Probing the Magnetic Field Structure in Gamma-Ray Bursts through Dispersive Plasma Effects on the Afterglow Polarization

The origin and structure of magnetic fields in gamma-ray burst (GRB) fireball plasmas are two of the most important open questions in all GRB models. It has been claimed that recent measurements of γ-ray polarization suggest the presence of a uniform field originating in the compact object driving the outflow. This interpretation is, however, controversial, since a high degree of linear polarization is also possible in the presence of a random magnetic field, arguably originating in electromagnetic instabilities that develop at the collisionless shock. We show that the structure and strength of the magnetic field may be constrained by radio and IR observations of the early afterglow, where plasma effects on the polarization of the propagating radiation are significant. We calculate these propagation effects for cold and relativistic plasmas, and find that in the presence of a uniform equipartition field the degree of linear polarization is suppressed and circular polarization prevails at low frequencies, below ~1-3 GHz (~a few × 10¹⁴ Hz) in the forward (reverse) shock, at the onset of fireball deceleration. At higher frequencies linear polarization dominates. At the frequency of the transition between circular and linear polarization, the net level of polarization is minimal, ~10%-20%. These features are nearly independent of the density of the environment into which the fireball expands. When the uniform field is much weaker than the equipartition value, the transition frequency is smaller by an order of magnitude. Depending on the geometry of the emitting region, oscillations of the polarization position angle may be observed from the optical reverse-shock emission, provided that the strength of the magnetic field is close to equipartition. The dependence of these results on viewing geometry, outflow collimation, and magnetic field orientation is discussed. When the field is entangled over length scales much smaller than the extent of the emitting plasma, the aforementioned effects should not be observed, and a linear polarization at the few percent level is expected. Polarimetric observations during the early afterglow, and particularly of the reverse-shock emission, may therefore place strong constraints on the structure and strength of the magnetic field within the fireball plasma.