Failure of the stochastic approach to inflation beyond slow-roll

TL;DR

This paper clarifies the limitations of the stochastic inflation approach beyond slow-roll, showing it is reliable only under specific conditions and demonstrating its validity in ultra-slow-roll scenarios.

Contribution

The paper identifies the precise conditions under which the stochastic approach to inflation is valid beyond slow-roll and demonstrates its consistency in ultra-slow-roll regimes.

Findings

Stochastic approach is reliable only at zeroth order in slow-roll parameter $\epsilon_1$ when $\epsilon_2^2 \\ll 6/\\epsilon_1$.

Standard stochastic $\\Delta N$ formalism fails beyond slow-roll but works in ultra-slow-roll.

In ultra-slow-roll, stochastic approach reproduces the linear theory power spectrum.

Abstract

After giving a pedagogical review we clarify that the stochastic approach to inflation is generically reliable only at zeroth order in the (geometrical) slow-roll parameter $\epsilon_1$ if and only if $\epsilon_2^2\ll 6/\epsilon_1$, with the notable exception of slow-roll. This is due to the failure of the stochastic $\Delta N$ formalism in its standard formulation. However, by keeping the formalism in its regime of validity, we showed that, in ultra-slow-roll, the stochastic approach to inflation reproduces the power spectrum calculated from the linear theory approach.