General brane cosmologies and their global spacetime structure

Starting from a completely general standpoint, we find the most general brane-Universe solutions for a three-brane in a five dimensional spacetime. The brane can border regions of spacetime with or without a cosmological constant. Making no assumptions other than the usual cosmological symmetries of the metric, we prove that the equations of motion form an integrable system, and find the exact solution. The cosmology is indeed a boundary of a (class II) Schwarzschild-AdS spacetime, or a Minkowski (class I) spacetime. We analyse the various cosmological trajectories focusing particularly on those bordering vacuum spacetimes. We find, not surprisingly, that not all cosmologies are compatible with an asymptotically flat spacetime branch. We comment on the role of the radion in this picture.