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 (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 MP1V '' > vertical bar V'vertical bar (or for a field rolling up a concave potential with MP1V '' < -vertical bar V'vertical bar). 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, 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.