On the merger rate of primordial black holes: effects of nearest neighbours distribution and clustering

One of the seemingly strongest constraints on the fraction of dark matter in the form of primordial black holes (PBH) of ${\cal O}$(10)$\,M_\odot$ relies on the merger rate inferred from the binary BH merger events detected by LIGO/Virgo. The robustness of these bounds depends however on the accuracy with which the formation of PBH binary in the early universe can be described. We revisit the standard estimate of the merger rate, focusing on a couple of key ingredients: the spatial distribution of nearest neighbours and the initial clustering of PBHs associated to a given primordial power spectrum. Overall, we confirm the robustness of the results presented in the literature (which constrain the PBH fraction of dark matter to be $f_{\rm PBH}\lesssim 0.001-0.01$), although a more careful treatment of the former aspect tightens the constraints by a factor $\sim 2$ with respect to existing calculations. The initial clustering of PBHs may have a further effect in the same direction, but only for broad PBH mass functions that extend over several decades.