A no-boundary proposal for braneworld perturbations

We propose a novel approach to the problem of cosmological perturbations in a braneworld model with induced gravity, which leads to a closed system of equations on the brane. We focus on a spatially closed brane that bounds the interior four-ball of the bulk space. The background cosmological evolution on the brane is now described by the normal branch, and the boundary conditions in the bulk become the regularity conditions for the metric everywhere inside the four-ball. In this approach, there is no spatial infinity or any other boundary in the bulk space since the spatial section is compact, hence, we term this setup as a no-boundary proposal. Assuming that the bulk cosmological constant is absent and employing the Mukohyama master variable, we argue that the effects of nonlocality on brane perturbations may be ignored if the brane is marginally closed. In this case, there arises a relation that closes the system of equations for perturbations on the brane. Perturbations of pressureless matter and dark radiation can now be described by a system of coupled second-order differential equations. Remarkably, this system can be exactly solved in the matter-dominated and de Sitter regimes. In this case, apart from the usual growing and decaying modes, we find two additional modes that behave monotonically on super-Hubble spatial scales and exhibit rapid oscillations with decaying amplitude on sub-Hubble spatial scales.