Primordial black hole formation in the radiative era: investigation of the critical nature of the collapse

Abstract: Following on after two previous papers discussing the formation of primordial black holes in the early universe, we present here results from an in-depth investigation of the extent to which primordial black hole formation in the radiative era can be considered as an example of the critical collapse phenomenon. We focus on initial supra-horizon-scale perturbations of a type which could have come from inflation, with only a growing component and no decaying component. In order to study perturbations with amplitudes extremely close to the supposed critical limit, we have modified our previous computer code with the introduction of an adaptive mesh refinement scheme. This has allowed us to follow black hole formation from perturbations whose amplitudes are up to eight orders of magnitude closer to the threshold than we could do before. We find that scaling-law behaviour continues down to the smallest black hole masses that we are able to follow and we see no evidence of shock production such as has been reported in some previous studies and which led there to a breaking of the scaling-law behaviour at small black-hole masses. We attribute this difference to the different initial conditions used. In addition to the scaling law, we also present other features of the results which are characteristic of critical collapse in this context.

• The paper demonstrates that PBH mass scales with the perturbation amplitude difference, confirming a gamma of approximately 0.357 over a broad mass range.
• It employs adaptive mesh refinement to simulate nearly-critical collapse, probing perturbations eight orders of magnitude closer to the critical threshold.
• The study reveals an intermediate, self-similar state with ultra-relativistic mass shedding, enhancing our understanding of early universe black hole formation.

Analytical Investigation of Primordial Black Hole Formation via Critical Collapse in the Early Universe

The study presented in the paper "Primordial black hole formation in the radiative era: investigation of the critical nature of the collapse" explores the intricate dynamics of primordial black hole (PBH) formation in the early universe. The research offers a comprehensive numerical analysis of PBH formation, using adaptive mesh refinement (AMR) techniques to explore nearly-critical collapse regimes with a focus on supra-horizon-scale perturbations originating from inflation. Here, I will summarize and analyze the paper’s major findings and implications.

The motivation for this work stems from the need to understand how cosmological perturbations in the early universe could lead to the formation of PBHs. These black holes are theorized to arise from significant overdensities collapsing under gravity, a hypothesis originally suggested by Zel'dovich and Novikov and later Hawking. The current study provides insights into how these black holes can be considered manifestations of the critical collapse phenomenon, a situation where the mass of the resultant black hole displays scaling-law behavior close to a critical perturbation amplitude, δ_c.

Significantly, this research uses an updated numerical framework incorporating an AMR scheme, enabling simulations to higher precision than previous codes could achieve. This enhancement allows the paper to probe perturbations with amplitudes eight orders of magnitude closer to the critical threshold than before, focusing on scenarios without a decaying mode, that is, with only the inflationary growing component present.

A major result of the study is the observed scaling-law behavior, represented by the relationship $M_{\text{BH}} \propto (\delta - \delta_c)^\gamma$ with $\gamma \approx 0.357$, congruent with theoretical predictions for a perfect fluid of radiative particles. This behavior remains consistent over a mass range exceeding a thousand-fold difference, with no evidence of a mass plateau as previously noted in research by Hawke and Stewart. This discrepancy is attributed to the differing initial conditions used across studies, particularly concerning the presence of shocks, which are absent in this paper’s setup.

The paper not only reaffirms the scaling law for PBH mass as a function of δ but also identifies features characteristic of critical collapse such as approximate self-similarity and an emergent scaling law related to the critical collapse time. For perturbations near δ_c, the study highlights the formation of an intermediate state where the collapsing region’s compactness remains around half that of a black hole. This intermediate state experiences mass shedding through ultra-relativistic winds, developing a semi-void that re-fills slowly in cosmological terms.

Theoretical and practical implications of this work are broad, especially considering the cosmological significance of PBHs as dark matter candidates or probes of early universe conditions. This paper extends our understanding of PBH formation, confirming that critical collapse is deeply rooted in the behavior of cosmological perturbations consistent with inflationary dynamics. Future developments may involve exploring quantum effects at smaller PBH masses and the potential incorporation of alternative inflationary models to examine universality across scaling behaviors.

In conclusion, this research presents a refined understanding of the critical collapse phenomenon specific to primordial black hole formation in the radiative era, with robust numerical results emphasizing the necessity of incorporating precise initial conditions derived from inflationary cosmology. This study contributes to unraveling the complexities of early universe phenomena and their potential long-term implications for cosmology and fundamental physics.