All You Need is not $\Omega_\mathrm{gw}$: Beyond the Mean of the Cross-Correlation Estimator when Searching for an Astrophysical Gravitational-Wave Background

Emanuele Berti; Francesco Iacovelli; Haowen Zhong; Vuk Mandic

REVIEW 3 major objections 5 minor 73 references

All You Need is not $\Omega_\mathrm{gw}$: Beyond the Mean of the Cross-Correlation Estimator when Searching for an Astrophysical Gravitational-Wave Background

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper claims that for Cosmic Explorer the cross-correlation estimator from binary black holes becomes non-Gaussian, breaking the Gaussian likelihood assumption used in stochastic background searches.

desk verdict A solid, useful forecast: the cross-correlation estimator stays Gaussian at O5 but becomes non-Gaussian for BBH at Cosmic Explorer, with the main caveat being an arbitrary low-redshift cutoff that deserves a sensitivity scan. read the letter →

arxiv 2608.11119 v1 pith:YUDXVXNL submitted 2026-08-11 gr-qc astro-ph.HEastro-ph.IM

Haowen Zhong , Francesco Iacovelli , Vuk Mandic , Emanuele Berti This is my paper · ORCID

classification gr-qcastro-ph.HEastro-ph.IM

keywords gravitational-wavebackgroundcross-correlationestimatornon-GaussianitycumulantscompoundPoissonprocessCosmicExplorercompactbinarycoalescencesEdgeworthexpansion

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 asks whether the cross-correlation estimator $\bar C_f$ used to search for the stochastic gravitational-wave background can still be treated as Gaussian when the background is produced by compact binary coalescences. The authors derive the estimator's cumulants up to fourth order from a compound-Poisson description of individual merger events, splitting each event's contribution into mean intensity, geometrical shot noise, and polarization leakage. They find that for LIGO's upcoming O5 run the Gaussian assumption holds for black-hole, neutron-star, and mixed populations, but for a two-detector Cosmic Explorer network the binary-black-hole background makes $\bar C_f$ measurably non-Gaussian: skewness $0.31^{+0.04}_{-0.03}$ and excess kurtosis $1.3^{+0.4}_{-0.3}$ near 20 Hz over one year. That matters because inference frameworks built on Gaussian likelihoods would misstate uncertainties, and the result shows the Edgeworth expansion is only a leading-order fix in the next-generation era.

What carries the argument

The engine is the compound-Poisson cumulant relation $\kappa_n(\bar C_f)=\langle N_{\rm tot}\rangle N_{\rm seg}^{-n}\langle (X_f^{(i)})^n\rangle_\theta + \kappa_n(\bar n_f)$, which reduces the difficult many-event statistics to single-event averages. The single-event response is decomposed via Stokes parameters into three physically distinct terms: the sky-averaged mean intensity (which reproduces the standard $\Omega_{\rm gw}$ spectrum), the geometrical shot noise from the deviation of each event's detector response from its sky average, and the polarization leakage from finite catalog size. Adaptive Monte Carlo integration over the population models evaluates the moments, and a lower-redshift cutoff at $z_{\min}=0.2$ makes the fourth moment converge.

What would settle it

Recompute the BBH cumulants at Cosmic Explorer with $z_{\min}$ varied from 0.05 to 0.5 under the same population model: if the excess kurtosis at 20 Hz moves from the quoted 1.3 to below 0.3 or above 3, or if a direct histogram of simulated $\bar C_f$ values at 20 Hz shows the Edgeworth density becoming negative in the bulk, the central claim about non-Gaussianity needs revision.

Watch

Extended reading notes

Core claim

The central claim is that the statistical distribution of $\bar C_f$ is not fully captured by its mean $\Omega_{\rm gw}$, and for the next generation of ground-based detectors the higher-order cumulants of the binary-black-hole background become too large to ignore. The key identity is $\kappa_n(\bar C_f)\simeq \langle N_{\rm tot}\rangle / N_{\rm seg}^n \langle (X_f^{(i)})^n\rangle_\theta + \kappa_n(\bar n_f)$, which ties each cumulant to the Poisson rate of mergers and the moments of a single event's detector response. Using this, the paper computes 90% credible intervals for the cumulants for the three CBC populations and finds effectively Gaussian behavior at O5 for all, but at CE a BBH-driven skewness of order 0.3 and excess kurtosis of order 1.3 in the most sensitive band, with the signal variance itself overtaking detector noise there.

Load-bearing premise

The quoted non-Gaussianity numbers rest on the $z_{\min}=0.2$ lower-redshift cutoff imposed to make the moment integrals converge; the loudest nearby mergers dominate the fourth-order cumulant, and the paper does not include a dedicated scan over this cutoff.

Editorial extensions

If this is right

For the LIGO O5 network, current Gaussian-likelihood stochastic searches remain valid for all three compact-binary populations.
For a two-detector Cosmic Explorer network, analyses of the BBH background that keep all data must go beyond a Gaussian likelihood; neglecting the signal variance would understate uncertainties.
The Edgeworth expansion can serve as a leading-order description for the moderate non-Gaussianity, but its truncated series has limited validity in the tails.
If one notches out individually resolvable BBH signals, the remaining background can be treated as effectively Gaussian again.
BNS and NSBH backgrounds stay effectively Gaussian even at CE, so no new formalism is needed for them.

Reading between the lines

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

The paper's compound-Poisson machinery also implies that $\bar C_f$ values at different frequencies are positively correlated through the chirp evolution, so a full likelihood should include off-diagonal frequency covariances; the paper flags this as future work.
Because the non-Gaussianity is driven by the loudest nearby mergers, the measured skewness and kurtosis could in principle be used as an independent handle on the local merger rate or on how imperfectly resolvable events are removed.
The z_min cutoff models a residual background left after notching; a realistic notching pipeline will produce a population-dependent residual, so the CE numbers should be interpreted as a full-background upper bound on non-Gaussianity rather than a prediction for a specific analysis.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Bottom line: this is a real contribution, not a repackaging. The paper gives the first systematic computation of third and fourth cumulants of the frequency-resolved cross-correlation estimator for all three CBC populations, across O5 and Cosmic Explorer, and the headline result—Cbar_f stays effectively Gaussian for O5 but the BBH background at CE has skewness 0.31 and excess kurtosis 1.3 at 20 Hz—is something the field needs to know. The derivation is parameter-free: compound Poisson moments fed by external LVK population fits, no fitting to the target non-Gaussianity. The three-component decomposition (mean intensity, geometrical shot noise, polarization leakage) is clean, and they check their approximations quantitatively via the Rtau calculation, showing where off-diagonal and self-interaction terms can be neglected. That's good work.

Soft spots: the z_min=0.2 cutoff is the real issue. The moments diverge as z->0, and the fourth-order moment is dominated by the nearest, loudest events, so the quoted gamma2=1.3 depends directly on where you put this cutoff. The authors assert robustness for z_min below ~0.5 and cite their earlier paper [34], but they do not show a sensitivity scan here. The 90% credible intervals they quote only cover the local merger rate, not the cutoff choice or population-model uncertainty. That's a genuine gap, though I don't think it overturns the qualitative conclusion. Also, z_min is only a proxy for the actual notching threshold; real resolvability is SNR- and mass-dependent, so the residual background after subtraction could be different. But the paper is transparent about this.

One more caveat: the neglect of frequency correlations is flagged but not quantified beyond the statement that posteriors will be overly wide. That's fine for this paper, but it's a warning for anyone using the results.

Who benefits: anyone building stochastic background likelihoods for next-generation detectors, and the popcorn/non-Gaussian background community. The paper deserves a serious referee. My recommendation: send it to review, and ask for a z_min scan (or at least a figure showing the dependence) before acceptance. That's a minor-to-moderate revision.

Referee Report

3 major / 5 minor

Summary. The paper computes the cumulants up to fourth order of the frequency-domain cross-correlation estimator \bar{C}_f for a stochastic gravitational-wave background produced by compact binary coalescences. The estimator is decomposed into mean intensity, geometrical shot noise, and polarization leakage, and the cumulants are obtained from a compound-Poisson master equation, Eq. (15), with population inputs taken from external LVK fits. Numerical results are presented for BBH, BNS, and NSBH populations under LIGO A+ (O5) and two-detector Cosmic Explorer configurations. The central claim is that \bar{C}_f remains effectively Gaussian for all three populations at O5 and for BNS/NSBH even at CE, while for the BBH background at CE the estimator becomes non-Gaussian, with skewness \gamma_1=0.31^{+0.04}_{-0.03} and excess kurtosis \gamma_2=1.3^{+0.4}_{-0.3} at f=20 Hz. The paper argues that an Edgeworth expansion is adequate as a leading-order description and that notching resolved BBH events is a viable alternative in the next-generation era.

Significance. If the quantitative claims hold, this is a timely and useful result for the stochastic-background community: it validates Gaussian likelihood frameworks for the upcoming O5 runs, quantifies when those frameworks break down for next-generation detectors, and identifies the physical origin of the non-Gaussianity in terms of shot noise and polarization leakage. The central derivation is parameter-free in the sense that the cumulant relation, Eq. (15), is an exact property of the compound-Poisson model and is not fitted to the target non-Gaussianity; the inputs are external LVK population fits. The paper also performs useful checks of the neglected off-diagonal and self-interaction terms through the R\tau criterion in Sec. III B and uses adaptive Vegas integration to handle heavy-tailed moments. The main weakness is the treatment of the lower-redshift cutoff z_min=0.2, which is not subjected to a dedicated sensitivity scan in this work and directly affects the headline BBH kurtosis value.

major comments (3)

[Sec. III C, Eqs. (15), (27)-(28); Fig. 5] The headline result \gamma_2=1.3^{+0.4}_{-0.3} for the BBH background at CE depends directly on the imposed lower-redshift cutoff z_min=0.2. As the paper itself notes, the moments \langle (X_f^{(i)})^n\rangle diverge as z\to 0, and the fourth moment is dominated by the nearest and loudest events. The manuscript asserts robustness for z_min below O(0.5) and cites Ref. [34], but it does not present a z_min sensitivity scan, nor does it quantify how the 90% credible intervals on \gamma_1 and \gamma_2 would shift for, say, z_min=0.1 or z_min=0.35. Because z_min is also only a proxy for the actual SNR- and mass-dependent resolvability/notching threshold, the quoted non-Gaussianity is a modeling choice rather than a prediction for a uniquely defined residual background. I request a dedicated scan over z_min (including the value z_min=0.35 studied in Ref. [34]) and a discussion of how the headline values and their credible intervals change.
[Sec. III E, Figs. 4-5] The fractional-contribution plots for \kappa_3 and \kappa_4 show that the diagonal terms (mean intensity, shot noise, polarization leakage) do not account for the full cumulant: below 25 Hz the sum of the three diagonal contributions is only about 45% of \kappa_3, with the remaining 55% attributed to cross-terms. The text describes this briefly, but it would strengthen the paper to state explicitly how the cross-terms are computed and to verify that the numerical evaluation of the full \kappa_n via Eq. (15) is stable in the frequency region where the cross-terms dominate. Without this detail, the reader cannot assess whether the reported \gamma_1(f) and \gamma_2(f) values, especially their oscillatory high-frequency behavior, are robust.
[Sec. III F and Sec. IV] For the BNS background, the paper acknowledges that the neglect of off-diagonal (overlap) and self-interaction terms may break down below about 8-12 Hz, where R\tau reaches the marginal and dangerous zones. The conclusions for BNS non-Gaussianity are nevertheless stated as robust, relying on the argument that the induced error would need to reach O(10^4) to affect the quoted \gamma_2 bound. This is a reasonable order-of-magnitude argument, but it is not a calculation; since the BNS background is the one most affected by the approximation, the caveat should be stated in the abstract or conclusions with the same prominence as the BBH result, rather than only in the body.

minor comments (5)

[Acknowledgments] The acknowledgment text contains a typo: the Minnesota Supercomputing Institute URL is followed by 'eduledgments' instead of 'Acknowledgments'.
[References [43]-[44]] References [43] and [44] appear to cite the same arXiv preprint (2209.01310 and 2209.01221) with different identifiers; please verify and consolidate to the published version if available.
[Sec. III A] The text says 'Planck18 parameters' but the word appears as 'planet18' in the PDF; please fix the capitalization.
[Fig. 4 and Fig. 5 captions] The right panels use a broken vertical axis, but the captions do not explain the break or the scale change; adding a note would improve readability.
[Sec. II A, Eq. (8)] The notation P_{a;f}\equiv P_{gw;f}+P_{na;f} is introduced but P_{gw;f} is not defined in Eq. (8) until later; please define it at first use for clarity.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the headline non-Gaussianity is a forward-model output from a standard compound-Poisson identity and external LVK population inputs; self-citations are supporting, not definitional.

full rationale

The derivation chain is self-contained rather than circular. Equation (15) is the standard compound-Poisson cumulant identity, with the Poisson rate and population moments taken from external inputs (PixelPop rates, LVK mass/spin models, IMRPhenomXAS waveforms, A+ and CE PSDs). The headline skewness and excess kurtosis, Eqs. (27)-(28), are outputs of this chain, not quantities fitted into it. The three-component decomposition in Eq. (19) is an algebraic identity, so it cannot smuggle in conclusions. The self-citations that do appear are not load-bearing in a circular sense: Ref. [38] supplies the standard estimator variance formula, Ref. [41] is a side remark on Stokes decomposition, and Ref. [34] is cited for robustness of the z_min=0.2 cutoff. The paper acknowledges that this cutoff is 'somewhat arbitrary' and that its robustness is discussed in Ref. [34]; while the absence of an in-paper z_min sensitivity scan is a legitimate limitation for the numerical values, the quoted cumulants are conditional on a stated modeling prior rather than being defined by the target result. There is no exhibited reduction of any prediction to its own input, so the circularity score is 0.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The central calculation rests on the compound Poisson model for events, the neglect of intra-chirp self-interaction and inter-chirp interference (valid at the 20 Hz BBH band), the Gaussian detector noise assumption, the external CBC population models, and the arbitrary z_min cutoff. No new physical entities are introduced.

free parameters (5)

z_min lower redshift cutoff = 0.2
Introduced in Sec. III C to render the higher moments <(X_f)^n> finite; the paper asserts robustness for z_min < 0.5 citing Ref [34], but the quoted skewness and kurtosis depend on this cutoff and no dedicated sensitivity study is shown.
Local BBH merger rate R0 = 18.0 (+4.9/-4.4) Gpc^-3 yr^-1
Taken from PixelPop/GWTC-5 [40]; drives the overall amplitude of all cumulants; 90% credible interval propagated into quoted errors.
Local BNS merger rate R0 = 23.4 (+54.7/-18.2) Gpc^-3 yr^-1
From [40]; large uncertainty and weak constraints; only rate uncertainty is propagated.
Local NSBH merger rate R0 = 15.9 (+16.9/-8.9) Gpc^-3 yr^-1
From [40]; used for the NSBH background.
BBH Madau-Dickinson parameters (alpha_z, beta_z, z_p) = 2.85, 4.14, 2.27
Maximum-likelihood values from [40]; fixed in this work, not varied in the uncertainty propagation.

assumptions (5)

domain assumption CBC mergers are independent events forming a Poisson process in time; N_tot is Poisson and each event's X_f is drawn independently from the population model.
Sec. II B, Eqs. (14)-(15): the compound Poisson cumulant formula kappa_n(Cbar_f) = <N_tot> N_seg^-n <(X_f)^n> + kappa_n(nbar_f) is the foundation of all numerical results.
domain assumption Each CBC chirp contributes to at most one time-frequency pixel; mutual interference between events and self-interaction across segments are negligible.
Sec. II B, Eqs. (11)-(15): off-diagonal and self-interaction terms are dropped; the paper checks this holds only above about 5-12 Hz depending on population (Sec. III B), and the headline 20 Hz BBH result is in the safe zone.
domain assumption Detector noise is Gaussian, stationary, uncorrelated between detectors, so kappa_3(nbar_f)=kappa_4(nbar_f)=0 and kappa_2(nbar_f) follows Eq. (9).
Sec. II A, Eq. (9); standard for SGWB searches but an assumption.
domain assumption The adopted CBC population models (mass spectra, Madau-Dickinson redshift evolution, zero spins, IMRPhenomXAS waveforms) represent the true populations.
Sec. III A; BBH parameters from GWTC-5, BNS and NSBH models described as 'purely representative'; results are conditional on these models.
domain assumption The local merger rates and their 90% credible intervals from PixelPop/GWTC-5 are correct.
Sec. III A; the quoted uncertainty bands on skewness and kurtosis only propagate these rate uncertainties.

how reviews work

0 comments

Cite this review

Pith. "Pith review of All You Need is not $\Omega_\mathrm{gw}$: Beyond the Mean of the Cross-Correlation Estimator when Searching for an Astrophysical Gravitational-Wave Background." pith.science (2026). https://pith.science/paper/YUDXVXNL

@misc{pith2026260811119,
  author       = {Pith},
  title        = {Pith review of: All You Need is not $\Omega_\mathrmgw$: Beyond the Mean of the Cross-Correlation Estimator when Searching for an Astrophysical Gravitational-Wave Background},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YUDXVXNL}},
  note         = {Machine review of arXiv:2608.11119}
}

abstract

Searches for the stochastic gravitational-wave background (SGWB) using ground-based detectors rely heavily on the cross-correlation estimator $\bar C_f$, whose statistical properties determine the theoretical foundation for inference frameworks built on it. Past analyses usually assume Gaussianity of $\bar C_f$, but this assumption has not been systematically tested for a background of astrophysical origin like the one produced by compact binary coalescences. In this work, we decompose $\bar C_f$ into three physically motivated components: (1) the mean intensity, (2) the geometrical shot noise, and (3) the polarization leakage, and we compute 90\% credible intervals for its cumulants up to fourth order for the binary black hole, binary neutron star, and neutron star-black hole populations by propagating local merger rate uncertainties. We find that for the upcoming LIGO O5 sensitivity, $\bar C_f$ remains effectively Gaussian for all three populations, validating current Gaussian likelihood frameworks. For Cosmic Explorer (CE), however, the $\bar C_f$ produced by binary black holes becomes non-Gaussian, with skewness and excess kurtosis of $0.31^{+0.04}_{-0.03}$ and $1.3^{+0.4}_{-0.3}$, respectively, in the most sensitive band over a one-year observation period. For binary neutron star and neutron star - black hole mergers, $\bar C_f$ remains largely Gaussian even at CE. These results indicate that the Edgeworth expansion provides an adequate leading-order description of the non-Gaussianity expected in the next-generation era, while also motivating the development of more sophisticated statistical methods for a more accurate treatment.

Figures

Figures reproduced from arXiv: 2608.11119 by the authors.

**Figure 2.** FIG. 2. Expected number of events per time-frequency bin [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 3.** FIG. 3. Statistical properties of the cross-correlation estimator [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

Figures from the paper (6 more)

**Figure 4.** Figure 4: FIG. 4. As in Fig [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]

**Figure 5.** Figure 5: FIG. 5. Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]

**Figure 6.** Figure 6: FIG. 6. Same as the right panel of Fig [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]

**Figure 7.** Figure 7: FIG. 7. Skewness [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]

**Figure 8.** Figure 8: FIG. 8. Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p012_8.png]

**Figure 9.** Figure 9: FIG. 9. Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p013_9.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

73 extracted references · 11 canonical work pages

[34]

Zhong and V

H. Zhong and V. Mandic, Phys. Rev. D113, 044048 (2026), arXiv:2509.15014 [gr-qc]

arXiv 2026
[1]

B. P. Abbottet al.(LIGO Scientific, Virgo), Phys. Rev. Lett.116, 061102 (2016), arXiv:1602.03837 [gr-qc]

arXiv 2016
[2]

B. P. Abbottet al.(LIGO Scientific, Virgo), Phys. Rev. 13 10−5 10−4 10−3 CE median CE 90% CI A+ median A+ 90% CI 101 102 103 f [Hz] 10−22 10−17 10−12 10−7 10−10 10−8 10−6 10−4 101 102 103 f [Hz] 10−30 10−26 10−22 10−18 |γ1| γ2 FIG. 9. Same as Fig. 8, for the NSBH case. X9, 031040 (2019), arXiv:1811.12907 [astro-ph.HE]

arXiv 2019
[3]

Abbottet al.(LIGO Scientific, Virgo), Phys

R. Abbottet al.(LIGO Scientific, Virgo), Phys. Rev. X 11, 021053 (2021), arXiv:2010.14527 [gr-qc]

arXiv 2021
[4]

Abbottet al.(KAGRA, Virgo, LIGO Scientific), Phys

R. Abbottet al.(KAGRA, Virgo, LIGO Scientific), Phys. Rev. X13, 041039 (2023), arXiv:2111.03606 [gr-qc]

arXiv 2023
[5]

A. G. Abacet al.(LIGO Scientific, Virgo, KAGRA), (2025), arXiv:2508.18082 [gr-qc]

arXiv 2025
[6]

(LIGO Scientific, Virgo, KAGRA), (2026), arXiv:2605.27225 [gr-qc]

arXiv 2026
[7]

Christensen, Rep

N. Christensen, Rep. Prog. Phys.82, 016903 (2019), arXiv:1811.08797 [gr-qc]

arXiv 2019

Show all 73 references

[8]

A. I. Renzini, B. Goncharov, A. C. Jenkins, and P. M. Meyers, Galaxies10, 34 (2022), arXiv:2202.00178 [gr-qc]

2022 arXiv
[9]

J. D. Romano and N. J. Cornish, Living Rev. Relativ. 20, 2 (2017), arXiv:1608.06889 [gr-qc]

2017 arXiv
[10]

A. G. Abacet al.(LIGO Scientific, Virgo, KAGRA), (2025), arXiv:2508.20721 [gr-qc]

2025 arXiv
[11]

Abbottet al.(KAGRA, Virgo, LIGO Scientific), Phys

R. Abbottet al.(KAGRA, Virgo, LIGO Scientific), Phys. Rev. D104, 022004 (2021), arXiv:2101.12130 [gr-qc]

2021
[12]

Allen and J

B. Allen and J. D. Romano, Phys. Rev. D59, 102001 (1999), arXiv:gr-qc/9710117

1999 arXiv
[13]

Mandic, E

V. Mandic, E. Thrane, S. Giampanis, and T. Regimbau, Phys. Rev. Lett.109, 171102 (2012), arXiv:1209.3847 [astro-ph.CO]

2012 arXiv
[14]

Matas and J

A. Matas and J. D. Romano, Phys. Rev. D103, 062003 (2021), arXiv:2012.00907 [gr-qc]

2021 arXiv
[15]

Drasco and E

S. Drasco and E. E. Flanagan, Phys. Rev. D67, 082003 (2003), arXiv:gr-qc/0210032

2003 arXiv
[16]

Smith and E

R. Smith and E. Thrane, Phys. Rev. X8, 021019 (2018), arXiv:1712.00688 [gr-qc]

2018 arXiv
[17]

Biscoveanu, C

S. Biscoveanu, C. Talbot, E. Thrane, and R. Smith, Phys. Rev. Lett.125, 241101 (2020), arXiv:2009.04418 [astro-ph.HE]

2020 arXiv
[18]

Lawrence, K

J. Lawrence, K. Turbang, A. Matas, A. I. Renzini, N. van Remortel, and J. D. Romano, Phys. Rev. D107, 103026 (2023), arXiv:2301.07675 [gr-qc]

2023 arXiv
[19]

X.-X. Kou, M. Saleem, V. Mandic, C. Talbot, and E. Thrane, Phys. Rev. D112, 084064 (2025), arXiv:2506.14179 [gr-qc]

2025 arXiv
[20]

Mukherjee and J

S. Mukherjee and J. Silk, Mon. Not. Roy. Astron. Soc. 491, 4690 (2020), arXiv:1912.07657 [gr-qc]

2020 arXiv
[21]

M. R. Sah and S. Mukherjee, Mon. Not. Roy. Astron. Soc.527, 4100 (2023), arXiv:2307.06405 [gr-qc]

2023 arXiv
[22]

M. R. Sah and S. Mukherjee, Phys. Rev. D113, 123058 (2026), arXiv:2511.03262 [astro-ph.HE]

2026 arXiv
[23]

Martellini and T

L. Martellini and T. Regimbau, Phys. Rev. D89, 124009 (2014), arXiv:1405.5775 [astro-ph.CO]

2014 arXiv
[24]

Martellini and T

L. Martellini and T. Regimbau, Phys. Rev. D92, 104025 (2015), [Erratum: Phys.Rev.D 97, 049903 (2018)], arXiv:1509.04802 [astro-ph.CO]

2015 arXiv
[25]

Liu and S

X. Liu and S. Kuroyanagi, (2026), arXiv:2601.10428 [gr- qc]

2026
[26]

Punturoet al., Class

M. Punturoet al., Class. Quantum Grav.27, 194002 (2010)

2010
[27]

Branchesiet al., J

M. Branchesiet al., J. Cosmology Astropart. Phys.2023, 068 (2023), arXiv:2303.15923 [gr-qc]

2023 arXiv
[28]

Abacet al.(ET), J

A. Abacet al.(ET), J. Cosmology Astropart. Phys. 2026, 081 (2026), arXiv:2503.12263 [gr-qc]

2026 arXiv
[29]

Reitzeet al., Bull

D. Reitzeet al., Bull. Am. Astron. Soc.51, 035 (2019), arXiv:1907.04833 [astro-ph.IM]

2019 arXiv
[30]

Evanset al., (2021), arXiv:2109.09882 [astro-ph.IM]

M. Evanset al., (2021), arXiv:2109.09882 [astro-ph.IM]

2021 arXiv
[31]

Evanset al., (2023), arXiv:2306.13745 [astro-ph.IM]

M. Evanset al., (2023), arXiv:2306.13745 [astro-ph.IM]

2023 arXiv
[32]

Zhong, R

H. Zhong, R. Ormiston, and V. Mandic, Phys. Rev. D 107, 064048 (2023), [Erratum: Phys.Rev.D 108, 089902 (2023)], arXiv:2209.11877 [gr-qc]

2023 arXiv
[33]

Zhong, B

H. Zhong, B. Zhou, L. Reali, E. Berti, and V. Mandic, Phys. Rev. D110, 064047 (2024), arXiv:2406.10757 [gr- qc]

2024 arXiv
[35]

Caprini and D

C. Caprini and D. G. Figueroa, Class. Quantum Grav. 35, 163001 (2018), arXiv:1801.04268 [astro-ph.CO]

2018 arXiv
[36]

Aghanimet al.(Planck), Astron

N. Aghanimet al.(Planck), Astron. Astrophys.641, A6 (2020), [Erratum: Astron.Astrophys. 652, C4 (2021)], arXiv:1807.06209 [astro-ph.CO]

2020 arXiv
[37]

C. M. F. Mingarelli, S. R. Taylor, B. S. Sathyaprakash, and W. M. Farr, arXiv preprint (2019), arXiv:1911.09745 [gr-qc]

2019 arXiv
[38]

Zhong, J

H. Zhong, J. D. Romano, V. Mandic, S. Kandhasamy, and A. I. Renzini, (2026), arXiv:2606.23816 [gr-qc]

2026 arXiv
[39]

A. I. Renziniet al., Astrophys. J.952, 25 (2023), arXiv:2303.15696 [gr-qc]

2023 arXiv
[40]

GWTC-5.0: Popula- tion Properties of Merging Compact Binaries,

LIGO Scientific Collaboration and Virgo Collabora- tion and KAGRA Collaboration, “GWTC-5.0: Popula- tion Properties of Merging Compact Binaries,” (2026), arXiv:2605.27226 [astro-ph.HE]. 14

2026 arXiv
[41]

Boybeyi and V

T. Boybeyi and V. Mandic, (2026), arXiv:2605.25772 [gr-qc]

2026 arXiv
[42]

Belgacem, F

E. Belgacem, F. Iacovelli, M. Maggiore, M. Mancarella, and N. Muttoni, J. Cosmology Astropart. Phys.04, 032 (2025), arXiv:2411.04028 [gr-qc]

2025 arXiv
[43]

B. Zhou, L. Reali, E. Berti, M. C ¸ alı¸ skan, C. Creque- Sarbinowski, M. Kamionkowski, and B. S. Sathyaprakash, Phys. Rev. D108, 064040 (2023), arXiv:2209.01310 [gr-qc]

2023 arXiv
[44]

B. Zhou, L. Reali, E. Berti, M. C ¸ alı¸ skan, C. Creque- Sarbinowski, M. Kamionkowski, and B. S. Sathyaprakash, arXiv (2022), arXiv:2209.01221 [gr-qc]

2022 arXiv
[45]

H. Song, D. Liang, Z. Wang, and L. Shao, Phys. Rev. D 109, 123014 (2024), arXiv:2401.00984 [gr-qc]

2024 arXiv
[46]

A. I. Renzini and J. Golomb, Astron. Astrophys.691, A238 (2024), arXiv:2407.03742 [astro-ph.CO]

2024 arXiv
[47]

Ebersold and T

M. Ebersold and T. Regimbau, Phys. Rev. D113, 044026 (2026), arXiv:2510.02163 [gr-qc]

2026
[48]

W. G. Lamb and S. R. Taylor, Astrophys. J. Lett.971, L10 (2024), arXiv:2407.06270 [gr-qc]

2024 arXiv
[49]

W. G. Lamb, J. M. Wachter, A. Mitridate, S. C. Sardesai, B. B´ ecsy, E. L. Hagen, S. R. Taylor, and L. Z. Kelley, Phys. Rev. D113, 123065 (2026), arXiv:2511.09659 [gr- qc]

2026 arXiv
[50]

Meacher, E

D. Meacher, E. Thrane, and T. Regimbau, Phys. Rev. D89, 084063 (2014), arXiv:1402.6231 [astro-ph.CO]

2014 arXiv
[51]

A. C. Jenkins, J. D. Romano, and M. Sakellariadou, Phys. Rev. D100, 083501 (2019), arXiv:1907.06642 [astro-ph.CO]

2019 arXiv
[52]

A. C. Jenkins and M. Sakellariadou, Phys. Rev. D100, 063508 (2019), arXiv:1902.07719 [astro-ph.CO]

2019 arXiv
[53]

Kouvatsos, A

N. Kouvatsos, A. C. Jenkins, A. I. Renzini, J. D. Ro- mano, and M. Sakellariadou, Phys. Rev. D109, 103535 (2024), arXiv:2312.09110 [astro-ph.CO]

2024 arXiv
[54]

Heinzel, M

J. Heinzel, M. Mould, S. ´Alvarez-L´ opez, and S. Vi- tale, Phys. Rev. D111, 063043 (2025), arXiv:2406.16813 [astro-ph.HE]

2025 arXiv
[55]

Madau and M

P. Madau and M. Dickinson, Annu. Rev. Astron. Astro- phys.52, 415 (2014), arXiv:1403.0007 [astro-ph.CO]

2014 arXiv
[56]

Abbottet al.(KAGRA, Virgo, LIGO Scientific), Phys

R. Abbottet al.(KAGRA, Virgo, LIGO Scientific), Phys. Rev. X13, 011048 (2023), arXiv:2111.03634 [astro- ph.HE]

2023 arXiv
[57]

Landry and J

P. Landry and J. S. Read, Astrophys. J. Lett.921, L25 (2021), arXiv:2107.04559 [astro-ph.HE]

2021 arXiv
[58]

Zhong, L

H. Zhong, L. Reali, B. Zhou, E. Berti, and V. Mandic, arXiv (2025), arXiv:2501.17717 [gr-qc]

2025 arXiv
[59]

Madau and T

P. Madau and T. Fragos, Astrophys. J.840, 39 (2017), arXiv:1606.07887 [astro-ph.GA]

2017 arXiv
[60]

Regimbauet al., Phys

T. Regimbauet al., Phys. Rev. D86, 122001 (2012), arXiv:1201.3563 [gr-qc]

2012 arXiv
[61]

Biscoveanu, P

S. Biscoveanu, P. Landry, and S. Vitale, Mon. Not. R. Astron. Soc.518, 5298 (2022), arXiv:2207.01568 [astro- ph.HE]

2022 arXiv
[62]

A. G. Abacet al.(LIGO Scientific, KAGRA, Virgo), Astrophys. J. Lett.970, L34 (2024), arXiv:2404.04248 [astro-ph.HE]

2024 arXiv
[63]

Pratten, S

G. Pratten, S. Husa, C. Garcia-Quiros, M. Colleoni, A. Ramos-Buades, H. Estelles, and R. Jaume, Phys. Rev. D102, 064001 (2020), arXiv:2001.11412 [gr-qc]

2020 arXiv
[64]

Buonanno, B

A. Buonanno, B. Iyer, E. Ochsner, Y. Pan, and B. S. Sathyaprakash, Phys. Rev. D80, 084043 (2009), arXiv:0907.0700 [gr-qc]

2009 arXiv
[65]

vegas: Adaptive multidimensional monte carlo integration,

G. P. Lepage, “vegas: Adaptive multidimensional monte carlo integration,”https://github.com/gplepage/ vegas(2024), python package

2024
[66]

G. P. Lepage, J. Comput. Phys.27, 192 (1978)

1978
[67]

G. P. Lepage, J. Comput. Phys.439, 110386 (2021), arXiv:2009.05112 [physics.comp-ph]

2021 arXiv
[68]

Aasiet al.(LIGO Scientific), Class

J. Aasiet al.(LIGO Scientific), Class. Quantum Grav. 32, 074001 (2015), arXiv:1411.4547 [gr-qc]

2015 arXiv
[69]

The A+ design curve,

L. Barsotti, L. McCuller, M. Evans,et al., “The A+ design curve,”https://dcc.ligo.org/LIGO-T1800042/ public(2018), LIGO Document T1800042-v5

2018
[70]

Chandra,https://github.com/koustavchandra/ gwforge/tree/main/GWForge/ifo/noise_curves (2025)

K. Chandra,https://github.com/koustavchandra/ gwforge/tree/main/GWForge/ifo/noise_curves (2025)

2025
[71]

Chandra, Class

K. Chandra, Class. Quantum Grav.42, 025003 (2025), arXiv:2407.21109 [gr-qc]

2025 arXiv
[72]

Thrane and J

E. Thrane and J. D. Romano, Phys. Rev. D88, 124032 (2013), arXiv:1310.5300 [astro-ph.IM]

2013 arXiv
[73]

Blinnikov and R

S. Blinnikov and R. Moessner, Astron. Astrophys. Sup. 130, 193 (1998), arXiv:astro-ph/9711239

1998 arXiv

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [34]

Zhong and V

H. Zhong and V. Mandic, Phys. Rev. D113, 044048 (2026), arXiv:2509.15014 [gr-qc]

arXiv 2026

[2] [1]

B. P. Abbottet al.(LIGO Scientific, Virgo), Phys. Rev. Lett.116, 061102 (2016), arXiv:1602.03837 [gr-qc]

arXiv 2016

[3] [2]

B. P. Abbottet al.(LIGO Scientific, Virgo), Phys. Rev. 13 10−5 10−4 10−3 CE median CE 90% CI A+ median A+ 90% CI 101 102 103 f [Hz] 10−22 10−17 10−12 10−7 10−10 10−8 10−6 10−4 101 102 103 f [Hz] 10−30 10−26 10−22 10−18 |γ1| γ2 FIG. 9. Same as Fig. 8, for the NSBH case. X9, 031040 (2019), arXiv:1811.12907 [astro-ph.HE]

arXiv 2019

[4] [3]

Abbottet al.(LIGO Scientific, Virgo), Phys

R. Abbottet al.(LIGO Scientific, Virgo), Phys. Rev. X 11, 021053 (2021), arXiv:2010.14527 [gr-qc]

arXiv 2021

[5] [4]

Abbottet al.(KAGRA, Virgo, LIGO Scientific), Phys

R. Abbottet al.(KAGRA, Virgo, LIGO Scientific), Phys. Rev. X13, 041039 (2023), arXiv:2111.03606 [gr-qc]

arXiv 2023

[6] [5]

A. G. Abacet al.(LIGO Scientific, Virgo, KAGRA), (2025), arXiv:2508.18082 [gr-qc]

arXiv 2025

[7] [6]

(LIGO Scientific, Virgo, KAGRA), (2026), arXiv:2605.27225 [gr-qc]

arXiv 2026

[8] [7]

Christensen, Rep

N. Christensen, Rep. Prog. Phys.82, 016903 (2019), arXiv:1811.08797 [gr-qc]

arXiv 2019