  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.
