Bulk gravitational field and dark radiation on the brane in dilatonic   brane world
  We discuss the connection between the dark radiation on the brane and the bulk gravitational field in a dilatonic brane world model proposed by Koyama and Takahashi where the exact solutions for the five dimensional cosmological perturbations can be obtained analytically. It is shown that the dark radiation perturbation is related to the non-normalizable Kaluza-Klein (KK) mode of the bulk perturbations. For the de Sitter brane in the anti-de Sitter bulk, the squared mass of this KK mode is $2 H^2$ where $H$ is the Hubble parameter on the brane. This mode is shown to be connected to the excitation of small black hole in the bulk in the long wavelength limit. The exact solution for an anisotropic stress on the brane induced by this KK mode is found, which plays an important role in the calculation of cosmic microwave background radiation anisotropies in the brane world. 
Hiroyuki Yoshiguchi and Kazuya Koyama