Observations of radio halos and relics in galaxy clusters indicate efficient electron acceleration. Protons should likewise be accelerated, suggesting that clusters may also be sources of very high-energy (VHE; E>100 GeV) gamma-ray emission. We report here on VHE gamma-ray observations of the Coma galaxy cluster with the VERITAS array of imaging Cherenkov telescopes, with complementing Fermi-LAT observations at GeV energies. No significant gamma-ray emission from the Coma cluster was detected. Integral flux upper limits at the 99% confidence level were measured to be on the order of (2-5)*10^-8 ph. m^-2 s^-1 (VERITAS, >220 GeV} and ~2*10^-6 ph. m^-2 s^-1 (Fermi, 1-3 GeV), respectively. We use the gamma-ray upper limits to constrain CRs and magnetic fields in Coma. Using an analytical approach, the CR-to-thermal pressure ratio is constrained to be < 16% from VERITAS data and < 1.7% from Fermi data (averaged within the virial radius). These upper limits are starting to constrain the CR physics in self-consistent cosmological cluster simulations and cap the maximum CR acceleration efficiency at structure formation shocks to be <50%. Assuming that the radio-emitting electrons of the Coma halo result from hadronic CR interactions, the observations imply a lower limit on the central magnetic field in Coma of (2 - 5.5) muG, depending on the radial magnetic-field profile and on the gamma-ray spectral index. Since these values are below those inferred by Faraday rotation measurements in Coma (for most of the parameter space), this {renders} the hadronic model a very plausible explanation of the Coma radio halo. Finally, since galaxy clusters are dark-matter (DM) dominated, the VERITAS upper limits have been used to place constraints on the thermally-averaged product of the total self-annihilation cross section and the relative velocity of the DM particles, <sigma v>. (abr.)
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