The abundance of primordial black holes depends on the shape of the
  inflationary power spectrum
  In this letter, combining peak theory and the numerical analysis of
gravitational collapse in the radiation dominated era, we show that the
abundance of primordial blacks holes, generated by an enhancement in the
inflationary power spectrum, is extremely dependent on the shape of the peak.
Given the amplitude of the power spectrum, we show that the density of
primordial black holes generated from a narrow peak, is exponentially smaller
than in the case of a broad peak. Specifically, for a top-hat profile of the
power spectrum in Fourier space, we find that for having primordial black holes
comprising all of the dark matter, one would only need a power spectrum
amplitude an order of magnitude smaller than suggested previously whereas in
the case of a narrow peak, one would instead need a much larger power spectrum
amplitude, which in many cases would invalidate the perturbative analysis of
cosmological perturbations. Finally, we show that, although critical collapse
gives a broad mass spectrum, the density of primordial black holes formed is
dominated by masses roughly equal to the cosmological horizon mass measured at
horizon crossing.
