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Two-step Bayesian test separates SMBHB signals from primordial GW noise

2026-07-09 09:32 UTC pith:4HJRPIQY

load-bearing objection Solid methodological framework for PTA source-population inference, but the validation is missing a false-positive test — worth a serious referee. the 1 major comments →

arxiv 2607.07477 v1 pith:4HJRPIQY submitted 2026-07-08 astro-ph.HE gr-qc

Population statistics of nanohertz gravitational wave sources

Jiming Yu , Zhen Pan , Xiao Xue , Liang Dai This is my paper
classification astro-ph.HE gr-qc PACS 04.80.Nn98.62.Js98.80.-k
keywords gravitational wave backgroundpulsar timing arrayssupermassive black hole binarieshierarchical Bayesian inferencenon-Gaussianitypopulation statisticsprimordial gravitational waves
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper introduces a two-stage hierarchical Bayesian inference framework designed to determine whether the nanohertz gravitational wave background detected by pulsar timing arrays originates from a finite population of supermassive black hole binaries or from a Gaussian primordial background. The central mechanism is a decomposition of evidence into two parts: first, a per-frequency-bin free-spectrum reconstruction tests whether an individual bright source can be distinguished from a smooth Gaussian background (H1 vs H0); second, a population-level likelihood tests whether bin-to-bin fluctuations in the recovered power spectrum are consistent with a finite number of discrete sources rather than an infinite ensemble obeying central limit theorem statistics (Hfin vs Hinf). The framework leverages non-Gaussianity in two distinct forms—the presence of individually bright sources within frequency bins and the statistical fluctuations of power across frequency bins—as a discriminant between astrophysical and primordial origins. Applied to mock PTA data with 200 pulsars and a 30-year observation span, the combined hypothesis H1+Hfin is favored over H0+Hinf with a log Bayes factor of 22.74 for the 15 lowest frequency bins, and the injected population parameters (GWB amplitude, spectral index, and source-amplitude power-law index) are recovered. The paper also demonstrates that even when individual bright sources are not detected in the first step (as in the gamma=2.5 case), the second step can still extract mild evidence for a finite population from power-spectrum fluctuations alone.

Core claim

The key result is that non-Gaussianity in the nanohertz gravitational wave background can be systematically decomposed into two orthogonal statistical signals—per-bin bright-source non-Gaussianity and cross-bin power-spectrum fluctuations—and that a hierarchical Bayesian framework combining both provides a quantitative Bayes-factor discriminant between a finite SMBHB population and a Gaussian primordial background. The proof-of-principle demonstration on mock data shows that this two-step approach correctly recovers injected population parameters and strongly favors the finite-source hypothesis when the source-amplitude distribution is steep enough (gamma=2.2) for individual sources to carry

What carries the argument

Two-stage hierarchical Bayesian inference: (1) free-spectrum reconstruction per frequency bin under H0 (Gaussian isotropic GWB) vs H1 (GWB + brightest individual SMBHB), and (2) population-level likelihood under Hinf (infinite sources, CLT applies, delta-function PDF) vs Hfin (finite sources, fluctuating PSD, characteristic-function-based PDF from Eq. 20). The total log Bayes factor decomposes as the sum of per-bin free-spectrum evidence ratios and per-bin population evidence ratios (Eq. 10).

Load-bearing premise

The framework explicitly models only the single brightest SMBHB in each frequency bin and treats all remaining sources as an unresolved Gaussian stochastic component. If the second-brightest source in a bin carries non-negligible power, the Gaussian residual assumption breaks, and the paper does not test sensitivity to this truncation.

What would settle it

If applied to real PTA data, the framework would fail to distinguish SMBHB from primordial origins if (a) the actual source-amplitude distribution is shallow (high gamma, many faint sources per bin) so that power-spectrum fluctuations are small, and (b) no individual bright sources are detectable, leaving both steps of the inference uninformative.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • If applied to real PTA data, this framework could provide a Bayesian model-comparison verdict on the astrophysical versus primordial origin of the nanohertz GW background without requiring confirmed individual source detections.
  • The decomposition of evidence into per-bin and population-level components allows astronomers to identify which frequency bins contribute most to the finite-source hypothesis, potentially guiding targeted searches for individual SMBHBs at specific frequencies.
  • The framework can constrain the power-law index gamma of the SMBHB strain-amplitude distribution, which encodes information about the astrophysical processes governing binary assembly and evolution in galaxy mergers.
  • Even in regimes where individual sources are too faint to detect (gamma=2.5), the cross-bin fluctuation signal alone retains discriminating power, suggesting the method scales favorably as PTA sensitivity improves over time.

Where Pith is reading between the lines

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

  • The framework's reliance on modeling only the single brightest source per bin means its discriminating power is bounded by the ratio of the brightest to second-brightest source in each bin. If this ratio approaches unity in multiple bins, the Gaussian residual assumption degrades and the method may lose sensitivity without an explicit multi-source extension.
  • The mock PTA configuration (200 pulsars, 30 years, 200 ns white noise) is more capable than current arrays. Applying the framework to present-day data (fewer pulsars, shorter baselines) would likely yield weaker Bayes factors, and the transition point at which real data becomes informative depends on the actual value of gamma, which is currently unknown.
  • The population model uses a single power-law index gamma for the strain-amplitude distribution. Real SMBHB populations may have more complex amplitude distributions shaped by stellar hardening, eccentricity, and environmental coupling, which could bias the recovered gamma or reduce the Bayes factor if the true distribution deviates substantially from a power law.
  • Extending the framework to jointly infer population parameters and nuisance noise parameters (rather than assuming noise is well-measured) would be necessary for real-data applications, as noise misestimation could mimic or mask the power-spectrum fluctuations that carry the finite-population signal.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 6 minor

Summary. This Letter introduces a two-stage hierarchical Bayesian inference framework for distinguishing a finite population of supermassive black hole binaries (SMBHBs) from a Gaussian primordial gravitational wave background (GWB) using pulsar timing array (PTA) data. In the first stage, a free-spectrum reconstruction is performed per frequency bin under two hypotheses: a Gaussian isotropic GWB (H0) or a GWB plus the brightest individual source (H1). In the second stage, the free-spectrum posteriors are used to construct a population-level likelihood comparing an infinite-source model (Hinf) against a finite-source model (Hfin). The framework is demonstrated on mock PTA data (200 pulsars, 30-year observation) generated from a known SMBHB population model, showing that the method recovers injected parameters and yields a Bayes factor of ΔlnZ=22.74 favoring H1+Hfin over H0+Hinf for a strongly non-Gaussian injection (γ=2.2). A mildly non-Gaussian case (γ=2.5) is also examined in the Supplemental Material.

Significance. The hierarchical decomposition of the evidence into free-spectrum (Δ^(1)lnZ) and population-level (Δ^(2)lnZ) contributions (Eq. 10) is a clean and useful formalism. The population PDF derivation (Eqs. 16-20) following the characteristic function approach is mathematically sound. The mock-data recovery of injected parameters (Fig. 3) is self-consistent. The framework's ability to extract population information from power-spectrum fluctuations even when individual sources are not detected (as shown in the γ=2.5 case, Table III) is a genuine strength. The consistency check with free h_min (Fig. 4) is a welcome robustness test. However, the central distinguishing claim—that the framework can separate an SMBHB population from a Gaussian primordial background—is not yet fully substantiated because no false-positive test on Gaussian-background mock data has been performed.

major comments (1)
  1. The central claim that the framework 'can distinguish a finite SMBHB population from a Gaussian primordial GWB' requires a false-positive test that is absent. The paper only tests true-positive scenarios: mock data generated from the Hfin model family (Eqs. 15, 18) with γ=2.2 (Table I) and γ=2.5 (Table III). The γ=2.5 case (ΔlnZ=−3.11 for 15 bins) provides partial evidence that the framework reduces its preference as the signal approaches Gaussian, but this data is still generated from the Hfin model family. A genuine false-positive test would generate data from H0+Hinf (smooth power-law spectrum, no individual sources) and verify that the framework does not prefer H1+Hfin. Without this test, it cannot be ruled out that the preference for Hfin reflects an inherent bias toward the more parameter-rich model (Hfin has the additional parameter γ relative to Hinf). The Occam penalty from the
minor comments (6)
  1. Table I caption: the parenthetical decomposition (Δ^(1)lnZ, Δ^(2)lnZ) is not explicitly defined in the caption; the reader must infer it from Eq. (10).
  2. Fig. 1 lower panel: the y-axis label 'ln Z_i' should be clarified as Δln(Z_{H1,i}/Z_{H0,i}).
  3. Fig. 2 lower panel: the y-axis label 'ln Z_i' should similarly be clarified as the per-bin population-level contribution.
  4. The paper states noise parameters are 'well measured' and held fixed. A brief comment on how this assumption would affect the framework when applied to real data with uncertain noise parameters would be useful.
  5. Reference [74] is cited as a note but appears in the reference list as 'Note1'; this should be formatted as a proper reference or footnote.
  6. The phrase 'semi-analythic' in the Simulations section contains a typo and should read 'semi-analytic'.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for a careful and constructive report. The referee correctly identifies that our central distinguishing claim requires a false-positive test on Gaussian-background mock data, and we agree this test is necessary. We will add it to the revised manuscript.

read point-by-point responses

  1. Referee: The central claim that the framework 'can distinguish a finite SMBHB population from a Gaussian primordial GWB' requires a false-positive test that is absent. The paper only tests true-positive scenarios: mock data generated from the Hfin model family (Eqs. 15, 18) with γ=2.2 (Table I) and γ=2.5 (Table III). The γ=2.5 case (ΔlnZ=−3.11 for 15 bins) provides partial evidence that the framework reduces its preference as the signal approaches Gaussian, but this data is still generated from the Hfin model family. A genuine false-positive test would generate data from H0+Hinf (smooth power-law spectrum, no individual sources) and verify that the framework does not prefer H1+Hfin. Without this test, it cannot be ruled out that the preference for Hfin reflects an inherent bias toward the more parameter-rich model (Hfin has the additional parameter γ relative to Hinf). The Occam penalty from the

    Authors: We fully agree with the referee that a false-positive test is essential to substantiate the central distinguishing claim of our framework. This is a fair and important point. In the revised manuscript, we will generate mock PTA data from the H0+Hinf model (a smooth power-law Gaussian isotropic background with no individual sources, using the same PTA configuration of 200 pulsars and 30-year observation) and run the full two-stage hierarchical inference pipeline on this data. We will report the resulting Bayes factor ΔlnZ(H1+Hfin vs. H0+Hinf) to verify that the framework does not spuriously prefer the more parameter-rich Hfin model when the data are genuinely Gaussian. We expect the Occam penalty to disfavor Hfin in this scenario, but we agree this must be demonstrated rather than assumed. We will add the results of this false-positive test as a new table and discussion in the revised manuscript, and we will temper the language of the central claim until this test confirms the framework's behavior. revision: yes

Circularity Check

0 steps flagged

No significant circularity: the hierarchical Bayesian framework is self-contained, and the mock-data validation is standard proof-of-principle, not a circular derivation.

full rationale

The paper introduces a two-stage hierarchical Bayesian inference framework: (1) free-spectrum reconstruction under H0/H1, and (2) population-level inference under Hinf/Hfin. The population PDF p_pop(S|Hfin,Λ) in Eq. 20 is derived from the power-law distribution g(h²) in Eq. 18 via the characteristic function method (Eq. 20), citing [25-27] for the method. The mock data is generated from the same population model family (Eqs. 15, 18) and then recovered. This is a standard proof-of-principle validation — injecting a known model and checking recovery — not a circular derivation. The framework's mathematical structure (the hierarchical likelihood Eq. 8, the Bayes factor decomposition Eq. 10) is derived independently of the specific mock data values. The ΔlnZ=22.74 result is a measurement on mock data, not a prediction forced by construction. The population model parameters (A_GWB, γ_HD, γ) are free parameters fitted to the mock data, not defined in terms of the data itself. The self-citation to [27] (Xue, Pan, Dai — overlapping authors) for the population PDF method is not load-bearing for circularity because the method is a standard characteristic-function calculation that can be independently verified, and the paper also cites [25, 26] (non-overlapping authors) for the same method. The absence of a false-positive test on Gaussian-background mock data is a correctness/completeness concern, not a circularity issue. The harmonic mean estimator (Eq. 37) is a known computational tool, not a circular definition. No step in the derivation chain reduces to its inputs by construction.

Axiom & Free-Parameter Ledger

7 free parameters · 5 axioms · 0 invented entities

The paper introduces no new physical entities, particles, or forces. It uses standard GR waveforms, standard PTA noise models, and standard SMBHB population models. The free parameters are population-level quantities (A_GWB, γ_HD, γ, h_min/max) that are standard in the SMBHB literature. The key axioms are modeling simplifications (single bright source, Gaussian residual, fixed noise, power-law distribution) that are appropriate for a proof-of-principle but would need relaxation for real-data analysis.

free parameters (7)
  • A_GWB = 2.4e-15 (injected)
    GWB amplitude at reference frequency, a population-level parameter inferred in the hierarchical step.
  • γ_HD = 13/3 (injected)
    Spectral index of the GWB, inferred in the hierarchical step.
  • γ = 2.2 or 2.5 (injected)
    Power-law index of the SMBHB strain-amplitude distribution g(h²), inferred in the hierarchical step.
  • h_min(fyr) = 2e-16 (fixed in main text)
    Lower cutoff of the strain-amplitude distribution; fixed in main text, varied in Supplemental with consistent results.
  • h_max(fyr) = 2e3 × h_min (fixed)
    Upper cutoff of the strain-amplitude distribution; fixed based on observational constraints.
  • A_CURN = 1e-15
    Common-spectrum red noise amplitude, fixed in simulation.
  • γ_CURN = 4.0
    Common-spectrum red noise spectral index, fixed in simulation.
axioms (5)
  • domain assumption The remaining signal after subtracting the brightest SMBHB in each frequency bin can be approximated as a Gaussian and isotropic background.
    Invoked in Eq. 13 and the surrounding text. This is the load-bearing modeling assumption; if the residual is non-Gaussian due to multiple bright sources, the likelihood is misspecified.
  • domain assumption The pulsar term in the SMBHB waveform can be neglected.
    Stated in Supplemental Material §'Response to the Brightest SMBHB'. Justified for this proof-of-principle but would introduce biases in real data analysis.
  • ad hoc to paper Noise parameters (white noise, CURN) are well-measured and can be held fixed during inference.
    Stated in the Inference section: 'we focus on inference of source parameters and assume that the noise parameters have been well measured.' This simplifies computation but is unrealistic for actual PTA data.
  • domain assumption The SMBHB strain-amplitude distribution follows a power law g(h²) = A h^(-2γ).
    Invoked in Eq. 18. The paper acknowledges in Note [74] that the high-amplitude tail universally behaves as h^(-5), but uses a general power law to account for the low-amplitude regime. This is a simplifying model choice.
  • domain assumption SMBHB orbital evolution is governed purely by GW emission.
    Invoked in the Simulations section to set h_min/max(f_i) scaling with frequency. Environmental coupling or stellar hardening would modify this.

pith-pipeline@v1.1.0-glm · 21267 in / 3238 out tokens · 219148 ms · 2026-07-09T09:32:39.342215+00:00 · methodology

0 comments
read the original abstract

The recent detection of a nanohertz gravitational wave (GW) background by pulsar timing arrays (PTA) has sparked extensive discussions regarding its origin-whether it arises from astrophysical supermassive black hole binaries (SMBHBs) or from primordial GWs generated by various early universe processes. Previous studies suggest that a key discriminant between these two origins is the non-Gaussianity of the GW background prior to the detection of any individual source. In this Letter, we introduce a hierarchical Bayesian inference framework for inferring population properties of GW sources. This approach enables not only the measurement of evidence for different GW origins using PTA data but also the inference of population properties of astrophysical SMBHBs, by optimally leveraging non-Gaussian information in individual bright sources and in power spectrum fluctuations of the GW background.

Figures

Figures reproduced from arXiv: 2607.07477 by Jiming Yu, Liang Dai, Xiao Xue, Zhen Pan.

Figure 1
Figure 1. Figure 1: FIG. 1: The upper panel presents the posterior distributions [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: The corner plots for [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Comparison of constraints of population model [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Same as Figs [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗

discussion (0)

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Reference graph

Works this paper leans on

91 extracted references · 91 canonical work pages · 72 internal anchors

  1. [1]

    package anddynesty[76] sampler to explore the pa- rameter space, with 2000 live points and a threshold of 10000 maximum MCMC steps. Fig. 1 shows the free spectrum reconstruction for𝛾=2.2. Nonzeroℎ c is favored in the lowest∼14 frequency bins, whereas the higher-frequency bins mainly yield upper lim- its where the white noise dominates over the signal. In ...

    work page doi:10.13039/501100011033 2000

  2. [2]

    F. B. Estabrook and H. D. Wahlquist, General Relativity and Gravitation6, 439 (1975)

    work page 1975

  3. [3]

    M. V. Sazhin, Soviet Ast.22, 36 (1978)

    work page 1978

  4. [4]

    Detweiler, Astrophys

    S. Detweiler, Astrophys. J.234, 1100 (1979)

    work page 1979

  5. [5]

    R. S. Foster and D. C. Backer, Astrophys. J.361, 300 (1990)

    work page 1990

  6. [6]

    The NANOGrav 15-year Data Set: Evidence for a Gravitational-Wave Background

    G. Agazie, A. Anumarlapudi, A. M. Archibald, Z. Arzoumanian, P. T. Baker, B. B ´ecsy, L. Blecha, A. Brazier, P. R. Brook, S. Burke-Spolaor, et al., ApJ951, L8 (2023), 2306.16213

    work page internal anchor Pith review Pith/arXiv arXiv 2023

  7. [7]

    The second data release from the European Pulsar Timing Array III. Search for gravitational wave signals

    EPTA Collaboration, InPTA Collaboration, J. Antoniadis, P. Arumugam, S. Arumugam, S. Babak, M. Bagchi, A. S. Bak Nielsen, C. G. Bassa, A. Bathula, et al., A&A678, A50 (2023), 2306.16214

    work page internal anchor Pith review Pith/arXiv arXiv 2023

  8. [8]

    D. J. Reardon, A. Zic, R. M. Shannon, G. B. Hobbs, M. Bailes, V. Di Marco, A. Kapur, A. F. Rogers, E. Thrane, J. Askew, et al., ApJ951, L6 (2023), 2306.16215

    work page internal anchor Pith review Pith/arXiv arXiv 2023

  9. [9]

    H. Xu, S. Chen, Y. Guo, J. Jiang, B. Wang, J. Xu, Z. Xue, R. Nicolas Caballero, J. Yuan, Y. Xu, et al., Research in Astron- omy and Astrophysics23, 075024 (2023), 2306.16216

    work page internal anchor Pith review Pith/arXiv arXiv 2023

  10. [10]

    M. T. Miles, R. M. Shannon, M. Bailes, D. J. Reardon, M. J. 6 Keith, A. D. Cameron, A. Parthasarathy, M. Shamohammadi, R. Spiewak, W. van Straten, et al., Mon. Not. R. Astron. Soc. 519, 3976 (2023), 2212.04648

    work page internal anchor Pith review Pith/arXiv arXiv 2023

  11. [11]

    M. T. Miles, R. M. Shannon, D. J. Reardon, M. Bailes, D. J. Champion, M. Geyer, P. Gitika, K. Grunthal, M. J. Keith, M. Kramer, et al., Mon. Not. R. Astron. Soc.536, 1467 (2025), 2412.01148

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  12. [12]

    R. W. Hellings and G. S. Downs, ApJ265, L39 (1983)

    work page 1983

  13. [13]

    Ultra-Low Frequency Gravitational Radiation from Massive Black Hole Binaries

    M. Rajagopal and R. W. Romani, Astrophys. J.446, 543 (1995), astro-ph/9412038

    work page internal anchor Pith review Pith/arXiv arXiv 1995

  14. [14]

    Gravitational Waves from Supermassive Black Hole Coalescence in a Hierarchical Galaxy Formation Model

    M. Enoki, K. T. Inoue, M. Nagashima, and N. Sugiyama, As- trophys. J.615, 19 (2004), astro-ph/0404389

    work page internal anchor Pith review Pith/arXiv arXiv 2004

  15. [15]

    Low-frequency gravitational radiation from coalescing massive black hole binaries in hierarchical cosmologies

    A. Sesana, F. Haardt, P. Madau, and M. Volonteri, Astrophys. J. 611, 623 (2004), astro-ph/0401543

    work page internal anchor Pith review Pith/arXiv arXiv 2004

  16. [16]

    L. Z. Kelley, L. Blecha, and L. Hernquist, Mon. Not. R. Astron. Soc.464, 3131 (2017), 1606.01900

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  17. [17]

    P. C. Peters and J. Mathews, Physical Review131, 435 (1963)

    work page 1963

  18. [18]

    P. C. Peters, Physical Review136, 1224 (1964)

    work page 1964

  19. [19]

    Maggiore, Gravitational Waves, Vol

    M. Maggiore, Gravitational Waves, Vol. 1 (Oxford University Press, 2007)

    work page 2007

  20. [21]

    C. M. F. Mingarelli, T. Sidery, I. Mandel, and A. Vecchio, Phys. Rev. D88, 062005 (2013), 1306.5394

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  21. [22]

    The NANOGrav 15-year Data Set: Search for Signals from New Physics

    A. Afzal et al. (NANOGrav), Astrophys. J. Lett.951, L11 (2023), [Erratum: Astrophys.J.Lett. 971, L27 (2024), Erratum: Astro- phys.J. 971, L27 (2024)], 2306.16219

    work page internal anchor Pith review Pith/arXiv arXiv 2023

  22. [23]

    The spatial correlations between pulsars for interfering sources in Pulsar Timing Array and evidence for gravitational-wave background in NANOGrav 15-year data set

    Y.-M. Wu, Y.-C. Bi, and Q.-G. Huang, Phys. Rev. D112, 123034 (2025), 2407.07319

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  23. [24]

    Raidal, J

    J. Raidal, J. Urrutia, V. Vaskonen, and H. Veerm ¨ae, Astron. Astrophys.706, A159 (2026), 2411.19692

    work page arXiv 2026

  24. [25]

    Gravitational Waves from SMBH Binaries in Light of the NANOGrav 15-Year Data

    J. Ellis, M. Fairbairn, G. H¨ utsi, J. Raidal, J. Urrutia, V. Vasko- nen, and H. Veerm ¨ae, Phys. Rev. D109, L021302 (2024), 2306.17021

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  25. [26]

    Prospects for Future Binary Black Hole GW Studies in Light of PTA Measurements

    J. Ellis, M. Fairbairn, G. H¨ utsi, M. Raidal, J. Urrutia, V. Vasko- nen, and H. Veerm ¨ae, Astron. Astrophys.676, A38 (2023), 2301.13854

    work page internal anchor Pith review Pith/arXiv arXiv 2023

  26. [27]

    The distribution of the gravitational-wave background from supermassive black holes

    G. Sato-Polito and M. Zaldarriaga, Phys. Rev. D111, 023043 (2025), 2406.17010

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  27. [28]

    X. Xue, Z. Pan, and L. Dai, Phys. Rev. D111, 043022 (2025), 2409.19516

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  28. [29]

    W. G. Lamb and S. R. Taylor, ApJ971, L10 (2024), 2407.06270

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  29. [30]

    From eccentric binaries to nonstationary gravitational wave backgrounds

    M. Falxa, H. Quelquejay Leclere, and A. Sesana, Phys. Rev. D 111, 023047 (2025), 2412.01899

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  30. [31]

    A practical theorem on gravitational-wave background statistics

    Y. Ali-Ha¨ımoud (2026), 2604.19701

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  31. [32]

    Higher-order statistics of the stochastic gravitational wave background from supermassive black hole binaries

    H. Hisamatsu and K. Kyutoku (2026), 2605.17983

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  32. [33]

    Cosmic Variance in Anisotropy Searches at Pulsar Timing Arrays

    V. Domcke, G. Franciolini, and M. Pieroni (2025), 2508.21131

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  33. [34]

    The impact of cosmic variance on PTAs anisotropy searches

    T. Konstandin, A.-M. Lemke, A. Mitridate, and E. Perboni, JCAP04, 059 (2025), 2408.07741

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  34. [36]

    The stochastic gravity-wave background: sources and detection

    B. Allen, in Relativistic Gravitation and Gravitational Radiation, edited by J.-A. Marck and J.-P. Lasota (1997), pp. 373–417, gr-qc/9604033

    work page internal anchor Pith review Pith/arXiv arXiv 1997

  35. [37]

    Detecting a stochastic background of gravitational radiation: Signal processing strategies and sensitivities

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

    work page internal anchor Pith review Pith/arXiv arXiv 1999

  36. [38]

    J. D. Romano and N. J. Cornish, Living Reviews in Relativity 20, 2 (2017), 1608.06889

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  37. [39]

    Cosmic variance in the nanohertz gravitational wave background

    E. Roebber, G. Holder, D. E. Holz, and M. Warren, Astrophys. J.819, 163 (2016), 1508.07336

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  38. [40]

    Variance of the Hellings-Downs Correlation

    B. Allen, Phys. Rev. D107, 043018 (2023), 2205.05637

    work page internal anchor Pith review Pith/arXiv arXiv 2023

  39. [41]

    What is the source of the PTA GW signal?

    J. Ellis, M. Fairbairn, G. Franciolini, G. H¨ utsi, A. Iovino, M. Lewicki, M. Raidal, J. Urrutia, V. Vaskonen, and H. Veerm¨ae, Phys. Rev. D109, 023522 (2024), 2308.08546

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  40. [42]

    S. C. Sardesai, J. Simon, and S. J. Vigeland, Astrophys. J.976, 212 (2024), 2408.10139

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  41. [43]

    R. C. Bernardo, S. Appleby, and K.-W. Ng, J. Cosmology As- tropart. Phys.2025, 017 (2025), 2407.17987

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  42. [44]

    The NANOGrav 15 yr Data Set: Looking for Signs of Discreteness in the Gravitational-wave Background

    G. Agazie, A. Anumarlapudi, A. M. Archibald, Z. Arzoumanian, J. G. Baier, P. T. Baker, B. B ´ecsy, L. Blecha, A. Brazier, P. R. Brook, et al., Astrophys. J.978, 31 (2025), 2404.07020

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  43. [45]

    W. G. Lamb, J. M. Wachter, A. Mitridate, S. C. Sardesai, B. B ´ecsy, E. L. Hagen, S. R. Taylor, and L. Z. Kelley (2025), 2511.09659

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  44. [46]

    S. R. Taylor and J. R. Gair, Phys. Rev. D88, 084001 (2013), 1306.5395

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  45. [47]

    Exploring realistic nanohertz gravitational-wave backgrounds

    B. B ´ecsy, N. J. Cornish, and L. Z. Kelley, Astrophys. J.941, 119 (2022), 2207.01607

    work page internal anchor Pith review Pith/arXiv arXiv 2022

  46. [48]

    The NANOGrav 15-year Data Set: Bayesian Limits on Gravitational Waves from Individual Supermassive Black Hole Binaries

    G. Agazie, A. Anumarlapudi, A. M. Archibald, Z. Arzoumanian, P. T. Baker, B. B ´ecsy, L. Blecha, A. Brazier, P. R. Brook, S. Burke-Spolaor, et al., ApJ951, L50 (2023), 2306.16222

    work page internal anchor Pith review Pith/arXiv arXiv 2023

  47. [50]

    The NANOGrav 11-Year Data Set: Limits on Gravitational Waves from Individual Supermassive Black Hole Binaries

    K. Aggarwal et al., Astrophys. J.880, 2 (2019), 1812.11585

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  48. [51]

    Fast Bayesian analysis of individual binaries in pulsar timing array data

    B. B ´ecsy, N. J. Cornish, and M. C. Digman, Phys. Rev. D105, 122003 (2022), 2204.07160

    work page internal anchor Pith review Pith/arXiv arXiv 2022

  49. [52]

    The NANOGrav 12.5-year data set: A computationally efficient eccentric binary search pipeline and constraints on an eccentric supermassive binary candidate in 3C 66B

    G. Agazie et al. (NANOGrav), Astrophys. J.963, 144 (2024), 2309.17438

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  50. [53]

    The second data release from the European Pulsar Timing Array V. Search for continuous gravitational wave signals

    J. Antoniadis et al. (EPTA, InPTA), Astron. Astrophys.690, A118 (2024), 2306.16226

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  51. [54]

    Zhao et al., Astrophys

    S.-Y. Zhao et al., Astrophys. J.992, 181 (2025), 2508.13944

    work page arXiv 2025

  52. [55]

    The MeerKAT Pulsar Timing Array: Maps of the gravitational-wave sky with the 4.5 year data release

    K. Grunthal, R. S. Nathan, E. Thrane, D. J. Champion, M. T. Miles, R. M. Shannon, A. D. Kulkarni, F. Abbate, S. Buchner, A. D. Cameron, et al., Mon. Not. R. Astron. Soc.536, 1501 (2025), 2412.01214

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  53. [56]

    Hyper-efficient model-independent Bayesian method for the analysis of pulsar timing data

    L. Lentati, P. Alexander, M. P. Hobson, S. Taylor, J. Gair, S. T. Balan, and R. van Haasteren, Phys. Rev. D87, 104021 (2013), 1210.3578

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  54. [57]

    TempoNest: A Bayesian approach to pulsar timing analysis

    L. Lentati, P. Alexander, M. P. Hobson, F. Feroz, R. van Haasteren, K. J. Lee, and R. M. Shannon, Mon. Not. R. As- tron. Soc.437, 3004 (2014), 1310.2120

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  55. [58]

    A. D. Johnson, P. M. Meyers, P. T. Baker, N. J. Cornish, J. S. Hazboun, T. B. Littenberg, J. D. Romano, S. R. Taylor, M. Val- lisneri, S. J. Vigeland, et al., Phys. Rev. D109, 103012 (2024), 2306.16223

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  56. [59]

    Planck 2018 results. VI. Cosmological parameters

    Planck Collaboration, N. Aghanim, Y. Akrami, M. Ashdown, J. Aumont, C. Baccigalupi, M. Ballardini, A. J. Banday, R. B. Barreiro, N. Bartolo, et al., A&A641, A6 (2020), 1807.06209

    work page internal anchor Pith review Pith/arXiv arXiv 2020

  57. [60]

    S. R. Taylor, arXiv e-prints arXiv:2105.13270 (2021), 2105.13270

    work page internal anchor Pith review Pith/arXiv arXiv 2021

  58. [61]

    E. S. Phinney, arXiv e-prints astro-ph/0108028 (2001), astro- ph/0108028

    work page internal anchor Pith review Pith/arXiv arXiv 2001

  59. [62]

    A. H. Jaffe and D. C. Backer, Astrophys. J.583, 616 (2003), astro-ph/0210148

    work page internal anchor Pith review Pith/arXiv arXiv 2003

  60. [63]

    J. S. B. Wyithe and A. Loeb, Astrophys. J.590, 691 (2003), astro-ph/0211556

    work page internal anchor Pith review Pith/arXiv arXiv 2003

  61. [64]

    An introduction to Bayesian inference in gravitational-wave astronomy: parameter estimation, model selection, and hierarchical models

    E. Thrane and C. Talbot, PASA36, e010 (2019), 1809.02293

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  62. [65]

    Extracting distribution parameters from multiple uncertain observations with selection biases

    I. Mandel, W. M. Farr, and J. R. Gair, Mon. Not. R. Astron. Soc. 486, 1086 (2019), 1809.02063

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  63. [66]

    Vitale, D

    S. Vitale, D. Gerosa, W. M. Farr, and S. R. Tay- lor, in Handbook of Gravitational Wave Astronomy, edited by C. Bambi, S. Katsanevas, and K. D. Kokkotas (2022), p. 45

    work page 2022

  64. [67]

    The population of merging compact binaries inferred using gravitational waves through GWTC-3

    R. Abbott, T. D. Abbott, F. Acernese, K. Ackley, C. Adams, N. Adhikari, R. X. Adhikari, V. B. Adya, C. Affeldt, D. Agarwal, 7 et al., Physical Review X13, 011048 (2023), 2111.03634

    work page internal anchor Pith review Pith/arXiv arXiv 2023

  65. [68]

    The stochastic gravitational-wave background from massive black hole binary systems: implications for observations with Pulsar Timing Arrays

    A. Sesana, A. Vecchio, and C. N. Colacino, Mon. Not. R. Astron. Soc.390, 192 (2008), 0804.4476

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  66. [69]

    Gravitational waves from resolvable massive black hole binary systems and observations with Pulsar Timing Arrays

    A. Sesana, A. Vecchio, and M. Volonteri, Mon. Not. R. Astron. Soc.394, 2255 (2009), 0809.3412

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  67. [70]

    Systematic investigation of the expected gravitational wave signal from supermassive black hole binaries in the pulsar timing band

    A. Sesana, Mon. Not. R. Astron. Soc.433, L1 (2013), 1211.5375

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  68. [71]

    Y. Chen, Q. Yu, and Y. Lu, Astrophys. J.897, 86 (2020), 2005.10818

    work page internal anchor Pith review Pith/arXiv arXiv 2020

  69. [72]

    Exploring Proxies for the Supermassive Black Hole Mass Function: Implications for Pulsar Timing Arrays

    J. Simon, ApJ949, L24 (2023), 2306.01832

    work page internal anchor Pith review Pith/arXiv arXiv 2023

  70. [73]

    The NANOGrav 15-year Data Set: Constraints on Supermassive Black Hole Binaries from the Gravitational Wave Background

    G. Agazie, A. Anumarlapudi, A. M. Archibald, P. T. Baker, B. B ´ecsy, L. Blecha, A. Bonilla, A. Brazier, P. R. Brook, S. Burke-Spolaor, et al., ApJ952, L37 (2023), 2306.16220

    work page internal anchor Pith review Pith/arXiv arXiv 2023

  71. [74]

    A. Lapi, F. Shankar, M. Bosi, D. Roberts, H. Fu, K. M. Varadara- jan, and L. Boco, J. Cosmology Astropart. Phys.2026, 001 (2026), 2507.15436

    work page arXiv 2026

  72. [75]

    [79] shows the universal behavior,𝑔(ℎ 2) → (ℎ 2) −5/2, independent of population models

    Note1, in the high-amplitude tail, Ref. [79] shows the universal behavior,𝑔(ℎ 2) → (ℎ 2) −5/2, independent of population models. We use a universal power law index that may be different from 2.5 for taking the low-amplitude regime into account

    work page

  73. [76]

    Bilby: A user-friendly Bayesian inference library for gravitational-wave astronomy

    G. Ashton, M. H¨ ubner, P. D. Lasky, C. Talbot, K. Ackley, S. Biscoveanu, Q. Chu, A. Divakarla, P. J. Easter, B. Goncharov, et al., ApJS241, 27 (2019), 1811.02042

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  74. [77]

    J. S. Speagle, Mon. Not. R. Astron. Soc.493, 3132 (2020), 1904.02180

    work page internal anchor Pith review Pith/arXiv arXiv 2020

  75. [78]

    Y. Chen, Q. Yu, and Y. Lu, Astrophys. J.974, 261 (2024), 2409.18029

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  76. [79]

    Host Galaxy Demographics Of Individually Detectable Supermassive Black-hole Binaries with Pulsar Timing Arrays

    K. Cella, S. R. Taylor, and L. Zoltan Kelley, Classical and Quantum Gravity42, 025021 (2025), 2407.01659

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  77. [80]

    The Heavy Tailed Non-Gaussianity of the Supermassive Black Hole Gravitational Wave Background

    J. Raidal, J. Urrutia, V. Vaskonen, and H. Veerm ¨ae (2026), 2604.08506

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  78. [81]

    A Joint Optimal Search for Gravitational Waves from Resolved and Unresolved Supermassive Binary Black Holes with Pulsar Timing Arrays

    B. Goncharov, G. Sato-Polito, X. Bi, and M. Zaldarriaga (2026), 2606.18241

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  79. [82]

    Astropy Collaboration, A. M. Price-Whelan, P. L. Lim, N. Earl, N. Starkman, L. Bradley, D. L. Shupe, A. A. Patil, L. Corrales, C. E. Brasseur, et al., Astrophys. J.935, 167 (2022), 2206.14220

    work page internal anchor Pith review Pith/arXiv arXiv 2022

  80. [83]

    Foreman-Mackey, The Journal of Open Source Software1, 24 (2016), URLhttps://doi.org/10.21105/joss.00024

    D. Foreman-Mackey, The Journal of Open Source Software1, 24 (2016), URLhttps://doi.org/10.21105/joss.00024

    work page doi:10.21105/joss.00024 2016

Showing first 80 references.