Scalar perturbations from brane-world inflation

We investigate the scalar metric perturbations about a de Sitter brane universe in a 5-dimensional anti de Sitter bulk. We compare the master-variable formalism, describing metric perturbations in a 5-dimensional longitudinal gauge, with results in a Gaussian normal gauge. For a vacuum brane (with constant brane tension) there is a continuum of normalizable Kaluza-Klein modes, with m>3H/2, which remain in the vacuum state. A light radion mode, with m=\sqrt{2}H, satisfies the boundary conditions for two branes but is not normalizable in the single-brane case. When matter is introduced (as a test field) on the brane, this mode, together with the zero-mode and an infinite ladder of discrete tachyonic modes, become normalizable. However, the boundary condition requires the self-consistent 4-dimensional evolution of scalar field perturbations on the brane and the dangerous growing modes are not excited. These normalizable discrete modes introduce corrections at first-order to the scalar field perturbations computed in a slow-roll expansion. On super-Hubble scales, the correction is smaller than slow-roll corrections to the de Sitter background. However on small scales the corrections can become significant.