Hidden universality in the merger rate distribution in the primordial
  black hole scenario
  It has been proposed that primordial black holes (PBHs) form binaries in the
radiation dominated era. Once formed, some fraction of them may merge within
the age of the Universe by gravitational radiation reaction. We investigate the
merger rate of the PBH binaries when the PBHs have a distribution of masses
around ${\cal O}(10) M_\odot$, which is a generalization of the previous
studies where the PBHs are assumed to have the same mass. After deriving a
formula for the merger time probability distribution in the PBH mass plane, we
evaluate it under two different approximations. We identify a quantity
constructed from the mass-distribution of the merger rate density per unit
cosmic time and comoving volume $\mathcal{R}(m_1,m_2)$, $\alpha =
-{(m_1+m_2)}^2\partial^2 \ln\mathcal{R}/\partial m_1\partial m_2 $, which
universally satisfies $0.97 \lesssim \alpha \lesssim 1.05$ for all binary
masses independently of the PBH mass function. This result suggests that the
measurement of this quantity is useful for testing the PBH scenario.
