Evolution of Cosmological Perturbations in the Brane World

The evolution of the cosmological perturbations is studied in the context of the Randall-Sundrum brane world scenario, in which our universe is realized on a three-brane in the five dimensional Anti-de Sitter(AdS) spacetime. We develop a formalism to solve the coupled dynamics of the cosmological perturbations in the brane world and the gravitational wave in the AdS bulk. Using our formalism, the late time evolution of the cosmological scalar perturbations at any scales larger than the AdS curvature scale $l$ is shown to be identical with the one obtained in the conventional 4D cosmology, provided the effect of heavy graviton modes may be neglected. Here the late time means the epoch when the Hubble horizon $H^{-1}$ in the 4D brane world is sufficiently larger than the AdS curvature scale $l$. If the inflation occurs sufficiently lower than $l^{-1}$, the scalar temperature anisotropies in the Cosmic Microwave Background at large scales can be calculated using the constancy of the Bardeen parameter as is done in the 4D cosmology. The assumption of the result is that the effect of the massive graviton with mass $m e^{-\alpha_0}>l^{-1}$ in the brane world is negligible, where $e^{\alpha_0}$ is the scale factor of the brane world. We also discuss the effect of these massive gravitons on the evolution of the perturbations.