  In order to gain new insights into the gauge field couplings in the early
universe, we consider the constraints on gauge field production during
inflation imposed by requiring that their effect on the CMB anisotropies are
subdominant. In particular, we calculate systematically the bispectrum of the
primordial curvature perturbation induced by the presence of vector gauge
fields during inflation. Using a model independent parametrization in terms of
magnetic non-linearity parameters, we calculate for the first time the
contribution to the bispectrum from the cross correlation between the inflaton
and the magnetic field defined by the gauge field. We then demonstrate that in
a very general class of models, the bispectrum induced by the cross correlation
between the inflaton and the magnetic field can be dominating compared with the
non-Gaussianity induced by magnetic fields when the cross correlation between
the magnetic field and the inflaton is ignored.
