Toward the primordial black hole formation threshold in a time-dependent equation-of-state background

TL;DR

This paper extends the calculation of the primordial black hole formation threshold to scenarios with a time-dependent equation-of-state, especially during the QCD phase transition, revealing potential enhancements in PBH production.

Contribution

It generalizes the critical density perturbation threshold calculation to a time-dependent EoS background, improving upon previous constant EoS assumptions.

Findings

The threshold $\delta_c$ varies significantly during the QCD phase transition.

Time-dependent EoS leads to an increased PBH formation probability.

Comparison shows differences from constant EoS models, highlighting the importance of dynamic backgrounds.

Abstract

The study of the primordial black hole (PBH) gravitational collapse process requires the determination of a critical energy density perturbation threshold $\delta_\mathrm{c}$, which depends on the equation of state of the universe at the time of PBH formation. Up to now, the majority of analytical and numerical techniques calculate $\delta_\mathrm{c}$ by assuming a constant equation-of-state (EoS) parameter $w$ at the time of PBH formation. In this work, after generalizing the constant $w$ prescription of [1] for the computation of $\delta_\mathrm{c}$ and commenting its limitations we give a first estimate for the PBH threshold in the case of a time-dependent $w$ background. In particular, we apply our formalism in the case of the QCD phase transition, where the EoS parameter varies significantly with time and one expects an enhanced PBH production due to the abrupt softening of $w$. At the end, we compare our results with analytic and numerical approaches for the determination of $\delta_\mathrm{c}$ assuming a constant EoS parameter.