  The magnetic field structure in GRB outflows is of great interest as it can
provide valuable clues that can help pin down the mechanism responsible for the
acceleration and collimation of GRB jets. The most promising way of probing it
is through polarization measurements of the synchrotron emission from the GRB
ejecta, which includes the gamma-ray emission and the reverse shock emission.
Measuring polarization in gamma-rays with current instruments is extremely
difficult: so far there is only one claim of detection (in GRB 021206) which
despite the favorable conditions remains very controversial. The emission from
the reverse shock that propagates into the ejecta as it is decelerated by the
ambient medium peaks in the optical after tens of seconds (the `optical flash')
and dominates the optical emission up to about ten minutes after the GRB.
Unfortunately, no polarization measurements of this optical emission have been
made to date. However, after the reverse shock finishes crossing the shell of
GRB ejecta, the shocked ejecta cools adiabatically and radiates at lower and
lower frequencies, peaking in the radio after $\sim 1$ day (the `radio flare').
We use VLA data of radio flares from GRBs to constrain the polarization of this
emission. We find only upper limits for both linear and circular polarization.
Our best limits are for GRB 991216, for which we find $3\sigma$ upper limits on
the linear and circular polarization of 7% and 9%, respectively. These limits
provide interesting constraints on existing GRB models. Specifically, our
results are hard to reconcile with a predominantly ordered toroidal magnetic
field in the GRB outflow together with a `structured' jet, where the energy per
solid angle drops as the inverse square of the angle from the jet axis, that is
expected in models where the outflow is Poynting flux dominated.

Astrophysics