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

The Fermi-LAT collaboration recently published deep upper limits to the gamma-ray emission of the Coma cluster, a cluster that hosts the prototype of giant radio halos. 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 radio halo is clearly disfavoured as it would require magnetic fields that are too strong. For instance $B_0 > 21 \mu$G is found in the cluster center assuming that the magnetic energy density scales with thermal density, to be compared with $B_0 \sim 4-5 \mu$G as inferred from Rotation Measures (RM) under the same assumption. 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. 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, $B \leq 2-3 \mu$G. Interestingly, we also find that the flux predicted by a large fraction of model configurations that assume a magnetic field consistent with RM 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.