Stochastic semiclassical equations for weakly inhomogeneous cosmologies

TL;DR

This paper derives stochastic semiclassical Einstein equations incorporating quantum fluctuations for weakly inhomogeneous cosmologies, revealing noise and dissipation effects and their impact on metric perturbations.

Contribution

It introduces a stochastic extension to semiclassical Einstein equations using influence functionals, capturing quantum fluctuation effects in cosmological models.

Findings

Derived fluctuation-dissipation relation for metric perturbations.

Calculated correlation functions of stochastic metric fluctuations.

Identified noise and dissipation kernels in the effective action.

Abstract

Semiclassical Einstein-Langevin equations for arbitrary small metric perturbations conformally coupled to a massless quantum scalar field in a spatially flat cosmological background are derived. Use is made of the fact that for this problem the in-in or closed time path effective action is simply related to the Feynman and Vernon influence functional which describes the effect of the ``environment'', the quantum field which is coarse grained here, on the ``system'', the gravitational field which is the field of interest. This leads to identify the dissipation and noise kernels in the in-in effective action, and to derive a fluctuation-dissipation relation. A tensorial Gaussian stochastic source which couples to the Weyl tensor of the spacetime metric is seen to modify the usual semiclassical equations which can be viewed now as mean field equations. As a simple application we derive the correlation functions of the stochastic metric fluctuations produced in a flat spacetime with small metric perturbations due to the quantum fluctuations of the matter field coupled to these perturbations.