Press-Schechter Formalism and The PBH Mass Distributions
Owais Farooq, Romana Zahoor, Balungi Francis
TL;DR
Addresses how to translate small-scale primordial curvature perturbations into PBH abundances during radiation domination. The authors develop an end-to-end pipeline based on Press--Schechter theory and excursion-set first-crossing with a sharp-$k$ filter, connecting the curvature spectrum $P_{\mathcal{R}}(k)$ to the PBH mass distribution via a Gaussian tail and a diffusion with an absorbing barrier, yielding a first-crossing density $f(S)$ and formation statistics. They provide explicit mappings from formation scales to present-day abundance, including the horizon-entry scaling $M \propto k^{-2}$ and the relation $f_{\rm PBH}(M) = (1/\Omega_{\rm DM}) \, d\Omega_{\rm PBH,0}/d\ln M$, and show how to propagate $P_{\mathcal{R}}(k)$ to $f_{\rm PBH}(M)$ with a constraint-ready framework; extensions to moving barriers, non-Gaussianities, and critical collapse are discussed. The framework offers a compact, transparent tool for testing PBH dark-matter scenarios against multi-channel observational constraints.
Abstract
Primordial black holes (PBHs) can form during radiation domination from rare primordial perturbations that re-enter the Hubble radius and undergo gravitational collapse. We derive PBH mass distributions using Press--Schechter theory completed by the excursion-set first-crossing construction. We define the smoothed density contrast $δ_R$ and its variance $S(R)=σ^2(R)$, and connect $S$ to the primordial curvature spectrum $\mathcal{P}_{\mathcal R}(k)$ through the radiation-era transfer. For Gaussian statistics and a constant collapse threshold $δ_c$, the formation fraction is an $\operatorname{erfc}$ tail with a controlled rare-event asymptotic. For a sharp-$k$ filter, $δ(S)$ is Markovian; solving the diffusion equation with an absorbing barrier yields the first-crossing density $f(S)=\frac{δ_c}{\sqrt{2π}}S^{-3/2}\exp\!\big(-δ_c^2/(2S)\big)$. This gives a differential formation fraction $\mathrm{d}β/\mathrm{d}\ln M=f(S)\,\big|\mathrm{d}S/\mathrm{d}\ln M\big|$ and a mass-conserving formation-era mass function $\mathrm{d}n_{\mathrm{PBH}}/\mathrm{d}M$. We then map to the present-day PBH dark-matter fraction per logarithmic mass, $f_{\mathrm{PBH}}(M)$, using horizon-entry scaling $M\propto k^{-2}$ and radiation-era redshifting.