Correlation function of high-threshold regions and application to the initial small-scale clustering of primordial black holes

TL;DR

This paper investigates the initial clustering of primordial black holes, showing that they are not significantly more clustered than random distributions at formation, and derives a new analytic correlation function applicable to large-threshold fluctuations.

Contribution

It provides a simple argument against significant small-scale clustering of PBHs at formation and introduces a new analytic expression for the two-point correlation function of large-threshold fluctuations.

Findings

Primordial black holes are not significantly clustered beyond Poisson expectations at formation.

Derived a new analytic formula for the two-point correlation function of large-threshold fluctuations.

Results have broader implications beyond PBH clustering, applicable to general large-threshold phenomena.

Abstract

Primordial black holes (PBHs) have been brought back into the spotlight by LIGO's first direct detection of a binary-black-hole merger. One of the poorly understood properties of PBHs is how clustered they are at formation. It has important implications on the efficacy of their merging in the early Universe, as well as on observational constraints. In this work we study the initial clustering of PBHs formed from the gravitational collapse of large density fluctuations in the early Universe. We give a simple and general argument showing that, in this scenario, we do not expect clustering on very small scales beyond what is expected from a random, Poisson distribution. We illustrate this result explicitly in the case where the underlying density field is Gaussian. We moreover derive a new analytic expression for the two-point correlation function of large-threshold fluctuations, generalizing previous results to arbitrary separation, and with broader implications than the clustering of PBHs.