Importance of the Resonant Cosmic-Ray Streaming Instability Upstream of Collisionless Shocks
Cosmic rays (CRs) escaping collisionless shocks form a dilute, relativistic population whose current amplifies the upstream magnetic field, a process widely attributed to the non-resonant (Bell) instability. Solving the dispersion relation for both cold and finite-spread CR distributions, we show that for the maximum-energy escaping population, the resonant mode can outgrow the non-resonant mode. At slower shocks, this dominance persists for broader CR distributions, and sufficient pitch-angle broadening can even stabilize the non-resonant mode while leaving the resonant mode unstable. Relativistic hybrid particle-in-cell simulations confirm the cold-beam linear theory predictions and, in the nonlinear regime, saturate at $δB/B_0 \sim 1$ with significant pitch-angle redistribution. Neglecting the resonant instability thus underestimates magnetic field growth at the very scale needed to confine the highest-energy escaping CRs.