6.4 Curvature perturbations and the Sachs–Wolfe effect

The curvature perturbation ℛ on uniform density surfaces is defined in Equation (273View Equation). The associated gauge-invariant quantity
--δρ---- ζ = ℛ + 3(ρ + p ) (314 )
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. (315 )
On large scales, the perturbed dark energy conservation equation is [271Jump To The Next Citation Point]
(δρ )⋅ + 4H δρ + 4ρ ℛ˙ = 0, (316 ) ℰ ℰ ℰ
which leads to
˙ζℰ = 0. (317 )
For adiabatic matter perturbations, by the perturbed matter energy conservation equation,
(δρ)⋅ + 3H (δρ + δp) + 3(ρ + p) ˙ℛ = 0, (318 )
we find
ζ˙= 0. (319 )
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 [271Jump To The Next Citation Point]: If ρℰ ⁄= 0 in the background, we have
[ ] ζ = ζ + ----------4ρℰ----------- (ζ − ζ ), (320 ) tot 3(ρ + p)(1 + ρ∕ λ) + 4ρℰ ℰ
and if ρℰ = 0 in the background, we get
ζtot = ζ + -------δρℰ-------- (321 ) 3(ρ + p)(1 + ρ∕λ) δC ℰ δρℰ = --4-, (322 ) 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 ˙ζtot = H ctot − 3- (ρ-+-p-)(1-+-ρ∕λ-). (323 )
The KK effects on the brane contribute a non-adiabatic mode, although ˙ζtot → 0 at low energies.

Although the density and curvature perturbations can be found on super-Hubble scales, the Sachs–Wolfe effect requires πℰ μν 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

( ) H ℛ˙ ζtot = ℛ − --- ---− ψ , (324 ) ˙H H ℛ + ψ = − κ2a2 δπ , (325 ) ℰ
where the radiation anisotropic stress on large scales is neglected, as in general relativity, and δπ ℰ is the scalar potential for π ℰμν, equivalent to the covariant quantity Π defined in Equation (295View Equation). In 4D general relativity, the right hand side of Equation (325View Equation) is zero. The (non-integrated) Sachs–Wolfe formula has the same form as in general relativity:
| δT-|| = (ζ + ψ − ℛ )| . (326 ) T now rad dec
The brane-world corrections to the general relativistic Sachs–Wolfe effect are then given by [271]
δT ( δT) 8 ( ρ ) 2κ2 ∫ ---= --- − -- -rad- S ℰ − κ2a2 δπℰ +---- daa7∕2δπ ℰ, (327 ) T T gr 3 ρcdm a5∕2
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 (327View Equation) has been generalized to a 2-brane model, in which the radion makes a contribution to the Sachs–Wolfe effect [244Jump To The Next Citation Point].)

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

A simple phenomenological approximation to δπ ℰ on large scales is discussed in [24], and the Sachs–Wolfe effect is estimated as

( ) ( )2 ∕3[ ] δT- ∼ δπℰ- teq- -ln(tin∕t4) , (328 ) 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 [245Jump To The Next Citation Point], using the low-energy 2-brane approximation [398Jump To The Next Citation Point, 428Jump To The Next Citation Point, 387Jump To The Next Citation Point, 399Jump To The Next Citation Point, 400Jump To The Next Citation Point] to find an effective 4D form for ℰ μν and hence for δπℰ. This is discussed below. In a single brane model in the AdS bulk, full numerical simulations were done to find the behaviour of δπℰ [69Jump To The Next Citation Point], as will be discussed in the next subsection.

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