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Comptes Rendus Mathématique
Volume 354, n° 1
pages 19-25 (janvier 2016)
Doi : 10.1016/j.crma.2015.09.023
Received : 5 Mars 2014 ;  accepted : 28 September 2015
Wave fluctuations near a de Sitter brane in an anti-de Sitter universe
Propagation des ondes près d'une brane de de Sitter dans un univers anti-de Sitter
 

Alain Bachelot
 Université de Bordeaux, Institut de mathématiques, UMR CNRS 5251, 33405 Talence cedex, France 

Abstract

We investigate the Klein–Gordon equation in the past causal domain of a de Sitter brane imbedded in an anti-de Sitter bulk. We solve the global mixed hyperbolic problem. We prove that any finite energy solution can be expressed as a Kaluza–Klein tower that is a superposition of free fields in the Steady State Universe, of which we study the asymptotic behaviors. We show that the leading term of a gravitational fluctuation is a massless graviton, i.e. the de Sitter brane is linearly stable.

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Résumé

Nous étudions l'équation de Klein–Gordon dans le domaine causal passé d'une brane de de Sitter incluse dans un espace anti-de Sitter. Nous résolvons le problème mixte hyperbolique et prouvons que les solutions d'énergie finie s'expriment comme superposition de champs libres dans le demi-espace de de Sitter et nous en étudions les comportements asymptotiques. Nous montrons que le terme dominant des fluctuations gravitationnelles est un graviton sans masse, ce qui traduit la stabilité linéaire de la brane.

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Version française abrégée

L'univers anti-de Sitter de dimension 5 est le cadre géométrique privilégié de la cosmologie branaire [[5], [7], [8], [9], [10]]. Sa carte de Poincaré est la variété lorentzienne
P:=Rt×Rx3×]0,∞[z,dsAdS2=z−2(dt2−dx2−dz2). Une brane de de Sitter dans   est juste la sous-variété   où  , qui est le bord de genre temps de  . On considère l'équation de Klein–Gordon de masse   dans   :[∂t2−Δx−∂z2+3z−1∂z+M2z−2]u=0,t<0,αt<z<−t,x∈R, complétée par la condition de Robin  , où ν est le quadrivecteur normal à  ,  , qui s'écrit :α∂tu+∂zu−c(αt)−1u1−α2=0,t<0,x∈R3,z=αt. On considère aussi la condition de Dirichlet  . Pour éviter la dépendance temporelle du domaine et de la condition au bord, on introduit les coordonnées   sur   :τ:=−2−1log⁡(t2−z2)∈R,ρ:=log⁡(t2−z2−t)−log⁡z∈]0,ρ0[,ρ0:=log⁡(1+1−α2)−log⁡(−α). Dans ce système, on a  ,  ,  ,  , et la dynamique prend la forme :[∂τ2+3∂τ−e−2τΔx−(sinh⁡ρ)−2∂ρ[(sinh⁡ρ)4∂ρ]+M2(sinh⁡ρ)2]u=0,τ∈R,x∈R3,0<ρ<ρ0,∂ρu+cu=0(ouu=0),τ∈R,x∈R3,ρ=ρ0. Étant données des conditions initiales spécifiées à un temps   fixé, on montre que le problème mixte hyperbolique est bien posé dans le cadre fonctionnel associé à l'énergie∫R3∫0ρ0(sinh⁡ρ)2|∂τu|2+e−2τ(sinh⁡ρ)2|∇xu|2+(sinh⁡ρ)4|∂ρu|2+M2(sinh⁡ρ)4|u|2dxdρ. Cette solution admet une décomposition en « tour de Kaluza–Klein »:u(τ,x,ρ)=∑juλj(τ,x)w(ρ;λj)+∫32∞um2(τ,x)w(ρ;m2)dm où   et   décrivent respectivement le spectre ponctuel et le spectre absolument continu de l'opérateur de Sturm–Liouville   sur   avec la condition  . De plus  , et   est une solution d'énergie finie de l'équation de Klein–Gordon de masse κ sur l'espace de de Sitter,[∂τ2+3∂τ−e−2τΔx+κ]uκ=0. L'étude du comportement asymptotique de   montre que le graviton sans masse   admet un profil asymptotique   quand  , tandis que   si  . On en déduit la stabilité linéaire d'une brane de de Sitter : les fluctuations gravitationnelles  , qui sont solutions pour  , sont asymptotes à un graviton sans masse  . Le détail des démonstrations est présenté dans [[4]].

English version

The 5-dimensional anti-de Sitter universe plays an important role in brane cosmology and has been deeply investigated by the physicists (see, e.g., [[5], [7], [8], [9], [10]]). Its Poincaré chard is the Lorentzian manifold
P:=Rt×Rx3×]0,∞[z,dsAdS2=z−2(dt2−dx2−dz2). Given  , the de Sitter brane   is just the time-like part of the boundary of the open set  . The other part of   is the light-like submanifold   that is the Cauchy horizon of our setting. We consider the Klein–Gordon equation with mass   in  :(0.1)[∂t2−Δx−∂z2+3z−1∂z+M2z−2]u=0,t<0,αt<z<−t,x∈R, with the Robin condition  , ν being the unit outgoing normal vector to  , and  :(0.2)α∂tu+∂zu−c(αt)−1u1−α2=0t<0,x∈R3,z=αt. We also treat the Dirichlet condition in [[4]]. In this Note, we construct a rigorous mathematical framework to solve the mixed hyperbolic problem, and to perform the spectral analysis of the solution in the spirit of the articles [[1], [2]] that were devoted to the Minkowski brane. The detail of the proofs is given in [[4]].

Klein–Gordon fields near a de Sitter brane

To avoid the time dependance of the domain and the boundary condition, we introduce new coordinates   on  :
τ:=−2−1log⁡(t2−z2)∈R,ρ:=log⁡(t2−z2−t)−log⁡z∈]0,ρ0[,ρ0:=log⁡(1+1−α2)−log⁡(−α). In this frame, we have  ,  ,  ,  , and the dynamics has the form:(1.1)[∂τ2+3∂τ−e−2τΔx−(sinh⁡ρ)−2∂ρ[(sinh⁡ρ)4∂ρ]+M2(sinh⁡ρ)2]u=0,τ∈R,x∈R3,0<ρ<ρ0,(1.2)∂ρu+cu=0,τ∈R,x∈R3,ρ=ρ0. Given a Cauchy data at time  ,(1.3)u(τ⁎,.)=u0(.),∂τu(τ⁎,.)=u1(.), we solve the global mixed hyperbolic problem in the functional framework associated with the energy∫R3∫0ρ0(sinh⁡ρ)2|∂τu|2+e−2τ(sinh⁡ρ)2|∇xu|2+(sinh⁡ρ)4|∂ρu|2+M2(sinh⁡ρ)4|u|2dxdρ. We introduce the following Hilbert spaces endowed with the natural norms:X0:=L2(Rx3×]0,ρ0[ρ,sinh2⁡(ρ)dρdx),X1:={u∈X0;∇xu,sinh⁡(ρ)∂ρu∈X0}, and to take into account the boundary condition:X2:={u∈X1;Δxu,(sinh⁡ρ)−2∂ρ[(sinh⁡ρ)4∂ρu]∈X0},Xc2:={u∈X2;∂ρu(ρ0)+cu(ρ0)=0}. We introduce also   and the Sturm–Liouville operatorLc:=−sinh−2⁡ρddρ(sinh4⁡ρddρ)+M2sinh2⁡ρ that is self-adjoint when it is endowed with the domain  . To define the weak solutions, we denote   the space of the vector distributions on  , which are  -valued, and we call finite energy solutions to ((1.1)) and ((1.2)), the distributions that are solutions to ((1.1)) and belong to  . Using the famous result of T. Kato on the propagator of the time-dependent hyperbolic evolution equations, we can establish the following theorem.

Theorem 1.1

Given  ,  ,  ,  ,   and  , there exists a unique finite energy solution u of ((1.1)), ((1.3)). There exists   such that for all   we have
‖u(τ)‖X1+‖∂τu(τ)‖X0≤f(|dτ−τ⁎|)(‖u0‖X1+‖u1‖X0), and there exists   such that for all  , ‖∫Θ(τ,x)u(τ,x,.)dτdx‖Dc≤C(τ⁎,Θ)(‖u0‖X1+‖u1‖X0). Here, f and C are independent of  ,  , and only depend on M, α and c.

If   and  , then u is a strong solution, i.e.  .

Kaluza–Klein tower

In this section, we show that any finite energy solution to ((1.1)) and ((1.2)) can be expressed as a Kaluza–Klein tower , which is a superposition of fields   solution to the Klein–Gordon equation:
(2.1)[∂τ2+3∂τ−e−2τΔx+κ]uκ=0, on the Steady State universe   that is isometric to the de Sitter brane  :(2.2)dS12:=Rτ×Rx3,dsdS2=dτ2−e2τdx2. The key of the proof is a complete spectral analysis of the operator   that we present now. In the sequel,   and   are the associated Legendre functions of first and second kinds.

Proposition 2.1

For all   the spectrum of the self-adjoint operator   has the following form:
σac(Lc)=[9/4,∞[,σsc(Lc)=∅,   is a finite subset included in   and when  ,  .

The eigenvalues are the solutions   of the transcendental equation
(c1−α2−2+M2+4)P−12+M2+4−94−λ(−α−1)+α(−12+M2+4−94−λ)P−32+M2+4−94−λ(−α−1)=0, and the corresponding eigenfunctions are given by: w(ρ;λ)=γ(sinh⁡ρ)−32P−12+M2+4−94−λ(cosh⁡ρ),γ∈C.

For all  , the generalized eigenfunction, solution to  ,  , is given by:
w(ρ;m2):=2mπ(4m2−9)14Γ(12+M2+4−im2−94)(sinh⁡ρ)−32×|(P−12+M2+4im2−94)′(−α−1)−α1−α2(c−321−α2)P−12+M2+4im2−94(−α−1)|−1×{[(P−12+M2+4im2−94)′(−α−1)−α1−α2(c−321−α2)P−12+M2+4im2−94(−α−1)]×Q−12+M2+4−im2−94(cosh⁡ρ)−[(Q−12+M2+4−im2−94)′(−α−1)−α1−α2(c−321−α2)Q−12+M2+4−im2−94(−α−1)]×P−12+M2+4im2−94(cosh⁡ρ)}.

The main result of this section states the existence of Kaluza–Klein towers:

Theorem 2.2

The finite-energy solutions u to ((1.1)) and ((1.2)) can be expressed as
u(τ,x,ρ)=∑λj∈σp(Lc)uλj(τ,x)w(ρ;λj)+limR→∞⁡∫32Rum2(τ,x)w(ρ;m2)dminC0(Rτ;X1)∩C1(Rτ;X0), where   is solution to ((2.1)), and uλj∈C0(Rτ;H1(Rx3))∩C1(Rτ;L2(Rx3)), and for any  um2∈L2(]32,∞[m;C0([−T,T]τ;H1(Rx3))∩C1([−T,T]τ;L2(Rx3))),mum2∈L2(]32,∞[m;C0([−T,T]τ;L2(Rx3))).

Since   is explicit and   is given by the formula ((3.1)) below, we have obtained an analytic expression of the finite-energy solutions to the mixed hyperbolic problem in term of special functions of Bessel and Legendre that could be useful to prove some decay estimates like in [[1]] and [[2]] in the case of the Minkowski branes.

Asymptotics in the Steady-State Universe

The Cauchy problem for the Klein–Gordon equation ((2.1)) in the Steady State Universe   is easily solved in the scale of the Sobolev spaces by using Fourier analysis. For any  ,  ,  ,  , there exists a unique   solution of ((2.1)) satisfying  ,  . The partial Fourier transform with respect to x of the solution,   is given by
(3.1)uˆ(τ,ξ)=π2e−32τ{[Yν′(|ξ|e−τ⁎)Jν(|ξ|e−τ)−Jν′(|ξ|e−τ⁎)Yν(|ξ|e−τ)]|ξ|e12τ⁎uˆ0(ξ)+[Yν(|ξ|e−τ⁎)Jν(|ξ|e−τ)−Jν(|ξ|e−τ⁎)Yν(|ξ|e−τ)]e32τ⁎(uˆ1(ξ)+32uˆ0(ξ))}, where   satisfies  ,   (for another representation with the fundamental solution, see [[6]]). When  ,   the natural energy defined asEκ(u,τ):=∫|∂τu(τ,x)|2+e−2τ|∇xu(τ,x)|2+κ|u(τ,x)|2dx is a decreasing function of τ and when  , we have for all  :∫|∂τu(τ,x)+32u(τ,x)|2dx≤2e3(|τ⁎|−τ)∫|u1|2+|∇xu0|2+κ|u0|2dx,∫|∂τu(τ,x)|2+κ|u(τ,x)|2dx≤min⁡(3κe−3τκ−94,1)e3|τ⁎|∫|u1|2+|∇xu0|2+κ|u0|2dx,∫|∇xu(τ,x)|2≤2e3|τ⁎|−τ∫|u1|2+|∇xu0|2+κ|u0|2dx.

An important point related to exponential expansion is the existence of a horizon: when   and   are compactly supported in  , then for all  ,   is supported in  . Another crucial consequence is the asymptotic behavior of the field as  :

Theorem 3.1

When  , u and   have an asymptotic profile at the time infinity: there exists   such that
(3.2)‖u(τ,.)−ϕ‖Hs+‖e2τ∂τu(τ,.)−Δϕ‖Hs−1⟶0,τ→+∞, and ϕ is given by: (3.3)ϕˆ(ξ)=π2e12τ⁎|ξ|−12{J12(e−τ⁎|ξ|)uˆ0(ξ)+J32(e−τ⁎|ξ|)eτ⁎|ξ|−1uˆ1(ξ)}.

When  , the solution is vanishing as  : for almost all  , when  ,  , for  ,  , and when  ,  . Moreover, given  , there exists   independent of   and   such that for all   we have:
0<κ<94,{‖u(τ,.)‖Hs+‖∂τu(τ,.)‖Hs−1≤Cemax⁡(94−κ−32,−1)τ(‖u0‖Hs+‖u1‖Hs−1),‖u(τ,.)‖Hs−12+‖∂τu(τ,.)‖Hs−32≤Ce(94−κ−32)τ(‖u0‖Hs+‖u1‖Hs−1),κ=94,{‖u(τ,.)‖Hs+‖∂τu(τ,.)‖Hs−1≤Cτe−τ(‖u0‖Hs+‖u1‖Hs−1),‖u(τ,.)‖Hs−12+‖∂τu(τ,.)‖Hs−32≤Cτe−32τ(‖u0‖Hs+‖u1‖Hs−1),κ>94,{‖u(τ,.)‖Hs+‖∂τu(τ,.)‖Hs−1≤Ce−τ(‖u0‖Hs+‖u1‖Hs−1),‖u(τ,.)‖Hs−12+‖∂τu(τ,.)‖Hs−32≤Ce−32τ(‖u0‖Hs+‖u1‖Hs−1),

When  , the solution can blow up at the time infinity: there exists Schwartz functions  ,   such that the solution satisfies
∀s∈R,‖u(τ,.)‖Hs∼e(94−κ−32)τ,τ→+∞.

We make some comments on this result. A consequence of this theorem is that when  , we have in  
u(τ,.)=O(e−(32−94−κ)τ) if 0≤κ≠94,u(τ,.)=O(τe−32τ) if κ=94. This is a consequence of the estimates of decay in  , the Sobolev embedding, and the horizon of radius   when   are supported in  . A. Vasy has established in [[11]] precise asymptotics in the much more large class of the asymptotically de Sitter space-times and we can deduce from his work that when  , there exists   such that as   we have:u(τ,x)=e−(32−94−κ)τv(x)+o(e−(32−94−κ)τ) if 0≤κ≠94,u(τ,x)=τe−32τv(x)+O(e−32τ) if κ=94. The main novelty of Theorem 3.1 is the explicit formula ((3.3)) for the trace at   when  . We remark also that, in this case, there exists “disappearing solutions” that tends to zero as   since ϕ can be equal to zero. We note also a somewhat unexpected loss of rate of decay in   when  .   is the critical mass for which the Klein–Gordon equation is conformal invariant, hence the solution can be expressed in this case from a free field on the static Einstein universe   by using the conformal method and the decay estimates simply follow from the conformal factor. Besides, since the proof is based on the Fourier analysis and the investigation of the differential equation[d2dτ2+3ddτ+e−2τλ2+κ]u=0,τ∈R,λ∈R, we could obtain a similar theorem for the Klein–Gordon equation on the exponentially expanding Friedmann–Robertson–Walker universeRτ×Kx,ds2=dτ2−e2τgijdxidxj where   is any Riemannian manifold. Parenthetically, we also could obtain several formula of products of Bessel functions, by using the propagator property of the map   and formulas ((3.1)) and ((3.3)).

Gravitational fluctuations

The gravitational waves are described by equation ((0.1)) with   and the Neumann condition on the brane, i.e. ((0.2)) with  . The crucial point is the existence of the sector of the massless graviton that is just the set of the solutions
uφ(t,x,z):=φ(−2−1log⁡(t2−z2),x), where   is any solution to ((2.1)) with  . Then   is solution to ((0.1)) with   in  ,  , and satisfies the Neumann condition   on  ,  , for any  . In the   coordinates, it is just a solution independent of ρ , i.e.   (by abuse of notation we write  ). When  , then  . Moreover, we have  ,  . A direct application of Theorem 3.1 and formula ((3.3)) allows us to investigate the behavior of this graviton along the de Sitter brane   as  , and when we approach its Cauchy horizon  . We introduce  .

Proposition 4.1

For any finite energy massless graviton  , there exists   such that when   with  , we have:
γ‖uφ(τ,.)−ϕ(.)‖X1=‖uφ(t,.,z)−ϕ‖H1(Rx3)→0,γ‖e2τ∂τuφ(τ,.)−Δϕ‖X0=‖t−1∂tuφ(t,.,z)+Δϕ‖L2(Rx3)=‖z−1∂zuφ(t,.,z)−Δϕ‖L2(Rx3)→0,‖(z∓t)−1(∂t±∂z)uφ(t,.,z)∓Δϕ‖L2(Rx3)→0, and the Fourier transform of ϕ is given by (4.1)ϕˆ(ξ)=π2|ξ|−12{J12(|ξ|)φˆ(0,ξ)+J32(|ξ|)|ξ|−1∂τφˆ(0,ξ)}.

The following consequence of Theorem 2.2 states that the usual de Sitter gravity is recovered on the brane: the leading term of any gravitational fluctuation in the ADS bulk   is a massless graviton propagating on the brane. The physical meaning of this result is that the brane is stable under the scalar perturbations when we omit the back-reaction to the metric (linearized gravity). This conclusion matches with numerous analyses of the linear stability of the de Sitter brane in the physical literature, e.g., [[5], [7], [9]].

Theorem 4.2

We assume  ,  . Then given  ,  , there exists a massless graviton   such that the solution   of ((1.1)) and ((1.3)), satisfies
‖(u−uφ)(τ)‖X1+‖∂τ(u−uφ)(τ)‖X0⟶0,τ→+∞. The initial data of φ are given by (4.2)φ(τ⁎,x)=γ2∫0ρ0u0(x,ρ)sinh2⁡ρdρ∈H1(Rx3),∂τφ(τ⁎,x)=γ2∫0ρ0u1(x,ρ)sinh2⁡ρdρ∈L2(Rx3).

In terms of the coordinates  , we have   as  ,  . We end this work by investigating the gravitational fluctuations beyond the Cauchy horizon  . We obviously have to add some constraint when  . Roughly speaking, we suppose that there is no field incoming from the past null infinity, i.e.   as  ,  . More precisely, we consider initial data  ,   and   the solution to ((1.1)), ((1.2)) and ((1.3)) in   ( ). Then, if  , u can be uniquely extended into a solution   of   in  , satisfying the Neumann condition   on  , and   for  ,  . We study the energy of   on the hypersurface  .

Theorem 4.3

We assume that ϕ defined by ((4.1)) and ((4.2)) is nonzero. Then we have
(4.3)∫αt∞∫R3|∇t,x,zu˜(t,x,z)|2z−3dxdz≳1t2,t→0−.

We conclude that the energy of   is infinite, hence the finite energy spaces used in [[2]] cannot be used if we want to continue the solution in  . Beyond the Cauchy horizon  , it would be natural to consider a boundary constraint on the time-like conformal infinity   in the new functional framework introduced in [[3]].

References

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Bachelot A. New dynamics in the anti-de Sitter universe   Commun. Math. Phys. 2013 ;  320 : 723-759 [cross-ref]
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