Joint population and cosmological properties inference with gravitational waves standard sirens and galaxy surveys

Gravitational wave (GW) sources at cosmological distances can be used to probe the expansion rate of the Universe. GWs directly provide a distance estimation of the source but no direct information on its redshift. The optimal scenario to obtain a redshift is through the direct identification of an electromagnetic (EM) counterpart and its host galaxy. With almost 100 GW sources detected without EM counterparts (dark sirens), it is becoming crucial to have statistical techniques able to perform cosmological studies in the absence of EM emission. Currently, only two techniques for dark sirens are used on GW observations: the spectral siren method, which is based on the source-frame mass distribution to estimate conjointly cosmology and the source's merger rate, and the galaxy survey method, which uses galaxy surveys to assign a probabilistic redshift to the source while fitting cosmology. It has been recognized, however, that these two methods are two sides of the same coin. In this paper, we present a novel approach to unify these two methods. We apply this approach to several observed GW events using the \textsc{glade+} galaxy catalog discussing limiting cases. We provide estimates of the Hubble constant, modified gravity propagation effects, and population properties for binary black holes. We also estimate the binary black hole merger rate per galaxy to be $10^{-6}-10^{-5} {\rm yr^{-1}}$ depending on the galaxy catalog hypotheses.