### 8.1 The low-energy approximation

The basic idea of the low-energy approximation [290, 298, 299, 300, 320] 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 .