Quasinormal mode content of binary black hole ringdowns

Technical Summary: Quasinormal Mode Content of Binary Black Hole Ringdowns

Problem Statement
Following the merger of binary black holes (BHs), the remnant settles into a stationary state by emitting gravitational waves (GWs) known as the ringdown. This phase is theoretically described as a perturbation of the final Kerr black hole, characterized by quasinormal modes (QNMs)—damped sinusoidal oscillations. While the fundamental quadrupolar QNM is routinely detected, the relative importance of subdominant modes (overtones, retrograde modes, and nonlinear modes) remains unclear and depends on progenitor binary properties.

Determining the precise mode content of numerical waveforms is challenging due to ambiguities in defining the ringdown start time ($t_0$), the presence of nonlinearities not captured by linear perturbation theory, and the risk of overfitting when including numerous QNMs. Previous approaches have largely relied on least-squares fits to individual harmonics, assessed via simple mismatch metrics, often requiring ad hoc adjustments like free-frequency fits or amplitude-stability checks.

Methodology
The authors present a fully Bayesian, data-driven framework for identifying QNM content in high-accuracy Cauchy–Characteristic Evolution (CCE) gravitational waveforms from the public SpECTRE CCE catalog.

• Waveform Source: The study utilizes simulations evolved to future null infinity ($\mathscr{I}^+$) using CCE, which provides an exterior solution based on null slices. Waveforms are transformed into the superrest frame and post-processed to account for numerical uncertainty using a physically-motivated Gaussian process model trained on the catalog.
• The Model: The GW signal is decomposed into spin-weighted spherical harmonics. The ringdown model is a sum of:
  • Linear QNMs: Damped sinusoids with frequencies and mode-mixing coefficients derived from BH perturbation theory, dependent on the remnant mass ($M_f$) and spin ($\chi_f$).
  • Nonlinear QNMs: Terms predicted by second- and third-order perturbation theory, including quadratic (pairs of linear modes) and cubic (triples of linear modes) interactions.
  • Offsets and Tails: Constant offsets for each harmonic and late-time power-law tails (Price tails), though the latter are treated separately.
• Model Building Algorithm:
  1. Candidate Selection: A set of candidate modes ($C$) includes linear QNMs (up to overtone $n=6$), retrograde modes, constant offsets, and specific nonlinear combinations.
  2. Iterative Detection: Starting with an empty model, the algorithm iteratively adds the candidate mode that maximizes the Bayes factor (significance $S_c$) relative to the current set of detected modes ($D$). This process continues until no remaining candidate exceeds a significance threshold ($S_{threshold} \approx 0.9999$, corresponding to $\sim 4\sigma$).
  3. Posterior Predictive Checks (PPC): To prevent over- or under-fitting, the final model undergoes a PPC. The amplitude of fit residuals is compared against the expected distribution derived from the Gaussian process model of numerical noise. A poor fit (large cumulative distribution function value) indicates the model is inadequate for that specific start time.
  4. Tail Search: A separate analysis searches for power-law tails by marginalizing over oscillatory parameters, fixing the oscillatory content to the modes detected by the primary algorithm.

Key Results
The framework was applied to 13 numerical simulations across a wide range of start times ($t_0$).

• High-Order Overtones: For early start times ($t_0 \lesssim 9M$), the algorithm confidently identifies multiple high-order overtones (up to $n=6$) across various spherical harmonics. As $t_0$ increases, these overtones drop out in a predictable manner, consistent with their faster decay rates.
• Nonlinear Modes: The study identifies nonlinear QNMs, including quadratic modes (e.g., $(2,2,0,+)^2$) and a cubic mode ($(2,2,0,+)^3$). These nonlinear modes persist longer than high-order overtones due to the slower decay of their sourcing fundamental linear modes. Notably, the cubic QNM is consistently identified with a stable amplitude for $t_0 \lesssim 20M$.
• Retrograde Modes: Retrograde modes are occasionally detected at very early start times but are more prominent in specific simulations (e.g., equal-mass systems with anti-aligned spins).
• Amplitude Stability: While the Bayesian approach correctly infers expected decay rates for fundamental and low-order modes, high-order overtone amplitudes show less stability with varying $t_0$. This is attributed to the short decay times of overtones and potential confusion with the prompt response.
• Power-Law Tails: No power-law tails were detected in the CCE waveforms. The posterior distributions for tail amplitudes were consistent with zero, placing tight upper bounds (e.g., $|\hat{A}_{(6,6)}| < 3.6 \times 10^{-6}$). This absence is attributed to the limited size of the Cauchy region in CCE simulations and the lack of back-reaction from outside this region.

Significance and Claims
The paper claims to provide a systematic, objective reference for the ringdown mode content across a wide range of start times for all available simulations in the catalog. By utilizing a fully Bayesian approach with evidence ratios and PPCs, the method avoids the ad hoc nature of previous least-squares techniques.

The detection of multiple high-order overtones near the merger, alongside nonlinear modes up to cubic order, reinforces the physical significance of these components for modeling the ringdown. The authors emphasize that their data-driven approach confirms the necessity of including these modes to accurately describe the waveform at early times, while the PPC ensures that the resulting models are statistically robust against numerical noise. The absence of tails in CCE data serves as a validation of the simulation methodology's limitations regarding late-time behavior, distinguishing it from Cauchy Characteristic Matching (CCM) simulations where tails have been previously observed.