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Working in the generalized longitudinal gauge (see appendix A for details), we introduce scalar perturbations Φ and Ψ and write the perturbed metric as 

Next we expand the N scalar fields to first-order in small perturbations ξ$_{a}$ ( x, y ) 

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and compute the linearized Einstein equations, yielding N + 1 dynamical equations for N + 1 scalar fields ( N fundamental scalars and one graviscalar, or radion). Since only N of these equations are independent, the Einstein constraint equations are used to eliminate 

