Primordial black hole formation in the early universe: critical behaviour and self-similarity

TL;DR

The paper investigates primordial black hole formation in the radiative early universe near the critical threshold, focusing on an intermediate state that may exhibit self-similarity. It derives self-similar equations in the Hernandez–Misner null foliation and compares them with high-resolution cosmological perturbation simulations using adaptive mesh refinement. The results show the intermediate state acts as an attractor described by a self-similar solution, with a growing mode driving departure toward final black hole formation and a measured γ ≈ 0.356 for w=1/3, in agreement with prior critical-collapse results. Varying the equation-of-state parameter w and perturbation shapes reveals systematic changes in δ_c and the maximal compactness, with implications for PBH production during EoS softening epochs.

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.

Figures (8)

• Figure 1: The self-similar solutions in null time for $U$, $\Phi$ and $\Omega$ plotted against $\log \xi$ for $w = 1/3$.
• Figure 2: In the left-hand frame, we show the behaviour of $2M/R$ for a nearly critical case plotted against $R/R_H$ at different time levels, where $R_H$ is the cosmological horizon scale at the moment of horizon crossing. The dashed curve indicates the initial conditions used by the null-time code. The right-hand frame shows the time evolution of the peak of $2M/R$ during the intermediate state with $t/t_H$ being the null time (normalised in the same way as for the similarity solution) measured in units of the horizon crossing time $t_H$ ($=R_H/2$). The horizontal dashed line indicates the value of $(2M/R)_{\rm peak}$ coming from the corresponding similarity solution, while the vertical dashed line indicates when the intermediate state ends for this case, very close to the critical time. In the collapse following this, $(2M/R)_{\rm peak}$ increases rapidly towards $1$ with the formation of the black hole.
• Figure 3: In the left-hand frame, $(2M/R)_{\rm peak}$ is plotted as a function of time for a succession of values of $(\delta-\delta_c)$ equally spaced in the log, showing the convergence of the black-hole formation time $t_{BH}$ as $(\delta-\delta_c)$ is reduced. The right-hand frame shows the scaling-law behaviour of $M_{BH}/M_H$ and $(t_c - t_{BH})/t_H$ as a function of $(\delta-\delta_c)$.
• Figure 4: Simulation results for the velocity $U$ (from the same run as in figure \ref{['fig.2']}) plotted against the similarity coordinate $\xi$. The top left-hand frame shows curves for a succession of times during the intermediate state (the higher peaks corresponding to the later ones). The similarity solution is marked with the short-dashed curve. The top right-hand frame shows only the last of these time levels, but together with a long-dashed curve indicating the unperturbed FRW solution mapped onto this same space-time slice. The bottom frame is a zoom of the first one, looking in detail at the range $U \leq 1$. The long-dashed curve marks the same mapping of the FRW solution as before and the truncated dotted lines show part of the corresponding mappings for the earlier time levels. See text for further details.
• Figure 5: Corresponding simulation results (to those of the previous figure) for $2M/R$ ($=2\Phi$) and $\Omega$ ($=4\pi R^2 e$). The similarity solutions are again marked with the short-dashed curves.