# Dissecting the growth of the power spectrum for primordial black holes

## TL;DR

This paper analyzes the maximum possible growth rate of the primordial power spectrum during inflation, showing a steeper growth than previously thought and discussing implications for primordial black hole formation.

## Contribution

It provides a simple explanation for the $k^4$ growth and demonstrates that a $k^5 ( ext{log} k)^2$ growth is possible, with implications for primordial black holes.

## Key findings

- Steepest growth rate is $k^4$, with a possible $k^5 ( ext{log} k)^2$ growth.
- Power spectrum remains large for modes exiting during ~2 e-folds after steep growth.
- Strong growth likely cannot be achieved through isocurvature mode conversion.

## Abstract

We consider the steepest rate at which the power spectrum from single field inflation can grow, with the aim of providing a simple explanation for the $k^4$ growth found recently. With this explanation in hand we show that a slightly steeper $k^5 (\log k )^2$ growth is in fact possible. Moreover, we argue that the power spectrum after a steep growth cannot immediately decay, but must remain large for the $k$ modes which exit during a $\sim2$ e-fold period. We also briefly consider how a strong growth can affect the spectral index of longer wavelengths preceding the growth, and show that even the conversion of isocurvature modes likely cannot lead to a stronger growth. These results have implications for the formation of primordial black holes, and other phenomena which require a large amplitude of power spectrum at short scales.