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

We investigate the second order energy density perturbation δ（2） 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 δr=δ（1）+1/2δ（2）.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 Aζ is constrained to be approximately Aζ～3×10-3,and for the local-type non-Gaussianity with fNL=10,the upper bound of Aζ is approximately 2.5×10-4.