(In)stability of de Sitter Quasinormal Mode spectra

TL;DR

This paper investigates how small potentials affect the quasinormal mode spectrum of the conformal wave operator in de Sitter space, providing explicit formulas and numerical verification for the spectrum's stability properties.

Contribution

It derives explicit formulas for the rate of change of quasinormal frequencies under perturbations and introduces a pseudospectrum framework to analyze their stability.

Findings

Explicit formulas for frequency shifts due to small potentials

Numerical verification using extended spectral hyperboloidal method

Proposed pseudospectra to characterize spectral instability

Abstract

We consider how the quasinormal spectrum for the conformal wave operator on the static patch of de Sitter changes in response to the addition of a small potential. Since the quasinormal modes and co-modes are explicitly known, we are able to give explicit formulae for the instantaneous rate of change of each frequency in terms of the perturbing potential. We verify these exact computations numerically using a novel technique extending the spectral hyperboloidal approach of Jaramillo et al. (2021). We propose a definition for a family of pseudospectra that we show capture the instability properties of the quasinormal frequencies.