Non-linear statistics of primordial black holes from gaussian curvature
  perturbations
  We develop the non-linear statistics of primordial black holes generated by a
gaussian spectrum of primordial curvature perturbations. This is done by
employing the compaction function as the main statistical variable under the
constraints that: a) the over-density has a high peak at a point $\vec{x}_0$,
b) the compaction function has a maximum at a smoothing scale $R$, and finally,
c) the compaction function amplitude at its maximum is higher than the
threshold necessary to trigger a gravitational collapse into a black hole of
the initial over-density. Our calculation allows for the fact that the patches
which are destined to form PBHs may have a variety of profile shapes and sizes.
The predicted PBH abundances depend on the power spectrum of primordial
fluctuations. For a very peaked power spectrum, our non-linear statistics, the
one based on the linear over-density and the one based on the use of curvature
perturbations, all predict a narrow distribution of PBH masses and comparable
abundance. For broader power spectra the linear over-density statistics
over-estimate the abundance of primordial black holes while the curvature-based
approach under-estimates it. Additionally, for very large smoothing scales, the
abundance is no longer dominated by the contribution of a mean over-density but
rather by the whole statistical realisations of it.
