in terms of the other two quantities: $${\tilde Q}_{\rm in}  \simeq  \left[ \frac{1}{2 N}\ln\left(\frac{1+\gamma_{\rm in}}{\gamma_{\rm in}}\right) \right]^{1/2}  \;\;. 
\label{getQin}$$ Next, consider eq. (\[def-gamma\]), and combine eqs. (\[s2seven\]) and (\[sl2\]). These relations, written in terms of dimensionless quantities, and evaluated at the initial time, give $$\begin{aligned}
&& {\tilde g} = \frac{ \gamma_{\rm in}^{1/2} {\tilde \partial } a_{\rm in} }{ {\tilde Q}_{\rm in} } \;\;, \nonumber\\ 
&& {\tilde \partial} {\tilde Q}_{\rm in} = - \delta_{\rm in} {\tilde Q}_{\rm in} {\tilde \partial a}_{\rm in}  \simeq - \frac{\gamma_{\rm in} \left( 1 + \gamma_{\rm in} \right) {\tilde Q}_{\rm in}^5}{6} \, 
 {\tilde \partial a}_{\rm in}  \;\; , 
\label{getgQp}\end{aligned}$$ These two equations, plus eq. (\[eom-00\]) are three relations in terms of three unknown quantities ${\tilde g} ,\,  
 {\tilde \partial } a_{\rm in} ,\, $ and $  {\tilde \partial} {\tilde Q}_{\rm in} $. In this way we obtain all the quantities needed for the background evolution.

To summarize: Gauge-flation is characterized by the two parameters $\kappa$ and $g$. A given inflationary evolution is specified by the number of e-folds $N$, the values of the inflaton ($Q$) and its time derivative at some initial time. One of the parameters ($\kappa$, in the current case) can be rescaled out of 