Clustering of Gravitational Wave and Supernovae events: a multitracer analysis in Luminosity Distance Space

We study the clustering of Gravitational Wave (GW) merger events and Supernovae IA (SN), as cosmic tracers in Luminosity Distance Space. We modify the publicly available CAMB code to numerically evaluate auto- and cross- power spectra for the different sources, including Luminosity Distance Space distortion effects generated by peculiar velocities and lensing convergence. We perform a multitracer Fisher analysis to forecast expected constraints on cosmological and GW bias coefficients, using outputs from hydrodinamical N-body simulations to determine the bias fiducial model and considering future observations from the Vera Rubin Observatory and Einstein Telescope (ET), both single and in a 3 detector network configuration. We find that adding SN to the GW merger dataset considerably improves the forecast, mostly by breaking significant parameter degeneracies, with final constraints comparable to those obtainable from a Euclid-like survey. GW merger bias is forecasted to be detectable with good significance even in the single ET case.