  Motivated by the interest in models of the early universe where statistical
isotropy is broken and can be revealed in cosmological observations, we
consider an SU(2) theory of gauge interactions in a single scalar field
inflationary scenario. We calculate the trispectrum of curvature perturbations,
as a natural follow up to a previous paper of ours, where we studied the
bispectrum in the same kind of models. The choice of a non-Abelian set-up turns
out to be very convenient: on one hand, gauge boson self-interactions can be
very interesting being responsible for extra non-trivial terms (naturally
absent in the Abelian case) appearing in the cosmological correlation
functions; on the other hand, its results can be easily reduced to the U(1)
case. As expected from the presence of the vector bosons, preferred spatial
directions arise and the trispectrum reveals anisotropic signatures. We
evaluate its amplitude tau_{NL}, which receives contributions both from scalar
and vector fields, and verify that, in a large subset of its parameter space,
the latter contributions can be larger than the former. We carry out a shape
analysis of the trispectrum; in particular we discuss, with some examples, how
the anisotropy parameters appearing in the analytic expression of the
trispectrum can modulate its profile and we show that the amplitude of the
anisotropic part of the trispectrum can be of the same order of magnitude as
the isotropic part.
