a formal level the most general perturbations of the background solution presented in Section \[sec:model\]. In Subsection \[subsesec:gauge\] we split the perturbations in three groups that are decoupled from each other at the linearized level (and that can be therefore studied independently), and we fix the gauge freedoms associated with general coordinate transformations and with the SU(2) group. In Subsection \[subsec:formal-deco\] we present our formalism to solve the linearized equations for these perturbations.

General decomposition and gauge choice {#subsesec:gauge}
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This discussion closely follows Subsection III A of [@Dimastrogiovanni:2012ew], where the reader is referred to for details. There are 22 perturbations in the model, $12$ of which in the SU(2) vector field, and $10$ in the metric. We decompose: $$\begin{aligned}
A_0^a & = & a \left( Y_a + \partial_a Y \right) \;\;, \nonumber\\ 
A_i^a & = & a \left[ \left( Q + \delta Q \right) \delta_{ai} + \partial_i \left( M_a + \partial_a M \right) 
+ \epsilon_{iab} \left( U_b + \partial_b U \right) + t_{ia} \right] \;\;, \nonumber\\
g_{00} & = & - a^2 \left( 1 - 2 \phi \right) \;\;, \nonumber\\
g_{0i} & = & a^2 \left( B_i + \partial_i B \right) \;\;, \nonumber\\
g_{ij} & = & a^2 \left[ 