Brane cosmological evolution in a bulk with cosmological constant

We consider the cosmology of a "3-brane universe" in a five dimensional (bu lk) space-time with a cosmological constant. We show that Einstein's equati ons admit a first integral, analogous to the first Friedmann equation, whic h governs the evolution of the metric in the brane, whatever the time evolu tion of the metric along the fifth dimension. We thus obtain the cosmologic al evolution in the brane for any equation of state describing the matter i n the brane, without needing the dependence of the metric on the fifth dime nsion. In the particular case p = wp (w = constant), we give explicit expre ssions for the time evolution of the brane scale factor, which show that st andard cosmological evolution can be obtained (after an early non conventio nal phase) in a scenario a la Randall and Sundrum, where a brane tension co mpensates the bulk cosmological constant. We also show that a tiny deviatio n from exact compensation leads to an effective cosmological constant at la te time. Moreover, when the metric along the fifth dimension is static, we are able to extend the solution found on the brane to the whole spacetime. (C) 2000 Published by Elsevier Science B.V. All rights reserved.