A unified approach to dark sirens in harmonic space

Dark siren methods aim to infer cosmological parameters from gravitational waves (GWs) without an electromagnetic counterpart by leveraging features in the mass distribution and statistically marginalizing over possible host galaxies. The latter method, known as the "galaxy catalog method", is uninformative at moderate to high redshifts where galaxy catalogs are highly incomplete. We show that cross-correlating GW sources and galaxies is an extension of the galaxy catalog method from one-point to two-point statistics, in which the galaxy-GW joint likelihood can be factored into an angular ``cross-correlation" part which encodes the two-point statistics, and a radial ``spectral sirens" part which encodes the rate evolution, mass distribution, and selection effects. We also show how to rigorously build GW measurement errors into the likelihood and highlight underlying assumptions. This method can be thought of as a hybrid of a full-shape power spectrum analysis in luminosity-distance and redshift space simultaneously, allowing for the constraining of cosmological parameters from both large-scale structure and the dL- z relation. We show that with a 2 ET + 1 CE setup, the cross-correlation part alone can measure H0 and Ωm,0 with a precision competitive with current precision cosmology experiments. We highlight the importance of analyzing GWs and galaxies as an ensemble in order to maximize the information extracted from the large-scale structure of the universe.