in studying the linearized theory of the perturbations. This amounts in expanding the equations of motion of the system up to first order in the perturbations, or in expanding the action of the system in quadratic order in the perturbation (this is equivalent, since the linearized equations for the perturbations are obtained by extremizing the quadratic action for the perturbations). In both these computations, the modes are decoupled from each other (since the perturbations are coupled to each other only at the nonlinear level). Therefore, without loss of generality in the linearized computation, we can choose the momentum $\vec{k}$ of the modes to be oriented along one given direction, which we choose to be the $3$rd one [^6]

We end up with the following perturbations: $$\begin{aligned}
{\rm scalar}: & &  \delta A_\mu^1 = a \left( 0 , \delta Q , 0 , 0 \right) \;\;,\;\;  
     \delta A_\mu^2 = a \left( 0 , 0 ,  \delta Q , 0   \right) \;\;,\;\;  
     \delta A_\mu^3 = a \left( \partial_z Y , 0 , 0 , \delta Q + \partial_z^2 M   \right) \;\;,  \nonumber\\
& & \delta g_{00} = a^2 2 \phi  \;\;,\;\; \delta g_{03} = a^2 \partial_z B  \;\;,  \nonumber\\
{\rm vector}: & 