Analytic approach to non-slow-roll inflation

Brief periods of non-slow-roll evolution during inflation can produce interesting observable consequences, as primordial black holes, or an inflationary gravitational wave spectrum enhanced at small scales. We develop a model independent, analytic approach for studying the predictions single-field scenarios which include short phases slow-roll violation. Our method is based on Taylor expanding equations cosmological fluctuations in quantity, parameterizes duration eras. The super-horizon perturbations described by few effective parameters, and characterized pronounced dip followed rapid growth its amplitude, typically found numerical analytical studies. position $k_{\rm dip}/k_*$ maximal enhancement $\Pi_{\rm max}$ towards scales are to be related law dip}/k_*\propto \Pi_{\rm max}^{-1/4}$, we determine proportionality constant. For single epoch violation confirm previous studies, finding that steepest slope well after has spectral index $n-1\,=\,4$. On other hand, with multiple violation, generally enhanced. example, when two epochs take place, separated phase quasi-de Sitter expansion, find reach value $n-1\,=\,8$. This phenomenon indicates keeps memory history occurred inflation.