Where does the hard X-ray diffuse emission in clusters of galaxies come from?

The surface brightness produced by synchrotron radiation in Clusters of Galaxies with a radio-halo sets a degenerate constraint on the magnetic field strength, the relativistic electron density and their spatial distributions. Using the Coma radio-halo as a case-study, predictions are made for the brightness profile expected in the 20-80 keV band due to ICS by the relativistic electrons on the CMB, for a range of central values of the magnetic field B_0 and models of its radial dependence. We show that the presence of B-field scalar fluctuations on small scales tends to systematically depress the electron density required by the radio data, hence to decrease the ICS brightness expected. These predictions are useful to evaluate the sensitivity required in future imaging HXR instruments, in order to obtain direct information on the spatial distribution and content of relativistic electrons, hence on the magnetic field properties. If compared with the flux in the Coma HXR tail - interpreted as ICS from within the radius R_h - the predictions lead to values of B_0 which are lower than those obtained from Faraday Rotation measurements. The discrepancy is somewhat reduced if the radio-halo profile is extrapolated out to R_{vir}, i.e. about 3 R_h, or if it is assumed that B(r) \propto n_{th}(r) (Dolag et al. 2002). Note that in the latter case, n_{rel}(r) has its minimum value at the center of the cluster. If real and from ICS, the bulk of the HXR tail should then be contributed by electrons other than those responsible for the bulk of the radio-halo emission. This case illustrates the need for spatially resolved spectroscopy in the HXR, in order to obtain solid information on the non-thermal content of Clusters of Galaxies