Quasi normal modes in Schwarzschild-DeSitter spacetime: A simple derivation of the level spacing of the frequencies

TL;DR

This work addresses the structure of quasinormal modes in Schwarzschild–DeSitter spacetime, a two-horizon geometry characterized by surface gravities $\kappa_-$ and $\kappa_+$. By applying the first Born approximation to the scalar wave equation and computing the scattering amplitude, the authors identify the QNM frequencies as poles of the amplitude. They find that the imaginary parts of the QNM frequencies are equally spaced with level spacing given by $\kappa_-$, and that this spacing is independent of the cosmological horizon $\kappa_+$. The analysis clarifies boundary-condition dependencies, aligns with monodromy results in the large-$n$ limit, and shows that pure DeSitter space has no QNMs for massless scalars; the work provides a tractable analytic handle on a two-horizon spacetime and discusses implications for horizon thermodynamics and future, more rigorous treatments of the real parts of the spectrum.

Abstract

It is known that the imaginary parts of the quasi normal mode (QNM) frequencies for the Schwarzschild black hole are evenly spaced with a spacing that depends only on the surface gravity. On the other hand, for massless minimally coupled scalar fields, there exist no QNMs in the pure DeSitter spacetime. It is not clear what the structure of the QNMs would be for the Schwarzschild-DeSitter (SDS) spacetime, which is characterized by two different surface gravities. We provide a simple derivation of the imaginary parts of the QNM frequencies for the SDS spacetime by calculating the scattering amplitude in the first Born approximation and determining its poles. We find that, for the usual set of boundary conditions in which the incident wave is scattered off the black hole horizon, the imaginary parts of the QNM frequencies have a equally spaced structure with the level spacing depending on the surface gravity of the black hole. Several conceptual issues related to the QNM are discussed in the light of this result and comparison with previous work is presented.