Collisionless shocks with low sonic Mach numbers , M _ { s } \lesssim 4 , are expected to accelerate cosmic ray ( CR ) protons via diffusive shock acceleration ( DSA ) in the intracluster medium ( ICM ) .
However , observational evidence for CR protons in the ICM has yet to be established .
Performing particle-in-cell simulations , we study the injection of protons into DSA and the early development of a nonthermal particle population in weak shocks in high \beta ( \approx 100 ) plasmas .
Reflection of incident protons , self-excitation of plasma waves via CR-driven instabilities , and multiple cycles of shock drift acceleration are essential to the early acceleration of CR protons in supercritical quasi-parallel shocks .
We find that only in ICM shocks with M _ { s } \gtrsim M _ { s } ^ { * } \approx 2.25 , a sufficient fraction of incoming protons are reflected by the overshoot in the shock electric potential and magnetic mirror at locally perpendicular magnetic fields , leading to efficient excitation of magnetic waves via CR streaming instabilities and the injection into the DSA process .
Since a significant fraction of ICM shocks have M _ { s } < M _ { s } ^ { * } , CR proton acceleration in the ICM might be less efficient than previously expected .
This may explain why the diffuse gamma-ray emission from galaxy clusters due to proton-proton collisions has not been detected so far .