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

TL;DR

The paper demonstrates that primordial black hole formation in the radiative era can exhibit genuine critical collapse when perturbations originate from inflation and contain only a growing mode. By employing an adaptive mesh refinement-enhanced numerical scheme, the authors resolve PBH formation across a mass range exceeding 10^3, revealing a persistent scaling law $M_{BH}/M_H = K(\delta-\delta_c)^\gamma$ with $\gamma\approx0.357$ and $K\approx4.02$, down to $\delta-\delta_c \sim 10^{-11}$. Near the critical limit, the evolution features a long-lived intermediate state with a central condensation that sheds matter via a relativistic wind, forming a semi-void and displaying a second scaling for the approach to collapse, indicative of approximate self-similarity. The results support the view that inflationary perturbations can physically realize critical collapse in the radiative era, with initial conditions playing a key role in shock formation observed in other studies.

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.

Figures (7)

• Figure 1: Scaling behaviour for $M_{BH}$ as function of $(\delta-\delta_c)^\gamma$ calculated for a radiative perfect fluid. For $M_{BH}\lesssim M_H$, the points are well-fitted by a scaling law with $\gamma = 0.357$ and $K=4.02$.
• Figure 2: The top two frames show the behaviour of $2M/R$ and the radial four-velocity $U$ for a nearly-critical case ($\delta \simeq \delta_c)$, plotted against $R/R_H$ at different time levels, with the dashed curve representing the initial conditions used by the observer-time code. $R_H$ is the horizon scale at the moment of horizon crossing. The bottom frame shows the equivalent plot of $2M/R$ for a sub-critical case with $(\delta - \delta_c) \simeq - 1.28 \cdot 10^{-6}$ which does not produce a black hole. Some information about timing is given later, in figure \ref{['fig.5']}.
• Figure 3: These plots, for a representative but fairly extreme case where a black hole is formed, show different views of the behaviour of $2M/R$ as a function of $R/R_H$ or $M/M_H$ at different time levels with the dashed curve representing the initial conditions used by the observer-time code. The perturbation has $(\delta - \delta_c) \simeq 2.63 \cdot 10^{-9}$ and the collapse gives rise to a black hole with a mass $M_{BH} \simeq 3.54 \cdot 10^{-3} M_H$. In the top two frames, $2M/R$ is plotted against $R/R_H$ with the right-hand frame being an enlargement of the inner parts of the left-hand one. The bottom two frames show the same data plotted as a function of $M/M_H$. Some information about timing is given later, in figure \ref{['fig.5']}.
• Figure 4: This figure shows further details for the same case as in figure \ref{['fig.3']}. The top two frames show the profiles of radial four velocity $U$ and energy density $e$ at two key moments: the time when the wind away from the central region reaches its maximum strength (solid line) and the moment when the void starts to refill from the outside (dot-dashed line). Note that the black hole is at a very small scale on the left-hand side of these plots. The bottom frame shows the mass profiles in the very central regions at different time levels, with the dashed curve representing the initial conditions used by the observer-time code and the dotted lines marking the mass and radius of the eventual black hole.
• Figure 5: The left-hand frame shows $(2M/R)_{peak}$ plotted against the time $t$ (given in units of the horizon-crossing time $t_H$) for seven different cases of collapse forming a black hole. The right-hand frame shows a second scaling behaviour for the same cases as in figure \ref{['fig.1']}: plotting $(t_c-t)/t_H$ as function of $(\delta-\delta_c)$. The time $t$ used here is measured when $(2M/R)_{peak}$ becomes larger than 0.5, 0.6, 0.7 and 0.8 respectively (the larger values corresponding to lower sets of points). Each of the four sets of data is fitted by a scaling law with $\gamma \simeq 0.36$.