Electron Preacceleration in Weak Quasi-perpendicular Shocks in High-beta Intracluster Medium

Giant radio relics in the outskirts of galaxy clusters are known to be lit up by the relativistic electrons produced via diffusive shock acceleration (DSA) in shocks with low sonic Mach numbers, $M_{\rm s}\lesssim3$. The particle acceleration at these collisionless shocks critically depends on the kinetic plasma processes that govern the injection to DSA. Here, we study the preacceleration of suprathermal electrons in weak, quasi-perpendicular ($Q_\perp$) shocks in the hot, high-$\beta$ ($\beta = P_{\rm gas}/P_{\rm B}$) intracluster medium (ICM) through two-dimensional particle-in-cell simulations. \citet{guo2014a,guo2014b} showed that in high-$\beta$ $Q_\perp$-shocks, some of incoming electrons could be reflected upstream and gain energy via shock drift acceleration (SDA). The temperature anisotropy due to the SDA-energized electrons then induces the electron firehose instability (EFI), and oblique waves are generated, leading to a Fermi-like process and multiple cycles of SDA in the preshock region. We find that such electron preacceleration is effective only in shocks above a critical Mach number $M_{\rm ef}^*\approx2.3$. This means that in ICM plasmas, $Q_\perp$-shocks with $M_{\rm s}\lesssim2.3$ may not efficiently accelerate electrons. We also find that even in $Q_\perp$-shocks with $M_{\rm s}\gtrsim2.3$, electrons may not reach high enough energies to be injected to the full Fermi-I process of DSA, because long-wavelength waves are not developed via the EFI alone. Our results indicate that additional electron preaccelerations are required for DSA in ICM shocks, and the presence of fossil relativistic electrons in the shock upstream region may be necessary to explain observed radio relics.