Impact of nonlinear overdensity statistics on primordial black hole abundance

It has been recently established that, if the nonlinear relationship between the overdensity perturbations and the curvature perturbations are taken into account, non-Gaussianity is introduced in the overdensity statistics which alters the expected primordial black hole abundance. This is explored by using the nonlinear relationship between the overdensities and curvature perturbations up to second order where a negative skewness and positive kurtosis aims at lowering and increasing the abundance while an abundance comparable to Gaussian perturbations is obtained by adjusting the amplitude of the curvature power spectrum. The effects of the nonvanishing skewness and kurtosis are studied using a toy model Dirac delta and lognormal curvature power spectra as well as one obtained from an $\alpha-$attractor model capable of primordial black hole production. Finally, the nonlinear calculations using Press-Schechter are compared with peaks theory.