  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, Mach < 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 <
gamma < 100 (10 < gamma < 10^4) 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 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.
