Classical approximation to quantum cosmological correlations

We investigate up to which order quantum effects can be neglected in calculating cosmological correlation functions after horizon exit. We study, as a toy model, phi(3) theory on a de Sitter background for a massless minimally coupled scalar field phi. We find that for tree level and one loop contributions in the quantum theory, a good classical approximation can be constructed, but for higher loop corrections this is in general not expected to be possible. The reason is that loop corrections get non-negligible contributions from loop momenta with magnitude up to the Hubble scale H, at which scale classical physics is not expected to be a good approximation to the quantum theory. An explicit calculation of the one loop correction to the two point function supports the argument that contributions from loop momenta of scale H are not negligible. Generalization of the arguments for the toy model to derivative interactions and the curvature perturbation leads to the conclusion that the leading orders of non-Gaussian effects generated after horizon exit can be approximated quite well by classical methods. Furthermore we compare with a theorem of Weinberg. We find that growing loop corrections after horizon exit are not excluded, even in single field inflation.