content_image/362523.jpg content_image/362522.jpg content_image/362521.jpg content_image/362519.jpg content_image/362520.jpg content_image/362518.jpg content_image/362517.jpg content_image/362516.jpg content_image/362515.jpg
where H ≡ a ′ $_{a}$, $^{(4)}$R is the 4D Ricci scalar with respect to the background 4D metric γ$_{µν}$ . To first-order in scalar perturbations, the 5D metric can be written 

where | indicates a covariant derivative with respect to the 4D slices of the bulk. Choosing to work in the generalized longitudinal gauge, we set B = E = 0, and the linearized metric simplifies to 

We also expand the scalar fields to first-order 

<img src='content_image/362523.jpg'>

<img src='content_image/362522.jpg'>

<img src='content_image/362521.jpg'>

where the fields ¯ χ$_{a}$ obey the background equations of motion (A2)-(A4) above and ξ$_{a}$ ( x, y ) are small perturbations. The linearized Einstein and field equations are 

<img src='content_image/362519.jpg'>

<img src='content_image/362520.jpg'>

<img src='content_image/362518.jpg'>

<img src='content_image/362517.jpg'>

<img src='content_image/362516.jpg'>

where $^{(4)}$□ ≡ γ $^{µν}$∂$_{µ}$ ∂$_{ν}$ is the 4D wave operator. Applying the constraint equation (A8) to equations (A9)-(A12), and making use of the background equations (A2)-(A4), yields 

<img src='content_image/362515.jpg'>

