Non-linear statistics of primordial black holes from gaussian curvature perturbations

Understand

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.