Validity of a factorizable metric ansatz in string cosmology

To support the validity of a factorizable metric ansatz used in string cosmology, we investigate a toy problem in the Randall-Sundrum I model. For this purpose, we revise the gradient expansion method to conform to the factorizable metric ansatz. By solving the five-dimensional equations of motion and substituting the results into the action, we obtain the four-dimensional effective action. It turns out that the resultant action is equivalent to that obtained by assuming the factorizable metric ansatz. Our analysis gives the support of the validity of the factorizable metric ansatz.