Investigating the impact of galaxies' compact binary hosting probability for gravitational wave cosmology

With the advent of future-generation interferometers, a huge number of gravitational wave (GW) signals without electromagnetic counterparts are expected to be measured. Although these signals do not allow a simultaneous measurement of the redshift and the luminosity distance, it is still possible to infer cosmological parameters. In this paper we focus on the systematic biases that could arise from mismodelling the GW host probability when inferring the Hubble constant (H-0) using GW dark sirens jointly with galaxy catalogues. We discuss the case in which the GW host probability is a function of galaxy luminosity and redshift, as it has been predicted by synthetic state-of-the-art compact binary coalescence catalogues. We show that, in the limiting case in which the analysis is done with a complete galaxy catalogue covering a footprint of similar to 10 deg(2), mismatching the host probability in terms of galaxy's luminosity will introduce a bias on H-0. In this case, the magnitude of the bias will depend on the distribution of the large-scale structure over the line of sight. Instead, in the limit of a complete wide-field-of-view galaxy catalogue and GW events localized at the O(Gpc) distance, mismatching the redshift dependence of the GW hosting probability is more likely to introduce a systematic bias.