Nonlinear statistics of primordial black holes from Gaussian curvature perturbations
Cristiano Germani, Ravi K. Sheth
Abstract
We develop the nonlinear 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 (i) the overdensity has a high peak at a point ${\stackrel{\ensuremath{\rightarrow}}{x}}_{0}$, (ii) the compaction function has a maximum at a smoothing scale $R$, and finally, (iii) 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 overdensity. 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 nonlinear statistics, the one based on the linear overdensity 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 overdensity statistics overestimate 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 overdensity but rather by the whole statistical realizations of it.