# Are primordial black holes produced by entropy perturbations in single field inflationary models?

## TL;DR

This paper clarifies that in single field inflation, the super-horizon growth of curvature perturbations leading to primordial black holes is driven by background evolution effects, not entropy perturbations, challenging previous assumptions.

## Contribution

It derives a general relation linking curvature and entropy perturbations valid on all scales, and explains super-horizon growth of curvature perturbations through background evolution effects.

## Key findings

- Super-horizon growth of curvature perturbations is due to background evolution, not entropy perturbations.
- The derived relation is valid on any scale, including sub-horizon scales.
- Entropy perturbations decrease while curvature perturbations grow during certain inflationary phases.

## Abstract

We show that in single field inflationary models the super-horizon evolution of curvature perturbations on comoving slices $\mathcal{R}$, which can cause the production of primordial black holes (PBH), is not due to entropy perturbations, but to the background evolution effect on the conversion between entropy and curvature perturbations. We derive a general relation between the time derivative of comoving curvature perturbations and entropy perturbations, in terms of a conversion factor depending on the background evolution. Contrary to previous results derived in the uniform density gauge assuming the gradient term can be neglected on super-horizon scales, the relation is valid on any scale for any minimally coupled single scalar field model, also on sub-horizon scales where gradient terms are large. We apply it to the case of quasi-inflection inflation, showing that while entropy perturbations are decreasing, $\mathcal{R}$ can grow on super-horizon scales, due to a large increase of the conversion factor. This happens in the time interval during which a sufficiently fast decrease of the equation of state $w$ transforms into a growing mode that in slow-roll models would be a decaying mode. The same mechanism also explains the super-horizon evolution of $\mathcal{R}$ in globally adiabatic systems, for which entropy perturbations vanish on any scale, such as ultra slow-roll inflation and its generalizations.