may arise from loops) can be safely suppressed. The authors of [@Maleknejad:2011jr] claim that this is the case within the parameter space of the theory.

The background evolution of Gauge-flation has been exhaustively studied in the literature, starting from the original proposal [@Maleknejad:2011jw]. We summarize it here for completeness, and to discuss the freedom in the choice of parameters and initial conditions in the theory.

We can immediately verify that slow roll inflation requires (the standard FRW equations for $H$ and $\dot{H}$ apply) $$\epsilon \equiv - \frac{\dot{H}}{H^2} = \frac{2 \rho_{YM}}{\rho_{YM} + \rho_\kappa}  \ll 1  \;\; \Leftrightarrow \;\; \rho_{\kappa}\gg \rho_{YM}  \;\;,
\label{epsilon}$$ confirming that the energy associated to the higher order term in (\[oneeq\]) must dominate over the one from the conventional Yang-Mills term. Besides the slow roll parameter $\epsilon$, it is useful to introduce the two additional slow-roll parameters: $$\label{s2seven}
  \eta\equiv -\frac{\ddot{H}}{2H\dot{H}}=\epsilon-\frac{\dot{\epsilon}}{2\epsilon H} \;\;\;,\;\;\;
 \delta\equiv -\frac{\dot{Q}}{HQ} \;\;.$$ The parameter $\eta$ is related to the time variation of $\epsilon$, so that $\eta \ll 1$ imposes that $\epsilon$ varies very slowly during inflation; the other slow-roll parameter is defined from the time variation of the physical field $$\label{s2eight}
 Q \equiv \frac{{\hat \phi } }{a}  \;\;,$$ so $  \delta \ll 1$ ensures that 