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Two-mode dominance and deterministic parameter bias bounds for equatorial Kerr-de Sitter ringdown

Ruiliang Li

Abstract

We study scalar waves on subextremal Kerr-de Sitter spacetimes in a compact slow-rotation regime and at a fixed overtone index. Working initially at a fixed cosmological constant $Λ>0$ and uniformly for $(M,a)$ in a compact slow-rotation set, using the meromorphic/Fredholm framework for quasinormal modes and a semiclassical equatorial labeling proved in a companion paper, we establish a quantitative two-mode dominance theorem in an equatorial high-frequency package: after exact azimuthal reduction, microlocal equatorial localization, and analytic pole selection by entire localization weights constructed from equatorial pseudopoles, the $k=\pm\ell$ sector signals are each governed by a single quasinormal exponential, up to an explicitly controlled tail and an $\mathcal O(\ell^{-\infty})$ contribution from all other poles. We then develop a fully deterministic frequency-extraction stability estimate based on time-shift invariance, and combine it with the two-mode dominance result and the companion paper's inverse stability theorem to obtain an explicit parameter bias bound for ringdown-based recovery of $(M,a)$. Finally, using the companion paper's three-parameter inverse theorem and a damping observable based on the scaled imaginary part of one equatorial mode, we propagate the same deterministic error chain to a local bias bound for recovery of $(M,a,Λ)$ on compact parameter sets with $|a|$ bounded away from $0$. As a further consequence, we obtain a localized pseudospectral stability statement for the equatorial resolvent package, quantifying how large microlocalized resolvent norms enforce proximity to the labeled equatorial poles. The resulting estimates clarify the conditioning mechanisms (start time, window length, shift step, and detector nondegeneracy) and provide a rigorous PDE-to-data interface for high-frequency black-hole spectroscopy.

Two-mode dominance and deterministic parameter bias bounds for equatorial Kerr-de Sitter ringdown

Abstract

We study scalar waves on subextremal Kerr-de Sitter spacetimes in a compact slow-rotation regime and at a fixed overtone index. Working initially at a fixed cosmological constant and uniformly for in a compact slow-rotation set, using the meromorphic/Fredholm framework for quasinormal modes and a semiclassical equatorial labeling proved in a companion paper, we establish a quantitative two-mode dominance theorem in an equatorial high-frequency package: after exact azimuthal reduction, microlocal equatorial localization, and analytic pole selection by entire localization weights constructed from equatorial pseudopoles, the sector signals are each governed by a single quasinormal exponential, up to an explicitly controlled tail and an contribution from all other poles. We then develop a fully deterministic frequency-extraction stability estimate based on time-shift invariance, and combine it with the two-mode dominance result and the companion paper's inverse stability theorem to obtain an explicit parameter bias bound for ringdown-based recovery of . Finally, using the companion paper's three-parameter inverse theorem and a damping observable based on the scaled imaginary part of one equatorial mode, we propagate the same deterministic error chain to a local bias bound for recovery of on compact parameter sets with bounded away from . As a further consequence, we obtain a localized pseudospectral stability statement for the equatorial resolvent package, quantifying how large microlocalized resolvent norms enforce proximity to the labeled equatorial poles. The resulting estimates clarify the conditioning mechanisms (start time, window length, shift step, and detector nondegeneracy) and provide a rigorous PDE-to-data interface for high-frequency black-hole spectroscopy.
Paper Structure (107 sections, 64 theorems, 378 equations)

This paper contains 107 sections, 64 theorems, 378 equations.

Table of Contents

  1. Introduction
  2. Geometric/PDE setup and functional framework
  3. Spacetime region, regular time coordinate, and a fixed spatial slice
  4. Parameter regime and geometric constants
  5. The wave operator and a convenient splitting
  6. Cauchy problem and energy spaces
  7. First-order formulation and the stationary family
  8. Meromorphic resolvent and quasinormal modes
  9. Functional-analytic conventions and mapping properties
  10. Forward Laplace transform and an explicit inversion formula
  11. Residues, higher-order poles, and time-domain contributions
  12. Exact symmetry reduction in the azimuthal number
  13. Microlocal equatorial cutoffs (angular semiclassical parameter $h_\ell$)
  14. Resolvent bounds on a shifted contour
  15. Choice of shifted contour and cutoff resolvents
  16. ...and 92 more sections

Key Result

Lemma 2.1

Let $P=\Box_g+V$ on $\mathcal{M}_{M,a}$ with $g$ stationary in $t_*$. Then there exist $t_*$--independent operators $Q\in\operatorname{Diff}^1(X_{M,a})$ and $L\in\operatorname{Diff}^2(X_{M,a})$ such that eq:splitting holds. Moreover, $Q$ and $L$ commute with $\Phi=\partial_{\varphi_*}$.

Theorems & Definitions (169)

  • Lemma 2.1: Existence of the splitting
  • proof
  • Remark 2.2: Normalization of the $\partial_{t_*}^2$ coefficient
  • Theorem 2.3: Well-posedness and an a priori bound
  • proof
  • Remark 2.4
  • Theorem 2.5: Meromorphic resolvent and QNMs
  • Remark 2.6
  • Remark 2.7: Two semiclassical parameters
  • Lemma 2.8: Distributional source at $t_*=0$
  • ...and 159 more