Detection probability of light compact binary mergers in future observing runs of the current ground-based gravitational wave detector network

With no binary neutron star (BNS) merger detected yet during the fourth observing run (O4) of the LIGO-Virgo-KAGRA (LVK) gravitational wave (GW) detector network, despite the time-volume (VT) surveyed with respect to the end of O3 increased by more than a factor of three, a pressing question is how likely the detection of at least one BNS merger is in the remainder of the run. I present here a simple and general method to address such a question, which constitutes the basis for the predictions that have been presented in the LVK Public Alerts User Guide during the hiatus between the O4a and O4b parts of the run. The method, which can be applied to neutron star - black hole (NSBH) mergers as well, is based on simple Poisson statistics and on an estimate of the ratio of the VT span by the future run to that span by previous runs. An attractive advantage of this method is that its predictions are independent from the mass distribution of the merging compact binaries, which is very uncertain at the present moment. The results, not surprisingly, show that the most likely outcome of the final part of O4 is the absence of any BNS merger detection. Still, the probability of a non-zero number of detections is 34-46\%. For NSBH mergers, the probability of at least one additional detection is 64-71\%. The prospects for the next observing run O5 are more promising, with predicted numbers $N_\mathrm{BNS,O5}=28_{-21}^{+44}$, and the NSBH detections to be $N_\mathrm{NSBH,O5}=65_{-38}^{+61}$ (median and 90\% symmetric credible range), based on the current LVK detector target sensitivities for the run. The calculations presented here also lead to an update of the LVK local BNS merger rate density estimate that accounts for the absence of BNS merger detections in O4 so far, that reads $2.8\,\mathrm{Gpc^{-3}\,yr^{-1}}\leq R_0\leq 480\,\mathrm{Gpc^{-3}\,yr^{-1}}$.