Primordial Black Hole Merger Rate in Self-Interacting Dark Matter Halo Models

We study the merger rate of primordial black holes (PBHs) in the self-interacting dark matter (SIDM) halo models. To explore a numerical description for the density profile of the SIDM halo models, we use the result of a previously performed simulation for the SIDM halo models with $\sigma/m=10~cm^{2}g^{-1}$. We also propose a concentration-mass-time relation that can explain the evolution of the halo density profile related to the SIDM models. Furthermore, we investigate the encounter condition of PBHs that may have been distributed in the medium of dark matter halos randomly. Under these assumptions, we calculate the merger rate of PBHs within each halo considering the SIDM halo models and compare the results with the one obtained for the cold dark matter (CDM) halo models. We indicate that the merger rate of PBHs for the SIDM halo models during the first epoch should be lower than the corresponding result for the CDM halo models, while by the time entering the second epoch sufficient PBH mergers in the SIDM halo models can be generated and even exceed the one resulted from the CDM halo models. By considering the spherical-collapse halo mass function, we obtain similar results for the cumulative merger rate of PBHs. Moreover, we calculate the redshift evolution of the PBH total merger rate. To determine a constraint on the PBH abundance, we study the merger rate of PBHs in terms of their fraction and masses and compare those with the black hole merger rate estimated by the Advanced LIGO (aLIGO) detectors during the third observing run. The results demonstrate that within the context of the SIDM halo models during the second epoch, the merger rate of $10~M_{\odot}-10~M_{\odot}$ events falls within the aLIGO window. We also estimate a relation between the fraction of PBHs and their masses, which is well consistent with our findings.