for the $f$-theory given by (\[fLQC\]), we will recover the results obtained in LQC using the [*deformed algebra approach*]{} [@we; @cw; @haa]. Anyway, this has to be properly checked by solving numerically the system of Eqs. (\[Z\]).

However, we want to stress that the system (\[Z\]) can in fact be solved iteratively, taking the right hand side term as a perturbation (when the pivot scale is well inside the Hubble radius, which happens at very early times, the right hand side term can be dismissed, since GR holds, and after the pivot scale leaves the Hubble radius, i.e. in the long wavelength approximation, the Laplacian terms can be dismissed too). In fact, at very early times, disregarding the right hand term, since the universe is matter-dominated, the equation for $u$ is, in Fourier space, $$\begin{aligned}
\label{short}
u''_k+k^2u_k-\frac{6}{\tau^2}u_k=0.\end{aligned}$$ Its solution satisfying the asymptotic condition $u_k= i\frac{e^{-ik\tau}}{k\sqrt{2k}}$ reads $$\begin{aligned}
\label{z1}
u^{(0)}_k=\frac{1}{k}\sqrt{\frac{-\pi\tau}{4}}H^{(1)}_{\frac{5}{2}}(-k\tau)=\frac{i e^{-ik\tau}}{k\sqrt{2k}}\left(1-\frac{3}{k^2\tau^2}-\frac{3i}{k\tau}   \right),\end{aligned}$$ where $H_{\frac{5}{2}}^{(1)}$ is a Hankel’s function, and the super-index $(0)$ means zero order approximation. On the other hand, when the pivot scale has left the Hubble radius, i.e. when $k^2\tau^2\ll 1$, one has $$\label{long}
u''_k-\frac{\theta''}{\theta}u_k=0.$$

Note that, in the case of the matter bounce scenario, given by the potential 