The Evolution of Diffuse Radio Sources in Galaxy Clusters

We investigate the evolution and number distribution of radio halos in galaxyclusters. Without re-acceleration or regeneration, the relativistic electronsresponsible for the diffuse radio emission will lose their energy viainverse-Compton and synchrotron losses in a rather short time, and radio haloswill have lifetimes $\sim$ 0.1 Gyr. Radio halos could last for $\sim$ Gyr if asignificant level of re-acceleration is involved. The lifetimes of radio haloswould be comparable with the cosmological time if the radio-emitting electronsare mainly the secondary electrons generated by pion decay followingproton-proton collisions between cosmic-ray protons and the thermalintra-cluster medium within the galaxy clusters. Adopting both observationaland theoretical constraints for the formation of radio halos, we calculate theformation rates and the comoving number density of radio halos in thehierarchical clustering scheme. Comparing with observations, we find that thelifetimes of radio halos are $\sim$ Gyr. Our results indicate that asignificant level of re-acceleration is necessary for the observed radio halosand the secondary electrons may not be a dominant origin for radio halos.Comment: 22 pages, 6 figures, ApJ, in press (v2:Corrected typos.