The Threshold for Primordial Black Hole Formation: a Simple Analytic
  Prescription
  Primordial black holes could have been formed in the early universe from non
linear cosmological perturbations re-entering the cosmological horizon when the
Universe was still radiation dominated. Starting from the shape of the power
spectrum on superhorizon scales, we provide a simple prescription, based on the
results of numerical simulations, to compute the threshold $\delta_c$ for
primordial black hole formation. Our procedure takes into account both the non
linearities between the Gaussian curvature perturbation and the density
contrast and, for the first time in the literature, the non linear effects
arising at horizon crossing, which increase the value of the threshold by about
a factor two with respect to the one computed on superhorizon scales.
