  In this paper, we examine the viability of inflation models with a spectator
axion field coupled to both gravitational and SU(2) gauge fields via
Chern-Simons couplings. Requiring phenomenological success of the axion-SU(2)
sector constrains the coupling strength of the gravitational Chern-Simons term.
We find that the impact of this term on the production and propagation of
gravitational waves can be as large as fifty percent enhancement for the
helicity that is not sourced by the gauge field, if the cut-off scale is as low
as $\Lambda$ = 20H. The effect becomes smaller for a larger value of $\Lambda$,
while the impact on the helicity sourced by the gauge field is negligible
regardless of $\Lambda$.
