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 .
Recent measurements of \gamma - ray polarization were claimed to 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 \sim 1 - 3 GHz ( \sim \textrm { few } \times 10 ^ { 14 } 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 , \sim 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 % 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 .