Bayesian framework to infer the Hubble constant from the cross-correlation of individual gravitational wave events with galaxies

Gravitational waves (GWs) from the inspiral of binary compact objects offer a one-step measurement of the luminosity distance to the event, which is essential for the measurement of the Hubble constant, $H_0$, which characterizes the expansion rate of the Universe. However, unlike binary neutron stars, the inspiral of binary black holes is not expected to be accompanied by electromagnetic radiation and a subsequent determination of its redshift. Consequently, independent redshift measurements of such GW events are necessary to measure $H_0$. In this study, we present a novel Bayesian approach to infer $H_0$ by measuring the overdensity of galaxies around individual binary black hole merger events in configuration space. We model the measured overdensity using the $3$D cross-correlation between galaxies and GW events, explicitly accounting for the GW event localization uncertainty. We demonstrate the efficacy of our method with $250$ simulated GW events distributed within $1$ Gpc in colored Gaussian noise of Advanced LIGO and Advanced Virgo detectors operating at O4 sensitivity. We show that such measurements can constrain the Hubble constant with a precision of $\lesssim 8 \%$ ($90\%$ highest density interval). We highlight the potential improvements that need to be accounted for in further studies before the method can be applied to real data.