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6.1 1 + 3-covariant perturbation equations on the brane

In the 1+3-covariant approach [201Jump To The Next Citation Point, 218Jump To The Next Citation Point, 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
Vμ = ∇⃗μV + ¯Vμ, (245 ) W = ∇⃗ ⃗∇ W + ⃗∇ W¯ + ¯W , (246 ) μν ⟨μ ν⟩ ⟨μ ν⟩ μν
where ¯W μν = ¯W ⟨μν⟩ and an overbar denotes a (3D) transverse quantity,
⃗∇ μ¯Vμ = 0 = ∇⃗νW¯μν. (247 )
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.,

¯Vμ = W¯μ = W¯ μν = 0. (248 )
Scalar perturbative quantities are formed from the potentials via the (3D) Laplacian, e.g., 𝒱 = ⃗∇ μ⃗∇ μV ≡ ⃗∇2V. Purely vector perturbations are characterized by
Vμ = V¯μ, W μν = ⃗∇ ⟨μ ¯W ν⟩, curl ⃗∇ μf = − 2f˙ωμ, (249 )
where ωμ is the vorticity, and purely tensor by
∇⃗μf = 0 = Vμ, W μν = W¯μν. (250 )

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:

qℰμ = ⃗∇ μqℰ + ¯qℰμ, (251 ) ℰ ⃗ ⃗ ℰ ⃗ ℰ ℰ πμν = ∇ ⟨μ∇ ν⟩π + ∇ ⟨μ¯πν⟩ + ¯πμν. (252 )
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 (253View Equation), (255View Equation), and (257View Equation) 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.


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