Improved constraint on the Hubble constant from dark sirens with LIGO/Virgo/KAGRA O4a

A new measurement of the Hubble constant $H_0$ is presented using the statistical dark siren method applied to a sample of seven well-localized gravitational-wave (GW) events from the fourth LIGO-Virgo-KAGRA (LVK) observing run and ten additional events from the first three runs. Galaxy catalogs from the DESI Legacy Imaging Survey (LS) are combined with a deep learning model to compute photometric redshift probability density functions. We extend our previous analysis by including the events GW230731_215307 and GW230927_153832, using sky maps from the fourth Gravitational-Wave Transient Catalog (GWTC-4), and introducing key methodological improvements: $r$-band luminosity weighting of host galaxies; an extended GW likelihood that incorporates information from the binary black hole component masses; and a consistent treatment of selection effects that accounts for the incompleteness of the magnitude-limited LS galaxy catalog. Using a total of 17 well-localized dark sirens (seven from the first part of the fourth observing run, O4a), we obtain $H_0 = 78.8^{+14.6}_{-12.2}$ km/s/Mpc without luminosity weighting and $H_0 = 78.2^{+12.0}_{-11.0}$ km/s/Mpc when applying $r$-band luminosity weighting. Finally, we combine the luminosity-weighted dark siren sample with the bright siren GW170817, including constraints on the jet viewing angle and corrections for the host galaxy peculiar velocity, to obtain a final constraint of $H_0 = 69.9^{+4.1}_{-4.0}$ km/s/Mpc, representing an improvement of approximately 11% in the uncertainty relative to the GW170817-only result.