upper) branch and $ \arcsin s\equiv \int_0^s \frac{1}{\sqrt{1-{\bar s}^2}}  {d\bar{s}}$ in the lower branch, whereas $\arcsin s\equiv \int_0^s \frac{1}{\sqrt{1-{\bar s}^2}}  {d\bar{s}}+\pi$, in the upper one, with the same criteria for the sign of the square root, thus obtaining that the function $\bar f$ is continuous in all the ellipse.

Note also that, a simple calculation shows that, for this particular $f$-theory, the modified Friedmann and Raychaudhuri equations read $$\begin{aligned}
\label{LQCstandard}
H^2=\frac{\rho}{3}\left(1-\frac{\rho}{\rho_c}\right) \quad \mbox{and}\quad \dot{H}=-\frac{\rho+P}{2}\left(1-\frac{2\rho}{\rho_c}\right),\end{aligned}$$ and coincide with the ones obtained in holonomy corrected LQC [@singh09; @sst06].

A final remark is in order. Since in our case the matter sector is depicted by a non-phantom scalar field $\bar{\phi}$ (see equation (\[Lagrangian\])), we will have infinitely many backgrounds because the conservation equation $\ddot{\bar{\phi}}+3H\dot{\bar\phi}+V_{\phi}(\bar\phi)=0$ is a second order differential equation. Effectively, for any initial condition $\bar{\phi}(0)=\alpha_0$ and $\dot{\bar{\phi}}(0)=\alpha_1$, one has a different background. Therefore, if we choose a curve $\rho=g(\bar\chi)$ in the plane $(H,\rho)$ containing a point of the form $(0, \tilde{\rho})$ with $\tilde{\rho}>0$ and initial conditions satisfying $ \frac{\alpha_1^2}{2}+V(\alpha_0)=\tilde{\rho}$, its corresponding solution will lead to a bouncing background provided that $V_{\phi}(\alpha_0)\not= 0$.

As an example one can consider the matter bounce scenario in LQC, which is given by the potential $V(\phi)=2\rho_c\frac{e^{-\sqrt{3}\phi}}{(1+e^{-\sqrt{3}\phi})^2}$ [@ha14a]. 