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

Primordial black holes from second order density perturbations as probes of the small-scale primordial power spectrum