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We forecast the potential of the forthcoming X-ray galaxy-cluster survey witheROSITA to constrain dark-energy models. We focus on spatially-flatcosmological scenarios with either constant or time-dependent dark-energyequation-of-state parameters. Fisher information is extracted from the numberdensity and spatial clustering of a photon-count-limited sample of clusters ofgalaxies up to z~2. We consider different scenarios for the availability of (i)X-ray follow-up observations, (ii) photometric and spectroscopic redshifts, and(iii) accurate knowledge of the observable -- mass relation down to the scaleof galaxy groups. With about 125,000 clusters (detected with more than 50photons and with mass M500c > $10^{13} h^{-1}$ Msun) from an average all-skyexposure of 1.6 ks, eROSITA will give marginalized, one-dimensional, 1-$\sigma$errors of $\Delta \sigma_8 = \pm~0.008$ ($\sim$1 per cent), $\Delta \Omega_{\rmM} = \pm~0.006$ (2.2 per cent), $\Delta w_0 = \pm~0.07$ (7 per cent), and$\Delta w_a = \pm~0.25$ (optimistic scenario) in combination with (and largelyimproving upon) current constraints from various cosmological probes (cosmicmicrowave background, BAOs, Type Ia SNe). Our findings correspond to adark-energy figure of merit in the range of $116-162$ (after the four years ofall-sky survey), making eROSITA one of the first Stage IV experiments to comeon line according to the classification of the Dark Energy Task Force. Tosecure improved mass calibrations and to include high-redshift clusters (z >0.5) as well as objects at the group-mass scale (M500c < 5 $\times~ 10^{13}h^{-1}$ Msun) will be vital to reach such accuracies.

Forecasts on dark energy from the X-ray cluster survey with eROSITA: constraints from counts and clustering