Constraining extended cosmologies with GW$\times$LSS cross-correlations

The rapid development of gravitational wave astronomy provides the unique opportunity of exploring the dynamics of the Universe using clustering properties of coalescing binary black hole mergers. Gravitational wave data, along with information coming from future galaxy surveys, have the potential of shedding light about many open questions in Cosmology, including those regarding the nature of dark matter and dark energy. In this work we explore which combination of gravitational wave and galaxy survey datasets are able to provide the best constraints both on modified gravity theories and on the nature of the very same binary black hole events. In particular, by using the public Boltzmann code \texttt{Multi\_CLASS}, we compare cosmological constraints on popular $\Lambda$CDM extensions coming from gravitational waves alone and in conjunction with either deep and localized or wide and shallow galaxy surveys. We show that constraints on extensions of General Relativity will be at the same level of existing limits from gravitational waves alone or one order of magnitude better when galaxy surveys are included. Furthermore, cross-correlating both kind of galaxy survey with gravitational waves datasets will allow to confidently rule in or out primordial black holes as dark matter candidate in the majority of the allowed parameter space.