Effects of the surrounding primordial black holes on the merger rate of primordial black hole binaries

We develop an analytic formalism for computing the merger rate of primordial black hole binaries with a general mass function by taking into account the torques by the surrounding primordial black holes and linear density perturbations. We find that $\alpha =-(m_i+m_j{)}^2{\partial }^2\ln \mathcal{R}(m_i,m_j)/\partial m_i\partial m_j=36/37$ is independent of the mass function. Moreover, the ratio of the merger rate density of primordial black hole binaries by taking into account the torques by the surrounding primordial black holes to the nearest primordial black hole is independent of the masses of binaries.

Figure 1

Event rate of mergers of monochromatic PBH binaries whose mass is $30M_{\odot }$ as a function of the dark matter in PBHs. The blue line is the case for taking into account the torques by all PBHs, and the red line is the case for taking into account the torque by the nearest PBH. The event rate as $R=12–213{\mathrm{Gpc}}^{-3}{\mathrm{yr}}^{-1}$ estimated by the LIGO-Virgo Collaboration is shown as the shaded region colored orange [26].

Figure 2

By taking $\beta =2.3$, $M=5M_{\odot }$, the merger rate of PBH binaries at present with $m_1$, $m_2\ge 5M_{\odot }$ and $m_1+m_2\le 100M_{\odot }$. The blue line is the case for taking into account the torques by all PBHs, and the red line is the case for taking into account the torque by the nearest PBH. The event rate as $R=12–213{\mathrm{Gpc}}^{-3}{\mathrm{yr}}^{-1}$ estimated by the LIGO-Virgo Collaboration is shown as the shaded region colored orange [26].