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 Δα_max of the temporal decay index α from its asymptotic value (at early and late times) is bounded (e.g, Δα_max < 0.4 for a = 10); Δα_max slowly increases with a, converging to Δα_max ≈ 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 α 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.