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 Multi_CLASS, we compare cosmological
constraints on popular ΛCDM 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.