Numerical analysis of quasinormal modes in nearly extremal Schwarzschild-de Sitter spacetimes

We calculate the high-order quasinormal modes with large imaginary frequencies for the electromagnetic and gravitational perturbations in nearly extremal Schwarzschild-de Sitter spacetimes. Our results show that for low-order quasinormal modes, analytical approximation formula in the extremal limit derived by Cardoso and Lemos is a quite good approximation for the quasinormal frequencies as long as the model parameter $r_1\kappa_1$ is small enough, where $r_1$ and $\kappa_1$ are the black hole horizon radius and the surface gravity, respectively. For high-order quasinormal modes, whose imaginary parts of the quasinormal frequencies are sufficiently large, on the other hand, this formula becomes inaccurate so much even for small values of $r_1\kappa_1$. We also find that the real parts of the quasinormal frequencies have oscillating behaviors in the limit of highly damped modes. The real parts therefore show a lot of local maximums as a function of the imaginary part of frequencies. Behaviors of these local maximums resemble those of the quasinormal modes of a Schwarzschild black hole. This means that for the gravitational perturbations, the quasinormal modes of the Schwarzschild-de Sitter spacetime appear not to have any asymptotic constant value. On the other hand, for the electromagnetic perturbations, the asymptotic value seems to be zero.