{\tilde \partial} a}{a}  {\tilde Q} \right)^2 + {\tilde g}^2 {\tilde Q}^4 \right] \;\; ,
\label{eom-Q-ii}\end{aligned}$$ while the background "00-Einstein” equation reads $$\left( \frac{{\tilde \partial} a}{a}  \right)^2 = \frac{1}{2} \left[ \left( {\tilde \partial} {\tilde Q} + \frac{ {\tilde \partial} a}{a}  {\tilde Q} \right)^2  \left( 1 + {\tilde g}^2 {\tilde Q}^4 \right)   + {\tilde g}^2 {\tilde Q}^4 \right] \;\;.
\label{eom-00}$$ These three equations are related to each other by a nontrivial Bianchi identity. A set of independent equations is provided by either one of (\[eom-Q-ii\]) and by (\[eom-00\]). Equivalently, one can solve the two equations (\[eom-Q-ii\]), but then eq. (\[eom-00\]) has to be imposed as an initial condition. Whatever the choice, the background evolution is completely determined by the initial values ${\tilde Q}_{\rm in}$ and ${\tilde \partial } {\tilde Q}_{\rm in}$ (for a flat universe, the normalization of the scale factor is arbitrary, and we set $a_{\rm in} = 1 $).

We now show that all quantities needed to specify a background evolution can be given in terms of $N$ and of $\gamma_{\rm in}$. Eliminating $\epsilon$ through (\[sl2\]), eq. (\[nN\]) becomes a relation in terms of $N, \gamma_{\rm in}$ and ${\tilde Q}_{\rm in}$, that we solve numerically to obtain ${\tilde Q}_{\rm in}$ 