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 $\phi$ in an attractor phase where the time derivative of $\phi$ is determined by a function of $\phi$, $\dot\phi=\dot\phi(\phi)$. 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 $\phi$, $N=N(\phi)$. 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 $\epsilon$, $\eta$, $c_s^2$ and $s$. Two illustrative examples are presented to discuss the non-slow-roll dynamics of the inflaton field consistent with observations.