The quasinormal modes, pseudospectrum and time evolution of Proca fields in quantum Oppenheimer-Snyder-de Sitter spacetime

TL;DR

The paper investigates axial Proca perturbations of a quantum-corrected black hole in de Sitter space (qOS-dS), focusing on quasinormal modes, pseudospectrum, and time-domain evolution. It introduces a parameterization by horizon ratios $p$ and $q$ and employs a hyperboloidal framework with Chebyshev spectral methods to compute QNMs, complemented by a Green-function approach for time dynamics. The study reveals parametric QNM instability driven by $p$ and $q$, supported by pseudospectrum analysis, and shows that Proca mass $m$ mainly shifts oscillation frequencies without inducing quasi-resonances, with no power-law tail due to exponential potential decay. These results provide a pathway to constrain quantum corrections in LQG-inspired spacetimes through ringdown and precursor signals, while highlighting limitations of the semi-classical treatment and the need for full quantum gravity analysis in the future.

Abstract

In this study, we investigate the quasinormal modes, pseudospectrum and time evolution of a massive vector field around a quantum corrected black hole in de-Sitter spacetime. We start by parameterization and using orthonormal tetrads to get the effective potential. Methodologically we use the hyperboloidal framework together with discretizing the non-selfadjoint operator through Chebyshev-Gauss-Labatto grid to attain the QNMs. We explore the parametric instability of QNMs caused by quantum correction, cosmological constant and Proca mass, and these three factors show very different influences on the QNMs' migration flow. On the other hand, we discuss the instability of QNMs with arbitrary-shape perturbation and the effectiveness of numerical results through pseudospectrum. We use high frequency approximation to attain the expression of the time domain Green function and clarify the origin of two different stages in time evolution. Through numerical methods we confirm that no power-law late time tail is expected, and the possible impact on time evolution caused by quantum correction is discussed.

Figures (12)

• Figure 1: Penrose diagram for the maximum extension of the spacetime with three horizons in the exterior area of the dust bull. However, only the areas $\mathrm{A}_{ext}$, $\mathrm{B}_{ext}$ and $\mathrm{C}_{ext}$ and $\mathrm{D}$ are real and the metric (\ref{['metric']}) can make sense.
• Figure 2: The relation between some given parameters and the relative ratio of the three horizon radius.
• Figure 3: The $\tau=$constant hypersurfaces in $t-\sigma$ diagram under the parameters $p=q=0.5$. It's shown that while $\tau$ remains constant $t$ approaches infinity on both sides.
• Figure 4: The $p-q$ diagram for the model, where the orange region corresponds to the permitted region for our method while the blue region corresponds to the forbidden region.
• Figure 5: The migration of the first 3 or 4 overtones under the parameter $m,l$ fixed and simply changing $p$ from 0 to 1 (left) or simply changing $q$ from 0 to 0.8 (right). Here and in the following calculation $q$ does not range from 0 to 1 to avoid being captured in the forbidden region.