1 + 3-covariant perturbation equations on the brane

6.1 1 + 3-covariant perturbation equations on the brane

In the 1+3-covariant approach [201

, 220 ], perturbative quantities are projected vectors, $V μ = V⟨μ⟩$ , and projected symmetric tracefree tensors, $W μν = W ⟨μν⟩$ , which are gauge-invariant since they vanish in the background. These are decomposed into (3D) scalar, vector, and tensor modes as

where $¯W μν = ¯W ⟨μν⟩$ and an overbar denotes a (3D) transverse quantity,

In a local inertial frame comoving with $uμ$ , i.e., $uμ = (1,⃗0)$ , all time components may be set to zero: $V μ = (0,Vi)$ , $W0 μ = 0$ , $⃗ ⃗ ∇ μ = (0,∇i)$ .

Purely scalar perturbations are characterized by the fact that vectors and tensors are derived from scalar potentials, i.e.,

Scalar perturbative quantities are formed from the potentials via the (3D) Laplacian, e.g., $𝒱 = ⃗∇ μ⃗∇ μV ≡ ⃗∇2V$ . Purely vector perturbations are characterized by

where $ωμ$ is the vorticity, and purely tensor by

The KK energy density produces a scalar mode $⃗∇ μρℰ$ (which is present even if $ρℰ = 0$ in the background). The KK momentum density carries scalar and vector modes, and the KK anisotropic stress carries scalar, vector, and tensor modes:

Linearizing the conservation equations for a single adiabatic fluid, and the nonlocal conservation equations, we obtain

˙ρ + Θ(ρ + p) = 0, (253 ) c2⃗∇ + (ρ + p)A = 0, (254 ) s μ μ ˙ρ + 4Θ ρ + ⃗∇ μqℰ = 0, (255 ) ℰ 3 ℰ μ ℰ ℰ 1 4 ν ℰ (ρ + p) q˙μ + 4Hq μ + --⃗∇ μρℰ + -ρ ℰAμ + ⃗∇ πμν = − -------⃗∇ μρ. (256 ) 3 3 λ

Linearizing the remaining propagation and constraint equations leads to

1 1 κ2 ρ Θ˙ + -Θ2 − ⃗∇ μA μ + -κ2(ρ + 3p) − Λ = − ---(2ρ + 3p)--− κ2 ρℰ, (257 ) 3 2 2 λ ˙ω + 2H ω + 1-curlA = 0, (258 ) μ μ 2 μ κ2 ℰ ˙σμν + 2H σμν + E μν − ∇⃗⟨μA ν⟩ =--πμν, (259 ) 2 2 2 E˙ + 3HE − curlH + κ-(ρ + p)σ = − κ--(ρ + p)ρσ μν μν μν 2 μν 2 λ μν κ2 [ ℰ ℰ ℰ] − --- 4ρ ℰσμν + 3˙πμν + 3H πμν + 3⃗∇ ⟨μqν⟩ , (260 ) 26 H˙μν + 3HH μν + curlEμν = κ-curl πℰ , (261 ) 2 μν ⃗∇ μωμ = 0, (262 ) ⃗∇ νσμν − curlωμ − 2-⃗∇ μΘ = − qℰμ, (263 ) 3 curlσμν + ⃗∇ ⟨μων⟩ − Hμν = 0, (264 ) 2 2 2 [ ] ∇⃗νE − κ-⃗∇ ρ = κ-ρ-⃗∇ ρ + κ-- 2⃗∇ ρ − 4Hq ℰ − 3⃗∇ νπℰ , (265 ) μν 3 μ 3 λ μ 6 μ ℰ μ μν ν 2 2 ρ κ2[ ℰ] ∇⃗ H μν − κ (ρ + p)ωμ = κ (ρ + p) -ω μ + ---8ρ ℰωμ − 3curl qμ . (266 ) λ 6

Equations (253

), (255

), and (257

) do not provide gauge-invariant equations for perturbed quantities, but their spatial gradients do.

These equations are the basis for a 1+3-covariant analysis of cosmological perturbations from the brane observer’s viewpoint, following the approach developed in 4D general relativity (for a review, see [92 ]). The equations contain scalar, vector, and tensor modes, which can be separated out if desired. They are not a closed system of equations until $π ℰμν$ is determined by a 5D analysis of the bulk perturbations. An extension of the 1+3-covariant perturbation formalism to 1+4 dimensions would require a decomposition of the 5D geometric quantities along a timelike extension $uA$ into the bulk of the brane 4-velocity field $u μ$ , and this remains to be done. The 1+3-covariant perturbation formalism is incomplete until such a 5D extension is performed. The metric-based approach does not have this drawback.