

6.2 Metric-based
perturbations
An alternative approach to brane-world cosmological perturbations
is an extension of the 4D metric-based gauge-invariant
theory [170, 241]. A review of this
approach is given in [40
, 269
]. In an arbitrary
gauge, and for a flat FRW background, the perturbed metric has the
form
where the background metric functions
are given by
Equations (181, 182). The scalars
represent scalar perturbations. The vectors
,
, and
are transverse, so
that they represent 3D vector perturbations, and the tensor
is transverse traceless, representing 3D tensor
perturbations.
In the Gaussian normal gauge, the brane
coordinate-position remains fixed under perturbation,
where
is the background metric,
Equation (180). In this gauge, we
have
In the 5D longitudinal gauge, one gets
In this gauge, and for an
background, the metric
perturbation quantities can all be expressed in terms of a “master
variable”
which obeys a wave equation [244, 246]. In the case of scalar
perturbations, we have for example
with similar expressions for the other quantities. All of the
metric perturbation quantities are determined once a solution is
found for the wave equation
The junction conditions (62) relate the off-brane
derivatives of metric perturbations to the matter
perturbations:
where
For scalar perturbations in the Gaussian normal gauge, this gives
where
is the scalar potential for the matter anisotropic
stress,
The perturbed KK energy-momentum tensor on the brane is given by
The evolution of the bulk metric perturbations is determined by the
perturbed 5D field equations in the vacuum bulk,
Then the matter perturbations on the brane enter via the perturbed
junction conditions (273).
For example, for scalar perturbations in Gaussian
normal gauge, we have
For tensor perturbations (in any gauge), the only nonzero
components of the perturbed Einstein tensor are
In the following, I will discuss various
perturbation problems, using either a
-covariant or a
metric-based approach.

