Analytical thresholds for black hole formation in general cosmological backgrounds

TL;DR

This paper derives analytical thresholds for black hole formation in cosmological backgrounds with different equations of state, extending previous results for radiation to a broader range of fluid types.

Contribution

It generalizes the critical threshold for black hole formation from radiation to all fluids with $w o 1$, providing accurate analytic approximations for these thresholds.

Findings

Threshold depends mainly on curvature for $w o 1/3$

Full shape of the compaction function is needed for $w < 1/3$

Analytic approximations are accurate for $w$ in [1/3, 1]

Abstract

We consider black holes which form from an initially spherically symmetric super-Hubble perturbation of a cosmological background filled by a perfect fluid $p=w \rho$ with $w\in (0,1]$. Previous work has shown that when $w = 1/3$ (radiation), there is a critical threshold for black hole formation ($\delta_c$), which, to a very good approximation, only depends upon the curvature of the compaction function around its peak value. We find that this generalizes to all $w\gtrsim 1/3$; for smaller $w$s the knowledge of the full shape of the compaction function is necessary. We provide analytic approximations for $\delta_c$ which are accurate for $w\in [1/3,1]$.