2004A&A...413...17P

We wish to constrain the cosmic-ray proton (CRp) population in galaxy clusters. By hadronic interactions with the thermal gas of the intra-cluster medium (ICM), the CRp produce γ-rays for which we develop an analytic formalism to deduce their spectral distribution. Assuming the CRp-to-thermal energy density ratio X_CRp and the CRp spectral index to be spatially constant, we derive an analytic relation between the γ-ray and bolometric X-ray fluxes, F_γ and FX. Based on our relation, we compile a sample of suitable clusters which are promising candidates for future detection of γ-rays resulting from hadronic CRp interactions. Comparing to EGRET upper limits, we constrain the CRp population in the cooling flow clusters Perseus and Virgo to X_CRp<20%. Assuming a plausible value for the CRp diffusion coefficient κ, we find the central CRp injection luminosity of M 87 to be limited to 10⁴³erg/sκ/(10²⁹cm2/s). The synchrotron emission from secondary electrons generated in CRp hadronic interactions allows even tighter limits to be placed on the CRp population using radio observations. We obtain excellent agreement between the observed and theoretical radio brightness profiles for Perseus, but not for Coma without a radially increasing CRp-to-thermal energy density profile. Since the CRp and magnetic energy densities necessary to reproduce the observed radio flux are very plausible, we propose synchrotron emission from secondary electrons as an attractive explanation of the radio mini-halos found in cooling flow clusters. This model can be tested with future sensitive γ-ray observations of the accompanying π⁰-decays. We identify Perseus (A 426), Virgo, Ophiuchus, and Coma (A 1656) as the most promising candidate clusters for such observations.