Probing particle acceleration at trans-relativistic shocks with off-axis gamma-ray burst afterglows

Particle acceleration is expected to be different between relativistic and non-relativistic collisionless shocks. We show that electromagnetic counterparts to gravitational waves (GWs), gamma-ray burst (GRB) afterglows, are ideal targets for observing trans-relativistic evolution of accelerated electron distribution because the GWs spot nearby GRBs with off-axis jets, otherwise missed in gamma-ray observations. We find that the relativistic spectral slope begins to change steeply near the peak time of the light curve and approaches the non-relativistic limit in about 10 times the peak time. The trans-relativistic evolution of the afterglow synchrotron spectrum is consistent with GRB 170817A observations within errors, and will be measurable in similar but more distant events at a GW horizon $\sim 200$ Mpc in a denser environment. We roughly estimate that such events represent a fraction of 10-50 per cent of the GRB 170817A-like off-axis short GRBs. We also find that the spectral evolution does not depend on the jet structure if their light curves are similar to each other.