High overtones of Schwarzschild-de-Sitter quasinormal spectrum

Abstract

We find the high overtones of gravitational and electromagnetic quasinormal spectrum of the Schwarzschild-de Sitter black hole. The calculations show that the real parts of the electromagnetic modes asymptotically approach zero. The gravitational modes show more peculiar behavior at large $n$: the real part oscillates as a function of imaginary even for very high overtones and these oscillations settles to some "profile" which just repeats itself with further increasing of the overtone number $n$. This lets us judge that $Re\omega$ is not a constant as $n\to \infty$ but rather some oscillating function. The spacing for imaginary part $Im{\omega }_{n+1}-Im{\omega }_n$ for both gravitational and electromagnetic perturbations at high $n$ oscillates as a function of $n$ around the value $k_e$, where $k_e$ is the surface gravity. In addition we find the lower QN modes for which the values obtained with numerical methods are in a very good agreement with those obtained through the 6th order WKB technique.

15 pages, 4figures, Latex format