Radion on the de Sitter Brane

The radion on the de Sitter brane is investigated at the linear perturbation level, using the covariant curvature tensor formalism developed by Shiromizu, Maeda and Sasaki.(1)) It is found that if there is only one de Sitter brane with positive tension, there is no radion, and thus ordinary Einstein gravity is recovered on the brane, with the exception of corrections due to the massive Kaluza-Klein modes. As a by-product of the covariant curvature tensor formalism, it is immediately seen that cosmological scalar, vector and tensor type perturbations all have the same Kaluza-Klein spectrum. On the other hand, if there are two branes, one with positive tension and another with negative tension, the gravity on each brane receives corrections from the radion mode in addition to the Kaluza-Klein modes, and the radion is found to have a negative mass-squared proportional to the curvature of the de Sitter brane. This is in contrast to the flat brane case, in which the radion mass vanishes and becomes degenerate with the 4-dimensional graviton modes. To relate our result with the metric perturbation approach, we derive the second-order action for the brane displacement. We find that the radion identified in our approach indeed corresponds to the relative displacement of the branes in the Randall-Sundrum gauge and describes the scalar curvature perturbations of the branes in Gaussian normal coordinates around the branes. The implications of our results with regard to the inflationary brane universe are briefly discussed.