Gravitational-wave dark siren cosmology systematics from galaxy weighting

The detection of GW170817 and the measurement of its redshift from the associated electromagnetic counterpart provided the first gravitational wave determination of the Hubble constant ($H_0$), demonstrating the potential power of standard-siren cosmology. In contrast to this bright siren approach, the dark siren approach can be utilized for gravitational-wave sources in the absence of an electromagnetic counterpart: one considers all galaxies contained within the localization volume as potential hosts. When statistically averaging over the potential host galaxies, weighting them by physically-motivated properties (e.g., tracing star formation or stellar mass) could improve convergence. Using mock galaxy catalogs, we explore the impact of these weightings on the measurement of $H_0$. We find that incorrect weighting schemes can lead to significant biases due to two effects: the assumption of an incorrect galaxy redshift distribution, and preferentially weighting incorrect host galaxies during the inference. The magnitudes of these biases are influenced by the number of galaxies along each line of sight, the measurement uncertainty in the gravitational-wave luminosity distance, and correlations in the parameter space of galaxies. We show that the bias may be overcome from improved localization constraints in future GW detectors, a strategic choice of priors or weighting prescription, and by restricting the analysis to a subset of high-SNR events. We propose the use of hierarchical inference as a diagnostic of incorrectly-weighted prescriptions. Such approaches can simultaneously infer the correct weighting scheme and the values of the cosmological parameters, thereby mitigating the bias in dark siren cosmology due to incorrect host-galaxy weighting.