Existence of Zero-damped Quasinormal Frequencies for Nearly Extremal Black Holes

It has been observed that many spacetimes which feature a near-extremal horizon exhibit the phenomenon of zero-damped modes. This is characterised by the existence of a sequence of quasinormal frequencies which all converge to some purely imaginary number $i\alpha$ in the extremal limit and cluster in a neighbourhood of the line $\Im s=\alpha$. In this paper, we establish that this property is present for the conformal Klein-Gordon equation on a Reissner-Nordstr\"om-de Sitter background. This follows from a similar result that we prove for a class of spherically symmetric black hole spacetimes with a cosmological horizon. We also show that the phenomenon of zero-damped modes is stable to perturbations that arise through adding a potential.