Using that at very early times we are in the framework of GR, in Fourier space, we will have $v_k\rightarrow \frac{e^{-ik\tau}}{\sqrt{2k}}$ when $\tau\rightarrow -\infty$. At very early times, equation $\Delta u=z\left(\frac{v}{z}\right)'$ holds and, since we are in a matter domination epoch, we have $u_k\rightarrow i\frac{e^{-ik\tau}}{k\sqrt{2k}}$. Finally, from the relation $$\begin{aligned}
v=\delta\sigma + 
\frac{({\bar\phi}')^2}{2{\mathcal H}}
u\cong 
\delta\sigma+
\frac{3}{2}{\mathcal H}u\cong\delta\sigma,\end{aligned}$$ one obtains the well-known result $\delta\sigma_k \rightarrow \frac{e^{-ik\tau}}{\sqrt{2k}}$ when $\tau\rightarrow -\infty$. Thus, the asymptotic conditions at very early times are $u_k= i\frac{e^{-ik\tau}}{k\sqrt{2k}}$ and $\delta\sigma_k=\frac{e^{-ik\tau}}{\sqrt{2k}}$.

A final remark is in order. In a bouncing scenario, as for instance the matter-ekpyrotic one, at early times GR holds, meaning that $\bar{f}_{\chi\chi}$ could be disregarded and thus obtaining for the variable $v$, in Fourier space, the classical equation $v_k''+(k^2-\frac{a''}{a})v_k=0$. Then, when the pivot scale leaves the Hubble radius, i.e. in the long wavelength approximation ($k^2\ll a^2 H^2$), one can safely disregard the Laplacian terms which appear on the right hand side of Eq. (\[v\]) (see the end of section $4$ in [@ha14a]). And maybe the same happens with the right hand side of our Eq. (\[v\]) because it contains a Laplacian. If so, Eq. (\[v\]), in the long wavelength approximation, will become, as usual, $v_k''-\frac{z''}{z}v_k=0$ and, 