The Luminosity Function of Cluster Radio Relics

In this paper we compute the luminosity function of radio relics. In our calculation we include only those relics that are produced by the compression of former radio cocoons. This compression is provided by shocks that are generated in the process of structure formation. Starting from an analytical model for the luminosity evolution of ageing radio cocoons, the luminosity function of radio galaxies and the statistics of shocks as inferred from cosmological simulations, we are able to make the first estimates of the brightness distribution of radio relics. The computed luminosity function is consistent with current observations and predicts that more than $10^3$ radio relics should be discovered with the upcoming generation of low-frequency radio telescopes. Moreover, we predict that radio relics are predominantly found in low-pressure regions outside the cores of clusters.