  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 constant now.
We rule out models with a self-tuned minimum of the four dimensional potential
as they would lead to rapid oscillations of the Hubble parameter now. A more
natural framework is provided by supergravity in singular spaces where the
brane coupling and the bulk potential are related by supersymmetry leading to a
four dimensional run-away potential. For late times we obtain an accelerating
universe due to the breaking of supersymmetry on the brane with an acceleration
parameter of q_0=-4/7 and associated equation of state omega=-5/7.
