Forming sub-horizon black holes at the end of inflation

TL;DR

The paper extends primordial black hole (PBH) formation analyses to sub-horizon scales at the end of inflation by treating BH formation as a measurement of high peaks in the Bardeen potential $\Psi$ with a threshold $\Psi_c\simeq 0.5$. It develops a framework combining the quantum origin of fluctuations during inflation, their evolution through the radiation era, and a Press-Schechter-like mass function based on the smoothed $\Psi$ field, yielding a PBH mass spectrum that depends on the first maximum of $\Psi$ after inflation and the amplitude of curvature perturbations ${\cal A}_R^{1/2}$. The resulting estimates show that the sub-horizon PBH abundance can be substantial under favorable conditions, with the total mass fraction highly sensitive to the UV extension and perturbation amplitude, highlighting the need for further work on UV effects and observational consequences. The work provides a concrete procedure to connect small-scale quantum fluctuations to PBH demographics on horizon-sized and sub-horizon scales at the end of inflation.

Abstract

Previous authors have calculated the mass function of primordial black holes only on scales which are well outside the horizon at the end of inflation. Here we extend the calculation to sub-horizon scales, on which the density perturbation never becomes classical. Regarding the formation of black holes as a `measurement' of the (high peaks) of the density perturbation, we estimate a mass function by assuming that black holes form as soon as inflation ends, in those rare regions where the Bardeen potential exceeds a threshold value of $Psi_c\simeq 0.5$.

Figures (3)

• Figure 1: The mass fraction $(m^2/\rho)dn/dm$ versus $y=\ln(m/m_{\rm e})$ for $\Psi_{\rm c}=0.5$ and ${\cal{A}_R}^{1/2}=0.09$. The dotted line correspond to the extension of the mass fraction in Eq. (\ref{['eq:massfracres']}) beyond the lower bound $y\simeq7$.
• Figure 2: $\log_{10}((m^2/\rho)dn/dm)$ versus $y=\ln(m/m_{\rm e})$ for $\Psi_{\rm c}=0.5$. The dotted line corresponds to the amplitude ${\cal{A}_R}^{1/2}=0.02$, the dashed line to ${\cal{A}_R}^{1/2}=0.05$, and the full line to ${\cal{A}_R}^{1/2}=0.09$.
• Figure 3: $\log_{10}((m^2/\rho)dn/dm)$ versus $y=\ln(m/m_{\rm e})$ for $\Psi_{\rm c}=0.5$ and ${\cal{A}_R}^{1/2}=0.02$, $0.05$ and $0.09$. Here the dotted lines correspond to the extension of the mass fraction in Eq. (\ref{['eq:massfracres']}) beyond the lower bound $m={\cal{A}_R}^{3/2}m_{\rm e}$.