Re-acceleration of Nonthermal Particles at Weak Cosmological Shock Waves
  We examine diffusive shock acceleration (DSA) of the pre-exisiting as well as
freshly injected populations of nonthermal, cosmic-ray (CR) particles at weak
cosmological shocks. Assuming simple models for thermal leakage injection and
Alfv\'enic drift, we derive analytic, time-dependent solutions for the two
populations of CRs accelerated in the test-particle regime. We then compare
them with the results from kinetic DSA simulations for shock waves that are
expected to form in intracluster media and cluster outskirts in the course of
large-scale structure formation. We show that the test-particle solutions
provide a good approximation for the pressure and spectrum of CRs accelerated
at these weak shocks. Since the injection is extremely inefficient at weak
shocks, the pre-existing CR population dominates over the injected population.
If the pressure due to pre-existing CR protons is about 5 % of the gas thermal
pressure in the upstream flow, the downstream CR pressure can absorb typically
a few to 10 % of the shock ram pressure at shocks with the Mach number $M \la
3$. Yet, the re-acceleration of CR electrons can result in a substantial
synchrotron emission behind the shock. The enhancement in synchrotron radiation
across the shock is estimated to be about a few to several for $M \sim 1.5$ and
$10^2-10^3$ for $M \sim 3$, depending on the detail model parameters. The
implication of our findings for observed bright radio relics is discussed.
