Analytic approach to non-slow-roll inflation

In order to produce primordial black holes, or to enhance the spectrum of primordial gravitational waves at interferometer scales, cosmological inflation should include brief periods of nonattractor evolution, with departures from slow-roll conditions. To better understand these systems we develop a model independent, analytic approach to study cosmological fluctuations in single field inflationary models with short periods of slow-roll violation. We Taylor expand the equations for fluctuations in a quantity parametrizing the duration of slow-roll violation, and find solutions at leading order in this quantity. The resulting spectrum of perturbations is described by few parameters, and presents a pronounced dip followed by a rapid growth in amplitude, as typically found in numerical as well as analytical studies of similar scenarios. We determine new analytic relations between the position and depth of the dip, and the maximal enhancement of the spectrum toward small scales. For a single epoch of slow-roll violation we confirm previous studies finding that the steepest slope of the spectrum has spectral index nₒ-1=4. On the other hand, when considering multiple phases of slow-roll violation, the slope of the spectrum is generally enhanced. When two epochs of slow-roll violation occur, the spectral index can reach the value nₒ-1=8, indicating that the slope of the spectrum has memory of the history of non-slow-roll phases during inflation.