Validity of semiclassical gravity in the stochastic gravity approach

In semiclassical gravity the back-reaction of the classical gravitational field interacting with quantum matter fields is described by the semiclassical Einstein equations. A criterion for the validity of semiclassical gravity based on the stability of the solutions of the semiclassical Einstein equations with respect to quantum metric perturbations is discussed. The two-point quantum correlation functions for the metric perturbations can be described by the Einstein-Langevin equation obtained in the framework of stochastic gravity. These correlation functions agree, to leading order in the large $N$ limit, with the quantum correlation functions of the theory of gravity interacting with $N$ matter fields. The Einstein-Langevin equations exhibit runaway solutions and methods to deal with these solutions are discussed. The validity criterion is used to show that flat spacetime as a solution of semiclassical gravity is stable and, consequently, a description based on semiclassical gravity is a valid approximation in that case.