Non-linear statistics of primordial black holes from gaussian curvature perturbations

TL;DR

This paper develops a non-linear statistical framework for primordial black hole formation from gaussian curvature perturbations, accounting for various profile shapes and sizes, and compares predictions based on different statistical approaches.

Contribution

It introduces a novel non-linear statistical method using the compaction function to predict PBH abundance from gaussian curvature perturbations, considering diverse profile shapes.

Findings

For peaked power spectra, all methods predict narrow PBH mass distributions.

Linear over-density statistics overestimate PBH abundance for broad spectra.

Curvature-based approach underestimates PBH abundance for broad spectra.

Abstract

We develop the non-linear statistics of primordial black holes generated by a gaussian spectrum of primordial curvature perturbations. This is done by employing the compaction function as the main statistical variable under the constraints that: a) the over-density has a high peak at a point $\vec{x}_0$, b) the compaction function has a maximum at a smoothing scale $R$, and finally, c) the compaction function amplitude at its maximum is higher than the threshold necessary to trigger a gravitational collapse into a black hole of the initial over-density. Our calculation allows for the fact that the patches which are destined to form PBHs may have a variety of profile shapes and sizes. The predicted PBH abundances depend on the power spectrum of primordial fluctuations. For a very peaked power spectrum, our non-linear statistics, the one based on the linear over-density and the one based on the use of curvature perturbations, all predict a narrow distribution of PBH masses and comparable abundance. For broader power spectra the linear over-density statistics over-estimate the abundance of primordial black holes while the curvature-based approach under-estimates it. Additionally, for very large smoothing scales, the abundance is no longer dominated by the contribution of a mean over-density but rather by the whole statistical realisations of it.