Extended primordial black hole mass functions with a spike

TL;DR

This paper develops a modified formalism to derive primordial black hole mass functions with spikes, exploring their properties, formation conditions, and implications for dark matter and supermassive black hole seeds.

Contribution

It introduces a new approach to model PBH mass functions with spikes from primordial power spectra, extending the Press-Schechter formalism for these cases.

Findings

Mass functions with spikes can contain significant small PBH populations.

The formalism is more accurate when spike mass is comparable to or larger than the cutoff mass.

Spiked mass functions can provide seeds for supermassive black holes.

Abstract

We introduce a modification of the Press-Schechter formalism aimed to derive general mass functions for primordial black holes (PBHs). In this case, we start from primordial power spectra (PPS) which include a monochromatic spike, typical of ultra slow-roll inflation models. We consider the PBH formation as being associated to the amplitude of the spike on top of the linear energy density fluctuations, coming from a PPS with a blue index. By modelling the spike with a log-normal function, we study the properties of the resulting mass function spikes, and compare these to the underlying extended mass distributions. When the spike is at PBH masses which are much lower than the exponential cutoff of the extended distribution, very little mass density is held by the PBHs within the spike, and it is not ideal to apply the Press-Schechter formalism in this case as the resulting characteristic overdensity is too different from the threshold for collapse. It is more appropriate to do so when the spike mass is similar to, or larger than the cutoff mass. Additionally, it can hold a similar mass density as the extended part. Such particular mass functions also contain large numbers of small PBHs, especially if stable PBH relics are considered, and they can provide $\sim 1000M_\odot$ seeds for the supermassive black holes at the centres of present-day galaxies. The constraints on the fraction of dark matter in PBHs for monochromatic mass functions are somewhat relaxed when there is an additional underlying extended distribution of masses.