  Braneworld cosmology supported by a bulk scalar field with an exponential
potential is developed. A general class of separable backgrounds for both
single and two-brane systems is derived, where the bulk metric components are
given by products of world-volume and bulk coordinates and the world-volumes
represent any anisotropic and inhomogeneous solution to an effective
four-dimensional Brans-Dicke theory of gravity. We deduce a cosmic no hair
theorem for all ever expanding, spatially homogeneous Bianchi world-volumes and
find that the spatially flat and isotropic inflationary scaling solution
represents a late-time attractor when the bulk potential is sufficiently flat.
The dependence of this result on the separable nature of the bulk metric is
investigated by applying the techniques of Hamilton-Jacobi theory to
five-dimensional Einstein gravity. We employ the spatial gradient expansion
method to determine the asymptotic form of the bulk metric up to third-order in
spatial gradients. It is found that the condition for the separable form of the
metric to represent the attractor of the system is precisely the same as that
for the four-dimensional world-volume to isotropize. We also derive the
fourth-order contribution to the Hamilton-Jacobi generating functional.
Finally, we conclude by placing our results within the context of the
holographic approach to braneworld cosmology.
