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.