Application of peaks theory to the abundance of primordial black holes

Abstract

We consider the application of peaks theory to the calculation of the number density of peaks relevant for primordial black hole (PBH) formation. For PBHs, the final mass is related to the amplitude and scale of the perturbation from which it forms, where the scale is defined as the scale at which the compaction function peaks. We therefore extend peaks theory to calculate not only the abundance of peaks of a given amplitude, but peaks of a given amplitude and scale. A simple fitting formula is given in the high-peak limit relevant for PBH formation. We also adapt the calculation to use a Gaussian smoothing function, ensuring convergence regardless of the choice of power spectrum.

Did NANOGrav see a signal from primordial black hole formation?

“…where γ = 0.36 corresponds to the universal critical exponent during radiation domination [47,50], and we must account for the non-linear relation δ m = δ ζ − 3 8 δ 2 ζ between the density contrast and the curvature perturba-tion δ ζ [51][52][53]. The critical threshold δ c for PBH formation and the κ-parameter depend on the procedure used to smooth the primordial perturbations [54][55][56] as well as on the shape of individual peaks [51,57,58]. Nonsphericity was found to have an insignificant effect on the threshold for large perturbations [59].…”

The First Three Seconds: a Review of Possible Expansion Histories of the Early Universe

“…[615,624,629] for nonlinear relations between curvature and density perturbations and Refs. [630][631][632][633] for possible extensions of peak theory.…”

Baryogenesis from ultra-slow-roll inflation

“…The curvature perturbation ζ is another natural option for the fundamental random field in the statistics of PBH abundance from models of inflation [69][70][71]. The compaction function is a good choice of F to explore the P (F, y i , σ F i ) correlation with the threshold value F c [68,80], and the inflaton perturbation, δφ, is a convenient candidate of F to study the effect of quantum diffusion in ultra-slow-roll scenarios [90].…”