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Periodic line-of-sight velocity-driven modulations to gravitational waves emitted by compact binaries in Keplerian outer orbits

T0 review · 3 major / 4 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read Gravitational waves from a binary whose center of mass circles a third body carry a periodic Doppler phase ripple at 4PN order, encoding the companion's mass and orbital radius.

desk verdict A real analytic extension of the LOS-acceleration waveform formalism to full periodic outer orbits, but the Fisher forecasts overreach in the low-tertiary-mass corner where the paper's own Shapiro-delay estimate dominates the omitted term. read the letter →

arxiv 2607.09644 v2 pith:LY6WQKDW submitted 2026-07-10 gr-qc astro-ph.HE

Avinash Tiwari , Shasvath J. Kapadia , Aditya Vijaykumar , Sourav Chatterjee This is my paper
classification gr-qcastro-ph.HE
keywords gravitationalwavesDopplershiftline-of-sightvelocityhierarchicaltriplepost-NewtonianwaveformmodelingFisherforecastcompactbinarycoalescence
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 derives the gravitational-wave phase and amplitude corrections produced when a compact binary's center of mass follows a circular or eccentric Keplerian orbit around a third body, rather than the previously treated constant-acceleration approximation. The corrections appear at 4 post-Newtonian order and take a closed sinusoidal form in the observed frequency, governed by the outer orbital frequency and the maximum Doppler shift. In the limit where the observation lasts much less than one outer orbital period, they reduce to the known constant line-of-sight acceleration and higher-derivative results. If correct, a single merger can reveal the mass of the third body, the size of the outer orbit, and (for eccentric outer orbits) its eccentricity, with forecasts across A+, ET, LISA, and DECIGO. The paper also shows the periodic corrections significantly improve constraints over the constant-acceleration approximation and fill parameter space that approximation cannot touch.

What carries the argument

The load-bearing object is the time-varying Doppler shift z_LC(t) = z_{L,0} cos(Ω_det(t_u − t_c) + θ_c) for circular outer orbits, and its true-anomaly counterpart for eccentric orbits. Under the stationary phase approximation, the frequency-domain phase is obtained by integrating the perturbed chirp equation dvo/dto, and the result organizes itself into sinusoidal functions of ξ/v^8, where ξ ≡ (5/(256η))(GMΩ_det/c^3) is proportional to the outer orbital angular frequency. This combination is what makes the outer orbital frequency measurable: the phase oscillates with a 1/f^8 dependence, and the modulation frequency is tied directly to the companion's orbital period. For eccentric outer orbi

What would settle it

For the configuration the paper's §III F flags as fragile—a 20 M⊙ binary with a 2 M⊙ companion in a near-edge-on outer orbit—compute the full phase shift from both the Doppler term (Eq. 16) and the Shapiro-delay term (Eq. 44). If the difference between the two exceeds the inverse signal-to-noise ratio for an A+ or ET detection, then the Doppler-only waveform of Eq. (20) is incomplete in a regime the paper's Fisher grids include, and the claimed measurability of M3 and a in that corner would be invalid.

Watch

Extended reading notes

Core claim

Central claim: a compact binary coalescing while its center of mass follows a Keplerian circular or eccentric orbit around a third body imprints a periodic Doppler phase correction on the emitted gravitational waves, given to leading order by ΔΨ_LC(f) = −(5 z_{L,0}/(128η)) (v^3/ξ)[sin(ξ/v^8 − θ_c) − sin(ξ/v_lso^8 − θ_c)] for circular outer orbits, with an analogous O(e_out^4) expansion for eccentric outer orbits. The corrections appear at 4PN order, arise from the time-varying Doppler redshift alone, and reduce to the known line-of-sight acceleration and higher-derivative results when ξ/v^8 ≪ 1. The paper further claims these modulations break the mass-redshift degeneracy and, through a Fish

Load-bearing premise

The waveform model assumes the Doppler shift from a fixed Keplerian outer orbit is the only non-negligible environmental phase effect; the paper itself shows the Shapiro delay can dominate for near-edge-on outer orbits with companions of only a few solar masses, which is exactly the regime of some of the reported Fisher constraints.

Editorial extensions

If this is right

  • For a CBC whose outer orbital period is comparable to or shorter than the observation time, the periodic Doppler modulation breaks the mass-redshift degeneracy, allowing the third body's mass to be measured from a single event.
  • Fisher forecasts show that A+, ET, DECIGO, and LISA can constrain M3 from 1 M⊙ companions up to 10^8 M⊙ supermassive black holes, with semi-major axes measurable to a few percent in favorable regions.
  • The new corrections fill the parameter space that the constant-acceleration approximation cannot cover and improve upon earlier constraints, especially for supermassive third bodies and high-mass binaries.
  • The time-domain waveform goes in and out of phase repeatedly when the outer period is shorter than the signal duration; the match between perturbed and unperturbed templates can be as low as 0.76, so template banks need to include this family to avoid missing such signals.

Reading between the lines

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

  • Because the phase correction is derived only at Newtonian order in the inner binary, extending to higher post-Newtonian orders should shift the 4PN coefficient and could change the Fisher forecasts for loud events; an immediate follow-up would compute the 1PN correction and test whether the measurable region shrinks or grows.
  • The paper's own §III F shows that for near-edge-on outer orbits and M3 ≲ MS, the Shapiro delay's frequency shift can exceed the Doppler term by up to a factor ~1.57 for M3 = 2 M⊙ with MS = 20 M⊙. A combined Doppler-plus-Shapiro waveform would likely modify the constraints in the bottom-left corner of the measurement grids, and could either widen or shrink the claimed detectable region for 1–5 M⊙ c
  • The same mechanism should apply to circum-binary exoplanets: the formalism only requires a periodic line-of-sight velocity, so a planet-mass companion on a sufficiently tight outer orbit would imprint an analogous 4PN ripple; the paper's companion work on exoplanets is a direct application of this idea.
  • In dense stellar environments, repeated three-body encounters might approximate a periodic line-of-sight velocity over short stretches; this waveform could be used to search for such encounters, though the coherence time of the periodicity would be the main uncertainty.
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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

3 major / 4 minor

Summary. This paper derives stationary-phase-approximation (SPA) corrections to the GW phase and amplitude for a compact binary whose centre of mass follows a periodic line-of-sight velocity, in either a circular (COO) or an eccentric (EOO) outer orbit around a third body. The phase and amplitude corrections are given in closed form, Eqs. (14)–(15) and (20)–(21), and are shown to reduce to the known constant line-of-sight acceleration (LOSA) results in the limit where the outer orbital period is much longer than the observation time (Appendix C). The authors then perform a Fisher-matrix analysis to forecast constraints on the tertiary mass M3, the outer semi-major axis a, and the outer eccentricity eout for several detector configurations (A+, ET, LISA, DECIGO) and source types (BNS, NSBH, BBH). The paper also discusses stability criteria, gravitational redshift, Shapiro delay, and dynamical effects such as Kozai–Lidov oscillations.

Significance. If the forecasts are reliable, this work fills a genuine gap: the existing constant-kinematic-parameter formalism is invalid when the outer orbital period is comparable to or shorter than the observation time, and this paper provides a periodic generalization that is transparently derived and reduces correctly to the earlier results. The analytic derivation is a useful contribution, and the paper is honest in discussing many of the caveats (stability, Shapiro delay, gravitational redshift, SPA validity). However, the statistical claims in the abstract depend on a waveform that omits Shapiro delay, and the paper's own §III.F shows that this omission is not subdominant in a portion of the parameter space that is explicitly included in the forecasts and highlighted in the Results. Because the detection claims for low-mass tertiaries are made in that regime, the forecasts are not yet supported as stated. The analytic derivation itself appears sound and is not affected by this issue.

major comments (3)
  1. [§III.F and §IV.A/IV.B (Figs. 3, 6)] The waveform used in the Fisher forecasts contains only the Doppler correction (Eq. 5), but §III.F shows that the Shapiro-delay-induced frequency shift dΔt_SE/dtu can exceed the Doppler term for near-edge-on outer orbits when M3 ≲ MS. Concretely, with M_S = 20 M⊙, M3 = 2 M⊙, zL0 = 0.05, and ι_out = 89°, Eq. (45)–(46) give (dΔt_SC/dtu)_max ≈ 1.57 zL0, and the ratio scales as (1 + M_S/M3)^2 for smaller M3. The Fisher grids in Figures 3 and 6 include M3 = 1 M⊙ tertiaries around BNS/BBH systems, and sin ι_out is fixed to 1 throughout, so the omitted term is not uniformly small in the plotted domain. The acknowledgment at the end of §III.F that 'a study of both effects combined would be essential' is not reflected in the Results or the abstract. Please either include the Shapiro term in the waveform used for forecasting, or mask/restrict the reported constraints to the region where the Shapir
  2. [§III.A (Eq. 23)] The assertion that 'the amplitude corrections have a negligible effect on the Fisher matrix' is made without quantitative support. Since Eq. (23) explicitly contains the factor (1 + ΔA/A)^2, omitting these terms is a modeling choice that could, in principle, change the covariance estimates at O(zL0). Please provide a quantitative check — for example, the maximum fractional change in Σ when including the amplitude terms over the full M3–a grids — or an analytic argument showing that the phase-correction contributions to the Fisher information dominate the amplitude-correction contributions everywhere in the relevant frequency band. This is needed to justify the Fisher results presented in later sections.
  3. [Abstract, §V, and Fig. 5] The abstract and discussion claim that 'constraints acquired using GW waveforms derived in this work improve significantly in comparison to those acquired from approximate methods valid for constant kinematic parameters.' The only direct comparison is in Fig. 5 (and Fig. 13), where the new method fills in parameter space that the old method did not cover because |Γ_n t_obs| ≪ 1 was not satisfied. It is therefore unclear how much of the shown improvement is an extension of the accessible parameter space rather than tighter constraints in the region of overlap. Please present the comparison restricted to the overlapping regime (e.g., overlay the old and new error contours only where the old approximation is valid), or explicitly state that the improvement is primarily an enlargement of the accessible parameter space. This is a central claim of the paper and should not remain ambiguous.
minor comments (4)
  1. [§II.A, after Eq. (11)] The phrase 'As an first order of approximation' should read 'As a first order of approximation.'
  2. [Abstract and §II] The paper states that the corrections 'lead to phase and amplitude modulations at 4 PN order,' but the limit in Eq. (C1) contains a zL0-only term (0PN, degenerate with mass) and a −4PN LOSA term. Please clarify precisely which term in the expansion gives the claimed 4PN order, to avoid confusion about the PN counting.
  3. [§III.F] When stating that F_C,max = tan ι_out and 'For ι_out = π/2, this becomes ∞,' it should be noted that the divergence is an artifact of the geometric-optics point-mass approximation; a real tertiary has finite size. The resulting conclusion about the Shapiro term being large near edge-on is unchanged, but the presentation would be more accurate with this caveat.
  4. [§IV and Fig. 5] The figure captions state that the dashed-dotted lines in the rightmost panels denote constant-SNR contours, but the lower-left region is described as where SNR < 4; the caption in Fig. 5 is slightly ambiguous about whether the dashed-dotted line is the SNR=4 contour. Please clarify.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: periodic waveform corrections are derived from standard Doppler mapping and chirp evolution; Fisher forecasts are model predictions, not fits.

full rationale

The central derivation is self-contained. Equations (7)-(10) are the standard Doppler mapping (attributed to [18], with overlapping authors, but parameter-free and independently derivable); substituting the periodic z_LC = z_L0 cos(...) into the chirp evolution equations of [49] and integrating yields the phase correction Eq. (14) without fitting any data. Equation (15) and the EOO results (20)-(21) follow the same analytic chain, using standard Keplerian expansions (e.g., dϑ/dtu is the usual Keplerian relation). The Fisher forecasts are genuine model predictions: they use the derived waveform, detector PSDs, and the exact Keplerian relations (32)-(33) to map z_L0 and Ω_det to M3 and a; no quantity is fit to a subset of data and then 'predicted'. Self-citations ([18], [31], [41]) supply the Doppler mapping, the Taylor-expansion formalism for the long-period limit, and comparison points; these are externally verifiable formulas, not an unverified uniqueness theorem or ansatz. The paper's own §III F flags a real model-completeness limitation: 'for outer orbits that are close to edge-on and M3 ≲ MS, the Shapiro delay can become dominant, and a study of both effects combined would be essential' — with the numerical example (dΔt_SE/dtu)/z_L0 ≈ 1.57 for MS = 20 M⊙, M3 ≈ 2 M⊙. This is an omitted physical effect affecting some low-tertiary-mass Fisher forecasts, but an omission is not circularity; the derivation chain itself is independent of the fitted values. Overall: no circular step; only a minor self-citation burden.

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

The central derivation introduces no new physical entities or fitted constants: it reuses standard inspiral equations, Keplerian motion, and the SPA. The only hand-chosen values are the fiducial forecast parameters (θc, ϑp, eout) and the linearization cutoff zL,0 ≤ 0.05. The main domain assumptions are the small-Doppler linearization, leading-Newtonian-order inspiral, SPA validity, and the neglect of Shapiro delay/dynamical effects; the Shapiro-delay neglect is the most fragile.

free parameters (3)
  • θc (fiducial) = 0.1 rad
    Chosen initial orbital phase for COO/EOO Fisher forecasts; constraints depend on it but formalism is valid for any value.
  • ϑp (fiducial) = 0.1 rad
    Chosen longitude of periapsis for EOO forecasts.
  • eout (fiducial) = 0.5
    Chosen outer-orbit eccentricity for EOO forecasts; series truncated at e^4 so eout ≲ 0.66 is required.
assumptions (7)
  • domain assumption zL,0/sqrt(1-e_out^2) ≪ 1 (bounded by 0.05) and all corrections linearized in zL,0
    Used throughout Section II to write M_LE = M(1+z_LE) etc.; quadratic and higher Doppler terms are neglected.
  • standard math Inner binary inspiral follows the leading-order (Newtonian) frequency evolution dvu/dtu from [49]
    Used in Eqs. (10)-(19) to invert dvo/dto and obtain the phase; higher PN corrections are not included.
  • domain assumption Stationary phase approximation holds for chosen frequency ranges
    Section III B imposes f_min ≥ f_SPA,C/E so that dfo/dto > 0; the waveform model (5) relies on SPA.
  • domain assumption Keplerian outer orbit with fixed eccentricity and no precession; cosϑ/sinϑ expanded in eout up to O(e_out^4), convergent for eout ≤ 0.6627434
    Section II B uses expansions (A1)-(A2) from [51]; the paper restricts EOO forecasts to eout=0.5.
  • domain assumption Gravitational redshift and Shapiro delay are subdominant to the Doppler shift in the reported parameter space
    Argued in Sections III E/F; the paper's own numbers show Shapiro delay can dominate for near-edge-on, low-mass tertiaries, which weakens this assumption.
  • domain assumption Dynamical effects (Kozai-Lidov, nodal precession, tidal dephasing) are negligible or suppressed for the systems considered
    Sections III D/G check stability and GR precession suppression; tidal dephasing is explicitly declared out of scope.
  • domain assumption Outer-orbit inclination fixed to sin ι_out = 1 (edge-on), with ι_out degenerate with M3 and a
    Section III C notes the degeneracy cannot be broken, so reported constraints are lower limits.

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Pith. "Pith review of Periodic line-of-sight velocity-driven modulations to gravitational waves emitted by compact binaries in Keplerian outer orbits." pith.science (2026). https://pith.science/paper/LY6WQKDW

@misc{pith2026260709644,
  author       = {Pith},
  title        = {Pith review of: Periodic line-of-sight velocity-driven modulations to gravitational waves emitted by compact binaries in Keplerian outer orbits},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LY6WQKDW}},
  note         = {Machine review of arXiv:2607.09644}
}
abstract

The centre of mass (CoM) of compact binary coalescences (CBCs) occurring in the vicinity of a supermassive black hole, through interaction with an arbitrary third body (e.g., of stellar mass), or in a dense stellar environment, will undergo a time-varying line-of-sight (LOS) velocity. This in turn leads to a time-varying Doppler shift and corresponding modulations in the shape of the gravitational waves (GWs). The phase and amplitude corrections arising from constant LOS acceleration and its higher-order time derivatives are already known. Specifically, these effects lead to corrections to the GW waveform at $-4n$ post-Newtonian (PN) order, where $n$ is the $n^{th}$ time derivative of the LOS velocity. In the context of a circular or eccentric outer orbit of the CoM of the CBC, these effects can be thought of as approximations to the LOS velocity in the limit: observation duration $\ll$ period of the outer orbit. However, this condition is not necessarily always satisfied. In this {\it paper}, we present phase and amplitude corrections to the GW waveforms arising from a periodic non-relativistic LOS velocity for circular and eccentric outer orbits of the CBC's CoM. Specifically, these lead to phase and amplitude modulations at 4 PN order, and reduce to the known corrections for constant kinematic parameters under appropriate limits mentioned above. We also perform a Fisher matrix analysis to forecast constraints on the environment that is sourcing the time-varying LOS velocity, for various future ground and space-based detectors. We further show that constraints acquired using GW waveforms derived in this work improve significantly in comparison to those acquired from approximate methods valid for constant kinematic parameters.

Figures

Figures reproduced from arXiv: 2607.09644 by the authors.

Figure 1
Figure 1. The schematic representation of a BH (M3) and a BBH (MS) orbiting in eccentric orbits around the system’s centre of mass (barycenter) O. ϑp is the longitude of periapsis, ϑ is the true anomaly of the outer orbit, and ιout is the angle between the angular momentum (along the Z-axis) of the outer orbit and the observer’s LOS ˆn. Let M = m1+m2 be the cosmologically redshifted total mass of the CBC, where m1 = m1,S(1 + … view at source ↗
Figure 2
Figure 2. Example Waveform: The top panel shows the time domain waveform of a non-spinning static BBH at 500 Mpc having component masses m1,S = m2,S = 10 M⊙, the middle panel shows the same when there is a 8 M⊙ BH in the vicinity of this BBH at 2.25 × 103 Rs in a COO perturbing the motion of its CoM — this configuration leads to zL,0 = 8 × 10−3 and Ωdet = 0.142 Hz, and the bottom panel shows the difference between the two wav… view at source ↗
Figure 3
Figure 3. SBH-IMBH: Left two panels show the relative errors in the measurement of mass of the tertiary M3 (top panels) and radius of the outer orbit a (bottom panels) over a grid of M3 and a for the A+: BNS and ET: BNS cases mentioned in Table I in COO scenario, while the right two panels show the same for A+: BBH and ET: BBH cases. The patches on the upper right represent the parameter space where either δX > 1 for paramete… view at source ↗
Figures from the paper (21 more)
Figure 4
Figure 4. Figure 4: SMBH: The left and right panels show the relative errors in the measurement of M3 (top panels) and a (bottom panels) over a grid of M3 and a for the A+: BNS and ET: BNS cases mentioned in Table I in COO scenario, respectively. The patches on the upper right and the dot…
Figure 5
Figure 5. Figure 5: SMBH: Left two panels show the relative errors in the measurement of M3 (top panels) and a (bottom panels) over a grid of M3 and a for the A+: NSBH and ET: BBH2 cases mentioned in Table I in COO scenario, while the right two panels show the same for DECIGO: BBH and LIS…
Figure 6
Figure 6. Figure 6: SBH-IMBH: Left two panels show the relative errors in the measurement of mass of the tertiary M3 (top panels), semi-major axis of the outer orbit a (middle panels), and eccentricity of the outer orbit eout (bottom panels) over a grid of M3 and a for the A+: BNS and ET:…
Figure 7
Figure 7. Figure 7: SMBH: left panel show the relative errors in the measurement of M3 (top panels), a (middle panels), and eout (bottom panels) over a grid of M3 and a for the ET: BNS case mentioned in Table I in EOO scenario, the middle panel from left shows the same for the DECIGO: BBH…
Figure 8
Figure 8. Figure 8: Example Waveform: The upper panel shows the time domain waveform of the non-spinning static BBH considered in [PITH_FULL_IMAGE:figures/full_fig_p020_8.png]
Figure 8
Figure 8. Figure 8: Example Waveform: The upper panel shows the time domain waveform of the non-spinning static BBH considered in [PITH_FULL_IMAGE:figures/full_fig_p021_8.png]
Figure 9
Figure 9. Figure 9: Example Waveform: The upper panel shows the time domain waveform of the perturbed BBH considered in [PITH_FULL_IMAGE:figures/full_fig_p021_9.png]
Figure 10
Figure 10. Figure 10: The upper panel shows the frequency domain waveforms of the A+: BBH system considered in Table I in presence of a 8 M⊙ BH in the vicinity at a = 3 × 104 Rs in COO scenario after incorporating only LOSA corrections (blue) in the waveform and full LOSV corrections (oran…
Figure 10
Figure 10. Figure 10: The upper panel shows the frequency domain waveforms of the A+: BBH system considered in Table I in presence of a 8 M⊙ BH in the vicinity at a = 3 × 104 Rs in COO scenario after incorporating only LOSA corrections (blue) in the waveform and full LOSV corrections (oran…
Figure 11
Figure 11. Figure 11: A comparison of the phase corrections due to LOSV, LOSA, and other terms appearing in the expansion of Equation [PITH_FULL_IMAGE:figures/full_fig_p022_11.png]
Figure 12
Figure 12. Figure 12: The variation of acrit,KL/acrit, equation (55), with the mass of the tertiary in the vicinity of the BNS and BBH systems considered in A+ and ET (see Table I) [PITH_FULL_IMAGE:figures/full_fig_p022_12.png]
Figure 12
Figure 12. Figure 12: The variation of acrit,KL/acrit, equation (55), with the mass of the tertiary in the vicinity of the BNS and BBH systems considered in A+ and ET (see Table I). 103 104 105 105 106 107 108 M3 [M ] DECIGO fmin = 0.01 Hz 103 104 105 105 106 107 108 LISA % = 4 fmin = 0.00…
Figure 13
Figure 13. Figure 13: SMBH: The left two panels show the relative errors in the measurement of M3 (top panels) and a (bottom panels) over a grid of M3 and a for the DECIGO: BBH and LISA: BBH cases corresponding to top panels of Figures 3 and 4 of [37] in EOO scenario and the right two pane…
Figure 14
Figure 14. Figure 14: SBH-IMBH: The left two panels show the relative errors in the measurement of zL,0 (top panels) and Ωdet (bottom panels) over a grid of M3 and a for the A+: BNS and ET: BNS cases corresponding to 3 in COO scenario, while the right two panels show the same for A+: BBH a…
Figure 14
Figure 14. Figure 14: SBH-IMBH: The left two panels show the relative errors in the measurement of zL,0 (top panels) and Ωdet (bottom panels) over a grid of M3 and a for the A+: BNS and ET: BNS cases corresponding to 3 in COO scenario, while the right two panels show the same for A+: BBH a…
Figure 15
Figure 15. Figure 15: SMBH: The left and right panels show the relative errors in the measurement of zL,0 (top panels) and Ωdet (bottom panels) over a grid of M3 and a for the A+: BNS and ET: BNS cases corresponding to [PITH_FULL_IMAGE:figures/full_fig_p024_15.png]
Figure 16
Figure 16. Figure 16: SMBH: The left two panels show the relative errors in the measurement of zL,0 (top panels) and Ωdet (bottom panels) over a grid of M3 and a for the A+: NSBH and ET: BBH2 cases corresponding to 5 in COO scenario, while the right two panels show the same for DECIGO: BBH…
Figure 16
Figure 16. Figure 16: SMBH: The left two panels show the relative errors in the measurement of zL,0 (top panels) and Ωdet (bottom panels) over a grid of M3 and a for the A+: NSBH and ET: BBH2 cases corresponding to 5 in COO scenario, while the right two panels show the same for DECIGO: BBH…
Figure 17
Figure 17. Figure 17: SBH-IMBH: The left two panels show the relative errors in the measurement of zL,0 (top panels) and Ωdet (bottom panels) over a grid of M3 and a for the A+: BNS and ET: BNS cases corresponding to [PITH_FULL_IMAGE:figures/full_fig_p025_17.png]
Figure 18
Figure 18. Figure 18: SMBH: The left, middle, and right panels show the relative errors in the measurement of zL,0 (top panels) and Ωdet (bottom panels) over a grid of M3 and a for the ET: BNS, DECIGO: BBH, and LISA: BBH cases, respectively, corresponding to 7 in EOO scenario. The patches …
Figure 18
Figure 18. Figure 18: SMBH: The left, middle, and right panels show the relative errors in the measurement of zL,0 (top panels) and Ωdet (bottom panels) over a grid of M3 and a for the ET: BNS, DECIGO: BBH, and LISA: BBH cases, respectively, corresponding to 7 in EOO scenario. The patches …

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Cited by 2 Pith papers

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    Future gravitational-wave detectors, especially ET and DECIGO, could in principle detect and characterize exoplanets around extragalactic compact binary mergers.

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Pith tools

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