Dynamics of Primordial Black Hole Formation

TL;DR

This work addresses primordial black hole formation from horizon-size density fluctuations in a radiation-dominated early universe by solving 1D general-relativistic hydrodynamics with the Hernandez–Misner formulation. It analyzes three representative perturbation shapes to determine the formation threshold $\delta_{\rm c}$ and the initial PBH mass $M_{\rm bh}$, finding $\delta_{\rm c} \approx 0.7$ and confirming a near-critical mass scaling $M_{\rm bh} = K M_h (\delta - \delta_{\rm c})^{\gamma}$ with $\gamma \approx 0.36$; accretion after formation is typically negligible. The results imply a revised PBH mass function and demonstrate a universal scaling behavior in an expanding FRW background, enhancing predictions for PBH abundances and observational signatures. Methodologically, the Hernandez–Misner approach allows robust long-time evolution beyond horizon formation, enabling reliable assessments of accretion and late-time dynamics. Overall, the study refines threshold estimates and confirms critical phenomena-like scaling in primordial collapse, with implications for cosmological constraints on PBHs.

Abstract

We present a numerical investigation of the gravitational collapse of horizon-size density fluctuations to primordial black holes (PBHs) during the radiation-dominated phase of the Early Universe. The collapse dynamics of three different families of initial perturbation shapes, imposed at the time of horizon crossing, is computed. The perturbation threshold for black hole formation, needed for estimations of the cosmological PBH mass function, is found to be $δ_{\rm c} \approx 0.7$ rather than the generally employed $δ_{\rm c} \approx 1/3$, if $δ$ is defined as $ΔM/\mh$, the relative excess mass within the initial horizon volume. In order to study the accretion onto the newly formed black holes, we use a numerical scheme that allows us to follow the evolution for long times after formation of the event horizon. In general, small black holes (compared to the horizon mass at the onset of the collapse) give rise to a fluid bounce that effectively shuts off accretion onto the black hole, while large ones do not. In both cases, the growth of the black hole mass owing to accretion is insignificant. Furthermore, the scaling of black hole mass with distance from the formation threshold, known to occur in near-critical gravitational collapse, is demonstrated to apply to primordial black hole formation.

Figures (8)

• Figure 1: Shapes of the critical perturbations as imposed at the onset of the simulations: Gaussian (solid line, Eq. \ref{['gauss']}), Mexican Hat (dotted line, Eq. \ref{['mexhat']}), and polynomial (dashed line, Eq. \ref{['poly']}).
• Figure 2: Time evolution of a near-critical Mexican-Hat density perturbation with initial $\delta = 0.6780$. A black hole with mass $M_{\rm bh} = 0.37$ forms in the interior.
• Figure 3: Time evolution of a near-critical polynomial perturbation with initial $\delta = 0.7175$, forming a black hole with mass $M_{\rm bh} = 0.36$.
• Figure 4: Time evolution of a near-critical Gaussian-curve perturbation with initial $\delta = 0.7015$, forming a black hole with mass $M_{\rm bh} = 0.37$.
• Figure 5: Time evolution of an undercritical Gaussian-curve perturbation with initial $\delta = 0.7006$. No black hole is formed.