Brane-World Cosmology: Perturbations

6 Brane-World Cosmology: Perturbations

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 [19 , 30 , 37 , 40 , 41 , 57 , 60 , 70 , 75 , 82 , 86 , 109 , 121 , 122 , 133 , 134 , 166 , 168 , 169 , 179 , 185 , 186 , 191 , 192 , 200 , 201 , 205 , 206 , 218 , 222 , 226 , 244 , 245 , 246 , 252 , 258 , 269 , 271 , 275 , 277 , 288 , 311 , 312 ] 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 [19 , 191 , 200 , 201 ], and approximation schemes have recently been developed [22 , 38 , 91 , 142 , 177 , 235 , 237 , 268 , 290 , 298 , 299 , 300 , 320 ] 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)

where $𝒢$ is the bulk retarded Green’s function evaluated on the brane, and $Ak,Bk$ are made up of brane metric and matter perturbations and their (brane) derivatives, and include high-energy corrections to the background dynamics. Solving for the bulk Green’s function, which then determines $𝒢$ , is the core of the 5D problem.

We can isolate the KK anisotropic stress $πℰ μν$ as the term that must be determined from 5D equations. Once $πℰ μν$ 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):

where the functional $Fk$ will be determined by the covariant brane perturbation quantities and their derivatives. It is known in the case of a Minkowski background [281 ], but not in the cosmological case.

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

In 4D general relativity, the right hand side is zero. In the brane-world, the first source term on the right is the KK term, and the second term is the high-energy modification. The other modification is a straightforward high-energy correction of the background quantities $H$ and $ρ$ via the modified Friedmann equations.

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.

6.1 1 + 3-covariant perturbation equations on the brane
6.2 Metric-based perturbations
6.3 Density perturbations on large scales
6.4 Curvature perturbations and the Sachs–Wolfe effect
6.5 Vector perturbations
6.6 Tensor perturbations