Modeling fluctuations in gamma-ray burst afterglow light curves

The fluctuations observed in the light curves of some gamma-ray burst (GRB) afterglows ( such as GRB 021004) provide a useful tool to probe the circumburst density pro. le and to probe the variations in the energy of the blast wave with time. We present a general formalism that reduces the calculation of the observed light curve from a Blandford-McKee blast wave to the evaluation of a one-dimensional integral. Using this formalism we obtain a simple approximation to the general light curve that arises in more complex situations where the afterglow's energy or the external density varies. The solution is valid for spherically symmetric profiles, and it takes full consideration of angular time delay effects. We present the light curves of several external density profiles and demonstrate the effects of density variations on the light curve. We also revisit the afterglow of GRB 021004 and find that the steep decay after the first bump ( similar to 4000 s) cannot result from a spherically symmetric density variation or from the passage of the synchrotron frequency through the optical band. This suggests that an angular structure is responsible for some of the observed features in the light curve. This may be the first evidence that an angular structure is important in the early stages of the afterglow.