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arxiv: 2605.15271 · v1 · submitted 2026-05-14 · 🌀 gr-qc · astro-ph.HE

Polarization Analysis of Ringdown Signals

Nicole Khusid , Will Farr , Max Isi This is my paper

Pith reviewed 2026-05-19 15:34 UTC · model grok-4.3

classification 🌀 gr-qc astro-ph.HE
keywords gravitational wavesringdownquasi-normal modespolarizationblack hole mergersLIGOinclination angle
0
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The pith

Ringdown signals from binary black hole mergers carry measurable circular polarization that directly infers the source inclination angle.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper demonstrates that when binary black holes merge with spins aligned or anti-aligned to their orbital plane, the ringdown quasi-normal modes obey equatorial reflection symmetry. This symmetry reduces each mode's four degrees of freedom by two, locking the relative amplitudes and phases of the two polarization states to the viewing inclination. Consequently, the degree of circular polarization and its handedness become measurable from the ringdown alone, even when only the two LIGO detectors record the signal, and this measurement translates into an inclination estimate. For precessing systems the symmetry is absent, so the same restricted model produces biased frequency and amplitude recoveries. The work therefore isolates a concrete condition under which ringdown data by itself can constrain geometry without needing the full inspiral-merger-ringdown waveform.

Core claim

When quasi-normal modes are excited with equatorial reflection symmetry, the relationship between polarization amplitudes and phases in each mode is fixed by the inclination angle to the equatorial plane. Fitting this constrained model to ringdown data from events like GW150914 extracts the degree of circular polarization and handedness using only the two LIGO detectors; the measured polarization then yields an independent inclination inference. The same model fails for precessing systems such as GW190521, producing systematic biases in recovered mode parameters.

What carries the argument

Equatorial reflection symmetry that constrains two of the four degrees of freedom per quasi-normal mode and fixes the polarization-amplitude and phase relations through the inclination angle.

If this is right

  • For aligned-spin mergers the ringdown alone suffices to measure circular polarization without reference to the earlier inspiral or merger phases.
  • Polarization-derived inclinations provide an independent cross-check on geometry parameters extracted from the complete waveform.
  • Detector networks with only two sites can still extract geometric information from sufficiently loud ringdown signals under the symmetry assumption.
  • Precessing mergers require the unrestricted four-degree-of-freedom polarization model to avoid biased quasi-normal-mode frequency and amplitude estimates.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Future higher-sensitivity detectors could apply the same polarization test to a statistical sample of events to verify the prevalence of aligned-spin mergers.
  • The method offers a post-merger-only route to study spin-orbit alignment, complementary to full-waveform analyses.
  • Deviations from the predicted polarization-inclination relation in high-signal-to-noise ringdowns could flag either precession or departures from Kerr quasi-normal modes.

Load-bearing premise

Merging black holes whose spins are aligned or anti-aligned with the orbital angular momentum excite quasi-normal modes that possess equatorial reflection symmetry.

What would settle it

A statistically significant mismatch between the inclination angle inferred from ringdown polarization and the inclination obtained from a full inspiral-merger-ringdown analysis of a confirmed non-precessing event would falsify the constrained model's applicability.

Figures

Figures reproduced from arXiv: 2605.15271 by Max Isi, Nicole Khusid, Will Farr.

Figure 1
Figure 1. Figure 1: Comparison of Mf and χf posteriors for GW150914 between ringdown and IMR analyses. The results of two ring￾down analyses, both fitting the signal with {(2,2,0),(2,2,1)}, are shown, with 90% credible contours; the dashed plum contour corresponds to the unconstrained, generic ringdown model, whereas the solid blue contour arises from fitting the signal with the constrained aligned-spin model. The NR￾Sur7dq4 … view at source ↗
Figure 2
Figure 2. Figure 2: The aligned-spin ringdown model (blue) measures [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Comparing cos ι posteriors between polarization models that fit the ringdown of GW150914 with the Kerr (2,2,0) and (2,2,1) modes. Top: The prior on cos ι in the aligned (blue) and generic (dark plum) models. There is no explicit cos ι parameter in the generic model, so the priors shown are derived using the inverse of Eq. (5) applied to this one single mode. As discussed at the beginning of this sec￾tion, … view at source ↗
Figure 3
Figure 3. Figure 3: Furthermore, the overtone posterior is nearly iden [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Various parameter posteriors from ringdown fits with both polarization models to GW190521 data. [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Posteriors (priors) in solid pink (dotted black) on [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Top: Generic ringdown model fit to the real GW190521 data, using Kerr modes {(2,2,0), (2,1,0)}. Both 90% credible contours and the histogram are equivalent to those in the leftmost panel of [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Various parameter posteriors from ringdown fits with both polarization models to GW190521-like reflection [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: For the set of GW190521-like reflection-asymmetric [PITH_FULL_IMAGE:figures/full_fig_p012_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: See Fig. 7 caption for figure conventions. Various parameter posteriors from ringdown fits with both polarization [PITH_FULL_IMAGE:figures/full_fig_p013_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: For the set of reflection-symmetric GW190521-like [PITH_FULL_IMAGE:figures/full_fig_p014_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Posteriors on parameters that quantify devia [PITH_FULL_IMAGE:figures/full_fig_p014_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Inferred joint polarization angle and 220 intrinsic [PITH_FULL_IMAGE:figures/full_fig_p019_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Mode ellipticities inferred by the generic and aligned-spin ringdown models from fits to reflection-symmetric [PITH_FULL_IMAGE:figures/full_fig_p021_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: We compare the degree of “aligned-ness” of each of [PITH_FULL_IMAGE:figures/full_fig_p022_14.png] view at source ↗
read the original abstract

Merging binary black holes exhibit a ringdown phase in which they primarily emit gravitational waves in the shape of damped sinusoids corresponding to quasi-normal modes of the Kerr remnant. In general, each mode carries four degrees of freedom encoding amplitude and phase information. When the modes are excited with equatorial reflection symmetry, as is the case for black hole mergers with spins (anti)aligned to the orbital angular momentum, the symmetry constrains two degrees of freedom. As a result, the relationship between polarization amplitudes and phases in each mode is fixed by the viewing (inclination) angle to the equatorial plane. We use such a constrained model to fit the ringdown signals of both non-precessing and precessing systems such as GW150914 and GW190521, respectively. We show that we can measure the degree of circular polarization and handedness of ringdown signals like those of GW150914, even if only the two LIGO detectors are available; such a polarization measurement can be translated into an inferred source inclination assuming the reflection symmetry above, again using the ringdown signal alone. On the other hand, the constrained polarization model is insufficient to capture the polarization structure of signals from precessing systems, leading to biases in the inferred mode frequencies and amplitudes. We explore the magnitude of this effect by fitting GW190521-like injections with the restricted model, finding weaker predictive accuracy relative to the arbitrary-polarization model and potentially significant systematic biases. As our detectors continue to improve, using the correct polarization model is increasingly important to avoid biased ringdown measurements.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper introduces a constrained polarization model for black hole ringdown signals that exploits equatorial reflection symmetry expected in non-precessing mergers. This symmetry reduces each quasi-normal mode from four to two degrees of freedom, fixing the relationship between polarization amplitudes/phases and the source inclination angle. The authors fit both this constrained model and an unconstrained arbitrary-polarization model to GW150914-like signals and to GW190521-like precessing injections, showing that circular polarization degree and handedness can be recovered with two LIGO detectors alone and translated into an inclination estimate under the symmetry assumption. For precessing systems the constrained model produces biases in recovered frequencies and amplitudes relative to the unconstrained fit.

Significance. If the symmetry assumption holds for the targeted systems, the work offers a ringdown-only route to inclination inference that could complement full IMR analyses and become more valuable with improving detector sensitivity. The explicit comparison of constrained versus arbitrary-polarization models on both real-event data and controlled injections quantifies the systematic risk of using an incorrect polarization ansatz, which is a useful cautionary result for the field.

major comments (2)
  1. [Section describing the constrained model and GW150914 fits] The central claim that polarization measurements from two detectors can be translated into an inclination estimate rests on the equatorial reflection symmetry assumption for non-precessing mergers. The manuscript should include a quantitative test (e.g., injections with small spin misalignments that mildly violate the symmetry) showing the size of the resulting bias in the recovered inclination when the constrained model is still applied.
  2. [Results section on precessing-system injections] For the GW190521-like injections, the reported biases in mode frequencies and amplitudes under the constrained model need to be compared directly to the statistical uncertainties of the fit; without this comparison it is difficult to judge whether the biases are large enough to affect science results at current or design detector sensitivities.
minor comments (2)
  1. [Abstract and results] Clarify the precise metric used for 'weaker predictive accuracy' when comparing the two models on injections.
  2. [Methods] Ensure all symbols for polarization amplitudes, phases, and inclination are defined consistently before first use and appear in a notation table if possible.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their positive assessment of our manuscript and for the constructive suggestions. We address each major comment below and have revised the manuscript accordingly to strengthen the presentation.

read point-by-point responses

  1. Referee: [Section describing the constrained model and GW150914 fits] The central claim that polarization measurements from two detectors can be translated into an inclination estimate rests on the equatorial reflection symmetry assumption for non-precessing mergers. The manuscript should include a quantitative test (e.g., injections with small spin misalignments that mildly violate the symmetry) showing the size of the resulting bias in the recovered inclination when the constrained model is still applied.

    Authors: We agree that a quantitative assessment of the inclination bias under mild violations of equatorial reflection symmetry would be a useful addition. Our existing analysis contrasts the exact symmetry case (aligned spins) with strong violations (highly precessing systems), but we will add a new set of injections with small spin misalignments (10–20 degrees) and report the resulting bias in the recovered inclination. These results will be included in a revised version of the manuscript. revision: yes

  2. Referee: [Results section on precessing-system injections] For the GW190521-like injections, the reported biases in mode frequencies and amplitudes under the constrained model need to be compared directly to the statistical uncertainties of the fit; without this comparison it is difficult to judge whether the biases are large enough to affect science results at current or design detector sensitivities.

    Authors: We thank the referee for highlighting this point. In the revised manuscript we will augment the GW190521-like injection results with a direct comparison of the observed biases in mode frequencies and amplitudes against the statistical uncertainties extracted from the posterior distributions. This will allow readers to evaluate the practical significance of the systematics relative to statistical errors at both current and design sensitivities. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper applies an external modeling assumption of equatorial reflection symmetry (valid for non-precessing aligned-spin mergers) to constrain two of the four degrees of freedom per quasi-normal mode. This constraint is used to fit ringdown signals directly to detector strain data, enabling extraction of circular polarization degree and handedness even with two LIGO detectors, followed by translation to inclination under the same assumption. No step reduces by construction to its inputs: the symmetry is stated as a prior physical expectation rather than derived from the fit, the polarization parameters are obtained from data fitting rather than being redefined in terms of inclination, and no self-citations or ansatzes are invoked as load-bearing. Analysis of precessing cases (e.g., GW190521) explicitly shows model mismatch and biases, confirming the framework is not tautological. The derivation remains self-contained against the strain data under the stated assumptions.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

The central claim rests on the equatorial reflection symmetry assumption for aligned-spin systems and on the ability of the two LIGO detectors to resolve polarization degrees of freedom in ringdown.

free parameters (2)
  • mode amplitudes and phases
    Fitted to data under the symmetry constraint; the constraint reduces the number of free parameters per mode from four to two.
  • inclination angle
    Inferred from the measured polarization after fitting; treated as a derived quantity rather than an independent fit parameter.
axioms (1)
  • domain assumption Merging binary black holes with spins (anti)aligned to orbital angular momentum excite quasi-normal modes with equatorial reflection symmetry.
    Invoked in the abstract to constrain two degrees of freedom per mode and fix polarization relationships via viewing angle.

pith-pipeline@v0.9.0 · 5804 in / 1368 out tokens · 39230 ms · 2026-05-19T15:34:18.419313+00:00 · methodology

discussion (0)

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    When the modes are excited with equatorial reflection symmetry, as is the case for black hole mergers with spins (anti)aligned to the orbital angular momentum, the symmetry constrains two degrees of freedom. As a result, the relationship between polarization amplitudes and phases in each mode is fixed by the viewing (inclination) angle

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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