# A universal threshold for primordial black hole formation

## TL;DR

This paper demonstrates that the threshold for primordial black hole formation is approximately universal when expressed via the compaction function, with a specific critical value supported by numerical simulations, and provides an analytic formula for this threshold.

## Contribution

It introduces a universal threshold for primordial black hole formation based on the compaction function and derives an analytic expression for it, supported by numerical evidence.

## Key findings

- Threshold value for black hole formation is approximately 0.40.
- The threshold depends mainly on the curvature of the compaction function at its maximum.
- Numerical simulations support the universality of the threshold.

## Abstract

In this letter, we argue and show numerically that the threshold to form primordial black holes from an initial spherically symmetric perturbation is, to an excellent approximation, universal, whenever given in terms of the compaction function averaged over a sphere of radius $r_m$, where $r_m$ is the scale on which the compaction function is maximum. This can be understood as the requirement that, for a black hole to form, each shell of the averaged compaction function should have an amplitude exceeding the so-called Harada-Yoo-Kohri limit. For a radiation dominated universe we argued, supported by the numerical simulations, that this limit is $\delta_c = 0.40$, which is slightly below the one quoted in the literature. Additionally, we show that the profile dependence of the threshold for the compaction function is only sensitive to its curvature at the maximum. We use these results to provide an analytic formula for the threshold amplitude of the compaction function at its maximum in terms of the normalised compaction function curvature at $r_m$.