Primordial black hole formation in the matter-dominated phase of the Universe

Abstract

We investigate primordial black hole formation in the matter-dominated phase of the Universe, where nonspherical effects in gravitational collapse play a crucial role. This is in contrast to the black hole formation in a radiation-dominated era. We apply the Zel'dovich approximation, Thorne's hoop conjecture, and Doroshkevich's probability distribution and subsequently derive the production probability $β_{0}$ of primordial black holes. The numerical result obtained is applicable even if the density fluctuation $σ$ at horizon entry is of the order of unity. For $σ\ll 1$, we find a semi-analytic formula $β_{0}\simeq 0.05556 σ^{5}$, which is comparable with the Khlopov-Polnarev formula. We find that the production probability in the matter-dominated era is much larger than that in the radiation-dominated era for $σ\lesssim 0.05$, while they are comparable with each other for $σ\gtrsim 0.05$. We also discuss how $σ$ can be written in terms of primordial curvature perturbations.

12 pages, 1 figure, minor correction, accepted for publication in Astrophysical Journal