Distribution of gamma-ray burst ejecta energy with Lorentz factor

The early X-ray afterglow for a significant number of gamma-ray bursts detected by the Swift satellite is observed to have a phase of very slow flux decline with time (F_ν αt^-α with 0.2 ≲ a ≲ 0.8) for 10^2.5 ≲ t ≲10⁴ s, while the subsequent decline is the usual 1 ≲α₃ ≲ 1.5 behaviour, which was seen in the pre-Swift era. We show that this behaviour is a natural consequence of a small spread in the Lorentz factor of the ejecta, by a factor of ~2-4, where the slower ejecta gradually catch up with the shocked external medium, thus increasing the energy of the forward shock and delaying its deceleration. The end of the 'shallow' flux decay stage marks the beginning of the Blandford-McKee self-similar external shock evolution. This suggests that most of the energy in the relativistic outflow is in material with a Lorentz factor of ~30-50.