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We investigate up to which order quantum effects can be neglected incalculating cosmological correlation functions after horizon exit. As a toymodel, we study $\phi^3$ theory on a de Sitter background for a masslessminimally coupled scalar field $\phi$. We find that for tree level and one loopcontributions in the quantum theory, a good classical approximation can beconstructed, but for higher loop corrections this is in general not expected tobe possible. The reason is that loop corrections get non-negligiblecontributions from loop momenta with magnitude up to the Hubble scale H, atwhich scale classical physics is not expected to be a good approximation to thequantum theory. An explicit calculation of the one loop correction to the twopoint function, supports the argument that contributions from loop momenta ofscale $H$ are not negligible. Generalization of the arguments for the toy modelto derivative interactions and the curvature perturbation leads to theconclusion that the leading orders of non-Gaussian effects generated afterhorizon exit, can be approximated quite well by classical methods. Furthermorewe compare with a theorem by Weinberg. We find that growing loop correctionsafter horizon exit are not excluded, even in single field inflation.

Classical approximation to quantum cosmological correlations