Primordial black hole formation in matter domination

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.

💡 Research Summary

The paper investigates the formation of primordial black holes (PBHs) during an early matter‑dominated era, focusing on how the shape of primordial density peaks influences the collapse threshold and the resulting spin of the black holes. Starting from the well‑established formalism for PBH formation in a perfect fluid with equation‑of‑state parameter (w=p/\rho), the authors review the Misner‑Sharp equations that govern the nonlinear evolution of spherically symmetric perturbations. They emphasize that in the limit (w\rightarrow0) (dust) the standard analytic result predicts a vanishing threshold (\zeta_{\rm th}\to0), suggesting that arbitrarily small overdensities could form black holes.

The authors argue that this conclusion rests on two unrealistic assumptions: perfect spherical symmetry and a perfect‑fluid description. In a dust‑dominated universe, pressure gradients are absent, but velocity dispersion and shell‑crossing become the dominant mechanisms that prevent collapse. By expanding the curvature profile (K(r)) of a peak as (K(0)