Testing gravity with gravitational waves times electromagnetic probes cross-correlations

In a General Relativistic framework, Gravitational Waves (GW) and Electromagnetic (EM) waves are expected to respond in the same way to the effects of matter perturbations between the emitter and the observer. A different behaviour might be a signature of alternative theories of gravity. In this work we study the cross-correlation of resolved GW events (from compact objects mergers detected by the Einstein Telescope, either assuming or excluding the detection of an EM counterpart) and EM signals (coming both from the Intensity Mapping of the neutral hydrogen distribution and resolved galaxies from the SKA Observatory), considering weak lensing, angular clustering and their cross term ($\mathrm{L \times C}$) as observable probes. Cross-correlations of these effects are expected to provide promising information on the behaviour of these two observables, hopefully shedding light on beyond GR signatures. We perform a Fisher matrix analysis with the aim of constraining the $\{\mu_0,\eta_0,\Sigma_0\}$ parameters, either opening or keeping fixed the background parameters $\{w_0,w_a\}$. We find that, although lensing-only forecasts provide significantly unconstrained results, the combination with angular clustering and the cross-correlation of all three considered tracers (GW, IM, resolved galaxies) leads to interesting and competitive constraints. This offers a novel and alternative path to both multi-tracing opportunities for Cosmology and the Modified Gravity sector.