  The electron energy distribution index, p, is a fundamental parameter of the
process by which electrons at shock fronts are accelerated to relativistic
speeds and by which they radiate, via synchrotron emission. This acceleration
process is applicable to a myriad of astronomical jet sources such as AGN,
X-ray binaries and gamma-ray bursts (GRBs), as well as to particle acceleration
in the solar wind and supernovae, and the acceleration of cosmic rays. The
accurate measurement of the distribution of p is of fundamental importance to
differentiate between the possible theories of electron acceleration at any
relativistic shock front; there is division as to whether p has a universal
value or whether it has a distribution, and if so, what that distribution is.
Here one such source of synchrotron emission is examined: the blast waves of
GRB afterglows observed by the Swift satellite. Within the framework of the GRB
blast wave model, the constraints placed on the distribution of p by the
observed X-ray spectral and temporal indices are examined and the distribution
parametrized. The results show that the observed distribution of spectral
indices is inconsistent with an underlying distribution of p composed of a
single discrete value but consistent with a Gaussian distribution centred at p
= 2.4 and having a width of 0.6. This finding disagrees with theoretical work
that argues for a single, universal value of p, but also demonstrates that the
width of the distribution is not as wide as has been suggested by some authors.
