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

TL;DR

This paper establishes a rigorous mathematical framework for analyzing black-hole ringdown signals, demonstrating two-mode dominance, stability of frequency extraction, and explicit parameter bias bounds for Kerr-de Sitter black holes, enhancing high-frequency black-hole spectroscopy.

Contribution

It introduces a novel two-mode dominance theorem, deterministic stability estimates, and explicit bias bounds for recovering black hole parameters from ringdown signals in Kerr-de Sitter spacetimes.

Findings

Two-mode dominance in high-frequency ringdown signals

Explicit parameter bias bounds for $(M,a)$ and $(M,a,bLambda)$

Quantitative stability estimates for frequency extraction

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 $\Lambda>0$ 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 $(M,a,\Lambda)$ 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.