Threshold and abundance of primordial black holes: dependence on the profile of the cosmological perturbations

Primordial black holes can form in the early Universe from the collapse of cosmological perturbations after the cosmological horizon crossing. They are possible candidates for the dark matter as well as for the seeds of supermassive black holes observed today in the centre of galaxies. In calculations of spherically symmetric collapse, using a large curvature perturbation in the super horizon regime, the initial conditions are specified using the gradient expansion approximation in the long wave length limit. The non linear evolution is then simulated using a Lagrangian relativistic hydrodynamical code. If the perturbation is larger than  a threshold depending on the equation of state and on the specific shape of the perturbation, a black hole is formed. In this talk I will discuss the dependence of PBH formation from the initial shape of the curvature profile showing the relation between the threshold amplitude and the steepness of the perturbation which is linked to the amplitude of the pressure gradients that are developing during the collapse. I will show how to derive the initial curvature profile form the shape of the inflationary power spectrum, which affects also the abundance of PBHs. Depending on the model, a proper calculation, using the shape of the power spectrum, shows that the abundance of PBHs is significantly increased by several order of magnitudes compared to previous estimations.