Integro-differential equation for brane-world cosmological perturbations -art. no. 064006

Cosmological perturbations in the brane-world cosmology with a positive ten sion brane in the AdS background bulk geometry are analyzed by using the do ubly gauge-invariant formalism. We derive four independent equations for sc alar perturbations in the plane symmetric (K = 0) background. Three of thes e 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 reta rded Green's function of the bulk gravitational waves. We compare these fou r 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 nume rical calculations.