Quadratic Quasinormal Modes Explained

• Quadratic quasinormal modes are nonlinear oscillations arising from second-order interactions of linear QNMs, with frequencies formed by sum and difference combinations of parent modes.
• They are computed using second-order perturbation theory and spectral methods such as Laplace transforms, residue analysis, and continued fraction techniques, ensuring precise amplitude extraction.
• QQNMs play a pivotal role in probing nonlinear dynamics, mode parity effects, and potential deviations from general relativity in both gravitational and photonic systems.

Quadratic quasinormal modes (QQNMs) are a distinct class of gravitational or electromagnetic oscillations emerging from second-order perturbative interactions of linear quasinormal modes (QNMs) on black hole backgrounds and, more generally, in dissipative resonant systems. Unlike linear QNMs, whose frequencies and decay rates are intrinsic properties of the background and associated boundary conditions, QQNMs arise from quadratic couplings among linear modes. Their frequencies are set by specific frequency combinations (typically sums and differences) of the parent linear modes, and their amplitudes encode information about nonlinear processes, mode parity mixing, and—in some contexts—features of the underlying theory of gravity beyond general relativity (GR).

1. Mathematical Definition and Origins

Quadratic quasinormal modes are generated when linear QNMs interact nonlinearly, e.g., in the post-merger ringdown of gravitational waves from binary black holes. If the first-order metric perturbation $h^{(1)}_{ab}$ excites ringdown at frequencies $\omega_{\ell m n}$, at second order the nonlinear source term

$L[h^{(2)}] = S[h^{(1)}, h^{(1)}]$

induces oscillations at frequencies

$$\omega^{\text{QQNM}} = \omega_{I} \pm \omega_{J}, \quad \tau^{-1}_{\text{QQNM}} = \tau^{-1}_I + \tau^{-1}_J$$

where $I, J$ index the linear modes. The general second-order solution thus contains damped sinusoids at these frequencies, forming a set of quadratic quasinormal modes whose amplitudes depend quadratically on the first-order amplitudes $A_I, A_J$, with proportionality factors sensitive to spectral overlaps, parity mixing, and background black hole parameters (Bourg et al., 2024, Yi et al., 2024, Bourg et al., 10 Mar 2025).

2. Perturbative Formalism and Spectral Expansion

Second-order perturbation theory

Black hole metric expansions yield:

$$g_{ab} = g_{ab}^{(0)} + \epsilon\, h^{(1)}_{ab} + \epsilon^2 h^{(2)}_{ab} + O(\epsilon^3)$$

where $g_{ab}^{(0)}$ is the background (typically Kerr or Schwarzschild), $h^{(1)}_{ab}$ is the linear perturbation, and $h^{(2)}_{ab}$ obeys

$\delta G[h^{(2)}] = -\delta^2 G[h^{(1)}, h^{(1)}]$

with $\delta^2 G$ quadratic in first-order fields and their derivatives. This structure induces second-order “driven” Teukolsky equations for the Weyl scalars (e.g., $\Psi_4^{(2)}$ for gravitational waves) with quadratic source terms, which select QQNM frequencies by spectral projection (Bourg et al., 10 Mar 2025).

QNM and QQNM spectral decomposition

In the Laplace-transform or frequency-domain approach,

$$\Psi^{(2)}(t, r, \theta, \phi) = \sum_{I, J} \mathcal A^{(2)}_{I \times J} e^{i (\omega_I + \omega_J)t} + \text{tails}$$

The amplitude $\mathcal A^{(2)}_{I \times J}$ can be extracted by residue methods, with the explicit dependence on parity ratios and black hole parameters. Notably, the second-order solution contains not only the sum frequency but also the difference frequency, producing a richer spectral content than at linear order (Fransen et al., 3 Sep 2025).

3. Parity and Selection Rules for QQNMs

The amplitude and detectability of QQNMs are strongly modulated by the parity composition of the parent linear modes. Even in highly symmetric scenarios (e.g., Schwarzschild, equatorial collisions), both even (polar-led) and odd (axial-led) modes contribute, but the QQNM amplitude at infinity depends on the progenitor's up–down symmetry (Bourg et al., 2024, Bourg et al., 10 Mar 2025). The key closed-form formula for the QQNM-to-QNM amplitude ratio, using $R$ as the parity ratio, is:

$$\mathcal R^{h}_{LM}(R) = -\frac{s_{mn}^2}{8} \left[ a^{LM}(0) + b^{LM}(0) \frac{1 + iR}{1 - iR} \right]$$

with $R = C^-_{mn}/C^+_{mn}$, reflecting the ratio of odd to even excitation in the linear metric perturbation. This explicit parity dependence explains discrepancies in reported QQNM amplitudes and informs templates for nonlinear ringdown analysis (Bourg et al., 2024, Bourg et al., 10 Mar 2025).

4. QQNMs in Alternative Gravitational Theories

Quadratic QNMs also serve as diagnostic features in gravitational theories beyond GR:

• Quadratic Gravity: In Einstein–Weyl and symmergent gravity, additional quadratic curvature terms (e.g., $R^2$, $C_{\mu\nu\rho\sigma}C^{\mu\nu\rho\sigma}$) propagate extra degrees of freedom, such as massive spin-2 ghosts. These yield novel QNM branches (“massive vector/tensor modes”) with frequencies distinct from GR, and are excited through matter stress-energy or coupling to massless modes. The amplitude and spectral content depend explicitly on the quadratic coupling constants, e.g., $\alpha$ or $c_O$, and non-GR “hairy” branches may dominate in specific parameter regions (Antoniou et al., 2024, Gogoi et al., 2023).
• Shift-symmetric Einstein-scalar-Gauss-Bonnet (EsGB) Theories: Perturbations of rotating black holes admit a quadratic expansion in the Gauss–Bonnet coupling $\alpha$, where the $O(\alpha^2)$ term $\omega_2(a)$ quantifies genuine non-GR “QQNM” corrections. These corrections can reach the percent level at moderate coupling and break axial/polar isospectrality, thus shifting ringdown observables in ways potentially measurable by future detectors (Khoo et al., 2024).

5. Analytical and Numerical Techniques for QQNM Calculation

Computation of QQNMs leverages several spectral and numerical methods:

• Frequency-domain spectral codes: These allow high-precision extraction of modal frequencies and amplitudes, using compactified coordinate systems (e.g., Chebyshev–Lobatto or hyperboloidal slices) and matrix eigenvalue problems (Khoo et al., 2024, Bourg et al., 10 Mar 2025).
• Laplace-transform and residue analysis: The use of Laplace transforms, along with analytic contour integration, isolates pole contributions from sum-frequency combinations and yields direct measures of QQNM amplitudes for arbitrary initial data (Bourg et al., 10 Mar 2025).
• Continued fraction (Leaver) and direct integration techniques: Particularly in static backgrounds, continued-fraction algorithms efficiently solve coupled mode equations for both massless and massive QQNM sectors. Direct integration with matching conditions at intermediate points complements the spectral approach, ensuring robust boundary condition enforcement (Antoniou et al., 2024).
• Regularized modal expansions in open systems: In photonics, resonance expansions for quadratic observables (e.g., energy or radiative power) utilize the Riesz projection method. This demands evaluation of regularized QNM fields at conjugate frequencies to maintain convergence outside the resonator (Betz et al., 2022).

6. Black Hole Spin and Mode Coupling Effects

The excitation mapping from linear to quadratic modes is modulated by spin:

• For Kerr black holes, the ratio $\alpha(a) = A_{(2)}/A_{(1)}^2$ in channels such as $(2,2,0) \times (2,2,0) \to (4,4)$ decreases with increasing spin, while off-diagonal channels (e.g., $(2,2,0) \times (2,2,0) \to (5,4)$) activate only at nonzero spin and rise monotonically (Ma et al., 2024).
• The geometric origin of such variations lies in the angular overlap functions between parent and child spheroidal harmonics, and frame-dragging induced by rotation enhances mixing into higher-$\ell$ modes.

7. Observational Relevance and Signal Extraction

Advanced detectors (Einstein Telescope, Cosmic Explorer, LISA) may achieve detection of QQNMs in gravitational wave ringdown signals (Yi et al., 2024):

• The dominant QQNM (e.g., $(220 \times 220)$ in Kerr) appears at twice the frequency of its parent linear mode, with amplitude proportional to the square of the linear excitation ($$A_{220\times 220} \approx \kappa (A_{220})^2,~\kappa\sim 0.3{-}0.4$$). Parameter-estimation studies find relative errors of 20–100% for amplitude and $\lesssim 10\%$ for frequency and phase in optimistic LISA scenarios.
• QQNMs are subdominant in time series unless the fundamental is filtered, but their presence constrains nonlinearities in the strong-field regime and supports “nonlinear spectroscopy” tests.
• Mischaracterization of QQNM amplitudes due to neglected parity ratios or non-GR corrections can bias no-hair tests and ringdown parameter inference.

8. QQNMs Beyond Gravity: Photonic Resonators

In nanophotonics, QQNMs enable resonance expansions of quadratic observables such as absorbed power or radiated far-field intensity, using regularized QNM fields to handle divergence outside the resonator. A handful of QQNMs generally suffice to capture quadratic observables to high accuracy (Betz et al., 2022).

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References:

• Quasinormal modes of rotating black holes in shift-symmetric Einstein-scalar-Gauss-Bonnet theory (Khoo et al., 2024)
• Quadratic quasinormal modes at null infinity on a Schwarzschild spacetime (Bourg et al., 10 Mar 2025)
• Gravitational quasinormal modes of black holes in quadratic gravity (Antoniou et al., 2024)
• Perturbations of Plane Waves and Quadratic Quasinormal Modes on the Lightring (Fransen et al., 3 Sep 2025)
• Quasinormal modes and greybody factors of symmergent black hole (Gogoi et al., 2023)
• Nonlinear quasinormal mode detectability with next-generation gravitational wave detectors (Yi et al., 2024)
• Resonance expansion of quadratic quantities with regularized quasinormal modes (Betz et al., 2022)
• Quadratic quasi-normal mode dependence on linear mode parity (Bourg et al., 2024)
• The excitation of quadratic quasinormal modes for Kerr black holes (Ma et al., 2024)