The background field approximation in (quantum) cosmology

Abstract

We analyse the Hamilton-Jacobi action of gravity and matter in the limit where gravity is treated at the background field approximation. The motivation is to clarify when the Wheeler-DeWitt equation leads to the Schrödinger equation in a background geometry. In particular, we investigate the problem of the choice of the background and that of the corrections to the Schrödinger equation associated with this choice. To this end, we first work classically. We determine when the total action, a solution of the constraint equations of general relativity, leads to the matter action in a given background. This is achieved by comparing neighbouring solutions differing in their matter-energy content. To first order in the change of the 3-geometries, the total action splits into an irrelevant background contribution and the usual matter action evaluated in the background. Higher-order terms are governed by the choice of the background geometry and by its `susceptibility'. These properties apply to quantum cosmology when it is described by WKB wavefunctions. We then show that the corrections to these WKB waves and to the background field approximation appear together and remain small when the susceptibility of the background is low.