Polarization of prompt GRB emission: evidence for electromagnetically-dominated outflow

Observations by the {\RHESSI} satellite of large polarization of the prompt $\gamma$-ray emission from the Gamma Ray Burst GRB021206 \citep{coburn03} imply that the magnetic field coherence scale is larger than the size of the visible emitting region, $\sim R/\Gamma$, where $R$ is the radius of the flow, $\Gamma$ is the associated Lorentz factor. Such fields cannot be generated in a causally disconnected, hydrodynamically dominated outflow. Electromagnetic models of GRBs \citep{lyutikov02}, in which large scale, dynamically dominant, magnetic fields are present in the outflow from the very beginning, provide a natural explanation of this large reported linear polarization. We derive Stokes parameters of synchrotron emission of a relativistically moving plasma with a given magnetic field configuration and calculate the pulse averaged polarization fraction of the emission from a relativistically expanding shell carrying global toroidal magnetic field. For viewing angles larger than $1/\Gamma$ the observed patch of the emitting shell has almost homogeneous magnetic field, producing a large fractional polarization ($56$ for a power-law energy distribution of relativistic particles $dn/d\epsilon \propto \epsilon^{-3}$). The maximum polarization is smaller than the theoretical upper limit for a stationary plasma in uniform magnetic field due to relativistic kinematic effects.