The background dynamics of brane-world cosmology are simple because the FRW symmetries simplify the bulk and rule out nonlocal effects. But perturbations on the brane immediately release the nonlocal KK modes. Then the 5D bulk perturbation equations must be solved in order to solve for perturbations on the brane. These 5D equations are partial differential equations for the 3D Fourier modes, with both initial and boundary conditions needed.
The theory of gauge-invariant perturbations in brane-world cosmology has been extensively investigated and
developed [303, 271
, 24
, 308, 99, 312, 369, 346, 286, 177
, 272
, 176
, 331
, 333
, 190, 235, 265, 416, 258, 332,
417, 231
, 266, 191, 123, 158, 234, 50, 127, 340, 385, 368
, 285, 85, 90, 116, 41, 54, 364, 53
, 362
, 282
, 281
]
and is qualitatively well understood. The key task is integration of the coupled brane-bulk perturbation
equations. Special cases have been solved, where these equations effectively decouple [271
, 24
, 282
, 281
], and
approximation schemes have been developed [398
, 428
, 387
, 399
, 400
, 245
, 361
, 51
, 201
, 136
, 321
, 323
, 27
]
for the more general cases where the coupled system must be solved. Below we will also present the results
of full numerical integration of the 5D perturbation equations in the RS case.
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 [331, 333
], 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 (131) yields
As in 4D general relativity, there are various different, but essentially equivalent, ways to formulate linear cosmological perturbation theory. First we describe the covariant brane-based approach.
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