$\Delta\mathcal{N}$ and the stochastic conveyor belt of Ultra Slow-Roll

TL;DR

This paper studies fluctuations during Ultra Slow-Roll inflation using stochastic methods, revealing how initial conditions influence the generation of curvature perturbations and the transition to eternal inflation.

Contribution

It extends the stochastic inflation framework to include gravitational backreaction and analyzes the probability distribution of e-folds in a flat potential, highlighting conditions for finite or divergent curvature perturbations.

Findings

High initial velocity leads to finite curvature perturbation.

Low initial velocity can cause eternal inflation with divergent perturbations.

Probability distribution of e-folds can be normalizable but with infinite moments.

Abstract

We analyse field fluctuations during an Ultra Slow-Roll phase in the stochastic picture of inflation and the resulting non-Gaussian curvature perturbation, fully including the gravitational backreaction of the field's velocity. By working to leading order in a gradient expansion, we first demonstrate that consistency with the momentum constraint of General Relativity prevents the field velocity from having a stochastic source, reflecting the existence of a single scalar dynamical degree of freedom on long wavelengths. We then focus on a completely level potential surface, $V=V_0$, extending from a specified exit point $\phi_{\rm e}$, where slow roll resumes or inflation ends, to $\phi\rightarrow +\infty$. We compute the probability distribution in the number of e-folds $\mathcal{N}$ required to reach $\phi_{\rm e}$ which allows for the computation of the curvature perturbation. We find that, if the field's initial velocity is high enough, all points eventually exit through $\phi_{\rm e}$ and a finite curvature perturbation is generated. On the contrary, if the initial velocity is low, some points enter an eternally inflating regime despite the existence of $\phi_{\rm e}$. In that case the probability distribution for $\mathcal{N}$, although normalizable, does not possess finite moments, leading to a divergent curvature perturbation.