Nonlinear QNM 220Q in Black Hole Ringdown

• Nonlinear QNMs (220Q) are second-order perturbative solutions that capture the quadratic, self-coupled response of black hole spacetimes during ringdown.
• The mode is characterized by frequency doubling (2ω220) and halved damping time, validated through analytic derivations and numerical relativity simulations.
• Detection of 220Q enhances black hole spectroscopy, offering precise tests of GR’s nonlinear dynamics and aiding in the discrimination of true signals from noise.

Nonlinear Quasi-Normal Mode (220Q)

Nonlinear quasi-normal modes (QNMs), typified by the second-order harmonic 220Q, represent the nonlinear, quadratic response of a black hole spacetime to perturbations—predicted and quantitatively characterized within general relativity. The 220Q mode arises from the nonlinear self-coupling of the fundamental linear (2,2,0) QNM and is a distinctive feature of post-merger ringdown gravitational waves. Its detection signals direct evidence of nonlinear dynamics in the strong-field regime and enables precision tests of general relativity’s nonlinear structure through black hole spectroscopy.

1. Perturbative Origin and Mathematical Structure

The 220Q mode is a second-order solution of black hole perturbation theory. The metric is expanded as $$g_{\mu\nu} = g^{(0)}_{\mu\nu} + h^{(1)}_{\mu\nu} + h^{(2)}_{\mu\nu}$$, where $h^{(1)}$ is the linear perturbation and $h^{(2)}$ is the second-order correction. For a Schwarzschild background, the first-order master function $\psi^{(1)}(t, r)$ obeys the Zerilli equation: $$\left[ -\frac{\partial^2}{\partial t^2} + \frac{\partial^2}{\partial r_*^2} - V_Z(r) \right]\psi^{(1)}(t, r) = 0$$ where $r_*$ is the tortoise coordinate.

At second order,

$$\left[ -\frac{\partial^2}{\partial t^2} + \frac{\partial^2}{\partial r_*^2} - V_Z(r) \right]\chi^{(2)}(t, r) = S(t, r)$$

with the source term $S(t, r)$ quadratic in $\psi^{(1)}$ and its derivatives. This quadratic source is regularized so as to decay physically (falling off as $O(r^{-2})$ at infinity and being regular at the horizon), ensuring the physicality of the resulting solutions (0708.0450).

The dominant nonlinear mode 220Q has a frequency and damping time given by: $$\omega_{220Q} = 2\omega_{220}, \qquad \tau_{220Q} = \tau_{220}/2$$ arising from the quadratic anharmonic coupling, with both analytical and numerical studies confirming these relations (0708.0450, Lagos et al., 4 Nov 2024, Yi et al., 14 Mar 2024, Kehagias et al., 12 Nov 2024, Kehagias et al., 12 Mar 2025).

2. Numerical Computation and Extraction of 220Q

Leaver’s continued fraction method is adapted to solve the inhomogeneous second-order Zerilli equation (0708.0450). The second-order wavefunction is expanded as a series around either the horizon or spatial infinity with boundary conditions determined by outgoing/ingoing wave requirements. For each coefficient in the expansion, modified three-term recurrence relations (now inhomogeneous due to the source $S$) are solved. The solution, in the frequency domain, yields the amplitude of $\chi^{(2)}$ at infinity (or at the horizon) in terms of the linear QNM: $$\chi^{(2)}(t, r\to\infty) = C_I \omega^{(1)} \psi^{(1)}(t, r\to\infty)$$ where $C_I$ is computed numerically for a given mode and regularization (0708.0450).

In full numerical relativity simulations (NR), multimode fitting—implemented as a greedy linear-least-squares algorithm—serves to extract the amplitudes and phases of both linear and quadratic (220Q-type) QNMs, even when the signal is a linear combination of spheroidal harmonics. Second-order excitation coefficients are modeled as proportional to the product of first-order excitation coefficients, i.e.,

$$A_{(l_1 m_1 n_1)(l_2 m_2 n_2)} = \mu_{(l_1 m_1 n_1)(l_2 m_2 n_2)} A_{l_1 m_1 n_1} A_{l_2 m_2 n_2}$$

with the coefficient $\mu$ fit numerically (London et al., 2014).

3. Analytic Insights and Universality

A key analytical insight is that, for perturbations localized near the photon ring, the Penrose limit reduces the spacetime to a pp-wave geometry, allowing the nonlinear coupling to be solved explicitly (Kehagias et al., 12 Nov 2024, Kehagias et al., 12 Mar 2025). In this limit, the quadratic nonlinearity in the wave amplitude is universal in the high-$\ell$ (eikonal) regime: the ratio $h^{(2)}/[h^{(1)}]^2$ becomes independent of the angular momentum number $\ell$ and asymptotes to a value ($\sim 0.24$ in Schwarzschild). Symmetry under rescalings confirms this result; the quadratic source and wave function scaling with $\ell$ cancel so that the quadratic-to-linear amplitude ratio in gravitational wave strain is constant for all large $\ell$ (Kehagias et al., 12 Mar 2025). This is confirmed by direct computation and symmetry analysis: $h^{(2)}/[h^{(1)}]^2 = \text{const}(\ell\to\infty)$ This universality extends to channels ($\ell\times\ell \to 2\ell$) and is robust under matching from the near-ring regime to asymptotic (far-field) amplitudes.

4. Parity Structure and Initial Perturbations

The amplitude of the 220Q mode is not solely a function of black hole parameters—it is also sensitive to the mixture of even- and odd-parity linear QNMs excited during the merger (Bourg et al., 16 May 2024). In the generalized solution,

$$A_{mn} = -\frac{s_{mn}^2}{4}\lambda_{,2}(C^{+}_{mn} - i C^{-}_{mn})$$

where $C^{+},C^{-}$ encode the even/odd decomposition of the initial perturbation. The 220Q amplitude thus reflects both intrinsic black hole properties and the formation history, being a function of the ratio of even- to odd-parity QNM components. The overall QQNM-to-QNM amplitude ratio can be written schematically as: $$\mathcal{R}^{(h_{\mathrm{SpEC}})} = -\frac{s_{mn}^2}{8} \mathcal{R}^{(\Psi_4)}\left(C_{mn}^{+}, C_{mn}^{-}\right)$$ This implies that ringdown signals carry information about the symmetry of the initial merger state, and that QQNM measurements can, in principle, inform on the details of the binary progenitor.

5. Astrophysical Detection and Data Analysis

The 220Q mode ("quadratic mode") may be detected by current and future gravitational wave observatories. Its amplitude relative to the linear (220) mode is predicted to be in the range 10–24% for binary black hole mergers, with a frequency given by $2\omega_{220}$ and a damping time half that of the linear mode (0708.0450, Lagos et al., 4 Nov 2024, Yi et al., 14 Mar 2024).

Detection and measurement strategies:

• Bayesian inference methods can distinguish between models including the 220Q (quadratic) mode and those including solely the linear 440 mode, with the quadratic mode often statistically favored (i.e., higher Bayes factors) (Yang et al., 19 Oct 2025).
• Inclusion of the 220Q mode in parameter estimation improves consistency between ringdown-derived remnant mass and spin and those from full inspiral-merger-ringdown analyses.
• Signal-to-noise ratio (SNR) forecasts suggest that ground-based detectors (e.g., ET, CE) can detect the 220Q mode in a modest number (order tens) of stellar-mass events per year, with dramatically larger numbers of massive black hole events for space-based detectors (LISA), where redshifting moves the quadratic mode frequency into optimal sensitivity bands (Yi et al., 14 Mar 2024).

A typical waveform model for the measured strain is

$$h_+ - i h_\times = \frac{M}{r} \Bigg\{ \sum_{(\ell m n)} A_{\ell m n} e^{-i\omega_{\ell m n}(t-t_0)} {}_{-2}Y_{\ell m}(\iota, \beta) + A_{(220)\times(220)} e^{-i\omega_{(220)\times(220)}(t-t_0)} {}_{-2}Y_{44}(\iota, \beta) \Bigg\}$$

with the QQNM amplitude and phase empirically related to the square of the (220) mode coefficients (Lagos et al., 4 Nov 2024): $$A_{(220)\times(220)} = 0.154\,e^{-i0.068}\, A_{220}^2$$

6. Physical Implications and Fundamental Tests

Detection of the 220Q mode enables several key tests:

• Strong-field nonlinearity: The frequency doubling and predictable amplitude scaling of the 220Q mode provide a direct probe of general relativity’s nonlinear regime. Deviations from the predicted relation $\omega^{(2)} = 2\omega^{(1)}$, or from the expected amplitude ratio, would signal the breakdown of GR or the presence of new physics (0708.0450, Lagos et al., 4 Nov 2024).
• No-hair theorem and black hole spectroscopy: The remnant mass and spin inferred from the second-order QNM must be consistent with the values inferred from linear modes, providing an independent check on the uniqueness property of the Kerr solution (0708.0450).
• Astrophysical applications: The 220Q amplitude can act as a “smoking gun” in distinguishing real ringdown signals from instrumental transients, enhance distance estimation in cases where the inspiral phase is inaccessible, and improve parameter constraints when “folded in” with linear modes (0708.0450).
• Sensitivity to progenitor details: Due to the dependence of the 220Q amplitude on even/odd-parity content, its detection carries imprints of the symmetry and geometry of the precursor system (Bourg et al., 16 May 2024).

7. Outlook and Future Directions

The analytic and numerical understanding of nonlinear QNMs, particularly 220Q, is now mature and supported both by perturbation theory and full NR. Current and next-generation detectors are approaching the sensitivity needed for routine detection of quadratic QNMs in high-SNR events, especially for massive black holes (Yi et al., 14 Mar 2024, Lagos et al., 4 Nov 2024).

Open directions include:

• Extension of analytic results to rotating (Kerr) black holes, where preliminary studies indicate modest spin dependence in the nonlinear amplitude (Kehagias et al., 12 Nov 2024, Kehagias et al., 12 Mar 2025).
• Further exploration of the full space of quadratic (and higher order) QNM couplings, including mixed modes and mode-mixing effects in asymmetric or generic merger configurations.
• Integration into gravitational wave data pipelines for improved classification and rejection of spurious events, and for more stringent consistency tests of general relativity.
• Assessment of possible degeneracies between the quadratic 220Q mode and other linear or overtone modes in realistic detector noise conditions (Yang et al., 19 Oct 2025).

Summary table: Characteristic relations for the 220Q mode

Quantity	Linear (220)	Nonlinear (220Q)
Frequency ($\omega$)	$\omega_{220}$	$2\omega_{220}$
Damping time ($\tau$)	$\tau_{220}$	$\tau_{220}/2$
Amplitude relation	$A_{220}$	$A_{220Q} \propto (A_{220})^2$
Typical amplitude ratio	1	0.1–0.24 (see text/paper)

Detection and precision measurement of the nonlinear (220Q) QNM stand as a direct probe of strong-field gravity, offering both signatures of GR’s nonlinear structure and new handles for black hole astrophysics.