Leading order corrections to the cosmological evolution of tensor perturbations in braneworld

In recent years there has been much interest in braneworld scenarios [1, 2, 3], where our four dimensional universe could be a brane embedded in a higher dimensional bulk spacetime. The basic ingredient of braneworld scenarios is that while ordinary matter fields are confined to the brane, only gravitational interactions can propagate in the bulk. The second Randall-Sundrum model (RS II) [4] with one brane in an anti de Sitter (AdS) bulk is particularly interesting in the point that, despite an infinite extra dimension, four dimensional general relativity (4D GR) can be recovered at low energies/long distances on the brane. Then, what is leading order corrections to the conventional gravitational theory? It was shown that the Newtonian potential in the RS II braneworld, including the correction due to the bulk gravitational effects with a precise numerical factor [5, 6], is given by