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

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 $A_{\zeta }\sim 3\times 10^{-3}$. And for local-type non-Gaussianity with $f_{\mathrm{NL}}=10$, the upper bound of $A_{\zeta }$ is about $2.5\times 10^{-4}$.

5 pages, 4 figures