Correlated Spectral and Temporal Behavior of Late-Time Afterglows of Gamma-Ray Bursts

The cannonball (CB) model of gamma-ray bursts (GRBs) predicts that the asymptotic behavior of the spectral energy density of GRB afterglows is a power law in time and in frequency, and the difference between the temporal and spectral power-law indices, α X − β X , is restricted to the values 0, 1/2, and 1. Here we report the distributions of the values α X and β X , and their difference for a sample of 315 Swift GRBs. This sample includes all Swift GRBs that were detected before 2012 August 1, whose X-ray afterglow extended well beyond 1 day and the estimated error in α X − β X was ⩽0.25. The values of α X were extracted from the CB-model fits to the entire light curves of their X-ray afterglow while the spectral index was extracted by the Swift team from the time-integrated X-ray afterglow of these GRBs. We found that the distribution of the difference α X − β X for these 315 Swift GRBs has three narrow peaks around 0, 1/2, and 1 whose widths are consistent with being due to the measurement errors, in agreement with the CB-model prediction.