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.
The late-time response of a perturbed black hole is often modeled, on a finite observation window, by a superposition of a small number of exponentially damped oscillations, (1.1) y(t * ) ≈ J j=1
A j e -iω j t * , t * ≥ t 0 , ℑω j < 0, where y(t * ) is a time series obtained from an observation of the field and the complex frequencies ω j are quasinormal modes (QNMs). In applications y is typically not the raw waveform: it may incorporate symmetry reduction, phase-space localization, or other preprocessing designed to isolate a particular dynamical channel before attempting a low-rank fit.
From a deterministic PDE viewpoint, using (1.1) for inference leads to two intertwined stability issues. Dominance and leakage. Even when a meromorphic resolvent and a resonance expansion are available, the time-domain representation is typically an infinite sum with a remainder. A finite-mode model on a prescribed window requires quantitative dominance of the selected poles together with quantitative control of leakage from all other poles and from the tail term on a shifted contour.
Nonnormality and extraction conditioning. The stationary problem is non-selfadjoint; residue projectors can be large (often called excitation factors), and pseudospectral and transient effects may interfere with naive truncations or fits that do not separate the relevant phase-space channels. Moreover, extracting complex frequencies from finite-time data is ill-conditioned in general and must be analyzed together with branch selection for the complex logarithm; this is classical in the Prony/matrix-pencil literature [25,26,27,28,29]. Recent work on black-hole ringdown has highlighted these mechanisms from several complementary angles, including pseudospectral analyses, non-modal dynamics, and transients [14,20,15,21,18]. Closely related phenomena arise near avoided crossings and exceptional points, where resonant excitation can amplify or reshuffle the observed mode hierarchy [16,24].
Setting and scope; two-mode dominance in the equatorial package. We work on subextremal Kerr-de Sitter spacetimes with fixed cosmological constant Λ > 0 and slow rotation |a| ≤ a 0 . In this setting the stationary resolvent is meromorphic, and contour deformation yields resonance expansions for wave solutions within the Fredholm/microlocal framework developed for asymptotically hyperbolic and Kerr-de Sitter geometries [2,5,10,11]. We isolate an equatorial high-frequency package: we fix an overtone index n ∈ N 0 and consider large angular momentum ℓ in the exact azimuthal sectors k = ±ℓ, using h ℓ = ℓ -1 as semiclassical parameter.
A key point of this paper is that, within this package, one can justify a two-mode model after an explicit sequence of localizations: exact azimuthal projection to k = ±ℓ, microlocal equatorial localization, and analytic pole selection by an entire weight. Our two-mode dominance theorem is therefore not a statement about a generic unfiltered ringdown waveform. Rather, it is a quantitative statement about two refined signals (one for each of k = ±ℓ) obtained by preprocessing, in a regime where the relevant QNMs admit a semiclassical labeling and can be isolated with uniform error bounds. More precisely, the dominance statement concerns the mode-separated, microlocalized signals obtained by applying the exact azimuthal projectors Π k ± and the equatorial cutoffs A ±,h ℓ to the solution, together with the entire weights g ±,ℓ inserted in the inverse Laplace representation; see Definition 5.11. This paper is the second part of a series. The companion paper [1] proves the semiclassical equatorial QNM labeling in the above package and establishes a quantitative local inversion map from the pair of equatorial frequencies (ω +,ℓ , ω -,ℓ ) to the parameters (M, a). The purpose of the present paper is to provide the time-domain mechanism needed to turn that frequency information into a deterministic inference procedure with explicit error bookkeeping on a finite window.
Strategy: microlocal/analytic mode selection plus deterministic extraction. Two ingredients are essential. First, we separate the relevant phase-space channel before fitting a low-rank time series: we perform exact azimuthal projection to k = ±ℓ and apply equatorial microlocal cutoffs. Second, we select a single overtone uniformly in ℓ by multiplying the resolvent integrand with an entire weight g ±,ℓ (ω) constructed from equatorial pseudopoles provided by [1]. This analytic window is designed to be compatible with contour deformation: it suppresses all poles except the labeled one while having controlled growth on a shifted contour, yielding a windowed resonance expansion with an explicitly controlled tail term.
A related approach-isolating individual QNM contributions using a mode prior-appears in recent data-analysis work under the name QNM filters [13]. The common theme is that, in a regime where QNM expansions are meaningful, prior frequency information can be used t
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