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 [303Jump To The Next Citation Point, 271Jump To The Next Citation Point, 24Jump To The Next Citation Point, 308, 99, 312, 369, 346, 286, 177Jump To The Next Citation Point, 272Jump To The Next Citation Point, 176Jump To The Next Citation Point, 331Jump To The Next Citation Point, 333Jump To The Next Citation Point, 190, 235, 265, 416, 258, 332, 417, 231Jump To The Next Citation Point, 266, 191, 123, 158, 234, 50, 127, 340, 385, 368Jump To The Next Citation Point, 285, 85, 90, 116, 41, 54, 364, 53Jump To The Next Citation Point, 362Jump To The Next Citation Point, 282Jump To The Next Citation Point, 281Jump To The Next Citation Point] 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 [271Jump To The Next Citation Point, 24Jump To The Next Citation Point, 282Jump To The Next Citation Point, 281Jump To The Next Citation Point], and approximation schemes have been developed [398Jump To The Next Citation Point, 428Jump To The Next Citation Point, 387Jump To The Next Citation Point, 399Jump To The Next Citation Point, 400Jump To The Next Citation Point, 245Jump To The Next Citation Point, 361Jump To The Next Citation Point, 51Jump To The Next Citation Point, 201Jump To The Next Citation Point, 136Jump To The Next Citation Point, 321Jump To The Next Citation Point, 323Jump To The Next Citation Point, 27Jump To The Next Citation Point] 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 [331Jump To The Next Citation Point, 333Jump To The Next Citation Point], which take the form (assuming no incoming 5D gravitational waves)

∫ ′ ′ ′ Ak(t) = dt𝒢 (t,t )Bk(t), (248 )
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):

∫ ℰ ′ ′ ′ πk(t) ∝ dt𝒢 (t,t )Fk(t), (249 )
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 [374], 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 (131View Equation) yields

κ2 κ2 κ2 ρ σ˙μν + 2H σ μν + E μν −-π μν − ⃗∇⟨μA ν⟩ = --πℰμν − ---(1 + 3w )-πμν. (250 ) 2 2 4 λ
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 we 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 Full numerical solutions
 6.6 Vector perturbations
 6.7 Tensor perturbations

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