Intrapulse spectral evolution in photospheric gamma-ray bursts

Photons that decouple from a relativistic jet do so over a range of radii, which leads to a spreading in arrival times at the observer. Therefore, changes to the comoving photon distribution across the decoupling zone are encoded in the emitted signal. In this paper, we study such spectral evolution occurring across a pulse. We track the radiation from the deep subphotospheric regions all the way to the observed time-resolved signal, accounting for emission at various angles and radii. We assume a simple power-law photon spectrum injection over a range of optical depths and let the photons interact with the local plasma. At high optical depths, we find that the radiation exists in one of three characteristic regimes, two of which exhibit a high-energy power law. Depending on the nature of the injection, this power law can persist to low optical depths and manifest itself during the rise time of the pulse with a spectral index $\beta \approx \alpha - 1$, where $\alpha$ is the low-energy spectral index. The results are given in the context of a gamma-ray burst jet but are general to optically thick, relativistic outflows.