From Primordial Black Holes Abundance to Primordial Curvature Power
  Spectrum (and back)
  In the model where Primordial Black Holes (PBHs) form from large primordial
curvature (C) perturbations, i.e., CPBHs, constraints on PBH abundance provide
in principle constraints on the primordial curvature power spectrum. This
connection however depends necessarily on the details of PBH formation
mechanism. In this paper we provide, for the first time, constraints on the
primordial curvature power spectrum from the latest limits on PBH abundance,
taking into account all the steps from gravitational collapse in real space to
PBH formation. In particular, we use results from numerical relativity
simulations and peak theory to study the conditions for PBH formation for a
range of perturbation shapes, including non-linearities, perturbation profile
and a careful treatment of smoothing and filtering scales. We then obtain
updated PBH formation conditions and translate that into primordial spectrum
constraints for a wide range of shapes and abundances. These updated
constraints cover a range of scales not probed by other cosmological
observables. Our results show that the correct and accurate modelling of
non-linearities, filtering and typical perturbation profile, is crucial for
deriving meaningful cosmological implications.
