Monte Carlo techniques are used to model nonlinear particle acceleration in parallel collisionless shocks of various speeds , including mildly relativistic ones .
When the acceleration is efficient , the backreaction of accelerated particles modifies the shock structure and causes the compression ratio , r , to increase above test-particle values .
Modified shocks with Lorentz factors , \gamma _ { 0 } \lesssim 3 , can have compression ratios considerably greater than 3 and the momentum distribution of energetic particles no longer follows a power law relation .
These results may be important for the interpretation of gamma-ray bursts if mildly relativistic internal and/or afterglow shocks play an important role accelerating particles that produce the observed radiation .
For \gamma _ { 0 } \gtrsim 10 , r approaches 3 and the so-called ‘ universal ’ test-particle result of N ( E ) \propto E ^ { -2.3 } is obtained for sufficiently energetic particles .
In all cases , the absolute normalization of the particle distribution follows directly from our model assumptions and is explicitly determined .