  (Abr) 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. 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 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, nu < 1-3 GHz, (2x10^11 Hz < nu < few x 10^14 Hz) in the forward
(reverse) shock. 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 circumburst density. The transition frequency is smaller by
a factor of ~10 when the uniform field is much weaker than equipartition. The
dependence of these results on viewing geometry, outflow collimation and
magnetic field orientation is discussed. When the configuration of 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 % 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.
