Stochastic inflation at all order in slow-roll parameters: foundations

TL;DR

This paper develops a comprehensive stochastic inflation formalism incorporating all orders of slow-roll parameters, compares it with quantum field theory results, and discusses limitations of common approximations in capturing quantum effects.

Contribution

It introduces a formalism for stochastic inflation at all orders in slow-roll parameters, including momentum and Hamiltonian constraints, and analyzes the validity of the Starobinski approximation.

Findings

Stochastic and QFT two-point functions agree under the Starobinski approximation.

Starobinski approximation does not capture loop effects of quantum scalar-gravity system.

Correlation functions are limited to linear regimes; non-perturbative statistics are not accessible.

Abstract

In this paper we develop the formalism for the stochastic approach to inflation at all order in slow-roll parameters. This is done by including the momentum and Hamiltonian constraints into the stochastic equations. We then specialise to the widely used Starobinski approximation where interactions between IR and UV modes are neglected. We show that, whenever this approximation holds, no significant deviations are observed when comparing the two-point correlation functions (power spectrum) calculated with stochastic methods, to the ones calculated with the QFT approach to linear theory. As a byproduct, we argue that: a) the approaches based on the Starobinski approximation, generically, do not capture any loop effects of the quantum scalar-gravity system; b) correlations functions can only be calculated in the linear theory regimes, thus, no non-perturbative statistics can be extracted within this approximation, as commonly claimed.