Primordial black hole formation in matter domination

Abstract

We study Primordial Black Holes (PBHs) formed by the collapse of rare primordial fluctuations during an early period of Matter Domination. The collapse threshold strongly depends on the shape of the peaks, decreasing as they become flatter and hence rarer. In the extreme limit of a top-hat perturbation, Harada, Kohri, Sasaki, Terada, and Yoo have argued that the growth of velocity dispersion prevents the formation of black holes unless the initial peak is larger than $ζ_{\rm th} \sim ζ_{\rm rms}^{2/5}$. Including the shape distribution of the peaks, we find that for a realistic cosmic abundance of PBHs, the effective threshold is larger, $ζ_{\rm th} \sim ζ_{\rm rms }^{1/10}$. And this model requires $ζ_{\rm rms}\sim 10^{-1}$, which is much larger than the observed value at the CMB scales. Hence, PBH formation during Matter Domination is barely more efficient than Radiation Domination. We estimate the dimensionless spin parameter to be $a_{\rm rms} \sim ζ_{\rm rms}^{7/4}\ll 1$, slightly larger than PBHs formed in Radiation Domination.

Figures (7)

• Figure 1: Evolution of a perturbation in MD era, $r_{\rm sh}$ is Schwarzschild radius, $r_{\rm min}$ is the raduis where the collapse stops, if $r_{\rm min}$ becomes smaller than $r_{\rm sh}$ then a black hole is formed, otherwise it will re-expand and reaches to $r_{\rm max}/2$ which is the equilibrium radius predicted by Virial Theorem.
• Figure 2: Three snapshots in the collapse of a Gaussian profile. The comoving radius of the central flat region $r_{\rm ed}$ shrinks from $r_m$ at $t_{\rm max}$ to $0$ at $t_c$.
• Figure 3: Schematic threshold for PBH formation for a typical high-amplitude profile. For $\zeta_{\rm rms}^{2/5}\ll w\ll 1$, the threshold is linearly proportional to $w$. However, for $w\lesssim \zeta_{\rm rms}^{2/5}$, the effect of profile shape become important then by decreasing $w$, the threshold increases until $\delta_{\rm th}$ reaches to order unity.
• Figure 4: Each $+$ shows a $b_{\rm min}$ for a certain $N$ and the solid line is $1.2 N^{-\frac{1}{3}}$.
• Figure 5: Each point on the plots shows the results of simulations with a different $s_v$, the red curve represents Eq.\ref{['eq:halting2']}, and the yellow line shows Eq.\ref{['eq:halting3']}.