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In this paper we derive the probability of the radial profiles of sphericallysymmetric inhomogeneities in order to provide an improved estimation of thenumber density of primordial black holes (PBHs). We demonstrate that theprobability of PBH formation depends sensitively on the radial profile of theinitial configuration. We do this by characterising this profile with twoparameters chosen heuristically: the amplitude of the inhomogeneity and thesecond radial derivative, both evaluated at the centre of the configuration. Wecalculate the joint probability of initial cosmological inhomogeneities as afunction of these two parameters and then find a correspondence between theseparameters and those used in numerical computations of PBH formation. Finally,we extend our heuristic study to evaluate the probability of PBH formationtaking into account for the first time the radial profile of curvatureinhomogeneities.

Probability of primordial black hole formation and its dependence on the radial profile of initial configurations