Time-Integrated Gamma-Ray Burst Synchrotron Spectra from Blast Wave/Cloud Interactions

We show that the spectral shape of the low energy tails found for the time-integrated spectra of gamma-ray bursts, even in the absence of strong synchrotron cooling, can be significantly softer than the $\nu F_\nu \propto \nu^{4/3}$ asymptote predicted by synchrotron shock models. As we have noted in a previous work, blast wave deceleration via interaction with ambient material causes the characteristic electron injection energy to decrease in proportion to the bulk Lorentz factor of the blast wave, and under certain conditions, this effect will at least partially account for the observed increase in pulse widths with decreasing energy. This spectral softening can also be reflected in the time-integrated pulse spectrum. Using a simple model for the blast wave interaction with a dense cloud of material, we show that just below the $\nu F_\nu$ spectral peak the integrated spectrum behaves as $\nu F_\nu \sim \nu^{1/2}$ and rolls over to a $\nu^{4/3}$ dependence at lower energies, thus a spectral shape arises which is similar to that predicted for the spectrum of a strongly synchrotron-cooled electron population. We discuss the implications of this work in the context of models of burst light curve variability which are based on blast wave/cloud interactions.