A new parameter in attractor single-field inflation

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, c_s² and s. Two illustrative examples are presented to discuss the non-slow-roll dynamics of the inflaton field consistent with observations.