Non-Gaussianity effects on the primordial black hole abundance for sharply-peaked primordial spectrum

TL;DR

This paper investigates how non-Gaussian features in the primordial curvature perturbation, specifically bispectrum and trispectrum effects, influence the abundance of primordial black holes formed from sharply-peaked spectra, providing simplified formulas for their impact.

Contribution

It derives formulas linking non-Gaussian parameters to PBH abundance, introducing an effective parameter that simplifies the analysis for sharply-peaked spectra.

Findings

Non-Gaussianity parameters linearly combine into an effective $f_{NL}^{eff}$.

Formulas explicitly relate bispectrum and trispectrum effects to PBH mass fraction.

Simplified expressions for non-Gaussian effects in narrow spectral peaks.

Abstract

We perturbatively study the effect of non-Gaussianities on the mass fraction of primordial black holes (PBHs) at the time of formation by systematically taking its effect into account in the one-point probability distribution function of the primordial curvature perturbation. We focus on the bispectrum and trispectrum and derive formulas that describe their effects on the skewness and kurtosis of the distribution function. Then considering the case of narrowly peaked spectra, we obtain simple formulas that concisely express the effect of the bi- and trispectra. In particular, together with the $g_{\rm NL}$ and $\tau_{\rm NL}$ parameters of the trispectrum, we find that non-Gaussianity parameters for various types of the bispectrum are linearly combined to give an effective parameter, $f_{\rm NL}^{\rm eff}$, that determines the PBH mass fraction in the narrow spectral shape limit.