Abstract
We argue that the observed correlation between the radio luminosity and the thermal X-ray luminosity of radio emitting galaxy clusters implies that the radio emission is due to secondary electrons that are produced by p-p interactions and lose their energy by emitting synchrotron radiation in a strong magnetic field, B > (8πaTCMB4)1/2 ≃ 3 μG. We construct a simple model that naturally explains the correlation, and show that the observations provide stringent constraints on cluster magnetic fields and cosmic rays (CRs): Within the cores of clusters, the ratio βcore between the CR energy (per logarithmic particle energy interval) and the thermal energy is βcore ∼ 2 
10−4; The source of these CRs is most likely the cluster accretion shock, which is inferred to deposit in CRs ∼ 0.1 of the thermal energy it generates; The diffusion time of 100 GeV CRs over scales ≳ 100 kpc is not short compared to the Hubble time; Cluster magnetic fields are enhanced by mergers to ≳ 1% of equipartition, and decay (to < 1 μG) on 1 Gyr time scales. The inferred value of βcore implies that high energy gamma-ray emission from secondaries at cluster cores will be difficult to detect with existing and planned instruments.