Non-Gaussian statistics of de Sitter spectators: A perturbative derivation of stochastic dynamics

Scalar fields interacting with the primordial curvature perturbation during inflation may communicate their statistics to the latter. This situation motivates the study of how the probability density function (PDF) of a light spectator field $\varphi$ in a pure de Sitter space-time, becomes non-Gaussian under the influence of a scalar potential ${\mathcal V(\varphi)}$. One approach to this problem is offered by the stochastic formalism introduced by Starobinsky and Yokoyama. It results in a Fokker-Planck equation for the time-dependent PDF $\rho (\varphi , t)$ describing the statistics of $\varphi$ which, in the limit of equilibrium gives one back the solution $\rho (\varphi) \propto \exp \big[ - \frac{8 \pi^2}{3 H^4} {\mathcal V(\varphi)} \big]$. We study the derivation of $\rho (\varphi , t)$ using quantum field theory tools. Our approach yields an almost Gaussian distribution function, distorted by minor corrections comprised of terms proportional to powers of $\mathcal O_\varphi {\mathcal V(\varphi)}$, where $\mathcal O_\varphi$ stands for a derivative operator acting on ${\mathcal V(\varphi)}$ proportional to $\Delta N$, the number of $e$-folds succeeding the Hubble-horizon crossing of $\varphi$'s wavelengths. This general form is obtained perturbatively and remains valid even with loop corrections. Our solution satisfies a Fokker-Planck equation that receives corrections with respect to the one found within the stochastic approach, allowing us to comment on the validity of the standard equilibrium solution for generic potentials. We posit that higher order corrections to the Fokker-Planck equation may become important towards the equilibrium.