Constraining the population of cosmic ray protons in cooling flow clusters with gamma-ray and radio observations: Are radio mini-halos of hadronic origin?

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 gamma-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 gamma-ray and bolometric X-ray fluxes, F-gamma and F-X. Based on our relation, we compile a sample of suitable clusters which are promising candidates for future detection of gamma-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 kappa, we find the central CRp injection luminosity of M 87 to be limited to 10(43) erg s(-1) kappa/(10(29) cm(2) s(-1)). 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 gamma-ray observations of the accompanying pi(0)-decays. We identify Perseus (A 426), Virgo, Ophiuchus, and Coma (A 1656) as the most promising candidate clusters for such observations.