Metric-based perturbations

6.2 Metric-based perturbations

An alternative approach to brane-world cosmological perturbations is an extension of the 4D metric-based gauge-invariant theory [170 , 241 ]. A review of this approach is given in [40

, 269

]. 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 2ℛ δij + 2∂i∂j𝒞 + 2∂(iFj ) + fij | A2 (∂iβ − χi) | , (267 ) |⌈ ------------------------------------------------------|----------------|⌉ N α A2 (∂ β − χ ) | 2ν j j

where the background metric functions $A, N$ are given by Equations (181

, 182

). 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,

where $g(0μ)ν$ is the background metric, Equation (180

). In this gauge, we have

In the 5D longitudinal gauge, one gets

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 [244 , 246 ]. In the case of scalar perturbations, we have for example

with similar expressions for the other quantities. All of the metric perturbation quantities are determined once a solution is found for the wave equation

The junction conditions (62 ) relate the off-brane derivatives of metric perturbations to the matter perturbations:

where

0 δT 0 = − δρ, (274 ) δT 0i = a2qi, (275 ) i i i δT j = δp δj + δπ j. (276 )

For scalar perturbations in the Gaussian normal gauge, this gives

2 ∂y ψ(x,0) = κ-5(2δρ + 3δp), (277 ) 6 ∂y ℬ(x,0) = κ25δp, (278 ) 2 ∂y𝒞(x,0) = − κ5-δπ, (279 ) 2 κ25- i ∂yℛ (x,0) = − 6 δρ − ∂i∂ 𝒞(x,0), (280 )

where $δπ$ is the scalar potential for the matter anisotropic stress,

The perturbed KK energy-momentum tensor on the brane is given by

0 2 δℰ 0 = κ δρℰ, (282 ) δℰ0i = − κ2a2qℰi , (283 ) κ2 δℰij = − --δρℰ δij − δπ ℰij. (284 ) 3

The evolution of the bulk metric perturbations is determined by the perturbed 5D field equations in the vacuum bulk,

Then the matter perturbations on the brane enter via the perturbed junction conditions (273

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

For tensor perturbations (in any gauge), the only nonzero components of the perturbed Einstein tensor are

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