Radiation from an expanding cocoon as an explanation of the steep decay observed in GRB early afterglow light curves

Observations of early afterglow emission from gamma-ray bursts (GRB's) with the Swift satellite show steep decay of the X-ray light curve, F_\nu(t) ~ t^{-\alpha} with \alpha ~ 2.5 - 4 at ~300-500 s after the burst trigger. The spectrum in this time interval is consistent with a spectrum F_\nu ~ \nu^{-\beta} with \beta ~1. Here, we show that these results can be explained as due to emission from the hot plasma "cocoon" associated with the jet, which expands relativistically after the jet has broken through the stellar envelope, if a substantial fraction of the coccon kinetic energy is dissipated at scattering optical depths \tau_T ~ 10^2-10^3. This results in the bulk of the coccon photons being observed at X-ray energies, after a delay of few hundreds of seconds relative to the gamma-ray photons from the jet. Multiple Compton scattering inside the cocoon causes a spread in the arrival times of the X-ray photons. We calculate numerically the observed light curve of photons emerging from the cocoon, and show that it exhibits a steep decay, which resembles that observed in many GRB afterglows. During the adiabatic expansion that follows the dissipation phase, photons lose energy to the expanding plasma, and as a result, the emerging photon energy distribution differs from a black-body spectrum, and can be approximated as a power law in the Swift XRT band. Comparison of the numerical results with the Swift XRT data of GRB050315 and GRB050421 shows good agreement between the light curves and spectra during the initial steep decay phase.