Compact Binary Merger Rate with Modified Gravity in Dark-Matter Spikes

In this study, we investigate the impact of modified gravity on the merger rate of compact binaries within dark-matter spikes surrounding super-massive black holes (SMBHs). Specifically, we calculate binary merger rates involving primordial black holes (PBHs) and/or neutron stars (NSs) in Hu-Sawicki $f(R)$ gravity and the normal branch of Dvali-Gabadadze-Porrati (nDGP) gravity, with three SMBH mass functions, Benson, Vika, and Shankar. The results show consistently higher merger rates predicted for PBH-PBH and PBH-NS binaries in these gravity models compared to general relativity (GR), in particular at lower SMBH masses and for steeper dark-matter spike density profiles. The predicted merger rates are compared to the LIGO-Virgo observations to constrain the parameters of the theory. In particular we find steeper dark-matter spike density profiles in the modified gravity scenarios compared to GR. When compared to current observational constraints on PBH abundance, the mass ranges allowed by Hu-Sawicki $f(R)$ models are found to be wider than by nDGP models, for given merger rates. The results are highly dependent on the choice of SMBH mass function, with Vika and Shankar mass functions predicting lower abundances. The considerable sensitivity of the results on the assumed gravity scenario and SMBH mass function demonstrates the necessity of incorporating the corresponding theoretical uncertainties in making relatively robust predictions on compact binary merger rates and, as a result on PBH properties.