Computations of primordial black hole formation

TL;DR

The paper investigates primordial black hole formation during the radiation-dominated era using general-relativistic simulations with growing-mode perturbations. It systematically analyzes supercritical and subcritical evolutions, confirming a scaling law $M_{BH}=K(\delta-\delta_c)^\gamma M_H(t_H)$ with $\gamma$ around 0.356 and showing that the threshold $\delta_c$ depends on initial data, decreasing when perturbations are chosen as pure growing modes. The study also explores the impact of a cosmological constant, finding linear shifts in both $\delta_c$ and $\gamma$, and reveals violent near-threshold subcritical dynamics with strong compressions and multiple bounces. These results help link inflationary fluctuation spectra to PBH abundances and highlight the sensitivity of thresholds to initial conditions and vacuum energy across cosmological contexts.

Abstract

Results are presented from general relativistic numerical computations of primordial black-hole formation during the radiation-dominated era of the universe. Growing-mode perturbations are specified within the linear regime and their subsequent evolution is followed as they become nonlinear. We use a spherically symmetric Lagrangian code and study both super-critical perturbations, which go on to produce black holes, and sub-critical perturbations, for which the overdensity eventually disperses into the background medium. For super-critical perturbations, we confirm the results of previous work concerning scaling-laws but note that the threshold amplitude for a perturbation to lead to black-hole formation is substantially reduced when the initial conditions are taken to represent purely growing modes. For sub-critical cases, where an initial collapse is followed by a subsequent re-expansion, strong compressions and rarefactions are seen for perturbation amplitudes near to the threshold. We have also investigated the effect of including a significant component of vacuum energy and have calculated the resulting changes in the threshold and in the slope of the scaling law.

Figures (5)

• Figure 1: Scaling behaviour for $M_{BH}$ as a function of $(\delta-\delta_c)$ calculated for growing-mode Mexican-hat perturbations specified within the linear regime. The filled circles refer to the standard calculation discussed in section 3.2, while the open circles are for a calculation including a non-zero cosmological constant $\Lambda$, as discussed in section 3.3, giving $y = 3.0 \times 10^{-3}$.
• Figure 2: A typical evolution leading to black hole formation: the initial perturbation had a Mexican-hat profile and gave $(\delta-\delta_c)=2.37 \times 10^{-3}$ at the horizon crossing time. The top left-hand panel shows the behaviour of the lapse function (the time sequence of the curves goes from bottom to top on the right hand side); the top right-hand panel shows the fluid-element worldlines (the time is measured in units of the horizon-crossing time $t_H$). The bottom left-hand panel shows the profile of $2M/R$ at different times, with the inset showing the approach of the maximum value of $2M/R \to 1$; the bottom right-hand panel shows the corresponding evolution of the mass-energy (in both of these panels, the time sequence of the curves goes from top to bottom on the right hand side).
• Figure 3: Worldlines for a Mexican-hat perturbation with $(\delta - \delta_c)=-3.0 \times 10^{-3}$). This plot shows alternating collapse and expansion of the perturbed region while the outer material continues to expand uniformly. The "cosmic" time is measured in units of the time at horizon crossing.
• Figure 4: Plots of local quantities as functions of $R/R_H(t_H)$: the velocity $U/c$ is shown in the left-hand column and the energy density $e/e_b$ in the right-hand column. The frames correspond to the following values of $(t-t_0)/t_H$: (a) 7.02; (b) 25.92, (c) 31.67; (d) 33.64; (e) 40.11 . Note that $R_H$ is increasing with time and so points with $R/R_H(t_H) > 1$ can be within the current horizon scale at times after horizon-crossing.
• Figure 5: Evolution of the energy density $e$ and radial velocity $U$ at three (comoving) locations: near to the centre of the perturbation, at an intermediate region and at the edge of the grid where the fluid is unperturbed. Each quantity is measured in units of its initial value at the same comoving location.