  Primordial black holes may have formed in the radiative era of the early
Universe from the collapse of large enough amplitude perturbations of the
metric. These correspond to non linear energy density perturbations
characterized by an amplitude larger than a certain threshold, measured when
the perturbations reenter the cosmological horizon. The process of primordial
black hole formation is studied here within spherical symmetry, using the
gradient expansion approximation in the long wavelength limit, where the
pressure gradients are small, and the initial perturbations are functions only
of a time-independent curvature profile. In this regime it is possible to
understand how the threshold for primordial black hole formation depends on the
shape of the initial energy density profile, clarifying the relation between
local and averaged measures of the perturbation amplitude. Although there is no
universal threshold for primordial black hole formation, the averaged mass
excess of the perturbation depends on the amplitude of the energy density peak,
and it is possible to formulate a well-defined criterion to establish when a
cosmological perturbation is able to form a black hole in terms of one of these
two key quantities. This gives understanding of how the abundance of primordial
black holes depends on the shape of the the inflationary power spectrum of
cosmological perturbations.
