Dark siren cross-correlations and the sensitivity of $H_0$ to methodological choices

Gravitational wave sources act as absolute distance indicators, making them powerful probes of the present-day expansion rate of the Universe, $H_0$. The cross-correlation method combines gravitational wave events with galaxy catalogues to constrain cosmological parameters through their shared large-scale structure. In this work, we investigate how key methodological choices -- including covariance treatment, bias parametrisation for galaxies and gravitational wave events, and distance and redshift binning width -- affect the inferred value of $H_0$. We also study catalogue incompleteness, showing that selection effects can be incorporated directly into the theoretical prediction, without the need to model the missing population explicitly, a key advantage over the standard galaxy catalogue approach. Our results indicate that, with appropriate modelling choices and a sufficiently large sample of precise gravitational wave events, the systematic biases considered here can be effectively mitigated, highlighting the potential of the cross-correlation method for future dark siren precision cosmology.