Pseudospectrum of de Sitter black holes

Abstract: Pseudospectral analyses have broadened our understanding of ringdown waveforms from binary remnants, by providing insight into both the stability of their characteristic frequencies under environmental perturbations, as well as the underlying transient and non-modal phenomenology that a mode analysis may miss. In this work we present the pseudospectrum of scalar perturbations on spherically-symmetric black holes in de Sitter spacetimes. We expand upon previous analyses in this setting by calculating the pseudospectrum of Reissner-Nordstr\"om-de Sitter black holes, and revisit results regarding the stability of quasinormal modes under perturbations in several cases. Of particular note is the case of scalar quasinormal modes with angular parameter $\ell=0$, which possess a zero mode related to the presence of a cosmological horizon. We show that the non-trivial eigenfunction associated to this mode has a vanishing energy norm which poses a challenge in quantifying the magnitude of external perturbations to the wave equation's potential, as well as in calculating the pseudospectrum. Nonetheless, we present results which suggest that the spectral instability manifestation of $\ell=0$ scalar quasinormal modes is qualitatively the same as in other cases, in contrast to recent claims. We also analyze the stability of the fundamental mode for $\ell\ge1$, finding it to be spectrally stable, except for certain configurations in which a perturbation leads to a discontinuous overtaking of the fundamental unperturbed purely-imaginary mode by a perturbed complex quasinormal mode.