Sub-photospheric, radiation mediated shocks in GRBs: Multiple shock emission and the Band spectrum

We compute the time integrated, thermal emission produced by a series of radiation mediated shocks that emerge from the photosphere of a GRB outflow. We show that for a sufficiently broad distribution of shock strengths, the overall shape of the time integrated spectral energy distribution below the peak is a power law, $\nu E_\nu\propto \nu^\alpha$, with a slope $1<\alpha<2$. A substructure in the SED can also be produced in this model for certain choices of the shock train distribution. In particular, we demonstrate that our model can reproduce the double-peak SED observed in some bursts, in events whereby a strong shock is followed by a sequence sufficiently weaker ones.