High-energy gamma-ray and neutrino backgrounds from clusters of galaxies and radio constraints

Cosmic-ray protons accumulate for cosmological times in clusters of galaxies because their typical radiative and diffusive escape times are longer than the Hubble time. Their hadronic interactions with protons of the intra-cluster medium generate secondary electrons, gamma rays, and neutrinos. In light of the high-energy neutrino events recently discovered by the IceCube neutrino observatory, for which galaxy clusters have been suggested as possible sources, and the forthcoming results from the Fermi gamma-ray survey, we here estimate the contribution from galaxy clusters to the diffuse gamma-ray and neutrino backgrounds. We modelled the cluster population by means of their mass function, using a phenomenological luminosity-mass relation applied to all clusters, as well as a detailed semi-analytical model. In the latter model, we divide clusters into cool-core/non-cool-core, and loud/quiet subsamples, as suggested by observations, and model the cosmic-ray proton population according to state-of-the-art hydrodynamic numerical simulations. Additionally, we consider observationally-motivated values for the cluster magnetic field. This is a crucial parameter since the observed radio counts of clusters need to be respected owing to synchrotron emission by secondary electrons. 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 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 indices <=-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 seems to suggest a spectral energy distribution of the neutrino flux that decreases with the particle energy. We prove that IceCube should be able to test our most optimistic scenarios for spectral indices >=-2.2 by stacking a few nearby massive galaxy clusters. In the case of proton-photon interactions in clusters, we find that very likely protons do not reach sufficiently high energies to produce neutrinos in these environments. We argue that our results are optimistic because of our assumptions and that clusters of galaxies cannot make any relevant contribution to the extragalactic gamma-ray and neutrino backgrounds in any realistic scenario. Finally, we find that the cluster contribution to the angular fluctuations in the gamma-ray background is subdominant, less than 10% on sub-degree scales.