ΔN and the stochastic conveyor belt of ultra slow-roll inflation

Abstract

We analyze 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 ${\ensuremath{\phi}}_{\mathrm{e}}$, where slow roll resumes or inflation ends, to $\ensuremath{\phi}\ensuremath{\rightarrow}+\ensuremath{\infty}$. We compute the probability distribution in the number of $e$-folds $\mathcal{N}$ required to reach ${\ensuremath{\phi}}_{\mathrm{e}}$, 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 ${\ensuremath{\phi}}_{\mathrm{e}}$ 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 ${\ensuremath{\phi}}_{\mathrm{e}}$. In that case, the probability distribution for $\mathcal{N}$, although normalizable, does not possess finite moments, leading to a divergent curvature perturbation.