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Radio relics are Mpc-sized synchrotron sources located in the peripheralregions of galaxy clusters. Models based on the diffuse shock acceleration(DSA) scenario have been widely accepted to explain the formation of radiorelics. However, a critical challenge to these models is that most observedshocks seem too weak to generate detectable emission, unless fossil electrons,a population of mildly energetic electrons that have been acceleratedpreviously, are included in the models. To address this issue, we present a newsemi-analytical model to describe the formation and evolution of radio relicsby incorporating fossil relativistic electrons into DSA theory, which isconstrained by a sample of 14 observed relics, and employ the Press-Schechterformalism to simulate the relics in a $20^{\circ} \times 20^{\circ}$ sky fieldat 50, 158, and 1400 MHz, respectively. Results show that fossil electronscontribute significantly to the radio emission, which can generate radiationfour orders of magnitude brighter than that solely produced by thermalelectrons at 158 MHz, and the power distribution of our simulated radio reliccatalog can reconcile the observed $P_{1400}-M_{\mathrm{vir}}$ relation. Wepredict that $7.1\%$ clusters with $M_{\mathrm{vir}} > 1.2\times10^{14}\,\mathrm{M}_{\odot}$ would host relics at 158 MHz, which is consistentwith the result of $10 \pm 6\%$ given by the LoTSS DR2. It is also found thatradio relics are expected to cause severe foreground contamination in futureEoR experiments, similar to that of radio halos. The possibility of AGNproviding seed fossil relativistic electrons is evaluated by calculating thenumber of radio-loud AGNs that a shock is expected to encounter during itspropagation.

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