Nonlinear perturbations from the coupling of the inflaton to a
  non-Abelian gauge field, with a focus on Chromo-Natural Inflation
  Several models of inflation employing a triplet of SU(2) vectors with
spatially orthogonal vacuum expectation values (VEVs) have been recently
proposed. One (tensor) combination $t$ of the vector modes is amplified in some
momentum range during inflation. Due to the vector VEVs, this combination mixes
with gravitational waves (GW) at the linear level, resulting in a GW
amplification that has been well studied in the literature. Scalar
perturbations in this class of models have been so far studied only at the
linear level. We perform a first step toward the nonlinear computation using as
an example the original model of Chromo-Natural Inflation. We compute the
contribution to the scalar power spectrum arising from the coupling of the
combination $t$ to the inflaton. This contribution is mostly controlled by a
single parameter of the model (namely, the ratio between the mass of the
fluctuations of the vector field and the Hubble rate), and, for a wide range of
this parameter, it can significantly affect the phenomenology obtained from the
linear theory. This nonlinear contribution is significantly blue, improving the
comparison between the two-point function and the Cosmic Microwave Background
(CMB) data. This growth can be also relevant for smaller scale phenomenology,
such as large scale structure, CMB distortions, and primordial black holes.
