Structure formation in the early universe is a key problem in modern cosmology. In this chapter we discuss stochastic gravity as an alternative framework for studying the generation of primordial inhomogeneities in inflationary models, which can easily incorporate effects that go beyond the linear perturbations of the inflaton field. We show that the correlation functions that follow from the Einstein–Langevin equation, which emerge in the framework of stochastic gravity, coincide with that obtained with the usual quantization procedures when both the metric perturbations and the inflaton fluctuations are linear. Stochastic gravity, however, can also deal very naturally with the fluctuations of the inflaton field beyond the linear approximation. Here, we illustrate the stochastic approach with one of the simplest chaotic inflationary models in which the background spacetime is a quasi de Sitter universe, and prove the equivalence of the stochastic and quantum correlations to the linear order.

As a second application of stochastic gravity, we discuss in this chapter the backreaction problem in cosmology when the gravitational field couples to a quantum conformal matter field, and derive the Einstein–Langevin equations describing the metric fluctuations on the cosmological background. Conformal matter may be a reasonable assumption, because matter fields in the standard model of particle physics are expected to become effectively conformally invariant in the very early universe. We consider a weakly perturbed spatially flat Friedman–Lemaitre–Robertson–Walker spacetime and derive the Einstein–Langevin equation for the metric perturbations off this spacetime, using the CTP functional formalism described in previous chapters. With this calculation we also obtain the probability for particle creation. The CTP effective action is also used to derive the renormalized expectation value of the quantum stress-energy tensor and the corresponding semiclassical Einstein equation.

Whereas semiclassical gravity is based on the semiclassical Einstein equation with sources given by the expectation value of the stress-energy tensor, stochastic gravity is based on the Einstein–Langevin equation, which in addition has sources due to the noise kernel. The noise kernel is a bitensor which describes the quantum stress-energy tensor fluctuations of the matter fields. In this chapter we describe the fundamentals of this theory using an axiomatic and a functional approach. In the axiomatic approach, the equation is introduced as an extension of semiclassical gravity motivated by the search for self-consistent equations describing the backreaction of the stress-energy fluctuations on the gravitational field. We then discuss the equivalence between the stochastic correlation functions for the metric perturbations and the quantum correlation functions in the 1/N expansion, and illustrate the equivalence with a simple model. Based on the stochastic formulation, a criterion for the validity of semiclassical gravity is proposed. Alternatively, stochastic gravity is formulated using the Feynman–Vernon influence functional based on the open quantum system paradigm, in which the system of interest (the gravitational field) interacts with an environment (the matter fields).