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 Hr(c), where H is the Hubble parameter and r(c) 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 (Hr(c)> 1), the spin-2 graviton has mass in the range 0 < m(2)< 2H(2) and there is a normalizable brane fluctuation mode with mass m(2)=2H(2). Although the brane fluctuation mode is healthy, the spin-2 graviton has a helicity-0 excitation that is a ghost. If we allow a negative cosmological constant on the brane, the brane fluctuation mode becomes a ghost for 1/2 < Hr(c)< 1. This confirms the results obtained by the boundary effective action that there exists a scalar ghost mode for Hr(c)> 1/2. In a self-accelerating universe Hr(c)=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 Hr(c)=1, although a careful calculation of the effective action is required to verify this rigorously.