How deep is the dip and how tall are the wiggles in inflationary power spectra?

TL;DR

This paper analyzes scalar perturbations in single-field inflation models with a non-attractor phase, revealing how peaks and dips in the power spectrum relate to the inflaton's dynamics and transition duration, with implications for primordial black hole formation.

Contribution

It introduces a transfer-matrix formalism to map perturbations during the transition, providing new insights into the spectral features and their universal relationships in non-attractor inflation models.

Findings

Damped oscillations occur at scales smaller than the peak.

A dip in the power spectrum appears if the inflaton's velocity does not flip sign.

The dip amplitude scales as the inverse square-root of the peak amplitude.

Abstract

We study linear scalar perturbations in single-field models of inflation featuring a non-attractor phase. These models lead to a peak in the curvature power spectrum that may result in the formation of primordial black holes. We develop a transfer-matrix formalism, analogous to the S-matrix program in quantum-field theory, that maps perturbations throughout the transitory phase. At scales smaller than the peak, the power spectrum features damped oscillations, and the duration of the transition sets the scale at which power-law damping switches to exponential damping. At scales larger than the peak, we demonstrate that a dip appears in the power spectrum if and only if the inflaton's velocity does not flip sign. We show that the amplitude at the dip always scales as the inverse square-rooted amplitude of the peak, and comment on the physical consequences of this universal relationship. We also test the robustness of our results with a few toy models and interpret them with an intuitive mechanical analogy.