Cosmological structure formation is a key problem in modern cosmology [190, 229] and inflation offers a natural solution to this problem. If an inflationary period is present, the initial seeds for the generation of the primordial inhomogeneities that lead to the large scale structure have their source in the quantum fluctuations of the inflaton field, the field which is generally responsible for driving inflation. Stochastic gravity provides a sound and natural formalism for the derivation of the cosmological perturbations generated during inflation.
In  it was shown that the correlation functions that follow from the Einstein–Langevin equation which emerges 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 both linearized. Stochastic gravity, however, can naturally deal with the fluctuations of the inflaton field even beyond the linear approximation.
Here we will illustrate the equivalence with the usual formalism, based on the quantization of the linear cosmological and inflaton perturbations, with one of the simplest chaotic inflationary models in which the background spacetime is a quasi-de Sitter universe [253, 254].
© Max Planck Society and the author(s)