Correlated Spectral And Temporal Behaviour Of Late-Time Afterglows Of Gamma Ray Bursts

The cannonball (CB) model of gamma ray bursts (GRBs) predicts that the asymptotic behaviour of the spectral energy density of the X-ray afterglow of GRBs is a power-law in time and in frequency where the difference between the temporal and spectral power-law indexes, ${\alpha }_X-{\beta }_X$, is restricted to the values 0, 1/2 and 1. Here we report the distributions of the values ${\alpha }_X$, ${\beta }_X$ and their difference for a sample of 315 Swift GRBs. This sample includes all Swift GRBs that were detected before August 1, 2012, whose X-ray afterglow extended well beyond 1 day and the estimated error in ${\alpha }_X-{\beta }_X$ was $\le 0.25$. The values of ${\alpha }_X$ 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 ${\alpha }_X-{\beta }_X$ 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.