Ringdown Mode Amplitudes and Phases

• The paper demonstrates that ringdown mode amplitudes and phases quantitatively encode pre-merger parameters and the remnant’s mass and spin.
• Analytical formulas, such as the relation between A21 and A22, capture the dependence on mass ratio and spin asymmetry, enabling indirect spin measurements.
• Bayesian inference applied to ringdown signals reconstructs binary configurations and provides consistency tests for the Kerr hypothesis in gravitational wave astronomy.

The amplitudes and phases of black hole ringdown modes quantitatively characterize the degree to which each quasi-normal mode (QNM) is excited in the post-merger radiation, and collectively encode detailed information not only about the remnant black hole’s mass and spin, but also about the binary progenitor’s masses, spins, and orbital geometry. While the QNM frequencies and damping times are functions of the remnant properties alone, the complex excitation amplitudes and phases are sensitive tracers of pre-merger parameters, embodying the “memory” of the binary history as it is imprinted on the post-merger gravitational wave signal.

1. Progenitor-Dependent Encoding in Ringdown Mode Excitation

The gravitational waveforms from binary black hole mergers are represented during the ringdown phase as sums over QNMs, each characterized by frequency $\omega_{\ell m}$, decay time $\tau_{\ell m}$, amplitude $A_{\ell m}$, and phase $\varphi_{\ell m}$. The general polarization content obeys

$$h_+(t) = \frac{M}{D} \sum_{\ell m} A_{\ell m} Y_+^{(\ell m)}(\iota) e^{-t/\tau_{\ell m}} \cos(\omega_{\ell m} t - m\phi + \varphi_{\ell m}), \ h_\times(t) = \frac{M}{D} \sum_{\ell m} A_{\ell m} Y_\times^{(\ell m)}(\iota) e^{-t/\tau_{\ell m}} \sin(\omega_{\ell m} t - m\phi + \varphi_{\ell m}).$$

While the frequencies and damping times are determined uniquely by the final mass and spin via the Kerr solution, the amplitudes $A_{\ell m}$ and phases $\varphi_{\ell m}$ are explicit functions of the progenitor parameters.

Key findings include:

• The relative amplitudes $A_{22}, A_{33}$ depend predominantly on the symmetric mass ratio $\nu = m_1 m_2/M^2$ and are weakly sensitive to the spins.
• The $(2,1)$ mode amplitude is a strong tracer of progenitor spin differences, capturing detailed spin configuration dependence.
• The mapping from binary parameters $(m_1, m_2, \chi_1, \chi_2)$ to $A_{\ell m}$ is highly nontrivial and encodes the progenitor properties in the observed ringdown multipole structure.

2. Analytical Formulas Linking Amplitudes to Progenitor Masses and Spins

A quantitatively accurate relation was found for the relative amplitude of the $(2,1)$ and $(2,2)$ modes for non-precessing binaries: $$\hat{A}_{21} \equiv \frac{A_{21}}{A_{22}} = 0.43\left[ \sqrt{1-4\nu} - \chi_\text{eff}\right], \qquad (1)$$ where the effective spin parameter is

$$\chi_\text{eff} = \frac{1}{2} \left[ \sqrt{1-4\nu}\, \chi_1 + \chi_{-}\right], \quad \chi_{-} = \frac{m_1\chi_1 - m_2\chi_2}{m_1 + m_2}.$$

Properties of this mapping:

• The factor $\sqrt{1-4\nu}$ is determined by the binary's mass difference (equal to zero for equal-mass binaries).
• The amplitude $A_{21}$ vanishes (or its phase jumps by $\pi$) exactly when $\chi_\text{eff} = \sqrt{1-4\nu}$, indicating mode suppression for specific spin alignments.
• Thus, while $A_{22}$ and $A_{33}$ encode the mass ratio, $A_{21}$ exposes spin asymmetry, enabling indirect spin measurements.

This formula mirrors similar structures found in the post-Newtonian inspiral regime, demonstrating a robust transfer of binary configuration memory into the ringdown.

3. Extraction of Progenitor Parameters from Ringdown Using Bayesian Inference

Because the ringdown signal is short-lived and exponentially damped, parameter estimation for $A_{\ell m}$ and $\varphi_{\ell m}$ requires careful statistical inference. The approach is:

• Construct a ringdown signal model in terms of $A_{22}$, $A_{21}$, $A_{33}$ (and others as required), $\omega_{\ell m}$, $\tau_{\ell m}$, and $\varphi_{\ell m}$.
• Use Bayesian inference (e.g., nested sampling) to obtain posterior distributions for the intrinsic parameters: total mass $M$, symmetric mass ratio $\nu$, and spins $\chi_1$, $\chi_2$.
• Extrinsic parameters such as distance, sky location, and orientation are either marginalized or assumed known, as they are largely uncorrelated with the intrinsic binary configuration.
• Simulated studies for proposed detectors (e.g., the Einstein Telescope) demonstrate that both effective spin and mass ratio can be inferred solely from the ringdown, even if the inspiral is undetectable.

This methodology enables the reconstruction of the binary progenitor properties by inverting the amplitude mapping, effectively using the ringdown as a probe of the pre-merger system.

4. Mapping to Final Remnant Spin and the Recovery of Binary Component Spins

A further consequence of the amplitude-phase relationships is that, since phenomenological fits exist between the progenitor parameters and the final black hole spin, one can:

• Measure the “effective” spin parameter using the amplitude formula for $A_{21}/A_{22}$.
• Apply the mapping from $(m_1, m_2, \chi_1, \chi_2)$ to $a_\text{final}$ (final Kerr spin), allowing for the recovery of the individual pre-merger black hole spins, even when only the ringdown is observed.
• This procedure is particularly useful in astrophysical scenarios where the inspiral signal is below detection threshold (e.g., very massive black holes merging at high redshift).

The activation or suppression of specific QNMs (and their amplitudes) thus enables the deconvolution of the progenitor spin content.

5. Physical Implications for Gravitational Astronomy and Tests of General Relativity

The above analysis has broad consequences and applications:

• It opens a pathway for determining the mass ratio and a key spin combination of a binary system via the post-merger ringdown signal alone.
• Independent measurements of the mode spectrum (frequencies/damping times) and excitation amplitudes furnish a strong consistency check of the Kerr hypothesis: in GR, deviations between the inferred remnant properties from amplitudes versus spectrum would indicate a breakdown of the no-hair theorem.
• Precision inferences of spin differences through $A_{21}/A_{22}$ facilitate population studies of black hole formation channels and evolutionary scenarios.
• This framework expands the scope of ringdown analysis from a tool for black hole spectroscopy to a diagnostic of the entire binary coalescence, including unseen inspiral events.
• In cases of significant disagreement between amplitude-derived and frequency-derived remnant parameters, this would signal possible new strong-field physics.

6. Summary Table: Main Ringdown Amplitude Dependencies

Mode	Dominant dependence	Formula (if present)
$(2,2), (3,3)$	Mass ratio $\nu$	$A_{22}$, $A_{33}$ increase with decreasing $\nu$
$(2,1)$	Effective spin difference	$$A_{21}/A_{22} = 0.43[\sqrt{1-4\nu}-\chi_\text{eff}]$$

Amplitudes for additional modes, and the phases $\varphi_{\ell m}$, similarly reflect the binary parameters but with less direct or more complicated dependencies, often requiring full numerical fits.

7. Future Directions

Further extension and generalization of these results include:

• Application to precessing binaries, as initial simulations suggest similar trends in the encoding of progenitor properties.
• Expansion to cases of higher harmonics and mode mixing, relevant for more sensitive future detectors.
• Integration of this amplitude-phase mapping into next-generation waveform models for gravitational-wave parameter estimation.

A precise understanding and modeling of ringdown mode amplitudes and phases are thus fundamental for reconstructing the dynamical history of black hole mergers and for leveraging ringdown signals in precision gravitational physics and astronomy.