The intra-cluster medium of several galaxy clusters hosts large-scale regions of diffuse synchrotron radio emission, known as radio halos and relics, which demonstrate the presence of magnetic fields and relativistic electrons in clusters. These relativistic electrons should also emit X-rays through inverse-Compton scattering off of cosmic microwave background photons. The detection of such a non-thermal X-ray component, together with the radio measurement, would permit to clearly separate the magnetic field from the relativistic electron distribution as the inverse-Compton emission is independent from the magnetic field in the cluster. However, non-thermal X-rays have not been conclusively detected from any cluster of galaxies so far. In this paper, for the first time, we model the synchrotron and inverse-Compton emission of all clusters hosting radio halos and relics for which the spectral index can be determined. We provide constraints on the volume-average magnetic field by comparing with current X-ray measurements. We then estimate the maximum volume-average magnetic field that will allow the detection of inverse-Compton hard X-rays by the ASTRO-H satellite. We found that several clusters are good targets for ASTRO-H to detect their inverse-Compton emission, in particular that corresponding to radio relics, and propose a list of promising targets for which ASTRO-H can test $ge1$~$mu$G magnetic fields. We conclude that future hard X-ray observations by the already-operating NuSTAR and the soon-to-be-launched ASTRO-H definitely have the potential to shed light on the long-sought non-thermal hard-X-ray emission in clusters of galaxies.
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