Primordial black holes in linear and non-linear regimes

TL;DR

This paper investigates primordial black hole formation in the early universe, demonstrating that linear perturbations do not lead to PBH formation, while non-linear collapse with specific density and velocity conditions can produce PBHs when density perturbations exceed a threshold.

Contribution

The study develops a gauge-invariant perturbation framework and models non-linear PBH formation, establishing a threshold for density perturbations necessary for horizon formation.

Findings

PBHs do not form in the linear regime.

A threshold density perturbation of δ > 0.7 is required for PBH formation.

PBH apparent horizons form when δ > 3.

Abstract

Using the concept of apparent horizon for dynamical black holes, we revisit the formation of primordial black holes (PBH) in the early universe for both linear and non-linear regimes. First, we develop the perturbation theory for spherically symmetric spacetimes to study the formation of spherical PBHs in linear regime and we fix two gauges. We also introduce a well defined gauge invariant quantity for the expansion. Using this quantity, we argue that PBHs do not form in the linear regime. Finally, we study the non-linear regime. We adopt the spherical collapse picture by taking a closed FRW model in the radiation dominated era to investigate PBH formation. Taking the initial condition of the spherical collapse from the linear theory of perturbations, we allow for both density and velocity perturbations. Our model gives a constraint on the velocity perturbation. This model also predicts that the apparent horizon of PBHs forms when $\delta > 3$. Applying the sound horizon constraint, we have shown the threshold value of density perturbations at horizon re-entry must be larger than $\delta_{th} > 0.7$ to overcome the pressure gradients.