Integro-differential equation for brane-world cosmological perturbations

TL;DR

This paper derives and analyzes a set of gauge-invariant equations, including an integro-differential equation, for scalar cosmological perturbations in brane-world models with a positive tension brane in AdS space, facilitating comparison with standard cosmology.

Contribution

It introduces a novel formalism for scalar perturbations in brane-world cosmology, deriving a new integro-differential equation involving the bulk gravitational Green's function.

Findings

Derived four independent equations for scalar perturbations.

Compared brane-world equations with standard cosmology.

Provided equations useful for numerical simulations.

Abstract

Cosmological perturbations in the brane-world cosmology with a positive tension brane in the AdS background bulk geometry is analyzed by using the doubly gauge-invariant formalism. We derive four independent equations for scalar perturbations in the plane symmetric (K=0) background. Three of these equations are differential equations written in terms of gauge invariant variables on the brane only, and another of them is an integro-differential equation whose kernel is constructed formally from the Z_2-symmetric retarded Green's function of the bulk gravitational waves. We compare these four equations with the corresponding equations in the standard cosmology. As a by-product, we also obtain a set of equations which may be useful in numerical calculations.