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, $B(r)$ and $n_{\rm rel}(r)$, in the intracluster medium. Using the radio-halo in the Coma Cluster as a case study, with the radio brightness profile and the spectral index as the only constraints, predictions are made for the brightness profiles expected in the 20–80 keV band due to Inverse Compton Scattering (ICS) by the relativistic electrons on the Cosmic Microwave Background. This is done for a range of central values of the magnetic field, B, and models of its radial dependence, $B(r)$ (of which two represent extreme situations, namely a constant value either of B or of $n_{\rm rel}(r)$, the third a more realistic intermediate case). It is shown that the possible presence of scalar fluctuations on small scales in the strength of B tends to systematically depress the electron density required by the radio data, hence to decrease the ICS brightness expected. These predictions should be 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 HXR tail, whose detection has been claimed in the Coma cluster (Fusco-Femiano et al. 2004, ApJ, 602, L73), when interpreted as ICS from within the radius Rh of the radio-halo, as measured so far, the predictions lead to central values B0 that are significantly lower than those which have been obtained (albeit still controversial) from Faraday Rotation measurements. The discrepancy is somewhat reduced if the radio-halo profile is hypothetically extrapolated out to Rvir, that is about three times Rh, or, as suggested by hydrodynamical simulations (Dolag et al. 2002, A&A, 387, 383), if it is assumed that $B(r) \propto n_{\rm th}(r)$. In the latter case $n_{\rm rel}(r)$ 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.