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