  We study a variant of the Chromo-Natural Inflation (CNI) mechanism in which
the inflaton interacts only gravitationally with the CNI fields. Integrating
out all the non-dynamical scalar fields of the model results in a coupling
between the perturbations of the inflaton and of the CNI pseudo-scalar which is
significantly greater than the one obtained in the absence of the gauge CNI
dynamics. We compute how this greater coupling impacts the power spectrum of
the inflaton perturbations that are sourced nonlinearly by the unstable
(tensor) gauge CNI modes, and we require that the amplitude of these modes is
well below that of the linear perturbations. Combining this result with various
constraints, including backreaction effects, the requirement of having
observable and dominant sourced gravitational waves (GW), and the current upper
bound on the tensor-to-scalar ratio, significantly constrains the range of
parameter space where this model can produce an interesting GW signal.
