Stochastic Inflation at NNLO

TL;DR

This paper advances the theoretical understanding of Stochastic Inflation by calculating NNLO corrections using SdSET, revealing the role of non-Gaussian fluctuations and confirming the formalism's consistency beyond tree level.

Contribution

It provides the first NNLO correction to Stochastic Inflation, derived from two-loop anomalous dimensions, and demonstrates the systematic incorporation of higher order effects within SdSET.

Findings

NNLO corrections significantly impact the equilibrium distribution.

Matching UV theory onto SdSET confirms the persistence of Wilson-coefficient corrections.

Logarithmic divergences affect the factorization in SdSET, leading to renormalization group flow.

Abstract

Stochastic Inflation is an important framework for understanding the physics of de Sitter space and the phenomenology of inflation. In the leading approximation, this approach results in a Fokker-Planck equation that calculates the probability distribution for a light scalar field as a function of time. Despite its successes, the quantum field theoretic origins and the range of validity for this equation have remained elusive, and establishing a formalism to systematically incorporate higher order effects has been an area of active study. In this paper, we calculate the next-to-next-to-leading order (NNLO) corrections to Stochastic Inflation using Soft de Sitter Effective Theory (SdSET). In this effective description, Stochastic Inflation manifests as the renormalization group evolution of composite operators. The leading impact of non-Gaussian quantum fluctuations appears at NNLO, which is presented here for the first time; we derive the coefficient of this term from a two-loop anomalous dimension calculation within SdSET. We solve the resulting equation to determine the NNLO equilibrium distribution and the low-lying relaxation eigenvalues. In the process, we must match the UV theory onto SdSET at one-loop order, which provides a non-trivial confirmation that the separation into Wilson-coefficient corrections and contributions to initial conditions persists beyond tree level. Furthermore, these results illustrate how the naive factorization of time and momentum integrals in SdSET no longer holds in the presence of logarithmic divergences. It is these effects that ultimately give rise to the renormalization group flow that yields Stochastic Inflation.