to what happens in Chromo-natural inflation, see eq. (\[cn-instab\])). Moreover, the scalar spectrum obtained in [@Maleknejad:2011jw] is significantly bluer than the one found for Chromo-natural inflation; compare for instance Figure 8 of the last work in [@Maleknejad:2011jw] with Figure 12 of [@Adshead:2013nka]. This different behavior is somewhat puzzling, given the analogy that the two models present at the background level. Specifically, ref. [@SheikhJabbari:2012qf] shows that, integrating out the axion $\varphi$ from the action (\[cn\]) of Chromo-natural inflation, one obtains the action (\[oneeq\]) of Gauge-flation, plus corrections. [^3] The procedure is conceptually simple, but it is not straightforward to compute the corrections explicitly. The integration of $\varphi$ is done in [@SheikhJabbari:2012qf] at the level of the one loop effective action; slow roll approximations are used, and in computing the functional determinant the laplacian has been replaced by its flat space value, under the assumption that the main contribution comes from sub-horizon momenta [@SheikhJabbari:2012qf]. One may wonder about the impact of the corrections that would emerge by going beyond these approximations. For instance, the departure of the perturbations from scale invariance is controlled by the full evolution (and not only the sub-horizon regime) and by the slow roll parameters. Barring mistakes, the 