Moduli instability in warped compactification - 4D effective theory approach

We consider a 5D BPS dilatonic two brane model which reduces to the Randall-Sundrum model or the Horava-Witten theory for a particular choice of parameters. Recently new dynamical solutions were found by Chen et al., which describe a moduli instability of the warped geometry. Using a 4D effective theory derived by solving the 5D equations of motion, based on the gradient expansion method, we show that the exact solution of Chen et. al. can be reproduced within the 4D effective theory and we identify the origin of the moduli instability. We revisit the gradient expansion method with a new metric ansatz to clarify why the 4D effective theory solution can be lifted back to an exact 5D solution. Finally we argue against a recent claim that the 4D effective theory allows a much wider class of solutions than the 5D theory and provide a way to lift solutions in the 4D effective theory to 5D solutions perturbatively in terms of small velocities of the branes.