
(23) $\displaystyle
T_{2}^{2}(\beta^{\prime})=\frac{T_{2}^{2}(\alpha)+T_{2}^{2}(\gamma)+2T_{2}(\alpha)T_{2}(\gamma)C_{1}(b)+k_{1}k_{2}T_{2}^{2}(\alpha)T_{2}^{2}(\gamma)S_{1}^{2}(b)}{(1-k_{2}T_{2}(\alpha)T_{2}(\gamma)C_{1}(b))^{2}},$
(24) $\displaystyle
T_{2}^{2}(\gamma)=\frac{T_{2}^{2}(\alpha)+T_{2}^{2}(\beta^{\prime})-2T_{2}(\alpha)T_{2}(\beta^{\prime})C_{1}(c)+k_{1}k_{2}T_{2}^{2}(\alpha)T_{2}^{2}(\beta^{\prime})S_{1}^{2}(c)}{(1+k_{2}T_{2}(\alpha)T_{2}(\beta^{\prime})C_{1}(c))^{2}}.$
(25)
As another example, consider $\mathbb{B}^{2}$ with characteristics
$k_{1},k_{2}=1$ and $ABC\in\mathbb{B}^{2}$ right triangle with catheti $a,b$,
hypotenuse $c$ and angles $\alpha$ and $\beta$. Equations of $ABC$ have the
same form for elliptic, euclidean and hyperbolic planes.
$\displaystyle T_{1}^{2}(c)$ $\displaystyle=$ $\displaystyle
T^{2}_{1}(a)+T_{1}^{2}(b)+k_{1}T^{2}_{1}(a)T_{1}^{2}(b),$ (26) $\displaystyle
T_{1}(b)$ $\displaystyle=$ $\displaystyle T_{1}(c)\cos\alpha,$ (27)
$\displaystyle T_{1}(a)$ $\displaystyle=$ $\displaystyle T_{1}(c)\cos\beta,$
(28) $\displaystyle S_{1}(a)$ $\displaystyle=$ $\displaystyle
S_{1}(c)\sin\alpha,$ (29) $\displaystyle S_{1}(b)$ $\displaystyle=$
$\displaystyle S_{1}(c)\sin\beta,$ (30) $\displaystyle T_{1}(a)$
$\displaystyle=$ $\displaystyle S_{1}(b)\tan\alpha,$ (31) $\displaystyle
T_{1}(b)$ $\displaystyle=$ $\displaystyle S_{1}(a)\tan\beta,$ (32)
$\displaystyle\cos\alpha$ $\displaystyle=$ $\displaystyle C_{1}(a)\sin\beta,$
(33) $\displaystyle\cos\beta$ $\displaystyle=$ $\displaystyle
C_{1}(b)\sin\alpha,$ (34) $\displaystyle C_{1}(c)$ $\displaystyle=$
$\displaystyle\cot\alpha\cot\beta.$ (35)
References
[1] Felix Klein, Vorlesungen Nicht-Euklidische Geometrie, B.G.Teubner, Leipzig
1890.
[2] Isaak Yaglom, A simple non-euclidean geometry and its physical basis,
Springer, New York 1979.
[3] Isaak Yaglom, Felix Klein and Sophus Lie, Birkhauser, 1988.
[4] Fenchel, Werner, Elementary geometry in hyperbolic space, De Gruyter
Studies in mathematics. 11. Berlin-New York: Walter de Gruyter & Co 1989.
[5] Naber, Gregory L., The Geometry of Minkowski Spacetime. New York,
Springer-Verlag 1992, ISBN 0387978488.
[6] Reynolds, William F, Hyperbolic Geometry on a Hyperboloid, American
Mathematical Monthly 1993, 100:442-455.
[7] Coxeter H. S. M., The Real Projective Plane, 3rd ed, Springer Verlag 1995.
[8] A. B. Khachaturean, Galilean geometry, MCNMO, Moskow, 2005.
[9] James W. Anderson, Hyperbolic Geometry, Springer 2005, ISBN 1852339349
Alexandru Popa
Department of Computer Sciences
Vest University of Timisoara
Address: Blvd. V. Parvan 4, Timisoara 300223, Timis, Romania
email:alpopa@gmail.com
|
arxiv-papers
| 2010-11-02T19:03:56 |
2024-09-04T02:49:14.440096
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Alexandru Popa",
"submitter": "Alexandru Popa",
"url": "https://arxiv.org/abs/1011.0706"
}
|
1011.0712
|
# Statistical and dynamical fluctuations in the ratios of higher net-proton
cumulants in relativistic heavy ion collisions
Lizhu Chen Institute of Particle Physics, Hua-Zhong Normal University, Wuhan
430079, China Brookhaven National Laboratory, Upton, NY 11973, U.S.A. Xue
Pan Institute of Particle Physics, Hua-Zhong Normal University, Wuhan 430079,
China Fengbo Xiong Institute of Particle Physics, Hua-Zhong Normal
University, Wuhan 430079, China Lin Li Institute of Particle Physics, Hua-
Zhong Normal University, Wuhan 430079, China Na Li Institute of Particle
Physics, Hua-Zhong Normal University, Wuhan 430079, China Zhiming Li
Institute of Particle Physics, Hua-Zhong Normal University, Wuhan 430079,
China Gang Wang Department of Physics and Astronomy, University of
California, Los Angeles, CA 90095, U.S.A. Yuanfang Wu Institute of Particle
Physics, Hua-Zhong Normal University, Wuhan 430079, China Brookhaven National
Laboratory, Upton, NY 11973, U.S.A. Key Laboratory of Quark $\&$ Lepton
Physics (Huazhong Normal University), Ministry of Education, China
###### Abstract
With the help of transport and statistical models, we find that the ratios of
higher net-proton cumulants measured at RHIC are dominated by the statistical
fluctuations. Future measurements should focus on the dynamical fluctuations,
which are relevant to the underlying mechanisms of particle production, the
critical phenomena in particular. We also demonstrate that a proton-antiproton
correlation directly show if protons and antiprotons are emitted
independently.
###### pacs:
25.75.Nq, 12.38.Mh, 21.65.Qr
## I Introduction
One of the main goals of relativistic heavy ion collisions is to locate the
critical point on the QCD phase diagram, spanned by the temperature ($T$) and
the baryon chemical potential ($\mu_{B}$). At the critical point, the
correlation length ($\xi$) goes to infinity and the long range correlations
become dominant. So the fluctuations of final state particles are expected to
be largely enhanced, if the $(T,\mu_{B})$ trajectory of the collision system
is close to the critical point. The $\xi$-related observables are therefore of
great interest in heavy ion collisions corr-fluc .
The thermodynamic quantities, such as order parameter, specific heat capacity
and susceptibility ($\chi$), diverge with the correlation length at the
critical point. The $i$th net-baryon cumulant is recently shown to be directly
related to the $i$th susceptibility ($\chi_{i}$) of the formed system antoniou
; stephanov ; koch ,
$\langle\delta N^{i}\rangle=VT\chi_{i},$ (1)
where $N$ is the net-baryon number, $\langle\delta
N^{i}\rangle=\langle(N-\langle N\rangle)^{i}\rangle$ is the $i$th net-baryon
cumulant, and $V$ is the volume. The third and fourth cumulants,
$K_{3}=\langle\delta N^{3}\rangle,\ \ K_{4}=\langle\delta
N^{4}\rangle-3\langle\delta N^{2}\rangle^{2},$ (2)
are argued to be more sensitive to the correlation length as they are
proportional to $\xi^{4.5}$ and $\xi^{7}$, respectively stephanov ; koch ;
rajargopal ; akasawa . Experimentally, the proton number is a good
approximation of the baryon number stephanov , and the properly normalized
ratios, net-proton Skewness and Kurtosis,
$\displaystyle S=K_{3}/K_{2}^{3/2}=\frac{\langle\delta
N^{3}\rangle}{\langle\delta N^{2}\rangle^{3/2}},$ $\displaystyle
K=K_{4}/K_{2}^{2}=\frac{\langle\delta N^{4}\rangle}{\langle\delta
N^{2}\rangle^{2}}-3,$ (3)
are preferred star-prl , measuring the symmetry and sharpness of the net-
proton distribution, respectively.
The STAR measurements star-prl show that both net-proton Skewness and
Kurtosis decrease with increasing number of participants (centrality), which
could be explained by the central limit theorem (CLT). On the other hand,
various model calculations luoxf reproduce the experimental results
surprisingly well, raising the suspicion that Skewness and Kurtosis are
insensitive to the mechanisms of particle production implemented in different
models. Recently, Karsch and Redlich karsch have derived simple relations
between the cumulant ratios and the thermal parameters ($T$ and $\mu_{B}$ at
the chemical freeze-out parameters ), based on the hadron resonance gas (HRG)
model HRG , with the system well thermalized and without phase transition.
They have shown that the HRG model results are well consistent with the STAR
data at different collision energies karsch .
However, before trying to understand the physics behind the measured Skewness
and Kurtosis, we need to take into account and properly remove the
contributions from the statistical fluctuations bialas ; rajargopal and all
non-thermal sources (minijets, resonance decay, initial size fluctuation,
etc.) gupta ; Giorgio . In this paper, we will focus on the elimination of the
statistical fluctuations. We first estimate the statistical fluctuations in
the cumulant ratios, which turn out to dominate the behavior of net-proton
Kurtosis at RHIC energies. Then we propose the dynamical ratios of higher net-
proton cumulants in Section III and the correlation between proton and
antiproton in Section IV. The centrality dependence of the dynamical ratios
and the correlations from two versions of AMPT ampt , UrQMD urqmd , and
Therminator therminator are presented and discussed. Finally, the summary and
conclusions are given in Section V.
## II Statistical fluctuations in the ratios of higher net-proton cumulants
The statistical fluctuation comes from the finite number of particles, usually
obeying a Poisson distribution bialas ; claude ; rajargopal . If we have two
independent Poisson distributions for protons and antiprotons with means
$\langle N_{p}\rangle$ and $\langle N_{\bar{p}}\rangle$, respectively, the
net-proton number ($N$) follows a Skellam (SK) distribution luoxf ; skellam ,
$\displaystyle f(N;\langle N_{p}\rangle,\langle N_{\bar{p}}\rangle)$
$\displaystyle=e^{-(\langle N_{p}\rangle+\langle N_{\bar{p}}\rangle)}(\langle
N_{p}\rangle/\langle
N_{\bar{p}}\rangle)^{\frac{N}{2}}I_{|N|}\left(2\sqrt{\langle
N_{p}\rangle\langle N_{\bar{p}}\rangle}\right),$ (4)
where $I_{|N|}(2\sqrt{\langle N_{p}\rangle\langle N_{\bar{p}}\rangle})$ is the
modified Bessel function of the first kind. Then the statistical fluctuations
of the ratios of higher net-proton cumulants can be directly deduced from the
Skellam distribution,
$\displaystyle S_{\rm stat}$ $\displaystyle=$ $\displaystyle\frac{\langle
N_{p}\rangle-\langle N_{\bar{p}}\rangle}{[\langle N_{p}\rangle+\langle
N_{\bar{p}}\rangle]^{3/2}},$ $\displaystyle K_{\rm stat}$ $\displaystyle=$
$\displaystyle\frac{1}{\langle N_{p}\rangle+\langle N_{\bar{p}}\rangle},$
$\displaystyle R_{2,1,{\rm stat}}$ $\displaystyle=$
$\displaystyle\frac{\langle N_{p}\rangle+\langle N_{\bar{p}}\rangle}{\langle
N_{p}\rangle-\langle N_{\bar{p}}\rangle},$ $\displaystyle R_{3,2,{\rm stat}}$
$\displaystyle=$ $\displaystyle\frac{\langle N_{p}\rangle-\langle
N_{\bar{p}}\rangle}{\langle N_{p}\rangle+\langle N_{\bar{p}}\rangle},$
$\displaystyle R_{4,2,{\rm stat}}$ $\displaystyle=$ $\displaystyle 1,$ (5)
where $R_{i,j}=\chi_{i}/\chi_{j}=K_{i}/K_{j}$. The HRG model results karsch
show that $R_{4,2}$ is unity, and now we find that it is completely
statistical. The other ratios are determined only by the mean numbers of
protons and antiprotons, which usually increase with the incident energy and
centrality. So the statistical fluctuations of Skewness and Kurtosis decrease
with the incident energy and centrality.
In Fig. 1(a), the model results of net-proton Kurtosis and its statistical
fluctuations (SK) are shown from APMT default, AMPT with string melting ampt
and Therminator therminator . Where the transverse momentum, $p_{\rm t}$, and
rapidity cuts are respectively $0.4<p_{\rm t}<0.8$ GeV/c, $|y|<0.5$, the same
as they are given at RHIC/STAR published paper star-prl . In both transport
(AMPT) and statistical (Therminator) models, the net-proton Kurtosis results
(solid points) are very close to the corresponding statistical fluctuations
(open points). The STAR measurements of net-proton Kurtosis (stars) star-prl
follow the same trend as a function of the number of participants ($N_{\rm
part}$). So the statistical fluctuations dominate the behavior of net-proton
Kurtosis at RHIC, which is why the results from various models closely
resemble the experimental data.
Figure 1: (Color online) (a) Centrality dependence of net-proton Kurtosis at
200 GeV, and (b) Energy dependence of $R_{2,1}$, $R_{4,2}$ and $R_{3,2}$, for
Au + Au collisions. The results are obtained from RHIC/STAR data star-prl ,
HRG karsch , two versions of AMPT ampt and Therminator therminator models,
and the corresponding statistical fluctuations are estimated with the Skellam
distribution (SK) from Eq. (II), respectively.
In Fig. 1(b), we show the incident-energy dependence of the net-proton ratios,
$R_{2,1}$, $R_{4,2}$, and $R_{3,2}$, obtained from RHIC/STAR data, HRG, AMPT
default, AMPT with string melting and Therminator, together with the
corresponding statistical fluctuations (SK) of the last three cases determined
by Eq. (5). At 200 GeV, the ratios from Therminator (squares) coincide with
the corresponding statistical fluctuations (crosses), because Therminator is a
statistical model with only constraints on kinetics. The HRG curves karsch
are given by $R_{2,1}=1/\tanh(\mu_{B}/T)$ and $R_{3,2}=\tanh(\mu_{B}/T)$.
Similarly, both Therminator and HRG models have protons and antiprotons
completely independently emitted with Poisson distributions, so their results
are close to the data from RHIC/STAR, where the statistical fluctuations
dominate. The ratios from two versions of AMPT (solid circles and triangles)
slightly deviate from the corresponding statistical fluctuations (open circles
and rhombi), and from the HRG lines, due to the more complicated particle
production mechanisms in AMPT.
We have demonstrated that the influence of statistical fluctuations is far
from being negligible in the ratios of higher net-proton cumulants at RHIC
energies. To investigate the underlying physics, we have to first remove the
statistical fluctuations.
## III Dynamical ratios of higher net-proton cumulants
There have been long efforts in eliminating the statistical fluctuations in
elementary collisions bialas ; kittel . For a single particle distribution,
the factorial moments are used to remove the statistical fluctuations antoniou
; bialas . But this method can not be directly generalized to the distribution
of the difference between two Poisson variables.
From previous discussions, the statistical fluctuations in the ratios of net-
proton cumulants are directly obtainable. The dynamical ratios of net-proton
cumulants can be simply defined as a deviation of the ratios from the
statistical fluctuations claude , e.g.,
$\displaystyle K_{\rm dyn}$ $\displaystyle=$ $\displaystyle K-K_{\rm stat},$
(6)
and so on, where the statistical parts are given by Eq. (II).
Figure 2: (Color online) Centrality dependence of dynamical Skewness (left)
and Kurtosis (right) for Au + Au collisions at 200 GeV, given by transport
models (AMPT and UrQMD) and a statistical model (Therminator).
The centrality dependence of dynamical net-proton Skewness and Kurtosis are
shown in Fig. 2(a) and (b), respectively, from AMPT default, AMPT with string
melting, UrQMD and Therminator models for Au + Au collisions at 200 GeV. Both
dynamical Skewness and Kurtosis from Therminator are zero at all centralities,
illustrating that the symmetry and sharpness of the net-proton distribution in
the model both follow the Skellam distribution. For the transport models (AMPT
and UrQMD), both dynamical Skewness and Kurtosis are larger than zero in
peripheral collisions, and approach zero in central collisions. Compared with
the Skellam distribution, the positive dynamical Kurtosis and Skewness implies
that the net-proton distribution has a sharper peak and a longer tail at the
large net-proton side, respectively. These deviations are caused by non-
thermal sources implemented in transport models.
To study how the results change with the incident energy, the centrality
dependence of dynamical net-proton Skewness and Kurtosis from AMPT default are
shown in Fig. 3 for Au+Au collisions at 3 incident energies. When the incident
energy changes from 200 GeV to 39 GeV, both dynamical Skewness and Kurtosis
remain positive. Dynamical Skewness shows a significant dependence on the
incident energy, especially in peripheral collisions, and dynamical Kurtosis
is almost independent of the incident energy.
Figure 3: (Color online) Centrality dependence of dynamical net-proton
Skewness (left) and Kurtosis (right) for Au + Au collisions at 3 incident
energies, obtained from AMPT default.
The behavior of dynamical ratios of higher cumulants are called for at the
RHIC beam energy scan. If the deviation from the statistical fluctuations is
zero, protons and antiprotons are emitted independently as the statistical
models assume. Otherwise if the non-zero deviation remains the same sign for
different incident energies, like what the transport model shows in Fig. 3,
then there is no critical related phenomena. However, if the deviation changes
dramatically with the variation of the incident energy, e.g. showing the sign
changes at the third and fourth cumulants, where the symmetry and sharpness of
the net-proton distribution deviate from the corresponding statistical
fluctuations in opposite directions akasawa ; Liuyx ; fs3 , it may reveal the
critical incident energy nearby fs3 .
## IV Correlations between proton and antiproton
To see if protons and antiprotons are emitted independently, we could also
directly measure the correlation between them,
$\displaystyle{}C(N_{p},N_{\bar{p}})=\frac{\langle
N_{p}N_{\bar{p}}\rangle}{\langle N_{p}\rangle\langle N_{\bar{p}}\rangle}-1.$
(7)
The correlation will be zero if protons and antiprotons are independent.
The centrality dependence of the correlations from two versions of AMPT, UrQMD
and Therminator models for Au + Au collisions at 200 GeV are presented in Fig.
4. The correlation is zero at all centralities in Therminator, another
illustration of the model’s assumption. In transport models, the correlation
decreases with centrality, following a similar trend as dynamical Skewness or
Kurtosis. The correlation is positive, indicating that protons and antiprotons
are not emitted independently. This leads to the difference between the net-
proton distribution in transport models and the pure statistical Skellam
distribution.
Figure 4: (Color online) Centrality dependence of proton antiproton
correlations for Au + Au collisions at 200 GeV, given by transport models
(AMPT and UrQMD) and a statistical model (Therminator).
## V Summary
We have demonstrated that the statistical fluctuations dominate the behavior
of the ratios of higher net-proton cumulants measured at RHIC, and this
explains why the results from various models are consistent with the
experimental data. We argue that before trying to understand the underlying
physics the statistical fluctuations should be taken into account.
To study the particle production mechanism, the dynamical ratios of higher
net-proton cumulants and the correlations between proton and antiproton have
been proposed and discussed. It is shown that the dynamical ratios and the
correlations are similarly zero at all centralities in a statistical model,
and positive in transport models. This indicates that protons and antiprotons
are not emitted independently in transport models.
The behaviors of the dynamical ratios of higher net-proton cumulants, as well
as that of the proton-antiproton correlation, are more relevant to the
location of the critical point, and the corresponding measurements during RHIC
beam energy scan will shed light on the study of the QCD phase transition.
We are grateful for stimulating discussions with Dr. Nu Xu, Xiaofeng Luo, Dr.
Fuqiang Wang and Dr. Zhangbu Xu. The first and last authors are grateful for
the hospitality of BNL STAR group. This work is supported in part by the NSFC
of China with project No. 10835005, 11005046, MOE of China with project No.
IRT0624, No. B08033 and a grant from U.S. Department of Energy, Office of
Nuclear Physics.
## References
* (1) M. A. Stephanov, K. Rajagopal, and E. Shuyak, Phys. Rev. Lett. 81, 4816(1998); S. Jeon and V. Koch, Phys. Rev. Lett. 85, 2076 (2000); M. Asakawa, U. Heinz and B. Müller, Phys. Rev. Lett. 85, 2072 (2000); H. Heiselberg, Phys. Rept. 351, 161(2001).
* (2) N. G. Antoniou, F. K. Diakonos,* and A. S. Kapoyannis, K. S. Kousouris, Phys. Rev. Lett.97, 032002 (2006); D. Bower and S. Gavin, Phys. Rev. C 64, 051902(R) (2001); N. G. Antoniou, Nucl. Phys. B, Proc. Suppl. 92, 26 (2001).
* (3) Y. Hatta and M. A. Stephanov, Phys. Rev. Lett. 91, 102003 (2003); Y. Hatta and T. Ikeda, Phys. Rev. D 67, 014028 (2003).
* (4) V. Koch, arXiv:0810.2520.
* (5) C. Athanasiou, K. Rajagopal, and M. Stephanov, arXiv:1006.4636; C. Athanasiou, K. Rajagopal, and M. Stephanov, arXiv:1008.3385.
* (6) M. Asakawa, S. Ejiri, M. Kitazawa, Phys. Rev. Lett. 103, 262301(2009).
* (7) M. M. Aggarwal et al. (STAR Coll.), Phys. Rev. Lett. 105, 022302(2010).
* (8) X. F. Luo, B. Mohanty, H. G. Ritter, N. Xu, J. Phys. G 37, 094061(2010).
* (9) F. Karsch and K. Redlich, arXiv:1007.2581.
* (10) J. Cleymans, and K. Redlich, Phys. Rev. Lett. 81, 5284 (1998).
* (11) P. Braun-Munzinger, K. Redlich and J. Stachel, nucl-th/0304013; A. Andronic, P. Braun-Munzinger and J. Stachel, Nucl. Phys. A 772, 167 (2006).
* (12) A. Bialas and R. Peschanski, Nucl. Phys. B 273, 703(1986); A. Bialas and R. Peschanski, Nucl. Phys. B 308, 857(1988); A. Bialas and R. Peschanski, Phys. Lett. B 207, 59(1988).
* (13) S. Gupta, arXiv:0909.4630.
* (14) V.P. Konchakovski, M. Hauer, G Torrieri, M.I. Gorenstein, E.L. Bratkovskaya, Phys. Rev. C 79, 034910(2009); Giorgio Torrieri, Rene Bellwied, Christina Markert, Gary Westfall, arXiv:1001.0087.
* (15) Zi-Wei Lin, Che Ming Ko, Bao-An Li, Bin Zhang and Subrata Pal, Phys. Rev. C72, 064901 (2005).
* (16) H. Petersen, et al., arXiv:0805.0567.
* (17) A. Kisiel et al., Comput. Phys. Commun. 174, 669(2006).
* (18) Skellam J G, Journal of the Royal Statistical Society 109, 296(1946).
* (19) C. Pruneau, S. Gavin, S. Voloshin, Phys. Rev. C 66, 044904(2002); STAR Coll. Phys. Rev. C 68, 044905(2003); STAR Coll. Phys. Rev. C 79, 024906(2009).
* (20) E. A. De Wolf, I. M. Dremin, W. Kittel, Phys. Report, 270, 1(1996).
* (21) Wei-jie Fu, Yu-xin Liu, Yue-Liang Wu, Phys. Rev. D 81, 014028(2010); Wei-jie Fu, and Yue-liang Wu, arXiv: 1008.3684.
* (22) Chen Lizhu, Pan Xue, X. S Chen, and Wu Yuanfang, arXiV:1010.1166.
|
arxiv-papers
| 2010-11-02T19:37:28 |
2024-09-04T02:49:14.445204
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Chen Lizhu, Pan Xue, Xiong Fengbo, Li Lin, Li Na, Li Zhiming, Gang\n Wang and Wu Yuanfang",
"submitter": "Yuanfang Wu",
"url": "https://arxiv.org/abs/1011.0712"
}
|
1011.0874
|
11institutetext: Instituto de Física Rosario (CONICET) and Universidad
Nacional de Rosario, Boulevard 27 de Febrero 210 bis, (2000) Rosario,
Argentina
Quantized spin models, including quantum spin frustration
# A test of the bosonic spinon theory for the triangular antiferromagnet
spectrum
A. Mezio C. N. Sposetti L. O. Manuel and A. E. Trumper
###### Abstract
We compute the dynamical structure factor of the spin-$\frac{1}{2}$ triangular
Heisenberg model using the mean field Schwinger boson theory. We find that a
reconstructed dispersion, resulting from a non trivial redistribution of the
spectral weight, agrees quite well with the spin excitation spectrum recently
found with series expansions. In particular, we recover the strong
renormalization with respect to linear spin wave theory along with the
appearance of roton-like minima. Furthermore, near the roton-like minima the
contribution of the two spinon continuum to the static structure factor is
about $40\%$ of the total weight. By computing the density-density dynamical
structure factor, we identify an unphysical weak signal of the spin excitation
spectrum with the relaxation of the local constraint of the Schwinger bosons
at the mean field level. Based on the accurate description obtained for the
static and dynamic ground state properties, we argue that the bosonic spinon
theory should be considered seriously as a valid alternative to interpret the
physics of the triangular Heisenberg model.
###### pacs:
75.10.Jm
## 1 Introduction
During a long time the magnetic ground state of the spin-$\frac{1}{2}$
triangular Heisenberg model (THM) has attracted the attention of many
researchers, due to the possible realization of the resonating valence bond
(RVB) ground state proposed by P. W. Anderson in $1973$ [1]. The revival of
the RVB theory for the cuprates [2] prompted the investigations of quantum
disordered ground states within large $N$ theories where the Heisenberg
interaction is naturally written in terms of singlet bond operators and
fractional spin-$\frac{1}{2}$ excitations with bosonic or fermionic character
[3]. The fermionic version leads to exotic disordered ground states [4] while
the bosonic one allows to describe disordered and ordered ground states [5] by
relating the magnetization with the condensation of bosons [6]. For this case,
using gauge field theoretical arguments, it has been conjectured that, when
short range spiral correlations are present in the disordered phases, the
bosonic spinons would be in a deconfined regime [5]. Therefore, a broad two
spinon continuum is expected in the spin excitation spectrum.
From the numerical side, instead, thanks to the enormous effort of the
community to develop unbiased techniques [7, 8, 9, 10], it has been firmly
established that the ground state of the spin-$\frac{1}{2}$ THM is a robust
$120^{\circ}$ Néel order. These numerical results precluded the fermionic
version of the RVB theory, giving support to both the linear spin wave theory
(LSWT) and the bosonic version of the RVB theory, namely the Schwinger boson
theory. In fact, both theories agree quite well with numerical results on
finite size systems [11, 12], although for spiral phases the singlet structure
of the mean field Schwinger bosons theory does not recover the spin wave
dispersion relation in the large $s$ limit [13]. Consequently, linear spin
wave theory seemed to capture the quantum and semiclassical features expected
for a $120^{\circ}$ Néel ground state of the THM. However, recent series
expansions studies [14, 10] challenged LSWT, showing that for
$s\\!=\\!\frac{1}{2}$ the functional form of the dispersion relation differs
considerably (points of fig. 2) from that of LSWT (solid line of fig. 2). In
particular, it was observed a strong downward renormalization of the high
energy part of the spectrum along with the appearance of roton-like minima at
the midpoints of edges of the hexagonal Brillouin zone (BZ) (B and D points of
the inset of fig. 1). The authors argued that the differences with LSWT could
be attributed, probably, to the presence of fermionic spinon excitations.
Nevertheless, further spin wave studies [15] showed that, to first order in
$1/s$, there appear non trivial corrections to the linear spin wave dispersion
due to the non collinearity of the ground state, giving a fairly accurate
description of the series expansion results. However, magnons are not well
defined for an ample region of the BZ [16]. Another question, regarding the
spectrum of the THM, is the nature of the multiparticle continuum above the
one magnon states. For instance, it is believed that the broad multiparticle
continuum measured in the $Cs_{2}CuCl_{4}$ compound is better described by an
interacting spinon picture than a magnon one [17]. In this sense, given
accurate predictions of the Schwinger boson theory for the static ground state
properties of the THM [11], it is important to investigate whether the
anomalous features of the spectrum found with series expansions can be
captured, or not, by this alternative theory that naturally incorporates
fractional spin-$\frac{1}{2}$ excitations.
In the present article we investigate the validity of the bosonic spinon
theory to interpret the spin excitation spectrum of the spin-$\frac{1}{2}$
THM. Our main finding is that the mean field Schwinger boson theory (intensity
curves of fig. 2), based on the two singlet operator scheme [18], reproduces
qualitatively and quantitatively quite well the recent series expansions
results. By computing the dynamical structure factor, we remarkably find that
the expected spin excitation spectrum is recovered by a reconstruction
resulting from a non trivial redistribution of the spectral weight located at
the spinonic branches shifted by $\pm\frac{\bf Q}{2}$, where ${\bf
Q}=(\frac{4}{3}\pi,0)$ is the magnetic wave vector. By computing the density-
density dynamical structure factor, we were able to identify, at the mean
field level, the remnant weaker signal of the spectrum with the relaxation of
the local constraint of the number of bosons. We also discuss the validity of
the alternative mean field decoupling based on one singlet operator scheme.
## 2 Mean field Schwinger bosons approximation
In the Schwinger boson representation [3] the spin operators are expressed as
${\hat{\bf S}}_{i}\\!\\!=\\!\frac{1}{2}{\bf b}^{\dagger}_{i}\vec{\sigma}\;{\bf
b}_{i}$, with the spinor ${\bf
b}^{\dagger}_{i}\\!=\\!(\hat{b}^{\dagger}_{i\uparrow};\hat{b}^{\dagger}_{i\downarrow})$
composed by the bosonic operators $\hat{b}^{\dagger}_{i\uparrow}$ and
$\hat{b}^{\dagger}_{i\downarrow}$, and
$\vec{\sigma}\\!\\!=\\!\\!(\sigma^{x},\sigma^{y},\sigma^{z})$ the Pauli
matrices. To fulfil the spin algebra the constraint of $2s$ bosons per site,
$\sum_{\sigma}\hat{b}^{\dagger}_{i\sigma}\hat{b}_{i\sigma}\\!=2s$, must be
imposed. Then, the spin-spin interaction of the Heisenberg Hamiltonian can be
written as
$\hat{{\bf S}}_{i}\\!\cdot\\!\hat{{\bf
S}}_{j}=\;:\hat{B}^{\dagger}_{ij}\hat{B}_{ij}:-\hat{A}^{\dagger}_{ij}\hat{A}_{ij},$
(1)
where $::$ means normal order and the singlet bond operators are defined as
$\hat{A}^{\dagger}_{ij}\\!=\\!\frac{1}{2}\sum_{\sigma}\sigma\hat{b}^{\dagger}_{i\sigma}\hat{b}^{\dagger}_{j\bar{\sigma}}$
and
$\hat{B}^{\dagger}_{ij}\\!=\\!\frac{1}{2}\sum_{\sigma}\hat{b}^{\dagger}_{i\sigma}\hat{b}_{j\sigma}$.
We will briefly describe the main steps of the mean field while the details of
the calculation can be found in our previous works [11, 18]. Introducing a
Lagrange multiplier $\lambda$ to impose the local constraint on average and
performing a mean field decoupling of eq. (1), such as
$A_{ij}=\langle\hat{A}_{ij}\rangle=\langle\hat{A}^{\dagger}_{ij}\rangle$ and
$B_{ij}=\langle\hat{B}_{ij}\rangle=\langle\hat{B}^{\dagger}_{ij}\rangle$, the
diagonalized mean field Hamiltonian results
$\hat{H}_{MF}=E_{\texttt{gs}}+\sum_{\bf k}\omega_{\bf
k}\left[\hat{\alpha}^{\dagger}_{{\bf k}\uparrow}\hat{\alpha}_{{\bf
k}\uparrow}+\hat{\alpha}^{\dagger}_{-{\bf k}\downarrow}\hat{\alpha}_{-{\bf
k}\downarrow}\right],$
where
$E_{\texttt{gs}}=\frac{1}{2}\sum_{\bf k}\omega_{\bf k}+\lambda
N(s+\frac{1}{2})$
is the ground state energy and
$\omega_{{\bf k}\uparrow}=\omega_{{\bf k}\downarrow}=\omega_{\bf
k}=[(\gamma^{B}_{\bf k}+\lambda)^{2}-(\gamma^{A}_{\bf k})^{2}]^{\frac{1}{2}},$
is the spinon dispersion relation with geometrical factors, $\gamma^{B}_{\bf
k}\\!=\\!\frac{1}{2}J\sum_{\delta}B_{\delta}\cos{\bf k}.\delta$ and
$\gamma^{A}_{\bf k}\\!=\\!\frac{1}{2}J\sum_{\delta}A_{\delta}\sin{\bf
k}.\delta$, and with the sums going over all the vectors $\delta$ connecting
the first neighbours of a triangular lattice. The mean field parameters has
been chosen real and satisfy the relations $B_{\delta}\\!=\\!B_{-\delta}$ and
$A_{\delta}\\!=\\!-A_{-\delta}$. The ground state wave function of
$\hat{H}_{MF}$ can be written in a Jastrow form [6],
$|\texttt{gs}\rangle=\exp\left[\sum_{ij}f_{ij}\hat{A}^{\dagger}_{ij}\right]|0\rangle_{b},$
(2)
where $|0\rangle_{b}$ represents the vacuum of Schwinger bosons and the odd
pairing function is defined as $f_{ij}\\!\\!=(\frac{1}{N})\sum_{\bf k}f_{\bf
k}e^{\imath{\bf k}({\bf r}_{i}-{\bf r}_{j})}$, with $f_{\bf
k}\\!\\!=\\!\\!-v_{\bf k}/u_{\bf k}$ , and Bogoliubov coefficients $u_{\bf
k}\\!=\\![\frac{1}{2}(1+\frac{\gamma^{B}_{\bf k}+\lambda}{\omega_{\bf
k}})]^{\frac{1}{2}}$ and $v_{\bf k}\\!=\\!\imath\ {\it sgn}(\gamma^{A}_{\bf
k})[\frac{1}{2}(-1+\frac{\gamma^{B}_{\bf k}+\lambda}{\omega_{\bf
k}})]^{\frac{1}{2}}$. The singlet bond structure of eq. (2) guarantees the
singlet behavior of $|\texttt{gs}\rangle$. Even if the Lieb-Mattis theorem
cannot be applied to non bipartite lattices, the singlet character of the
ground state for cluster sizes with an even number of sites $N$ has been
confirmed numerically [7, 8]. It should be noted, however, that
$|\texttt{gs}\rangle$ is not a true RVB state because the constraint is only
satisfied on average. Furthermore, by solving the self consistent mean field
equations at zero temperature,
$\displaystyle A_{\delta}$ $\displaystyle=$
$\displaystyle\frac{1}{2N}\sum_{\bf k}\frac{\gamma^{A}_{\bf k}}{\omega_{\bf
k}}\sin{\bf k}.{\delta}$ $\displaystyle B_{\delta}$ $\displaystyle=$
$\displaystyle\frac{1}{2N}\sum_{\bf k}\frac{(\gamma^{B}_{\bf
k}+\lambda)}{\omega_{\bf k}}\cos{\bf k}.{\delta}$ (3) $\displaystyle
s+\frac{1}{2}$ $\displaystyle=$ $\displaystyle\frac{1}{2N}\sum_{\bf
k}\frac{(\gamma^{B}_{\bf k}+\lambda)}{\omega_{\bf k}},$
it is found that as the system size $N$ increases the singlet ground state
$|\texttt{gs}\rangle$ develops $120^{\circ}$ Néel correlations signalled by
the minimum gap of the spinon dispersion located at $\pm\frac{\bf Q}{2}$,
where ${\bf Q}\\!=\\!(\frac{4}{3}\pi,0)$ is the magnetic wave vector [19]. As
the spinon gap behaves as $\omega_{\pm\frac{\bf Q}{2}}\\!\sim\\!1/N$, for
large system sizes the singular modes of eq. (3) can be treated apart,
analogously to a Bose condensation phenomena [6]. In particular, the local
magnetization $m({\bf Q})$ can be derived from the last line of eq. (3),
yielding the relation [20]
$\frac{1}{2N}\frac{(\gamma^{B}_{\frac{\bf
Q}{2}}+\lambda)^{2}}{\omega^{2}_{\frac{\bf Q}{2}}}=S({\bf
Q})=\frac{N}{2}m^{2}({\bf Q}),$
where $S({\bf k})\\!\\!=\\!\\!\\!\sum_{\bf R}e^{\imath{\bf k}.{\bf
R}}\langle\texttt{gs}|\hat{S}_{0}\\!\\!\cdot\\!\\!\hat{S}_{\bf
R}|\texttt{gs}\rangle$ is the static structure factor. Formally, it can be
shown that in the thermodynamic limit $|\texttt{gs}\rangle$ is degenerated
with a manifold of Bose condensate ground states, each one corresponding to
all the possible orientations, in spin space, of the $120^{\circ}$ Néel order.
In the Schwinger boson language the condensate of the up/down bosons at
$\pm\frac{\bf Q}{2}$ and the normal fluids of bosons corresponds to the
spiralling magnetization $m({\bf Q})$ and the zero point quantum fluctuations,
respectively [6]. For the triangular lattice the present mean field
approximation [21] gives a local magnetization $m=0.275$.
An alternative procedure, is to use the operator identity
$:\\!\\!\hat{B}^{\dagger}_{ij}\hat{B}_{ij}\\!\\!:\\!\\!+\hat{A}^{\dagger}_{ij}\hat{A}_{ij}\\!\\!=\\!\\!S^{2}$,
and write the spin-spin interaction (1) in terms of the singlet operator
$\hat{A}_{ij}$ [3, 22, 24]:
$\hat{\bf S}_{i}\\!\cdot\\!\hat{\bf
S}_{j}=-2\hat{A}^{\dagger}_{ijd}\hat{A}_{ij}+S^{2}.$ (4)
Even if eqs. (1) and (4) are equivalent, the latter leads to a different mean
field decoupling with parameters $A_{\delta}$ and $\lambda$ [22, 24]. In table
1 it is shown the values of the ground state energy and magnetization for the
THM obtained with the two mean field Schwinger boson decouplings along with
Gaussian fluctuations [11], linear spin wave theory, non linear spin wave
theory (LSWT$+1/s$) [23]; and quantum Monte Carlo [8] (QMC) results [25]. Even
though it has not yet been calculated, we expect that Gaussian fluctuations
above the mean field will reduce the magnetization, as has already been found
for the spin stiffness in the THM [11]. From table 1 it is seen that the two
singlet scheme describe quantitatively better the static properties of the
THM.
Table 1: Energy and magnetization of the $120^{\circ}$ Néel ground state of
the spin-$\frac{1}{2}$ Heisenberg antiferromagnet on the triangular lattice as
obtained with mean field Schwinger bosons within one [22] ($A$) and two [21]
($AB$) singlet scheme; Gaussian fluctuations [11] above the $AB$ mean field
($AB+\textrm{Fluct}$), Quantum Monte Carlo [8] (QMC), linear spin wave theory
(LSWT) and non linear spin wave theory (LSWT$+1/s$) [23]
.
| | $E/JN$ | | $m$
---|---|---|---|---
$A$ | | -0.7119 | | 0.328
$AB$ | | -0.5697 | | 0.275
$AB+$Fluct | | -0.5533 | |
QMC | | -0.5458(1) | | 0.205(1)
LSWT | | -0.5388 | | 0.2387
LSWT$+1/s$ | | -0.5434 | | 0.2497
## 3 Dynamical structure factor
### 3.1 Spin-spin correlation functions
We study the spectrum through the dynamical structure factor at $T=0$, defined
as
$S^{\alpha\alpha}\\!({\bf k},\omega)=\sum_{n}|\langle\texttt{gs}|\hat{\bf
S}^{\alpha}_{\bf
k}(0)|n\rangle|^{2}\delta(\omega-(\epsilon_{n}-E_{\texttt{gs}})),$
where $\alpha$ denotes $x,y,z$, $|n\rangle$ are the excited states, and
$\hat{\bf S}^{\alpha}_{\bf k}$ is the Fourier transform of $\hat{\bf
S}^{\alpha}_{i}$. As we work on finite systems the $SU(2)$ symmetry is not
broken explicitly and $S^{xx}\\!\\!=\\!\\!S^{yy}\\!\\!=\\!\\!S^{zz}$ (in what
follows the $\alpha\alpha$ indices are discarded). A straightforward
calculation leads to the expression
$S\\!({\bf k},\omega)\\!=\\!\frac{1}{4N}\\!\\!\sum_{{\bf q}}|u_{{\bf k}+{\bf
q}}v_{\bf q}-u_{{\bf q}}v_{{\bf k}+{\bf q}}|^{2}\delta(\omega-(\omega_{-{\bf
q}}+\omega_{{\bf k}+{\bf q}})),$ (5)
which satisfies the correct sum rule $\int\\!\sum_{{\bf
k}\alpha}S^{\alpha\alpha}({\bf k},\omega)d\omega=Ns(s+1)$.
As at the mean field level the triplet excitations are made of two
spin-$\frac{1}{2}$ free spinons a broad two spinon continuum is expected.
Nevertheless, as the $120^{\circ}$ long range Néel order is developed there
can be distinguished three distinct contributions in the spectrum. Following
the interpretation of the spectra of [26], it is instructive to split eq. (5)
as
$S({\bf k},\omega)=S^{sing}_{{\bf k},\omega}+S^{cont}_{{\bf k},\omega},$
by using the fact that $u_{\pm\frac{\bf Q}{2}}\\!=\\!|v_{\pm\frac{\bf
Q}{2}}|\\!\sim\\!(\frac{Nm}{2})^{\frac{1}{2}}$ and $\omega_{\pm{\frac{\bf
Q}{2}}}\sim 0$. For ${\bf k}=\pm{\bf Q}$, the spectrum is dominated by zero
energy processes that create two spinons in the condensate. This gives rise to
the magnetic Bragg peaks which, to leading order, behave as $S^{sing}_{\pm{\bf
Q},\omega}\\!\\!\sim Nm^{2}\delta(\omega)$. For ${\bf k}\neq\pm{\bf Q}$, the
spectrum is dominated by low energy processes that create one spinon in the
condensate and another one in the normal fluid. This gives rise to a double
peaked signal proportional to $m$, represented by
$\displaystyle S^{sing}_{{\bf k},\omega}\\!$ $\displaystyle=$
$\displaystyle\\!\frac{m}{4}|\imath\;u_{{\bf k}+\frac{\bf Q}{2}}\\!-\\!v_{{\bf
k}+\frac{\bf Q}{2}}|^{2}\delta(\omega-\omega_{{\bf k}\\!+\\!\frac{\bf
Q}{2}}\\!)+$ $\displaystyle+$ $\displaystyle\frac{m}{4}|\imath\;u_{{\bf
k}\\!-\\!\frac{\bf Q}{2}}\\!+\\!v_{{\bf k}-\frac{\bf
Q}{2}}|^{2}\delta(\omega-\omega_{{\bf k}-\frac{\bf Q}{2}}\\!).$
Then, the shifted spinon dispersion $\omega_{{\bf k}\pm\frac{\bf Q}{2}}$ can
be identified with the low energy physical magnetic excitations. Finally, at
high energy, the spectrum is dominated by the processes of creating two
spinons in the normal fluid. This gives rise to a broad continuum represented
by
$S^{cont}_{{\bf k},\omega}\\!=\\!\\!\frac{1}{4N}\\!\\!\sum_{\bf
q}{}^{{}^{\prime}}\\!|u_{{\bf k}+{\bf q}}v_{\bf q}-u_{{\bf q}}v_{{\bf k}+{\bf
q}}|^{2}\delta(\omega-(\omega_{-{\bf q}}+\omega_{{\bf k}+{\bf q}})),$
where the prime means that sum goes over the triangular BZ except for ${\bf
q}=\pm\frac{\bf Q}{2}$ or $\pm\frac{\bf Q}{2}-{\bf k}$.
[width=0.3angle=-90]fig1.eps
Figure 1: Dynamical structure factor, $S({\bf k},\omega)$, for momentum
$M=(\frac{5}{6}\pi,\frac{\sqrt{3}}{2}\pi)$. Inset: path of the triangular BZ
along which the spectrum has been investigated. $O\\!\\!=\\!\\!(0,0)$,
$A\\!\\!=\\!\\!(\pi,0)$, $Q\\!=\\!(\frac{4}{3}\pi,0)$,
$D\\!=\\!(2\pi,0),B\\!=\\!(\pi,\frac{1}{\sqrt{3}}\pi)$, and
$C\\!=\\!(\frac{2}{3}\pi,\frac{2}{\sqrt{3}}\pi)$. $\omega$ is measured in
units of $J$.
In fig. 1 we have plotted eq. (5) for the $M$ point of the BZ (see inset of
fig. 1). As noticed above, the low energy double peaked structure comes from
$S^{sing}_{{\bf k},\omega}$ while the high energy tail corresponds to the
continuum $S^{cont}_{{\bf k},\omega}$. In order to get the spectrum in the
energy-momentum space we have plotted in fig. 2 the intensity curves of
$S({\bf k},\omega)$ (eq. (5)) along the path shown in the inset of fig. 1. The
yellow and red curves are the shifted spinon dispersion $\omega_{{\bf
k}\mp\frac{\bf Q}{2}}$ of $S^{sing}_{{\bf k},\omega}$ while the blue zone
corresponds to $S^{cont}_{{\bf k},\omega}$. In the figure we compare with the
dispersion relations obtained with LSWT (solid line) and the recent series
expansion calculations [14] (points). At low energies the dispersion agrees
quite well with LSWT and series expansions, being the spectral weight mostly
located around ${\bf k}\sim\pm{\bf Q}$ (points Q and C). In this regime the
physical excitations correspond to long range transverse distorsions of the
local magnetization which are correctly described by both, LSWT and mean field
Schwinger bosons. At higher energies LSWT is not valid any more since the true
spin excitations show a strong downward renormalization along with the
appearance of roton-like minima (points). Remarkably, the mean field Schwinger
boson theory predicts a non trivial redistribution of the spectral weight
between the two spinon branches modulated by the form factor of eq. (5). The
reconstructed dispersion, resulting from those pieces of spinon dispersion
with the dominant spectral weight, reproduces quite well the series expansions
results. In particular, the crossing of the spinon dispersions at points $B$
and $D$ can be identified with the roton-like minima observed in series
expansions.
[width=0.29angle=-90]fig2.eps
Figure 2: Intensity curves for the dynamical structure factor, $S({\bf
k},\omega)$, calculated with the mean field Schwinger bosons theory within the
two singlet scheme. Solid green line and blue points are the dispersion
relations obtained with LSWT and series expansions [14], respectively. The
path along the BZ is shown in the inset of fig. 1.
Regarding the interpretation of the roton minima, the singlet bond structure
of the Schwinger boson theory takes naturally into account the collinear spin
fluctuations even in the presence of the $120^{\circ}$ Néel order of the THM.
For instance, the roton minimum located at $B$ can be interpreted as the
development of magnetic correlations modulated by the magnetic wave vector
$(\pi,\frac{1}{\sqrt{3}}\pi)$ which corresponds to certain collinear
correlations pattern, while the other two non equivalent midpoints of the
edges of the hexagonal BZ corresponds to different collinear fluctuations
patterns. In fact, if these fluctuations are favoured by introducing spatially
anisotropic or second neighbours exchange interactions the roton minima
soften, giving rise to the new Goldstone mode structure of the stabilized
collinear ground state [21, 27].
[width=0.3angle=-90]fig3.eps
Figure 3: Static structure factor (upper panel) and relative weight of the two
spinon continuum, $\int S^{cont}_{{\bf k},\omega}/S({\bf k})d\omega$, (bottom
panel) along the same path of the BZ.
Performing the frequency integration it is possible to analyze the relative
weight of the two spinon continuum (blue zone of fig. 2) to the static
structure factor $S({\bf k})$ [28]. In fig. 3 we plot $S({\bf k})$ with
diverging peaks located at the expected magnetic wave vectors $\pm{\bf Q}$
(upper panel), along with the relative weight of the two spinon continuum,
$\int S^{cont}_{{\bf k},\omega}/S({\bf k})d\omega$ (bottom panel).
Interestingly, the contribution of the two spinon continuum to $S({\bf k})$ is
neglegible around $\pm{\bf Q}$ while outside their neighbourhood, and in
particular at the roton position, the contribution to $S({\bf k})$ is about
$40\%$.
### 3.2 Density-density correlation functions
The small peak of fig. 1 leads to the remnant weak signal of fig. 2 which can
be traced back to the local density fluctuation of Schwinger bosons. In fact,
to describe the physical Hilbert space of the spin operators the local
constraint of the Schwinger bosons must be satisfied exactly, $\hat{{\bf
S}}^{2}_{i}=\frac{n_{i}}{2}(\frac{n_{i}}{2}+1)$. Then, no fluctuations on the
number of boson per site should be observed. However, since the constraint is
taken into account on average there are unphysical spin fluctuations in
$S({\bf k},\omega)$ coming from such density fluctuations. In order to
identify them we have computed the density-density dynamical structure factor
defined as
$\emph{N}({\bf k},\omega)=\sum_{n}\\!|\langle\texttt{gs}|\hat{n}_{\bf
k}(0)|n\rangle|^{2}\delta(\omega-(\epsilon_{n}-E_{\texttt{gs}})),$
where $\hat{n}_{\bf k}$ is the Fourier transform of the number of bosons per
site, $\hat{n}_{i}=\sum_{\sigma}\hat{b}^{\dagger}_{i\sigma}\hat{b}_{i\sigma}$.
A little of algebra leads to the expression
$\emph{N}({\bf k},\omega)\\!=\frac{1}{N}\\!\\!\sum_{{\bf q}}|u_{{\bf k}+{\bf
q}}v_{\bf q}+u_{{\bf q}}v_{{\bf k}+{\bf q}}|^{2}\delta(\omega-(\omega_{-{\bf
q}}+\omega_{{\bf k}+{\bf q}})),$ (6)
[width=0.29angle=-90]fig4.eps
Figure 4: Intensity curves for the density-density dynamical structure factor,
$\emph{N}({\bf k},\omega)$, calculated within the mean field Schwinger bosons
based on the two singlet scheme. The path along the BZ is shown in the inset
of fig. 1.
which is similar to eq. (5), except to the plus sign within the form factor.
If we split the two spinon contributions as $\emph{N}({\bf
k},\omega)=\emph{N}^{sing}_{{\bf k},\omega}+\emph{N}^{cont}_{{\bf k},\omega}$
it is easy to show that the main signal is located again at the shifted spinon
dispersions $\omega_{{\bf k}\mp\frac{\bf Q}{2}}$. But now, due to the
different form factor, there is an important spectral weight transfer between
such spinon dispersions. This is shown in fig. 4 where we have plotted the
intensity curves of $\emph{N}({\bf k},\omega)$ (eq. (6)). It can be clearly
observed that now the dominant signal is gapped at Q and C points, while most
of the spectral weight is located around ${\bf k}\sim 0$. Such a soft mode can
be identified with a spurious tendency of the bosonic system to phase
separation. Given the notable resemblance with the strong signal of
$\emph{N}({\bf k},\omega)$, we suggest that the low energy weak signal of
figs. 1 and 2 could be ascribed with the unphysical density fluctuation
effects which we expect to disappear once they are projected out. For the
unfrustrated square lattice $\omega_{{\bf
k}+(\frac{\pi}{2},\frac{\pi}{2})}=\omega_{{\bf
k}-(\frac{\pi}{2},\frac{\pi}{2})}$ so both, the unphysical and the physical
spin excitations, overlap in energy-momentum space, giving rise only to one
low energy band in $S({\bf k},\omega)$ [3, 28].
### 3.3 Comparison with the one singlet scheme
So far we have found that the mean field Schwinger boson within the two
singlet scheme reproduces quite well the series expansions spectrum. It is
also interesting to compare with the predictions of the one singlet scheme,
since it is widely used in the literature. The first difference is the
incorrect sum rule $\int\\!\sum_{{\bf k}\alpha}S^{\alpha\alpha}({\bf
k},\omega)d\omega=\frac{3}{2}Ns(s+1)$ which implies the well known
$\frac{2}{3}$ factor of Arovas and Auerbach [3]. Furthermore, in fig. 5, we
have computed $S({\bf k},\omega)$ after solving the corresponding self
consistent equations for the parameters $A_{\delta}$, and $\lambda$. At very
low energies the spectrum seems to be correct around points $C$, $O$ and $Q$.
However, at higher energies it is impossible to discern a reconstructed
dispersion that fit the series expansion results along the whole path of the
BZ, besides the factor about $3$ in the energy scale. Therefore, we conclude
that the two singlet scheme turns out the proper framework to describe
correctly the spectrum of the THM. Besides its quantitative accuracy, there
are symmetry arguments that give further support to the two singlet scheme. In
the literature, the one singlet scheme has been justified as the saddle point
of a symplectic $Sp(N)$ theory, originally adapted to extend previous large
$N$ works [3] to non bipartite lattices [5]. More recently, however, Flint and
Coleman [29] demonstrated that if the $\hat{B}_{ij}$ and $\hat{A}_{ij}$
operators are kept the corresponding large $N$ extension preserves the time
reversal properties of the spins, in contrast to the $Sp(N)$ theory. Finally,
it is worth to stress that the two singlet scheme is the basis of the $Z_{2}$
spin liquid theory, specially formulated to describe magnetically disordered
phases [30].
[width=0.3angle=-90]fig5.eps
Figure 5: Intensity curves for the dynamical structure factor calculated
within the mean field Schwinger bosons based on the one singlet scheme. The
path along the BZ is shown in the inset of fig. 1. Solid line and points are
the same as in fig. 2.
## 4 Conclusions
We have demonstrated that the singlet structure of the mean field ground state
along with the fractional character of the spin excitations of the Schwinger
boson theory take naturally into account the anomalous excitations of the
spin-$\frac{1}{2}$ triangular Heisenberg model recently observed [14, 10]. The
appearance of the roton-like minima can be attributed to the tendency of the
magnetic ground state to be correlated collinearly, even in the presence of
$120^{\circ}$ Néel order. By computing the density-density dynamical structure
factor, and thanks to the series expansion results, we were able for the first
time to discern, at the mean field level, between the physical and the
spurious fluctuations coming from the relaxation of the local constraint. A
further investigation within the context of the Schwinger boson theory reveals
that the correct description of the spectrum depends crucially on the mean
field decoupling. In particular, the two singlet scheme turns out more
appropriate than the one singlet scheme. Based on the accurate description of
the ground state static properties [11] (see table 1) and in the light of the
present results for the spectrum, we think that the bosonic spinon hypothesis
should be considered seriously as an alternative viewpoint to interpret the
physics of the triangular Heisenberg model. At the mean field level the
triplet excitations consist of two spin-$\frac{1}{2}$ free spinons and,
besides the low energy bands due to the onset of the long range order, there
is a broad two spinon continuum, which could be related with the magnon decay
found in the literature [23]. In this sense, it would be important to improve
the present mean field theory by deriving an effective interaction between
spinons resulting from $1/N$ corrections or a better implementation of the
constraint. We would expect a picture of tightly bound spinons near the
Goldstone modes while at high energies they would be weakly bound. Work in
this direction is in progress. Finally, we hope our present analysis in terms
of bosonic spinons could help for a better understanding of the unconventional
neutron scattering spectra of the $Cs_{2}CuCl_{4}$ compound [17].
###### Acknowledgements.
We thank W. Zheng and R. Coldea for sending us their series expansions
results, and C. Lhuillier and C. Batista for very useful discussions. This
work was supported by PIP2009 under grant No. $1948$.
## References
* [1] Anderson P. W. Mater. Res. Bull. 8 (1973) 153; Fazekas P. and Anderson P. W. Philos. Mag. 30 (1974) 423.
* [2] Anderson P. W. Science 235 (1987) 1196.
* [3] Auerbach A. and Arovas D. P. Phys. Rev. Lett. 61 (1988) 617; Arovas D. P. and Auerbach A. Phys. Rev. B 38 (1988) 316.
* [4] Affleck I. and Marston J. B. Phys. Rev. B 37 (1988) 3774.
* [5] Read N. and Sachdev S. Phys. Rev. Lett. 66 (1991) 1773; Sachdev S. and Read N. Int. J. Mod. Phys. B 5 (1991) 219.
* [6] Chandra P., Coleman P. and Larkin A. I. J. Phys. Condens. Matter 2 (1990) 7933.
* [7] Bernu B., Lhuillier C. and Pierre L. Phys. Rev. Lett. 69 (1992) 2590.
* [8] Capriotti L., Trumper A. E. and Sorella S. Phys. Rev. Lett. 82 (1999) 3899.
* [9] Huse D. A. and Elser V. Phys. Rev. Lett. 60 (1988) 2531; Leung P. W. and Runge K. J. Phys. Rev. B 47 (1993) 5861; Kruger S. E., Darradi R., Richter J. and Farnell D. J. J. Phys. Rev. B 73 (2006) 094404; White S. R. and Chernyshev A. L. Phys. Rev. Lett. 99 (2007) 127004.
* [10] Zheng W., Fjaerestad J. O., Singh R. R. P., McKenzie R. H. and Coldea R. Phys. Rev. B 74 (2006) 224420.
* [11] Manuel L. O., Trumper A. E. and Ceccatto H. A. Phys. Rev. B 57 (1998) 8348.
* [12] Lecheminant P., Bernu B., Lhuillier C. and Pierre L. Phys. Rev. B 52 (1995) 9162; Trumper A. E., Capriotti L. and Sorella S. Phys. Rev. B 61 (2000) 11529.
* [13] Chandra P., Coleman P. and Ritchey L. Int. J. Mod. Phys. B 5 (1991) 171.
* [14] Zheng W., Fjaerestad J. O., Singh R. R. P., McKenzie R. H. and Coldea R. Phys. Rev. Lett. 96 (2006) 057201.
* [15] Starykh O. A., Chubukov A. V. and Abanov A. G. Phys. Rev. B 74 (2006) 180403.
* [16] Chernyshev A. L. and Zhitomirsky M. E. Phys. Rev. Lett. 97 (2006) 207202 .
* [17] Coldea R., Tennant D. A. and Tylczynski Z. Phys. Rev. B 68 (2003) 134424.
* [18] Ceccatto H. A., Gazza C. J. and Trumper A. E. Phys. Rev. B 47 (1993) 12329.
* [19] Note that, in contrast to ref. [18], the self consistent equations (3) do not depend on ${\bf Q}$ explicitly because we have transformed as $\hat{b}_{{\bf k}\sigma}=\frac{1}{\sqrt{N}}\sum_{i}\hat{b}_{i\sigma}e^{\imath{\bf k}\cdot{\bf R}_{i}}$. Both procedures lead to the same results.
* [20] Hirsch J. E. and Tang S. Phys. Rev. B 39 (1989) 2850.
* [21] Gazza C. J. and Ceccatto H. A. J. Phys.: Condens. Matter 5 (1993) L135.
* [22] Yoshioka D. and Miyazaki J. J. Phys. Soc. Jpn. 60 (1991) 614.
* [23] Chernyshev A. L. and Zhitomirsky M. E. Phys. Rev. B 79, (2009) 144416.
* [24] Sachdev S. Phys. Rev. B 45, (1992) 12377.
* [25] A complete summary of the results for the THM derived by different methods can be found in ref. [10].
* [26] Lefmann K. and Hedegård P. Phys. Rev. B 50 (1994) 1074; Messio L., Cepas O. and Lhuillier C. Phys. Rev. B 81 (2010) 064428.
* [27] Manuel L. O. and Ceccatto H. A. Phy. Rev. B 60 (1999) 489; Trumper A. E. Phys. Rev. B 60 (1999) 2987.
* [28] Capriotti L., Läuchli A. and Paramekanti A. Phys. Rev. B 72 (2005) 214433.
* [29] Flint R. and Coleman P. Phys. Rev. B 79 (2009) 014424.
* [30] Wang F. and Vishwanath A. Phys. Rev. B 74 (2006) 174423.
|
arxiv-papers
| 2010-11-03T13:26:58 |
2024-09-04T02:49:14.454244
|
{
"license": "Public Domain",
"authors": "A. Mezio, C. N. Sposetti, L. O. Manuel, and A. E. Trumper",
"submitter": "Adolfo Emilio Trumper",
"url": "https://arxiv.org/abs/1011.0874"
}
|
1011.0884
|
∎
11institutetext: Changjin Zhang 22institutetext: Lei Zhang 33institutetext:
Langsheng Ling 44institutetext: Wei Tong 55institutetext: Yuheng Zhang
66institutetext: High Magnetic Field Laboratory, Chinese Academy of Sciences,
Hefei 230031, People’s Republic of China
Tel.: +86-551-559-1672
Fax: +86-551-559-1149
66email: zhangcj@hmfl.ac.cn 77institutetext: Shun Tan 88institutetext: Yuheng
Zhang 99institutetext: Hefei National Laboratory for Physical Sciences at
Microscale, University of Science and Technology of China, Hefei 230026,
People’s Republic of China
# Single crystal growth of BaFe2-xCoxAs2 without fluxing agent
Changjin Zhang Lei Zhang Chuanying Xi Langsheng Ling Wei Tong Shun Tan
Yuheng Zhang
(Received: date / Accepted: date)
###### Abstract
We report a simple, reliable method to grow high quality BaFe2-xCoxAs2 single
crystal samples without using any fluxing agent. The starting materials for
the single crystal growth come from well-crystallized polycrystalline samples
and the highest growing temperature can be 1230∘C. The as-grown crystals have
typical dimensions of 4$\times$3$\times$0.5 mm3 with $c$-axis perpendicular to
the shining surface. We find that the samples have very large current carrying
ability, indicating that the samples have good potential technological
applications.
###### Keywords:
Single crystal growth magnetism critical current density
###### pacs:
81.10.Dn 74.70.Xa 74.25.Jb 74.25.Dw
††journal: Journal of Superconductivity and Novel Magnetism
## 1 Introduction
The discovery of superconductivity in iron-pnictide systems has attracted
tremendous interests not only due to its scientific value but also its
potential industrial applications [1]. The relatively high transition
temperature, highly flexibility, very high upper-critical magnetic field and
other physical qualities make the iron-pnictide systems very useful in
industry [2-4]. The existing challenges, such as optimizing synthesis methods
for technological applications and clarifying the ambiguity in the
superconducting mechanism, will keep iron-pnictide systems on the frontiers of
research for a long time, in parallel to high-$T_{c}$ cuprates [4].
In order to determine the application parameters which are important to
commercial use, great efforts have been made to grow high-quality single
crystals of iron-pnictide superconductors [5-8] A lot of physical properties,
such as the transition temperature, the upper critical field, the vortex
structure, etc., have been determined using single crystal samples. However,
due to the relatively high melting temperature of the iron-pnictide samples,
the single crystals of iron-pnictides are generally grown by using self-flux
method or flux method where excess FeAs mixture or Sn is used as the fluxing
agent. The advantage of these methods is that the melting temperature can be
significantly decreased comparing to the melting point of the crystal itself.
While the disadvantage of these methods is unnegligible. For example, if we
grow BaFe2-xCoxAs2 single samples using excess Fe(Co)As mixture as fluxing
agent, the actual Fe/Co ratio can not be accurately controlled in the growth
procedure. If one uses Sn as fluxing agent, the problem is that one can not
remove the Sn from the surface of the sample easily [5]. In this paper we
report a simple, reliable method to grow high quality BaFe2-xCoxAs2 single
crystal samples without using any fluxing agent. The samples have typical
dimensions of 4$\times$3$\times$0.5mm3 with $c$-axis perpendicular to the
shining surface. The critical current density of the samples are also
determined. The critical current density without external magnetic field is
quite high, meaning large current carrying ability of the samples, which
points to optimistic applications.
## 2 Experimental detail
Single crystal samples were grown using well-crystalized BaFe2-xCoxAs2
polycrystalline samples as the starting materials. The polycrystalline samples
with nominal composition BaFe2-xCoxAs2 were prepared by conventional solid-
state reaction method using high-purity Ba (crystalline dendritic solid,
99.9%, Alfa-Aesar), Fe (powder, 99.9%, Alfa-Aesar), Co (powder, 99.9%, Alfa-
Aesar), and As (powder, 99%, Alfa-Aesar) as starting materials. The
crystalline dendritic solid Ba was pressed into thin pellet using an agate
mortar and was cut into very small size (typically less than 0.5$\times$0.5
mm2). The raw materials were mixed and wrapped up by Ta foil and sealed in an
evacuated quartz tube. They were pre-heated at 600∘C for 12 hours and cooled
down slowly to room temperature. The mixture was then ground and pressed into
pellets and heated at 900∘C for 24 hours. When the furnace was cooled down,
the pellets were taken out and placed in an argon-filled glove box. We
performed powder x-ray diffraction measurements on these samples and found
that the samples were all in single phase.
The polycrystalline powder was pressed into pellets and placed in a quartz
tube in an Argon-filled glove box. The quartz tube was sealed after it was
evacuated by a molecular pump. Then the quartz tube was placed into a box
furnace. The furnace was heated to 1230∘C at a rate of 60∘C per hour. After
holding at 1230∘C for 12 hours, it was cooled to 850∘C at 2∘C per hour
followed by furnace cooling to room temperature. The quartz tube was found
almost intact after the whole procedure. When we break the quartz tube and
pick out the sample, slides of samples with shining surfaces can be easily
cleaved. It should be noted that we have tried to melt the samples at even
higher temperature using a double-wall quartz tube. However, we find that the
samples begin to decompose at temperature higher than 1240∘C.
X-ray diffraction (XRD) was carried out by a Rigaku-D/max-gA diffractometer
using high-intensity Cu-K$\alpha$ radiation to screen for the presence of an
impurity phase and the changes in structure. The homogeneity and chemical
compositions of the samples were examined using an energy dispersive x-ray
spectrometer (EDXS). The resistivity was measured using a standard four-probe
method in a closed-cycle helium cryostat. The magnetic susceptibility and the
magnetic hysteresis loops of the samples were determined by a SQUID
magnetometer (Quantum Design, MPMS).
## 3 Results and discussion
Figure 1: (a) Photograph of a as-grown BaFe2As2 single crystal. (b) X-ray
diffraction pattern at room temperature for the BaFe2-xCoxAs2 single crystals.
(c) An enlarge view of the (004) reflection.
Figure 1(a) shows a picture of a single crystal sample which has dimensions of
about 4.5$\times$3$\times$0.5mm3. We select several pieces of crystal and
perform EDXS measurement and find that the Co-contents in all pieces are close
to the nominal compositions, indicating that the samples having shining
surface are chemically homogeneous. The nominal and measured compositions of
the selected samples are summarized in table 1. In order to judge the
orientation of the samples, we perform x-ray diffraction (XRD) measurement on
the as-grown samples. Figure 1(b) gives the typical XRD patterns of the
BaFe2-xCoxAs2 ($x$=0, 0.06, 0.12, 0.18, 0.25, and 0.35) samples. Only the
(00$l$) diffraction peaks with even $l$ are observed, confirming that the
crystallographic $c$-axis is perpendicular to the shining surface. For all the
diffraction peaks, the full width at half maximum (FWHM) is less than 0.06∘,
indicating the excellent quality of the single crystals. In order to see the
shift of the peaks clearly, we plot in Fig. 1(c) the enlarged view of the
(004) reflection. One can see that all the reflections are splitted into two
shoulder peaks. The shoulder peak at lower angle is the reflection of the
Cu-K$\alpha$1 radiation and the one at higher angle is the reflection of the
Cu-K$\alpha$2 radiation. It can be seen that the (004) peak slightly shifts to
higher angle with increasing Co content, meaning that the $c$-axis constant
decreases monotonously as the Co content is increased. The calculated $c$-axis
lattice contents for the samples are given in Table 1.
Figure 2: Temperature dependence of in-plane resistivity for the BaFe2-xCoxAs2
samples.
The superconducting properties of the BaFe2-xCoxAs2 single crystals are given
in Fig. 2. The superconductivity emerges in the $x$$\geq$0.06 samples. And the
maximum critical transition temperature $T_{c,\rho=0}$ reaches to 23.3 K at
the optimal doping concentration $x$=0.15. With further increasing Co doping
content, $T_{c}$ decreases monotonously. The superconductivity disappears when
$x$$>$0.35.
Figure 3: (a) Temperature dependence of magnetic susceptibility for
BaFe1.80Co0.20As2 both under zero-field cooling condition and under field-
cooling condition at 10 Oe. (b) The magnetization as the function of
temperature under 1 Tesla magnetic field.
Figure 3(a) gives the temperature dependence of magnetic susceptibility below
$T_{c}$ for the $x$=0.20 sample both under zero-field cooling condition and
under field-cooling condition at 10 Oe. It is found that the superconducting
transition occurs at 23.1 K, consistent with the resistivity results. For the
magnetic susceptibility at $T$$>$$T_{c}$, the susceptibility signal is almost
undetectable within the accuracy limit of the $Quantum$ $Design$ MPMS
magnetometer (about 10-8 emu). In order to know the magnetic state at the
normal state, we measure the temperature dependence of magnetic susceptibility
under 1 Tesla. The result is shown in Fig. 3(b). From Fig. 3(b) we notice that
the magnetic susceptibility exhibits almost temperature-independent behavior
above $T_{c}$, indicating that the magnetic state of the BaFe1.80Co0.20As2
system can not be the Curie paramagnetism. The fact that the magnetization is
very weak and temperature-independent suggest that the paramagnetic state is a
Pauli-paramagnetic state, which is consistent with the metallic behavior of
the BaFe1.80Co0.20As2 system. The predominant Pauli-paramagnetic state in the
Co-doped sample suggest that the magnetic moment of the electrons near the
Fermi surface should be delocalized. Previous neutron scattering experiments
on CaFe2As2 have suggested that the magnetism is neither purely local nor
purely itinerant and that it is a complicated mix of the two [9]. Here the
predominant Pauli-paramagnetic state in the Co-doped sample suggest that the
itinerant moments might be dominate in the superconducting sample.
Figure 4: The magnetization as the function of external magnetic field below
$T_{c}$ for the $x$=0.20 sample. (b) The critical current density as the
function of magnetic field for the $x$=0.20 sample at different temperatures.
(c) The temperature dependence of critical current density for the $x$=0.20
sample under zero-field and under external magnetic field.
Figure 4(a) shows magnetic hysteresis loops at various temperatures below
$T_{c}$ calculated by applying the magnetic field up to 6 T. The $M$$\sim$$H$
curves exhibit a central peak at zero magnetic field and the magnetization
decreases continuously with increasing magnetic fields. The sharp peak around
$\mu$0$H$ = 0 is similarly observed in other iron-pnictide materials [8,
10-11]. Figure 4(b) shows the magnetic field dependence of the critical
current density $J_{c}$ derived from the hysteresis loop width by Bean
critical state model using the relation $J_{c}$ = 20$\triangle$$M$$/$$a$(1-
$a/3b$) [12], where $a$ and $b$ are the width and length of the sample,
respectively ($a$$<$$b$), and $\triangle$$M$ is the difference between the
upper and the lower branches in the $M$$\sim$$H$ loops. It is found from Fig.
4(b) that the critical current density $J_{c}$ of the sample reaches to
1.2$\times$106 A/cm2 without external magnetic field. We notice that this
$J_{c}$ value is higher than previous reported $J_{c}$ value of BaFe2-xCoxAs2
single crystal samples, either grown using self-flux method or using flux
method [8,13-15]. For example, the $J_{c}$ values of recent grown Co-doped
BaFe2As2 single crystal thin films are within the range of 60-100 kA/cm2 at 12
K (without external magnetic field) [8], which is less than the value of 280
kA/cm2 in present sample. The $J_{c}$ value of a BaFe1.80Co0.20As2 sample
grown by self-flux method is about 6$\times$105 A/cm2 at 5 K [13]. For a
BaFe1.852Co0.148As2 single crystal grown using Sn flux, the $J_{c}$ value at
16 K under 6 Tesla is about 5 kA/cm2 [14], which is also less than the value
of 26 kA/cm2 in present case. Based on these facts we suggest that the samples
grown without any fluxing agent may have better current carrying ability
comparing to those from flux growth. But this value is slightly less than the
highest critical current density of 4 MA/cm2 in Co-Doped BaFe2As2 epitaxial
films which was recently grown on (La,Sr)(Al,Ta)O3 substrates [16]. The
comparison between single crystals grown using different methods reveals that
further improvement of critical current density is still possible. Considering
that the BaFe2-xCoxAs2 samples have upper critical field as high as 60 T,
critical temperatures of above 20 K, low anisotropy, and, as shown here, high
intrinsic critical current density, these materials can be considered as good
candidates for applications.
The $J_{c}$ value decreases both with increasing temperature and with
increasing external magnetic field, as can be seen from Figs. 4(b) and (c). At
low temperatures ($\leq$20 K), the trend of $J_{c}$ decay is similar to that
of conventional high-$T_{c}$ cuprates [17]. At high temperature ($>$20 K), the
flux creep effect is evident by showing relatively strong dependence of
critical current density on the external magnetic field [18].
## 4 Conclusions
In summary, we have grown large-size Co-doped BaFe2As2 single crystals without
using any fluxing agent. We find that the as-grown samples have larger current
carrying ability comparing to those grown with the aid of fluxing agent,
indicating promising industrial applications.
Table 1: The comparison between nominal and real compositions and the $c$-axis lattice parameters of the BaFe2-xCoxAs2 samples Nominal composition | real composition | $c$ (Å)
---|---|---
BaFe2As2 | BaFe2As2 | 13.018(4)
BaFe1.9Co0.1As2 | BaFe1.9Co0.1As2 | 13.004(4)
BaFe1.8Co0.2As2 | BaFe1.81Co0.19As2 | 12.983(2)
BaFe1.7Co0.3As2 | BaFe1.71Co0.29As2 | 12.956(5)
###### Acknowledgements.
This work was supported by the State Key Project of Fundamental Research of
China through Grant 2010CB923403 and 2011CBA00111, and the Hundred Talents
Program of the Chinese Academy of Sciences.
## References
* (1) Y. Kamihara, T. Watanabe, M. Hirano, and H. Hosono, J. Am. Chem. Soc. 130, 3296 (2008).
* (2) F. Hunte, J. Jaroszynski, A. Gurevich, D. C. Larbalestier, R. Jin, A. S. Sefat, M. A. McGuire, B. C. Sales, D. K. Christen, and D. Mandrus, Nature 453, 903 (2008).
* (3) K. Ishida, Y. Nakai, and H. Hosono, J. Phys. Soc. Jpn. 78, 062001 (2009).
* (4) P. M. Aswathy, J. B. Anooja. P. M. Sarun, and U. Syamaprasad, Supercond. Sci. Technol. 23, 073001 (2010).
* (5) N. Ni, S. L. Bud’ko, A. Kreyssig, S. Nandi, G. E. Rustan, A. I. Goldman, S. Gupta, J. D. Corbett, A. Kracher, and P. C. Canfield, Phys. Rev. B 78, 014507 (2008).
* (6) F. Hardy, P. Adelmann, T. Wolf, Hilbert v. Lohneysen, and C. Meingast, Phys. Rev. Lett. 102, 187004 (2009).
* (7) J. S. Kim, E. G. Kim, and G. R. Stewart, J. Phys.: Condens. Matter 21, 252201 (2009).
* (8) C. Tarantini, S. Lee, Y. Zhang, J. Jiang, C. W. Bark, J. D. Weiss, A. Polyanskii, C. T. Nelson, H. W. Jang, C. M. Folkman, S. H. Baek, X. Q. Pan, A. Gurevich, E. E. Hellstrom, C. B. Eom, and D. C. Larbalestier, Appl. Phys. Lett. 96, 142510 (2010).
* (9) J. Zhao, D. T. Adroja, D.-X Yao, R. Bewley, S. L. Li, X. F. Wang, G. Wu, X. H. Chen, J. P. Hu, and P. C. Dai, Nature Phys. 5, 555 (2009).
* (10) R. Prozorov, M. A. Tanatar, N. Ni, A. Kreyssig, S. Nandi, S. L. Bud’ko, A. I. Goldman, and P. C. Canfield, Phys. Rev. B 80, 174517 (2009).
* (11) F. Kametani, P. Li, D. Abraimov, A. A. Polyanskii, A. Yamamoto, J. Jiang, E. E. Hellstrom, A. Gurevich, D. C. Larbalestier, Z. A. Ren, J. Yang, X. L. Dong, W. Lu, Z. X. Zhao, Appl. Phys. Lett. 95, 142502 (2009).
* (12) C. P. Bean, Rev. Mod. Phys. 36, 31 (1964).
* (13) Y. Nakajima, T. Taen, and T. Tamegai, J. Phys. Soc. Jpn. 78, 023702 (2009).
* (14) M. A. Tanatar, N. Ni, S. L. Bud’ko, P. C. Canfield, and R. Prozorov, Supercond. Sci. Technol. 23, 054002 (2010).
* (15) R. Prozorov, M. A. Tanatar, N. Ni, A. Kreyssig, S. Nandi, S. L. Bud’ko, A. I. Goldman, and P. C. Canfield, Phys. Rev. B 80, 174517 (2009).
* (16) T. Katase, H. Hiramatsu, T. Kamiya, and H. Hosono, Appl. Phys. Exp. 3, 063101 (2010).
* (17) C. Cai, B. Holzapfel, J. Hänisch, L. Fernández and L. Schultz, Phys. Rev. B 69, 104531 (2004).
* (18) T. Matsushita, T.Fujiyoshi, K.Toko and K.Yamafuji, Appl. Phys. Lett. 56, 2039 (1990).
|
arxiv-papers
| 2010-11-03T14:01:03 |
2024-09-04T02:49:14.461239
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Changjin Zhang, Lei Zhang, Chuanying Xi, Langsheng Ling, Shun Tan, and\n Yuheng Zhang",
"submitter": "Changjin Zhang",
"url": "https://arxiv.org/abs/1011.0884"
}
|
1011.0935
|
11institutetext: Interdisciplinary Center for Security, Reliability and Trust
University of Luxembourg, Luxembourg
11email: Jianguo.Ding@ieee.org
# Probabilistic Inferences in Bayesian Networks
Jianguo Ding
###### Abstract
Bayesian network is a complete model for the variables and their
relationships, it can be used to answer probabilistic queries about them. A
Bayesian network can thus be considered a mechanism for automatically applying
Bayes’ theorem to complex problems. In the application of Bayesian networks,
most of the work is related to probabilistic inferences. Any variable updating
in any node of Bayesian networks might result in the evidence propagation
across the Bayesian networks. This paper sums up various inference techniques
in Bayesian networks and provide guidance for the algorithm calculation in
probabilistic inference in Bayesian networks.
## 1 Introduction
Because a Bayesian network is a complete model for the variables and their
relationships, it can be used to answer probabilistic queries about them. For
example, the network can be used to find out updated knowledge of the state of
a subset of variables when other variables (the evidence variables) are
observed. This process of computing the posterior distribution of variables
given evidence is called probabilistic inference. A Bayesian network can thus
be considered a mechanism for automatically applying Bayes’ theorem to complex
problems.
In the application of Bayesian networks, most of the work is related to
probabilistic inferences. Any variable updating in any node of Bayesian
networks might result in the evidence propagation across the Bayesian
networks. How to examine and execute various inferences is the important task
in the application of Bayesian networks.
This chapter will sum up various inference techniques in Bayesian networks and
provide guidance for the algorithm calculation in probabilistic inference in
Bayesian networks. Information systems are of discrete event characteristics,
this chapter mainly concerns the inferences in discrete events of Bayesian
networks.
## 2 The Semantics of Bayesian Networks
The key feature of Bayesian networks is the fact that they provide a method
for decomposing a probability distribution into a set of local distributions.
The independence semantics associated with the network topology specifies how
to combine these local distributions to obtain the complete joint probability
distribution over all the random variables represented by the nodes in the
network. This has three important consequences.
Firstly, naively specifying a joint probability distribution with a table
requires a number of values exponential in the number of variables. For
systems in which interactions among the random variables are sparse, Bayesian
networks drastically reduce the number of required values.
Secondly, efficient inference algorithms are formed in that work by
transmitting information between the local distributions rather than working
with the full joint distribution.
Thirdly, the separation of the qualitative representation of the influences
between variables from the numeric quantification of the strength of the
influences has a significant advantage for knowledge engineering. When
building a Bayesian network model, one can focus first on specifying the
qualitative structure of the domain and then on quantifying the influences.
When the model is built, one is guaranteed to have a complete specification of
the joint probability distribution.
The most common computation performed on Bayesian networks is the
determination of the posterior probability of some random variables, given the
values of other variables in the network. Because of the symmetric nature of
conditional probability, this computation can be used to perform both
diagnosis and prediction. Other common computations are: the computation of
the probability of the conjunction of a set of random variables, the
computation of the most likely combination of values of the random variables
in the network and the computation of the piece of evidence that has or will
have the most influence on a given hypothesis.
A detailed discussion of inference techniques in Bayesian networks can be
found in the book by Pearl [Pearl, 2000].
* •
Probabilistic semantics. Any complete probabilistic model of a domain must,
either explicitly or implicitly, represent the joint distribution which the
probability of every possible event as defined by the values of all the
variables. There are exponentially many such events, yet Bayesian networks
achieve compactness by factoring the joint distribution into local,
conditional distributions for each variable given its parents. If $x_{i}$
denotes some value of the variable $X_{i}$ and $\pi(x_{i})$ denotes some set
of values for $X_{i}$’s parents $\pi(x_{i})$, then $P(x_{i}|\pi(x_{i}))$
denotes this conditional distribution. For example, $P(x_{4}|x_{2},x_{3})$ is
the probability of wetness given the values of sprinkler and rain. Here
$P(x_{4}|x_{2},x_{3})$ is the brief of $P(x_{4}|\\{x_{2},x_{3}\\})$. The set
parentheses are omitted for the sake of readability. We use the same
expression in this thesis. The global semantics of Bayesian networks specifies
that the full joint distribution is given by the product
$P(x_{1},\ldots,x_{n})=\prod_{i}P(x_{i}|\pi(x_{i}))$ (1)
Equation 1 is also called the chain rule for Bayesian networks.
Figure 1: Causal Influences in A Bayesian Network.
In the example Bayesian network in Figure 1, we have
$P(x_{1},x_{2},x_{3},x_{4},x_{5})=P(x_{1})P(x_{2}|x_{1})P(x_{3}|x_{1})P(x_{4}|x_{2},x_{3})P(x_{5}|x_{4})$
(2)
Provided the number of parents of each node is bounded, it is easy to see that
the number of parameters required grows only linearly with the size of the
network, whereas the joint distribution itself grows exponentially. Further
savings can be achieved using compact parametric representations, such as
noisy-OR models, decision tress, or neural networks, for the conditional
distributions [Pearl, 2000].
There are also entirely equivalent local semantics, which assert that each
variable is independent of its non-descendants in the network given its
parents. For example, the parents of $X_{4}$ in Figure 1 are $X_{2}$ and
$X_{3}$ and they render $X_{4}$ independent of the remaining non-descendant,
$X_{1}$. That is,
$P(x_{4}|x_{1},x_{2},x_{3})=P(x_{4}|x_{2},x_{3})$ (3)
The collection of independence assertions formed in this way suffices to
derive the global assertion in Equation 2, and vice versa. The local semantics
are most useful in constructing Bayesian networks, because selecting as
parents the direct causes of a given variable automatically satisfies the
local conditional independence conditions. The global semantics lead directly
to a variety of algorithms for reasoning.
* •
Evidential reasoning. From the product specification in Equation 2, one can
express the probability of any desired proposition in terms of the conditional
probabilities specified in the network. For example, the probability that the
sprinkler was on, given that the pavement is slippery, is
$\displaystyle P(X_{3}=on|X_{5}=true)$ (4)
$\displaystyle=\frac{P(X_{3}=on,X_{5}=true)}{P(X_{5}=true)}$
$\displaystyle=\frac{\sum_{x_{1},x_{2},x_{4}}P(x_{1},x_{2},X_{3}=on,x_{4},X_{5}=true)}{\sum_{x_{1},x_{2},x_{3},x_{4}}P(x_{1},x_{2},x_{3},x_{4},X_{5}=true)}$
$\displaystyle=\frac{\sum_{x_{1},x_{2},x_{4}}P(x_{1})P(x_{2}|x_{1})P(X_{3}=on|x_{1})P(x_{4}|x_{2},X_{3}=on)P(X_{5}=true|x_{4})}{\sum_{x_{1},x_{2},x_{3},x_{4}}P(x_{1})P(x_{2}|x_{1})P(x_{3}|x_{1})P(x_{4}|x_{2},x_{3})P(X_{5}=true|x_{4})}$
These expressions can often be simplified in the ways that reflect the
structure of the network itself.
It is easy to show that reasoning in Bayesian networks subsumes the
satisfiability problem in propositional logic and hence reasoning is NP-hard
[Cooper, 1990]. Monte Carlo simulation methods can be used for approximate
inference [Pearl, 1987], given that estimates are gradually improved as the
sampling proceeds. (Unlike join-tree methods, these methods use local message
propagation on the original network structure.) Alternatively, variational
methods [Jordan et al., 1998] provide bounds on the true probability.
* •
Functional Bayesian networks. The networks discussed so far are capable of
supporting reasoning about evidence and about actions. Additional refinement
is necessary in order to process counterfactual information. For example, the
probability that ”the pavement would not have been slippery had the sprinkler
been OFF, given that the sprinkler is in fact ON and that the pavement is in
fact slippery” cannot be computed from the information provided in Figure 1
and Equation 2. Such counterfactual probabilities require a specification in
the form of functional networks, where each conditional probability
$P(x_{i}|\pi(i))$ is replaced by a functional relationship
$x_{i}=f_{i}(\pi(i),\epsilon_{i})$, where $\epsilon_{i}$ is a stochastic
(unobserved) error term. When the functions $f_{i}$ and the distributions of
$\epsilon_{i}$ are known, all counterfactual statements can be assigned unique
probabilities, using evidence propagation in a structure called a ”twin
network”. When only partial knowledge about the functional form of $f_{i}$ is
available, bounds can be computed on the probabilities of counterfactual
sentences [Balke & Pearl, 1995] [Pearl, 2000].
* •
Causal discovery. One of the most exciting prospects in recent years has been
the possibility of using Bayesian networks to discover causal structures in
raw statistical data [Pearl & Verma, 1991] [Spirtes et al., 1993] [Pearl,
2000], which is a task previously considered impossible without controlled
experiments. Consider, for example, the following pattern of dependencies
among three events: $A$ and $B$ are dependent, $B$ and $C$ are dependent, yet
$A$ and $C$ are independent. If you ask a person to supply an example of three
such events, the example would invariably portray $A$ and $C$ as two
independent causes and $B$ as their common effect, namely, $A\rightarrow
B\leftarrow C$. Fitting this dependence pattern with a scenario in which $B$
is the cause and $A$ and $C$ are the effects is mathematically feasible but
very unnatural, because it must entail fine tuning of the probabilities
involved; the desired dependence pattern will be destroyed as soon as the
probabilities undergo a slight change.
Such thought experiments tell us that certain patterns of dependency, which
are totally void of temporal information, are conceptually characteristic of
certain causal directionalities and not others. When put together
systematically, such patterns can be used to infer causal structures from raw
data and to guarantee that any alternative structure compatible with the data
must be less stable than the one(s) inferred; namely, slight fluctuations in
parameters will render that structure incompatible with the data.
* •
Plain beliefs. In mundane decision making, beliefs are revised not by
adjusting numerical probabilities but by tentatively accepting some sentences
as ”true for all practical purposes”. Such sentences, called plain beliefs,
exhibit both logical and probabilistic characters. As in classical logic, they
are propositional and deductively closed; as in probability, they are subject
to retraction and to varying degrees of entrenchment. Bayesian networks can be
adopted to model the dynamics of plain beliefs by replacing ordinary
probabilities with non-standard probabilities, that is, probabilities that are
infinitesimally close to either zero or one [Goldszmidt & Pearl, 1996].
* •
Models of cognition. Bayesian networks may be viewed as normative cognitive
models of propositional reasoning under uncertainty [Pearl, 2000]. They handle
noise and partial information by using local, distributed algorithm for
inference and learning. Unlike feed forward neural networks, they facilitate
local representations in which nodes correspond to propositions of interest.
Recent experiments [Tenenbaum & Griffiths, 2001] suggest that they capture
accurately the causal inferences made by both children and adults. Moreover,
they capture patterns of reasoning that are not easily handled by any
competing computational model. They appear to have many of the advantages of
both the “symbolic” and the “subsymbolic” approaches to cognitive modelling.
Two major questions arise when we postulate Bayesian networks as potential
models of actual human cognition.
Firstly, does an architecture resembling that of Bayesian networks exist
anywhere in the human brain? No specific work had been done to design neural
plausible models that implement the required functionality, although no
obvious obstacles exist.
Secondly, how could Bayesian networks, which are purely propositional in their
expressive power, handle the kinds of reasoning about individuals, relations,
properties, and universals that pervades human thought? One plausible answer
is that Bayesian networks containing propositions relevant to the current
context are constantly being assembled as needed to form a more permanent
store of knowledge. For example, the network in Figure 1 may be assembled to
help explain why this particular pavement is slippery right now, and to decide
whether this can be prevented. The background store of knowledge includes
general models of pavements, sprinklers, slipping, rain, and so on; these must
be accessed and supplied with instance data to construct the specific Bayesian
network structure. The store of background knowledge must utilize some
representation that combines the expressive power of first-order logical
languages (such as semantic networks) with the ability to handle uncertain
information.
## 3 Reasoning Structures in Bayesian Networks
### 3.1 Basic reasoning structures
#### 3.1.1 d-Separation in Bayesian Networks
d-Separation is one important property of Bayesian networks for inference.
Before we define d-separation, we first look at the way that evidence is
transmitted in Bayesian Networks. There are two types of evidence:
* •
Hard Evidence (instantiation) for a node $A$ is evidence that the state of $A$
is definitely a particular value.
* •
Soft Evidence for a node $A$ is any evidence that enables us to update the
prior probability values for the states of $A$.
d-Separation (Definition):
Two distinct variables $X$ and $Z$ in a causal network are d-separated if, for
all paths between $X$ and $Z$, there is an intermediate variable $V$ (distinct
from $X$ and $Z$) such that either
* •
the connection is serial or diverging and $V$ is instantiated or
* •
the connection is converging, and neither $V$ nor any of $V$’s descendants
have received evidence.
If $X$ and $Z$ are not d-separated, we call them d-connected.
#### 3.1.2 Basic structures of Bayesian Networks
Based on the definition of d-seperation, three basic structures in Bayesian
networks are as follows:
1. 1.
Serial connections
Consider the situation in Figure 2. $X$ has an influence on $Y$, which in turn
has an influence on $Z$. Obviously, evidence on $Z$ will influence the
certainty of $Y$, which then influences the certainty of $Z$. Similarly,
evidence on $Z$ will influence the certainty on $X$ through $Y$. On the other
hand, if the state of $Y$ is known, then the channel is blocked, and $X$ and
$Z$ become independent. We say that $X$ and $Z$ are d-separated given $Y$, and
when the state of a variable is known, we say that it is instantiated (hard
evidence).
We conclude that evidence may be transmitted through a serial connection
unless the state of the variable in the connection is known.
Figure 2: Serial Connection. When $Y$ is Instantiated, it blocks the
communication between $X$ and $Z$.
2. 2.
Diverging connections
The situation in Figure 3 is called a diverging connection. Influence can pass
between all the children of $X$ unless the state of $X$ is known. We say that
$Y_{1},Y_{2},\ldots,Y_{n}$ are d-separated given $X$.
Evidence may be transmitted through a diverging connection unless it is
instantiated.
Figure 3: Diverging Connection. If $X$ is instantiated, it blocks the
communication between its children.
3. 3.
Converging connections
Figure 4: Converging Connection. If $Y$ changes certainty, it opens for the
communication between its parents.
A description of the situation in Figure 4 requires a little more care. If
nothing is known about $Y$ except what may be inferred from knowledge of its
parents $X_{1},\ldots,X_{n}$, then the parents are independent: evidence on
one of the possible causes of an event does not tell us anything about other
possible causes. However, if anything is known about the consequences, then
information on one possible cause may tell us something about the other
causes.
This is the explaining away effect illustrated in Figure 1. $X_{4}$ (pavement
is wet) has occurred, and $X_{3}$ (the sprinkler is on) as well as $X_{2}$
(it’s raining) may cause $X_{4}$. If we then get the information that $X_{2}$
has occurred, the certainty of $X_{3}$ will decrease. Likewise, if we get the
information that $X_{2}$ has not occurred, then the certainty of $X_{3}$ will
increase.
The three preceding cases cover all ways in which evidence may be transmitted
through a variable.
## 4 Classification of Inferences in Bayesian Networks
In Bayesian networks, 4 popular inferences are identified as:
1. 1.
Forward Inference
Forward inferences is also called predictive inference (from causes to
effects). The inference reasons from new information about causes to new
beliefs about effects, following the directions of the network arcs. For
example, in Figure 2, $X\rightarrow Y\rightarrow Z$ is a forward inference.
2. 2.
Backward Inference
Backward inferences is also called diagnostic inference (from effects to
causes). The inference reasons from symptoms to cause, Note that this
reasoning occurs in the opposite direction to the network arcs. In Figure 2 ,
$Z\rightarrow Y$ is a backward inference. In Figure 3 , $Y_{i}\rightarrow
X(i\in[1,n])$ is a backward inference.
3. 3.
Intercausal Inference
Intercausal inferences is also called explaining away (between parallel
variables). The inference reasons about the mutual causes (effects) of a
common effect (cause). For example, in Figure 4, if the $Y$ is instantiated,
$X_{i}$ and $X_{j}(i,j\in[1,n])$ are dependent. The reasoning
$X_{i}\leftrightarrow X_{j}(i,j\in[1,n])$ is an intercausal inference. In
Figure 3, if $X$ is not instantiated, $Y_{i}$ and $Y_{j}(i,j\in[1,n])$ are
dependent. The reasoning $Y_{i}\leftrightarrow Y_{j}(i,j\in[1,n])$ is an
intercausal inference.
4. 4.
Mixed inference
Mixed inferences is also called combined inference. In complex Bayesian
networks, the reasoning does not fit neatly into one of the types described
above. Some inferences are a combination of several types of reasoning.
### 4.1 Inference in Bayesian Networks
#### 4.1.1 inference in basic models
* •
in Serial Connections
* –
the forward inference executes with the evidence forward propagation. For
example, in Figure 5, consider the inference $X\rightarrow Y\rightarrow Z$.
111Note: In this chapter, $P(X^{+})$ is the abbreviation of $P(X=true)$,
$P(X^{-})$ is the abbreviation of $P(|X=false)$. For simple expression, we use
$P(Y|X)$ to denote $P(Y=true|X=true)$ by default. But in express $P(Y^{+}|X)$,
$X$ denotes both situations $X=true$ and $X=false$.
Figure 5: Inference in Serial Connection
If Y is instantiated, X and Z are independent, then we have following example:
$P(Z|XY)=P(Z|Y)$;
$P(Z^{+}|Y^{+})=0.95$;
$P(Z^{-}|Y^{+})=0.05$;
$P(Z^{+}|Y^{-})=0.01$;
$P(Z^{-}|Y^{-})=0.99$;
if Y is not instantiated, X and Z are dependent, then
$P(Z^{+}|X^{+}Y)=P(Z^{+}|Y^{+})P(Y^{+}|X^{+})+P(Z^{+}|Y^{-})P(Y^{-}|X^{+})$
$=0.95*0.85+0.01*0.15=0.8075+0.0015=0.809$;
$P(Z^{-}|X^{-}Y)=P(Z^{-}|Y^{+})P(Y^{+}|X^{-})+P(Z^{-}|Y^{-})P(Y^{-}|X^{-})$
$=0.05*0.03+0.99*0.97=0.0015+0.9603=0.9618$.
* –
the backward inference executes the evidence backward propagation. For
example, in Figure 5, consider the inference $Z\rightarrow Y\rightarrow X$.
1. 1.
If $Y$ is instantiated ($P(Y^{+})=1$ or $P(Y^{-})=1)$, $X$ and $Z$ are
independent, then
$\displaystyle P(X|YZ)=P(X|Y)=\frac{P(X)P(Y|X)}{P(Y)}$ (5)
$P(X^{+}|Y^{+}Z)=P(X^{+}|Y^{+})=\frac{P(X^{+})P(Y^{+}|X^{+})}{P(Y^{+})}=\frac{09*0.85}{1}=0.765$;
$P(X^{+}|Y^{-}Z)=P(X^{+}|Y^{-})=\frac{P(X^{+})P(Y^{-}|X^{+})}{P(Y^{-})}=\frac{09*0.15}{1}=0.135$.
2. 2.
If $Y$ is not instantiated, $X$ and $Z$ are dependent (See the dashed lines in
Figure 5). Suppose $P(Z^{+})=1$ then
$P(X^{+}|YZ^{+})=\frac{P(X^{+}YZ^{+})}{P(YZ^{+})}=\frac{P(X^{+}YZ^{+})}{\sum_{X}P(XYZ^{+})}$;
$P(X^{+}YZ^{+})=P(X^{+}Y^{+}Z^{+})+P(X^{+}Y^{-}Z^{+})=0.9*0.85*0.95+0.9*0.15*0.05=0.72675+0.00675=0.7335$;
$\sum_{X}P(XYZ^{+})=P(X^{+}Y^{+}Z^{+})+P(X^{+}Y^{-}Z^{+})+P(X^{-}Y^{+}Z^{+})+P(X^{-}Y^{-}Z^{+})\\\
=0.9*0.85*0.95+0.9*0.15*0.99+0.1*0.03*0.95+0.1*0.97*0.01\\\
=0.72675+0.13365+0.00285+0.00097=0.86422$;
$P(X^{+}|YZ^{+})=\frac{P(X^{+}YZ^{+})}{\sum_{X}P(XYZ^{+})}=\frac{0.7335}{0.86422}=0.8487.$
In serial connections, there is no intercausal inference.
* •
in Diverging Connections
* –
the forward inference executes with the evidence forward propagation. For
example, in Figure 6, consider the inference $Y\rightarrow X$ and
$Y\rightarrow Z$, the goals are easy to obtain by nature.
Figure 6: Inference in Diverging Connection
* –
the backward inference executes with the evidence backward propagation, see
the dashed line in Figure 6, consider the inference $(XZ)\rightarrow Y$, $X$
and $Z$ are instantiated by assumption, suppose $P(X^{+}=1)$, $P(Z^{+}=1)$.
Then,
$\displaystyle
P(Y^{+}|X^{+}Z^{+})=\frac{P(Y^{+}X^{+}Z^{+})}{P(X^{+}Z^{+})}=\frac{P(Y^{+})P(X^{+}|Y^{+})P(Z^{+}|Y^{+})}{P(X^{+}Z^{+})}$
$\displaystyle=\frac{0.98*0.95*0.90}{1}=0.8379$ (6)
* –
the intercausal inference executes between effects with a common cause. In
Figure 6, if $Y$ is not instantiated, there exists intercausal inference in
diverging connections. Consider the inference $X\rightarrow Z$,
$P(X^{+}|YZ^{+})=\frac{P(X^{+}YZ^{+})}{P(YZ^{+})}=\frac{P(X^{+}Y^{+}Z^{+})+P(X^{+}Y^{-}Z^{+})}{P(Y^{+}Z^{+})+P(Y^{-}Z^{+})}$;
$=\frac{0.98*0.95*0.90+0.02*0.01*0.03}{0.98*0.90+0.02*0.03}=0.94936$.
* •
in Converging Connections,
* –
the forward inference executes with the evidence forward propagation. For
example, in Figure 7, consider the inference $(XZ)\rightarrow Y$, $P(Y|XZ)$ is
easy to obtain by the definition of Bayesian Network in by nature.
Figure 7: Inference in Converging Connection
* –
the backward inference executes with the evidence backward propagation. For
example, in Figure 7, consider the inference $Y\rightarrow(XZ)$.
$P(Y)=\sum_{XZ}P(XYZ)=\sum_{XZ}(P(Y|XZ)P(XZ))$,
$P(XZ|Y)=\frac{P(Y|XZ)P(XZ)}{P(Y)}=\frac{P(Y|XZ)P(X)P(Z)}{\sum_{XZ}(P(Y|XZ)P(XZ))}$.
Finally,
$P(X|Y)=\sum_{Z}P(XZ|Y)$,
$P(Z|Y)=\sum_{X}P(XZ|Y)$.
* –
the intercausal inference executes between causes with a common effect, and
the intermediate node is instantiated, then $P(Y^{+})=1$ or $P(Y^{-})=1$. In
Figure 7, consider the inference $X\rightarrow Z$, suppose $P(Y^{+})=1$,
$P(Z^{+}|X^{+}Y^{+})=\frac{P(Z^{+}X^{+}Y^{+})}{P(X^{+}Y^{+})}=\frac{P(Z^{+}X^{+}Y^{+})}{\sum_{Z}P(X^{+}Y^{+}Z)}$;
$P(Z^{+}X^{+}Y^{+})=P(X^{+})P(Z^{+})P(Y^{+}|X^{+}Z^{+})$;
$\sum_{Z}P(X^{+}YZ)=P(X^{+}Y^{+}Z^{+})+P(X^{+}Y^{+}Z^{-})$;
$P(Z^{+}|X^{+}Y^{+})=\frac{P(Z^{+}X^{+}Y^{+})}{\sum_{Z}P(X^{+}Y^{+}Z)}=\frac{P(X^{+})P(Z^{+})P(Y^{+}|X^{+}Z^{+})}{P(X^{+}Y^{+}Z^{+})+P(X^{+}Y^{+}Z^{-})}.$
#### 4.1.2 inference in complex model
For complex models in Bayesian networks, there are single-connected networks,
multiple-connected, or event looped networks. It is possible to use some
methods, such as Triangulated Graphs, Clustering and Join Trees [Bertele &
Brioschi, 1972] [Finn & Thomas, 2007] [Golumbic, 1980], etc., to simplify them
into a polytree. Once a polytree is obtained, the inference can be executed by
the following approaches.
Polytrees have at most one path between any pair of nodes; hence they are also
referred to as singly-connected networks.
Suppose $X$ is the query node, and there is some set of evident nodes
$E,X\notin E$. The posterior probability (belief) is denoted as
$\mathbb{B}(X)=P(X|E)$, see Figure 8.
Figure 8: Evidence Propagation in Polytree
$E$ can be splitted into 2 parts: $E^{+}$ and $E^{-}$. $E^{-}$ is the part
consisting of assignments to variables in the subtree rooted at $X$, $E^{+}$
is the rest of it.
$\pi_{X}(E^{+})=P(X|E^{+})$
$\lambda_{X}(E^{-})=P(E^{-}|X)$
$\mathbb{B}(X)=P(X|E)=P(X|E^{+}E^{-})=\frac{P(E^{-}|XE^{+})P(X|E^{+})}{P(E^{-}|E^{+})}=\frac{P(E^{-}|X)P(X|E^{+})}{P(E^{-}|E^{+})}=\alpha\pi_{X}(E^{+})\lambda_{X}(E^{-})$
(7)
$\alpha$ is a constant independent of $X$.
where
$\lambda_{X}(E^{-})=\\{\begin{array}[]{cc}1&if\ evidence\ is\ X=x_{i}\\\ 0&if\
evidence\ is\ for\ another\ x_{j}\\\ \end{array}$ (8)
$\pi_{X}(E^{+})=\sum_{u_{1},...,u_{m}}P(X|u_{1},...,u_{m})\prod_{i}\pi_{X}(u_{i})$
(9)
1. 1.
Forward inference in Polytree
Node $X$ sends $\pi$ messages to its children.
$\pi_{X}(U)=\\{\begin{array}[]{cc}1&if\ x_{i}\in X\ is\ entered\\\ 0&if\
evidentce\ is\ for\ another\ value\ x_{j}\\\
\sum_{u_{1},...u_{m}}P(X|u_{1},...u_{m})\prod_{i}\pi_{X}(u_{i})&otherwise\end{array}$
(10)
2. 2.
Backward inference in Polytree Node $X$ sends new $\lambda$ messages to its
parents.
$\lambda_{X}(Y)=\prod_{y_{j}\in Y}[\sum_{j}P(y_{j}|X)\lambda_{X}(y_{j})]$ (11)
### 4.2 Related Algorithms for Probabilistic Inference
Various types of inference algorithms exist for Bayesian networks [Lauritzen &
Spiegelhalter, 1988] [Pearl, 1988] [Pearl, 2000] [Neal, 1993]. Each class
offers different properties and works better on different classes of problems,
but it is very unlikely that a single algorithm can solve all possible problem
instances effectively. Every resolution is always based on a particular
requirement. It is true that almost all computational problems and
probabilistic inference using general Bayesian networks have been shown to be
NP-hard by Cooper [Cooper, 1990].
In the early 1980’s, Pearl published an efficient message propagation
inference algorithm for polytrees [Kim & Pearl, 1983] [Peal, 1986]. The
algorithm is exact, and has polynomial complexity in the number of nodes, but
works only for singly connected networks. Pearl also presented an exact
inference algorithm for multiple connected networks called loop cutset
conditioning algorithm [Peal, 1986]. The loop cutset conditioning algorithm
changes the connectivity of a network and renders it singly connected by
instantiating a selected subset of nodes referred to as a loop cutset. The
resulting single connected network is solved by the polytree algorithm, and
then the results of each instantiation are weighted by their prior
probabilities. The complexity of this algorithm results from the number of
different instantiations that must be considered. This implies that the
complexity grows exponentially with the size of the loop cutest being
$O(d^{c})$, where $d$ is the number of values that the random variables can
take, and $c$ is the size of the loop cutset. It is thus important to minimize
the size of the loop cutset for a multiple connected network. Unfortunately,
the loop cutset minimization problem is NP-hard. A straightforward application
of Pearl’s algorithm to an acyclic digraph comprising one or more loops
invariably leads to insuperable problems [Koch & Westphall, 2001] [Neal,
1993].
Another popular exact Bayesian network inference algorithm is Lauritzen and
Spiegelhalter’s clique-tree propagation algorithm [Lauritzen & Spiegelhalter,
1988]. It is also called a ”clustering” algorithm. It first transforms a
multiple connected network into a clique tree by clustering the triangulated
moral graph of the underlying undirected graph and then performs message
propagation over the clique tree. The clique propagation algorithm works
efficiently for sparse networks, but still can be extremely slow for dense
networks. Its complexity is exponential in the size of the largest clique of
the transformed undirected graph.
In general, the existent exact Bayesian network inference algorithms share the
property of run time exponentiality in the size of the largest clique of the
triangulated moral graph, which is also called the induced width of the graph
[Lauritzen & Spiegelhalter, 1988].
## 5 Conclusion
This chapter summarizes the popular inferences methods in Bayesian networks.
The results demonstrates that the evidence can propagated across the Bayesian
networks by any links, whatever it is forward or backward or intercausal
style. The belief updating of Bayesian networks can be obtained by various
available inference techniques. Theoretically, exact inferences in Bayesian
networks is feasible and manageable. However, the computing and inference is
NP-hard. That means, in applications, in complex huge Bayesian networks, the
computing and inferences should be dealt with strategically and make them
tractable. Simplifying the Bayesian networks in structures, pruning unrelated
nodes, merging computing, and approximate approaches might be helpful in the
inferences of large scale Bayeisan networks.
## References
* [Balke & Pearl, 1995] A. Balke and J. Pearl. Counterfactuals and policy analysis in structural models. Proceedings of the Eleventh Conference on Uncertainty in Artificial Intelligence, pages 11-18, 1995. Morgan Kaufmann.
* [Bertele & Brioschi, 1972] Bertele, U. and Brioschi, F. (1972). Nonserial Dynamic Programming. Academic Press, London, ISBN-13: 978-0120934508.
* [Cooper, 1990] G. Cooper. Computational complexity of probabilistic inference using Bayesian belief networks. Artificial Intelligence, 42:393-405, 1990.
* [Finn & Thomas, 2007 ] Finn V. Jensen and Thomas D. Nielsen (2007). Bayesian Networks and Decision Graphs. Springer, ISBN-13:978-0-387-68281-5.
* [Golumbic, 1980] Golumbic, M. C. (1980). Algorithmic Graph Theory and Perfect Graphs. Academic Press, London,ISBN-13: 978-0122892608.
* [Goldszmidt & Pearl, 1996] M. Goldszmidt and J. Pearl. Qualitative Probabilities for Default Reasoning, Belief Revision, and Causal Modeling. Artificial Intelligence, 84(1-2): 57-112, July 1996.
* [Jordan et al., 1998] M. I. Jordan, Z. Ghahramani, T. S. Jaakkola and L. K. Saul. An Introduction to Variational Methods for Graphical Models. M. I. Jordan (Ed.), Learning in Graphical Models. Kluwer, Dordrecht, The Netherlands, 1998.
* [Kim & Pearl, 1983] Jin H. Kim and Judea Pearl. A computational model for combined causal and diagnostic reasoning in inference systems. In Proceedings of the Eighth International Joint Conference on Artificial Intelligence (IJCAI-83), pages 190-193, 1983. Morgan Kaufmann.
* [ Koch & Westphall, 2001] F. L. Koch, and C. B. Westphall. Decentralized Network Management Using Distributed Artificial Intelligence. Journal of Network and systems management, Vol. 9, No. 4, December 2001.
* [Lauritzen & Spiegelhalter, 1988] S. L. Lauritzen and D. J. Spiegelhalter. Local Computations with Probabilities on Graphical Structures and Their Application to Expert Systems. Journal of the Royal Statistical Society, Series B 50:157-224, 1988.
* [Neal, 1993] R. M. Neal, Probabilistic Inference Using Markov Chain Monte Carlo methods, Tech. Rep. CRG-TR93-1, University of Toronto, Department of Computer Science, 1993.
* [Peal, 1986] J. Pearl. A constraint-propagation approach to probabilistic reasoning, Uncertainty in Artificial Intelligence. North-Holland, Amsterdam, pages 357-369, 1986.
* [Pearl, 1987] J. Pearl. Evidential Reasoning Using Stochastic Simulation of Causal Models. Artificial Intelligence, 32:247-257, 1987.
* [Pearl, 1988] J. Pearl. Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference. Morgan Kaufmann, San Mateo, CA, 1988.
* [Pearl, 2000] J. Pearl. Causality: Models, Reasoning, and Inference. Cambridge, England: Cambridge University Press. New York, NY, 2000.
* [Pearl & Verma, 1991] J. Pearl and T. Verma. A theory of inferred causation. J. A. Allen, R. Fikes and E. Sandewall (Eds.), Principles of Knowledge Representation and Reasoning. Proceedings of the Second International Conference, pages 441-452. Morgan Kaufmann, San Mateo, CA, 1991.
* [Spirtes et al., 1993] P. Spirtes, C. Glymour and R. Scheines. Causation, Prediction, and Search. Springer-Verlag, New York, 1993.
* [Tenenbaum & Griffiths, 2001] J. B. Tenenbaum and T. L. Griffiths. Structure learning in human causal induction. Advances in Neural Information Processing Systems, volume 13, Denver, Colorado, 2001. MIT Press.
|
arxiv-papers
| 2010-11-03T16:50:22 |
2024-09-04T02:49:14.469980
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jianguo Ding",
"submitter": "Jianguo Ding",
"url": "https://arxiv.org/abs/1011.0935"
}
|
1011.0939
|
# Issues with $J$-dependence in the LSDA$+U$ method for non-collinear magnets
Eric Bousquet1,2 Nicola Spaldin3 1Materials Department, University of
California, Santa Barbara, CA 93106, USA 2Physique Théorique des Matériaux,
Université de Liège, B-4000 Sart Tilman, Belgium 3 Department of Materials,
ETH Zurich, Wolfgang-Pauli-Strasse 10 CH-8093 Zurich, Switzerland
###### Abstract
We re-examine the commonly used density functional theory plus Hubbard U
(DFT$+U$) method for the case of non-collinear magnets. While many studies
neglect to explicitly include the exchange correction parameter J, or consider
its exact value to be unimportant, here we show that in the case of non-
collinear magnetism calculations the J parameter can strongly affect the
magnetic ground state. We illustrate the strong J-dependence of magnetic
canting and magnetocrystalline anisotropy by calculating trends in the
magnetic lithium orthophosphate family LiMPO4 (M = Fe and Ni) and difluorite
family MF2 (M = Mn, Fe, Co and Ni). Our results can be readily understood by
expanding the usual DFT$+U$ equations within the spinor scheme, in which the J
parameter acts directly on the off-diagonal components which determine the
spin canting.
first-principles, LDA+U, non-collinear magnetism, magnetocrystalline
anisotropy
Density functional theory (DFT) within the local density (LDA) and generalized
gradient (GGA) approximations is widely used to describe a large variety of
materials with good accuracy. The LDA and GGA functionals often fail, however,
to correctly reproduce the properties of strongly correlated materials
containing d and f electrons. The LDA$+U$ approach – in which a Hubbard U
repulsion term is added to the LDA functional for selected orbitals – was
introduced in response to this problem, and often improves drastically over
the LDA or GGA. Indeed, it provides a good description of the electronic
properties of a range of exotic magnetic materials, such as the Mott insulator
KCuF3liechtenstein1995 and the metallic oxide LaNiO2 lee2004 .
Two main LDA$+U$ schemes are in widespread use today: The Dudarev dudarev1998
approach in which an isotropic screened on-site Coulomb interaction
$U_{eff}=U-J$ is added, and the Liechtenstein liechtenstein1995 approach in
which the $U$ and exchange ($J$) parameters are treated separately. The
Dudarev approach is equivalent to the Liechtenstein approach with $J=0$
baettig2005 . Both the effect of the choice of LDA+$U$ scheme on the orbital
occupation and subsequent properties ylvisaker2009 , as well as the dependence
of the magnetic properties on the value of $U$ savrasov2005 , have recently
been analyzed. There has been no previous systematic study, however, of the
effect of the $J$ parameter of the Liechtenstein approach in non-collinear
magnetic materials. Here we show that neither the approach of not explicitly
considering the $J$ parameter (as in the Dudarev implementation), nor the
assumption that its importance is borderline – a common approximation is to
use $J\simeq 10\%\ U$ without careful testing – within the Liechtenstein
implementation are justified in the case of non-collinear magnets. We
demonstrate that in the case of non-collinear antiferromagnets, the choice of
$J$ can strongly change the amplitude of the spin canting angle (LiNiPO4) or
even modify the easy axis of the system (LiFePO4 and FeF2), with consequent
drastic effects on the magnetic susceptibilities and magnetoelectric
responses.
First we remind the reader how the $U$ and $J$ parameters appear in the usual
collinear spin LSDA$+U$ formalism. The LSDA$+U$ reformulation of the LSDA
Hamiltonian is usually written as:
$\displaystyle H_{LSDA+U}=H_{LSDA}+H_{U}\quad,$ (1)
whith
$\displaystyle
H_{U}^{\sigma}=\displaystyle\sum_{m_{1},m_{2}}P_{m_{1},m_{2}}V^{\sigma}_{m_{2},m_{1}}\quad,$
(2)
where $P$ is the projection operator, $\sigma$ is the spin index, and (on a
given atomic site):
$\displaystyle V^{\uparrow(\downarrow)}_{m_{2},m_{1}}=$
$\displaystyle\displaystyle\sum_{3,4}\left(V^{ee}_{1,3,2,4}-U\delta_{1,2}-V^{ee}_{1,3,4,2}+J\delta_{1,2}\right)n^{\uparrow(\downarrow)}_{3,4}$
$\displaystyle+\left(V^{ee}_{1,3,2,4}-U\delta_{1,2}\right)n^{\downarrow(\uparrow)}_{3,4}+\frac{1}{2}(U-J)\delta_{1,2}$
(3)
Here $V^{ee}_{1,3,2,4}=\left\langle
m_{1},m_{3}\left|V^{ee}_{m_{1},m_{3},m_{2},m_{4}}\right|m_{2},m_{4}\right\rangle$
are the elements of the screened Coulomb interaction (which can be viewed as
the sum of Hartree (direct) contributions $V^{ee}_{1,3,2,4}$ and Fock
(exchange) contributions $V^{ee}_{1,3,4,2}$ and $n^{\sigma}_{i,j}$ are the
$d$-orbital occupancies.
In the case of non-collinear magnetism, the formalism is extended and the
density is expressed in a two-component spinor formulation:
$\displaystyle\rho=$
$\displaystyle\begin{pmatrix}\rho^{\uparrow\uparrow}&\rho^{\uparrow\downarrow}\\\
\rho^{\downarrow\uparrow}&\rho^{\downarrow\downarrow}\end{pmatrix}=\begin{pmatrix}n+m_{z}&m_{x}-im_{y}\\\
m_{x}+im_{y}&n-m_{z}\end{pmatrix}$ (4)
where $n$ is the charge density and $m_{\alpha}$ the magnetization density
along the $\alpha$ direction ($\alpha=x,y,z$). Using the double-counting
proposed by Bultmark et al.bultmark2009 , the LSDA$+U$ potential is then also
expressed in the two-component spin space as:
$\displaystyle V_{i,j}=$
$\displaystyle\begin{pmatrix}V^{\uparrow\uparrow}_{i,j}&V^{\uparrow\downarrow}_{i,j}\\\
V^{\downarrow\uparrow}_{i,j}&V^{\downarrow\downarrow}_{i,j}\end{pmatrix}$ (5)
where $V^{\uparrow\uparrow}$ and $V^{\downarrow\downarrow}$ are equal to Eqs.3
and
$\displaystyle
V^{\uparrow\downarrow(\downarrow\uparrow)}_{m_{2},m_{1}}=\displaystyle\sum_{3,4}\left(-V^{ee}_{1,3,4,2}+J\delta_{1,2}\right)n^{\uparrow\downarrow(\downarrow\uparrow)}_{3,4}$
(6)
For collinear magnets, only $V^{\uparrow\uparrow}$ and
$V^{\downarrow\downarrow}$ (Eqs. 3) are relevant since
$n^{\uparrow\downarrow}$ and $n^{\downarrow\uparrow}$ are equal to zero, and
$J$ affects the potential mainly through an effective $U-J$. However, in the
case of non-collinear magnetism, the $n^{\uparrow\downarrow}$ and
$n^{\downarrow\uparrow}$ and hence the $V^{\uparrow\downarrow}$ and
$V^{\downarrow\uparrow}$ (Eqs. 6) are non-zero. Then it is clear from Eqs. 6
that $J$ acts explicitly on the off-diagonal potential components.
Next, we show the effect of the choice of $J$ parameter in the family of
lithium orthophosphates, LiMPO4 (M = Ni and Fe) and in the family of
difluorites MF2 (M = Mn, Co, Fe and Ni). The orthophosphates crystallize in
the orthorhombic Pnma space group with $C$-type antiferromagnetic (AFM) order.
The difluorites crystalize in the tetragonal P42/mnm rutile structure with AFM
order. We performed calculations within the Liechtenstein approach of the
DFT$+U$ as implemented in the VASP code vasp1 ; vasp2 111We note that
LSDA$+U$ double-couting term taking into accound the magnetization density as
proposed by Bultmark et al.bultmark2009 is mandatory within non-collinear
magnetism calculations. This is not necessarily done in the present
implementation of other codes. with $U$ and $J$ corrections applied to the 3d
orbitals of the M cations. In all cases we relaxed the atomic positions until
the residual forces on each atom were lower than 10 $\mu$eV/Å at the
experimental volume and cell shape reported in Tab. 1, taking into account the
spin-orbit interaction. We found good convergence of the non-collinear spin
ground state with a cutoff energy of 500 eV on the plane wave expansion and a
k-point grid of $2\times 4\times 4$ for the orthophophates and $4\times
4\times 6$ for the difluorites.
| $a$ | $b$ | $c$ | Ref.
---|---|---|---|---
LiFePO4 | 10.332 | 6.010 | 4.692 | streltsov1993,
LiNiPO4 | 10.032 | 5.854 | 4.677 | abrahams1993,
NiF2 | 4.650 | 4.650 | 3.084 | hutchings1970,
FeF2 | 4.700 | 4.700 | 3.310 | dealmeida1989,
MnF2 | 4.650 | 4.650 | 3.084 | oguchi1958,
CoF2 | 4.695 | 4.695 | 3.179 | otoole2001,
Table 1: Experimental cell parameters (Å) used in the simulations of LiMPO4
phosphates and MF2 difluorites.
Figure 1: (a) Calculated LSDA$+U$ canting angle of LiNiPO4 versus $J$ for
$U=5$ eV. The experimental value of the canting angle is equal to 7.8∘
jensen2009 . (b) Energy versus canting angle in LiNiPO4 for $U=5$ eV and $J=0$
eV (red circles), $U=5$ eV and $J=1$ eV (blue triangles), $U_{eff}=4$ eV
(green crosses) and $U=5$ eV and $J=1$ eV but by fixing $J=$ 0 eV in Eqs.6
(pink squares). The zero energy reference is chosen at zero canting angle. (c)
Magnetocrystaline anisotropy energy (MCAE) between the $a$ and $b$
orientations of the magnetic moments of LiFePO4. The experimental $b$
orientations is taken as energy reference.
First, we focus on LiNiPO4, which is known experimentally to be $C$-type AFM,
with an easy-axis along the $c$ direction and a small A-type AFM canting of
the spins along the $a$ direction ($C_{z}A_{x}$ ground state with mm’m
magnetic point group) jensen2009 . Performing calculations within the LSDA$+U$
method with $J=0$, we find that we correctly reproduce the $C_{z}A_{x}$ ground
state with a rather small $U$ sensitivity of the magnetocrystalline anisotropy
energy (MCAE) and the spin canting; this finding is consistent with a previous
report using the GGA functional yamauchi2010 . However, our calculated canting
angle of 1.6∘ for $U=5$ eV and $J=0$ eV severely underestimates the
experimental value of 7.8∘jensen2009 . In Fig.1 (a) we show the evolution of
the canting angle with $J$ at $U=5$ eV. We find that the canting angle is
extremely sensitive to the value of $J$ – in fact it is $\propto J^{3}$ –
changing from 1.6∘ at $J=0$ eV to 7.8∘ at $J=1.7$ eV. To reproduce the
experimental value of the canting angle we need to use the rather large $J$
value of 1.7 eV. The dependence of the canting angle on $J$ is consistent with
Eqs. 6, as the off-diagonal elements $n^{\uparrow\downarrow}$ and
$n^{\downarrow\uparrow}$ are non-zero when the spins cant away from the easy
axis.
In Fig. 1 (b) we report the energy versus the canting angle in LiNiPO4 for
$U=5$ eV and different values of $J$. We see that as $J$ is increased from
$J=0$ eV to $J=1$ eV (red circles and blue triangles) the minimum of the
energy shifts to larger canting angle, with a stronger gain of energy with
respect to the uncanted reference. When performing the same calculation with
$U_{eff}=$4 eV (green crosses in Fig. 1) we obtain results that are very
similar to the case $U=5$ eV and $J=0$ eV, which is formally equivalent to the
Dudarev approach with $U_{eff}=5$ eV. These comparisons confirm that varying
$U$ has a minimal effect on the canting angle in LiNiPO4 and also that the use
of the Liechtenstein treatment of $J$ is extremly important. To further
confirm the direct relationship between the spin canting and the $J$
parameter, we performed the same calculations with $U=5$ eV and $J=1$ eV but
we artificially fixed $J=0$ eV only in Eqs. 6 (pink squares in Fig.1 (b)). We
clearly see that the energy versus canting angle is strongly affected by this
modification and in fact the canting is almost removed.
Similar $J$ dependence of the canting angle was also reported previously for
Ni2+ in BaNiF4 ederer2006 ; in Ref. ederer2006, it was found that at $U=5$
eV, the canting varies from 2∘ to 3∘ when $J$ is varied from 0 eV to 1 eV. In
both LiNiPO4 and BaNiF4 the Ni ion is divalent, with a $d^{8}$ configuration,
and octahedrally coordinated. To investigate the generality of this behavior,
we next consider the case of the canted-spin antiferromagnet NiF2, in which
the Ni ion is in the same coordination environnement as in BaNiF4.
Experimentally, NiF2 has the spins aligned preferentially in the plane
perpendicular to the c axis with a slight canting from antiparallel alignment
by an estimated $\sim$0.5∘ at low temperatures hutchings1970 . Performing
LSDA$+U$ calculations at the experimental volume and with $U=5$ eV and $J=0$
eV we indeed obtain the easy axis perpendicular to the c axis and a small
canting of 0.3∘, in excellent agreement with the experiments. In contrast to
the case of LiNiPO4, however, we find that the amplitude of the canting angle
is almost insensitive to the value of $J$ with just a small tendency to be
reduced when $J$ increased. This insensitivity of the canting angle to the
value of $J$ in NiF2 can be understood from the fact that in this compound the
magnetism is almost collinear, and therefore the off-diagonal elements of the
occupation matrix, $n^{\uparrow\downarrow}$ and $n^{\downarrow\uparrow}$, are
close to zero. Inspection of Eqs. 3 then shows that the effect of $J$ is
reduced largely to the diagonal part of the potential where the $U$ parameter
is dominant.
To summarize our findings for the Ni-based compounds, in cases where the
experimental canting is large (2-3∘) we find a strong $J$-dependence of the
canting angle, which increases with increasing $J$; when the canting is weak
experimentally the $J$-dependence is much weaker.
Figure 2: Magnetocrystaline anisotropy energy versus the $J$ parameter of (a)
FeF2 (Experimental value from Ref.lines1967, ), (b) NiF2, (c) MnF2
(Experimental value from Ref.gafver1977, ) and (d) CoF2 (“sc“ are calculations
with Co semi-cores while ”no sc” are calculations without Co semi-cores). The
MCAE reported here is the energy between the $a$ and $c$ orientation of the
spins, the energy of the $c$ orientation is taken as reference.
Next we analyse the effect of $J$ on the behavior on the corresponding
divalent iron compounds. We begin with LiFePO4, which is known experimentally
to be a $C$-type AFM with an easy axis along the $b$ direction and no observed
canting of the spins zimmermann2009 ; liang2008 ($C_{y}$ ground state with
mmm’ magnetic point group). Our calculations within the LSDA$+U$ functional at
the commonly used values of $U=4$ eV and $J=0$ eV for Fe2+ yield the correct
$C$-type AFM order but find the easy axis incorrectly along the $a$ direction.
Now we switch to $J\neq 0$ eV and report in Fig. 1.c the MCAE between the $b$
and $a$ directions, calculated by turning all the spins homogenously from the
$C_{y}$ to the $C_{x}$ direction. We find that the MCAE is approximately
linear with $J$, but with rather dramatic qualitative dependence: while at
$J=0$ eV the easy axis is along the $a$ direction (negative MCAE) the MCAE is
almost reduced to zero around $J=0.5$ eV and the easy axis changes to the $b$
direction for $J\gtrsim 0.5$ eV (positive MAE). To reproduce the experimental
easy axis ($C_{y}$) a value of $J$ greater than 0.58 eV is required. In the
cases where the correct easy axis is reproduced ($C_{y}$) we do not observe
any canting of the spins, in agreement with the experimental magnetic point
group mmm’.
As a second example with Fe2+, we analyse the effect of $J$ on the MCAE of
FeF2. Experimentally FeF2 is known to have its spin magnetization parallel to
the tetragonal $c$ axis with a rather large MCAE of about +4800 $\mu$eV
rudowicz1977 ; ohlmann1961 . In Fig. 2.a we report the LSDA$+U$ MCAE energies
with respect to $J$ at four different values of $U$ (3, 4, 5 and 6 eV). All
the calculations with $J=0$ eV give the wrong easy axis (spins are
perpendicular to $c$) with a huge error in the MCA energy (MCAE from -16000 to
-26000 $\mu$eV for $U$ going from 3 to 6 eV). Increasing the value of $J$ in
the range of 0–0.5 eV has the tendency to strongly reduce this error with a
linear increase of the MCAE with $J$ as we found above for LiFePO4. However
beyond $J\simeq 0.5$ the increase of the MCAE is reduced and the evolution
becomes more complex with the appearance of two maxima before a drastic
decrease beyond $J\simeq 1.3$ eV. The correct easy axis (MCAE$>0$) is only
obtained for a very small range of $U$ and $J$ values, and the amplitude of
the MCAE is correct over an even smaller range. This $J$ dependence of the
MCAE is again consistent with Eqs.3-6. From Eq.4 it is clear that when
changing the orientation of the spins from the $z$ axis to the $x$ or $y$ axis
the off-diagonal parts of Eq.4 become non-zero resulting in a direct effect of
$J$ on the MCAE from Eqs.6.
We also performed the same analysis of the MCAE for NiF2 (Fig.2.b), MnF2
(Fig.2.c) and CoF2 (Fig.2.d). MnF2 and CoF2 have the same easy axis as FeF2
while NiF2 has its easy axis perpendicular to the $c$ direction. The easy axis
is well reproduced for all three compounds at $J=0$ eV. As for FeF2, the
amplitudes of the MCAE depend strongly on $J$ but with a completely different
trend in each compound. For MnF2 and FeF2 the experimental value can be
reproduced by adjusting the values of $U$ and $J$. In the case of CoF2 and
NiF2 no experimental values are available. For CoF2 we also performed
calculations with and without Co semi-cores states (Fig.2.d) and find a strong
difference in the magnitude of the MCAE for the two cases. For FeF2 we also
performed calculations within the GGA functional (black pentagons in Fig.2.a)
and obtained a completely different $J$ dependence than those calculated with
the LDA functional. These comparisons illustrate the difficulty of extracting
a general rule about the $J$ dependence of the MCAE.
Our results reveal a problem with the predictability of the LSDA$+U$ method
for non-collinear magnetic materials: A strong dependence of the MCAE and spin
canting angles on the values of $U$ and particularly $J$ that are used in the
calculation. Since properties such as magnetostriction, piezomagnetic
response, magnetoelectric response and exchange bias coupling are directly
related to MCAEs and spin canting, it is of primary importance to reproduce
these quantities accurately. At the moment, the most reliable, although not
entirely satisfactory, option appears to be a fine tuning of the $U$ and $J$
parameters by adjustment to reproduce experimentally measured anisotropies and
canting angles; there is some evidence to suggest that properties such as
magnetoelectric responses are then in turn well reproduceddelaney2010 . Future
studies might explore methodologies for self-consistent calculation of the $J$
parameter, or the predictions of new descriptions of the exchange and
correlation such as the hybrid functionals heyd2004 . On the flip side, it is
clear that non-collinear magnetic systems provide a challenging case for
testing the correctness of new exchange correlation functionals within the
density functional formalism.
Acknowledgments: This work was supported by the Department of Energy SciDAC
DE-FC02-06ER25794. We made use of computing facilities of TeraGrid at the
National Cen-ter for Supercomputer Applications and of the California
Nanosystems Institute with facilities provided by NSF grant No. CHE-0321368
and Hewlett-Packard. EB also acknowledges FRS-FNRS Belgium and the ULg SEGI
supercomputer facilities.
## References
* (1) A. I. Liechtenstein, V. I. Anisimov, and J. Zaanen, Phys. Rev. B 52, R5467 (Aug 1995)
* (2) K.-W. Lee and W. E. Pickett, Phys. Rev. B 70, 165109 (Oct 2004)
* (3) S. L. Dudarev, G. A. Botton, S. Y. Savrasov, C. J. Humphreys, and A. P. Sutton, Phys. Rev. B 57, 1505 (Jan 1998)
* (4) P. Baettig, C. Ederer, and N. A. Spaldin, Phys. Rev. B 72, 214105 (2005)
* (5) E. R. Ylvisaker, W. E. Pickett, and K. Koepernik, Phys. Rev. B 79, 035103 (Jan 2009)
* (6) S. Y. Savrasov, A. Toropova, Kat, K. M. I., L. A. I., A. V., and G. Kotliar, Z. Kristallogr. 220, 473 (2005)
* (7) F. Bultmark, F. Cricchio, O. Grånäs, and L. Nordström, Phys. Rev. B 80, 035121 (2009)
* (8) G. Kresse and J. Furthmüller, Phys. Rev. B 54, 11169 (Oct 1996)
* (9) G. Kresse and D. Joubert, Phys. Rev. B 59, 1758 (Jan 1999)
* (10) We note that LSDA$+U$ double-couting term taking into accound the magnetization density as proposed by Bultmark et al.bultmark2009 is mandatory within non-collinear magnetism calculations. This is not necessarily done in the present implementation of other codes.
* (11) V. A. Streltsov, E. L. Belokoneva, V. G. Tsirelson, and N. K. Hansen, Acta Cryst. B 49, 147 (Apr 1993)
* (12) I. Abrahams and K. S. Easson, Acta Cryst. C 49, 925 (1993)
* (13) M. T. Hutchings, M. F. Thorpe, R. J. Birgeneau, P. A. Fleury, and H. J. Guggenheim, Phys. Rev. B 2, 1362 (Sep 1970)
* (14) M. J. M. de Almeida, M. M. R. Costa, and J. A. Paixão, Acta Cryst. B 45, 549 (1989)
* (15) T. Oguchi, Phys. Rev. 111, 1063 (Aug 1958)
* (16) N. J. O’Toole and V. A. Streltsov, Acta Cryst. B 57, 128 (2001)
* (17) T. B. S. Jensen, N. B. Christensen, M. Kenzelmann, H. M. Rønnow, C. Niedermayer, N. H. Andersen, K. Lefmann, J. Schefer, M. v. Zimmermann, J. Li, J. L. Zarestky, and D. Vaknin, Phys. Rev. B 79, 092412 (2009)
* (18) K. Yamauchi and S. Picozzi, Phys. Rev. B 81, 024110 (2010)
* (19) C. Ederer and N. A. Spaldin, Phys. Rev. B 74, 020401 (Jul 2006)
* (20) M. E. Lines, Phys. Rev. 156, 543 (Apr 1967)
* (21) U. Gäfvert, L. Lundgren, P. Nordblad, B. Westerstrandh, and . Beckman, Sol. State Comm. 23, 9 (1977)
* (22) Zimmermann, A. S., Van Aken, B. B., Schmid, H., Rivera, J.-P., Li, J., Vaknin, D., and Fiebig, M., Eur. Phys. J. B 71, 355 (2009)
* (23) G. Liang, K. Park, J. Li, R. E. Benson, D. Vaknin, J. T. Markert, and M. C. Croft, Phys. Rev. B 77, 064414 (2008)
* (24) C. Rudowicz, J. Phys. Chem. Solids 38, 1243 (1977)
* (25) R. C. Ohlmann and M. Tinkham, Phys. Rev. 123, 425 (1961)
* (26) K. Delaney, E. Bousquet, and N. A. Spaldin, arXiv:0912.1335v2(2010)
* (27) J. Heyd and G. E. Scuseria, The Journal of Chemical Physics 120, 7274 (2004)
|
arxiv-papers
| 2010-11-03T16:59:38 |
2024-09-04T02:49:14.477323
|
{
"license": "Public Domain",
"authors": "Eric Bousquet and Nicola Spaldin",
"submitter": "Eric Bousquet",
"url": "https://arxiv.org/abs/1011.0939"
}
|
1011.1019
|
# Warm Spitzer Photometry of the Transiting Exoplanets CoRoT-1 and CoRoT-2 at
Secondary Eclipse
Drake Deming11affiliation: Planetary Systems Laboratory, NASA’s Goddard Space
Flight Center, Greenbelt MD 20771 , Heather Knutson22affiliation: Department
of Astronomy, University of California at Berkeley, Berkeley CA 94720
33affiliation: Miller Research Fellow , Eric Agol44affiliation: Department of
Atronomy, University of Washington, Box 351580, Seattle, WA 98195 , Jean-
Michel Desert55affiliation: Harvard-Smithsonian Center for Astrophysics,
Cambridge, MA 02138 , Adam Burrows66affiliation: Department of Astrophysical
Sciences, Princeton University, Princeton, NJ 05844 , Jonathan J.
Fortney77affiliation: Department of Astronomy and Astrophysics, University of
California at Santa Cruz,
Santa Cruz, CA 95064 , David Charbonneau55affiliation: Harvard-Smithsonian
Center for Astrophysics, Cambridge, MA 02138 , Nicolas B. Cowan44affiliation:
Department of Atronomy, University of Washington, Box 351580, Seattle, WA
98195 88affiliation: Currently: CIERA Fellow, Department of Physics &
Astronomy, Northwestern University, 2131 Tech Drive, Evanston, IL 60208 ,
Gregory Laughlin77affiliation: Department of Astronomy and Astrophysics,
University of California at Santa Cruz,
Santa Cruz, CA 95064 , Jonathan Langton99affiliation: Department of Physics,
Principia College, Elsah, IL 62028 , Adam P. Showman1010affiliation: Lunar and
Planetary Laboratory, University of Arizona, Tucson, AZ 85721 , and Nikole K.
Lewis1010affiliation: Lunar and Planetary Laboratory, University of Arizona,
Tucson, AZ 85721
###### Abstract
We measure secondary eclipses of the hot giant exoplanets CoRoT-1 at 3.6 and
4.5 $\mu$m, and CoRoT-2 at 3.6 $\mu$m, both using Warm Spitzer. We find that
the Warm Spitzer mission is working very well for exoplanet science. For
consistency of our analysis we also re-analyze archival cryogenic Spitzer data
for secondary eclipses of CoRoT-2 at 4.5 and 8 $\mu$m. We compare the total
data for both planets, including optical eclipse measurements by the CoRoT
mission, and ground-based eclipse measurements at 2 $\mu$m, to existing
models. Both planets exhibit stronger eclipses at 4.5 than at 3.6 $\mu$m,
which is often indicative of an atmospheric temperature inversion. The
spectrum of CoRoT-1 is best reproduced by a 2460K blackbody, due either to a
high altitude layer that strongly absorbs stellar irradiance, or an isothermal
region in the planetary atmosphere. The spectrum of CoRoT-2 is unusual because
the 8 $\mu$m contrast is anomalously low. Non-inverted atmospheres could
potentially produce the CoRoT-2 spectrum if the planet exhibits line emission
from CO at 4.5 $\mu$m, caused by tidal-induced mass loss. However, the
viability of that hypothesis is questionable because the emitting region
cannot be more than about 30% larger than the planet’s transit radius, based
on the ingress and egress times at eclipse. An alternative possibility to
account for the spectrum of CoRoT-2 is an additional opacity source that acts
strongly at wavelengths less than 5 $\mu$m, heating the upper atmosphere while
allowing the deeper atmosphere seen at 8 $\mu$m to remain cooler. We obtain a
similar result as Gillon et al. (2010) for the phase of the secondary eclipse
of CoRoT-2, implying an eccentric orbit with $e\,cos(\omega)=-0.0030\pm
0.0004$.
stars: planetary systems - eclipses - techniques: photometric
††slugcomment: Accepted for the Astrophysical Journal
## 1 Introduction
An especially interesting class of giant extrasolar planets, the ‘very hot
Jupiters’ (hereafter, VHJs), orbit extremely close to solar-type stars, within
0.03 AU in several cases. The temperature structure in the atmosphere of such
a planet is likely to be significantly perturbed by the strong stellar
irradiation. Absorption of stellar radiation is one possible energy source
that may drive atmospheric temperature inversions. Temperature inversions with
height appear to be common in hot Jupiter atmospheres; they occur over a wide
range of stellar irradiation level (Knutson et al., 2008; Machalek et al.,
2008; Christiansen et al., 2010; Todorov et al., 2010), but are not well
understood. The emergent spectra of VHJs are an important key to this problem.
The emergent spectrum of a transiting planet can often be measured by
observing the decrease in total light as the planet passes behind the star
during secondary eclipse (Charbonneau et al., 2005; Deming et al., 2005).
Eclipses of the VHJs offer the opportunity to determine their emergent spectra
at wavelengths as short as visible light (Alonso et al., 2009a; Snellen et
al., 2009). Fortunately, VHJs have high transit probabilities, and are
represented by transiting planets such as WASP-12 (Hebb et al., 2009), WASP-19
(Hebb et al., 2010), CoRoT-1 (Barge et al., 2008), and CoRoT-2 (Alonso et al.,
2008).
The CoRoT planets are particularly important in the study of VHJ temperature
structure. Their emergent flux has been measured at secondary eclipse using
infrared (IR) wavelengths, and also in the visible by the CoRoT mission. The
currently available secondary eclipse measurements for the CoRoT planets are
summarized in Table 1, including the results from this paper. While eclipses
of CoRoT-2 have been measured at 4.5 $\mu$m and 8.0 $\mu$m using the Spitzer
Space Telescope (Gillon et al., 2010), no Spitzer measurements have been
reported for CoRoT-1. In this paper, we report measurements of CoRoT-1 using
Warm Spitzer (Deming et al., 2007) at $3.6$\- and 4.5 $\mu$m, and we complete
Spitzer’s measurement of CoRoT-2 by adding the 3.6 $\mu$m observation. These
additional data allow us to address the existence and nature of the inversion
phenomenon in these planets. Moreover, because we measure near the peak of the
VHJ’s spectral energy distribution, we can speak to whether the visible
wavelength eclipse measurements are sensing primarily thermal radiation, as
opposed to reflected light.
Our results, together with those of Hebrard et al. (2010), are among the first
to be reported for transiting exoplanets using Warm Spitzer. The Warm phase of
Spitzer refers to operation of the observatory after the loss of cryogen, with
only the 3.6- and 4.5 $\mu$m channels of the IRAC instrument remaining
operational. The InSb detectors used at these wavelengths are now functioning
at a temperature of approximately 29 Kelvins, cooled by passive radiation.
This very different operating temperature regime may have significant
implications for the observatory performance as regards high precision
photometry. Therefore, we comment on the performance of the observatory,
within the limits allowed by the fact that we have observed relatively faint
stars.
In Sec. 2 we describe the observations, aperture photometry, and linear
regression procedures to derive eclipse depths and central phases. Sec. 3
discusses the implications of our results for the orbital and atmospheric
properties of these giant CoRoT planets, and in the Appendix we discuss some
details concerning the performance of the Warm Spitzer observatory for this
type of exoplanet science.
## 2 Observations and Photometry
### 2.1 CoRoT-1
We observed CoRoT-1 at 4.5 $\mu$m on 23 November 2009, starting at 11:06 UT
(orbital phase 0.380), for a duration of 465.7 minutes, yielding 888 30-second
exposures. Among transiting systems, CoRoT-1 is relatively faint, having
V=13.6 and K=12.1, and a short orbital period of P=1.509 days (Barge et al.,
2008). We observed this system at 3.6 $\mu$m on 26 November 2009, starting at
11:30 UT (orbital phase 0.379) for the same duration, and the same exposure
time per frame. The CoRoT-1 observations at both wavelengths used full frame
($256\times 256$-pixel) mode. Following the eclipse observations, we acquired
9 minutes of additional data (17 exposures) by offsetting the telescope to
view blank sky using the same detector pixels as for CoRoT-1.
The detectors in the Warm mission are significantly affected by an artifact
called column pull-down111see
http://ssc.spitzer.caltech.edu/irac/warmfeatures/, wherein the presence of a
bright star reduces the signal level for an entire detector column. This, as
well as other artifacts, are significantly mitigated in the cBCD files
produced by Spitzer’s pipeline processing. However, neither CoRoT-1 nor
CoRoT-2 lie on columns affected by pull-down, and in any case we would want to
remove any such artifacts as an integral part of our photometry, so that we
could fully judge their impact. We therefore extracted photometry using the
Basic Calibrated Data (BCD) files produced by version S18.12.0 of the Spitzer
pipeline, not the cBCD files. We calculated orbital phase using the UTC-based
HJD values for the start of each observation from the FITS headers of the BCD
files, and we correct the values to the time of mid-exposure.
As a first step, we stack the blank sky images and median-filter each pixel in
time to construct an average blank sky frame. We subtract this sky frame from
each CoRoT-1 image immediately after reading each BCD file. In principle, this
subtraction of a sky-nod will remove the background radiation, but we
nevertheless fit and remove residual background anyway, as described below.
Although the true sky background should be constant to an excellent
approximation, we find that the background does vary significantly from frame
to frame. This is one significant difference from the cryogenic mission, as we
discuss in the Appendix.
We locate and correct energetic particle events by comparing the time history
of each pixel to a 5-point median filter of that pixel intensity vs. time, and
we replace $>4\sigma$ outliers with the median value. The fraction of pixels
we correct varies between 0.45% and 0.55%, depending on which planet and
wavelength are analyzed. We perform aperture photometry on the images, after
first applying corrections for variations in pixel solid angle, and for
slightly different flat-field response for point sources vs. extended
sources222see Secs. 5.3, and 5.6.2 of the IRAC Data Handbook, V3.0. Prior to
subtracting the residual background and performing aperture photometry, we
convert the pixel intensities to electrons, using the calibration information
given in the FITS headers. This facilitates the evaluation of the photometric
errors.
Our photometry code locates the centroid of the stellar point spread function
(PSF) by fitting a symmetric 2-D Gaussian to the PSF-core (Agol et al., 2010).
We calculate the flux within a centered circular aperture, of variable radius,
using radii of 2.0 to 4.5 pixels, in 0.5-pixel steps. To determine the
residual background intensity, we fit a Gaussian to a histogram of pixel
intensities for each frame. The center histogram bin, defined to fractional
precision by the Gaussian fit, is adopted as the residual background
intensity. Subtracting the resultant background from the raw aperture
photometry, yields 6 photometric time series for the star corresponding to
aperture radii from 2.0 to 4.5 pixels. We tabulate the magnitude of the point-
to-point scatter in our photometry, and errors in our final results, as a
function of aperture radius. We find that both the scatter and final parameter
errors depend only weakly on aperture radius, with best values near 2.5 to 3.0
pixels. We adopt a radius of 3.0 pixels for all of our photometry.
The aperture photometry for CoRoT-1 at 3.6 $\mu$m, uncorrected for instrument
systematic effects, is shown in the top panel of Figure 1. The corresponding
time series at 4.5 $\mu$m is shown in the top panel of Figure 2.
### 2.2 CoRoT-2
CoRoT-2 (V=12.6, K=10.3) observations at 3.6 $\mu$m began on 24 November 2009
at 18:22 UT (orbital phase 0.4), for a duration of 467.6 minutes. CoRoT-2
being brighter than CoRoT-1, these observations used subarray mode. We
collected 215 data cubes, each comprising $64$ 2-second exposures of $32\times
32$ pixels, followed by 3 data cubes of blank sky.
We perform photometry on the CoRoT-2 data cubes in a similar manner to the
full-frame data for CoRoT-1. We inspect the aperture photometry for the 64
frames within each data cube, and zero-weight outliers exceeding the average
by more than $4\sigma$. The first frame in each data cube is consistently
found to be an outlier, and is always ignored. We analyze the remaining
63-frame data cubes so as to produce two distinct versions of the photometry,
and we perform the entire eclipse-fitting and error analysis for each version.
In the first (default) version, we average the background-subtracted aperture
photometry for all 63 frames in each data cube, to produce a single
photometric point. For the second version, we use each of the 63 frames as a
separate photometric point. Using these individual frames potentially exploits
the short-term pointing jitter to better define the intra-pixel effect.
However, in practice the frame-to-frame fluctuations within a data cube are
dominated by photon noise for these relatively faint stars. The eclipse
results and errors from these two versions of the photometry are close to
being identical (difference much less than $1\sigma$). Note that the default
method is essentially just a binning of the data. We prefer the default
version because the eclipse plot (Figure 3) is visually clearer.
We also explored a third version of the photometry, wherein we average the
actual data frames in each data cube, omitting the first frame and using a
median filter to reject outlying pixels. We then perform aperture photometry
on the averaged frame. This method gives essentially the same result as our
default method: the eclipse amplitude (see below) differed by 0.4$\sigma$ and
the phase differed by 1.14$\sigma$.
### 2.3 Eclipse Amplitudes
CoRoT-1 and -2 have well defined transit parameters (planetary and stellar
radii, orbit inclination, etc.). We adopt these parameters from Barge et al.
(2008) and Alonso et al. (2008), and we calculate eclipse curves numerically,
following Todorov et al. (2010). We maintain the known durations of ingress
and egress, but we vary the central phase and eclipse depth when fitting to
the photometry.
Both the 3.6- and 4.5 $\mu$m channels show the well known intra-pixel
sensitivity variation (Morales-Calderon et al., 2006). We fit for the eclipse
depth and the coefficients of the intra-pixel correction using linear
regression. The details of the fitting procedure vary with wavelength, but at
all wavelengths we search for the best central phase by repeating the linear
regressions at many phase values in a dense grid (spacing 0.0002 in phase),
and we adopt the central phase yielding the best $\chi^{2}$. We always perform
this grid-search in phase when fitting for eclipse amplitude, for both planets
at all wavelengths and also for our Monte-Carlo trials to define errors (see
below).
We apply the linear regressions using an iterative procedure. We first
decorrelate the photometry to remove the intra-pixel effect, while ignoring
the eclipse, and then we fit for the eclipse depth using a second regression
on the decorrelated data. After removing the fitted eclipse depth from the
original photometry, we then re-fit and decorrelate the intra-pixel variation,
then re-fit the eclipse. This procedure converges in two cycles. In principle,
iteration is unnecessary because the regressions are linear and an identical
result can be achieved by solving simultaneously for both the intra-pixel
coefficients and the eclipse depth. (We verified this by actually doing the
simultaneous fit for a simple case.) Nevertheless, we use the iterative
procedure because in actual practice it is more flexible and it affords the
opportunity to use variants of the fit that would be awkward to implement in a
simultaneous solution. This should become apparent from the description below.
At 3.6 $\mu$m the intra-pixel signature in the photometry ($\sim 2\%$ peak-to-
peak) is larger than the eclipse (see top panel of Figure 1). Our first step
is to solve for a provisional intra-pixel decorrelation. The provisional
decorrelation function is assumed to be linear in both $\delta X$ and $\delta
Y$, which are defined as the change in X- and Y-pixel position of the image
centroid after removing a trend in $X$ and $Y$ with time. The approximately
1-hour quasi-periodic jitter in position has peak-to-peak amplitude in $\delta
X$ and $\delta Y$ of about 0.03 and 0.08-pixels, respectively. The trends
(slow drifts) are smaller, about 0.005-pixels in X over the entire dataset,
and 0.06-pixels in Y. The provisional intra-pixel decorrelation function is
linear in both $\delta X$ and $\delta Y$, and includes a term linear in time
that accounts for both the slow drift in position as well as possible change
in detector sensitivity. We solve for the coefficients using linear regression
(matrix inversion), and correct the original photometry using this
decorrelation function.
Following the provisional intra-pixel decorrelation, we solve for the eclipse
depth, again using linear regression. This regression formally allows a linear
baseline in time, but that term is effectively accounted for by the intra-
pixel decorrelation of the previous step. We remove the fitted eclipse from
the original photometry, and begin the second cycle of the iteration. This
implements a more sophisticated version of the intra-pixel decorrelation,
expressing the decorrelation function as linear in both time and the radial
distance of the image from pixel center (called pixel phase). Because there is
slow drift of the image toward pixel center by about $0.06$ pixels over the
duration of the observations, intrinsic spatial variation in the intra-pixel
sensitivity (i.e., a change of spatial slope) may be manifest as a change in
the decorrelation coefficient of pixel phase. In this particular case (CoRoT-1
at 3.6 $\mu$m), visual inspection of the data indeed suggested a change in the
slope of the intrapixel effect. To allow for this change in slope, we divide
the decorrelation into two halves, the first half before mid-eclipse and the
second half after mid-eclipse. In effect, this is a minimalist implementation
of using a quadratic term in the intrapixel fit. Although it is
unconventional, we judge it to be the best approach to this particular case.
The coefficients of both halves are found via linear regression on the
eclipse-removed data. The separate decorrelation functions for the first and
second halves of the data can be discerned on the top panel of Figure 1. Note
that they are almost continuous at the break near phase 0.5. None of the
conclusions of this paper would be different if we restricted the
decorrelation to more conventional methodology, but the quality of the 3.6
$\mu$m eclipse fit for CoRoT-1 would be degraded.
After this decorrelation, we again remove the intra-pixel effect from the
original photometry, and re-solve for the final eclipse depth and a possible
linear baseline via regression. The eclipse fit uses all of the data, not
breaking it into halves. Decorrelated CoRoT-1 data and the best-fit eclipse
are shown in the middle panel of Figure 1, and are binned (to 100 bins) in the
bottom panel of Figure 1.
We use a nearly identical procedure to fit the 3.6 $\mu$m eclipse of CoRoT-2,
shown in Figure 3, except that we do not break the decorrelation at mid-
eclipse. The first $\sim 30$ minutes of these data (not illustrated in Figure
3) exhibit a transient decrease in flux, similar to the ramp effect seen at
longer wavelength, but decreasing instead of increasing, and not correlated
with the image position on the detector. Transient effects at this wavelength
are not well understood, so we simply omit the 19 data cubes prior to orbital
phase 0.41.
Another difference for CoRoT-2 is that a correction is needed for diffracted
light from an M-dwarf lying 4 arc-sec distant (Gillon et al., 2010). Since we
also re-analyze archival data at 4.5 and 8 $\mu$m for CoRoT-2 (see below), we
need to estimate the diffracted light contributed by the M-dwarf in the
CoRoT-2 aperture at 3.6, 4.5, and 8 $\mu$m. We calculated the flux ratio
(M-dwarf to CoRoT-2) in the IRAC bands, using the the flux estimation tool
(STAR-PET) on the Spitzer website, and the 2MASS K-magnitudes and J-K colors
of the two stars. Knowing their relative brightness, we also need to know the
fraction of the M-dwarf flux that is diffracted into the photometry aperture
for CoRoT-2. We estimated this by placing the aperture at a symmetric location
on the other side of CoRoT-2, where the diffracted light is contributed almost
exclusively by CoRoT-2 itself. Using that diffracted fraction together with
the flux ratio of M-dwarf to CoRoT-2, we infer that the diffracted light from
the M-dwarf contributes 5.9%, 5.0%, and 8.3% to CoRoT-2 at 3.6, 4.5 and 8.0
$\mu$m, respectively. The eclipse photometry and derived parameters for
CoRoT-2 in our Figures and Tables have all been corrected for this diffracted
light. Gillon et al. (2010) inferred 16.4% and 14.3% at 4.5 and 8 $\mu$m,
respectively, but he used aperture radii of 4.0 and 3.5 pixels, respectively,
vs. 3.0 pixels in our case.
As a check, we repeated our diffracted light correction using apertures having
the same size as Gillon et al. (2010). Because the diffracted light is not
uniform, the values do not simply scale as the area of the aperture. For the
same apertures as Gillon et al. (2010), we obtain corrections of 14.2% and
12.0% at 4.5 and 8.0 $\mu$m, respectively, in reasonably good agreement with
the independent determination of Gillon et al. (2010). Uncertainty in the
diffracted light correction is not included in our eclipse amplitude error
estimates. Given our good agreement with the diffracted light corrections of
Gillon et al. (2010), and given that we use smaller photometric apertures than
Gillon et al. (2010), we conclude that uncertainty in the diffracted light
correction does not contribute significantly to the errors on our measured
eclipse depths.
The best-fit eclipse depths and errors are listed in Table 1, and the central
phases and errors are listed in Table 2.
### 2.4 Error Estimation
The ideal method to calculate errors would be to repeat all of the
observations and analysis, and compare the results from analyzing many
independent sets of observations. This is obviously impractical, so we mimic
some key aspects of that ideal procedure. We generate fake photometric
datasets having the same properties as the real photometry, and we repeat the
entire iterative fitting process - including intra-pixel corrections and ramp
fitting - on each fake dataset. We calculate the standard deviation of the
collection of eclipse depths and central phases resulting from the repetitions
of the analysis on the fake data.
To generate each fake dataset, we subtract the best-fit eclipse curve (plus
baseline and intra-pixel decorrelation function) from the original photometry
to produce a set of photometric residuals. We likewise produce a set of image
position residuals by subtracting a multi-point running average of the X and
Y-pixel positions from each individual (X,Y) position measurement. We permute
all of the residuals and add them back to the best-fit function (photometry)
or running average coordinate (position) to make an individual fake dataset.
We permute the residuals using two methods, to make two distinct collections
of fake data. The first permutation method scrambles the residuals randomly,
which is equivalent to the conventional bootstrap Monte Carlo technique (Press
et al., 1992). We generate $10^{4}$ bootstrap datasets (trials) using this
method, and calculate the standard deviation of eclipse depth and central
phase from the distributions of these parameters over the $10^{4}$ trials.
These distributions are close to Gaussian.
A second method to permute the residuals preserves their relative order but
shifts their initial phase; this is sometimes called the ‘prayer-bead’ method
(Gillon et al., 2009b). In this case, the number of trials equals the number
of original photometry points. This is 888 for CoRoT-1, and 13,545 for version
2 of the CoRoT-2 subarray photometry. These are adequate to define the
distributions of eclipse depth and phase. The prayer-bead method is more
sensitive to the presence of red noise in the data. Nevertheless we find that
the distribution of eclipse depth remains consistent with a Gaussian, but for
CoRoT-1 the distribution of eclipse phase shows about $7\%$ of the central
phases lie below the $3\sigma$ point in the distribution. We attribute this to
the presence of some red noise before mid-eclipse, visible in the bottom panel
of Figure 1.
For CoRoT-2, the distributions of eclipse depth and phase were close to
Gaussian, so errors from the prayer-bead method were quite close to the values
from the bootstrap method. This indicates relatively little red noise in the
CoRoT-2 data (after we omitted the first 19 data points, as noted above). For
both CoRoT-1 and -2, we adopted the greater of the bootstrap and prayer-bead
errors for each parameter. CoRoT-1 errors are uniformly larger than for
CoRoT-2 because the star is fainter and the red noise is slightly greater.
Tables 1 & 2 list the errors on eclipse depth and central phase for all three
eclipses, plus our results from re-analysis of CoRoT-2 at 4.5 and 8 $\mu$m
(see below).
### 2.5 CoRoT-2 at 4.5 and 8 $\mu$m
We check our methodology by analyzing archival Spitzer data for CoRoT-2 at
4.5- and 8 $\mu$m, for comparison to the results of Gillon et al. (2010). Our
analysis at 4.5 $\mu$m proceeds as described above for CoRoT-1. At 8 $\mu$m
our eclipse fitting procedure uses a ‘ramp’ baseline (Deming et al., 2006;
Knutson et al., 2009) that is fit simultaneously with the eclipse depth by
linear regression. The ramp is comprised of a term linear in time, a term
linear in the logarithm of time, with a zero-point on the time axis as
described by Todorov et al. (2010). We also find that the photometry exhibits
a rather rapid decrease in flux during the first 100 data points.
Investigating this, we find an approximately 0.1-pixel change in the image
Y-position during those first 100 points. This transient positional drift is
not in sync with the well known telescope pointing oscillation. Although the
pointing oscillation has not (to our knowledge) been shown to affect 8 $\mu$m
Spitzer photometry, the 0.1-pixel transient drift apparently does. We
therefore include a Y-position term in the linear regression fit for the
eclipse depth. Without this term, the eclipse depth would be $0.42\%$, versus
our result of $0.446\%$ (Table 1).
We also perform trial fits using the double-exponential ramp of Agol et al.
(2010). These fits, like the log ramp discussed above, omit the first 100
points and include a Y-position term. The ramp observed in the 8 $\mu$m data
(illustrated by Gillon et al., 2010) is very shallow, and the scatter is
relatively large compared to the ramp-related flux change. For this reason, we
use a single exponential ramp, not a double exponential ramp. We experimented
with double-exponential fits, but our Levenberg-Marquardt fitting procedure
produced degeneracies when attempting to fit two exponentials to such a
shallow ramp. We believe that only one exponential is warranted in this case.
Moreoever, the best-fit exponential ramp is close to a straight line, since
the ramp curvature is minimal. As will become apparent in Sec. 3.2, the 8
$\mu$m eclipse depth of CoRoT-2 is crucial to the interpretation of our
results, so we will return to the implications of fitting the exponential ramp
during that discussion.
Our results for CoRoT-2 at 4.5 and 8 $\mu$m are included in Tables 1 and 2.
The eclipse depth using the exponential ramp at 8 $\mu$m is included in Table
1, but the phase results for that ramp are the same as the log ramp, and are
not listed separately in Table 2. Overall, we find excellent agreement with
Gillon et al. (2010).
## 3 Results and Discussion
### 3.1 Orbital Phase
For CoRoT-1, we compute the weighted average of the central eclipse phase
using both 3.6- and 4.5 $\mu$m eclipses, adopting weights equal to the inverse
of the variance of each measurement. This yields a central phase of $0.4994\pm
0.0013$, and $|e\,cos(\omega)|<0.006$ to $3\sigma$. Our limit indicates that
the orbit is close to circular, but a small non-zero eccentricity (such as we
infer for CoRoT-2, see below) is not excluded.
For CoRoT-2, Gillon et al. (2010) found $e\,cos(\omega)=-0.00291\pm 0.00063$.
Our result for the 3.6 $\mu$m eclipse (central phase at $0.4994\pm 0.0007$) is
displaced in the same direction as Gillon et al. (2010) infer, but with
insufficient precision to confirm or reject the Gillon et al. (2010) claim.
Combining our 3.6 $\mu$m result with the eclipses analyzed by Gillon et al.
(2010) could increase the significance of the total result. For maximum
consistency, we re-analyzed the 4.5- and 8 $\mu$m eclipse data, as described
above. We verified that our adopted transit ephemeris (see Table 2) should not
be a significant source of error when propagated to the eclipse times.
Weighting each eclipse phase (3.6, 4.5 and 8, see Table 2) by the inverse of
its variance yields an average central phase of $0.49809\pm 0.00028$.
Including the 28 seconds for light to cross the planetary orbit, we expect to
find the eclipse at phase $0.500019$ if the orbit is circular. Hence, we
derive $e\,cos(\omega)=-0.0030\pm 0.0004$. The excellent agreement with Gillon
et al. (2010) is in part because we are analyzing much of the same data.
However, the result is heavily weighted by the single eclipse at 4.5 $\mu$m,
which is a reason to be cautious concerning a claim of non-zero eccentricity.
Nevertheless, at face value we are able to reproduce the result of Gillon et
al. (2010) using an independent analysis, and improve the precision slightly.
Gillon et al. (2010) point out that a non-zero eccentricity does not require
an additional planet in the system, since incomplete two-body tidal
circularization is a plausible alternative for this system.
### 3.2 Atmospheric Temperature Structure
Our results for both planets are summarized in Figure 4, which shows all
available eclipse data in comparison to various models. The caption of Figure
4 gives reduced $\chi^{2}$ values for the comparison between each model and
the eclipse data. Since Figure 4 is a comparison of the data to model
predictions, not a fit involving adjustable parameters, we take the degrees of
freedom to equal the number of data points when calculating the reduced
$\chi^{2}$.
The model comparison for CoRoT-1 (top panels of Figure 4) suggest an inverted
atmospheric temperature structure. The best overall account of the data is
actually produced using a $T=2460K$ blackbody spectrum (Rogers et al., 2009,
green line, see reduced $\chi^{2}$ values in Figure 4 caption). However, this
likely indicates the presence of a high altitude absorbing layer, and such
layers are implicated in driving the inversion phenomenon (Burrows et al.,
2007; Knutson et al., 2008). The nature of the absorber is the subject of
current debate (Fortney et al., 2008; Spiegel et al., 2009). The conventional
model (black line, Burrows et al., 2008) shows significant absorption due to
the CO bandhead that occurs near 4.7 $\mu$m, and the Spitzer data show no sign
of being affected by this feature. An inverted model using TiO absorption
(blue line) shows much better agreement with the data than the non-inverted
model, but does not account particularly well for the ground-based (2 $\mu$m)
measurements. An atmosphere with a nearly isothermal region over extended
heights will produce a blackbody-like spectrum, and can be regarded as a
special case of an inverted temperature structure. The inverted and blackbody
model for CoRoT-1 both give good agreement with the Spitzer data, as well as
the CoRoT optical eclipse measurements (Snellen et al., 2009; Alonso et al.,
2009b). This indicates that the optical emission is predominately thermal in
origin. The models that account for our Spitzer data, when compared to the
optical eclipses (Figure 4), leave little room for a reflected light
component. Based on the models of Seager, Whitney, & Sasselov (2000), a
geometric albedo near unity would produce a reflected light eclipse depth of
approximately 520 ppm, whereas the difference between the CoRoT-1 observations
(Snellen et al., 2009) and the inverted model (blue curve on Figure 4) is 84
and 21 ppm at 0.6 and 0.71 $\mu$m, respectively. Also, Cowan & Agol (2010)
inferred a Bond albedo of $<10$% for CoRoT-1. Our results therefore support
the conclusion of Snellen et al. (2009) and Cowan & Agol (2010) that CoRoT-1
is a dark planet.
CoRoT-2 (bottom panels of Figure 4) is more complex than CoRoT-1. A
conventional model (black line, Burrows et al., 2008) produces excellent
agreement with all of the data except for the 4.5 $\mu$m point, where the
disagreement is substantial. Since the 4.5- to 3.6 $\mu$m contrast ratio is
even greater than for CoRoT-1, a temperature inversion is suggested. But
inverted models do not reproduce the 8 $\mu$m contrast and, based on the
reduced $\chi^{2}$ values (Figure 4 caption), no model gives a reasonable
account of the total data. Both the 4.5 and 8 $\mu$m observed values are in
good agreement between our analysis and Gillon et al. (2010), so the problem
does not seem to lie with the observations. We first mention some caveats, and
then we suggest two hypotheses to account for the contrast values of this
unusual planet.
One caveat that applies to CoRoT-2 is the fact that the star is active (Alonso
et al., 2008). However, because the planet passes behind the star during
eclipse, there is no time-variable blocking of active regions on the stellar
disk. The primary consequence of stellar activity is the photometric variation
of the star itself. This variation can manifest itself in two ways. First,
stellar variations can appear directly in the eclipse curve. The dominant
stellar variation will be due to rotational modulation of active regions, with
a 4.5-day period (Lanza et al., 2009). This time scale is more than an order
of magnitude longer than the 2.2-hour eclipse duration. Although rotation of
active regions can still affect eclipse data (e.g., by perturbing the
photometric baseline) we do not discern any indications of it, so we interpret
our data at face value. The second way in which stellar variations can affect
eclipse depth is through the normalization. When the star is fainter, the
disappearance of the planet during eclipse translates to a larger fraction of
the stellar flux. This effect can alter eclipse depths on long time scales.
However, the 4.5 and 8 $\mu$m observations made by Gillon et al. (2010) were
simultaneous, so long-term stellar variability cannot be a factor in the
puzzling spectrum of CoRoT-2.
A final caveat concerns the ramp effect for CoRoT-2 at 8 $\mu$m. We find that
fitting the exponential model of Agol et al. (2010) increases the eclipse
depth to 0.51% (Table 1). However, this does not alter the situation
concerning the interpretation of the CoRoT-2 results, so we now discuss two
hypotheses to account for the totality of the CoRoT-2 data as summarized in
Tables 1 & 2.
### 3.3 Possible Mass Loss for CoRoT-2
Our first hypothesis for CoRoT-2 is that the planetary atmosphere is well
described by a conventional (non-inverted) model, but the 4.5 $\mu$m eclipse
appears anomalously deep because it contains carbon monoxide emission lines
due to mass loss. We find that a conventional model lacking CO absorption (see
Figure 4) does not increase the contrast sufficiently in the 4.5 $\mu$m band
to account for the data - the reduced $\chi^{2}$ is 13.5 (Figure 4). Actual
emission from mass loss would be required. Mass loss for close-in giant
exoplanets can occur via tidal stripping (Li et al., 2010), and also via
energy deposition from stellar UV flux. The latter process is particularly
important for planets orbiting young, UV-bright stars (Baraffe et al., 2004;
Hubbard et al., 2007). CoRoT-2 orbits very close-in, where the tidal force is
strong (0.026 AU, Barge et al., 2008). Moreover, the star is young and active
(Bouchy et al., 2008), possibly as young as 30 Ma (Guillot & Havel, 2010).
Hence both mass loss mechanisms are potentially important for this planet.
Li et al. (2010) have predicted significant CO emission in the $\Delta{V}=2$
overtone bands near 2.29 $\mu$m, due to tidally-stripped mass loss from
WASP-12. This mass should also emit in the CO $\Delta{V}=1$ bands, which are
intrinsically stronger than the overtone bands, and arise from upper levels
that are easier to excite. Emission from the $\Delta{V}=1$ bands will fall
within the 4.5 $\mu$m bandpass, increasing the eclipse depth. Tidal-induced
mass loss is at least qualitatively consistent with the apparent non-zero
eccentricity of the orbit. However, recent results show that the orbit of
WASP-12b is likely to be more circular than Li et al. (2010) suppose (Campo et
al., 2010; Husnoo et al., 2010). The evidence for non-circularity is better in
the CoRoT-2 case than for WASP-12, so we explore whether a mass loss and CO
emission scenario might be profitably applied to CoRoT-2.
We calculate what mass loss rate is required to increase the 4.5 $\mu$m
contrast sufficiently over the conventional model to account for the observed
eclipse depth. We compare the requisite mass loss rate with model calculations
for both tidal-stripping, and evaporation by stellar UV flux. If the required
mass loss rate is (for example) so large that the planet would disappear
within an unacceptably small time scale, then we could discard the mass loss
hypothesis.
Prior to calculating the mass loss required to account for the 4.5 $\mu$m
eclipse, we mention a potentially serious problem with this hypothesis. This
problem derives from the eclipse curve itself. In a variant of our bootstrap
error analysis, we allowed the ingress and egress times of the eclipse to
vary. We implemented variations in ingress/egress time by applying linear
transformations to the time axis prior to second contact, and subsequent to
third contact. We find that the 1$\sigma$ precision of the observed
ingress/egress time is about 10%. This implies that the radius of any CO-
emitting volume cannot be more than about 30% larger (3$\sigma$ limit) than
the radius of the planet. Given the requisite mass loss rate (see below), we
calculated a synthetic spectrum for the resultant CO column density of
$10^{19}$ cm-2, adopting excitation temperatures from 3000K to 15,000K. Many
individual lines in this spectrum are optically thick, and attain intensities
closely equal to the Planck function at the excitation temperature. However,
the line density in the 4.5 $\mu$m Spitzer bandpass is insufficient to produce
the required eclipse flux unless the excitation temperature exceeds 15,000K.
Since CO is primarily dissociated at such temperatures, we cannot easily match
the required eclipse flux using such a compact source of CO emission.
Nevertheless, the details of mass loss in the Roche lobe and through the inner
Lagrangian point are not completely understood, so we present our calculation
of the mass loss required to account for the 4.5 $\mu$m eclipse depth.
Let the continuum flux from the star, integrated over the 4.5 $\mu$m band be
denoted $F_{s}$, in ergs cm-2 sec-1. Let the flux from the hypothetical CO
cloud be denoted $F_{CO}$ in the same units. Then the excess over the standard
model atmosphere for the planet (Figure 4) requires:
$F_{CO}\approx 0.005F_{s}$ (1)
A Phoenix model atmosphere for the star (Hauschildt et al., 1999), integrated
over the 4.5 $\mu$m bandpass, gives the same flux as blackbody having
$T=5237$K, so $F_{s}={\Delta\nu}\Omega_{s}B_{\nu}$, where $B_{\nu}=5.17\times
10^{-6}$ is the Planck function (in cgs units) at 5237K, $\Omega_{s}$ is the
solid angle of the star as seen from Spitzer, and $\Delta\nu$ is the bandwidth
of the 4.5 $\mu$m band in Hz. We also have $F_{CO}=L/(4\pi d^{2})$, where $L$
is the luminosity of the CO-emitting cloud within the 4.5 $\mu$m band (ergs
sec-1), and $d$ is the distance to the system. The solid angle $\Omega_{s}=\pi
R^{2}/d^{2}$, where $R$ is the radius of the star. We substitute for $d^{2}$
in the expression for $F_{CO}$, and then (1) becomes:
$L\approx 0.2\Delta\nu B_{\nu}R^{2}\approx 6.2\times 10^{28}ergs~{}sec^{-1}$
(2)
The number of CO molecules required to produce this luminosity depends on
their excitation state and on the Einstein-A values for the emission. We first
adopt a thermal distribution at $T=3000$K for the CO vibrational levels, and
we use the rotationless Einstein A-values $A_{ji}$ for $\Delta V=1$ from Okada
et al. (2002). Summing over the vibrational levels, we find that the effective
emitting rate is $28$ sec-1 per CO molecule. Since $h\nu\approx 4.42\times
10^{-13}$, $L\approx 1.4\times 10^{41}$ photons sec-1. This requires
$4.9\times 10^{39}$ CO molecules in the emitting volume. Adopting a solar
carbon abundance ($10^{-3.5}$), and stipulating that all of the carbon appears
in CO, the total mass in the emitting volume is approximately $1.5\times
10^{-11}$ Jupiters.
To determine a mass loss rate from the total mass in the emitting volume, we
must estimate the transit time of CO molecules. This has been discussed by Li
et al. (2010), who conclude that mass flows through the Roche lobe at the
sound speed, and forms a disk around the star. Most of that disk emission will
not be modulated by the secondary eclipse, so our observations refer only to
the mass flowing out of the Roche lobe itself. The relevant time is therefore
the Roche lobe radius $a(M_{p}/3M_{s})^{1/3}$ divided by the sound speed
$(\gamma P/\rho)^{1/2}$. We calculate a Roche lobe radius for CoRoT-2 of
$4.3\times 10^{5}$ km, and a sound speed of $4.5$ km sec-1. These values yield
a mass loss rate of $\sim 5\times 10^{-9}M_{J}$ per year. This value is in
close accord with a minimum value for WASP-12, calculated by Lai, Helling &
van den Heuvel (2010). It is also a reasonable value for a giant planet close-
in to a young active star (Hubbard et al., 2007).
The greatest uncertainty in the above calculation is the excitation state of
the CO molecules. Because the population of the vibrational levels varies
exponentially with vibrational temperature, the effective emitting rate could
vary by orders of magnitude and still be consistent with our ignorance. If CO
lost from the planet is vibrationally cold (T=300K, for example), as will tend
to happen in the absence of collisional excitation, then the effective
emission rate drops by over 4 orders of magnitude, and the required mass loss
rate increases by that factor, and becomes unacceptably large. Indeed, in the
arguably applicable limit of no collisional excitation, each CO molecule would
emit approximately one photon as it expanded from the planetary atmosphere
through the Roche lobe. That limit would require a mass loss rate as high as
$10^{-2}M_{J}$ per year, which is unacceptably high.
Although the requisite mass loss rate is within the range for tidal-stripping
and UV-energy deposition models, we conclude that this CO-emission hypothesis
is an unlikely interpretation of the Spitzer data, due to the difficulty with
the ingress/egress time and the necessity of maintaining collisional
excitation. However, it cannot be absolutely ruled out without more detailed
models as well as observed high resolution spectroscopy of the system. If this
hypothesis could be confirmed, the consequent lack of an atmospheric
temperature inversion for this planet - orbiting an active star - would be
consistent with the emerging anti-correlation between the presence of
inversions and stellar activity levels (Knutson et al., 2010).
### 3.4 An Inverted Atmosphere Variant for CoRoT-2
A second hypothesis to account for CoRoT-2b is a variant of an inverted
atmospheric structure. The 8 $\mu$m radiation may hypothetically emerge from
deeper and cooler atmospheric layers, whereas the shorter wavelengths are
formed in a high altitude layer that is heated by absorption. Absorption in a
high altitude layer has been implicated (Burrows et al., 2007) as driving
atmospheric temperature inversions, by absorbing stellar irradiance and
heating the planetary atmosphere at altitude. Radiative equilibrium of a high
altitude absorbing layer that is optically thick in the optical and near-IR
could potentially shield lower levels of the atmosphere from radiative
heating. A high altitude layer would re-emit both to space and to lower levels
of the atmosphere, but the net downward flux would be reduced by upward
emission to space. If the opacity of the absorbing layer is high in the
optical and near-IR ($\lambda<5\mu$m), eclipse observations at those
wavelengths may sense only the absorbing layer, whereas longer wavelengths
(e.g., 8 $\mu$m) may penetrate and sense the cooler lower atmosphere.
Recently, Guillot & Havel (2010) have concluded that the IR opacity of
CoRoT-2’s atmosphere is greater than normal. We are here hypothesizing exactly
the opposite of that conclusion, but based in part on the additional 3.6
$\mu$m eclipse result that was not available to Guillot & Havel (2010).
One immediate problem with this hypothesis is that 8 $\mu$m radiation is not
believed to be formed any deeper than the shorter wavelength IRAC bands
(Burrows et al., 2007). Hence some additional source of short wavelength
opacity is required. Scattering by micron-sized haze particles or aerosols is
a potential source of the required opacity if such particles can be lofted and
maintained at high altitudes. Haze due to smaller particles at high altitudes
has been inferred for other planets (Pont et al., 2008). However, several
caveats should be cited with regard to this hypothesis. First, most scattering
opacities from small particles have a very broad dependence on wavelength,
whereas a sharper long-wavelength cutoff might be required. If the extra
opacity is from absorption (as opposed to scattering) then it might perturb
the atmospheric temperature gradient so that the cooler lower atmosphere we
envision might not exist.
This hypothesis of a heated high altitude layer and a cooler lower atmosphere
brings to mind the situation with respect to the global energy budget of HD
189733b. Barman (2008) pointed out that the efficiency of zonal heat
redistribution can be highly depth dependent. Deeper layers can redistribute
heat more efficiently because their radiative time constant (Iro & Deming,
2010) is comparable to or exceeds the time for advection of heat by zonal
winds. In that case the lower atmosphere responds primarily to the day-night
average irradiation, whereas the upper atmosphere comes to radiative
equilibrium with day-side irradiation on a short time scale. If 8 $\mu$m
radiation from CoRoT-2 arises from deeper layers, then this effect can in
principle act to reinforce the presence of a temperature inversion.
If this second hypothesis is correct, then high opacity at optical and near-
infrared wavelengths could produce a blackbody spectrum at these wavelengths.
An 1866K blackbody (green line on Figure 4) produces a reasonable agreement
with the 3.6 and 4.5 $\mu$m data, but is below the optical CoRoT measurements
(Alonso et al., 2009a; Snellen et al., 2010). Cowan & Agol (2010) invoked a
simple analytic model of the published photometric observations of close-in
exoplanets, and inferred $T=1866$K and a Bond albedo of $16\%\pm 7\%$ for
CoRoT-2b. This is qualitatively consistent with our second hypothesis for this
planet. Ground-based JHK eclipse measurements of this unusual planet would be
very useful in defining the blackbody shape and temperature of the near-
infrared spectrum.
This work is based on observations made with the Spitzer Space Telescope,
which is operated by the Jet Propulsion Laboratory, California Institute of
Technology, under a contract with NASA. Support for this work was provided by
NASA. Eric Agol acknowledges support under NSF CAREER grant no. 0645416. Adam
Burrows was supported by NASA grant NNX07AG80G and under JPL/Spitzer
Agreements 1328092, 1348668, and 1312647. He is also pleased to note that part
of this work was performed while in residence at the Kavli Institute for
Theoretical Physics, funded by the NSF through grant no. PHY05-51164. We thank
Dr. Rory Barnes for informative conversations regarding the tidal evolution of
CoRoT-2, and an anonymous referee for a very thorough review that improved
this paper significantly.
Because our results are among the first for exoplanets using Warm Spitzer, we
take this opportunity to comment on the photometric quality of the Warm
mission exoplanet data. The loss of cryogen has increased the operating
temperature of the InSb detectors from 15K (cryogenic) to 29K (warm), and that
has altered some characteristics of the detectors. For example, the ‘column
pull down’ effect has become more prominent. Bright stars cause the signal
levels to drop for all pixels in the column they overlap. None of our target
stars happen to lie on columns that are noticeably affected by pull-down. Our
photometry code calculates the theoretical limiting signal-to-noise ratio
based on the Poisson statistics of the total number of electrons recorded from
the star, and we include a read noise of 10 electrons for each pixel within
the numerical aperture. After fitting the photometric time series to remove
the intra-pixel variations and the eclipse, we calculate the scatter of the
residuals and compare this to the theoretical limiting noise. For CoRoT-1 at
3.6 and 4.5 $\mu$m we achieve 87% and 92% of the theoretical signal-to-noise,
respectively. However, this seemingly excellent performance may be mis-leading
because these are relatively faint stars, where the stellar photon noise is
high and will tend to dominate instrumental noise. A more sensitive test for
possible instrumental red noise is to calculate the reduced $\chi^{2}$ of the
binned data, after removing the best fit eclipse (bottom panels of Figs. 1-3).
We base the predicted error of each bin (error bars on the figures) on the
observed scatter of the unbinned points, reduced by the square-root of the
number of points in each bin (typically, 9). On this basis, the reduced
$\chi^{2}$ values are 1.10 and 1.31 for CoRoT-1 at 3.6 and 4.5 $\mu$m,
respectively. This indicates that a small amount of red noise occurs for time
scales longer than about 5 minutes. In the case of CoRoT-2, the only binning
we used was the averaging over 64-frames in each data cube. Measuring the
observed scatter after removing the fit, we find a ratio of 83% when using all
individual frames of each 63-frame data cube, but this reduces to 75% of the
theoretical signal-to-noise when we average the frames in each data cube
before fitting the eclipse. Like CoRoT-1, this indicates the presence of a
small amount of red noise. We are interested in whether the column pull-down
effect causes enhanced noise for stars that lie on affected columns.
Unfortunately, there are no suitably bright stars that overlie pulled-down
columns in our CoRoT data, nor did we find any optimal test stars in several
other Warm Spitzer data sets that we examined. The best test star we located
was HD 189314, lying in the Kepler field (D. Charbonneau, PID 60028). This
relatively bright star (K=9.3) is above the 1% nonlinearity limit for the
12-sec exposures we examined. Because pointing jitter moves the star toward
and away from pixel-center, it modulates the nonlinearity effect
simultaneously with the intrapixel effect. We were unable to effectively
decorrelate these mixed instrumental effects. However, we were able to
evaluate the point-to-point scatter in the photometry, by removing a smoothing
function (high-pass filtering). We find that the point-to-point scatter in the
photometry achieves 76% of the theoretical signal-to-noise. We tentatively
conclude that the column pull-down effect does not add short-term noise to
Warm Spitzer photometry, even for stars overlying affected columns. We are
unable to evaluate whether it causes increased red noise, but we anticipate
that this will become clear as additional Warm Spitzer observations are
accumulated. Finally, we draw attention to another important difference
between the cryogenic and Warm missions. With cryogenic data, we sometimes
evaluated the background for subarray photometry by considering a median over
all pixels in a data cube (fitting to a distribution), and using this single
best-fit background value for each of the 64-frames in the data cube. This had
the advantage that the larger number of pixels over the entire data cube
resulted in a more precisely determined value, but it was premised on the
background being constant within each data cube. We find that this premise is
no longer accurate for the Warm mission: the background value varies
significantly from frame to frame within a subarray data cube. (The background
is probably not due to impinging IR radiation, but is more likely to be
electronic in nature.) The statistical penalty of having fewer pixels
available when measuring the background in individual frames is offset by the
necessity of following these frame-to-frame variations. The background
variations are illustrated in Figure 5, where we show the 3.6 $\mu$m
background per frame as a function of the frame number within a data cube, and
compare the cryogenic mission (represented by HD 189733) to the Warm mission
(represented by CoRoT-2). Note that the 58th frame continues to exhibit a
higher background value in the Warm mission, as it did in the cryogenic
mission (Harrington et al., 2007; Agol et al., 2010). We find, in agreement
with Agol et al. (2010), that the photometry from the 58th frame is well-
behaved if the higher background is accounted for. Because the Warm mission
will inevitably observe fainter exoplanet host stars than during the cryogenic
mission, accurate background subtraction becomes a high priority. Our 3.6
$\mu$m photometry for CoRoT-2 used the ‘per-frame’ method that we now find to
be necessary, and achieved the 83% of theoretical signal-to-noise as described
above. Facilities: Spitzer.
## References
* Agol et al. (2010) Agol, E., Cowan, N. B., Knutson, H. A., Deming, D., Steffen, J. H., Henry, G. W., & Charbonneau, D. 2010, ApJ, 721, 1861.
* Alonso et al. (2008) Alonso, R., & 42 co-authors, 2008, A&A, 482, L21.
* Alonso et al. (2009a) Alonso, R., Guillot, T., Mazeh, T., Aigrain, S., Barge, P., Hatzes, A., & Pont, F. 2009a, A&A, 501, L23.
* Alonso et al. (2009b) Alonso, R., & 35 co-authors, 2009b, A&A, 506, 353.
* Alonso et al. (2010) Alonso, R., Deeg, H. J., Kabath, P., & Rabus, M., 2010, AJ, 139, 1481.
* Barge et al. (2008) Barge, P., & 37 co-authors, 2008, A&A, 482, L17.
* Baraffe et al. (2004) Baraffe, I., Selsis, F., Chabrier, G., Barman, T. S., Allard, F., Hauschildt, P. H., & Lammer, H., 2004, A&A, 419, L13.
* Barman (2008) Barman, T., 2008, ApJ, 676, L61.
* Bouchy et al. (2008) Bouchy, F., & 36 co-authors, 2008, A&A, 482, L25.
* Burrows et al. (2007) Burrows, A., Hubeny, I., Budaj, J., Knutson, H. A., & Charbonneau, D. 2007, ApJ, 668, L171.
* Burrows et al. (2008) Burrows, A., Budaj, J., & Hubeny, I., 2008, ApJ, 678, 1436.
* Campo et al. (2010) Campo, C. J., & 18 co-authors, 2010, ApJ, in press, astro-ph/1003.2763.
* Charbonneau et al. (2005) Charbonneau, D., Allen, L. E., Megeath, S. T., Torres, G., Alonso, R., Brown, T. M., Gilliland, R. L., Latham, D. W., Mandushev, G., O’Donovan, F., & Sozetti, A. 2005, ApJ 626, 523.
* Charbonneau et al. (2008) Charbonneau, D., Knutson, H. A., Barman, T., Allen, L. E., Mayor, M., Megeath, S. T., Queloz, D., & Udry, S., 2008, ApJ, 686, 1341.
* Christiansen et al. (2010) Christiansen, J., Ballard, S., Charbonneau, D., Madhusudhan, N., Seager, S., Holman, M. J., Wellnitz, D. D., Deming, D., A’Hearn, M. F., & the EPOXI Team, 2010, ApJ, 710, 97.
* Cowan & Agol (2010) Cowan, N., & Agol, E., 2010, submitted to ApJ, astro-ph/1001.0012.
* Deming et al. (2005) Deming, D., Seager, S., Richardson, L. J., & Harrington, J. 2005, Nature 434, 740.
* Deming et al. (2006) Deming, D., Harrington, J., Seager, S., & Richardson, L. J., 2006, ApJ, 644, 560.
* Deming et al. (2007) Deming, D., Agol, E., Charbonneau, D., Cowan, N., Knutson, H., & Marengo, M. 2007, in The Science Opportunities for the Warm Spitzer Mission, A.I.P. Conf. Proc., eds. L. J. Storrie-Lombardi & N. A. Silbermann, p. 89.
* Fazio et al. (2004) Fazio, G. G., and 64 co-authors, 2004, ApJ(Suppl), 154, 10.
* Fortney et al. (2005) Fortney, J. J., Marley, M. S., Lodders, K., Saumon, D., & Freedman, R. S., 2005, ApJ, 627, L69.
* Fortney et al. (2006) Fortney, J. J., Saumon, D., Marley, M. S., Lodders, K., & Freedman, R. S., 2006, ApJ, 642, 495.
* Fortney et al. (2008) Fortney, J. J., Lodders, K., Marley, M. S., & Freedman, R. S., 2008, ApJ, 678, 1419.
* Gillon et al. (2009a) Gillon, M., Demory, B.-O., Triuad, A. H. M. J., Barman, T., Hebb, L., Montalban, J., Maxted, P. F. L., Queloz, D., Medeuil, M., & Magain P., 2009, A&A, 506, 359.
* Gillon et al. (2009b) Gillon, M., Smalley, B., Hebb, L., Anderson, D. R., Triaud, A. H. M. J., Hellier, C., Maxted, P. F. L., Queloz, D., & Wilson, D. M. 2009, A&A, 496, 259.
* Gillon et al. (2010) Gillon, M., & 18 co-authors, 2010, A&A, 511, A3.
* Guillot & Havel (2010) Guillot, T. & Havel, M. 2010, A&A, in press (astro-ph/1010.1078).
* Harrington et al. (2007) Harrington, J., Luszcz, S., Seager, S., Deming, D., & Richardson, L. J., 2007, Nature, 447, 691.
* Hauschildt et al. (1999) Hauschiltd, P. H., Allard, F., Ferguson, J., Baron, E., & Alexander, D. R., 1999, ApJ, 525, 871.
* Hebb et al. (2010) Hebb, L., & 21 co-authors, 2010, ApJ, 708, 224.
* Hebb et al. (2009) Hebb, L., & 34 co-authors, 2009, ApJ, 693, 1920.
* Hebrard et al. (2010) Hebrard, G., and 26 co-authors, 2010, A&A, 516, A95.
* Hubbard et al. (2007) Hubbard, W. B., Hattori, M. F., Burrows, A., Hubeny, I., & Sudarsky, D., 2007, Icarus, 187, 358.
* Husnoo et al. (2010) Husnoo, N., & 11 co-authors, 2010, submitted to MNRAS, astro-ph/1004.1809.
* Iro & Deming (2010) Iro, N., & Deming, D., 2010, ApJ, 712, 218.
* Knutson et al. (2007) Knutson, H. A., Charbonneau, D., Allen, L. E., Fortney, J. J., Agol, E., Cowan, N. B., Showman, A. P., Cooper, C. S., & Megeath, S. T., 2007, Nature, 447, 183.
* Knutson et al. (2008) Knutson, H. A., Charbonneau, D., Allen, L. E., Burrows, A., & Megeath, S. T. 2008, ApJ, 673, 526.
* Knutson et al. (2009) Knutson, H. A., Charbonneau, D., Burrows, A., O’Donovan, F. T., & Mandushev, G. 2009, ApJ, 691, 866.
* Knutson et al. (2010) Knutson, H. A., Howard, A. W., Isaacson, H., et al., 2010, ApJ, 720, 1569.
* Lai, Helling & van den Heuvel (2010) Lai, D., Helling, Ch., & van den Heuvel, E. P. J., 2010, ApJ, 721, 923.
* Lanza et al. (2009) Lanza, A. F., & 20 ao-authors, 2009, A&A, 493, 193.
* Li et al. (2010) Li, S.-L., Miller, N., Lin, D. N. C., & Fortney, J. J., 2010, Nature, 463, 1054.
* Machalek et al. (2008) Machalek, P., McCullough, P. R., Burke, C. J., Valenti, J. A., Burrows, A., & Hora, J. L. 2008, ApJ, 684, 1427.
* Morales-Calderon et al. (2006) Morales-Calderon, M., and 12 co-authors 2006, ApJ, 653, 1454.
* Okada et al. (2002) Okada, K., Aoyagi, M., & Iwata, S., 2002, JQSRT, 72, 813.
* Pont et al. (2008) Pont, F., Knutson, H. A., Gilliland, R. L., Moutou, C., & Charbonneau, D. 2008, MNRAS, 385, 109.
* Press et al. (1992) Press, W. H., Teukolsky, S. A., Vetterling, W. T., & Flannery B. P. 1992, Numerical Recipes, Cambridge University Press.
* Rogers et al. (2009) Rogers, J. C., Apai, D., Lopez-Morales, M., Sing, D. K., & Burrows, A. 2009, ApJ, 707, 1707.
* Seager, Whitney, & Sasselov (2000) Seager, S., Whitney, B. A., & Sasselov, D. D. 2000, ApJ, 540, 504.
* Snellen et al. (2009) Snellen, I. A. G., de Mooij, E.J.W., & Albrecht, S. 2009, Nature, 459, 543.
* Snellen et al. (2010) Snellen, I. A. G., de Mooij, E. J. W., & Burrows, A., 2010, A&A, 513, A76.
* Spiegel et al. (2009) Spiegel, D. S., Silverio, K., & Burrows, A. 2009, ApJ, 699, 1487.
* Todorov et al. (2010) Todorov, K., Deming, D., Harrington, J., Stevenson, K. B., Bowman, W. C., Nymeyer, S., Fortney, J. J., & Bakos, G. A. 2010, ApJ, 708, 498.
Figure 1: Upper Panel: Photometry of CoRoT-1, vs. orbital phase, at 3.6 $\mu$m
(points), with the decorrelation function overplotted (red line). Middle
Panel: Photometry after correction with the decorrelation function, with the
best-fit eclipse curve overlaid (blue line). Bottom Panel: Decorrelated
photometry binned to a resolution of approximately 0.002 in orbital phase (100
bins), with the best fit eclipse curve overlaid (blue line). The error bars
are based on the scatter of individual points within each bin. The best-fit
central phase is $0.5012\pm 0.0024$.
Figure 2: Upper Panel: Photometry of CoRoT-1, vs. orbital phase, at 4.5 $\mu$m
(points), with the decorrelation function overplotted (red line). Bottom
Panel: Photometry with the decorrelation function removed, and binned to a
resolution of approximately 0.002 in orbital phase (100 bins), with the best
fit eclipse curve overlaid (blue line). The error bars are based on the
scatter of individual points within each bin. The best-fit central phase is
$0.4992\pm 0.0014$.
Figure 3: Upper Panel: Photometry of CoRoT-2, vs. orbital phase, at 3.6 $\mu$m
(points), with the decorrelation function overplotted (red line). Each point
is the average of 63 temporal frames in a data cube of $32\times 32$ pixels
times 64 temporal frames (dropping the first). Bottom Panel: Photometry with
the decorrelation function removed, with the best fit eclipse curve overlaid
(blue line). The error bars are the theoretical limit based on the photon and
read noise. The best-fit central phase is $0.4994\pm 0.0007$.
Figure 4: Planet to star contrast ratios for CoRoT-1 and CoRoT-2 versus
wavelength, from Table 1. The short wavelength data are on left panels
(contrast on log scale) and longer wavelength data on the right (contrast on
linear scale). Data from CoRoT, ground-based at 2 $\mu$m, and Spitzer are all
plotted with red points. Error bars on the abscissa give the half-intensity
wavelength limits of the bandpasses. For CoRoT-2 we plot our re-analysis of
the Gillon et al. (2010) data at 4.5 and 8 $\mu$m, but the original Gillon et
al. (2010) values are similar. The square point at 8.0 $\mu$m is the eclipse
depth using the exponential ramp (see table 1). The black curves are non-
inverted Burrows models having 30% redistribution of stellar irradiance to the
night side, with no extra absorbing layers at high altitude. For CoRoT-2, the
black dotted portion near 4.5 $\mu$m is the same Burrows model, only lacking
CO absorption. The blue lines are inverted models from Fortney and
collaborators (Fortney et al., 2005, 2006, 2008) having TiO absorption, and no
re-distribution of stellar irradiance. The green lines are blackbodies having
$T=2460$K (CoRoT-1, Rogers et al., 2009) and $T=1866$K (CoRoT-2, Cowan & Agol,
2010). The reduced $\chi^{2}$ values for the CoRoT-1 data as compared to the
different models are: conventional model (black line) = 12.6, inverted model
(blue line) = 2.4, blackbody (green line) = 1.9. For CoRoT-2, the reduced
$\chi^{2}$ values for those models are 61.4, 30.4, and 12.5, respectively.
(These values use the log ramp point at 8 $\mu$m, not the exponential ramp.)
The reduced $\chi^{2}$ value for CoRoT-2 compared to the non-inverted model
without CO absorption (dotted portion) is 13.5.
Figure 5: Number of electrons per pixel in the background of CoRoT-2 at 3.6
$\mu$m (points with line connecting), shown as a function of the frame number
in each 64-frame subarray data cube observed using Warm Spitzer. These results
are averaged over all 215 data cubes that were acquired, and the exposure time
per frame was 2 seconds. The line without points shows the background for
subarray photometry of HD 189733, using observations acquired during the
cryogenic mission (Charbonneau et al., 2008). Since background contains both
real infrared radiation as well as electronic effects, it is not proportional
to exposure time. The short-exposure (0.1-sec) HD 189733 observations were
scaled upward by an arbitrary factor for this plot.
Table 1: Summary of Secondary Eclipse Measurements for CoRoT-1 and CoRoT-2 Planet | Wavelength | Eclipse Depth | Reference
---|---|---|---
CoRoT-1 | 0.60(0.42) $\mu$m | $0.016\%\pm 0.006\%$ | Alonso et al.(2009b)
– | 0.71(0.25) | $0.0126\%\pm 0.0033\%$ | Snellen et al.(2009)
– | 2.10(0.02) | $0.278\%^{+0.043\%}_{-0.066\%}$ | Gillon et al.(2009)
– | 2.15(0.32) | $0.336\%\pm 0.042\%$ | Rogers et al.(2010)
– | 3.6(0.75) | $0.415\%\pm 0.042\%$ | This paper
– | 4.5(1.0) | $0.482\%\pm 0.042\%$ | This paper
CoRoT-2 | 0.60(0.42) $\mu$m | $0.006\%\pm 0.002\%$ | Alonso et al.(2009a)
– | 0.71(0.25) | $0.0102\%\pm 0.002\%$ | Snellen et al.(2010)
– | 2.15 (0.32) | $0.16\%\pm 0.09\%$ | Alonso et al.(2010)
– | 3.6(0.75) | $0.355\%\pm 0.020\%$ | This paper
– | 4.5(1.0) | $0.510\%\pm 0.042\%$ | Gillon et al.(2010)
– | 4.5(1.0) | $0.500\%\pm 0.020\%$ | This paper
– | 8.0(2.9) | $0.41\%\pm 0.11\%$ | Gillon et al.(2010)
– | 8.0(2.9) | $0.446\%\pm 0.10\%$ | This paper - log ramp
– | 8.0(2.9) | $0.510\%\pm 0.059\%$ | This paper - exponential ramp
Table 2: Eclipse Central Times and Phase for CoRoT-1 and CoRoT-2.
Planet | Wavelength | HJD | Phase
---|---|---|---
CoRoT-1 | 3.6 $\mu$m | $2455162.1643\pm 0.0036$ | $0.5012\pm 0.0024$
| 4.5 | $2455159.1433\pm 0.0021$ | $0.4992\pm 0.0014$
CoRoT-2 | 3.6 | $2455160.4496\pm 0.0012$ | $0.4994\pm 0.0007$
| 4.5 | $2454771.7598\pm 0.0007$ | $0.4976\pm 0.0004$
| 8.0 | $2454771.7633\pm 0.0033$ | $0.4992\pm 0.0019$
Note: Orbital phase for CoRoT-1 used $T_{0}=2454524.62324$ and $P=1.5089686$
days (Gillon et al., 2009a). For CoRoT-2 we used $T_{0}=2454237.53562$ (Alonso
et al., 2008) and $P=1.7429935$ days (Gillon et al., 2010).
|
arxiv-papers
| 2010-11-03T22:03:12 |
2024-09-04T02:49:14.485426
|
{
"license": "Public Domain",
"authors": "Drake Deming, Heather Knutson, Eric Agol, Jean-Michel Desert, Adam\n Burrows, Jonathan J. Fortney, David Charbonneau, Nicolas B. Cowan, Gregory\n Laughlin, Jonathan Langton, Adam P. Showman, and Nikole K. Lewis",
"submitter": "Drake Deming",
"url": "https://arxiv.org/abs/1011.1019"
}
|
1011.1182
|
# Tunable subpicosecond electron bunch train generation
using a transverse-to-longitudinal phase space exchange technique
Y.-E Sun Accelerator Physics Center, Fermi National Accelerator Laboratory,
Batavia, IL 60510, USA P. Piot Accelerator Physics Center, Fermi National
Accelerator Laboratory, Batavia, IL 60510, USA Northern Illinois Center for
Accelerator & Detector Development and Department of Physics, Northern
Illinois University, DeKalb IL 60115, USA A. Johnson Accelerator Division,
Fermi National Accelerator Laboratory, Batavia, IL 60510, USA Northern
Illinois Center for Accelerator & Detector Development and Department of
Physics, Northern Illinois University, DeKalb IL 60115, USA A. H. Lumpkin
Accelerator Division, Fermi National Accelerator Laboratory, Batavia, IL
60510, USA T. J. Maxwell Accelerator Physics Center, Fermi National
Accelerator Laboratory, Batavia, IL 60510, USA Northern Illinois Center for
Accelerator & Detector Development and Department of Physics, Northern
Illinois University, DeKalb IL 60115, USA J. Ruan Accelerator Division,
Fermi National Accelerator Laboratory, Batavia, IL 60510, USA R. Thurman-Keup
Accelerator Division, Fermi National Accelerator Laboratory, Batavia, IL
60510, USA
###### Abstract
We report on the experimental generation of a train of subpicosecond electron
bunches. The bunch train generation is accomplished using a beamline capable
of exchanging the coordinates between the horizontal and longitudinal degrees
of freedom. An initial beam consisting of a set of horizontally-separated
beamlets is converted into a train of bunches temporally separated with
tunable bunch duration and separation. The experiment reported in this Letter
unambiguously demonstrates the conversion process and its versatility.
###### pacs:
29.27.-a, 41.85.-p, 41.75.Fr
Recent applications of electron accelerators have spurred the demand for
precise phase-space control schemes. In particular, electron bunches with a
well-defined temporal distribution are often desired. An interesting class of
temporal distribution consists of trains of bunches with subpicosecond
duration and separation. Applications of such trains include the generation of
super-radiant radiation gover ; bosco ; ychuang and the resonant excitation
of wakefields in novel beam-driven acceleration methods muggliprstab ; jing .
To date there are very few methods capable of providing this class of beams
reliably muggli . We have recently explored an alternative technique based on
the use of a transverse-to-longitudinal phase space exchange method piotAAC08
; yineLINAC08 . The method consists of shaping the beam’s transverse density
to produce the desired horizontal profile, the horizontal profile is then
mapped onto the longitudinal profile by a beamline capable of exchanging the
phase spaces between the horizontal and longitudinal degrees of freedom.
Therefore the production of a train of bunches simply relies on generating a
set of horizontally-separated beamlets upstream of the beamline, e.g., using a
masking technique. Considering an electron with coordinates
${\bf\widetilde{X}}\equiv(x,x^{\prime}\equiv p_{x}/p_{z},z,\delta\equiv
p_{z}/\mbox{$\langle{p_{z}}\rangle$}-1)$ (here $p_{x}$, $p_{z}$ are
respectively the horizontal and longitudinal momenta, $\langle{p_{z}}\rangle$
represents the average longitudinal momentum) in the four dimensional trace
space, the $4\times 4$ transfer matrix $R$ associated to an ideal transverse-
to-longitudinal phase-space-exchanging beamline is $2\times 2$-block anti-
diagonal. Thus the beamline exchanges the emittances between the transverse
and longitudinal degrees of freedom. The normalized horizontal root-mean-
square (rms) emittance is defined as
$\varepsilon_{x}^{n}\equiv\gamma\beta[\mbox{$\langle{x^{2}}\rangle$}\mbox{$\langle{x^{\prime
2}}\rangle$}-\mbox{$\langle{xx^{\prime}}\rangle$}^{2}]^{1/2}$, where $\gamma$
is the Lorentz factor and $\beta\equiv\sqrt{1-\gamma^{-2}}$. A similar
definition holds for the longitudinal degree of freedom. Phase-space-
exchanging [or emittance-exchanging (EEX)] beamlines were initially considered
as a means to increase the luminosity in the B-factories orlov , mitigate
instabilities in high-brightness electron beams emma , and improve the
performance of single-pass free-electron lasers emma2 .
A simple configuration capable of performing such a phase-space exchange
consists of a horizontally-deflecting resonant cavity, operating in the TM110
mode, flanked by two horizontally-dispersive sections henceforth referred to
as “doglegs” kim . Describing the beamline elements with their thin-lens-
matrix approximation, an electron with initial trace space coordinates ${\bf
X_{0}}$ will have its final coordinates ${\bf X}=R{\bf X_{0}}$. In particular
the electron’s final longitudinal coordinates $(z,\delta)$ are solely
functions of its initial transverse coordinates $(x_{0},x^{\prime}_{0})$ yine
$\displaystyle\left\\{\begin{array}[]{ll}z=&-\frac{\xi}{\eta}x_{0}-\frac{L\xi-\eta^{2}}{\eta}x_{0}^{\prime}\\\
\delta=&-\frac{1}{\eta}x_{0}-\frac{L}{\eta}x_{0}^{\prime}\end{array},\right.$
(3)
where $L$ is the distance between the dogleg’s dipoles, and $\eta$ and $\xi$
are respectively the horizontal and longitudinal dispersions generated by one
dogleg. Here the deflecting cavity is operated at the zero-crossing phase,
i.e., the center of the bunch is not affected while the head and tail are
horizontally deflected in opposite directions. The deflecting strength of the
cavity $\kappa\equiv 2\pi|e|V_{x}/(\lambda c\mbox{$\langle{p_{z}}\rangle$})$
where $e$ is the electron charge, $\lambda$ is the wavelength of the TM110
mode, and $V_{x}$ is the integrated maximum deflecting voltage, is chosen as
$\kappa=-1/\eta$. The coupling described by Eq. 3 can be used to arbitrarily
shape the current or energy profile of an electron beam piotPRSTAB .
Figure 1: Top view of the experimental setup displaying elements pertinent to
the present experiment. The “X” refers to diagnostic stations (beam viewers
and/or multi-slit masks location), “Q” the quadrupole magnets and “D” the
dipole magnets. Distances are in millimeters and referenced to the
photocathode surface. The spectrometer dipole magnet downstream of the EEX
beamline bends the beam in the vertical direction.
The experiment reported in this Letter uses the $\sim 14$-MeV electron bunches
produced by a radiofrequency (rf) photoemission electron source and
accelerated in an rf superconducting cavity at Fermilab’s A0 Photoinjector
carneiro . Downstream of the cavity, the beamline includes a set of quadrupole
and steering dipole magnets, and beam diagnostics stations before splitting
into two beamlines as shown in Fig. 1.
The “straight ahead” beamline incorporates a horizontally-bending spectrometer
equipped with a Cerium-doped Yttrium Aluminum Garnet (Ce:YAG) scintillating
screen (labeled as XS3 in Fig. 1) to measure the beam’s energy distribution.
The horizontal dispersion at the XS3 location is $317$ mm.
Table 1: Typical initial beam parameters measured before emittance exchange. The Courant-Snyder (C-S) parameters are $\alpha_{x}\equiv-\mbox{$\langle{xx^{\prime}}\rangle$}/\varepsilon_{x}$ and $\beta_{x}\equiv\mbox{$\langle{x^{2}}\rangle$}/\varepsilon_{x}$, where $\varepsilon_{x}\equiv\varepsilon_{x}^{n}/(\beta\gamma)$ is the geometric emittance. Parameter | Symbol | Value | Units
---|---|---|---
energy | $E$ | 14.3 $\pm$ 0.1 | MeV
charge | $Q$ | $550\pm 30$ | pC
rms duration | $\sigma_{t}$ | 4.0 $\pm$ 0.3 | ps
horizontal emit. | $\varepsilon_{x}^{n}$ | $4.7\pm 0.3$ | $\mu$m
rms frac. energy spread | $\sigma_{\delta}$ | 0.06 $\pm$ 0.01 | %
horizontal C-S param. | $(\alpha_{x},\beta_{x})$ | ($1.2\pm 0.3$,$14.3\pm 1.6$) | (–,m)
The other beamline, referred to as the EEX beamline, implements the double-
dogleg setup described above koeth0 and has been used to explore emittance
exchange amber . The doglegs consist of dipole magnets with $\pm 22.5^{\circ}$
bending angles and each generates horizontal and longitudinal dispersion of
$\eta\simeq-33$ cm and $\xi\simeq-12$ cm, respectively footnote . The
deflecting cavity is a liquid-Nitrogen–cooled five-cell copper cavity
operating on the TM110 $\pi$-mode at 3.9 GHz koeth . The section downstream of
the EEX beamline includes three quadrupoles, beam diagnostics stations and a
vertical spectrometer. The dispersion generated by the spectrometer at the XS4
Ce:YAG screen is $944$ mm. The temporal distribution of the electron bunch is
diagnosed via the coherent transition radiation (CTR) transmitted through a
single-crystal quartz window as the beam impinges an aluminum foil at X24. The
CTR is sent through a Michelson autocorrelator tr and the autocorrelation
function is measured by a liquid helium-cooled bolometer which is used as the
detector of the autocorrelator. The CTR spectrum is representative of the
bunch temporal distribution provided $\sigma_{\perp}\ll\gamma\sigma_{z}$ where
$\sigma_{z}$ and $\sigma_{\perp}$ are respectively the rms bunch length and
transverse size at the CTR radiator location (the beam is assumed to be
cylindrically symmetric at this location). In the present experiment the beam
was focused to an rms spot size of $\sigma_{\perp}\simeq 400$ $\mu$m at X24.
Imperfections due to the frequency-dependent transmissions of the THz beamline
components alter the spectrum of the detected CTR and limit the resolution to
$\sim 200$ fs.
Figure 2: Transverse initial beam density at X5 (a), XS3 (b) and corresponding
final beam density at X23 with deflecting cavity off (c) and on (e), and at
XS4 with deflecting cavity off (d) and on (f). The corresponding relevant
intensity-normalized horizontal profile at X23 (g) and fractional energy
spread (h) profiles obtained from XS3 (red) and XS4 for the cases when the
cavity is on (green) and off (dashed blue) are also displayed.
For the proof-of-principle experiment reported here, a number of horizontally-
separated beamlets were generated by passing the beam through a set of
vertical slits at X3. The measured parameters for the incoming beam are
gathered in Table 1. The multislit mask, nominally designed for single-shot
transverse emittance measurements, consists of 48 $\mu$m-wide slits made out
of a 3-mm-thick tungsten plate. The slits are separated by 1 mm. Less than 5 %
of the incoming beam is transmitted through the mask. Up to 50 electron
bunches repeated at 1 MHz were used to increase the signal-to-noise ratio of
the measurements.
The beam was first diagnosed in the straight-ahead line to ensure that
horizontal modulations are clearly present and there are no energy modulations
[Fig. 2 (a) and (b)]. It was then transported through the EEX beamline with
the deflecting cavity turned off. The transverse modulation was still
observable at X23 but no energy modulation could be seen at XS4 as shown in
Fig. 2 (c), (d), (g) and (h). Powering the cavity to its nominal deflecting
voltage ($V_{x}\simeq 720$ kV) resulted in the suppression of the transverse
modulation at X23 and the appearance of an energy modulation at XS4 [Fig. 2
(e), (f), (g) and (h)]. These observations clearly demonstrate the ability of
the EEX beamline to convert an incoming transverse density modulation into an
energy modulation. In the present measurement the incoming horizontal Courant-
Snyder (C-S) parameters at the EEX beamline entrance were empirically tuned
for energy and time modulation in the beam by setting the current of
quadrupole magnets $Q_{1}$ and $Q_{2}$ to respectively 1.6 A and -0.6 A.
Figure 3: Total normalized CTR energy detected at X24 as a function of
quadrupole magnets currents $I_{Q_{1}}$ and $I_{Q_{2}}$ with X3 slits out(a)
and in (b) the beamline. The bolometer signal is representative of the inverse
of the bunch duration $\sigma_{t}$. The white dots in (b) indicate loci where
more detailed measurements were performed; see Fig. 5.
To characterize the expected temporal modulations we detect and analyze the
CTR emitted as the beam impinges the X24 aluminum foil saxon . The total CTR
energy detected within the detector bandwidth $[\omega_{l},\omega_{u}]$ and
angular acceptance increases as the bunch duration
$\sigma_{t}\equiv\sigma_{z}/c$ decreases. In the limit
$\omega_{l}\ll\sigma_{t}^{-1}\ll\omega_{u}$, the total radiated energy is
inversely proportional to the rms bunch duration piotvelo . The final
longitudinal C-S parameters downstream of the EEX beamline can be varied by
altering the initial horizontal C-S parameters using the quadrupole magnets
$Q_{1}$ and $Q_{2}$. Figure 3 shows the detected CTR energy as a function of
quadrupole magnet currents for the cases without (a) and with (b) intercepting
the beam with the X3 multislit mask. The two plots illustrate the ability to
control the final bunch length (as monitored by the CTR power detected at X24)
using the EEX technique. The insertion of the multislit mask results in the
appearance of a small island of coherent radiation at the lower right corner
of Fig. 3 (b). The corresponding autocorrelation functions $\Gamma(\tau)$
(where $\tau$ is the optical path difference) recorded by the bolometer for
the quadrupole magnets currents $(I_{Q1},I_{Q2})=$(1.6 A,-0.6 A) are shown in
Fig. 4 (a) with and without inserting the multislit mask. When the multislit
mask is inserted the autocorrelation function is multipeaked indicating a
train of bunches is produced. For this particular case a train of $N=6$
bunches with unequal peak intensity are produced resulting in an
autocorrelation function with $2N-1=11$ peaks. The measured separation between
the bunches is $\Delta z=762\pm 44$ $\mu$m. It should be noted that the two
autocorrelations shown in Fig. 4 correspond to very different charges and
longitudinal space charge effects influence the bunch dynamics and result in
different final longitudinal C-S parameters. In addition, the low frequency
limit of the CTR detection system prevents the accurate measurement of
autocorrelation functions of bunches with rms length larger than $\sim
500$$\mu$m TimM .
Figure 4: (a) Normalized autocorrelation function $\Gamma(\tau)/\Gamma(0)$ of
the CTR signal (a) recorded with (solid) and without (dashed) the X3 slits
inserted as a function of the optical path difference $\tau$. The
corresponding beam transverse densities at XS4 appear in (b) and (c). The
vertical axis on the bottom image is proportional to the beam’s fractional
momentum spread ($\delta$). The nominal bunch charge is $550\pm 30$ pC and
reduces to $\sim 15\pm 3$ pC when the slits are inserted.
In addition, varying the settings of the quadrupole magnets provide control
over the final longitudinal phase space time-energy correlation. The
correlation can be measured as the ratio of the peak separation along the
longitudinal coordinate and the energy ${\cal
C}=\mbox{$\langle{z\delta}\rangle$}/\mbox{$\langle{z^{2}}\rangle$}\simeq\Delta\delta/\Delta
t$. These measurements are presented in Fig. 5 for different quadrupole magnet
settings. As shown in Fig. 5, the technique can provide a tunable bunch
spacing ranging from $\sim 350$ to $760$ $\mu$m given an initial slit spacing
of 1 mm by adjusting one quadrupole magnet strength (Q1) only. For $\Delta
z\simeq 350$ $\mu$m (corresponding to $\Delta t=\Delta z/c\simeq 1.2$ ps), the
autocorrelation has a 100% modulation implying that the bunches within the
train are fully separated. Assuming the bunches follow a Gaussian
distribution, their estimated rms duration is $<300$ fs (this estimate
includes the finite resolution of our diagnostics). Variation of both Q1 and
Q2 quadrupole magnet strengths can generate even shorter bunch separations,
however our current measurement system has limited sensitivity in the shorter
wavelength region, resulting in less than 100% modulation in the
autocorrelation curve.
Figure 5: Fractional momentum spread separation $\Delta\delta$ versus time
separation $\Delta t$ between the bunches within the train for different
initial beam conditions. The different data points are obtained from the
autocorrelation functions recorded for settings $I_{Q1}=1.0$, 1.2, 1.4, 1.6,
and 1.8 A (from left to right) shown as white dots in Fig. 3. The current
$I_{Q2}$ is kept constant at $-0.6$ A.
In summary we have experimentally demonstrated that an incoming phase space
modulation in the horizontal coordinate can be converted into the longitudinal
phase space using an EEX beamline. The method was shown to produce energy- and
time-modulated bunches arranged as a train of subpicosecond bunches with
variable spacing. This proof-of-principle experiment also provides an
unambiguous demonstration of the main property of the EEX beamline to exchange
the phase space coordinates between the horizontal and longitudinal degrees of
freedom. The technique experimentally demonstrated in this Letter can be used
to tailor the current and energy profile of electron beams and could have
applications in novel beam-driven acceleration techniques, compact short-
wavelength accelerator-based light sources, and ultra-fast electron
diffraction.
We are indebted to E. Harms, E. Lopez, R. Montiel, W. Muranyi, J. Santucci, C.
Tan and B. Tennis for their technical supports. We thank M. Church, H.
Edwards, and V. Shiltsev for their interest and encouragement. The work was
supported by the Fermi Research Alliance, LLC under the U.S. Department of
Energy Contract No. DE-AC02-07CH11359, and by Northern Illinois University
under the US Department of Energy Contract No. DE-FG02-08ER41532.
## References
* (1) A. Gover, Phys. Rev. ST Accel. Beams 8, 030701 (2005).
* (2) M. Bolosco, I. Boscolo, F. Castelli, S. Cialdi, M. Ferrario, V. Petrillo and C. Vaccarezza, Nucl. Instr. Meth. A 577, 409 (2007).
* (3) Y.-C. Huang, Int. Jour. Mod. Phys. B 21, 287 (2007).
* (4) P. Muggli, B. Allen, V. E. Yakimenko, J. Park, M. Babzien, K. P. Kusche, and W. D. Kimura, Phys. Rev. ST Accel. Beams 13, 052803 (2010).
* (5) C. Jing, A. Kanareykin, J. G. Power, M. Conde, Z. Yusof, P. Schoessow, and W. Gai, Phys. Rev. Lett. 98, 144801 (2007).
* (6) P. Muggli, V. Yakimenko, M. Babzien, E. K. Kallos, K. P. Kusche, Phys. Rev. Lett. 101, 054801 (2008).
* (7) P. Piot, Y.-E Sun, and M. Rihaoui, Proceedings of the 13th Advanced Accelerator Concept workshop (AAC08), AIP Conf. Proc. 1086, 677 (2009).
* (8) Y.-E Sun, and P. Piot, Proceedings of the 2008 International Linac Conference (LINAC08), Victoria BC, 498 (2009).
* (9) Y. Orlov, C. M. O’Neill, J. J. Welch, and R. Sieman, Proceedings of the 1991 Particle Accelerator Conference (PAC91), San Francisco CA, 2838 (1991).
* (10) M. Cornacchia, and P. Emma, Phys. Rev. ST Accel. Beams 5 084001 (2002).
* (11) P. Emma, Z. Huang, K.-J. Kim, and P. Piot, Phys. Rev. ST Accel. Beams 9, 100702 (2006).
* (12) K.-J. Kim and A. Sessler, Proceedings of the 2006 Electron Cooling Workshop, Galena IL (ECOOL06), AIP Conf. Proc. 821, 115 (2006).
* (13) Y.-E Sun, et al, Proceedings of the 2007 Particle Accelerator Conference, Albuquerque NM (PAC07), 3441 (2007).
* (14) P. Piot, et al., submitted to Phys. Rev. ST Accel. Beams (2010); see electronic preprint arXiv:1007.4499v1 (2010).
* (15) J.-P. Carneiro, et al., Phys. Rev. ST Accel. Beams 8, 040101 (2005).
* (16) T. Koeth, et al, Proceedings of the 2009 Particle Accelerator Conference, Vancouver BC (PAC09), FR5PFP020 (2009).
* (17) A. Johnson, et al., Proceedings of the 2010 International Particle Accelerator Conference, Kyoto Japan (IPAC10), 4614 (2010).
* (18) In our convention, the head of the bunch is for $z<0$.
* (19) T. Koeth, et al, ibid yine , 3663 (2007).
* (20) U. Happek, A. J. Sievers, and E. B. Blum, Phys. Rev. Lett. 67, 2962 (1991).
* (21) J. S. Nodvick and D. S. Saxon, Phys. Rev. 96, 180 (1954).
* (22) P. Piot, L. Carr, W. S. Graves, and H. Loos, Phys. Rev. ST Accel. Beams 6, 033503 (2003).
* (23) T. J. Maxwell, D. Mihalcea and P. Piot, Proceedings of the 13th Beam Instrumentation Workshop (BIW08), Tahoe City CA, 225 (2008).
|
arxiv-papers
| 2010-11-04T15:08:33 |
2024-09-04T02:49:14.499418
|
{
"license": "Public Domain",
"authors": "Y.-E Sun, P. Piot, A. Johnson, A. H. Lumpkin, T. J. Maxwell, J. Ruan,\n R. Thurman-Keup",
"submitter": "Yine Sun",
"url": "https://arxiv.org/abs/1011.1182"
}
|
1011.1256
|
# The Distribution of Coalescing Compact Binaries in the Local Universe:
Prospects for Gravitational-Wave Observations
Luke Zoltan Kelley11affiliation: Department of Astronomy and Astrophysics,
University of California, Santa Cruz, CA 95064 , Enrico Ramirez-
Ruiz11affiliation: Department of Astronomy and Astrophysics, University of
California, Santa Cruz, CA 95064 , Marcel Zemp22affiliation: Department of
Astronomy, University of Michigan, Ann Arbor, MI 48109 , Jürg
Diemand33affiliation: Institute for Theoretical Physics, University of Zurich,
8057 Zurich, Switzerland , and Ilya Mandel44affiliation: NSF Astronomy and
Astrophysics Postdoctoral Fellow, MIT Kavli Institute, Cambridge, MA 02139
lzkelley@ucsc.edu
###### Abstract
Merging compact binaries are the most viable and best studied candidates for
gravitational wave (GW) detection by the fully operational network of ground-
based observatories. In anticipation of the first detections, the expected
distribution of GW sources in the local universe is of considerable interest.
Here we investigate the full phase space distribution of coalescing compact
binaries at $z=0$ using dark matter simulations of structure formation. The
fact that these binary systems acquire large barycentric velocities at birth
(“kicks”) results in merger site distributions that are more diffusely
distributed with respect to their putative hosts, with mergers occurring out
to distances of a few Mpc from the host halo. Redshift estimates based solely
on the nearest galaxy in projection can, as a result, be inaccurate. On the
other hand, large offsets from the host galaxy could aid the detection of
faint optical counterparts and should be considered when designing strategies
for follow-up observations. The degree of isotropy in the projected sky
distributions of GW sources is found to be augmented with increasing kick
velocity and to be severely enhanced if progenitor systems possess large kicks
as inferred from the known population of pulsars and double compact binaries.
Even in the absence of observed electromagnetic counterparts, the differences
in sky distributions of binaries produced by disparate kick-velocity models
could be discerned by GW observatories, within the expected accuracies and
detection rates of advanced LIGO–in particular with the addition of more
interferometers.
###### Subject headings:
gravitational waves — stars: neutron — binaries: general
## 1\. Introduction
The merger of double compact objects represents the first identified and most
predictable source of gravitational wave (GW) radiation (e.g. Phinney, 1991).
Only recently have the first GW observatories come online, and the first
detection events are expected in the next few years. Over the past three
decades the merger rate within the local universe has been thoroughly examined
(see e.g. Abadie et al., 2010; Mandel & O’Shaughnessy, 2010, for recent
reviews).
The merger rates are expected to be dominated by mergers of neutron-star
binaries, with $\langle{\Re}\rangle\sim
1\,\textrm{Mpc}^{-3}\,\textrm{Myr}^{-1}$. However, these rates are
significantly uncertain, since they come either from extrapolations from the
small observed sample of Galactic binary pulsars whose luminosity distribution
is not well constrained, or from population-synthesis models that have many
ill-determined parameters such as common-envelope efficiencies. In particular,
Abadie et al. (2010) estimate the confidence bounds on the neutron-star binary
merger rates as $\Re\approx 0.01-10\,\textrm{Mpc}^{-3}\,\textrm{Myr}^{-1}$.
The horizon distances111The horizon distance is the maximum distance at which
a signal can be detected with a given signal-to-noise threshold (e.g., 8); for
a single detector, this is the distance at which gravitational waves from a
face-on, overhead binary can be detected. for the initial and advanced
LIGO/Virgo detector networks are estimated as $\mathcal{D}\sim 30$ and $\sim$
400 Mpc, respectively, based on the distance at which a single detector could
detect gravitational waves from a neutron-star binary at a signal-to-noise
ratio of 8. Abadie et al. (2010) estimate that the advanced LIGO/Virgo network
could plausibly detect between 0.4 and 400 neutron-star binaries per year,
with a likely rate of approximately 40 detections per year.
The prospects for detection and characterization of GW sources are thus
sensitive to the distribution of compact binaries in the local universe (i.e.
at distances $\leq\mathcal{D}$). The fact that these systems must have large
systemic velocities at birth (Brandt & Podsiadlowski, 1995; Fryer & Kalogera,
1997) implies that by the time they merge, after approximately a Hubble time,
they will be far from their birth sites. The locations of merging sites
depends critically on the binary’s natal kick velocity and the temporal
evolution of the gravitational potential of the host halo as well as that of
its nearby neighbors (Zemp et al., 2009).
In this Letter, we study the evolving distribution of compact binary systems
from formation until coalescence at $z=0$ using cosmological simulations of
structure formation. This allows us to examine the full radial and angular
distributions of merging compact binaries in the local universe. The
organization is as follows. In §2, we describe the numerical methods and
initial setup and the criteria used to select a local-like universe. The
distributions of compact binaries at $z=0$ are presented in §3 for three
different kick velocity scenarios, and in §4 we examine the ability of GW
observatories to discern between them experimentally. Finally, §5 discusses
the implications of our findings.
## 2\. Methods and Initial Model
### 2.1. Simulation
The focus of this work is to understand the distribution of compact binaries
in the local universe using cosmological simulations. To this end, we have
performed a dark matter only cosmological structure formation simulation
following the numerical procedure outlined in Zemp et al. (2009). A comoving
80 Mpc periodic box is initialized at redshift $z=22.4$ (161 Myr) and uses
WMAP3 cosmological parameters (Spergel et al., 2007). The initial conditions
are evolved using the parallel tree code PKDGRAV2 (Stadel, 2001) until
$z=1.60$ (4.24 Gyr). At this time, we populate each halo with mass greater
than $2.15\times 10^{11}M_{\odot}$ (of which there are 2461 in the simulation)
with 2000 massless tracers.
Each tracer particle is meant to represent a compact binary system, which, on
average, forms around the peak of the star formation epoch (Madau et al.,
1996, 1998). In general, the local merger rate is given by the convolution of
the star formation rate with the probability distribution $P(\tau)$ of the
merging time delays $\tau$. Compact binaries formed at the peak of the star
formation history, merging after delays consistent with the orbital
separations of known relativistic binary pulsars (O’Shaughnessy et al., 2008),
dominate the local merger rate222For $P(\tau)\propto 1/\tau$ this early-
assembled population could increase the local event rate by at least $\sim 3$
(Guetta & Piran, 2005)..
Tracers are injected into the center of their halo, with an isotropic Maxwell-
Boltzmann velocity distribution with mean speeds $\bar{v}=360$, 180, and
90$\textrm{ km s}^{-1}$ and dispersions $\sigma=150$, 75, and 37.5 $\textrm{
km s}^{-1}$ (hereafter denoted as models $M_{360}$, $M_{180}$ and $M_{90}$).
This is consistent with the magnitude of the natal kicks required to explain
the observed parameters of binary NS systems (Brandt & Podsiadlowski, 1995;
Fryer & Kalogera, 1997) — only when the center of mass kicks have magnitudes
exceeding 200 km s-1 can the progenitor orbits be sufficiently wide to
accommodate evolved helium stars and still produce the small separations
measured in these systems.
The contribution of individual tracers to the overall population is weighted
linearly with their progenitor halo’s mass (at $z=1.6$) in all of our
calculations.
Finally, the cosmological box together with the tracer particle populations
are evolved until redshift $z=0$ (13.8 Gyr). This results in diverse
predictions of compact binary demographics at $z=0$ in the case of an
isotropic kick velocity distribution whose properties do not vary with the
initial binary separation. Merger times in population synthesis models are
found to be relatively insensitive to the initial kick velocity (e.g., Bloom
et al., 1999). This not only justifies our assumption but, when taken together
with the progenitors’ long time delays (O’Shaughnessy et al., 2010), also
reinforces the validity of a single injection time.
### 2.2. Local-Like Universe Selection
Once the tracers and DM are evolved to $z=0$, a local-like universe is
selected based on criteria adapted from Hoffman et al. (2008). The local-like
universe is characterized here by the following:
1. i.
There are two dark matter halos, representing the Milky-Way and Andromeda
pair, with maximum circular velocities $V_{c}\in[125,270]$ $\textrm{ km
s}^{-1}$.
2. ii.
These halos are separated by $d\leq 1.4$ $h^{-1}\textrm{Mpc}$, and approaching
each-other (i.e. $\dot{d}\leq 0.0\textrm{ km s}^{-1}$).
3. iii.
There is a Virgo-like halo at a distance $d\in[5,12]$ $h^{-1}$Mpc, and
$V_{c}\in[500,1500]$ $\textrm{ km s}^{-1}$
4. iv.
No halos with comparable or higher maximum circular velocities than either of
the pair exist within $3$ $h^{-1}$Mpc, and no other Virgo-like halos exist
within $12$ $h^{-1}$Mpc.
The first three constraints resulted in three local-like groups. Inclusion of
the fourth criterion resulted in a single, optimal environment for our
analysis. These criteria ensure that the evolutionary environment is similar
to that of the actual Milky Way and local group galaxies.
## 3\. The Local Distribution of Compact Binaries
We now examine the local, three-dimensional distribution of merging compact
binaries, which are characterized here by the massless tracer particle
population at $z=0$ centered on the Milky Way-like galaxy as defined in §2.2.
Figure 1 shows the radial distribution of tracers and dark matter within our
local-like universe. In models $M_{180}$ and $M_{90}$, tracer particles
closely follow the dark matter central-density peaks, just like the galaxies
themselves in CDM cosmology (Blumenthal et al., 1984).
As the natal, barycentric kick-velocity becomes comparable to the escape
velocity of the progenitor halos, an increasing fraction of tracers escape.
These unbound tracers pollute the intergalactic region, thus forming a
particle background which closely follows the overall dark matter
distribution, as seen for model $M_{360}$ in Figure 1. In super-galactic
regions with more continuous gravitational potentials (i.e. more densely and
uniformly populated halo environments, such as clusters), the tracer
background becomes more heavily populated. The relative isolation of the Milky
Way-Andromeda pair contributes to the enduring presence of strong native
tracer peaks located at each halo center. As a result, the extended background
distributions of tracers–centered on the pair–are only apparent in the highest
kick velocity model.
As seen in Figure 2, the number of tracers within a sphere encompassing the
Milky Way and Andromeda halos is noticeably depleted at higher kick-
velocities. At these velocities, the host halos are unable to effectively
retain most of their tracers. As the sphere’s volume approaches the Virgo
cluster, the number of (sub)halos becomes so large that the mean separation
between central peak densities decreases below the characteristic size of the
background tracer population. As a result, the variation with kick-velocity in
the tracer distributions is drowned-out. It should be noted that the expected
event rate in such a small volume is negligible (Abadie et al., 2010), and
thus the effects of varying kick-velocity will be indiscernible in the
integrated merger-rate of compact objects within LIGO/Virgo detection
horizons.
Although the integrated tracer distribution is insensitive to the model, the
angular distribution of tracers depends strongly on the binary’s kick
velocity. This is evident in Figure 3, which plots sky maps of tracers and
dark matter within a given volume (resolved to 4 square degree pixels). As
expected, high velocity kicks lead to more pronounced isotropies when compared
to the low velocity kick scenarios. At 10 Mpc, $\sim$40% of the $M_{360}$
weighted tracers lie in pixels outside those of $M_{90}$; this fraction falls
to 15% and 10% for 40 and 80 Mpc respectively. This trend results from the
increasing isotropy of dark matter in projection at progressively larger
scales.
For large velocities, the distribution of GW sources forms a sky continuum
(Figure 3) rather than well-isolated substructures — complicating host galaxy
identification and thus redshift determination. On the other hand, the
extension of the particle tracer distributions, which grows with increasing
kick velocity, could aid the detection of photonic counterparts, especially at
optical wavelengths. This is because at large kick velocities the majority of
the mergers will take place well outside the host galaxy’s half-light radius.
## 4\. Predictions for Gravitation Wave Observations
The number of detections required for GW observatories to be able to
reconstruct the kick-velocity distribution is examined here. Timing
triangulation from relative GW phase shifts333For a review of GW emission from
compact binaries see Hughes (2009). between widely separated detectors is the
primary source of sky localization (Fairhurst, 2009), and Fisher matrix or
Markov Chain Monte Carlo techniques can be used to compute error estimates
(van der Sluys et al., 2008). The intricacies of parameter determination and
error estimation can be extensive, as correlations between waveform parameters
mean that some parameters (such as distance and inclination) are partially
degenerate (see, e.g. Cutler & Flanagan, 1994, and references therein).
Typically, with a three-detector network, the distance to the GW event could
be determined to within a $\sim$20–50% uncertainty, and the angular location
of the event to $\sim 5-50$ square degrees, depending on source location,
masses, and signal-to-noise ratio (Fairhurst, 2009; van der Sluys et al.,
2008). The large uncertainty in the distance determination can be understood
when considering its dependence on the signal amplitude, which is much more
uncertain than the phase-space information.
To estimate the number of events required to distinguish between different
kick velocity models, we apply a Bayesian approach similar to that used by
Mandel (2010) to approximate the efficacy of population reconstruction from GW
signals. Data sets $D_{j}\in D_{360},D_{180}\;\textrm{and}\;D_{90}$ are drawn
from each model $M_{i}\in M_{360},M_{180}\;\textrm{and}\;M_{90}$ respectively.
Each data set contains $n$ independently drawn data points (i.e. tracers),
characterized by 3 position coordinates; i.e.
$D_{i}(n)=[x_{i,1}(r,\alpha,\delta),x_{i,2}(r,\alpha,\delta),\ldots
x_{i,n}(r,\alpha,\delta)]$. The probability of a tracer being selected for a
given data set is linearly proportional to the halo mass of the progenitor, in
accordance with the weighting scheme of §2.1. The probability that a
particular model $i$ fits a data set $j$ can be rewritten using Bayes’
formula:
$P(M_{i}|D_{j}(n))=\frac{P(D_{j}(n)|M_{i})\cdot P(M_{i})}{P(D_{j}(n))}.$ (1)
Throughout our analysis we assume flat priors [$P(M_{i})=P(M_{j})$], and
equivalent evidence [$P(D_{i})=P(D_{j})$]. A comparison between models then
yields:
$\frac{P(M_{i}|D_{i}(n))}{P(M_{j}|D_{i}(n))}=\frac{P(D_{i}(n)|M_{i})}{P(D_{i}(n)|M_{j})}=\displaystyle\prod_{k=1}^{n}\frac{P(x_{i,k}|M_{i})}{P(x_{i,k}|M_{j})},$
(2)
where the probability of a particular data point given a specific model,
$P(x_{i,k}|M_{j})$, is described by the convolution of the point spread
function (PSF–$S$) of the detector with the probability distribution function
of the model in question. That is
$P(x_{i,k}|M_{j})=\displaystyle\sum_{l=1}^{q}S(x_{i,k}|\textrm{\small{pixel}}_{l})\cdot
P(\textrm{\small{pixel}}_{l}|M_{j}),$ (3)
where the sum is being performed on each pixel ($\textrm{\small{pixel}}_{l}$)
for all $q$ pixels.
The PSFs are assumed here to be gaussian in each coordinate direction,
characterized by standard deviations in distance, right ascension, and
declination: $\sigma_{\rm high}=[5\%,1^{\circ},1^{\circ}]$, $\sigma_{\rm
med}=[30\%,2^{\circ},2^{\circ}]$, $\sigma_{\rm
low}=[50\%,4^{\circ},4^{\circ}]$. These reflect different assumptions for the
high, medium, and low accuracy of positional reconstruction for gravitational-
wave detections. The exact parameter-estimation accuracy is difficult to
predict, since it will depend both on the details of the detector network
(e.g., the relative sensitivity of detectors and their calibration accuracy)
and on the specifics of individual events (their signal-to-noise ratio, and
the sky location and orientation of the binary). Therefore, these three
assumptions should be considered only as possible predictions for typical
accuracies. Thus, low accuracies may be typical for events detected with a
three-detector LIGO/Virgo network at the threshold of detectability.
Meanwhile, the addition of a fourth interferometer, such as a possible AIGO
detector in Australia or LGCT in Japan, could significantly enhance the sky
localization accuracy and moderately improve distance sensitivity (Fairhurst
et al. 2010), making medium-accuracy measurements typical and high-accuracy
measurements possible.
In these calculations, we compare hypothetical GW observations with models of
compact binary distributions. This comparison is being made assuming that the
local dark matter distribution is perfectly known. In reality, this is not the
case; and the results presented here are thus optimistic. In the future, the
comparison between model and observation should be refined to include the
local distribution of light (e.g. galaxies) rather than dark matter halos.
Table 1 summarizes the ability of GW observatories to discern the kick
velocity distribution of the merging binaries from the reconstructed angular
positions and distances (assumed to be determined without a galaxy host
association). Two sample volumes are considered: 40 and 80 Mpc. This is done
in order to understand the sensitivity of our results to the uncertainty in
physical separation which, for a fixed angular resolution, varies with
distance. We find that $\sim 50$ events are required to distinguish between
the lowest and highest kick velocity scenarios for moderate detector
accuracies, irrespective of which sample volume is examined. For low detector
accuracies, $50-350$ detections are necessary444 It is important to note that
the number of detections required is highly sensitive to the model from which
the data is drawn, not simply on which models are being contrasted.. Thus, a
distinction between the two extreme models is possible once advanced detectors
come online, with an expected event rate of $\sim$ 40 per year for detections
of binary neutron star mergers (Abadie et al., 2010).555The event rate
estimates have significant uncertainties, and range from pessimistic estimates
of $\sim$0.4 events per year to optimistic estimates of $\sim$400 events per
year (Abadie et al., 2010). Meanwhile, distinguishing between the two low-kick
scenarios is very difficult, if not impossible, until the era of third-
generation detectors. The addition of a fourth GW detector to the LIGO/Virgo
network would significantly improve source localization, and thereby the
accuracy with which event distributions could be distinguished.
Assuming a LIGO/Virgo horizon of $\sim 400$ Mpc, only $\sim$ 10% of all
detected mergers would take place within 80 Mpc. With a constant angular
resolution, the uncertainty in physical position is proportional to the
event’s distance, suggesting that using events at greater distances leads to a
degradation in the ability to distinguish between kick-velocity models.
Although we find no clear increase in the number of required detections
between the 40 and 80 Mpc samples, further investigation is required to assess
the effects of a larger sample volume.
## 5\. Summary
In this Letter, we use dark matter cosmological simulations to examine the
full three-dimensional distribution of coalescing compact binaries in the
local universe under the following assumptions. First, we assume a single
epoch of star formation and a simple star formation recipe; that is, the
contribution of a particular halo to the total star formation is directly
proportional to its dark matter mass. Although a more realistic treatment of
star formation should be considered, we do not expect that our qualitative
results will change significantly. Second, we assume an isotropic natal kick
velocity distribution, whose properties are invariant of initial binary
separation. Under this assumption, the merging time is independent of the kick
velocity. This is found to be a reasonable approximation in binary population
synthesis models, which helps justify our single epoch of tracer injection.
Third, our comparisons between kick velocity models in §4 assume a perfect
knowledge of the local dark matter distribution, when in actuality this
distribution would have to be deduced from the observable, local universe.
Finally, due to computational constraints, only an 80 Mpc region of the
expected 400 Mpc horizon of advanced LIGO/Virgo has been modeled. Despite the
increased uncertainty in the true-distance offset between host and merger at
such distances, the difference between our 40 and 80 Mpc results (Table 1)
suggests that our methods could remain effective in deducing the kick velocity
distribution with a reasonable number of detections. Keeping these assumptions
in mind, it is still evident that the use of static, non-evolving potentials
for individual hosts at the time of binary formation severely overestimates
the retention of all but the lowest barycentric velocity systems (Fryer et
al., 1999; Belczyński et al., 2000; Rosswog et al., 2003; Bloom et al., 1999;
Bulik et al., 1999; Portegies Zwart & Yungelson, 1998).
Static calculations predict that the distribution of gravitational wave
sources in the sky should closely trace the distribution of galaxies. An
accurate inclusion of evolving host halo potentials in cosmological
simulations have shown this to be inaccurate (Zemp et al., 2009). In fact, we
show that not only do the distributions of merging compact binaries extend
well beyond their birth halo, but variations in kick velocity lead to marked
differences in their sky distributions. The repercussions of this result are
twofold. On one hand, we find that the variation in the projected distribution
of double compact objects with different natal kick-velocities should be
distinguishable with the expected accuracies of GW observatories. In
principle, this will allow important information on the formation and
evolution of the binary progenitor to be deciphered from the distribution of
GW detections alone. On the other hand, the fact that the distribution of
merging binaries does not accurately trace the locations of their birth halos
complicates redshift determination. Having said this, the presence of a binary
distribution extending well beyond the half-light radius of their hosts
suggests that associating optical counterparts to GW events could be easier as
they are less likely to be drowned out by their host galaxy’s light. This is
particularly important as the optical counterparts are predicted to be
relatively dim (Li & Paczyński, 1998; Rosswog & Ramirez-Ruiz, 2002; Kulkarni,
2005; Metzger et al., 2010).
Gravitational waves offer the possibility of casting proverbial light on
otherwise invisible phenomena; they will – by their very nature – tell us
about events where large quantitites of mass move in such small regions that
they are utterly opaque and forever hidden from direct electromagnetic probing
(see, e.g. Lee & Ramirez-Ruiz, 2007). A time-integrated luminosity of the
order of a fraction of a solar rest mass is predicted from merging compact
binaries. Ground-based facilities, like LIGO, GEO600 and Virgo, will be
searching for these stellar-remnant mergers in the local universe. The
distribution of merger sites is thus of considerable importance to GW
observatories. The proposed use of galaxy catalogs as priors when passing
triggers from possible GW detections to point telescopes for electromagnetic
follow-ups will need to account for the possibility of mergers away from the
observed galaxies. Using cosmological simulations of structure formation, the
local sky distributions are found to vary with the kick velocity distributions
of the progenitor systems, allowing a determination of the cosmography of
massive binary stars. Despite the fact that individual detections lack the
positional accuracy of electromagnetic observations, it may be possible to
strengthen the case for (or against) high natal kick velocities based solely
on GW observations. The addition of more gravitational-wave detectors to the
LIGO/Virgo network will greatly improve our ability to distinguish between
models with different kick velocity distributions by improving the positional
reconstruction of individual events.
We thank C. Fryer, V. Kalogera and R. O Shaughnessy for useful discussions and
the referee for constructive comments. We acknowledge support from NASA
NNX08AN88G and NNX10AI20G (L.Z.K. and E.R.), the David and Lucile Packard
Foundation (E.R.); NSF grants: AST-0847563 (L.Z.K. and E.R.), AST-0708087
(M.Z.), AST-0901985 (I.M.); and the Swiss National Science Foundation (J.D.).
Computations were performed on the Pleaides UCSC computer cluster.
## References
* Abadie et al. (2010) Abadie, J., et al. 2010, Classical and Quantum Gravity, 27, 173001
* Belczyński et al. (2000) Belczyński, K., Bulik, T., & Zbijewski, W. 2000, A&A, 355, 479
* Bloom et al. (1999) Bloom, J. S., Sigurdsson, S., & Pols, O. R. 1999, MNRAS, 305, 763
* Blumenthal et al. (1984) Blumenthal, G. R., Faber, S. M., Primack, J. R., & Rees, M. J. 1984, Nature, 311, 517
* Brandt & Podsiadlowski (1995) Brandt, N., & Podsiadlowski, P. 1995, MNRAS, 274, 461
* Bulik et al. (1999) Bulik, T., Belczyński, K., & Zbijewski, W. 1999, MNRAS, 309, 629
* Cutler & Flanagan (1994) Cutler, C., & Flanagan, É. E. 1994, Phys. Rev. D, 49, 2658
* Fairhurst (2009) Fairhurst, S. 2009, New Journal of Physics, 11, 123006
* Fairhurst, S., et. al. (2010) Fairhurst, S., et. al. 2010, in preparation
* Fryer & Kalogera (1997) Fryer, C., & Kalogera, V. 1997, ApJ, 489, 244
* Fryer et al. (1999) Fryer, C. L., Woosley, S. E., & Hartmann, D. H. 1999, ApJ, 526, 152
* Guetta & Piran (2005) Guetta, D., & Piran, T. 2005, A&A, 435, 421
* Hoffman et al. (2008) Hoffman, Y., Martinez-Vaquero, L. A., Yepes, G., & Gottlöber, S. 2008, MNRAS, 386, 390
* Hughes (2009) Hughes, S. A. 2009, ARA&A, 47, 107
* Kulkarni (2005) Kulkarni, S. R. 2005, ArXiv Astrophysics e-prints (arXiv:astro-ph/0510256)
* Lee & Ramirez-Ruiz (2007) Lee, W. H., & Ramirez-Ruiz, E. 2007, New Journal of Physics, 9, 17
* Li & Paczyński (1998) Li, L., & Paczyński, B. 1998, ApJ, 507, L59
* Madau et al. (1996) Madau, P., Ferguson, H. C., Dickinson, M. E., Giavalisco, M., Steidel, C. C., & Fruchter, A. 1996, MNRAS, 283, 1388
* Madau et al. (1998) Madau, P., Pozzetti, L., & Dickinson, M. 1998, ApJ, 498, 106
* Mandel (2010) Mandel, I. 2010, Phys. Rev. D, 81, 084029
* Mandel & O’Shaughnessy (2010) Mandel, I., & O’Shaughnessy, R. 2010, Classical and Quantum Gravity, 27, 114007
* Metzger et al. (2010) Metzger, B. D., et al. 2010, MNRAS, 406, 2650
* O’Shaughnessy et al. (2010) O’Shaughnessy, R., Kalogera, V., & Belczynski, K. 2010, ApJ, 716, 615
* O’Shaughnessy et al. (2008) O’Shaughnessy, R., Kim, C., Kalogera, V., & Belczynski, K. 2008, ApJ, 672, 479
* Phinney (1991) Phinney, E. S. 1991, ApJ, 380, L17
* Portegies Zwart & Yungelson (1998) Portegies Zwart, S. F., & Yungelson, L. R. 1998, A&A, 332, 173
* Rosswog & Ramirez-Ruiz (2002) Rosswog, S., & Ramirez-Ruiz, E. 2002, MNRAS, 336, L7
* Rosswog et al. (2003) Rosswog, S., Ramirez-Ruiz, E., & Davies, M. B. 2003, MNRAS, 345, 1077
* Spergel et al. (2007) Spergel, D. N., et al. 2007, ApJS, 170, 377
* Stadel (2001) Stadel, J. G. 2001, PhD thesis, University of Washington
* van der Sluys et al. (2008) van der Sluys, M. V., et al. 2008, ApJ, 688, L61
* Zemp et al. (2009) Zemp, M., Ramirez-Ruiz, E., & Diemand, J. 2009, ApJ, 705, L186
| | $M_{360}$($D_{360}$) | $M_{180}$($D_{180}$) | $M_{90}$($D_{90}$)
---|---|---|---|---
Dist | PSF Accuracy | vs. $M_{180}$ | vs. $M_{90}$ | vs. $M_{360}$ | vs. $M_{90}$ | vs. $M_{360}$ | vs. $M_{180}$
$\leq$ 80Mpc | High | 22 | 16 | 26 | $>1000$ | 22 | 282
Med | 73 | 39 | 35 | $>1000$ | 31 | 384
Low | $>1000$ | 349 | 52 | $>1000$ | 50 | 881
$\leq$ 40Mpc | High | 27 | 17 | 34 | $>1000$ | 23 | $>1000$
Med | 78 | 46 | 40 | $>1000$ | 37 | $>1000$
Low | 146 | 137 | 56 | $>1000$ | 56 | $>1000$
Table 1Number of detections required to achieve 99% confidence in the correct
model for $90\%$ ($\frac{45}{50}$) of data sets. These results are compared
between two different sample radii, and three different detector accuracies
characterized by standard deviations (in distance, right ascension,
declination) of: $\sigma_{high}=\\{5\%,1^{\circ},1^{\circ}\\}$,
$\sigma_{med}=\\{30\%,2^{\circ},2^{\circ}\\}$,
$\sigma_{low}=\\{50\%,4^{\circ},4^{\circ}\\}$. Entries marked ‘$>1000$’ failed
to reach the desired confidence in the required number of data sets within the
1000 data points used. Figure 1.— Tracer vs. dark matter distribution in a
local-like universe as a function of barycentric kick-velocity. Integrated
particle mass in uniform radial-width shells is plotted versus distance from a
solar-equivalent offset from the Milky Way center. The vertical axes are
plotted in arbitrary units of number per unit length, with tracers normalized
with respect to the total tracer population as described in §2.1. As the kick-
velocity increases from 90 $\textrm{km s}^{-1}$ (top panel) to 360 $\textrm{km
s}^{-1}$ (bottom panel), a noticeable portion of tracers becomes delocalized,
forming a background and mixing populations. Figure 2.— Cumulative
distribution of tracers and dark matter within a given distance as a function
of kick-velocity. The vertical axes are plotted in arbitrary units of number
per unit volume, with tracers normalized self-consistently. Although the
number of tracers in the central halo is noticeably lower for the highest
kick-velocity model, the difference is negligible once the volume reaches the
Virgo-like cluster, where the background distribution of tracers outweighs
changes in local distributions. The number of merging binaries is assumed to
be proportional to the mass of the host halo. Figure 3.— Sky maps of dark
matter (first column) and tracers with highest and lowest kick velocity
scenarios (second and third columns, respectively) as a function of distance.
Figures make use of Hammer projections with $2^{\circ}$x $2^{\circ}$ bins.
Densities are plotted in units of column density, scaled to the maximum
densities of the dark matter and normalized tracer distributions
independently. Pixels with no tracers or dark matter are white, corresponding
to densities less than the resolution of the simulation. Although tracer peak
densities remain relatively unchanged, a tracer-background forms as the kick
velocity approaches the escape velocity. At 90 $\textrm{km s}^{-1}$ the
tracers follow only the dark matter overdensities, as does the light-
distrubtion. The distributions approach isotropy slower than the dark matter
distribution. Differences in distribution are clearly apparent. Note the
logarithmic color scale.
|
arxiv-papers
| 2010-11-04T20:00:01 |
2024-09-04T02:49:14.506676
|
{
"license": "Public Domain",
"authors": "Luke Zoltan Kelley, Enrico Ramirez-Ruiz, Marcel Zemp, J\\\"urg Diemand,\n and Ilya Mandel",
"submitter": "Luke Zoltan Kelley",
"url": "https://arxiv.org/abs/1011.1256"
}
|
1011.1349
|
# An upper bound on the total inelastic cross-section as a function of the
total cross-section
Tai Tsun Wu ttwu@seas.harvard.edu ,tai.tsun.wu@cern.ch Harvard University,
Cambridge, Massachusetts,and CERN,Geneva André Martin martina@mail.cern.ch
Theoretical Physics Division,CERN, Geneva Shasanka Mohan Roy
shasanka1@yahoo.co.in Homi Bhabha Centre for Science Education, TIFR, V. N.
Purav Marg, Mankhurd, Mumbai - 400 088. Virendra Singh
vsingh@theory.tifr.res.in Tata Institute of Fundamental Research, Mumbai
400005
###### Abstract
Recently André Martin has proved a rigorous upper bound on the inelastic
cross-section $\sigma_{inel}$ at high energy which is one-fourth of the known
Froissart-Martin-Lukaszuk upper bound on $\sigma_{tot}$. Here we obtain an
upper bound on $\sigma_{inel}$ in terms of $\sigma_{tot}$ and show that the
Martin bound on $\sigma_{inel}$ is improved significantly with this added
information.
###### pacs:
03.67.-a, 03.65.Ud, 42.50.-p
## 1\. Introduction
The total cross-section $\sigma_{tot}(s)$ for two particles to go to anything
at c.m. energy $\sqrt{s}$ must obey the Froissart-Martin bound,
$\sigma_{tot}(s)\leq_{s\rightarrow\infty}C\>[\ln(s/s_{0})]^{2}$ (1)
proved at first from the Mandelstam representation by Froissart Froissart1961
and later from the basic principles of axiomatic field theory by Martin
Martin1966 . Of the two unknown constants the constant $C$ was fixed by
Lukaszuk-Martin1967 to obtain,
$\sigma_{tot}(s)\leq_{s\rightarrow\infty}4\pi/t_{0}\>[\ln(s/s_{0})]^{2},$ (2)
where, $t=t_{0}$ is the lowest singularity in the $t$-channel. For many
physically interesting cases such as $\pi\pi,KK,K\overline{K},\pi K,\pi
N,\pi\Lambda$ scattering $t_{0}=4m_{\pi}^{2}-\epsilon$, $\epsilon$ being an
arbitrary small positive constant, and $m_{\pi}$ the pion-mass Bessis-
Glaser1967 . In some cases we can take $\epsilon=0$ , e.g. for pion-pion
scattering if the D-wave scattering length is finite Colangelo2000 . It will
be convenient to denote the right-hand side of the bound on $\sigma_{tot(s)}$
as
$\sigma_{max}(s)=4\pi/t_{0}\>[\ln(s/s_{0})]^{2}.$ (3)
In equation (3) $s_{0}$ is unknown. However, if one assumes that the total and
elastic cross-sections are increasing beyond a certain energy, or if one works
with cross-sections averaged over a certain energy interval, one can, using
fixed $t$ dispersion relations, fix the scale Martin2010a . A reasonable guess
is that $s_{0}$ lies between the square of the pion mass and the square of the
nucleon mass. This means an uncertainty of $\pm$ 10% at the present energy of
the LHC.
The Froissart-Martin bound has been seminal both to the development of the
field of high energy theorems in axiomatic field theory (see e.g. the review
Roy1972 )and to that of phenomenological models leading to accurate
predictions of total and elastic cross sections before their experimental
measurements Cheng-Wu1970 . Remarkably, one of us (A. M.) has recently
obtained a bound on the total inelastic cross section at high energy
Martin2009 ,
$\sigma_{inel}(s)\leq_{s\rightarrow\infty}\pi/t_{0}\>[\ln(s/s_{0})]^{2},$ (4)
which is one-fourth of the bound $\sigma_{max}(s)$ on the total cross-section,
thus improving the simple bound $\sigma_{inel}\leq\sigma_{tot}$.
The present paper is inspired by Martin’s bound on the inelastic cross-
section. In fact T. T. Wu Wu2009 by extending Martin’s variational
calculation to incorporate a given total cross-section and independently S.M
.Roy and Virendra Singh Roy2009 , by exploiting their previous upper bound on
the differential cross section in terms of elastic cross-section, Singh-
Roy1970 ,Roy1972a realized that one could solve a more general problem: find
a bound on the inelastic cross-section as a function of the value of the total
cross-section. It is obvious that if the total cross section vanishes the
inelastic cross section also vanishes. but it is also extremely plausible that
if one maximizes the total cross section, the important partial wave
amplitudes will be imaginary and maximal so that, from the unitarity
condition, there is no room left for the inelastic cross section which will
receive only negligible contributions from the tail of the partial wave
distribution.
The net result exhibiting both these features is the bound we present in this
paper,
$\Sigma_{inel}(s)\leq_{s\rightarrow\infty}\Sigma_{tot}(s)\bigl{(}1-\Sigma_{tot}(s)\bigr{)},$
(5)
where,
$\Sigma_{tot}(s)\equiv\sigma_{tot}(s)/\sigma_{max}(s),$ (6)
and
$\Sigma_{inel}(s)\equiv\sigma_{inel}(s)/\sigma_{max}(s).$ (7)
Maximizing wth respect to $\sigma_{tot}$ we get the factor 1/4 announced at
the beginning of this paper,i.e.
$\sigma_{inel}(s)\leq_{s\rightarrow\infty}\sigma_{max}(s)/4.$ (8)
In Sec. 2 we summarise our notations and recall the basic results from
axiomatic field theory. We then present two possible derivations of the bound
on the inelastic cross-section in terms of total cross-section, the direct
variational approach in Sec. 3, and the approach using the 1970 bound on the
differential cross section in terms of the elastic cross-section Singh-Roy1970
, Roy1972a in Sec. 4. Sec. 5 contains concluding remarks including directions
for future work on high energy phenomenology.
## 2\. Basic Results from Axiomatic Field Theory
Let $F(s,t)$ be the elastic scattering amplitude for $ab\rightarrow ab$ at
c.m. energy $\sqrt{s}$ and momentum transfer squared $t$ and be normalized
such that the differential cross-section is given by
$\frac{d\sigma}{d\Omega}(s,t)=\bigl{|}\frac{F(s,t)}{\sqrt{s}}\bigr{|}^{2}$ (9)
with $t$ being given in terms of the c.m. momentum $k$ and the scattering
angle $\theta$ by the relation,
$t=-2k^{2}(1-\cos\theta).$ (10)
Then, for fixed $s$ larger than the physical $s-$channel threshold,
$F(s;\cos\theta)\equiv F(s,t)$ is analytic in the complex $\cos\theta$ -plane
inside the Lehmann-Martin ellipse Lehmann1958 , Martin1966 , with foci -1 and
+1 and semi-major axis $\cos\theta_{0}=1+t_{0}/(2k^{2})$, where $t_{0}$ is
independent of $s$. In fact, as mentioned already,
$t_{0}=4m_{\pi}^{2}-\epsilon$ for many interesting cases. Within the ellipse
$F(s,t)$ has the partial wave expansion,
$F(s,t)=\frac{\sqrt{s}}{k}\sum_{l=0}^{\infty}(2l+1)a_{l}(s)P_{l}(1+t/(2k^{2})),$
(11)
which converges absolutely and uniformly in $t$ for $|t|<t_{0}$ ; hence
$F(s,t)$ is analytic in $t$ for $|t|<t_{0}$ . Unitarity implies that,
$Ima_{l}(s)\geq|a_{l}(s)|^{2}$ (12)
in the physical region. Further, Jin-Martin1964 for fixed $t$ in the region
$|t|<t_{0}$ , $F(s,t)$ satisfies dispersion relations in $s$ with two
subtractions. This implies, in particular, that the $s$-channel absorptive
part for $0\leq t<t_{0}$ has the convergent partial wave expansion,
$\displaystyle A(s,t)\equiv ImF(s,t)$
$\displaystyle=\frac{\sqrt{s}}{k}\sum_{l=0}^{\infty}(2l+1)Ima_{l}(s)P_{l}(1+t/(2k^{2})),$
(13)
and obeys
$\int_{C}^{\infty}dsA(s,t)/s^{3}<\infty,\>0\leq t<t_{0}.$ (14)
Hence, if we assume that $A(s,t)$ is continuous in $s$, there exist sequences
of $s\rightarrow\infty$ such that
$A(s,t)<Const.\frac{s^{2}}{\ln(s/s_{0})},\>0\leq t<t_{0}.$ (15)
For simplicity, in this paper, we deduce asymptotic bounds on
$\sigma_{inel}(s)$ only for such sequences. Bounds on energy averages will be
considered later to avoid this restriction.
## 3\. Variational Bound on Inelastic Cross-section in terms of Total Cross-
section
Since $\sigma_{inel}=\sigma_{tot}-\sigma_{el}$, this problem is equivalent to
finding a lower bound on $\sigma{el}$. Further,
$\displaystyle\sigma_{el}(s)=\frac{4\pi}{k^{2}}\sum_{l=0}^{\infty}(2l+1)|a_{l}(s)|^{2}$
$\displaystyle\geq\frac{4\pi}{k^{2}}\sum_{l=0}^{\infty}(2l+1)(Ima_{l}(s))^{2}\equiv\sigma_{el,im}(s).$
(16)
So, it suffices to find a variational lower bound on $\sigma_{el,im}$. We vary
the $Ima_{l}(s)$ subject to the unitarity constraints
$Ima_{l}(s)\geq\>0\>,$ (17)
to a given value of,
$\sigma_{tot}(s)=\frac{4\pi}{k^{2}}\sum_{l=0}^{\infty}(2l+1)Ima_{l}(s)\>,$
(18)
and to the constraint
$\displaystyle
A(s,t_{0})\equiv\frac{\sqrt{s}}{k}\sum_{l=0}^{\infty}(2l+1)Ima_{l}(s)P_{l}(1+t_{0}/(2k^{2}))$
$\displaystyle<Const.\frac{s^{2}}{\ln(s/s_{0})}\>.$ (19)
For simplicity, since we work at a fixed-$s$ , we suppress the $s$-dependence
of $Ima_{l}(s)$, $\sigma_{el,im}(s)$ and $\sigma_{tot}(s)$ . Denoting ,
$z_{0}=1+t_{0}/(2k^{2}),$ (20)
the lower bound on $\sigma_{el,im}$ is obtained by choosing,
$Ima_{l}=\alpha\bigl{(}1-P_{l}(z_{0})/P_{L+r}(z_{0})\bigr{)},for\>0\leq l\leq
L\>,$ (21)
and,
$Ima_{l}=0,for\>l>L\>,$ (22)
with the constants $0\leq r<1,\alpha>0$ and the positive integer $L$ being
fixed from the given value of $\sigma_{tot}$ and the given upper bound on
$A(s,t_{0})$. We omit the straight forward proof which is by direct
subtraction of a $\sigma_{el,im}$ with arbitrary partial waves obeying the
given constraints from the variational result. After carrying out the
summations over $l$, the constraint equations become,
$\displaystyle\sigma_{tot}\>k^{2}/(4\pi\alpha)=$
$\displaystyle(L+1)^{2}-(P_{L+1}^{{}^{\prime}}(z_{0})+P_{L}^{{}^{\prime}}(z_{0}))/P_{L+r}(z_{0})\>,$
(23)
and
$\displaystyle
A(s,t_{0})\frac{k}{\alpha\sqrt{s}}=P_{L+1}^{{}^{\prime}}(z_{0})+P_{L}^{{}^{\prime}}(z_{0})-$
$\displaystyle\frac{(L+1)^{2}P_{L}^{2}(z_{0})-(z_{0}^{2}-1)(P_{L}^{{}^{\prime}}(z_{0}))^{2}}{P_{L+r}(z_{0})},$
(24)
and the bound on $\sigma_{el}$ becomes,
$\displaystyle
k^{2}/(4\pi\alpha^{2})\>\sigma_{el}\geq\>k^{2}/(4\pi\alpha^{2})\>\sigma_{el,im}\geq$
$\displaystyle(L+1)^{2}-2(P_{L+1}^{{}^{\prime}}(z_{0})+P_{L}^{{}^{\prime}}(z_{0}))/P_{L+r}(z_{0})+$
$\displaystyle\frac{(L+1)^{2}P_{L}^{2}(z_{0})-(z_{0}^{2}-1)(P_{L}^{{}^{\prime}}(z_{0}))^{2}}{(P_{L+r}(z_{0}))^{2}}.$
(25)
At high energies, using $s/\sigma_{tot}\rightarrow\infty$, the two constraint
equations yield easily that $L=O(\sqrt{s}\ln(s/s_{0}))$ ; we may therefore set
$r=0$ and use the following approximations for the Legendre polynomials,
$\displaystyle P_{L}(z_{0})$ $\displaystyle=$ $\displaystyle
I_{0}(\xi)(1+O(L/s)),\xi\equiv(2L+1)\sqrt{(z_{0}-1)/2}$ $\displaystyle
P_{L}^{{}^{\prime}}(z_{0})$ $\displaystyle=$
$\displaystyle(1/2)L\sqrt{s/t_{0}}I_{1}(\xi)(1+O(L/s)),$ $\displaystyle
I_{\nu}(\xi)$ $\displaystyle=$
$\displaystyle\frac{\exp{\xi}}{\sqrt{2\pi\xi}}(1-(4\nu^{2}-1)/(8\xi)+...),\xi\rightarrow\infty,$
$\displaystyle for$ $\displaystyle
s\rightarrow\infty,\>L/\sqrt{s}\rightarrow\infty,\>L/s\rightarrow 0\>,$ (26)
where the $I_{\nu}(\xi)$ denote the modified Bessel functions. We then have,
$\displaystyle\sigma_{tot}\>k^{2}/(4\pi\alpha)\approx\>L^{2}-\frac{I_{1}(\xi)}{I_{0}(\xi)}L\sqrt{s/t_{0}}$
(27) $\displaystyle\approx\>L^{2}(1+O(\sqrt{s}/L)),$ (28) $\displaystyle
A(s,t_{0})\frac{k}{\alpha\sqrt{s}}\approx
I_{1}(\xi)L\sqrt{s/t_{0}}+L^{2}\frac{I_{1}(\xi)^{2}-I_{0}(\xi)^{2}}{I_{0}(\xi)}$
(29) $\displaystyle\approx
I_{0}(\xi)\frac{L\sqrt{s}}{2\sqrt{t_{0}}}(1+O(\sqrt{s}/L)).$ (30)
The asymptotic bounds on elastic and inelastic cross-sections become, with
these approximations,
$k^{2}/(4\pi\alpha^{2})\>\sigma_{el}\geq\>L^{2}-2L\sqrt{s/t_{0}}\frac{I_{1}(\xi)}{I_{0}(\xi)}+L^{2}(1-(\frac{I_{1}(\xi)}{I_{0}(\xi)})^{2}),$
(31)
and
$\displaystyle
k^{2}/(4\pi\alpha)\>\sigma_{inel}\leq\>(1-2\alpha))(L^{2}-L\sqrt{s/t_{0}}\frac{I_{1}(\xi)}{I_{0}(\xi)})+$
$\displaystyle\alpha L^{2}(\frac{I_{1}(\xi)}{I_{0}(\xi)})^{2}.$ (32)
We now use the assumed upper bound on $A(s,t_{0})$ to evaluate $L,\alpha$ for
high energies. We have,
$Const.s/(\sigma_{tot}\ln(s/s_{0}))=I_{0}(\xi)\frac{\sqrt{s}}{2L\sqrt{t_{0}}}(1+O(\sqrt{s}/L)),$
(33)
which yields ,
$\displaystyle\frac{L}{\sqrt{s}}=(1/(2\sqrt{t_{0}}))\ln(\frac{s}{s_{0}^{2}\sigma_{tot}})(1+O(\ln(s/s_{0}))^{-1})$
(34)
$\displaystyle\alpha=\frac{\sigma_{tot}(s)}{\hat{\sigma}_{tot}(s)}(1+O(\ln(s/s_{0}))^{-1})$
(35)
where,
$\hat{\sigma}_{tot}(s)\equiv
4\pi/t_{0}\>[\ln(\frac{s}{s_{0}^{2}\sigma_{tot}})]^{2}.$ (36)
Hence, we have the lower bound on elastic cross-sections,
$\sigma_{el}(s)\geq\frac{(\sigma_{tot}(s))^{2}}{\hat{\sigma}_{tot}(s)}(1+O(\ln(s/s_{0}))^{-1}).$
(37)
Note that $\hat{\sigma}_{tot}(s)$ can be replaced by $\sigma_{max}(s)$ for
$s\rightarrow\infty$, except in the unrealistic case $\sigma_{tot}\rightarrow
0,fors\rightarrow\infty$ which leads to a small inelastic cross-section
$\sigma_{inel}(s)\rightarrow 0$. Hence, using equation (37), the upper bound
on the inelastic cross-section valid in all cases is ,
$\sigma_{inel}(s)\leq_{s\rightarrow\infty}\sigma_{tot}(s)\bigl{(}1-\Sigma_{tot}(s)\bigr{)},$
(38)
which leads to the announced bound on $\sigma_{inel}(s)/\sigma_{max}(s)$,
given by equation (5).
## 4\. Upper Bound on Inelastic Cross-section from an Upper Bound on
Differential Cross-section in terms of Elastic Cross-section
We show here that the inelastic cross-section bound can also be derived as a
corollary of an upper bound on the differential cross section in terms of the
elastic cross-section, established by two of us Singh-Roy1970 many years ago,
$\frac{d\sigma}{dt}(s,t=0)\leq_{s\rightarrow\infty}\frac{\sigma_{el}(s)}{4t_{0}}[\ln(\frac{s}{s_{0}^{2}\sigma_{el}})]^{2}.$
(39)
This bound can also be written as Roy1972a ,
$\displaystyle\sigma_{tot}[1+\bigl{(}\frac{ReF(s,t=0)}{ImF(s,t=0)}\bigr{)}^{2}]$
$\displaystyle\leq_{s\rightarrow\infty}\frac{4\pi\sigma_{el}}{t_{0}\sigma_{tot}}[\ln(\frac{s}{s_{0}^{2}\sigma_{el}})]^{2}.$
(40)
If the real part $ReF(s,t=0)$ is unknown we have the weaker bound,
$\displaystyle\sigma_{tot}$ $\displaystyle\leq_{s\rightarrow\infty}$
$\displaystyle\sqrt{\frac{4\pi}{t_{0}}}\bigl{[}\sqrt{\sigma_{el}}\bigl{(}\ln(\frac{s}{s_{0}^{2}\sigma_{tot}})-\ln(\frac{\sigma{el}}{\sigma_{tot}})\bigr{)}\bigr{]}$
(41) $\displaystyle\leq_{s\rightarrow\infty}$
$\displaystyle\sqrt{\sigma_{el}\hat{\sigma}_{tot}}+(2/e)\sqrt{\frac{4\pi\sigma_{tot}}{t_{0}}}\>,$
where, in the last line we have used the elementary inequality,$\sqrt{x}\ln
x\geq-2/e,for\>0<x<1$. This equation yields a lower bound on $\sigma_{el}(s)$
for any asymptotic behaviour of $\sigma_{tot}(s)$. As noted in the last
section,for deducing an upper bound on $\sigma_{inel}(s)$, it suffices to
assume that $\sigma_{tot}$ does not vanish for ${s\rightarrow\infty}$. In
particular, if $\sigma_{tot}(s)>16\pi/(e^{2}t_{0})$, we have
$\displaystyle\sigma_{tot}(\sqrt{\sigma_{tot}}-(2/e)\sqrt{\frac{4\pi}{t_{0}}}\>)^{2}$
$\displaystyle\leq_{s\rightarrow\infty}$
$\displaystyle\sigma_{el}\hat{\sigma}_{tot}$ (42)
$\displaystyle\approx_{s\rightarrow\infty}$
$\displaystyle\sigma_{el}\sigma_{max}$
and hence the upper bound on the inelastic cross-section,
$\sigma_{inel}\leq_{s\rightarrow\infty}\sigma_{tot}\bigl{[}1-\Sigma_{tot}(1-(2/e)\sqrt{\frac{4\pi}{t_{0}\sigma_{tot}}}\>)^{2}\bigr{]},$
(43)
which yields the desired bound (5) on the inelastic cross-section if
$\sigma_{tot}(s)\rightarrow\infty,for\>s\rightarrow\infty$.
## 5\. Conclusion
We have derived an asymptotic upper bound on the inelastic cross-section in
terms of the total cross-section which improves Martin’s recent bound
Martin2009 when $\sigma_{tot}(s)\sim C(\ln(s/s_{0}))^{2}$. Varying
$\sigma_{tot}(s)$ over its allowed range we recover Martin’s result
$\sigma_{inel}<\sigma_{max}/4$ for some sequences of $s\rightarrow\infty$
mentioned before.For applications to high energy phenomenology, it is
desirable to remove the unknown scale factor $s_{0}$ in these bounds, as well
as the restriction to special sequences of $s\rightarrow\infty$. One way
forward is to derive bounds on energy averages of $\sigma_{inel}(s)$ given
energy averages of $\sigma_{tot}(s)$ and $A(s,t_{0})$. One of us now has
definitive results on the analogous problem of finding bounds on energy
averages of the inelastic cross-section, as well as of the total cross-section
Martin2010b .
Acknowledgements
S.M.R. is Raja Ramanna Fellow of the Department of Atomic Energy , and V. S.
is INSA Senior Scientist. S. M. R. and V. S. acknowledge support from the
project # 3404 of the Indo-French Centre for promotion of advanced research
(IFCPAR/CEFIPRA); S.M.R., V. S. and T. T. W. thank Luis Alvarez Gaume for
hospitality at CERN. We thank Tullio Basaglia for help in the submission and
the revision of the manuscript.
## References
* (1) M. Froissart, Phys. Rev. 123, 1053 (1961).
* (2) A. Martin, Nuov. Cimen. 42, 930 (1966).
* (3) L. Lukaszuk and A. Martin, Nuov. Cimen. 52A, 122 (1967).
* (4) J. D. Bessis and V. Glaser, Nuov. Cimen. 50, 568 (1967).
* (5) G. Colangelo, J. Gasser, and H. Leutwyler, Phys. Lett.B488,261 (2000).
* (6) A. Martin, talk given at ITEP, Moscow, October 2010, and to be published.
* (7) S. M. Roy, Phys. Reports, 5C, 125 (1972).
* (8) H. Cheng and T. T. Wu, Phys. Rev. Letters 24,1456 (1970); C. Bourrely, J. Soffer, and T. T. Wu, Phys. Rev. D19, 3249 (1979) and Nucl. Phys. B247, 15 (1984). See also, for instance, A. D. Kaidalov, L. A. Ponomarev, and K. A. Ter-Martirosyan, Sov. J. Nucl. Phys. 44, 468 (1986), and A.Donnachie, H.G. Dosch, P.V. Landshoff and O.Nachmann, Pomeron Physics and QCD, Cambridge University Press (2002).
* (9) A. Martin, Phys. Rev. D80, 065013 (2009).
* (10) T. T. Wu, private communication to A. Martin, April 2009, and presentation at Martin’s 80th birthday Fest, Aug. 27, 2009, CERN-GENEVA.
* (11) S. M. Roy, private communication to A. Martin, July 2009.
* (12) V. Singh and S. M. Roy, Ann. Phys. 57, 461 (1970).
* (13) S. M. Roy, Phys. Reports 5C, 125 (1972), p.146, Eq. (4.6b).
* (14) H. Lehmann, Nuovo Cimen. 10, 579 (1958); Fortschr. Physik 6 159 (1959).
* (15) Y. S. Jin and A. Martin, Phys. Rev. 135B, 1375(1964).
* (16) A. Martin, in preparation.
|
arxiv-papers
| 2010-11-05T09:05:47 |
2024-09-04T02:49:14.519010
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Tai Tsun Wu, Andr\\'e Martin, Shasanka Mohan Roy, Virendra Singh",
"submitter": "Andre Martin J",
"url": "https://arxiv.org/abs/1011.1349"
}
|
1011.1410
|
# High-spatial-resolution imaging of thermal emission
from debris disks
Margaret M. Moerchen11affiliation: European Southern Observatory, Alonso de
Córdova 3107, Santiago, Chile 22affiliation: University of Florida, Department
of Astronomy, 211 Bryant Space Science Center, Gainesville, FL 32611, USA
mmoerche@eso.org Charles M. Telesco22affiliation: University of Florida,
Department of Astronomy, 211 Bryant Space Science Center, Gainesville, FL
32611, USA and Christopher Packham22affiliation: University of Florida,
Department of Astronomy, 211 Bryant Space Science Center, Gainesville, FL
32611, USA
###### Abstract
We have obtained sub-arcsec mid-IR images of a sample of debris disks within
100 pc. For our sample of nineteen A-type debris disk candidates chosen for
their IR excess, we have resolved, for the first time, five sources plus the
previously resolved disk around HD 141569. Two other sources in our sample
have been ruled out as debris disks since the time of sample selection. Three
of the six resolved sources have inferred radii of 1–4 AU (HD 38678, HD 71155,
and HD 181869), and one source has an inferred radius $\sim$10–30 AU (HD
141569). Among the resolved sources with detections of excess IR emission, HD
71155 appears to be comparable in size (r$\sim$2 AU) to the solar system’s
asteroid belt, thus joining $\zeta$ Lep (HD 38678, reported previously) to
comprise the only two resolved sources of that class. Two additional sources
(HD 95418 and HD 139006) show spatial extent that implies disk radii of
$\sim$1–3 AU, although the excess IR fluxes are not formally detected with
better than 2-$\sigma$ significance. For the unresolved sources, the upper
limits on the maximum radii of mid-IR disk emission are in the range
$\sim$1–20 AU, four of which are comparable in radius to the asteroid belt. We
have compared the global color temperatures of the dust to that expected for
the dust in radiative equilibrium at the distances corresponding to the
observed sizes or limits on the sizes. In most cases, the temperatures
estimated via these two methods are comparable, and therefore, we see a
generally consistent picture of the inferred morphology and the global mid-IR
emission. Finally, while our sample size is not statistically significant, we
notice that the older sources ($>$200 Myr) host much warmer dust
(T$\gtrsim$400 K) than younger sources (in the 10s of Myr).
circumstellar matter – infrared: stars – planetary systems
## 1 The Search for Resolved Disks
Studying the structure of circumstellar debris disks is proving to be a
valuable technique for inferring the presence of planets and understanding
physical processes associated with the early evolution of planetary systems
(see Wyatt 2008 for review). The detection of an infrared excess for a main-
sequence star is a strong indication that a debris disk is present. Through
relatively large IR-photometric surveys, we can assess trends in excess IR
luminosity for samples of debris disks that span a large range of ages (e.g.,
Su et al. 2006). By going a step further to image single sources with high
spatial resolution (with currently available facilities, $<$0.5” is considered
“high”), we may discover structures in the dust disk that are indicative of
the physical processes occurring in them.
There are presently several hundred photometric detections of debris disks
(e.g., Oudmaijer et al. 1992; Mannings & Barlow 1998; Zuckerman & Song 2004),
but only a small number ($\sim$20) have been spatially resolved. Debris disks
can be imaged in scattered light at optical or near-IR wavelengths, but such
observations suffer from strong photospheric contamination from the central
star, and a coronagraph typically must be employed (e.g., Weinberger et al.
1999). This problem is largely avoided by observing at mid-IR wavelengths
where there is relatively less emission from the photosphere, and thermal dust
emission dominates.
Recent observations by Currie et al. (2008) demonstrate the expected decay in
disk brightness with age. That work supports the models of Kenyon & Bromley
(2004), in which the disk luminosity rises sharply before slowly declining,
with the peak luminosity occurring at $\sim$10–15 Myr. Although such general
trends in disk brightness with age can be inferred through survey photometric
observations, the location of the dust and the physical processes that sculpt
the disks are poorly understood. For example, HR 4796A and $\beta$ Pic, both A
stars, have nearly the same 18 $\mu$m/25 $\mu$m color (Moerchen et al.,
2007a), and Wyatt (2008) notes that they occupy similar positions in the age-
vs.-24-$\mu$m-excess plot presented by Currie et al. (2008). Despite such
apparent similarities, resolved near-IR and mid-IR images of both sources
reveal significantly different dust distributions: HR 4796A has a well-defined
dust annulus at $\sim$70 AU that is 17 AU wide (Schneider et al., 1999; Wyatt
et al., 1999; Telesco et al., 2000) and the dominant mid-IR-emitting dust disk
of Beta Pic spans 20-120 AU (Lagage & Pantin, 1994; Telesco et al., 2005)
(although optically scattered light reveals a disk extending out to nearly
1500 AU [e.g., Larwood & Kalas 2001]). Thus, the excess flux levels tell only
a small part of the story, but degeneracies such as the example above can
sometimes be broken through imaging observations.
The initial goal of this research was to explore how morphological asymmetries
(or lack thereof) are generated by various physical processes such as
collisions and orbital resonances. Imaging at several wavelengths permits some
assessment of which process may dominate a particular disk or region therein.
The mid-IR regime offers the additional benefit of near-diffraction-limited
observing, and by imaging from the ground, as in this work, we can exploit
large telescopes to achieve the desired high angular resolution. For example,
the $\lambda/D$ diffraction limits at 11.7 $\mu$m and 18.3 $\mu$m at the
7.9-meter Gemini Observatory telescopes are 0.24” and 0.39”, respectively.
## 2 Observations
### 2.1 Source Sample
Seventeen of the nineteen debris disk candidates observed in this program are
associated with the stars listed in Table 2.1. They are main-sequence stars
(with the exception of HD 141569), essentially A-type (B8–A5), all within 100
pc. In the literature, HD 141569 is considered to be a transition disk, since
a significant amount of gas (e.g., CO) has been detected within it (Brittain &
Rettig, 2002; Brittain et al., 2003). All sources were observed either with
MIPS on $Spitzer$ at 24 $\mu$m or with $IRAS$ at 25 $\mu$m. The sources in
this sample were chosen both for their high disk-to-star ratio of excess
emission ($>$1.1) at 24 $\mu$m (Rieke et al., 2005) as well as their high
estimated flux densities at 10 $\mu$m ($>$10 mJy) and 18 $\mu$m ($>$40 mJy)
attributable to dust emission. Two sources, HD 172555 and HD 181296, were
chosen for their $IRAS$-discovered high fractional dust luminosities
(Oudmaijer et al., 1992; Mannings & Barlow, 1998) that were confirmed with
MIPS and/or ISO (Moór et al., 2006). The ages of the sample stars are in the
range $\sim$5–600 Myr. Since the time of sample selection, other works have
demonstrated that two of the targets that we observed (not listed in Table 1)
have infrared excesses that cannot be attributed to debris disk processes.
These instances are reviewed individually in this section, and will not be
discussed in future sections regarding debris disk analysis.
Table 1: Summary of Imaging Observations
Object | Gemini | Program ID | Filter | Timea [s] | Dates Observed
---|---|---|---|---|---
HD 38206 | S | 2005A-Q-2 | N | 900 | 19 Sep 2005
| | | Qa | 900 | 5 Feb 2006
HD 38678 | S | 2005A-Q-2 | N | 900 | 3 Feb 2005
| | | Qa | 900 | 3 Feb 2005
HD 56537 | N | 2006A-Q-10 | N′ | 900 | 4 Apr 2006
| | | Qa | 900 | 4 Apr 2006, 7 Apr 2006
HD 71155 | S | 2005A-Q-2 | N | 900 | 4 Mar 2006
| | | Qa | 900 | 5 Feb 2006
HD 75416 | S | 2005A-Q-2 | Si-5 | 300 | 22 May 2005
| | | Qa | 900 | 22 May 2005
HD 80950 | S | 2005A-Q-2 | N | 900 | 8 Mar 2006
| | | Qa | 900 | 5 Feb 2006
HD 83808 | N | 2006A-Q-10 | N′ | 900 | 4 Apr 2006
| | | Qa | 900 | 6 Apr 2006
HD 95418 | N | 2006A-Q-10 | N′ | 900 | 10 Jun 2006
| | | Qa | 900 | 7 Apr 2006
HD 102647 | N | 2006A-Q-10 | N′ | 960 | 11 Jun 2006
| | | Qa | 900 | 10 May 2006, 15 May 2006
HD 115892 | S | 2005A-Q-2 | Si-5 | 900 | 22 May 2005
| | | Qa | 1500 | 21 May 2005, 22 May 2005
HD 139006 | N | 2006A-Q-10 | N′ | 900 | 29 May 2006
| | | Qa | 900 | 30 Apr 2006, 12 May 2006
HD 141569 | N | 2006A-Q-10 | N′ | 900 | 12 Jun 2006
| | | Qa | 900 | 10 May 2006, 14 May 2006
HD 161868 | N | 2006A-Q-10 | N′ | 900 | 29 May 2006
| | | Qa | 900 | 30 Apr 2006, 20 May 2006
HD 172555 | S | 2007A-Q-23 | Si-5 | 680 | 1 Jul 2007
| | | Qa | 680 | 27 Jun 2007, 1 Jul 2007
HD 178253 | S | 2005A-Q-2 | Si-5 | 900 | 22 May 2005
| | | Qa | 600 | 22 May 2005
HD 181296 | S | 2007A-Q-23 | Si-5 | 700 | 28 Apr 2007
| | | Qa | 700 | 28 Apr 2007
HD 181869 | S | 2005A-Q-2 | N | 900 | 20 Aug 2005
| | | Qa | 1200 | 20 Aug 2005, 19 Mar 2006
Notes– aTime is on-source integration time.
HD 21362– The IRS (Infrared Spectrograph) instrument on $Spitzer$ obtained
low-resolution spectra of HD 21362 that showed several hydrogen emission lines
that are indicative of free-free radiation from an ionized stellar wind. Su et
al. (2006) concluded that the infrared excess previously thought to be due to
a debris disk presence is actually due to a fast-rotating B-type star with a
strong stellar wind creating a circumstellar gas disk (also known as the Be
phenomenon.) HD 21362 is not resolved in our images, and its measured flux
densities are 683 $\pm$ 68 mJy at 11.2 $\mu$m and 456 $\pm$ 68 mJy at 18.1
$\mu$m.
HD 74956– Recent $Spitzer$ MIPS images at 24 $\mu$m have demonstrated that the
observed infrared excess associated with HD 74956 (e.g., Aumann 1985, 1988;
Cote 1987; Chen et al. 2006; Su et al. 2006) is the result of this multiple-
star system traveling through an interstellar cloud and producing a bow shock
(Gáspár et al., 2008). The dust in this overdense region ($\sim$15 times the
Local Bubble density) is compressed at the shock front generated by photon
pressure and is heated by the star, which gives rise to an arc-shaped
morphology that is responsible for the infrared excess. More recently,
Kervella et al. (2009) studied the system in greater detail at both near-IR
and mid-IR wavelengths (with NACO and VISIR, respectively, at the VLT) to
determine whether some of the observed excess might still be attributable to a
disk around one of the system members. However, their final result
corroborates that of Gáspár et al. (2008): the bow shock alone is likely to be
responsible for the IR excess. HD 74956 is resolved only in our 10.4-$\mu$m
images, and the level of extension implies a dust disk radius of 1.4 AU
(simply estimated by quadratically subtracting the PSF FWHM from the source
FWHM). The measured flux densities are 8.61 $\pm$ 0.86 mJy at 10.4 $\mu$m and
2.38 $\pm$ 0.36 mJy at 10.4 $\mu$m
Table 2: Debris Disk Candidate Target List
Name | Type | Age | Age | $d$ | 24 or 25um $F_{\nu}$ | Flux | Excess RatioaaFor flux density measurements at 24 or 25 $\mu$m.
---|---|---|---|---|---|---|---
| | [Myr] | Ref. | [pc] | [Jy] | Ref. | [$F_{total}/F_{\star}$]
HD 38206 | A0V | 9 | 2 | 69 | 0.115 | 4 | 3.34
HD 38678 | A2Vann | 231, 330 | 1, 3 | 22 | 1.160 | 9 | 2.43
HD 56537 | A3V | 560 | 3 | 29 | 0.586 | 9 | 1.32
HD 71155 | A0V | 169, 240 | 1, 3 | 38 | 0.321 | 4 | 1.54
HD 75416 | B8V | 5 | 4 | 97 | 0.128 | 4 | 3.51
HD 80950 | A0V | 80 | 2 | 81 | 0.121 | 4 | 3.79
HD 83808 | A5V+ | 400 | 3 | 41 | 1.140 | 9 | 1.16
HD 95418 | A1V | 300, 358, 380 | 5, 1, 3 | 24 | 1.400 | 9 | 1.21
HD 102647 | A3V | 50, 520 | 1, 3 | 11 | 2.320 | 9 | 1.42
HD 115892 | A2V | 350 | 3 | 18 | 0.705 | 4 | 1.2
HD 139006 | A0V | 314, 350 | 1, 3 | 23 | 1.686 | 9 | 1.29
HD 141569 | B9.5e | 5 (PMS) | 6, 7 | 99 | 1.819 | 9 | 162.4bbThese excess ratios were computed with $IRAS$ 25 $\mu$m flux density measurements and 25 $\mu$m photospheric flux densities estimated with the same method as described in §3.1. All other excess ratio values are taken from Rieke et al. 2005.
HD 161868 | A0V | 184, 305 | 1, 3 | 29 | 0.525 | 9 | 1.47
HD 172555 | A5IV-V | 12 | 8 | 29 | 1.092 | 9 | 9.66bbThese excess ratios were computed with $IRAS$ 25 $\mu$m flux density measurements and 25 $\mu$m photospheric flux densities estimated with the same method as described in §3.1. All other excess ratio values are taken from Rieke et al. 2005.
HD 178253 | A2V | 254, 320 | 1, 3 | 40 | 0.348 | 9 | 1.45bbThese excess ratios were computed with $IRAS$ 25 $\mu$m flux density measurements and 25 $\mu$m photospheric flux densities estimated with the same method as described in §3.1. All other excess ratio values are taken from Rieke et al. 2005.
HD 181296 | A0Vn | 12 | 8 | 48 | 0.491 | 9 | 8.32
HD 181869 | B8V | 110 | 3 | 52 | 0.280 | 9 | 1.46
References. — (1) Song et al. 2001; (2) Gerbaldi et al. 1999; (3) Rieke et al.
2005; (4) de Zeeuw et al. 1999; (5) King et al. 2003; (6) Weinberger et al.
2000; (7) Merín et al. 2004; (8) Zuckerman et al. 2001; (9) Moshir et al. 1989
### 2.2 The Images: General Comments
We obtained mid-IR images of 19 debris disk candidates (including HD 21362 and
HD 74956 mentioned above) in 2005, 2006, and 2007 at the Gemini North and
South facilities (program IDs: GS-2005A-Q-2, GN-2006A-Q-10, GS-2006A-Q-5,
GS-2007A-Q-23) with Michelle and T-ReCS (Thermal Region Camera and
Spectrograph), respectively. The log of the observation dates and program IDs
is given in Table 1 in the appendix. With Michelle, we used the narrowband N′
($\lambda_{c}$ = 11.2 $\mu$m, $\Delta\lambda$ = 2.4 $\mu$m) and narrowband Qa
($\lambda_{c}$ = 18.1 $\mu$m, $\Delta\lambda$ = 1.9 $\mu$m) filters. With
T-ReCS, we used the broadband N ($\lambda_{c}$ = 10.36 $\mu$m, $\Delta\lambda$
= 5.27 $\mu$m), narrowband Si-5 ($\lambda_{c}$ = 11.66 $\mu$m, $\Delta\lambda$
= 1.13 $\mu$m), and narrowband Qa ($\lambda_{c}$ = 18.30 $\mu$m,
$\Delta\lambda$ = 1.51 $\mu$m) filters. These filters were chosen to sample
the disk emission in the two atmospheric transmission windows in the mid-IR
regime, at $\sim$10 (N band) and $\sim$20 $\mu$m (Q band). Among the filters
available in the N band, the N′ filter in Michelle and the broadband N and
Si-5 filters in T-ReCS have the best sensitivity, as documented in Gemini-
provided tables. The Qa filter ($\sim$18 $\mu$m) was chosen for its relative
lack of water absorption lines in its wavelength range compared to Qb filters
at $\sim$25 $\mu$m, thus decreasing the importance of low atmospheric water
vapor content for execution of the observations. Both Michelle and T-ReCS
utilize Raytheon Si:As blocked-impurity-band (BIB) detectors with 320 x 240
pixels. With Michelle, each pixel subtends 0.10”, and the total field of view
is 32” x 24”, and with T-ReCS, each pixel subtends 0.09”, and the total field
of view is 29” x 22”.
A point-spread-function (PSF) comparison star was observed before and after
each science target observation, with three exceptions where the PSF star was
only observed either before or after (but not both) the disk target. The PSF
reference star was typically a Cohen IR standard (Cohen et al., 1999) that
also served as a flux calibrator. The observations used the standard mid-IR
technique of chopping with a chop throw of 15” and nodding (parallel to the
chopping direction) to remove time-variable sky background, telescope thermal
emission, and low-frequency detector noise. The data were reduced with the
Gemini IRAF package.
The total (disk + star) flux densities of the sources observed in this study
are given in Table 3. The flux densities were measured with aperture
photometry. The average sky background was measured in an annulus centered on
the star with a radius range of 1.5–2 times the main (source) aperture radius,
and the measured source flux density was corrected for the sky according to
the number of pixels in the aperture. Observations of the flux standards were
not repeated on each night, so we adopt nominal calibration uncertainties of
10% at 11.7 $\mu$m and 15% at 18.3 $\mu$m, which are typical for photometric
variations in the mid-IR (e.g. De Buizer et al. 2005, Packham et al. 2005).
Such variations dominate the uncertainties associated with the background shot
noise in all cases. The 1-$\sigma$ uncertainties presented in Table 3
represent the dispersion in the measurements due to fluctuations in the level
of thermal emission of the sky and the shot noise in the background photon
stream. The total 1-$\sigma$ uncertainties (the quadratic addition of both
background and photometric uncertainties) are given in Tables 4 and 5.
Table 3: Flux DensitiesaaThe uncertainties given here are those associated with the measured background noise. Additionally, nominal calibration uncertainties of 10% at 11.7 $\mu$m and 15% at 18.3 $\mu$m were adopted, which are typical for photometric variations in the mid-IR (e.g. De Buizer et al. 2005, Packham et al. 2005). These photometric uncertainties dominate the background noise in all cases. (in mJy) of Debris Disk Candidates Source | Gemini | $F_{\nu}(10.4~{}\mu$m) | $F_{\nu}(11.2~{}\mu$m) | $F_{\nu}(11.7~{}\mu$m) | $F_{\nu}(18.1~{}\mu$m) | $F_{\nu}(18.3~{}\mu$m)
---|---|---|---|---|---|---
HD # | | (N) | (N′) | (Si-5) | (Qa) | (Qa)
38206 | S | 202 $\pm$ 1 | … | … | … | 116 $\pm$ 7
38678 | S | 2147 $\pm$ 2 | … | … | … | 960 $\pm$ 10
56537 | N | | 1432 $\pm$ 1 | … | 597 $\pm$ 12 | …
71155 | S | 1083 $\pm$ 2 | … | … | … | 375 $\pm$ 10
75416 | S | … | … | 229 $\pm$ 3 | … | 92 $\pm$ 7
80950 | S | 191 $\pm$ 13 | | … | … | 109 $\pm$ 10
83808 | N | … | 3614 $\pm$ 3 | … | 1567 $\pm$ 13 | …
95418 | N | … | 3588 $\pm$ 2 | … | 1743 $\pm$ 10 | …
102647 | N | … | 5822 $\pm$ 3 | … | 2317 $\pm$ 17 | …
115892 | S | 2540 $\pm$ 3 | … | 2521 $\pm$ 2 | … | 1026 $\pm$ 17
139006 | N | … | 4077 $\pm$ 2 | … | 1795 $\pm$ 15 | …
141569 | N | … | 338 $\pm$ 1 | … | 883 $\pm$ 13 | …
161868 | N | … | 1105 $\pm$ 1 | … | 443 $\pm$ 11 | …
172555 | S | … | … | 1155 $\pm$ 2 | … | 1094 $\pm$ 11
178253 | S | … | … | 770 $\pm$ 2 | … | 360 $\pm$ 12
181296 | S | … | … | 395 $\pm$ 2 | … | 343 $\pm$ 16
181869 | S | 695 $\pm$ 2 | … | … | … | 202 $\pm$ 14
Table 4: Excess Emission of Debris Disk Candidates (Michelle)
| $F_{\nu}$(11.2 $\mu$m) [mJy] | | $F_{\nu}$(18.1 $\mu$m) [mJy]
---|---|---|---
HD | Total | Star | Excess | | Total | Star | Excess
56537 | 1432 $\pm$ 143 | 1154 | 278 $\pm$ 143 | | 597 $\pm$ 91 | 440 | 157 $\pm$ 91
83808 | 3614 $\pm$ 361 | 2985 | 628 $\pm$ 361 | | 1567 $\pm$ 235 | 1138 | 429 $\pm$ 235
95418 | 3588 $\pm$ 359 | 3650 | -62 $\pm$ 359 | | 1743 $\pm$ 262 | 1392 | 351 $\pm$ 262
102647 | 5822 $\pm$ 582 | 5297 | 525 $\pm$ 582 | | 2317 $\pm$ 348 | 2020 | 297 $\pm$ 348
139006 | 4077$\pm$ 408 | 3924 | 153 $\pm$ 408 | | 1795 $\pm$ 262 | 1496 | 299 $\pm$ 262
141569 | 338 $\pm$ 34 | 56 | 282 $\pm$ 34 | | 883 $\pm$ 147 | 22 | 861 $\pm$ 147
161868 | 1105 $\pm$ 111 | 1064 | 41 $\pm$ 111 | | 443 $\pm$ 72 | 406 | 37 $\pm$ 72
Notes– The 1-$\sigma$ uncertainties given here are the quadratic addition of
both background and photometric uncertainties. For the measurement
uncertainties alone, see Table 3.
Table 5: Excess Emission of Debris Disk Candidates (T-ReCS)
| $F_{\nu}$(10.4 $\mu$m) [mJy] | | $F_{\nu}$(11.7 $\mu$m) [mJy] | | $F_{\nu}$(18.3 $\mu$m) [mJy]
---|---|---|---|---|---
HD | Total | Star | Excess | | Total | Star | Excess | | Total | Star | Excess
38206 | 202 $\pm$ 20 | 169 | 33 $\pm$ 20 | | … | … | … | | 116 $\pm$ 19 | 54 | 62 $\pm$ 19
38678 | 2130 $\pm$ 190 | 1388 | 742 $\pm$ 190 | | … | … | … | | 960 $\pm$ 60 | 484 | 476 $\pm$ 60
71155 | 1083 $\pm$ 108 | 815 | 268 $\pm$ 108 | | … | … | … | | 375 $\pm$ 57 | 261 | 114 $\pm$ 57
75416 | … | … | … | | 229 $\pm$ 23 | 141 | 88 $\pm$ 23 | | 92 $\pm$ 15 | 58 | 34 $\pm$ 15
80950 | 191 $\pm$ 23 | 162 | 29 $\pm$ 23 | | … | … | … | | 109 $\pm$ 19 | 52 | 57 $\pm$ 19
115892 | 2540 $\pm$ 254 | 2748 | -208 $\pm$ 254 | | 2521 $\pm$ 252 | 2155 | 366 $\pm$ 252 | | 1026 $\pm$ 155 | 881 | 145 $\pm$ 155
172555 | … | … | … | | 1155 $\pm$ 116 | 520 | 635 $\pm$ 116 | | 1094 $\pm$ 164 | 213 | 881 $\pm$ 164
178253 | … | … | … | | 770 $\pm$ 77 | 655 | 115 $\pm$ 77 | | 360 $\pm$ 55 | 268 | 92 $\pm$ 55
181296 | … | … | … | | 395 $\pm$ 40 | 271 | 124 $\pm$ 40 | | 343 $\pm$ 54 | 111 | 232 $\pm$ 54
181869 | 695 $\pm$ 70 | 731 | -36 $\pm$ 70 | | … | … | … | | 202 $\pm$ 34 | 234 | -33 $\pm$ 34
Notes– The 1-$\sigma$ uncertainties given here are the quadratic addition of
both background and photometric uncertainties. For the measurement
uncertainties alone, see Table 3.
Table 6: Summary of Debris Disk Candidate IR Excess Detections
HD | N (10.4 $\mu$m) | N′ (11.2 $\mu$m) | Si-5 (11.7 $\mu$m) | Qaa (18.1 $\mu$m) | Qab (18.3 $\mu$m)
---|---|---|---|---|---
38206 | none | … | … | … | significant
38678 | significant | … | … | … | significant
56537 | … | none | … | none | …
71155 | marginal | … | … | … | marginal
75416 | … | … | significant | … | marginal
80950 | none | … | … | … | significant
83808 | … | none | … | none | …
95418 | … | none∗ | … | none∗ | …
102647 | … | none | … | none | …
115892 | … | … | none | … | none
139006 | … | none∗ | … | none∗ | …
141569 | … | significant | … | significant | …
161868 | … | none | … | none | …
172555 | … | … | significant | … | significant
178253 | … | … | none | … | none
181296 | … | … | significant | … | significant
181869 | none | … | … | … | none
Notes– a Michelle, Gemini North. b T-ReCS, Gemini South.
∗ While there is no statistically significant detection of excess emission for
these sources in our data, they do appear to be spatially resolved. This point
is discussed in §3.1 and §5.2.
Table 7: FWHM of Debris Disk Candidates & PSF Reference Stars (Michelle) | N′ FWHM [arcsec] | | Qa FWHM [arcsec]
---|---|---|---
Name | Source | PSF | | Source | PSF
HD 56537 | 0.365 $\pm$ 0.002 | 0.398 $\pm$ 0.013 | | 0.545 $\pm$ 0.019 | 0.539 $\pm$ 0.004
| | | | 0.609 $\pm$ 0.025 | 0.554 $\pm$ 0.006
HD 83808 | 0.347 $\pm$ 0.001 | 0.380 $\pm$ 0.009 | | 0.537 $\pm$ 0.004 | 0.529 $\pm$ 0.005
HD 95418 | 0.339 $\pm$ 0.001 | 0.328 $\pm$ 0.002 | | 0.539 $\pm$ 0.003 | 0.543 $\pm$ 0.003
HD 102647 | 0.361 $\pm$ 0.002 | 0.353 $\pm$ 0.008 | | 0.533 $\pm$ 0.005 | 0.535 $\pm$ 0.002
| | | | 0.531 $\pm$ 0.003 | 0.533 $\pm$ 0.003
HD 139006 | 0.419 $\pm$ 0.003 | 0.364 $\pm$ 0.006 | | 0.574 $\pm$ 0.007 | 0.556 $\pm$ 0.004
| | | | 0.516 $\pm$ 0.004 | 0.544 $\pm$ 0.003
HD 141569 | 0.436 $\pm$ 0.004 | 0.374 $\pm$ 0.016 | | 0.807 $\pm$ 0.029 | 0.539 $\pm$ 0.005
| | | | 0.818 $\pm$ 0.066 | 0.523 $\pm$ 0.003
HD 161868 | 0.386 $\pm$ 0.003 | 0.356 $\pm$ 0.006 | | 0.518 $\pm$ 0.031 | 0.528 $\pm$ 0.003
| | | | 0.537 $\pm$ 0.017 | 0.525 $\pm$ 0.006
Table 8: FWHM of Debris Disk Candidates & PSF Reference Stars (T-ReCS) | N FWHM [arcsec] | | Si-5 FWHM [arcsec] | | Qa FWHM [arcsec]
---|---|---|---|---|---
Name | Source | PSF | | Source | PSF | | Source | PSF
HD 38206 | 0.442 $\pm$ 0.008 | 0.427 $\pm$ 0.014 | | | | | 0.592 $\pm$ 0.042 | 0.534 $\pm$ 0.006
HD 38678 | 0.311 $\pm$ 0.001 | 0.308 $\pm$ 0.001 | | | | | 0.605 $\pm$ 0.015 | 0.536 $\pm$ 0.016
HD 71155 | 0.348 $\pm$ 0.003 | 0.332 $\pm$ 0.003 | | | | | 0.647 $\pm$ 0.040 | 0.583 $\pm$ 0.025
HD 75416 | | | | 0.494 $\pm$ 0.014 | 0.471 $\pm$ 0.008 | | 0.875 $\pm$ 0.097 | 0.648 $\pm$ 0.024
HD 80950 | 0.372 $\pm$ 0.008 | 0.412 $\pm$ 0.007 | | | | | 0.738 $\pm$ 0.125 | 0.617 $\pm$ 0.015
HD 115892 | | | | 0.412 $\pm$ 0.010 | 0.467 $\pm$ 0.006 | | 0.580 $\pm$ 0.016 | 0.597 $\pm$ 0.012
| | | | | | | 0.600 $\pm$ 0.011 | 0.616 $\pm$ 0.022
HD 172555 | | | | 0.369 $\pm$ 0.002 | 0.378 $\pm$ 0.006 | | 0.591 $\pm$ 0.014 | 0.559 $\pm$ 0.013
HD 178253 | | | | 0.439 $\pm$ 0.003 | 0.446 $\pm$ 0.008 | | 0.530 $\pm$ 0.053 | 0.584 $\pm$ 0.019
HD 181296 | | | | 0.384 $\pm$ 0.002 | 0.380 $\pm$ 0.006 | | 0.584 $\pm$ 0.192 | 0.519 $\pm$ 0.012
HD 181869 | 0.378 $\pm$ 0.003 | 0.354 $\pm$ 0.011 | | | | | 0.506 $\pm$ 0.024 | 0.613 $\pm$ 0.037
## 3 Source Measurements
### 3.1 Statistical Significance of IR Excesses
The photospheric flux densities at 10–12 $\mu$m and 18 $\mu$m were estimated
by extrapolating the $2MASS$ K-band (2.2 $\mu$m) flux densities (Cutri et al.,
2003) to 10 $\mu$m. The flux density was assumed to vary as $\nu^{1.88}$ over
this wavelength range, as is estimated by Kurucz (1979) to be appropriate for
an A0 star (e.g., Jura et al. 1998). Beyond 10 $\mu$m, we assumed a Rayleigh-
Jeans relation ($\nu^{2}$) for the photosphere. The photospheric flux density
estimate and the corresponding excess flux density estimate are given for each
source in Tables 4 and 5. As discussed in §2.2, photometric uncertainties of
10% and 15% are assumed for the 10- and 18-$\mu$m windows, respectively. We
have compared our flux density estimates for the photosphere with those
determined via Kurucz models for several sources (e.g., Smith et al. 2008),
and the difference is less than 4%, which is well below the photometric
uncertainty.
The photometric measurements, when combined with estimates of the photospheric
contribution, permit assessment of the level of excess emission attributable
to dust. For some of the sources that are known to have 24-$\mu$m excess
emission, we do not detect statistically significant excess emission at 10-
and/or 18-$\mu$m. We comment on these cases specifically later in this
section. We further confirm that the excess emission (when present) is
spatially coincident with the star and does not originate from a background
object. These sources were chosen based on space-based observations of their
infrared excess, and the lower resolution of those images (due to the
$\sim$10x smaller primary mirror) is more prone to confusion within the beam.
In §5, we consider whether the implications for the location of the dust from
photometric measurements and measurements of spatial extent present a
consistent picture for each of the sources.
The following definitions apply to our characterization of the measured IR
excesses:
• total uncertainty: The total uncertainty is the quadratic addition of the
measurement uncertainty (given in Table 3) and the photometric uncertainty for
a calibrated image. This value is referred to as $\sigma_{phot}$.
• no detected excess: A source with no detected excess is defined as having an
IR excess of less than two times the total uncertainty of the flux density
measurement.
• marginal excess: A source with marginal excess is defined as having an IR
excess of two to three times the total uncertainty of the flux density
measurement.
• significant excess: A source with significant excess is defined as having an
IR excess greater than or equal to three times the total uncertainty of the
flux density measurement.
For nine (HD 56537, HD 83808, HD 95418, HD 102647, HD 115892, HD 139006, HD
161868, HD 178253, and HD 181869) of our 17 debris disk sources, we do not
detect statistically significant excess emission in either of the bandpasses
used in this work. Two sources (HD 38206 and HD 80950) have an excess detected
in only one bandpass. The detections of excess emission are summarized in
Table 6. As discussed in §2.1, the debris disk candidates were chosen based on
excess emission observed at 24 or 25 $\mu$m. In the following sections, we
report the detection of spatial extension for several sources (HD 56537, HD
95418, HD 139006, and HD 161868) that do not have statistically significant
excess IR emission. These results are not necessarily inconsistent, due to the
uncertainties in both quantities.
### 3.2 Source Extent
Only a handful of circumstellar debris disks have been spatially resolved at a
level that permits examination of detailed structure. However, it is important
to keep in mind that valuable information is still obtained when only the
scale size is determined. A “disk” can consist of several components that
reflect the complex relationships among the dust population, the dust parent
bodies, and the planetary system, with the proposed (but still unresolved)
asteroid-belt and Kuiper-belt dust zones in the triple-planet system HD 8799
being a spectacular example (Marois et al., 2008; Chen et al., 2009;
Reidemeister et al., 2009). Establishing the existence of any of these
subsystems by constraining the emitting-region size permits assessment of
broader system properties, as illustrated in our analysis of Zeta Lep Moerchen
et al. (2007b) where the resolved dust may well betoken an asteroid belt and,
consequently, planets.
To check for the presence of an extended disk source in our targets, we
observed a source that is known to be not extended, a PSF star, in close
temporal proximity to the target observation, as described in §2.2. In most
cases, there was no obvious 2-D structure to the disk source, and the PSF
references often showed asymmetric features. Examples of asymmetric PSFs,
whose causes may be associated with chopping and nodding, are shown in Fig. 1.
A key measure of scale size is the full-width at half-maximum (FWHM) intensity
of the emitting source. Especially for fainter sources, the FWHM may be the
only available measure of source size. Given the small source sizes
anticipated in this study, we have focused exclusively on the use of the FWHM
to characterize their extent, while remaining open to the possibility that
more extended lower-level emission might be present. We used as our primary
metric the FWHM measurement from a 1-D Moffat profile fitted to the azimuthal
average of each source with an IRAF routine (i.e., we do not simply measure
directly the FWHM, which, due to noise on the profile, would be a much less
accurate measurement). Moffat profile fitting has been used frequently in mid-
IR image analysis (e.g., Radomski et al. 2008), because we find that the true
profile width at the half-maximum level is better approximated by a Moffat
profile than by a Gaussian, regardless of the overall goodness-of-fit as
evaluated by, e.g., chi-squared analysis. We have again verified this
hypothesis for several sources in the dataset presented in this work,
including the known resolved sources HD 38678 and HD 141569, and the FWHM of
the Moffat profile fit is closer to the true FWHM value in $>$90% of the
images measured. This is likewise the finding of a report on PSF image quality
in the mid-IR at Gemini South (which is summarized in Li, Telesco & Varosi
2010). We acknowledge that, in practice, one could choose a different metric
such as the full-width at quarter-maximum, for which a Moffat profile fit may
not be the optimal choice.
Figure 1: PSF observations taken at 11.2 $\mu$m with Michelle before and after
HD 141569, shown to illustrate typical asymmetries encountered. Contours on
the two PSF images are drawn logarithmically to show the structure, with the
lowest contour drawn at 0.3% of the peak surface brightness. Contours on the
residual image are drawn at linear intervals of the normalized peak surface
brightness of the two images, every 5% from 0–20%. Note that the first PSF
(left) shows slight N-S elongation in the core and a partial trefoil pattern
in the wings. The second PSF appears slightly cross-shaped in the core,
resulting in contours at the center that are more square-shaped. The residual
difference between these two peak-normalized PSFs is up to 20% of the original
peak flux within the central arcsecond.
We then compared the PSF and the disk candidate FWHM values to make an
assessment of the extent of the emission. However, the science targets are
typically at least an order of magnitude fainter than the reference star and
must be observed for correspondingly longer integration times to achieve
similar signal-to-noise ratios. Pupil rotation, incorrect guiding correction,
and changes in the quality of seeing during long observations on a science
target can result in a final image degraded by lower-frequency components that
are not accurately represented in the PSF determined from generally shorter
integration times. Minor variations may sometimes average out, but in the
majority of cases these effects broaden the source profile in a final stacked
image (see also Li, Telesco & Varosi 2010).
To estimate more robustly the profile widths and thereby assess spatial
extent, we examined the FWHM of the PSF star and the target sources throughout
their integration sequences. Usually, the smallest unit of integration time
was that corresponding to the so-called saveset image111A saveset is a stack
of chopped images, on- and off-source, taken within one telescope nod
position. Each saveset corresponds to $\sim$10 s of integration time, and
there are typically three savesets in one nod position before the telescope
switches to the opposite nod position., but in some cases it was necessary to
bin two or more savesets to achieve a signal-to-noise ratio high enough to
perform a Moffat fit to the source profile. In this way, we were able to
determine a mean FWHM and corresponding uncertainty from the set of subdivided
images for both the PSF and the debris disk target. Using all of the data
points, and taking into account the number of points in the set, we determined
the standard deviation of the mean (the standard deviation divided by
$\sqrt[]{n}$ number of measurements) for each set of FWHM measurements, and we
adopt this value as the uncertainty. The standard deviation (and standard
deviation of the mean) is valid for a Gaussian distribution of independent
values extracted from a population that does not vary with time, but the
assumption of a stationary Gaussian distribution of values for our data sets
was not always valid. The variation in FWHMs for each of our data sets in this
work sometimes revealed outlying data points, or an obvious deterioration or
improvement in image quality due to factors like seeing. However, such time-
variant changes are not observed for the majority of sources in this work, and
in particular we do not observe obvious changes in image quality in the data
sets of sources that we claim to have spatially resolved. Likewise, in an
effort to make our data quality transparent, plots of the FWHM measurements
for each source are shown in the appendix.
We have also applied a Student’s $t$-test (assuming unknown and unequal
variances) to compare the FWHM values of the source and PSF for each dataset
to better assess whether the source and PSF data are drawn from the same image
quality distribution. We quote these results for individual sources (§5) when
relevant, which occurs in two cases: (1) the PSF profile data are
systematically broader than the source profile data, and the $t$-test confirms
that the two sets are not drawn from the same population, in which case the
data are rejected, or (2) the source profile data are significantly broader
than the PSF profile data, and the $t$-test confirms that the two sets are not
drawn from the same population, in which case we consider the source spatially
resolved.
The following terms are defined, for use in characterization of the measured
extent of the sources:
• combined standard deviation of the mean: The combined standard deviation of
the mean is the combination of the standard deviation of the mean of the PSF
FWHM measurements and the standard deviation of the mean of the source FWHM
measurements, given by
$\sigma_{ext}=\sqrt[]{\sigma_{PSF}^{2}+\sigma_{source}^{2}}.$ (1)
This value is referred to as $\sigma_{ext}$ in the following discussion of
extension measurements.
• unresolved: An unresolved source is defined as having an average source FWHM
value that is greater than its corresponding PSF FWHM by less than three times
the combined standard deviation of the mean of those two measurements:
$FWHM_{source}-FWHM_{PSF}<3\sigma_{ext}$ (2)
• resolved: A resolved source is defined as having an average source FWHM
value that is greater than its corresponding PSF FWHM by three or more times
the combined standard deviation of the mean of those two measurements:
$3\sigma_{ext}\leq FWHM_{source}-FWHM_{PSF}$ (3)
Based on these FWHM measurements (Tables 7 and 8), several sources appear to
be extended. The statistical significance of these extended sources has been
assessed by breaking up the full integration time into individual images, as
described in §3.2. A list of the sources that appear to be resolved (based on
FWHM measurements of profile fits to the data) and their sizes is given in
Table 9. The images of the resolved sources, their corresponding PSFs, and the
residuals from peak-normalized subtraction of the PSFs are shown for reference
in Fig 2.
The profile widths of Moffat fits to the disk sources, their PSF stars, and
the associated uncertainties (one standard deviation of the mean) are listed
in Tables 7 and 8. The statistical significance of the difference between
target and PSF profile width is given in Table 9. Of the 17 debris disk
candidates (i.e., excluding the two sources mentioned in §2.1) that we imaged,
five sources near 10 $\mu$m ($\lambda_{c}$ = 10.7, 11.2, or 11.7 $\mu$m) and
two sources near 20 $\mu$m ($\lambda_{c}$ = 18.1 or 18.3 $\mu$m) had source
FWHM values bigger than the PSF FWHM by more than three times the combined
standard deviation of the mean for the two measurements. These sources are
discussed further in §5.
Figure 2: Images of the sources resolved in this work, the corresponding PSF stars, and the residual from peak-normalized subtraction. Contours are drawn at 3, 6, and 9$\sigma_{bkd}$ in all images. Table 9: Sizes of Extended Sources | 10.4 $\mu$m | | 11.2 $\mu$m |
---|---|---|---|---
| $\Delta FWHM\tablenotemark{a}$ | r | | $\Delta FWHM\tablenotemark{a}$ | r |
HD | [arcsec] | [AU] | | [arcsec] | [AU] |
71155 | 0.10 $\pm$ 0.01 | 1.96 $\pm$ 0.02 | | … | … |
95418 | … | … | | 0.09 $\pm$ 0.01 | 1.09 $\pm$ 0.01 |
139006 | … | … | | 0.20 $\pm$ 0.01 | 2.31 $\pm$ 0.04 |
141569 | … | … | | 0.22 $\pm$ 0.02 | 10.89 $\pm$ 0.48 |
181869 | 0.16 $\pm$ 0.01 | 4.12 $\pm$ 0.11 | | … | … |
| 18.1 $\mu$m | | 18.3 $\mu$m |
| $\Delta FWHM\tablenotemark{a}$ | r | | $\Delta FWHM\tablenotemark{a}$ | r |
HD | [arcsec] | [AU] | | [arcsec] | [AU] |
38678 | … | … | | 0.28 $\pm$ 0.02 | 3.01 $\pm$ 0.24 |
139006 | 0.12 $\pm$ 0.06 | 1.36 $\pm$ 0.09 | | … | … |
141569 | 0.60 $\pm$ 0.03 | 29.7 $\pm$ 1.1 | | … | … |
| 0.63 $\pm$ 0.07 | 31.1 $\pm$ 2.5 | | … | … |
## 4 Expected detectability based on comparison to archetypes
Here we consider a simple assessment of how many sources from our sample we
would expect to spatially resolve if the source morphologies were similar to
those of certain archetypal disks that have already been resolved with mid-IR
imaging. We use two archetypes for comparison: (1) a possible asteroid belt
analog, $\zeta$ Lep, and (2) a possible Kuiper Belt analog, $\beta$ Pic. The
$\zeta$ Lep emission is very compact, and the evidence for a disk in this type
of a source would be found in the FWHM. In contrast, the central disk of
$\beta$ Pic is very extended but relatively faint and for the most part not
evident in a measurement of the FWHM, but rather in the wings of the profile.
### 4.1 $\zeta$ Lep Analog
$\zeta$ Lep was resolved in 18.3-$\mu$m images, and the measured spatial
extent implied a dust disk radius of 3 AU (Moerchen et al., 2007b). The disk
radius was inferred from a quadratic subtraction of the PSF FWHM from the
azimuthally averaged source FWHM:
${FWHM}_{source}^{2}-{FWHM}_{PSF}^{2}={D}_{disk}^{2}$. (Such a relationship is
strictly true for Gaussian functions, and another example of combination in
quadrature is given in Eq. 1.) Therefore, we consider whether each of the
sources in our sample (besides $\zeta$ Lep) would be spatially resolved if it
were described by a point source (the size of which is dictated by the
corresponding PSF) and a dust disk of radius 3 AU with the same brightness
relative to its host star as that of the $\zeta$ Lep system. We calculate the
FWHM for each source as the quadratic addition of its corresponding PSF FWHM
and a disk with a 3-AU radius:
${FWHM}_{PSF}^{2}+D_{6~{}AU}^{2}={FWHM}_{source}^{2}$. The point source
considered here may correspond to stellar emission and/or emission from dust
from an unresolved region near the star.
We consider a source to be spatially resolved if the value $FWHM_{source}$
(convolved PSF and 3-AU-radius-disk) is greater than its corresponding value
for $FWHM_{PSF}$ by more than 3$\sigma_{ext}$, where $\sigma_{ext}$ is the
total uncertainty associated with the FWHM measurements (discussed further in
§3.2). Based on the PSF observations associated with each source and the
assumption that they have a morphology like that of $\zeta$ Lep, we estimate
that eleven sources should have been resolved in images near 10 $\mu$m and
five sources should have been resolved in images near 18 $\mu$m. In reality,
five sources were resolved in 10-$\mu$m images, and two sources were resolved
in 18-$\mu$m images (results discussed further in §3.2). HD 141569 is in both
of these sets, and it is known from prior observations that this dust disk is
more extended (r$>$30 AU) and is not comparable in nature to $\zeta$ Lep.
Removing HD 141569 from the set of resolved sources leaves three sources
resolved in 10-$\mu$m images and one source resolved in 18-$\mu$m images. The
fact that only approximately half of the projected number of resolved Zeta-
Lep-like sources were actually resolved suggests that not all of the sources
are comparable in size to the asteroid-belt-like analog associated with
$\zeta$ Lep. However, it is also possible that some sources have a disk size
comparable to that of $\zeta$ Lep but remain undetected due to lower disk
brightness. This is a reasonable possibility, because $\zeta$ Lep has a higher
fractional IR luminosity than most of the sources in our sample.
### 4.2 $\beta$ Pic Analog
We also assess the number of sources we would expect to resolve if the disk
morphology of each target were comparable to that of the well known disk
$\beta$ Pic. For each target, we check whether extended emission (r$\sim$100
AU) like that seen around $\beta$ Pic would be significantly detected above
the background noise level in our actual source images. To be clear, this is
not spatial extent sought at the FWHM level, but emission farther out in the
brightness profile that is significantly above the level of the background. We
acknowledge that edge-on disks like $\beta$ Pic may be more detectable than
face-on disks because of the higher line-of-sight column densities, and so we
consider the disk emission levels for both an edge-on and a face-on case. We
used unaltered mid-IR images of $\beta$ Pic (Telesco et al., 2005) for the
edge-on case, since it is nearly edge-on already. For the face-on case, we
constructed simple model images of a face-on $\beta$ Pic by integrating the
measured flux density as a function of radius and then redistributing it
azimuthally. By generating this face-on model, we assume that the disk is
optically thin and its MIR emission is azimuthally symmetric. In images of
$\beta$ Pic taken from the ground with subarcsecond resolution (e.g., Telesco
et al. 2005), we see that the assumption of azimuthal symmetry is a
simplification of the disk structure but is reasonable for our purposes here.
We generated profile cuts of the edge-on and face-on images to compare the
flux density levels to the background noise measured in our target images.
Profile cuts were made by sampling the line of pixels along the major axis of
the disk for the edge-on case and along an axis bisecting the disk in the
face-on case. To account for the different central star brightness and
accordingly different disk brightness of our target sources, we scaled the
brightness of the $\beta$ Pic profiles according to the stellar flux density
of each of the sources. We expect that the 2-$\mu$m flux density is
predominantly from the photosphere, and we therefore used the $2MASS$
measurement for each target as a metric for the stellar brightness and,
accordingly, its dust heating ability. We scaled the profiles for each of the
sources by the ratio of the target source $2MASS$ flux density to that of
$\beta$ Pic, $F_{target}(2~{}\mu m)/F_{\beta~{}Pic}(2~{}\mu m)$. The width of
the extended emission profiles was also scaled for each source according to
the source distance.
We characterize a $\beta$-Pic-type source as one that we could have detected
if the extended emission of the test profile is (1) above the 5-$\sigma_{bkd}$
level (five times the per-pixel background noise measured in the actual source
images) and (2) beyond the first Airy null, since some simulated profiles
might have been bright enough for detection but were not spatially extended
beyond the Airy disk. Five-$\sigma$ levels are used to assess extension
because the mid-IR background is noisy enough that 3-$\sigma$ “blobs” that are
not associated with disk emission are relatively common. If all of the sources
in our sample had the same morphology and fractional IR luminosity as $\beta$
Pic, we would expect the following number of detections: for an edge-on
orientation, eleven at 10 $\mu$m and six at 18 $\mu$m, and for a face-on
orientation, five at 10 $\mu$m and none at 18 $\mu$m. (We note that the disk
of $\beta$ Pic itself is not detected in a face-on orientation in this test,
keeping in mind that the considerable observed levels of optically thin
emission have been redistributed azimuthally for the face-on model, resulting
in much lower surface brightnesses than in its true nearly edge-on
orientation.) In fact, our images reveal that only one source, HD 141569,
shows significant extended emission (in both bandpasses). However, a random
distribution of disks would only have approximately 10% of disks with
inclinations within ten degrees of edge-on; therefore, the number of
detections that we expect in a realistic distribution of disk inclinations
should be $\sim$10% of our predicted number of edge-on detections, $\sim$1 at
10$\mu$m and $<$1 at 18 $\mu$m. Therefore, perhaps surprisingly, our observed
results are consistent with a population composed entirely of $\beta$-Pic-type
disks with a random distribution of orientations. However, this statement is
weakened by the fact that our sample contains a relatively small number of
sources.
Our general conclusion based on comparison to $\zeta$ Lep and $\beta$ Pic
archetypes is that our results are generally consistent with the expected
detection rate of several more asteroid belt analogs like $\zeta$ Lep but only
one (HD 141569) Kuiper Belt analog like $\beta$ Pic. We examine this issue
more below.
## 5 Consistency of Spatial and Photometric Measurements
Here we assess the consistency of the observed color temperature of the excess
emission with the temperature of dust in radiative equilibrium at the distance
implied by the observed spatial disk extent. The color temperature is an upper
limit to the true temperature because it is the unique solution to the
equation
$\frac{F_{{\nu}_{1}}}{F_{{\nu}_{2}}}=\frac{Q_{{\nu}_{1}}}{Q_{{\nu}_{2}}}\frac{B_{{\nu}_{1}}(T)}{B_{{\nu}_{2}}(T)}$
(4)
where $B_{{\nu}_{1}}(T)/B_{{\nu}_{2}}(T)$ is the ratio of two points on the
Planck function for a temperature $T$, and $Q_{{\nu}_{1}}/Q_{{\nu}_{2}}$, the
ratio of the two emission efficiencies, is unity. If the particles behave as
blackbodies, then the ratio $Q_{{\nu}_{1}}/Q_{{\nu}_{2}}$ is unity. For
particles comparable in size or smaller than the emission wavelength, the
emission efficiency is sometimes described as $Q_{em}\propto\nu^{n}$, with
n=1–2, and if $\nu_{1}>\nu_{2}$, then $Q_{{\nu}_{1}}/Q_{{\nu}_{2}}$ will be
greater than unity. In this non-blackbody case, the ratio of the two points on
the Planck function $B_{{\nu}_{1}}(T)/B_{{\nu}_{2}}(T)$ would have to be
lower, and thus originate from a lower-temperature source, in order to produce
the same observed flux density ratios. Real dust particles are generally not
blackbodies, and the computed color temperature is therefore an overestimate
of the true physical temperature.
While the relationship between a disk s observed color temperature and a
“true” dust temperature depends on the distributions of particle sizes and
locations, numerous examples suggest that the observed global color
temperature of a disk can give an indication, albeit a rough one, of the
typical distances of the mid-infrared-emitting particles from the star, and
therefore of the disk size. For example, the 100-K mid-IR color temperature of
$\beta$ Pic implies that dust with blackbody behavior would be located at
$\sim$25 AU, which is within the bounds of the $\sim$100 AU radial extent of
the mid-IR disk emission (Telesco et al., 2005); likewise, the 327-K mid-IR
color temperature of $\zeta$ Lep implies a dust distance of 2.9 AU, which is
consistent with the 3-AU radial extent determined from 18-$\mu$m images
(Moerchen et al., 2007b).
Disk extent ($r_{AU}$) is estimated by quadratic subtraction of the PSF FWHM
from the source FWHM. The resulting estimate for the dust temperature $T_{d}$,
for blackbody particles at that distance from the star, is given by
$T_{d}=278~{}L_{\star}^{\frac{1}{4}}~{}r_{AU}^{-\frac{1}{2}}$ (5)
where the stellar luminosity $L_{\star}$ is in units of $L_{\odot}$, solar
luminosity, $r_{AU}$ is the radius of the dust annulus in AU, and the
equilibrium temperature at 1 AU (Earth) is 278 K. Uncertainties for this
temperature estimate are calculated by propagating the uncertainty in the disk
extent ($r_{AU}$) through Equation 5. This temperature estimate from Equation
5 is a lower limit to the true temperature, because we have assumed that the
dust particles are blackbodies. In reality, as noted above, “small” particles
are heated to higher temperatures than blackbodies at the same distance from
the star. For example, we have plotted in Figure 3 the relationship between
temperature and distance from a $\sim$7-L⊙ star (representative of an A-type
main-sequence star) for both blackbody-type particles and less efficient
emitters with characteristic sizes of 0.05, 0.075, and 0.25 $\mu$m. The
temperatures for these inefficient emitters were calculated based on equations
from Backman & Paresce (1993), which estimate particle temperature as a
function of distance based on assumptions regarding particle size and
composition, and thereby radiation efficiency. The case adopted for our purely
demonstrative calculations is that of a particle which absorbs efficiently but
emits inefficently, such as graphite or amorphous silicate. It can be seen in
this plot that for a given observed dust temperature, the implied distance
from the star depends significantly on the particle properties, especially at
the lowest temperatures, and this should be kept in mind when considering our
calculations in the following sections.
Figure 3: The expected blackbody temperature as a function of distance from a
$\sim$7-L⊙ main-sequence star (solid line), and the predicted temperatures for
efficiently absorbing but inefficiently emitting particles (as discussed in
Backman & Paresce 1993) with characteristic grain sizes of (from left to
right) 0.25 $\mu$m (long dash), 0.075 $\mu$m (medium dash), and 0.05 $\mu$m
(short dash).
In cases where we detect excess emission that is not spatially extended, we
estimate a temperature with Equation 5 and the 2-$\sigma_{ext}$ limit for the
observation. That is, we assume that any extension (source FWHM minus the PSF
FWHM) that is less than 2$\sigma_{ext}$ could escape detection. Thus, we
estimate the upper limit for disk extent as
$r_{limit}=\frac{1}{2}~{}\sqrt[]{(FWHM_{PSF}+2\sigma_{ext})^{2}-FWHM_{PSF}^{2}}.$
(6)
This estimate also yields a lower limit to the true dust temperature, because
the temperature will increase if (1) the dust is any closer to the star, or
(2) the particles are small enough that they do not behave like blackbody
emitters. The radius limits calculated with this method are summarized in
Table 10.
Table 10: Dust Radius Limit Estimates (in AU) for Unresolved Sources with IR
Excess
HD | Si-5 (11.7 $\mu$m) | Qaa (18.3 $\mu$m)
---|---|---
38206 | … | 10.8
71155 | … | 6.2
75416 | 8.6 | 26.5
80950 | … | 8.1
172555 | 1.4 | 3.1
178253 | 2.4 | 7.5
181296 | 2.3 | 17.6
181869 | … | 8.9
Notes– a T-ReCS, Gemini South. All limits were estimated with Equation 6.
### 5.1 Sources Unresolved at Both Wavelengths
HD 38206– There was no excess emission detected at 10.4 $\mu$m. We estimated a
temperature for the unresolved 18.3-$\mu$m-emitting dust by calculating the
blackbody temperature at the orbital distance corresponding to the
2-$\sigma_{ext}$ limit of the source FWHM at 18.3 $\mu$m, as described by
Equation 6. This temperature limit is 201 K.
HD 56537– HD 56537 was not resolved at either wavelength, and we did not
detect significant excess emission.
HD 75416– The equilibrium temperature at the orbital distance corresponding to
the 2-$\sigma_{ext}$ limits of the source FWHM at 11.7 $\mu$m and 18.3 $\mu$m
are 307 K and 175 K, respectively. To estimate the color temperature of 653
$\pm$ 272 K, we assumed that the particles behave as efficient absorbers and
inefficient emitters, because the color could not be fitted by a single-
temperature perfect blackbody. Since the color temperature is an upper limit
to the true dust temperature, the temperature estimates based on spatial
extension measurements are consistent.
HD 80950– For the purpose of assessing spatial resolution via FWHM
measurements, we rejected the 10.4-$\mu$m data, because the PSF was
systematically broader than the disk candidate. This was quantitatively
confirmed by a Student’s $t$-test with $p$-value of 0.0001, or greater than
99.9% certainty that the datasets are drawn from distinct image quality
distributions.
In 10.4-$\mu$m images of HD 80950, we detected two sources separated by 1.6”.
The flux density of the primary is 170 $\pm$ 18 mJy, and the flux density of
the secondary is 8 $\pm$ 1 mJy (uncertainties include both photometric and
background uncertainty). We did not detect any excess emission associated with
the primary at 10.4 $\mu$m. If both sources are measured within the same
aperture, their measured flux density is 190 $\pm$ 23 mJy, where we see a
greater contribution to the background noise due to the larger aperture. Our
flux density measurements are consistent with the $IRAS$ 12-$\mu$m flux
density (Faint Source Catalog) of 211 $\pm$ 25 mJy, and $IRAS$ would not have
distinguished the two sources due to its large beam size. Indeed, no record of
another source within the T-ReCS field is found in the $IRAS$ or $2MASS$
catalogs. It is therefore possible that the IR excess previously thought to be
associated with HD 80950 arises from this apparent companion object. We intend
to characterize this object with further mid-IR and near-IR photometry and/or
spectroscopy, but we note that the frequency of unassociated background
objects encountered in the mid-IR is low. If the unresolved 18.3-$\mu$m
emission originates with dust at an orbital distance corresponding to the
2-$\sigma_{ext}$ limit of the source FWHM, then the implied blackbody
temperature is at least 244 K.
HD 83808– While no excess emission was detected in either bandpass, the
uncertainties for the observed excess are large (Table 4). $Spitzer$ detected
excess emission at 24 $\mu$m, and there are therefore two possibilities for
why we do not detect any excess at 12 or 18 $\mu$m: (1) the dust emission is
diffuse enough that we do not have sufficient sensitivity to detect it above
the background noise, or (2) the dust is too cool to emit significantly at 12
and 18 $\mu$m. We also reject the 11.2-$\mu$m data for use in assessing
spatial extent due to the PSF being systematically broader than the disk
target, which is confirmed by a Student’s $t$-test $p$-value of 0.008.
HD 102647 ($\beta$ Leo)– We did not detect any IR excess emission associated
with the source, within our error bars. Again, the $Spitzer$ detection of
excess emission indicates that there is dust present associated with the
source, so there exist the same two possibilities as in the case of HD 83808:
low surface brightness or cool dust temperatures.
HD 115892– We rejected the 11.7-$\mu$m data for use in assessment of spatial
extent, because the PSF profile is broader than the disk source profile; this
is confirmed by a Student’s $t$-test $p$-value of 1.5 x 10-6.
We detected no excess emission within our error bars. $Spitzer$ observations
do show excess emission at 24 $\mu$m. If this excess is real, then we do not
detect the dust emission either because of low surface brightness or because
of dust temperatures that only yield emission longward of 18 $\mu$m.
HD 161868– We did not detect excess emission in either bandpass, and the
source FWHM is not greater than the PSF FWHM by more than 3$\sigma_{ext}$.
Su et al. (2008) recently resolved this debris disk with $Spitzer$ MIPS images
and estimated a disk radius of $\sim$520 AU from the 24 $\mu$m images and a
disk radius of $\gtrsim$260 AU from the 70 $\mu$m images. The color
temperature based on the 24 $\mu$m and 70 $\mu$m photometry is 81 K, which
overestimates the flux density at 28–35 $\mu$m and 55–65 $\mu$m, and the
authors note that this suggests a range of dust temperatures. Nonetheless, the
81 K blackbody fits the available SED points reasonably well. The IRS spectrum
is also shown, which indicates flux densities of $<$100 mJy at wavelengths
$<$20 $\mu$m; these spectral measurements confirm that our estimated mid-IR
excess levels are within expectations. Given that the color temperature and
the IRS spectrum (and MIPS images) indicate the presence of predominantly cold
dust, it is not surprising that we do not find strong evidence of excess
emission or resolved spatial structure in mid-IR images.
HD 172555– We detected significant IR excess in both bandpasses. We estimated
a minimum dust temperature by assuming that the dust must lie interior to the
disk size corresponding to the 2-$\sigma_{ext}$ limit of the source FWHM. The
temperature estimates based on extension measurements are 407 K (based on the
11.7 $\mu$m data) and 278 K (based on the 18.3 $\mu$m data). The color
temperature calculated from the excess emission measurements from the two
bandpasses is 274 $\pm$ 34 K. Therefore, the temperature estimate based on the
source size limit at 18.3 $\mu$m is consistent with the color temperature, but
the temperature estimate based on the 11.7-$\mu$m source size is not. It is
possible for nonspherical particles to have temperatures lower than those
expected for blackbody emission (e.g., Greenberg & Shah 1971; Voshchinnikov &
Semenov 2000), and this may be the case for the 11.7-$\mu$m-emitting particles
in HD 172555.
Wyatt et al. (2007) noted that the fractional IR luminosity of HD 172555 is
anomalously high, or eighty-six times the expected maximum fractional IR
luminosity based on steady-state evolution models. An estimate of 6 AU is
given for the disk radius, based on 24 $\mu$m and 70 $\mu$m $Spitzer$
photometry; this radius corresponds to a blackbody equilibrium temperature of
$\sim$200 K. This temperature is $\sim$75 K lower than the color temperature
estimate based on mid-IR color, but it is possible that the dust present in
the disk spans a range of radii, and therefore temperatures, and (at least)
two different populations of dust are being sampled by the 12 $\mu$m/18 $\mu$m
color and the 24 $\mu$m/70 $\mu$m color.
HD 178253– We did not detect statistically significant excess emission
associated with this source. However, if there is unresolved emission
originating from dust within the region corresponding to the size of the
2-$\sigma_{ext}$ limit of the source FWHM at each of the observed bandpasses,
then the emitting dust temperatures should be at least 432 K (based on the
11.7-$\mu$m FWHM) and 246 K (based on the 18.3-$\mu$m FWHM).
HD 181296– There is significant IR excess emission detected in both
bandpasses. The temperature estimates for dust within a region the size of the
2-$\sigma_{ext}$ limit of the source FWHM are 397 K (based on the 11.7-$\mu$m
FWHM) and 143 K (based on the 18.3-$\mu$m FWHM). Only the estimate based on
the source FWHM at 18.3 $\mu$m is consistent with the color temperature, which
is 229 $\pm$ 23 K.
Since our initial observations, HD 181296 has been resolved in Q-band images
taken in July 2007 at mid-IR wavelengths with longer (6.5x) integration times
(Smith et al., 2009a). It is likely that the initial observations did not
resolve the source because the integration time was not sufficient to detect
the surface brightness of the disk. The extended emission peaks detected by
Smith et al. (2009a) total $90~{}\pm~{}5$ mJy, and if we assume that the flux
is approximately equally distributed between the two peaks, then this emission
is less than three times our background noise level. Smith et al. (2008)
discuss in greater detail the relationship between disk morphology and the
predicted length of integration required to resolved a given source.
Models of the 18.3-$\mu$m images of the source indicate that $\sim$50% of the
excess emission originates in the resolved component in an apparently edge-on
disk with a 24-AU radius, which can be fit by a modified blackbody of
temperature $\sim$100 K. The rest of the excess emission resides in an
unresolved component of temperature 310 K, consistent with dust at 3.9 AU.
Discrepancies in temperature estimates based on the initial and follow-up data
sets exist primarily because of a 13% lower flux density measurement at 11.7
$\mu$m (from the more recent images); however, these two measurements are
still within $\sim$1$\sigma$ of each other.
### 5.2 Resolved Sources
For reference, we have plotted the azimuthally averaged radial profiles of the
resolved sources and their corresponding PSFs in Figure 5. However, we note
that the images used to generate these plots are only median stacks of all of
the subset frames that were used to assess their spatial resolution.
HD 141569– We review HD 141569 first, because its spatial extent has been
detected previously in mid-IR images (Fisher et al., 2000; Marsh et al., 2002)
and this data set can therefore provide a benchmark for comparison as a
consistent body of evidence that describes the source size.
The source FWHM appears extended in comparison to the PSF FWHM at both 11.2
$\mu$m and 18.1 $\mu$m, and the statistical significance of these extensions
are 3.6$\sigma_{ext}$ (11.2$\mu$m) and 9.1$\sigma_{ext}$ and 4.5$\sigma_{ext}$
(18.1-$\mu$m, from two nights of data). These data were not combined, because
the location of the central star could not be determined accurately, and this
location is necessary to register and stack images. Student’s $t$-tests on the
data yield the following results: at 11.2 $\mu$m, the $p$-value is 0.005 and
there is a 95% confidence interval that the source FWHM is greater than the
PSF FWHM by 0.023” to 0.100”; at 18.1 $\mu$m, the $p$-values are 6 x 10-5 and
6 x 10-4, and the 95% confidence intervals are 0.195”–0.340” and
0.151”–0.439”.
The blackbody dust temperatures at the radii inferred from the 11.2-$\mu$m
images and the 18.1-$\mu$m images are 186 $\pm$ 4 K at $\sim 11$ AU and 112
$\pm$ 3 K at $\sim$30 AU, respectively. (For comparison, Fisher et al. (2000)
determined disk radii of 17 AU and 34 AU at 10.8 $\mu$m and 18.2 $\mu$m,
respectively.) These temperatures are consistent with the color temperature
estimate for HD 141569 of 191 $\pm$ 16 K, in that the extension-implied
temperatures are a lower limit and the color temperature is an upper limit.
Indeed, the models of both Fisher et al. (2000) and Marsh et al. (2002) for
this disk suggest that the dominant particle population is composed of
inefficient emitters that would be heated to temperatures above those expected
for perfect blackbody emitters, which implies that our extent-implied
temperatures should also be higher.
HD 38678, $\zeta$ Lep– $\zeta$ Lep is unresolved at 10.4 $\mu$m but is
resolved at 18.3 $\mu$m, with a significance level of 3.1$\sigma_{ext}$. These
results are discussed in detail in a prior work (Moerchen et al., 2007b). We
have further confirmed the extent with a Student’s $t$-test, which yields a
95% confidence interval that the source FWHM is greater than the PSF FWHM by
0.035”–0.114”. The spatial extent of the 18.3-$\mu$m disk profile implies a
disk radius of 3 AU, which is comparable in location to the asteroid belt in
the solar system. Prior to the spatial resolution of HD 71155 (discussed
below), this was the only resolved debris disk spanning only a few AU (Chen &
Jura, 2001; Moerchen et al., 2007b). The color temperature for the excess
emission associated with $\zeta$ Lep is 323${}^{+27}_{-30}$ K, which is
consistent with the blackbody temperature of 320 K for dust grains at 3.0 AU.
The implied presence of large (r $\sim$ few microns) grains that emit like
blackbodies is supported by $Spitzer$ IRS spectra taken by Chen et al. (2006),
which show no silicate emission feature. Relatively large grains may not cause
silicate emission, as shown by Przygodda et al. (2003).
HD 71155– HD 71155 is resolved at 10.4 $\mu$m, at a significance level of
4.4$\sigma_{ext}$. The 95% confidence interval from a Student’s $t$-test is
0.008”-0.025”, with a $p$-value of 1 x 10-4. The extent (as computed with the
full data set) implies a disk radius of 2.0 $\pm$ 0.1 AU, at which the
blackbody temperature is 499 $\pm$ 3 K. The marginal excess emission for this
source could not be fitted by a simple single-temperature blackbody. The
excess was instead fitted with emitting particles that are efficient absorbers
but inefficient emitters. This may be the case for particles which are larger
than the peak wavelength of stellar emission but smaller than the peak
wavelength of particle thermal emission (Backman & Paresce, 1993). In the
literature, an emission efficiency of the form $Q_{em}\propto\nu^{n}$, with
n=1–2, is often assumed. The color temperature for such inefficient emitters
with n=1 is 487 $\pm$ 129 K, which is consistent with the temperature implied
by the 10.4-$\mu$m extension.
The unresolved 18.3-$\mu$m result is consistent if this emission originates
from the same location as the 10.4-$\mu$m emission; the quadratic sum of the
10.4-$\mu$m-based source size and the 18.3-$\mu$m PSF FWHM size is less than
the nominal detection limit for extension, $FHWM_{PSF}+2\sigma_{ext}$. If this
unresolved 18.3-$\mu$m-emitting dust is assumed to lie within a region
corresponding to this 2-$\sigma_{ext}$ limit of the source FWHM, then its
equilibrium temperature is at least 281 K, which is also consistent with the
upper limit set by the color temperature.
In Fig. 6, we note the possibility that the image quality has systematically
degraded around savesets #40 and #65. We tested the impact of the data in this
region by removing 25 contiguous savesets that include the peaks in FWHM and
then recomputing the mean FWHM value for the source, an exercise which
demonstrates the utility of measuring the source profile in every saveset
rather than solely in the final stacked image. We found that the source FWHM
decreased from 0.348” to 0.339”, but that the standard deviation likewise
decreases, such that the source FWHM is still greater than the PSF FWHM at a
3.1-$\sigma_{ext}$ level. When the $t$-test is repeated with the truncated
data set, the $p$-value is 0.035, and there is a 95% confidence level that the
source FWHM is greater than the PSF FWHM by 0.001”–0.015”. Nonetheless, this
result should be confirmed with deeper imaging observations.
HD 95418– The source is resolved at 11.2 $\mu$m with a statistical
significance of 6.8$\sigma_{ext}$. The results of a Student’s $t$-test on the
data are a $p$-value of 3 x 10-5 and a 95% confidence interval of
0.008”–0.016”. No excess emission associated with HD 95418 is detected in
either bandpass, but the measurement of the excess emission at 11.2$\mu$m is
within 2$\sigma_{phot}$ of a statistically significant excess detection, so
the spatial resolution at that wavelength is consistent. The dust temperature
at the orbital distance implied by this resolved result is 764 $\pm$ 2 K.
Figure 4: 11.2-$\mu$m images of HD 139006, its PSF reference star, and the
residual emission following PSF subtraction, where the PSF was scaled to match
the peak emission of the disk source. Contours are drawn at 3-$\sigma_{bkd}$
intervals from 6$\sigma_{bkd}$ to 15$\sigma_{bkd}$. Note that HD 139006 is
visibly elongated compared to the PSF in the first night of data, and less so
in the second night of data (in which the PSF also appears elongated). We
believe that these images may suffer from so-called “chop tails,” an issue
that has since been resolved at Gemini.
HD 139006– This source is resolved at 11.2 $\mu$m with a statistical
significance of 8.6$\sigma_{ext}$. A Student’s $t$-test yields a $p$-value of
2 x 10-7, with a 95% confidence interval that the source FWHM should be
broader than the PSF FWHM by 0.041”– 0.068”. However, in two sets of
18.1-$\mu$m images from two different nights, one is resolved and one is not.
We believe that this discrepancy arises from image elongation associated with
chopping and nodding procedures, which can be seen clearly in a comparison
images of HD 139006 at 18.1 $\mu$m on two different nights in Fig. 4. Indeed,
in one set, the PSF profile is broader than that of the source (confirmed by a
Student’s $t$-test $p$-value of 6 x 10-7), and thus we do not claim that the
source is spatially resolved at 18.1 $\mu$m. While the large photometric
uncertainty $\sigma_{phot}$ renders the observed excess emission statistically
insignificant, the measurements are well within 2$\sigma_{phot}$ of a formal
excess detection, so the results are consistent.
HD 181869– HD 181869 is resolved at 10.4 $\mu$m (3.1$\sigma_{ext}$) and
unresolved at 18.3 $\mu$m. A Student’s $t$-test on the 10.4-$\mu$m data yields
a $p$-value of 0.041, and a 95% confidence interval that the source FWHM
should be 0.001”–0.048” greater than the PSF FWHM. We have rejected the
profile measurement data at 18.3 $\mu$m on the basis of the PSF being broader
than the source, which was confirmed by a Student’s $t$-test with a $p$-value
of 0.029. No excess emission associated with HD 181869 was detected at either
10.4 $\mu$m or 18.3 $\mu$m, and the simultaneous resolution and lack of
detected excess seem contradictory. However, the measured excess emission at
10.4 $\mu$m is within 2$\sigma_{phot}$ of a marginal excess detection, which
is consistent with the apparent spatial resolution at that wavelength.
Figure 5: Normalized radial brightness profiles of resolved sources (open
circles) and their corresponding PSF stars (filled circles). Sources: (a) HD
71155 (10.4 $\mu$m), (b) HD 95418 (11.2 $\mu$m), (c) HD 141569 (11.2 $\mu$m),
(d) HD 139006 (11.2 $\mu$m), (e) HD 38678 (18.3 $\mu$m), (f) HD 141569 (18.1
$\mu$m), (g) HD 181869 (10.4 $\mu$m). Figure 6: FWHM of profile fits to the
sources at 10.4 $\mu$m, per saveset. Open circles represent the PSF reference
star, and filled circles represent the debris disk target. Sources: (a) HD
38206, (b) HD 38678, (c) HD 71155, (d) HD 80950, (e) HD 181869.
## 6 Discussion
To better understand the nature of the unresolved sources (among those that
are still considered debris disk candidates), their colors were used to
compare these sources to spatially resolved debris disks whose structure is
relatively well known. In Figure 7, the mid-IR color temperature is plotted
against the age for each debris disk candidate. The resolved sources are
represented by star symbols, and the unresolved sources are represented by
filled circles. Most of the sources that have been resolved with ground-based
mid-IR imaging observations also have the cooler color temperatures. We expect
cooler dust to be more distant from the star and therefore be part of a more
extended disk that is easier to resolve. Thus, the fact that most of the
resolved sources have relatively cool dust is not surprising.
Of course, our sample is limited in number (eight, after non-disk sources and
null excesses have been culled) and biased. The sources were chosen on the
basis of their 24-$\mu$m excess, so it is already known that they have some
warm dust, although possibly not hot enough to emit significantly at 12 or 18
$\mu$m, as the null excess detections suggest. In addition, it is well known
that there are more sources with high fractional luminosities at younger ages,
especially less than 20 Myr (Rieke et al., 2005; Su et al., 2006; Currie et
al., 2008), and our sample, which was chosen with a brightness criterion,
reflects that trend. The cluster of sources toward the left of the plot, at
ages less than 100 Myr, must be considered with these biases in mind.
Figure 7: Mid-IR color temperature of dust versus system age. Age values are
the average of all estimates quoted by Rieke et al. (2005), with the exception
of HD 38678 (230 Myr). Sources represented by filled circles have standard
color temperatures as estimated for unresolved sources (see text). Sources
represented by star-shaped points have been spatially resolved by mid-IR
images from this study (and in the case of HD 141569, also by prior works).
For reference, $\beta$ Pic has an age of 12 Myr and a color temperature of
$\sim$180 K.
Although our sample is not statistically significant, the lack of sources in
Figure 7 with ages in the range $\sim$50–200 Myr is thought-provoking. In the
cluster of sources at young ages, we have resolved one source with a cool dust
temperature that is comparable in extent to the Kuiper Belt (along with
$\beta$ Pic, HR 4796A, and HD 32297, as further examples). In contrast, with
the exception of HD 75416 (5 Myr), which has a very large uncertainty in the
dust temperature, the two sources in our sample with significantly hotter dust
populations are also significantly older: HD 38678 ($\zeta$ Lep) at $\sim$230
Myr and HD 71155 at $\sim$200 Myr. HD 71155 is a new spatially resolved source
with a dust disk radius implied by its 10.4-$\mu$m extent of 2.0 $\pm$ 0.8 AU,
and this size is also comparable to the size of the solar system’s asteroid
belt. Our sample also includes two sources (HD 95418 and HD 139006) that do
not have a statistically significant detection of excess IR emission from our
data set, but whose spatial extents imply disk radii similar to that of the
asteroid belt. As mentioned in §3.1, the measurements of the excess emission
and the spatial extension are not necessarily inconsistent due to the error
bars associated with each value.
While low-resolution surveys show that the mid- and far-IR emission from disks
generally diminishes with time as the inverse of the system lifetime (Rieke et
al., 2005; Su et al., 2006), our observations of two apparent asteroid-belt
analogs in our sample imply that somewhat older ($>$ few 100 Myr) sources can
sustain significant mid-IR emission above the average levels by ongoing
production of dust in asteroid-belt-type collections of planetesimals
relatively close to the star. Whether the collisions have been occurring in a
steady state is not obvious, but the amount of the IR excess may help to
answer that question. Wyatt et al. (2007) distinguish disks as having
potentially transient dust-producing events if they have greater than 1000
times the maximum fractional IR luminosity predicted for their age that have
experienced only steady-state collisions. In the case of $\zeta$ Lep, the
24-$\mu$m excess exceeds the expected level due to steady-state collisions
alone by more than a factor of 10. However, the excess level for HD 71155
falls within the envelope of expected values for disks experiencing solely
steady-state collisions (Wyatt et al., 2007). Therefore, it may be more
plausible (but not imperative) to invoke a process such as delayed stirring or
an event analogous to the Earth and moon-progenitor collision for $\zeta$ Lep,
whereas observations of HD 71155 seem to be consistent with steady-state
evolution.
It is worth noting that the fractional IR luminosities of sources in our
sample (with formal detections of excess emission) that we consider to be
Kuiper Belt analogs (e.g., HD 32297, HR 4796A; $L_{IR}/L_{\star}\sim$10-3) are
$\sim$100 times higher than that of sources that we consider to be asteroid
belt analogs (e.g., HD 38678 [$\zeta$ Lep], HD 71155;
$L_{IR}/L_{\star}\sim$10-5). Thus, in a system that hosts both asteroid-belt-
like and Kuiper-Belt-like structures, the presence of a Kuiper Belt with a
significantly larger amount of dust may make it difficult to discern the
emission from an asteroid belt (see also Liou & Zook 1999). However, with the
advent of the next generation of ground-based telescopes ($>$30-m), the
improvement in diffraction-limited resolving power should enable MIR cameras
to distinguish both belts in such systems.
For the remaining unresolved sources that sustain statistically significant IR
excesses, what can the presence of the warm dust tell us? Ultimately, we would
like to know the distribution of the dust both radially and azimuthally in
order to investigate the planetary system’s architecture. That will hold clues
to the production of the dust (e.g. steady-state or catastrophic collisions)
and what maintains it in its current location (e.g., shepherding planets).
There are (as of the time of writing) approximately 10 disks that are known to
harbor planets (e.g., Wyatt 2008). It is currently easier to make radial
velocity planet detections (the primary detection technique) around FGK-type
stars, while it is easier to spatially resolve the thermal emission from dust
disks around the much more luminous A-type stars, and so there is
unfortunately little overlap in the detections. If information about the
planetary orbits is known, however, dynamical simulations may indicate where
the dust is likely to be stable and how and where the dust was initially
produced. Such simulations have been made for the K-star debris disk HD 69830,
which sustains a surprisingly high amount of dust for its 4–10 Gyr age in
addition to three Neptune-mass planets (Beichman et al., 2005; Lovis et al.,
2006; Lisse et al., 2007). Lovis et al. (2006) showed that, given the
locations of the planets as determined by radial velocity measurements, there
are two stable radii for dust annuli. The recent direct detection of an
orbiting body apparently sculpting the sharp inner edge of the debris disk of
Fomalhaut (Kalas et al., 2008) highlights this relationship.
In future work, observations at 8-meter facilities with longer integration
times and tighter constraints on image quality may reveal more details of the
disk structure for some of the sources in this sample, particularly for the
“borderline” cases. For example, four sources in our sample (HD 161868, HD
172555, HD 178253, and HD 181296) are not spatially resolved, but the color
temperature of their excess IR emission corresponds to that of dust particles
emitting like blackbodies in the approximate region of the asteroid belt
($\sim$1–3 AU). High-resolution imaging at other wavelengths such as the near-
IR or submillimeter may also provide a more complete picture of the disk
morphology (e.g., Maness et al. 2008, Debes et al. 2009, Fitzgerald et al.
2007). For disks with especially small angular sizes ($\lesssim$0.1”),
interferometric observations in the near-IR and mid-IR have also yielded
useful constraints on disk morphologies (e.g., Smith et al. 2009, Akeson et
al. 2009).
MMM gratefully acknowledges fellowship support from the Michelson Science
Center. This work was performed [in part] under contract with JPL funded by
NASA through the Michelson Fellowship Program. JPL is managed for NASA by the
California Institute of Technology. This research was partially supported by
NSF Grant AST 0098392 to CMT. Observations were obtained at the Gemini
Observatory, operated by AURA, Inc., under agreement with the NSF on behalf of
the Gemini partnership: NSF (US), PPARC (UK), NRC (Canada), CONICYT (Chile),
ARC (Australia), CNPq (Brazil), and CONICET (Argentina). Facilities:
Gemini:North (Michelle), Gemini:South (T-ReCS).
## Appendix A Appendix: Detailed Profile Width Measurements
Here we provide the details of the FWHM measurements of the debris disk
candidates and their corresponding PSF reference stars (Figures 6, 8, 9, 10,
and 11). As discussed in §3.2, the total integration time for an image was
broken up into sub-images each corresponding to a fraction of the total time,
such that the FWHM of the source could be sampled as frequently as possible.
When S/N levels allowed, the smallest unit of time for a sub-image was that
corresponding to a saveset, $\sim$10 s. A saveset is a stack of chopped images
(on- and off-source), and there are typically three savesets per nod position.
For formal and final image stacking, images from both nod positions must be
combined to remove the radiative offset. However, the S/N of the sources was
high enough that the radiative offset did not affect the profile fits to the
sources. Measuring the FWHM in single savesets had the additional benefit of
not incorporating positional errors arising from telescope motion. When images
are taken at two nod positions, we expect that the source location is the same
in both images. However, there may be a slight positional inaccuracy occurring
between each nod switch, and this is avoided by not combining images from two
nod positions.
When the S/N levels were not high enough to perform a reasonable profile fit
to the source in a single saveset, these frames were binned up until a
sufficient S/N level was reached. In the following plots, the total number of
savesets is shown as a temporal series along the x-axis. If the FWHM was
measured in each saveset image, then the number of data points equals the
number of savesets. If, for example, six savesets had to be binned for a FWHM
measurement, then there will only be one data point for every six savesets,
and the data point will be shown at the center of the binned saveset group,
e.g., savesets 1–6 are binned, so the FWHM value is plotted above the “saveset
#3” tick mark.
Figure 8: FWHM of profile fits to the sources at 11.2 $\mu$m, per saveset.
Open circles represent the PSF reference star, and filled circles represent
the debris disk target. Sources: (a) HD 56537, (b) HD 83808, (c) HD 95418, (d)
HD 102647, (e) HD 139006, (f) HD 141569, (g) HD 161868. Figure 9: FWHM of
profile fits to the sources at 11.7 $\mu$m, per saveset. Open circles
represent the PSF reference star, and filled circles represent the debris disk
target. Sources: (a) HD 75416, (b) HD 115892, (c) HD 178253. Figure 10: FWHM
of profile fits to the sources at 18.1 $\mu$m, per saveset. Open circles
represent the PSF reference star, and filled circles represent the debris disk
target. Sources: (a) HD 56537, (b) HD 56537, (c) HD 83808, (d) HD 95418, (e)
HD 102647, (f) HD 102647, (g) HD 139006, (h) HD 139006, (i) HD 141569, (j) HD
141569, (k) HD 161868, (l) HD 161868. Figure 11: FWHM of profile fits to the
sources at 18.3 $\mu$m, per saveset. Open circles represent the PSF reference
star, and filled circles represent the debris disk target. Sources: (a) HD
38206, (b) HD 38678, (c) HD 71155, (d) HD 75416, (e) HD 80950, (f) HD 115892,
(g) HD 115892, (h) HD 178253, (i) HD 181869. Figure 12: FWHM of profile fits
to the sources, per saveset. Open circles represent the PSF reference star at
11.7 $\mu$m, and filled circles represent the debris disk target at 11.7
$\mu$m. Open squares represent the PSF reference star at 18.3 $\mu$m, and
filled squares represent the debris disk target at 18.3 $\mu$m. Sources: (a)
HD 172555, (b) HD 181296.
## References
* Akeson et al. (2009) Akeson, R. L., Ciardi, D. R., Millan-Gabet, R., Merand, A., Folco, E. D., Monnier, J. D., Beichman, C. A., Absil, O., Aufdenberg, J., McAlister, H., Brummelaar, T. t., Sturmann, J., Sturmann, L., & Turner, N. 2009, ApJ, 691, 1896
* Aumann (1985) Aumann, H. H. 1985, PASP, 97, 885
* Aumann (1988) —. 1988, AJ, 96, 1415
* Backman & Paresce (1993) Backman, D. E. & Paresce, F. 1993, in Protostars and Planets III, ed. E. H. Levy & J. I. Lunine, 1253–1304
* Beichman et al. (2005) Beichman, C. A., Bryden, G., Gautier, T. N., Stapelfeldt, K. R., Werner, M. W., Misselt, K., Rieke, G., Stansberry, J., & Trilling, D. 2005, ApJ, 626, 1061
* Brittain & Rettig (2002) Brittain, S. D. & Rettig, T. W. 2002, Nature, 418, 57
* Brittain et al. (2003) Brittain, S. D., Rettig, T. W., Simon, T., Kulesa, C., DiSanti, M. A., & Dello Russo, N. 2003, ApJ, 588, 535
* Chen & Jura (2001) Chen, C. H. & Jura, M. 2001, ApJ, 560, L171
* Chen et al. (2006) Chen, C. H., Sargent, B. A., Bohac, C., Kim, K. H., Leibensperger, E., Jura, M., Najita, J., Forrest, W. J., Watson, D. M., Sloan, G. C., & Keller, L. D. 2006, ApJS, 166, 351
* Chen et al. (2009) Chen, C. H., Sheehan, P., Watson, D. M., Manoj, P., & Najita, J. R. 2009, ApJ, 701, 1367
* Cohen et al. (1999) Cohen, M., Walker, R. G., Carter, B., Hammersley, P., Kidger, M., & Noguchi, K. 1999, AJ, 117, 1864
* Cote & Waters (1987) Cote, J. & Waters, L. B. F. M. 1987, A&A, 176, 93
* Currie et al. (2008) Currie, T., Kenyon, S. J., Balog, Z., Rieke, G., Bragg, A., & Bromley, B. 2008, ApJ, 672, 558
* Cutri et al. (2003) Cutri, R. M., Skrutskie, M. F., van Dyk, S., Beichman, C. A., Carpenter, J. M., Chester, T., Cambresy, L., Evans, T., Fowler, J., Gizis, J., Howard, E., Huchra, J., Jarrett, T., Kopan, E. L., Kirkpatrick, J. D., Light, R. M., Marsh, K. A., McCallon, H., Schneider, S., Stiening, R., Sykes, M., Weinberg, M., Wheaton, W. A., Wheelock, S., & Zacarias, N. 2003, 2MASS All Sky Catalog of point sources. (The IRSA 2MASS All-Sky Point Source Catalog, NASA/IPAC Infrared Science Archive. http://irsa.ipac.caltech.edu/applications/Gator/)
* de Zeeuw et al. (1999) de Zeeuw, P. T., Hoogerwerf, R., de Bruijne, J. H. J., Brown, A. G. A., & Blaauw, A. 1999, AJ, 117, 354
* Debes et al. (2009) Debes, J. H., Weinberger, A. J., & Kuchner, M. J. 2009, ApJ, 702, 318
* Fisher et al. (2000) Fisher, R. S., Telesco, C. M., Piña, R. K., Knacke, R. F., & Wyatt, M. C. 2000, ApJ, 532, L141
* Fitzgerald et al. (2007) Fitzgerald, M. P., Kalas, P. G., Duchêne, G., Pinte, C., & Graham, J. R. 2007, ApJ, 670, 536
* Gáspár et al. (2008) Gáspár, A., Su, K. Y. L., Rieke, G. H., Balog, Z., Kamp, I., Martínez-Galarza, J. R., & Stapelfeldt, K. 2008, ApJ, 672, 974
* Gerbaldi et al. (1999) Gerbaldi, M., Faraggiana, R., Burnage, R., Delmas, F., Gómez, A. E., & Grenier, S. 1999, A&AS, 137, 273
* Jura et al. (1998) Jura, M., Malkan, M., White, R., Telesco, C., Pina, R., & Fisher, R. S. 1998, ApJ, 505, 897
* Kalas et al. (2008) Kalas, P., Graham, J. R., Chiang, E., Fitzgerald, M. P., Clampin, M., Kite, E. S., Stapelfeldt, K., Marois, C., & Krist, J. 2008, Science, 322, 1345
* Kenyon & Bromley (2004) Kenyon, S. J. & Bromley, B. C. 2004, ApJ, 602, L133
* Kervella et al. (2009) Kervella, P., Thévenin, F., & Petr-Gotzens, M. G. 2009, A&A, 493, 107
* King et al. (2003) King, J. R., Villarreal, A. R., Soderblom, D. R., Gulliver, A. F., & Adelman, S. J. 2003, AJ, 125, 1980
* Kurucz (1979) Kurucz, R. L. 1979, ApJS, 40, 1
* Lagage & Pantin (1994) Lagage, P. O. & Pantin, E. 1994, Nature, 369, 628
* Liou & Zook (1999) Liou, J.-C. & Zook, H. A. 1999, AJ, 118, 580
* Lisse et al. (2007) Lisse, C. M., Beichman, C. A., Bryden, G., & Wyatt, M. C. 2007, ApJ, 658, 584
* Lovis et al. (2006) Lovis, C., Mayor, M., Pepe, F., Alibert, Y., Benz, W., Bouchy, F., Correia, A. C. M., Laskar, J., Mordasini, C., Queloz, D., Santos, N. C., Udry, S., Bertaux, J.-L., & Sivan, J.-P. 2006, Nature, 441, 305
* Maness et al. (2008) Maness, H. L., Fitzgerald, M. P., Paladini, R., Kalas, P., Duchene, G., & Graham, J. R. 2008, ApJ, 686, L25
* Mannings & Barlow (1998) Mannings, V. & Barlow, M. J. 1998, ApJ, 497, 330
* Marois et al. (2008) Marois, C., Macintosh, B., Barman, T., Zuckerman, B., Song, I., Patience, J., Lafrenière, D., & Doyon, R. 2008, Science, 322, 1348
* Marsh et al. (2002) Marsh, K. A., Silverstone, M. D., Becklin, E. E., Koerner, D. W., Werner, M. W., Weinberger, A. J., & Ressler, M. E. 2002, ApJ, 573, 425
* Merín et al. (2004) Merín, B., Montesinos, B., Eiroa, C., Solano, E., Mora, A., D’Alessio, P., Calvet, N., Oudmaijer, R. D., de Winter, D., Davies, J. K., Harris, A. W., Cameron, A., Deeg, H. J., Ferlet, R., Garzón, F., Grady, C. A., Horne, K., Miranda, L. F., Palacios, J., Penny, A., Quirrenbach, A., Rauer, H., Schneider, J., & Wesselius, P. R. 2004, A&A, 419, 301
* Moerchen et al. (2007a) Moerchen, M. M., Telesco, C. M., De Buizer, J. M., Packham, C., & Radomski, J. T. 2007a, ApJ, 666, L109
* Moerchen et al. (2007b) Moerchen, M. M., Telesco, C. M., Packham, C., & Kehoe, T. J. J. 2007b, ApJ, 655, L109
* Moór et al. (2006) Moór, A., Ábrahám, P., Derekas, A., Kiss, C., Kiss, L. L., Apai, D., Grady, C., & Henning, T. 2006, ApJ, 644, 525
* Moshir (1989) Moshir, M. 1989, IRAS Faint Source Survey, Explanatory supplement version 1 and tape (Pasadena: Infrared Processing and Analysis Center, California Institute of Technology, 1989, edited by Moshir, M.)
* Oudmaijer et al. (1992) Oudmaijer, R. D., van der Veen, W. E. C. J., Waters, L. B. F. M., Trams, N. R., Waelkens, C., & Engelsman, E. 1992, A&AS, 96, 625
* Przygodda et al. (2003) Przygodda, F., van Boekel, R., Àbrahàm, P., Melnikov, S. Y., Waters, L. B. F. M., & Leinert, C. 2003, A&A, 412, L43
* Radomski et al. (2008) Radomski, J. T., Packham, C., Levenson, N. A., Perlman, E., Leeuw, L. L., Matthews, H., Mason, R., De Buizer, J. M., Telesco, C. M., & Orduna, M. 2008, ApJ, 681, 141
* Reidemeister et al. (2009) Reidemeister, M., Krivov, A. V., Schmidt, T. O. B., Fiedler, S., Müller, S., Löhne, T., & Neuhäuser, R. 2009, A&A, 503, 247
* Rieke et al. (2005) Rieke, G. H., Su, K. Y. L., Stansberry, J. A., Trilling, D., Bryden, G., Muzerolle, J., White, B., Gorlova, N., Young, E. T., Beichman, C. A., Stapelfeldt, K. R., & Hines, D. C. 2005, ApJ, 620, 1010
* Schneider et al. (1999) Schneider, G., Smith, B. A., Becklin, E. E., Koerner, D. W., Meier, R., Hines, D. C., Lowrance, P. J., Terrile, R. J., Thompson, R. I., & Rieke, M. 1999, ApJ, 513, L127
* Smith et al. (2009a) Smith, R., Churcher, L. J., Wyatt, M. C., Moerchen, M. M., & Telesco, C. M. 2009a, A&A, 493, 299
* Smith et al. (2008) Smith, R., Wyatt, M. C., & Dent, W. R. F. 2008, A&A, 485, 897
* Smith et al. (2009b) Smith, R., Wyatt, M. C., & Haniff, C. A. 2009b, A&A, 503, 265
* Song et al. (2001) Song, I., Caillault, J.-P., Barrado y Navascués, D., & Stauffer, J. R. 2001, ApJ, 546, 352
* Su et al. (2006) Su, K. Y. L., Rieke, G. H., Stansberry, J. A., Bryden, G., Stapelfeldt, K. R., Trilling, D. E., Muzerolle, J., Beichman, C. A., Moro-Martin, A., Hines, D. C., & Werner, M. W. 2006, ApJ, 653, 675
* Su et al. (2008) Su, K. Y. L., Rieke, G. H., Stapelfeldt, K. R., Smith, P. S., Bryden, G., Chen, C. H., & Trilling, D. E. 2008, ApJ, 679, L125
* Telesco et al. (2000) Telesco, C. M., Fisher, R. S., Piña, R. K., Knacke, R. F., Dermott, S. F., Wyatt, M. C., Grogan, K., Holmes, E. K., Ghez, A. M., Prato, L., Hartmann, L. W., & Jayawardhana, R. 2000, ApJ, 530, 329
* Telesco et al. (2005) Telesco, C. M., Fisher, R. S., Wyatt, M. C., Dermott, S. F., Kehoe, T. J. J., Novotny, S., Mariñas, N., Radomski, J. T., Packham, C., De Buizer, J., & Hayward, T. L. 2005, Nature, 433, 133
* Voshchinnikov & Semenov (2000) Voshchinnikov, N. V. & Semenov, D. A. 2000, Astronomy Letters, 26, 679
* Weinberger et al. (1999) Weinberger, A. J., Becklin, E. E., Schneider, G., Smith, B. A., Lowrance, P. J., Silverstone, M. D., Zuckerman, B., & Terrile, R. J. 1999, ApJ, 525, L53
* Weinberger et al. (2000) Weinberger, A. J., Rich, R. M., Becklin, E. E., Zuckerman, B., & Matthews, K. 2000, ApJ, 544, 937
* Wyatt (2008) Wyatt, M. C. 2008, Annual Review of Astronomy and Astrophysics, 46
* Wyatt et al. (1999) Wyatt, M. C., Dermott, S. F., Telesco, C. M., Fisher, R. S., Grogan, K., Holmes, E. K., & Piña, R. K. 1999, ApJ, 527, 918
* Wyatt et al. (2007) Wyatt, M. C., Smith, R., Su, K. Y. L., Rieke, G. H., Greaves, J. S., Beichman, C. A., & Bryden, G. 2007, ApJ, 663, 365
* Zuckerman & Song (2004) Zuckerman, B. & Song, I. 2004, ApJ, 603, 738
* Zuckerman et al. (2001) Zuckerman, B., Song, I., Bessell, M. S., & Webb, R. A. 2001, ApJ, 562, L87
## References
* Akeson et al. (2009) Akeson, R. L., Ciardi, D. R., Millan-Gabet, R., Merand, A., Folco, E. D., Monnier, J. D., Beichman, C. A., Absil, O., Aufdenberg, J., McAlister, H., Brummelaar, T. t., Sturmann, J., Sturmann, L., & Turner, N. 2009, ApJ, 691, 1896
* Aumann (1985) Aumann, H. H. 1985, PASP, 97, 885
* Aumann (1988) —. 1988, AJ, 96, 1415
* Backman & Paresce (1993) Backman, D. E. & Paresce, F. 1993, in Protostars and Planets III, ed. E. H. Levy & J. I. Lunine, 1253–1304
* Beichman et al. (2005) Beichman, C. A., Bryden, G., Gautier, T. N., Stapelfeldt, K. R., Werner, M. W., Misselt, K., Rieke, G., Stansberry, J., & Trilling, D. 2005, ApJ, 626, 1061
* Brittain & Rettig (2002) Brittain, S. D. & Rettig, T. W. 2002, Nature, 418, 57
* Brittain et al. (2003) Brittain, S. D., Rettig, T. W., Simon, T., Kulesa, C., DiSanti, M. A., & Dello Russo, N. 2003, ApJ, 588, 535
* Chen & Jura (2001) Chen, C. H. & Jura, M. 2001, ApJ, 560, L171
* Chen et al. (2006) Chen, C. H., Sargent, B. A., Bohac, C., Kim, K. H., Leibensperger, E., Jura, M., Najita, J., Forrest, W. J., Watson, D. M., Sloan, G. C., & Keller, L. D. 2006, ApJS, 166, 351
* Chen et al. (2009) Chen, C. H., Sheehan, P., Watson, D. M., Manoj, P., & Najita, J. R. 2009, ApJ, 701, 1367
* Cohen et al. (1999) Cohen, M., Walker, R. G., Carter, B., Hammersley, P., Kidger, M., & Noguchi, K. 1999, AJ, 117, 1864
* Cote & Waters (1987) Cote, J. & Waters, L. B. F. M. 1987, A&A, 176, 93
* Currie et al. (2008) Currie, T., Kenyon, S. J., Balog, Z., Rieke, G., Bragg, A., & Bromley, B. 2008, ApJ, 672, 558
* Cutri et al. (2003) Cutri, R. M., Skrutskie, M. F., van Dyk, S., Beichman, C. A., Carpenter, J. M., Chester, T., Cambresy, L., Evans, T., Fowler, J., Gizis, J., Howard, E., Huchra, J., Jarrett, T., Kopan, E. L., Kirkpatrick, J. D., Light, R. M., Marsh, K. A., McCallon, H., Schneider, S., Stiening, R., Sykes, M., Weinberg, M., Wheaton, W. A., Wheelock, S., & Zacarias, N. 2003, 2MASS All Sky Catalog of point sources. (The IRSA 2MASS All-Sky Point Source Catalog, NASA/IPAC Infrared Science Archive. http://irsa.ipac.caltech.edu/applications/Gator/)
* de Zeeuw et al. (1999) de Zeeuw, P. T., Hoogerwerf, R., de Bruijne, J. H. J., Brown, A. G. A., & Blaauw, A. 1999, AJ, 117, 354
* Debes et al. (2009) Debes, J. H., Weinberger, A. J., & Kuchner, M. J. 2009, ApJ, 702, 318
* Fisher et al. (2000) Fisher, R. S., Telesco, C. M., Piña, R. K., Knacke, R. F., & Wyatt, M. C. 2000, ApJ, 532, L141
* Fitzgerald et al. (2007) Fitzgerald, M. P., Kalas, P. G., Duchêne, G., Pinte, C., & Graham, J. R. 2007, ApJ, 670, 536
* Gáspár et al. (2008) Gáspár, A., Su, K. Y. L., Rieke, G. H., Balog, Z., Kamp, I., Martínez-Galarza, J. R., & Stapelfeldt, K. 2008, ApJ, 672, 974
* Gerbaldi et al. (1999) Gerbaldi, M., Faraggiana, R., Burnage, R., Delmas, F., Gómez, A. E., & Grenier, S. 1999, A&AS, 137, 273
* Jura et al. (1998) Jura, M., Malkan, M., White, R., Telesco, C., Pina, R., & Fisher, R. S. 1998, ApJ, 505, 897
* Kalas et al. (2008) Kalas, P., Graham, J. R., Chiang, E., Fitzgerald, M. P., Clampin, M., Kite, E. S., Stapelfeldt, K., Marois, C., & Krist, J. 2008, Science, 322, 1345
* Kenyon & Bromley (2004) Kenyon, S. J. & Bromley, B. C. 2004, ApJ, 602, L133
* Kervella et al. (2009) Kervella, P., Thévenin, F., & Petr-Gotzens, M. G. 2009, A&A, 493, 107
* King et al. (2003) King, J. R., Villarreal, A. R., Soderblom, D. R., Gulliver, A. F., & Adelman, S. J. 2003, AJ, 125, 1980
* Kurucz (1979) Kurucz, R. L. 1979, ApJS, 40, 1
* Lagage & Pantin (1994) Lagage, P. O. & Pantin, E. 1994, Nature, 369, 628
* Liou & Zook (1999) Liou, J.-C. & Zook, H. A. 1999, AJ, 118, 580
* Lisse et al. (2007) Lisse, C. M., Beichman, C. A., Bryden, G., & Wyatt, M. C. 2007, ApJ, 658, 584
* Lovis et al. (2006) Lovis, C., Mayor, M., Pepe, F., Alibert, Y., Benz, W., Bouchy, F., Correia, A. C. M., Laskar, J., Mordasini, C., Queloz, D., Santos, N. C., Udry, S., Bertaux, J.-L., & Sivan, J.-P. 2006, Nature, 441, 305
* Maness et al. (2008) Maness, H. L., Fitzgerald, M. P., Paladini, R., Kalas, P., Duchene, G., & Graham, J. R. 2008, ApJ, 686, L25
* Mannings & Barlow (1998) Mannings, V. & Barlow, M. J. 1998, ApJ, 497, 330
* Marois et al. (2008) Marois, C., Macintosh, B., Barman, T., Zuckerman, B., Song, I., Patience, J., Lafrenière, D., & Doyon, R. 2008, Science, 322, 1348
* Marsh et al. (2002) Marsh, K. A., Silverstone, M. D., Becklin, E. E., Koerner, D. W., Werner, M. W., Weinberger, A. J., & Ressler, M. E. 2002, ApJ, 573, 425
* Merín et al. (2004) Merín, B., Montesinos, B., Eiroa, C., Solano, E., Mora, A., D’Alessio, P., Calvet, N., Oudmaijer, R. D., de Winter, D., Davies, J. K., Harris, A. W., Cameron, A., Deeg, H. J., Ferlet, R., Garzón, F., Grady, C. A., Horne, K., Miranda, L. F., Palacios, J., Penny, A., Quirrenbach, A., Rauer, H., Schneider, J., & Wesselius, P. R. 2004, A&A, 419, 301
* Moerchen et al. (2007a) Moerchen, M. M., Telesco, C. M., De Buizer, J. M., Packham, C., & Radomski, J. T. 2007a, ApJ, 666, L109
* Moerchen et al. (2007b) Moerchen, M. M., Telesco, C. M., Packham, C., & Kehoe, T. J. J. 2007b, ApJ, 655, L109
* Moór et al. (2006) Moór, A., Ábrahám, P., Derekas, A., Kiss, C., Kiss, L. L., Apai, D., Grady, C., & Henning, T. 2006, ApJ, 644, 525
* Moshir (1989) Moshir, M. 1989, IRAS Faint Source Survey, Explanatory supplement version 1 and tape (Pasadena: Infrared Processing and Analysis Center, California Institute of Technology, 1989, edited by Moshir, M.)
* Oudmaijer et al. (1992) Oudmaijer, R. D., van der Veen, W. E. C. J., Waters, L. B. F. M., Trams, N. R., Waelkens, C., & Engelsman, E. 1992, A&AS, 96, 625
* Przygodda et al. (2003) Przygodda, F., van Boekel, R., Àbrahàm, P., Melnikov, S. Y., Waters, L. B. F. M., & Leinert, C. 2003, A&A, 412, L43
* Radomski et al. (2008) Radomski, J. T., Packham, C., Levenson, N. A., Perlman, E., Leeuw, L. L., Matthews, H., Mason, R., De Buizer, J. M., Telesco, C. M., & Orduna, M. 2008, ApJ, 681, 141
* Reidemeister et al. (2009) Reidemeister, M., Krivov, A. V., Schmidt, T. O. B., Fiedler, S., Müller, S., Löhne, T., & Neuhäuser, R. 2009, A&A, 503, 247
* Rieke et al. (2005) Rieke, G. H., Su, K. Y. L., Stansberry, J. A., Trilling, D., Bryden, G., Muzerolle, J., White, B., Gorlova, N., Young, E. T., Beichman, C. A., Stapelfeldt, K. R., & Hines, D. C. 2005, ApJ, 620, 1010
* Schneider et al. (1999) Schneider, G., Smith, B. A., Becklin, E. E., Koerner, D. W., Meier, R., Hines, D. C., Lowrance, P. J., Terrile, R. J., Thompson, R. I., & Rieke, M. 1999, ApJ, 513, L127
* Smith et al. (2009a) Smith, R., Churcher, L. J., Wyatt, M. C., Moerchen, M. M., & Telesco, C. M. 2009a, A&A, 493, 299
* Smith et al. (2008) Smith, R., Wyatt, M. C., & Dent, W. R. F. 2008, A&A, 485, 897
* Smith et al. (2009b) Smith, R., Wyatt, M. C., & Haniff, C. A. 2009b, A&A, 503, 265
* Song et al. (2001) Song, I., Caillault, J.-P., Barrado y Navascués, D., & Stauffer, J. R. 2001, ApJ, 546, 352
* Su et al. (2006) Su, K. Y. L., Rieke, G. H., Stansberry, J. A., Bryden, G., Stapelfeldt, K. R., Trilling, D. E., Muzerolle, J., Beichman, C. A., Moro-Martin, A., Hines, D. C., & Werner, M. W. 2006, ApJ, 653, 675
* Su et al. (2008) Su, K. Y. L., Rieke, G. H., Stapelfeldt, K. R., Smith, P. S., Bryden, G., Chen, C. H., & Trilling, D. E. 2008, ApJ, 679, L125
* Telesco et al. (2000) Telesco, C. M., Fisher, R. S., Piña, R. K., Knacke, R. F., Dermott, S. F., Wyatt, M. C., Grogan, K., Holmes, E. K., Ghez, A. M., Prato, L., Hartmann, L. W., & Jayawardhana, R. 2000, ApJ, 530, 329
* Telesco et al. (2005) Telesco, C. M., Fisher, R. S., Wyatt, M. C., Dermott, S. F., Kehoe, T. J. J., Novotny, S., Mariñas, N., Radomski, J. T., Packham, C., De Buizer, J., & Hayward, T. L. 2005, Nature, 433, 133
* Voshchinnikov & Semenov (2000) Voshchinnikov, N. V. & Semenov, D. A. 2000, Astronomy Letters, 26, 679
* Weinberger et al. (1999) Weinberger, A. J., Becklin, E. E., Schneider, G., Smith, B. A., Lowrance, P. J., Silverstone, M. D., Zuckerman, B., & Terrile, R. J. 1999, ApJ, 525, L53
* Weinberger et al. (2000) Weinberger, A. J., Rich, R. M., Becklin, E. E., Zuckerman, B., & Matthews, K. 2000, ApJ, 544, 937
* Wyatt (2008) Wyatt, M. C. 2008, Annual Review of Astronomy and Astrophysics, 46
* Wyatt et al. (1999) Wyatt, M. C., Dermott, S. F., Telesco, C. M., Fisher, R. S., Grogan, K., Holmes, E. K., & Piña, R. K. 1999, ApJ, 527, 918
* Wyatt et al. (2007) Wyatt, M. C., Smith, R., Su, K. Y. L., Rieke, G. H., Greaves, J. S., Beichman, C. A., & Bryden, G. 2007, ApJ, 663, 365
* Zuckerman & Song (2004) Zuckerman, B. & Song, I. 2004, ApJ, 603, 738
* Zuckerman et al. (2001) Zuckerman, B., Song, I., Bessell, M. S., & Webb, R. A. 2001, ApJ, 562, L87
|
arxiv-papers
| 2010-11-05T14:45:37 |
2024-09-04T02:49:14.533135
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Margaret Moerchen (European Southern Observatory, Univ. of Florida),\n Charles Telesco, Christopher Packham (Univ. of Florida)",
"submitter": "Margaret Moerchen",
"url": "https://arxiv.org/abs/1011.1410"
}
|
1011.1437
|
11institutetext: European Southern Observatory, Alonso de Córdova 3107,
Casilla 19001, Vitacura, Santiago 19, Chile
11email: mmoerche@eso.org 22institutetext: Department of Astronomy, University
of Florida, Gainesville, FL 32611, USA
33institutetext: Institute of Astronomy, University of Cambridge, Madingley
Road, Cambridge CB3 0HA, UK
44institutetext: National Science Foundation, Division of Astronomical
Sciences, 4201 Wilson Blvd., Suite 1045, Arlington, VA 22230, USA
55institutetext: Gemini Observatory, Northern Operations Center, 670 N. A
ohoku Place, Hilo, HI 96720, USA
# Asymmetric Heating of the HR 4796A Dust Ring
Due to Pericenter Glow
Margaret M. Moerchen 1122 Laura J. Churcher 33 Charles M. Telesco 22 Mark
Wyatt 33 R. Scott Fisher & Christopher Packham 445522
###### Abstract
Context. We have obtained new resolved images of the well-studied HR 4796A
dust ring at 18 and 25 $\mu$m with the 8-meter Gemini telescopes. These images
confirm the previously observed spatial extent seen in mid-IR, near-IR, and
optical images of the source. We detect brightness and temperature asymmetries
such that dust on the NE side is both brighter and warmer than dust in the SW.
We show that models of so-called pericenter glow account for these
asymmetries, thus both confirming and extending our previous analyses. In this
scenario, the center of the dust ring is offset from the star due to
gravitational perturbations of a body with an eccentric orbit that has induced
a forced eccentricity on the dust particle orbits. Models with 2-$\mu$m
silicate dust particles and a forced eccentricity of 0.06 simultaneously fit
the observations at both wavelengths. We also show that parameters used to
characterize the thermal-emission properties of the disk can also account for
the disk asymmetry observed in shorter-wavelength scattered-light images.
Aims.
Methods.
Results.
###### Key Words.:
circumstellar matter, planetary systems, Stars: individual: HR 4796A,
Infrared: planetary systems
## 1 Introduction
HR 4796A is an A0V star with the highest fractional IR luminosity
($L_{IR}$/$L_{\star}$= 5 x $10^{-3}$) yet discovered among debris disks. Its
recently revised distance estimate is 73 pc (van Leeuwen, 2007) (updated from
67 pc [Perryman et al. 1997]), and it has an M-star T Tauri companion with a
separation of 7.7” (Jura et al., 1993). Its age of 8$\pm$2 Myr (Stauffer et
al., 1995) (based on the lithium abundance of HR 4796B) places the disk of HR
4796A in the very interesting early phase of chaotic disk evolution dominated
by collisions (e.g., Kenyon & Bromley 2005).
Based on the dust temperature, Jura et al. (1993) first tentatively suggested
40 AU (now 44 AU) as the main location of dust responsible for the HR 4796A
excess thermal emission. Indeed, the mid-IR discovery images of the source at
CTIO and Keck revealed a highly inclined ringlike disk with a dust
distribution peaking near 70 AU (now 76 AU) (Jayawardhana et al., 1998;
Koerner et al., 1998). Follow-up coronagraphic near-IR images with
$HST$/NICMOS tightly constrained the width of the ring to $\sim$17 AU (now
$\sim$18.5 AU), with the dust population severely depleted both interior and
exterior to this region (Schneider et al., 1999).
Further imaging observations of the thermal emission confirmed a strong peak
in the dust density at 70 AU (Telesco et al., 2000; Wahhaj et al., 2005).
Telesco et al. (2000) also noted a 1.8-$\sigma$ brightness asymmetry at 18
$\mu$m in which the northeast side of the disk is brighter than the southwest
side. In a companion paper, Wyatt et al. (1999) demonstrated that such an
asymmetry could arise from the phenomenon of pericenter glow. A second body in
the system on an eccentric orbit about the star may cause pericenter glow
when, through secular perturbations, it effectively shifts the center of the
dust ring away from the star and closer to the apastron of the perturbing
companion’s orbit. The side of the ring shifted closer to the star becomes
relatively warmer and more luminous (Dermott et al., 1998), giving rise to the
“glow”. At present, while several works have acknowledged the existence of a
brightness asymmetry (e.g., Wahhaj et al. 2005, Debes et al. 2008), no
alternative explanations for the brightness asymmetry have been proposed.
We have obtained mid-IR images of HR 4796A with the Michelle and T-ReCS
cameras at the Gemini North and Gemini South telescopes, respectively. These
permit us to confirm unambiguously the brightness asymmetry, which was
previously characterized at only the 1.8-$\sigma$ level of significance. By
imaging thermally emitting dust at two wavelengths, we also examine the
spatial distribution of dust temperatures within the resolved disk, and we
conclude that the dust population in the NE ansa of the disk is not only
brighter, but also warmer, than the dust in the SW ansa. With these new data
and with the consideration of shorter-wavelength images of scattered light, we
re-examine the possibility that these observed asymmetric characteristics are
caused by pericenter glow.
## 2 Mid-IR images of HR 4796A
We observed HR 4796A with T-ReCS at Gemini South (using the Qb filter) in May
2004 and with Michelle at Gemini North (using the Qa filter) in April 2005.
The central wavelengths for these two filters are 18.1 $\mu$m for Qa and 24.5
$\mu$m for Qb. A PSF and flux standard star were observed close in time to the
target object for both sets of images. We reduced the images with an IDL
reduction package developed at the University of Florida, and we performed
photometric measurements with the Starlink GAIA program. The images in both
bandpasses show that the disk of HR 4796A is clearly resolved, in the form of
an elongated disk $\sim$3” in extent with two prominent peaks in brightness
(Figure 1).
Figure 1: Top: 18.1-$\mu$m image contours drawn at 2-$\sigma$ intervals
starting at 5$\sigma$ (background $\sigma$ = 0.295 mJy/pixel). Bottom:
24.5-$\mu$m image contours drawn at 1-$\sigma$ intervals starting at
3-$\sigma$ (background $\sigma$ = 1.463 mJy/pixel). Note that the 18.1-$\mu$m
image was taken with Michelle (plate scale: 0.1005”/pixel) and the 24.5-$\mu$m
image was taken with T-ReCS (plate scale: 0.089”/pixel). The images have been
scaled to show the same area in square arcseconds, and both images are
photosphere-subtracted.
Table 1 lists results of the sky-subtracted aperture photometry of HR 4796A at
18.1 $\mu$m and 24.5 $\mu$m. The uncertainties reflect only measurement
errors. Photometric uncertainties are driven primarily by variable sky
transmission throughout the night, and lacking multiple standard star
measurements, we adopt typical photometric uncertainties of 15% for both
filters.
Table 1: Photometry of the HR 4796A Disk
$\lambda_{c}$ | Total Flux Density | Photosphere | Disk Flux Density
---|---|---|---
$[\mu$m] | [mJy] | [mJy] | [mJy]
18.1 | 1106 $\pm$ 7 | 48 | 1058 $\pm$ 7
24.5 | 3307 $\pm$ 47 | 33 | 3274 $\pm$ 47
Notes– Uncertainties listed in this table are measurement uncertainties.
Photometric uncertainties are assumed to be 15% for both bandpasses.
The photospheric contribution in the bandpasses of the imaging observations
are based on a 10.8-$\mu$m value derived by Jura et al. (1998) by
extrapolating the $K$-band (2.2 $\mu$m) flux density using a $\nu^{1.88}$
power law as given by the Kurucz (1979) model atmosphere with T = 9500 K and
log $g$ = 4.0. Photospheric values at longer wavelengths were estimated with
the assumption that HR 4796A and Vega have the same 10-to-18 $\mu$m flux
ratio, since they are both A0V stars. The photospheric contribution to the
total flux density of the source is 4% at 18.3 $\mu$m and 1% at 24.5 $\mu$m.
The PSF was scaled such that the integrated flux matched the photosphere
level, and its center was shifted to the position assumed for the star in the
target source images. The maximum intensities of the scaled PSF and of the
disk (prior to the photosphere subtraction) at the stellar position are,
respectively, 1.4 mJy/pixel and 5.0 mJy/pixel at 18.1 $\mu$m and 0.3 mJy/pixel
and 9.8 mJy/pixel at 24.5 $\mu$m. For the 18.3-$\mu$m image, the adopted
stellar position was the centroid peak of the 2-pixel-smoothed central lobe of
emission (which we assume arises from the presence of the star) between the NE
and SW ansae, where the peaks were also determined from a centroid measurement
of the 2-pixel-smoothed image. The stellar centroid peak position in the
18.1-$\mu$m image differs from the actual midpoint between the NE and SW peaks
of emission by $<$1 pixel, which is insignificant as this is approximately the
level of certainty in the determination of the peak location. One pixel in the
18.1-$\mu$m image corresponds to 0.1”, or $\sim$7 AU. For the 24.5-$\mu$m
image, the stellar position is not obvious, and so the midpoint between the
two centroid peaks of the NE and SW ansae was adopted as the stellar position.
The distance from the NE peak to the SW peak in the 2-pixel-smoothed image is
17.3 pixels (1.74”) at 18.1 $\mu$m and 18.2 pixels (1.62”) at 24.5 $\mu$m.
The scaled PSF image in each bandpass was then subtracted from the
corresponding image of HR 4796A. The total photosphere-subtracted flux density
of the source was measured at each wavelength, and the results are given in
Table 1.
## 3 Disk asymmetry
The global dust color temperature is 102 $\pm$ 11 K, based on the
18.1-$\mu$m/24.5$\mu$m flux density ratio determined from aperture photometry
of the entire photosphere-subtracted disk. This temperature is consistent with
the initial color temperature estimated by Jura et al. (1993) based on $IRAS$
photometry (T$\sim$110 K). The color temperature of the dust as a function of
disk radius was also calculated using the approach described below.
Figure 2: Color temperature profile of the HR 4796A disk based on photometric
measurements at 18.1 $\mu$m and 24.5 $\mu$m (images “cross-convolved” to
achieve the same spatial resolution). The NE profile is represented by
circles, and the SW profile is represented by squares. Combined measurement
uncertainties from both bandpasses are smaller than the size of the data
points. The temperatures at the location of the peak dust density in the ring
(labeled) are the key values to note. Figure 3: Brightness profiles of the HR
4796A observations and model at 18.1 $\mu$m (lower curves) and 24.5 $\mu$m
(upper curves). The brightness profile for each observation or disk model was
generated by summing the central 3” along the long axis of the disk. Model
curves are shown for four different angles of pericenter, and we have
concluded that a pericenter angle of zero provides the best fit.
The position angle (PA), east of north, is defined as the angle between the
vertical image axis and the line connecting the two central peaks of emission
of the two ansae: 28.1∘ in the 18.1-$\mu$m image and 29.9∘ in the 24.5-$\mu$m
image. We estimate an uncertainty for the PA of 0.9 degrees, half the
difference between the values measured in the stacked image at each
wavelength. We adopt a PA value of 29∘, the average of the two values. For
convenient display, the images of HR 4796A were rotated counter-clockwise by
an angle of 61∘, to orient the disk plane parallel to the x-axis.
Quantifying the asymmetric structure– The asymmetric structure is quantified
in both the observations and in the model distributions (§4) by finding the
brightness peak for each ansal “lobe” and its offset from the star. We
calculated the following values for each image:
* •
$D_{mean}$, the mean offset of the lobes’ peak brightness from the center, in
pixels
* •
$\frac{\Delta D}{D_{mean}}$, the difference between the distance of the lobe
brightness peaks ($D_{NE}-D_{SW}$), divided by the mean offset of the
brightness peaks from the center, given as a percentage value
* •
$F_{mean}$, the mean peak brightness of the lobes
* •
$\frac{\Delta F}{F_{mean}}$, the difference in lobe peak brightness,
($F_{NE}-F_{SW}$), divided by the mean peak brightness of the lobe
($F_{mean}$), given as a percentage value
The level of the brightness asymmetry, $\Delta F/F_{mean}$, is of primary
interest, and it is 15.3$\leavevmode\nobreak\ \pm$ 2.6% at 18.1 $\mu$m and
13.0$\leavevmode\nobreak\ \pm$ 3.8% at 24.5 $\mu$m. The uncertainty in $\Delta
F/F_{mean}$ was calculated by taking a circular symmetrical fit to the
observations (i.e., repeating the modeling but imposing zero forced
eccentricity) and then using Monte Carlo methods to add noise to the model at
the observed level (0.007 mJy/pixel at 18.3 $\mu$m, 0.135 mJy/pixel at 24.5
$\mu$m) to work out the asymmetry that would have been detected purely due to
noise.
Assessing the temperature asymmetry– After removal of the photospheric
contribution, the angular resolution of each image was degraded to achieve the
same resolution in both images for accurate spatial comparison in the color
temperature calculation. Using the $gauss$ routine in IRAF, each of the target
images was convolved with a Gaussian profile having the same FWHM as the PSF
from the other bandpass. The PSF FWHM was estimated by a Gaussian fit to the
azimuthally averaged profile. The FWHM values were 5.19 pixels (0.52”) in the
18.1-$\mu$m image and 8.08 pixels (0.72”) in the 24.5-$\mu$m image. The PSF
images themselves were convolved in the same manner, which confirmed that the
resulting resolution (assessed with FWHM measurements) was identical for both
images. The final resolution of the cross-convolved images is $\sim$0.9”,
which corresponds to $\sim$65 AU for the source distance of 73 pc.
The rotated images were cropped to a swath of vertical width
$\Delta$y$\sim$3”. The pixel values in these images were summed along the
y-axis, resulting in a one-dimensional brightness profile for the disk at each
wavelength. From these brightness profiles, a color profile and a color
temperature profile (Figure 2) were constructed to illustrate variations in
temperature along the extent of the disk. As a test of the robustness of the
temperature estimates, the 24.5-$\mu$m brightness profile was shifted by
$\pm$1 pixel ($\sim$6 AU) along the x-axis relative to the 18.1-$\mu$m
brightness profile before the two brightness profiles were combined to
determine the color profile. The two resulting color profiles that
incorporated these offsets yielded color temperature profiles that did not
differ from the results presented in Figure 2 within 60 AU of the star by more
than 2 K.
The NE and SW profiles are overplotted on the same radius scale (along the
x-axis) to better show differences in temperature at the same distances from
the central star. We see no significant variation in color temperature within
a radius of 15 AU from the star. While that result is expected for such a
narrow annulus, this result is in fact attributable principally to the
degradation of resolution that we carried out to make the temperature
estimates. This effect is also seen in the separation of the two ansae peaks,
which decreased for both images following the cross-convolution, from 1.74” to
1.1” at 18.1 $\mu$m, and from 1.62” to 1.58” at 24.5 $\mu$m (where the lesser
effect for the 24.5 $\mu$m image is likely due to being convolved by a smaller
kernel).
There is a clear temperature asymmetry, with the dust in the NE side of the
disk having significantly higher color temperatures at greater than 30 AU from
the star and in particular at the location of the emission peaks corresponding
to the two ansae of the dust ring. The temperature difference peaks at $\sim$9
K near 60 AU. The uncertainties due to measurement error in the temperature
estimates are less than 2 K. The color temperature uncertainties that
incorporate photometric uncertainties are $\pm$8 K, but in this analysis we
are most interested in the relative temperatures of the two ansae and are
therefore most concerned with uncertainties within the images and not absolute
photometric uncertainties.
## 4 Pericenter glow modeling
### 4.1 Pericenter glow scenario and prior work
One motivation for observing debris disks is the potential to reveal as-yet
unseen planetary companions through the disk morphology. A massive orbiting
body can gravitationally perturb the orbits of smaller bodies like dust
particles over long-period (“secular”) timescales, and these perturbations may
be observed in the global characteristics of the disk. For example, a planet
on an eccentric orbit eventually imposes a forced eccentricity on the dust
particles, which results in an offset of the disk’s center of symmetry from
the host star. While this offset may be observable directly, other
manifestations of the offset may be more prominent. In particular, if the dust
population is azimuthally homogenous in size distribution and composition, the
side of the disk (the pericenter side) that is offset toward the star will
experience enhanced heating, increasing both its temperature and brightness.
The resulting asymmetry in brightness between the apocenter and pericenter
sides of the disk is referred to as pericenter glow (Dermott et al., 1998).
The asymmetric morphology of the HR 4796A disk observed in previous thermal
emission images at 18 $\mu$m was well approximated by models that invoked
pericenter glow (Wyatt et al., 1999; Telesco et al., 2000), but the
statistical significance of the asymmetry in those earlier images was
relatively low. We note that the brightness asymmetry was also previously
detected at 10.8 $\mu$m (Telesco et al., 2000), but the much greater
prominence of the starlight at those wavelengths makes such shorter wavelength
observations less useful for our analysis. With new 18.1-$\mu$m images and the
addition of images at 24.5 $\mu$m in this dataset, we have re-examined this
hypothesis by using the same models to reconstruct the observed thermal
emission and color temperature profiles.
### 4.2 Thermal emission reconstruction
Model inputs– The physics of pericenter glow and the associated modeling of it
are outlined in detail by Wyatt et al. (1999), and here we review only the key
parameters involved in that analysis. The model for the disk’s density
distribution of dust was generated with three distinct inputs: the physical
structure of the disk, the combination of the optical properties and size
distribution of the dust particle population, and the orientation of the disk
along the line of sight. We recall that the mass of the perturbing body itself
does not come into play and is not constrained by this approach. The physical
structure of the disk is modelled by a radial density distribution of dust
with defining parameters as follows: $a_{min}$ and $a_{max}$, the inner and
outer semimajor axes of the dust ring (which are roughly equivalent to the
inner and outer radii), $\gamma$, the power-law index of the semimajor axis
distribution, $e_{f}$, the forced eccentricity, and $\sigma_{tot}$, the total
surface area of the dust particles in the disk. The forced eccentricity is
that imposed on all particles in the disk by the planet’s secular
perturbations (Murray & Dermott, 1999); as such, the planet’s orbit is not
affected by its own perturbations. The eccentricities of particle orbits are
the vector sum of this forced eccentricity and their own proper eccentricity.
To first order, the disk remains circular but is just offset. The total
surface area of the dust is effectively a scaling factor that can be adjusted
to approximate the overall brightness level of the dust emission. While the
intrinsic disk offset and asymmetry (i.e., if viewed face-on) are determined
by the forced eccentricity $e_{f}$, the measured offset and asymmetry are
determined by the additional variables of disk inclination, $i$, and the angle
of pericenter, ${\omega}_{f}$. For example, the forced eccentricity required
to replicate a given measured asymmetry increases as the angle of pericenter
deviates from being perpendicular to the line of sight.
The optical properties of dust were calculated using Mie theory, Rayleigh-Gans
theory, and geometric optics (Li & Greenberg, 1997; Augereau et al., 1999;
Bohren & Huffman, 1983) for astronomical silicate spheres the diameter of
which ($D_{typ}$=2 $\mu$m) was constrained by a simultaneous fit to the images
at both wavelengths. The resulting dielectric constant was 6.7x$10^{-5}$, and
the absorption coefficients were 0.54 and 0.30 at 18.1 and 24.5 $\mu$m,
respectively. While a single particle size is used to represent a particle
population that has a range of sizes and composition, it reasonably
encapsulates key mid-IR emission characteristics of the disk. We adopted
values for the stellar luminosity and effective temperature of 21 L⊙ and 9500
K, respectively (e.g., Debes et al., 2008).
The output from the disk radial distribution model and the calculation of the
dust optical properties were combined with the specified orientation to
calculate the line-of-sight brightness and generate images of the model disk.
The model images at each wavelength were rotated to the same position angle as
the observed disk, and each image was convolved by a PSF kernel with the
profile width of the observed PSF reference star in the same filter. In the
generation of color temperature profiles for the model data, the images were
also convolved by the profile width corresponding to the other filter to match
the treatment of the observational data (§3).
Figure 4: Color temperature profiles of the HR 4796A observations and models.
Colors were calculated by summing the flux within a 3”-wide swath along the
long axis of the disk for each image. Model curves are shown for two different
angles of pericenter, and we have concluded that a pericenter angle of zero
provides the best fit.
Optimizing the model– With the images of the model disk, we generated
brightness profiles and color temperature profiles in the same way that we did
with the observational data (§3). We compared the observed and model profiles,
and we adjusted the model parameters until the model disk yielded a good
approximation to the observations. First, we varied the inner and outer disk
radii and the total disk surface area until the brightness profile width and
magnitude were well approximated. We then adjusted the forced eccentricity
until the brightness asymmetry level was matched for a fixed value of the
angle of pericenter. The fit was repeated for several values of the angle of
pericenter to assess the corresponding forced eccentricity necessary to
reproduce the same asymmetry (Table 2). The forced eccentricity and the angle
of pericenter define the magnitude of the radial offset asymmetry in the disk
model, which in turn defines the brightness asymmetry and the dust color
temperature asymmetry. The initial assessment of how changing these parameters
affected the resulting brightness profiles was performed by chi-squared
minimization on the brightness profiles and images. The brightness profiles of
the observations and the model disk with varying values for the angle of
pericenter are shown in Figure 3, and the color temperature profiles are shown
in Figure 4. The quantitative measurements used to assess the asymmetry in the
model and in the observed disk, as described above, are summarized in Table 2.
Table 2: Asymmetry parameters for observations and models
| | 18.1 | | 24.5 |
---|---|---|---|---|---
| $e_{f}$ | $\frac{\Delta D}{D_{mean}}$ [%] | $\frac{\Delta F}{F_{mean}}$ [%] | | $\frac{\Delta D}{D_{mean}}$ [%] | $\frac{\Delta F}{F_{mean}}$ [%] | reduced $\chi^{2}$
observed | – | 0 | 15.28 | | 0 | 13.00 | –
model $\omega_{f}=0^{\circ}$ | 0.06 | -11.71 | 15.26 | | -6.11 | 13.10 | 2.17
model $\omega_{f}=40^{\circ}$ | 0.10 | -10.81 | 15.31 | | -7.01 | 13.30 | 2.31
model $\omega_{f}=55^{\circ}$ | 0.13 | -11.18 | 15.35 | | -7.15 | 13.05 | 2.21
model $\omega_{f}=75^{\circ}$ | 0.30 | -14.50 | 10.55 | | -7.39 | 12.70 | 2.26
111 All models were simultaneously fitted to 18.1- and 24.5-$\mu$m brightness
profile linecuts of 0.5” and share the following parameter values:
$a_{min}$=70 AU, $a_{max}$=84 AU, $\gamma$=-1.5, $D_{typ}$=2 $\mu$m, and
$i$=14.1∘. Also recall that the value $\frac{\Delta D}{D_{mean}}$ for the
observations is 0 because we defined the stellar position at 24.5 $\mu$m as
the midpoint between the brightness peaks of the two ansae (with the measured
difference between the midpoint and the stellar peak at 18.1 $\mu$m to be $<$1
pixel.)
Results of the modelling– We have determined that a forced eccentricity of
0.06 imposed on a ring of dust with $D_{typ}$= 2 $\mu$m spanning 74–84 AU can
reproduce the thermal IR images, fitting the observed brightness asymmetry
well at both 18.1 and 24.5 $\mu$m. The disk density power-law exponent was set
to -1.5. The same result for the characteristic particle size was inferred
previously in order to simultaneously fit 10-$\mu$m and 18-$\mu$m images of HR
4796A (Wyatt et al., 1999; Telesco et al., 2000). We have estimated the
uncertainty in the forced eccentricity to be 0.01, which is the change
required to effect a 1-$\sigma$ change in the brightness asymmetry of the
model at 18.3 $\mu$m (the more limiting case). The angle of pericenter was
0$\pm$30∘ (perpendicular to the line of sight), and the disk inclination was
14.1∘ (Schneider et al., 2009). We note that these results differ from those
found by Wyatt et al. (1999) and Telesco et al. (2000) ($e_{f}$=0.02,
$\omega_{f}$=$75^{\circ}$) likely because of the different levels of observed
asymmetry, which are nonetheless within 3$\sigma$ of one another (5.1 $\pm$
3.2% in the previous work, 15.3 $\pm$ 2.6% in this work).
We have also used the model to quantify the limits set by these data on the
presence of hot dust inside the ring. We compared the photometry of the
photosphere-subtracted images with that of the model within an aperture radius
equal to the size of the PSF FWHM (0.52” at 18.3 $\mu$m, 0.72” at 24.5
$\mu$m), and then we repeated the measurement with the same aperture on the
models, where an unresolved flux component could be added at the position of
the star. Excluding calibration uncertainties, the measured flux density
within a 0.52”-radius aperture on the 18.3-$\mu$m photosphere-subtracted image
was 289 $\pm$ 3 mJy. The same measurement on the model with no added
unresolved flux component was 290 mJy. Therefore, the maximum unresolved
contribution we can add to the model that is 3-$\sigma$ consistent with the
observations is 8 mJy. Repeating the same procedure at 24.5 um with an
aperture of radius 0.72”, we measured a flux density of 1290 $\pm$ 12 mJy for
the photosphere-subtracted images and 1289 mJy for the model. Therefore, the
3-$\sigma$ upper limit of 1326 mJy for this aperture allows the presence of an
unresolved component at a level of 37 mJy. However, these measurements do not
account for the assumed 15% calibration uncertainty. Since a change in
calibration would also result in a corresponding change to the model (to fit
the peaks at the correct level), it is instructive to consider what would have
been derived with a revised calibration factor. In the extreme situation that
the calibration factor was 45% higher, the flux measured in the previously
described apertures at 18.1 and 24.5 $\mu$m respectively would have been 438
$\pm$ 4 mJy and 1881 $\pm$ 17 mJy, with corresponding model fluxes of 290*1.45
= 421 mJy and 1289*1.45 = 1869 mJy, and so a flux excess of 17 $\pm$ 4 and 12
$\pm$ 17 mJy. Therefore, a 3-$\sigma$ deviation in both calibration and
statistical uncertainty puts the limits on the unresolved flux at $<$29 mJy
and $<$63 mJy at 18.1 and 24.5 $\mu$m. Wahhaj et al. (2005) estimated an
exozodiacal contribution of 87 $\pm$ 22 mJy at 24.5 $\mu$m, which is
consistent to within 3$\sigma$ with the upper limit we derive.
### 4.3 Scattered light comparison
A brightness asymmetry has also been detected at wavelengths that are
dominated by light scattered by the dust disk. Images obtained by Schneider et
al. (1999) at 1.1 $\mu$m were suggestive of a brightness imbalance favoring
the NE that was later noted to be at the 10–15% level (Schneider et al.,
2001). The disk ansae were again compared in the analysis of data sets taken
with $HST$ NICMOS at 1.71–2.22 $\mu$m, but the NE ansa was not consistently
brighter in all bandpasses (Debes et al., 2008). Most recently, $HST$ STIS
images at 0.57 $\mu$m show that the SW ansa is 0.74 $\pm$ 0.07 times as bright
as the NE ansa, as measured in “lobes” within $\pm$20∘ of the major axis and
between 0.8” and 1.5” from the star. Schneider et al. (2009) measured an
offset asymmetry, $(D_{NE}-D_{SW})/D_{mean}\leavevmode\nobreak\ =-3.7\%$, such
that the center of the disk is offset from the star by 1.4 $\pm$ 0.4 AU (19
$\pm$ 6 mas) in the plane perpendicular to the line of sight.
Thus, another test for our pericenter glow model is whether it can replicate
the observations in scattered light. To examine this, we modified the model’s
particle properties to correctly calculate the scattered light fluxes, using a
combination of Mie theory, the Rayleigh-Gans approximation, and geometric
optics, which yielded a mean albedo of 0.05. We assumed that the same particle
population (in this case, modeled as a first approximation by 2-$\mu$m
silicate spheres) is responsible for both the thermal emission and the
scattered light reflection. We adopted a Henyey-Greenstein phase function to
model the non-isotropic scattering properties of the small grains (Augereau et
al., 1999), with the form
$f(\alpha)=\frac{1-g^{2}}{1-g^{2}-2gcos(\alpha)}$ (1)
where $\alpha$ is the scattering angle to the line of sight and $g$ is the
asymmetry parameter. The parameter $g$ indicates the relative level of front
or back scattering, such that $g=1$ for 100% forward scattering, $g=-1$ for
100% backward scattering, and $g=0$ for isotropic scattering. We adopted the
value $g=0.16$ as determined from the optical $STIS$ images (Debes et al.,
2008; Schneider et al., 2009), which is consistent within 2$\sigma$ of the
values for $g$ determined by Debes et al. (2008) for the near-IR $NICMOS$
images.
To check that our model produced realistic flux levels, we compared them to
those measured by Schneider et al. (2009) at a wavelength of 0.57 $\mu$m. Due
to the presence of coronagraphic spikes in the image, the flux density
measurement had to be corrected for the missing area, yielding a total flux
9.4$\pm$0.8 mJy. The total flux in our model scattered light image is 10.8
mJy. As a secondary check, we used the formula given by Weinberger et al.
(1999) to predict the surface brightness of a disk in scattered light (in mJy
arcsec-2)
$S=\frac{F\tau\omega}{4\pi\phi^{2}}$ (2)
where $F$ is the flux received from the star in mJy, $\tau$ is the optical
depth of the scattering material, $\omega$ is the albedo (from Mie theory,
where $Q_{sca}$=0.08), and the factor $\tau\omega$ can be approximated by
$L_{disk}/L_{\star}$ if $\omega=1$. The expected flux density calculated with
this relation agrees with the flux density values both from observations and
from our model to within 20% at 0.57, 1.1, and 1.6 $\mu$m.
We also replicated the lobe asymmetry measurement by Schneider et al. (2009)
with our disk model. Within the same apertures (within $\pm$20∘ of the major
axis and between 0.8” and 1.5” from the star), we determine a SW/NE brightness
ratio of 0.92 at 0.57 $\mu$m and 0.93 at 1.1 $\mu$m. Our model values are
therefore consistent within 3$\sigma$ with the Schneider et al. (2009) value
of 0.74$\leavevmode\nobreak\ \pm\leavevmode\nobreak\ $0.07, and are also
consistent with a simple 1/$r^{2}$ attenuation of starlight due to the disk
ansae being at different distances from the star. Our model also replicates
the radial offset asymmetry at a level of 3.3%, which is consistent with the
Schneider et al. (2009) value of 3.7 $\pm$ 1.1%.
Schneider et al. (2009) acknowledge that dynamical perturbations such as those
we propose may be responsible for part of the brightness asymmetry, also
suggesting that a segregation of the particle population or an azimuthal
density variation in a homogenous dust ring could produce the asymmetry.
However, whether such a model could also reproduce the observed temperature
asymmetry has yet to be tested. Additionally, we note that particles in the
disk should be moving slightly faster at pericenter, relative to their
velocity at apocenter, and the collisional timescale at pericenter should be
relatively shorter in turn. Therefore, the population of small particles that
scatter near-IR light may be enhanced due to the increased collision rate near
pericenter, thus yielding a density variation in the dust ring and a greater
brightness imbalance in scattered light. There is the potential to further
develop the pericenter glow model with such an enhanced particle population,
which may help to determine whether factors beyond dynamical perturbations
must be invoked to explain the observed morphology. Such refinements to our
model would bear greater consideration especially if the large near-IR
brightness asymmetry is confirmed.
## 5 Discussion and conclusions
Using new mid-IR images at 18 and 25 $\mu$m, we have confirmed the previously
observed brightness asymmetry discovered by Telesco et al. (2000) in the HR
4796A disk to a high level of statistical significance. We have determined
that a model disk with forced eccentricity of 0.06 that is dynamically induced
by a companion can replicate that brightness asymmetry. After incorporating
the dust scattering properties, we can also reproduce, to within 3$\sigma$,
the level of brightness asymmetry seen in near-IR scattered-light images . We
believe that near-IR imaging in additional bandpasses will further constrain
the level of the asymmetry observed in scattered light, and that such
constraints will assist in confirming whether the pericenter glow model is
sufficient to explain the disk asymmetry observed throughout the spectrum.
The model disk is comprised of particles 2 $\mu$m in diameter ($D_{typ}$), in
agreement with previous analysis of the disk (Wyatt et al., 1999; Telesco et
al., 2000), and the particles are assumed to be spherical silicates. Dust
particles smaller than $\sim$8 $\mu$m in diameter should be blown out roughly
within an orbital timescale ($\sim$400 yr). In reality the dust population is
not this simple, but we have adopted this value to characterize the overall
population. We must consider that the particles may have a more complex set of
optical properties that allows them to be heated to the temperatures we infer
and to remain in the system without being rapidly blown out. More complex dust
properties (e.g., Li & Lunine 2003) could be incorporated into the pericenter
glow model in future iterations.
Finally, we recall that a key result for a system like HR 4796A that exhibits
pericenter glow is the implied presence of a perturbing companion. In fact,
the principal perturbing body may be the nearby M-star companion HR 4796B.
However, the orbital parameters of HR 4796B are unknown and may not be
commensurate with its being the primary perturber. The sharp inner edge of the
ring points to the possibility that the perturbing object responsible for the
asymmetry is also truncating the inner edge, as has been suggested for
Fomalhaut (Kalas et al., 2005; Quillen, 2006). In this case, the putative
planet responsible would be expected to lie just interior to the ring. For
example, a Jupiter-mass planet would be expected to orbit at $\sim$60 AU with
an eccentricity of $\sim$0.06 (maximum projected separation 0.87”), but it
could be closer to the dust ring if it were more massive. Given the ambiguity
in potential perturbers and the fact that the level of forced eccentricity
induced by the companion does not constrain its mass, the continued search for
the putative companion is warranted, if not with the current generation of
telescopes then with the next, such as the space-based $JWST$ or the ground-
based TMT or E-ELT facilities. Kalas et al. (2008) have demonstrated a
successful direct planet detection that was pursued following similar evidence
of a forced stellar offset, as in the case of Fomalhaut, which is nearer and
brighter than HR 4796A.
###### Acknowledgements.
MMM gratefully acknowledges fellowship support from the Michelson Science
Center. This work was performed [in part] under contract with JPL funded by
NASA through the Michelson Fellowship Program. JPL is managed for NASA by the
California Institute of Technology. LJC is grateful for the support of an STFC
studentship. Observations were obtained at the Gemini Observatory, operated by
AURA, Inc., under agreement with the NSF on behalf of the Gemini partnership:
NSF (US), PPARC (UK), NRC (Canada), CONICYT (Chile), ARC (Australia), CNPq
(Brazil), and CONICET (Argentina).
## References
* Augereau et al. (1999) Augereau, J. C., Lagrange, A. M., Mouillet, D., Papaloizou, J. C. B., & Grorod, P. A. 1999, A&A, 348, 557
* Bohren & Huffman (1983) Bohren, C. F. & Huffman, D. R. 1983, Absorption and scattering of light by small particles, ed. Bohren, C. F. & Huffman, D. R.
* Debes et al. (2008) Debes, J. H., Weinberger, A. J., & Schneider, G. 2008, ApJ, 673, L191
* Dermott et al. (1998) Dermott, S. F., Grogan, K., Holmes, E. K., & Wyatt, M. C. 1998, in Exozodiacal Dust Workshop:, ed. D. E. Backman, L. J. Caroff, S. A. Sandford, & D. H. Wooden, 59–+
* Jayawardhana et al. (1998) Jayawardhana, R., Fisher, S., Hartmann, L., Telesco, C., Pina, R., & Fazio, G. 1998, ApJ, 503, L79+
* Jura et al. (1998) Jura, M., Malkan, M., White, R., Telesco, C., Pina, R., & Fisher, R. S. 1998, ApJ, 505, 897
* Jura et al. (1993) Jura, M., Zuckerman, B., Becklin, E. E., & Smith, R. C. 1993, ApJ, 418, L37+
* Kalas et al. (2008) Kalas, P., Graham, J. R., Chiang, E., Fitzgerald, M. P., Clampin, M., Kite, E. S., Stapelfeldt, K., Marois, C., & Krist, J. 2008, Science, 322, 1345
* Kalas et al. (2005) Kalas, P., Graham, J. R., & Clampin, M. 2005, Nature, 435, 1067
* Kenyon & Bromley (2005) Kenyon, S. J. & Bromley, B. C. 2005, AJ, 130, 269
* Koerner et al. (1998) Koerner, D. W., Ressler, M. E., Werner, M. W., & Backman, D. E. 1998, ApJ, 503, L83+
* Kurucz (1979) Kurucz, R. L. 1979, ApJS, 40, 1
* Li & Greenberg (1997) Li, A. & Greenberg, J. M. 1997, A&A, 323, 566
* Li & Lunine (2003) Li, A. & Lunine, J. I. 2003, ApJ, 590, 368
* Murray & Dermott (1999) Murray, C. D. & Dermott, S. F. 1999, Solar system dynamics
* Perryman et al. (1997) Perryman, M. A. C., Lindegren, L., Kovalevsky, J., Hoeg, E., Bastian, U., Bernacca, P. L., Crézé, M., Donati, F., Grenon, M., van Leeuwen, F., van der Marel, H., Mignard, F., Murray, C. A., Le Poole, R. S., Schrijver, H., Turon, C., Arenou, F., Froeschlé, M., & Petersen, C. S. 1997, A&A, 323, L49
* Quillen (2006) Quillen, A. C. 2006, MNRAS, 372, L14
* Schneider et al. (2001) Schneider, G., Nicmos/Eons, T., & Go/8624 Teams. 2001, in Astronomical Society of the Pacific Conference Series, Vol. 244, Young Stars Near Earth: Progress and Prospects, ed. R. Jayawardhana & T. Greene, 203–+
* Schneider et al. (1999) Schneider, G., Smith, B. A., Becklin, E. E., Koerner, D. W., Meier, R., Hines, D. C., Lowrance, P. J., Terrile, R. J., Thompson, R. I., & Rieke, M. 1999, ApJ, 513, L127
* Schneider et al. (2009) Schneider, G., Weinberger, A. J., Becklin, E. E., Debes, J. H., & Smith, B. A. 2009, AJ, 137, 53
* Stauffer et al. (1995) Stauffer, J. R., Hartmann, L. W., & Barrado y Navascues, D. 1995, ApJ, 454, 910
* Telesco et al. (2000) Telesco, C. M., Fisher, R. S., Piña, R. K., Knacke, R. F., Dermott, S. F., Wyatt, M. C., Grogan, K., Holmes, E. K., Ghez, A. M., Prato, L., Hartmann, L. W., & Jayawardhana, R. 2000, ApJ, 530, 329
* van Leeuwen (2007) van Leeuwen, F., ed. 2007, Astrophysics and Space Science Library, Vol. 350, Hipparcos, the New Reduction of the Raw Data
* Wahhaj et al. (2005) Wahhaj, Z., Koerner, D. W., Backman, D. E., Werner, M. W., Serabyn, E., Ressler, M. E., & Lis, D. C. 2005, ApJ, 618, 385
* Weinberger et al. (1999) Weinberger, A. J., Becklin, E. E., Schneider, G., Smith, B. A., Lowrance, P. J., Silverstone, M. D., Zuckerman, B., & Terrile, R. J. 1999, ApJ, 525, L53
* Wyatt et al. (1999) Wyatt, M. C., Dermott, S. F., Telesco, C. M., Fisher, R. S., Grogan, K., Holmes, E. K., & Piña, R. K. 1999, ApJ, 527, 918
|
arxiv-papers
| 2010-11-05T15:49:02 |
2024-09-04T02:49:14.547418
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Margaret Moerchen (European Southern Observatory, Univ. Florida),\n Laura Churcher (Univ. Cambridge), Charles Telesco (Univ. Florida), Mark Wyatt\n (Univ. Cambridge), R. Scott Fisher (Gemini Observatory, NSF) and Christopher\n Packham (Univ. Florida)",
"submitter": "Margaret Moerchen",
"url": "https://arxiv.org/abs/1011.1437"
}
|
1011.1458
|
# Ionizing wave via high-power HF acceleration
Evgeny Mishin and Todd Pedersen
Space Vehicles Directorate Air Force Research Laboratory Hanscom AFB MA
USA
###### Abstract
Recent ionospheric modification experiments with the 3.6 MW transmitter at the
High Frequency Active Auroral Research Program (HAARP) facility in Alaska led
to discovery of artificial ionization descending from the nominal interaction
altitude in the background F-region ionosphere by $\sim$60 km. This paper
presents a physical model of an ionizing wavefront created by suprathermal
electrons accelerated by the HF-excited plasma turbulence.
Submitted to GRL, 1 November 2010 MISHIN AND PEDERSEN HF-INDUCED IONIZATION
Space Vehicles Directorate, Air Force Research Laboratory, Hanscom AFB, MA
01731 (e-mail: Evgeny.Mishin@hanscom.af.mil; Todd.Pedersen@hanscom.af.mi)
## 1 Introduction
High-power HF radio waves can excite electrostatic waves in the ionosphere
near altitudes where the injected wave frequency $f_{0}$ matches either the
local plasma frequency $f_{p}\approx 9\sqrt{n_{e}}$ kHz (the density $n_{e}$
in cm-3) or the upper hybrid resonance $f_{uhr}=\sqrt{f_{p}^{2}+f_{c}^{2}}$
($f_{c}$ is the electron cyclotron frequency) [e.g., Gurevich, 1978]. The
generated waves increase the bulk electron temperature to $T_{e}=$0.3-0.4 eV,
while some electrons are accelerated to suprathermal energies
$\varepsilon=\frac{1}{2}mv^{2}$ up to a few dozen eV [e.g., Carlson et al.,
1982; Rietveld et al., 2003]. Upon impact with neutrals ($N_{2}$, $O_{2}$,
$O$), suprathermal electrons excite optical emissions termed Artificial Aurora
(AA) [e.g., Bernhardt et al., 1989; Gustavsson and Eliasson, 2008].
Heating-induced plasma density modifications are usually described in terms of
chemical and transport processes [e.g., Bernhardt et al., 1989; Djuth et al.,
1994; Dhillon and Robinson, 2005; Ashrafi et al., 2006]. However, the Pedersen
et al. [2009; 2010] discovery of rapidly descending plasma layers seems to
point to additional mechanisms. Pedersen et al. [2010, hereafter P10]
suggested that the artificial plasma is able to sustain interaction with the
transmitted HF beam and that the interaction region propagates (downward) as
an ionizing wavefront. In this paper, the formation of such downward-
propagating ionizing front is ascribed to suprathermal electrons accelerated
by the HF-excited plasma turbulence.
## 2 Ionizing wave
The descending feature is evident in Figure 1, which is representative of P10
Fig. 3 with the regions of ion-line (IL) radar echoes from the MUIR incoherent
scatter radar (courtesy of Chris Fallen) overlaid [c.f. Oyama et al., 2006].
Shown are sequential altitude profiles of the green-line emissions
($\lambda=$557.7 nm, excitation potential $\varepsilon_{g}\approx$4.2 eV)
observed at the HAARP facility on 17 March 2009\. Here, the O-mode radio beam
was injected into the magnetic zenith (MZ), i.e. along the magnetic field
$\mathbf{B}_{0}$, at the effective radiative power
$P_{0}[\mathrm{MW}]\approx$440 and frequency $f_{0}=$2.85 MHz (2$f_{c}$ at
$h_{2f_{c}}\approx$230 km). The contours of $f_{p}=f_{0}$ or
$n_{e}=n_{c}\approx 10^{5}$ cm-3 (cyan) and $f_{uhr}=f_{0}$ (violet) are
inferred from ionograms acquired at 1 min intervals [P10 Fig. 3]. The regions
of enhanced IL are shown in green color. Note, the blue-line emissions at
427.8 nm (not shown) coincided with the green-line emissions, as seen looking
from the HAARP site [P10].
Figure 1: Time-vs-altitude plot of 557.7 nm optical emissions (black color)
along B0. Blue (violet) lines indicate the matching altitudes $f_{0}=$ $f_{p}$
($f_{uh}$). The dashed line indicates $h_{2f_{c}}$. The transmitter on periods
are indicated. Shown in green is the MUIR IL intensity (courtesy of Chris
Fallen). Horizontal blips are stars passing through the view.
During the first 2 min in the heating, the artificial plasma is confined to
the bottomside of the F layer at altitudes $h>$180 km. The corresponding
descent of the IL scatter is similar to that described by Dhillon and Robinson
[2005] and Ashrafi et al. [2006]. A sudden brightening of AA and increased
speed of descent of the artificial plasma ‘layer’ (patch) in the HF-beam
center occurs near 180 km, while its peak plasma frequency $foFa$ reaches
$f_{0}$. In fact, the optical data shows [P10] that this patch is fairly
uniform near $\sim$180 km but then becomes a $\sim$20-km collection of
($\parallel$B0) filaments a few km in diameter. While the degree of
inhomogeneity of the descending patch increases, its speed, $V_{obs}\simeq$0.3
km/s, appears to be constant until $\approx$160 km. Then, the artificial
plasma slows down, staying near the terminal altitude $h_{\min}\approx$150 km
before the emissions retreat in altitude near the end of 4-min injection
pulse. During a continuous ‘on’ period, the artificial plasma near $h_{\min}$
was quenched several times, initiating the process over again from higher
altitudes.
Hereafter, we focus on the descending feature at $h\leq$180 km where $foFa\geq
f_{0}$ or $n_{e}\geq n_{c}$. Enhanced 427.8-nm emissions indicate the presence
of electrons with energies $\varepsilon>\varepsilon_{b}\approx$18.7 eV,
exceeding the ionization energies of $N_{2}$, $O_{2}$, and $O$. The ionization
rate $q_{a}$ is given by
$dn_{e}/dt=q_{a}=n_{a}\cdot\left\langle\nu_{ion}(\varepsilon)\right\rangle$
(1)
where $\nu_{ion}$ is the ionization frequency, and
$\left\langle...\right\rangle$ means averaging over the accelerated
distribution of the density $n_{a}$. Hereafter, we employ Majeed and
Strickland’s [1997] electron impact cross-sections and the Hedin [1991] MSIS90
model for the densities $[N_{2}]$, $[O]$, and $[O_{2}]$ on 17 March 2009.
At each time step $t_{i}$, artificial ionization occurs near the critical
altitude $h_{c}(t_{i})$, defined from the condition $n_{e}(h_{c})=n_{c}$. The
density profile just below $h_{c}$ is represented as follows
$n_{e}(x,t_{i})=n_{c}\cdot\Psi\left(x\right)$ (2)
Here $x=\xi/L_{\parallel}$, $\xi=\left(h_{c}-h\right)/\cos\alpha_{0}$ is the
distance along $\mathbf{B}_{0}$, $\alpha_{0}$ is the conjugate of the magnetic
dip angle ($\approx$15∘ at HAARP), and $L_{\parallel}$ is the ($\parallel$B0)
extent of the ionization region. $\Psi(x)$ is a monotonic function satisfying
the conditions $\Psi(0)\geq$1 and $\Psi(x)\ll$1 at $x>$1, since the ambient
plasma density $n_{0}\ll n_{c}$ at $h\leq$180 km. As the ratio
$\delta_{e}(\varepsilon)$ of inelastic ($\nu_{il}$) to elastic ($\nu_{el})$
collision frequencies is small, the accelerated electrons undergo fast
isotropization due to elastic scattering and thus
$L_{\parallel}\simeq\left\langle l_{ion}\sqrt{\delta_{e}/2}\right\rangle$,
where $l_{ion}=v/\nu_{ion}$ [c.f. Gurevich et al., 1985].
Evidently, as soon as at some point $x_{i}\leq 1$ the density
$n_{e}(x_{i},t_{i}+\Delta t)\simeq q_{a}(\xi_{i})\cdot\Delta t$ reaches
$n_{c}$, the critical height shifts to this point, i.e. $h_{c}(t_{i+1})\simeq
h_{c}(t_{i})-L_{\parallel}\cdot x$. These conditions define the ionization
time, $T_{ion}^{-1}\simeq q_{a}/n_{c}$, and the speed of descent
$V_{d}=\left|dh_{c}/dt\right|\simeq
L_{\parallel}T_{ion}^{-1}\simeq\left\langle v\sqrt{\delta_{e}/2}\right\rangle
n_{a}/n_{c}$ (3)
Note, eq. (3) contains no dependence on the total neutral density $N_{n}$ and
hence predicts $V_{d}\simeq\mathrm{const}(h)$, if the same distribution of
accelerated electrons is created at each step. As
$\left\langle\delta_{e}^{1/2}v\right\rangle\simeq$1.5$\cdot 10^{6}$ m/s, we
get from eq. (3) that the value of $V_{d}$ (3) matches $V_{obs}$ at
$n_{a}=n_{a}^{(d)}\simeq$6$\cdot 10^{-4}n_{c}$ or $n_{a}^{(d)}\simeq$60 cm-3.
## 3 Discussion and conclusions
We now turn to justify this acceleration-ionization-descent scenario. Enhanced
IL echoes, like in Figure 1, usually result from the parametric decay
instability (PDIl) and oscillating two stream instability (OTSI) of the pump
wave near the plasma resonance [e.g., Mjølhus et al., 2003]. The latter
develops if the relative pump wave energy density
$\widetilde{W}_{0}=\left|E_{0}^{2}\right|/8\pi n_{c}T_{e}$ exceeds
$\widetilde{W}_{th}\simeq\frac{2}{kL_{n}}+\frac{4\nu_{T}}{\omega_{0}}$, where
$k$ is the plasma wave number, $\nu_{T}$ is the collision frequency of thermal
electrons, and $L_{n}^{-1}=\left|\nabla\ln n_{e}\right|$. The free space field
of the pump wave is $E_{fs}\approx$5.5$\sqrt{P_{0}}/r\approx$0.65 V/m at
$r=$180 km (at the HF-beam center) or $\widetilde{W}_{fs}\simeq$5$\cdot
10^{-4}$ at $T_{e}=$0.2 eV. For incidence angles
$\theta<\arcsin\left(\sqrt{f_{c}/(f_{c}+f_{0})}\sin\alpha_{0}\right)$, the
amplitude in the first Airy maximum is
$E_{A}\approx(2\pi/\sin\alpha_{0})^{2/3}(f_{0}L_{n}/c)^{1/6}E_{fs}$ [e.g.,
Mjølhus et al., 2003] or $\widetilde{W}_{A}\approx 0.1(L_{n}/L_{0})^{1/3}$,
where $L_{0}=$30 km. For injections at MZ, following Mjølhus et al. [2003] one
gets $\widetilde{W}_{A}^{(mz)}\approx\widetilde{W}_{A}/4$.
As $\widetilde{W}_{A}^{(mz)}\gg\mu\ $(the electron-to-ion mass ratio), we get
$k\simeq r_{D}^{-1}\left(\mu\widetilde{W}_{A}^{(mz)}\right)^{1/4}$ [e.g.,
Alterkop et al., 1973] or $kr_{D}\simeq 1/40$ ($r_{D}$ is the Debye radius)
yielding $\widetilde{W}_{th}\simeq 10^{-4}(L_{0}/L_{n})$. The ‘instant’
gradient-scale of the artificial layer is $L_{n}\simeq$3$\rightarrow$1 km at
180$\rightarrow$150 km (see below) gives
$\widetilde{W}_{th}\approx$(1$\rightarrow$3)$\cdot 10^{-3}$. Thus, OTSI can
easily develop in the first Airy maximum. In turn, PDIl can develop in as many
as $\simeq$30 Airy maxima over a distance $l_{a}\simeq$1 km [c.f. Djuth, 1984;
Newman et al., 1998]. At $T_{e}/T_{i}<4$, PDIl is saturated via induced
scattering of Langmuir ($l$) waves, piling them up into ‘wave condensate’
($k\rightarrow 0$) [e.g., Zakharov et al., 1976]. The condensate is subject to
OTSI, thereby leading to strong (cavitating) turbulence and electron
acceleration [e.g., Galeev et al., 1977].
At $W_{l}/n_{0}T_{e}<\left(f_{c}/f_{p}\right)^{2}$, the acceleration results
in a power-law ($\parallel$B0) distribution
at$\mathrm{\,}\varepsilon_{\max}\geq\varepsilon_{\parallel}=\frac{1}{2}mu^{2}\geq\varepsilon_{\min}$
[Galeev et al., 1983; Wang et al., 1997]
$F_{a}^{\parallel}(\varepsilon_{\parallel})\simeq
n_{a}(2p_{a}-1)/v_{\min}\cdot\left(\varepsilon_{\min}/\varepsilon_{\parallel}\right)^{p_{a}}$
(4)
where $p_{a}\simeq$0.75-1. The density $n_{a}$ and $\varepsilon_{\min}$ are
determined by the wave energy $W_{l}$ trapped by cavitons and the joining
condition with the ambient electron distribution
$F_{a}^{\parallel}(\varepsilon_{\min})=F_{0}(\varepsilon_{\min})$. If $F_{0}$
is a Maxwellian distribution, this gives $\varepsilon_{\min}^{m}\approx
10T_{e}$ and $n_{a}^{m}\approx 10^{-4}n_{e}$. When background suprathermal
($s$) electrons of the density $n_{s}$ are present, then $F_{0}(\varepsilon\gg
T_{e})\rightarrow F_{s}(\varepsilon)$ and $\varepsilon_{\min}\simeq
30\left(n_{s}T_{e}/W_{l}\right)^{2/5}\ $eV [e.g., Mishin et al., 2004],
yielding $\varepsilon_{\min}\leq 10$ eV at $n_{e}=n_{c}$,
$\widetilde{W}_{l}\simeq 10^{-3}$, and $n_{s}\leq$10 cm-3. In the ionizing
wave, a natural source of the $s$-electrons is ionization by those accelerated
electrons that can propagate from $\xi\sim 0$ to $\xi\sim L_{\parallel}$ (see
Figure 2).
We can now evaluate the excitation and ionization rates. The column 427.8-nm
intensity in Rayleighs (R) is given by
$I\approx 10^{-6}A_{b}\int
d\xi\int\sigma_{b}(\varepsilon)\Phi_{a}(\varepsilon\mathbf{,}\xi)d\varepsilon\cdot[N_{2}(\xi)]$
(5)
Here $\sigma_{b}\ $is the excitation cross section of the $N_{2}^{+}(^{1}N)$
state, $A_{b}\approx$0.19, $\Phi_{a}=\frac{2\varepsilon}{m^{2}}F_{a}$ is the
differential number flux, and $F_{a}(\varepsilon)\simeq
n_{a}\frac{p_{a}-0.5}{2\pi}v_{\min}^{-3}\left(\varepsilon_{\min}/\varepsilon\right)^{p_{a}+1}$
is an isotropic distribution to which the accelerated distribution
$F_{a}^{\parallel}$ (4) is transformed at distances
$\left|\xi\right|>v/\nu_{el}$ due to elastic scattering [c.f. Gurevich et al.,
1985].
Integrating eq. (5) over the energy range
$\varepsilon_{b}\leq\varepsilon\leq$102 eV at $p_{a}=$0.85 yields the
brightness of a $\Delta h$-km column $\Delta I(h_{c})\approx 2.5\cdot
10^{-12}n_{a}[N_{2}(h_{c})]\cdot\Delta h$ R near altitude $h_{c}$, given that
$\Delta h\ll H_{n}\simeq$8 km (the atmosphere scale-height). The total
intensity $I$ is defined by the vertical extent of the (excitation) layer
$\Delta_{b}$, where $\varepsilon(\xi)\geq\varepsilon_{b}$. It can be evaluated
using the Majeed and Strickland [1997] loss function
$L(\varepsilon)=\sum_{j}L_{j}(\varepsilon)$ with $j$ designating $N_{2}$,
$O_{2}$, and $O$. Outside the acceleration layer, i.e.
$\left|\xi\right|>l_{a}$, the energy of an electron of the initial energy
$\varepsilon_{0}$ at a distance $\xi$ from the origination point $h_{0}$ is
$\varepsilon(\varepsilon_{0},\xi)\simeq\varepsilon_{0}-\int_{h_{0}}^{h_{0}+\xi}L(\varepsilon(z))\sqrt{2/\delta_{e}(\varepsilon(z))}dz$
(6)
Figure 2: (a) Altitude profiles $\varepsilon(\varepsilon_{0},\xi)$ at
$\varepsilon_{0}=$10, 15, …100 eV and $h_{0}=$150, …200 km. (b) Half-widths
$\Delta_{g}\ $(thin lines) and $\Delta_{b}$ (thick) of the green- and blue-
line excitation layers near $h_{c}=$160 (circles) and 180 (solid lines) km.
Figure 2a presents the results of calculations of eq. (6) for
$\varepsilon_{0}=$10, 15, … 102 eV and $h_{0}=$150, 160, 180, and 200 km. The
altitude profiles at $\varepsilon\geq$5 eV and hence the layers of
excitation/ionization are nearly symmetric about $h_{0}$ at $h\leq$180 km.
Panel b shows the half-widths $\Delta_{g}$ and $\Delta_{b}$ of the green- and
blue-line excitation layers about $h_{c}=$180 and 160 km as function of
$\varepsilon_{0}$. The half-width of the ionization layer $\Delta_{ion}$ (not
shown) is $\approx\Delta_{b}$ at $\varepsilon_{0}>$20 eV. Since
$\Delta_{b}<H_{n}$, we can estimate the 427.8-nm intensity at
$h_{c}=$180$\rightarrow$160 km as $\left.I\right|_{h_{c}}\simeq$5$\cdot
10^{-12}n_{a}\cdot[N_{2}(h_{c})]\left\langle\Delta_{b}(h_{c})\right\rangle\simeq$(0.16$\rightarrow$0.2)$\cdot
n_{a}$ R. Comparing $\left.I\right|_{h_{c}}$ with the spatially-averaged
intensities $\widehat{I_{b}}\approx$10$\rightarrow$5 R [P10 Fig. 1] yields
$\widehat{n}_{a}\simeq$60$\rightarrow$25 cm-3. Note that
$\widehat{n}_{a}\approx n_{a}^{(d)}$ at 180 km, in agreement with a uniform
structure, while spatial averaging underestimates $n_{a}$ inside the $\sim$km-
scale filaments at 160 km.
Calculating the ionization frequency in eq. (1) with $F_{a}(\varepsilon)$
gives
$\left\langle\nu_{ion}\right\rangle\approx\kappa_{ion}^{\ast}\cdot\left([N_{2}]+\frac{1}{2}[O]+0.95[O_{2}]\right)$
s-1, where $\kappa_{ion}^{\ast}=\left\langle
v\sigma_{ion}\right\rangle/n_{a}\approx$1.8$\cdot 10^{-8}$ cm3s-1 is the
coefficient of ionization of $N_{2}$. The total ionization rate
$q_{a}^{(d)}\sim 10^{4}$ cm-3s-1 greatly exceeds recombination losses $\approx
10^{-7}n_{c}^{2}\approx 10^{3}$ cm-3s-1 (the main ion component at these
altitudes is $NO^{+}$). This justifies the use of eq. (1) for evaluating the
artificial plasma density. Taking an average energy loss per ionization
$\sim$20 eV results in the column dissipation rate $<$0.1 mW/m2 or $<$10% of
the 440-MW Poynting flux, consistent with P10’s estimates.
Figure 3: The ionization coefficients of $N_{2}$, $O_{2}$, and $O$ and the
excitation coefficient of the $N_{2}^{+}(^{1}N)$ state vs.
$\varepsilon_{\max}$.
As shows Figure 3, the coefficients of ionization and blue-line excitation by
accelerated electrons decrease by a factor of $\sim$2 (10) between
$\varepsilon_{\max}=$102 and 50 (30) eV. The Liouville theorem predicts
$F(\varepsilon_{0}-\Delta\varepsilon(\varepsilon_{0},\xi),h_{0}+\xi)=F_{0}(\varepsilon_{0},h_{0})$,
where $\Delta\varepsilon(\varepsilon_{0},\xi)$ is given by the integral in eq.
(6). Thus, the gradient scale-length $L_{n}$ of the artificial plasma is about
the distance $\xi_{50}$, defined by the condition
$\Delta\varepsilon(10^{2},\xi_{50})\approx$50 eV. Numerically, we get
$\xi_{50}\approx\Delta_{b}(50)$ or $L_{n}\approx$3$\rightarrow$1.5 km near
$h_{c}=$180$\rightarrow$160 km and $q_{a}L_{n}/n_{c}\simeq V_{obs}$, as
predicted by eq. (3). Note that the artificial plasma density profiles derived
from ionograms indeed have $\sim$1-km gradient scale-lengths near 150 km [c.f.
P10 Fig. 2].
Figure 1 shows that the descent slows down below 160 km and ultimately stops
at $h_{\min}\approx$150 km. The presence of IL and bright green-line emissions
indicate that plasma turbulence is still excited and efficiently accelerates
electrons above 4 eV. However, the blue-line emissions almost vanish [P10],
thereby indicating only few accelerated electrons at
$\varepsilon\geq\varepsilon_{b}$. That this is in no way contradictory follows
from the fact that inelastic losses increase tenfold between 10 and 20 eV.
Acceleration stops at $\varepsilon=\varepsilon_{\max}\ll$100 eV when
$\nu_{il}(\varepsilon_{\max})$ exceeds the acceleration rate
$mD_{\parallel}(u_{\max})/8\pi\varepsilon_{\max}$, where
$D_{\parallel}(u)\approx\frac{\omega_{p}^{2}}{4n_{e}mu}\left|E_{k_{\parallel}}\right|^{2}$
and $k_{\parallel}=\omega_{p}/u$ [Volokitin and Mishin, 1979]. The critical
neutral density is roughly estimated as $\sim$5$\cdot 10^{11}$ cm-3, i.e
$N_{n}$ at $\sim$150 km. The fact that the artificial plasma stays near
$h_{\min}$ indicates that ionization is balanced by recombination or
$q_{a}^{\min}\sim 10^{-7}n_{c}^{2}\approx 0.1q_{a}^{(d)}$, which at $n_{a}\sim
n_{a}^{(d)}$ corresponds to $\varepsilon_{\max}\approx$30 eV (Figure 3).
A mechanism for generating km-sized filaments below 180 km could be the
thermal self-focusing instability (SFI) near $h_{c}$, resulting in a broad
spectrum of plasma irregularity scale sizes [e.g., Guzdar et al., 1998].
Significantly, $\sim$km-scale plasma irregularities grow initially but within
10s of seconds thermal self-focusing leads to smaller (10s to 100s meters)
scale sizes. During descent, the critical altitude moves downward by several
km within 10 s, thereby precluding further development of SFI, while the
$\sim$km-scale irregularities have sufficient time to develop. When the
descent rate drops, small-scale irregularities can fully develop and scatter
the HF beam, thereby impeding the development of OTSI/PDIL and hence
ionization. As soon as the artificial plasma decays, SFI falls away and hence
irregularities gradually disappear. Then, the artificial plasma can be created
again. This explains why the artificial layer ceases and then reappears
(Figure 1).
In conclusion, we have shown that the artificial plasma sustaining interaction
with the transmitted HF beam can be created via enhanced ionization by
suprathermal electrons accelerated by Langmuir turbulence near the critical
altitude. As soon as the interaction region is ionized, it shifts toward the
upward-propagating HF beam, thereby creating an ionizing wavefront, which
resembles Pedersen et al.’s [2010] descending artificial ionospheric layers.
###### Acknowledgements.
This research was supported by Air Force Office of Scientific Research. We
thank Chris Fallen for providing the MUIR IL data.
## References
* [1] Alterkop, B., A. Volokitin, V. Shapiro, and V. Shevchenko (1973), Contribution to the nonlinear theory of the ”modified” decay instability, JETP Letters, 18, 24.
* [2] Ashrafi, M., M. Kosch and F. Honary (2006), Heater-induced altitude descent of the EISCAT UHF ion-line enhancements: Observations and modeling, Adv. Space Res., 38, 2645.
* [3] Bernhardt, P., C. Tepley, and L. Duncan (1989), Airglow enhancements associated with plasma cavities formed during ionospheric heating experiments, J. Geophys. Res., 94, 9071.
* [4] Carlson, H., V. Wickwar, and G. Mantas (1982), Observations of fluxes of suprathermal electrons accelerated by HF excited Langmuir instabilities, J. Atm. Terr. Phys., 12, 1089.
* [5] Djuth, F., (1984), HF-enhanced plasma lines in the lower ionosphere, Radio Sci., 19, 383.
* [6] Djuth., F., P. Stubbe, M. Sulzer, H. Kohl, M. Rietveld, and J. Elder (1994), Altitude characteristics of plasma turbulence excited with Tromsø superheater, J. Geophys. Res., 99, 333.
* [7] Dhillon, R. S., and T. R. Robinson (2005), Observations of time dependence and aspect sensitivity of regions of enhanced UHF backscatter associated with RF heating, Ann. Geophys., 23, 75.
* [8] Galeev, A., R. Sagdeev, V. Shapiro, and V. Shevchenko (1977), Langmuir turbulence and dissipation of high-frequency energy, Sov. Phys. JETP, 46, 711.
* [9] Galeev, A., R. Sagdeev, V. Shapiro, and V. Shevehenko (1983), Beam plasma discharge and suprathermal electron tails, in Active Experiments in Space (Alpbach, Austria), SP-195, pp. 151, ESA, Paris.
* [10] Gurevich, A., Y. Dimant, G. Milikh, and V. Vaskov (1985), Multiple acceleration of electrons in the regions high-power radio-wave reflection in the ionosphere, J. Atmos. Terr. Phys., 47, 1057.
* [11] Gustavsson, B., and B. Eliasson (2008), HF radio wave acceleration of ionospheric electrons: Analysis of HF-induced optical enhancements, J. Geophys. Res., 113, A08319, doi:10.1029/2007JA012913.
* [12] Guzdar, P. N., P. K. Chaturvedi, K. Papadopoulos, and S. L. Ossakow (1998), The thermal self-focussing instability near the critical surface in the high-latitude ionosphere, J. Geophys. Res., 103, 2231.
* [13] Hedin, A. (1991), Extension of the MSIS thermospheric model into the middle and lower atmosphere, J. Geophys. Res., 96, 1159\.
* [14] Majeed, T., and D. J. Strickland (1997), New survey of electron impact cross sections for photoelectron and auroral electron energy loss calculations, J. Phys. Chem. Ref. Data, 26, 335.
* [15] Mishin, E., W. Burke, and T. Pedersen (2004), On the onset of HF-induced airglow at magnetic zenith, J. Geophys. Res., 109, A02305, doi: 10.1029/2003JA010205.
* [16] Mjølhus, E. Helmersen, and D. DuBois (2003), Geometric aspects of HF driven Langmuir turbulence in the ionosphere, Nonl. Proc. Geophys., 10, 151.
* [17] Newman, D., M. Goldman, F. Djuth, and P. Bernhardt (1998), Langmuir turbulence associated with ionospheric modification: Challenges associated with recent observations during a sporadic-E event, in: Phys. of Space Plasmas., ed. by T. Chang and J. Jaasperse, v. 15, p. 259, MIT, Cambridge, MA.
* [18] Oyama, S., B. J. Watkins, F. T. Djuth, M. J. Kosch, P. A. Bernhardt, and C. J. Heinselman (2006), Persistent enhancement of the HF pump-induced plasma line measured with a UHF diagnostic radar at HAARP, J. Geophys. Res., 111, A06309, doi:10.1029/2005JA011363.
* [19] Pedersen, T., B. Gustavsson, E. Mishin, E. MacKenzie, H. C. Carlson, M. Starks, and T. Mills (2009), Optical ring formation and ionization production in high-power HF heating experiments at HAARP, Geophys. Res. Lett., 36, L18107, doi:10.1029/2009GL040047.
* [20] Pedersen, T., B. Gustavsson, E. Mishin, E. Kendall, T. Mills, H. C. Carlson, and A. L. Snyder (2010), Creation of artificial ionospheric layers using high-power HF waves, Geophys. Res. Lett., 37, L02106, doi:10.1029/ 2009GL041895.
* [21] Rietveld, M., M. Kosch, N. Blagoveshchenskaya, V. Kornienko, T. Leyser, and T. Yeoman (2003), Ionospheric electron heating, optical emissions and striations induced by powerful HF radio waves at high latitudes: Aspect angle dependence, J. Geophys. Res., 108, 1141, doi: 10.1029/2002JA009543.
* [22] Wang, J., D. Newman, and M. Goldman (1997), Vlasov simulations of electron heating by Langmuir turbulence near the critical altitude in the radiation-modified ionosphere, J. A. S. -T. P., 59, 2461\.
* [23] Volokitin, A., and E. Mishin (1979), Relaxation of an electron beam in a plasma with infrequent collisions, Sov. J. Plasma Phys _._ 5, 654.
* [24] Zakharov, V., S. Musher, and A. Rubenchik (1976), Weak Langmuir turbulence of an isothermal plasma, Sov. Phys. JETP, 42, 80.
|
arxiv-papers
| 2010-11-05T16:58:48 |
2024-09-04T02:49:14.556005
|
{
"license": "Public Domain",
"authors": "Evgeny Mishin and Todd Pedersen",
"submitter": "Evgeny Mishin",
"url": "https://arxiv.org/abs/1011.1458"
}
|
1011.1509
|
# Nucleon, Delta and Omega excited state spectra at three pion mass values
John Bulava
NIC, DESY, Platanenallee 6, D-15738, Zeuthen, Germany
Email john.bulava@desy.de Robert G. Edwards, Bálint Joó, David G. Richards
Thomas Jefferson National Accelerator Facility, Newport News, VA 23606, USA
Email edwards@jlab.org bjoo@jlab.org dgr@jlab.org Eric Engelson
In-Depth Engineering Corp., 11350 Random Hills Road, Fairfax, VA 22030, USA
Email engelson@gmail.com Huey-Wen Lin
Department of Physics, University of Washington, Seattle, WA 98195, USA
Email hwlin@phys.washington.edu Colin Morningstar
Department of Physics, Carnegie Mellon University,Pittsburgh, PA 15213, USA
Email colin_morningstar@cmu.edu
Department of Physics, University of Maryland, College Park, MD 20742, USA
E-mail
###### Abstract:
The energies of the excited states of the Nucleon, $\Delta$ and $\Omega$ are
computed in lattice QCD, using two light quarks and one strange quark on
anisotropic lattices. The calculations are performed at three values of the
pion mass: $m_{\pi}$ = 392(4), 438(3) and 521(3) MeV. We employ the
variational method with a basis of about ten interpolating operators enabling
six energies to be distinguished clearly in each irreducible representation of
the octahedral group. We compare our calculations of nucleon excited states
with the low-lying experimental spectrum. There is reasonable agreement for
the pattern of states.
## 1 Introduction
The goal of determining the spectrum of hadron masses from lattice QCD is
addressed in this work by calculations of the excited state spectrum of the
nucleon, $\Delta$ and $\Omega$ using anisotropic lattices.[1] Earlier excited-
baryon analyses were based on quenched QCD [2] and two-light-flavor
($N_{f}=2$) QCD [3]. In this work we use ensembles of gauge configurations
developed in Ref. [4] for $N_{f}=2+1$ QCD with two dynamical light quarks and
one strange quark. The lattices are $16^{3}\times 128$ with spatial and
temporal lattice spacings $a_{s}$= 0.122 fm and $a_{t}$ = 0.035 fm.
For families of particles with given isospin and strangeness, spectra are
calculated in the six double-valued irreducible representations (irreps) of
the octahedral group. There are three irreps for even-parity that are labeled
with a $g$ subscript (gerade) and three for odd-parity that are labeled with a
$u$ subscript (ungerade). They are: $G_{1g},H_{g},G_{2g},G_{1u},H_{u}$ and
$G_{2u}$.
Sets of seven to eleven three-quark operators are used in each irrep and the
variational method [5, 6] is used to extract energies of six states. Most
operators incorporate gauge-covariant displacements of the quarks relative to
one another in order to obtain nontrivial shapes. [7] The recently developed
“distillation” method [8] is used for quark smearing. We start with a large
set of operators in each irrep and then “prune” them to sets of about 10 that
have the lowest condition numbers. That yields sets of approximately linearly-
independent operators that are suitable for calculations based on
diagonalizing a matrix of correlation functions.
## 2 Results
A detailed presentation of all of our results is given in Ref. [1]. Here we
present selected results for the nucleon and $\Delta$ excited states.
Plots of the nucleon effective energies, calculated as
$E_{\rm
eff}(t)=\frac{1}{2}ln\left(\frac{\widetilde{\lambda}(t-1)}{\widetilde{\lambda}(t+1)}\right),$
(1)
where $\widetilde{\lambda}(t)$ is an eigenvalue of the generalized eigenvalue
problem, are shown in Figure 1 for the $G_{1g}$ and $G_{1u}$ irreps. These
plots show the values of $E_{\rm eff}$ obtained from Eq. (1) as vertical bars
and $E_{\rm eff}$ calculated using the fit function,
$\lambda_{fit}(t)=(1-A)e^{-E(t-t_{0})}+Ae^{-E^{\prime}(t-t_{0})},$ (2)
in place of $\widetilde{\lambda}(t)$ in Eq. (1) as curved dashed lines.
Comparison of the curved dashed lines with the bars from the lattice ensembles
shows the usefulness of two-exponential fits. The term
$Ae^{-E^{\prime}(t-t_{0})}$ models the contributions of higher energy states
at early times allowing the exponential term $(1-A)e^{-E(t-t_{0})}$ to be
determined over a larger fit window $(t_{i},t_{f})$ than would be possible
using a single exponential. Fit energy $E$ and uncertainty of the fit energy,
$\sigma$, are shown by dashed horizontal lines at $E+\sigma$ and $E-\sigma$
extending over the fit window. Note that the statistics allow credible
determinations of six energy levels in each irrep. This provides evidence that
quark smearing based on “distillation” is effective with regard to suppressing
high-frequency fluctuations in the gauge ensembles.
Figure 1: Nucleon $G_{1g}$ effective energies are shown for the lowest states
in the upper six graphs. The effective energy increases from left to right
along the first row and continues to increase from left to right along the
second row. The lower six graphs show nucleon $G_{1u}$ effective energies
increasing in the same pattern. Calculations are for $m_{\pi}=392(4)$ MeV.
Vertical bars show the effective energy and the curved dashed line shows the
effective energy calculated from the fit function. Horizontal dashed lines
show the fit results for $E\pm\sigma$ and their extent shows the fitting
interval $(t_{i},t_{f})$.
The energies obtained from the $G_{1g}$ and $G_{1u}$ effective mass plots of
Fig. 1 are shown as boxes extending from $E-\sigma$ to $E+\sigma$ in Fig. 2.
We show nucleon energies that are obtained in the same manner as shown in Fig.
1 for all irreps of the octahedral group and three pion masses. Experimental
spectra are shown to the left of lattice energies for the spins and parities
that have subductions to the lattice irreps. Lattice and experimental spectra
are shown in Fig. 3 for the $\Delta$ family. See Ref. [1] for the $\Omega$
spectra.
In the nucleon spectra, there is good evidence for a spin $\frac{5}{2}^{-}$
state. We find nearly degenerate $H_{u}$ and $G_{2u}$ partner states, which is
the signature of spin $\frac{5}{2}^{-}$. However, other spins are difficult to
identify because there are many nearly-degenerate states, within
uncertainties. It is a near-term goal within the collaboration to address spin
identification by using operators that are subduced from continuum spins.
Our lattice spectra show scant evidence for multiparticle states even though
many energies lie above the relevant thresholds. This is probably because
single-hadron operators are used. There is an inference for a multiparticle
contribution in that we find four low-lying states in $H_{u}$ while there are
three low-lying experimental states that have subductions to $H_{u}$. There is
a threshold for a multiparticle state in the same energy range. Our lattice
results agree with the experimental pattern if one of the four low-lying
$H_{u}$ states is multiparticle. However, we cannot identify the multiparticle
states in the spectrum. It is a near-term goal within the collaboration to
incorporate multiparticle operators that couple directly to such states.
Some lattice states appear to be “squeezed” by the small lattice volume used.
They show up at higher energies than would be the case in a larger volume. The
$G_{2}$ states require partner states in other irreps, such as $H$, in order
to realize all the magnetic substates for a given spin. The partners should be
close to the same energy. However, in the $\Delta$ spectra of Fig. 3 we find
$G_{2}$ states at high energies without suitable partners being evident.
Possibly they have been “squeezed”. It is also a goal to perform calculations
of spectra at larger volumes.
Although we do not attempt to extrapolate energies to $m_{\pi}$ = 140 MeV, it
is evident from Figs. 2 and 3 that the lowest-energy states on the lattice
tend toward the energies of the physical resonances as the pion mass
decreases. Decreasing the pion mass is an obvious goal but we recognize that
it entails a more complex analysis for excited states that can decay.
Figure 2: Spectra for isospin $\frac{1}{2}$ (nucleon family) at three values
of $m_{\pi}$ in each irrep of the cubic group are compared with experimental
spectra. Columns labeled by $m_{\pi}$ = 392, 438 and 521 MeV show lattice
spectra. The $G_{1g}$ and $G_{1u}$ spectra in the $m_{\pi}$ = 392 MeV column
are obtained from the plots of Fig. 1. Boxes extend from $E-\sigma$ to
$E+\sigma$. Two, three and four-star experimental resonances are shown to the
left of lattice spectra in columns labeled by their $J^{P}$ values. Each
$J^{P}$ value listed has a subduction to the lattice irrep shown. Each box for
an experimental resonance has height equal to the full decay width and an
inner box (color aqua) showing the uncertainty in the Breit-Wigner energy.
Open boxes show the thresholds for multiparticle states. Figure 3: Spectra
for isospin $\frac{3}{2}$ ($\Delta$ family) at three values of $m_{\pi}$ in
each irrep of the cubic group are compared with experimental spectra. Columns
labeled by $m_{\pi}$ = 392, 438 and 521 MeV show lattice spectra. Boxes extend
from $E-\sigma$ to $E+\sigma$. Two, three and four-star experimental
resonances are shown to the left of lattice spectra in columns labeled by
their $J^{P}$ values. Each $J^{P}$ value listed has a subduction to the
lattice irrep shown. Each box for an experimental resonance has height equal
to the full decay width and an inner box (color aqua) showing the uncertainty
in the Breit-Wigner energy.
## 3 Summary
This work represents a milestone in our long-term research program aimed at
determining the spectra of baryons in QCD. It provides the first spectrum for
N, $\Delta$ and $\Omega$ baryons based on $N_{f}=2+1$ QCD with high
statistics. A large number of baryon operators is used to calculate matrices
of correlation functions. They are analyzed using the variational method with
fixed eigenvectors. The analysis provides spectra at three pion masses:
$m_{\pi}$ = 392(4) MeV, 438(3) MeV and 521(3) MeV.
The lattice volume and pion masses used give considerably higher energies than
the experimental resonance energies. However, there is reasonable agreement of
the overall pattern of lattice and experimental states. One exception is that
almost all $\Delta$ states in the $G_{2}$ irrep are too high. That may be
caused by a volume that is too small for highly excited states.
###### Acknowledgments.
This work was done using the Chroma software suite [9] on clusters at
Jefferson Laboratory and the Fermi National Accelerator Laboratory using time
awarded under the USQCD Initiative. JB and CM acknowledge support from U.S.
National Science Foundation Award PHY-0653315. EE and SW acknowledge support
from U.S. Department of Energy contract DE-FG02-93ER-40762. HL acknowledges
support from U.S. Department of Energy contract DE-FG03-97ER4014. BJ, RE and
DR acknowledge support from U.S. Department of Energy contract DE-
AC05-060R23177, under which Jefferson Science Associates, LLC, manages and
operates Jefferson Laboratory. BJ and RE acknowledge support under U.S. Dept.
of Energy SciDAC contracts DE-FC02-06ER41440 and DE-FC02-06ER41449. The U.S.
Government retains a non-exclusive, paid-up, irrevocable, world-wide license
to publish or reproduce this manuscript for U.S. Government purposes.
## References
* [1] J. M. Bulava et al., _Nucleon, $\Delta$ and $\Omega$ excited state spectra in $N_{f}$ = 2+1 lattice QCD_, _Phys. Rev. D_ 82, 014507 (2010) [arXiv:1004.5072].
* [2] S. Basak et al., _Lattice QCD determination of patterns of excited baryon states_ , _Phys. Rev. D_ 76, 074504 (2007) [arXiv:0709.0008].
* [3] J. M. Bulava et al. , _Excited State Nucleon Spectrum with Two Flavors of Dynamical Fermions_ , _Phys. Rev. D_ 79 034505 (2009), [arXiv:0901.0027].
* [4] H-W. Lin et al., _First results from 2+1 dynamical quark flavors on an anisotropic lattice: light-hadron spectroscopy and setting the strange-quark mass_ , _Phys. Rev. D_ 79, 034502 (2009) [arXiv:0810.3588].
* [5] C. Michael, _Adjoint sources in lattice gauge theory_ , _Nucl. Phys. B_ 259, 58 (1985).
* [6] M. Lüscher and U. Wolff, _How to calculate the elastic scattering matrix in two-dimensional quantum field theories by numerical simulation_ , _Nucl. Phys. B_ 339, 222 (1990).
* [7] S. Basak et al., _Group-theoretical construction of extended baryon operators in lattice QCD_ , _Phys. Rev. D_ 72 094506 (2005) [arXiv:hep-lat/0506029].
* [8] M. Peardon et al., _Novel quark-field creation operator construction for hadronic physics in lattice QCD_ , _Phys. Rev. D_ 80, 054506 (2009) [arXiv:0905.2160].
* [9] R. G. Edwards and B. Joó , _The Chroma Software System for Lattice QCD_ , _Nucl. Phys. B. Proc. Suppl._ 140, 832 (2005) [hep-lat/0409003].
|
arxiv-papers
| 2010-11-05T20:51:12 |
2024-09-04T02:49:14.564227
|
{
"license": "Public Domain",
"authors": "John Bulava, Robert G. Edwards, B\\'alint Jo\\'o, David G. Richards,\n Eric Engelson, Huey-Wen Lin, Colin Morningstar and Stephen J. Wallace",
"submitter": "Stephen J. Wallace",
"url": "https://arxiv.org/abs/1011.1509"
}
|
1011.1596
|
# A universal étale lift of a proper local embedding
Anca M. Mustaţǎ and Andrei Mustaţǎ School of Mathematical Sciences,
University College Cork, Cork, Ireland a.mustata@ucc.ie,
andrei.mustata@ucc.ie
###### Abstract.
To any finite local embedding of Deligne–Mumford stacks $g:Y\to X$ we
associate an étale, universally closed morphism $F_{Y/X}\to X$ such that for
the complement $Y^{2}_{X}$ of the image of the diagonal $Y\to Y\times_{X}Y$,
the stack $F_{Y^{2}_{X}/Y}$ admits a canonical closed embedding in $F_{Y/X}$,
and $F_{Y/X}\times_{X}Y$ is a disjoint union of copies of $F_{Y^{2}_{X}/Y}$.
The stack $F_{Y/X}$ has a natural functorial presentation, and the morphism
$F_{Y/X}\to X$ commutes with base-change. The image of $Y^{2}_{X}$ in $Y$ is
the locus of points where the morphism $Y\to g(Y)$ is not smooth. Thus for
many practical purposes, the morphism $g$ can be replaced in a canonical way
by copies of the closed embedding $F_{Y^{2}_{X}/Y}\to F_{Y/X}$.
## Introduction
Local embeddings of Deligne-Mumford stacks constitute a natural extension of
the notion of closed embeddings of schemes. For example, the diagonal of a
Deligne-Mumford a stack, and the natural morphism from its inertia stack to
the stack itself, both belong to this class.
Many difficulties in extending classical algebraic geometry constructions from
the category of schemes to stacks stem from the existence of such local
embeddings. To solve this problem, one can rely on the local nature for the
étale topology of these morphisms. Indeed, given a local embedding of
algebraic stacks $g:Y\to X$, there exist étale atlases $V_{0}$ and $U$ of $Y$
and $X$ respectively, and a closed embedding $V_{0}\hookrightarrow U$
compatible with the morphism $g$. This local construction yields the notions
of normal bundle of a local embedding as introduced by A. Vistoli ([V]), and
deformation to the normal cone as introduced by A. Kresch ([K]), and
consequently an intersection theory on smooth Deligne-Mumford stacks.
In [MM] we argued that a more refined étale presentation of the morphism
$g:Y\to X$ is needed if such ubiquitous constructions like blow-ups are to be
defined for local embeddings. For this purpose, given a proper local embedding
$g:Y\to X$, we constructed an étale atlas $U$ of $X$ such that the fibre
product $Y\times_{X}U$ is a union of étale atlases $V_{i}$ of $Y$, each of
which is embedded as a closed subscheme in $U$. The locus where the images
$W_{i}$ of $V_{i}$-s intersect pairwise is an étale atlas for the stack of
non-smooth values of $g$. Moreover, the stratification determined by the
number of intersecting components $W_{i}$ indicates how far the morphism $g$
is from being étale on the image over each point in $g(Y)$. The étale atlas
$U$ thus encodes essential information about the structure of $g$. In [MM] we
set out to translate this information from étale atlases to stacks amenable to
global constructions like e.g. blow-ups, or intersection rings. For a proper
$g:Y\to X$, we found a pair of stacks $Y^{\prime}$ and $X^{\prime}$ with
étale, universally closed morphisms $Y^{\prime}\to Y$ and $X^{\prime}\to X$,
and a morphism $g^{\prime}:Y^{\prime}\to X^{\prime}$ such that
$Y^{\prime}=Y\times_{X}X^{\prime}$ is a disjoint union of stacks each embedded
as a closed substack of $X^{\prime}$ via $g^{\prime}$. However, our
construction was not unique. Indeed, it depends on the choice of a suitable
étale atlas $U$ of $X$.
In this paper we introduce an étale, universally closed morphism $F_{Y/X}\to
X$ which is intrinsically associated to the proper local embedding $g:Y\to X$,
which has the desired properties listed above, and which commutes with base
change. We give a functorial presentation of this stack and study its
properties in more detail. As applications, this canonical definition provides
grounds for extending other constructions from schemes to stacks. For example,
we can now define compactifications of configuration spaces for stacks by
extending W. Fulton and R. MacPherson’s [FMcP] constructions in a coherent,
natural way. Also, in our opinion the stack $F_{X/X\times X}$ provides a
natural context for orbifold products like the ones defined by Edidin, Jarvis
and Kimura in [EJK] for quotient Deligne-Mumford stacks. We will explore such
applications in more detail in a sequel to this paper.
We start this article by discussing the case when $g:Y\to X$ is a morphism of
Deligne–Mumford stacks which is finite and étale on its image. In [MM] we
showed that such a morphism can be factored into an étale, universally closed
morphism $F_{Y/X}\to X$ and an embedding $Y\hookrightarrow F_{Y/X}$, which
identifies $Y$ with the preimage of $g(Y)$ in $F_{Y/X}$, and such that
$F_{Y/X}\setminus Y\cong X\setminus g(Y)$. In Proposition 1.2 we provide a
detailed list of properties for $F_{Y/X}$, some of which will prove very
useful in more general set-up. For example, property (10) will lead to a
natural definition of a lift $F_{Y/X}$ in the case when $g$ is a general
proper local embedding, and $Y$ is reducible.
For any morphism of Deligne–Mumford stacks $g:Y\to X$, the fibered product
$Y\times_{X}Y$ represents the functor of isomorphisms in $X$ of objects coming
from $Y$: its objects over a scheme $S$ are tuples $(\xi_{1},\xi_{2},f)$,
where $\xi_{1},\xi_{2}$ are objects in $Y(S)$, and $f$ is an isomorphism
between $g(\xi_{1})$ and $g(\xi_{2})$. Let $\Delta:Y\to Y\times_{X}Y$ denote
the diagonal morphism and let $Y^{2}_{X}$ denote the complement of its image
in $Y\times_{X}Y$. If $g$ is finite and unramified, then so are the
projections $Y^{2}_{X}\to Y$, and their image is the locus of points where $g$
is not étale on its image. We can reiterate this construction with
$(Y^{2}_{X})^{2}_{Y}\to Y^{2}_{X}$. Here $(Y^{2}_{X})^{2}_{Y}$ is isomorphic
to the complement $Y^{3}_{X}$ of all diagonals in $Y\times_{X}Y\times_{X}Y$,
and as such it admits three different projections to $Y^{2}_{X}$. By
successively reiterating this construction until we reach
$Y^{n+1}_{X}=\emptyset,$ we obtain a canonical network $\mathcal{N}^{n}(Y/X)$
of local embeddings, the last one of which is étale on its image. This network
commutes with base change, and it encapsulates the local étale structure of
the morphism $g:Y\to X$ in a way which is simultaneously comprehensive and
non-redundant.
In a sequence of steps, the network $\mathcal{N}^{n}(Y/X)$ can be replaced by
another network $\mathcal{N}^{0}(Y/X)$ where all morphisms are closed
embeddings, and the objects admit étale, universally closed surjections to the
objects of $\mathcal{N}^{n}(Y/X)$. The target of the new network is $F_{Y/X}$.
Moreover, the other objects of $\mathcal{N}^{0}(Y/X)$ are also canonical lifts
for the local embeddings contained in $\mathcal{N}^{n}(Y/X)$. Thus, for
practical purposes the morphism $g$ can be replaced by a set of copies of the
closed embedding $F_{Y^{2}_{X}/Y}\to F_{Y/X}$. The functorial presentation and
properties of $F_{Y/X}$ are listed in Theorem 1.21 and the Definition
preceding it.
In [R], David Rydh constructed a different canonical lift $E_{Y/X}$ for any
unramified morphism $g:Y\to X$: he showed that $g$ has a universal
factorization $Y\to E_{Y/X}\to X$, where the first morphism is a closed
embedding $i$ and the second is étale; moreover, $E_{Y/X}$ comes with an open
immersion $j:X\to E_{Y/X}$ such that $i(Y)$ is the complement of $j(X)$ in
$E_{Y/X}$. His construction works in a more general context than ours, and
indeed it was meant to address the lack of an intrinsic presentation for our
étale lift in [MM]. However $E_{Y/X}$ differs from $F_{Y/X}$ in its range of
applicability. We would like to thank David Rydh for his useful observations.
The authors were supported by a Science Foundation Ireland grant.
## 1\. The universal lift of a local embedding
The stacks in this article are assumed to be algebraic in the sense of
Deligne–Mumford, Noetherian, and all morphisms considered between them are of
finite type.
### 1.1. The lift of a local embedding étale on its image.
###### Definition 1.1.
Following [V], we will call local embedding any representable unramified
morphism of finite type of stacks. A regular local embedding is a local
embedding which is also locally a complete intersection.
###### Proposition 1.2.
Let $g:Y\to X$ be a proper morphism of stacks étale on its image. There exists
an étale morphism $e_{g}:F_{Y/X}\to X$ together with an isomorphism
$\phi:g(Y)\times_{X}F_{Y/X}\to Y$, such that
* (0) i)
the triangles in the following diagram are commutative
where the upper horizontal arrow is the projection on $Y$ and the lower
horizontal arrow is the restriction of $e_{g}$ to $g(Y)\times_{X}F_{Y/X}$;
* (0) ii)
Let $p_{2}:g(Y)\times_{X}F_{Y/X}\to F_{Y/X}$ be the projection on the second
factor and consider the closed embedding $i:=p_{2}\circ\phi^{-1}$. The
restriction of $e_{g}$ induces an isomorphism $F_{Y/X}\setminus i(Y)\to
X\setminus g(Y)$.
The following properties also hold:
* (1)
For any stack $Z$, there is an equivalence of categories between
$\operatorname{Hom}(Z,F_{Y/X})$ and the category of morphisms $Z\to X$ endowed
with a section
$s:g(Y)\times_{X}Z\to Y\times_{X}Z$
for the étale map $Y\times_{X}Z\to g(Y)\times_{X}Z$.
* (2)
The triple $(F_{Y/X},e_{g},\phi)$, with $e_{g}$ étale and satisfying (0)i) and
(0)ii) is uniquely defined up to unique 2-isomorphism.
* (3)
The morphism $e_{g}:F_{Y/X}\to X$ is universally closed.
* (4)
If $g:Y\to X$ is a closed embedding, then $F_{Y/X}\cong X$.
* (5)
If $g:Y\to X$ is étale and proper, and $X$ is connected then $F_{Y/X}\cong Y$.
* (6)
For any morphism of stacks $u:X^{\prime}\to X$ and
$Y^{\prime}:=Y\times_{X}X^{\prime}$, there exists a morphism
$F_{u}:F_{Y^{\prime}/X^{\prime}}\to F_{Y/X}$ making the squares in the
following diagram Cartesian:
* (7)
If $h:Z\to Y$ is proper and étale on its image, and $g:Y\to X$ is a closed
embedding, then $F_{Z/Y}\cong Y\times_{X}F_{Z/X}$. In particular, there exists
a natural étale morphism $g_{*}:F_{Z/Y}\to F_{Z/X}$.
* (8)
For any morphism $h:Z\to Y$ proper and étale on its image, the composition
morphism $g\circ i_{h}:F_{Z/Y}\to X$, universally closed and étale on its
image, comes with an étale map $F_{F_{Z/Y}/X}\to X$ satisfying properties (0)
and (1). Moreover,
$\displaystyle F_{F_{Z/Y}/X}\cong F_{Z/F_{Y/X}}.$
In particular, if the morphism $h:Z\to Y$ is étale, then there exists a
natural morphism $h_{*}:F_{Z/X}\to F_{Y/X}$.
* (9)
If $h:Z\to Y$ and $g:Y\to X$ are proper and étale on their images, and if
$g(h(Z))\times_{X}Y\cong h(Z)$ over $Y$, then there exists a morphism
$g_{*}:F_{Z/Y}\to F_{Z/X}$ such that $e_{g\circ h}\circ\bar{g}=g\circ e_{h}$.
In this case $F_{F_{Z/Y}/F_{Z/X}}\cong F_{F_{Z/Y}/X}\cong F_{Z/F_{Y/X}}.$
* (10)
Given any proper local embeddings $g:Y\to X$ and $f:T\to X$, and
$Z:=Y\times_{X}T$, there are natural isomorphisms
$\displaystyle F_{Y/X}\times_{X}F_{T/X}\cong F_{F_{Z/T}/F_{Y/X}}\cong
F_{F_{Z/Y}/F_{T/X}}\cong F_{F_{Z/Y}\bigcup_{Z}F_{Z/T}/F_{Z/X}},$
where $F_{Z/Y}\bigcup_{Z}F_{Z/T}$ denotes the stack obtained by gluing the
stacks $F_{Z/Y}$ and $F_{Z/T}$ along $Z$.
###### Proof.
An explicit étale groupoid presentation for a functor $F_{Y/X}$ which
satisfies properties (0) and (1) was found in [MM], section 1.1. We briefly
recall it here. One chooses an étale cover by a scheme $p:U\to X$ such that
$Y\times_{X}U\cong V=V_{1}\bigsqcup V_{2}$ where $V_{1}=g(Y)\times_{X}U$. Let
$S_{ij}:=\mbox{ Im }(\phi_{ij}:V_{i}\times_{Y}V_{j}\to U\times_{X}U),$
for the map $\phi_{ij}$ given as a composition
$V_{i}\times_{Y}V_{j}\hookrightarrow
V\times_{Y}V=V\times_{Y}(Y\times_{X}U)\cong V\times_{X}U\to U\times_{X}U.$
A groupoid presentation of $F_{Y/X}$ is given by
$\left[R^{\prime}\rightrightarrows U\right]$
$R^{\prime}:=(U\times_{X}U)\setminus(S_{12}\cup S_{21}\cup(S_{22}\setminus
S_{11}))\cup\mbox{ Im }e.$
To prove property (2), we note that for any triple
$(F^{\prime},e^{\prime},\phi^{\prime})$ satisfying the properties (0), there
is a canonical section $g(Y)\times_{X}F^{\prime}\to Y\times_{X}F^{\prime}$
which, together with the map $e^{\prime}$, determine a unique morphism
$u:F^{\prime}\to F_{Y/X}$ such that $e\circ u=e^{\prime}$, due to condition
(1). Both $e$ and $e^{\prime}$ are étale, and so $u$ must be étale as well. On
the other hand, $e$ and $e^{\prime}$ induce isomorphisms
$g(Y)\times_{X}F_{Y/X}\cong Y\cong g(Y)\times_{X}F^{\prime}$ and
$F_{Y/X}\setminus i(Y)\cong X\setminus g(Y)\cong F^{\prime}\setminus
i^{\prime}(Y)$. Thus $u$ is both étale and bijective, and so an isomorphism.
Property (3) follows from the valuation criterium in conjunction with property
(1). Consider a complete discrete valuation ring $R$ with field of fractions
$K$, a commutative diagram
$\displaystyle\begin{CD}\operatorname{Spec}(K)@>{{u}}>{}>F_{Y/X}\\\
@V{}V{{\rho}}V@V{}V{{p}}V\\\ \operatorname{Spec}(R)@>{v}>{}>X.\end{CD}$
The closed embedding
$g(Y)\times_{X}\operatorname{Spec}(K)\to\operatorname{Spec}(K)$ is either the
empty embedding or an isomorphism. If empty, then $u$ factors through
$\operatorname{Spec}(K)\to F_{Y/X}\setminus Y\cong X\setminus g(Y)$ and so $v$
also naturally yields $\operatorname{Spec}(R)\to X\setminus g(Y)\cong
F_{Y/X}\setminus Y$. If an isomorphism, then the map $v$ induces a natural
morphism $\operatorname{Spec}(K)\cong g(Y)\times_{X}\operatorname{Spec}(K)\to
Y$ whose composition with $g$ is $u\rho$. As $g$ is proper, there is a lift
$\operatorname{Spec}(R)\to Y$, which yields a section
$g(Y)\times_{X}\operatorname{Spec}(R)\to Y\times_{X}\operatorname{Spec}(R)$.
This, together with the map $u\to\operatorname{Spec}(R)\to X$ give the data
for a unique morphism $\operatorname{Spec}(R)\to F_{Y/X}$ as required.
Properties (4) and (5) are direct consequences of (2).
Property (6) was proven in [MM], Corollary 1.8. Alternatively, it follow
immediately from (2). Indeed, consider a morphism of stacks $f:X^{\prime}\to
X$ and let $Y^{\prime}:=Y\times_{X}X^{\prime}$, with the morphism
$g^{\prime}:Y^{\prime}\to X^{\prime}$ induced by $g$. Then the étale morphism
$X^{\prime}\times_{X}F_{Y/X}\to X^{\prime}$ induced by $e_{g}$, together with
the composition
$\displaystyle
g^{\prime}(Y^{\prime})\times_{X^{\prime}}(X^{\prime}\times_{X}F_{Y/X})\cong(g(Y)\times_{X}X^{\prime})\times_{X^{\prime}}(X^{\prime}\times_{X}F_{Y/X})\cong$
$\displaystyle\cong g(Y)\times_{X}X^{\prime}\times_{X}F_{Y/X}\cong
X^{\prime}\times_{X}Y\cong Y^{\prime},$
satisfy properties (0) for the morphism $g^{\prime}:Y^{\prime}\to X^{\prime}$
and so $X^{\prime}\times_{X}F_{Y/X}\cong F_{Y^{\prime}/X^{\prime}}$. Thus
$\displaystyle Y^{\prime}\cong
g^{\prime}(Y^{\prime})\times_{X^{\prime}}F_{Y^{\prime}/X^{\prime}}\cong(g(Y)\times_{X}X^{\prime})\times_{X^{\prime}}F_{Y^{\prime}/X^{\prime}}\cong$
$\displaystyle\cong
g(Y)\times_{X}F_{Y^{\prime}/X^{\prime}}\cong(g(Y)\times_{X}F_{Y/X})\times_{F_{Y/X}}F_{Y^{\prime}/X^{\prime}}\cong
Y\times_{F_{Y/X}}F_{Y^{\prime}/X^{\prime}}.$
To prove (7), we will construct a canonical morphism $F_{Z/Y}\to
Y\times_{X}F_{Z/X}$, together with its inverse. To construct $F_{Z/Y}\to
F_{Z/X}$, we first consider the composition $g\circ e_{h}:F_{Z/Y}\to Y\to X$.
The canonical isomorphisms
$\displaystyle
g(h(Z))\times_{X}F_{Z/Y}\cong(h(Z)\times_{X}Y)\times_{Y}F_{Z/Y}\cong
h(Z)\times_{Y}F_{Z/Y}\cong Z,$ and $\displaystyle
Z\times_{X}F_{Z/Y}\cong(Z\times_{X}Y)\times_{Y}F_{Z/Y}\cong
Z\times_{Y}F_{Z/Y},$
together with the embedding $Z\to Z\times_{Y}F_{Z/Y}$ give a section
$g(h(Z))\times_{X}F_{Z/Y}\to Z\times_{Y}F_{Z/Y}$, and thus, according to (1),
a map $F_{Z/Y}\to F_{Z/X}$. This, together with the étale map $F_{Z/Y}\to Y$
generate the desired morphism ${}_{Z/Y}\to Y\times_{X}F_{Z/X}$. Its inverse is
also constructed via property (1) as follows: We consider the projection
$Y\times_{X}F_{Z/X}\to Y$ together with the canonical section
$\displaystyle h(Z)\times_{Y}(Y\times_{X}F_{Z/X})\cong
h(Z)\times_{X}F_{Z/X}\to Z\times_{X}F_{Z/X}\cong
Z\times_{Y}(Y\times_{X}F_{Z/X}).$
Proof of (8): Consider now a morphism $h:Z\to Y$ proper and étale on its
image. We will show that $e_{g}\circ e_{i\circ h}:F_{Z/F_{Y/X}}\to X$ is an
’etale lift for the composition $g\circ i_{h}:F_{Z/Y}\to X$. Note that $g\circ
i_{h}$ is universally closed and étale on its image $\mbox{ Im }g\circ
i_{h}=g(Y)$, though not necessarily separated. Let $i:Y\to F_{Y/X}$ be the
natural embedding induced by $g$. Then $i\circ h:Z\to F_{Y/X}$ is the
composition of a proper morphism étale on its image and a closed embedding.
Due to (7) applied to this composition, there are Cartesian diagrams
whose composition implies that $F_{Z/Y}\cong F_{Z/F_{Y/X}}\times_{X}g(Y)$
canonically and that $e_{g}\circ e_{i\circ h}$ induces $F_{Z/F_{Y/X}}\setminus
F_{Z/Y}\cong X\setminus g(Y)$. We also check that $e_{g}\circ e_{i\circ h}$
satisfies property (1), namely that any map $T\to F_{Z/F_{Y/X}}$ is uniquely
determined by a pair of maps $f:T\to X$ and a section $s:g(Y)\times_{X}T\to
F_{Z/Y}\times_{X}T$. Indeed, such a pair, together with the composition
$g(Y)\times_{X}T\to F_{Z/Y}\times_{X}T\to Y\times_{X}T$, determine in a first
instance a map $T\to F_{Y/X}$, whose composition with $e_{g}$ yields $f$. The
restriction of $s$ also yields a section $g(h(Z))\times_{X}T\to Z\times_{X}T$
and so a sequence of morphisms over $F_{Y/X}$:
$\displaystyle i(h(Z))\times_{F_{Y/X}}T\to g(h(Z))\times_{X}T\to Z,$
and so a section $i(h(Z))\times_{F_{Y/X}}T\to Z\times_{F_{Y/X}}T$. This
determines a map $T\to F_{Z/F_{Y/X}}$ whose composition with $e_{g}\circ
e_{i\circ h}$ yields $f$.
Proof of (9): The isomorphisms
$\displaystyle
g(h(Z))\times_{X}F_{Z/Y}\cong(g(h(Z))\times_{X}Y)\times_{Y}F_{Z/Y}\cong
h(Z)\times_{Y}F_{Z/Y}\cong Z,$
together with the composition $g\circ e_{h}$, give a morphism $F_{Z/Y}\to
F_{Z/X}$. Clearly $F_{F_{Z/Y}/F_{Z/X}}$ satisfies properties (0) as an étale
lift of $g\circ e_{h}$, hence the isomorphism $F_{F_{Z/Y}/F_{Z/X}}\cong
F_{F_{Z/Y}/X}$.
Proof of (10): The first two isomorphisms are direct consequences of property
(6). Indeed,
$\displaystyle F_{F_{Y\times_{X}T/T}/F_{Y/X}}\cong
F_{F_{Y/X}\times_{X}T/F_{Y/X}}\cong F_{Y/X}\times_{X}F_{T/X},$
and similarly for $F_{F_{Y\times_{X}T/Y}/F_{T/X}}$. To prove the last
isomorphism, we will first need to pinpoint the existence of a natural local
embedding $F_{Z/Y}\bigcup_{Z}F_{Z/T}\to F_{Z/X}$. Indeed, via property (9),
there exist compositions
$\displaystyle F_{Z/T}\hookrightarrow F_{F_{Z/T}/X}\cong F_{Z/F_{T/X}}\to
F_{Z/X}\mbox{ and }F_{Z/Y}\hookrightarrow F_{F_{Z/Y}/X}\cong F_{Z/F_{Y/X}}\to
F_{Z/X}.$
Indeed, the hypotheses necessary for property (9) hold because
$Z=Y\times_{X}Y$. Moreover, the compositions above, together with the
embeddings of $Z$ into $F_{Z/T}$ and $F_{Z/Y}$, respectively, form a
commutative diagram, which insure the existence of the morphism
$F_{Z/Y}\bigcup_{Z}F_{Z/T}\to F_{Z/X}$ (conform [AGV], Appendix 1). Moreover,
by construction this morphism is proper and a local embedding.
In a similar way we can check the existence of a closed embedding
$\displaystyle j:F_{Z/Y}\bigcup_{Z}F_{Z/T}\to F_{Y/X}\times_{X}F_{T/X}.$
Indeed, the isomorphisms $F_{Y/X}\times_{X}F_{T/X}\cong
F_{F_{Z/T}/F_{Y/X}}\cong F_{F_{Z/Y}/F_{T/X}}$ implicitly state the existence
of closed embeddings of $F_{Z/Y}$ and $F_{Z/T}$ into
$F_{Y/X}\times_{X}F_{T/X}$, which commute with the embeddings of $Z$ into
$F_{Z/T}$ and $F_{Z/Y}$ respectively, and thus define the closed embedding
$j$.
Furthermore, property (9) implies the existence of natural morphisms from
$F_{Z/Y}$ and $F_{Z/T}$ to $F_{Z/X}$, which induce a natural morphism
$F_{Z/Y}\bigcup_{Z}F_{Z/T}\to F_{Z/X}$.
The existence of a natural morphism $e:F_{Y/X}\times_{X}F_{T/X}\to F_{Z/X}$
follows from the universal property (1) of $F_{Z/X}$. Indeed, via the
canonical étale morphism $F_{Y/X}\times_{X}F_{T/X}\to X$, there are natural
morphisms
$\displaystyle\mbox{ Im }(Z\to
X)\times_{X}F_{Y/X}\times_{X}F_{T/X}\hookrightarrow\mbox{ Im
}f\times_{X}F_{T/X}\times_{X}F_{Y/X}\to T\times_{X}F_{Y/X}\times_{X}F_{T/X},$
and similarly
$\displaystyle\mbox{ Im }(Z\to
X)\times_{X}F_{T/X}\times_{X}F_{T/X}\hookrightarrow\mbox{ Im
}g\times_{X}F_{Y/X}\times_{X}F_{T/X}\to Y\times_{X}F_{Y/X}\times_{X}F_{T/X},$
forming a commutative diagram with the projections to
$F_{Y/X}\times_{X}F_{T/X}$, and thus inducing a section
$\displaystyle\mbox{ Im }(Z\to X)\times_{X}F_{Y/X}\times_{X}F_{T/X}\to
Z\times_{X}F_{Y/X}\times_{X}F_{T/X}.$
This proves the existence of the natural morphism
$e:F_{Y/X}\times_{X}F_{T/X}\to F_{Z/X}$, which is étale because the natural
maps from both its target and source to $X$ are étale. We have thus obtained a
diagram
which is commutative due to the natural choices of the morphisms and property
(1).
By (0) and (2), it remains to show that the complement of
$F_{Z/Y}\bigcup_{Z}F_{Z/T}$ in $F_{Y/X}\times_{X}F_{T/X}$ is naturally
isomorphic to the complement of $\mbox{ Im }(F_{Z/Y}\bigcup_{Z}F_{Z/T}\to
F_{Z/X})$ in $F_{Z/X}$, and that there is a natural isomorphism
$\displaystyle\mbox{ Im }(F_{Z/Y}\bigcup_{Z}F_{Z/T}\to
F_{Z/X})\times_{F_{Z/X}}(F_{Y/X}\times_{X}F_{T/X})\cong
F_{Z/Y}\bigcup_{Z}F_{Z/T}.$
These properties follow canonically from the definitions of the objects and
morphisms involved.
∎
###### Lemma 1.3.
Let $g:Y\to X$ be a proper local embedding of Noetherian stacks, with $Y$
integral. Then there exists a stack $D_{Y/X}$ together with an étale
epimorphism $e:Y\to D_{Y/X}$ and a proper local embedding $g_{1}:D_{Y/X}\to X$
of generic degree 1, such that $g=g_{1}\circ e$. Moreover, $D_{Y/X}$ is unique
up to an isomorphism.
###### Proof.
A factorization of the morphism $g:Y\to X$ into an étale epimorphism $e:Y\to
D_{Y/X}$ and a proper local embedding $g_{1}:D_{Y/X}\to X$ of generic degree
one was constructed in [MM], Lemma 1.10. It remains to prove uniqueness up to
an isomorphism. For this, we first recall the étale local structure of
$D_{Y/X}$: There exists an étale cover of $X$ by a scheme $U$ such that
$Y\times_{X}U=\bigsqcup_{i,a}V_{i}^{a}$, and for each $i,a$, the morphism
$g_{U}:Y\times_{X}U\to U$ restricts to a closed embedding
$V_{i}^{a}\hookrightarrow U$, with image $W_{i}$, such that $W_{i}\not=W_{j}$
if $i\not=j$. Let $W=\bigcup_{i}W_{i}$. There exists a canonical groupoid
structure
$\left[s_{e},t_{e}:R_{e}:=\bigsqcup_{i}W_{i}\times_{X}U\rightrightarrows\bigsqcup_{i}W_{i}\right]$,
and $D_{Y/X}$ is defined as its associated stack. The morphisms $e:Y\to
D_{Y/X}$ and $g_{1}:D_{Y/X}\to X$, respectively, are determined by the
canonical choice of maps $e_{U}:\bigsqcup_{i,a}V_{i}^{a}\to\bigcup_{i}W_{i}$
and $g_{1U}:\bigcup_{i}W_{i}\to U$, together with
$e_{R}:\bigsqcup_{i,a,j,b}V_{i}^{a}\times_{Y}V_{j}^{b}\to\bigsqcup_{i}W_{i}\times_{X}U$
and $g_{1R}:\bigsqcup_{i}W_{i}\times_{X}U\to U\times_{X}U$ at the level of
relations.
Assume that $e^{\prime}:Y\to Y^{\prime}$ is another étale epimorphism, and
that $f^{\prime}:Y^{\prime}\to X$ is a proper local embedding of generic
degree one such that $f^{\prime}\circ e^{\prime}=g$. Let
$\bigcup_{i}V^{\prime}_{i}:=Y^{\prime}\times_{X}U$, with the induced morphism
$f^{\prime}_{U}:\bigcup_{i}V^{\prime}_{i}\to U$, such that each
$V^{\prime}_{i}$ is the preimage of $W_{i}$. As the induced morphism
$e^{\prime}_{U}:\bigsqcup_{i,a}V_{i}^{a}\to\bigcup_{i}V^{\prime}_{i}$ is étale
and surjective and sends each $V_{i}^{a}$ to $V^{\prime}_{i}$, the components
$V^{\prime}_{i}$ must be pairwise disjoint. We will construct an isomorphism
of groupoids
$\displaystyle\phi:\left[s_{e},t_{e}:R_{e}:=\bigsqcup_{i}W_{i}\times_{X}U\rightrightarrows\bigsqcup_{i}W_{i}\right]\rightarrow\left[s^{\prime},t^{\prime}:R^{\prime}:=\bigsqcup_{i}V^{\prime}_{i}\times_{Y^{\prime}}\bigsqcup_{i}V^{\prime}_{i}\rightrightarrows\bigsqcup_{i}V^{\prime}_{i}\right].$
First consider any section
$\sigma:\bigsqcup_{i}W_{i}\to\bigsqcup_{i,a}V_{i}^{a}$ of $e_{U}$ and define
$\phi_{U}:=e^{\prime}_{U}\circ\sigma$. As section of the étale morphism
$e_{U}$, the map $\sigma$ must be étale itself. In fact, it consists of a
choice of an index $a$ for each $i$, and an isomorphism $W_{i}\to V_{i}^{a}$.
The map $e^{\prime}_{U}$ is étale and surjective, and it maps each $V_{I}^{a}$
onto $V^{\prime}_{i}$. Indeed, $\deg g_{1U}\circ e_{U}=\deg
f^{\prime}_{U}\circ e^{\prime}_{U}$ while $\deg g_{1U}$, $f^{\prime}_{U}$ are
both of generic degree one, so the image of each $V_{I}^{a}$ under
$e^{\prime}_{U}$ must be a dense open subset of $V^{\prime}_{i}$. On the other
hand, $e^{\prime}_{U}$ is also proper, so
$e^{\prime}_{U}(V_{I}^{a})=V^{\prime}_{i}$.
It follows that $\phi_{U}=e^{\prime}_{U}\circ\sigma$ is étale and surjective
as well. Moreover, $f^{\prime}_{U}\circ\phi_{U}=g_{1U}$, so the degree of
$\phi_{U}$ must be one. Thus $\phi_{U}$ is an isomorphism.
Let $R:=U\times_{X}U$, and consider the first projection $s:R\to U$. As
$R^{\prime}\cong\bigsqcup_{i}V^{\prime}_{i}\times_{U}R$, we can construct
$\phi_{R}:R_{e}\to R^{\prime}$ as the morphism uniquely defined by the
conditions
$\displaystyle f^{\prime}_{R}\circ\phi_{R}=g_{1R}\mbox{ and
}s^{\prime}\circ\phi_{R}=\phi_{U}\circ s_{e}.$
Similarly, a morphism $\psi_{R}:R^{\prime}\to R_{e}$ can be defined by the
conditions
$\displaystyle g_{1R}\circ\psi_{R}=f^{\prime}_{R}\mbox{ and
}s_{e}\circ\psi_{R}=\phi^{-1}_{U}\circ s^{\prime}.$
We note that $\psi_{R}\circ\phi_{R}=\mbox{ id}_{R_{e}}$, as
$g_{1R}\circ\psi_{R}\circ\phi_{R}=g_{1R}$ and
$s_{e}\circ\psi_{R}\circ\phi_{R}=s_{e}$, and
$R_{e}\cong\bigsqcup_{i}W_{i}\times_{U}R$. Similarly,
$\phi_{R}\circ\psi_{R}=\mbox{ id}_{R^{\prime}}$.
It remains to prove that the pair $(\phi_{U},\phi_{R})$ is a morphism of
groupoids. This is a slightly long, but direct check. Here we will prove the
equality:
(1.5) $\displaystyle\phi_{U}\circ t_{e}=t^{\prime}\circ\phi_{R}.$
Let $\begin{array}[]{ll}i_{e}:R_{e}\to R_{e},&i^{\prime}:R^{\prime}\to
R^{\prime}\end{array}$ and $i:R\to R$ denote the inverting maps of the
groupoids
$\begin{array}[]{ll}[R_{e}\rightrightarrows\bigsqcup_{i}W_{i}],&[R^{\prime}\rightrightarrows\bigsqcup_{i}V^{\prime}_{i}]\end{array}$
and $[R\rightrightarrows U]$ respectively, so that $i_{e}\circ s_{e}=t_{e},$
$i^{\prime}\circ s^{\prime}=t^{\prime}$ and $i\circ s=t$. Composition with
$f^{\prime}_{U}$ of the two terms in the equation (1.5) yields:
$\displaystyle f^{\prime}_{U}\circ\phi_{U}\circ t_{e}=g_{1U}\circ t_{e},\mbox{
and }$ $\displaystyle f^{\prime}_{U}\circ
t^{\prime}\circ\phi_{R}=f^{\prime}_{U}\circ i^{\prime}\circ
s^{\prime}\circ\phi_{R}=f^{\prime}_{U}\circ i^{\prime}\circ\phi_{U}\circ
s_{e}=$ $\displaystyle=i\circ f^{\prime}_{U}\circ\phi_{U}\circ s_{e}=i\circ
g_{1U}\circ s_{e}=g_{1U}\circ i_{e}\circ s_{e}=g_{1U}\circ t_{e}.$
As $f^{\prime}_{U}$ is generically injective, the closed subset
$\begin{array}[]{ll}\\{x\in R_{e};&\phi_{U}\circ
t_{e}(x)=t^{\prime}\circ\phi_{R}(x)\\}\end{array}$ contains an open dense
subset of $R_{e}$, so it must be the entire $R_{e}$.
All other compatibility relations follow directly by the same method as above.
∎
###### Remark 1.4.
We note that if $g:Y\to X$ was not separable, the uniqueness of a possible
split $Y\to D_{Y/X}\to X$ would not be guaranteed. For example, if $p\not=q$
are natural numbers and $Y$ is obtained by gluing $pq$ copies of the space $X$
along the complement of a point, then two possible choices for $D_{Y/X}$ would
be obtained by gluing $p$, respectively $q$ copies of the space $X$ along the
complement of that same point.
###### Proposition 1.5.
Let $g:Y\to X$ be a proper local embedding of Noetherian stacks, with $Y$
integral. The stack $D_{Y/X}$ constructed above satisfies the following
properties:
* (1)
For any morphism of stacks $u:X^{\prime}\to X$ and
$Y^{\prime}:=Y\times_{X}X^{\prime}$, there exists a morphism
$D_{u}:D_{Y^{\prime}/X^{\prime}}\to D_{Y/X}$ making the squares in the
following diagram Cartesian:
* (2)
If $h:Z\to Y$ is another proper local embedding of integral Noetherian stacks,
then there exists a natural isomorphism
$\displaystyle D_{D_{Z/Y}/D_{Y/X}}\cong D_{Z/X},$
where $D_{D_{Z/Y}/D_{Y/X}}$ is the stack associated to the composition $e\circ
h_{1}$ of the local embedding of generic degree one $h_{1}:D_{Z/Y}\to Y$ and
the étale map $e:Y\to D_{Y/X}$.
* (3)
There is an equivalence of categories between the category of commutative
diagrams
with $f$ étale, and that of pairs in
$\operatorname{Hom}(Z,D_{Y/X})\times\operatorname{Hom}(T,Z\times_{D_{Y/X}}Y)$
such that the induced morphism $T\to Z$ is étale.
The morphisms in the first category are given by Cartesian diagrams
such that $t$ and $z$ commute with the given morphisms to $Y$ and $X$,
respectively.
###### Proof.
Properties (1) and (2) are direct consequences of the definition of $D_{Y/X}$
and Lemma 1.3. The proof of property (3) is based on arguments also employed
in the proof of same Lemma. Indeed, given an étale cover of $X$ by a scheme
$U$ such that $Y\times_{X}U=\bigsqcup_{i,a}V_{i}^{a}$, and for each $i,a$, the
morphism $g_{U}:Y\times_{X}U\to U$ restricts to a closed embedding
$V_{i}^{a}\hookrightarrow U$, with image $W_{i}$, such that $W_{i}\not=W_{j}$
if $i\not=j$. Then $D_{Y/X}\times_{X}U\cong\bigsqcup_{i}W_{i}$. Also,
$T\times_{Y}(\bigsqcup_{i,a}V_{i}^{a})$, and as the morphism $f:T\to Z$ is
étale, then $Z\times_{X}U\cong\bigsqcup V^{\prime}_{i}$ for some
$V^{\prime}_{i}$-s such that for each $i$, the pullback of $f$ restricts to
maps $\bigsqcup_{a}V_{i}^{a}\to\bigsqcup V^{\prime}_{i}$, and the pullback of
$u$ restricts to $V^{\prime}_{i}\to W_{i}$. In particular, this induces a map
$\bigsqcup V^{\prime}_{i}\to\bigsqcup_{i}W_{i}$. A morphism of groupoids
$\displaystyle[(\bigsqcup V^{\prime}_{i})\times_{Z}(\bigsqcup
V^{\prime}_{i})\rightrightarrows\bigsqcup
V^{\prime}_{i}]\to[(\bigsqcup_{i}W_{i})\times_{D_{Y/X}}(\bigsqcup_{i}W_{i})\rightrightarrows\bigsqcup_{i}W_{i}]$
can then be constructed by the exact same method as in the proof of the
previous Lemma. ∎
### 1.2.
In the next paragraphs we will work with simple categories whose objects are
Noetherian stacks, and such that there exists at most one morphism between
each pair of objects. We will discuss some additional properties below.
###### Definition 1.6.
By extending the terminology of [L], we can define a poset of stacks as
follows. We regard any poset $\mathcal{P}$ as a category, such that for any
elements $I,J\in\mathcal{P}$, the set of morphisms $\mbox{ Morph}(I,J)$
consists of a unique element if $I\leq J$, and is empty otherwise. Then a
poset of stacks is a contravariant functor from $\mathcal{P}$ to the category
of sets. Any such poset of stacks $\mathcal{N}=\\{\phi_{J}^{I}:Y_{J}\to
Y_{I}\\}_{I\subseteq J\in\mathcal{P}}$ where $\mathcal{P}$ is the power set of
a finite set $\Lambda$, and the partial order is given by inclusion, will be
called simply a network. In particular, a network will include a unique target
$Y_{\emptyset}$, (and a source $Y_{\Lambda}$, possibly empty).
###### Definition 1.7.
Let $\mathcal{N}=\\{\phi_{J}^{I}:Y_{J}\to Y_{I}\\}_{I\subseteq
J\in\mathcal{P}}$ be a network of morphisms with target $X=Y_{\emptyset}$, and
let $\mathcal{N}^{\prime}=\\{\phi^{\prime I}_{J}:Y^{\prime}_{J}\to
Y^{\prime}_{I}\\}_{I,J\in\mathcal{P}^{\prime}}$ be another network with target
$X^{\prime}=Y_{\emptyset}$, where $\mathcal{P}^{\prime}\subseteq\mathcal{P}$.
A morphism of networks $F:\mathcal{N}^{\prime}\to\mathcal{N}$ is a fully
faithful functor from the category $\mathcal{N}^{\prime}$ to the category
$\mathcal{N}$, given by a set of morphisms $\\{f_{I}:Y^{\prime}_{I}\to
Y_{I}\\}_{I\in\mathcal{P}}$, such that $f_{I}\circ\phi^{\prime
I}_{J}=\phi_{J}^{I}\circ f_{J}$. In particular, $F$ includes a morphism
between targets $f:X^{\prime}\to X$.
We say that $\mathcal{N}^{\prime}\cong\mathcal{N}\times_{X}X^{\prime}$ if each
of the commutative diagrams corresponding to the equalities
$f_{I}\circ\phi^{\prime I}_{J}=\phi_{J}^{I}\circ f_{J}$ is Cartesian.
Given a network of closed embeddings, there is a natural way to glue any
subset of objects $\\{Y_{I}\\}_{I\in\mathcal{Q}}$ into a stack
$S_{\mathcal{Q}}$ as follows:
###### Lemma 1.8.
Let $\mathcal{N}=\\{\phi_{J}^{I}:Y_{J}\to Y_{I}\\}_{I\subseteq
J\in\mathcal{P}}$ be a network of closed embeddings, with target $X$. Consider
$\mathcal{Q}\subseteq\mathcal{P}$.
a) There exists a stack $S_{\mathcal{Q}}$, and commutative diagrams
for all $I,J\in{\mathcal{Q}}$, such that for any stack $T$, the natural
functor
$\displaystyle\operatorname{Hom}(S_{\mathcal{Q}},T)\to\times_{\\{\operatorname{Hom}(Y_{I\cup
J},T)\\}_{I,J\in{\mathcal{Q}}}}\\{\operatorname{Hom}(Y_{J},T)\\}_{J\in{\mathcal{Q}}}$
is an equivalence of categories.
In particular, there exists a natural morphism $S_{\mathcal{Q}}\to
Y_{(\bigcap_{I\in\mathcal{Q}}I)}$ compatible with the morphisms
$\phi_{I}^{(\bigcap_{I\in\mathcal{Q}}I)}$, for all $I\in\mathcal{Q}$.
Such a stack is unique up to unique isomorphism.
b) For any morphism $X^{\prime}\to X$, consider the network
$\mathcal{N}^{\prime}:=\mathcal{N}\times_{X}X^{\prime}$, with objects
$Y^{\prime}_{I}=Y_{I}\times_{X}X^{\prime}$. Consider the stack
$S^{\prime}_{\mathcal{Q}}$, obtained by gluing the objects
$\\{Y^{\prime}_{I}\\}_{I\in\mathcal{Q}}$ in the network
$\mathcal{N}^{\prime}$. Then for all $I\in\mathcal{Q}$, the squares in the
following diagram are Cartesian:
###### Proof.
a) We will proceed by induction on the cardinality of ${\mathcal{Q}}$. If
$|{\mathcal{Q}}|=1$, then $S_{\mathcal{Q}}=Y_{I}$ for $I\in{\mathcal{Q}}$.
Assume now that $S_{\mathcal{Q}}$ exists for any ${\mathcal{Q}}$ of a given
cardinality. Fix such ${\mathcal{Q}}$ and let $J\not\in{\mathcal{Q}}$. Note
that if $J\supseteq I$ for some $I\in{\mathcal{Q}}$, then
$S_{{\mathcal{Q}}\cup\\{J\\}}=S_{\mathcal{Q}}$. If this is not the case, let
${\mathcal{Q}}^{J}:=\begin{array}[]{ll}\\{I\bigcup
J;&I\in{\mathcal{Q}}\\}\end{array}$. Then by induction,
$S_{{\mathcal{Q}}^{J}}$ exists and, moreover, there is a unique closed
embedding $S_{{\mathcal{Q}}^{J}}\to S_{\mathcal{Q}}$ determined by the
compositions $Y_{J\cup K}\to Y_{K}\to S_{\mathcal{Q}}$ for all
$K\in{\mathcal{Q}}$. Gluing $Y_{J}$ and $S_{\mathcal{Q}}$ along
$S_{{\mathcal{Q}}^{J}}$ yields a stack satisfying the required properties
(conform [AGV], Proposition A.1.1).
Part b) follows by standard category theoretical arguments. Indeed, since
$Y^{\prime}_{I}\cong
Y_{I}\times_{Y_{(\bigcap_{I\in\mathcal{Q}}I)}}Y^{\prime}_{(\bigcap_{I\in\mathcal{Q}}I)}$,
it is enough to show that the right side of the diagram is Cartesian. Given
two morphisms $T\to S_{\mathcal{Q}}$ and $T\to
Y^{\prime}_{(\bigcap_{I\in\mathcal{Q}}I)}$ commuting to the respective
morphisms to $Y_{(\bigcap_{I\in\mathcal{Q}}I)}$, we can think of $T$ as being
obtained by gluing the objects
$\\{T\times_{S_{\mathcal{Q}}}Y_{I}\\}_{I\in\mathcal{Q}}$ within the network
whose objects are $\\{T\times_{S_{\mathcal{Q}}}Y_{J}\\}_{J\supseteq I\mbox{
for some }I\in\mathcal{Q}}$, and the target $T$. Each such object
$T\times_{S_{\mathcal{Q}}}Y_{J}$ admits two natural morphisms, to $Y_{I}$ and
$Y^{\prime}_{(\bigcap_{I\in\mathcal{Q}}I)}$, respectively, commuting to the
respective morphisms to $Y_{(\bigcap_{I\in\mathcal{Q}}I)}$, and thus admits
natural morphisms $T\times_{S_{\mathcal{Q}}}Y_{J}\to Y^{\prime}_{I}\to
S^{\prime}_{\mathcal{Q}}$, for $J\supseteq I\in\mathcal{Q}$. By part a), there
exists a unique natural morphism $T\to S^{\prime}_{Q}$ compatible with $T\to
S_{\mathcal{Q}}$ and $T\to Y^{\prime}_{(\bigcap_{I\in\mathcal{Q}}I)}$.
∎
Consider a proper local embedding of Noetherian stacks $g:Y\to X$, with $Y$
integral. Starting from the flat stratification of $g$, in [MM], we
constructed a network of local embeddings associated to $g$, and an étale lift
$F_{Y/X}\to X$ which reflected the local étale structure of the morphism $g$.
However, this lift was not canonical, as it depended of the choice of étale
cover by a nice scheme $U$ of $X$. We recall here the properties of $U$ which
were essential for the construction of $F_{Y/X}$.
Let $Y_{n}\hookrightarrow Y_{n-1}\hookrightarrow...\hookrightarrow
Y_{1}\hookrightarrow Y_{0}=Y$ be a filtration of $Y$ consisting of the
closures $\overline{g^{-1}(S_{i})}\subseteq Y$, where $\\{S_{i}\\}_{i}$ is the
flattening stratification for the morphism $Y\to g(Y)$.
###### Definition 1.9.
Let $g:Y\to X$ be a proper local embedding of Noetherian stacks. An étale
cover $U$ of $X$ is called suitable for the morphism $g$ if the following
properties hold:
1. (1)
$g(Y)\times_{X}U=\bigcup_{l\in L}W_{l}$, where $W_{l}$ are isomorphic, for all
$l\in L$.
2. (2)
For all $k=0,...,n$, and for some suitable choices of subsets
$\mathcal{P}_{k}\subset\mathcal{P}:=\mathcal{P}(L)$, we have
$\bigcup_{I\in\mathcal{P}_{k}}W_{I}=g(Y_{k})\times_{X}U$, where
$W_{I}=\bigcap_{l\in I}W_{l}$, and $W_{I}\cong W_{I^{\prime}}$ for all
$I,I^{\prime}\in\mathcal{P}_{k}$.
3. (3)
For each $I$ as above, there exist sets $\\{V_{I}^{a}\\}_{a\in A_{I}}$ mapping
onto $Y_{k}$, with isomorphisms $V_{I}^{a}\to W_{I}$ standing over
$g_{k}:Y_{k}\to g(Y_{k})$, and satisfying
$Y_{k}\times_{X}U=\bigcup_{I\in\mathcal{P}_{k},a\in A_{I}}V_{I}^{a}.$
Here $A_{I}=\bigsqcup_{k\in I}A_{k}$ and $V_{I}^{a}\subseteq V_{k}^{a}$ if
$k\in I$ and $a\in A_{k}$.
###### Definition 1.10.
Let $X$ and $Y$ be Noetherian stacks, with $Y$ integral. Let $g:Y\to X$ be a
proper local embedding of generic degree one. Let $U\to X$ be a suitable étale
cover for $g$. We associate to $g$ and $U$ a network of local embeddings
$\phi_{J}^{I}:Y_{J}\to Y_{I}$, one for each pair $I\subseteq J$,
$I\in\mathcal{P}_{i}$ and $J\in\mathcal{P}_{j}$, as follows. For each
$I\subseteq L$, for each distinct $i,j\in L$ and the uniquely associated
indices $a\in A_{i}$, $b\in A_{j}$, we define
$\displaystyle\begin{array}[]{lll}R_{\emptyset}:=U\times_{X}U,&R_{i}:=V^{a}_{i}\times_{Y}V^{a}_{i}&\mbox{
and }R_{I}:=R_{I}=(\prod_{i\in I})_{R_{\emptyset}}R_{i},\end{array}$
and $Y_{I}$ as the stack with groupoid presentation
$\left[R_{I}\rightrightarrows V_{I}^{a}\right].$ We consider by convention
$V_{\emptyset}^{a}=U$, such that $Y_{\emptyset}=X$. We note that
$\displaystyle R_{I}\cong W_{I}\times_{X}W_{I}\setminus\bigcup_{j\not=i\in
I}S_{ij}^{ab},\mbox{ where }S_{ij}^{ab}:=\mbox{ Im
}(V_{i}^{a}\times_{Y}V_{j}^{b}\to W_{i}\times_{X}W_{j}).$
Whenever $J\supseteq I$, the natural morphism between the groupoid
presentations $\left[R_{J}\rightrightarrows V^{a}_{J}\right]$ and
$\left[R_{I}\rightrightarrows V^{a}_{I}\right]$ induces the morphism of stacks
$\phi_{J}^{I}:Y_{J}\to Y_{I}$. In particular,
$\phi_{I}^{I}=\mbox{id}_{Y_{I}}$. The space $Y_{\emptyset}=X$ will be called
the target of the network.
###### Definition 1.11.
Consider a proper local embedding of Noetherian stacks $g:Y\to X$, with $Y$
integral. If $g$ factors through an étale epimorphism $e:Y\to D_{Y/X}$ and a
proper local embedding $g_{1}:D_{Y/X}\to X$ of generic degree 1, then we
define $Y_{I}:=D_{Y/X,I}\times_{D_{Y/X}}Y$, for the network consisting of
$\\{D_{Y/X,I},\varphi_{J}^{I}\\}_{I\subseteq J\not=\emptyset}$ constructed as
in the preceding definition, a target $Y_{\emptyset}=X$ and the morphisms
$g_{i}:Y_{i}\to X$. The morphisms $\phi_{J}^{I}:Y_{J}\to Y_{I}$ are also
obtained by pull-back from the network of $D_{Y/X}$.
###### Remark 1.12.
Even though each space $Y_{I}$ in the network of $g$ and $U$ is intrinsic to
the morphism $g$ ([MM], Corollary 2.8), the network itself depends on the
choice of the suitable cover $U$, inasmuch as the number of copies of the same
space $Y_{I}$ can vary from network to network. For example, if we replace $U$
by a disjoint union of $m$ copies of $U$, where $m$ is a positive integer,
then the network of $g,U$ is replaced by $m$ of its copies, with the exception
of the final target $X$ which is unique. In the next proposition we will show
that there is, however, a canonical choice of a minimal network for the
morphism $g$, which will make the subsequent construction of an étale lift of
$g$ canonical, too.
###### Notation.
Consider now a proper local embedding of Noetherian stacks $g:Y\to X$ of
generic degree one, with $Y$ integral. For every natural number $n$, we denote
by $\prod_{X}^{n}Y$ the fibered product over $X$ of $n$ copies of $Y$. We
denote by $\Delta_{n}$ the union of the images of all diagonal morphisms
$\prod_{X}^{m}Y\to\prod_{X}^{n}Y$ for $m\leq n$, and by $Y^{n}$ the complement
of $\Delta_{n}$ in $\prod_{X}^{n}Y$.
###### Lemma 1.13.
$Y^{n}$ is a closed substack of $\prod_{X}^{n}Y$.
###### Proof.
We only need to check that the image of the diagonal morphism $Y\to
Y\times_{X}Y$ is both open and closed in $Y\times_{X}Y$. Then, by induction on
$n$ we obtain that $\Delta_{n}$ is a union of connected components of
$\prod_{X}^{n}Y$ for any $n>1$. Indeed, since $g:Y\to X$, then so is $Y\to
Y\times_{X}Y$. On the other hand, to prove that the image of this morphism is
open, we choose any cover $U$ of $X$ suitable for the morphism $g$. Let
$V=\bigsqcup_{i}V_{i}:=Y\times_{X}U$, such that each $V_{i}$ is imbedded as a
closed subscheme of $U$. For any indices $i,j$ as above,
$V_{i}\times_{X}V_{j}$ is an étale cover of $Y\times_{X}Y$, and
$(V_{i}\times_{X}V_{j})\times_{Y\times_{X}Y}Y\cong V_{i}\times_{Y}V_{j}$. On
the other hand,
$\displaystyle\bigsqcup_{j}(V_{i}\times_{Y}V_{j})=V_{i}\times_{Y}V\cong
V_{i}\times_{Y}(Y\times_{X}U)\cong
V_{i}\times_{X}U\cong\bigcup_{j}(V_{i}\times_{X}V_{j}),$
and so
$V_{i}\times_{X}V_{j}\cong(V_{i}\times_{Y}V_{j})\bigsqcup(\bigsqcup_{k}(V_{i}\times_{Y}V_{k})\bigcap(V_{i}\times_{X}V_{j}))$.
∎
###### Definition 1.14.
Let $n_{g}$ be the largest integer such that $Y^{n_{g}}$ is non-empty. We
denote by $\mathcal{N}(Y/X)$ the network made out of stacks $Y_{J}:=Y^{|J|}$,
for any $J\subseteq\\{1,...,n_{g}\\}$, and of morphisms $\phi_{J}^{I}:Y_{J}\to
Y_{I}$, defined by restrictions of the natural projections, for $I\subseteq
J$. Here $Y_{\emptyset}=X$ and $\phi_{i}^{\emptyset}=g$ for any
$i\in\\{1,...,n_{g}\\}$. For a general proper local embedding $g$, let
$\mathcal{N}(Y/X):=\mathcal{N}(D_{Y/X}/X)\times_{D_{Y/X}}Y$.
$\mathcal{N}(Y/X)$ will be called the canonical network of the the finite
local embedding $g:Y\to X$.
###### Proposition 1.15.
a) There exists an étale cover $U$ of $X$ suitable for $g$ such that
$\mathcal{N}(Y/X)$ is the network of local embeddings associated to $g,U$.
b) If $X^{\prime}\to X$ is a morphism and $Y^{\prime}\cong
Y\times_{X}X^{\prime}$, then
$\mathcal{N}(Y^{\prime}/X^{\prime})\cong\mathcal{N}(Y/X)\times_{X}X^{\prime}$.
###### Proof.
Consider any étale covering $U^{\prime}$ of $X$ suitable for $g$. At least one
such cover exists, by Proposition 1.9 in [MM]. Let $\phi^{\prime
I^{\prime}}_{J^{\prime}}:Y_{J^{\prime}}\to Y_{I^{\prime}}$ denote the
morphisms in the associated network. By examining the respective groupoid
presentations it can be proven ([MM], Corollary 2.8.) that the spaces
$Y_{J^{\prime}}$ are isomorphic to $Y^{n}$ and, moreover, by the same proof,
the morphisms are restrictions of projections as above. It remains to check
that, after possibly ”pruning” $U^{\prime}$ , the associated network has the
required set of nodes and morphisms. Indeed, assume that
$g(Y)\times_{X}U^{\prime}=\bigcup_{l\in\\{1,...,m\\}}W_{l}$, with $W_{l}$ as
in Definition 1.9. If $m>n_{g}$, let
$U:=U^{\prime}\setminus(\bigcup_{l=n_{g}+1}^{m}W_{l})$. The induced map $U\to
X$ is étale and also surjective, due to the maximality of $n_{g}$ and to
property (2) in Definition 1.9. As $W_{I}\cong W_{I^{\prime}}$ whenever
$|I|=|I^{\prime}|$, then the network associated to $U$ has exactly the right
number of nodes and morphisms as $\mathcal{N}(Y/X)$. The second statement is
due to the definition of $\mathcal{N}(Y/X)$ and Proposition 1.3. ∎
###### Lemma 1.16.
Let $g:Y\to X$ be a finite local embedding of Noetherian stacks, and let
$\mathcal{N}(Y/X)=\\{\phi_{J}^{I}:Y_{J}\to Y_{I}\\}_{I\subseteq
J\subseteq\\{1,...,n_{g}\\}}$ be its canonical network. Then for any integer
$k$ with $0\leq k<n_{g}$, the projection morphism and for any $K\subseteq
L\subseteq\\{1,...,n_{g}\\}$ with $|K|=k$ and $|L|=k+1$, the morphism
$\phi^{K}_{L}:Y_{L}\to Y_{K}$ is a finite local embedding with associated
canonical network
$\displaystyle\mathcal{N}(Y_{L}/Y_{K})=\\{\phi_{J}^{I}:Y_{J}\to
Y_{I}\\}_{K\subseteq I\subseteq J\subseteq\\{1,...,n_{g}\\}}.$
Here by convention $Y_{0}=X$.
###### Proof.
The lemma is due to the existence of canonical isomorphisms
$\prod^{l}_{\prod^{k}_{X}Y}\prod^{k+1}_{X}Y\cong\prod^{l+k}_{X}Y$, which
commute with the projections
$\prod^{l}_{\prod^{k}_{X}Y}\prod^{k+1}_{X}Y\to\prod^{l-1}l_{\prod^{k}_{X}Y}\prod^{k+1}_{X}Y$
and $\prod^{l+k}_{X}Y\to\prod^{l+k-1}_{X}Y$, respectively, and with the
respective diagonal morphisms.
∎
###### Definition 1.17.
Consider a network $\mathcal{N}$ of proper local embeddings
$\phi_{J}^{I}:Y_{J}\to Y_{I}$ for $I\subseteq J\in\mathcal{P}$ associated to a
proper local embedding $g:Y\to X$, where by convention $Y_{\emptyset}=X$. We
will briefly describe here an étale lift $\mathcal{N}^{0}$ of $\mathcal{N}$
which is a configuration stack, namely a network of closed embeddings
$\mathcal{N}^{0}=\\{\phi_{J}^{I0}:Y_{J}^{0}\hookrightarrow Y_{I}^{0}\\}$, and
a morphism $p^{0}:\mathcal{N}^{0}\to\mathcal{N}$ which is étale, in the sense
that each of the constituent morphisms is étale. A detailed proof of the
existence of this network based on étale coverings can be found in [MM],
(Theorem 1.5). $\mathcal{N}^{0}$ is in fact the last of a sequence of networks
$\\{\mathcal{N}^{i}\\}_{n_{g}\geq i\geq 0}$ constructed inductively, where
$\mathcal{N}^{n_{g}}:=\mathcal{N}$, and for each index $i$,
1. (1)
the morphisms $\phi_{J}^{Ii}:Y_{J}^{i}\hookrightarrow Y_{I}^{i}$ are closed
embeddings for all $I,J$ such that $J\supseteq I\in\mathcal{P}_{k}$ with
$k\leq i$;
2. (2)
there is an étale morphism $\mathcal{N}^{i-1}\to\mathcal{N}^{i}$.
The sequence is constructed as follows. Assume that $\mathcal{N}^{i}$ with the
property $(2)$ above has been defined. For each $I\in\mathcal{P}$, we denote
by $S_{I}^{i}$ the stack obtained by gluing all stacks $Y^{i}_{J}$ satisfying
$J\supset I$ like in Lemma 1.8. Each $S_{I}^{i}$ comes with a map
$S_{I}^{i}\to Y_{I}^{i}$ which is in fact proper and étale on its image, and
the set of all these maps defines a map of networks
$\mathcal{S}^{i}\to\mathcal{N}^{i}$. With the notations from Proposition 1.2,
we define $\mathcal{N}^{i-1}:=F_{\mathcal{S}^{i}/\mathcal{N}^{i}}$ in the
obvious sense: each $Y_{I}^{i-1}=F_{S_{I}^{i}/Y_{I}^{i}}$, the étale lift of
$S_{I}^{i}\to Y_{I}^{i}$. Thus there exists a natural étale map
$\mathcal{N}^{i-1}\to\mathcal{N}^{i}$. We note that for $I\in\mathcal{P}_{k}$
with $k\geq i$, the morphism $S_{I}^{i}\to Y_{I}^{i}$ is already a closed
embedding, so $Y_{I}^{i-1}=Y_{I}^{i}$, while for $I\in\mathcal{P}_{i-1}$ and
$J\supseteq I$, we have $Y_{J}^{i-1}=Y_{J}^{i}\hookrightarrow
S_{I}^{i}\hookrightarrow Y_{I}^{i-1}$, a closed embedding.
###### Definition 1.18.
Consider a proper local embedding $g:Y\to X$. Let $\\{Y^{a}\\}_{a}$ denote the
set of irreducible components of $Y$, and for each $a$ let
$\mathcal{N}^{i}(Y^{a}/X)=\\{\phi_{J}^{a,I,i}:Y_{J}^{a,i}\hookrightarrow
Y_{I}^{a,i}\\}_{J\supseteq I;I,J\in\mathcal{P}(\Lambda^{a})}$ denote the
networks associated to the restriction of $g$ on $Y^{a}$, as in the previous
definition. Here $0\leq i\leq|\Lambda^{a}|$, and $|\Lambda^{a}|$ is the
largest number such that $(Y^{a})^{n}$ is nonempty for all
$n\leq|\Lambda^{a}|$. We define the canonical network of the local embedding
$g$ by
$\displaystyle\mathcal{N}(Y/X)=\mathcal{N}^{\sum_{a}|\Lambda^{a}|}(Y/X):=\times_{X}\\{\mathcal{N}(Y^{a}/X)\\}_{a}=\times_{X}\\{\mathcal{N}^{|\Lambda^{a}|}(Y^{a}/X)\\}_{a}.$
The objects of this network are fibered products over $X$ of factors
$Y^{a}_{I^{a}}$ for all $a$, where $I^{a}\subseteq\Lambda^{a}$. All the
networks $\mathcal{N}^{i}(Y/X)$ for $0\leq i\leq|\Lambda^{a}|$ are constructed
inductively by the process outlined in the previous definition.
###### Proposition 1.19.
Consider a proper local embedding $g:Y\to X$, and let $\\{Y^{a}\\}_{a}$ denote
the set of irreducible components of $Y$. With the notations from the previous
definitions,
$\displaystyle\mathcal{N}^{0}(Y/X)\cong\times_{X}\\{\mathcal{N}^{0}(Y^{a}/X)\\}_{a},$
the fiber product over $X$ of all the networks $\mathcal{N}^{0}(Y^{a}/X)$.
###### Proof.
The proof relies on induction after the number of irreducible components, as
well as decreasing induction after the step $i$ in the construction of the
networks $\mathcal{N}^{i}(Y/X)$, and largely on property (10) in Proposition
1.2. The induction after the number of irreducible components reduces to
proving the proposition for $Y=Z\bigcup T$. Denote
$\displaystyle\mathcal{N}(Y/X)=\mathcal{N}^{m+n}(Y/X):=\mathcal{N}(Z/X)\times_{X}\mathcal{N}(T/X)=\mathcal{N}^{m}(Z/X)\times_{X}\mathcal{N}^{n}(T/X),$
and $\mathcal{N}^{i}(Z/X)=\\{\phi_{J}^{I,i}:Z_{J}^{i}\hookrightarrow
Z_{I}^{i}\\}_{J\supseteq I;I,J\in\mathcal{P}(\Lambda)}$, with $|\Lambda|=m$,
while $\mathcal{N}^{j}(Z/X)=\\{\phi_{B}^{A,j}:T_{B}^{j}\hookrightarrow
T_{A}^{j}\\}_{B\supseteq A;A,B\in\mathcal{P}(\Gamma)}$ with $|\Gamma|=n$. Our
induction hypothesis will be that for a fixed $k$ integer, $0\leq k\leq m+n$,
the objects of the network $\mathcal{N}^{k}(Y/X)$ are of the form
* (1)
$Z^{|I|}_{I}\times_{X}T^{|A|}_{A}$, if $|I|+|A|\geq k$, and
* (2)
$F_{S_{I\cup A}^{k+1}/(Z_{I}\times_{X}T_{A})}$ otherwise.
Here the notations are consistent with Definition 1.17, and the index $(k+1)$
refers to the naturally corresponding objects in the $k+1$-th network of
$g:Y\to X$. Thus (2) is a direct consequence of the definition of the objects
in $\mathcal{N}^{k}(Y/X)$, together with property (8) in Proposition 1.2
applied successively to the compositions $S_{I\cup A}^{l+1}\hookrightarrow
S_{I\cup A}^{l}\to F_{S_{I\cup A}^{l+1}/Z_{I}\times_{X}T_{A}}$ for $l\geq
k+1>|I|+|A|$ . Condition (1) is clearly satisfied when $k=m+n$. If satisfied
for a fixed $k$, then for any $I\in\mathcal{P}(\Lambda)$ and
$A\in\mathcal{P}(\Gamma)$ there is a Cartesian diagram
whenever $|I|+|A|=k-1$. Gluing in the network $\mathcal{N}^{k}(Y/X)$ yields
$\displaystyle S_{I\bigcup
A}^{k}=(S^{|I|+1}_{I}\times_{X}T^{|A|}_{A})\bigcup_{S^{|I|+1}_{I}\times_{X}S^{|A|+1}_{A}}(Y^{|I|}_{I}\times_{X}S^{|A|+1}_{A})=$
$\displaystyle=(F_{S^{|I|+1}_{I}\times_{X}S^{|A|+1}_{A}/S^{|I|+1}_{I}\times_{X}T_{A}})\bigcup_{S^{|I|+1}_{I}\times_{X}S^{|A|+1}_{A}}(F_{S^{|I|+1}_{I}\times_{X}S^{|A|+1}_{A}/Y_{I}\times_{X}S^{|A|+1}_{A}}),$
(in accord with property (8), Proposition 1.2). Thus by (2) above and
properties (10), (8) in Proposition 1.2, when $|I|+|A|=k-1$,
$\displaystyle(Y_{I}\times_{X}T_{A})^{k-1}\cong F_{S_{I\bigcup
A}^{k}/Y_{I}\times_{X}T_{A}}\cong$
$\displaystyle\cong(F_{S^{|I|+1}_{I}\times_{X}T_{A}/Y_{I}\times_{X}T_{A}})\times_{Y_{I}\times_{X}T_{A}}(F_{Y_{I}\times_{X}S^{|A|+1}_{A}/Y_{I}\times_{X}T_{A}})\cong$
$\displaystyle\cong
F_{S^{|I|+1}_{I}/Y_{I}}\times_{X}F_{S^{|A|+1}_{A}/T_{A}}\cong
Y^{|I|}_{I}\times_{X}T^{|A|}_{A}.$
(Here $F_{S^{|I|+1}_{I}/Y_{I}}\cong Y^{|I|}_{I}$ due to property (8) in
Proposition 1.2, applied successively to the compositions
$S^{l+1}_{I}\hookrightarrow S^{l}_{I}\to Y^{l}_{I}$ for $l>|I|$.) This ends
the proof of the induction step.
∎
###### Definition 1.20.
Let $g:Y\to X$ be a proper local embedding. For each positive integer $k$, let
$Y^{k}_{X}$ denote the complement of all the diagonals in the $k$-th fibered
product of $Y$ over $X$. Let $n$ be the largest integer such that $Y^{n}_{X}$
is non-empty. We define the functor
$F_{Y/X}:\operatorname{\mathbf{Sch}}_{/X}\to\operatorname{\mathbf{Sets}}$ as
follows: For any scheme $T$ and any morphism $T\to X$ given by an object
$\alpha\in X(T)$, we consider the set of all tuples
$((T_{i},\beta_{i},f_{i})_{i})_{i\in\\{1,...,n\\}}$, where
* (1)
$T_{i}$ are closed subschemes of $T$ such that for $I\subseteq\\{1,...,n\\}$,
the intersections $T_{I}=\bigcap_{i\in I}T_{i}$ (where by convention
$T_{\emptyset}=T$) satisfy
$\displaystyle T_{I}\times_{X}g(Y^{k}_{X})=\bigcup_{J\supseteq
I;|J|=k+|I|}T_{J},$
* (2)
$\beta_{i}\in Y(T_{i})$ are objects whose pullbacks to any of the subsets
$T_{I}$ are pairwise distinct (non-isomorphic), and
* (3)
$f_{i}$ is an isomorphism between $g(\beta_{i})$ and $\alpha_{|T_{i}}$.
###### Theorem 1.21.
Let $g:Y\to X$ be a proper local embedding. The functor $F_{Y/X}$ is a stack.
Moreover, there exists a unique morphism $F_{Y/X}\to X$, étale and universally
closed, with the following properties:
1. (1)
$F_{Y/X}\times_{X}g(Y)\cong S_{Y/X},$ where $S_{Y/X}=S_{\\{1,2,...,n\\}}$ is
the stack constructed by gluing the stacks
$\\{F_{Y_{ij}/Y_{i}}\\}_{i\not=j;i,j\in\\{1,...,n\\}}$ within the network
$\mathcal{N}^{0}(Y/X)$. Furthermore, $F_{Y/X}\setminus S_{Y/X}\cong X\setminus
Y$, and the étale morphism $F_{Y/X}\to X$ is uniquely (up to a unique
isomorphism) defined by these properties.
2. (2)
For each object $Y_{I}$ in $\mathcal{N}(Y/X)$,
$\displaystyle
Y_{I}\times_{X}F_{Y/X}\cong\bigsqcup_{|I_{0}|=|I|}F_{Y_{I_{1}}/Y_{I_{0}}},$
where $I_{1}\supset I_{0}$ is a fixed choice such that $|I_{1}|=|I_{0}|+1$.
3. (3)
If $g:Y\to X$ is a closed embedding, then $F_{Y/X}\cong X$.
4. (4)
If $g:Y\to X$ is étale and proper, and $X$ is connected then $F_{Y/X}\cong Y$.
5. (5)
For any morphism of stacks $u:X^{\prime}\to X$ and
$Y^{\prime}:=Y\times_{X}X^{\prime}$, there exists a morphism
$F_{u}:F_{Y^{\prime}/X^{\prime}}\to F_{Y/X}$ making the squares in the
following diagram Cartesian:
6. (6)
If $h:Z\to Y$ is proper and étale on its image, and $g:Y\to X$ is a closed
embedding, then $F_{Z/Y}\cong Y\times_{X}F_{Z/X}$. In particular, there exists
a natural étale morphism $g_{*}:F_{Z/Y}\to F_{Z/X}$.
7. (7)
If $g:Y\bigcup T\to X$ is a local embedding, then
$\displaystyle F_{Y/X}\times_{X}F_{T/X}\cong F_{Y\bigcup T/X}.$
###### Proof.
With the notations from Definitions 1.14 and 1.17, we will show that
$F_{Y/X}=X^{0}$, the target of the network $\mathcal{N}^{0}(Y/X)$. More
generally, if $I,J\subset\\{1,...,n\\}$ are such that $J\supset I$ and
$|J|=|I|+1$, then we will show by decreasing induction on $I$ that
$F_{Y_{J}/Y_{I}}=Y_{I}^{|I|}$, the target of the network
$\mathcal{N}^{0}(Y_{J}/Y_{I})$. We recall that $\mathcal{N}^{0}(Y_{J}/Y_{I})$
is a subnetwork of $\mathcal{N}^{0}(Y/X)$, due to Lemma 1.16 and Definition
1.17.
As a first step, we notice that when $|I|=n-1$, the functor $F_{Y_{J}/Y_{I}}$,
coincides with the one in Proposition 1.2, and it is thus a stack satisfying
all the required properties. Consider now a general $I$ and assume that
$F_{Y_{K}/Y_{J}}=Y_{J}^{|J|}$ for all $K\supset J\supset I$ with
$|K|=|J|+1=|I|+2$. Consider now a scheme $T_{I}$ with a morphism $T_{I}\to
Y_{I}$ and a set of data
$(T_{I\bigcup\\{i\\}},\beta_{I\bigcup\\{i\\}},f_{I\bigcup\\{i\\}})_{i\not\in
I}$ like in Definition 1.20. Thus
(1.14) $\displaystyle
T_{I}\times_{Y_{I}}\phi_{J}^{I}(Y_{J})\cong\bigcup_{i\not\in
I}T_{I\bigcup\\{i\\}}.$
Moreover, for each $J$ as above, the set of data consisting in
$\displaystyle\beta_{J}\mbox{ together with the
tuple}(T_{J\bigcup\\{i\\}},\phi_{J\bigcup\\{i\\}}^{I\bigcup\\{i\\}*}\beta_{I\bigcup\\{i\\}},\phi_{J\bigcup\\{i\\}}^{I\bigcup\\{i\\}*}f_{I\bigcup\\{i\\}})_{i\not\in
J}$
determine an element in $F_{Y_{K}/Y_{J}}(T)$ and thus by the induction
hypothesis, a morphism $T_{J}\to Y_{J}^{|J|}$, making the following diagrams
commutative:
for all $i\not\in J$. Composition with the closed embeddings
$Y_{J}^{|J|}\hookrightarrow S_{I}^{|J|}$, for $S_{I}^{|J|}$ like in Definition
1.17, yields maps $T_{J}\to S_{I}^{|J|}$ for all $J$ as above, which glue to
$\bigcup_{i\not\in I}T_{I\bigcup\\{i\\}}\to S_{I}^{|J|}$. This, together with
equation (1.14) and Proposition 1.2, insure the existence of a natural
morphism $T_{I}\to Y_{I}^{|I|}$ compatible with the data
$(T_{I\bigcup\\{i\\}},\beta_{I\bigcup\\{i\\}},f_{I\bigcup\\{i\\}})_{i\not\in
I}$. This ends the induction step.
From here, properties (1) and (2), and (6) follow from the construction of the
network $\mathcal{N}(Y/X)$, together with Proposition 1.2. Properties (2) and
(3) are direct consequences of (1). Property (4) also follows the construction
of the network $\mathcal{N}(Y/X)$, together with properties listed in
Proposition 1.15 b), Lemma 1.8, b) and Proposition 1.5, part (1). Property (7)
is a consequence of Proposition 1.19.
∎
###### Example 1.22.
If $g:Y\to X$ is proper and étale on its image, then $F_{Y/X}$ coincides with
the stack defined in Proposition 1.2.
###### Example 1.23.
For a separated Deligne-Mumford stack $X$, the diagonal morphism $\Delta:X\to
X\times X$ is a finite local embedding. Then $X\times_{X\times X}X$ is the
inertia stack $I^{1}(X)$ of $X$, representing objects of $X$ with their
isomorphisms. Similarly, the higher inertia stack $I^{n}(X)$ is defined as the
$n$-th order product of $X$ over $X\times X$. With notations from 1.14, the
objects of the canonical stack of $\Delta:X\to X\times X$ are the components
$I^{n}_{0}(X)$ of the inertia stacks obtained after removing all the previous
components which are images of $I^{k}(X)$ for $k<n$, as well as $X$ itself,
through diagonal morphisms.
## References
* [AGV] Dan Abramovich, Tom Graber, Angelo Vistoli, _Gromov-Witten theory of Deligne-Mumford stacks_ , Amer. Journal of Math., Volume 130, Number 5, October 2008, 1337–1398
* [EJK] Edidin, Dan; Jarvis, Tyler J.; Kimura, Takashi _Logarithmic trace and orbifold products._ Duke Math. J. 153 (2010), no. 3, 427 473.
* [FMcP] Fulton, William; MacPherson, Robert _A compactification of configuration spaces. Ann. of Math._ (2) 139 (1994), no. 1, 183 225.
* [G] Alexandre Grothendieck, _Séminaire de Géométrie Algébrique du Bois Marie I - 1960-61 - Rev tements tales et groupe fondamental - (SGA 1)_ (Lecture notes in mathematics 224). Berlin; New York: Springer-Verlag, xxii+447.
* [K] Andrew Kresch, _Canonical rational equivalence of intersections of divisors_ , Invent. Math. 136 (1999) 483 -496.
* [L] Valery Lunts, _Coherent Sheves on Configuration Schemes_ , Journal of Algebra, Vol. 244, No.2, 379-406, 2001
* [MM] A. Mustata, A. Mustata, _The structure of a local embedding and Chern classes of weighted blow-ups_ , arXiv:0812.3101
* [R] David Rydh, _The canonical embedding of an unramified morphism in an étale morphism_ , arXiv:0910.0056, to appear in Math. Z.
* [S] _Stack Theory and Applications_. Notes taken by H. Clemens at an informal seminar run by A. Bertram, H. Clemens and A. Vistoli; available from www.math.utah.edu/ bertram/lectures.
* [V] Angelo Vistoli, _Intersection theory on algebraic stacks and their moduli spaces_ in Inv. Math. 1989, volume 97, page 613–670
|
arxiv-papers
| 2010-11-06T23:13:00 |
2024-09-04T02:49:14.571982
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Anca Mustata, Andrei Mustata",
"submitter": "Anca Mustata",
"url": "https://arxiv.org/abs/1011.1596"
}
|
1011.1665
|
11institutetext: Leiden Observatory, Leiden University, P.O. Box 9513, 2300
RA Leiden, The Netherlands.
11email: fumagalli@strw.leidenuniv.nl 22institutetext: Dipartimento di Fisica
G. Occhialini, Università di Milano- Bicocca, Piazza della Scienza 3, 20126
Milano, Italy
22email: giuseppe.gavazzi@mib.infn.it 33institutetext: INAF, Osservatorio
Astronomico di Roma, via Frascati 33, 0040 Monte Porzio Catone (RM), Italy
33email: roberto.scaramella@oa-roma.inaf.it 44institutetext: INAF, IASF-
Milano, Via Bassini 15, I-20133, Milano, Italy
44email: paolo@lambrate.inaf.it
# Constraining the ages of the fireballs in the wake of the dIrr galaxy
VCC1217 / IC3418
Mattia Fumagalli 1122 Giuseppe Gavazzi 22 Roberto Scaramella 33 Paolo
Franzetti 44
(Received …; accepted …)
###### Abstract
Context. A complex of H$\alpha$ emitting blobs with strong FUV excess is
associated to the dIrr galaxy VCC1217 / IC3418 (Hester et al. 2010), and
extends up to 17 Kpc in the South-East direction. These outstanding features
can be morphologically divided into diffuse filaments and compact knots, where
most of the star formation activity traced by H$\alpha$ takes place.
Aims. We investigate the properties of the galaxy and the blobs using a
multiwavelength approach in order to constrain their origin.
Methods. We collect publicly available data in UV and H$\alpha$ and observe
the scene in the optical U,g,r,i bands with LBT. The photometric data allows
to evaluate the star formation rate and to perform a SED fitting separately of
the galaxy and the blobs in order to constrain their stellar population age.
Moreover we analyze the color and luminosity profile of the galaxy and its
spectrum to investigate its recent interaction with the Virgo cluster.
Results. Our analysis confirms that the most plausible mechanism for the
formation of the blobs is ram pressure stripping by the Virgo cluster IGM. The
galaxy colors, luminosity profile and SED are consistent with a sudden gas
depletion in the last few hundred Myr. The SED fitting of the blobs constrain
their ages in $<400$ Myr.
###### Key Words.:
Galaxies: clusters: individual: Virgo; Galaxies evolution; Galaxies irregular
## 1 Introduction
Recent studies of the Virgo Cluster (Boselli et al. 2008), Coma Supercluster
(Gavazzi et al. 2010), Perseus Cluster (Penny & Conselice, 2010) and Shapley
Supercluster (Haines et al. 2006) invoke ram pressure stripping (Gunn & Gott
1972) as the responsible process for a significant migration of galaxies from
the Blue Cloud to the Red Sequence, via suppression of the star formation due
to gas ablation of low mass galaxies in regions of high galactic density. The
necessary ingredient of these ”near-field Cosmology” approaches is that
significant infall of low mass star forming objects exists along the
filamentary structures onto the densest clusters. These galaxies have their
star formation truncated in a short timescale due to the interaction with the
IGM.
Observations of stripped gas are frequent in the local Universe. Long narrow
H$\alpha$ tails, stretching up to 150 Kpc, are reported in the Virgo Cluster
(Kenney & Koopmann 1999), Abell 1367 (Gavazzi et al. 2001, Cortese et al.
2006) and Coma Cluster (Yagi et al. 2010) associated to infalling galaxies.
There is however little evidence that star formation ignites in the stripped
wakes, except in a few cases. Cortese et al. (2007) discovers for the first
time a complex of star forming blobs in the trails of two spiral galaxies
belonging to two clusters at z=0.2, and Yoshida et al. (2008) finds a unusual
complex of blue ”fireballs” associated to the Coma galaxy RB 199. These cases
of star formation in the wakes of stripped galaxies are remarkably similar to
the hydrodynamical simulations by Kapferer et al. 2009.
Recently Hester et al. (2010) reports the discovery of a similar object in the
Virgo cluster, associated to the dIrr galaxy VCC 1217 / IC 3418. We have been
independently studying the same system with deep LBT photometry in addition to
public H$\alpha$ and GALEX-UV data, aimed at constraining the ages of the
galaxy and the fireballs via SED fitting.
Through the paper we assume a standard cosmology and a distance module of 31
mag for the Virgo Cluster A corresponding to a distance of 17 Mpc, as in
Gavazzi et al. (1999).
## 2 The data
VCC1217 has been observed by GALEX in March 2004 in the Near UltraViolet (NUV,
1750-2750 $\AA$) and in the Far UltraViolet (FUV, 1350-1750 $\AA$) bands, with
an exposure time of $\approx 4000$ and $\approx 1600$ s
respectively111Significantly shorter than quoted by Hester et al. (2010) who
used additional GALEX observations that are not yet public.. A narrow
H$\alpha$ band image has been taken at the ESO 3.6m telescope in 2004 (see
Gavazzi et al. 2006). Sources with a H$\alpha$ surface brightness higher than
$\sigma_{min}=3.16\cdot 10^{-17}\rm erg/s/cm^{2}/arcsec^{2}$ have been
detected (2$\sigma$ of the background). The galaxy is undetected at 21 cm, as
reported by Hoffman et al. (1989), who used the Arecibo telescope to put a
stringent upper limit of $M_{\odot}\approx 3.46\cdot 10^{6}$ on the HI gas in
the
Figure 1: High contrast RGB picture of VCC 1217 obtained from the LBT images (Uspec, g-SDSS, i-SDSS), highlighting the morphology and the color of the blobs. Table 1: Observation Log Instrument | Filter | Seeing | Date (yy/mm/dd) | Exposure Time
---|---|---|---|---
LBT | U-spec | 1.80 arcsec | 2008-02-01,02 | 25 x 240 sec
| | | 2008-04-03,04 |
LBT | g-SDSS | 1.35 arcsec | 2008-04-03,04 | 28 x 150 sec
| | | 2009-02-22 |
| | | 2009-05-28 |
LBT | r-SDSS | 1.47 arcsec | 2008-02-02 | 32 x 150 sec
| | | 2008-04-03,04 |
| | | 2009-05-28 |
LBT | i-SDSS | 1.29 arcsec | 2009-02-01,02 | 34 x 240 sec
| | | 2008-04-03,04 |
ESO 3.6 | r Gunn | 1.13 arcsec | 2005-04-21 | 240 sec
ESO 3.6 | 692 | 1.13 arcsec | 2005-04-21 | 1800 sec
GALEX | NUV | 4.95 arcsec | 2004-03-11 | 1597 + 1161 +1690 sec
GALEX | FUV | 4.05 arcsec | 2004-03-11 | 1597 sec
ESO 3.6 | EFOSC spectrometer | | 2002-03-17 | 2400 sec
a. c.
b. d.
Figure 2: (a) Low contrast RGB picture highlighting the galaxy structure and
colors. (b) u-i image of the galaxy obtained from the ratio of the u and i
images, each thresholded above $2\sigma$ of the sky (outer black regions). In
regions above the threshold the grey scale goes from white (u-i=1.5) to black
(u-i=1.9) with increasing u-i index. (c) Superposed to the H$\alpha$ image,
the regions on which the photometry of the individual blobs has been evaluated
and the concentric elliptical rings used to obtain the color profile of the
galaxy (see Figure 4) (d) NUV contours superposed to a low contrast g image
Table 2: Photometry of VCC1217 and its associated blobs.
ID | Distance | u | g | r | i | NUV | FUV
---|---|---|---|---|---|---|---
| (arcmin) | (mag) | (mag) | (mag) | (mag) | (mag) | (mag)
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8)
GAL | 0 | 15.98 $\pm$ 0.03 | 14.74 $\pm$ 0.02 | 14.39 $\pm$ 0.02 | 14.19 $\pm$ 0.02 | 17.36 $\pm$ 0.022 | 18.14 $\pm$ 0.06
K8 | 1.13 | 20.79 $\pm$ 0.12 | 20.42 $\pm$ 0.03 | 20.41 $\pm$ 0.04 | 20.16 $\pm$ 0.05 | 21.05 $\pm$ 0.11 | 20.92 $\pm$ 0.19
F4 | 1.28 | 21.36 $\pm$ 0.16 | 21.20 $\pm$ 0.04 | 21.17 $\pm$ 0.05 | 21.01 $\pm$ 0.08 | 21.63 $\pm$ 0.15 | 22.03 $\pm$ 0.29
K7 | 1.83 | 21.35 $\pm$ 0.16 | 21.31 $\pm$ 0.04 | 21.71 $\pm$ 0.06 | 21.98 $\pm$ 0.11 | 21.39 $\pm$ 0.13 | 21.27 $\pm$ 0.21
F3 | 2.14 | 20.92 $\pm$ 0.13 | 20.76 $\pm$ 0.04 | 20.49 $\pm$ 0.04 | 20.43 $\pm$ 0.06 | 21.52 $\pm$ 0.14 | 21.86 $\pm$ 0.27
K6 | 2.39 | 20.83 $\pm$ 0.13 | 20.94 $\pm$ 0.04 | 20.74 $\pm$ 0.04 | 20.96 $\pm$ 0.07 | 21.23 $\pm$ 0.12 | 21.28 $\pm$ 0.22
F2 | 2.40 | 21.62 $\pm$ 0.18 | 21.04 $\pm$ 0.04 | 21.07 $\pm$ 0.05 | 20.91 $\pm$ 0.07 | 21.75 $\pm$ 0.15 | 21.69 $\pm$ 0.26
K5 | 2.55 | 20.84 $\pm$ 0.13 | 20.78 $\pm$ 0.04 | 20.82 $\pm$ 0.04 | 20.90 $\pm$ 0.07 | 20.90 $\pm$ 0.11 | 20.80 $\pm$ 0.18
F1 | 2.74 | 21.00 $\pm$ 0.14 | 20.58 $\pm$ 0.03 | 20.67 $\pm$ 0.04 | 20.65 $\pm$ 0.07 | 21.25 $\pm$ 0.12 | 21.41 $\pm$ 0.23
K4 | 3.00 | 20.94 $\pm$ 0.13 | 21.04 $\pm$ 0.04 | 20.64 $\pm$ 0.04 | 20.65 $\pm$ 0.06 | 21.31 $\pm$ 0.13 | 21.17 $\pm$ 0.21
K2 | 3.16 | 22.51 $\pm$ 0.26 | 22.83 $\pm$ 0.08 | 22.67 $\pm$ 0.09 | 23.22 $\pm$ 0.20 | 23.13 $\pm$ 0.27 | 22.92 $\pm$ 0.42
K3 | 3.30 | 21.93 $\pm$ 0.20 | 22.21 $\pm$ 0.06 | 22.03 $\pm$ 0.07 | 22.43 $\pm$ 0.14 | 22.19 $\pm$ 0.19 | 21.84 $\pm$ 0.27
K1 | 3.55 | 22.41 $\pm$ 0.25 | 22.44 $\pm$ 0.07 | 22.45 $\pm$ 0.08 | 22.6 $\pm$ 0.15 | 23.01 $\pm$ 0.26 | 22.37 $\pm$ 0.34
(1) GAL = VCC1217, K-# Knots, F-# Filaments (2) Projected distance from the
center of VCC1217 (3) to (8) The photometric uncertainties are quadratic sum
of the ZP error and Poisson error
galaxy. Optical observations have been obtained at the LBT in different nights
in the first semester of 2008, using the prime focus LBC cameras
(http://lbc.mporzio.astro.it/), with 4 filters: Uspec and g,r,i in the SDSS
system. The seeing ranged between 1.4 and 2 arcsec. Data has been reduced
using the LBC standard pipeline by the LBC support team and final images are
typically composed by stacks of $<$ 20 dithers.
The photometric calibration was performed cross-correlating the flux of 27
stars in the field with those in the SDSS, resulting in a zero points with an
error of less than 0.02 mags. A summary of observations is found in Table 1.
Magnitudes are given in the AB system throughout the paper.
## 3 The Galaxy
### 3.1 Morphology and Photometry
VCC 1217 is a dIrr (Nilson et al. 1973) low surface brightness galaxy located
near the center of the Virgo Cluster, approximately 1 degree South of M87
(0.31 Mpc projected distance) and with a redshift of 38 km/s, $\sim$ 1000 km/s
lower than the mean redshift of Virgo Cluster A. At this location the emission
from the hot IGM is at its peak intensity (Boehringer et al. 1994) and the
density of IGM is $\rho=10^{-27}g/cm^{3}$ (Schindler et al. 1999). The
GOLDMine database (Gavazzi et al. 2003) reports for VCC 1217 an infrared
luminosity of $H_{lum}=8.70\cdot 10^{8}\rm L_{\odot}$ and a color $NUV-H=3.02$
mag. With this characteristics VCC 1217 lies in the blue sequence of the Virgo
Cluster (see Figure 3).
VCC 1217 consists of a low surface brightness disk, whose structure is
contaminated by several foreground stars. The central brightest envelope,
shaped as an elongated ellipse, contains an excess of light in the North
Eastern part of the object (Figure 2a): this is the brightest region of the
galaxy and we will center all our subsequent radial analysis there. A
secondary peak is found more South West, offset by 15 arcsec.
Figure 3: NUV-H Color Luminosity diagram of the Virgo Cluster (from the
GOLDMine database, Gavazzi et al. 2003). Objects are color coded according to
morphological classification: Red for Early Type Galaxies (Elliptical and S0),
Blue for disk dominated Late Type Galaxies (Sc to Sd), Green for Disk+Bulge
Galaxies (Sa to Sb). Dashed lines represent the best fit to the Red and Blue
sequences. The position of VCC1217 is highlighted. Figure 4: NUV-g (top), u-g
(centre), u-i (bottom) color profile of VCC1217, obtained in the concentric
annuli of Figure 2c. A significant positive color gradient is evident in all
bands with increasing distance from the center up to 60 arcsec where
contamination by the blue blobs is null.
Figure 5: Spectrum of VCC1217 obtained in drift scan mode at ESO 3.6m
telescope. The flux is normalized to F(5500 $\AA$). Dashed lines highlight the
position of the Balmer series.
The surface brightness profile has been evaluated (after the masking the four
most luminous foreground stars) using a modified version of the ellipse task
in IRAF, centering the ellipses on the brightest spot, and fitting the profile
with an exponential law, with a scale length of 19.1 arcsec.
As revealed by both the color map u-i (Figure 2b) and the RGB image (Figure
2a), the brightest spot has a blue color of $u-i=1.5$ (see Figure 4), the
secondary peak in the South West of the galaxy is redder ($U-i=1.87$). We have
performed a radial color analysis, integrating the u, g, i and NUV images on
12 elliptical annuli with major axis of 1 arcmin, an axis ratio of 1.5 and an
inclination of the major axis of 50 degrees clockwise. Again we have masked
the 4 more luminous stars superposed to the disk. The color profiles (Figure
4) show a gradient of increasing color index by 0.4 mag with increasing
distance from the center to 1 arcmin.
### 3.2 Spectroscopy
A spectrum of the galaxy has been published by Gavazzi et al. (2004) and it is
publicly available through GOLDMine (Gavazzi et al. 2003). It has been
obtained at the ESO 3.6m telescope in drift scan mode, i.e. with the slit
sliding over the whole galaxy area, thus representing the mean spectral
characteristics of the object. We have smoothed the spectrum by 5 $\AA$
(Figure 5) and measured the equivalent widths of the Balmer lines which result
all stronger than 5 $\AA$ in absorption (as reported in Table 3), with no
emission lines. In particular, the H$\delta$ line has an EW=13.4$\AA$
(adopting the convention that positive EW mean absorption), i.e. stronger than
the threshold in the diagnostic diagrams of k+a galaxies (e.g. Poggianti et
al. 2004, Dressler et al 1999). k+a galaxies are interpreted to be Post Star-
Burst (PSB) galaxies that underwent a sudden truncation of the star formation
in the past 0.5-1.5 Gyr (Couch & Sharples 1987).
Table 3: Spectroscopy of VCC1217 | Wavelength | Continuum | EW
---|---|---|---
| $\AA$ | F/F(5500$\AA$) |
H$\epsilon$ | 3969.1 | 1.02 | 8.0
H$\delta$ | 4109.7 | 1.18 | 13.4
H$\gamma$ | 4339.8 | 1.06 | 7.4
H$\beta$ | 4861.0 | 1.07 | 9.0
H$\alpha$ | 6568.3 | 0.78 | 6.2
## 4 The Fireballs
A complex of faint blue knots and filaments extends from the galaxy in South
East direction, up to 3.5 arcmin (17 Kpc). They are outstanding in both the
GALEX data (Figure 2d) and in the RGB image prepared with the u, g, i LBT
images (Figure 1, where the galaxy is saturated). We identify the brightest
and clumpiest structures as ”knots” (enumerating them from K-1 to K-8 East to
West). Other structures with a lower surface brightness and visually more
diffuse are labeled as ”filaments”, from F-1 to F-4.
Regions (highlighted in Figure 1 and 2a, c) were selected on the basis of
their detection on the GALEX images (with a resolution $>$ 5 arcsec), even
though the LBT images offer a better resolution. This choice is dictated by
the fact that we wanted to obtain for each feature the set of photometric
measurements over the full spectral range, from UV to i-band, necessary for a
SED fitting analysis (see Section 5), For instance, the region K-5 appears to
be a bright knot in the NUV image (Figure 2 d), while it is resolved into two
distinct blobs by the LBT. The same holds for filaments (see for instance
F-3), which might consist of fainter knots connected by diffuse regions.
Although our regions do not exactly coincide with the ones in Hester et al.
(2010)222The rationale for the small discrepancy between our regions and the
ones in Hester et al. (2010), beside nomenclature, is twofold: 1. our analysis
initiated before the appearence of Hester et al. (2010); 2\. we positioned the
regions on the basis of the NUV detection, but the fine tuning and the
division into knots and filaments was aided by our high resolution LBT image,
the photometry in the FUV and NUV bands shows a general consistence.
Figure 6: Colors of the knots (blue) and filaments (red)
Figure 6 shows the fireballs in the two color differences FUV-NUV and u-g. The
filaments appear marginally redder than the knots, having a mean color
$\rm<FUV-NUV>_{F}=0.21$ and $\rm<u-g>_{F}=0.31$, while knots have $\rm<FUV-
NUV>_{K}=-0.21$ and $\rm<u-g>_{K}=-0.05$. Only one knot is as red as the
filaments (K-6 in FUV-NUV, K-8 in u-g). Note that all the structures in the
wake are much bluer than the galaxy (FUV-NUV=0.66 and u-g=1.4). Moreover there
is a slight dependence of the color of knots/filaments on the distance from
the galaxy, i.e. the ones located farther away are $\approx 0.5$ mags bluer
than the closest ones.
Various blobs display an H$\alpha$ emission, as shown in Figure 2c and in
Table 4, with luminosities ranging from $8\cdot 10^{36}\rm
erg\phantom{x}s^{-1}\rm$ (K-1) to $8\cdot 10^{37}\rm erg\phantom{x}s^{-1}\rm$
(K-6), consistently with the faint end of the HII luminosity function
(Kennicutt et al. 1989). The signal-to-noise ratio of the image (see Section
2) is such that every star forming region with a flux larger than $2.51\cdot
10^{-16}\rm erg\phantom{x}s^{-1}cm^{-2}str^{-1}$ can be detected (integrating
2$\sigma$ counts of the sky on the typical dimension of a blob, 10 arcsec2).
The H$\alpha$ emission is concentrated in the knots farther than 2.3 Kpc from
the galaxy and only in one filament (F-3), which was said to consist of
smaller knots. Consistently with the case of RB199 (Yoshida et al. 2008) the
star formation resides in the most compact regions in the wake. According to
Kennicutt (1998), the ongoing SFR is evaluated to be of the order of
$10^{-3/-4}M_{\odot}/yr$ in each blob (see Table 4), with a cumulative SF in
the entire wake of $\approx 1.9\cdot 10^{-3}M_{\odot}/yr$.
## 5 SED Fitting
In order to reconstruct the star formation history of the blobs and of the
galaxy, we generate spectral evolution models with the PEGASE2.0 code (Fioc &
Rocca-Volmerange 1997) and perform a SED-fitting using GOSSIP (Franzetti et
al. 2008). For both the galaxy and the filaments/knots, which are analyzed
separately, the procedure consists in:
* •
Giving as an input to PEGASE one or more Star Formation Histories, i.e. SFR(t)
* •
Generating synthetic spectra at fixed times
* •
Running in GOSSIP the SED-fitting between the synthetic spectra and the
photometric points of the objects
### 5.1 The Galaxy
We model the spectral evolution of the galaxy assuming a Salpeter IMF with an
upper mass of $120M_{\odot}$, a null initial metallicity and a star formation
history ’a la Sandage’ (as reported in Gavazzi et al. 2003), with a discrete
set of $\tau$ parameters. Since we want to test the hypothesis of ram pressure
stripping on the galaxy, in this phase we simulate the effect of the gas
depletion including in the models a truncation of the star formation at a
given characteristic time (Truncation Time, $t_{trunc}$). The Star Formation
History becomes:
$\rm
SFR(t)=\left\\{\begin{array}[]{rl}\frac{t}{\tau^{2}}e^{-\frac{t^{2}}{\tau^{2}}}&\mbox{
if $t<t_{trunc}$}\\\ 0&\mbox{ if $t>t_{trunc}$}\end{array}\right.$
Figure 7: A sample SFH from the library, with $\tau$=4 Gyr, $t_{age}$= 8 Gyr
and $t_{trunc}$ = 5 Gyr
In Figure 7 we show a sample Star Formation History in the models library and
highlight the different timescales referred to in the article: $t_{age}$ is
the time from the onset of star formation to now, $t_{trunc}$ the time from
the formation until the end of star formation activity, $t_{age}$-$t_{trunc}$
the period between the previous two. The grid of parameters is built with
$\tau$ ranging from 1 to 20 Gyr with 1 Gyr step and $t_{trunc}$ from 1 to 13
Gyr with 1 Gyr step, while $t_{age}$ spans from 0 to 13.5 Gyr with an step of
100 Myr, for a total of 35K spectra. We remark that in the models building we
don’t include any fixed age for the start of star formation activity. We don’t
include any correction for dust extinction, since for low mass galaxies it is
negligible (see Figure 8 of Cortese et al. 2008 and 3-4 of Lee et al. 2009).
We run GOSSIP and evaluate the parameters and their probability distribution
functions (PDFs). We obtain that the $t_{age}$, the $t_{trunc}$ and $\tau$ are
not well costrained. However, we compute the probability distribution function
of $t_{age}-t_{trunc}$, representing the lookback time at which the truncation
of star formation occurred, this parameter results very well constrained in
$t_{age}-t_{trunc}=200^{+90}_{-90}\rm Myr$. A halting of the star formation
approximately 200 Myr ago is in accordance with the PSB signature in the
galaxy spectrum (the PDF is given in Figure 8). The stellar mass of the galaxy
evaluated from the normalization to the fit ($M_{star}=3\cdot
10^{8}M_{\odot}$) turns out to be in fair agreement with the stellar mass
evaluation from the optical data (Bell et al. 2007), $M_{star}=3.8\cdot
10^{8}M_{\odot}$.
Figure 8: Probability Distribution Functions of the $t_{age}-t_{trunc}$ parameter for VCC1217. Table 4: Analysis: (1) Distance from the galaxy in arcmin (2) H$\alpha$ Flux in units of 10-16 erg/s/cm2 (3) Star Formation rate from the Kennicutt law, in units of 10-3 M⊙ / yr (4) Mass computed from the normalization to the SED fitting, in units of 105 M⊙ (5) Age (in Myr) for the best fit of a model with an exponential star formation rate. The error is computed from the probability distribution function. Name | Distance | H$\alpha$ Flux | SFR | Mass | Age
---|---|---|---|---|---
| (1) | (2) | (3) | (4) | (5)
K8 | 1.13 | - | - | 3.74 | 130${}^{+454}_{-150}$
F4 | 1.28 | - | - | 2.45 | 620${}^{+160}_{-90}$
K7 | 1.83 | - | - | 0.93 | 80${}^{+23}_{-23}$
F3 | 2.14 | 7.76 | 0.21 | 5.49 | 1400${}^{+350}_{-150}$
K6 | 2.39 | 0.25 | 0.69 | 2.03 | 390${}^{+87}_{-87}$
F2 | 2.40 | - | - | 2.20 | 780${}^{+192}_{-186}$
K5 | 2.55 | 6.92 | 0.19 | 1.18 | 170${}^{+39}_{-39}$
F1 | 2.74 | - | | 3.48 | 640${}^{+248}_{-144}$
K4 | 3.00 | 0.16 | 0.10 | 3.79 | 740${}^{+122}_{-122}$
K2 | 3.16 | 6.22 | 0.16 | 0.35 | 330${}^{+123}_{-123}$
K3 | 3.30 | 8.31 | 0.23 | 0.55 | 140${}^{+52}_{-52}$
K1 | 3.55 | 2.34 | 0.06 | 0.40 | 260${}^{+146}_{-146}$
### 5.2 The Fireballs
For the fireballs we compute with PEGASE2.0 a sample of synthetic spectra
assuming a Salpeter IMF, a subsolar initial metallicity and the following set
of SFH models:
* •
Single burst
$\rm SFR(t)=\delta(0)$
* •
Constant Star Formation rate
$\rm SFR(t)=SFR_{0}$
* •
Exponential decrement of SFR rate
$\rm SFR(t)=SFR_{0}\cdot exp(-t/\tau)$
with a finite set of $\tau$ parameters ($\tau=$ 10, 20, 40, 60, 80, 100, 200,
300, 400, 500, 600, 700, 800, 900, 1000, 1200, 1400, 1600, 1800, 2000 Myr).
Again we run the SED fitting with GOSSIP, and extract the best fits (Table 4)
and evaluate their PDFs. The stellar masses of the fireballs (derived from the
normalization to the SED) range from $3.9\cdot 10^{4}$ to $5.0\cdot 10^{5}$
M⊙, which are typical dimensions of a Giant Molecular Cloud / HII region
(Kennicutt et al. 1989). For such small objects we neglect the internal
absorption.
We want to stress that we prefer not to improve the quality of the fit by
replacement of our SFHs models (with at most one free parameter) with
additional ad-hoc bursts.
Table 4 contains the parameters of the SEDs. Most of the knots SEDs are well
described by a simple exponential SFH with age $\rm t<400$ Myr (excluding K4).
We refer to Age as the time from the ingnition of the star formation to now.
Filaments appear to be all together older than blobs ($<Age>_{F}=850$ Myr)
with star formation activity stretched over a longer period of time.
While the age parameter is well constrained, as shown by the spiky shape of
the PDFs in Figure 9 (left panels), not much can be said about $\tau$,
generally just a lower limit. From the PDFs of $\tau$ we can also conclude
that a Single Burst Model (which corresponds to an exponential model with
$\tau\rightarrow 0$) can be ruled out, while a model with a constant star
formation rate (which corresponds to an exponential model with
$\tau\rightarrow\infty$) is consistent with the data, especially for the young
knots. The fact that for the knots $\tau$ is longer than the age indicates
that star formation activity is still ongoing, in accordance with the
H$\alpha$ emission in most of the knots.
For such young objects it is impossible to constrain more precisely the star
formation history by simply letting $\tau$ vary. This fact is illustrated in a
color-color diagram (Figure 10) where evolutionary tracks obtained with
various $\tau$ are plotted together with the data. At young ages different
tracks are indistinguishable until the stellar population is more evolved
(compare with Fig. 13-14 of Yoshida et. al. 2008: the fireballs associated
with RB199 are older than the ones in the present study). Figure 11 collects
some SEDs of the blobs and the galaxy, showing the quality of the fits. The
features in the wake of VCC 1217 look all very similar, and extremely
different from the galaxy.
Figure 9: Probability Distribution Functions of the Age and $\tau$ parameters
for a sample knot and a sample blob. Figure 10: Color-Color diagram of FUV-
NUV and g-i for the blobs in the wake of VCC1217. The solid lines are the
predictions of some SFH models: constant star formation rate and exponential
star formation rate with $\tau$ 100, 200, 500 and 1000 Myr, color coded as in
the legend. Figure 11: Photometric points for the blobs (blue) and their best
fit SEDs. Photometric points (red), observed optical spectrum (green) for the
galaxy VCC 1217 and best fit model with a truncated Sandage SFH. The dashed
line represents for comparison the spectrum of an old quiescent elliptical
galaxy with a Sandage SFH with parameters Age = 13 Gyr and $\tau=4$ Gyr
## 6 Ram pressure stripping
Adopting the classical Gunn & Gott (1972) criterion for ram pressure :
$\rho_{IGM}v^{2}\geq 2\pi G\Sigma_{star}\Sigma_{gas}$ (1)
where $\rho_{IGM}$ is the intracluster density, $v$ the infall velocity and
$\Sigma_{star,gas}$ the density of the star and gas components in the galaxy,
and assuming an exponential profile for the stellar and gas component, the
radius at which ram pressure becomes efficient can be estimated as (Domainko
et al, 2006):
$R_{strip}=0.5R_{0}ln\left(\frac{GM_{star}M_{gas}}{v^{2}\rho_{IGM}2\pi
R_{0}^{4}}\right)$ (2)
and the stripped mass as:
$M_{strip}=M_{gas}\left(\frac{R_{strip}}{R_{0}}+1\right)exp\left(-\frac{R_{strip}}{R_{0}}\right)$
(3)
Adopting $M_{star}=3.8\cdot 10^{8}M_{\odot}$, $M_{star}/M_{gas}\approx 1$ (for
late type galaxies, e.g. Boselli 2002), $\rho=6\cdot 10^{-4}cm^{-3}$ for the
IGM density of the Virgo cluster at the projected distance of 0.3Mpc from M87
(Schindler et al. 1999), an infall velocity of 1000 km/s, and the typical
scale length computed in Section 3.1, Using these parameters we obtain that
VCC 1217 is unable to retain its gas at any radius if subjected to ram
pressure stripping ($R_{strip}=0.0Kpc$) and therefore it results totally
depleted of gas ($M_{strip}=M_{gas}$).
## 7 Discussion
Although looking just at UV data333See
``http://www.galex.caltech.edu/media/images/—
``glx2010-02f_img01.jpg— it can’t be excluded that the system of blobs is the
remnant of a dwarf irregular galaxy in some stage of merging with VCC1217, the
LBT data resolves the tail feature into separated compact (at most
filamentary) blobs, revealing that the morphology of the system is not
consistent with the merging scenario. Several aspects of our analysis indicate
that the ram pressure stripping picture is the most favorable one (in
agreement with Hester and al. 2010), suggesting that VCC1217 has been recently
stripped by the interaction with the Virgo Cluster IGM. The color profiles,
the spectroscopy and SED fitting of the galaxy all support the scenario
consisting in a truncation of the star formation in the last few hundreds Myr.
The ensemble of blue knots and filaments stretching more than 17 Kpc South
East of the galaxy is a remarkable feature. Their SEDs are consistent with
very young stellar objects, born in the last few hundreds Myr, consistent with
the timing of the gas depletion from the galaxy.
Similar objects have been observed in other clusters, but the phenomenon
appears to be rare. Cortese et al (2007) report the discovery of two complexes
of stellar tails and blue bright blobs associated with two spiral galaxies
infalling in massive clusters at $z\approx 0.2$. Yoshida et al (2008) analyze
a complex of H${\alpha}$ emitting fireballs extending from the Coma cluster
galaxy RB199, up to 80 Kpc. Like VCC1217, also RB199 and 131124-012040 in
Abell 1689 have k+a spectra, while 235144-260358 in Abell 2667 is still a star
forming galaxy. Notice however the different scale between the objects in this
study and the ones in the literature, both for the galaxies and the blobs.
RB199 is estimated to have a mass of 3-4 $\cdot 10^{9}M_{\odot}$, i.e. 10
times more massive than the one of VCC 1217. Yoshida et al. compute a typical
mass of the fireballs associated to RB199 of $\approx 10^{7-8}M_{\odot}$ and a
total $L_{H\alpha}=2\cdot 10^{39}\rm erg\phantom{x}s^{-1}$, while the blobs in
the present study range from 104.6 to 105.7 M⊙ in mass and have a total
H$\alpha$ luminosity of $2\cdot 10^{38}\rm erg\phantom{x}s^{-1}$. Also the
computed ages of the blobs are different, 500-1000 Myr for the complex
stretching from RB199 and less than 400 Myr for most of the ones in the
complex South West of VCC 1217.
The complex in the present study appears in conclusion to be a scaled down
version of the one in Yoshida et al. (2008) because all its characteristic
dimensions (galaxy mass, blobs masses, H$\alpha$ luminosities) are
approximately 10 times smaller than the ones in RB199.
Besides the limited number statistics, this occurance might arise because the
wake associated to VCC 1217 would result too faint to be seen at the distance
of Coma, but also because in a lower density environment, such as Virgo
compared to Coma, low mass galaxies are primarily affected by ram pressure
(Bekki, 2009).
We note that this kind of events is extremely rare in the Universe, but the
phenomenon happens at a variety of mass scales. Different simulations (e.g.
Tommesen et al 2010, Kapferer et al. 2009) have studied the impact of ram
pressure in the distribution of gas in a galaxy, producing mock observations
in HI, H$\alpha$ (and X-rays) that are similar to the observed tails. Various
mechanisms are proposed for the H$\alpha$ emission in the wakes. Kenney et al.
(2008) suggest that H$\alpha$ emission in the wake can be caused by thermal
conduction from the IGM and turbulent shock heating. In the current case
however it is more likely to be associated with star formation, since the
blobs are seen also optically and have the SEDs typical of young stars
complexes.
The simulations by Kapferer et al. (2009) conclude that turbulence in the wake
can bring to gravitational instability and to the formation of stars up to 100
Kpc behind the stripped galaxy. The observed pictures of VCC1217 (Fig.2 a,b,c)
are remarkably similar to the mock observations (see Figures 9-12 in Kapferer
et al. 2009). We can’t compare directly the amount of new stars formed in the
wake since the simulations have been run assuming for the test galaxy a
stellar mass of $2\cdot 10^{10}M_{\odot}$ and a dark matter halo of
$10^{12}M_{\odot}$, while VCC1217 is significantly smaller and the dependence
on mass of the combined effect of ram pressure, turbulence and gravitational
instability has to be investigated further.
We quantify that after approximately 200 Myr from the stripping event the
amount of new stars formed in the wake is $\approx 2.5\cdot 10^{6}\rm
M_{\odot}$ (i.e. the sum of the masses of the blobs, see Table 4), 1/100 of
the whole mass of the stripped galaxy. The non-detection in HI (Hoffman et al.
1989) is not surprising: assuming a residual gas mass similar to the mass of
the blobs, it lies under the Arecibo detection limit.
## 8 Conclusion
We propose that VCC 1217 has just interacted for the first time with the Virgo
cluster, and underwent a sudden truncation of its gas content due to ram
pressure stripping. Turbulence in the wake can bring to gravitational
instability and the densest parts of the wake into collapse, with subsequent
birth of stars. The analysis by Hester et al. (2010) is confirmed and
reinforced by the spectroscopic inspection of the galaxy and more notably by
the deep LBT imaging.
It is already known that in the first stage of the interaction with the IGM a
galaxy can form a tail of ionized gas stretching up to $>50$ Kpc. Examples of
H$\alpha$ tails can be found in the Virgo Cluster (NGC 4522, Kenney & Koopmann
1999) and in Abell 1367 (97-079 and 97-073, Gavazzi et al. 2001; the BIG
group, Cortese et al. 2006). A recent survey of the Coma cluster by Yagi et
al. (2010) shows that almost all blue galaxies in the core of this cluster
reveal tails and distorted H$\alpha$ profiles when observed with a 10-m class
telescope.
In spite of the high frequency of cometary H$\alpha$ structures associated
with IGM-interacting galaxies, the inset of star formation in the wakes is
less common and not fully understood. For instance within the Yagi et al.
(2010) sample only two objects show signs of star formation along the trail
(GMP3016 and RB199). VCC 1217 represents so far the closest and smallest known
object with this unusual feature.
## 9 Acknowledgments
We thank Alessandro Boselli and Luca Cortese for the useful discussions and
the LBT Survey Center (LBC) for carrying out the observation and for technical
support during the reduction and analysis. This research has made use of the
GOLDMine Database. We acknowledge the anonymous referee for constructive
criticism.
## References
* Bekki (2009) Bekki, K. 2009, MNRAS, 399, 2221
* Bell et al. (2003) Bell, E. F., McIntosh, D. H., Katz, N., & Weinberg, M. D. 2003, ApJS, 149, 289
* Böhringer et al. (1994) Böhringer, H., Briel, U.G., Schwarz, R.A., Voges, W., Hartner, G., & Trumper, J. 1994, Nature, 368, 828
* Boselli (2001) Boselli, A. 2001, Astrophysics and Space Science Supplement, 277, 40
* Boselli & Gavazzi (2006) Boselli, A., & Gavazzi, G. 2006, PASP, 118, 517
* Boselli et al. (2008) Boselli, A., Boissier, S., Cortese, L., & Gavazzi, G. 2008, ApJ, 674, 742
* Couch & Sharples (1987) Couch, W. J., & Sharples, R. M. 1987, MNRAS, 229, 423
* Cortese et al. (2006) Cortese, L., Gavazzi, G., Boselli, A., Franzetti, P., Kennicutt, R. C., O’Neil, K., & Sakai, S. 2006,aap, 453,
* Cortese et al. (2007) Cortese, L., et al. 2007, MNRAS, 376, 157
* Cortese et al. (2008) Cortese, L., Boselli, A., Franzetti, P., Decarli, R., Gavazzi, G., Boissier, S., & Buat, V. 2008, MNRAS, 386, 1157
* Domainko et al. (2006) Domainko, W., et al. 2006, A&A, 452, 795
* Draine & Lee (1984) Draine, B. T., & Lee, H. M. 1984, ApJ, 285, 89
* Dressler et al. (1999) Dressler, A., Smail, I., Poggianti, B. M., Butcher, H., Couch, W. J., Ellis, R. S., & Oemler, A. J. 1999, ApJS, 122, 51
* Fioc & Rocca-Volmerange (1997) Fioc, M., & Rocca-Volmerange, B. 1997, A&A, 326, 950
* Franzetti et al. (2008) Franzetti, P., Scodeggio, M., Garilli, B., Fumana, M., & Paioro, L. 2008, Astronomical Data Analysis Software and Systems XVII, 394, 642
* Gavazzi et al. (1996) Gavazzi, G., Pierini, D., & Boselli, A. 1996, A&A, 312, 397
* Gavazzi et al. (1999) Gavazzi, G., Boselli, A., Scodeggio, M., Pierini, D., & Belsole, E. 1999, MNRAS, 304, 595
* Gavazzi et al. (2001) Gavazzi, G., Boselli, A., Mayer, L., Iglesias-Paramo, J., Vílchez, J. M., & Carrasco, L. 2001, ApJ, 563, L23
* Gavazzi et al. (2003) Gavazzi, G., Boselli, A., Donati, A., Franzetti, P., & Scodeggio, M. 2003, A&A, 400, 451
* Gavazzi et al. (2004) Gavazzi, G., Zaccardo, A., Sanvito, G., Boselli, A., & Bonfanti, C. 2004, A&A, 417, 499
* Gavazzi et al. (2006) Gavazzi, G., Boselli, A., Cortese, L., Arosio, I., Gallazzi, A., Pedotti, P., & Carrasco, L. 2006, A&A, 446, 839
* Gavazzi et al. (2010) Gavazzi, G., Fumagalli, M., Cucciati, O., & Boselli, A. 2010, arXiv:1003.3795
* Gunn & Gott (1972) Gunn, J. E., & Gott, J. R., III 1972, ApJ, 176, 1
* Haines et al. (2006) Haines, C. P., Merluzzi, P., Mercurio, A., Gargiulo, A., Krusanova, N., Busarello, G., La Barbera, F., & Capaccioli, M. 2006, MNRAS, 371, 55
* Hester et al. (2010) Hester, J. A., et al. 2010, ApJ, 716, L14
* Hoffman et al. (1989) Hoffman, G. L., Williams, H. L., Salpeter, E. E., Sandage, A., & Binggeli, B. 1989, ApJS, 71, 701
* Kapferer et al. (2009) Kapferer, W., Sluka, C., Schindler, S., Ferrari, C., & Ziegler, B. 2009, A&A, 499, 87
* Kenney & Koopmann (1999) Kenney, J. D. P., & Koopmann, R. A. 1999, AJ, 117, 181
* Kenney et al. (2008) Kenney, J. D. P., Tal, T., Crowl, H. H., Feldmeier, J., & Jacoby, G. H. 2008, ApJ, 687, L69
* Kennicutt et al. (1989) Kennicutt, R. C., Jr., Edgar, B. K., & Hodge, P. W. 1989, ApJ, 337, 761
* Kennicutt (1998) Kennicutt, R. C., Jr. 1998, ApJ, 498, 541
* Lee et al. (2009) Lee, J. C., et al. 2009, ApJ, 706, 599
* Nilson (1973) Nilson, P. 1973, Nova Acta Regiae Soc. Sci. Upsaliensis Ser. V, 0
* Penny & Conselice (2010) Penny, S. J., & Conselice, C. J. 2010, arXiv:1009.2922
* Poggianti et al. (2004) Poggianti, B. M., Bridges, T. J., Komiyama, Y., Yagi, M., Carter, D., Mobasher, B., Okamura, S., & Kashikawa, N. 2004, ApJ, 601, 197
* Schindler et al. (1999) Schindler, S., Binggeli, B., Boehringer, H. 1999, A&A, 343, 420
* Tonnesen & Bryan (2010) Tonnesen, S., & Bryan, G. L. 2010, ApJ, 709, 1203
* Yagi et al. (2010) Yagi, M., et al. 2010, arXiv:1005.3874
* Yoshida et al. (2008) Yoshida, M., et al. 2008, ApJ, 688, 918
|
arxiv-papers
| 2010-11-07T18:03:10 |
2024-09-04T02:49:14.583867
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Mattia Fumagalli, Giuseppe Gavazzi, Roberto Scaramella, Paolo\n Franzetti",
"submitter": "Mattia Fumagalli",
"url": "https://arxiv.org/abs/1011.1665"
}
|
1011.1947
|
2010 Vol. 10 No. XX, 000–000
11institutetext: LPTA, Université Montpellier 2 - CNRS/IN2P3, 34095
Montpellier, France 22institutetext: Department of Astronomy, Nanjing
University, Nanjing 210093, P. R. China 33institutetext: European Southern
Observatory, Alonso de Córdova 3107, Santiago, Chile 44institutetext:
Department of Astronomy and Astrophysics, P. Universidad Catolica de Chile,
Casilla 306, Santiago, Chile
Received [year] [month] [day]; accepted [year] [month] [day]
# Low-ionization galaxies and evolution in a pilot survey up to z = 1
00footnotetext: Based on observations obtained in service mode at the European
Southern Observatory at Paranal
E. Giraud 11 Q.-S. Gu 22 J. Melnick 33 H. Quintana 44 F. Selman 33 I. Toledo
44 P. Zelaya 44
###### Abstract
We present galaxy spectroscopic data on a pencil beam of $10.75^{\prime}\times
7.5^{\prime}$ centered on the X-ray cluster RXJ0054.0-2823 at $z=0.29$. We
study the spectral evolution of galaxies from $z=1$ down to the cluster
redshift in a magnitude-limited sample at $\rm R\leq 23$, for which the
statistical properties of the sample are well understood. We divide emission-
line galaxies in star-forming galaxies, LINERs, and Seyferts by using
emission-line ratios of [OII], $\rm H\beta$, and [OIII], and derive stellar
fractions from population synthesis models. We focus our analysis on
absorption and low-ionization galaxies. For absorption-line galaxies we
recover the well known result that these galaxies have had no detectable
evolution since $z\sim 0.6-0.7$, but we also find that in the range $z=0.65-1$
at least 50% of the stars in bright absorption systems are younger than
2.5Gyr. Faint absorption-line galaxies in the cluster at $z=0.29$ also had
significant star formation during the previous 2-3Gyr, while their brighter
counterparts seem to be composed only of old stars. At $z\sim 0.8$, our
dynamically young cluster had a truncated red-sequence. This result seems to
be consistent with a scenario where the final assembly of E/S0 took place at
$z<1$. In the volume-limited range $0.35\leq z\leq 0.65$ we find that 23% of
the early-type galaxies have LINER-like spectra with $\rm H\beta$ in
absorption and a significant component of A stars. The vast majority of LINERs
in our sample have significant populations of young and intermediate-aged
stars and are thus not related to AGN, but to the population of ‘retired
galaxies’ recently identified by Cid-Fernandes et al. (2010) in the SDSS.
Early-type LINERs with various fractions of A stars, and E+A galaxies appear
to play an important role in the formation of the red sequence.
###### keywords:
cosmology: observations – galaxies: evolution - large scale structures -
evolution – RX J0054.0-2823
## 1 Introduction
In the course of an investigation of the diffuse intergalactic light in X-ray
emitting clusters at intermediate redshifts (Melnick et al., 1999), we
detected a puzzling S-shaped arc-like structure in the ROSAT cluster RX
J0054.0-2823 (Faure et al., 2007), which we tentatively identified as the
gravitationally lensed image of a background galaxy at a redshift between
z=0.5 and z=1.0. The cluster, however, is characterized by having three
dominant D or cD galaxies in the center, two of which are clearly interacting.
We designed an observing strategy that allowed us at the same time to observe
the arc, the diffuse Intra-Cluster Light (ICL), and a magnitude limited sample
of individual galaxies in the field taking advantage of the multi-object
spectroscopic mode of the FORS2 instrument on Paranal. By optimizing the mask
design (see below) we were able to obtain: (a) very deep observations of the
arc; (b) very deep long-slit observations of the ICL; and (c) redshifts and
flux distributions for 654 galaxies of which 550 are in the pencil beam and at
$0.275\leq z\leq 1.05$.
Our pencil beam sample covers a redshift range up to z = 1 (with some galaxies
up to z = 1.7). In standard cosmology with $H_{o}=75$ $\rm
km~{}s^{-1}~{}Mpc^{-1}$, $\Omega_{0,m}=0.30$, and $\Omega_{0,\Lambda}=0.70$,
this range provides a large leverage of about 3000 Mpc or 7 Gyr, which should
be sufficient to extract some of the most conspicuous characteristics on
galaxy evolution at $z\leq 1$. About half of all stars seem to be still
forming, mostly in disks, in this redshift range (Dickinson et al., 2003;
Hammer et al., 2005). Our spectroscopy provides a 50-60% complete sample of
the galaxies in a pencil beam of $\sim 10^{\prime}\times 10^{\prime}$,
centered on the cluster, uniformly down to R=23. Our sample compares in size
with the DEEP1 spectroscopic pilot survey (Weiner et al., 2005) but is smaller
than large surveys such as DEEP2 (e.q. Lin et al. 2008; Yan et al. 2009), VVDS
(e.q. Franzetti et al. 2007; Garilli et al. 2008), GOODS (e.q. Bell et al.
2005; Weiner et al. 2006). The advantage of a pilot survey is that it can be
handled rather easily by a single (or a few) researcher(s) to test new
methods, new ideas before applying these new methods to large samples.
The vast majority of our individual spectra reduced to zero redshift have S/N
ratios per $\rm 2.6\AA$ pixel larger than 3 at $\rm 4200\AA$. This resolution
is very well adapted to the detection of small equivalent width [OII]
emission, which is expected to be found in bulge dominated galaxies with small
disks, in some LINERs, in “mixed” mergers between E/S0 and star-forming
objects, and perhaps in some post-starbursts galaxies. The line of sight of
our field crosses three main structures: a dynamically young cluster at
$z=0.29$, an over-dense region with layers at $z=0.4-0.5$, and a mixed region
of field and possible layers from $z=0.6$ to $z=1$. According to morphology-
density relations (Dressler, 1980; Dressler et al., 1997; Melnick & Sargent,
1977; Smith et al., 2005; Postman et al., 2005; Cooper et al., 2006; Scoville
et al., 2007), we expect that over-dense regions will provide a rather large
number of red objects available to our study. Therefore red objects with or
without star formation, or with low photo-ionization is the subject which we
will focus on, having in mind the possible roles of E+A galaxies (Dressler &
Gunn, 1983; Norton et al., 2001; Blake et al., 2004; Goto, 2007; Yang et al.,
2008, and references therein) and of LINERs (Yan et al., 2006) in the
building-up of the red sequence.
We focus on galaxies with either low star-formation or low ionization which
appear at $z\leq 0.6$. We use line ratio diagnosis based upon [OII], $\rm
H\beta$, and [OIII], from Yan et al. (2006), to classify galaxies in LINERs,
star-forming galaxies, and Seyferts. This method, combined with visible
morphology, allow us to isolate a significant population of early-type LINERs,
and galaxies with diluted star-formation in later morphological types at
$z=0.35-0.6$.
Several studies suggest that the bulk of stars in early-type cluster galaxies
had a formation redshift of $z\geq 3$, while those in lower density
environments may have formed later, but still at $z\geq 1.5-2$ (for reviews
see Renzini, 2006, 2007). This may be in contradiction with the rise in the
number of massive red galaxies found by Faber et al. (2007) who concluded that
most early types galaxies reached their final form below $z=1$. Our data
include a clear red sequence at $z=0.29$ and a quite large number of
absorption systems up to $z\sim 1$ which we fit with population synthesis
models in order to search for age variations with $z$ and luminosity.
The paper is structured as follows. Section 2 presents details of the
observations and the data reduction procedures. Section 3 is on the resulting
redshift catalog. Section 4 presents an overview of variations in spectral
energy distribution with redshift for absorption and emission systems. Section
5 is dedicated to population variations with $z$ and luminosity in absorption
systems. Low-ionization galaxies are in 5.3. In Section 5.4 we suggest a
scenario in which early-type LINERs will become E/S0 galaxies once the A stars
die, and photo-ionization disappear. Summary and Conclusions are in Section 6.
## 2 Observations and data reduction
The observations (ESO program 078A-0456(A) were obtained with the FORS2
instrument (fors:2005, 2005) on the Cassegrain focus of the VLT UT1 telescope
in multi-object spectroscopy mode with the exchangeable mask unit (MXU). They
were acquired in service observing and were spread over two periods 78 and 80
to satisfy our observing conditions. FORS2 was equipped with two $\rm 2k\times
4k$ MIT CCDs with $15\rm\mu m$ pixels. These CCDs have high efficiency in the
red combined with very low fringe amplitudes. We used the grisms 300V and
600RI, both with the order sorting filter GG435. With this filter, the 300V
grism has a central wavelength at 5950 Å and covers a wavelength between
$4450-8700$ Å at a resolution of 112 Å$~{}{\rm mm}^{-1}$. The 600RI grism has
a central wavelength of 6780 Å and covers the 5120-8450 Å region at a
resolution of 55 Å ${\rm mm}^{-1}$. Combined with a detector used in binned
mode, the 300V grism has a pixel resolution of 3.36 Å pixel-1. The grisms were
used with a slit width of 1′′. In order to match the major and minor axis of
the ICL and the prominent arc-like feature rotation angles of $-343^{o}$,
$-85^{o}$, and $-55^{o}$ were applied. The slit lenghts used for the ICL
spectra are 56.5′′, 32.5′′, and 24.5′′, while those of typical galaxies vary
between 7′′ and 12′′. The ICL was located either on the master CCD or the
second one, resulting in a combined pencil beam field of $\rm
10.75^{\prime}\times 7.5^{\prime}$ (Figure 1).
A total of 30 hours of observing time including field acquisition, mask
positioning, and integration time were dedicated to our pencil beam redshift
survey of the J0054.0-2823 field. Each mask was filled with 39-49 slitlets in
addition to the ICL slits. In order to trace some of the apparent structures
connected to J0054.0-2823, and to reach beyond its Virial radius, we also
obtained MXU exposures of 8 FORS2 fields of $\rm 7^{\prime}\times 5^{\prime}$
adjacent to the pencil beam, so in total we obtained spectra of 730 individual
sources.
Figure 1: The central (pencil beam) field from R images obtained with the wide
field camera at the 2.2m telescope in La Silla
### 2.1 Mask preparation
Tables for preparing the masks and instrument setups were obtained with the
FORS Instrumental Mask Simulator111
http://www.eso.org/sci/observing/phase2/FORS/FIMS.html (FIMS, 2006). The
selection of the objects for the preparation of the slit masks of the pencil
beam field was done by using a photometric catalog in V and I which we had
derived from deep images obtained in a previous NTT run (Faure et al., 2007),
and pre-images in R from the VLT. The selection of the objects in the fields
adjacent to the pencil beam were obtained by using images taken with the WIde
Field Imager (WIFI; $34^{\prime}\times 33^{\prime}$) at the 2.2m telescope on
La Silla. Photometry in V and R from the WFI images are used throughout the
paper. The allocated time was divided in observing blocks (OBs) to be executed
in service mode. A typical OB of 1h execution time had a science integration
time of 2900s in two exposures of 1450s.
We estimated exposure times for E to Sb galaxies in the range z = 0.3 - 0.8.
Using the exposure time calculator of FORS, we obtained magnitude limits, the
major steps of which are given in Table LABEL:maglimit, which we used to
optimize the distribution of slitlets in the masks. After isolating bright
objects which did not require long exposure time, we prepared a grid with an
exposure time step $\rm 2\times 1450~{}s$ which we filled with galaxies having
V magnitudes such that the expected S/N ratio would be better than 2.8 (1
pixel along the dispersion). After receiving VLT pre-images in the red band,
we did a similar grid in R and adjusted the two grids. The masks were prepared
interactively with the FIMS tool and the R pre-images. We started to fill
masks with objects that require an exposure time $\rm\leq 2\times 1450~{}s$,
then moved to $\rm\leq 4,~{}6,~{}and~{}8\times 1450~{}s$. Because we prepared
sets of masks with slits in very different directions (those of the ICL long
and short axis in particular), objects that could not be targeted with a mask
in a given direction (i.e. such as any mask with running name ICL-s in Table
2) were targeted in a perpendicular one (i.e. masks with running name ICL-L),
an approach which made the mask filling quite efficient, in particular in
over-dense areas and field edges. Objects which were close to a predicted S/N
of 2.8 in an OB, were selected to be also observed in another OB as often as
possible. Some objects with expected good S/N in an OB, were re-observed in
another OB when there was no other target in the corresponding slit strip.
They provides a set of high S/N $(\sim 20)$ ratio spectra.
A total of 973 slitlets were selected, 621 in 14 different masks in the pencil
beam field, and 352 in 8 masks in the adjacent fields. Thirty five percent of
the sources of the pencil beam field were observed through different masks,
whereas the slitlets of the adjacent fields are all for different sources.
Table 1: Table used for preparing MXU plates of multiple Observing Blocks Number of OBs of 1h | Integration time | Magnitude limit in V | S/N for S0-Sb at $0.3\leq z\leq 0.8$
---|---|---|---
1 | 2900s | 24.4 - 24.8 | 2.8 - 5.2
2 | 5800s | 24.8 - 25.2 | 2.8 - 5.2
4 | 11600s | 25.2 - 25.6 | 2.8 - 5.2
The resulting list of masks and OBs, and the journal of observations are given
in Table 2. Spectra of the pencil beam field were obtained through masks with
running names Bright, ICL-L, ICL-s, and arc. ICL-L and ICL-s were obtained
with rotator angle $-343^{o}$ and $-85^{o}$ respectively, and arc with a
rotation of $-55^{o}$. Masks with names SE, E, NE, N, NW, W, SW1 & SW2 are on
adjacent fields. The observations were obtained during clear nights, with
seeing between $0.7^{\prime\prime}$ and $1.5^{\prime\prime}$ and dark sky.
Table 2: Journal of the MXU Observations Name | OB ID | Date | Exp. time (s) | # slitlets | Grism
---|---|---|---|---|---
Bright1 | 255728 | 20 Oct. 06 | $3\times 550$ | 34 | 600RI
Bright2 | 255726 | 23 Oct. 06 | $3\times 550$ | 38 | 600RI
SW1 | 255710 | 18 Oct. 06 | $3\times 550$ | 45 | 300V
SW2 | 255708 | 15 Oct. 06 | $3\times 550$ | 40 | 300V
W | 255712 | 19 Oct. 06 | $3\times 550$ | 49 | 300V
SE | 255706 | 3 Oct. 07 | $3\times 550$ | 42 | 300V
N | 255716 | 5 Oct. 07 | $3\times 550$ | 42 | 300V
NW | 255714 | 5 Oct. 07 | $3\times 550$ | 48 | 300V
NE | 255718 | 14 Oct. 07 | $3\times 550$ | 47 | 300V
E | 255704 | 15 Oct. 06 | $3\times 710$ | 39 | 600RI
ICL-s1 | 255750 | 12 Dec. 06 | $2\times 1450$ | 46 | 300V
ICL-s2 | 255748 | 15 Nov. 06 | $2\times 1450$ | 48 | 300V
ICL-L1 | 255761 | 12 Dec. 07 | $2\times 1450$ | 41 | 300V
ICL-L2 | 255763 | 9 Jan. 07 | $2\times 1450$ | 39 | 300V
arc2 | 255734 | 24 Nov. 06 | $2\times 1450$ | 48 | 300V
arc1 | 255736, 38 | 9 Jan. 07, 11 Sept. 07 | $4\times 1450$ | 43 | 300V
ICL-s3 | 255744, 46, 47 | 27 Oct. 06, 9 Nov. 06 | $6\times 1450$ | 47 | 300V
ICL-L3 | 255752, 59, 60 | 21 Sept. 07, 31 Oct. 07 | $6\times 1450$ | 49 | 300V
arc3 | 255730, 32, 33 | 17 Aug. 07 | $6\times 1450$ | 46 | 300V
ICL-s4 | 255739, 41, 42, 43 | 23 Oct. 06, 13 Nov. 06 | $8\times 1450$ | 46 | 300V
ICL-L4 | 255754, 56, 57, 58 | 15, 17 & 20 Nov. 06 | $8\times 1450$ | 49 | 300V
RI | 255720, 22, 23, 24, 25 | 13 Nov. 06, 12 & 14 Sept. 07, | | |
| | & 3 Oct. 07 | $10\times 1450$ | 47 | 600RI
### 2.2 Spectral extraction
The data were reduced by the ESO quality control group who provided us with
science products (i.e. sky subtracted, flat fielded and wavelength calibrated
spectra of our objects), together with calibration data: master bias (bias and
dark levels, read-out noise), master screen flats (high spatial frequency
flat, slit function), wavelength calibration spectra from He-Ar lamps, and a
set of spectrophotometric standards, which were routinely observed. The sky
subtracted and wavelength calibrated 2D spectra allowed a very efficient
extraction of about 60 % of the spectra. Nevertheless the pipeline lost a
significant fraction of objects, in particular when they were located on the
edges of the slitlets. To increase the efficiency of the spectral extraction
we performed a new reduction starting from frames that were dark subtracted,
flat-fielded and wavelength calibrated, but not sky subtracted, using a list
of commands taken from the LONG context of the MIDAS package. For each
slitlet, the position of the object spectrum was estimated by averaging 500
columns in the dispersion direction between the brightest sky lines and
measuring the maximum on the resulting profile. The sky background was
estimated on one side of the object, or on both, depending on each case.
Spatial distortion with respect to the columns was measured on the sky line at
5577 Å and used to build a 2D sky which was subtracted to the 2D spectrum.
Multiple exposures where then aligned and median averaged. The 1D spectra of
objects were extracted from 2D medians by using the optimal extraction method
in MIDAS.
### 2.3 Redshift identification
The identification of lines for determining the redshifts was done
independently by two methods and three of the authors. The 2D spectrum was
visually scanned to search for a break in the continuum, or an emission-line
candidate (e.g. [OII] $\lambda$3728.2 Å). A plot of the 1D spectrum was
displayed in the corresponding wavelength region to search for [OII], the Ca H
& K lines, and/or Balmer lines H$\epsilon$, H9 $\lambda$3835.4 Å, H8
$\lambda$3889.1 Å, H10 $\lambda$3797.9 Å, and H$\delta$. The redshift was then
confirmed by searching for the [OIII] doublet $\lambda$4958.9 & 5006.8 Å, and
H$\beta$ in emission if [OII] had been detected, or G and the Mgb band, if the
4000 Å break and (or) the H and K lines had been identified. The MgII
$\lambda$2799 Å line in absorption and, in some cases AlII $\lambda$ 3584 Å,
were searched to confirm a potential redshift $z\geq 0.65$, while in the cases
of low redshift candidates we searched for $\rm H\beta$, the NaD doublet
$\lambda$5890 & 5896 Å, and in a few cases H$\alpha$. The H$\gamma$ line, the
E (FeI+CaI $\lambda$5270 Å) absorption feature and, in some bright galaxies
the Fe $\lambda$4383 Å, Ca $\lambda$4455 Å, Fe $\lambda$4531 Å absorption
lines, were used to improve the redshift value. The resulting identification
ratio of galaxy redshifts is of the order of 90%. The 10% of so-called
unidentified include stars, objects with absorption lines which were not
understood, a few objects with low signal, and defects. Six QSO’s were also
found. An example of good spectrum of red galaxy, with its main absorption
lines identified, is shown in Figure 2.
Figure 2: Example of a spectrum of a red and bright galaxy with [OII] and the
main absorption lines identified
A second independent visual identification was performed using Starlink’s
Spectral Analysis Tool (SPLAT-VO), matching an SDSS reference table of
emission and absorption
lines222http://www.sdss.org/dr5/algorithms/linestable.html to the spectra.
After a first estimate of the redshift a cross-correlation was performed using
the FXCOR task on the RV package of IRAF333IRAF is distributed by the National
Optical Astronomy Observatory, which is operated by the Association of
Universities for Research in Astronomy, Inc., under cooperative agreement with
the National Science Foundation.. Due to the large span of redshifts, two sets
of templates were used. The first one consisting of 3 template spectra of
galaxies ($\lambda=3500-9000$Å with emission and absorption lines and a
dispersion of 3Å/pix) from the SDSS
survey444http://www.sdss.org/dr2/algorithms/spectemplates/index.html with
continuum subtraction using a spline3 order 5 fitting function. The second set
of templates were two average composite spectra of early type and intermediate
type galaxies ($\lambda=2000-7000$Å with only absorption lines and a
dispersion of 2Å/pix) from the K20
survey555http://www.arcetri.astro.it/$\sim$k20/spe_release_dec04/index.html
using a spline3 order 7 function for continuum subtraction. An interactive
selection of the wavelength range used in the cross correlation was done on
each spectrum avoiding contamination by sky lines. The spectra were re-binned
to the template dispersion (smaller for 300V spectra and larger for 600RI
spectra) , which gave the best results. Velocity errors were determined from
the quality of the cross-correlation, by using standard R value of Tonry &
Davies (Tonry:1979p5170, 1979). Here we used $R_{T}$ to differentiate it from
the R band magnitudes symbol. These values are provided in the IRAF task FXCOR
and explained in the reference quoted. In brief, $R_{T}$ is proportional to
the ratio of the fitted peak height and the antisymmetric noise as defined by
Tonry & Davies (1979). The redshifts, $R_{T}$ values, and velocity errors are
given in Table 5, which also includes the list of visually identified lines.
A third independent visual inspection was carried out when a discrepancy was
seen between the previous two sets of measurements, and also in the very few
cases were no redshift could be measured. For these spectra we first tried to
detect emission or absorption lines and then used Gaussian fits to establish
the line centroids and their errors and shifts. The redshift of each line was
measured independently and the galaxy redshift was obtained from the weighted
average of all lines. This third inspection resolved nearly all the few
remaining discrepancies so we have retained the cross-correlation values
whenever possible. We note that Xcorr failed in two instances: 1) for $z>0.8$
galaxies with low S/N and few weak absorption lines, and, 2) when no
absorption lines, but 1, 2 or 3 clear emission lines were present. In these
cases we used the visual line identifications and assigned a conservative
error of 300 km/s.
Spectra from more than one mask were obtained for 94 objects. Their final
velocities and velocity errors were calculated as error-weighted means from
multiple observations, although no significant disagreements were found. These
repeated observations serve as a check on the internal errors. Figure 3
presents the differences between the cross-correlation velocity measurements
for all galaxies with multiple observations. The representative full width
half maximum (FWHM) error is 200 km/s. In Figure 4 we have plotted the
relation between velocity errors and the Tonry $R_{T}$ value obtained in our
cross correlations. Most errors are $<300$ km/s even for $4>R_{T}>2$ and the
typical error is of order 80 km/s with the vast majority of the radial
velocities have errors below 200 km/s. We have only discarded a few values
with $R_{T}<1$ when there were no measurable emission lines.
Figure 3: Radial velocity differences for galaxies with multiple
observations. The objects with velocity discrepancies larger than $400$ km/s
are broad line QSO’s and one high-z galaxy. Figure 4: Relation between radial
velocity errors ($V_{err}$) and the Tonry $R_{T}$ parameter (Tonry:1979p5170,
1979) in redshifts obtained by cross-correlation. The points away from the
general trend (5 points with $R_{T}>10$ and 5 with $V_{err}>500~{}\rm
km~{}s^{-}1$) are 5 QSO’s and distant weak spectrum galaxies with emission
lines. Objects with $V_{err}>500~{}\rm km~{}s^{-}1$, marked in red, were not
used in combined spectra.
### 2.4 Flux Calibrations
The 1D spectra were divided by the response curve of the detector, which had
been determined from 4 spectrophotometric standard stars observed along the
runs, and reduced by the same method (bias, flat field, wavelength
calibration, and extraction) as the galaxy spectra. The thick absorption
telluric band of O2 centered at 7621 Å (unresolved line series) was not
removed from the observation response curve and was considered as a feature of
the global wavelength dependent efficiency.
The relative fluxes per wavelength of the corrected spectra can be compared
with stellar population models, in arbitrary unit, but are not calibrated in
flux. The spectra were re-binned to the z = 0 rest frame with relative flux
conservation. Because a significant fraction of spectra have a too low S/N
ratio for a meaningful comparison with population synthesis models, one may
either select the brightest objects or combine spectra of similar types. The
spectra taken at different locations of the MXU masks have different lengths
along the dispersion direction. In order to merge them the spectra were
normalized to have the same flux in the region 4050-4250 Å (see below).
### 2.5 Quality of the spectra
The final S/N ratio of the extracted spectra, corrected for the response
curve, and re-binned to zero redshift depends on a number of parameters:
seeing, night sky transparency and background, magnitude of the object and
integration time, wavelength of the S/N measurement, and redshift. To give an
idea of the final products we present in Table 3 a representative set of 28
spectra at various z, magnitudes, number of OBs and resulting S/N ratio
measured on zero redshift spectra in the wavelength range 4150-4250 Å which
corresponds well to the location where we will measure the main indexes of
this work. S/N ratios of spectra re-binned to zero redshift are for a pixel
element of $\rm 2.6~{}\AA$ throughout the paper. Table 3 gives also the names
of the OBs.
Table 3: Signal-to-noise ratio of representative spectra. The columns indicate respectively: the redshift $(z)$ of a selected object, its V and R Petrosian magnitudes, the number of observing blocks, N(OB), from which its spectrum is extracted, the S/N ratio measured in the wavelength range 4150-4250Å of the spectrum rebinned to zero redshift, the name of observing blocks from Table 2, and the grism used. Spectra from OB’s with running name “arc” have on the average higher S/N ratio than those with name “ICL” as illustrated by the two objects marked (*). $z$ | V | R | N(OB) | S/N | Name of OBs | Grism
---|---|---|---|---|---|---
0.2923 | 19.2 | 18.6 | 2 | 14 | arc1 | 300V
0.2932 | 20.3 | 19.3 | 2 | 18 | ICL-L1 & L2 | 300V
0.2928 | 21.6 | 20.5 | 3 | 17 | arc2 & ICL-L1 | 300V
0.2905 | 22.8 | 22.1 | 2 | 10 | ICL-L1 & L2 | 300V
0.2910 | 23.5 | 22.8 | 3 | 9 | arc1 & 2 | 300V
0.4486 | 21.3 | 20.0 | 1 | 6 | Bright2 | 600RI
0.4477 | 22.3 | 21.3 | 2 | 20 | arc1 | 300V
0.4148 | 23.0 | 22.3 | 2 | 7 | ICL-s1 & s2 | 300V
0.4538 | 23.1 | 22.0 | 5 | 14 | ICL-L4 & arc2 | 300V
0.5355 | 22.3 | 20.9 | 1 | 8 | arc2 | 300V
0.6309 | 22.7 | 21.4 | 4 | 11 | arc2 & 3 | 300V
0.6553 | 22.3 | 21.5 | 4 | 9 | ICL-s4 & arc2 | 300V
0.6282 | 23.5 | 22.5 | 5 | 9 | arc1 & 3 | 300V
0.6267 | 23.9 | 22.9 | 4 | 10 | arc3 | 300V
0.6864 | 22.6 | 21.9 | 1 | 4 | ICL-s2 | 300V
0.6886 | 23.0 | 22.0 | 7 | 13 | ICL-L3 & L4 | 300V
0.6861 | 23.0 | 22.1 | 4 | 8 | ICL-s4 | 300V
0.6864 | 23.5 | 22.3 | 4 | 10 | ICL-s3 & arc2 | 300V
0.6879 | 23.8 | 22.8 | 4 | 7 | ICL-s4 | 300V
0.8222 | 20.7 | 20.0 | 1 | 10 | ICL-s1 | 300V
0.8287 | 22.7 | 22.4 | 5 | 10 | ICL-s4 & arc2 | 300V
0.8249 | 23.2 | 22.6 | 3 | 8 | arc3 | 300V
0.8823 | 23.8 | 23.4 | 4 | 3.5 | ICL-s4 | 300V
0.9792 | 23.3 | 22.7 | 3 | 8 | arc3 (*) | 300V
0.9626 | 23.2 | 22.7 | 4 | 5 | ICL-L4 (*) | 300V
0.9637 | 23.4 | 23.2 | 3 | 6 | ICL-s3 | 300V
0.9809 | 23.8 | 23.7 | 5 | 6 | RI | 600RI
1.0220 | 24.1 | 23.3 | 4 | 3 | ICL-s4 | 300V
### 2.6 Spectral indexes
The 4000 Å break amplitude definition used in the present paper is the
‘narrow’ 4000 Å break defined by Balogh et al. (1999) as the flux ratio in the
range 4000-4100Å over 3850-3950Å (e.g. Kauffmann et al., 2003). The error in
D(4000) is calculated from the spectral noise in the two passbands. The
equivalent widths of [OII] and of $\rm H\delta$ were measured by using the
MIDAS context ALICE as follows: the continuum was obtained by linear
interpolation through two passbands each side of the line, a Gaussian was
fitted to the emission or absorption line, and an integration was done over
the resulting Gaussian profile above or below the continuum. The continuum and
line fits, and the integration were done interactively on a graphic window in
which the spectral region of the line was displayed. Table LABEL:integr lists
the wavelength ranges of the sidebands used to define the fluxes and continua.
Table 4: Wavelength bands used in the measurement of 4000 Å break amplitude, and in the determination of the continua of the [OII] and $\rm H\delta$ indexes (equivalent widths). Index | Blue band | Red band
---|---|---
D(4000) | 3850 - 3950 Å | 4000 - 4100 Å
EQW([OII]) | 3650 - 3700 Å | 3750 - 3780 Å
$\rm EQW(H\delta)$ | 4030 - 4070 Å | 4130 - 4180 Å
Uncertainties in equivalent widths were deduced from simple Monte Carlo: the
values of the equivalent widths are the average of 20 continuum determinations
and best Gaussian fits to the absorption or emission lines, and the errors in
equivalent widths are deduced from the Monte Carlo dispersion. The largest
index errors are for spectra in which $\rm H\delta$ is both in absorption and
in emission. In such cases the emission line was removed after fitting the
spectrum of an A star onto all Balmer lines to estimate the depth of $\rm
H\delta$ in absorption, and this step was added to the Monte Carlo. The errors
on indexes given in Tables of combined spectra throughout the paper are those
which were measured on combined spectra. They do not take into account the
astrophysical dispersions in the distributions of individual galaxies which
were used to build combined spectra. Those astrophysical dispersions are given
in relevant Tables concerning spectral variations.
Full observational measurement errors on indexes of individual spectra were
obtained by measuring $\rm D(4000)$ and $\rm EQW([OII])$ on spectra with
multiple observations. Thus $17\%$ of the spectra have typical errors of $4\%$
in D(4000) and $10\%$ in EQW([OII]); $54\%$ have typical errors of $8\%$ in
D(4000) and $20\%$ in EQW([OII]); and $14\%$ have poorer spectra with typical
errors of $16\%$ in D(4000) and $40\%$ in EQW([OII]).
### 2.7 Stellar Population Analysis
In order to study the stellar population quantitatively, we applied a modified
version of the spectral population synthesis code,
starlight666http://www.starlight.ufsc.br/ (Cid Fernandes et al., 2004; Gu et
al., 2006) to fit the observed and combined spectra. The code does a search
for the best-fitting linear combination of 45 simple stellar populations
(SSPs), 15 ages, and 3 metallicities ($0.2\,Z_{\odot}$, $1\,Z_{\odot}$,
$2.5\,Z_{\odot}$) provided by Bruzual & Charlot (2003) to match a given
observed spectrum $O_{\lambda}$. The model spectrum $M_{\lambda}$ is:
$M_{\lambda}(x,M_{\lambda_{0}},A_{V},v_{\star},\sigma_{\star})=M_{\lambda_{0}}\left[\sum_{j=1}^{N_{\star}}x_{j}b_{j,\lambda}r_{\lambda}\right]\otimes
G(v_{\star},\sigma_{\star})$ (1)
where
$b_{j,\lambda}=L_{\lambda}^{SSP}(t_{j},Z_{j})/L_{\lambda_{0}}^{SSP}(t_{j},Z_{j})$
is the spectrum of the $j^{\rm th}$ SSP normalized at $\lambda_{0}$,
$r_{\lambda}=10^{-0.4(A_{\lambda}-A_{\lambda_{0}})}$ is the reddening term,
$x$ is the population vector, $M_{\lambda_{0}}$ is the synthetic flux at the
normalization wavelength, and $G(v_{\star},\sigma_{\star})$ is the line-of-
sight stellar velocity distribution modeled as a Gaussian centered at velocity
$v_{\star}$ and broadened by $\sigma_{\star}$. The match between model and
observed spectra is calculated as
$\chi^{2}(x,M_{\lambda_{0}},A_{V},v_{\star},\sigma_{\star})=\sum_{\lambda=1}^{N_{\lambda}}\left[\left(O_{\lambda}-M_{\lambda}\right)w_{\lambda}\right]^{2}$,
where the weight spectrum $w_{\lambda}$ is defined as the inverse of the noise
in $O_{\lambda}$. The code yields a table with input and ouput parameters for
each component. Input parameters include individual stellar masses, ages,
metallicities, L/M, … and ouput parameters include luminosity fractions, mass
fractions, fit parameters of individual components …, and global parameters
such as velocity dispersion and extinction. For more details we refer to the
paper by Cid Fernandes et al. (2005). In the present work we use the standard
luminosity fraction in the rest frame of normalized spectra at $\rm
4050~{}\AA$, which we compare in different redshift bins.
Figure 5: Spectral fitting results with SSP models for the redshift $<z>=0.29$
bin. (a): Observed (thin black line), model (red line) and residuals for the
absorption spectrum. Points indicate bad pixels and emission-line windows that
were masked out during fitting. (b): Emission-line spectrum; (c): Total
spectrum.
Figure 5 shows an example of the fit for the averaged spectrum at $<z>=0.29$.
Panels (a), (b), and (c) correspond to absorption-line, emission-line, and all
spectra respectively. After fitting the spectra, we rebin the 45 SSPs into 5
components according to their age: I ($10^{6}\leq t<10^{8}$ yr), II
($10^{8}\leq t<5\times 10^{8}$ yr), III ($5\times 10^{8}\leq t<10^{9}$ yr), IV
($10^{9}\leq t<2.5\times 10^{9}$ yr), and V ($t\geq 2.5\times 10^{9}$ yr).
Components with the same age and different metallicities are combined
together.
## 3 The Catalog of Galaxies and Large Scale Structures in the Line of Sight
in the Pencil Beam
Table 5 presents positions, redshifts, Petrossian R-magnitudes ($m_{R}$), and
line identifications for the full sample of 654 galaxies observed in our
program. The radial velocities and the corresponding measurement errors are
also given. The full Catalogue from which Table 5 is extracted will be sent as
a public database to CDS. The rough data are presently in the public domain at
ESO.
Table 5: Properties of galaxies in the field of RX J0054.0-2823 obj | RA ($\alpha$) | DEC ($\delta)$ | z | $m_{R}$ | V | Verr | $R_{T}$ | Nobs | lines
---|---|---|---|---|---|---|---|---|---
| J2000 | J2000 | | | km/s | km/s | | |
23 | 13.598707 | -28.434965 | 0.79304 | 22.78 | 237912 | 161 | 4.7 | 1 | K–H
26 | 13.590379 | -28.416917 | 0.77636 | 22.26 | 232908 | 77 | 8.1 | 1 | [OII]–H10–H9–H
27 | 13.584442 | -28.394515 | 0.41463 | 22.29 | 124389 | 22 | 17.3 | 1 | [OII]–H9–H–H$\beta$–[OIII]
28 | 13.586628 | -28.438063 | 0.44877 | 22.02 | 134631 | 73 | 6.7 | 1 | K–H–G
30 | 13.580301 | -28.435437 | 0.29032 | 21.08 | 87096 | 68 | 11.3 | 1 | H9–K–H–H$\delta$–H$\alpha$
31 | 13.572009 | -28.380385 | 0.63267 | 21.32 | 189801 | 49 | 11.5 | 1 | [OII]–K–H
32 | 13.579012 | -28.439414 | 0.45335 | — | 136005 | 22 | 19.5 | 1 | [OII]–H$\gamma$–H$\beta$–[OIII]
33 | 13.574997 | -28.439377 | 0.44741 | 19.21 | 134223 | 80 | 11.4 | 1 | K–H–G–H$\beta$
34 | 13.573913 | -28.442855 | 0.63013 | 20.63 | 189039 | 87 | 7.9 | 1 | K–H–H$\delta$–G
35 | 13.571781 | -28.435423 | 0.44862 | 20.26 | 134586 | 73 | 9.8 | 1 | H9–K–H–G–H$\beta$
Figure 6 shows the R-magnitude histogram of the galaxies with measured
redshifts superimposed on the magnitude histogram of all galaxies in our
pencil-beam indicating that our observations sample uniformly at a rate of
50-60% the population of galaxies down to $\rm R=22.5$. The sampling seems
fairly representative in the magnitude bin $\rm R=22.5-23.0$, and sparse at
$\rm R>23$. The apparent increase in incompleteness toward brighter magnitudes
is due to a selection bias in the observations, which were designed to avoid
bright galaxies at redshifts $z\leq 0.25$.
Figure 6: R-magnitude histogram of galaxies with measured redshift in the
central beam.
Figure 7 presents the magnitude redshift relation and the cone diagrams for
the full sample. The points are color coded according to the presence or
absence of emission lines.
Figure 7: (a) Magnitude redshift relation for the full sample. The three
lines overploted over the measured points correspond to absolute R magnitudes
of -22.5, -20.5, and -18.5. The distances have been calculated using a
cosmology with $\Omega_{0,\Lambda}=0.70$, $\Omega_{0,m}=0.30$, $w=-1$, and
$\rm H_{0}=75km~{}s^{-1}~{}Mpc^{-1}$ (h = $\rm
H_{0}/75km~{}s^{-1}~{}Mpc^{-1}$). Red dots are galaxies with no emission lines
and blue dots are galaxies with emission lines. (b) Cone diagrams in Dec for
all the galaxies measured in the field of RX J0054.0-2823. The scales is in
Mpc calculated using the angular distance for the standard cosmology. The
detection threshold for emission-lines is $\rm EQW([OII])\sim 2-3$Å. (c) Same
as (b) but for RA.
A cursory inspection of Figure 7 reveals the presence of several conspicuous
structures - walls of objects spanning almost the entire field of view - over
the full range of redshifts covered by our observations. Ignoring objects with
$z<0.28$, we see structures centered at $z=0.29$ (our prime target); two
distinct structures at $z\sim 0.4$, which we will denote $z=0.415$ and
$z=0.447$; a rather complex structure at $z\sim 0.6$, with two main over-
densities at $z=0.58-0.63$, and $z=0.68$; a single rather sparsely populated
layer at $z=0.82$. In what follows, we will refer to these groups (including
the main cluster at $z=0.29$) as our pencil beam structures. Making bins
centered on the peaks of the redshift distribution maximizes the number of
objects in each bin and minimizes its redshift dispersion. So using the
apparent structures rather than a blind slicing appears well adapted to our
sample. If the structures are real, the objects of a given structure may have
a common history and this may also help to reduce the cosmic scatter.
The numbers of spectra observed in each structure are given in Table
LABEL:zdist. The redshift bins given in column (2) are chosen a posteriori to
fit the structures. The numbers of redshifts measured in each bin are given in
column (3), with the respective numbers of absorption and emission systems in
columns (4) & (5). We have determined the median distance to the nearest
object $\triangle\delta$ in each of the apparent structures in column (6), and
the median velocity dispersion, $\rm\sigma(V)\equiv c\sigma(z)/(1+z)$ in
column (7). We give a rough morphology of the structures in column (1). The
cluster at $z=0.293$ appears to have small projected separation and velocity
dispersion. Layers or filaments have a comoving velocity dispersion (dynamical
and cosmological) less than $\rm\sim 1500~{}km~{}s^{-1}$; clouds have
$\rm\sigma(V)>2000~{}km~{}s^{-1}$. The structures marked “filaments” are the
arc layers seen in Figure 7. Projected on the sky they seem to be filamentary,
but the median distances $\triangle\delta$ to the nearest object are
approximately 2/3 those expected for uniform distributions, so they are not
clearly different from 2D layers. The names of the bins that are used to
combine spectra are given in column (8). Large scale arc structures, as seen
in cone diagrams, are expected to be formed by infall of galaxies on
gravitational potentials: galaxies which are on the far side have a negative
infall velocity, while those on the nearby side have a positive infall
component, which when superimposed on the Hubble flow reduces the velocity
dispersion. This is presumably what we observe in the two filaments or layers
with low velocity dispersion at z = 0.4.
We combined the spectra in each structure using the median. This results in a
slightly lower total S/N (by $\sqrt{2}$), but allows to eliminate spurious
features.
Table 6: Apparent structures in the field of RX J0054.0-2823 Apparent Structure | z range | N | N(abs) | N(em) | $\triangle\delta~{}(kpc)$ | $c\sigma(z)/(1+z)$ | Composite name
---|---|---|---|---|---|---|---
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8)
| 0.275 - 0.285 | 0 | 0 | 0 | | |
Cluster | 0.285 - 0.298 | 91 | 60 | 31 | 165 | 527 | SPEC029
| 0.298 - 0.320 | 5 | 1 | 4 | | |
| 0.320 - 0.330 | 12 | 7 | 5 | | |
| 0.330 - 0.390 | 28 | 7 | 21 | | |
| 0.390 - 0.430 | 35 | 6 | 29 | 490 | 3250 | SPEC0415
Filament (layer) | 0.432 - 0.440 | 29 | 5 | 24 | 500 | 350 | SPEC0415
| 0.440 - 0.444 | 1 | 1 | 0 | | |
Filament (layer) | 0.444 - 0.456 | 46 | 21 | 25 | 470 | 450 | SPEC0447
| 0.456 - 0.465 | 1 | 1 | 0 | | |
Cloud | 0.465 - 0.550 | 53 | 15 | 38 | 480 | 3370 |
Cloud | 0.550 - 0.620 | 56 | 15 | 41 | 520 | 3550 | SPEC063
Filament (layer) | 0.620 - 0.657 | 48 | 12 | 36 | 580 | 1450 | SPEC063
| 0.657 - 0.673 | 1 | 0 | 1 | | |
Filament (layer) | 0.673 - 0.696 | 43 | 12 | 31 | 650 | 1010 | SPEC068
| 0.710 - 0.790 | 22 | 4 | 18 | | |
Filament & cloud | 0.790 - 0.850 | 33 | 6 | 27 | 820 | 2050 | SPEC082
| 0.850 - 0.880 | 0 | 0 | 0 | | |
Cloud | 0.880 - 0.930 | 11 | 1 | 10 | | |
| 0.930 - 0.946 | 0 | 0 | 0 | | |
Cloud | 0.946 - 1.046 | 25 | 4 | 21 | 1080 | 4110 | SPEC099
### 3.1 Magnitudes
The R-band average magnitudes of galaxies in each redshift bin are given in
Table LABEL:magnitudes separately for absorption, “red” and “blue” emission-
line galaxies, together with the adopted distance moduli. The partition “red”
versus “blue” is defined by the median spectral slope in each redshift bin. In
a study on emission line galaxies (Giraud et al., 2010) we divided the sample
of emission-line galaxies in two halves: those with continuum slopes bluer
than the average and those with continuum slopes redder than the average in
each redshift bin. This was done interactively by displaying reduced 1D
spectra and using MIDAS. While a median partition is not necessarily a
physical partition, we showed that, in the present case, it divides ”young”
galaxies, for which the evolution is dominated by on-going star formation from
”old” galaxies where the evolution is dominated by changes in the older
stellar populations.
Table 7: Average R-band magnitudes of absorption systems (abs), and red and blue emission-line galaxies. The adopted distance moduli $(m-M)_{0}$ and the 4150-4250Å fluxes $f$ normalized to the blue galaxies at $z=0.9$ are also tabulated. $<z>$ | R(abs) | R(red) | R(blue) | $(m-M)_{0}$ | $f$(abs) | $f$(red) | $f$(blue)
---|---|---|---|---|---|---|---
0.29 | 19.80 | 20.12 | 20.97 | 40.18 | 0.74 | 0.72 | 0.48
$0.43$ | 20.24 | 20.59 | 20.95 | 40.86 | 1.08 | 0.92 | 0.75
$0.65$ | 21.50 | 21.60 | 21.94 | 41.51 | 1.42 | 1.28 | 0.86
$0.9$ | 22.45 | 22.13 | 22.35 | 41.98 | 2.08 | 1.70 | 1
We used the R-band photometry to calibrate individual spectra by convolving
each spectrum with a box filter 1290 Å wide, centered at $\lambda=6460$ Å.
Once the spectra were calibrated in the observer R-band, we measured the
average fluxes in the wavelength range 4150-4250Å of the galaxies, which we
normalized to the flux of blue emission galaxies at $<z>=0.9$ to compute the
luminosity index $f$. Thus $f$ (that is equal to 1 for blue galaxies at
$<z>=0.9$) is an indicator of AB(4200) that allows us to compare the
luminosities of red and blue galaxies at a given redshift and to investigate
luminosity variations with $z$. Thus Table LABEL:magnitudes clearly shows that
in each redshift bin, absorption-line and red emission-line galaxies are more
luminous than blue galaxies.
## 4 Composite spectra
Each galaxy spectrum was wavelength calibrated, corrected for instrument
response, re-binned to zero redshift, and normalized to have the same flux in
the wavelength range $\rm\triangle\lambda=4050-4250\AA$. Normalizing spectra
gives the same weight to all galaxies. As a consequence stellar fractions must
be understood as average stellar fractions per galaxy.
We have truncated the sample at $z=1.05$ and assembled the spectra in bins
centered on (pseudo) structures at 0.29, 0.41, 0.45, 0.63, 0.68, 0.82, and
0.99 to build high S/N composite spectra for each bin. In order to compensate
(or at least alleviate) for Malmquist bias we rejected objects fainter than
$\rm M_{R}=-18.8$ mostly at $z\leq 0.45$ (Figure 7a). A sample completely free
of Malmquist bias would require a cutoff at $\rm M_{R}\sim-20.5$. For clarity
of the figures, we often combined the mean spectra at $z=0.41$ & $0.45$ into a
single bin at $<z>=0.43$, the spectra at $z=0.63$ & $0.68$ into a bin at
$<z>=0.65$, and in some cases the spectra at $z=0.82$ & $0.99$ into a bin at
$<z>=0.9$. The spectra of galaxies in these four bins are presented in Figure
8 where we show the spectra of absorption systems (top) and emission line
galaxies (bottom) separately. The corresponding $4000\AA$ break amplitudes are
given in Table LABEL:D4000
Figure 8: Composite spectra of absorption systems (top); and emission line galaxies (bottom) normalized in the wavelength range $\rm\triangle\lambda=4050-4250$ Å. All individual galaxies are brighter than $\rm M_{R}=-18.8$ Table 8: 4000Å break amplitudes for absorption (abs) and emission (em) galaxies, and equivalent width of H$\delta$ for absorption galaxies with measurement errors. The S/N ratios of the combined spectra were measured in the interval 4050Å–4250Å. The magnitude cutoff is $\rm M_{R}=-18.8$ for all redshift bins. | Absorption systems | Emission systems
---|---|---
$<z>$ | D(4000) | $\rm EQW(H\delta)$ | S/N | D(4000) | S/N
0.29 | $1.67\pm 0.065$ | $-1.5\pm 0.2$ | 23 | $1.22\pm 0.02$ | 32
$0.43$ | $1.70\pm 0.06$ | $-1.5\pm 0.2$ | 22 | $1.22\pm 0.01$ | 52
$0.65$ | $1.60\pm 0.055$ | $-1.8\pm 0.2$ | 24 | $1.14\pm 0.01$ | 35
0.82 | $1.57\pm 0.06$ | $-2.4\pm 0.5$ | 18 | $1.07\pm 0.02$ | 28
0.99 | $1.43\pm 0.05$ | $-2.9\pm 0.3$ | 23 | $1.08\pm 0.02$ | 25
The most conspicuous spectral change with redshift is a decrease in flux
redward of the G-band from $<z>=0.29$ and $<z>=0.43$ to higher $z$ coupled to
an increase to the blue of [OII] from $<z>=0.65$ to $<z>=0.82$ and higher $z$
in emission-line galaxies. This systematic change of the continuum implies
that the galaxy population varies as a function of redshift: more star forming
galaxies at higher $z$ and more galaxies with old stars at lower $z$. This
spectral change, which is known, will not be studied further in this paper
except to quantify (in 5.3.1) the impact of LINER-like galaxies at $z=0.4--
0.9$. In the following section we concentrate on absorption systems and low-
ionization galaxies.
## 5 Absorption line systems
The spectral resolution of the 300V grism allows us to detect [OII] emission
down to $\rm EQW([OII])\sim 2-3~{}\AA$. We will call absorption-line galaxies
those for which any mechanism of ionization is low enough to preclude [OII]
detection at our detection level. Thus, our pure absorption-line sample
comprises mostly E, E+A, and S0 galaxies with no on-going star formation,
nuclear activity, or other mechanism of ionization.
### 5.1 Absorption systems as function of redshift
The normalized and combined spectra of absorption line systems presented in
Figure 8 (top) do not show any obvious change in their continuum and 4000Å
break amplitude up to $z\approx 0.6$ (Table LABEL:D4000). There is a moderate
decrease in the 4000Å break at $z\geq 0.65$ ranging from $5\%$ at $z\sim 0.65$
to $7\%$ at $z\sim 0.82$ and up to $15\%$ at $z\sim 1$, while the $\rm
H\delta$ absorption line becomes stronger at $z\geq 0.65$ (Table LABEL:D4000),
suggesting the presence of increasing numbers of A stars at higher redshifts.
The indexes suggest that these galaxies had the bulk of their star formation
at $z\geq 1$, while some of the systems at $z>0.8$ still had clearly
detectable star formation about 1 Gyr ago.
We have compared our spectral indexes at $z\sim 0.82$ with those measured by
Tran et al. (2007) in the rich cluster MS 1054-03 at z = 0.83 using the same
index definitions from Kauffmann et al. (2003). The average break amplitude
and $\rm H\delta$ index of absorption systems in MS 1054-03 are respectively
$\rm D(4000)(abs)=1.67\pm 0.00$ and $\rm EQW(H\delta)(abs)=-1.7\pm 0.0$ (Tran
et al., 2007, Table 4). Our absorption systems at $z\sim 0.82$ appear to have
younger stellar populations as indicated both by $\rm D(4000)$ and $\rm
EQW(H\delta)$ (Table LABEL:D4000). Therefore our absorption systems contain A
stars, but clearly less than composite field E+A galaxies at $<z>=0.6$ for
which $\rm D(4000)(abs)=1.36\pm 0.02$ and $\rm EQW(H\delta)(abs)=-4.6\pm 0.2$
(quoted in Tran et al. (2007, Table 4) from data in Tran et al. (2004)).
Consequently our average spectrum at $z\sim 0.82$ is intermediate between pure
E and pure E + A. In fact, our SSP models (Table LABEL:poptable) indicate that
absorption-line systems at $z\geq 0.65$ contain on average more than $50\%$ of
stars younger than 2.5Gyr per galaxy, while those at $z\geq 0.8$ had
significant star formation as recently as one Gyr ago (Table LABEL:poptable).
Table 9: Stellar population properties of normalized average absorption (abs) spectra in each redshift bin. The magnitude cutoff is at $\rm M_{R}=-18.8$, except for the 10 faintest absorption systems at $z=0.29$ where we used all the observed objects. The fractions indicated in all SSP Tables are standard luminosity fractions at $\rm 4050\AA$, as in Cid-Fernandes et al. (2010, and references therein) $<z>$ | log(Age): | $<8$ | $8-8.7$ | $8.7-9$ | $9-9.4$ | $>9.4$
---|---|---|---|---|---|---
$0.29$ | abs | 0.0% | 0.0% | 30.1% | 0.1% | 69.8%
| abs (10 brightest) | 0 | 0 | 17.4 | 0 | 82.4
| abs (10 faintest) | 0 | 0 | 12.0 | 66.4 | 21.7
$0.43$ | abs | 0.0 | 0.7 | 11.7 | 6.9 | 80.7
$0.65$ | abs | 0.0 | 0.0 | 18.2 | 38.5 | 43.3
$0.82$ | abs | 0.0 | 0.0 | 86.8 | 3.3 | 9.8
$0.99$ | abs | 0.0 | 0.0 | 42.3 | 0.0 | 57.7
Post-starburst E+A galaxies are thought to be in a transition phase between a
star-forming period and a passively evolving period. Being close to the phase
of shutdown or quenching of star formation, they probably play an important
role in the build-up of early-type systems (e.g. Wild et al, 2009; Yan et al.,
2009). Studies of intermediate redshift clusters at $0.3\leq z\leq 0.6$ have
found either a higher fraction of post-starburst galaxies in clusters than in
the field (Dressler et al., 1999; Tran et al., 2003, 2004), or a similar
fraction (Balogh et al., 1999). In fact, there is a strong variation in the
E+A fraction between the SSDS low redshift survey at $z\sim 0.07-0.09$, and
high $z$ surveys at $z\approx 0.5-1$ (VVDS, Wild et al, 2009), or $z\approx
0.7-0.9$ (DEEP2, Yan et al., 2009).
In order to search for E+A galaxies in our sample we built template spectra by
combining a pure E spectrum from our sample with various fractions of an A
stellar template. We then compared our models with absorption-line systems in
the range $\rm 1.2\leq D(4000)\leq 1.5$ assuming, by definition, that E+A
galaxies contain at least $\rm 25\%$ A stars. Using this (standard) definition
we searched our sample at $0.35\leq z\leq 1$ and found only 6 bona-fide E+A
galaxies. In fact all the objects were found at $0.68\leq z\leq 1$, which
makes our small number consistent with the VVDS and the DDEP2 surveys within a
factor of 2. The median of the normalized spectra of these 6 (as far as we can
judge from our images) elliptical galaxies is presented in Figure 9 (a). We
were surprised to find no E+As at $z\sim 0.4$, but we did find 4 objects with
early-type morphology and very small $\rm<EQW([OII])>\approx 3.5\AA$, which
probably would have been classified as E+As on lower resolution spectra. The
average spectrum of these 4 objects is shown in Figure 9 (b). Their $\rm R\sim
22$ magnitudes place them at the faint end of absorption line systems at the
corresponding redshifts.
Figure 9: (Top) Average spectrum of 6 E+A galaxies at $0.68\leq z\leq 1$. The
Balmer series $\rm H\delta$, $\rm H+H\epsilon$, H8, H9, H10, and H11 is very
promintent and $\rm<D(4000)>=1.40$. (Botton) Average spectrum of 4 galaxies in
the intermediate redshift range $0.4\leq z\leq 0.5$ with E/S0 morphological
type, showing a poststarburst E+A spectrum with still some star formation.
$\rm<EQW([OII])>\approx 3.5\AA$ and $\rm<D(4000)>=1.41$. Probably this
spectrum would have been classified as E+A at lower resolution.
### 5.2 Absorption-line galaxies as function of luminosity at $z=0.29$
Having tested bright absorption galaxies at various redshifts (with cut-off at
$\rm M_{R}=-18.8$), we now turn to faint absorption galaxies in the cluster at
$<z>=0.29$ by combining the spectra of the 10 faintest galaxies without
emission lines. Their average R-band magnitude is $\rm R=22$, which at a
distance modulus of 40.18 corresponds to $\rm M_{R}=-18.2$, and the faintest
object has $\rm M_{R}=-17.44$. Their mean indexes, $\rm D(4000)=1.55\pm 0.01$;
$\rm H\delta=-2.27\pm 0.04$, measured on the spectrum shown in Figure 10, are
consistent with a younger age than absorption-line galaxies with $\rm
M_{R}\leq-18.8$ (Table LABEL:D4000) in the same cluster. This is in agreement
with the well known evidence that the stellar populations in absorption
systems tend to be younger in low mass galaxies than in the more massive ones
(e.g. Renzini, 2006). The index values are in fact very close to those of our
absorption systems at z = 0.8 (Table LABEL:D4000) which, by selection effects,
are bright (Table LABEL:magnitudes and Figure 7).
The SSP models indicate that on average about 80% of the stars in the 10
faintest galaxies are younger than 2.5 Gyr (Table LABEL:poptable), i.e. were
born at $z<1$. In comparison, 80% of the stars contributing to the spectrum of
the brightest absorption galaxies in the cluster are older than 2.5 Gyr (Table
LABEL:poptable). To illustrate the spectral differences between bright and
faint systems at $z=0.29$, and the striking similarity between the spectra of
faint galaxies at $z=0.29$ and those of bright galaxies at $z=0.8$, we have
plotted in Figure 10 the average spectra of the 10 brightest and the 10
faintest absorption systems at $z=0.29$, and the average spectrum of
absorption galaxies at $z=0.82$. The effect of downsizing, (in the present
case the so-called ’archeological dowsizing’) where star formation shifts from
high mass galaxies at high redshifts, to low mass galaxies at low redshifts is
clearly exemplified in this figure.
Figure 10: Normalized spectra of the 10 brightest (in red) and the 10 faintest
(in blue) absorption-line galaxies in the cluster at $z=0.29$, and the full
sample of absorption-line systems at $z=0.82$ (in cyan).
At a redshift of $z\sim 0.8$ (i.e. $\sim 4$ Gy earlier), the red-sequence of
our unrelaxed (merging central system; elongated intra-cluster light and
galaxy distribution) cluster at z=$0.29$ was already in place, but was
truncated at brighter magnitudes because the faint absorption-line galaxies
were still copiously forming stars. This seems consistent with the observation
that some clusters at $z\simeq 1$ have red sequences truncated at faint limits
(Kodama et al., 2004; Koyama et al., 2007), and supports the picture of an
environmental dependence of red-sequence truncation presented by Tanaka et al.
(2005). This is also in agreement with scenarios where the final assembling of
the red-sequence can be observed well below $z=1$ (Faber et al., 2007).
As discussed above, the strict definition of E+A galaxy requires a mix of an
E-type spectrum with at least 25% A stars and no traces of star formation,
which in our sample implies no emission lines with equivalent widths larger
than 2-3Å. With this definition our $z=0.29$ cluster contains only one E+A
galaxy while the dense layers at $z\sim 0.4$, where the red-sequence is
already in place (layer in Figure 7), contains none. However, both in the
cluster and in the intermediate redshift layers we find plenty of galaxies
with early type morphologies, A stars, and very weak emission lines. In the
next section we present a closer look at these low-ionization emission line
galaxies.
### 5.3 Galaxy evolution and low-ionization emission-line galaxies (LINERs).
In an extensive work based on the SDSS survey, Yan et al. (2006) determined
the extent to which [OII] emission produced by mechanisms other than recent
star formation introduces biases in galaxy evolution studies based upon [OII]
only. They showed that the $\rm[OII]/H\beta$ ratio separates LINERs from star-
forming galaxies, while $\rm[OIII]/[OII]$ and $\rm[OIII]/H\beta$ separate
Seyferts from LINERs and star-forming galaxies. Using the classification
scheme of Yan et al. (2006) we divided our spectra in 3 main classes: LINERs,
with clearly detected [OII], but no ($3\sigma$) detection of [OIII] and $\rm
H\beta$ in emission after subtracting an E+A profile; Seyferts, with
$\rm[OIII]/H\beta\geq 3$; and star-forming galaxies, which are the objects
with clearly detected [OII] that are neither Seyferts nor LINERs after
subtracting an E+A profile. Typical spectra of low-ionization objects, star-
forming galaxies and Seyferts are shown in Figure 11.
Figure 11: Typical average spectra of low-ionization objects, star-forming
galaxies and Seyferts from the sample in the $<z>=0.415$ layer.
#### 5.3.1 The impact of LINERs in our previous results on red emission-line
galaxies
Because the spectral coverage in a rather large fraction of our objects at
$<z>=0.68$ and higher is truncated below $\rm 5000\AA$ in the rest frame, we
applied our classification scheme only to objects in the range $0.29\leq z\leq
0.65$. To extract $\rm H\beta$ in emission we built a series of E+A models,
combining an observed E spectrum with different fractions of an A stellar
template, ranging form $0.05\%$ to $80\%$ of the total luminosity. To
determine the best-fit model we minimized the continuum slope of the
difference between the spectrum and the E+A model. Thus, in the range
$0.35-0.55$ our sample contains 23% LINERs, 51% star-forming galaxies, 8%
Seyferts, and 14% uncertain types. The layer at $z=0.63$ has 18% LINERs, 50%
star-forming galaxies, 7% Seyferts, 13% uncertain types and 11% of truncated
spectra. Altogether, the fraction of LINERs among emission-line galaxies up to
$z=0.65$ in our pencil beam is $\rm\approx 22\%$. With an average
$\rm<D(4000)>=1.39\pm 0.18$. LINERs at $z\leq 0.65$ have a potentially
significant impact on the conclusions of Giraud et al. (2010) about the
evolution of red emission-line galaxies. To quantify this impact, we have
subtracted all LINERs from the sample of emission-line spectra in the $z=0.43$
bin, determined the new blue-to-red partition (as in Giraud et al. 2010;
section 5.1), and computed a new average spectrum for the red galaxies. This
(also cleaned of rare red Seyferts) is shown in Figure 12 where it is compared
with the mean red spectrum at $z=0.9$. We find that the differences in
continuum slope and D(4000) between $<z>=0.43$ and $<z>=0.9$ is reduced by a
factor of 2/3. The main difference between red galaxies with LINERs and those
without is the presence of young stellar population.
Figure 12: Average spectra of red emission-line galaxies after subtracting
early-type LINERs and galaxies with diluted star formation (and rare Seyferts)
from the sample in the $<z>=0.43$ bin and recalculating the median blue-to-red
partition, and of the red half of emission-line galaxies at $<z>=0.9$. The
spectra at $<z>=0.9$ were not classified because $\rm H\beta$ and [OIII] are
missing in most cases.
#### 5.3.2 Early-type LINERs
The fraction of nearby early-type galaxies hosting bona-fide (i.e. nuclear)
LINERs in the Palomar survey (Filippenko & Sargent, 1985; Ho et al. 1997a, )
was found to be $\sim 30\%$ (Ho et al. 1997b, ), but LINER-like emission line
ratios are also observed in extended regions (Phillips et al., 1986;
Goudfrooij et al., 1994; Zeilinger et al., 1996; Sarzi et al., 2006, and
references therein). A similar fraction of LINER-like ratios is found in the
SDSS at $0.05\leq z\leq 0.1$ in color-selected red galaxies (Yan et al.,
2006).
Because it is very difficult to disentangle early-type LINERs from spirals
with extended and diluted star formation by using only [OII] and $\rm H\beta$,
we make use of morphology to distinguish compact objects with low ellipticity
and profiles consistent with early type galaxies, from other morphologies:
apparent disks, high ellipticity, and irregular or distorted morphologies.
Images of early-type galaxies with low ionization spectra are shown in Figure
13.
Figure 13: Examples of early-type galaxies having low-ionization spectra, and
indicated redshifts.
Our visual early-type morphologies are the same as ZEST type T=1 (Scarlata et
al., 2007, Figure 4 (b), (c), (d)). In the $<z>=0.43$ bin we find that 92% of
the galaxies classified as star-forming objects have morphologies inconsistent
with early-types. At $<z>=0.43$ and in the $z=0.63$ layer, about half of the
LINERs have compact morphology while the other half are mainly bulge-dominated
disk galaxies, or “early disks” of ZEST type T=2.0 ((Bundy et al, 2009, Figure
4)). At $z=0.29$ all LINERs have disks. The spectra of galaxies with apparent
disks have an extended [OII] emission suggesting that they do have extended
star-formation. Average spectra of 11 early-type galaxies (E) and 10 later
types (hereafter S) resulting from our morphological classification are shown
in Figure 14.
Figure 14: Median spectra of 11 early-type galaxies and of 10 galaxies with
later type morphology (S) with low ionization at $z=0.4-0.5$.
The absence of $\rm H\beta$ in the S sample suggests that $\rm H\beta$ in
emission resulting from star formation is diluted in $\rm H\beta$ in
absorption from A and older stars. The closest spectral comparison in the
atlas of galaxy spectra (Kennicutt, 1992) is with an Sb galaxy. The rather
strong $\rm H\beta$ in absorption in early-types (E) combined with [OII]
suggests either a low fraction of young stars or a mechanism of photo-
ionization other than young stars as in (Fillipenko, 2003; Ho, 2004, and
references therein). In fact, the recent work by the SEAGAL collaboration (Cid
Fernandes et al., 2010, and references therein) has shown that the majority of
galaxies with LINER spectra in the SDSS can be explained as retired galaxies,
that is, galaxies that have stopped forming stars but still contain
appreciable amounts of gas that is being photoionized by intermediate-aged
post-AGB stars. In fact, the SEAGAL models with no young stars, but with
significant populations of 100Myr-1Gyr stars resemble remarkably well our
average LINER spectrum shown in Figure 11.
We calculated population synthesis models for our average spectra of LINERs
with early-type and late-type morphologies. The results, shown in Figure 15
and Table LABEL:earlylinerpop, indicate that both early-type and late-type
LINERs have significant populations of young and intermediate age stars, but
late-type (S) LINERs have much younger populations. In fact, the residuals of
the S-LINER fit show $\rm H\beta$ in emission stronger than [OIII], consistent
with the idea that they are red spirals with diluted star formation.
Table 10: Stellar population properties of an average of LINERs with early-type morphology and with morphology of later types Type | log Age | $\rm\chi^{2}$
---|---|---
| $<8$ | $8-8.7$ | $8.7-9$ | $9-9.4$ | $>9.4$ |
Early | 18.2 | 0 | 57.6 | 0 | 24.2 | 1.4
Later-type | 32.7 | 0 | 53.4 | 0 | 13.9 | 1.3
Figure 15: Spectral fitting with SSP models for an average spectrum of LINERs
with early-type morphology (left), and with morphology of later type (right).
Thus, our results are consistent with the interpretation that most early-type
LINERs at intermediate redshifts are in fact post-starburst galaxies, as
postulated by the SEAGAL collaboration for lower redshift objects. These
results indicate that LINERs and E+As depict the quenching phase in the
evolution of galaxies massive enough to retain significant amounts of gas
after the stellar-wind and supernova phases of the most massive stars.
### 5.4 The red limit of emission-line galaxies
At each $z$ we have selected galaxies with the reddest continuum (the reddest
quartile at each redshift bin) to construct the combined spectra of the red
envelope or red limit of emission line galaxies. Since we are working with
small numbers of galaxies, typically 5-10, it was necessary to combine the
samples at $z=0.82$ and $z=0.99$ to improve statistics. Nevertheless, because
our red emission-line galaxies are rather luminous, the combined spectra still
have high continuum S/N ratios (Table LABEL:redest). The common parts of the
red envelopes of spectra at $<z>=0.29$, $<z>=0.43$, $<z>=0.65$ are similar,
while the red limit at $<z>=0.9$ has noticeably stronger UV continuum. The
spectra in different bins are shown in Figure 7 of Giraud et al. (2010). Both
the continuum and the indexes of the red limit at $z\leq 0.65$ (i.e. $\rm
D(4000)\sim 1.35-1.45;EQW([OII])\sim 4-8$), are typical of nearby spirals with
prominent bulges and low star formation (Kennicutt, 1992; Kinney et al., 1996;
Balogh et al., 1999), or early-type LINERs. Up to $z\sim 0.7$ the populations
of red spirals and early-types can be well separated by their morphology. The
higher UV continuum and lower $\rm D(4000)$ of the limit spectrum at $<z>=0.9$
indicate that such red objects become rare at $z\geq 0.68$ in our sample.
Absorption systems have (by definition) already lost enough gas to suppress
any detected star formation by the time they first appear in our sample at
$z\simeq 1$. At $z=0.8-1$ emission-line galaxies in our sample are found to
have very strong star formation, which declines at lower $z$, the reddest
quartile being bluer than at lower $z$. Therefore the evolutionary paths of
bright absorption and emission systems might have been more separated at
$z\simeq 1$ than at lower redshift suggesting two different physical processes
of different time scales. In one we have early-type LINERs and E+A galaxies
that define the ”entrance gate” to the red sequence of passively evolving
galaxies. In the other we have red spirals with diluted star formation, that
are in a final phase of smooth star formation, possibly of a “main sequence”
(Noeske et al., 2007).
In Section 5.3 we found a large fraction of LINERs in layers at intermediate
$z$. More precisely, in the volume-limited range $0.35\leq z\leq 0.65$, we
find, gathering the counts of Section 5.3, that LINERs are $\rm 23\%$ of all
early-type galaxies with measured redshifts.
Table 11: Equivalent width of [OII], 4000 Å break amplitude, $\rm H\delta$ index, and the G-step of the red envelope of emission-line galaxies. The continuum S/N ratios are given in the last column. $<z>$ | EQW([OII]) | D(4000) | $\rm EQW(H\delta)$ | G step | $\rm S/N$
---|---|---|---|---|---
0.29 | $4.2\pm 0.3$ | $1.35\pm 0.07$ | $-1.8\pm 0.3$ | $1.224\pm 0.017$ | $\rm 19$
$0.43$ | $8.5\pm 0.2$ | $1.39\pm 0.06$ | $-2.5\pm 0.2$ | $1.268\pm 0.014$ | $\rm 24$
$0.65$ | $8.3\pm 0.2$ | $1.44\pm 0.06$ | $-3.0\pm 0.2$ | $1.273\pm 0.012$ | $\rm 29$
$0.9$ | $9.0\pm 0.3$ | $1.30\pm 0.07$ | $-3.9\pm 0.2$ | - | $\rm 18$
## 6 Summary and Conclusions
We have presented a catalogue of galaxy spectra in a pencil beam survey of
$\sim 10.75^{\prime}\times 7.5^{\prime}$, and used these data to make an
analysis of the spectral energy distribution of a magnitude limited sample up
to $z\sim 1$, concentrating on absorption and low ionization emission-line
systems. The redshift range has been divided in bins centered on the
structures that were detected in the (RA, Dec, $z$) pseudo-volume, and
corresponding to cosmic time slices of $\rm\sim 1Gyr$. Our sample is
reasonably complete for galaxies brighter than $\rm M_{R}=-18.8$ up to
$z\approx 0.5$; at $z\geq 0.75$ the cutoff is at -20.5.
From this analysis we reach the following conclusions:
1. 1.
We confirm in our pencil-beam sample the well known result Hamilton (1985)
that absorption-line galaxies do not show significant variations in their
continuum energy distributions up to $z=0.6$, and a moderate decrease of the
4000 Å break amplitude of 5% at $z\sim 0.65$, 7% at $z\sim 0.82$, and up to
15% at $z\sim 1$. Using stellar population synthesis models we find that
absorption-line galaxies at $z\geq 0.65$ show more than 50% of stars younger
than 2.5Gyrs, while those at $z\geq 0.8$ had star formation as recently as
1Gyr ago. This suggests that the red sequence is still in a buildup phase at
$z\leq 1$.
The faint absorption-line galaxies in our dynamically young cluster at
$z=0.29$ have indexes similar to those of bright absorption-line systems at
$z=0.8$, suggesting that faint galaxies without emission lines tend to be
younger than more massive galaxies with similar spectra. Our population
synthesis models indicate that about 50% of the stars contributing to the
luminosity of faint absorption-line galaxies at z = 0.29 were formed at $z<1$.
This is consistent with cases of truncated red sequences observed in some
high-$z$ clusters and suggests that clusters with truncated red-sequences may
be dynamically young.
2. 2.
Combining simple emission-line diagnostics with galaxy morphology we identify
a significant population of early-type LINERs at $0.35\leq z\leq 0.65$. In
that redshift range early-type LINERs constitute about 23% of all early-types
galaxies, a much larger fraction than E+A post-starburst galaxies. However,
our population synthesis models show that early-type LINERs contain
substantial populations of intermediate age stars that can easily explain the
observed line emission, as recently proposed by Cid-Fernandes et al. (2010).
This led us to conclude that most LINERs in our sample are in fact post-
starburst galaxies.
3. 3.
The red limit in the spectral energy distribution of emission-line galaxies at
$z\leq 0.65$ is typical of bulge-dominated spirals with moderate star
formation, and of early-type LINERs. Thus, early-type LINERs and E+As define
the “entrance gate” of the red sequence of passively evolving galaxies, while
bulge-dominated spirals have diluted star formation.
###### Acknowledgements.
EG thanks the hospitality of ESO and Universidad Catolica in Santiago during
the initial phase of this work. JME thanks the hospitality of Nanjing
University during the initial phase of this research. QGU would like to
acknowledge the financial support from the China Scholarship Council (CSC),
the National Natural Science Foundation of China under grants 10878010,
10221001, and 10633040, and the National Basic Research Program (973 program
No. 2007CB815405). HQU thanks partial support from FONDAP “Centro de
Astrofísica”. PZE acknowledge a studentship from CONICYT. We thank S. di
Serego Alighieri for reading a preliminary version of the manuscript and for
his suggestions, and R. Cid-Fernandes for fruitful discussions.
## References
* Adelman-McCarthy et al (2006) Adelman-McCarthy, J.K., et al, 2006, ApJS 162, 38
* Baldwin et al. (1981) Baldwin, J.A., Phillips, M.M., & Terlevitch, R., 1981, PASP 93, 5
* Balogh et al. (1999) Balogh, M. L., Morris, S. L., Yee, H. K. C., Carlberg, R. G., & Ellingson, E., 1999, ApJ 527, 54
* Baugh et al. (1996) Baugh, C.M.,Cole, S., & Frenk, C.S., 1996, MNRAS.283, 1361
* Bell et al. (2005) Bell, E. F., et al, 2005, ApJ 625, 23
* Blake et al. (2004) Blake, C., et al., 2004, MNRAS 355, 713
* Bundy et al (2009) Bundy, K., et al., 2009, astro-ph 0912.1077v1
* Bruzual & Charlot (2003) Bruzual G., & Charlot S., 2003, MNRAS, 344, 1000
* Cid Fernandes et al. (2004) Cid Fernandes R. , et al., 2004, MNRAS, 355, 273
* Cid Fernandes et al. (2005) Cid Fernandes R., Mateus A., Sodre L., Stasinska G., Gomes J., 2005, MNRAS 358, 363
* Cid Fernandes et al. (2010) Cid Fernandes R., et al., 2010, MNRAS 403, 1036
* Cooper et al. (2006) Cooper, M. C. et al, 2006, MNRAS 370, 198
* Cucciati et al. (2006) Cucciati, O., et al (the VVDS Team), 2006, A&A 458, 39
* Dickinson et al. (2003) Dickinson, M., Papovich, C., Fergusson, H.C., Budavari, T., 2003, ApJ 587, 25
* Dressler et al. (1999) Dressler, A., et al, 1999, ApJS 122, 51
* Dressler et al. (1997) Dressler, A., et al, 1997, ApJ 490, 577
* Dressler & Gunn (1983) Dressler A., & Gunn J.E., 1983 ApJ 270 7
* Dressler (1980) Dressler, A., 1980, ApJ 236, 351
* Faber et al. (2007) Faber, S. et al, 2007, ApJ 665, 265
* Faure et al. (2007) Faure, C., Giraud, E., Melnick, J., Quintana, H., Selman, F., & Wambganss, J., 2007, A&A, 463, 833
* Filippenko & Sargent (1985) Filippenko, A.V., & Sargent, W.L., 1985, ApJS 57, 503
* Fillipenko (2003) Filippenko, A.V., 2003 in ASP Conf. Ser. 290 Active Galactic Nuclei: From Central Engine to Host Galaxy, 369
* (23) FORS1+2 User Manual VLT-MAN-ESO-13100-1543 Issue 4, 2005
* FIMS (2006) FORS1+2 FIMS Manual VLT-MAN-ESO-13100-2308 Issue 78, 2006
* Franzetti et al. (2007) Franzetti, P., et al (the VVDS Team), 2007, A&A 465, 711
* Garilli et al. (2008) Garilli, B., et al (the VVDS Team), 2008, astro-ph 0804.4568
* Giraud et al. (2010) Giraud E., Melnick J., Gu Q.S., et al., 2010, in press
* Goto (2007) Goto, T., 2007, MNRAS 377, 1222
* Goudfrooij et al. (1994) Goudfrooij, P., et al., 1994, A & AS 105, 341
* Gu et al. (2006) Gu Q. et al. 2006, MNRAS, 366, 480
* Hamilton (1985) Hamilton, D., 1985 ApJ, 297, 371
* Hammer et al. (2005) Hammer, F., et al, 2005, A&A 430, 115
* (33) Ho, L.C.W., et al., 1997a, ApJS 112, 315
* (34) Ho, L.C.W., et al., 1997b, ApJ 487, 568
* Ho (2004) Ho, L.C.W,. 2004, in Coevolution of Black Holes and Galaxies, 293
* Huang & Gu (2009) Huang, S., & Gu Q., 2009, MNRAS 398, 165
* Kauffmann et al. (2003) Kauffmann, G., et al, 2003, MNRAS 341, 33
* Kauffmann et al. (2003) Kauffmann, G., et al, 2003, MNRAS 346, 1055
* Kennicutt (1992) Kennicutt, R.C.., 1992, ApJS 79, 255
* Kinney et al. (1996) Kinney, A.L., Calzetti, D., Bohlin, R.C., McQuade, K., Storchi-Bergmann, T., & Schmitt, H.R., 1996, ApJ 467, 38
* Kodama et al. (2004) Kodama, T., et al, 2004, MNRAS 350, 1005
* Koyama et al. (2007) Koyama, Y., Kodama, T., Tanaka, M, Shimasaku, K., & Okamura, S., 2007, MNRAS 382, 1719
* Lin et al. (2008) Lin, L., et al., 2008, ApJ 681, 232
* Melnick & Sargent (1977) Melnick, J., & Sargent, L. W. L., 1977, ApJ 215, 401
* Melnick et al. (1999) Melnick, J., Selman, F., & Quintana, H., 1999, PASP 111, 1444
* Neinstein et al. (2006) Neinstein, E., van den Bosch, F.C., & Dekel, A., 2006, MNRAS 372, 933
* Noeske et al. (2007) Noeske, K.G., et al., 2007, ApJ 660, L43
* Norton et al. (2001) Norton, S.A., Gebhardt, K., Zabludoff, A.I., & Zaritsky, D., 2001, ApJ 557, 150
* Pei & Fall (1995) Pei, Y. C., & Fall, S. M., 1995, ApJ 454, 69
* Phillips et al. (1986) Phillips, M.M., et al., 1986, AJ 91, 1062
* (51) p2pp, 2006: http://www.eso.org/observing/p2pp/
* Postman et al. (2005) Postman, M., et al, 2005, ApJ 623, 721
* Renzini (2006) Renzini, A., 2006, ARA&A 44, 141
* Renzini (2007) Renzini, A., 2007, ASPC 390, 309
* Sarzi et al. (2006) Sarzi, M., et al., 2006, MNRAS 366, 1151
* Scarlata et al. (2007) Scarlata, C., et al., 2007, ApJS 172, 406
* Scoville et al. (2007) Scoville, N., et al, 2007a, ApJS 172, 1
* Scoville et al. (2007) Scoville, N., et al, 2007b, ApJS 172, 150
* Smith et al. (2005) Smith, G. P., Treu, T., Ellis, R. S., Moran, S. M., & Dressler, A., 2005, ApJ 620, 78
* Tanaka et al. (2005) Tanaka, M., et al, 2005, MNRAS 362, 268
* Toledo et al. (2010) Toledo, I., et al. 2010 in preparation
* (62) Tonry, J., & Davis, M., 1979, AJ 84, 1511.
* Tran et al. (2007) Tran, K.-V. H., et al, 2007, ApJ 661, 750
* Tran et al. (2004) Tran, K.-V. H., et al, 2004, ApJ 609, 683
* Tran et al. (2003) Tran, K.-V. H,. et al, 2003, ApJ 599, 865
* Veilleux & Osterbrock (1987) Veilleux, S., & Osterbrock, D.E., 1987, ApJS 63, 295
* Weiner et al. (2005) Weiner, B.J., et al, 2005, ApJ 620, 595
* Weiner et al. (2006) Weiner, B.J., et al, 2006, ApJ 653, 1027
* Wild et al (2009) Wild V. et al. 2009 MNRAS 395 144
* Yan et al. (2009) Yan, R., et al., 2009, MNRAS 398, 735
* Yan et al. (2006) Yan, R., Newman, J.A., Faber, S., Konidaris, N., Koo, D., & Davis, M., 2006 ApJ 648 281
* Yang et al. (2008) Yang Y., Zabludoff, A.I., Zaritsky, D., & Mihos, J.C., 2008, ApJ 688, 945
* Zeilinger et al. (1996) Zeilinger, W.W., et al., 1996, A & AS 120, 257
|
arxiv-papers
| 2010-11-09T00:16:11 |
2024-09-04T02:49:14.598354
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "E. Giraud (1), Q.-S. Gu (2), J. Melnick (3), H. Quintana (4), F.\n Selman (3), I. Toledo (4), P. Zelaya (4) ((1) LPTA, Universit\\'e Montpellier\n France, (2) Nanjing University, China, (3) ESO, Chile, (4) P. Universidad\n Catolica de Chile)",
"submitter": "Qiusheng Gu",
"url": "https://arxiv.org/abs/1011.1947"
}
|
1011.1990
|
# Fluid Dynamic Limit to the Riemann Solutions of Euler Equations: I.
Superposition of rarefaction waves and contact discontinuity
fhuang@amt.ac.cn;wangyi@amss.ac.cn;matyang@cityu.edu.hk
###### Abstract.
Fluid dynamic limit to compressible Euler equations from compressible Navier-
Stokes equations and Boltzmann equation has been an active topic with limited
success so far. In this paper, we consider the case when the solution of the
Euler equations is a Riemann solution consisting two rarefaction waves and a
contact discontinuity and prove this limit for both Navier-Stokes equations
and the Boltzmann equation when the viscosity, heat conductivity coefficients
and the Knudsen number tend to zero respectively. In addition, the uniform
convergence rates in terms of the above physical parameters are also obtained.
It is noted that this is the first rigorous proof of this limit for a Riemann
solution with superposition of three waves even though the fluid dynamic limit
for a single wave has been proved.
###### Key words and phrases:
Compressible Navier-Stokes equations, Boltzmann equation, rarefaction wave,
contact discontinuity, fluid dynamic limit.
###### 1991 Mathematics Subject Classification:
Primary: 35Q30, 35Q20, 76N15, 76P05; Secondary: 35L65, 82B40, 82C40.
Feimin Huang and Yi Wang
Institute of Applied Mathematics, AMSS and
Hua Loo-Keng Key Laboratory of Mathematics, Academia Sinica
Beijing 100190, China
Tong Yang
Department of Mathematics, City University of HongKong
HongKong, China
(Communicated by Seiji Ukai)
###### Contents
1. 1 Introduction
2. 2 Main results
1. 2.1 Compressible Navier-Stokes equations
1. 2.1.1 Contact discontinuity
2. 2.1.2 Rarefaction waves
3. 2.1.3 Superposition of rarefaction waves and contact discontinuity
4. 2.1.4 Main result to the compressible Navier-Stokes equations
2. 2.2 Boltzmann equation
1. 2.2.1 Contact discontinuity
2. 2.2.2 Rarefaction waves
3. 2.2.3 Superposition of rarefaction waves and contact discontinuity
4. 2.2.4 Main result to Boltzmann equation
3. 3 Proof of Theorem 2.4: Zero dissipation limit of Navier-Stokes equations
4. 4 Proof of Theorem 2.5: Hydrodynamic limit of Boltzmann equation
## 1\. Introduction
This paper concerns the fluid dynamic limit to the compressible Euler
equations for two physical models, that is, the compressible Navier-Stokes
equations and the Boltzmann equation. In the first part, we consider zero
dissipation limit of the compressible Navier-Stokes system for viscous and
heat conductive fluid in the Lagrangian coordinates:
$\left\\{\begin{array}[]{ll}\displaystyle v_{t}-u_{x}=0,\\\ \displaystyle
u_{t}+p_{x}=\varepsilon(\frac{u_{x}}{v})_{x},\\\
\displaystyle\bigl{(}e+\frac{u^{2}}{2}\bigr{)}_{t}+(pu)_{x}=(\kappa\frac{\theta_{x}}{v}+\varepsilon\frac{uu_{x}}{v})_{x},\end{array}\right.$
(1.1)
where the functions $v(t,x)>0,u(t,x),\theta(t,x)>0$ represent the specific
volume, velocity and the absolute temperature of the gas respectively. And
$p=p(v,\theta)$ is the pressure, $e=e(v,\theta)$ is the internal energy,
$\varepsilon>0$ is the viscosity coefficient, $\kappa>0$ is the coefficient of
the heat conductivity. Here, both $\varepsilon$ and $\kappa$ are taken as
positive constants. And we consider the perfect gas where
$p=\frac{R\theta}{v}=Av^{-\gamma}\exp\big{(}\frac{\gamma-1}{R}s\big{)},\qquad
e=\frac{R\theta}{\gamma-1},$ (1.2)
with $s$ denoting the entropy of the gas and $A,R>0$ , $\gamma>1$ being the
gas parameters.
Formally, as the coefficients $\kappa$ and $\varepsilon$ tend to zero, the
limiting system of (1.1) is the compressible Euler equations
$\left\\{\begin{array}[]{ll}v_{t}-u_{x}=0,\\\ u_{t}+p_{x}=0,\\\
(e+\frac{u^{2}}{2})_{t}+(pu)_{x}=0.\end{array}\right.$ (1.3)
The study of this limiting process of viscous flows when the viscosity and
heat conductivity coefficients tend to zero, is one of the important problems
in the theory of the compressible fluid. When the solution of the inviscid
flow is smooth, the zero dissipation limit can be solved by classical scaling
method. However, the inviscid compressible flow usually contains
discontinuities, such as shock waves and contact discontinuities. Therefore,
how to justify the zero dissipation limit to the Euler equations with basic
wave patterns is a natural and difficult problem.
Keeping in mind that the Navier-Stokes equations can be derived from the
Boltzmann equation through the Chapman-Enskog expansion when the Knudsen
number is close to zero, we assume the following condition on the viscosity
constant $\varepsilon$ and the heat conductivity coefficient $\kappa$ in the
system (1.1), cf. also [17]:
$\left\\{\begin{array}[]{l}\displaystyle\kappa=O(\varepsilon)\qquad\qquad\rm
as\qquad\varepsilon\rightarrow 0;\\\
\displaystyle\nu\doteq\frac{\kappa(\varepsilon)}{\varepsilon}\geq
c>0\qquad{\rm for~{}some~{}positive~{}constant}~{}c,\quad\rm
as\quad\varepsilon\rightarrow 0.\end{array}\right.$ (1.4)
Now we briefly review some recent results on the zero dissipation limit of the
compressible fluid with basic wave patterns. For the hyperbolic conservation
laws with artificial viscosity
$u_{t}+f(u)_{x}=\varepsilon u_{xx},$
Goodman-Xin [9] verified the viscous limit for piecewise smooth solutions
separated by non-interacting shock waves using a matched asymptotic expansion
method. For the compressible isentropic Navier-Stokes equations, Hoff-Liu [12]
first proved the vanishing viscosity limit for piecewise constant solutions
separated by non-interacting shocks even with initial layer. Later Xin [30]
obtained the zero dissipation limit for rarefaction waves and Wang [28]
generalized the result of Goodmann-Xin [9] to the isentropic Navier-Stokes
equations.
For the inviscid limit of the full compressible Navier-Stokes equations (1.1),
Jiang-Ni-Sun [17] justified the zero dissipation limit of the system (1.1) for
centered rarefaction waves. Wang [29] proved the zero dissipation limit of the
system (1.1) for piecewise smooth solutions separated by shocks using the
matched asymptotic expansion method introduced in [9]. Recently, Xin-Zeng [31]
considered the zero dissipation limit of the system (1.1) for single
rarefaction wave with well prepared initial data and obtained a uniform decay
rate in terms of the dissipation coefficients. And Ma [22] obtained the zero
dissipation limit of a single strong contact discontinuity in any fixed time
interval with a decay rate.
However, to our knowledge, so far there is no result on the zero dissipation
limit of the system (1.1) for superposition of different types of basic wave
patterns. In the first part of this paper, we investigate the fluid dynamic
limit of the compressible Navier-Stokes equations when the corresponding Euler
equations have the Riemann solution as a superposition of two rarefaction
waves and a contact discontinuity. For this, we need to study the interaction
between the rarefaction waves and contact discontinuity.
In the second part of the paper, we study the hydrodynamic limit of the
Boltzmann equation [2] with slab symmetry
$f_{t}+\xi_{1}f_{x}=\frac{1}{\varepsilon}Q(f,f),~{}(f,t,x,\xi)\in{\mathbf{R}}\times{\mathbf{R}}^{+}\times{\mathbf{R}}\times{\mathbf{R}}^{3},$
(1.5)
where $\xi=(\xi_{1},\xi_{2},\xi_{3})\in{\mathbf{R}}^{3}$, $f(t,x,\xi)$ is the
density distribution function of particles at time $t$ with location $x$ and
velocity $\xi$, and $\varepsilon>0$ is called the Knudsen number which is
proportional to the mean free path. Remark that the notation $\varepsilon$
here is same as the viscosity of the compressible Navier-Stokes equations
(1.1), but it has different physical meanings from (1.1) in different
equations and related contexts.
For monatomic gas, the rotational invariance of the particles leads to the
following bilinear form for the collision operator
$\begin{array}[]{ll}\displaystyle
Q(f,g)(\xi)=\frac{1}{2}\int_{{\mathbf{R}}^{3}}\\!\\!\int_{{\mathbf{S}}_{+}^{2}}\Big{(}f(\xi^{\prime})g(\xi_{*}^{\prime})+f(\xi_{*}^{\prime})g(\xi^{\prime})-f(\xi)g(\xi_{*})-f(\xi_{*})g(\xi)\Big{)}\\\
\displaystyle\hskip 227.62204pt\qquad
B(|\xi-\xi_{*}|,\hat{\theta})\;d\xi_{*}d\Gamma,\end{array}$
where $\xi^{\prime},\xi_{*}^{\prime}$ are the velocities after an elastic
collision of two particles with velocities $\xi,\xi_{*}$ before the collision.
Here, $\hat{\theta}$ is the angle between the relative velocity $\xi-\xi_{*}$
and the unit vector $\Gamma$ in
${\mathbf{S}}^{2}_{+}=\\{\Gamma\in{\mathbf{S}}^{2}:\
(\xi-\xi_{*})\cdot\Gamma\geq 0\\}$. The conservation of momentum and energy
gives the following relation between the velocities before and after
collision:
$\left\\{\begin{array}[]{l}\xi^{\prime}=\xi-[(\xi-\xi_{*})\cdot\Gamma]\;\Gamma,\\\\[8.53581pt]
\xi_{*}^{\prime}=\xi_{*}+[(\xi-\xi_{*})\cdot\Gamma]\;\Gamma.\end{array}\right.$
In this paper, we consider the Boltzmann equation for two basic models, that
is, the hard sphere model and the hard potential including Maxwellian
molecules under the assumption of angular cut-off. For this, we assume that
the collision kernel $B(|\xi-\xi_{*}|,\hat{\theta})$ takes one of the
following two forms,
$B(|\xi-\xi_{*}|,\hat{\theta})=|(\xi-\xi_{*},\Gamma)|=|\xi-\xi_{*}|\cos\hat{\theta},$
and
$B(|\xi-\xi_{*}|,\hat{\theta})=|\xi-\xi_{*}|^{\frac{n-5}{n-1}}b(\hat{\theta}),\quad
b(\hat{\theta})\in L^{1}([0,\pi]),~{}n\geq 5.$
Here, $n$ is the index in the potential of inverse power law which is
proportional to $r^{1-n}$ with $r$ being the distance between two concerned
particles.
Formally, when the Knudsen number $\varepsilon$ tends to zero, the limit of
the Boltzmann equation (1.5) is the classical system of Euler equations
$\left\\{\begin{array}[]{l}\displaystyle\rho_{t}+(\rho u_{1})_{x}=0,\\\
\displaystyle(\rho u_{1})_{t}+(\rho u_{1}^{2}+p)_{x}=0,\\\ \displaystyle(\rho
u_{i})_{t}+(\rho u_{1}u_{i})_{x}=0,~{}i=2,3,\\\
\displaystyle[\rho(E+\frac{|u|^{2}}{2})]_{t}+[\rho
u_{1}(E+\frac{|u|^{2}}{2})+pu_{1}]_{x}=0,\end{array}\right.$ (1.6)
where
$\left\\{\begin{array}[]{l}\displaystyle\rho(t,x)=\int_{\mathbf{R}^{3}}\varphi_{0}(\xi)f(t,x,\xi)d\xi,\\\
\displaystyle\rho
u_{i}(t,x)=\int_{\mathbf{R}^{3}}\varphi_{i}(\xi)f(t,x,\xi)d\xi,~{}i=1,2,3,\\\
\displaystyle\rho(E+\frac{|u|^{2}}{2})(t,x)=\int_{\mathbf{R}^{3}}\varphi_{4}(\xi)f(t,x,\xi)d\xi.\end{array}\right.$
(1.7)
Here, $\rho$ is the density, $u=(u_{1},u_{2},u_{3})$ is the macroscopic
velocity, $E$ is the internal energy of the gas, and $p=R\rho\theta$ with $R$
being the gas constant is the pressure. Note that the temperature $\theta$ is
related to the internal energy by $E=\frac{3}{2}R\theta$, and
$\varphi_{i}(\xi)(i=0,1,2,3,4)$ are the collision invariants given by
$\left\\{\begin{array}[]{l}\varphi_{0}(\xi)=1,\\\ \varphi_{i}(\xi)=\xi_{i}\ \
{\textrm{for}}\ \ i=1,2,3,\\\
\varphi_{4}(\xi)=\frac{1}{2}|\xi|^{2},\end{array}\right.$
that satisfy
$\int_{{\mathbf{R}}^{3}}\varphi_{i}(\xi)Q(h,g)d\xi=0,\quad{\textrm{for}}\ \
i=0,1,2,3,4.$
How to justify the above limit, that is, the Euler equation (1.6) from
Boltzmann equation (1.5) when Knudsen number $\varepsilon$ tends to zero is an
open problem going way back to the time of Maxwell. For this, Hilbert
introduced the famous Hilbert expansion to show formally that the first order
approximation of the Boltzmann equation gives the Euler equations. On the
other hand, it is important to verify this limit process rigorously in
mathematics. For the case when the Euler equation has smooth solutions, the
zero Knudsen number limit of the Boltzmann equation has been studied even in
the case with an initial layer, cf. Ukai-Asano [26], Caflish [3], Lachowicz
[18] and Nishida [24] etc. However, as is well-known, solutions of the Euler
equations (1.6) in general develop singularities, such as shock waves and
contact discontinuities. Therefore, how to verify the fluid limit from
Boltzmann equation to the Euler equations with basic wave patterns becomes an
natural problem. In this direction, Yu [32] showed that when the solution of
the Euler equations (1.6) contains only non-interacting shocks, there exists a
sequence of solutions to the Boltzmann equation that converge to a local
Maxwellian defined by the solution of the Euler equations (1.6) uniformly away
from the shock in any fixed time interval. In this work, the inner and outer
expansions developed by Goodman-Xin [9] for conservation laws and the Hilbert
expansion were skillfully and cleverly used. Recently, Huang-Wang-Yang [15]
proved the fluid dynamic limit of the Boltzmann equation to the Euler
equations for a single contact discontinuity where the uniform decay rate was
also obtained. And Xin-Zeng [31] proved the fluid dynamic limit of the
compressible Navier-Stokes equations and Boltzmann equation to the Euler
equations with non-interacting rarefaction waves. About the detailed
introductions of the Boltzmann equation and its hydrodynamic limit, see the
books [4], [7] etc.
In this paper, we will study the hydrodynamic limit of the Boltzmann equation
when the corresponding Euler equations have a Riemann solution as a
superposition of two rarefaction waves and a contact discontinuity. More
precisely, given a Riemann solution of the Euler equations (1.6) with
superposition of two rarefaction waves and a contact discontinuity, we will
show that there exists a family of solutions to the Boltzmann equation that
converge to a local Maxwellian defined by the Euler solution uniformly away
from the contact discontinuity for strictly positive time as
$\varepsilon\rightarrow 0$. Moreover, a uniform convergence rate in
$\varepsilon$ is also given.
As mentioned above for the compressible Navier-Stokes equations, we also need
to study the detailed wave interactions through this limiting process.
For later use, we now briefly present the micro-macro decomposition around the
local Maxwellian defined by the solution to the Boltzmann equation, cf. [19]
and [21]. For a solution $f(t,x,\xi)$ of the Boltzmann equation (1.5), set
$f(t,x,\xi)=\mathbf{M}(t,x,\xi)+\mathbf{G}(t,x,\xi),$
where the local Maxwellian
$\mathbf{M}(t,x,\xi)=\mathbf{M}_{[\rho,u,\theta]}(\xi)$ represents the
macroscopic (fluid) component of the solution, which is naturally defined by
the five conserved quantities, i.e., the mass density $\rho(t,x)$, the
momentum $\rho u(t,x)$, and the total energy $\rho(E+\frac{1}{2}|u|^{2})(t,x)$
in (1.7), through
$\mathbf{M}=\mathbf{M}_{[\rho,u,\theta]}(t,x,\xi)=\frac{\rho(t,x)}{\sqrt{(2\pi
R\theta(t,x))^{3}}}e^{-\frac{|\xi-u(t,x)|^{2}}{2R\theta(t,x)}}.$ (1.8)
And $\mathbf{G}(t,x,\xi)$ being the difference between the solution and the
above localMaxwellian represents the microscopic (non-fluid) component.
For convenience, we denote the inner product of $h$ and $g$ in
$L^{2}_{\xi}({\mathbf{R}}^{3})$ with respect to a given Maxwellian
$\tilde{\mathbf{M}}$ by:
$\langle
h,g\rangle_{\tilde{\mathbf{M}}}\equiv\int_{{\mathbf{R}}^{3}}\frac{1}{\tilde{\mathbf{M}}}h(\xi)g(\xi)d\xi.$
If $\tilde{\mathbf{M}}$ is the local Maxwellian $\mathbf{M}$ defined in (1.8),
with respect to the corresponding inner product, the macroscopic space is
spanned by the following five pairwise orthogonal base
$\left\\{\begin{array}[]{l}\chi_{0}(\xi)\equiv{\displaystyle\frac{1}{\sqrt{\rho}}\mathbf{M}},\\\\[5.69054pt]
\chi_{i}(\xi)\equiv{\displaystyle\frac{\xi_{i}-u_{i}}{\sqrt{R\theta\rho}}\mathbf{M}}\
\ {\textrm{for}}\ \ i=1,2,3,\\\\[5.69054pt]
\chi_{4}(\xi)\equiv{\displaystyle\frac{1}{\sqrt{6\rho}}(\frac{|\xi-u|^{2}}{R\theta}-3)\mathbf{M}},\\\
\langle\chi_{i},\chi_{j}\rangle=\delta_{ij},~{}i,j=0,1,2,3,4.\end{array}\right.$
In the following, if $\tilde{\mathbf{M}}$ is the local Maxwellian
$\mathbf{M}$, we just use the simplified notation $\langle\cdot,\cdot\rangle$
to denote the inner product $\langle\cdot,\cdot\rangle_{\mathbf{M}}$. The
macroscopic projection $\mathbf{P}_{0}$ and microscopic projection
$\mathbf{P}_{1}$ can be defined as follows
$\left\\{\begin{array}[]{l}\mathbf{P}_{0}h={\displaystyle\sum_{j=0}^{4}\langle
h,\chi_{j}\rangle\chi_{j},}\\\
\mathbf{P}_{1}h=h-\mathbf{P}_{0}h.\end{array}\right.$
The projections $\mathbf{P}_{0}$ and $\mathbf{P}_{1}$ are orthogonal and
satisfy
$\mathbf{P}_{0}\mathbf{P}_{0}=\mathbf{P}_{0},\mathbf{P}_{1}\mathbf{P}_{1}=\mathbf{P}_{1},\mathbf{P}_{0}\mathbf{P}_{1}=\mathbf{P}_{1}\mathbf{P}_{0}=0.$
Note that a function $h(\xi)$ is called microscopic or non-fluid if
$\int h(\xi)\varphi_{i}(\xi)d\xi=0,~{}i=0,1,2,3,4,$
where $\varphi_{i}(\xi)$ is the collision invariants.
Under the above micro-macro decomposition, the solution $f(t,x,\xi)$ of the
Boltzmann equation (1.5) satisfies
$\mathbf{P}_{0}f=\mathbf{M},~{}\mathbf{P}_{1}f=\mathbf{G},$
and the Boltzmann equation (1.5) becomes
$(\mathbf{M}+\mathbf{G})_{t}+\xi_{1}(\mathbf{M}+\mathbf{G})_{x}=\frac{1}{\varepsilon}[2Q(\mathbf{M},\mathbf{G})+Q(\mathbf{G},\mathbf{G})].$
(1.9)
By multiplying the equation (1.9) by the collision invariants
$\varphi_{i}(\xi)(i=0,1,2,3,4)$ and integrating the resulting equations with
respect to $\xi$ over ${\mathbf{R}}^{3}$, one has the following fluid-type
system for the fluid components:
$\left\\{\begin{array}[]{lll}\displaystyle\rho_{t}+(\rho u_{1})_{x}=0,\\\
\displaystyle(\rho u_{1})_{t}+(\rho
u_{1}^{2}+p)_{x}=-\int\xi_{1}^{2}\mathbf{G}_{x}d\xi,\\\ \displaystyle(\rho
u_{i})_{t}+(\rho
u_{1}u_{i})_{x}=-\int\xi_{1}\xi_{i}\mathbf{G}_{x}d\xi,~{}i=2,3,\\\
\displaystyle[\rho(E+\frac{|u|^{2}}{2})]_{t}+[\rho
u_{1}(E+\frac{|u|^{2}}{2})+pu_{1}]_{x}=-\int\frac{1}{2}\xi_{1}|\xi|^{2}\mathbf{G}_{x}d\xi.\end{array}\right.$
(1.10)
Note that the above fluid-type system is not closed and one more equation for
the non-fluid component ${\mathbf{G}}$ is needed and it can be obtained by
applying the projection operator $\mathbf{P}_{1}$ to the equation (1.9):
$\mathbf{G}_{t}+\mathbf{P}_{1}(\xi_{1}\mathbf{M}_{x})+\mathbf{P}_{1}(\xi_{1}\mathbf{G}_{x})=\frac{1}{\varepsilon}\left[\mathbf{L}_{\mathbf{M}}\mathbf{G}+Q(\mathbf{G},\mathbf{G})\right].$
(1.11)
Here $\mathbf{L}_{\mathbf{M}}$ is the linearized collision operator of
$Q(f,f)$ with respect to the local Maxwellian $\mathbf{M}$:
$\mathbf{L}_{\mathbf{M}}h=2Q(\mathbf{M},h)=Q(\mathbf{M},h)+Q(h,\mathbf{M}).$
Note that the null space $\mathfrak{N}$ of $\mathbf{L}_{\mathbf{M}}$ is
spanned by the macroscopic variables:
$\chi_{j}(\xi),~{}j=0,1,2,3,4.$
Furthermore, there exists a positive constant $\sigma_{0}>0$ such that for any
function $h(\xi)\in\mathfrak{N}^{\bot}$, cf. [10],
$<h,\mathbf{L}_{\mathbf{M}}h>\leq-\sigma_{0}<\nu(|\xi|)h,h>,$
where $\nu(|\xi|)$ is the collision frequency. For the hard sphere model and
the hard potential including Maxwellian molecules with angular cut-off, the
collision frequency $\nu(|\xi|)$ has the following property
$0<\nu_{0}<\nu(|\xi|)\leq c(1+|\xi|)^{\kappa_{0}},$
for some positive constants $\nu_{0},c$ and $0\leq\kappa_{0}\leq 1$.
Consequently, the linearized collision operator $\mathbf{L}_{\mathbf{M}}$ is a
dissipative operator on $L^{2}({\mathbf{R}}^{3})$, and its inverse
$\mathbf{L}_{\mathbf{M}}^{-1}$ exists in $\mathfrak{N}^{\bot}$.
It follows from (1.11) that
$\mathbf{G}=\varepsilon\mathbf{L}_{\mathbf{M}}^{-1}[\mathbf{P}_{1}(\xi_{1}\mathbf{M}_{x})]+\Pi,$
(1.12)
with
$\Pi=\mathbf{L}_{\mathbf{M}}^{-1}[\varepsilon(\mathbf{G}_{t}+\mathbf{P}_{1}(\xi_{1}\mathbf{G}_{x}))-Q(\mathbf{G},\mathbf{G})].$
(1.13)
Plugging the equation (1.12) into (1.10) gives
$\left\\{\begin{array}[]{l}\displaystyle\rho_{t}+(\rho u_{1})_{x}=0,\\\
\displaystyle(\rho u_{1})_{t}+(\rho
u_{1}^{2}+p)_{x}=\frac{4\varepsilon}{3}(\mu(\theta)u_{1x})_{x}-\int\xi_{1}^{2}\Pi_{x}d\xi,\\\
\displaystyle(\rho u_{i})_{t}+(\rho
u_{1}u_{i})_{x}=\varepsilon(\mu(\theta)u_{ix})_{x}-\int\xi_{1}\xi_{i}\Pi_{x}d\xi,~{}i=2,3,\\\
\displaystyle[\rho(E+\frac{|u|^{2}}{2})]_{t}+[\rho
u_{1}(E+\frac{|u|^{2}}{2})+pu_{1}]_{x}=\varepsilon(\lambda(\theta)\theta_{x})_{x}+\frac{4\varepsilon}{3}(\mu(\theta)u_{1}u_{1x})_{x}\\\
\displaystyle\qquad\qquad+\varepsilon\sum_{i=2}^{3}(\mu(\theta)u_{i}u_{ix})_{x}-\int\frac{1}{2}\xi_{1}|\xi|^{2}\Pi_{x}d\xi,\end{array}\right.$
(1.14)
where the viscosity coefficient $\mu(\theta)>0$ and the heat conductivity
coefficient $\lambda(\theta)>0$ are smooth functions of the temperature
$\theta$. Here, we normalize the gas constant $R$ to be $\frac{2}{3}$ so that
$E=\theta$ and $p=\frac{2}{3}\rho\theta$. The explicit formula of
$\mu(\theta)$ and $\lambda(\theta)$ can be found for example in [5], we omit
it here for brevity.
Since the problem considered in this paper is one dimensional in the space
variable $x\in{\bf R}$, in the macroscopic level, it is more convenient to
rewrite the equation (1.5) and the system (1.6) in the Lagrangian coordinates
as in the study of conservation laws. That is, set the coordinate
transformation:
$x\Rightarrow\int_{0}^{x}\rho(t,y)dy,\qquad t\Rightarrow t.$
We will still denote the Lagrangian coordinates by $(t,x)$ for simplicity of
notation. Then (1.5) and (1.6) in the Lagrangian coordinates become,
respectively,
$f_{t}-\frac{u_{1}}{v}f_{x}+\frac{\xi_{1}}{v}f_{x}=\frac{1}{\varepsilon}Q(f,f),$
(1.15)
and
$\left\\{\begin{array}[]{llll}\displaystyle v_{t}-u_{1x}=0,\\\ \displaystyle
u_{1t}+p_{x}=0,\\\ \displaystyle u_{it}=0,~{}i=2,3,\\\
\displaystyle(\theta+\frac{|u|^{2}}{2}\bigr{)}_{t}+(pu_{1})_{x}=0.\\\
\end{array}\right.$ (1.16)
Also, (1.10)-(1.14) take the form
$\left\\{\begin{array}[]{llll}\displaystyle v_{t}-u_{1x}=0,\\\ \displaystyle
u_{1t}+p_{x}=-\int\xi_{1}^{2}\mathbf{G}_{x}d\xi,\\\ \displaystyle
u_{it}=-\int\xi_{1}\xi_{i}\mathbf{G}_{x}d\xi,~{}i=2,3,\\\
\displaystyle\bigl{(}\theta+\frac{|u|^{2}}{2}\bigr{)}_{t}+(pu_{1})_{x}=-\int\frac{1}{2}\xi_{1}|\xi|^{2}\mathbf{G}_{x}d\xi,\\\
\end{array}\right.$ (1.17)
$\mathbf{G}_{t}-\frac{u_{1}}{v}\mathbf{G}_{x}+\frac{1}{v}\mathbf{P}_{1}(\xi_{1}\mathbf{M}_{x})+\frac{1}{v}\mathbf{P}_{1}(\xi_{1}\mathbf{G}_{x})=\frac{1}{\varepsilon}(\mathbf{L}_{\mathbf{M}}\mathbf{G}+Q(\mathbf{G},\mathbf{G})),$
(1.18)
with
$\mathbf{G}=\varepsilon\mathbf{L}^{-1}_{\mathbf{M}}(\frac{1}{v}\mathbf{P}_{1}(\xi_{1}\mathbf{M}_{x}))+\Pi_{1},$
(1.19)
$\Pi_{1}=\mathbf{L}_{\mathbf{M}}^{-1}[\varepsilon(\mathbf{G}_{t}-\frac{u_{1}}{v}\mathbf{G}_{x}+\frac{1}{v}\mathbf{P}_{1}(\xi_{1}\mathbf{G}_{x}))-Q(\mathbf{G},\mathbf{G})].$
(1.20)
and
$\left\\{\begin{array}[]{llll}\displaystyle v_{t}-u_{1x}=0,\\\ \displaystyle
u_{1t}+p_{x}=\frac{4\varepsilon}{3}(\frac{\mu(\theta)}{v}u_{1x})_{x}-\int\xi_{1}^{2}\Pi_{1x}d\xi,\\\
\displaystyle
u_{it}=\varepsilon(\frac{\mu(\theta)}{v}u_{ix})_{x}-\int\xi_{1}\xi_{i}\Pi_{1x}d\xi,~{}i=2,3,\\\
\displaystyle\bigl{(}\theta+\frac{|u|^{2}}{2}\bigr{)}_{t}+(pu_{1})_{x}=\varepsilon(\frac{\lambda(\theta)}{v}\theta_{x})_{x}+\frac{4\varepsilon}{3}(\frac{\mu(\theta)}{v}u_{1}u_{1x})_{x}\\\
\displaystyle\qquad+\varepsilon\sum_{i=2}^{3}(\frac{\mu(\theta)}{v}u_{i}u_{ix})_{x}-\int\frac{1}{2}\xi_{1}|\xi|^{2}\Pi_{1x}d\xi.\end{array}\right.$
(1.21)
With the above preparation, the main results in this paper for both the
compressible Navier-Stokes equations and the Boltzmann equation will be given
in the next section. And the proof of the zero dissipation limit for the
compressible Navier-Stokes equations will be given in Section 3 while the
proof of hydrodynamic limit for the Boltzmann equation will be given in the
last section.
## 2\. Main results
### 2.1. Compressible Navier-Stokes equations
It is well known that for the Euler equations, there are three basic wave
patterns, shock, rarefaction wave and contact discontinuity. And the Riemann
solution to the Euler equations has a basic wave pattern consisting the
superposition of these three waves with the contact discontinuity in the
middle. For later use, let us firstly recall the wave curves for the two types
of basic waves studied in this paper.
Given the right end state $(v_{+},u_{+},\theta_{+})$, the following wave
curves in the phase space $(v,u,\theta)$ are defined with $v>0$ and $\theta>0$
for the Euler equations.
$\bullet$ Contact discontinuity wave curve:
$CD(v_{+},u_{+},\theta_{+})=\\{(v,u,\theta)|u=u_{+},p=p_{+},v\not\equiv
v_{+}\\}.$ (2.1)
$\bullet$ $i$-Rarefaction wave curve $(i=1,3)$:
$R_{i}(v_{+},u_{+},\theta_{+}):=\Bigg{\\{}(v,u,\theta)\Bigg{|}v<v_{+},~{}u=u_{+}-\int^{v}_{v_{+}}\lambda_{i}(\eta,s_{+})\,d\eta,~{}s(v,\theta)=s_{+}\Bigg{\\}}$
(2.2)
where $s_{+}=s(v_{+},\theta_{+})$ and $\lambda_{i}=\lambda_{i}(v,s)$ is $i$-th
characteristic speed of the Euler system (1.3) or (1.16).
Accordingly, when we study the Navier-Stokes equations, the corresponding wave
profiles can be defined approximately as follows, cf. [16], [30].
#### 2.1.1. Contact discontinuity
If $(v_{-},u_{-},\theta_{-})\in CD(v_{+},u_{+},\theta_{+})$, i.e.,
$u_{-}=u_{+},~{}p_{-}=p_{+},$
then the following Riemann problem of the Euler system (1.3) with Riemann
initial data
$(v,u,\theta)(t=0,x)=\left\\{\begin{array}[]{ll}(v_{-},u_{-},\theta_{-}),&x<0,\\\
(v_{+},u_{+},\theta_{+}),&x>0\end{array}\right.$
admits a single contact discontinuity solution
$(v^{cd},u^{cd},\theta^{cd})(t,x)=\left\\{\begin{array}[]{ll}(v_{-},u_{-},\theta_{-}),&x<0,~{}t>0,\\\
(v_{+},u_{+},\theta_{+}),&x>0,~{}t>0.\end{array}\right.$ (2.3)
As in [14], the viscous version of the above contact discontinuity, called
viscous contact wave $(V^{CD},U^{CD},\Theta^{CD})(t,x)$, can be defined as
follows. Since we expect that
$P^{CD}\approx p_{+}=p_{-},\quad\rm{and}\quad|U^{CD}|\ll 1,$
the leading order of the energy equation $(1.1)_{3}$ is
$\frac{R}{\gamma-1}\Theta_{t}+p_{+}U_{x}=\kappa(\frac{\Theta_{x}}{V})_{x}.$
Thus, we can get the following nonlinear diffusion equation
$\Theta_{t}=a\varepsilon(\frac{\Theta_{x}}{\Theta})_{x},\quad\Theta(t,\pm)=\theta_{\pm},\quad
a=\frac{\nu p_{+}(\gamma-1)}{R^{2}\gamma},$
which has a unique self-similar solution
$\hat{\Theta}(t,x)=\hat{\Theta}(\eta),~{}\eta=\frac{x}{\sqrt{\varepsilon(1+t)}}$.
Now the viscous contact wave $(V^{CD},U^{CD},\Theta^{CD})(t,x)$ can be defined
by
$\begin{array}[]{ll}\displaystyle
V^{CD}(t,x)=\frac{R\hat{\Theta}(t,x)}{p_{+}},\\\ \displaystyle
U^{CD}(t,x)=u_{+}+\frac{\kappa(\gamma-1)}{R\gamma}\frac{\hat{\Theta}_{x}(t,x)}{\hat{\Theta}(t,x)},\\\
\displaystyle\Theta^{CD}(t,x)=\hat{\Theta}(t,x)+\frac{\varepsilon[R\gamma-\nu(\gamma-1)]}{\gamma
p_{+}}\hat{\Theta}_{t}.\end{array}$ (2.4)
Here, it is straightforward to check that the viscous contact wave defined in
(2.4) satisfies
$|\Theta^{CD}-\theta_{\pm}|+[\varepsilon(1+t)]^{\frac{1}{2}}|\Theta^{CD}_{x}|+\varepsilon(1+t)|\Theta^{CD}_{xx}|=O(1)\delta^{CD}e^{-\frac{C_{0}x^{2}}{\varepsilon(1+t)}},$
(2.5)
as $|x|\rightarrow+\infty$, where $\delta^{CD}=|\theta_{+}-\theta_{-}|$
represents the strength of the viscous contact wave and $C_{0}$ is a positive
generic constant. Note that in the above definition, the higher order term
$\frac{\varepsilon[R\gamma-\nu(\gamma-1)]}{\gamma p_{+}}\hat{\Theta}_{t}$ is
used in $\Theta^{CD}(t,x)$ so that the viscous contact wave
$(V^{CD},U^{CD},\Theta^{CD})(t,x)$ satisfies the momentum equation exactly.
Precisely, $(V^{CD},U^{CD},\Theta^{CD})(t,x)$ satisfies the system
$\left\\{\begin{array}[]{l}\displaystyle V^{\scriptscriptstyle
CD}_{t}-U^{CD}_{x}=0,\\\ \displaystyle
U^{CD}_{t}+P^{CD}_{x}=\varepsilon(\frac{U^{CD}_{x}}{V^{CD}})_{x},\\\
\displaystyle\frac{R}{\gamma-1}\Theta^{CD}_{t}+P^{CD}U^{CD}_{x}=\kappa(\frac{\Theta^{CD}_{x}}{V^{CD}})_{x}+\varepsilon\frac{(U^{CD}_{x})^{2}}{V^{CD}}+Q^{CD},\end{array}\right.$
(2.6)
where $\displaystyle P^{CD}=\frac{R\Theta^{CD}}{V^{CD}}$ and the error term
$Q^{CD}$ has the property that
$\begin{array}[]{ll}\displaystyle
Q^{CD}&\displaystyle=O(1)\delta^{CD}\varepsilon(1+t)^{-2}e^{-\frac{C_{0}x^{2}}{\varepsilon(1+t)}},\qquad{\rm
as}~{}~{}|x|\rightarrow+\infty.\end{array}$ (2.7)
###### Remark 1.
The viscous contact wave $(V^{CD},U^{CD},\Theta^{CD})(t,x)$ defined in (2.4)
is different from the one used in [14] and [16]. Here, this ansatz is chosen
such that the mass equation and the momentum equation are satisfied exactly
while the error term occurs only in the energy equation. However, note that
the approximate energy equation that the viscous contact wave satisfies is not
in the conservative form.
#### 2.1.2. Rarefaction waves
We now turn to the rarefaction waves. Since there is no exact rarefaction wave
profile for either the Navier-Stokes equations or the Boltzmann equation, the
following approximate rarefaction wave profile satisfying the Euler equations
was motivated by [23] and [30]. For the completeness of the presentation, we
include its definition and the properties in this subsection.
If $(v_{-},u_{-},\theta_{-})\in R_{i}(v_{+},u_{+},\theta_{+})(i=1,3)$, then
there exists a $i$-rarefaction wave
$(v^{r_{i}},u^{r_{i}},\theta^{r_{i}})(x/t)$ which is a global solution of the
following Riemann problem
$\displaystyle\left\\{\begin{array}[]{l}\displaystyle v_{t}-u_{x}=0,\\\
\displaystyle u_{t}+p_{x}(v,\theta)=0,\\\
\displaystyle\frac{R}{\gamma-1}\theta_{t}+p(v,\theta)u_{x}=0,\\\
\displaystyle(v,u,\theta)(t=0,x)=\left\\{\begin{array}[]{l}\displaystyle(v_{-},u_{-},\theta_{-}),x<0,\\\
\displaystyle(v_{+},u_{+},\theta_{+}),x>0.\end{array}\right.\end{array}\right.$
(2.14)
Consider the following inviscid Burgers equation with Riemann data
$\left\\{\begin{array}[]{l}w_{t}+ww_{x}=0,\\\
w(t=0,x)=\left\\{\begin{array}[]{ll}w_{-},&x<0,\\\
w_{+},&x>0.\end{array}\right.\end{array}\right.$ (2.15)
If $w_{-}<w_{+}$, then the above Riemann problem admits a rarefaction wave
solution
$w^{r}(t,x)=w^{r}(\frac{x}{t})=\left\\{\begin{array}[]{ll}w_{-},&\frac{x}{t}\leq
w_{-},\\\ \frac{x}{t},&w_{-}\leq\frac{x}{t}\leq w_{+},\\\
w_{+},&\frac{x}{t}\geq w_{+}.\end{array}\right.$ (2.16)
Obviously, we have the following Lemma,
###### Lemma 2.1.
For any shift $t_{0}>0$ in the time variable, we have
$|w^{r}(t+t_{0},x)-w^{r}(t,x)|\leq\frac{C}{t}t_{0},$
where $C$ is a positive constant depending only on $w_{\pm}$.
Remark that Lemma 2.1 plays an important role in the wave interaction
estimates for the rarefaction waves.
As in [30], the approximate rarefaction wave $(V^{R},U^{R},\Theta^{R})(t,x)$
to the problem (1.1) can be constructed by the solution of the Burgers
equation
$\displaystyle\left\\{\begin{array}[]{l}\displaystyle w_{t}+ww_{x}=0,\\\
\displaystyle
w(0,x)=w_{\sigma}(x)=w(\frac{x}{\sigma})=\frac{w_{+}+w_{-}}{2}+\frac{w_{+}-w_{-}}{2}\tanh\frac{x}{\sigma},\end{array}\right.$
(2.19)
where $\sigma>0$ is a small parameter to be determined. Note that the solution
$w^{r}_{\sigma}(t,x)$ of the problem (2.19) is given by
$w^{r}_{\sigma}(t,x)=w_{\sigma}(x_{0}(t,x)),\qquad
x=x_{0}(t,x)+w_{\sigma}(x_{0}(t,x))t.$
And $w^{r}_{\sigma}(t,x)$ has the following properties:
###### Lemma 2.2.
([30]) Let $w_{-}<w_{+}$, $(\ref{(2.11)})$ has a unique smooth solution
$w^{r}_{\sigma}(t,x)$ satisfying
1. (1)
$w_{-}<w^{r}_{\sigma}(t,x)<w_{+},~{}(w^{r}_{\sigma})_{x}(t,x)\geq 0$;
2. (2)
For any $p$ $(1\leq p\leq+\infty)$, there exists a constant $C$ such that
$\begin{array}[]{ll}\|\frac{\partial}{\partial
x}w^{r}_{\sigma}(t,\cdot)\|_{L^{p}(\mathbf{R})}\leq
C\min\big{\\{}(w_{+}-w_{-})\sigma^{-1+1/p},~{}(w_{+}-w_{-})^{1/p}t^{-1+1/p}\big{\\}},\\\\[5.69054pt]
\|\frac{\partial^{2}}{\partial
x^{2}}w^{r}_{\sigma}(t,\cdot)\|_{L^{p}(\mathbf{R})}\leq
C\min\big{\\{}(w_{+}-w_{-})\sigma^{-2+1/p},~{}\sigma^{-1+1/p}t^{-1}\big{\\}};\end{array}$
3. (3)
If $x-w_{-}t<0$ and $w_{-}>0$, then
$\begin{array}[]{l}|w^{r}_{\sigma}(t,x)-w_{-}|\leq(w_{+}-w_{-})e^{-\frac{2|x-w_{-}t|}{\sigma}},\\\\[5.69054pt]
|\frac{\partial}{\partial
x}w^{r}_{\sigma}(t,\cdot)|\leq\frac{2(w_{+}-w_{-})}{\sigma}e^{-\frac{2|x-w_{-}t|}{\sigma}};\end{array}$
If $x-w_{+}t>0$ and $w_{+}<0$, then
$\begin{array}[]{l}|w^{r}_{\sigma}(t,x)-w_{+}|\leq(w_{+}-w_{-})e^{-\frac{2|x-w_{+}t|}{\sigma}},\\\\[5.69054pt]
|\frac{\partial}{\partial
x}w^{r}_{\sigma}(t,\cdot)|\leq\frac{2(w_{+}-w_{-})}{\sigma}e^{-\frac{2|x-w_{+}t|}{\sigma}};\end{array}$
4. (4)
$\sup\limits_{x\in\mathbf{R}}|w^{r}_{\sigma}(t,x)-w^{r}(\frac{x}{t})|\leq\min\big{\\{}(w_{+}-w_{-}),\frac{\sigma}{t}[\ln(1+t)+|\ln\sigma|]\big{\\}}$.
Then the smooth approximate rarefaction wave profile denoted by
$(V^{R_{i}},U^{R_{i}},\Theta^{R_{i}})(t,x)~{}(i=1,3)$ can be defined by
$\displaystyle\left\\{\begin{array}[]{l}\displaystyle
S^{R_{i}}(t,x)=s(V^{R_{i}}(t,x),\Theta^{R_{i}}(t,x))=s_{+},\\\ \displaystyle
w_{\pm}=\lambda_{i\pm}:=\lambda_{i}(v_{\pm},\theta_{\pm}),\\\ \displaystyle
w_{\sigma}^{r}(t+t_{0},x)=\lambda_{i}(V^{R_{i}}(t,x),s_{+}),\\\ \displaystyle
U^{R_{i}}(t,x)=u_{+}-\int^{V^{R_{i}}(t,x)}_{v_{+}}\lambda_{i}(v,s_{+})dv,\end{array}\right.$
(2.24)
where $t_{0}$ is the shift used to control the interaction between waves in
different families with the property that $t_{0}\rightarrow 0$ as
$\varepsilon\rightarrow 0$. In the following, we choose
$t_{0}=\varepsilon^{\frac{1}{5}},\qquad\mbox{\rm
and}\qquad\sigma=\varepsilon^{\frac{2}{5}}.$ (2.25)
Note that $(V^{R_{i}},U^{R_{i}},\Theta^{R_{i}})(t,x)$ defined above satisfies
$\displaystyle\begin{cases}V^{R_{i}}_{t}-U^{R_{i}}_{x}=0,\cr
U^{R_{i}}_{t}+P^{R_{i}}_{x}=0,\cr\frac{R}{\gamma-1}\Theta^{R_{i}}_{t}+P^{R_{i}}U^{R_{i}}_{x}=0,\end{cases}$
(2.26)
where $P^{R_{i}}=p(V^{R_{i}},\Theta^{R_{i}})$.
By Lemmas 2.1 and 2.2, the properties on the rarefaction waves can be
summarized as follows.
###### Lemma 2.3.
The approximate rarefaction waves
$(V^{R_{i}},U^{R_{i}},\Theta^{R_{i}})(t,x)~{}(i=1,3)$ constructed in (2.24)
have the following properties:
1. (1)
$U^{R_{i}}_{x}(t,x)>0$ for $x\in\mathbf{R}$, $t>0$;
2. (2)
For any $1\leq p\leq+\infty,$ the following estimates holds,
$\begin{array}[]{ll}\|(V^{R_{i}},U^{R_{i}},\Theta^{R_{i}})_{x}\|_{L^{p}(dx)}\leq
C(t+t_{0})^{-1+\frac{1}{p}},\\\
\|(V^{R_{i}},U^{R_{i}},\Theta^{R_{i}})_{xx}\|_{L^{p}(dx)}\leq
C\sigma^{-1+\frac{1}{p}}(t+t_{0})^{-1},\\\
\|(V^{R_{i}},U^{R_{i}},\Theta^{R_{i}})_{xxx}\|_{L^{p}(dx)}\leq
C\sigma^{-2+\frac{1}{p}}(t+t_{0})^{-1},\\\ \end{array}$
where the positive constant $C$ only depends on $p$ and the wave strength;
3. (3)
If $x\geq\lambda_{1+}(t+t_{0})$, then
$\begin{array}[]{l}|(V^{R_{1}},U^{R_{1}},\Theta^{R_{1}})(t,x)-(v_{-},u_{-},\theta_{-})|\leq
Ce^{-\frac{2|x-\lambda_{1+}(t+t_{0})|}{\sigma}},\\\\[5.69054pt]
|(V^{R_{1}},U^{R_{1}},\Theta^{R_{1}})_{x}(t,x)|\leq\frac{C}{\sigma}e^{-\frac{2|x-\lambda_{1+}(t+t_{0})|}{\sigma}};\end{array}$
If $x\leq\lambda_{3-}(t+t_{0})$, then
$\begin{array}[]{l}|(V^{R_{3}},U^{R_{3}},\Theta^{R_{3}})(t,x)-(v_{+},u_{+},\theta_{+})|\leq
Ce^{-\frac{2|x-\lambda_{3-}(t+t_{0})|}{\sigma}},\\\\[5.69054pt]
|(V^{R_{3}},U^{R_{3}},\Theta^{R_{3}})_{x}(t,x)|\leq\frac{C}{\sigma}e^{-\frac{2|x-\lambda_{3-}(t+t_{0})|}{\sigma}};\end{array}$
4. (4)
There exist positive constants $C$ and $\sigma_{0}$ such that for
$\sigma\in(0,\sigma_{0})$ and $t,t_{0}>0,$
$\sup_{x\in\mathbf{R}}|(V^{R_{i}},U^{R_{i}},\Theta^{R_{i}})(t,x)-(v^{r_{i}},u^{r_{i}},\theta^{r_{i}})(\frac{x}{t})|\leq\frac{C}{t}[\sigma\ln(1+t+t_{0})+\sigma|\ln\sigma|+t_{0}].$
#### 2.1.3. Superposition of rarefaction waves and contact discontinuity
In this subsection, we will define the solution profile that consists of the
superposition of two rarefaction waves and a contact discontinuity. Let
$(v_{-},u_{-},\theta_{-})\in$ $R_{1}$-$CD$-$R_{3}(v_{+},u_{+},\theta_{+})$.
Then there exist uniquely two intermediate states $(v_{*},u_{*},\theta_{*})$
and $(v^{*},u^{*},\theta^{*})$ such that $(v_{-},u_{-},\theta_{-})\in
R_{1}(v_{*},u_{*},\theta_{*})$, $(v_{*},u_{*},\theta_{*})\in
CD(v^{*},u^{*},\theta^{*})$ and $(v^{*},u^{*},\theta^{*})\in
R_{3}(v_{+},u_{+},\theta_{+})$.
So the wave pattern $(\bar{V},\bar{U},\bar{\Theta})(t,x)$ consisting of
1-rarefaction wave, 2-contact discontinuity and 3-rarefaction wave that solves
the corresponding Riemann problem of the Euler system (1.3) can be defined by
$\displaystyle\left(\begin{array}[]{cc}\bar{V}\\\ \bar{U}\\\
\bar{\Theta}\end{array}\right)(t,x)=\left(\begin{array}[]{cc}v^{r_{1}}+v^{cd}+v^{r_{3}}\\\
u^{r_{1}}+u^{cd}+u^{r_{3}}\\\
\theta^{r_{1}}+\theta^{cd}+\theta^{r_{3}}\end{array}\right)(t,x)-\left(\begin{array}[]{cc}v_{*}+v^{*}\\\
u_{*}+u^{*}\\\ \theta_{*}+\theta^{*}\end{array}\right),$ (2.36)
where $(v^{r_{1}},u^{r_{1}},\theta^{r_{1}})(t,x)$ is the 1-rarefaction wave
defined in (2.14) with the right state $(v_{+},u_{+},\theta_{+})$ replaced by
$(v_{*},u_{*},\theta_{*})$, $(v^{cd},u^{cd},\theta^{cd})(t,x)$ is the contact
discontinuity defined in (2.3) with the states $(v_{-},u_{-},\theta_{-})$ and
$(v_{+},u_{+},\theta_{+})$ replaced by $(v_{*},u_{*},\theta_{*})$ and
$(v^{*},u^{*},\theta^{*})$ respectively, and
$(v^{r_{3}},u^{r_{3}},\theta^{r_{3}})(t,x)$ is the 3-rarefaction wave defined
in (2.14) with the left state $(v_{-},u_{-},\theta_{-})$ replaced by
$(v^{*},u^{*},\theta^{*})$.
Correspondingly, the approximate wave pattern $(V,U,\Theta)(t,x)$ of the
compressible Navier-Stokes equations can be defined by
$\displaystyle\left(\begin{array}[]{cc}V\\\ U\\\
\Theta\end{array}\right)(t,x)=\left(\begin{array}[]{cc}V^{R_{1}}+V^{CD}+V^{R_{3}}\\\
U^{R_{1}}+U^{CD}+U^{R_{3}}\\\
\Theta^{R_{1}}+\Theta^{CD}+\Theta^{R_{3}}\end{array}\right)(t,x)-\left(\begin{array}[]{cc}v_{*}+v^{*}\\\
u_{*}+u^{*}\\\ \theta_{*}+\theta^{*}\end{array}\right),$ (2.46)
where $(V^{R_{1}},U^{R_{1}},\Theta^{R_{1}})(t,x)$ is the approximate
1-rarefaction wave defined in (2.24) with the right state
$(v_{+},u_{+},\theta_{+})$ replaced by $(v_{*},u_{*},\theta_{*})$,
$(V^{CD},U^{CD},\Theta^{CD})(t,x)$ is the viscous contact wave defined in
(2.4) with the states $(v_{-},u_{-},\theta_{-})$ and
$(v_{+},u_{+},\theta_{+})$ replaced by $(v_{*},u_{*},\theta_{*})$ and
$(v^{*},u^{*},\theta^{*})$ respectively, and
$(V^{R_{3}},U^{R_{3}},\Theta^{R_{3}})(t,x)$ is the approximate 3-rarefaction
wave defined in (2.24) with the left state $(v_{-},u_{-},\theta_{-})$ replaced
by $(v^{*},u^{*},\theta^{*})$.
Thus, from the construction of the contact wave and Lemma 2.3, we have the
following relation between the approximate wave pattern $(V,U,\Theta)(t,x)$ of
the compressible Navier-Stokes equations and the exact inviscid wave pattern
$(\bar{V},\bar{U},\bar{\Theta})(t,x)$ to the Euler equations
$\begin{array}[]{ll}\displaystyle|(V,U,\Theta)(t,x)-(\bar{V},\bar{U},\bar{\Theta})(t,x)|\\\
\displaystyle\quad\leq\frac{C}{t}[\sigma\ln(1+t+t_{0})+\sigma|\ln\sigma|+t_{0}]+C\delta^{CD}e^{-\frac{cx^{2}}{\varepsilon(1+t)}},\end{array}$
(2.47)
with $t_{0}=\varepsilon^{\frac{1}{5}}$ and $\sigma=\varepsilon^{\frac{2}{5}}$.
Moreover, $(V,U,\Theta)(t,x)$ satisfies the following system
$\displaystyle\begin{cases}V_{t}-U_{x}=0,\cr
U_{t}+P_{x}=\varepsilon(\frac{U_{x}}{V})_{x}+Q_{1},\cr\frac{R}{\gamma-1}\Theta_{t}+PU_{x}=\kappa(\frac{\Theta_{x}}{V})_{x}+\varepsilon\frac{U_{x}^{2}}{V}+Q_{2},\end{cases}$
(2.48)
where $P=p(V,\Theta)$, and
$\begin{array}[]{ll}\displaystyle
Q_{1}&\displaystyle=(P-P^{R_{1}}-P^{CD}-P^{R_{3}})_{x}-\varepsilon(\frac{U_{x}}{V}-\frac{U^{CD}_{x}}{V^{CD}})_{x},\\\
\displaystyle
Q_{2}&\displaystyle=(PU_{x}-P^{R_{1}}U^{R_{1}}_{x}-P^{CD}U^{CD}_{x}-P^{R_{3}}U^{R_{3}}_{x})-\kappa(\frac{\Theta_{x}}{V}-\frac{\Theta^{CD}_{x}}{V^{CD}})_{x}\\\
&\displaystyle-\varepsilon(\frac{U_{x}^{2}}{V}-\frac{(U^{CD}_{x})^{2}}{V^{CD}})-Q^{CD}.\end{array}$
Direct calculation shows that
$\begin{array}[]{lll}\displaystyle Q_{1}&=&\displaystyle
O(1)\Big{[}|(V^{R_{1}}_{x},\Theta^{R_{1}}_{x})||(V^{CD}-v_{*},\Theta^{CD}-\theta_{*},V^{R_{3}}-v^{*},\Theta^{R_{3}}-\theta^{*})|\\\
&&\displaystyle+|(V^{R_{3}}_{x},\Theta^{R_{3}}_{x})||(V^{R_{1}}-v_{*},\Theta^{R_{1}}-\theta_{*},V^{CD}-v^{*},\Theta^{CD}-\theta^{*})|\\\
&&\displaystyle+|(V^{CD}_{x},\Theta^{CD}_{x},U^{CD}_{xx})||(V^{R_{1}}-v_{*},\Theta^{R_{1}}-\theta_{*},V^{R_{3}}-v^{*},\Theta^{R_{3}}-\theta^{*})|\\\
&&\displaystyle+\varepsilon|(U^{CD}_{x},V^{CD}_{x})||(U^{R_{1}}_{x},V^{R_{1}}_{x},U^{R_{3}}_{x},V^{R_{3}}_{x})|+\varepsilon|(U^{R_{1}}_{x},V^{R_{1}}_{x})||(U^{R_{3}}_{x},V^{R_{3}}_{x})|\Big{]}\\\
&&\displaystyle+O(1)\varepsilon\Big{[}|U^{R_{1}}_{xx}|+|U^{R_{3}}_{xx}|+|U^{R_{1}}_{x}||V^{R_{1}}_{x}|+|U^{R_{3}}_{x}||V^{R_{3}}_{x}|\Big{]}\\\
&:=&\displaystyle Q_{11}+Q_{12}.\end{array}$ (2.49)
Similarly, we have
$\begin{array}[]{lll}\displaystyle Q_{2}&=&\displaystyle
O(1)\Big{[}|U^{R_{1}}_{x}||(V^{CD}-v_{*},\Theta^{CD}-\theta_{*},V^{R_{3}}-v^{*},\Theta^{R_{3}}-\theta^{*})|\\\
&&\displaystyle+|U^{R_{3}}_{x}||(V^{R_{1}}-v_{*},\Theta^{R_{1}}-\theta_{*},V^{CD}-v^{*},\Theta^{CD}-\theta^{*})|\\\
&&\displaystyle+|(U^{CD}_{x},V^{CD}_{x},\Theta^{CD}_{x})||(V^{R_{1}}-v_{*},\Theta^{R_{1}}-\theta_{*},V^{R_{3}}-v^{*},\Theta^{R_{3}}-\theta^{*})|\\\
&&\displaystyle+\varepsilon|(U^{CD}_{x},V^{CD}_{x},\Theta^{CD}_{x})||(U^{R_{1}}_{x},V^{R_{1}}_{x},\Theta^{R_{1}}_{x},U^{R_{3}}_{x},V^{R_{3}}_{x},\Theta^{R_{1}}_{x})|\\\
&&\displaystyle+\varepsilon|(U^{R_{1}}_{x},V^{R_{1}}_{x},\Theta^{R_{1}}_{x})||(U^{R_{3}}_{x},V^{R_{3}}_{x},\Theta^{R_{3}}_{x})|\Big{]}\\\
&&\displaystyle+O(1)\varepsilon\Big{[}|\Theta^{R_{1}}_{xx}|+|\Theta^{R_{3}}_{xx}|+|(U^{R_{1}}_{x},V^{R_{1}}_{x},\Theta^{R_{1}}_{x},U^{R_{3}}_{x},V^{R_{3}}_{x},\Theta^{R_{3}}_{x})|^{2}\Big{]}+|Q^{CD}|\\\
&:=&\displaystyle Q_{21}+Q_{22}+|Q^{CD}|.\end{array}$ (2.50)
Here $Q_{11}$ and $Q_{21}$ represent the interactions coming from different
wave patterns, $Q_{12}$ and $Q_{22}$ represent the error terms coming from the
approximate rarefaction wave profiles, and $Q^{CD}$ is the error term defined
in (2.7) due to the viscous contact wave.
Firstly, we estimate the interaction terms $Q_{11}$ and $Q_{21}$ by dividing
the whole domain $\Omega=\\{(t,x)|(t,x)\in\mathbf{R}^{+}\times\mathbf{R}\\}$
into three regions:
$\begin{array}[]{l}\Omega_{-}=\\{(t,x)|2x\leq\lambda_{1*}(t+t_{0})\\},\\\
\Omega_{CD}=\\{(t,x)|\lambda_{1*}(t+t_{0})<2x<\lambda_{3}^{*}(t+t_{0})\\},\\\
\Omega_{+}=\\{(t,x)|2x\geq\lambda_{3}^{*}(t+t_{0})\\},\end{array}$
where $\lambda_{1*}=\lambda_{1}(v_{*},\theta_{*})$ and
$\lambda_{3}^{*}=\lambda_{3}(v^{*},\theta^{*})$.
Now from Lemma 2.3, we have the following estimates in each section:
* •
In $\Omega_{-}$,
$\begin{array}[]{ll}|V^{R_{3}}-v^{*}|&=O(1)e^{-\frac{2|x|+2\lambda_{3}^{*}(t+t_{0})}{\sigma}}\\\
&=O(1)e^{-\lambda_{3}^{*}\varepsilon^{-1/5}}e^{-\frac{2|x|+\lambda_{3}^{*}(t+t_{0})}{\varepsilon^{2/5}}},\end{array}$
$\begin{array}[]{ll}|(V^{CD}-v_{*},V^{CD}-v^{*})|&=O(1)\delta^{CD}e^{-\frac{C[\lambda_{1*}(t+t_{0})]^{2}}{4\varepsilon(1+t)}}\\\
&=O(1)e^{-\frac{Ct_{0}(t+t_{0})}{\varepsilon}}\\\
&=O(1)e^{-\frac{Ct_{0}(|x|+t+t_{0})}{\varepsilon}}\\\
&=O(1)e^{-C\varepsilon^{-3/5}}e^{-\frac{C(|x|+t+t_{0})}{\varepsilon^{4/5}}};\end{array}$
* •
In $\Omega_{CD}$,
$\begin{array}[]{ll}|V^{R_{1}}-v_{*}|&=O(1)e^{-\frac{2|x|+2|\lambda_{1*}|(t+t_{0})}{\sigma}}\\\
&=O(1)e^{-|\lambda_{1*}|\varepsilon^{-1/5}}e^{-\frac{2|x|+|\lambda_{1*}|(t+t_{0})}{\varepsilon^{2/5}}},\end{array}$
$\begin{array}[]{ll}|V^{R_{3}}-v^{*}|&=O(1)e^{-\frac{2|x|+2\lambda_{3}^{*}(t+t_{0})}{\sigma}}\\\
&=O(1)e^{-\lambda_{3}^{*}\varepsilon^{-1/5}}e^{-\frac{2|x|+\lambda_{3}^{*}(t+t_{0})}{\varepsilon^{2/5}}};\end{array}$
* •
In $\Omega_{+}$,
$\begin{array}[]{ll}|V^{R_{1}}-v_{*}|&=O(1)e^{-\frac{2|x|+2|\lambda_{1*}|(t+t_{0})}{\sigma}}\\\
&=O(1)e^{-|\lambda_{1*}|\varepsilon^{-1/5}}e^{-\frac{2|x|+2|\lambda_{1*}|(t+t_{0})}{\varepsilon^{2/5}}},\end{array}$
$\begin{array}[]{ll}|(V^{CD}-v_{*},V^{CD}-v^{*})|&=O(1)\delta^{CD}e^{-\frac{C[\lambda_{3}^{*}(t+t_{0})]^{2}}{4\varepsilon(1+t)}}\\\
&=O(1)e^{-\frac{Ct_{0}(t+t_{0})}{\varepsilon}}\\\
&=O(1)e^{-\frac{Ct_{0}(|x|+t+t_{0})}{\varepsilon}}\\\
&=O(1)e^{-C\varepsilon^{-3/5}}e^{-\frac{C(|x|+t+t_{0})}{\varepsilon^{4/5}}}.\end{array}$
Hence, in summary, we have
$|(Q_{11},Q_{21})|=O(1)e^{-C\varepsilon^{-1/5}}e^{-\frac{C(|x|+t+t_{0})}{\varepsilon^{2/5}}},$
(2.51)
for some positive constants $C$.
Now we consider the system (1.1) with the initial values
$(v,u,\theta)(t=0,x)=(V,U,\Theta)(t=0,x).$ (2.52)
Introduce the following scaled variables
$y=\frac{x}{\varepsilon},\quad\tau=\frac{t}{\varepsilon}.$ (2.53)
In the following, we will use the notations $(v,u,\theta)(\tau,y)$ and
$(V,U,\Theta)(\tau,y)$ for the unknown functions and the approximate wave
profiles in the scaled variables. Set the perturbation around the composite
wave pattern $(V,U,\Theta)(\tau,y)$ by
$(\phi,\psi,\zeta)(\tau,y)=(v-V,u-U,\theta-\Theta)(\tau,y).$
Then the perturbation $(\phi,\psi,\zeta)(\tau,y)$ satisfies the system
$\left\\{\begin{array}[]{ll}\displaystyle\phi_{\tau}-\psi_{y}=0,\\\
\displaystyle\psi_{\tau}+(p-P)_{y}=(\frac{u_{y}}{v}-\frac{U_{y}}{V})_{y}-\varepsilon
Q_{1},\\\
\displaystyle\frac{R}{\gamma-1}\zeta_{\tau}+(pu_{y}-PU_{y})=\nu(\frac{\theta_{y}}{v}-\frac{\Theta_{y}}{V})_{y}+(\frac{u_{y}^{2}}{v}-\frac{U^{2}_{y}}{V})-\varepsilon
Q_{2},\\\ \displaystyle(\phi,\psi,\zeta)(\tau=0,y)=0.\end{array}\right.$
(2.54)
And this system will be studied in Section 3.
#### 2.1.4. Main result to the compressible Navier-Stokes equations
We are now ready to state the main result on the compressible Navier-Stokes
equations as follows.
###### Theorem 2.4.
Given a Riemann solution $(\bar{V},\bar{U},\bar{\Theta})(t,x)$ defined in
(2.36), which is a superposition of two rarefaction waves and a contact
discontinuity for the Euler system (1.3), there exist small positive constants
$\delta_{0}$ and $\varepsilon_{0}$ such that if the contact wave strength
$\delta^{CD}\leq\delta_{0}$ and the viscosity coefficient
$\varepsilon\leq\varepsilon_{0}$, then the compressible Navier-Stokes
equations (1.1) with (1.2) and (1.4) admits a unique global solution
$(v^{\varepsilon},u^{\varepsilon},\theta^{\varepsilon})(t,x)$ satisfying
$\sup_{(t,x)\in\Sigma_{h}}|(v^{\varepsilon},u^{\varepsilon},\theta^{\varepsilon})(t,x)-(\bar{V},\bar{U},\bar{\Theta})(t,x)|\leq
C_{h}~{}\varepsilon^{\frac{1}{5}},\quad\forall h>0,$ (2.55)
where $\Sigma_{h}=\\{(t,x)|t\geq h,\frac{x}{\sqrt{1+t}}\geq
h\varepsilon^{\alpha},0<\alpha<\frac{1}{2}\\}$, and the positive constant
$C_{h}$ depends only on $h$ but is independent of $\varepsilon$.
###### Remark 2.
Theorem 2.4 shows that, away from the initial time $t=0$ and the contact
discontinuity located at $x=0$ with the expansion rate
$\frac{x^{2}}{\varepsilon(1+t)}$, for the viscosity coefficient
$\varepsilon<\varepsilon_{0}$, there exists a unique global solution
$(v^{\varepsilon},u^{\varepsilon},\theta^{\varepsilon})(t,x)$ of the
compressible Navier-Stokes equations (1.1) which tends to the Riemann solution
$(\bar{V},\bar{U},\bar{\Theta})(t,x)$ consisting of two rarefaction waves and
a contact discontinuity when $\varepsilon\rightarrow 0$ and
$\kappa=O(\varepsilon)\rightarrow 0$. Moreover, a uniform convergence rate
$\varepsilon^{\frac{1}{5}}$ holds on the set $\Sigma_{h}$ for any $h>0$.
###### Remark 3.
Theorem 2.4 holds uniformly when $(t,x)\in\Sigma_{h}$ for any fixed $h>0$ if
the contact wave strength $\delta^{CD}$ and the viscosity coefficient
$\varepsilon$ are suitably small. However, if we restrict the problem to a set
$\Sigma_{h}\cap\\{t\leq T\\}$ for any fixed $T>0$, then we do not need to
impose the smallness condition on the contact wave strength $\delta^{CD}$
because one can apply the Gronwall inequality to get an estimate depending on
time $T$ rather than the uniform estimate in time.
### 2.2. Boltzmann equation
We now turn to the Boltzmann equation. Similarly, we also define individual
wave pattern, and then the superposition and finally state the main result in
this subsection.
#### 2.2.1. Contact discontinuity
We first recall the construction of the contact wave
$(V^{CD},U^{CD},\Theta^{CD})(t,x)$ for the Boltzmann equation in [16].
Consider the Euler system (1.16) with a Riemann initial data
$(v,u,\theta)(t=0,x)=\left\\{\begin{array}[]{l}(v_{-},u_{-},\theta_{-}),~{}~{}~{}x<0,\\\
(v_{+},u_{+},\theta_{+}),~{}~{}~{}x>0,\end{array}\right.$ (2.56)
where $u_{\pm}=(u_{1\pm},0,0)$ and $v_{\pm}>0,\theta_{\pm}>0,u_{1\pm}$ are
given constants. It is known (cf. [25]) that the Riemann problem (1.16),
(2.56) admits a contact discontinuity solution
$(v^{cd},u^{cd},\theta^{cd})(t,x)=\left\\{\begin{array}[]{l}(v_{-},u_{-},\theta_{-}),~{}~{}~{}x<0,\\\
(v_{+},u_{+},\theta_{+}),~{}~{}~{}x>0,\end{array}\right.$ (2.57)
provided that
$u_{1+}=u_{1-},\qquad
p_{-}:=\frac{2\theta_{-}}{3v_{-}}=p_{+}:=\frac{2\theta_{+}}{3v_{+}}.$ (2.58)
Motivated by (2.57) and (2.58), we expect that for the contact wave
$(V^{CD},U^{CD},\Theta^{CD})(t,x)$,
$P^{CD}=\frac{2\Theta^{CD}}{3V^{CD}}\approx p_{+},~{}~{}~{}|U^{CD}|^{2}\ll 1.$
Then the leading order of the energy equation $(\ref{(1.22)})_{4}$ is
$\theta_{t}+p_{+}u_{1x}=\varepsilon(\frac{\lambda(\theta)\theta_{x}}{v})_{x}.$
(2.59)
By using the mass equation $(\ref{(1.22)})_{1}$ and
$v\approx\frac{R\theta}{p_{+}}$, we obtain the following nonlinear diffusion
equation
$\theta_{t}=\varepsilon(a(\theta)\theta_{x})_{x},~{}~{}~{}a(\theta)=\frac{9p_{+}\lambda(\theta)}{10\theta}.$
(2.60)
From [1] and [6], we know that the nonlinear diffusion equation (2.60) admits
a unique self-similar solution
$\hat{\Theta}(\eta),~{}\eta=\frac{x}{\sqrt{\varepsilon(1+t)}}$ with the
following boundary conditions
$\hat{\Theta}(-\infty,t)=\theta_{-},~{}~{}\hat{\Theta}(+\infty,t)=\theta_{+}.$
Let $\delta=|\theta_{+}-\theta_{-}|$. $\hat{\Theta}(t,x)$ has the property
$\hat{\Theta}_{x}(t,x)=\frac{O(1)\delta^{CD}}{\sqrt{\varepsilon(1+t)}}e^{-\frac{cx^{2}}{\varepsilon(1+t)}},~{}~{}~{}~{}~{}~{}{\rm
as}~{}~{}~{}x\rightarrow\pm\infty,$ (2.61)
with some positive constant $c$ depending only on $\theta_{\pm}$.
Now the contact wave $(V^{CD},U^{CD},\Theta^{CD})(t,x)$ can be defined by
$\begin{array}[]{ll}\displaystyle V^{CD}=\frac{2}{3p_{+}}\hat{\Theta},\\\
\displaystyle U^{CD}_{1}=u_{1+}+\frac{2\varepsilon
a(\hat{\Theta})}{3p_{+}}\hat{\Theta}_{x},~{}~{}~{}~{}U^{CD}_{i}=0,(i=2,3),\\\\[8.53581pt]
~{}~{}~{}\Theta^{CD}=\hat{\Theta}+\frac{2\varepsilon}{3p_{+}}\hat{\Theta}_{t}[\frac{4}{3}\mu(\hat{\Theta})-\frac{3}{5}\lambda(\hat{\Theta})].\end{array}$
(2.62)
Note that the contact wave $(V^{CD},U^{CD},\Theta^{CD})(t,x)$ satisfies the
following system
$\left\\{\begin{array}[]{llll}\displaystyle V^{CD}_{t}-U^{CD}_{1x}=0,\\\
\displaystyle
U^{CD}_{1t}+P^{CD}_{x}=\frac{4\varepsilon}{3}(\frac{\mu(\Theta^{CD})}{V^{CD}}U^{CD}_{1x})_{x}+Q^{CD}_{1},\\\
\displaystyle
U^{CD}_{it}=\varepsilon(\frac{\mu(\Theta^{CD})}{V^{CD}}U^{CD}_{ix})_{x},i=2,3,\\\
\displaystyle\Theta^{CD}_{t}+P^{CD}U^{CD}_{1x}=\varepsilon(\frac{\lambda(\Theta^{CD})}{V^{CD}}\Theta^{CD}_{x})_{x}+\frac{4\varepsilon}{3}\frac{\mu(\Theta^{CD})}{V^{CD}}(U^{CD}_{1x})^{2}\\\
\quad\displaystyle+\varepsilon\sum_{i=2}^{3}\frac{\mu(\Theta^{CD})}{V^{CD}}(U^{CD}_{ix})^{2}+Q^{CD}_{2},\end{array}\right.$
(2.63)
where
$Q^{CD}_{1}=\frac{4\varepsilon}{3}(\frac{\mu(\Theta^{CD})-\mu(\hat{\Theta})}{V^{CD}}U^{CD}_{1x})_{x}=\displaystyle
O(1)\delta^{CD}\varepsilon^{\frac{3}{2}}(1+t)^{-\frac{5}{2}}e^{-\frac{cx^{2}}{\varepsilon(1+t)}},$
(2.64)
$\begin{array}[]{ll}Q^{CD}_{2}&\displaystyle=[\frac{2\varepsilon}{3p_{+}}\hat{\Theta}_{t}(\frac{4}{3}\mu(\hat{\Theta})-\frac{3}{5}\lambda(\hat{\Theta}))]_{t}+\frac{2\varepsilon}{3p_{+}V^{CD}}\hat{\Theta}_{t}[\frac{4}{3}\mu(\hat{\Theta})-\frac{3}{5}\lambda(\hat{\Theta})]U^{CD}_{1x}\\\
&\displaystyle\quad+\frac{\varepsilon}{V^{CD}}(\lambda(\hat{\Theta})\hat{\Theta}_{x}-\lambda(\Theta^{CD})\Theta^{CD}_{x})_{x}-\frac{4\varepsilon\mu(\Theta^{CD})}{3V^{CD}}(U^{CD}_{1x})^{2}\\\
&\displaystyle=O(1)\delta^{CD}\varepsilon(1+t)^{-2}e^{-\frac{cx^{2}}{\varepsilon(1+t)}},\end{array}$
(2.65)
with some positive constant $c>0$ depending only on $\theta_{\pm}$.
###### Remark 4.
The viscous contact wave $(V^{CD},U^{CD},\Theta^{CD})(t,x)$ for the Boltzmann
equation (1.5) defined in (2.62) is different from the one used in [16]. Here,
this ansatz is chosen such that the momentum equation is satisfied with a
higher order error term. This is also different from the compressible Navier-
Stokes equations where the ansatz satisfies the momentum equation exactly. But
similar to the compressible Navier-Stokes cases, the approximate energy
equation that the viscous contact wave satisfies is not in the conservative
form.
From (2.61), we have
$\left\\{\begin{array}[]{l}|\hat{\Theta}-\theta_{-}|=O(1)\delta^{CD}e^{-\frac{cx^{2}}{2\varepsilon(1+t)}},~{}~{}~{}~{}~{}{\rm
if}~{}x<0,\\\
|\hat{\Theta}-\theta_{+}|=O(1)\delta^{CD}e^{-\frac{cx^{2}}{2\varepsilon(1+t)}},~{}~{}~{}~{}~{}{\rm
if}~{}x>0.\end{array}\right.$ (2.66)
Therefore,
$|(V^{CD},U^{CD},\Theta^{CD})(t,x)-(v^{cd},u^{cd},\theta^{cd})(t,x)|=O(1)\delta^{CD}e^{-\frac{cx^{2}}{2\varepsilon(1+t)}}.\\\
$ (2.67)
#### 2.2.2. Rarefaction waves
The construction of the $i$-rarefaction wave
$(V^{R_{i}},U^{R_{i}},\Theta^{R_{i}})(t,x)~{}(i=1,3)$ to the Boltzmann
equation is almost same as the one defined in (2.26) for the compressible
Navier-Stokes equations in the previous section. By setting $U^{R_{i}}_{j}=0$
for $i=1,3$ and $j=2,3$, all the properties of the approximate rarefaction
waves $(V^{R_{i}},U_{1}^{R_{i}},\Theta^{R_{i}})(t,x)~{}(i=1,3)$ given in Lemma
2.3 will also be used later.
#### 2.2.3. Superposition of rarefaction waves and contact discontinuity
We now consider the superposition of two rarefaction waves and a contact
discontinuity. Set $(v_{-},u_{-},\theta_{-})\in$
$R_{1}$-$CD$-$R_{3}(v_{+},u_{+},\theta_{+})$. Then there exist uniquely two
intermediate states $(v_{*},u_{*},\theta_{*})$ and $(v^{*},u^{*},\theta^{*})$
such that $(v_{*},u_{*},\theta_{*})\in R_{1}(v_{-},u_{-},\theta_{-})$,
$(v_{*},u_{*},\theta_{*})\in CD(v^{*},u^{*},\theta^{*})$ and
$(v^{*},u^{*},\theta^{*})\in R_{3}(v_{+},u_{+},\theta_{+})$.
So the wave pattern $(\bar{V},\bar{U},\bar{\Theta})(t,x)$ consisting of
1-rarefaction wave, 2-contact discontinuity and 3-rarefaction wave as a
Riemann solution to the Euler system (1.16) can be defined by
$\begin{array}[]{l}\left(\begin{array}[]{cc}\bar{V}\\\ \bar{U}_{1}\\\
\bar{\Theta}\end{array}\right)(t,x)=\left(\begin{array}[]{cc}v^{r_{1}}+v^{cd}+v^{r_{3}}\\\
u_{1}^{r_{1}}+u_{1}^{cd}+u_{1}^{r_{3}}\\\
\theta^{r_{1}}+\theta^{cd}+\theta^{r_{3}}\end{array}\right)(t,x)-\left(\begin{array}[]{cc}v_{*}+v^{*}\\\
u_{1*}+u_{1}^{*}\\\ \theta_{*}+\theta^{*}\end{array}\right),\\\\[19.91692pt]
\displaystyle\bar{U}_{i}=0,(i=2,3).\end{array}$ (2.68)
where $(v^{r_{1}},u_{1}^{r_{1}},\theta^{r_{1}})(t,x)$ is the approximate
1-rarefaction wave defined in (2.14) with the right state
$(v_{+},u_{+},\theta_{+})$ replaced by $(v_{*},u_{1*},\theta_{*})$,
$(v^{cd},u_{1}^{cd},\theta^{cd})(t,x)$ is the contact discontinuity defined in
(2.57) with the states $(v_{-},u_{-},\theta_{-})$ and
$(v_{+},u_{+},\theta_{+})$ replaced by $(v_{*},u_{*},\theta_{*})$ and
$(v^{*},u^{*},\theta^{*})$ respectively, and
$(v^{r_{3}},u_{1}^{r_{3}},\theta^{r_{3}})(t,x)$ is the 3-rarefaction wave
defined in (2.14) with the left state $(v_{-},u_{-},\theta_{-})$ replaced by
$(v^{*},u_{1}^{*},\theta^{*})$.
Correspondingly, the approximate superposition wave $(V,U,\Theta)(t,x)$ can be
defined by
$\begin{array}[]{l}\left(\begin{array}[]{cc}V\\\ U_{1}\\\
\Theta\end{array}\right)(t,x)=\left(\begin{array}[]{cc}V^{R_{1}}+V^{CD}+V^{R_{3}}\\\
U_{1}^{R_{1}}+U_{1}^{CD}+U_{1}^{R_{3}}\\\
\Theta^{R_{1}}+\Theta^{CD}+\Theta^{R_{3}}\end{array}\right)(t,x)-\left(\begin{array}[]{cc}v_{*}+v^{*}\\\
u_{1*}+u_{1}^{*}\\\ \theta_{*}+\theta^{*}\end{array}\right),\\\\[19.91692pt]
\displaystyle U_{i}=0,(i=2,3).\end{array}$ (2.69)
where $(V^{R_{1}},U_{1}^{R_{1}},\Theta^{R_{1}})(t,x)$ is the 1-rarefaction
wave defined in (2.24) with the right state $(v_{+},u_{+},\theta_{+})$
replaced by $(v_{*},u_{1*},\theta_{*})$,
$(V^{CD},U_{1}^{CD},\Theta^{CD})(t,x)$ is the viscous contact wave defined in
(2.62) with the states $(v_{-},u_{-},\theta_{-})$ and
$(v_{+},u_{+},\theta_{+})$ replaced by $(v_{*},u_{*},\theta_{*})$ and
$(v^{*},u^{*},\theta^{*})$ respectively, and
$(V^{R_{3}},U_{1}^{R_{3}},\Theta^{R_{3}})(t,x)$ is the approximate
3-rarefaction wave defined in (2.24) with the left state
$(v_{-},u_{-},\theta_{-})$ replaced by $(v^{*},u_{1}^{*},\theta^{*})$.
Thus, from the construction of the contact wave and Lemma 2.3, we have the
following relation between the approximate wave pattern $(V,U,\Theta)(t,x)$ of
the Boltzmann equation and the exact inviscid wave pattern
$(\bar{V},\bar{U},\bar{\Theta})(t,x)$ to the Euler equations
$\begin{array}[]{ll}\displaystyle|(V,U,\Theta)(t,x)-(\bar{V},\bar{U},\bar{\Theta})(t,x)|\\\
\displaystyle\quad\leq\frac{C}{t}[\sigma\ln(1+t+t_{0})+\sigma|\ln\sigma|+t_{0}]+C\delta^{CD}e^{-\frac{cx^{2}}{\varepsilon(1+t)}},\end{array}$
(2.70)
with $t_{0}=\varepsilon^{\frac{1}{5}}$ and $\sigma=\varepsilon^{\frac{2}{5}}$.
Then we have
$\left\\{\begin{array}[]{l}V_{t}-U_{1x}=0,\\\
U_{1t}+P_{x}=\varepsilon(\frac{\mu(\Theta)U_{1x}}{V})_{x}+Q_{1},\\\
U_{it}=\varepsilon(\frac{\mu(\Theta)U_{ix}}{V})_{x},~{}~{}i=2,3,\\\
\Theta_{t}+PU_{1x}=\varepsilon(\frac{\lambda(\Theta)\Theta_{x}}{V})_{x}+\varepsilon\frac{\mu(\Theta)U_{1x}^{2}}{V}+Q_{2},\end{array}\right.$
(2.71)
where $P=p(V,\Theta)$ and
$\begin{array}[]{ll}\displaystyle
Q_{1}&\displaystyle=(P-P^{R_{1}}-P^{CD}-P^{R_{3}})_{x}-\varepsilon(\frac{\mu(\Theta)U_{1x}}{V}-\frac{\mu(\Theta^{CD})U^{CD}_{1x}}{V^{CD}})_{x}-Q^{CD}_{1},\\\
\displaystyle
Q_{2}&\displaystyle=(PU_{1x}-P^{R_{1}}U^{R_{1}}_{1x}-P^{CD}U^{CD}_{1x}-P^{R_{3}}U^{R_{3}}_{1x})-\varepsilon(\frac{\lambda(\Theta)\Theta_{x}}{V}-\frac{\lambda(\Theta^{CD})\Theta^{CD}_{x}}{V^{CD}})_{x}\\\
&\displaystyle\qquad-\varepsilon(\frac{\mu(\Theta)U_{1x}^{2}}{V}-\frac{\mu(\Theta^{CD})(U^{CD}_{1x})^{2}}{V^{CD}})-Q_{2}^{CD}.\end{array}$
Direct computation yields
$\begin{array}[]{lll}\displaystyle Q_{1}&=&\displaystyle
O(1)\Big{[}|(V^{R_{1}}_{x},\Theta^{R_{1}}_{x})||(V^{CD}-v_{*},\Theta^{CD}-\theta_{*},V^{R_{3}}-v^{*},\Theta^{R_{3}}-\theta^{*})|\\\
&&\displaystyle+|(V^{R_{3}}_{x},\Theta^{R_{3}}_{x})||(V^{R_{1}}-v_{*},\Theta^{R_{1}}-\theta_{*},V^{CD}-v^{*},\Theta^{CD}-\theta^{*})|\\\
&&\displaystyle+|(V^{CD}_{x},\Theta^{CD}_{x},U^{CD}_{xx})||(V^{R_{1}}-v_{*},\Theta^{R_{1}}-\theta_{*},V^{R_{3}}-v^{*},\Theta^{R_{3}}-\theta^{*})|\\\
&&\displaystyle+\varepsilon|(U^{CD}_{x},V^{CD}_{x})||(U^{R_{1}}_{x},V^{R_{1}}_{x},U^{R_{3}}_{x},V^{R_{3}}_{x})|+\varepsilon|(U^{R_{1}}_{x},V^{R_{1}}_{x})||(U^{R_{3}}_{x},V^{R_{3}}_{x})|\Big{]}\\\
&&\displaystyle+O(1)\varepsilon\Big{[}|U^{R_{1}}_{xx}|+|U^{R_{3}}_{xx}|+|U^{R_{1}}_{x}||V^{R_{1}}_{x}|+|U^{R_{3}}_{x}||V^{R_{3}}_{x}|\Big{]}+|Q^{CD}_{1}|\\\
&:=&\displaystyle Q_{11}+Q_{12}+|Q^{CD}_{1}|,\end{array}$ (2.72)
and
$\begin{array}[]{lll}\displaystyle Q_{2}&=&\displaystyle
O(1)\Big{[}|U^{R_{1}}_{x}||(V^{CD}-v_{*},\Theta^{CD}-\theta_{*},V^{R_{3}}-v^{*},\Theta^{R_{3}}-\theta^{*})|\\\
&&\displaystyle+|U^{R_{3}}_{x}||(V^{R_{1}}-v_{*},\Theta^{R_{1}}-\theta_{*},V^{CD}-v^{*},\Theta^{CD}-\theta^{*})|\\\
&&\displaystyle+|(U^{CD}_{x},V^{CD}_{x},\Theta^{CD}_{x})||(V^{R_{1}}-v_{*},\Theta^{R_{1}}-\theta_{*},V^{R_{3}}-v^{*},\Theta^{R_{3}}-\theta^{*})|\\\
&&\displaystyle+\varepsilon|(U^{CD}_{x},V^{CD}_{x},\Theta^{CD}_{x})||(U^{R_{1}}_{x},V^{R_{1}}_{x},\Theta^{R_{1}}_{x},U^{R_{3}}_{x},V^{R_{3}}_{x},\Theta^{R_{1}}_{x})|\\\
&&\displaystyle+\varepsilon|(U^{R_{1}}_{x},V^{R_{1}}_{x},\Theta^{R_{1}}_{x})||(U^{R_{3}}_{x},V^{R_{3}}_{x},\Theta^{R_{3}}_{x})|\Big{]}\\\
&&\displaystyle+O(1)\varepsilon\Big{[}|\Theta^{R_{1}}_{xx}|+|\Theta^{R_{3}}_{xx}|+|(U^{R_{1}}_{x},V^{R_{1}}_{x},\Theta^{R_{1}}_{x},U^{R_{3}}_{x},V^{R_{3}}_{x},\Theta^{R_{3}}_{x})|^{2}\Big{]}+|Q_{2}^{CD}|\\\
&:=&\displaystyle Q_{21}+Q_{22}+|Q_{2}^{CD}|.\end{array}$ (2.73)
Here, $Q_{11}$ and $Q_{21}$ represent the interaction of waves in different
families, $Q_{12}$ and $Q_{22}$ represent the error terms coming from the
approximate rarefaction wave profiles, and $Q_{i}^{CD}(i=1,2)$ are the error
terms defined in (2.64) and (2.65) due to the viscous contact wave.
Similar to the compressible Navier-Stokes equations case, for the interaction
terms, we have
$|(Q_{11},Q_{21})|=O(1)e^{-C\varepsilon^{-1/5}}e^{-\frac{C(|x|+t+t_{0})}{\varepsilon^{2/5}}},$
(2.74)
for some positive constants $C$.
We now reformulate the system by introducing a scaling for the independent
variables. Set
$y=\frac{x}{\varepsilon},~{}~{}\tau=\frac{t}{\varepsilon}$
as in the previous section for the compressible Navier-Stokes equations. We
also use the notations
$(v,u,\theta)(\tau,y),\mathbf{G}(\tau,y,\xi),\Pi_{1}(\tau,y,\xi)$ and
$(V,U,\Theta)(\tau,y)$ in the scaled independent variables. Set the
perturbation around the composite wave $(V,U,\Theta)(\tau,y)$ by
$(\phi,\psi,\zeta)(\tau,y)=(v-V,u-U,\theta-\Theta)(\tau,y).$
Under this scaling, the hydrodynamic limit problem is reduced to a time
asymptotic stability problem of the composite wave to the Boltzmann equation.
Notice that the hydrodynamic limit proved here is global in time compared to
the case on shock profile studied in [32] which is locally in time.
From (1.21) and (2.72), we have the following system for the perturbation
$(\phi,\psi,\zeta)$
$\left\\{\begin{array}[]{ll}\displaystyle\phi_{\tau}-\psi_{1y}=0,\\\
\displaystyle\psi_{1\tau}+(p-P)_{y}=\frac{4}{3}(\frac{\mu(\theta)u_{1y}}{v}-\frac{\mu(\Theta)U_{1y}}{V})_{y}-\int\xi_{1}^{2}\Pi_{1y}d\xi-\varepsilon
Q_{1},\\\
\displaystyle\psi_{i\tau}=(\frac{\mu(\theta)u_{i1y}}{v}-\frac{\mu(\Theta)U_{iy}}{V})_{y}-\int\xi_{1}\xi_{i}\Pi_{1y}d\xi,~{}~{}i=2,3,\\\
\displaystyle\zeta_{\tau}+(pu_{1y}-PU_{1y})=(\frac{\lambda(\theta)\theta_{y}}{v}-\frac{\lambda(\Theta)\Theta_{y}}{V})_{y}+\frac{4}{3}(\frac{\mu(\theta)u_{1y}^{2}}{v}-\frac{\mu(\Theta)U_{1y}^{2}}{V})\\\
\displaystyle\qquad+\sum_{i=2}^{3}\frac{\mu(\theta)u_{iy}^{2}}{v}+\sum_{i=1}^{3}u_{i}\int\xi_{1}\xi_{i}\Pi_{1y}d\xi-\int\xi_{1}\frac{|\xi|^{2}}{2}\Pi_{1y}d\xi-\varepsilon
U_{1}Q_{1}-\varepsilon Q_{2},\\\ \end{array}\right.$ (2.75)
where the error terms $Q_{i}~{}(i=1,2)$ are given in (2.72) and (2.73)
respectively.
We now derive the equation for the non-fluid component
$\mathbf{G}(\tau,y,\xi)$ in the scaled independent variables. From (1.18), we
have
$\displaystyle\mathbf{G}_{\tau}-\frac{u_{1}}{v}\mathbf{G}_{y}+\frac{1}{v}\mathbf{P}_{1}(\xi_{1}\mathbf{M}_{y})+\frac{1}{v}\mathbf{P}_{1}(\xi_{1}\mathbf{G}_{y})=\mathbf{L}_{\mathbf{M}}\mathbf{G}+Q(\mathbf{G},\mathbf{G}).$
(2.76)
Thus, we obtain
$\mathbf{G}=\frac{1}{v}\mathbf{L}^{-1}_{\mathbf{M}}[\mathbf{P}_{1}(\xi_{1}\mathbf{M}_{y})]+\Pi_{1},$
(2.77)
and
$\Pi_{1}(\tau,y,\xi)=\mathbf{L}_{\mathbf{M}}^{-1}[\mathbf{G}_{\tau}-\frac{u_{1}}{v}\mathbf{G}_{y}+\frac{1}{v}\mathbf{P}_{1}(\xi_{1}\mathbf{G}_{y})-Q(\mathbf{G},\mathbf{G})].$
(2.78)
Let
$\mathbf{G}_{0}(\tau,y,\xi)=\frac{3}{2v\theta}\mathbf{L}^{-1}_{\mathbf{M}}\\{\mathbf{P}_{1}[\xi_{1}(\frac{|\xi-u|^{2}}{2\theta}{\Theta}_{y}+\xi\cdot{U}_{y})\mathbf{M}]\\},$
(2.79)
and
$\mathbf{G}_{1}(\tau,y,\xi)=\mathbf{G}(\tau,y,\xi)-\mathbf{G}_{0}(\tau,y,\xi).$
(2.80)
Then $\mathbf{G}_{1}(\tau,y,\xi)$ satisfies
$\begin{array}[]{ll}\mathbf{G}_{1\tau}-\mathbf{L}_{\mathbf{M}}\mathbf{G}_{1}=&\displaystyle-\frac{3}{2v\theta}\mathbf{P}_{1}[\xi_{1}(\frac{|\xi-u|^{2}}{2\theta}\zeta_{y}+\xi\cdot\psi_{y})\mathbf{M}]\\\
&\displaystyle+\frac{u_{1}}{v}\mathbf{G}_{y}-\frac{1}{v}\mathbf{P}_{1}(\xi_{1}\mathbf{G}_{y})+Q(\mathbf{G},\mathbf{G})-\mathbf{G}_{0\tau}.\end{array}$
(2.81)
Notice that in (2.80) and (2.81), $\mathbf{G}_{0}$ is subtracted from
$\mathbf{G}$ because
$\|(\Theta_{y},U_{y})\|^{2}\sim(1+\varepsilon^{\frac{1}{2}}\tau)^{-1/2}$ is
not integrable globally in $\tau$.
Finally, from (1.15) and the scaling transformation (2.53), we have
$\displaystyle f_{\tau}-\frac{u_{1}}{v}f_{y}+\frac{\xi_{1}}{v}f_{y}=Q(f,f).$
(2.82)
The estimation on the fluid and non-fluid components governed by the above
systems will be given in the last section.
#### 2.2.4. Main result to Boltzmann equation
With the above preparation, we are now ready to state the main result on the
Boltzmann equation as follows.
###### Theorem 2.5.
Given a Riemann solution $(\bar{V},\bar{U},\bar{\Theta})(t,x)$ defined in
(2.68), which is a superposition of two rarefaction waves and a contact
discontinuity to the Euler system (1.16), there exist small positive constants
$\delta_{0}$, $\varepsilon_{0}$ and a global Maxwellian
$\mathbf{M}_{\star}=\mathbf{M}_{[v_{\star},u_{\star},\theta_{\star}]}$, such
that if the contact wave strength $\delta^{CD}\leq\delta_{0}$, and the Knudsen
number $\varepsilon\leq\varepsilon_{0}$, then the Boltzmann equation (1.5)
admits a unique global solution $f^{\varepsilon}(t,x,\xi)$ satisfying
$\sup_{(t,x)\in\Sigma_{h}}\|f^{\varepsilon}(t,x,\xi)-\mathbf{M}_{[\bar{V},\bar{U},\bar{\Theta}]}(t,x,\xi)\|_{L_{\xi}^{2}(\frac{1}{\sqrt{\mathbf{M}_{\star}}})}\leq
C_{h}~{}\varepsilon^{\frac{1}{5}},\qquad\forall h>0,$ (2.83)
where $\Sigma_{h}=\\{(t,x)|t\geq h,\frac{x}{\sqrt{1+t}}\geq
h\varepsilon^{\alpha},0<\alpha<\frac{1}{2}\\}$, the norm
$\|\cdot\|_{L_{\xi}^{2}(\frac{1}{\sqrt{\mathbf{M}_{\star}}})}$ is
$\|\frac{\cdot}{\sqrt{\mathbf{M}_{\star}}}\|_{L_{\xi}^{2}(\mathbf{R}^{3})}$
and the positive constant $C_{h}$ depends only on $h$ but is independent of
$\varepsilon$.
###### Remark 5.
Theorem 2.5 shows that, away from the initial time $t=0$ and the contact
discontinuity located at $x=0$ with the expansion rate
$\frac{x^{2}}{\varepsilon(1+t)}$, for Knudsen number
$\varepsilon<\varepsilon_{0}$, there exists a unique global solution
$f^{\varepsilon}(t,x,\xi)$ of the Boltzmann equation (1.5) which tends to the
Maxwellian $\mathbf{M}_{[\bar{V},\bar{U},\bar{\Theta}]}(t,x,\xi)$ with
$(\bar{V},\bar{U},\bar{\Theta})(t,x)$ being the Riemann solution to the Euler
equation with the combination of two rarefaction waves and a contact
discontinuity when $\varepsilon\rightarrow 0$. Moreover, a uniform convergence
rate $\varepsilon^{\frac{1}{5}}$ in the norm
$L_{\xi}^{2}(\frac{1}{\sqrt{\mathbf{M}_{\star}}})$ holds on the set
$\Sigma_{h}$ for any fixed $h>0$.
###### Remark 6.
Theorem 2.5 holds uniformly on the $(t,x)\in\Sigma_{h}$ for any $h>0$ if the
contact wave strength $\delta^{CD}$ and Knudsen number $\varepsilon$ are
suitably small. But if we restrict the problem to the set
$\Sigma_{h}\cap\\{t\leq T\\}$ for any fixed $T>0$, then we don’t need the
smallness condition on the contact wave strength $\delta^{CD}$ by using
Gronwall inequality to get a time dependent estimate rather than the uniform
estimation in time.
Notations: Throughout this paper, the positive generic constants which are
independent of $T,\varepsilon$ are denoted by $c$, $C$ or $C_{0}$. For
function spaces, $H^{l}(\mathbf{R})$ denotes the $l$-th order Sobolev space
with its norm
$\|f\|_{l}=(\sum^{l}_{j=0}\|\partial^{j}_{y}f\|^{2})^{\frac{1}{2}},\quad{\rm
and}~{}\|\cdot\|:=\|\cdot\|_{L^{2}(dy)},$
where $L^{2}(dz)$ means the $L^{2}$ integral over $\mathbf{R}$ with respect to
the Lebesgue measure $dz$, and $z=x$ or $y$.
## 3\. Proof of Theorem 2.4: Zero dissipation limit of Navier-Stokes
equations
We will prove Theorem 2.4 about the fluid dynamic limit for the compressible
Navier-Stokes equations to the Riemann solution of the Euler equations in this
section. The proof is based on the energy estimates on the perturbation in the
scaled independent variables. In fact, to prove Theorem 2.4, it is sufficient
to prove the following theorem.
###### Theorem 3.1.
There exist small positive constants $\delta_{1}$ and $\varepsilon_{1}$ such
that if the initial values and the contact wave strength $\delta^{CD}$ satisfy
$\mathcal{N}(\tau)|_{\tau=0}+\delta^{CD}\leq\delta_{1},$ (3.1)
and the Knudsen number $\varepsilon$ satisfies
$\varepsilon\leq\varepsilon_{1}$, then the problem (2.54) admits a unique
global solution
$(v^{\varepsilon},u^{\varepsilon},\theta^{\varepsilon})(\tau,y)$ satisfying
$\begin{array}[]{l}\displaystyle\sup_{\tau,y}|(v^{\varepsilon},u^{\varepsilon},\theta^{\varepsilon})(\tau,y)-(V,U,\Theta)(\tau,y)|\leq
C\varepsilon^{\frac{1}{5}}.\\\ \end{array}$ (3.2)
Here $\mathcal{N}(\tau)$ is defined by (3.3) below.
We will focus on the reformulated system (2.54). Since the local existence of
the solution to (2.54) is standard, to prove the global existence, we only
need to close the following a priori estimate by the continuity argument
$\begin{array}[]{ll}\displaystyle\mathcal{N}(\tau)=&\displaystyle\sup_{0\leq\tau^{\prime}\leq\tau}\|(\phi,\psi,\zeta)(\tau^{\prime},\cdot)\|_{1}^{2}\leq\chi^{2},\end{array}$
(3.3)
where $\chi$ is a small positive constant depending only on the initial values
and the strength of the contact wave. And the proof of the above a priori
estimate is given by the following energy estimations.
Firstly, multiplying $\eqref{(2.24)}_{2}$ by $\psi$ yields
$\begin{array}[]{ll}{\displaystyle(\frac{1}{2}\psi^{2})_{\tau}-(p-P)\psi_{y}+(\frac{u_{y}}{v}-\frac{U_{y}}{V})\psi_{y}=-\varepsilon
Q_{1}\psi+\left[(\frac{u_{y}}{v}-\frac{U_{y}}{V})\psi-(p-P)\psi\right]_{y}.}\end{array}$
(3.4)
Since $p-P=R\Theta(\frac{1}{v}-\frac{1}{V})+\frac{R\zeta}{v}$ and
$\phi_{\tau}=\psi_{y}$, we get
$\begin{array}[]{l}\displaystyle(\frac{1}{2}\psi^{2})_{\tau}-R\Theta(\frac{1}{v}-\frac{1}{V})\phi_{\tau}-\frac{R}{v}\zeta\psi_{y}+\frac{\psi^{2}_{y}}{v}\\\
\displaystyle=-(\frac{1}{v}-\frac{1}{V})U_{y}\psi_{y}-\varepsilon
Q_{1}\psi+\left[(\frac{u_{y}}{v}-\frac{U_{y}}{V})\psi-(p-P)\psi\right]_{y}.\end{array}$
(3.5)
Set
$\Phi(z)=z-1-\ln z.$ (3.6)
It is easy to check that $\Phi(1)=\Phi^{\prime}(1)=0$ and $\Phi(z)$ is
strictly convex around $z=1$. Moreover,
$\displaystyle[R\Theta\Phi(\frac{v}{V})]_{\tau}=R\Theta_{\tau}\Phi(\frac{v}{V})-R\Theta(\frac{1}{v}-\frac{1}{V})\phi_{\tau}-\frac{PV_{\tau}}{vV}\phi^{2}.$
(3.7)
On the other hand, note that
$[\frac{R}{\gamma-1}\Theta\Phi(\frac{\theta}{\Theta})]_{\tau}=\frac{R}{\gamma-1}(1-\frac{\Theta}{\theta})\zeta_{\tau}+\frac{R}{\gamma-1}\Phi(\frac{\theta}{\Theta})\Theta_{\tau}-\frac{R}{\gamma-1}\frac{\Theta_{\tau}\zeta^{2}}{\theta\Theta},$
(3.8)
and
$\begin{array}[]{ll}&\displaystyle\frac{R}{\gamma-1}(1-\frac{\Theta}{\theta})\zeta_{\tau}\\\
&\displaystyle=(1-\frac{\Theta}{\theta})[-(pu_{y}-PU_{y})+\nu(\frac{\theta_{y}}{v}-\frac{\Theta_{y}}{V})_{y}+(\frac{u_{y}^{2}}{v}-\frac{U_{y}^{2}}{V})-\varepsilon
Q_{2}]\\\
&\displaystyle=-\frac{R}{v}\zeta\psi_{y}-\frac{\zeta}{\theta}(p-P)U_{y}-\nu(\frac{\zeta}{\theta})_{y}(\frac{\theta_{y}}{v}-\frac{\Theta_{y}}{V})+\frac{\zeta}{\theta}(\frac{u_{y}^{2}}{v}-\frac{U_{y}^{2}}{V})\\\
&\displaystyle\quad-\varepsilon\frac{\zeta}{\theta}Q_{2}+\left[\nu\frac{\zeta}{\theta}(\frac{\theta_{y}}{v}-\frac{\Theta_{y}}{V})\right]_{y}\\\
&\displaystyle=-\frac{R}{v}\zeta\psi_{y}-\frac{\zeta}{\theta}(p-P)U_{y}-\frac{\nu\zeta_{y}^{2}}{v\theta}-\nu\frac{\zeta_{y}}{\theta}(\frac{1}{v}-\frac{1}{V})\Theta_{y}\\\
&\displaystyle\quad+\frac{\nu\zeta\theta_{y}}{\theta^{2}}(\frac{\theta_{y}}{v}-\frac{\Theta_{y}}{V})+\frac{\zeta}{\theta}(\frac{u_{y}^{2}}{v}-\frac{U_{y}^{2}}{V})-\varepsilon
Q_{2}\frac{\zeta}{\theta}+\left[\frac{\nu\zeta}{\theta}(\frac{\theta_{y}}{v}-\frac{\Theta_{y}}{V})\right]_{y}.\end{array}$
(3.9)
Substituting (3.7)-(3.9) into (3.5) gives
$\begin{array}[]{l}\displaystyle[\frac{1}{2}\psi^{2}+R\Theta\Phi(\frac{v}{V})+\frac{R}{\gamma-1}\Theta\Phi(\frac{\theta}{\Theta})]_{\tau}+\frac{\psi_{y}^{2}}{v}\displaystyle+\frac{\nu\zeta_{y}^{2}}{v\theta}+J_{1}\\\
\displaystyle=-U_{y}(\frac{1}{v}-\frac{1}{V})\psi_{y}-\nu\frac{\zeta_{y}}{\theta}(\frac{1}{v}-\frac{1}{V})\Theta_{y}+\frac{\nu\zeta\theta_{y}}{\theta^{2}}(\frac{\theta_{y}}{v}-\frac{\Theta_{y}}{V})\\\
\displaystyle\quad+\frac{\zeta}{\theta}(\frac{u_{y}^{2}}{v}-\frac{U_{y}^{2}}{V})-\varepsilon
Q_{1}\psi-\varepsilon Q_{2}\frac{\zeta}{\theta}+(\cdots)_{y},\end{array}$
(3.10)
where
$J_{1}=\frac{\zeta}{\theta}(p-P)U_{y}-R\Theta_{\tau}\Phi(\frac{v}{V})-\frac{R}{\gamma-1}\Theta_{\tau}\Phi(\frac{\theta}{\Theta})+\frac{PV_{\tau}}{vV}\phi^{2}+\frac{R}{\gamma-1}\frac{\Theta_{\tau}\zeta^{2}}{\theta\Theta}.$
(3.11)
Direct calculation shows that
$\begin{array}[]{ll}J_{1}&\displaystyle=PU_{y}[\Phi(\frac{\theta
V}{v\Theta})+\gamma\Phi(\frac{v}{V})]-[\frac{U_{y}^{2}}{V}+\nu(\frac{\Theta_{y}}{V})_{y}+\varepsilon
Q_{2}][(\gamma-1)\Phi(\frac{v}{V})-\Phi(\frac{\Theta}{\theta})]\\\
&\displaystyle=PU_{y}[\Phi(\frac{\theta
V}{v\Theta})+\gamma\Phi(\frac{v}{V})]-[\frac{U_{y}^{2}}{V}+\nu(\frac{\Theta_{y}}{V})_{y}+\varepsilon
Q_{2}][(\gamma-1)\Phi(\frac{v}{V})-\Phi(\frac{\Theta}{\theta})].\end{array}$
(3.12)
Thus, substituting (3.12) into (3.10) gives
$\begin{array}[]{l}\displaystyle[\frac{1}{2}\psi^{2}+R\Theta\Phi(\frac{v}{V})+\frac{R}{\gamma-1}\Theta\Phi(\frac{\theta}{\Theta})]_{\tau}+\frac{\psi_{y}^{2}}{v}\displaystyle+\frac{\nu\zeta_{y}^{2}}{v\theta}\\\
\displaystyle+P(U^{R_{1}}_{y}+U^{R_{3}}_{y})[\Phi(\frac{\theta
V}{v\Theta})+\gamma\Phi(\frac{v}{V})]=J_{2}-\varepsilon Q_{1}\psi-\varepsilon
Q_{2}\frac{\zeta}{\theta}+(\cdots)_{y},\end{array}$ (3.13)
where
$\begin{array}[]{ll}\displaystyle J_{2}=\displaystyle-
PU^{CD}_{y}[\Phi(\frac{\theta
V}{v\Theta})+\gamma\Phi(\frac{v}{V})]+[\frac{U_{y}^{2}}{V}+\nu(\frac{\Theta_{y}}{V})_{y}+\varepsilon
Q_{2}][(\gamma-1)\Phi(\frac{v}{V})-\Phi(\frac{\Theta}{\theta})]\\\
\qquad\displaystyle-
U_{y}(\frac{1}{v}-\frac{1}{V})\psi_{y}-\nu\frac{\zeta_{y}}{\theta}(\frac{1}{v}-\frac{1}{V})\Theta_{y}+\frac{\nu\zeta\theta_{y}}{\theta^{2}}(\frac{\theta_{y}}{v}-\frac{\Theta_{y}}{V})+\frac{\zeta}{\theta}(\frac{u_{y}^{2}}{v}-\frac{U_{y}^{2}}{V}).\end{array}$
(3.14)
Here, $(\cdots)_{y}$ represents the conservative terms which vanishes after
integrating in $y$ over $\mathbf{R}$.
By the strict convexity of $\Phi(z)$ around $z=1$, under the a priori
assumption (3.3) with sufficiently small $\chi>0$, there exist positive
constants $c_{1}$ and $c_{2}$ such that,
$\begin{array}[]{l}\displaystyle c_{1}\phi^{2}\leq\Phi(\frac{v}{V})\leq
c_{2}\phi^{2},\quad
c_{1}\zeta^{2}\leq\Phi(\frac{\Theta}{\theta}),\Phi(\frac{\theta}{\Theta})\leq
c_{2}\zeta^{2},\\\ \displaystyle
c_{1}(\phi^{2}+\zeta^{2})\leq\Phi(\frac{\theta V}{v\Theta})\leq
c_{2}(\phi^{2}+\zeta^{2}).\end{array}$ (3.15)
Thus, we have
$\begin{array}[]{l}\displaystyle\int_{\mathbf{R}}|J_{2}|dy\leq\int_{\mathbf{R}}(\frac{\psi_{y}^{2}}{4v}+\frac{\nu\zeta_{y}^{2}}{4v\theta})dy+C(\tau+\tau_{0})^{-2}\|(\phi,\zeta)\|^{2}\\\
\displaystyle+C\int_{\mathbf{R}}\delta^{CD}\varepsilon(1+\varepsilon\tau)^{-1}e^{-\frac{c\varepsilon
y^{2}}{1+\varepsilon\tau}}|(\phi,\zeta)|^{2}dy+\int_{\mathbf{R}}\varepsilon|Q_{2}||(\phi,\zeta)|^{2}dy.\end{array}$
(3.16)
Notice that the last term $\varepsilon|Q_{2}||(\phi,\zeta)|^{2}$ on the right
hand side of (3.16) can be estimated similarly as for the terms $\varepsilon
Q_{1}\psi$ and $\varepsilon Q_{2}\frac{\zeta}{\theta}$ under the a priori
assumption (3.3). Now we estimate the terms $\varepsilon Q_{1}\psi$ and
$\varepsilon Q_{2}\frac{\zeta}{\theta}$ on the right hand side of (3.13).
First,
$\int_{\mathbf{R}}\varepsilon|Q_{1}||\psi|dy=\int_{\mathbf{R}}\varepsilon(|Q_{11}|+|Q_{12}|)|\psi|dy.$
From the estimation on the interaction given in (2.51), we get
$\begin{array}[]{ll}&\displaystyle\int_{0}^{\tau}\int_{\mathbf{R}}\varepsilon|Q_{11}||\psi|d\tau
dy\\\
&\displaystyle\leq\int_{0}^{\tau}\|\psi\|_{L^{\infty}_{y}}\int_{\mathbf{R}}|Q_{11}|dxd\tau\\\
&\displaystyle\leq
C\int_{0}^{\tau}e^{-C\varepsilon^{-1/5}}e^{-\frac{C(t+t_{0})}{\varepsilon^{2/5}}}\|\psi\|^{\frac{1}{2}}\|\psi_{y}\|^{\frac{1}{2}}d\tau\\\
&\displaystyle\leq\beta\int_{0}^{\tau}\|\psi_{y}\|^{2}d\tau+C_{\beta}\int_{0}^{\tau}e^{-C\varepsilon^{-1/5}}e^{-C\varepsilon^{3/5}(\tau+\tau_{0})}\|\psi\|^{\frac{2}{3}}d\tau\\\
&\displaystyle\leq\beta\int_{0}^{\tau}\|\psi_{y}\|^{2}d\tau+C_{\beta}e^{-C\varepsilon^{-1/5}}\sup_{[0,\tau]}\|\psi(\tau)\|^{\frac{2}{3}}\\\
&\displaystyle\leq\beta\int_{0}^{\tau}\|\psi_{y}\|^{2}d\tau+\beta\sup_{[0,\tau]}\|\psi(\tau)\|^{2}+C_{\beta}e^{-C\varepsilon^{-1/5}},\end{array}$
(3.17)
and
$\begin{array}[]{ll}\displaystyle&\displaystyle\int_{0}^{\tau}\int_{\mathbf{R}}\varepsilon|Q_{12}||\psi|d\tau
dy\\\
&\displaystyle\leq\varepsilon^{2}\int_{0}^{\tau}\int_{\mathbf{R}}(|(w_{\delta}^{r})_{xx}|,|(w_{\delta}^{r})_{x}|^{2})|\psi|d\tau
dy\\\
&\displaystyle\leq\varepsilon\int_{0}^{\tau}(\|(w_{\delta}^{r})_{xx}\|_{L^{1}(dx)},\|(w_{\delta}^{r})_{x}\|_{L^{2}(dx)}^{2})\|\psi\|_{L^{\infty}_{y}}d\tau\\\
&\displaystyle\leq\int_{0}^{\tau}(\tau+\tau_{0})^{-1}\|\psi\|^{\frac{1}{2}}\|\psi_{y}\|^{\frac{1}{2}}d\tau\\\
&\displaystyle\leq\beta\int_{0}^{\tau}\|\psi_{y}\|^{2}d\tau+C_{\beta}\int_{0}^{\tau}(\tau+\tau_{0})^{-\frac{4}{3}}\|\psi\|^{\frac{2}{3}}d\tau\\\
&\displaystyle\leq\beta\int_{0}^{\tau}\|\psi_{y}\|^{2}d\tau+3C_{\beta}\tau_{0}^{-\frac{1}{3}}\sup_{[0,\tau]}\|\psi(\tau)\|^{\frac{2}{3}}\\\
&\displaystyle\leq\beta\int_{0}^{\tau}\|\psi_{y}\|^{2}d\tau+\beta\sup_{[0,\tau]}\|\psi(\tau)\|^{2}+C_{\beta}\varepsilon^{\frac{2}{5}},\end{array}$
(3.18)
where $\tau_{0}=\frac{t_{0}}{\varepsilon}=\varepsilon^{-\frac{4}{5}}$, and
$\beta>0$ is a small constant to be determined later and $C_{\beta}$ is a
positive constant depending on $\beta$.
The term $\varepsilon Q_{2}\frac{\zeta}{\theta}$ can be estimated similarly
because the only difference is about the error term $Q^{CD}$ coming from the
viscous contact wave in $Q_{2}$. For this, we have
$\begin{array}[]{ll}\displaystyle\varepsilon\int_{0}^{\tau}\int_{\mathbf{R}}|Q^{CD}||\zeta|dyd\tau\\\
\displaystyle\leq\varepsilon^{2}\int_{0}^{\tau}\Big{[}\|\zeta\|_{L^{\infty}_{y}}\int_{\mathbf{R}}(1+\varepsilon\tau)^{-2}e^{-\frac{c\varepsilon
y^{2}}{1+\varepsilon\tau}}dy\Big{]}d\tau\\\
\displaystyle\leq\varepsilon^{\frac{3}{2}}\int_{0}^{\tau}\Big{[}\|\zeta\|_{L^{2}_{y}}^{\frac{1}{2}}\|\zeta_{y}\|^{\frac{1}{2}}_{L^{2}_{y}}(1+\varepsilon\tau)^{-\frac{3}{2}}\Big{]}d\tau\\\
\displaystyle\leq\beta\int_{0}^{\tau}\|\zeta_{y}\|^{2}d\tau+C_{\beta}\varepsilon^{2}\sup_{[0,\tau]}\|\zeta\|_{L^{2}_{y}}^{\frac{2}{3}}\int_{0}^{\tau}(1+\varepsilon\tau)^{-2}d\tau\\\
\displaystyle\leq\beta\|\zeta_{y}\|^{2}+\beta\sup_{[0,\tau]}\|\zeta\|_{L^{2}_{y}}^{2}+C_{\beta}\varepsilon^{\frac{3}{2}}.\end{array}$
(3.19)
By substituting (3.15)-(3.19) into (3.13) and choosing $\beta$ suitably small,
we can get
$\begin{array}[]{ll}&\displaystyle\|(\phi,\psi,\zeta)(\tau,\cdot)\|^{2}+\int_{0}^{\tau}\Big{[}\|(\psi_{y},\zeta_{y})\|^{2}+\|\sqrt{(U^{R_{1}}_{y},U^{R_{3}}_{y})}(\phi,\zeta)\|^{2}\Big{]}d\tau\\\
&\displaystyle\leq
C\int_{0}^{\tau}(\tau+\tau_{0})^{-2}\|(\phi,\zeta)\|^{2}d\tau+C\varepsilon^{\frac{2}{5}}\\\
&\displaystyle\quad+C\delta^{CD}\varepsilon\int_{0}^{\tau}\int_{\mathbf{R}}(1+\varepsilon\tau)^{-1}e^{-\frac{C_{0}\varepsilon
y^{2}}{1+\varepsilon\tau}}|(\phi,\zeta)|^{2}dyd\tau.\end{array}$ (3.20)
Now we need to estimate $\|\phi_{y}\|^{2}$. Let $\tilde{v}=\frac{v}{V}$, then
$\frac{\tilde{v}_{\tau}}{\tilde{v}}=\frac{u_{y}}{v}-\frac{U_{y}}{V}.$
Rewrite the equation $\eqref{(2.24)}_{2}$ as
$(\frac{\tilde{v}_{y}}{\tilde{v}})_{\tau}-\psi_{\tau}-(p-P)_{y}-\varepsilon
Q_{1}=0.$ (3.21)
By multiplying (3.21) by $\frac{\tilde{v}_{y}}{\tilde{v}}$ and noticing that
$-(p-P)_{y}=\frac{R\theta}{v}\frac{\tilde{v}_{y}}{\tilde{v}}-\frac{R\zeta_{y}}{v}+(p-P)\frac{V_{y}}{V}+R\Theta_{y}(\frac{1}{v}-\frac{1}{V}),$
(3.22)
we get
$\begin{array}[]{ll}&\displaystyle\left[\frac{1}{2}(\frac{\tilde{v}_{y}}{\tilde{v}})^{2}-\psi\frac{\tilde{v}_{y}}{\tilde{v}}\right]_{\tau}+\left[\psi\frac{\tilde{v}_{\tau}}{\tilde{v}}\right]_{y}+\frac{R\theta}{v}(\frac{\tilde{v}_{y}}{\tilde{v}})^{2}\\\\[14.22636pt]
=&\displaystyle\psi_{y}(\frac{u_{y}}{v}-\frac{U_{y}}{V})+\left[\frac{R\zeta_{y}}{v}-(p-P)\frac{V_{y}}{V}-R\Theta_{y}(\frac{1}{v}-\frac{1}{V})+\varepsilon
Q_{1}\right]\frac{\tilde{v}_{y}}{\tilde{v}}.\end{array}$
Integrating the above equality over $[0,\tau]\times\mathbf{R}$ in $\tau$ and
$y$, we obtain
$\begin{array}[]{ll}&\displaystyle\int_{\bf
R}\left[\frac{1}{2}(\frac{\tilde{v}_{y}}{\tilde{v}})^{2}-\psi\frac{\tilde{v}_{y}}{\tilde{v}}\right](\tau,y)dy+\int_{0}^{\tau}\int_{\bf
R}\frac{R\theta}{2v}(\frac{\tilde{v}_{y}}{\tilde{v}})^{2}dyd\tau\\\\[11.38109pt]
\leq&\displaystyle
C\int_{0}^{\tau}\bigg{[}\|(\psi_{y},\zeta_{y})\|^{2}+\varepsilon^{2}\|Q_{1}\|^{2}\bigg{]}d\tau+C\int_{0}^{\tau}\int_{\bf
R}|(V_{y},U_{y},\Theta_{y})|^{2}|(\phi,\zeta)|^{2}dyd\tau.\end{array}$ (3.23)
The by using the equality
$\frac{\tilde{v}_{y}}{\tilde{v}}=\frac{v_{y}}{v}-\frac{V_{y}}{V}=\frac{\phi_{y}}{v}-\frac{V_{y}\phi}{vV},$
we have
$C^{-1}(|\phi_{y}|^{2}-|V_{y}\phi|^{2})\leq(\frac{\tilde{v}_{y}}{\tilde{v}})^{2}\leq
C(|\phi_{y}|^{2}+|V_{y}\phi|^{2}).$ (3.24)
By the estimation on $Q_{11}$ in (2.51) and Lemma 2.3, we have
$\begin{array}[]{ll}\displaystyle\int_{0}^{\tau}\varepsilon^{2}\|Q_{1}\|^{2}d\tau&\displaystyle\leq
C\int_{0}^{\tau}\int_{\mathbf{R}}\varepsilon^{2}(|Q_{11}|^{2}+|Q_{12}|^{2})dyd\tau\\\
&\displaystyle\leq
C\int_{0}^{t}\int_{\mathbf{R}}(|Q_{11}|^{2}+\varepsilon^{2}|(w_{\delta}^{r})_{xx}|^{2}+\varepsilon^{2}|(w_{\delta}^{r})_{x}|^{4})dxdt\\\
&\displaystyle\leq
Ce^{-C\varepsilon^{-1/5}}+C\varepsilon^{2}(t_{0}^{-2}+\delta^{-1}t_{0}^{-1})\\\
&\displaystyle\leq C\varepsilon^{\frac{7}{5}}.\end{array}$ (3.25)
Moreover, we have
$\begin{array}[]{ll}\displaystyle|(V_{y},U_{y},\Theta_{y})|^{2}=\varepsilon^{2}|(V_{x},U_{x},\Theta_{x})|^{2}\\\
\qquad\displaystyle\leq\varepsilon^{2}\sum_{i=1,3}|(V^{R_{i}}_{x},U^{R_{i}}_{x},\Theta^{R_{i}}_{x})|^{2}+\varepsilon^{2}|(V_{x}^{CD},U^{CD}_{x},\Theta_{x}^{CD})|^{2}\\\
\qquad\displaystyle\leq
C\varepsilon^{2}(t+t_{0})^{-2}+C\delta^{CD}\varepsilon(1+t)^{-1}e^{-\frac{C_{0}x^{2}}{\varepsilon(1+t)}}\\\
\qquad\displaystyle=C(\tau+\tau_{0})^{-2}+C\delta^{CD}\varepsilon(1+\varepsilon\tau)^{-1}e^{-\frac{C_{0}\varepsilon
y^{2}}{1+\varepsilon\tau}}.\end{array}$ (3.26)
Substituting (3.24)-(3.26) into (3.23) gives
$\begin{array}[]{ll}&\displaystyle\|\phi_{y}(\tau,\cdot)\|^{2}+\int_{0}^{\tau}\|\phi_{y}\|^{2}d\tau\leq
C\|(\phi,\psi)(\tau,\cdot)\|^{2}\\\
&\displaystyle\quad+C\int_{0}^{\tau}\|(\psi_{y},\zeta_{y})\|^{2}d\tau+C\int_{0}^{\tau}(\tau+\tau_{0})^{-2}\|(\phi,\zeta)\|^{2}d\tau+C\varepsilon^{\frac{7}{5}}\\\
&\displaystyle\quad+C\delta^{CD}\int_{0}^{\tau}\int_{\bf
R}\varepsilon(1+\varepsilon\tau)^{-1}e^{-\frac{C_{0}\varepsilon
y^{2}}{1+\varepsilon\tau}}|(\phi,\zeta)|^{2}dyd\tau.\end{array}$ (3.27)
Now we estimate the higher order derivatives of $(\psi,\zeta)$. Multiplying
$\eqref{(2.24)}_{2}$ by $-\psi_{yy}$ and $\eqref{(2.24)}_{3}$ by
$-\zeta_{yy}$, and then adding the resulting equations together yield
$\begin{array}[]{l}\displaystyle[\frac{1}{2}\psi_{y}^{2}+\frac{R}{2(\gamma-1)}\zeta_{y}^{2}]_{\tau}+\frac{\psi_{yy}^{2}}{v}+\nu\frac{\zeta_{yy}^{2}}{v}\\\
\displaystyle=\big{\\{}(p-P)_{y}+\frac{v_{y}}{v^{2}}\psi_{y}+[U_{y}(\frac{1}{v}-\frac{1}{V})]_{y}+\varepsilon
Q_{1}\big{\\}}\psi_{yy}\\\ \displaystyle+\big{\\{}(pu_{y}-PU_{y})+\frac{\nu
v_{y}}{v^{2}}\zeta_{y}+[\nu\Theta_{y}(\frac{1}{v}-\frac{1}{V})]_{y}+(\frac{u_{y}^{2}}{v}-\frac{U_{y}^{2}}{V})+\varepsilon
Q_{2}\big{\\}}\zeta_{yy}.\end{array}$ (3.28)
The right hand side of (3.28) will be estimated terms by terms as follows.
From (3.22) and (3.26), we get
$\begin{array}[]{ll}\displaystyle\int_{0}^{\tau}\int_{\mathbf{R}}(p-P)_{y}\psi_{yy}dyd\tau\\\
\displaystyle\leq
C\int_{0}^{\tau}\int_{\mathbf{R}}\Big{[}|(\phi_{y},\zeta_{y})|+|(V_{y},\Theta_{y})||(\phi,\zeta)|\Big{]}|\psi_{yy}|dyd\tau\\\
\displaystyle\leq\beta\int_{0}^{\tau}\|\psi_{yy}\|^{2}d\tau+C_{\beta}\int_{0}^{\tau}\|(\phi_{y},\zeta_{y})\|^{2}d\tau+C_{\beta}\int_{0}^{\tau}(\tau+\tau_{0})^{-2}\|(\phi,\zeta)\|^{2}d\tau\\\
\displaystyle+C_{\beta}\delta^{CD}\int_{0}^{\tau}\int_{\bf
R}\varepsilon(1+\varepsilon\tau)^{-1}e^{-\frac{C\varepsilon
y^{2}}{1+\varepsilon\tau}}|(\phi,\zeta)|^{2}dyd\tau.\end{array}$ (3.29)
Similar estimate holds for the term
$\int_{0}^{\tau}\int_{\mathbf{R}}(pu_{y}-PU_{y})\zeta_{yy}dyd\tau.$
Notice that
$\begin{array}[]{ll}\displaystyle\int_{0}^{\tau}\int_{\mathbf{R}}\frac{v_{y}}{v^{2}}\psi_{y}\psi_{yy}dyd\tau\\\
\displaystyle\leq
C\int_{0}^{\tau}\int_{\mathbf{R}}(|\phi_{y}|+|V_{y}|)|\psi_{y}||\psi_{yy}|dyd\tau\\\
\displaystyle\leq
C\int_{0}^{\tau}(\|\phi_{y}\|\|\psi_{yy}\|\|\psi_{y}\|_{L_{y}^{\infty}}+\|V_{y}\|_{L^{\infty}_{y}}\|\|\psi_{y}\|\psi_{yy}\|)d\tau\\\
\displaystyle\leq
C\int_{0}^{\tau}\|\psi_{yy}\|^{\frac{3}{2}}\|\psi_{y}\|^{\frac{1}{2}}\|\phi_{y}\|d\tau+C\varepsilon^{\frac{1}{2}}\int_{0}^{\tau}\|\psi_{y}\|\psi_{yy}\|d\tau\\\
\displaystyle\leq\beta\int_{0}^{\tau}\|\psi_{yy}\|^{2}d\tau+C_{\beta}(\sup_{[0,\tau]}\|\phi_{y}\|^{4}+\varepsilon)\int_{0}^{\tau}\|\psi_{y}\|^{2}d\tau\\\
\displaystyle\leq\beta\int_{0}^{\tau}\|\psi_{yy}\|^{2}d\tau+C_{\beta}(\chi^{4}+\varepsilon)\int_{0}^{\tau}\|\psi_{y}\|^{2}d\tau,\end{array}$
(3.30)
where in the third inequality we have used the fact that
$\|V_{y}\|_{L^{\infty}}\leq C\varepsilon^{\frac{1}{2}}$ because of (3.26).
Similarly, we have
$\begin{array}[]{ll}\displaystyle\int_{0}^{\tau}\int_{\mathbf{R}}\nu\frac{v_{y}}{v^{2}}\zeta_{y}\zeta_{yy}dyd\tau\\\
\displaystyle\leq\beta\int_{0}^{\tau}\|\zeta_{yy}\|^{2}d\tau+C_{\beta}(\chi^{4}+\varepsilon)\int_{0}^{\tau}\|\zeta_{y}\|^{2}d\tau.\end{array}$
(3.31)
The remaining terms can be estimated directly by using (3.25) and the fact
that
$[U_{y}(\frac{1}{v}-\frac{1}{V})]_{y}=O(1)[|(U_{yy},U_{y}V_{y})||\phi|+|U_{y}||\phi_{y}|],$
$[\nu\Theta_{y}(\frac{1}{v}-\frac{1}{V})]_{y}=O(1)[|(\Theta_{yy},\Theta_{y}V_{y})||\phi|+|\Theta_{y}||\phi_{y}|].$
Hence, if we take $\beta$ suitably small, then we obtain
$\begin{array}[]{ll}\displaystyle\|(\psi_{y},\zeta_{y})(\tau,\cdot)\|^{2}+\int_{0}^{\tau}\|(\psi_{yy},\zeta_{yy})\|^{2}d\tau\\\
\displaystyle\quad\leq
C\int_{0}^{\tau}\|(\phi_{y},\psi_{y},\zeta_{y})\|^{2}d\tau+C\int_{0}^{\tau}(\tau+\tau_{0})^{-2}\|(\phi,\zeta)\|^{2}d\tau+C\varepsilon^{\frac{7}{5}}\\\
\displaystyle\quad+C\delta^{CD}\int_{0}^{\tau}\int_{\bf
R}\varepsilon(1+\varepsilon\tau)^{-1}e^{-\frac{C\varepsilon
y^{2}}{1+\varepsilon\tau}}|(\phi,\zeta)|^{2}dyd\tau.\end{array}$ (3.32)
The combination of (3.20), (3.27) and (3.32) yields that
$\begin{array}[]{ll}\displaystyle\|(\phi,\psi,\zeta)(\tau,\cdot)\|_{1}^{2}+\int_{0}^{\tau}\Big{[}\|\phi_{y}\|^{2}+\|(\psi_{y},\zeta_{y})\|_{1}^{2}\Big{]}d\tau\\\
\displaystyle\quad\leq
C\int_{0}^{\tau}(\tau+\tau_{0})^{-2}\|(\phi,\zeta)\|^{2}d\tau+C\varepsilon^{\frac{2}{5}}\\\
\displaystyle\quad+C\delta^{CD}\int_{0}^{\tau}\int_{\bf
R}\varepsilon(1+\varepsilon\tau)^{-1}e^{-\frac{C_{0}\varepsilon
y^{2}}{1+\varepsilon\tau}}|(\phi,\zeta)|^{2}dyd\tau.\end{array}$ (3.33)
In order to close the estimate, we only need to control the last term in
(3.33), which comes from the viscous contact wave. For this, we will apply the
following technique by using the heat kernel motivated by [13].
###### Lemma 3.2.
Suppose that $h(\tau,y)$ satisfies
$h\in L^{\infty}(0,+\infty;L^{2}(\mathbf{R})),~{}~{}h_{y}\in
L^{2}(0,+\infty;L^{2}(\mathbf{R})),~{}~{}h_{\tau}\in
L^{2}(0,+\infty;H^{-1}(\mathbf{R})),$
Then
$\begin{array}[]{ll}&\displaystyle\int_{0}^{\tau}\int_{\mathbf{R}^{+}}\varepsilon(1+\varepsilon\tau)^{-1}e^{-\frac{2a\varepsilon
y^{2}}{1+\varepsilon\tau}}h^{2}(\tau,y)dyd\tau\\\\[5.69054pt]
&\displaystyle\leq
C_{a}\bigg{[}\|h(0,y)\|^{2}+\int_{0}^{\tau}\|h_{y}\|^{2}d\tau+\int_{0}^{\tau}\langle
h_{\tau},hg_{a}^{2}\rangle_{H^{-1}\times H^{1}}d\tau\bigg{]}\end{array}$
(3.34)
where
$g_{a}(\tau,y)=\varepsilon^{\frac{1}{2}}(1+\varepsilon\tau)^{-\frac{1}{2}}\int^{y}_{-\infty}e^{-\frac{a\varepsilon\eta^{2}}{1+\varepsilon\tau}}d\eta,$
and $a>0$ is the constant to be determined later.
The proof of Lemma 3.2 is similar to the one given in [13]. The only
difference here is that we need to be careful about the parameter
$\varepsilon$ in the estimation. Therefore, we omit its proof for brevity.
Based on Lemma 3.2, we can obtain
###### Lemma 3.3.
There exists a constant $C>0$ such that if $\delta^{CD}$ and $\varepsilon_{0}$
are small enough, then we have
$\begin{array}[]{ll}&\displaystyle\int_{0}^{\tau}\int_{{\bf
R}}\varepsilon(1+\varepsilon\tau)^{-1}e^{-\frac{C_{0}\varepsilon
y^{2}}{1+\varepsilon\tau}}|(\phi,\psi,\zeta)|^{2}dyd\tau\\\\[8.53581pt]
&\displaystyle\leq
C\|(\phi,\psi,\zeta)(\tau,\cdot)\|^{2}+C\int_{0}^{\tau}\|(\phi_{y},\psi_{y},\zeta_{y})\|^{2}d\tau\\\
&\displaystyle+C\int_{0}^{\tau}(\tau+\tau_{0})^{-\frac{3}{2}}\|(\phi,\psi,\zeta)\|^{2}d\tau+C\varepsilon^{\frac{2}{5}}.\end{array}$
(3.35)
###### Proof.
From the equation $\eqref{(2.24)}_{2}$ and the fact that
$p-P=\frac{R\zeta-P\phi}{v}$, we have
$\psi_{\tau}+(\frac{R\zeta-P\phi}{v})_{y}=(\frac{u_{y}}{v}-\frac{U_{y}}{V})_{y}-\varepsilon
Q_{1}.$
Then
$(R\zeta-P\phi)_{y}=\frac{R\zeta-P\phi}{v}(V_{y}+\phi_{y})-v\psi_{\tau}+v(\frac{u_{y}}{v}-\frac{U_{y}}{V})_{y}-v\varepsilon
Q_{1}.$ (3.36)
Let
$G_{b}(\tau,y)=\varepsilon(1+\varepsilon\tau)^{-1}\int^{y}_{-\infty}e^{-\frac{b\varepsilon\eta^{2}}{1+\varepsilon\tau}}d\eta,$
where $b$ is a positive constant to be determined later. Multiplying the
equation (3.36) by $G_{b}(R\zeta-P\phi)$ gives
$\begin{array}[]{ll}&\displaystyle\left[\frac{G_{b}(R\zeta-P\phi)^{2}}{2}\right]_{y}-(G_{b})_{y}\frac{(R\zeta-P\phi)^{2}}{2}\\\\[5.69054pt]
=&\displaystyle\frac{G_{b}(R\zeta-P\phi)^{2}}{v}(V_{y}+\phi_{y})-G_{b}v(R\zeta-P\phi)\psi_{\tau}\\\\[5.69054pt]
&\displaystyle+G_{b}v(R\zeta-P\phi)(\frac{u_{y}}{v}-\frac{U_{y}}{V})_{y}-\varepsilon
G_{b}v(R\zeta-P\phi)Q_{1}.\end{array}$ (3.37)
Note that
$\begin{array}[]{ll}\displaystyle-
G_{b}v(R\zeta-P\phi)\psi_{\tau}&\displaystyle=-[G_{b}v(R\zeta-P\phi)\psi]_{\tau}+[G_{b}v(R\zeta-P\phi)\psi]_{y}\\\\[8.53581pt]
&\displaystyle\quad+(G_{b}v)_{\tau}(R\zeta-P\phi)\psi+G_{b}v\psi(R\zeta-P\phi)_{\tau},\end{array}$
(3.38) $\begin{array}[]{ll}\displaystyle(R\zeta-P\phi)_{\tau}\\\\[2.84526pt]
\displaystyle=R\zeta_{\tau}-P_{\tau}\phi-P\phi_{\tau}\\\\[2.84526pt]
\displaystyle=(\gamma-1)\bigg{[}-(p-P)(U_{y}+\psi_{y})+(\frac{u_{y}^{2}}{v}-\frac{U_{y}^{2}}{V})+\nu(\frac{\theta_{y}}{v}-\frac{\Theta_{y}}{V})_{y}-\varepsilon
Q_{2}\bigg{]}\\\ \displaystyle\quad-\gamma P\psi_{y}-P_{\tau}\phi.\end{array}$
(3.39)
By using the equality
$\displaystyle-G_{b}v\gamma P\psi_{y}\psi=-[\gamma
G_{b}vP\frac{\psi^{2}}{2}]_{y}+\gamma
vP(G_{b})_{y}\frac{\psi^{2}}{2}+\gamma(vP)_{y}G_{b}\frac{\psi^{2}}{2},$ (3.40)
we have
$\varepsilon(1+\varepsilon\tau)^{-1}e^{-\frac{b\varepsilon
y^{2}}{1+\varepsilon\tau}}[(R\zeta-P\phi)^{2}+\gamma
Pv\psi^{2}]=[G_{b}v(R\zeta-P\phi)\psi]_{\tau}+(\cdots)_{y}+Q_{4},$ (3.41)
where
$\begin{array}[]{ll}\displaystyle
Q_{4}=&\displaystyle-(vG_{b})_{\tau}v(R\zeta-P\phi)\psi-\frac{\gamma\psi^{2}}{2}(Pv)_{y}G_{b}+G_{b}v\psi
P_{\tau}\phi\\\\[5.69054pt]
&\displaystyle+(\gamma-1)G_{b}v\psi\left[(p-P)(U_{y}+\psi_{y})-(\frac{u_{y}^{2}}{v}-\frac{U_{y}^{2}}{V})+\varepsilon
Q_{2}\right]\\\\[5.69054pt]
&\displaystyle+[G_{b}v(R\zeta-P\phi)]_{y}(\frac{u_{y}}{v}-\frac{U_{y}}{V})+(\gamma-1)\nu(G_{b}v\psi)_{y}(\frac{\theta_{y}}{v}-\frac{\Theta_{y}}{V})\\\\[5.69054pt]
&\displaystyle-\frac{G_{b}(R\zeta-P\phi)^{2}}{v}(V_{y}+\phi_{y})+\varepsilon
G_{b}v(R\zeta-P\phi)Q_{1}.\end{array}$ (3.42)
Note that
$\|G_{b}(\tau,\cdot)\|_{L^{\infty}}\leq
C_{\alpha}\varepsilon^{\frac{1}{2}}(1+\varepsilon\tau)^{-\frac{1}{2}}.$
Thus, integrating (3.41) over $(0,\tau)\times\mathbf{R}$ gives
$\begin{array}[]{ll}&\displaystyle\int_{0}^{\tau}\int_{{\bf
R}}\varepsilon(1+\varepsilon\tau)^{-1}e^{-\frac{b\varepsilon
y^{2}}{1+\varepsilon\tau}}[(R\zeta-P\phi)^{2}+\psi^{2}]dyd\tau\\\\[8.53581pt]
&\displaystyle\leq
C\|(\phi,\psi,\zeta)(\tau,\cdot)\|^{2}+C\int_{0}^{\tau}\|(\phi_{y},\psi_{y},\zeta_{y})(\tau,\cdot)\|^{2}d\tau\\\\[5.69054pt]
&\displaystyle+C\int_{0}^{\tau}(\tau+\tau_{0})^{-\frac{3}{2}}\|(\phi,\psi,\zeta)(\tau,\cdot)\|^{2}d\tau+C\varepsilon^{\frac{7}{5}}\\\\[5.69054pt]
&\displaystyle+C\delta^{CD}\int_{0}^{\tau}\int_{{\bf
R}}\varepsilon(1+\varepsilon\tau)^{-1}e^{-\frac{C_{0}\varepsilon
y^{2}}{1+\varepsilon\tau}}|(\phi,\zeta)|^{2}dyd\tau.\end{array}$ (3.43)
In order to get the desired estimate stated in Lemma 3.3, set
$h=\frac{R}{\gamma-1}\zeta+P\phi$
in Lemma 3.2. We only need to compute the last term on the right hand side of
(3.34) for this given function $h$. From the energy equation
$\eqref{(2.24)}_{3}$, we have
$\begin{array}[]{ll}\displaystyle
h_{\tau}&\displaystyle=-(p-P)\psi_{y}+[P_{\tau}\phi-(p-P)U_{y}]+\nu(\frac{\theta_{y}}{v}-\frac{\Theta_{y}}{V})_{y}+(\frac{u_{y}^{2}}{v}-\frac{U_{y}^{2}}{V})-\varepsilon
Q_{2}\\\ &\displaystyle:=\sum_{i=1}^{5}H_{i}.\end{array}$ (3.44)
Thus
$\begin{array}[]{ll}\displaystyle\int_{0}^{\tau}\langle
h_{\tau},hg_{a}^{2}\rangle_{H^{1}\times
H^{-1}}d\tau=\sum_{i=1}^{5}\int_{0}^{\tau}\int_{\mathbf{R}}hg_{a}^{2}H_{i}dyd\tau.\end{array}$
(3.45)
By noticing that
$\|g_{a}(\tau,\cdot)\|_{L^{\infty}}\leq C_{a},$
we can estimate
$\displaystyle\int_{0}^{\tau}\int_{\mathbf{R}}hg_{a}^{2}H_{i}dyd\tau(i=2,\cdots,6)$
directly. The estimation on
$\displaystyle\int_{0}^{\tau}\int_{\mathbf{R}}hg_{a}^{2}H_{1}dyd\tau$ is more
subtle. Firstly, by using the mass equation $\eqref{(2.24)}_{1}$, we have
$\begin{array}[]{ll}\displaystyle
hg_{a}^{2}H_{1}&\displaystyle=-(p-P)\psi_{y}hg_{a}^{2}\\\\[5.69054pt]
&\displaystyle=-\frac{(\gamma-1)h+\gamma
P\phi}{v}hg_{a}^{2}\phi_{\tau}\\\\[5.69054pt]
&\displaystyle=-\frac{(\gamma-1)h^{2}g_{a}^{2}}{v}\phi_{\tau}-\frac{\gamma
Phg_{a}^{2}}{2v}(\phi^{2})_{\tau}\\\\[5.69054pt]
&\displaystyle=-\big{[}\frac{(\gamma-1)h^{2}\phi g_{a}^{2}}{v}+\frac{\gamma
Ph\phi^{2}g_{a}^{2}}{2v}\big{]}_{\tau}+\frac{2(\gamma-1)h^{2}\phi+\gamma
Ph\phi^{2}}{v}g_{a}(g_{a})_{\tau}\\\\[8.53581pt]
&\displaystyle\quad-\frac{2(\gamma-1)h^{2}\phi+\gamma
Ph\phi^{2}}{2v^{2}}g_{a}^{2}v_{\tau}+\frac{\gamma
h\phi^{2}g_{a}^{2}}{2v}P_{\tau}+\big{[}\frac{2(\gamma-1)\phi
h}{v}+\frac{\gamma P\phi^{2}}{2v}\big{]}g_{a}^{2}h_{\tau}\\\
&\displaystyle:=\sum_{i=1}^{5}J_{i}.\end{array}$
Now the terms $J_{i}(i=1,\cdots,4)$ can be estimated directly, cf. [13]. Here
we only calculate the term $J_{5}.$ From (3.44), we have
$J_{5}=\sum_{i=1}^{6}\big{[}\frac{2(\gamma-1)\phi h}{v}+\frac{\gamma
P\phi^{2}}{2v}\big{]}g_{a}^{2}H_{i}:=\sum_{i=1}^{5}J_{5}^{i}.$
Now $J_{5}^{1}$ can be estimated as follows:
$\begin{array}[]{ll}\displaystyle\int_{0}^{\tau}\int|J_{5}^{1}|dyd\tau&\displaystyle\leq
C\int_{0}^{\tau}\int|\psi_{y}||(\phi,\zeta)|^{3}dyd\tau\\\ &\displaystyle\leq
C\int_{0}^{\tau}\|(\phi,\zeta)\|^{2}_{L_{\infty}}\|\psi_{y}\|\|(\phi,\zeta)\|d\tau\\\
&\displaystyle\leq
C\int_{0}^{\tau}\|(\phi,\zeta)_{y}\|\|\psi_{y}\|\|(\phi,\zeta)\|^{2}d\tau\\\
&\displaystyle\leq
C\sup_{[0,\tau]}\|(\phi,\zeta)(\tau,\cdot)\|^{2}\int_{0}^{\tau}\|(\phi,\psi,\zeta)_{y}\|^{2}d\tau\\\
&\displaystyle\leq
C\chi^{2}\int_{0}^{\tau}\|(\phi,\psi,\zeta)_{y}\|^{2}d\tau.\end{array}$
Note that the other terms $J_{5}^{i}(i=2,\cdots,5)$ can be estimated directly,
we omit the details for brevity.
Therefore, by taking the constant $a=\frac{C_{0}}{2}$, we obtain
$\begin{array}[]{ll}&\displaystyle\int_{0}^{\tau}\int_{{\bf
R}}\varepsilon(1+\varepsilon\tau)^{-1}e^{-\frac{C_{0}\varepsilon
y^{2}}{1+\varepsilon\tau}}h^{2}dyd\tau\\\ &\displaystyle\leq
C\|(\phi,\psi,\zeta)(\tau,\cdot)\|^{2}+C\int_{0}^{\tau}\|(\phi_{y},\psi_{y},\zeta_{y})\|^{2}d\tau+C\int_{0}^{\tau}(\tau+\tau_{0})^{-\frac{3}{2}}\|(\phi,\psi)\|^{2}d\tau\\\
&\displaystyle+C\varepsilon^{\frac{2}{5}}+C(\delta^{CD}+\chi)\int_{0}^{\tau}\int_{{\bf
R}^{+}}\varepsilon(1+\varepsilon\tau)^{-1}e^{-\frac{C_{0}\varepsilon
y^{2}}{1+\varepsilon\tau}}|(\phi,\zeta)|^{2}dyd\tau.\end{array}$ (3.46)
By taking $b=C_{0}$ in (3.43) and by combining the estimates (3.43) with
(3.46), we yield the desired estimation in Lemma 3.3 if we choose suitably
small positive constants $\delta^{CD}$, $\varepsilon_{0}$ and $\chi$. ∎
Now from (3.33) and Lemma 3.3, if the strength of the contact wave
$\delta^{CD}$ and the parameter $\chi$ on the a priori estimate are suitably
small, we can get
$\begin{array}[]{ll}\displaystyle\|(\phi,\psi,\zeta)(\tau,\cdot)\|_{1}^{2}&\displaystyle+\int_{0}^{\tau}\Big{[}\|\phi_{y}\|^{2}+\|(\psi_{y},\zeta_{y})\|_{1}^{2}\Big{]}d\tau\\\
&\displaystyle\leq
C\Big{[}\int_{0}^{\tau}(\tau+\tau_{0})^{-\frac{3}{2}}\|(\phi,\psi,\zeta)\|^{2}d\tau+\varepsilon^{\frac{2}{5}}\Big{]}.\end{array}$
With this, the Gronwall inequality gives
$\|(\phi,\psi,\zeta)(\tau,\cdot)\|_{1}^{2}+\int_{0}^{\tau}\Big{[}\|\phi_{y}\|^{2}+\|(\psi_{y},\zeta_{y})\|_{1}^{2}\Big{]}d\tau\leq
C\varepsilon^{\frac{2}{5}}.$
And then we complete the proof of Theorem 3.1 by Sobolev imbedding.
## 4\. Proof of Theorem 2.5: Hydrodynamic limit of Boltzmann equation
In the last section, we will prove the fluid dynamic limit for the Boltzmann
equation to the Riemann solution for the Euler equations as stated in Theorem
2.5. Again, the proof is based on energy estimates for the Boltzmann equation
(2.82) in the scaled independent variables. For this, it is sufficient to
prove the following theorem.
###### Theorem 4.1.
There exist two small positive constants $\delta_{1}$, $\varepsilon_{1}$, and
a globalMaxwellian
$\mathbf{M}_{\star}=\mathbf{M}_{[v_{\star},u_{\star},\theta_{\star}]}$ such
that if the initial data and the strength of the contact wave $\delta^{CD}$
satisfy
$\mathcal{N}(\tau)|_{\tau=0}+\delta^{CD}\leq\delta_{1},$ (4.1)
and the Knudsen number $\varepsilon\leq\varepsilon_{1}$, then the problem
(2.82) admits a unique global solution $f^{\varepsilon}(\tau,y,\xi)$
satisfying
$\begin{array}[]{l}\displaystyle\sup_{\tau,y}\|f^{\varepsilon}(\tau,y,\xi)-\mathbf{M}_{[V,U,\Theta]}(\tau,y,\xi)\|_{L^{2}_{\xi}(\frac{1}{\sqrt{\mathbf{M}_{\star}}})}\leq
C\varepsilon^{\frac{1}{5}}.\\\ \end{array}$ (4.2)
Here, $\mathcal{N}(\tau)$ is defined by (4.5) below.
###### Remark 7.
If we choose the initial data for the Boltzmann equation (2.82) as
$f^{\varepsilon}(0,y,\xi)=\mathbf{M}_{[V,U,\Theta]}(0,y,\xi)=\mathbf{M}_{[V(0,y),U(0,y),\Theta(0,y)]}(\xi),$
(4.3)
then
$\mathcal{N}(\tau)|_{\tau=0}=O(1)\bigg{[}\|(\Theta_{y},U_{y})\|^{2}+\|(V_{yy},\Theta_{yy},U_{yy})\|^{2}\bigg{]}\bigg{|}_{\tau=0}=O(1)\varepsilon^{\frac{1}{2}}.$
(4.4)
In this case, the functional measuring the perturbation $\mathcal{N}(\tau)$ at
$\tau=0$ is smaller than the estimate given in Theorem 4.1 that is of the
order of $O(\varepsilon^{\frac{2}{5}})$ because $\varepsilon$ is small.
Consider the reformulated system (2.75) and (2.81). Since the local existence
of solution to (2.75) and (2.81) is now standard, cf. [11] and [27], to prove
the global existence, we only need to close the following a priori estimate by
the continuity argument:
$\begin{array}[]{ll}\displaystyle\mathcal{N}(\tau)=&\displaystyle\sup_{0\leq\tau^{\prime}\leq\tau}\Bigg{\\{}\|(\phi,\psi,\zeta)(\tau^{\prime},\cdot)\|_{1}^{2}+\int\int\frac{|\mathbf{G}_{1}|^{2}}{\mathbf{M}_{\star}}d\xi
dy\\\
&\displaystyle+\sum_{|\alpha^{\prime}|=1}\int\int\frac{|\partial^{\alpha^{\prime}}\mathbf{G}|^{2}}{\mathbf{M}_{\star}}d\xi
dy+\sum_{|\alpha|=2}\int\int\frac{|\partial^{\alpha}f|^{2}}{\mathbf{M}_{\star}}d\xi
dy\Bigg{\\}}\leq\chi^{2},\end{array}$ (4.5)
where $\partial^{\alpha},\partial^{\alpha^{\prime}}$ denote the derivatives
with respect to $y$ and $\tau$ respectively, and $\chi$ is a small positive
constant depending on the initial data and the strength of the contact wave,
and $\mathbf{M}_{\star}$ is a global Maxwellian to be chosen later.
Note that the a priori assumption (4.5) implies that
$\|(\phi,\psi,\zeta)\|^{2}_{L_{\infty}}\leq C\chi^{2},$ (4.6)
and
$\|\int\frac{\mathbf{G}_{1}^{2}}{\mathbf{M}_{\star}}d\xi\|_{L_{\infty}^{y}}\leq
C\left(\int\int\frac{\mathbf{G}_{1}^{2}}{\mathbf{M}_{\star}}d\xi
dy\right)^{\frac{1}{2}}\cdot\left(\int\int\frac{|\mathbf{G}_{1y}|^{2}}{\mathbf{M}_{\star}}d\xi
dy\right)^{\frac{1}{2}}\leq C(\varepsilon+\chi^{2}),$ (4.7)
and for $|\alpha|=1$,
$\|\int\frac{|\partial^{\alpha}\mathbf{G}|^{2}}{\mathbf{M}_{\star}}d\xi\|_{L_{\infty}^{y}}\leq
C\left(\int\int\frac{|\partial^{\alpha}\mathbf{G}|^{2}}{\mathbf{M}_{\star}}d\xi
dy\right)^{\frac{1}{2}}\cdot\left(\int\int\frac{|\partial^{\alpha}\mathbf{G}_{y}|^{2}}{\mathbf{M}_{\star}}d\xi
dy\right)^{\frac{1}{2}}\leq C(\varepsilon+\chi^{2}).$ (4.8)
From (1.17) and (2.71), we have
$\left\\{\begin{array}[]{l}\displaystyle\phi_{\tau}-\psi_{1y}=0,\\\
\displaystyle\psi_{1\tau}+(p-P)_{y}=-\frac{4}{3}(\frac{\mu(\Theta)}{V}U_{1y})_{y}-\varepsilon
Q_{1}-\int\xi_{1}^{2}\mathbf{G}_{y}d\xi,\\\
\displaystyle\psi_{i\tau}=-(\frac{\mu(\Theta)}{V}U_{iy})_{y}-\int\xi_{1}\xi_{i}\mathbf{G}_{y}d\xi,~{}~{}i=2,3,\\\
\displaystyle\zeta_{\tau}+(pu_{1y}-PU_{1y})=-(\frac{\lambda(\Theta)}{V}\Theta_{y})_{y}-\frac{4}{3}\frac{\mu(\Theta)}{V}U_{1y}^{2}-\varepsilon
Q_{2}\\\ \displaystyle\qquad-\varepsilon
Q_{1}U_{1}-\frac{1}{2}\int\xi_{1}|\xi|^{2}\mathbf{G}_{y}d\xi+\sum_{i=1}^{3}u_{i}\int\xi_{1}\xi_{i}\mathbf{G}_{y}d\xi.\end{array}\right.$
(4.9)
Thus
$\|(\phi_{\tau},\psi_{\tau},\zeta_{\tau})\|^{2}\leq C(\varepsilon+\chi^{2}).$
(4.10)
Hence, we have
$\|(v_{\tau},u_{\tau},\theta_{\tau})\|^{2}\leq
C\|(\phi_{\tau},\psi_{\tau},\zeta_{\tau})\|^{2}+C\|(V_{\tau},U_{\tau},\Theta_{\tau})\|^{2}\leq
C(\varepsilon+\chi^{2}).$ (4.11)
In addition, (4.5) also implies that
$\|(v_{y},u_{y},\theta_{y})\|^{2}\leq
C\|(\phi_{y},\psi_{y},\zeta_{y})\|^{2}+C\|(V_{y},U_{y},\Theta_{y})\|^{2}\leq
C(\varepsilon+\chi^{2}).$ (4.12)
Since
$\|\partial^{\alpha}\left(\rho,\rho
u,\rho(E+\frac{|u|^{2}}{2})\right)\|^{2}\leq
C\int\int\frac{|\partial^{\alpha}f|^{2}}{\mathbf{M}_{\star}}d\xi dy\leq
C\chi^{2},$ (4.13)
the inequalities (4.11)-(4.13) give
$\begin{array}[]{ll}\displaystyle\|\partial^{\alpha}(v,u,\theta)\|^{2}&\displaystyle\leq
C\|\partial^{\alpha}\left(\rho,\rho
u,\rho(E+\frac{|u|^{2}}{2})\right)\|^{2}\\\
&\displaystyle\quad\quad+C\sum_{|\alpha|=1}\int|\partial^{\alpha}\left(\rho,\rho
u,\rho(E+\frac{|u|^{2}}{2})\right)|^{4}dy\\\ &\displaystyle\leq
C(\varepsilon+\chi^{2}).\end{array}$ (4.14)
Thus, for $|\alpha|=2$, we have
$\|\partial^{\alpha}(\phi,\psi,\zeta)\|^{2}\leq
C(\|\partial^{\alpha}(v,u,\theta)\|^{2}+\|\partial^{\alpha}(V,U,\Theta)\|^{2})\leq
C(\varepsilon+\chi^{2}).$ (4.15)
Finally, from the fact that $f=\mathbf{M}+\mathbf{G}$, we can obtain for
$|\alpha|=2$,
$\begin{array}[]{l}\displaystyle\int\int\frac{|\partial^{\alpha}\mathbf{G}|^{2}}{\mathbf{M}_{\star}}d\xi
dy\leq C\int\int\frac{|\partial^{\alpha}f|^{2}}{\mathbf{M}_{\star}}d\xi
dy+C\int\int\frac{|\partial^{\alpha}\mathbf{M}|^{2}}{\mathbf{M}_{\star}}d\xi
dy\\\ \displaystyle\quad\leq
C\int\int\frac{|\partial^{\alpha}f|^{2}}{\mathbf{M}_{\star}}d\xi
dy+C\|\partial^{\alpha}(v,u,\theta)\|^{2}+C\sum_{|\alpha^{\prime}|=1}\int|\partial^{\alpha^{\prime}}(v,u,\theta)|^{4}dy\\\
\quad\leq C(\varepsilon+\chi^{2}).\end{array}$ (4.16)
Before proving the a priori estimate (4.5), we list some basic lemmas based on
the celebrated H-theorem for later use. The first lemma is from [8].
###### Lemma 4.2.
There exists a positive constant $C$ such that
$\int\frac{\nu(|\xi|)^{-1}Q(f,g)^{2}}{\tilde{\mathbf{M}}}d\xi\leq
C\left\\{\int\frac{\nu(|\xi|)f^{2}}{\tilde{\mathbf{M}}}d\xi\cdot\int\frac{g^{2}}{\tilde{\mathbf{M}}}d\xi+\int\frac{f^{2}}{\tilde{\mathbf{M}}}d\xi\cdot\int\frac{\nu(|\xi|)g^{2}}{\tilde{\mathbf{M}}}d\xi\right\\},$
where $\tilde{\mathbf{M}}$ can be any Maxwellian so that the above integrals
are well defined.
Based on Lemma 4.2, the following three lemmas are taken from [20]. And the
proofs are straightforward by using Cauchy inequality.
###### Lemma 4.3.
If $\theta/2<\theta_{\star}<\theta$, then there exist two positive constants
$\sigma=\sigma(v,u,\theta;\break v_{\star},u_{\star},\theta_{\star})$ and
$\eta_{0}=\eta_{0}(v,u,\theta;v_{\star},u_{\star},\theta_{\star})$ such that
if $|v-v_{\star}|+|u-u_{\star}|+|\theta-\theta_{\star}|<\eta_{0}$, we have for
$h(\xi)\in\mathfrak{N}^{\bot}$,
$-\int\frac{h\mathbf{L}_{\mathbf{M}}h}{\mathbf{M}_{\star}}d\xi\geq\sigma\int\frac{\nu(|\xi|)h^{2}}{\mathbf{M}_{\star}}d\xi.$
###### Lemma 4.4.
Under the assumptions in Lemma 4.3, we have for each
$h(\xi)\in\mathfrak{N}^{\bot}$,
$\left\\{\begin{array}[]{l}\displaystyle\int\frac{\nu(|\xi|)}{\mathbf{M}}|\mathbf{L}_{\mathbf{M}}^{-1}h|^{2}d\xi\leq\sigma^{-2}\int\frac{\nu(|\xi|)^{-1}h^{2}}{\mathbf{M}}d\xi,\\\
\displaystyle\int\frac{\nu(|\xi|)}{\mathbf{M}_{\star}}|\mathbf{L}_{\mathbf{M}}^{-1}h|^{2}d\xi\leq\sigma^{-2}\int\frac{\nu(|\xi|)^{-1}h^{2}}{\mathbf{M}_{\star}}d\xi.\end{array}\right.$
###### Lemma 4.5.
Under the conditions in Lemma 4.3, for any positive constants $k$ and
$\lambda$, it holds that
$|\int\frac{g_{1}\mathbf{P}_{1}(|\xi|^{k}g_{2})}{\mathbf{M}_{\star}}d\xi-\int\frac{g_{1}|\xi|^{k}g_{2}}{\mathbf{M}_{\star}}d\xi|\leq
C_{k,\lambda}\int\frac{\lambda|g_{1}|^{2}+\lambda^{-1}|g_{2}|^{2}}{\mathbf{M}_{\star}}d\xi,$
where the constant $C_{k,\lambda}$ depends on $k$ and $\lambda$.
With the above preparation, we are ready to perform the energy estimation as
follows. Firstly, similar to (3.13), we can get
$\begin{array}[]{l}\displaystyle\left(\sum_{i=1}^{3}\frac{1}{2}\psi_{i}^{2}+R\Theta\Phi(\frac{v}{V})+\Theta\Phi(\frac{\theta}{\Theta})\right)_{\tau}+\frac{4}{3}\frac{\mu(\theta)}{v}\psi_{1y}^{2}+\sum_{i=2}^{3}\frac{\mu(\theta)}{v}\psi_{iy}^{2}+\frac{\lambda(\theta)}{v\theta}\zeta_{y}^{2}\\\
\displaystyle+P(U^{R_{1}}_{1y}+U^{R_{3}}_{1y})\bigg{[}\Phi(\frac{\theta
V}{v\Theta})+\frac{5}{3}\Phi(\frac{v}{V})\bigg{]}=-PU^{CD}_{1y}\bigg{[}\Phi(\frac{\theta
V}{v\Theta})+\frac{5}{3}\Phi(\frac{v}{V})\bigg{]}\\\
\displaystyle+\bigg{[}(\frac{\lambda(\Theta)\Theta_{y}}{V})_{y}+\frac{4}{3}\frac{\mu(\Theta)U_{1y}^{2}}{V}+\varepsilon
Q_{2}\bigg{]}\bigg{[}\frac{2}{3}\Phi(\frac{v}{V})-\Phi(\frac{\Theta}{\theta})\bigg{]}-\frac{4}{3}(\frac{\mu(\theta)}{v}-\frac{\mu(\Theta)}{V})U_{1y}\psi_{1y}\\\
\displaystyle-\frac{\zeta_{y}}{\theta}(\frac{\lambda(\theta)}{v}-\frac{\lambda(\Theta)}{V})\Theta_{y}+\frac{\zeta\theta_{y}}{\theta^{2}}(\frac{\lambda(\theta)\theta_{y}}{v}-\frac{\lambda(\Theta)\Theta_{y}}{V})+\frac{4\zeta}{3\theta}(\frac{\mu(\theta)}{v}u_{1y}^{2}-\frac{\mu(\Theta)}{V}U_{1y}^{2})\\\\[8.5359pt]
\displaystyle+\frac{\zeta}{\theta}\sum_{i=2}^{3}\frac{\mu(\theta)}{v}u_{iy}^{2}-\frac{\zeta}{\theta}(\varepsilon
Q_{2}-\varepsilon Q_{1}U_{1})-\varepsilon
Q_{1}\psi_{1}+N_{1}+(\cdots)_{y},\end{array}$ (4.17)
where
$N_{1}=-\sum_{i=1}^{3}\psi_{i}\int\xi_{1}\xi_{i}\Pi_{1y}d\xi+\frac{\zeta}{\theta}(\sum_{i=1}^{3}u_{i}\int\xi_{1}\xi_{i}\Pi_{1y}d\xi-\frac{1}{2}\int\xi_{1}|\xi|^{2}\Pi_{1y}d\xi).$
(4.18)
The estimation on the macroscopic terms in (4.17) is almost same as (3.20) for
the compressible Navier-Stokes equations so that we have
$\begin{array}[]{ll}&\displaystyle\|(\phi,\psi,\zeta)(\tau,\cdot)\|^{2}+\int_{0}^{\tau}\Big{[}\|(\psi_{y},\zeta_{y})\|^{2}+\|\sqrt{(U^{R_{1}}_{1y},U^{R_{3}}_{1y})}(\phi,\zeta)\|^{2}\Big{]}d\tau\\\
&\displaystyle\leq
C\int_{0}^{\tau}(\tau+\tau_{0})^{-2}\|(\phi,\zeta)\|^{2}d\tau+C\varepsilon^{\frac{2}{5}}\\\
&\displaystyle\quad+C\delta^{CD}\varepsilon\int_{0}^{\tau}\int_{\mathbf{R}}(1+\varepsilon\tau)^{-1}e^{-\frac{C_{0}\varepsilon
y^{2}}{1+\varepsilon\tau}}(\phi^{2}+\zeta^{2})dyd\tau+\int_{0}^{\tau}\int
N_{1}dyd\tau.\end{array}$ (4.19)
Now we estimate the microscopic term $\displaystyle\int_{0}^{\tau}\int
N_{1}dyd\tau$ in (4.19). For this, we only estimate the term $\displaystyle
T_{1}=:-\int_{0}^{\tau}\int\psi_{1}\int\xi_{1}^{2}\Pi_{1y}d\xi dyd\tau$
because other terms in $\displaystyle\int_{0}^{\tau}\int N_{1}dyd\tau$ can be
estimated similarly.
For $T_{1}$, integration by parts with respect to $y$ and Cauchy inequality
yield
$\begin{array}[]{ll}T_{1}&\displaystyle=\int_{0}^{\tau}\int\psi_{1y}\int\xi_{1}^{2}\Pi_{1}d\xi
dyd\tau\\\
&\displaystyle\leq\beta\int_{0}^{\tau}\|\psi_{1y}\|^{2}d\tau+C_{\beta}\int_{0}^{\tau}\int|\int\xi_{1}^{2}\Pi_{1}d\xi|^{2}dyd\tau.\end{array}$
(4.20)
By (2.79), we have
$\begin{array}[]{ll}\displaystyle\int_{0}^{\tau}\int|\int\xi_{1}^{2}\Pi_{1}d\xi|^{2}dyd\tau\\\
\displaystyle\leq
C\int_{0}^{\tau}\int|\int\xi_{1}^{2}\mathbf{L}_{\mathbf{M}}^{-1}(\mathbf{G}_{\tau})d\xi|^{2}dyd\tau+C\int_{0}^{\tau}\int|\int\xi_{1}^{2}\mathbf{L}_{\mathbf{M}}^{-1}(\frac{u_{1}}{v}\mathbf{G}_{y})d\xi|^{2}dyd\tau\\\
\displaystyle+C\int_{0}^{\tau}\int|\int\xi_{1}^{2}\mathbf{L}_{\mathbf{M}}^{-1}[\frac{1}{v}\mathbf{P}_{1}(\xi_{1}\mathbf{G}_{y})]d\xi|^{2}dyd\tau+C\int_{0}^{\tau}\int|\int\xi_{1}^{2}\mathbf{L}_{\mathbf{M}}^{-1}[Q(\mathbf{G},\mathbf{G})]d\xi|^{2}dyd\tau\\\
\displaystyle:=\sum_{i=1}^{4}T_{1}^{i}.\end{array}$ (4.21)
Let $\mathbf{M}_{\star}$ be a global Maxwellian with its state
$(v_{\star},u_{\star},\theta_{\star})$ satisfying
$\frac{1}{2}\theta<\theta_{\star}<\theta$ and
$|v-v_{\star}|+|u-u_{\star}|+|\theta-\theta_{\star}|\leq\eta_{0}$ so that
Lemma 4.3 holds. Then we can obtain
$\begin{array}[]{ll}T_{1}^{1}&\displaystyle\leq
C\int_{0}^{\tau}\int|\int\frac{\nu(|\xi|)|\mathbf{L}_{\mathbf{M}}^{-1}\mathbf{G}_{\tau}|^{2}}{\mathbf{M}_{\star}}d\xi\cdot\int\nu^{-1}(|\xi|)\xi_{1}^{4}\mathbf{M}_{\star}d\xi|dyd\tau\\\
&\displaystyle\leq
C\int_{0}^{\tau}\int\int\frac{\nu^{-1}(|\xi|)|\mathbf{G}_{\tau}|^{2}}{\mathbf{M}_{\star}}d\xi
dyd\tau.\end{array}$ (4.22)
Similarly,
$T_{1}^{2}\leq
C\int_{0}^{\tau}\int\int\frac{\nu^{-1}(|\xi|)|\mathbf{G}_{y}|^{2}}{\mathbf{M}_{\star}}d\xi
dyd\tau.$ (4.23)
Moreover,
$\begin{array}[]{ll}\displaystyle T_{1}^{3}\leq
C\int_{0}^{\tau}\int|\int\frac{\nu(|\xi|)|\mathbf{L}_{\mathbf{M}}^{-1}[\frac{1}{v}\mathbf{P}_{1}(\xi_{1}\mathbf{G}_{y})]|^{2}}{\mathbf{M}_{[2v_{\star},2u_{\star},2\theta_{\star}]}}d\xi\cdot\int\nu^{-1}(|\xi|)\xi_{1}^{4}\mathbf{M}_{[2v_{\star},2u_{\star},2\theta_{\star}]}d\xi|dyd\tau\\\
\quad\displaystyle\leq
C\int_{0}^{\tau}\int\int\frac{\nu^{-1}(|\xi|)|\frac{1}{v}\mathbf{P}_{1}(\xi_{1}\mathbf{G}_{y})|^{2}}{\mathbf{M}_{[2v_{\star},2u_{\star},2\theta_{\star}]}}d\xi
dyd\tau\\\ \displaystyle\quad\leq
C\int_{0}^{\tau}\int\int\frac{\nu^{-1}(|\xi|)|\mathbf{G}_{y}|^{2}}{\mathbf{M}_{\star}}d\xi
dyd\tau.\end{array}$ (4.24)
From Lemma 4.2, we have
$\begin{array}[]{ll}T_{1}^{4}&\displaystyle\leq
C\int_{0}^{\tau}\int\int\frac{\nu^{-1}(|\xi|)|Q(\mathbf{G},\mathbf{G})|^{2}}{\mathbf{M}_{\star}}d\xi
dyd\tau\\\ &\displaystyle\leq
C\int_{0}^{\tau}\int\int\frac{\nu(|\xi|)|\mathbf{G}|^{2}}{\mathbf{M}_{\star}}d\xi\cdot\int\frac{|\mathbf{G}|^{2}}{\mathbf{M}_{\star}}d\xi
dyd\tau\\\ &\displaystyle\leq
C\int_{0}^{\tau}\int\int\frac{\nu(|\xi|)(|\mathbf{G}_{0}|^{2}+|\mathbf{G}_{1}|^{2})}{\mathbf{M}_{\star}}d\xi\cdot\int\frac{|\mathbf{G}_{0}|^{2}+|\mathbf{G}_{1}|^{2}}{\mathbf{M}_{\star}}d\xi
dyd\tau\\\ &\displaystyle\leq
C(\varepsilon+\chi^{2})\int_{0}^{\tau}\int\int\frac{\nu(|\xi|)|\mathbf{G}_{1}|^{2}}{\mathbf{M}_{\star}}d\xi
dyd\tau+C\varepsilon^{\frac{1}{2}}.\end{array}$ (4.25)
Substituting (4.20)-(4.25) into (4.19) yields that
$\begin{array}[]{ll}&\displaystyle\|(\phi,\psi,\zeta)(\tau,\cdot)\|^{2}+\int_{0}^{\tau}\Big{[}\|(\psi_{y},\zeta_{y})\|^{2}+\|\sqrt{(U^{R_{1}}_{1y},U^{R_{3}}_{1y})}(\phi,\zeta)\|^{2}\Big{]}d\tau\\\
&\displaystyle\leq
C\int_{0}^{\tau}(\tau+\tau_{0})^{-2}\|(\phi,\zeta)\|^{2}d\tau+C\varepsilon^{\frac{2}{5}}\\\
&\displaystyle\quad+C\delta^{CD}\varepsilon\int_{0}^{\tau}\int_{\mathbf{R}}(1+\varepsilon\tau)^{-1}e^{-\frac{C_{0}\varepsilon
y^{2}}{1+\varepsilon\tau}}(\phi^{2}+\zeta^{2})dyd\tau\\\
&\displaystyle\quad+C\sum_{|\alpha^{\prime}|=1}\int_{0}^{\tau}\int\int\frac{\nu^{-1}(|\xi|)|\partial^{\alpha^{\prime}}\mathbf{G}|^{2}}{\mathbf{M}_{\star}}d\xi
dyd\tau\\\
&\displaystyle\quad+C(\chi^{2}+\varepsilon)\int_{0}^{\tau}\int\int\frac{\nu(|\xi|)|\mathbf{G}_{1}|^{2}}{\mathbf{M}_{\star}}d\xi
dyd\tau.\end{array}$ (4.26)
To recover the term $\|\phi_{y}\|^{2}$ in the integral
$\displaystyle\int_{0}^{\tau}\cdots d\tau$ in (4.26), as in the previous
subsection for the compressible Navier-Stokes equations, we firstly rewrite
the equation $(\ref{(2.47)})_{2}$ as
$\begin{array}[]{l}\quad\displaystyle\frac{4}{3}\frac{\mu(\Theta)}{V}\phi_{y\tau}-\psi_{1\tau}-(p-P)_{y}\\\
\displaystyle=-\frac{4}{3}(\frac{\mu({\Theta})}{V})_{y}\psi_{1y}-\frac{4}{3}[(\frac{\mu({\theta})}{v}-\frac{\mu({\Theta})}{V})u_{1y}]_{y}+\varepsilon
Q_{1}+\int\xi_{1}^{2}\Pi_{1y}d\xi,\end{array}$ (4.27)
by using the equation of conservation of the mass $(\ref{(2.47)})_{1}$.
Since
$-(p-P)_{y}=\frac{P}{V}\phi_{y}-\frac{2}{3V}\zeta_{y}+(\frac{p}{v}-\frac{P}{V})v_{y}-\frac{2}{3}(\frac{1}{v}-\frac{1}{V})\theta_{y},$
and
$\phi_{y}\psi_{1\tau}=(\phi_{y}\psi_{1})_{\tau}-(\phi_{\tau}\psi_{1})_{y}+\psi_{1y}^{2},$
by multiplying (4.27) by $\phi_{y}$, we get
$\begin{array}[]{l}\displaystyle(\frac{2\mu(\Theta)}{3V}\phi_{y}^{2}-\phi_{y}\psi_{1})_{\tau}+\frac{P}{V}\phi_{y}^{2}=(\frac{2\mu(\Theta)}{3V})_{\tau}\phi_{y}^{2}+\psi_{1y}^{2}+\frac{2}{3V}\zeta_{y}\phi_{y}\\\
\quad\displaystyle-(\frac{p}{v}-\frac{P}{V})v_{y}\phi_{y}+\frac{2}{3}(\frac{1}{v}-\frac{1}{V})\theta_{y}\phi_{y}-\frac{4}{3}(\frac{\mu(\Theta)}{V})_{y}\psi_{1y}\phi_{y}\\\
\quad\displaystyle-\frac{4}{3}[(\frac{\mu({\theta})}{v}-\frac{\mu({\Theta})}{V})u_{1y}]_{y}\phi_{y}+\varepsilon
Q_{1}\phi_{y}+\int\xi_{1}^{2}\Pi_{1y}d\xi\phi_{y}.\end{array}$ (4.28)
Integrating (4.28) with respect to $\tau,y$ and using the Cauchy inequality
yield
$\begin{array}[]{l}\displaystyle\|\phi_{y}(\tau,\cdot)\|^{2}+\int_{0}^{\tau}\|\phi_{y}\|^{2}d\tau\leq
C\|\psi_{1}(\tau,\cdot)\|^{2}+C\int_{0}^{\tau}\|(\psi_{y},\zeta_{y})\|^{2}d\tau\\\
\displaystyle~{}~{}+C\delta^{CD}\varepsilon\int_{0}^{\tau}\int_{\mathbf{R}}(1+\varepsilon\tau)^{-1}e^{-\frac{C_{0}\varepsilon
y^{2}}{1+\varepsilon\tau}}|(\phi,\psi,\zeta)|^{2}dyd\tau+C\int_{0}^{\tau}(\tau+\tau_{0})^{-2}\|(\phi,\zeta)\|^{2}d\tau\\\
\displaystyle~{}~{}+C\varepsilon^{\frac{7}{5}}+C\chi\int_{0}^{\tau}\|\psi_{1yy}\|^{2}d\tau+\int_{0}^{\tau}\int|\int\xi_{1}^{2}\Pi_{1y}d\xi|^{2}dyd\tau.\end{array}$
(4.29)
For the microscopic term
$\displaystyle\int_{0}^{\tau}\int|\int\xi_{1}^{2}\Pi_{1y}d\xi|^{2}dyd\tau$, by
(2.80), we have
$\begin{array}[]{l}\displaystyle\int_{0}^{\tau}\int|\int\xi_{1}^{2}\Pi_{1y}d\xi|^{2}dyd\tau\\\
\displaystyle\leq
C\Big{[}\int_{0}^{\tau}\int|\int\xi_{1}^{2}(\mathbf{L}_{\mathbf{M}}^{-1}\mathbf{G}_{\tau})_{y}d\xi|^{2}dyd\tau+\int_{0}^{\tau}\int|\int\xi_{1}^{2}(\mathbf{L}_{\mathbf{M}}^{-1}\frac{u_{1}}{v}\mathbf{G}_{y})_{y}d\xi|^{2}dyd\tau\\\
\displaystyle+\int_{0}^{\tau}\int|\int\xi_{1}^{2}[\mathbf{L}_{\mathbf{M}}^{-1}\frac{1}{v}\mathbf{P}_{1}(\xi_{1}\mathbf{G}_{y})]_{y}d\xi|^{2}dyd\tau+\int_{0}^{\tau}\int|\int\xi_{1}^{2}[\mathbf{L}_{\mathbf{M}}^{-1}Q(\mathbf{G},\mathbf{G})]_{y}d\xi|^{2}dyd\tau\Big{]}\\\
\displaystyle:=\sum_{i=1}^{4}T_{2}^{i}.\end{array}$ (4.30)
Note that the inverse of the linearized operator
$\mathbf{L}_{\mathbf{M}}^{-1}$ satisfies that , for any
$h\in\mathcal{N}^{\bot}$,
$\begin{array}[]{l}(\mathbf{L}_{\mathbf{M}}^{-1}h)_{\tau}=\mathbf{L}_{\mathbf{M}}^{-1}(h_{\tau})-2\mathbf{L}_{\mathbf{M}}^{-1}\\{Q(\mathbf{L}_{\mathbf{M}}^{-1}h,\mathbf{M}_{\tau})\\},\\\\[5.69054pt]
(\mathbf{L}_{\mathbf{M}}^{-1}h)_{y}=\mathbf{L}_{\mathbf{M}}^{-1}(h_{y})-2\mathbf{L}_{\mathbf{M}}^{-1}\\{Q(\mathbf{L}_{\mathbf{M}}^{-1}h,\mathbf{M}_{y})\\}.\end{array}$
(4.31)
Then we have
$\begin{array}[]{ll}\displaystyle T_{2}^{1}&\displaystyle\leq
C\int_{0}^{\tau}\int|\int\xi_{1}^{2}\mathbf{L}_{\mathbf{M}}^{-1}\mathbf{G}_{y\tau}d\xi|^{2}dyd\tau\\\
&\displaystyle\qquad+C\int_{0}^{\tau}\int|\int\xi_{1}^{2}\mathbf{L}_{\mathbf{M}}^{-1}\\{Q(\mathbf{L}_{\mathbf{M}}^{-1}\mathbf{G}_{\tau},\mathbf{M}_{y})\\}d\xi|^{2}dyd\tau\\\
&\displaystyle\leq
C\sum_{|\alpha|=2}\int_{0}^{\tau}\int\int\frac{\nu^{-1}(|\xi|)}{\mathbf{M}_{\star}}|\partial^{\alpha}\mathbf{G}|^{2}d\xi
dyd\tau\\\
&\displaystyle\qquad+C\int_{0}^{\tau}\int\int\frac{\nu(|\xi|)|\mathbf{G}_{\tau}|^{2}}{\mathbf{M}_{\star}}d\xi\int\frac{\nu(|\xi|)|\mathbf{M}_{y}|^{2}}{\mathbf{M}_{\star}}d\xi
dyd\tau\\\ &\displaystyle\leq
C\sum_{|\alpha|=2}\int_{0}^{\tau}\int\int\frac{\nu^{-1}(|\xi|)}{\mathbf{M}_{\star}}|\partial^{\alpha}\mathbf{G}|^{2}d\xi
dyd\tau\\\
&\displaystyle\qquad+C\int_{0}^{\tau}\int|(v_{y},u_{y},\theta_{y})|^{2}\int\frac{\nu(|\xi|)|\mathbf{G}_{\tau}|^{2}}{\mathbf{M}_{\star}}d\xi
dyd\tau\\\ &\displaystyle\leq
C\sum_{|\alpha|=2}\int_{0}^{\tau}\int\int\frac{\nu^{-1}(|\xi|)}{\mathbf{M}_{\star}}|\partial^{\alpha}\mathbf{G}|^{2}d\xi
dyd\tau\\\
&\displaystyle\qquad+C(\varepsilon+\chi^{2})\int_{0}^{\tau}\int\int\frac{\nu(|\xi|)|\mathbf{G}_{\tau}|^{2}}{\mathbf{M}_{\star}}d\xi
dyd\tau.\end{array}$ (4.32)
Similar estimates hold for $T_{2}^{i}~{}(i=2,3)$. Moreover,
$\begin{array}[]{ll}\displaystyle T_{2}^{4}&\displaystyle\leq
C\int_{0}^{\tau}\int|\int\xi_{1}^{2}\mathbf{L}_{\mathbf{M}}^{-1}Q(\mathbf{G},\mathbf{G}_{y})d\xi|^{2}dyd\tau\\\
&\displaystyle\quad+C\int_{0}^{\tau}\int|\int\xi_{1}^{2}\mathbf{L}_{\mathbf{M}}^{-1}\\{Q(\mathbf{L}_{\mathbf{M}}^{-1}Q(\mathbf{G},\mathbf{G}),\mathbf{M}_{y})\\}d\xi|^{2}dyd\tau\\\
&\displaystyle\leq
C\int_{0}^{\tau}\int\int\frac{\nu(|\xi|)|\mathbf{G}_{y}|^{2}}{\mathbf{M}_{\star}}d\xi\int\frac{|\mathbf{G}|^{2}}{\mathbf{M}_{\star}}d\xi
dyd\tau\\\
&\displaystyle\quad+C\int_{0}^{\tau}\int\int\frac{\nu(|\xi|)|\mathbf{G}|^{2}}{\mathbf{M}_{\star}}d\xi\int\frac{|\mathbf{G}_{y}|^{2}}{\mathbf{M}_{\star}}d\xi
dyd\tau\\\
&\displaystyle\quad+C\int_{0}^{\tau}\int|(v_{y},u_{y},\theta_{y})|^{2}\int\frac{\nu(|\xi|)|\mathbf{G}|^{2}}{\mathbf{M}_{*}}d\xi\int\frac{|\mathbf{G}|^{2}}{\mathbf{M}_{\star}}d\xi
dyd\tau\\\ &\displaystyle\leq
C(\chi^{2}+\varepsilon)\int_{0}^{\tau}\int\int\frac{\nu(|\xi|)(|\mathbf{G}_{1}|^{2}+|\mathbf{G}_{y}|^{2})}{\mathbf{M}_{\star}}d\xi
dyd\tau.\end{array}$ (4.33)
Substituting (4.30)-(4.33) into (4.29) gives
$\begin{array}[]{l}\displaystyle\|\phi_{y}(\tau,\cdot)\|^{2}+\int_{0}^{\tau}\|\phi_{y}\|^{2}d\tau\leq
C\|\psi_{1}(\tau,\cdot)\|^{2}+C\int_{0}^{\tau}\|(\psi_{y},\zeta_{y})\|^{2}d\tau\\\
\displaystyle+C\delta^{CD}\varepsilon\int_{0}^{\tau}\int_{\mathbf{R}}(1+\varepsilon\tau)^{-1}e^{-\frac{C_{0}\varepsilon
y^{2}}{1+\varepsilon\tau}}|(\phi,\psi,\zeta)|^{2}dyd\tau+C\int_{0}^{\tau}(\tau+\tau_{0})^{-2}\|(\phi,\zeta)\|^{2}d\tau\\\
\displaystyle+C\varepsilon^{\frac{2}{5}}+C\sum_{|\alpha|=2}\int_{0}^{\tau}\int\int\frac{\nu^{-1}(|\xi|)}{\mathbf{M}_{\star}}|\partial^{\alpha}\mathbf{G}|^{2}d\xi
dyd\tau+C\chi\int_{0}^{\tau}\|\psi_{1yy}\|^{2}d\tau\\\
\displaystyle+C(\varepsilon+\chi^{2})\int_{0}^{\tau}\int\int\frac{\nu(|\xi|)(\sum_{|\alpha^{\prime}|=1}|\partial^{\alpha^{\prime}}\mathbf{G}|^{2}+|\mathbf{G}_{1}|^{2})}{\mathbf{M}_{\star}}d\xi
dyd\tau.\end{array}$ (4.34)
We now turn to the time derivatives. To estimate
$\|(\phi_{\tau},\psi_{\tau},\zeta_{\tau})\|^{2}$, we need to use the system
(4.9). By multiplying $(\ref{(4.18)})_{1}$ by $\phi_{\tau}$,
$(\ref{(4.18)})_{2}$ by $\psi_{1\tau}$, $(\ref{(4.18)})_{3}$ by
$\psi_{i\tau}~{}(i=2,3)$ and $(\ref{(4.18)})_{4}$ by $\zeta_{\tau}$
respectively, and adding them together, after integrating with respect to
$\tau$ and $y$, we have
$\begin{array}[]{l}\displaystyle\int_{0}^{\tau}\|(\phi_{\tau},\psi_{\tau},\zeta_{\tau})(\tau,\cdot)\|^{2}d\tau\leq
C\int_{0}^{\tau}\|(\phi_{y},\psi_{y},\zeta_{y})\|^{2}d\tau+C\varepsilon^{\frac{2}{5}}\\\
\displaystyle\qquad+\int_{0}^{\tau}(\tau+\tau_{0})^{-2}\|(\phi,\psi,\zeta)\|^{2}d\tau+C\int_{0}^{\tau}\int\int\frac{\nu(|\xi|)}{\mathbf{M}_{\star}}|\mathbf{G}_{y}|^{2}d\xi
dyd\tau\\\
\qquad\displaystyle+C\delta^{CD}\varepsilon\int_{0}^{\tau}\int_{\mathbf{R}}(1+\varepsilon\tau)^{-1}e^{-\frac{C_{0}\varepsilon
y^{2}}{1+\varepsilon\tau}}|(\phi,\psi,\zeta)|^{2}dyd\tau.\end{array}$ (4.35)
The microscopic component $\mathbf{G}_{1}$ can be estimated by using the
equation (2.82). Multiplying (2.82) by
$\frac{v\mathbf{G}_{1}}{\mathbf{M}_{\star}}$ gives
$\begin{array}[]{ll}\displaystyle(v\frac{\mathbf{G}_{1}^{2}}{2\mathbf{M}_{\star}})_{\tau}-\frac{v\mathbf{G}_{1}}{\mathbf{M}_{\star}}\mathbf{L}_{\mathbf{M}}\mathbf{G}_{1}&\displaystyle=v_{\tau}\frac{|\mathbf{G}_{1}|^{2}}{2\mathbf{M}_{\star}}+\bigg{\\{}-\frac{3}{2v\theta}\mathbf{P}_{1}[\xi_{1}(\frac{|\xi-u|^{2}}{2\theta}\zeta_{y}+\xi\cdot\psi_{y})\mathbf{M}]\\\
&\displaystyle\qquad+\frac{u_{1}}{v}\mathbf{G}_{y}-\frac{1}{v}\mathbf{P}_{1}(\xi_{1}\mathbf{G}_{y})+Q(\mathbf{G},\mathbf{G})-\mathbf{G}_{0\tau}\bigg{\\}}\frac{v\mathbf{G}_{1}}{\mathbf{M}_{\star}}.\end{array}$
(4.36)
Integrating (4.36) with respect to $\tau,\xi$ and $y$ and using the Cauchy
inequality and Lemma 4.2-4.5 yield that
$\begin{array}[]{l}\displaystyle\int\int\frac{\mathbf{G}_{1}^{2}}{\mathbf{M}_{\star}}(\tau,y,\xi)d\xi
dy+\int_{0}^{\tau}\int\int\frac{\nu(|\xi|)|\mathbf{G}_{1}|^{2}}{\mathbf{M}_{\star}}d\xi
dyd\tau\\\ \leq\displaystyle
C\varepsilon^{\frac{2}{5}}+C\sum_{|\alpha^{\prime}|=1}\int_{0}^{\tau}\|\partial^{\alpha^{\prime}}(\phi,\psi,\zeta)\|^{2}d\tau+C\int_{0}^{\tau}\int\int\frac{\nu(|\xi|)}{\mathbf{M}_{\star}}|\mathbf{G}_{y}|^{2}d\xi
dyd\tau,\end{array}$ (4.37)
where we have used the fact that
$\begin{array}[]{ll}\displaystyle\int\int\frac{v\mathbf{G}_{1}^{2}}{\mathbf{M}_{\star}}(\tau=0,y,\xi)d\xi
dy&\displaystyle=\int\int\frac{v\mathbf{G}_{0}^{2}}{\mathbf{M}_{\star}}(\tau=0,y,\xi)d\xi
dy\\\ &\displaystyle\leq C\|(\Theta_{y},U_{y})(\tau=0,\cdot)\|^{2}\leq
C\varepsilon^{\frac{1}{2}}.\end{array}$
Next we derive the estimate on the higher order derivatives. By multiplying
$(\ref{(2.46)})_{2}$ by $-\psi_{1yy}$, $(\ref{(2.46)})_{3}$ by
$-\psi_{iyy}~{}(i=2,3)$, $(\ref{(2.46)})_{4}$ by $-\zeta_{yy}$, and adding
them together, we obtain
$\begin{array}[]{l}\displaystyle(\sum_{i=1}^{3}\frac{\psi_{iy}^{2}}{2}+\frac{\zeta_{y}^{2}}{2})_{\tau}+\frac{4}{3}\frac{\mu(\theta)}{v}\psi_{1yy}^{2}+\sum_{i=2}^{3}\frac{\mu(\theta)}{v}\psi_{iyy}^{2}+\frac{\lambda(\theta)}{v}\zeta_{yy}^{2}=\\\
\displaystyle-\frac{4}{3}(\frac{\mu(\theta)}{v})_{y}\psi_{1y}\psi_{1yy}-\sum_{i=2}^{3}(\frac{\mu(\theta)}{v})_{y}\psi_{iy}\psi_{iyy}-(\frac{\lambda(\theta)}{v})_{y}\zeta_{y}\zeta_{yy}\\\
\displaystyle-\frac{4}{3}[(\frac{\mu(\theta)}{v}-\frac{\mu(\Theta)}{V})U_{1y}]_{y}\psi_{1yy}-[(\frac{\lambda(\theta)}{v}-\frac{\lambda(\Theta)}{V})\Theta_{y}]_{y}\zeta_{yy}+(p-P)_{y}\psi_{1yy}\\\
\displaystyle+\varepsilon
Q_{1}\psi_{1yy}+(pu_{1y}-PU_{1y})\zeta_{yy}-[\frac{4}{3}(\frac{\mu(\theta)}{v}u_{1y}^{2}-\frac{\mu(\Theta)}{V}U_{1y}^{2})\\\\[8.5359pt]
\displaystyle+\sum_{i=2}^{3}\frac{\mu(\theta)}{v}u_{iy}^{2}-(\varepsilon
Q_{2}-\varepsilon
Q_{1}U_{1})]\zeta_{yy}+\sum_{i=1}^{3}\psi_{iyy}\int\xi_{1}\xi_{i}\Pi_{1y}d\xi\\\
\displaystyle-\zeta_{yy}(\sum_{i=1}^{3}u_{i}\int\xi_{1}\xi_{i}\Pi_{1y}d\xi-\frac{1}{2}\int\xi_{1}|\xi|^{2}\Pi_{1y}d\xi).\end{array}$
(4.38)
Integrating (4.38) with respect to $\tau,y$ and $\xi$ yields
$\begin{array}[]{l}\displaystyle\|(\psi_{y},\zeta_{y})(\tau,\cdot)\|^{2}+\int_{0}^{\tau}\|(\psi_{yy},\zeta_{yy})\|^{2}d\tau\\\
\displaystyle\leq
C\int_{0}^{\tau}\|(\phi_{y},\psi_{y},\zeta_{y})\|^{2}d\tau+C\int_{0}^{\tau}(\tau+\tau_{0})^{-2}\|(\phi,\psi,\zeta)\|^{2}d\tau+C\varepsilon^{\frac{2}{5}}\\\
\quad\displaystyle+C\delta^{CD}\varepsilon\int_{0}^{\tau}\int_{\mathbf{R}}(1+\varepsilon\tau)^{-1}e^{-\frac{C\varepsilon
y^{2}}{1+\varepsilon\tau}}|(\phi,\psi,\zeta)|^{2}dyd\tau\\\
\quad\displaystyle+C(\varepsilon^{\frac{1}{2}}+\chi)\int_{0}^{\tau}\int\int\frac{\nu(|\xi|)}{\mathbf{M}_{\star}}|\mathbf{G}_{1}|^{2}d\xi
dyd\tau+C\sum_{|\alpha|=2}\int_{0}^{\tau}\int\int\frac{\nu(|\xi|)}{\mathbf{M}_{\star}}|\partial^{\alpha}\mathbf{G}|^{2}d\xi
dyd\tau\\\
\quad\displaystyle+C(\varepsilon^{\frac{1}{2}}+\chi)\sum_{|\alpha^{\prime}|=1}\int_{0}^{\tau}\int\int\frac{\nu(|\xi|)}{\mathbf{M}_{\star}}|\partial^{\alpha^{\prime}}\mathbf{G}|^{2}d\xi
dyd\tau.\end{array}$ (4.39)
Again, to recover $\|\phi_{yy}\|^{2}$ in the time integral in (4.39), by
applying $\partial_{y}$ to $(\ref{(2.46)})_{2}$, we get
$\psi_{1y\tau}+(p-P)_{yy}=-\frac{4}{3}(\frac{\mu(\Theta)}{V}U_{1y})_{yy}-\varepsilon
Q_{1y}-\int\xi_{1}^{2}\mathbf{G}_{yy}d\xi.$ (4.40)
Note that
$(p-P)_{yy}=-\frac{p}{v}\phi_{yy}+\frac{R}{v}\zeta_{yy}-\frac{1}{v}(p-P)V_{yy}-\frac{\phi}{v}P_{yy}-\frac{2v_{y}}{v}(p-P)_{y}-\frac{2P_{y}}{v}\phi_{y}.$
(4.41)
Multiplying (4.40) by $-\phi_{yy}$ and using (4.41) imply
$\begin{array}[]{l}\displaystyle-\int\psi_{1y}\phi_{yy}(\tau,y)dy+\int_{0}^{\tau}\int\frac{p}{2v}\phi_{yy}^{2}dyd\tau\leq
C\int_{0}^{\tau}\|(\psi_{1yy},\zeta_{yy})\|^{2}d\tau+C\varepsilon^{\frac{2}{5}}\\\
\quad\displaystyle+C\int_{0}^{\tau}(\tau+\tau_{0})^{-2}\|(\phi,\psi,\zeta)\|^{2}d\tau+C(\varepsilon^{\frac{1}{2}}+\chi)\int_{0}^{\tau}\|(\phi_{y},\psi_{y},\zeta_{y})\|^{2}d\tau\\\
\quad\displaystyle+C\sum_{|\alpha|=2}\int_{0}^{\tau}\int\int\frac{\nu(|\xi|)}{\mathbf{M}_{\star}}|\partial^{\alpha}\mathbf{G}|^{2}d\xi
dyd\tau.\end{array}$ (4.42)
To estimate $\|(\phi_{y\tau},\psi_{y\tau},\zeta_{y\tau})\|^{2}$ and
$\|(\phi_{\tau\tau},\psi_{\tau\tau},\zeta_{\tau\tau})\|^{2}$, we use the
system (4.9) again. By applying $\partial_{y}$ to (4.9), and multiplying the
four equations of (4.9) by $\phi_{y\tau}$, $\psi_{1y\tau}$, $\psi_{iy\tau}$
$(i=2,3)$, $\zeta_{y\tau}$ respectively, then adding them together and
integrating with respect to $\tau$ and $y$ give
$\begin{array}[]{l}\displaystyle\int_{0}^{\tau}\|(\phi_{y\tau},\psi_{y\tau},\zeta_{y\tau})\|^{2}d\tau\leq
C\int_{0}^{\tau}\|(\phi_{yy},\psi_{yy},\zeta_{yy})\|^{2}d\tau+C\varepsilon^{\frac{2}{5}}\\\
\quad\displaystyle+C\int_{0}^{\tau}(\tau+\tau_{0})^{-2}\|(\phi,\psi,\zeta)\|^{2}d\tau+C(\varepsilon^{\frac{1}{2}}+\chi)\int_{0}^{\tau}\|(\phi_{y},\psi_{y},\zeta_{y})\|^{2}d\tau\\\
\quad\displaystyle+C\int_{0}^{\tau}\int\int\frac{\nu(|\xi|)}{\mathbf{M}_{\star}}|\mathbf{G}_{y}|^{2}d\xi
dyd\tau+C\sum_{|\alpha|=2}\int_{0}^{\tau}\int\int\frac{\nu(|\xi|)}{\mathbf{M}_{\star}}|\partial^{\alpha}\mathbf{G}|^{2}d\xi
dyd\tau.\end{array}$ (4.43)
Similarly, we have
$\begin{array}[]{l}\displaystyle\int_{0}^{\tau}\|(\phi_{\tau\tau},\psi_{\tau\tau},\zeta_{\tau\tau})\|^{2}d\tau\leq
C\int_{0}^{\tau}\|(\phi_{y\tau},\psi_{y\tau},\zeta_{y\tau})\|^{2}d\tau+C\varepsilon^{\frac{2}{5}}\\\
\quad\displaystyle+C\int_{0}^{\tau}(\tau+\tau_{0})^{-2}\|(\phi,\psi,\zeta)\|^{2}d\tau+C(\varepsilon^{\frac{1}{2}}+\chi)\sum_{|\alpha^{\prime}|=1}\int_{0}^{\tau}\|\partial^{\alpha^{\prime}}(\phi,\psi,\zeta)\|^{2}d\tau\\\
\quad\displaystyle+C\int_{0}^{\tau}\int\int\frac{\nu(|\xi|)}{\mathbf{M}_{\star}}|\mathbf{G}_{y}|^{2}d\xi
dyd\tau+C\sum_{|\alpha|=2}\int_{0}^{\tau}\int\int\frac{\nu(|\xi|)}{\mathbf{M}_{\star}}|\partial^{\alpha}\mathbf{G}|^{2}d\xi
dyd\tau.\end{array}$ (4.44)
A suitable linear combination of (4.39) - (4.44) gives
$\begin{array}[]{l}\displaystyle\|(\psi_{y},\zeta_{y},\phi_{yy})(\tau,\cdot)\|^{2}+\sum_{|\alpha|=2}\int_{0}^{\tau}\|\partial^{\alpha}(\phi,\psi,\zeta)\|^{2}d\tau\\\
\displaystyle\leq
C\sum_{|\alpha|=2}\int_{0}^{\tau}\int\int\frac{\nu(|\xi|)}{\mathbf{M}_{\star}}|\partial^{\alpha}\mathbf{G}|^{2}d\xi
dyd\tau+C\sum_{|\alpha^{\prime}|=1}\int_{0}^{\tau}\int\int\frac{\nu(|\xi|)}{\mathbf{M}_{\star}}|\partial^{\alpha^{\prime}}\mathbf{G}|^{2}d\xi
dyd\tau\\\
\displaystyle\quad+C(\varepsilon^{\frac{1}{2}}+\chi)\int_{0}^{\tau}\int\int\frac{\nu(|\xi|)}{\mathbf{M}_{\star}}|\mathbf{G}_{1}|^{2}d\xi
dyd\tau+C\int_{0}^{\tau}(\tau+\tau_{0})^{-2}\|(\phi,\psi,\zeta)\|^{2}d\tau\\\
\displaystyle\quad+C(\varepsilon^{\frac{1}{2}}+\chi)\sum_{|\alpha^{\prime}|=1}\int_{0}^{\tau}\|\partial^{\alpha^{\prime}}(\phi,\psi,\zeta)\|^{2}d\tau+C\varepsilon^{\frac{2}{5}}.\end{array}$
(4.45)
To close the a priori estimate, we also need to estimate the derivatives on
the non-fluid component $\mathbf{G}$, i.e.,
$\partial^{\alpha}\mathbf{G},(|\alpha|=1,2)$. Applying $\partial_{y}$ on
(2.77), we have
$\begin{array}[]{l}\quad\displaystyle\mathbf{G}_{y\tau}-(\frac{u_{1}}{v}\mathbf{G}_{y})_{y}+\\{\frac{1}{v}\mathbf{P}_{1}(\xi_{1}\mathbf{M}_{y})\\}_{y}+\\{\frac{1}{v}\mathbf{P}_{1}(\xi_{1}\mathbf{G}_{y})\\}_{y}\\\
\displaystyle=\mathbf{L}_{\mathbf{M}}\mathbf{G}_{y}+2Q(\mathbf{M}_{y},\mathbf{G})+2Q(\mathbf{G}_{y},\mathbf{G}).\end{array}$
(4.46)
Since
$\mathbf{P}_{1}(\xi_{1}\mathbf{M}_{y})=\frac{3}{2v\theta}\mathbf{P}_{1}[\xi_{1}(\frac{|\xi-u|^{2}}{2\theta}\theta_{y}+\xi\cdot
u_{y})\mathbf{M}],$
we have
$|\\{\frac{1}{v}\mathbf{P}_{1}(\xi_{1}\mathbf{M}_{y})\\}_{y}|\leq
C(v_{y}^{2}+u_{y}^{2}+\theta_{y}^{2}+|\theta_{yy}|+|u_{yy}|)|\hat{B}(\xi)|\mathbf{M},$
where $\hat{B}(\xi)$ is a polynomial of $\xi$. This yields that
$\begin{array}[]{ll}\displaystyle\int_{0}^{\tau}\int\int|\\{\frac{1}{v}\mathbf{P}_{1}(\xi_{1}\mathbf{M}_{y})\\}_{y}\frac{\mathbf{G}_{y}}{\mathbf{M}_{\star}}|d\xi
dyd\tau\leq\frac{\sigma}{8}\int_{0}^{\tau}\int\int\frac{\nu(|\xi|)}{\mathbf{M}_{\star}}|\mathbf{G}_{y}|^{2}d\xi
dyd\tau\\\
\displaystyle\qquad\qquad+C\int_{0}^{\tau}\|(\psi_{yy},\zeta_{yy})\|^{2}d\tau+C(\varepsilon^{\frac{1}{2}}+\chi)\int_{0}^{\tau}\|(\phi_{y},\psi_{y},\zeta_{y})\|^{2}d\tau+C\varepsilon^{\frac{2}{5}}.\end{array}$
Thus, multiplying (4.46) by $\frac{v\mathbf{G}_{y}}{\mathbf{M}_{\star}}$ and
using the Cauchy inequality and Lemmas 4.2-4.5 yield
$\begin{array}[]{l}\displaystyle\int\int\frac{|\mathbf{G}_{y}|^{2}}{2\mathbf{M}_{\star}}(\tau,y,\xi)d\xi
dy+\int_{0}^{\tau}\int\int\frac{\nu(|\xi|)}{\mathbf{M}_{\star}}|\mathbf{G}_{y}|^{2}d\xi
dyd\tau\\\ \displaystyle\leq
C(\varepsilon^{\frac{1}{2}}+\chi)\int_{0}^{\tau}\int\int\frac{\nu(|\xi|)}{\mathbf{M}_{\star}}|\mathbf{G}_{1}|^{2}d\xi
dyd\tau+C(\varepsilon^{\frac{1}{2}}+\chi)\int_{0}^{\tau}\|(\phi_{y},\psi_{y},\zeta_{y})\|^{2}d\tau\\\
\quad\displaystyle+C\int_{0}^{\tau}\int\int\frac{\nu(|\xi|)}{\mathbf{M}_{\star}}|\mathbf{G}_{yy}|^{2}d\xi
dyd\tau+C\int_{0}^{\tau}\|(\phi_{yy},\zeta_{yy})\|^{2}d\tau+C\varepsilon^{\frac{2}{5}}.\end{array}$
(4.47)
Similarly,
$\begin{array}[]{l}\displaystyle\int\int\frac{|\mathbf{G}_{\tau}|^{2}}{2\mathbf{M}_{\star}}(\tau,y,\xi)d\xi
dy+\int_{0}^{\tau}\int\int\frac{\nu(|\xi|)}{\mathbf{M}_{\star}}|\mathbf{G}_{\tau}|^{2}d\xi
dyd\tau\\\ \displaystyle\leq
C\int_{0}^{\tau}\int\int\frac{\nu(|\xi|)}{\mathbf{M}_{\star}}|\mathbf{G}_{y\tau}|^{2}d\xi
dyd\tau+C(\varepsilon^{\frac{1}{2}}+\chi)\int_{0}^{\tau}\int\int\frac{\nu(|\xi|)}{\mathbf{M}_{\star}}|\mathbf{G}_{1}|^{2}d\xi
dyd\tau\\\
\displaystyle+C(\varepsilon^{\frac{1}{2}}+\chi)\int_{0}^{\tau}\int\int\frac{\nu(|\xi|)}{\mathbf{M}_{\star}}|\mathbf{G}_{y}|^{2}d\xi
dyd\tau\\\
\displaystyle+C(\varepsilon^{\frac{1}{2}}+\chi)\sum_{|\alpha^{\prime}|=1}\int_{0}^{\tau}\|\partial^{\alpha^{\prime}}(\phi,\psi,\zeta)\|^{2}d\tau+C\int_{0}^{\tau}\|(\psi_{y\tau},\zeta_{y\tau})\|^{2}d\tau+C\varepsilon^{\frac{2}{5}},\end{array}$
(4.48)
where we have used the fact that
$\begin{array}[]{ll}\displaystyle\int\int\frac{v|\mathbf{G}_{\tau}|^{2}}{2\mathbf{M}_{\star}}(\tau=0,y,\xi)d\xi
dy&\displaystyle=\int\int\frac{|\mathbf{P}_{1}(\xi_{1}\mathbf{M}_{y})|^{2}}{2v\mathbf{M}_{\star}}(\tau=0,y,\xi)d\xi
dy\\\ &\displaystyle\leq C\|(v,u,\theta)_{y}(\tau=0,\cdot)\|^{2}\\\
&\displaystyle=C\|(V,U,\Theta)_{y}(\tau=0,\cdot)\|^{2}\leq
C\varepsilon^{\frac{1}{2}}.\end{array}$
Finally, we estimate the highest order derivatives, that is,
$\int\psi_{1y}\phi_{yy}dy$ and
$\int_{0}^{\tau}\int\int\frac{\nu(|\xi|)|\partial^{\alpha}\mathbf{G}|^{2}}{\mathbf{M}_{\star}}d\xi
dyd\tau$ with $|\alpha|=2$ in (4.45). To do so, it is sufficient to study
$\int\int\frac{|\partial^{\alpha}f|^{2}}{\mathbf{M}_{\star}}d\xi
dy~{}(|\alpha|=2)$ in view of (4.13)- (4.16). For this, from (2.83) we have
$vf_{\tau}-u_{1}f_{y}+\xi_{1}f_{y}=vQ(f,f)=v[\mathbf{L}_{\mathbf{M}}\mathbf{G}+Q(\mathbf{G},\mathbf{G})].$
Applying $\partial^{\alpha}$ $(|\alpha|=2)$ to the above equation gives
$\begin{array}[]{ll}\displaystyle
v(\partial^{\alpha}f)_{\tau}-v\mathbf{L}_{\mathbf{M}}\partial^{\alpha}\mathbf{G}-u_{1}(\partial^{\alpha}f)_{y}+\xi_{1}(\partial^{\alpha}f)_{y}\\\\[8.53581pt]
\displaystyle=-\partial^{\alpha}vf_{\tau}+\partial^{\alpha}u_{1}f_{y}-\sum_{|\alpha^{\prime}|=1}[\partial^{\alpha-\alpha^{\prime}}v\partial^{\alpha^{\prime}}f_{\tau}-\partial^{\alpha-\alpha^{\prime}}u_{1}\partial^{\alpha^{\prime}}f_{y}]\\\
\displaystyle\quad+[\partial^{\alpha}(v\mathbf{L}_{\mathbf{M}}\mathbf{G})-v\mathbf{L}_{\mathbf{M}}\partial^{\alpha}\mathbf{G}]+\partial^{\alpha}[vQ(\mathbf{G},\mathbf{G})].\end{array}$
(4.49)
Multiplying (4.49) by
$\frac{\partial^{\alpha}f}{\mathbf{M}_{\star}}=\frac{\partial^{\alpha}\mathbf{M}}{\mathbf{M}_{\star}}+\frac{\partial^{\alpha}\mathbf{G}}{\mathbf{M}_{\star}}$
yields
$\begin{array}[]{l}\quad\displaystyle(\frac{v|\partial^{\alpha}f|^{2}}{2\mathbf{M}_{\star}})_{\tau}-v\mathbf{L}_{\mathbf{M}}\partial^{\alpha}\mathbf{G}\cdot\frac{\partial^{\alpha}\mathbf{G}}{\mathbf{M}_{\star}}\\\
\displaystyle=\frac{\partial^{\alpha}f}{\mathbf{M}_{\star}}\bigg{\\{}-\partial^{\alpha}vf_{\tau}+\partial^{\alpha}u_{1}f_{y}-\sum_{|\alpha^{\prime}|=1}[\partial^{\alpha-\alpha^{\prime}}v\partial^{\alpha^{\prime}}f_{\tau}-\partial^{\alpha-\alpha^{\prime}}u_{1}\partial^{\alpha^{\prime}}f_{y}]\\\
\displaystyle\quad+[\partial^{\alpha}(v\mathbf{L}_{\mathbf{M}}\mathbf{G})-v\mathbf{L}_{\mathbf{M}}\partial^{\alpha}\mathbf{G}]+\partial^{\alpha}[vQ(\mathbf{G},\mathbf{G})]\bigg{\\}}+v\mathbf{L}_{\mathbf{M}}\partial^{\alpha}\mathbf{G}\cdot\frac{\partial^{\alpha}\mathbf{M}}{\mathbf{M}_{\star}}+(\cdots)_{y}.\end{array}$
(4.50)
Hence,
$\begin{array}[]{l}\displaystyle\int_{0}^{\tau}\int\int|\partial^{\alpha}vf_{\tau}\frac{\partial^{\alpha}f}{\mathbf{M}_{\star}}|d\xi
dyd\tau\\\
\displaystyle\leq\int_{0}^{\tau}\int\bigg{[}|\partial^{\alpha}v|\int(|\mathbf{M}_{\tau}|+|\mathbf{G}_{\tau}|)\frac{|\partial^{\alpha}\mathbf{M}|+|\partial^{\alpha}\mathbf{G}|}{\mathbf{M}_{\star}}d\xi\bigg{]}dyd\tau\\\
\displaystyle\leq
C(\varepsilon+\chi^{2})\int_{0}^{\tau}\|\partial^{\alpha}(\phi,\psi,\zeta)\|^{2}d\tau+\frac{\sigma}{16}\int_{0}^{\tau}\int\int\frac{v|\partial^{\alpha}\mathbf{G}|^{2}}{\mathbf{M}_{\star}}d\xi
dyd\tau\\\
\displaystyle\quad+C(\varepsilon+\chi^{2})\int_{0}^{\tau}\int\int\frac{|\mathbf{G}_{\tau}|^{2}}{\mathbf{M}_{\star}}d\xi
dyd\tau\\\
\displaystyle\quad+C(\varepsilon^{\frac{1}{2}}+\chi)\sum_{|\alpha^{\prime}|=1}\int_{0}^{\tau}\|\partial^{\alpha^{\prime}}(\phi,\psi,\zeta)\|^{2}d\tau+C\varepsilon^{\frac{2}{5}},\end{array}$
and
$\begin{array}[]{l}\displaystyle\sum_{|\alpha^{\prime}|=1}\int_{0}^{\tau}\int\int|\partial^{\alpha-\alpha^{\prime}}v\partial^{\alpha^{\prime}}f_{\tau}\frac{\partial^{\alpha}f}{\mathbf{M}_{\star}}|d\xi
dyd\tau\\\
\displaystyle\leq\sum_{|\alpha^{\prime}|=1}\int_{0}^{\tau}\int|\partial^{\alpha-\alpha^{\prime}}v|\int(|\partial^{\alpha^{\prime}}\mathbf{M}_{\tau}|+|\partial^{\alpha^{\prime}}\mathbf{G}_{\tau}|)\frac{|\partial^{\alpha}\mathbf{M}|+|\partial^{\alpha}\mathbf{G}|}{\mathbf{M}_{\star}}d\xi
dyd\tau\\\
\displaystyle\leq\frac{\sigma}{16}\int_{0}^{\tau}\int\int\frac{v|\partial^{\alpha}\mathbf{G}|^{2}}{\mathbf{M}_{\star}}d\xi
dyd\tau+C(\delta+\gamma)\int_{0}^{\tau}\|\partial^{\alpha}(\phi,\psi,\zeta)\|^{2}d\tau+C\varepsilon^{\frac{2}{5}}.\end{array}$
Notice that similar estimates can be obtained for the terms
$\partial^{\alpha}u_{1}f_{y}\frac{\partial^{\alpha}f}{\mathbf{M}_{\star}}$ and
$\sum_{|\alpha^{\prime}|=1}\partial^{\alpha-\alpha^{\prime}}u_{1}\partial^{\alpha^{\prime}}f_{y}\frac{\partial^{\alpha}f}{\mathbf{M}_{\star}}$.
Furthermore, we have
$\begin{array}[]{l}\displaystyle\partial^{\alpha}(v\mathbf{L}_{\mathbf{M}}\mathbf{G})-v\mathbf{L}_{\mathbf{M}}\partial^{\alpha}\mathbf{G}=(\partial^{\alpha}v)\mathbf{L}_{\mathbf{M}}\mathbf{G}+2vQ(\partial^{\alpha}\mathbf{M},\mathbf{G})\\\
\displaystyle~{}~{}+\sum_{|\alpha^{\prime}|=1}\bigg{\\{}2vQ(\partial^{\alpha-\alpha^{\prime}}\mathbf{M},\partial^{\alpha^{\prime}}\mathbf{G})+\partial^{\alpha-\alpha^{\prime}}v[\mathbf{L}_{\mathbf{M}}\partial^{\alpha^{\prime}}\mathbf{G}+2Q(\partial^{\alpha^{\prime}}\mathbf{M},\mathbf{G})]\bigg{\\}},\end{array}$
and
$\begin{array}[]{l}\displaystyle\partial^{\alpha}[vQ(\mathbf{G},\mathbf{G})]=(\partial^{\alpha}v)Q(\mathbf{G},\mathbf{G})+2vQ(\partial^{\alpha}\mathbf{G},\mathbf{G})\\\
\displaystyle\qquad+\sum_{|\alpha^{\prime}|=1}\bigg{\\{}vQ(\partial^{\alpha-\alpha^{\prime}}\mathbf{G},\partial^{\alpha^{\prime}}\mathbf{G})+2(\partial^{\alpha-\alpha^{\prime}}v)Q(\partial^{\alpha^{\prime}}\mathbf{G},\mathbf{G})]\bigg{\\}}.\end{array}$
For illustration, we only estimate one of the above terms in the following
because the other terms can be discussed similarly.
$\begin{array}[]{l}\quad\displaystyle\int_{0}^{\tau}\int\int\frac{v\partial^{\alpha}\mathbf{G}\cdot
Q(\partial^{\alpha}\mathbf{G},\mathbf{G})}{\mathbf{M}_{\star}}d\xi dyd\tau\\\
\displaystyle\leq\frac{\sigma}{16}\int_{0}^{\tau}\int\int\frac{v|\partial^{\alpha}\mathbf{G}|^{2}}{\mathbf{M}_{\star}}d\xi
dyd\tau\\\
\displaystyle~{}+C\int_{0}^{\tau}\int\bigg{(}\int\frac{\nu(|\xi|)|\partial^{\alpha}\mathbf{G}|^{2}}{\mathbf{M}_{\star}}d\xi\cdot\int\frac{|\mathbf{G}|^{2}}{\mathbf{M}_{\star}}d\xi+\int\frac{|\partial^{\alpha}\mathbf{G}|^{2}}{\mathbf{M}_{\star}}d\xi\cdot\int\frac{\nu(|\xi|)|\mathbf{G}|^{2}}{\mathbf{M}_{\star}}d\xi\bigg{)}dyd\tau\\\
\leq\displaystyle\frac{\sigma}{8}\int_{0}^{\tau}\int\int\frac{\nu(|\xi|)}{\mathbf{M}_{\star}}v|\partial^{\alpha}\mathbf{G}|^{2}d\xi
dyd\tau\\\
\qquad\qquad\qquad\displaystyle+C\int_{0}^{\tau}|\sup_{y}\int\frac{\nu(|\xi|)|\mathbf{G}_{1}|^{2}}{\mathbf{M}_{\star}}d\xi\cdot\int\int\frac{|\partial^{\alpha}\mathbf{G}|^{2}}{\mathbf{M}_{\star}}d\xi
dy|d\tau\\\
\displaystyle\leq\frac{\sigma}{8}\int_{0}^{\tau}\int\int\frac{\nu(|\xi|)}{\mathbf{M}_{\star}}v|\partial^{\alpha}\mathbf{G}|^{2}d\xi
dyd\tau\\\
\qquad\qquad\qquad\displaystyle+C(\varepsilon+\chi^{2})\int_{0}^{\tau}\int\int\frac{\nu(|\xi|)[|\mathbf{G}_{1y}|^{2}+|\mathbf{G}_{1}|^{2}]}{\mathbf{M}_{\star}}d\xi
dyd\tau\\\
\displaystyle\leq\frac{\sigma}{8}\int_{0}^{\tau}\int\int\frac{\nu(|\xi|)}{\mathbf{M}_{\star}}v|\partial^{\alpha}\mathbf{G}|^{2}d\xi
dyd\tau+C(\varepsilon+\chi^{2})\int_{0}^{\tau}\|(\phi_{y},\psi_{y},\zeta_{y})\|^{2}d\tau+C\varepsilon^{\frac{2}{5}}\\\\[8.53581pt]
\displaystyle\quad\quad\qquad+C(\varepsilon+\chi^{2})\int_{0}^{\tau}\int\int\frac{\nu(|\xi|)[|\mathbf{G}_{y}|^{2}+|\mathbf{G}_{1}|^{2}]}{\mathbf{M}_{\star}}d\xi
dyd\tau.\end{array}$
Now we estimate the term $\displaystyle\int_{0}^{\tau}\int\int
v\mathbf{L}_{\mathbf{M}}\partial^{\alpha}\mathbf{G}\cdot\frac{\partial^{\alpha}\mathbf{M}}{\mathbf{M}_{\star}}d\xi
dyd\tau$ in (4.50). First, note that
$\mathbf{P}_{1}(\partial^{\alpha}\mathbf{M})$ does not contain the term
$\partial^{\alpha}(v,u,\theta)$ for $|\alpha|=2$. Thus, we have
$\begin{array}[]{l}\quad\displaystyle\int_{0}^{\tau}\int\int\frac{v\mathbf{L}_{\mathbf{M}}\partial^{\alpha}\mathbf{G}\cdot\partial^{\alpha}\mathbf{M}}{\mathbf{M}}d\xi
dyd\tau=\int_{0}^{\tau}\int\int\frac{v\mathbf{L}_{\mathbf{M}}\partial^{\alpha}\mathbf{G}\cdot\mathbf{P}_{1}(\partial^{\alpha}\mathbf{M})}{\mathbf{M}}d\xi
dyd\tau\\\
\displaystyle\leq\frac{\sigma}{16}\int_{0}^{\tau}\int\int\frac{v|\partial^{\alpha}\mathbf{G}|^{2}}{\mathbf{M}_{\star}}d\xi
dyd\tau+C(\varepsilon^{\frac{1}{2}}+\chi)\sum_{|\alpha^{\prime}|=1}\int_{0}^{\tau}\|\partial^{\alpha^{\prime}}(\phi,\psi,\zeta)\|^{2}d\tau+C\varepsilon^{\frac{2}{5}}.\end{array}$
(4.51)
Also we can get
$\begin{array}[]{l}\displaystyle\int_{0}^{\tau}\int\int
v\mathbf{L}_{\mathbf{M}}\partial^{\alpha}\mathbf{G}\cdot\partial^{\alpha}\mathbf{M}(\frac{1}{\mathbf{M}_{\star}}-\frac{1}{\mathbf{M}})d\xi
dyd\tau\leq\frac{\sigma}{16}\int_{0}^{\tau}\int\int\frac{\nu(|\xi|)}{\mathbf{M}_{\star}}v|\partial^{\alpha}\mathbf{G}|^{2}d\xi
dyd\tau\\\
\quad\displaystyle+C\eta_{0}^{2}~{}\int_{0}^{\tau}\|\partial^{\alpha}(\phi,\psi,\zeta)\|^{2}d\tau+C(\varepsilon^{\frac{1}{2}}+\chi)\sum_{|\alpha^{\prime}|=1}\int_{0}^{\tau}\|\partial^{\alpha^{\prime}}(\phi,\psi,\zeta)\|^{2}d\tau+C\varepsilon^{\frac{2}{5}},\end{array}$
(4.52)
where the small constant $\eta_{0}$ is defined in Lemma 4.3. The combination
of (4.51) and (4.52) gives the estimation on
$\displaystyle\int_{0}^{\tau}\int\int
v\mathbf{L}_{\mathbf{M}}\partial^{\alpha}\mathbf{G}\cdot\frac{\partial^{\alpha}\mathbf{M}}{\mathbf{M}_{\star}}d\xi
dyd\tau$.
Thus, integrating (4.50) and using the above estimates give
$\begin{array}[]{l}\displaystyle\int\int\frac{v|\partial^{\alpha}f|^{2}}{2\mathbf{M}_{\star}}(\tau,y,\xi)d\xi
dy+\frac{\sigma}{2}\int_{0}^{\tau}\int\int\frac{\nu(|\xi|)}{\mathbf{M}_{\star}}v|\partial^{\alpha}\mathbf{G}|^{2}d\xi
dyd\tau\\\ \displaystyle\leq
C(\varepsilon^{\frac{1}{2}}+\chi)\sum_{|\alpha^{\prime}|=1}\int_{0}^{\tau}\|\partial^{\alpha^{\prime}}(\phi,\psi,\zeta)\|^{2}d\tau+C(\eta_{0}+\delta+\gamma)\sum_{|\alpha|=2}\int_{0}^{\tau}\|\partial^{\alpha}(\phi,\psi,\zeta)\|^{2}d\tau\\\
\quad\displaystyle+C(\varepsilon^{\frac{1}{2}}+\chi)\sum_{|\alpha^{\prime}|=1}\int_{0}^{\tau}\int\int\frac{\nu(|\xi|)}{\mathbf{M}_{\star}}|\partial^{\alpha^{\prime}}\mathbf{G}|^{2}d\xi
dyd\tau+C\varepsilon^{\frac{2}{5}}\\\
\quad\displaystyle+C(\varepsilon^{\frac{1}{2}}+\chi)\int_{0}^{\tau}\int\int\frac{\nu(|\xi|)}{\mathbf{M}_{\star}}|\mathbf{G}_{1}|^{2}d\xi
dyd\tau,\end{array}$
where we have used the fact that
$\begin{array}[]{ll}&\displaystyle\int\int\frac{v|\partial^{\alpha}f|^{2}}{2\mathbf{M}_{\star}}(\tau=0,y,\xi)d\xi
dy=\int\int\frac{v|\partial^{\alpha}\mathbf{M}_{[V,U,\Theta]}|^{2}}{2\mathbf{M}_{\star}}(\tau=0,y,\xi)d\xi
dy\\\ &\qquad\displaystyle\leq
C\|(V,U,\Theta)_{yy}(\tau=0,\cdot)\|^{2}+C\|(V,U,\Theta)_{y}(\tau=0,\cdot)\|_{L^{4}}^{4}\\\
&\displaystyle\qquad\leq C\varepsilon^{\frac{3}{2}}.\end{array}$
Finally, similar to Lemma 3.3 in the previous section, we can get
$\begin{array}[]{ll}&\displaystyle\int_{0}^{\tau}\int_{{\bf
R}}\varepsilon(1+\varepsilon\tau)^{-1}e^{-\frac{C_{0}\varepsilon
y^{2}}{1+\varepsilon\tau}}|(\phi,\psi,\zeta)|^{2}dyd\tau\\\\[8.53581pt]
&\displaystyle\leq
C\|(\phi,\psi,\zeta)(\tau,\cdot)\|^{2}+C\int_{0}^{\tau}\|(\phi_{y},\psi_{y},\zeta_{y})\|^{2}d\tau+C\varepsilon^{\frac{2}{5}}\\\\[8.53581pt]
&\displaystyle+C\int_{0}^{\tau}(\tau+\tau_{0})^{-\frac{4}{3}}\|(\phi,\psi,\zeta)\|^{2}d\tau+C(\varepsilon^{\frac{1}{2}}+\chi)\int_{0}^{\tau}\int\int\frac{\nu(|\xi|)|\mathbf{G}_{1}|^{2}}{\mathbf{M}_{\star}}d\xi
dyd\tau\\\
&\displaystyle+C\sum_{|\alpha^{\prime}|=1}\int_{0}^{\tau}\int\int\frac{\nu^{-1}(|\xi|)|\partial^{\alpha^{\prime}}\mathbf{G}|^{2}}{\mathbf{M}_{\star}}d\xi
dyd\tau.\end{array}$
Note that here we need to estimate the microscopic terms.
In summary, by combining all the above estimates and by choosing the strength
of the contact wave $\delta^{CD}$, the bound on the a priori estimate $\chi$
and the Knudsen number $\varepsilon$ to be suitably small, we obtain
$\begin{array}[]{ll}\displaystyle\mathcal{N}(\tau)+\int_{0}^{\tau}\Big{[}\sum_{1\leq|\alpha|\leq
2}\|\partial^{\alpha}(\phi,\psi,\zeta)\|^{2}+\|\sqrt{(U^{R_{1}}_{1y},U^{R_{3}}_{1y})}(\phi,\zeta)\|^{2}\Big{]}d\tau\\\
\displaystyle+\int_{0}^{\tau}\int\int\frac{\nu(|\xi|)\mathbf{G}_{1}^{2}}{\mathbf{M}_{\star}}d\xi
dyd\tau+\sum_{|\alpha^{\prime}|=1}\int\int\frac{\nu(|\xi|)|\partial^{\alpha^{\prime}}\mathbf{G}|^{2}}{\mathbf{M}_{\star}}(\tau,y,\xi)d\xi
dy\\\
\displaystyle+\sum_{|\alpha|=2}\int\int\frac{\nu(|\xi|)|\partial^{\alpha}f|^{2}}{\mathbf{M}_{\star}}(\tau,y,\xi)d\xi
dy\leq C\varepsilon^{\frac{2}{5}}.\end{array}$
With the energy estimate, we complete the proof of Theorem 4.1.
## Acknowledgments
The authors would like to thank the referee for the valuable comments on
revision of the paper. The research of F. M. Huang was supported in part by
NSFC Grant No. 10825102 for Outstanding Young scholars, NSFC-NSAF Grant No.
10676037 and 973 project of China, Grant No. 2006CB805902. The research of Y.
Wang was supported by the NSFC Grant No. 10801128. The research of T. Yang was
supported by the General Research Fund of Hong Kong, CityU #104310, and the
NSFC Grant No. 10871082.
## References
* [1] (MR0344559) [10.1007/BF00249197] F. V. Atkinson and L. A. Peletier, _Similarity solutions of the nonlinear diffusion equation_ , Arch. Rat. Mech. Anal., 54 (1974), 373–392.
* [2] (MR0158708) L. Boltzmann, (translated by Stephen G. Brush), “Lectures on Gas Theory,” Dover Publications, Inc. New York, 1964.
* [3] (MR0586416) R. E. Caflish, _The fluid dynamical limit of the nonlinear Boltzmann equation_ , Comm. Pure Appl. Math., 33 (1980), 491–508.
* [4] (MR1307620) C. Cercignani, R. Illner and M. Pulvirenti, “The Mathematical Theory of Dilute Gases,” Springer-Verlag, Berlin, 1994.
* [5] (MR1148892) S. Chapman and T. G. Cowling, “The Mathematical Theory of Non-Uniform Gases,” 3rd edition, Cambridge University Press, 1990.
* [6] (MR0511671) C. T. Duyn and L. A. Peletier, _A class of similarity solution of the nonlinear diffusion equation_ , Nonlinear Analysis, T.M.A., 1 (1977), 223-233.
* [7] (MR2099033) R. Esposito and M. Pulvirenti, _From particle to fluids_ , in “Handbook of Mathematical Fluid Dynamics,” Vol. III, North-Holland, Amsterdam, (2004), 1–82.
* [8] (MR0946970) [10.1007/BF00292921] F. Golse, B. Perthame and C. Sulem, _On a boundary layer problem for the nonlinear Boltzmann equation_ , Arch. Ration. Mech. Anal., 103 (1986), 81–96.
* [9] (MR1188982) [10.1007/BF00410614] J. Goodman and Z. P. Xin, _Viscous limits for piecewise smooth solutions to systems of conservation laws_ , Arch. Rational Mech. Anal., 121 (1992), 235–265.
* [10] (MR0156656) H. Grad, “Asymptotic Theory of the Boltzmann Equation II,” in “Rarefied Gas Dynamics” (J. A. Laurmann, ed.), Vol. 1, Academic Press, New York, (1963), 26–59.
* [11] (MR2095473) [10.1512/iumj.2004.53.2574] Y. Guo, _The Boltzmann equation in the whole space_ , Indiana Univ. Math. J., 53 (2004), 1081–1094.
* [12] (MR1029681) D. Hoff and T. P. Liu, _The inviscid limit for the Navier-Stokes equations of compressible isentropic flow with shock data_ , India. Univ. Math. J., 36 (1989), 861–915.
* [13] (MR2646815) [10.1007/s00205-009-0267-0] F. M. Huang, J. Li and A. Matsumura, _Stability of the combination of the viscous contact wave and the rarefaction wave to the compressible Navier-Stokes equations_ , Arch. Ration. Mech. Anal., 197 (2010), 89–116.
* [14] (MR2208289) [10.1007/s00205-005-0380-7] F. M. Huang, A. Matsumura and Z. P. Xin, _Stability of contact discontinuities for the 1-D compressible Navier-Stokes equations_ , Arch. Rat. Mech. Anal., 179 (2006), 55–77.
* [15] (MR2594329) [10.1007/s00220-009-0966-2] F. M. Huang, Y. Wang and T. Yang, _Hydrodynamic limit of the Boltzmann equation with contact discontinuities_ , Comm. Math. Phy., 295 (2010), 293–326.
* [16] (MR2450610) [10.1016/j.aim.2008.06.014] F. M. Huang, Z. P. Xin and T. Yang, _Contact discontinuities with general perturbation for gas motion_ , Adv. Math., 219 (2008), 1246–1297.
* [17] (MR2237152) [10.1137/050626478] S. Jiang, G. X. Ni and W. J. Sun, _Vanishing viscosity limit to rarefaction waves for the Navier-Stokes equiations of one-dimensional compressible heat-conducting fluids_ , SIAM J. Math. Anal., 38 (2006), 368–384.
* [18] (MR0908596) [10.1002/mma.1670090127] M. Lachowicz, _On the initial layer and existence theorem for the nonlinear Boltzmann equation_ , Math. Methods Appl.Sci., 9 (1987), 342–366.
* [19] (MR2043729) [10.1016/j.physd.2003.07.011] T. Liu, T. Yang, and S. H. Yu, _Energy method for the Boltzmann equation_ , Physica D, 188 (2004), 178–192.
* [20] (MR2221210) [10.1007/s00205-005-0414-1] T. Liu, T. Yang, S. H. Yu and H. J. Zhao, _Nonlinear stability of rarefaction waves for the Boltzmann equation_ , Arch. Rat. Mech. Anal., 181 (2006), 333–371.
* [21] (MR2044894) [10.1007/s00220-003-1030-2] T. Liu and S. H. Yu, _Boltzmann equation: Micro-macro decompositions and positivity of shock profiles_ , Commun. Math. Phys., 246 (2004), 133–179.
* [22] (MR2557896) S. X. Ma, _Zero dissipation limit to strong contact discontinuity for the 1-D compressible Navier-Stokes equations_ , J. Diff. Eqs., 248 (2010), 95–110.
* [23] (MR0899210) [10.1007/BF03167088] A. Matsumura and K. Nishihara, _Asymptotics toward the rarefaction wave of the solutions of a one-dimensional model system for compressible viscous gas_ , Japan J. Appl. Math., 3 (1986), 1–13.
* [24] (MR0503305) [10.1007/BF01609490] T. Nishida, _Fluid dynamical limit of the nonlinear Boltzmann equation to the level of the compressible Euler equation_ , Commun. Math. Phys., 61 (1978), 119–148.
* [25] (MR1301779) J. Smoller, “Shock Waves and Reaction-Diffusion Equations,” 2nd edition, Springer-Verlag, New York, 1994.
* [26] (MR0719971) S. Ukai and K. Asano, _The Euler limit and the initial layer of the nonlinear Boltzmann equation_ , Hokkaido Math. J., 12 (1983), 303–324.
* [27] (MR2134449) [10.1142/S0219530505000522] S. Ukai, T. Yang and H. J. Zhao, _Global solutions to the Boltzmann equation with external forces_ , Analysis and Applications, 3 (2005), 157–193.
* [28] (MR2098251) [10.1016/j.jmaa.2004.03.064] H. Y. Wang, _Viscous limits for piecewise smooth solutions of the p-system_ , J. Math. Anal. Appl., 299 (2004), 411–432.
* [29] (MR2462917) [10.1016/S0252-9602(08)60074-0] Y. Wang, _Zero dissipation limit of the compressible heat-conducting Navier-Stokes equations in the presence of the shock_ , Acta Mathematica Scientia Ser. B, 28 (2008), 727–748.
* [30] (MR1213990) [10.1002/cpa.3160460502] Z. P. Xin, _Zero dissipation limit to rarefaction waves for the one-dimentional Navier-Stokes equations of compressible isentropic gases_ , Commun. Pure Appl. Math, XLVI (1993), 621–665.
* [31] Z. P. Xin and H. H. Zeng, _Convergence to the rarefaction waves for the nonlinear Boltzmann equation and compressible Navier-Stokes equations_ , J. Diff. Eqs., 249 (2010), 827–871.
* [32] (MR2116619) [10.1002/cpa.20027] S. H. Yu, _Hydrodynamic limits with shock waves of the Boltzmann equations_ , Commun. Pure Appl. Math, 58 (2005), 409–443.
Received August 2010; revised October 2010.
|
arxiv-papers
| 2010-11-09T07:25:29 |
2024-09-04T02:49:14.614010
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Feimin Huang, Yi Wang and Tong Yang",
"submitter": "Yi Wang",
"url": "https://arxiv.org/abs/1011.1990"
}
|
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