# Primordial black hole formation and abundance: contribution from the non-linear relation between the density and curvature perturbation

## TL;DR

This paper investigates how the non-linear relationship between curvature perturbations and density contrast affects primordial black hole formation, revealing that non-linear effects significantly influence PBH abundance and related cosmological constraints.

## Contribution

It introduces a formalism incorporating non-linear effects into PBH formation calculations, showing that these effects alter the predicted abundance and weaken constraints on the small-scale power spectrum.

## Key findings

- PBH abundance is highly sensitive to non-linear effects.
- The power spectrum must be roughly twice as large to produce the same PBH abundance as in linear models.
- Constraints on small-scale power spectrum are weaker when non-linearities are considered.

## Abstract

The formation and abundance of primordial black holes (PBHs) arising from the curvature perturbation $\zeta$ is studied. The non-linear relation between $\zeta$ and the density contrast $\delta$ means that, even when $\zeta$ has an exactly Gaussian distribution, significant non-Gaussianities affecting PBH formation must be considered. Numerical simulations are used to investigate the critical value and the mass of PBHs which form, and peaks theory is used to calculate the mass fraction of the universe collapsing to form PBHs at the time of formation. A formalism to calculate the total present day PBH abundance and mass function is also derived. It is found that the abundance of PBHs is very sensitive to the non-linear effects, and that the power spectrum $\mathcal{P}_\zeta$ must be a factor of $\sim2$ larger to produce the same number of PBHs as the linear model (where the exact value depends on the critical value for a region to collapse and form a PBH). This also means that the derived constraints on the small-scale power spectrum from constraints on the abundance of PBHs are weaker by the same factor.