The attractive behaviour of ultra-slow-roll inflation

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, $V(\varphi )$. 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{P}\mathrm{l}$}} V’‘>|V’|$(orforafieldrollingupaconcavepotentialwith$M_{\scriptscriptstyle{$\mathrm{P}\mathrm{l}$}} V’‘<-|V’|$). 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, $\dot{\varphi }$ is not a unique function of $\varphi$ as it is in slow roll and dependence on initial conditions is retained. We confirm our analytical results with numerical examples.