Cosmic no hair for braneworlds with a bulk dilaton field

Braneworld cosmology supported by a bulk scalar field with an exponentialpotential is developed. A general class of separable backgrounds for bothsingle and two-brane systems is derived, where the bulk metric componentsare given by products of world volume and bulk coordinates and the world-volumesrepresent any anisotropic and inhomogeneous solution to an effective four-dimensionalBrans-Dicke theory of gravity. We deduce a cosmic no hair theorem for allever-expanding, spatially homogeneous Bianchi world volumes and find thatthe spatially flat and isotropic inflationary scaling solution representsa late-time attractor when the bulk potential is sufficiently flat. The dependenceof this result on the separable nature of the bulk metric is investigatedby applying the techniques of Hamilton-Jacobi theory to five-dimensional Einsteingravity. We employ the spatial gradient expansion method to determine theasymptotic 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 representthe attractor of the system is precisely the same as that for the four-dimensionalworld-volume to isotropize. We also derive the fourth–order contributionto the Hamilton-Jacobi generating functional. Finally, we conclude by placingour results within the context of the holographic approach to braneworld cosmology.