Threshold for primordial black holes. II. A simple analytic prescription

Primordial black holes could have been formed in the early universe from nonlinear cosmological perturbations reentering 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 δc for primordial black hole formation. Our procedure takes into account both the nonlinearities between the Gaussian curvature perturbation and the density contrast and, for the first time in the literature, the nonlinear 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.