Large slow roll parameters in single field inflation

We initially consider two simple situations where inflationary slow roll parameters are large and modes no longer freeze out shortly after exiting the horizon, treating both cases analytically. By modes, we refer to the comoving curvature perturbation R. We then consider applications to transient phases where the slow roll parameters can become large, especially in the context of the common 'fast-roll' inflation frequently used as a mechanism to explain the anomalously low scalar power at low l in the CMB. These transient cases we treat numerically. We find when ε, the first slow roll parameter, and only ε is large, modes decay outside the horizon, and when δ, the second slow roll parameter, is large, modes grow outside the horizon. When multiple slow roll parameters are large the behavior in general is more complicated, but we nevertheless show in the 'fast-roll' inflation case, modes grow outside the horizon