The Kramers-Moyal Equation of the Cosmological Comoving Curvature Perturbation

TL;DR

This paper derives a Kramers-Moyal equation for the probability distribution of the cosmological comoving curvature perturbation, providing a new framework to analyze infrared effects and late-time correlators in early universe cosmology.

Contribution

It introduces a novel derivation of the Kramers-Moyal equation for curvature perturbations using path-integral and random walk methods, extending the analysis beyond the Fokker-Planck approximation.

Findings

Derivation of a generalized Kramers-Moyal equation for curvature perturbations.

Provides an alternative approach to study infrared effects on late-time correlators.

Offers insights into the stochastic dynamics of super-horizon fluctuations.

Abstract

Fluctuations of the comoving curvature perturbation with wavelengths larger than the horizon length are governed by a Langevin equation whose stochastic noise arise from the quantum fluctuations that are assumed to become classical at horizon crossing. The infrared part of the curvature perturbation performs a random walk under the action of the stochastic noise and, at the same time, it suffers a classical force caused by its self-interaction. By a path-interal approach and, alternatively, by the standard procedure in random walk analysis of adiabatic elimination of fast variables, we derive the corresponding Kramers-Moyal equation which describes how the probability distribution of the comoving curvature perturbation at a given spatial point evolves in time and is a generalization of the Fokker-Planck equation. This approach offers an alternative way to study the late time behaviour of the correlators of the curvature perturbation from infrared effects.