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We investigate the effect of the bulk gravitational field on the cosmologicalperturbations on a brane embedded in the 5D Anti-de Sitter (AdS) spacetime. Theeffective 4D Einstein equations for the scalar cosmological perturbations onthe brane are obtained by solving the perturbations in the bulk. Then thebehaviour of the corrections induced by the bulk gravitational field to theconventional 4D Einstein equation are determined. Two types of the correctionsare found. First we investigate the corrections which become significant atscales below the AdS curvature scales and in the high energy universe with theenergy density larger than the tension of the brane. The evolution equation forthe perturbations on the brane is found and solved. Another type of thecorrections is induced on the brane if we consider the bulk perturbations whichdo not contribute to the metric perturbations but do contribute to the matterperturbations. At low energies, they have imaginary mass $m^2=-(2/3) \k^2$ inthe bulk where $\k$ is the 3D comoving wave number of the perturbations. Theydiverge at the horizon of the AdS spacetime. The induced density perturbationsbehave as sound waves with sound velocity $1/\sqrt{3}$ in the low energyuniverse. At large scales, they are homogeneous perturbations that depend onlyon time and decay like radiation. They can be identified as the perturbationsof the dark radiation. They produce isocurvature perturbations in the matterdominated era. Their effects can be observed as the shifts of the location andthe height of the acoustic peak in the CMB spectrum.

Bulk gravitational field and cosmological perturbations on the brane