6.2 Metric-based perturbations

An alternative approach to brane-world cosmological perturbations is an extension of the 4D metric-based gauge-invariant theory [236, 329]. A review of this approach is given in [53Jump To The Next Citation Point, 362Jump To The Next Citation Point]. In an arbitrary gauge, and for a flat FRW background, the perturbed metric has the form
⌊ | ⌋ − 2N 2ψ A2 (∂iℬ − Si) | N α || { } | || δ(5)gAB = | A2 (∂jℬ − Sj) A2 2 ℛδij + 2∂i∂j𝒞 + 2∂(iFj) + fij | A2 (∂iβ − χi) | , (273 ) |⌈ ------------------------------------------------------|----------------|⌉ N α A2 (∂ β − χ ) | 2ν j j
where the background metric functions A, N are given by Equations (187View Equation, 188View Equation). The scalars ψ, ℛ, 𝒞,α, β,ν represent scalar perturbations. The vectors S i, F i, and χ i are transverse, so that they represent 3D vector perturbations, and the tensor fij is transverse traceless, representing 3D tensor perturbations.

In the Gaussian normal gauge, the brane coordinate-position remains fixed under perturbation,

(5)ds2 = [g(0)(x,y) + δg (x,y)] dxμdx ν + dy2, (274 ) μν μν
where g(0μ)ν is the background metric, Equation (186View Equation). In this gauge, we have
α = β = ν = χi = 0. (275 )

In the 5D longitudinal gauge, one gets

− ℬ + ˙𝒞 = 0 = − β + 𝒞′. (276 )
In this gauge, and for an AdS5 background, the metric perturbation quantities can all be expressed in terms of a “master variable” Ω which obeys a wave equation [331, 333]. In the case of scalar perturbations, we have for example
( ) 1 ′′ 1 Λ5 ℛ = 6A- Ω − N-2 ¨Ω − -3-Ω , (277 )
with similar expressions for the other quantities. All of the metric perturbation quantities are determined once a solution is found for the wave equation
( 1 )⋅ ( Λ k2 ) N ( N )′ -----誽 + -5-+ --- ---Ω = ---Ω′ . (278 ) N A3 6 A2 A3 A3

The junction conditions (68View Equation) relate the off-brane derivatives of metric perturbations to the matter perturbations:

[ ] 2 1( (0)) 1-(0) ∂yδg μν = − κ 5 δT μν + 3 λ − T δgμν − 3gμνδT , (279 )
where
0 δT 0 = − δρ, (280 ) δT 0i = a2qi, (281 ) i i i δT j = δp δj + δπ j. (282 )
For scalar perturbations in the Gaussian normal gauge, this gives
2 ∂yψ(x,0 ) = κ5(2δρ + 3δp), (283 ) 6 ∂yℬ(x,0 ) = κ25δp, (284 ) 2 ∂y𝒞(x,0 ) = − κ-5δπ, (285 ) 2 κ25 i ∂yℛ (x,0 ) = − 6 δρ − ∂i∂ 𝒞(x,0), (286 )
where δπ is the scalar potential for the matter anisotropic stress,
1- k δπij = ∂i∂jδπ − 3 δij∂k∂ δπ. (287 )
The perturbed KK energy-momentum tensor on the brane is given by
0 2 δℰ 0 = κ δρℰ, (288 ) δℰ0i = − κ2a2qℰi , (289 ) κ2 δℰ ij = − --δρ ℰδij − δπ ℰij. (290 ) 3
The evolution of the bulk metric perturbations is determined by the perturbed 5D field equations in the vacuum bulk,
δ(5)GA = 0. (291 ) B
Then the matter perturbations on the brane enter via the perturbed junction conditions (279View Equation).

For example, for scalar perturbations in Gaussian normal gauge, we have

{ ( ) [ ( ) ] } (5) y ′ A′ N ′ ′ A2 ′ A˙ N˙ ′ δ G i = ∂i − ψ + A--− N-- ψ − 2ℛ − 2N-2- ℬ˙ + 5A-− N-- ℬ . (292 )
For tensor perturbations (in any gauge), the only nonzero components of the perturbed Einstein tensor are
{ 2 ( ˙ ˙) ( ′ ′) } δ(5)Gij = − 1- − -1-f¨ij + f′′ij −-k-f ij + -1- N-− 3A- f˙ij + N--+ 3 A-- f′ij . (293 ) 2 N 2 A2 N 2 N A N A

In the following, we will discuss various perturbation problems, using either a 1+3-covariant or a metric-based approach.


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