\gamma}{\left( 1 - \delta \right)^2} \, \frac{2-\epsilon}{\epsilon} \;\;,
\label{relation-kappa}$$ which will be used in the following Sections to eliminate the parameter $\kappa$ from the explicit form of the equations of the perturbations that we solve. Another useful exact relation that we will use in the study of the perturbations is $$M_p = Q \sqrt{\frac{\left( 1 - \delta \right)^2 + \gamma}{\epsilon}} \;\;.
\label{relation-Mp}$$ To prove this relation, we rewrite (\[epsilon\]) as $\epsilon = \frac{2 \rho_{YM}}{3 H^2 M_p^2}$, we express $\rho_{YM}$ through eqs. (\[s2four\]), (\[s2eight\]), and, finally, we trade $\dot{Q}$ for $\delta$ - using (\[s2seven\]) - and $g^2$ for $\gamma$ - using (\[def-gamma\]).

Differentiating (\[def-gamma\]), and using the definition of the slow roll parameters, we obtain that $$\dot{\gamma}= 2 \gamma \left(\epsilon-\delta \right) H \;\;,$$ so that $\gamma$ is a slowly rolling quantity. More in general, from the slow roll conditions, one can show [@Maleknejad:2011jw] that the quantities $\gamma H^2 \propto Q^2 \propto \epsilon / \left( 1 + \gamma \right)$ are constant at leading order in slow roll. Therefore, $$\label{s2fourteen}
\frac{\epsilon}{\epsilon_{\rm in}}\simeq\frac{1+\gamma}{1+\gamma_{\rm in}}
\;\;,\;\;
\frac{H^{2}}{H_{\rm in}^{2}}\simeq \frac{\gamma_{\rm in}}{\gamma} \;\;,$$ where the subscript “in” indicates some initial time, which we take to correspond to $N$ e-folds before the end of inflation.

The number of 