Abundance of Primordial Black Holes in Peak Theory for an Arbitrary Power Spectrum

TL;DR

This paper introduces a modified peak theory method for estimating primordial black hole (PBH) abundance that applies to arbitrary power spectra, improves accuracy by decoupling environmental effects, and refines mass spectrum predictions.

Contribution

The authors develop a new procedure for PBH abundance estimation that extends peak theory to arbitrary power spectra and reduces environmental variance effects.

Findings

Narrower PBH mass spectrum compared to previous methods.

Effective estimation of PBH abundance for scale-invariant flat power spectra.

Identification of the $k$-space tophat window as optimal for minimizing spectrum reduction.

Abstract

We modify the procedure to estimate PBH abundance proposed in arXiv:1805.03946 so that it can be applied to a broad power spectrum such as the scale-invariant flat power spectrum. In the new procedure, we focus on peaks of the Laplacian of the curvature perturbation $\triangle \zeta$ and use the values of $\triangle \zeta$ and $\triangle \triangle \zeta $ at each peak to specify the profile of $\zeta$ as a function of the radial coordinate while the values of $\zeta$ and $\triangle \zeta$ are used in arXiv:1805.03946. The new procedure decouples the larger-scale environmental effect from the estimate of PBH abundance. Because the redundant variance due to the environmental effect is eliminated, we obtain a narrower shape of the mass spectrum compared to the previous procedure in arXiv:1805.03946. Furthermore, the new procedure allows us to estimate PBH abundance for the scale-invariant flat power spectrum by introducing a window function. Although the final result depends on the choice of the window function, we show that the $k$-space tophat window minimizes the extra reduction of the mass spectrum due to the window function. That is, the $k$-space tophat window has the minimum required property in the theoretical PBH estimation. Our procedure makes it possible to calculate the PBH mass spectrum for an arbitrary power spectrum by using a plausible PBH formation criterion with the nonlinear relation taken into account.