The trichotomy of primordial black holes initial conditions

We show that the threshold to form a black hole, in an asymptotically flat and radiation-dominated Friedman-Robertson-Walker (FRW) Universe, is not solely (mainly) determined by the behaviour of the compaction function at its extrema, as earlier thought, but also by the Ricci scalar of the spatial geometry at smaller (but super-horizon) scales, which we call ``the core''. We introduce three classes of initial conditions characterised by an open (O), closed (C), or flat (F) FRW core surrounded by a shell with higher three-dimensional curvature. In the C case, the core helps the collapse so that the black hole formation threshold is the lowest among all cases. Type-II black holes might only be generated by Type-O or F (each of those with different thresholds, with O being the highest) or by a Type-C with a negligible Ricci scalar at the centre, which we call an effective F core. Finally, we argue that an F core is typically more probable for a sharp power spectrum, however, it is also more likely related to non-spherical initial conditions. On the other hand, a very broad power spectrum, which might be related to the observed NanoGrav signal, would favour the formation of Type-I black holes with a mass spectrum peaked at the Infra-Red scale.