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The NANOGrav 15 yr Data Set: Harmonic Analysis of the Pulsar Angular Correlations

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Pith's one-line read A flexible shape analysis of 15 years of pulsar timing data finds the quadrupolar angular correlations predicted by general relativity, but a monopolar signal near 4 nHz substantially weakens the quadrupole evidence when it is included.

desk verdict Clean, careful Bayesian harmonic analysis of the NG15 angular correlations; the quadrupole result is solid, but the headline monopole-induced evidence drop is prior-dependent and untested. read the letter →

arxiv 2411.13472 v1 pith:4GSXXZ6F submitted 2024-11-20 astro-ph.HE astro-ph.GAgr-qc

Gabriella Agazie , Jeremy G. Baier , Paul T. Baker , Bence Becsy , Laura Blecha , Kimberly K. Boddy , Adam Brazier , Paul R. Brook
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Sarah Burke-Spolaor Rand Burnette J. Andrew Casey-Clyde Maria Charisi Shami Chatterjee Tyler Cohen James M. Cordes Neil J. Cornish Fronefield Crawford H. Thankful Cromartie Megan E. DeCesar Paul B. Demorest Heling Deng Lankeswar Dey Timothy Dolch Elizabeth C. Ferrara William Fiore Emmanuel Fonseca Gabriel E. Freedman Emiko C. Gardiner Kyle A. Gersbach Joseph Glaser Deborah C. Good Kayhan Gultekin Jeffrey S. Hazboun Ross J. Jennings Aaron D. Johnson David L. Kaplan Luke Zoltan Kelley Joey S. Key Nima Laal Michael T. Lam William G. Lamb Bjorn Larsen T. Joseph W. Lazio Natalia Lewandowska Tingting Liu Jing Luo Ryan S. Lynch Chung-Pei Ma Dustin R. Madison Alexander McEwen James W. McKee Maura A. McLaughlin Patrick M. Meyers Chiara M. F. Mingarelli Andrea Mitridate Jonathan Nay David J. Nice Stella Koch Ocker Ken D. Olum Timothy T. Pennucci Polina Petrov Nihan S. Pol Henri A. Radovan Scott M. Ransom Paul S. Ray Jessie C. Runnoe Alexander Saffer Shashwat C. Sardesai Kai Schmitz Xavier Siemens Joseph Simon Magdalena S. Siwek Tristan L. Smith Sophia V. Sosa Fiscella Ingrid H. Stairs Daniel R. Stinebring Abhimanyu Susobhanan Joseph K. Swiggum Jacob Taylor Stephen R. Taylor Jacob E. Turner Caner Unal Michele Vallisneri Rutger van Haasteren Joris Verbiest Sarah J. Vigeland Caitlin A. Witt David Wright Olivia Young
This is my paper · ORCID
classification astro-ph.HEastro-ph.GAgr-qc
keywords gravitationalwavebackgroundpulsartimingarrayHellings-DownscorrelationsLegendrepolynomialsangularquadrupolemonopoleBayesfactor
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

The paper asks whether the angular correlations seen across pulsar pairs in 15 years of timing data really have the quadrupolar shape that general relativity predicts for a gravitational-wave background. It expands the correlation function in Legendre polynomials and lets the data choose the multipole coefficients. When only multipoles with ℓ ≥ 2 are included, the data show a quadrupole with amplitude consistent with the Hellings-Downs curve and a Bayes factor of 200 against an uncorrelated common-noise model. When monopole and dipole terms are added, the quadrupole evidence drops by more than an order of magnitude because the data prefer a monopolar signal at about 4 nHz. The headline verification of the gravitational-wave background's shape therefore comes with an unexplained monopole that must be understood before the quadrupole interpretation is secure.

What carries the argument

The load-bearing object is the Legendre-polynomial expansion of the angular-correlation function, Γab = (1 − δab) ∑_{ℓ=2}^{ℓmax} cℓ Pℓ(cos θab) + δab, which separates each pulsar's auto-correlation, fixed to one, from the cross-correlations between distinct pulsar pairs. The coefficients cℓ are the quantities constrained by the data, with the Hellings-Downs curve corresponding to c2 = 0.3125 and higher multipoles falling as $ℓ^{{-3}}$. To keep the covariance positive definite, the analysis imposes cℓ ≥ 0 and ∑ cℓ ≤ 1, and it models monopole and dipole terms with a separate free-spectrum parameter per frequency bin because those terms are not expected to share the gravitational-wave power spectrum. The argument works by comparing Bayes factors between models with different sets of cℓ and by reconstructing the full correlation function from the posterior distributions of c2 and c3.

What would settle it

Re-running the harmonic analysis on the same 15-year data with the auto-correlation left free and negative Legendre coefficients allowed; if the quadrupole Bayes factor then stays near 200 even when a monopole is included, the 4 nHz monopole is an artifact of the prior, not a signal in the data.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is that the 15-year pulsar timing array data, analyzed with a flexible Legendre expansion, confirm the quadrupolar angular correlations predicted by general relativity and reject multipoles ℓ ≥ 4, while also revealing that the evidence for the quadrupole depends on whether a monopolar component is modeled. The quadrupole coefficient is measured at c2/cHD2 = 1.088+0.32−0.45, nonzero at about 95% confidence and consistent with the HD value; the Bayes factor for quadrupole-only correlations over common uncorrelated red noise is 200, matching the earlier fixed-shape HD analysis. Adding a monopole free-spectrum model reduces the Savage-Dickey Bayes factor for the quadrupole from 90 to 5, driven by power at the second frequency bin, about 4 nHz, a signal with no accepted explanation. The paper therefore establishes both the expected quadrupole and a separate low-multipole anomaly that limits how strongly the quadrupole interpretation can be claimed.

Load-bearing premise

The argument assumes that each pulsar's auto-correlation is exactly one and that the Legendre coefficients are non-negative and sum to at most one; if the true auto-correlation is not one, the inferred quadrupole amplitude and the monopole's effect on the quadrupole evidence could be different.

Editorial extensions

If this is right

  • If the quadrupole measurement is right, the pulsar-pair angular correlations independently confirm the Hellings-Downs shape expected from a gravitational-wave background, with the HD value inside the 68% credible interval.
  • Multipoles with ℓ ≥ 4 are not needed to explain the data, and the octupole shows only mild evidence, so the angular power spectrum is effectively quadrupole-dominated.
  • The jump in quadrupole evidence from the 12.5-year to the 15-year data set comes mostly from longer observing time on existing pulsars, not from the added pulsars, matching the predicted scaling with observation time.
  • The about-4-nHz monopolar signal, if real, is a separate physical or systematic effect that must be identified before the quadrupole detection can be taken at face value.

Reading between the lines

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

  • If the monopole persists in future data sets, a clean test is whether its amplitude grows with observation time and whether it appears in other pulsar timing array data sets.
  • The prior that fixes the auto-correlation to unity and forbids negative multipole coefficients is a modeling choice; relaxing it could change the monopole-quadrupole tradeoff and should be explored before concluding the monopole is astrophysical.
  • A basis that is statistically orthogonal to the HD curve could separate the monopole effect from the quadrupole more cleanly than Legendre polynomials, since the monopole and quadrupole appear correlated in this fit.
  • The monopole's frequency dependence, peaking near 4 nHz rather than following the gravitational-wave power law, offers a way to distinguish it from a genuine gravitational-wave polarization effect.
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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

4 major / 6 minor

Summary. The paper applies a Legendre-polynomial 'harmonic analysis' to the NANOGrav 15-year pulsar-timing data set, using the full PTA likelihood with the standard NANOGrav noise models. The angular cross-correlation is expanded as sum_{ℓ} c_ℓ P_ℓ, with the auto-correlation fixed to unity (Eq. 9) and with priors on c_ℓ constrained by positivity and positive-definiteness (Eq. 14). The main results are: a Bayes factor of ~200 for a quadrupole-only model over a common-uncorrelated-red-noise model, a measured quadrupole amplitude c2/cHD2 = 1.088(+0.32, -0.45) consistent with the Hellings-Downs value, no evidence for multipoles ℓ >= 4, and a reduction of the quadrupole evidence by more than an order of magnitude when a monopole free-spectrum model is added, with monopolar power at approximately 4 nHz. The paper also compares the harmonic-analysis results with the optimal-statistic (MCOS) and spline-based angular-correlation reconstructions.

Significance. If the central claims survive scrutiny, the paper provides a useful and more flexible characterization of the angular correlations in the NANOGrav 15-year data set, complementing the standard HD-curve analysis. Its strengths are that it uses the full PTA likelihood rather than an approximate estimator, reports standard convergence diagnostics (Gelman-Rubin R-1 < 0.05), and compares its results to MCOS and spline reconstructions. The quadrupole measurement is not circular: c2 is a free parameter whose posterior is compared with the externally derived HD coefficient. However, the headline claim that a monopolar signal reduces the quadrupole evidence is conditional on the specific prior and parameterization choices in Eqs. (9) and (14), and the paper does not currently test whether that conclusion is robust to those choices. Because the paper's main new conclusion is the monopole-induced reduction of quadrupole evidence, this prior sensitivity is load-bearing rather than cosmetic.

major comments (4)
  1. [§3.1, Eq. (14); Table 3] The effective prior on each Legendre coefficient is p(x) = N_l (1-x)^{N_l-1}, which follows from requiring c_l >= 0 and sum c_l <= 1. For models with more multipoles this prior becomes strongly concentrated near zero; for N_l=4 the prior density at zero is 4 and the prior mean is 0.2. The sequence of Bayes factors 200, 55, 0.9, 0.1 in Table 3 as higher multipoles are added therefore reflects not only whether the data demand an octupole or higher multipoles but also the increasing prior-volume penalty on nonzero coefficients. The conclusion that there is 'no evidence for multipoles ℓ>=4' is thus conditional on this particular shrinkage prior. Please report a robustness check with an alternative prior (for example, independent uniform priors on each c_l with positivity enforced only at the covariance level, or a prior on the correlation matrix that does not impose the same simplex shrinkage) and state whether the evidence ordering in Table 3 survives.
  2. [§3.4, Figures 3 and 4; abstract] The abstract states that including multipoles ℓ<=1 reduces the quadrupole Bayes factor 'due to evidence for a monopolar signal at approximately 4 nHz'. This causal statement is not established by the presented model comparison. In HAγ(c2)+MONOfree, the monopole is implemented with five additional free-spectrum amplitudes log10 Φ(f_i) with wide uniform priors, while the quadrupole is modeled as a power-law. Adding flexible extra components will generically reduce the evidence for a fixed-form component even if the extra components are not truly present, and the observed drop in the c2 Savage-Dickey factor from 90 to 5 is therefore not by itself evidence that the data contain a physical monopole. To support the abstract's causal claim, the authors should show that the evidence drop persists when (a) the quadrupole is also allowed a free spectrum, or (b) the monopole is modeled with a single power law with a prior on its spectral index, or (c) the angular correlation is fit with unconstrained angular bins instead of Legendre polynomials. Without such a check, the central conclusion is conditional on the prior and parameterization choices.
  3. [§4, Figure 5] The comparison with the MCOS in Figure 5 is not apples-to-apples because the harmonic analysis imposes c_l >= 0 and sum c_l <= 1, while the MCOS allows negative coefficients and the authors state they 'only show positive values for the MCOS results for simplicity'. The opposite signs of the c0-c2 correlation between the two methods (positive for MCOS, negative for harmonic analysis) are attributed in the text to the harmonic-analysis constraints, which is precisely a prior effect rather than a data-driven feature. For the claim that the monopole has a larger impact on the quadrupole posterior in the harmonic analysis than in the MCOS, the paper should either repeat the MCOS with the same non-negativity and sum constraints, or repeat the harmonic analysis without the simplex prior, so that the comparison isolates the data content from the prior support.
  4. [§3.2 and §3.4; Eq. (13)] The paper reports a hypermodel Bayes factor HAγ(c2)/CURNγ ≈ 200 (Table 3) but a Savage-Dickey Bayes factor for c2 of 90 (Section 3.4). For the nested model comparison in which c2=0 reduces HAγ(c2) to CURNγ and the prior density at c2=0 is unity (Eq. 13 with c_l in [0,1] and N_l=1), the Savage-Dickey factor should equal the Bayes factor up to sampling noise. The discrepancy of more than a factor of two is not discussed. Please clarify how these two numbers are defined (for example, different priors, different treatment of the spectral-index parameter, or convergence issues) and state which one corresponds to the 'Bayes factor of 200' quoted in the abstract.
minor comments (6)
  1. [§3.1, Table 2 and Eq. (14)] Table 2 lists c_l as U[0,1], but Eq. (14) shows that the actual prior is not uniform when more than one multipole is present. Please clarify that Table 2 gives the nominal support and that Eq. (14) is the effective marginal prior.
  2. [Abstract and §3.4] The abstract and Section 3.4 refer to including 'multipoles ℓ<=1', but the models in Table 1 implement monopole and dipole as separate free-spectrum models (Γ=1 and Γ=cosθ) rather than as Legendre coefficients c0 and c1 in Eq. (9). The only place c0 and c1 appear in a Legendre expansion is the MCOS comparison in Section 4. Please make the relationship between these two implementations explicit, as the current wording suggests the same ℓ<=1 multipoles are being added to Eq. (9).
  3. [§4] In the sentence describing the Gersbach et al. (2024) method, 'proscription' should be 'prescription'.
  4. [§5] In the discussion of the MCOS, 'frequentest' should be 'frequentist'.
  5. [§3.1] The sentence near Eq. (14) contains a grammatical error: 'the red-process a covariance matrix' should be 'the red-process covariance matrix'.
  6. [References] The Anholm et al. (2009) reference appears twice with identical bibliographic information; please remove the duplicate.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the measured quadrupole coefficient is a free fit compared with the externally derived Hellings-Downs coefficient.

full rationale

The paper's central claims are data-driven measurements, not quantities forced by the model. Equation 9 parameterizes the cross-correlation as a free Legendre expansion with coefficients c_l, and the analysis fits c_2 to the NANOGrav 15-year likelihood; the comparison value c_2^HD = 0.3125 is computed analytically from the general-relativity Hellings-Downs prediction in Equation 8, which is independent of the fit. The prior in Equation 14 is an explicit positive-definiteness constraint on the Gaussian-process covariance, and the paper discloses its effect on the parameter space; it does not encode the quadrupole amplitude or the HD curve. The only notable self-citation, to Nay et al. (2024), supplies the harmonic-analysis framework and the Savage-Dickey evidence prescription, but that methodology does not contain the measured c_2 or the monopole evidence drop, which are recomputed here on the NG15 data. The reduction of quadrupole evidence when a monopole free spectrum is added is an empirical model-comparison result conditional on the stated priors; its sensitivity to the positivity constraint is a legitimate robustness concern, but it is not a circular reduction of the output to the input.

Assumptions & free parameters 7 free parameters · 6 assumptions · 0 invented entities

The central measured parameters are the Legendre coefficients c_l; A_gw, gamma_gw, and the noise amplitudes are standard spectrum parameters fit simultaneously. The free-spectrum monopole and dipole amplitudes are auxiliary but important for the monopole claim. No new physical entities are introduced; the monopolar signal is an empirical feature of the data. Most axioms are standard pulsar timing array modeling assumptions drawn from the cited literature.

free parameters (7)
  • c2 (quadrupole Legendre coefficient) = 0.34 mean; 68% CL [0.20, 0.44]; 95% CL [0.11, 0.58]
    Central measured quantity in the harmonic analysis; fit to the NG15 data set.
  • c3, c4, c5 (higher multipole Legendre coefficients) = constrained; no significant detection
    Higher-multipole coefficients in the HA models; posteriors are consistent with zero and reduce the Bayes factor through the Occam penalty.
  • log10 A_gw (GWB strain amplitude) = not quoted in the text
    GWB power-law amplitude fit jointly with the c_l coefficients; standard model parameter.
  • gamma_gw (GWB spectral index) = not quoted in the text
    GWB power-law spectral index fit jointly with the c_l coefficients; prior U[0, 7].
  • Monopole free-spectrum amplitudes log10 Phi(f_i) = power in the second bin at about 4 nHz
    Five frequency-bin amplitudes for the MONO_free model; the second bin has a non-zero posterior.
  • Dipole free-spectrum amplitudes log10 Phi(f_i) = consistent with zero
    Five frequency-bin amplitudes for the DIP_free model; no significant dipole power found.
  • Pulsar intrinsic red noise parameters (per pulsar) = marginalized; not reported individually
    A_RN,a and gamma_RN,a for each pulsar are included in every analysis as noise parameters; they are auxiliary to the central claim.
assumptions (6)
  • domain assumption An isotropic stochastic gravitational wave background in general relativity produces the Hellings and Downs angular correlation, expanded as Legendre polynomials with coefficients c_HD_l = 3/2 (2l+1)(l-2)!/(l+2)! for l >= 2.
    Used as the external theoretical benchmark in Eq. 7 and Figure 1; the comparison of c2 to c_HD_2 is the core consistency test.
  • domain assumption The pulsar timing residuals and the correlated red-noise processes are Gaussian with the covariance structure of Eqs. 1 through 5.
    Standard pulsar timing array likelihood from Johnson et al. 2023; all inferences depend on this assumed likelihood.
  • domain assumption For multipoles l >= 2, all angular correlation components share the same frequency power spectrum P(f) as in Eq. 12.
    Standard assumption for an isotropic gravitational wave background; if higher multipoles have different frequency spectra, the c_l constraints could be biased.
  • ad hoc to paper The auto-correlation is fixed to unity, the c_l are non-negative, and the sum of c_l is bounded by 1 to keep the covariance positive definite.
    This is the parameterization choice in Eqs. 9 and 14; it restricts the model space and can affect the inferred cross-correlation coefficients and evidence.
  • domain assumption Monopole and dipole correlations are modeled with free-spectrum amplitudes in the first five frequency bins only.
    Covers the frequencies where the gravitational wave background evidence is strongest; assumes the monopole and dipole have no significant power beyond those bins.
  • domain assumption White noise parameters are fixed to their single-pulsar maximum-likelihood values rather than marginalized in the joint analysis.
    Standard NANOGrav approximation; if white noise parameters are significantly correlated with the angular correlation parameters, the quoted evidence could change.

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Cite this review

Pith. "Pith review of The NANOGrav 15 yr Data Set: Harmonic Analysis of the Pulsar Angular Correlations." pith.science (2026). https://pith.science/paper/4GSXXZ6F

@misc{pith2026241113472,
  author       = {Pith},
  title        = {Pith review of: The NANOGrav 15 yr Data Set: Harmonic Analysis of the Pulsar Angular Correlations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4GSXXZ6F}},
  note         = {Machine review of arXiv:2411.13472}
}
abstract

Pulsar timing array observations have found evidence for an isotropic gravitational wave background with the Hellings-Downs angular correlations, expected from general relativity. This interpretation hinges on the measured shape of the angular correlations, which is predominately quadrupolar under general relativity. Here we explore a more flexible parameterization: we expand the angular correlations into a sum of Legendre polynomials and use a Bayesian analysis to constrain their coefficients with the 15-year pulsar timing data set collected by the North American Nanohertz Observatory for Gravitational Waves (NANOGrav). When including Legendre polynomials with multipoles $\ell \geq 2$, we only find a significant signal in the quadrupole with an amplitude consistent with general relativity and non-zero at the $\sim 95\%$ confidence level and a Bayes factor of 200. When we include multipoles $\ell \leq 1$, the Bayes factor evidence for quadrupole correlations decreases by more than an order of magnitude due to evidence for a monopolar signal at approximately 4 nHz which has also been noted in previous analyses of the NANOGrav 15-year data. Further work needs to be done in order to better characterize the properties of this monopolar signal and its effect on the evidence for quadrupolar angular correlations.

Figures

Figures reproduced from arXiv: 2411.13472 by the authors.

Figure 1
Figure 1. Marginalized 1D and 2D posterior distributions for quadrupole-only GWB harmonic analysis of the NG15 data set for the model HAγ (c2). The HD value for the quadrupole coefficient, c HD 2 = 0.3125 from Equation 8, is shown as the black dashed line. The GWB amplitude and spectral index from model HDγ , which has angular correla￾tions fixed to the theoretical HD values, are shown in gray. theoretical HD values, is shown… view at source ↗
Figure 2
Figure 2. Left panel: Marginalized 1D and 2D posterior distributions of c2 and c3 for the harmonic analysis HAγ (c2, c3) of the NANOGrav 12.5 and 15 yr data sets. The dashed black lines show the HD value of each Legendre coefficient. Right panel: Reconstructed angular correlation function from the same HAγ (c2, c3) model with the NANOGrav 12.5 and 15 yr data sets. The dark- and light-shaded regions denote the 68% and 95% CL r… view at source ↗
Figure 3
Figure 3. Marginalized 1D posterior distributions for the monopole (top plot) and dipole (bottom plot) free-spectrum parameters log10 Φi of the first five frequency components fi = i/Tobs where i = 1, ..., 5. The green (right-hand) portion of the split violin plots show the results from a model that include both monopole and dipole free-spectrum models. For the NG15 data set, Tobs = 16.03 years, which gives frequency componen… view at source ↗
Figures from the paper (2 more)
Figure 5
Figure 5. Figure 5 [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]
Figure 6
Figure 6. Figure 6 [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]

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Reference graph

Works this paper leans on

56 extracted references · 11 canonical work pages · cited by 4 Pith papers

  1. [1]

    , " * write output.state after.block = add.period write newline

    ENTRY address archivePrefix author booktitle chapter doi edition editor eprint howpublished institution journal key month number organization pages publisher school series title misctitle type volume year version url label extra.label sort.label short.list INTEGERS output.state before.all mid.sentence after.sentence after.block FUNCTION init.state.consts ...

  2. [2]

    write newline

    " write newline "" before.all 'output.state := FUNCTION format.url url empty "" new.block "" url * "" * if FUNCTION format.eprint eprint empty "" archivePrefix empty "" archivePrefix "arXiv" = new.block " " eprint * " " * new.block " " eprint * " " * if if if FUNCTION format.doi doi empty "" " " doi * " " * if FUNCTION format.pid doi empty eprint empty ur...

  3. [3]

    o tv\" o s Lor\' a nd University, funded by the E\

    thebibliography [1] 20pt to REFERENCES 6pt =0pt -12pt 10pt plus 3pt =0pt =0pt =1pt plus 1pt =0pt =0pt -12pt =13pt plus 1pt =20pt =13pt plus 1pt \@M =10000 =-1.0em =0pt =0pt 0pt =0pt =1.0em @enumiv\@empty 10000 10000 `\.\@m \@noitemerr \@latex@warning Empty `thebibliography' environment \@ifnextchar \@reference \@latexerr Missing key on reference command E...

    arXiv 2021

  4. [4]

    2023, The Astrophysical Journal Letters, 951, L11, 10.3847/2041-8213/acdc91

    Afzal, A., Agazie, G., Anumarlapudi, A., et al. 2023, The Astrophysical Journal Letters, 951, L11, 10.3847/2041-8213/acdc91

    doi:10.3847/2041-8213/acdc91 2023

  5. [5]

    M., et al

    Agazie, G., Anumarlapudi, A., Archibald, A. M., et al. 2023 a , The Astrophysical Journal Letters, 951, L8, 10.3847/2041-8213/acdac6

    doi:10.3847/2041-8213/acdac6 2023

  6. [6]

    2023 b , Astrophys

    Agazie, G., et al. 2023 b , Astrophys. J. Lett., 952, L37, 10.3847/2041-8213/ace18b

    doi:10.3847/2041-8213/ace18b 2023

  7. [7]

    2023 c , Astrophys

    ---. 2023 c , Astrophys. J. Lett., 951, L10, 10.3847/2041-8213/acda88

    doi:10.3847/2041-8213/acda88 2023

  8. [8]

    2023 d , Astrophys

    ---. 2023 d , Astrophys. J. Lett., 951, L50, 10.3847/2041-8213/ace18a

    doi:10.3847/2041-8213/ace18a 2023

Show all 56 references
  1. [9]

    2024, Astrophys

    ---. 2024, Astrophys. J. Lett., 964, L14, 10.3847/2041-8213/ad2a51

    2024 doi

  2. [10]

    L., & Mingarelli , C

    Ali-Ha \" moud , Y., Smith , T. L., & Mingarelli , C. M. F. 2020, , 102, 122005, 10.1103/PhysRevD.102.122005

    2020 doi

  3. [11]

    Allen, B., Dhurandhar, S., Gupta, Y., et al. 2023. 2304.04767

    2023 arXiv

  4. [13]

    Anholm, M., Ballmer, S., Creighton, J. D. E., Price, L. R., & Siemens, X. 2009, Phys. Rev. D, 79, 084030, 10.1103/PhysRevD.79.084030

    2009 doi

  5. [14]

    2023, Astron

    Antoniadis, J., et al. 2023, Astron. Astrophys., 678, A50, 10.1051/0004-6361/202346844

    2023 doi

  6. [15]

    2020, Astrophys

    Arzoumanian, Z., et al. 2020, Astrophys. J. Lett., 905, L34, 10.3847/2041-8213/abd401

    2020 doi

  7. [16]

    2021, Astrophys

    ---. 2021, Astrophys. J. Lett., 923, L22, 10.3847/2041-8213/ac401c

    2021 doi

  8. [17]

    K., et al

    Boddy, K. K., et al. 2022, JHEAp, 35, 112, 10.1016/j.jheap.2022.06.005

    2022 doi

  9. [18]

    2022, Gen

    Caldwell, R., et al. 2022, Gen. Rel. Grav., 54, 156, 10.1007/s10714-022-03027-x

    2022 doi

  10. [19]

    J., Creighton , J

    Chamberlin , S. J., Creighton , J. D. E., Siemens , X., et al. 2015, , 91, 044048, 10.1103/PhysRevD.91.044048

    2015 doi

  11. [20]

    J., & Siemens, X

    Chamberlin, S. J., & Siemens, X. 2012, Phys. Rev. D, 85, 082001, 10.1103/PhysRevD.85.082001

    2012 doi

  12. [21]

    J., O'Beirne, L., Taylor, S

    Cornish, N. J., O'Beirne, L., Taylor, S. R., & Yunes, N. 2018, Phys. Rev. Lett., 120, 181101, 10.1103/PhysRevLett.120.181101

    2018 doi

  13. [22]

    B., Ferdman , R

    Demorest , P. B., Ferdman , R. D., Gonzalez , M. E., et al. 2013, , 762, 94, 10.1088/0004-637X/762/2/94

    2013 doi

  14. [23]

    Detweiler, S. L. 1979, Astrophys. J., 234, 1100, 10.1086/157593

    1979 doi

  15. [24]

    Dickey, J. M. 1971, The Annals of Mathematical Statistics, 42, 204. http://www.jstor.org/stable/2958475

    1971

  16. [25]

    2017, jellis18/PTMCMCSampler: Official Release, 1.0.0, Zenodo, 10.5281/zenodo.1037579

    Ellis, J., & van Haasteren, R. 2017, jellis18/PTMCMCSampler: Official Release, 1.0.0, Zenodo, 10.5281/zenodo.1037579

    2017 doi

  17. [26]

    A., Vallisneri, M., Taylor, S

    Ellis, J. A., Vallisneri, M., Taylor, S. R., & Baker, P. T. 2020, ENTERPRISE: Enhanced Numerical Toolbox Enabling a Robust PulsaR Inference SuitE , v3.0.0, Zenodo, 10.5281/zenodo.4059815

    2020 doi

  18. [27]

    D., Taylor, S., & Mingarelli, C

    Gair, J., Romano, J. D., Taylor, S., & Mingarelli, C. M. F. 2014, Phys. Rev. D, 90, 082001, 10.1103/PhysRevD.90.082001

    2014 doi

  19. [28]

    Gelman, A., & Rubin, D. B. 1992, Statist. Sci., 7, 457, 10.1214/ss/1177011136

    1992

  20. [29]

    A., Taylor, S

    Gersbach, K. A., Taylor, S. R., Meyers, P. M., & Romano, J. D. 2024. 2406.11954

    2024 arXiv

  21. [30]

    Green, D., et al. 2022. 2209.06854

    2022 arXiv

  22. [31]

    W., & Downs , G

    Hellings , R. W., & Downs , G. S. 1983, The Astrophysical Journal Letters, 265, L39, 10.1086/183954

    1983 doi

  23. [32]

    N., et al

    Hobbs, G., Coles, W., Manchester, R. N., et al. 2012, Monthly Notices of the Royal Astronomical Society, 427, 2780, 10.1111/j.1365-2966.2012.21946.x

    2012

  24. [33]

    N., et al

    Hobbs, G., Guo, L., Caballero, R. N., et al. 2019, Monthly Notices of the Royal Astronomical Society, 491, 5951, 10.1093/mnras/stz3071

    2019 doi

  25. [34]

    C., Kamionkowski, M., & Jaffe, A

    Hotinli, S. C., Kamionkowski, M., & Jaffe, A. H. 2019, Open J. Astrophys., 2, 8, 10.21105/astro.1904.05348

    2019 arXiv

  26. [35]

    1998, The theory of probability (OuP Oxford)

    Jeffreys, H. 1998, The theory of probability (OuP Oxford)

    1998

  27. [36]

    D., et al

    Johnson, A. D., et al. 2023. 2306.16223

    2023 arXiv

  28. [37]

    J., Jenet , F

    Lee , K. J., Jenet , F. A., & Price , R. H. 2008, , 685, 1304, 10.1086/591080

    2008 doi

  29. [38]

    Madison, D. R. 2024. 2410.23381

    2024 arXiv

  30. [39]

    2018, Gravitational Waves

    Maggiore, M. 2018, Gravitational Waves. Vol. 2: Astrophysics and Cosmology (Oxford University Press)

    2018

  31. [40]

    N., Taylor , J

    Matsakis , D. N., Taylor , J. H., & Eubanks , T. M. 1997, , 326, 924

    1997

  32. [41]

    Mingarelli , C. M. F., & Casey-Clyde , J. A. 2022, Science, 378, 592, 10.1126/science.abq1187

    2022 doi

  33. [42]

    Mingarelli, C. M. F., Sidery, T., Mandel, I., & Vecchio, A. 2013, Phys. Rev. D, 88, 062005, 10.1103/PhysRevD.88.062005

    2013 doi

  34. [43]

    K., Smith, T

    Nay, J., Boddy, K. K., Smith, T. L., & Mingarelli, C. M. F. 2024, Phys. Rev. D, 110, 044062, 10.1103/PhysRevD.110.044062

    2024 doi

  35. [44]

    Phinney, E. S. 2001. astro-ph/0108028

    2001 arXiv

  36. [45]

    K., Kamionkowski, M., & Dai, L

    Qin, W., Boddy, K. K., Kamionkowski, M., & Dai, L. 2019, Phys. Rev. D, 99, 063002, 10.1103/PhysRevD.99.063002

    2019 doi

  37. [46]

    J., et al

    Reardon, D. J., et al. 2023, Astrophys. J. Lett., 951, L6, 10.3847/2041-8213/acdd02

    2023 doi

  38. [47]

    2019, Astrophys

    Roebber, E. 2019, Astrophys. J., 876, 55, 10.3847/1538-4357/ab100e

    2019 doi

  39. [48]

    2017, Astrophys

    Roebber, E., & Holder, G. 2017, Astrophys. J., 835, 21, 10.3847/1538-4357/835/1/21

    2017 doi

  40. [49]

    C., Vigeland, S

    Sardesai, S. C., Vigeland, S. J., Gersbach, K. A., & Taylor, S. R. 2023, Phys. Rev. D, 108, 124081, 10.1103/PhysRevD.108.124081

    2023 doi

  41. [50]

    Sazhin , M. V. 1978, , 22, 36

    1978

  42. [51]

    R., Baker, P

    Taylor, S. R., Baker, P. T., Hazboun, J. S., Simon, J., & Vigeland, S. J. 2021, enterprise-extensions. https://github.com/nanograv/enterprise_extension

    2021

  43. [52]

    R., & Gair, J

    Taylor, S. R., & Gair, J. R. 2013, Phys. Rev. D, 88, 084001, 10.1103/PhysRevD.88.084001

    2013 doi

  44. [53]

    R., Gair, J

    Taylor, S. R., Gair, J. R., & Lentati, L. 2013, Phys. Rev. D, 87, 044035, 10.1103/PhysRevD.87.044035

    2013 doi

  45. [54]

    2016, Mon

    Tiburzi, C., Hobbs, G., Kerr, M., et al. 2016, Mon. Not. Roy. Astron. Soc., 455, 4339, 10.1093/mnras/stv2143

    2016 doi

  46. [55]

    R., Simon , J., et al

    Vallisneri , M., Taylor , S. R., Simon , J., et al. 2020, , 893, 112, 10.3847/1538-4357/ab7b67

    2020 doi

  47. [56]

    J., Islo, K., Taylor, S

    Vigeland, S. J., Islo, K., Taylor, S. R., & Ellis, J. A. 2018, Phys. Rev. D, 98, 044003, 10.1103/PhysRevD.98.044003

    2018 doi

  48. [57]

    2023, Res

    Xu, H., et al. 2023, Res. Astron. Astrophys., 23, 075024, 10.1088/1674-4527/acdfa5

    2023 doi

Pith tools

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