Neutrinos from Clusters of Galaxies and Radio Constraints
  Cosmic-ray (CR) protons can accumulate for cosmological times in clusters of
galaxies. Their hadronic interactions with protons of the intra-cluster medium
(ICM) generate secondary electrons, gamma-rays and high-energy neutrinos. In
light of the high-energy neutrino events recently discovered by the IceCube
observatory, we estimate the contribution from galaxy clusters to the diffuse
gamma-ray and neutrino backgrounds. For the first time, we consistently take
into account the synchrotron emission generated by secondary electrons and
require the clusters radio counts to be respected. For a choice of parameters
respecting current constraints from radio to gamma-rays, and assuming a proton
spectral index of -2, we find that hadronic interactions in clusters contribute
by less than 10% to the IceCube flux, and much less to the total extragalactic
gamma-ray background observed by Fermi. They account for less than 1% for
spectral indexes <-2. The high-energy neutrino flux observed by IceCube can be
reproduced without violating radio constraints only if a very hard (and
speculative) spectral index >-2 is adopted. However, this scenario is in
tension with the high-energy IceCube data, which seem to suggest a spectral
energy distribution of the neutrino flux that decreases with the particle
energy. We stress that our results are valid for all kind of sources injecting
CR protons into the ICM, and that, while IceCube can test the most optimistic
scenarios for spectral indexes >=-2.2 by stacking few nearby massive objects,
clusters of galaxies cannot give any relevant contribution to the extragalactic
gamma-ray and neutrino backgrounds in any realistic scenario.
