Exploring Critical Overdensity Thresholds in Inflationary Models of Primordial Black Holes Formation

In this paper we study the production of Primordial Black Holes (PBHs) from inflation in order to explain the Dark Mater (DM) in the Universe. The evaluation of the fractional PBHs abundance to DM is sensitive to the value of the threshold $\mathrm{\delta_c}$ and the exact value of $\mathrm{\delta_c}$ is sensitive to the specific shape of the cosmological fluctuations. Different mechanisms producing PBHs lead to different thresholds and hence to different fractional abundances of PBHs. In this study, we examine various classes of inflationary models proposed in the existing literature to elucidate the formation of PBHs and we evaluate numerically the associated threshold values. Having evaluated the thresholds we compute the abundances of PBHs to DM using the Press Schecter approach and the Peak Theory. Given the influence of different power spectra on the thresholds, we investigate whether these inflationary models can successfully account for a significant fraction of DM. Moreover, we provide suggested values for the critical threshold. By examining the interplay between inflationary models, threshold values, and PBH abundances, our study aims to shed light on the viability of PBHs as a candidate for DM and contributes to the ongoing discussion regarding the nature of DM in the Universe