The luminosity of the unseen: luminosity function in dark sirens

Gravitational waves (GWs) offer a novel avenue for probing the Universe. An especially compelling application is an independent determination of the Hubble constant, H₀, using dark standard sirens, which associate GW signals with galaxy catalogues under the assumption that GW events are hosted by galaxies in the catalogue. However, due to the limited reach of telescopes, galaxy catalogues are largely incomplete at high redshifts. The commonly used Galaxy List for the Advanced Detector Era (GLADE+) catalogue, for example, is complete only up to redshift $z=0.1$. Handling this incompleteness necessitates the use of a model for the galaxy luminosity distribution, typically described by the Schechter function. In this paper, we examine the influence of the luminosity function model on dark sirens, focusing on its redshift evolution and its impact on H₀ and merger rate parameter measurements. We find that neglecting the evolution of the luminosity function can influence the prior in redshift on GWs, which has particularly high impact for distant GW events with limited galaxy catalogue support. Moreover, conducting a joint estimation of H₀ and the rate parameters, we find that allowing them to vary fixes the bias in H₀, but the rate evolution parameter γ depends on the evolving luminosity function. Our results underscore the importance of incorporating the correct luminosity function evolution model to account for changes in galaxy populations over cosmic time, as this impacts rate parameters to which H₀ is sensitive.