6.1 1 + 3-covariant perturbation equations on the brane

In the 1+3-covariant approach [303Jump To The Next Citation Point, 282Jump To The Next Citation Point, 305], 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, (251 ) W μν = ∇⃗⟨μ⃗∇ ν⟩W + ⃗∇ ⟨μW¯ν⟩ + ¯W μν, (252 )
where ¯W = ¯W μν ⟨μν⟩ and an overbar denotes a (3D) transverse quantity,
⃗∇ μ¯V = 0 = ∇⃗νW¯ . (253 ) μ μν
In a local inertial frame comoving with uμ, i.e., uμ = (1,⃗0), all time components may be set to zero: V = (0,V ) μ i, W = 0 0μ, ⃗∇ = (0, ⃗∇ ) μ i.

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

¯V = W¯ = W¯ = 0. (254 ) μ μ μν
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, (255 )
where ω μ is the vorticity, and purely tensor by
∇⃗ f = 0 = V , W = W¯ . (256 ) μ μ μν μν

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:

ℰ ℰ ℰ qμ = ⃗∇ μq + ¯qμ, (257 ) ℰ ⃗ ⃗ ℰ ⃗ ℰ ℰ πμν = ∇ ⟨μ∇ ν⟩π + ∇ ⟨μ¯πν⟩ + ¯πμν. (258 )
Linearizing the conservation equations for a single adiabatic fluid, and the nonlocal conservation equations, we obtain
˙ρ + Θ(ρ + p) = 0, (259 ) c2s⃗∇ μ + (ρ + p)Aμ = 0, (260 ) 4 ˙ρℰ + -Θ ρℰ + ⃗∇ μqℰμ = 0, (261 ) 3 ℰ ℰ 1-⃗ 4- ⃗ ν ℰ (ρ-+-p)⃗ q˙μ + 4Hq μ + 3 ∇ μρℰ + 3ρ ℰAμ + ∇ πμν = − λ ∇ μρ. (262 )
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, (263 ) 3 2 2 λ ˙ωμ + 2H ω μ + 1-curlAμ = 0, (264 ) 2 κ2 ℰ ˙σμν + 2H σμν + E μν − ∇⃗⟨μA ν⟩ = 2-πμν, (265 ) 2 2 E˙μν + 3HE μν − curlH μν + κ-(ρ + p)σμν = − κ--(ρ + p)ρσ μν 2 2 λ κ2-[ ℰ ℰ ℰ] − 6 4ρ ℰσμν + 3˙πμν + 3H πμν + 3⃗∇ ⟨μqν⟩ , (266 ) 2 H˙μν + 3HH μν + curlEμν = κ-curl πℰμν, (267 ) 2 ⃗∇ μωμ = 0, (268 ) 2 ⃗∇ νσμν − curlωμ − --⃗∇ μΘ = − qℰμ, (269 ) 3 curlσμν + ⃗∇ ⟨μων⟩ − Hμν = 0, (270 ) 2 2 2 [ ] ∇⃗νE μν − κ-⃗∇ μρ = κ-ρ-⃗∇ μρ + κ-- 2⃗∇ μρℰ − 4Hq ℰ − 3⃗∇ νπℰ , (271 ) 3 3 λ 6 μ μν ν 2 2 ρ- κ2[ ℰ] ∇⃗ H μν − κ (ρ + p)ωμ = κ (ρ + p) λω μ + 6 8ρ ℰωμ − 3curl qμ . (272 )
Equations (259View Equation), (261View Equation), and (263View 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 [137]). The equations contain scalar, vector, and tensor modes, which can be separated out if desired. They are not a closed system of equations until ℰ πμν 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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