Bulk gravitational field and cosmological perturbations on the brane

We investigate the effect of the bulk gravitational field on the cosmological perturbations on a brane embedded in 5D anti–de Sitter (AdS) spacetime. The effective 4D Einstein equations for the scalar cosmological perturbations on the brane are obtained by solving the perturbations in the bulk. Then the behavior of the corrections caused by the bulk gravitational field to the conventional 4D Einstein equation is determined. Two types of correction are found. First we investigate the corrections that become significant at scales below the AdS curvature scales and in the high energy universe with an energy density larger than the tension of the brane. The evolution equation for the perturbations on the brane is found and solved. Another type of correction is induced on the brane if we consider the bulk perturbations, which do not contribute to the metric perturbations but do contribute to matter perturbations. At low energy, they have an imaginary mass $m^2=-\mathrm{}(2/3)\mathrm{}{\mathbf{k}}^2$ in the bulk where $\mathbf{k}$ is the 3D comoving wave number of the perturbations. They diverge at the horizon of the AdS spacetime. The induced density perturbations behave as sound waves with a sound velocity of $1/\sqrt{3}$ in the low energy universe. At large scales, they are homogeneous perturbations that depend only on time and decay like radiation. They can be identified as the perturbations of dark radiation. They produce isocurvature perturbations in the matter dominated era. Their effects can be observed as shifts of the location and the height of the acoustic peak in the cosmic microwave background spectrum.