Cosmological perturbations from stochastic gravity

In inflationary cosmological models driven by an inflaton field the origin of the primordial inhomogeneities which are responsible for large scale structure formation are the quantum fluctuations of the inflaton field. These are usually computed using the standard theory of cosmological perturbations, where both the gravitational and the inflaton fields are linearly perturbed and quantized. The correlation functions for the primordial metric fluctuations and their power spectrum are then computed. Here we introduce an alternative procedure for computing the metric correlations based on the Einstein-Langevin equation which emerges in the framework of stochastic semiclassical gravity. We show that the correlation functions for the metric perturbations that follow from the Einstein-Langevin formalism coincide with those obtained with the usual quantization procedures when the scalar field perturbations are linearized. This method is explicitly applied to a simple model of chaotic inflation consisting of a Robertson-Walker background, which undergoes a quasi-de-Sitter expansion, minimally coupled to a free massive quantum scalar field. The technique based on the Einstein-Langevin equation can, however, deal naturally with the perturbations of the scalar field even beyond the linear approximation, as is actually required in inflationary models which are not driven by an inflaton field such as Starobinsky's trace-anomaly driven inflation or when calculating corrections due to non-linear quantum effects in the usual inflaton driven models.Comment: 29 pages, REVTeX; minor changes, additional appendix with an alternative proof of the equivalence between stochastic and quantum correlation functions as well as an exact argument showing that the correlation function of curvature perturbations remains constant in time for superhorizon modes, which clarifies a recent claim in arXiv:0710.5342v