  We study a single field axion inflation model in the presence of an SU(2)
gauge field with a small vev. In order to make the analysis as
model-independent as possible, we consider an arbitrary potential for the axion
that is able to support the slow-roll inflation. The gauge field is coupled to
the axion with a Chern-Simons interaction $\frac{\lambda}{f}F_{\mu\nu}^a\tilde
F_a^{\mu\nu}$ where $\frac{\lambda}{f}\sim\mathcal{O}(10)$. It has a negligible
effect on the background evolution, $\frac{\rho_{_{\rm YM}}}{M_{\rm
Pl}^2H^2}\lesssim\epsilon^2$. However, its quantum fluctuations make a
significant contribution to the cosmic perturbation. In particular, the gauge
field has a spin-2 fluctuation which explicitly breaks the parity between the
left- and right-handed polarization states. The chiral tensor modes are
linearly coupled to the gravitational waves and lead to a circularly polarized
tensor power spectrum comparable to the unpolarized vacuum power spectrum.
Moreover, the scalar sector is modified by the linear scalar fluctuations of
the gauge field. Since the spin-0 and spin-2 fluctuations of the SU(2) gauge
field are independent, the gauge field can, at the same time, generate a
detectable chiral gravitational wave signal and have a negligible contribution
to the scalar fluctuations, in agreement with the current CMB observations.
