Quantitative local recovery of Kerr-de Sitter parameters from high-frequency equatorial quasinormal modes

TL;DR

This paper demonstrates that high-frequency equatorial quasinormal modes of Kerr-de Sitter black holes can be used to quantitatively recover the black hole's mass, rotation, and cosmological constant parameters with stability estimates.

Contribution

It provides the first quantitative inverse spectral results for Kerr-de Sitter parameters using high-frequency quasinormal modes, including explicit geometric corrections.

Findings

Finite high-frequency mode data determines (M,a) with stability.

Adding a damping observable determines (M,a,Λ) locally.

Explicit second-order corrections to photon-orbit invariants are computed.

Abstract

We study an inverse resonance problem for the scalar wave equation on the Kerr-de Sitter family. In a compact subextremal slow-rotation regime and at a fixed overtone index, high-frequency quasinormal modes admit semiclassical quantization and a real-analytic labeling by angular momentum indices. Using this structure, we first prove that a finite equatorial high-frequency package of quasinormal-mode frequencies determines the mass and rotation parameter $(M,a)$ (for fixed cosmological constant $\Lambda>0$), with a quantitative stability estimate. As a key geometric input we compute explicit second-order (in $a$) corrections to the equatorial photon-orbit invariants which control the leading real and imaginary parts of the quasinormal modes. Finally, allowing $\Lambda$ to vary in a compact interval, we show that adding one damping observable (the scaled imaginary part of a single equatorial mode) yields a three-parameter inverse theorem: a finite package of three independent real observables determines $(M,a,\Lambda)$ locally in the slow-rotation regime away from $a=0$.