

6.4 Curvature perturbations
and the Sachs-Wolfe effect
The curvature perturbation
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
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
in the background, we have
and if
in the background, we get
where
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
even for
adiabatic matter perturbations; for example, if
in the background, then
The KK effects on the brane contribute a non-adiabatic mode,
although
at low energies.
Although the density and curvature perturbations
can be found on super-Hubble scales, the Sachs-Wolfe effect
requires
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
where the radiation anisotropic stress on large scales is
neglected, as in general relativity, and
is the scalar
potential for
, equivalent to the covariant quantity
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
is the KK entropy perturbation (determined by
). The KK term
cannot be determined by the
4D brane equations, so that
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
that is similar to an extra isocurvature
contribution.
- The KK anisotropic stress
also contributes to the CMB anisotropies. In the
absence of anisotropic stresses, the curvature perturbation
would be sufficient to determine the metric
perturbation
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
, can affect the relation between
,
, and
, and hence can affect the
CMB anisotropies at large angles.
A simple phenomenological approximation to
on large scales is discussed in [19], and the Sachs-Wolfe effect is estimated
as
where
is the 4D Planck time, and
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 [298
, 320
, 290
, 299
, 300
] to find an
effective 4D form for
and hence for
. This is discussed below.

