The attractive behaviour of ultra-slow-roll inflation

TL;DR

This paper analyzes the stability of ultra-slow-roll inflation, showing it can be a stable attractor under certain potential conditions, especially near flat inflection points, with implications for inflationary dynamics.

Contribution

It provides a phase-space analysis demonstrating conditions under which ultra-slow-roll inflation is stable, extending understanding beyond the traditional view of it as a transient regime.

Findings

Ultra-slow-roll is stable for convex potentials with specific curvature conditions.

Near flat inflection points, ultra-slow-roll can produce many e-folds.

Dependence on initial conditions is retained in ultra-slow-roll.

Abstract

It is often claimed that the ultra-slow-roll regime of inflation, where the dynamics of the inflaton field are friction dominated, is a non-attractor and/or transient. In this work we carry out a phase-space analysis of ultra-slow roll in an arbitrary potential, $V(\phi)$. We show that while standard slow roll is always a dynamical attractor whenever it is a self-consistent approximation, ultra-slow roll is stable for an inflaton field rolling down a convex potential with $M_{\scriptscriptstyle{\mathrm{Pl}}} V''>|V'|$ (or for a field rolling up a concave potential with $M_{\scriptscriptstyle{\mathrm{Pl}}} V''<-|V'|$). In particular, when approaching a flat inflection point, ultra-slow roll is always stable and a large number of $e$-folds may be realised in this regime. However, in ultra-slow roll, $\dot{\phi}$ is not a unique function of $\phi$ as it is in slow roll and dependence on initial conditions is retained. We confirm our analytical results with numerical examples.