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We analyze the impact of the Fermi non-detection of gamma-ray emission fromclusters of galaxies on hadronic models for the origin of cluster radio halos.In hadronic models, the inelastic proton-proton collisions responsible for theproduction of the electron-positron population fueling the observed synchrotronradio emission yield a gamma-ray flux, from the decay of neutral pions, whosespectrum and normalization depend on the observed radio emissivity and on thecluster magnetic field. We thus infer lower limits on the average clustermagnetic field in hadronic models from the Fermi gamma-ray limits. We alsocalculate the corresponding maximal energy density in cosmic rays and theminimal-guaranteed gamma-ray flux from hadronic radio-halo models. We find thatthe observationally most interesting cases correspond to clusters with largeradio emissivities featuring soft spectra. Estimates of the central magneticfield values for those clusters are larger than, or close, to the largestmagnetic field values inferred from Faraday rotation measures of clusters,placing tension on the hadronic origin of radio halos. In most cases, however,we find that the Fermi data do not per se rule out hadronic models for clusterradio halos as the expected gamma-ray flux can be pushed below the Fermisensitivity for asymptotically large magnetic fields. We also find that cosmicrays do not contribute significantly to the cluster energy budget for nearbyradio halo clusters.

Implications of Fermi Observations for Hadronic Models of Radio Halos in Clusters of Galaxies