2013MNRAS.435.1061P

Many bright radio relics in the outskirts of galaxy clusters have low inferred Mach numbers, defying expectations from shock acceleration theory and heliospheric observations that the injection efficiency of relativistic particles plummets at low Mach numbers. With a suite of cosmological simulations, we follow the diffusive shock acceleration as well as radiative and Coulomb cooling of cosmic ray electrons during the assembly of a cluster. We find a substantial population of fossil electrons. When reaccelerated at a shock (through diffusive shock acceleration), they are competitive with direct injection at strong shocks and overwhelmingly dominate by many orders of magnitude at weak shocks, M\lesssim 3, which are the vast majority at the cluster periphery. Their relative importance depends on cooling physics and is robust to the shock acceleration model used. While the abundance of fossils can vary by a factor of ∼ 10, the typical reaccelerated fossil population has radio brightness in excellent agreement with observations. Fossil electrons with 1 ≲ γ ≲ 100 (10 ≲ γ ≲ 10⁴) provide the main seeds for reacceleration at strong (weak) shocks; we show that these are well resolved by our simulation. We construct a simple self-similar analytic model which assumes steady recent injection and cooling. It agrees well with our simulations, allowing rapid estimates and physical insight into the shape of the distribution function. We predict that the Low-Frequency Array (LOFAR) should find many more bright steep-spectrum radio relics, which are inconsistent with direct injection. A failure to take fossil cosmic ray electrons into account will lead to erroneous conclusions about the nature of particle acceleration at weak shocks; they arise from well-understood physical processes and cannot be ignored.