Relativistic protons in the Coma galaxy cluster: first gamma-ray constraints ever on turbulent reacceleration

The Fermi-LAT (Large Area Telescope) collaboration recently published deep upper limits to the gamma-ray emission of the Coma cluster, a cluster hosting the prototype of giant radio haloes. In this paper, we extend previous studies and use a formalism that combines particle reacceleration by turbulence and the generation of secondary particles in the intracluster medium to constrain relativistic protons and their role for the origin of the radio halo. We conclude that a pure hadronic origin of the halo is clearly disfavoured as it would require excessively large magnetic fields. However, secondary particles can still generate the observed radio emission if they are reaccelerated. For the first time the deep gamma-ray limits allow us to derive meaningful constraints if the halo is generated during phases of reacceleration of relativistic protons and their secondaries by cluster-scale turbulence. In this paper, we explore a relevant range of parameter space of reacceleration models of secondaries. Within this parameter space, a fraction of model configurations is already ruled out by current gamma-ray limits, including the cases that assume weak magnetic fields in the cluster core, B <= 2-3 mu G. Interestingly, we also find that the flux predicted by a large fraction of model configurations assuming magnetic fields consistent with Faraday rotation measures (RMs) is not far from the limits. This suggests that a detection of gamma-rays from the cluster might be possible in the near future, provided that the electrons generating the radio halo are secondaries reaccelerated and the magnetic field in the cluster is consistent with that inferred from RM.