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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 asmodel-independent as possible, we consider an arbitrary potential for the axionthat is able to support the slow-roll inflation. The gauge field is coupled tothe axion with a Chern-Simons interaction $\frac{\lambda}{f}F_{\mu\nu}^a\tildeF_a^{\mu\nu}$ where $\frac{\lambda}{f}\sim\mathcal{O}(10)$. It has a negligibleeffect on the background evolution, $\frac{\rho_{_{\rm YM}}}{M_{\rmPl}^2H^2}\lesssim\epsilon^2$. However, its quantum fluctuations make asignificant contribution to the cosmic perturbation. In particular, the gaugefield has a spin-2 fluctuation which explicitly breaks the parity between theleft- and right-handed polarization states. The chiral tensor modes arelinearly coupled to the gravitational waves and lead to a circularly polarizedtensor power spectrum comparable to the unpolarized vacuum power spectrum.Moreover, the scalar sector is modified by the linear scalar fluctuations ofthe gauge field. Since the spin-0 and spin-2 fluctuations of the SU(2) gaugefield are independent, the gauge field can, at the same time, generate adetectable chiral gravitational wave signal and have a negligible contributionto the scalar fluctuations, in agreement with the current CMB observations.

Axion Inflation with an SU(2) Gauge Field: Detectable Chiral Gravity Waves