Stochastic parameters for scalar fields in de Sitter spacetime

The stochastic effective theory approach, often called stochastic inflation, is widely used in cosmology to describe scalar field dynamics during inflation. The existing formulations are, however, more qualitative than quantitative because the connection to the underlying quantum field theory (QFT) has not been properly established. A concrete sign of this is that the QFT parameters depend on the renormalisation scale, and therefore the relation between the QFT and stochastic theory must have explicit scale dependence that cancels it. In this paper we achieve that by determining the parameters of the second-order stochastic effective theory of light scalar fields in de Sitter to linear order in the self-coupling constant $\lambda$. This is done by computing equal-time two-point correlators to one-loop order both in QFT using dimensional regularisation and the $\overline{\rm MS}$ renormalisation scheme and the equal-time four-point correlator to leading order in both theories, and demanding that the results obtained in the two theories agree. With these parameters, the effective theory is valid when $m\lesssim H$ and $\lambda^2\ll m^4/H^4$, and therefore it is applicable in cases where neither perturbation theory nor any previously proposed stochastic effective theories are.