We may write this more compactly as $^{(4)}$□ Ψ + D y $_{1}$Ψ − N − 1 ∑ a =1 ( D y $_{2}$)$_{a}$ ξ$_{a}$ = 0 (A21) $^{(4)}$□ ξ$_{a}$ +$^{N −}$$^{1}$ ∑ b =1 ( D y $_{3}$)$_{ab}$ ξ$_{b}$ − ( D y $_{4}$)$_{a}$ Ψ = 0 , (A22) where D y 1 ≡ − ∂ 2 y − ( 9 H − 2 a 2 1 ¯ χ ′ N ∂W ∂χ$_{N}$ ∣ ¯ χ ) ∂$_{y}$ − ( 12 H 2 + 4 H ′ − 1 2$^{(4)}$ R − 4 a $^{2}$H 1 ¯ χ ′ N ∂W ∂χ$_{N}$ ∣ ¯ χ ) ( D y $_{2}$)$_{a}$ ≡ − 4 a 2 3 M 3 ∗ ( ∂W ∂χ$_{a}$ ∣ ¯ χ − ¯ χ ′ a ¯ χ ′ N ∂W ∂χ$_{N}$ ∣ ¯ χ ) ( D y $_{3}$)$_{ab}$ ≡ − δ$_{ab}$ ( ∂ 2 y + 3 H ∂$_{y}$ ) + M ab M ab ≡ − a 2 ( ∂ $^{2}$W ∂χ$_{b}$∂χ$_{a}$ ∣ ¯ χ + ¯ χ ′ b ¯ χ ′ N ∂ $^{2}$W ∂χ$_{N}$ ∂χ$_{a}$ ∣ ¯ χ ) ( D y $_{4}$)$_{a}$ ≡ − 3 ( ¯ χ ′ a − M 3 ∗ a 2 2¯ χ ′ N ∂ $^{2}$W ∂χ$_{N}$ ∂χ$_{a}$ ∣ ¯ χ ) ∂$_{y}$ − 2 a 2 ( ∂W ∂χ$_{a}$ ∣ ¯ χ + 3 M 3 ∗ H 2¯ χ ′ N ∂ $^{2}$W ∂χ$_{N}$ ∂χ$_{a}$ ∣ ¯ χ ) . (A23) If we suppress the discrete indices in equations (A21) and (A22) they take an even simpler form $^{(4)}$□ Ψ + D y $_{1}$Ψ = D y $_{2}$ξ (A24) $^{(4)}$□ ξ + D y $_{3}$ξ = D y $_{4}$Ψ . (A25) This is the generic form that the scalar perturbation equations of motion take for N − 1 coupled scalar fields and a graviscalar. 31