Reviving fossil radio plasma in clusters of galaxies by adiabatic compression in environmental shock waves

We give for a plasma with a history of several expansion and contraction phases an analytical model of the evolution of a contained relativistic electron population under synchrotron, inverse Compton and adiabatic energy losses or gains. This is applied to different scenarios for evolution of radio plasma inside the cocoons of radio galaxies, after the activity of the central engine has ceased. It is demonstrated that fossil radio plasma with an age of even up to 2 Gyr can be revived by compression in a shock wave of large-scale structure formation, caused during the merging events of galaxy clusters, or by the accretion onto galaxy clusters. We argue, that this is a highly plausible explanation for the observed cluster radio relics, which are the regions of diffuse radio emission found in clusters of galaxies, without any likely parent radio galaxy seen nearby. An implication of this model is the existence of a population of diffuse, ultra-steep spectrum, very low frequency radio sources located inside and possibly outside of clusters of galaxies, tracing the revival of aged fossil radio plasma by the shock waves associated with large-scale structure formation