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 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 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 29 Ψ ′ + 2 H Ψ = 2 3 M 3 ∗ ∑ a ¯ χ ′ $_{a}$ξ$_{a}$ (A13) $^{(4)}$□ ξ$_{a}$ − ξ ′′ a − 3 H ξ ′ a + a 2 ∑ b ∂ $^{2}$W ∂χ$_{a}$∂χ$_{b}$ ∣ ¯ χ ξ$_{b}$ = − 2¯ χ ′′ $_{a}$Ψ − ¯ χ ′ $_{a}$(Ψ ′ − 4Φ ′ + 6 H Ψ) , (A12) $^{(4)}$□ Φ − 4 H Φ ′ + 4 H $^{2}$Ψ + 1 3$^{(4)}$ R Ψ = − 1 3 M 3 ∗ ∑ a ( ¯ χ ′ $_{a}$ξ ′ a − ¯ χ ′ 2 $_{a}$Ψ − a 2 ∂W ∂χ$_{a}$ ∣ ¯ χ ξ$_{a}$ ) (A11) $^{(4)}$□ (2Φ + Ψ) − 4Φ ′′ + 8 H $^{′}$Ψ + 8 H $^{2}$Ψ + 4 H (Ψ ′ − 3Φ $^{′}$) + 2 3$^{(4)}$ R Φ = 4 3 M 3 ∗ ∑ a ( ¯ χ ′ $_{a}$ξ ′ a − ¯ χ ′ 2 $_{a}$Ψ + a 2 ∂W ∂χ$_{a}$ ∣ ¯ χ ξ$_{a}$ ) (A10) Φ ′ − H Ψ = − 1 3 M 3 ∗ ∑ a ¯ χ ′ $_{a}$ξ$_{a}$ (A9) 2Φ + Ψ = 0 (A8) χ$_{a}$ ( x, y ) = ¯ χ$_{a}$ ( y ) + ξ$_{a}$ ( x, y ) , (A7) ds 2 = a $^{2}$( y ) [ (1 + 2Φ( x, y )) γ$_{µν}$ ( x ) dx $^{µ}$dxν − (1 + 2Ψ( x, y )) dy $^{2}$] . (A6) ds 2 = a $^{2}$( y ) [{ (1 + 2Φ( x, y )) γ$_{µν}$ ( x ) + 2 E ( x, y )$_{|}$$_{µν}$ } dx $^{µ}$dxν + 2 B ( x, y )$_{|}$$_{µ}$ dx $^{µ}$dy − (1 + 2Ψ( x, y )) dy $^{2}$] (A5)