Stochastic semiclassical equations for weakly inhomogeneous cosmologies.

Semiclassical Einstein-Langevin equations for arbitrary small metric perturbations conformally coupled to a massless quantum scalar eld in a spatially at cosmological background are derived. Use is made of the fact that for this problem the in-in or closed time path e ective action is simply related to the Feynman and Vernon in uence functional which describes the e ect of the \environment", the quantum eld which is coarse grained here, on the \system", the gravitational eld which is the eld of interest. This leads to identify the dissipation and noise kernels in the in-in e ective action, and to derive a uctuation-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 eld equations. As a simple application we derive the correlation functions of the stochastic metric uctuations produced in a at spacetime with small metric perturbations due to the quantum uctuations of the matter eld coupled to these perturbations. 04.62.+v, 98.80.Cq, 05.40.+j Typeset using REVTEX Also at Institut de F sica d'Altes Energies (IFAE)