of the model were carried out for example in [@Maleknejad:2011jr], where the cosmic no-hair theorem is tested within Gauge-flation with a Bianchi I background, in [@Ghalee:2012gg], where the background solutions of Gauge-flation were explored, and in [@Noorbala:2012fh], where it is shown that the theory can be embedded in the gravi-leptogenesis scenario of [@Alexander:2004us]. Recently, it was also shown [@Adshead:2012qe; @SheikhJabbari:2012qf] that Gauge-flation shares some background trajectories with Chromo-natural inflation [@Adshead:2012kp], another recent model which assigns to non-Abelian gauge fields a crucial role in the dynamics of inflation and in the generation of primordial fluctuations (see [@review4] for a comprehensive review). [^2]

In Chromo-natural inflation an SU(2) gauge field is coupled to an axion $\varphi$ which plays the role of the inflaton: $$\label{cn}
S=\int d^{4}x\sqrt{-g}\left[\frac{M_{p}^{2}}{2}R-\frac{1}{4}F_{\mu\nu}^{a}F^{a,\mu\nu}-\frac{1}{2}\left(\partial\varphi\right)^{2}-V(\varphi)+\frac{\lambda}{4f}\varphi F_{\mu\nu}^{a}\tilde{F}^{a,\mu\nu}\right].$$ It turns out that, if sufficiently close to the bottom of its potential, the $\varphi$ field can be integrated out, leading precisely to the action in Eq. (\[oneeq\]) plus corrections [@SheikhJabbari:2012qf].

The main goal of Chromo-natural model, and Gauge-flation as well, is to solve an issue that affects most scalar field models of inflation, i.e. the difficulty in ensuring a quasi-flat potential for the inflaton. The condition of quasi-flatness for the potential is important in 