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

TL;DR

This work develops a perturbative quantum-field-theory framework to derive the time-dependent PDF $\rho(\varphi,N)$ for a light spectator field in de Sitter space, capturing non-Gaussian features induced by a scalar potential $\mathcal{V}(\varphi)$. By leveraging Schwinger–Keldysh diagrammatics, the authors relate $\rho(\varphi,N)$ to the full set of connected $n$-point functions and show that, to leading order, the PDF remains nearly Gaussian with gradient-driven corrections encoded by a differential operator $\mathcal{O}_{\varphi}$ acting on powers of $\mathcal{V}$; the resulting form can be resummed to express the PDF in terms of an effective potential $\mathcal{V}_{\rm eff}$ and, including loops, $\mathcal{V}_{\rm ren}$. A key finding is that the stochastic-Fokker–Planck equation emerges with corrections that depend on the e-folding time $\Delta N$, signaling that the standard equilibrium distribution $\rho \propto e^{- (8\pi^2/3H^4)\mathcal{V}}$ is not universally valid and requires higher-order resummations near equilibrium. The analysis also clarifies the infrared structure of $n$-point functions, showing how IR logarithms drive temporal evolution and how loop effects renormalize the potential without altering the qualitative PDF structure, paving the way for systematic beyond-Gaussian resummations in inflationary contexts.

Abstract

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 $ρ(\varphi , t)$ describing the statistics of $\varphi$ which, in the limit of equilibrium gives one back the solution $ρ(\varphi) \propto \exp \big[ - \frac{8 π^2}{3 H^4} {\mathcal V(\varphi)} \big]$. We study the derivation of $ρ(\varphi , t)$ 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 $ΔN$, 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.