Cosmological evolution on self-tuned branes and the cosmological constant

We consider the cosmological evolution of a bulk scalar field and ordinary matter living on the brane world in the light of the constraints imposed by the matter dominated cosmological evolution and a small cosmological const ant now. We rule out models with a self-tuned minimum of the four dimension al potential as they would lead to rapid oscillations of the Hubble paramet er now. A more natural framework is provided by supergravity in singular sp aces where the brane coupling and the bulk potential are related by supersy mmetry, leading to a four dimensional run-away potential. For late times we obtain an accelerating universe due to the breaking of supersymmetry on th e brane with an acceleration parameter of q(0) = -4/7 and associated equati on of state omega = 5/7.