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Shock Waves and Cosmic Ray Acceleration in the Outskirts of Galaxy Clusters

The outskirts of galaxy clusters are continuously disturbed by mergers and gas infall along filaments, which in turn induce turbulent flow motions and shock waves. We examine the properties of shocks that form within $r_{200}$ in sample galaxy clusters from structure formation simulations. While most of these shocks are weak and inefficient accelerators of cosmic rays (CRs), there are a number of strong, energetic shocks which can produce large amounts of CR protons via diffusive shock acceleration. We show that the energetic shocks reside mostly in the outskirts and a substantial fraction of them are induced by infall of the warm-hot intergalactic medium from filaments. As a result, the radial profile of the CR pressure in the intracluster medium is expected to be broad, dropping off more slowly than that of the gas pressure, and might be even temporarily inverted, peaking in the outskirts. The volume-integrated momentum spectrum of CR protons inside $r_{200}$ has the power-law slope of $4.25 - 4.5$, indicating that the average Mach number of the shocks of main CR production is in the range of $\left< M_s \right>_{\rm CR} \approx 3 - 4$. We suggest that some radio relics with relatively flat radio spectrum could be explained by primary electrons accelerated by energetic infall shocks with $M_s >~ 3$ induced in the cluster outskirts.