Is the rapid decay phase from high latitude emission?

There is good observationnal evidence that the Steep Decay Phase (SDP) that is observed in most Swift GRBs is the tail of the prompt emission. The most popular model to explain the SDP is Hight Latitude Emission (HLE). Many models for the prompt emission give rise to HLE, like the popular internal shocks (IS) model, but some models do not, such as sporadic magnetic reconnection events. Knowing if the SDP is consistent with HLE would thus help distinguish between different prompt emission models. In order to test this, we model the prompt emission (and its tail) as the sum of independent pulses (and their tails). A single pulse is modeled as emission arising from an ultra-relativistic thin spherical expanding shell. We obtain analytic expressions for the flux in the IS model with a Band function spectrum. We find that in this framework the observed spectrum is also a Band function, and naturally softens with time. The decay of the SDP is initially dominated by the tail of the last pulse, but other pulses can dominate later. Modeling several overlapping pulses as a single broader pulse would overestimates the SDP flux. One should thus be careful when testing the HLE.