Quantization of scalar perturbations in brane-world inflation

We consider a quantization of scalar perturbations about a de Sitter brane in a 5-dimensional anti-de Sitter (AdS) bulk spacetime. We first derive the second order action for a master variable $\Omega$ for 5-dimensional gravitational perturbations. For a vacuum brane, there is a continuum of normalizable Kaluza-Klein (KK) modes with $m>3H/2$. There is also a light radion mode with $m=\sqrt{2}H$ which satisfies the junction conditions for two branes, but is non-normalizable for a single brane model. We perform the quantization of these bulk perturbations and calculate the effective energy density of the projected Weyl tensor on the barne. If there is a test scalar field perturbation on the brane, the $m^2 = 2H^2$ mode together with the zero-mode and an infinite ladder of discrete tachyonic modes become normalizable in a single brane model. This infinite ladder of discrete modes as well as the continuum of KK modes with $m>3H/2$ introduce corrections to the scalar field perturbations at first-order in a slow-roll expansion. We derive the second order action for the Mukhanov-Sasaki variable coupled to the bulk perturbations which is needed to perform the quantization and determine the amplitude of scalar perturbations generated during inflation on the brane.