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-Nordström-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.

Figures (8)

• Figure 1: Eigenvalues of the discretized representation of the energy operator $\hat{H}$ (see Eqs. \ref{['energy norm scalar']} and \ref{['matrix evolution']}) for scalar perturbations on a SdS spacetime with $\Lambda M^2 = 0.01$ and $N=100$ grid points. Scalar perturbations with $\ell=0$ (red circles) have a vanishing fundamental eigenvalue $\uplambda_0\sim 10^{-65}$ (up to round-off error) associated with the non-trivial zero mode. For $\ell > 0$, i.e. $\ell=1$ (black squares), $\ell=2$ (blue 'x' crosses) and $\ell=3$ (purple '+' crosses), $\hat{H}$ is positive-definite and $\uplambda_n >0$. Due to the scaling of the plot, all these modes are bunched together at the top.
• Figure 2: Numerical convergence of the normalized perturbation amplitude $\epsilon$ from Eq. \ref{['perturbed operator redefinition']} for scalar fields on SdS spacetime with $\Lambda M^2 = 0.01$ and frequency $k=10$ for a deterministic perturbation added to the BH potential. For $\ell > 0$, i.e. $\ell=1$ (black squares), $\ell=2$ (blue 'x' crosses) and $\ell=3$ (purple '+' crosses), $\|\delta L_1 \|_{_E}\sim 1$ and the normalized amplitudes converge to values with the same order of magnitude of the target scale (here $10^{-1}$). For $\ell=0$ (red circles), $\epsilon$ tends to zero as $N$ increases, due to the norm $\|\delta L_1 \|_{_E}$ being ill-defined. These extremely small values of the normalized perturbation amplitude explain why Ref. Sarkar:2023rhp did not observe the same spectral instabilities for $\ell=0$ as those for $\ell>0$.
• Figure 3: Top panel: Scalar $\ell=0$ normal modes (red dots) and pseudospectra (white contour curves) of a SdS BH with $\Lambda M^2=0.01$. The operator $L$ in this test case is self-adjoint, since $L_2=0$. The $\epsilon$-pseudospectra contours range from $-1.8$ (innermost contour) to $-0.4$ (outermost contour) with steps of $0.2$. On top of figure, the condition number is shown for various modes. The question mark for the non-trivial zero mode corresponds to a divergent condition number as the grid points are increased. Bottom panel: Same as top panel for scalar $\ell=1$ normal modes. For both figures, $N=150$ grid points were used.
• Figure 4: Left: Scalar $\ell=1$ QNMs (red dots) and pseudospectra (white contour curves) of a SdS BH with $\Lambda M^2=0.01$. The $\epsilon$-pseudospectra contours range from $-27.5$ (uppermost contour) to $-2.5$ (bottom contour) with steps of $2.5$ and $N=150$. Right: Same as left with $\Lambda M^2=0.001$. The $\epsilon$-pseudospectra contours range from $-30$ (uppermost contour) to $-2.5$ (bottom contour) with steps of $2.5$ and $N=250$.
• Figure 5: Left: Scalar $\ell=2$ QNMs (red dots) and pseudospectra (white contour curves) of a RNdS BH with $\Lambda M^2=0.06$ and $Q/M=0.5$. The $\epsilon$-pseudospectra contours range from $-28$ (uppermost contour) to $0$ (bottom contour) with steps of $2$ and $N=150$. Right: Same as left for scalar $\ell=1$ QNMs of a RNdS BH with $\Lambda M^2=0.001$ and $Q/M=0.3$. The $\epsilon$-pseudospectra contours range from $-34$ (uppermost contour) to $0$ (bottom contour) with steps of $2$ and $N=250$.