e-folds can be related to the initial values of $\gamma$ and $\epsilon$ by [@Maleknejad:2011jw] $$\label{nN}
N\simeq \frac{1+\gamma_{\rm in}}{2\epsilon_{\rm in}}\ln\left[\frac{1+\gamma_{\rm in}}{\gamma_{\rm in}}\right].$$ up to subleading terms in slow roll.

We introduce the dimensionless quantities $${\tilde Q} \equiv \frac{Q}{M_p} \;\;,\;\; {\tilde t} \equiv \frac{t}{\sqrt{\kappa} M_p} \;\;,\;\;
{\tilde g} \equiv \sqrt{\kappa} M_p^2 g \;\;.
\label{tilde-combo}$$ and we denote ${\tilde \partial} \equiv \frac{\partial }{ \partial {\tilde t} } $. In terms of these quantities, the only nontrivial SU(2) background equation and the background “ii-Einstein” equation read, respectively $$\begin{aligned}
& & 
 {\tilde \partial}^2 {\tilde Q} + 2 \frac{ {\tilde \partial} a}{a}  {\tilde \partial} {\tilde Q} +  \frac{ {\tilde \partial}^2 a}{a} {\tilde Q}  + \frac{  2 {\tilde g}^2 }{  1 + {\tilde g}^2 {\tilde Q}^4  }
\left[ 1 + \left( {\tilde \partial} {\tilde Q} + \frac{ {\tilde \partial} a}{a}  {\tilde Q} \right)^2 \right] {\tilde Q}^3 
+ \frac{ {\tilde \partial} a }{a} \frac{  1 - 3 {\tilde g}^2 {\tilde Q}^4  }{  1 + {\tilde g}^2 {\tilde Q}^4   } 
 \left(  {\tilde \partial} {\tilde Q} + \frac{ {\tilde \partial} a}{a}  {\tilde Q} \right)  = 0 \;\; , \nonumber\\ 
& & 
\frac{{\tilde \partial}^2 a}{a}  - \left( \frac{{\tilde \partial} a}{a}  \right)^2 = - \left[  \left(  {\tilde \partial} {\tilde Q} + \frac{ 