Primordial black holes and scalar induced density perturbations: the effects of probability density functions

TL;DR

This paper examines how second order density perturbations, influenced by Gaussian and non-Gaussian primordial fluctuations, significantly tighten constraints on small-scale power spectra relevant for primordial black hole formation.

Contribution

It introduces a detailed analysis of second order energy density perturbations and their impact on primordial black hole abundance constraints, considering different probability density functions.

Findings

Second order perturbations reduce upper bounds on small-scale power spectra by 1-2 orders of magnitude.

For Gaussian spectra, the amplitude $A_{\zeta}$ is constrained to about $3 \times 10^{-3}$.

For non-Gaussian spectra with $f_{\mathrm{NL}}=10$, the constraint tightens to about $2.5 \times 10^{-4}$.

Abstract

We investigate the second order energy density perturbation $\delta^{(2)}$ induced by small-scale Gaussian and local-type non-Gaussian primordial curvature perturbations. The relative abundance of primordial black hole is calculated in terms of the probability density function of total energy density perturbation $\delta_r=\delta^{(1)}+\frac{1}{2}\delta^{(2)}$. The effects of second order density perturbation greatly reduce the upper bounds of small-scale power spectra of primordial curvature perturbations by one to two orders of magnitude. For log-normal primordial power spectrum, its amplitude $A_{\zeta}$ is constrained to be about $A_{\zeta}\sim 3\times10^{-3}$. And for local-type non-Gaussianity with $f_{\mathrm{NL}}=10$, the upper bound of $A_{\zeta}$ is about $2.5\times10^{-4}$.