Smooth light curves from a bumpy ride: relativistic blast wave encounters a density jump

In the standard forward shock model for gamma-ray burst (GRB) afterglow, the observed afterglow emission is synchrotron radiation from a quasi-spherical, adiabatic, self-similar, relativistic blast wave, that propagates into the external medium. This model predicts a smooth light curve where the flux scales as a power law in time, and may at most smoothly transition to a different power law. However, some GRB afterglow light curves show significant variability, which often includes episodes of rebrightening. Such temporal variability had been attributed in several cases to a large enhancement in the external density, or a density 'bump', that is encountered by the self-similar adiabatic blast wave. Here we examine the effect of a sharp increase in the external density on the afterglow light curve in this scenario by considering, for the first time, a full treatment of both the hydrodynamic evolution and the radiation. To this end we develop a semi-analytic model for the light curve and carry out numerical simulations using a one-dimensional hydrodynamic code together with a synchrotron radiation code. Two spherically symmetric cases are explored in detail - a density jump in a uniform external medium (which is used to constrain the effect of a density clump) and a wind termination shock. We find that even a very sharp (modelled as a step function) and large (by a factor of a > 1) increase in the external density does not produce sharp features in the light curve, and cannot account for significant temporal variability in GRB afterglows in the forward shock model. For a wind termination shock, the light curve smoothly transitions between the asymptotic power laws over about one decade in time, and there is no rebrightening in the optical or X-rays that could serve as a clear observational signature. For a sharp jump in a uniform density profile, we find that the maximal deviation Delta alpha(max) of the temporal decay index alpha from its asymptotic value (at early and late times) is bounded (e.g, Delta alpha(max) < 0.4 for a = 10); Delta alpha(max) slowly increases with a, converging to Delta alpha(max) approximate to 1 at very large a values. Therefore, no optical rebrightening is expected in this case as well. In the X-rays, while the asymptotic flux is unaffected by the density jump, the fluctuations in alpha are found to be comparable to those in the optical. Finally, we discuss the implications of our results for the origin of the observed fluctuations in several GRB afterglows.