Quantum radion on de Sitter branes

The quantum fluctuation of the relative location of two (n-1)-dimensional de Sitter branes (which have n spacetime dimensions) embedded in an (n+1)-dimensional anti-de Sitter bulk, which we call the "quantum radion", is investigated at the linear order in perturbation theory. The quantization of the radion is carried out by deriving the effective action of the radion. Assuming that the positive tension brane represents our universe, the effect of the quantum radion is evaluated by using the effective Einstein equations on the brane, in which the radion contributes to the effective energy momentum tensor at the linear order of the radion amplitude. Specifically, the rms effective energy density arising from the quantum radion is compared with the background energy density. It is found out that this ratio remains small for reasonable values of the parameters of the model even without introducing a stabilizing mechanism for the radion, although the radion itself has a negative mass squared and is unstable. The reason behind this phenomenon is also discussed.