Determining the Hubble Constant through Cross-Correlation of Galaxies and Gravitational Waves

Gravitational wave (GW) standard sirens have the potential to measure the Hubble constant $H_0$ in the local universe independently of the distance ladder, and thus offer unique new insights into the Hubble tension. A key challenge with standard sirens is detecting their electromagnetic counterparts, and therefore assigning redshifts to the measured distances. One promising way to proceed is to utilize GW `dark sirens' -- events without an identified electromagnetic counterpart -- and cross-correlate their angular distribution with that of galaxies. We present a quantitative study of how precisely the Hubble constant can be measured using tomographic cross-correlation between galaxies and GW sources. Overall, we find that the constraints on $H_0$ will be limited by the quality and quantity of GW data. We find that percent-level constraints on $H_0$ will primarily depend on achieving small distance uncertainties ($\sigma_{d_L}=0.1\,d_L$), obtaining a large number of GW dark sirens ($\gtrsim$$5{,}000$), and accurate sky localization in the tomographic analysis.