Formation of Primordial Black Holes in the Subhorizon k-Space of Inflation

We investigate a high momentum regime of inflation where the cosmological perturbation breaks down due to large inflationary quantum fluctuations ( \(\sim \) large k) to form the primordial black holes(PBHs) in the radiation-dominated era after inflation. In our study, we have found that, in this region, the values of the Bardeen potential is large negative causing a gravitational instability conducive to the formation of the PBHs. We have used three self-consistent differential equations to study the dynamical evolution in the k-space, in the spatially flat gauge i.e. \(\delta \phi \ne 0\) , which shows the role of the inflaton perturbation as well as that of the background metric in the formation of PBHs. We have found that the \(\alpha \) -attractor potentials which are favored by the PLANCK-18 data help to create the gravitational instability i.e. the large value of Bardeen potential in the region \(k \sim 10^{13}\) M \({\text {pc}}^{-1}\) and the density contrast exceeds the value of 0.41 which indicates the formation of the PBHs. Further we have calculated the values of the \(\sigma (M)\) , \(\beta (M)\) and \(f_{PBH}(M)\) and masses of these PBHs in a new method and get the values which are consistent with LISA, WD, NS, DECIGO/AI, FL, SIGWs forecasts. The calculated masses of the PBHs in the range of \(1.35\times 10^{-13}\) – \(2.60\times 10^{-16}\) \(M_{\odot }\) and \(f_{PBH}\) are 0.3– \(36\%\) are in the range of which indicates the fraction of dark matter present in PBH.