Synchrotron emission in small-scale magnetic fields as a possible explanation for prompt emission spectra of gamma-ray bursts

Synchrotron emission is believed to be a major radiation mechanism during gamma-ray bursts' (GRBs) prompt emission phase. A significant drawback of this assumption is that the theoretical predicted spectrum, calculated within the framework of the "internal shocks" scenario using the standard assumption that the magnetic field maintains a steady value throughout the shocked region, leads to a slope Fν ∝ ν^-1/2 below 100 keV, which is in contradiction to the much harder spectra observed. This is due to the electron cooling time being much shorter than the dynamical time. In order to overcome this problem, we propose here that the magnetic field created by the internal shocks decays on a length scale much shorter than the comoving width of the plasma. We show that under this assumption synchrotron radiation can reproduce the observed prompt emission spectra of the majority of the bursts. We calculate the required decay length of the magnetic field, and find it to be ∼10⁴-10⁵ cm (equivalent to 10⁵-10 ⁶ skin depths), much shorter than the characteristic comoving width of the plasma, ∼3 × 10⁹ cm. We implement our model to the case of GRB 050820A, where a break at ≲4 keV was observed, and show that this break can be explained by synchrotron self-absorption. We discuss the consequences of the small-scale magnetic field scenario on current models of magnetic field generation in shock waves.