Press-Schechter Formalism and The PBH Mass Distributions

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.