Inferring cosmological parameters from galaxy and dark sirens cross-correlation

The number of observed gravitational wave (GW) events is growing fast thanks to rapidly improving detector sensitivities. GWs from compact binary coalescences like Black Holes or Neutron Stars behave like standard sirens and can be used as cosmological probes. To this aim, generally, the observation of an electromagnetic counterpart and the measurement of the redshift are needed. However, even when those are not available, it is still possible to exploit these "dark sirens" via statistical methods. In this work, we explore a method that exploits the information contained in the cross-correlation of samples of GW events with matter over-density tracers like galaxy catalogues. Contrary to other currently employed dark-sirens methods, this approach does not suffer from systematic errors related to the incompleteness of the galaxy catalogue. To further enhance the technique, we implement tomography in redshift space for the galaxy catalogue and luminosity distance space for the GWs. We simulate future data collected by the array of currently existing detectors, namely LIGO, Virgo, and Kagra, as well as planned third-generation ones such as the Einstein Telescope and Cosmic Explorers. We cross-correlate these data with those from upcoming photometric galaxy surveys such as Euclid. We perform a sensitivity forecast employing a full-likelihood approach and explore the parameter space with Monte Carlo Markov Chains. We find that with this method, third-generation detectors will be able to determine the Hubble constant $H_0$ with an error of only 0.7%, which is enough to provide decisive information to shed light on the Hubble tension. Furthermore, for the other cosmological parameters, we find that the GWs and galaxy surveys information are highly complementary, and the use of both significantly improves the ability to constrain the underlying cosmology.