Calculating the mass spectrum of primordial black holes

TL;DR

This paper revises the calculation of primordial black hole abundance, highlighting the overestimation caused by standard methods and proposing the use of density contrast over curvature perturbation for more accurate results.

Contribution

It introduces a correction to PBH formation calculations by advocating for the density contrast over the curvature perturbation, and compares different theoretical approaches.

Findings

Standard calculations overestimate PBH abundance by orders of magnitude.

Using density contrast $\Delta$ aligns results between Press-Schechter and peaks theory.

Varying spectral index affects PBH abundance significantly.

Abstract

We reinspect the calculation for the mass fraction of primordial black holes (PBHs) which are formed from primordial perturbations, finding that performing the calculation using the comoving curvature perturbation $\mathcal{R}_{c}$ in the standard way vastly overestimates the number of PBHs, by many orders of magnitude. This is because PBHs form shortly after horizon entry, meaning modes significantly larger than the PBH are unobservable and should not affect whether a PBH forms or not - this important effect is not taken into account by smoothing the distribution in the standard fashion. We discuss alternative methods and argue that the density contrast, $\Delta$, should be used instead as super-horizon modes are damped by a factor $k^{2}$. We make a comparison between using a Press-Schechter approach and peaks theory, finding that the two are in close agreement in the region of interest. We also investigate the effect of varying the spectral index, and the running of the spectral index, on the abundance of primordial black holes.