The theory of gauge-invariant perturbations in
brane-world cosmology has been extensively investigated and
developed [218, 191
, 19
, 222, 70, 226, 277, 258, 206, 122
, 192
, 121
, 244
, 246
, 133, 169, 187, 311, 179, 245, 312, 166, 188, 134, 82, 108, 168, 38, 86, 252, 288, 275, 205, 57, 60, 75, 30, 41, 271, 40
, 269
, 201
, 200
] and is
qualitatively well understood. The key remaining task is
integration of the coupled brane-bulk perturbation equations.
Special cases have been solved, where these equations effectively
decouple [191
, 19
, 201
, 200
], and approximation
schemes have recently been developed [298
, 320
, 290
, 299
, 300
, 177
, 268
, 37
, 142
, 91
, 235
, 237
, 22
] for the more
general cases where the coupled system must be solved. From the
brane viewpoint, the bulk effects, i.e., the high-energy
corrections and the KK modes, act as source terms for the brane
perturbation equations. At the same time, perturbations of matter
on the brane can generate KK modes (i.e., emit 5D gravitons into
the bulk) which propagate in the bulk and can subsequently interact
with the brane. This nonlocal interaction amongst the perturbations
is at the core of the complexity of the problem. It can be
elegantly expressed via integro-differential equations [244
, 246
], which take the
form (assuming no incoming 5D gravitational waves)
We can isolate the KK anisotropic stress as the term that must be determined from 5D
equations. Once
is determined in this way, the
perturbation equations on the brane form a closed system. The
solution will be of the form (expressed in Fourier modes):
The KK terms act as source terms modifying the
standard general relativity perturbation equations, together with
the high-energy corrections. For example, the linearization of the
shear propagation equation (125) yields
As in 4D general relativity, there are various different, but essentially equivalent, ways to formulate linear cosmological perturbation theory. First I describe the covariant brane-based approach.