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6.4 Curvature perturbations and the Sachs–Wolfe effect

The curvature perturbation ℛ on uniform density surfaces is defined in Equation (267View Equation). The associated gauge-invariant quantity
--δρ---- ζ = ℛ + 3(ρ + p ) (308 )
may be defined for matter on the brane. Similarly, for the Weyl “fluid” if ρℰ ⁄= 0 in the background, the curvature perturbation on hypersurfaces of uniform dark energy density is
δρℰ- ζℰ = ℛ + 4ρℰ. (309 )
On large scales, the perturbed dark energy conservation equation is [191Jump To The Next Citation Point]
(δρ )⋅ + 4H δρ + 4ρ ℛ˙ = 0, (310 ) ℰ ℰ ℰ
which leads to
˙ζℰ = 0. (311 )
For adiabatic matter perturbations, by the perturbed matter energy conservation equation,
(δρ)⋅ + 3H (δρ + δp) + 3(ρ + p) ˙ℛ = 0, (312 )
we find
ζ˙= 0. (313 )
This is independent of brane-world modifications to the field equations, since it depends on energy conservation only. For the total, effective fluid, the curvature perturbation is defined as follows [191Jump To The Next Citation Point]: If ρℰ ⁄= 0 in the background, we have
[ ] ζ = ζ + ----------4ρℰ----------- (ζ − ζ ), (314 ) tot 3(ρ + p)(1 + ρ∕ λ) + 4ρℰ ℰ
and if ρℰ = 0 in the background, we get
ζtot = ζ + -------δρℰ-------- (315 ) 3(ρ + p)(1 + ρ∕λ) δC ℰ δρℰ = --4-, (316 ) a
where δCℰ is a constant. It follows that the curvature perturbations on large scales, like the density perturbations, can be found on the brane without solving for the bulk metric perturbations.

Note that ˙ζ ⁄= 0 tot even for adiabatic matter perturbations; for example, if ρ = 0 ℰ in the background, then

( ) 2 1 δρℰ ζ˙tot = H ctot − 3- (ρ +-p)(1-+-ρ∕-λ). (317 )
The KK effects on the brane contribute a non-adiabatic mode, although ζ˙tot → 0 at low energies.

Although the density and curvature perturbations can be found on super-Hubble scales, the Sachs–Wolfe effect requires ℰ πμν in order to translate from density/curvature to metric perturbations. In the 4D longitudinal gauge of the metric perturbation formalism, the gauge-invariant curvature and metric perturbations on large scales are related by

( ˙ ) ζtot = ℛ − H-- ℛ--− ψ , (318 ) ˙H H 2 2 ℛ + ψ = − κ a δπℰ, (319 )
where the radiation anisotropic stress on large scales is neglected, as in general relativity, and δπ ℰ is the scalar potential for π ℰ μν, equivalent to the covariant quantity Π defined in Equation (289View Equation). In 4D general relativity, the right hand side of Equation (319View Equation) is zero. The (non-integrated) Sachs–Wolfe formula has the same form as in general relativity:
| δT-| T |now= (ζrad + ψ − ℛ )|dec. (320 )
The brane-world corrections to the general relativistic Sachs–Wolfe effect are then given by [191]
( ) ( ) ∫ δT- δT- 8- ρrad-- 2 2 2κ2- 7∕2 T = T − 3 ρ Sℰ − κ a δπℰ + a5∕2 da a δπℰ, (321 ) gr cdm
where Sℰ is the KK entropy perturbation (determined by δ ρℰ). The KK term δπ ℰ cannot be determined by the 4D brane equations, so that δT∕T cannot be evaluated on large scales without solving the 5D equations. (Equation (321View Equation) has been generalized to a 2-brane model, in which the radion makes a contribution to the Sachs–Wolfe effect [176Jump To The Next Citation Point].)

The presence of the KK (Weyl, dark) component has essentially two possible effects:

A simple phenomenological approximation to δπℰ on large scales is discussed in [19], and the Sachs–Wolfe effect is estimated as

δT ( δπ ) ( t )2 ∕3[ ln(t ∕t) ] --- ∼ --ℰ- -eq- ----in--4 , (322 ) T ρ in tdec ln(teq∕t4)
where t4 is the 4D Planck time, and tin is the time when the KK anisotropic stress is induced on the brane, which is expected to be of the order of the 5D Planck time.

A self-consistent approximation is developed in [177Jump To The Next Citation Point], using the low-energy 2-brane approximation [290Jump To The Next Citation Point, 298Jump To The Next Citation Point, 299Jump To The Next Citation Point, 300Jump To The Next Citation Point, 320Jump To The Next Citation Point] to find an effective 4D form for ℰμν and hence for δπℰ. This is discussed below.

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