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. The abundance of PBHs is very
sensitive to the nonlinear effects, and the power spectrum P? must be a factor of ~
2 larger to produce the same number of PBHs as the linear model (where the exact
value depends on the critical value for a region to collapse and form a PBH). This
also means that the derived constraints on the smallscale power spectrum from
constraints on the abundance of PBHs are weaker by the same factor.
