Implications of Fermi Observations for Hadronic Models of Radio Halos in
  Clusters of Galaxies
  We analyze the impact of the Fermi non-detection of gamma-ray emission from
clusters of galaxies on hadronic models for the origin of cluster radio halos.
In hadronic models, the inelastic proton-proton collisions responsible for the
production of the electron-positron population fueling the observed synchrotron
radio emission yield a gamma-ray flux, from the decay of neutral pions, whose
spectrum and normalization depend on the observed radio emissivity and on the
cluster magnetic field. We thus infer lower limits on the average cluster
magnetic field in hadronic models from the Fermi gamma-ray limits. We also
calculate the corresponding maximal energy density in cosmic rays and the
minimal-guaranteed gamma-ray flux from hadronic radio-halo models. We find that
the observationally most interesting cases correspond to clusters with large
radio emissivities featuring soft spectra. Estimates of the central magnetic
field values for those clusters are larger than, or close, to the largest
magnetic 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 cluster
radio halos as the expected gamma-ray flux can be pushed below the Fermi
sensitivity for asymptotically large magnetic fields. We also find that cosmic
rays do not contribute significantly to the cluster energy budget for nearby
radio halo clusters.
