\left( 1 + 2 \psi \right) \delta_{ij} + 2 \partial_i \partial_j E + \partial_i E_j + \partial_j E_i + h_{ij} \right]
\;\;,\end{aligned}$$ where $a$ denotes both the SU(2) index and the scale factor (as we believe that this does not cause ambiguity), while $i=1,2,3$ ranges over the spatial coordinates. The modes $t_{ia}$ and $h_{ij}$ are transverse and traceless ($\partial_i h_{ij} = \partial_i t_{ia} = \partial_a t_{ia} = t_{ii} = h_{ii} = 0$) and we denote them as “tensor modes”. The modes $Y_a , M_a, U_a, B_i , E_i$ are transverse ($\partial_i Y_i = \dots = \partial_i E_i = 0$), and we denote them as “vector modes”. We denote the remaining modes as “scalar modes”. These three groups of modes are separate from each other at the linearized level, and can be studied independently [@Dimastrogiovanni:2012ew]. [^5]

As shown in [@Dimastrogiovanni:2012ew], we can always choose the gauge $\psi=E=E_i=U=U_i=0$. This gauge choice completely fixes the freedom associated to general coordinate and SU(2) transformations. We Fourier transform each perturbation according to $$\delta \left( t , \vec{x} \right) = \int \frac{d^3 k}{\left( 2 \pi \right)^{3/2} } \, {\rm e}^{i \vec{k} \cdot \vec{x}} \, \delta \left( t , \vec{k} \right) \;\;.
\label{FT}$$

We are only interested 