Probing deviations from general relativity via cross-correlation of galaxy catalogs and gravitational waves

In 2015, the first detection of gravitational waves (GWs) by LIGO opened a new window in cosmology, providing a self-calibrating way to measure luminosity distances directly from the GW signal. Combined with redshift information, this enables the construction of the Hubble diagram and constraints on the Hubble constant $H_0$, whose current measurements from early- and late-universe probes are in significant tension. At the same time, the discovery of cosmic acceleration has motivated theories extending General Relativity (GR). To systematically test these models, the Effective Field Theory (EFT) framework parameterizes deviations from the standard $\Lambda$CDM cosmology. This thesis explores the potential of using the cross-correlation between gravitational-wave events and galaxy catalogs to constrain such deviations. Since GWs and galaxies trace the same large-scale structure---albeit in different observational spaces---their two-point correlation can be exploited to infer cosmological parameters. While previous works focused on $\Lambda$CDM, this study extends the method to modified gravity scenarios within the EFT of dark energy. Forecasts show that combining GWs and galaxies yields significantly tighter constraints on modified gravity parameters---up to an order of magnitude improvement compared to single-tracer analyses. Depending on detector sensitivity, this synergy could either uncover small deviations from GR or further confirm its validity on cosmological scales. The results highlight gravitational-wave cosmology, in conjunction with large-scale galaxy surveys, as a powerful tool to test the $\Lambda$CDM paradigm and probe the physics driving cosmic acceleration.