of the scalar perturbations. Fortunately in the matter (or matter-ekpyrotic) scenario, in the contracting phase, the pivot scale leaves the Hubble radius at rather low energy densities, as compared to the Planck one, so the corrections due to $f$ can be safely disregarded and, thus, $v$ satisfies approximately the usual equation $v''-\Delta v-\frac{z''}{z}v=0$. This finally means that the calculation of the spectral quantities, such as the spectral index, its running, and the ratio of tensor to scalar perturbations, can safely be done using GR in the contracting phase, as we will show in next section.

On the other hand, to calculate the evolution of the scalar perturbations through time, we need two equations, since the equations for $\Phi$ and $\delta\phi$ are coupled. In this case, solving the system of Eqs. (\[lqcperturbed\]) and (\[linearization\]), we will obtain such evolution. The best suited variables for that are $u$ and $\delta\sigma\equiv a\delta\phi$, with the corresponding dynamical equations being $$\begin{aligned}
 \label{Z}
\left\{\begin{array}{ccc}
u''-\Omega\Delta u-\frac{\theta''}{\theta}u & = &\bar{\phi}'\sqrt{\Omega}\left( \frac{\bar{f}_{\chi\chi}\Delta\delta\sigma}{({\bar{\phi}')}^3 }  \right)' , \\
 & & \\
\delta\sigma''-\Delta \delta\sigma+a^2V_{\phi\phi}\delta\sigma-2{\bar{\phi}'}a\left(\frac{{\mathcal H}u}{a^2\theta} \right)'+\frac{a{\mathcal H}V_{\phi}u}{\theta} &=&
{\Omega{\bar{\phi}'}}\Delta\left( \frac{\bar{f}_{\chi\chi}\Delta\delta\sigma}{{({\bar{\phi}}')}^3} \right). \end{array}\right.\end{aligned}$$ We now have to look for the initial conditions for a matter or matter-ekpyrotic bouncing scenario [@we; @cw; @haa]. 