On the ICS interpretation of the Hard X-Ray Excesses in Galaxy Clusters: the case of Ophiuchus
(Abridged) High-E electrons produce Hard X-Ray (HXR) emission in galaxy clusters by via Inverse Compton Scattering (ICS) of CMB photons. We derive the ICS HXR emission of Ophiuchus under various scenarios: primary cosmic ray model, secondary cosmic rays model and neutralino DM annihilation scenario. We further discuss the predictions of the Warming Ray model for the cluster atmosphere. Under the assumption to fit the observed HXR emission, we find that the high-E electrons induce various consequences on the cluster atmosphere: i) primary electrons can be marginally consistent with the data provided that their spectrum is cutoff at E~30(90) MeV for spectral index of 3.5 (4.4); ii) secondary electron models from pp collisions are inconsistent with gamma-ray limits, cosmic ray protons produce too much heating of the IC gas and their pressure at the cluster center largely exceeds the thermal one; iii) secondary electron models from DM annihilation are inconsistent with gamma-ray and radio limits and electrons produce too much heating of the IC gas at the cluster center, unless the neutralino annihilation cross section is much lower than the proposed value. We conclude that ICS by secondary electrons from both neutralino DM annihilation and pp collisions cannot be the mechanism responsible for the HXR excess emission; primary electrons are still a marginally viable solution provided that their spectrum has a low-energy cutoff at E~30-90 MeV. The WR model offers, so far, the best description of the cluster in terms of temperature distribution, heating, pressure and spectral energy distribution. Fermi observations of Ophiuchus will set further constraints to this model.