Integro-differential equation for brane-world cosmological perturbations

Cosmological perturbations in the brane-world cosmology with a positive tension brane in the AdS background bulk geometry are 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.