Primordial black hole formation in the early universe: critical behaviour and self-similarity (1201.2379v3)

Abstract: Following on after three previous papers discussing the formation of primordial black holes during the radiative era of the early universe, we present here a further investigation of the critical nature of the process involved, aimed at making contact with some of the basic underlying ideas from the literature on critical collapse. We focus on the intermediate state, which we have found appearing in cases with perturbations close to the critical limit, and examine the connection between this and the similarity solutions which play a fundamental role in the standard picture of critical collapse. We have derived a set of self-similar equations for the null-slicing form of the metric which we are using for our numerical calculations, and have then compared the results obtained by integrating these with the ones coming from our simulations for collapse of cosmological perturbations within an expanding universe. We find that the similarity solution is asymptotically approached in a region which grows to cover both the contracting matter and part of the semi-void which forms outside it. Our main interest is in the situation relevant for primordial black hole formation in the radiative era of the early universe, where the relation between the pressure $p$ and the energy density $e$ can be reasonably approximated by an expression of the form $p = we$ with $w=1/3$. However, we have also looked at other values of $w$, both because these have been considered in previous literature and also because they can be helpful for giving further insight into situations relevant for primordial black hole formation. As in our previous work, we have started our simulations with initial supra-horizon scale perturbations of a type which could have come from inflation.

• The paper demonstrates that near-critical perturbations trigger black hole formation, following a scaling law governed by the critical exponent.
• It employs numerical simulations and analytic techniques within general relativistic hydrodynamics to reveal the attractor nature of self-similar solutions.
• Results show that variations in the equation of state significantly alter the threshold perturbation and the likelihood of primordial black hole genesis.

Primordial Black Hole Formation in the Early Universe: Insights into Critical Behavior and Self-Similarity

This paper, authored by Musco and Miller, presents a detailed investigation into the formation processes of primordial black holes (PBHs) in the early universe, particularly during the radiative era. It extends on their prior work by focusing on the critical behavior and self-similar aspects of black hole formation, topics that are pivotal for understanding the mass distribution and characteristics of such cosmological phenomena.

The research explores the critical collapse theory originally formulated by Choptuik, exploring the conditions under which perturbations lead to black hole formation. This paper is situated within the framework of general relativistic hydrodynamics, using a perfect fluid approximation characterized by the equation of state $p = we$ with $w = 1/3$ for radiation. The authors examine how varying this equation's parameter $w$ impacts the critical value $\delta_c$ — the threshold perturbation amplitude necessary for black hole formation — and the critical exponent $\gamma$, which governs the scaling law $M_{\text{BH}} \propto (\delta - \delta_c)^\gamma$.

The methodology employs both numerical simulations and analytic approaches to derive self-similar solutions within the null-slicing metric formulation. The authors successfully demonstrate that the similarity solution emerges as an attractor in the system's dynamics, confirming the theoretical predictions of critical collapse behavior in a cosmological context. Their numerical results reveal that as the perturbation amplitude $\delta$ approaches $\delta_c$, an intermediate state of constant compactness $2M/R$ manifests, persisting over multiple e-foldings of contraction, which supports previous findings on the self-similar nature of critical collapse.

Additionally, this paper investigates the influence of different initial perturbation profiles on the formation of PBHs. It shows that the growth and decay modes of perturbations around the critical solution align with the linear stability analysis predictions, with the growing mode index closely matching $1/\gamma$. These modes contribute significantly to both the approach toward and deviation from the similarity solution, thereby affecting the PBH formation process.

The paper also explores the consequences of varying $w$ outside the $w = 1/3$ range. This is particularly insightful given that, during certain epochs of the early universe, phase transitions could alter the effective equation of state. The results indicate that both the critical threshold $\delta_c$ and the scaling exponent $\gamma$ are sensitive to changes in $w$, highlighting the increased likelihood of PBH formation when the equation of state softens.

In conclusion, the findings of Musco and Miller contribute valuable insights into the critical collapse phenomena associated with PBH formation in the early universe. By elucidating the role of self-similar solutions and examining the impacts of various equations of state and initial perturbation shapes, the paper enhances our theoretical framework for understanding the primordial conditions leading to PBH genesis. Future research could expand upon these results by considering more complex equations of state and investigating other perturbative influences during different cosmic epochs.