An analytic approach to non-slow-roll inflation

TL;DR

This paper presents an analytic, model-independent method to study the effects of short non-slow-roll phases during inflation on primordial perturbations, revealing how these phases influence the spectrum's features and slopes.

Contribution

Develops a Taylor expansion-based analytic framework for single-field inflation with brief slow-roll violations, linking spectrum features to non-slow-roll phase history.

Findings

Spectrum exhibits a dip followed by rapid growth during non-slow-roll phases.

The dip position and maximum enhancement are related by a specific power law.

Multiple non-slow-roll phases can produce steeper spectral slopes, up to n-1=8.

Abstract

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