A new calculation of the mass fraction of primordial black holes

TL;DR

This work reframes primordial black hole formation by using a metric-perturbation criterion ($\psi$) from Shibata and Sasaki, linking it to the curvature perturbation threshold ($\zeta$) and re-evaluating PBH abundance. It compares the traditional Press–Schechter-like approach based on the density contrast with a peaks-theory calculation grounded in the initial curvature field, showing broad agreement within a specific threshold range but highlighting theoretical advantages of the peaks framework. A key result is the approximate mapping between $\Delta_{\rm th}$ and $\zeta_{\rm th}$ in radiation domination, aiding cross-method consistency. The paper thus advocates adopting the peaks-based mass function for PBH abundance calculations and clarifies threshold uncertainties impacting constraints on the primordial power spectrum.

Abstract

We revisit the calculation of the abundance of primordial black holes (PBHs) formed from primordial density perturbations, using a formation criterion derived by Shibata and Sasaki which refers to a metric perturbation variable rather than the usual density contrast. We implement a derivation of the PBH abundance which uses peaks theory, and compare it to the standard calculation based on a Press--Schechter-like approach. We find that the two are in reasonable agreement if the Press--Schechter threshold is in the range $Δ_{\rm th} \simeq 0.3$ to 0.5, but advocate use of the peaks theory expression which is based on a sounder theoretical footing.

Figures (1)

• Figure 1: PBH abundance as a function of horizon mass for power-law power spectra with $n=1.25$ and $n=1.5$ (left and right-hand sets of curves respectively) calculated using the Press--Schechter formalism with ${\Delta_{\rm{th}}}= 1/3$ (solid line) and the peaks formalism with $\zeta_{{\rm th}} = 0.7$ and $1.2$ (dotted and dashed respectively).