

8.1 The low-energy
approximation
The basic idea of the low-energy approximation [298
, 320
, 290
, 299
, 300
] is to use a
gradient expansion to exploit the fact that, during most of the
history of the universe, the curvature scale on the observable
brane is much greater than the curvature scale of the bulk (
):
These conditions are equivalent to the low energy regime, since
and
:
Using Equation (368) to neglect
appropriate gradient terms in an expansion in
, the low-energy equations can be solved. However,
two boundary conditions are needed to determine all functions of
integration. This is achieved by introducing a second brane, as in
the RS 2-brane scenario. This brane is to be thought of either as a
regulator brane, whose backreaction on the observable brane is
neglected (which will only be true for a limited time), or as a
shadow brane with physical fields, which have a gravitational
effect on the observable brane.
The background is given by low-energy FRW branes
with tensions
, proper times
, scale factors
, energy densities
and pressures
, and dark radiation
densities
. The physical distance between the
branes is
, and
Then the background dynamics is given by
(see [28, 196] for the general
background, including the high-energy regime). The dark energy
obeys
, where
is a constant. From
now on, we drop the +-subscripts which refer to the physical,
observed quantities.
The perturbed metric on the observable (positive
tension) brane is described, in longitudinal gauge, by the metric
perturbations
and
, and the perturbed radion is
. The approximation for the KK (Weyl) energy-momentum
tensor on the observable brane is
and the field equations on the observable brane can be written in
scalar-tensor form as
where
The perturbation equations can then be derived as
generalizations of the standard equations. For example, the
equation is [175]
The trace part of the perturbed field equation shows that the
radion perturbation determines the crucial quantity,
:
where the last equality follows from Equation (319). The radion
perturbation itself satisfies the wave equation
A new set of variables
turns out be very
useful [176
, 177
]:
Equation (377) gives
The variable
determines the metric shear in the
bulk, whereas
give the brane displacements in
transverse traceless gauge. The latter variables have a simple
relation to the curvature perturbations on large scales [176, 177
] (restoring the
+-subscripts):
where
.

