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Two of the most rapidly growing observables in cosmology and astrophysics aregravitational waves (GW) and the neutral hydrogen (HI) distribution. In thiswork, we investigate the cross-correlation between resolved gravitational wavedetections and HI signal from intensity mapping (IM) experiments. By using atomographic approach with angular power spectra, including all projectioneffects, we explore possible applications of the combination of the EinsteinTelescope and the SKAO intensity mapping surveys. We focus on three maintopics: \textit{(i)} statistical inference of the observed redshiftdistribution of GWs; \textit{(ii)} constraints on dynamical dark energy modelsas an example of cosmological studies; \textit{(iii)} determination of thenature of the progenitors of merging binary black holes, distinguishing betweenprimordial and astrophysical origin. Our results show that: \textit{(i)} the GWredshift distribution can be calibrated with good accuracy at low redshifts,without any assumptions on cosmology or astrophysics, potentially providing away to probe astrophysical and cosmological models; \textit{(ii)} theconstrains on the dynamical dark energy parameters are competitive with IM-onlyexperiments, in a complementary way and potentially with less systematics;\textit{(iii)} it will be possible to detect a relatively small abundance ofprimordial black holes within the gravitational waves from resolved mergers.Our results extend towards $\mathrm{GW \times IM}$ the promising field ofmulti-tracing cosmology and astrophysics, which has the major advantage ofallowing scientific investigations in ways that would not be possible bylooking at single observables separately.

Gravitational waves $\times$ HI intensity mapping: cosmological and astrophysical applications