Ghosts in the self-accelerating brane universe

We study the spectrum of gravitational perturbations about a vacuum de Sitter brane with the induced 4D Einstein-Hilbert term, in a 5D Minkowski spacetime (DGP model). We consider solutions that include a self-accelerating universe, where the accelerating expansion of the universe is realized without introducing a cosmological constant on the brane. The mass of the discrete mode for the spin-2 graviton is calculated for various Hr₂, where H is the Hubble parameter and r₂ is the crossover scale determined by the ratio between the 5D Newton constant and the 4D Newton constant. We show that, if we introduce a positive cosmological constant on the brane (Hr₂>1), the spin-2 graviton has mass in the range 01/2. In a self-accelerating universe Hr₂=1, the spin-2 graviton has mass m^2=2H^2, which coincides with the mass of the brane fluctuation mode. Then there arises a mixing between the brane fluctuation mode and the spin-2 graviton. We argue that this mixing presumably gives a ghost in the self-accelerating universe by continuity across Hr₂=1, although a careful calculation of the effective action is required to verify this rigorously.