In Randall-Sundrum-type brane-world cosmologies, the dynamical equations on the three-brane differ from the general relativity equations by terms that carry the effects of embedding and of the free gravitational field in the five-dimensional bulk. Instead of starting from an ansatz for the metric, we derive the covariant nonlinear dynamical equations for the gravitational and matter fields on the brane, and then linearize to find the perturbation equations on the brane. The local energy-momentum corrections are significant only at very high energies. The imprint on the brane of the nonlocal gravitational field in the bulk is more subtle, and we provide a careful decomposition of this effect into nonlocal energy density, flux and anisotropic stress. The nonlocal energy density determines the tidal acceleration in the off-brane direction, and can oppose singularity formation via the generalized Raychaudhuri equation. Unlike the nonlocal energy density and flux, the nonlocal anisotropic stress is not determined by an evolution equation on the brane, reflecting the fact that brane observers cannot in general make predictions from initial data. In particular, isotropy of the cosmic microwave background may no longer guarantee a Friedmann geometry. Adiabatic density perturbations are coupled to perturbations in the nonlocal bulk field, and in general the system is not closed on the brane. But on super-Hubble scales, density perturbations satisfy a decoupled third-order equation, and can be evaluated by brane observers. Tensor perturbations on the brane are suppressed by local bulk effects during inflation, while nonlocal effects can serve as a source or a sink. Vorticity on the brane decays as in general relativity, but nonlocal bulk effects can source the gravito-magnetic field, so that vector perturbations can be generated in the absence of vorticity.

• Received 25 April 2000

DOI:https://doi.org/10.1103/PhysRevD.62.084023