Critical collapse and the primordial black hole initial mass function

TL;DR

This work investigates how critical gravitational collapse, which yields $M_{\rm BH} \propto (p-p_c)^{\gamma}$ with $\gamma\approx 0.37$, shapes the primordial black hole (PBH) initial mass function when PBHs can form across a range of horizon masses. Using the excursion set formalism with a sharp $k$-space window, the authors simulate PBH-formation trajectories for both power-law and spike perturbation spectra, linking PBH masses to the horizon scale through $M_{\rm BH}=k M_H (\delta-\delta_c)^{\gamma}$. They find that, in the low-abundance limit, the mass function tends toward the Niemeyer–Jedamzik form (single-horizon-mass approximation), and the horizon-mass distribution narrows as the spectral index $n$ decreases for power-law spectra. In spike scenarios, the PBH mass function remains broadly NJMF-like but centers at larger $M_{\rm BH}$ and—for plausible spike parameters—becomes significantly broader than the observed MACHO mass function, challenging a simple PBH explanation for MACHOs. Overall, the results validate the single-horizon-mass approximation under current observational constraints while highlighting that spike-induced PBH populations would imprint a detectable broad mass spectrum, offering a potential discriminant for microlensing surveys.

Abstract

It has normally been assumed that primordial black holes (PBHs) always form with mass approximately equal to the mass contained within the horizon at that time. Recent work studying the application of critical phenomena in gravitational collapse to PBH formation has shown that in fact, at a fixed time, PBHs with a range of masses are formed. When calculating the PBH initial mass function it is usually assumed that all PBHs form at the same horizon mass. It is not clear, however, that it is consistent to consider the spread in the mass of PBHs formed at a single horizon mass, whilst neglecting the range of horizon masses at which PBHs can form. We use the excursion set formalism to compute the PBH initial mass function, allowing for PBH formation at a range of horizon masses, for two forms of the density perturbation spectrum. First we examine power-law spectra with $n>1$, where PBHs form on small scales. We find that, in the limit where the number of PBHs formed is small enough to satisfy the observational constraints on their initial abundance, the mass function approaches that found by Niemeyer and Jedamzik under the assumption that all PBHs form at a single horizon mass. Second, we consider a flat perturbation spectrum with a spike at a scale corresponding to horizon mass $\sim 0.5 M_{\odot}$, and compare the resulting PBH mass function with that of the MACHOs (MAssive Compact Halo Objects) detected by microlensing observations. The predicted mass spectrum appears significantly wider than the steeply-falling spectrum found observationally.

Figures (7)

• Figure 1: A typical PBH forming trajectory resulting from a power-law spectrum with $n=1.3$. The dotted points are the values of $\delta(M_{{\rm H}})$, the long-dashed line shows the threshold, at the fixed time for PBH formation $\delta_{{\rm c, ft}}(M_{{\rm H}})$, and the short-dashed line shows the typical size of perturbations on each scale, $\sigma(M_{{\rm H}})$.
• Figure 2: The smoothed distributions of the actual PBH masses, $\chi(M_{{\rm BH}})$, (top panel) and the horizon masses at their formation, $\chi(M_{{\rm H}})$, (lower panel). From top to bottom on the right-hand side of each diagram, the solid lines are for $n=1.310$, 1.305, 1.300 and 1.295. In the top panel the dotted line shows the Niemeyer and Jedamzik mass function, evaluated for $\sigma(10^{10}{\rm g})=0.032$ and smoothed on the same scale, while in the bottom panel the dotted line is a smoothed delta function, centered on $M_{{\rm H}}=10^{10}$g.
• Figure 3: The form of $\sigma(M_{{\rm H}})$ produced by a spike in the density perturbation spectrum with $C= 1 \times 10^{10} {\rm Mpc}^{-1}$ and $\Sigma= 2 \times 10^{9}{\rm Mpc}^{-1}$.
• Figure 4: A PBH forming trajectory produced when $\cal{P}_{\cal{R}}\sl$ has a gaussian spike with $\Sigma=10^{9} {\rm Mpc}^{-1}$ and $C=10^{10} {\rm Mpc}^{-1}$. The amplitude jumps dramatically when the spike is encountered. The dotted points are the values of $\delta(M_{{\rm H}})$, the long-dashed line shows the threshold $\delta_{{\rm c, ft}}$ for PBH formation, and the short-dashed line shows the typical size of perturbations on each scale $\sigma(M_{{\rm H}})$.
• Figure 5: The smoothed distribution of the actual masses, $\chi(M_{{\rm BH}})$, (upper panel) and horizon masses, $\chi(M_{{\rm H}})$, (lower panel) of PBHs formed due to a spike in the density perturbation spectrum with $C= 1 \times 10^{10} \,{\rm Mpc}^{-1}$. From right to left on the right-hand side of each diagram, the solid lines show $\Sigma= 1 \times 10^{8}, 5 \times 10^{8}, 1 \times 10^{9}, 1.5 \times 10^{9}$ and $2 \times10^{9} {\rm Mpc}^{-1}$. The dotted line in the upper panel shows the smoothed Niemeyer and Jedamzik mass function evaluated for $\sigma(10^{33}{\rm g})=0.053$, and in the lower panel a smoothed delta function centered on $M_{{\rm H}}=10^{33}$g.