Δ N and the stochastic conveyor belt of ultra slow-roll inflation

• Published in: Physical review. D 2021-10, Vol.104 (8), p.1, Article 083505
• Main Authors: Prokopec, Tomislav, Rigopoulos, Gerasimos
• Format: Article
• Language: English
• Subjects: Belt conveyors; Curvature; Perturbation; Probability distribution; Probability theory; Relativity; Velocity
• ISSN: 2470-0010; 2470-0029
• DOI: 10.1103/PhysRevD.104.083505

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 = V0, extending from a specified exit point ϕe, where slow roll resumes or inflation ends, to ϕ → +∞. We compute the probability distribution in the number of e -folds N required to reach ϕe, 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 ϕe 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 ϕe. In that case, the probability distribution for N, although normalizable, does not possess finite moments, leading to a divergent curvature perturbation.