Exploring galaxies-gravitational waves cross-correlations as an astrophysical probe

Gravitational waves astronomy has opened a new opportunity to study the Universe. Full exploitation of this window can especially be provided by combining data coming from gravitational waves experiments with luminous tracers of the Large Scale Structure, like galaxies. In this work we investigate the cross-correlation signal between gravitational waves resolved events, as detected by the Einstein Telescope, and actively star-forming galaxies. The galaxies distribution is computed through their UV and IR luminosity functions and the gravitational waves events, assumed to be of stellar origin, are self-consistently computed from the aforementioned galaxies distribution. We provide a state-of-the-art treatment both on the astrophysical side, {taking} into account the impact of the star formation and chemical evolution histories of galaxies, and in computing the cross-correlation signal, for which we include lensing and relativistic effects. {We find that the measured cross-correlation signal can be sufficiently strong to overcome the noise and provide a clear signal. As a possible application of this methodology, we consider a proof-of-concept case in which we aim at discriminating a metallicity dependence on the compact objects merger efficiency against a reference case with no metallicity dependence.} When considering galaxies with a Star Formation Rate ψ > 10 Mo˙/yr, a Signal-to-Noise ratio around a value of 2-4 is gained after a decade of observation time, depending on the observed fraction of the sky. This formalism can be exploited as an astrophysical probe and could potentially allow to test and compare different astrophysical scenarios.