the gauge inflaton $Q$ sustains inflation long enough. The condition $\epsilon\ll 1$ in turn translates into $\kappa g^{2}Q^{4}\gg 1$ and $\kappa H^{2}Q^{2}\left(1-\delta\right)^{2}\simeq\kappa H^{2}Q^{2}\gg 1$. The slow-roll parameters satisfy the relations $$\label{sl2}
\epsilon\simeq\frac{Q^{2}}{M_{p}^{2}}\left(1+\gamma\right),\quad\quad\eta\simeq\frac{\epsilon}{1+\gamma}\simeq\frac{Q^{2}}{M_{p}^{2}},\quad\quad\delta\simeq\frac{\gamma}{6\left(1+\gamma\right)}\epsilon^{2},$$ where $\gamma $ is defined in (\[def-gamma\]). The relations (\[sl2\]) have been derived in [@Maleknejad:2011jw] and for brevity we do not repeat the derivation here. We note that, for the slow-roll conditions to be satisfied, $Q\ll M_{p}$ is needed, so that, according to standard terminology, this is a “small-field” inflationary model. However, as we show below, contrary to what happens in standard small-field models, Gauge-flation can lead to too large an amount of gravitational waves for some choice of parameters.

Following the notation of [@Maleknejad:2011jw], in eq. (\[sl2\]) we have introduced the quantity $$\gamma \equiv \frac{g^{2}Q^{2}}{H^{2}} \;\;, 
\label{def-gamma}$$ which we see corresponds to the same combination (\[cn-instab\]) bounded by the stability of the scalar modes in Chromo-natural inflation. One of the goals of the current work is to study whether a bound analogous to (\[cn-instab\]) is also present in Gauge-flation.

By combining the definition (\[def-gamma\]) with eqs. (\[epsilon\]) and (\[s2seven\]) we can obtain the exact relation $$\kappa   = \frac{1}{H^2 \gamma Q^2} \;  \frac{\left( 1 - \delta \right)^2 + 