Press-Schechter Formalism and The PBH Mass Distributions

TL;DR

This paper develops a detailed theoretical framework using Press-Schechter formalism to derive primordial black hole mass distributions from early universe density perturbations, connecting initial conditions to present-day PBH abundance.

Contribution

It introduces a comprehensive method combining Press-Schechter theory with excursion-set formalism to accurately predict PBH mass functions from primordial spectra.

Findings

Derived PBH mass functions from primordial curvature spectra.

Connected initial density fluctuations to current PBH abundance.

Provided analytical solutions for the first-crossing distribution in the Markovian case.

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 $\delta_R$ and its variance $S(R)=\sigma^2(R)$, 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 $\delta_c$, the formation fraction is an $\operatorname{erfc}$ tail with a controlled rare-event asymptotic. For a sharp-$k$ filter, $\delta(S)$ is Markovian; solving the diffusion equation with an absorbing barrier yields the first-crossing density $f(S)=\frac{\delta_c}{\sqrt{2\pi}}S^{-3/2}\exp\!\big(-\delta_c^2/(2S)\big)$. This gives a differential formation fraction $\mathrm{d}\beta/\mathrm{d}\ln M=f(S)\,\big|\mathrm{d}S/\mathrm{d}\ln M\big|$ 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.