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

We investigate the second order energy density perturbation induced by small-scale Gaussian and local-type non-Gaussian primordial curvature perturbations. The relative abundance of primordial black holes (PBHs) is calculated in terms of the probability density functions (PDFs) of the total energy density perturbation . The effects of the 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 a log-normal primordial power spectrum, its amplitude is constrained to be approximately , and for the local-type non-Gaussianity with , the upper bound of is approximately .