Curvature perturbations and the Sachs

6.4 Curvature perturbations and the Sachs–Wolfe effect

The curvature perturbation $ℛ$ on uniform density surfaces is defined in Equation (267

). The associated gauge-invariant quantity

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

On large scales, the perturbed dark energy conservation equation is [191

]

which leads to

For adiabatic matter perturbations, by the perturbed matter energy conservation equation,

we find

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 [191

]: If $ρℰ ⁄= 0$ in the background, we have

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

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 (289

). In 4D general relativity, the right hand side of Equation (319

) is zero. The (non-integrated) Sachs–Wolfe formula has the same form as in general relativity:

The brane-world corrections to the general relativistic Sachs–Wolfe effect are then given by [191 ]

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 (321

) has been generalized to a 2-brane model, in which the radion makes a contribution to the Sachs–Wolfe effect [176

].)

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

A contribution from the KK entropy perturbation $Sℰ$ that is similar to an extra isocurvature contribution.
The KK anisotropic stress $δπ ℰ$ also contributes to the CMB anisotropies. In the absence of anisotropic stresses, the curvature perturbation $ζ tot$ would be sufficient to determine the metric perturbation $ℛ$ and hence the large-angle CMB anisotropies via Equations (318 , 319 , 320 ). However, bulk gravitons generate anisotropic stresses which, although they do not affect the large-scale curvature perturbation $ζtot$ , can affect the relation between $ζtot$ , $ℛ$ , and $ψ$ , and hence can affect the CMB anisotropies at large angles.

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

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 [177 ], using the low-energy 2-brane approximation [290 , 298 , 299 , 300 , 320 ] to find an effective 4D form for $ℰμν$ and hence for $δπℰ$ . This is discussed below.