Number count of gravitational waves and supernovae in luminosity distance space for CDM and scalar-tensor theories

The clustering of gravitational waves in luminosity distance space is emerging as a promising probe of the growth of structure. Just like for galaxies, its observation is subject to a number of relativistic corrections that affect the measured signal and need to be accounted for when fitting theoretical models to the data. We derive the full expression for the number count of gravitational waves in luminosity distance space, including all relativistic corrections, in Lambda CDM and in scalar -tensor theories with luminal propagation of tensors. We investigate the importance of each relativistic effect and the detectability of the total signal by current and planned GW detectors. We consider also supernovae in luminosity distance space, highlighting the differences with gravitational waves in the case of scalar -tensor theories. We carry out a thorough comparison among the number count of gravitational waves and supernovae in luminosity distance space, and that of galaxies in redshift space. We show how the relativistic corrections contain useful complementary information on the growth of perturbations and on the underlying theory of gravity, highlighting the synergy with other cosmological probes.