Wave function of the radion in the dS and AdS brane worlds

We study the linearized metric perturbation corresponding to the radion for the generalization of the five dimensional two brane setup of Randall and Sundrum to the case when the curvature of each brane is locally constant but nonzero. We find the wave function of the radion in a coordinate system where each brane is sitting at a fixed value of the extra coordinate. We find that the radion now has a ${\mathrm{mass}}^2,$ which is negative for the case of de Sitter branes but positive for anti–de Sitter branes. We also determine the couplings of the radion to matter on the branes, and construct the four dimensional effective theory for the radion valid at low energies. In particular we find that in AdS space the wave function of the radion is always normalizable and hence its effects, though small, remain finite at arbitrarily large brane separations.