Cosmological perturbation theory revisited

Increasingly accurate observations are driving theoretical cosmology towards the use of more sophisticated descriptions of matter and the study of nonlinear perturbations of Friedmann-Lemaitre cosmologies, whose governing equations are notoriously complicated. Our goal in this paper is to formulate the governing equations for linear perturbation theory in a particularly simple and concise form in order to facilitate the extension to nonlinear perturbations. Our approach has several novel features. We show that the use of so-called intrinsic gauge invariants has two advantages. It naturally leads to (i) a physically motivated choice of a gauge invariant associated with the matter density, and (ii) two distinct and complementary ways of formulating the evolution equations for scalar perturbations, associated with the work of Bardeen and of Kodama and Sasaki. In the first case, the perturbed Einstein tensor gives rise to a second-order (in time) linear differential operator, and in the second case to a pair of coupled first-order (in time) linear differential operators. These operators are of fundamental importance in cosmological perturbation theory, since they provide the leading order terms in the governing equations for nonlinear perturbations.