D. Inverse Compton
Inverse Compton (IC) scattering may modify our analysis in several ways. IC can influence the spectrum even if the system is optically thin (as it must be) to Compton scattering (see e.g. Rybicki and Lightman ). In view of the high energies involved a photon is IC scattered only once. After a single IC scattering the photon's energy is so high that in the electron's rest frame it is above the Klein-Nishina energy (me c2 ~ 0.5 Mev), and the decrease in the Compton cross section in this energy range makes a second scattering unlikely. Note that in some cases (e.g. in forward external shocks) even the first scattering may suffer from this problem. The effect of IC depends on the Comptonization parameter Y = 2 e. For fast cooling one can show  that Y satisfies:
where Ue and UB are the energy densities of the electron's and of the magnetic field respectively. IC is unimportant if Y < 1 and in this case it can be ignored.
If Y > 1, which corresponds to Ue > UB (or to e > B) and to Y = (Ue / UB)1/2, then a large fraction of the low energy synchrotron radiation will be up scattered by IC and a large fraction of the energy will be emitted via the IC processes. Those IC photons might be too energetic, that is their energy may be far beyond the observed energy range. In this case IC will not influence the observed spectra directly. However, as IC will take a significant fraction of the energy of the cooling electrons it will influence the observations in two ways: it will shorten the cooling time (the emitting electrons will be cooled by both synchrotron and IC process). Second, assuming that the observed -ray photons results from synchrotron emission, IC will influence the overall energy budget and reduce the efficiency of the production of the observed radiation. I turn now to each of this cases.
An IC scattering boosts the energy of the photon by a factor e2. Typical synchroton photon that have been scattered once by IC will be observed at the energy:
The electrons are cooled both by synchrotron and by IC. The latter is more efficient and the cooling is enhanced by the Compton parameter Y. The cooling time scale is:
The conditions needed to produce the observed emission using IC are probably not fulfilled in either external or internal shocks (see however Ghisellini and Celotti  and the discussion in Section VE below). However even if IC does not produce the observed -ray photons it still influences the process if Y > 1. First it will add an ultra high energy component to the GRB spectrum. This component will typically be at around e2 times the observed ~ 100 KeV photons, namely at the GeV-TeV range (see e.g. Bottcher and Dermer , Vietri  and the discussion in Section VIIIA). This component might have been already observed in some GRBs during the early afterglow (see Section IIA1). Inverse Compton will also speed up the cooling of the emitting regions and shorten the cooling time, tsyn estimated earlier (Eq. 19) by a factor of Y. At the same time this also reduces the efficiency (for producing the observed -rays) by the same factor.