Threshold of primordial black hole formation

TL;DR

This work derives a new analytic threshold for primordial black hole formation in a flat FRW universe with p = w ρ c^2 (w ≥ 0), showing δ^{UH}_{H c} = sin^{2}(π sqrt(w)/(1+3w)) and a corresponding comoving threshold tildeδ_c = [3(1+w)/(5+3w)] sin^{2}(π sqrt(w)/(1+3w)). The authors validate the result against state-of-the-art numerical simulations, finding qualitative and quantitative agreement that improves upon Carr's traditional estimates, particularly for small w and near w = 1/3. They connect horizon-crossing perturbations in the uniform Hubble slice to comoving-density perturbations, and extend the analysis to the statistical distribution of perturbations via curvature variables ζ and bar{ζ}, with implications for PBH abundance. The approach—centered on a three-zone model and a sound-crossing vs collapse timescale criterion—offers a physically motivated, relatively robust threshold across a range of equations of state, informing early-Universe scenario constraints and PBH production estimates.

Abstract

Based on a physical argument, we derive a new analytic formula for the amplitude of density perturbation at the threshold of primordial black hole formation in the universe dominated by a perfect fluid with the equation of state $p=wρc^{2}$ for $w\ge 0$. The formula gives $δ^{\rm UH}_{H c}=\sin^{2}[π\sqrt{w}/(1+3w)]$ and $\tildeδ_{c}=[3(1+w)/(5+3w)]\sin^{2}[π\sqrt{w}/(1+3w)]$, where $δ^{\rm UH}_{H c}$ and $\tildeδ_{c}$ are the amplitude of the density perturbation at the horizon crossing time in the uniform Hubble slice and the amplitude measure used in numerical simulations, respectively, while the conventional one gives $δ^{\rm UH}_{H c}=w$ and $\tildeδ_{c}=3w(1+w)/(5+3w)$. Our formula shows a much better agreement with the result of recent numerical simulations both qualitatively and quantitatively than the conventional formula. For a radiation fluid, our formula gives $δ^{\rm UH}_{H c}=\sin^{2}(\sqrt{3}π/6)\simeq 0.6203$ and $\tildeδ_{c}=(2/3)\sin^{2}(\sqrt{3}π/6)\simeq 0.4135$. We also discuss the maximum amplitude and the cosmological implications of the present result.

Figures (4)

• Figure 1: The schematic figure of the three-zone model of density perturbation.
• Figure 2: The trajectories of the sound waves and apparent horizons in the $\eta\chi$ plane for the formation threshold. The sound wave just crosses over the radius of the overdense region from the big bang to the maximum expansion, which is denoted by a thick solid line. The stronger and weaker conditions are also shown by thin dashed lines.
• Figure 3: The threshold values and the maximum value of the density perturbation variable $\tilde{\delta}$ in the comoving slice for different values of $w$. The crosses plot the result of numerical simulations by Musco and Miller Musco:2012au for the profile parameter $\alpha=0$ or a Gaussian curvature profile. The solid, long-dashed and dashed lines denote the analytic formula obtained in Sec. \ref{['subsec:threshold_derivation']}, Carr's original formula and its gauged version, respectively. We also plot our stronger and weaker conditions with thin dotted-dashed lines, which are discussed in Sec. \ref{['subsec:threshold_derivation']}. The short-dashed line denotes the geometrical maximum value, corresponding to a three-hemisphere.
• Figure 4: The threshold values of the curvature perturbations $\bar{\zeta}$ and $\zeta$ for different values of $w$. The lower thick and upper thin solid lines denote our analytic formula for the threshold $\bar{\zeta}_{c}$ and the value $\bar{\zeta}_{h}$ for a three-hemisphere, respectively. The lower thick and upper thin dashed lines denote our analytic formula for the threshold $\zeta_{c}$ and the value $\zeta_{h}$ for a three-hemisphere, respectively, under the approximation described in the text. The regions below and above the three-hemisphere line correspond to type I and II fluctuations, respectively, for each of $\bar{\zeta}$ and $\zeta$.