Where does the rate of mergers per halo in the "Did LIGO detect dark matter?" paper come from?

Question

It was recently proposed that dark matter (DM) could be made of primordial black holes (PBHs) of $\sim 30 M_{\odot}$, like the first ones detected by LIGO. This idea was written in "Did LIGO detect dark matter?" by Bird, et al. (2016). They compare the LIGO merger rate with the one calculated by them. In Equation 2, they write that the rate of mergers per halo would be $$N \sim (1/2) V (\rho/M_{pbh})^2\sigma v$$ where $V$ is the volume of the halo (assumed to have a uniform density $\rho$), $M_{pbh}$ is the mass of a PBH, $\sigma$ is the cross section for the merger rate, and $v$ the relative velocity of two PBHs. My question is: where does this equation come from? There's no derivation and I don't understand how it was obtained!

Answer 1

$\rho/M$ is the number volume density $n$ of the black holes. $\sigma v n$ gives you the rate of mergers $R$ per black hole. So, the number of mergers is, to first guess, $\frac{1}{2}R * \left({\rm number\, of\, black\, holes}\right)$, where the factor of 1/2 removes the double counting of black holes. Plugging all of this in gives the expression you cite.