A Semi-Analytical Model for the Formation and Evolution of Radio Relics in Galaxy Clusters

Radio relics are Mpc-sized synchrotron sources located in the peripheral regions of galaxy clusters. Models based on the diffuse shock acceleration (DSA) scenario have been widely accepted to explain the formation of radio relics. However, a critical challenge to these models is that most observed shocks seem too weak to generate detectable emission, unless fossil electrons, a population of mildly energetic electrons that have been accelerated previously, are included in the models. To address this issue, we present a new semi-analytical model to describe the formation and evolution of radio relics by incorporating fossil relativistic electrons into DSA theory, which is constrained by a sample of 14 observed relics, and employ the Press-Schechter formalism to simulate the relics in a $20^{\circ} \times 20^{\circ}$ sky field at 50, 158, and 1400 MHz, respectively. Results show that fossil electrons contribute significantly to the radio emission, which can generate radiation four orders of magnitude brighter than that solely produced by thermal electrons at 158 MHz, and the power distribution of our simulated radio relic catalog can reconcile the observed $P_{1400}-M_{\mathrm{vir}}$ relation. We predict that $7.1\%$ clusters with $M_{\mathrm{vir}} > 1.2\times 10^{14}\,\mathrm{M}_{\odot}$ would host relics at 158 MHz, which is consistent with the result of $10 \pm 6\%$ given by the LoTSS DR2. It is also found that radio relics are expected to cause severe foreground contamination in future EoR experiments, similar to that of radio halos. The possibility of AGN providing seed fossil relativistic electrons is evaluated by calculating the number of radio-loud AGNs that a shock is expected to encounter during its propagation.