Threshold for PBH formation in the type-II region and its analytical estimation

We numerically simulate the formation of primordial black holes (PBHs) in a radiation-dominated Universe under the assumption of spherical symmetry, driven by the collapse of adiabatic fluctuations, for different curvature profiles $\ensuremath{\zeta}$. Our results show that the threshold for PBH formation, defined as the peak value of the critical compaction function ${\mathcal{C}}_{c}({r}_{m})$ (where ${r}_{m}$ is the scale at which the peak occurs), does not necessarily asymptotically saturate to its maximum possible value in the type-I region for sufficiently sharp profiles. Instead, the threshold is found in the type-II region with ${\mathcal{C}}_{c}({r}_{m})$ being a minimum. We find, for the cases tested, that this is a general trend associated with profiles that exhibit extremely large curvatures in the linear component of the compaction function ${\mathcal{C}}_{l}(r)\ensuremath{\equiv}\ensuremath{-}4r{\ensuremath{\zeta}}^{\ensuremath{'}}(r)/3$ shape around its peak ${r}_{m}$ (spiky shapes). To measure this curvature at ${r}_{m}$, we define a dimensionless parameter, $\ensuremath{\kappa}\ensuremath{\equiv}\ensuremath{-}{r}_{m}^{2}{\mathcal{C}}_{l}^{\ensuremath{'}\ensuremath{'}}({r}_{m})$, and we find that the thresholds observed in the type-II region occur for sufficiently large $\ensuremath{\kappa}$ for the profiles we have used, contrary to expectations. By defining the threshold in terms of ${\mathcal{C}}_{l,c}({r}_{m})$, we extend previous analytical estimations to the type-II region, which is shown to be accurate within a few percent when compared to the numerical simulations for the tested profiles. Our results suggest that current PBH abundance calculations for models where the threshold lies in the type-II region may have been overestimated due to the general assumption that it should saturate at the boundary between the type-I and type-II regions.