Scalar perturbations from brane-world inflation

We investigate the scalar metric perturbations about a de Sitter brane universe in a five-dimensional anti-de Sitter bulk. We compare the master variable formalism, describing metric perturbations in a five-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 > 3/2H, which remain in the vacuum state. A light radion mode, with m = root2H, 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, becomes normalizable. However, the boundary condition requires the self-consistent four-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.