A new parameter in attractor single-field inflation

Abstract

We revisit the notion of slow-roll in the context of general single-field inflation. As a generalization of slow-roll dynamics, we consider an inflaton ϕ in an attractor phase where the time derivative of ϕ is determined by a function of ϕ, ϕ˙=ϕ˙(ϕ). In other words, we consider the case when the number of e-folds N counted backward in time from the end of inflation is solely a function of ϕ, N=N(ϕ). In this case, it is found that we need a new independent parameter to properly describe the dynamics of the inflaton field in general, in addition to the standard parameters conventionally denoted by ϵ, η, cs2 and s. Two illustrative examples are presented to discuss the non-slow-roll dynamics of the inflaton field consistent with observations.