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Ringdown and lensing of triple systems

T0 review · 2 major / 4 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read A companion black hole shifts the ringdown of a merger by about 2%.

desk verdict Solid NR evidence for companion-induced ringdown shifts, lensing amplification, and echoes—worth a serious referee despite the frequency control being cleaner for amplitudes than for frequencies. read the letter →

arxiv 2605.20320 v2 pith:7DJ4TUJP submitted 2026-05-19 gr-qc astro-ph.HE

Vitor Cardoso , Giuseppe Ficarra , Jaime Redondo-Yuste , João Sieiro dos Santos This is my paper · ORCID
classification gr-qcastro-ph.HE
keywords gravitationalwavesblackholemergerstriplesystemsringdownquasinormalmodeslensingnumericalrelativityDopplershift
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper claims that when two black holes merge in the vicinity of a third compact object, the ringdown signal from the merger is measurably modified by the companion. Using fully nonlinear numerical simulations of head-on collisions in a triple system, the authors find that the ringdown frequency is Doppler-shifted and gravitationally redshifted by up to about 2%, in a direction consistent with the remnant's motion and the companion's potential. The companion also acts as a gravitational lens, amplifying the signal on the far side by up to a factor of about two, and producing a delayed, weaker second image whose time delay matches a simple geometric-optics estimate to about 10%. If these results hold, a third-body environment can be quantitatively modeled in the strong-field regime within general relativity, and black-hole spectroscopy must account for such companions.

What carries the argument

The analysis hinges on comparing the nonlinear waveforms with the standard quasinormal-mode expansion of an isolated Schwarzschild black hole, corrected by two analytic factors: a Doppler factor from the remnant's velocity and a gravitational-redshift factor from the companion's potential. A second element is the frequency-dependent lensing amplification factor, which the paper compares with the measured mode-by-mode amplification. A third is the geometric-optics time-delay model, in which the second image corresponds to a null geodesic that travels from the first remnant to the companion's light ring, orbits half or fully around it, and returns; the delay integral is evaluated in a glued-Sc

What would settle it

Measure the remnant's velocity directly from the apparent-horizon trajectory in the simulation and recompute the Doppler correction; if the corrected ringdown frequencies still deviate from the isolated-Schwarzschild values beyond the numerical uncertainty in a systematic way, the claimed attribution fails.

Watch

Extended reading notes

Core claim

The central claim is that the ringdown of the first merger in a hierarchical triple system is not that of an isolated Schwarzschild black hole. In the fully nonlinear evolutions, the real part of the dominant quadrupolar mode deviates from the textbook value by up to about 2%, with the direction of the shift matching the Doppler and gravitational-redshift corrections computed from Newtonian estimates of the remnant's motion and the companion's potential. The companion also magnifies the signal seen on the opposite side, with the ratio of triple to binary amplitudes reaching about 2 for the dominant mode; the subdominant (higher-frequency) mode is amplified more, qualitatively consistent with

Load-bearing premise

The Newtonian estimate of the remnant's velocity during ringdown must be accurate enough for the Doppler correction to account for the observed frequency shift; if that estimate is significantly off, the attribution to Doppler and gravitational redshift is left open.

Editorial extensions

If this is right

  • Black-hole spectroscopy measurements of ringdown frequencies must include Doppler and gravitational-redshift corrections when a companion is present, otherwise inferred remnant masses and spins will be biased.
  • Gravitational-wave lensing by a third compact object produces a delayed second image of the ringdown that can be searched for in events from hierarchical triples.
  • Frequency-dependent amplification means higher-order modes are magnified more than the dominant mode, affecting mode-amplitude ratios and tests of the Kerr hypothesis.
  • Hierarchical triple systems can serve as probes of strong-field lensing in a regime where the thin-lens approximation fails, and the absence of collapse from focused radiation constrains related scenarios.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If companions can shift ringdown frequencies by ~2%, future gravitational-wave detectors might constrain the presence of a third body from ringdown-only data, even when the inspiral is not observed.
  • The geometric-optics time-delay model used here could be extended to predict higher-order images; only the first echo is likely observable, but stacking many events could reveal systematic signatures.
  • A quasi-circular inspiral in a triple system, rather than a head-on collision, may be a better 'tuning fork' to resonantly excite the companion's modes, which is a testable prediction for future simulations.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper studies, via fully nonlinear numerical relativity, the gravitational wave signal from hierarchical triple black hole systems in head-on collisions. It focuses on the ringdown of the first merger of the inner binary and its modification by a third companion. The authors report three main effects: (i) ringdown frequencies deviate from isolated Schwarzschild quasinormal mode values by up to ~2%, in a direction attributed to Doppler plus gravitational redshift; (ii) the companion lenses the radiation, producing amplification of the direct image by factors up to ~2, with a frequency-dependent scaling; (iii) a delayed, demagnified second image ('echo') appears with time delays claimed to match a geometric-optics estimate to ~10%, and with tentative evidence for resonant mode excitation of the companion. They also search for, but do not find, enhanced nonlinearities or collapse from focused gravitational radiation.

Significance. If the results hold, the paper demonstrates that third-body environments can measurably alter ringdown signals in the strong-field regime, beyond the reach of standard weak-field lensing approximations. This is relevant for gravitational-wave spectroscopy and for modeling mergers in dense environments such as AGN disks and globular clusters. The work is anchored by several independent analytical comparisons -- isolated Schwarzschild QNM frequencies, Newtonian free-fall dynamics, null geodesic time delays, and a wave-optics amplification scaling -- and it makes explicit use of companion-free reference runs as controls for the lensing amplitude analysis. The numerical infrastructure is open-source and the authors provide public movies, which are strength and reproducibility. The frequency-shift claim, however, currently lacks a companion-free frequency control, and the quantitative accuracy of the geometric-optics time-delay match is overstated in the text.

major comments (2)
  1. [IIIB1, Figs. 3-4, Table IV, Appendix B] The Doppler/redshift interpretation of the first-ringdown frequency shift lacks a companion-free frequency control. The fits are compared only to the analytic isolated Schwarzschild value (Eq. 8), not to the ID AB/UB runs that serve as amplitude controls in Fig. 5. Table IV uncertainties reflect only t0 scatter; Appendix B explicitly says numerical and systematic errors (N=1 vs N=2 vs N=3) are ignored. After the Newtonian correction, residuals up to ~2.5% remain, outside the 1% band used in Fig. 4. The correction inputs v_M1 (Eq. 1) and r_em (Eq. 5) are not measured from the simulation; v_M1 is a Newtonian estimate the paper calls an underestimate. The residual could thus be absorbed by acceleration, extraction-radius bias, or fitting systematics. Running the same fits on ID AB/UB at the same observers would calibrate the systematic floor and is the missing control for the central claim
  2. [Table II / Sec. IIIC1] The statement that the geometric-optics time-delay model is 'always within ~10% of the interval' is not supported by Table II. For UE top, Δt_num = 37 ± 11 M while Δt_go = 50 M; for UU right, Δt_num = 95 ± 8 M while Δt_go = 107 M. These are deviations of ~35% and ~13%, respectively, and in the first case the model lies outside the quoted 1σ interval. The qualitative conclusion that the echo is a lensed second image with the correct order-of-magnitude delay is unchanged, but the claimed quantitative accuracy should be restated, or the comparison should be made against the mismatch-fit values td (33.2 M and 90.8 M, which are closer). Please clarify which quantity is being compared and revise the accuracy claim.
minor comments (4)
  1. [Fig. 4 caption] Caption states 'ID UU and G'; this appears to be a typo for 'ID UU and UE'.
  2. [IIIB2, Eq. (12), Fig. 5] The dashed bands in the bottom panel rely on F = κ m1 ω, but the value or range of κ and how the band is generated are not specified. Please provide this information so the 'good agreement' with Eq. (12) is reproducible.
  3. [IIB, Eq. (5)] The frequency correction f_Grav depends on r_em and r_obs, but the paper does not state the values of r_em used for each observer and configuration. Since the corrected frequencies in Fig. 4 depend on these choices, the adopted values should be given explicitly.
  4. [Appendix A] Convergence tests are reported for ID AE only. For a paper whose quantitative claims are based on several different configurations, it would be useful to state whether the same convergence behavior is expected for the other runs and whether any run-specific differences were seen.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the NR ringdown, lensing-amplitude, and echo time-delay results are benchmarked against independent QNM frequencies, Newtonian dynamics, and geometric-optics predictions.

full rationale

The paper's derivation chain is not circular. The central claims—Doppler/gravitational frequency shifts of the first-merger ringdown, lensing amplification, and a delayed second image—are extracted from fully nonlinear numerical relativity waveforms and compared with independent standards: isolated-Schwarzschild QNM frequencies (Eqs. (8)–(9), standard values from Refs. [14,62]), Newtonian free-fall timescales (Eq. (1), validated against simulated merger times), wave-optics amplification scaling (Eq. (11), from Refs. [101,102]), and geometric-optics time delays (Appendix C, Schwarzschild null geodesics with Newtonian initial separations). The Doppler/redshift corrections (Eqs. (4)–(6)) use Newtonian estimates of v_M1 and observer geometry, not values fitted to the ringdown frequencies, so the corrected frequencies are not forced by construction. The amplitude ratios in Fig. 5 are measured against dedicated companion-free runs ID AB/UB, an external control for the same numerical setup; Eq. (12) is a consistency relation derived from the measured dominant-mode ratio and known QNM frequencies, not a fit to the subdominant mode. The only self-referential element is the perturbative lensing model of Ref. [67] (by some of the present authors), used as a cross-check; the paper explicitly states it fails to reproduce the frequency scaling in ID UU/UE, showing it is not load-bearing. The absence of a companion-free frequency control for Fig. 4 and the uncorrected ~2.5% residual are calibration/accuracy caveats, explicitly acknowledged in Appendix B, but they are not circular reductions of the predictions to their inputs. Ref. [107] is cited as 'in preparation' and therefore provides no independent support, but it is not load-bearing for any central claim.

Assumptions & free parameters 3 free parameters · 7 assumptions · 0 invented entities

No new particles, fields, or forces are introduced. The paper's added content over its inputs is numerical: NR waveforms and fitted QNM parameters. The free parameters are analysis choices and one empirically fitted proportionality constant κ; the axioms are standard GR, standard lensing/geometric-optics approximations, and one clearly stated ad hoc gluing model (Appendix C). The load-bearing external inputs (isolated QNM frequencies, Newtonian estimates) come from prior literature or from the initial data itself.

free parameters (3)
  • κ (amplification constant in F = κ m1 ω) = Not quoted directly; implied by dominant-mode amplification ratios (Fig. 5)
    Introduced in Sec. IIIB2 to explain the frequency scaling of lensing amplification. Eq. (12) cancels κ, so the subdominant-mode prediction depends only on the assumed linear-in-ω form, which was itself inferred from the same dataset.
  • Number of damped sinusoids N in QNM fits = 2
    Model choice for the ringdown fits (Sec. IIIB, Fig. 3); the authors note in Appendix B that recovered frequencies would shift slightly with N=1 or N=3.
  • QNM fit start-time window t0 = (t0 - t_RD1)/M1 in [10,20] for first image; [6,20] for second image
    Analysis choice; the spread of recovered frequencies over the window is used as the quoted uncertainty, excluding numerical and systematic errors (Sec. IIIB1, Appendix B).
assumptions (7)
  • domain assumption Isolated Schwarzschild QNM frequencies Mω20 = 0.373672 - 0.088962 i and Mω40 = 0.809178 - 0.094164 i are the correct unperturbed benchmark
    Used in Eqs. (8)-(9) as the reference for all frequency-shift claims (Sec. IIIB1). Standard result from Refs. [14,62].
  • domain assumption Newtonian free-fall dynamics (Eq. (1)) give accurate merger time, remnant velocity, and separation used in Doppler/redshift and time-delay predictions
    Table I and Sec. IIB. The paper notes the velocity estimate is an underestimate, so the assumption is only partially valid; the residual is visible in Fig. 4.
  • domain assumption Weak-field slow-motion gravitational redshift formula (Eq. (5)) applies at the observer and emission locations used
    Sec. IIB; both factors are within ~1% of unity, so the approximation is mild but unchecked in the strong-field region near the lens.
  • domain assumption Wave-optics magnification scaling F^2 ∝ m1 ω (Eq. (11)) from thin-lens theory is the baseline expectation
    Sec. IIIB2; the authors note the approximations behind it do not hold here, and the measured scaling indeed differs.
  • ad hoc to paper Geometric-optics time-delay model (Eqs. (13)-(14), Appendix C): stationary BHs, radial null geodesics, Schwarzschild metrics glued at the Newtonian equipotential point
    Appendix C states assumptions (i)-(iv) explicitly; used for the echo time-delay comparison in Table II.
  • domain assumption Brill-Lindquist initial data with BHs at rest faithfully represent the astrophysical setup
    Sec. II C; standard numerical-relativity choice, but the head-on-from-rest configuration is idealized and not the quasi-circular inspiral expected in most astrophysical channels.
  • standard math General relativity / BSSN evolution equations are correct
    Background theory; not questioned by the paper.

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Cite this review

Pith. "Pith review of Ringdown and lensing of triple systems." pith.science (2026). https://pith.science/paper/7DJ4TUJP

@misc{pith2026260520320,
  author       = {Pith},
  title        = {Pith review of: Ringdown and lensing of triple systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7DJ4TUJP}},
  note         = {Machine review of arXiv:2605.20320}
}
read the original abstract

Triple systems have progressively been recognized as ubiquitous in our Universe and provide a good testing ground for wave generation and propagation in nontrivial environments. We study the dynamics of triple systems in a fully nonlinear setting. In particular, we analyze numerical relativity simulations of head-on collisions of black holes in the presence of a companion. We show evidence for Doppler and gravitational redshift in the ringdown, and clear signs of amplification by lensing. In certain cases, we also show the appearance of a second image, with hints of resonant mode excitation. Our results pave the way for the understanding of mergers in the vicinity of massive companions. Even in extreme setups we do not find collapse to black holes from lensed gravitational radiation.

Figures

Figures reproduced from arXiv: 2605.20320 by the authors.

Figure 1
Figure 1. A cartoon of the setup studied in this work and [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Summary of two of the configurations considered in this work, corresponding to initial data ID AU ( [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Direct ringdown following the first merger, from ID AU. [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (7 more)
Figure 5
Figure 5. Figure 5: Ratio of amplitudes of the two free damped sinu [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]
Figure 4
Figure 4. Figure 4: Fractional deviation from the theoretical value of [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 6
Figure 6. Figure 6: Snapshot of |Ψ4| in the run ID UU. The two black holes, m1 and M1 are marked by black dots and circles in￾dicate different extraction points. The outer spherical front is the direct wave from the first merger. On the left, we see interference fringes (black dashed line…
Figure 7
Figure 7. Figure 7: First ringdown (blue), and lensed ringdown (dashed, red), after finding the best-fit magnification [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]
Figure 8
Figure 8. Figure 8: Convergence test for run ID AE: in Table [PITH_FULL_IMAGE:figures/full_fig_p016_8.png]
Figure 9
Figure 9. Figure 9: Same content as in Fig [PITH_FULL_IMAGE:figures/full_fig_p017_9.png]
Figure 10
Figure 10. Figure 10: Same content as Fig [PITH_FULL_IMAGE:figures/full_fig_p019_10.png]

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Cited by 1 Pith paper

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  1. Three-dimensional wave optics for weak-field lensing of gravitational waves

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    Doppler and gravitational shifts We expect the QNM frequencies observed to be grav- itationally and Doppler shifted. In order to test if this is an observable effect in our simulations, we ex- tract the frequency and damping time of the two dom- inant QNMs, at different observing points, for the four tripletconfigurationsofTableI.Weaveragetheextracted fre...

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    2009 arXiv

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    Universal interferometric signatures of a black hole’s photon ring,

    Michael D. Johnsonet al., “Universal interferometric signatures of a black hole’s photon ring,” Sci. Adv.6, eaaz1310 (2020), arXiv:1907.04329 [astro-ph.IM]

    2020 arXiv

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    Moving black holes: energy extraction, absorption cross-section and the ring of fire,

    Vitor Cardoso and Rodrigo Vicente, “Moving black holes: energy extraction, absorption cross-section and the ring of fire,” Phys. Rev. D100, 084001 (2019), arXiv:1906.10140 [gr-qc]

    2019 arXiv

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    2025 arXiv

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    Thiago Assumpcao, Vitor Cardoso, Akihiro Ishibashi, MauricioRichartz, andMiguelZilhao,“Blackholebina- ries: ergoregions, photon surfaces, wave scattering, and quasinormal modes,” Phys. Rev. D98, 064036 (2018), arXiv:1806.07909 [gr-qc]

    2018 arXiv

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    Spin depen- dence of black hole ringdown nonlinearities,

    Jaime Redondo-Yuste, Gregorio Carullo, Justin L. Rip- ley, Emanuele Berti, and Vitor Cardoso, “Spin depen- dence of black hole ringdown nonlinearities,” Phys. Rev. D109, L101503 (2024), arXiv:2308.14796 [gr-qc]

    2024 arXiv

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    Gregorio Carullo, Marina De Amicis, and Jaime Redondo-Yuste, “bayring,” github.com/GCArullo/bayRing (2023)

    2023

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    Quasinormal modes from numerical relativity with Bayesian infer- ence,

    Richard Dyer and Christopher J. Moore, “Quasinormal modes from numerical relativity with Bayesian infer- ence,” (2025), arXiv:2510.11783 [gr-qc]. 16 Appendix A: Convergence of numerical simulations 0 100 200 300 400 500 600 10-13 10-11 10-9 10-7 |∆ [Re[Ψ4]]| |Ψ4, med − Ψ4, low...

    2025 arXiv

Pith tools

Reviewed August 2, 2026 · model on record in the stance chip above.