We reexamine nonlinear diffusive shock acceleration ( DSA ) at cosmological shocks in the large scale structure of the Universe , incorporating wave-particle interactions that are expected to operate in collisionless shocks .
Adopting simple phenomenological models for magnetic field amplification ( MFA ) by cosmic-ray ( CR ) streaming instabilities and Alfvénic drift , we perform kinetic DSA simulations for a wide range of sonic and Alfvénic Mach numbers and evaluate the CR injection fraction and acceleration efficiency .
In our DSA model the CR acceleration efficiency is determined mainly by the sonic Mach number M _ { s } , while the MFA factor depends on the Alfvénic Mach number and the degree of shock modification by CRs .
We show that at strong CR modified shocks , if scattering centers drift with an effective Alfvén speed in the amplified magnetic field , the CR energy spectrum is steepened and the acceleration efficiency is reduced significantly , compared to the cases without such effects .
As a result , the postshock CR pressure saturates roughly at \sim 20 % of the shock ram pressure for strong shocks with M _ { s } \gtrsim 10 .
In the test-particle regime ( M _ { s } \lesssim 3 ) , it is expected that the magnetic field is not amplified and the Alfvénic drift effects are insignificant , although relevant plasma physical processes at low Mach number shocks remain largely uncertain .