Primordial Black Hole Formation from the Upward Step Model: Avoiding Overproduction

We investigate the formation of primordial black holes (PBHs) in an upward step inflationary model, where nonlinearities between curvature perturbations and field fluctuations introduce a cutoff, deviating from the Gaussian case. This necessitates a reevaluation of PBH formation, as $\mathcal{R}$ is not the optimal variable for estimating abundance. Using the extended Press-Schechter formalism, we show that non-Gaussianity modifies both the curvature perturbation profile $\mathcal{R}(r)$ and the integration path in probability space, significantly impacting PBH abundance. Our results reveal that the abundance initially increases with the parameter $h$, which characterizes the relaxation stage after the step. However, beyond a critical value ($h \simeq 5.9$), it sharply declines before rising again. Furthermore, we demonstrate that non-Gaussianity introduces uncertainties in indirect PBH observations via gravitational waves. Notably, we present an example where a positive $f_{\rm NL}$ does not necessarily enhance PBH production, contrary to conventional expectations. Finally, by accounting for non-perturbative effects, we resolve the overproduction of PBHs suggested by pulsar timing array (PTA) data, underscoring the critical importance of incorporating non-Gaussianity in future studies.