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 $H{r}_c$, 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 ($H{r}_c>1$), 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 $1/2<H{r}_c<1$. This confirms the results obtained by the boundary effective action that there exists a scalar ghost mode for $H{r}_c>1/2$. In a self-accelerating universe $H{r}_c=1$, 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 $H{r}_c=1$, although a careful calculation of the effective action is required to verify this rigorously.