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