Two of the most rapidly growing observables in cosmology and astrophysics are gravitational waves (GW) and the neutral hydrogen (HI) distribution. In this work, we investigate the cross-correlation between resolved gravitational wave detections and HI signal from intensity mapping (IM) experiments. By using a tomographic approach with angular power spectra, including all projection effects, we explore possible applications of the combination of the Einstein Telescope and the SKAO intensity mapping surveys. We focus on three main topics: textit{(i)} statistical inference of the observed redshift distribution of GWs; textit{(ii)} constraints on dynamical dark energy models as an example of cosmological studies; textit{(iii)} determination of the nature of the progenitors of merging binary black holes, distinguishing between primordial and astrophysical origin. Our results show that: textit{(i)} the GW redshift distribution can be calibrated with good accuracy at low redshifts, without any assumptions on cosmology or astrophysics, potentially providing a way to probe astrophysical and cosmological models; textit{(ii)} the constrains on the dynamical dark energy parameters are competitive with IM-only experiments, in a complementary way and potentially with less systematics; textit{(iii)} it will be possible to detect a relatively small abundance of primordial black holes within the gravitational waves from resolved mergers. Our results extend towards $mathrm{GW times IM}$ the promising field of multi-tracing cosmology and astrophysics, which has the major advantage of allowing scientific investigations in ways that would not be possible by looking at single observables separately.
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