Computations of primordial black-hole formation

Abstract

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.

Primordial black hole formation

“…In this situation, the coupled system of Einstein and hydrodynamical equations can be solved analytically, and one finds that the small perturbations in energy density, velocity, etc are generated uniquely by the curvature profile, which is then the only quantity that needs to be specified. In [12] the numerical scheme developed in [11] was modified so as to use initial data coming from this quasi-homogeneous solution, and the values of δ c found in [11] were confirmed. The crucial point of this argument is that the main perturbations of interest in this cosmological context are ones coming from inflation which then re-enter the cosmological horizon at later times.…”

“…In the 1980s, attention shifted mainly to the cosmological consequences that a population of PBHs would have, and this was extensively studied in various scenarios (see, for example, the recent review by Carr [7]). In 1999 Niemeyer & Jedamzik [8] returned to the issue of making numerical calculations of the PBH formation process, and they were subsequently followed in this by other groups [9,10] including ourselves [11,12,13]. All of these studies confirmed the overall picture of PBH formation, showing that the precise value of δ c depends on the initial perturbation profile.…”

“…During the subsequent evolution, in fact, the decaying component of the perturbation had time to become negligible before the perturbation passed inside the cosmological horizon (at the "horizon-crossing" time) and started its non-linear evolution, making the treatment of the eventual black-hole formation a better representation of what would happen for perturbations coming from inflation. The simulations in [11] gave values of δ c in the range 0.43 − 0.47 (instead of 0.67 − 0.71) for the same types of perturbation profile used in [8], explaining so the disagreement pointed out in [15].…”

“…However it is also necessary to have a consistent understanding of the collapse process and of the dependence of the PBH mass on the perturbation amplitude. An important contribution to this was made by Niemeyer & Jedamzik [8,18] (reproduced by us in [11]) who showed also that the masses of PBHs produced in the radiative era by perturbations with a given profile type, follow the scaling-law behaviour of critical collapse, first discovered for idealized conditions by Choptuik [19], i.e. the masses of the black holes produced follow a power law M BH ∝ (δ − δ c ) γ if δ is close enough to δ c .…”

“…In a recent paper [15] this analysis has been converted in terms of the perturbation amplitude δ , showing that the Shibata and Sasaki results, corresponding to a wide choice of perturbation shapes, are consistent with δ c in the range 0.3 < ∼ δ c < ∼ 0.5. Bearing all this in mind, we ourselves then made calculations in 2005 [11] specifying the initial conditions within the linear regime of energy density and velocity field, giving only growing mode solutions at the horizon crossing time. During the subsequent evolution, in fact, the decaying component of the perturbation had time to become negligible before the perturbation passed inside the cosmological horizon (at the "horizon-crossing" time) and started its non-linear evolution, making the treatment of the eventual black-hole formation a better representation of what would happen for perturbations coming from inflation.…”