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

Gabriella Agazie, Akash Anumarlapudi, Anne M. Archibald, Zaven Arzoumanian, Paul T. Baker, Bence Becsy, Laura Blecha, Adam Brazier, Paul R. Brook, Sarah Burke-Spolaor, Rand Burnette, Robin Case, Maria Charisi, Shami Chatterjee, Katerina Chatziioannou, Belinda D. Cheeseboro, Siyuan Chen, Tyler Cohen, James M. Cordes, Neil J. Cornish, Fronefield Crawford, H. Thankful Cromartie, Kathryn Crowter, Curt J. Cutler, Megan E. DeCesar, Dallas DeGan, Paul B. Demorest, Heling Deng, Timothy Dolch, Brendan Drachler, Justin A. Ellis, Elizabeth C. Ferrara, William Fiore, Emmanuel Fonseca, Gabriel E. Freedman, Nate Garver-Daniels, Peter A. Gentile, Kyle A. Gersbach, Joseph Glaser, Deborah C. Good, Kayhan Gultekin, Jeffrey S. Hazboun, Sophie Hourihane, Kristina Islo, Ross J. Jennings, Aaron D. Johnson, Megan L. Jones, Andrew R. Kaiser, David L. Kaplan, Luke Zoltan Kelley, Matthew Kerr, Joey S. Key, Tonia C. Klein, Nima Laal, Michael T. Lam, William G. Lamb, T. Joseph W. Lazio, Natalia Lewandowska, Tyson B. Littenberg, Tingting Liu, Andrea Lommen, Duncan R. Lorimer, Jing Luo, Ryan S. Lynch, Chung-Pei Ma, Dustin R. Madison, Margaret A. Mattson, Alexander McEwen, James W. McKee, Maura A. McLaughlin, Natasha McMann, Bradley W. Meyers, Patrick M. Meyers, Chiara M. F. Mingarelli, Andrea Mitridate, Priyamvada Natarajan, Cherry Ng, David J. Nice, Stella Koch Ocker, Ken D. Olum, Timothy T. Pennucci, Benetge B. P. Perera, Polina Petrov, Nihan S. Pol, Henri A. Radovan, Scott M. Ransom, Paul S. Ray, Joseph D. Romano, Shashwat C. Sardesai, Ann Schmiedekamp, Carl Schmiedekamp, Kai Schmitz, Levi Schult, Brent J. Shapiro-Albert, Xavier Siemens, Joseph Simon, Magdalena S. Siwek, Ingrid H. Stairs, Daniel R. Stinebring, Kevin Stovall, Jerry P. Sun, Abhimanyu Susobhanan, Joseph K. Swiggum, Jacob Taylor, Stephen R. Taylor, Jacob E. Turner, Caner Unal, Michele Vallisneri, Rutger van Haasteren, Sarah J. Vigeland, Haley M. Wahl, Qiaohong Wang, Caitlin A. Witt, Olivia Young

arXiv:2306.16213v1astro-ph.HEgr-qc

TL;DR

The paper asks whether a common nanohertz signal in pulsar-timing data has the spatial correlations expected from a gravitational-wave background rather than uncorrelated or intrinsic noise. Using Bayesian and frequentist analyses of the NANOGrav 15-year data set, it finds evidence for Hellings–Downs correlations and estimates an astrophysically compatible background amplitude. Alternative astrophysical and cosmological sources remain possible, and spectral inference depends on modeling choices.

  • Problem

    The central problem is distinguishing a gravitational-wave background from intrinsic pulsar processes or common systematic noise that can produce similar excess power and spectral shape.

    Source: §1 · p. 5

  • Method

    The analysis combines Bayesian model comparison and noise-marginalized optimal-statistic tests on 67 pulsars, using phase-shifted data to calibrate false-alarm probabilities.

    Source: §1 · p. 5, §2 · p. 7

  • Results

    The analysis finds compelling Hellings–Downs correlations, with false-alarm probabilities of p = 10^-3 for Bayesian statistics and p = 5 × 10^-5–1.9 × 10^-4 for frequentist statistics, and estimates AGWB = 2.4+0.7_-0.6 × 10^-15 at 1 yr^-1.

    Source: §1 · p. 5, §6 · p. 16

  • Takeaways & Limitations

    The observed Hellings–Downs pattern points to a gravitational-wave origin and is consistent with a population of supermassive-black-hole binaries.

    Source: §1 · pp. 5–6, §6 · p. 16

    The inferred spectral slope is sensitive to minor details of the data model, and alternative astrophysical or cosmological sources cannot be excluded.

    Source: §5.2 · p. 11

Abstract

from arXiv · show

We report multiple lines of evidence for a stochastic signal that is correlated among 67 pulsars from the 15-year pulsar-timing data set collected by the North American Nanohertz Observatory for Gravitational Waves. The correlations follow the Hellings-Downs pattern expected for a stochastic gravitational-wave background. The presence of such a gravitational-wave background with a power-law-spectrum is favored over a model with only independent pulsar noises with a Bayes factor in excess of $10^{14}$, and this same model is favored over an uncorrelated common power-law-spectrum model with Bayes factors of 200-1000, depending on spectral modeling choices. We have built a statistical background distribution for these latter Bayes factors using a method that removes inter-pulsar correlations from our data set, finding $p = 10^{-3}$ (approx. $3\sigma$) for the observed Bayes factors in the null no-correlation scenario. A frequentist test statistic built directly as a weighted sum of inter-pulsar correlations yields $p = 5 \times 10^{-5} - 1.9 \times 10^{-4}$ (approx. $3.5 - 4\sigma$). Assuming a fiducial $f^{-2/3}$ characteristic-strain spectrum, as appropriate for an ensemble of binary supermassive black-hole inspirals, the strain amplitude is $2.4^{+0.7}_{-0.6} \times 10^{-15}$ (median + 90% credible interval) at a reference frequency of 1/(1 yr). The inferred gravitational-wave background amplitude and spectrum are consistent with astrophysical expectations for a signal from a population of supermassive black-hole binaries, although more exotic cosmological and astrophysical sources cannot be excluded. The observation of Hellings-Downs correlations points to the gravitational-wave origin of this signal.

1. INTRODUCTION

Nanohertz gravitational-wave searches use pulsar timing to identify a stochastic excess across pulsars and distinguish it from intrinsic or common terrestrial noise through Hellings–Downs spatial correlations. The NANOGrav 15-year analysis combines Bayesian and frequentist evidence for such correlations and infers an amplitude compatible with supermassive-black-hole-binary expectations.

1. INTRODUCTION

  • A stochastic background from supermassive-black-hole binaries is expected to produce broadband nanohertz power with an f^-2/3 characteristic-strain spectrum.This expectation assumes binaries evolve through gravitational-radiation losses from circular orbital energy.Source: §1 · p. 3
  • The NANOGrav 15-year analysis presents multiple Bayesian and frequentist lines of evidence for Hellings–Downs correlations in a 67-pulsar data set.The analyses include Bayes factors, an optimal statistic, and correlation reconstructions.Source: Figure 1 · p. 4, §1 · p. 5
  • Hellings–Downs correlations are needed to distinguish a gravitational-wave background from intrinsic pulsar processes or common systematic noise with similar amplitude and spectrum.For an isotropic background, the pattern is a universal quasi-quadrupolar function of pulsar angular separation.Source: §1 · p. 5
  • Bayes factors favor the Hellings–Downs power-law model over an uncorrelated common-spectrum model by 200–1,000, while the optimal statistic reaches a signal-to-noise ratio of approximately 5.Phase-shift calibration gives false-alarm probabilities of p = 10^-3 for the Bayes factor and p = 5 × 10^-5 for the optimal statistic.Source: §1 · p. 5
  • The fiducial model estimates AGWB = 2.4+0.7_-0.6 × 10^-15 at 1 yr^-1, with a spectral exponent consistent with γ = 13/3 but sensitive to data-model details.The amplitude is compatible with current astrophysical estimates, while smaller γ values are preferred in the more general model.Source: §1 · pp. 5–6
  • The paper documents robustness studies and notes that cross-validation with independent pulsar-timing data sets will provide further insight.The authors also identify possible alternative astrophysical and cosmological sources of nanohertz backgrounds.Source: §1 · pp. 3, 6

2. THE 15-YEAR DATA SET AND DATA MODEL

The 15-year NANOGrav data set analyzes narrowband pulse arrival times from 67 pulsars with timing baselines of at least three years. Its probabilistic data model jointly represents timing uncertainties, intrinsic pulsar noise, and potentially correlated common processes under alternative spatial and spectral assumptions.

2. THE 15-YEAR DATA SET AND DATA MODEL

  • The data set contains observations of 68 pulsars from Arecibo, GBT, and VLA, with analysis restricted to 67 pulsars having timing baselines of at least three years.It extends the earlier data set with 2.9 years for 47 pulsars and 21 new pulsars.Source: §2 · p. 6
  • The timing analysis fits pulsar spin, astrometric, binary, dispersion-measure, white-noise, and intrinsic red-noise components to pulse arrival times.Parallax and proper-motion parameters are included because their signals can overlap with gravitational-wave frequencies.Source: §2 · p. 6
  • The Bayesian model represents timing residuals as a finite Gaussian process using Fourier bases for time-correlated fluctuations and a design matrix for timing-model uncertainties.The Fourier frequencies are fi = i/T with T = 16.03 yr.Source: §2 · p. 7
  • A joint normal prior couples pulsar likelihoods through intrinsic-noise and common-process covariance parameters, which are inferred after analytically marginalizing Fourier coefficients and timing-model coefficients.Markov chain Monte Carlo estimates the posterior over hyperparameters for alternative noise and common-spectrum models.Source: §2 · p. 7
  • The models distinguish intrinsic noise, an uncorrelated common spectrum, and an isotropic gravitational-wave background through different structures for Φab,i.The Hellings–Downs model uses Φab,i = Γ(ξab)ΦHD,i, whereas the uncorrelated model uses δabΦCURN,i.Source: §2 · p. 7
  • The fiducial hd13/3 model fixes γHD = 13/3, while free-spectrum and variable-exponent models allow the common-process spectrum to vary across frequencies or in slope.Common-spectrum analyses use restricted frequencies to reduce correlations with high-frequency white noise.Source: §2 · p. 7

3. BAYESIAN ANALYSIS

The Bayesian analysis finds strong evidence for a common-spectrum signal with Hellings–Downs inter-pulsar correlations, while additional monopole or dipole correlations are unsupported. The inferred amplitude is compatible with supermassive-black-hole-binary expectations, though spectral-index estimation is model-sensitive.

Common-spectrum signal

  • The common-spectrum signal is a loud time-correlated stochastic process with common amplitude and spectrum across pulsars.The very peaked amplitude posterior supports the large Bayes factor between intrinsic red noise alone and the common-spectrum model.Source: §3 · pp. 7–8

Signal amplitude

  • AHD,13/3 = 2.4^+0.7_-0.6 × 10^-15 for the fixed γ = 13/3 model, consistent with astrophysical expectations for inspiraling SMBHBs.The posterior has essentially no support below 10^-15.Source: §3 · p. 8

Spectral modeling

  • The inferred spectral index is consistent with γ = 13/3 but smaller values are preferred, and its posterior is sensitive to intrinsic-red-noise and interstellar-medium modeling in a few pulsars.More expansive data sets or sophisticated chromatic-noise models are needed to assess possible systematic errors in γHD.Source: §3 · p. 8

Bayesian model comparison

  • Bayes factors of 200–1,000 favor hdγ over curnγ, while the 15-year data set also favors hdγ over monopolar or dipolar correlations.The data are inconclusive about adding further spatially correlated signals to hdγ.Source: §3 · p. 8, Figure 2 · p. 8

Hypothesis testing

  • The observed hdγ-to-curnγ Bayes factor has p = 10^-3, approximately 3σ, under phase-shifted data with inter-pulsar correlations removed.Only five of 5,000 phase shifts exceeded the observed value; sky scrambles give p = 1.6 × 10^-3.Source: §3 · pp. 8–9, Figure 3 · p. 9

Correlation pattern

  • A minimally modeled spline reconstruction is consistent with the Hellings–Downs pattern, with both zero-crossing knot posteriors consistent with (0, 0) within 1σ.The reconstruction uses seven knots chosen around the theoretical extrema, zero crossings, and intermediate angles.Source: §3 · p. 9

4. OPTIMAL STATISTIC ANALYSIS

The optimal statistic measures Hellings–Downs-correlated excess power as a weighted combination of inter-pulsar correlations. Applied to the 15-year data, it yields S/N values of 5 ± 1 or 4 ± 1 depending on the common-spectrum noise model, with empirical tests producing low p-values.

Statistic construction

  • The optimal statistic is a variance-weighted, Hellings–Downs-template-matched average of inter-pulsar correlation coefficients.It estimates the squared Hellings–Downs amplitude and can be interpreted as a least-squares fit to the correlation template.Source: §4 · pp. 9–10

Optimal-statistic results

  • S/N is 5 ± 1 for curnγ and 4 ± 1 for curn13/3 when measuring Hellings–Downs correlations.These values are means ± standard deviations across noise-parameter posteriors.Source: Figure 4 · p. 10, §4 · p. 10

Statistical interpretation

  • The optimal-statistic S/N cannot be interpreted directly as a sigma level because its approximately normal amplitude distribution does not yield an analytic p-value.P-values are instead computed empirically.Source: §4 · p. 10

Empirical significance

  • Phase-shift tests find p = 5 × 10^-5, while simulation and analytic backgrounds give p = 1.8 × 10^-4 and 1.9 × 10^-4.The phase-shift distribution used 400,000 shifts; the simulation comparison used 27,000 curnγ realizations.Source: §4 · p. 10

Correlation pattern

  • The measured cross-correlations follow the Hellings–Downs pattern across 15 angular-separation bins.The reconstruction has χ2 = 8.1, with p = 0.75 from hdγ simulations and p = 0.92 under a canonical χ2 model; alternative binnings give p > 0.3.Source: §4 · pp. 10–11

5. CHECKS AND VALIDATION

Validation studies show that Hellings–Downs evidence is robust to dispersion-measure modeling and appears across frequencies, pulsars, time slices, and telescopes, while alternative-correlation analyses identify unresolved monopole power. Spectral features and modeling choices constrain interpretation of the recovered slope and additional correlations.

5.1. Alternative DM models

  • DMGP shifts the common-red-noise slope toward γCURN = 13/3, while Hellings–Downs correlations remain equally significant as with DMX.Source: §5.1 · p. 11, Figure 5 · p. 11

5.2. Spectral analysis

  • Correlated power appears mainly in bins 1–5 and 8, with missing power in bins 6–7 and bin-dependent slope estimates that favor shallower γHD values.A turnover below 10 nHz is favored, but its Bayes-factor comparison with the standard model is inconclusive.Source: §5.2 · p. 12, Figure 6 · p. 12

5.3. Alternative correlation patterns

  • MCOS finds the expected quadrupolar structure plus a small monopolar contribution, but simulations show that a pure Hellings–Downs signal can produce such monopole significance with p = 0.11.Bayesian searches find no significant additional monopole- or dipole-correlated red noise, except monopole power in the 3.95 nHz bin.Source: §5.3 · p. 13
  • The additional 3.95 nHz monopole feature has low Bayesian significance and disappears after subtracting sinusoid realizations, while Hellings–Downs S/N remains unchanged.Its persistence in future data and comparisons with other PTAs are needed to clarify whether it is astrophysical or systematic.Source: §5.3 · pp. 13–14

5.4. Dropout and cross-validation

  • Dropout analyses find twenty pulsars favoring an uncorrelated common process and seven favoring Hellings–Downs participation, while three disfavor the latter.Negative dropout factors may reflect noise fluctuations, chromatic-noise modeling, or the correlated and uncorrelated components induced by the GWB.Source: §5.4 · p. 14, Figure 8 · p. 14
  • Leave-one-out predictive validation gives PBF15 yr = 1,400 for hd13/3 over curn13/3, with log10 PBF displaced by approximately three standard deviations in simulations.Source: §5.4 · p. 14
  • Hellings–Downs S/N grows essentially monotonically as observing time and pulsar count increase, consistent with the expected sensitivity scaling.Source: §5.4 · pp. 14–15, Figure 9 · p. 15
  • Independent Arecibo and GBT subsets recover comparable Hellings–Downs S/N values of 2.9 and 3.3 despite having roughly half the full array’s pulsars.Their common-spectrum posteriors are broadly consistent with each other and with the full-data posterior.Source: §5.4 · p. 15, Figure 10 · p. 15

6. DISCUSSION

The 15-year analysis finds compelling Hellings–Downs correlations across 67 pulsars, while spectral features and possible monopolar contamination remain incompletely explained. The signal is consistent with a supermassive-black-hole-binary background, but alternative astrophysical and cosmological origins cannot be excluded.

6. DISCUSSION

  • p = 10^-3 and p = 5 × 10^-5–1.9 × 10^-4 for Bayesian and frequentist tests, respectively, provide compelling evidence of Hellings–Downs inter-pulsar correlations.The analysis uses 67 pulsars and finds the common-spectrum signal in more pulsars than before.Source: §6 · pp. 15–16
  • The correlated signal extends across frequencies, but spectral-slope estimates vary with frequency-bin anomalies and alternative dispersion-measure modeling.The optimal statistic confirms a quasi-quadrupolar Hellings–Downs pattern, with inconclusive evidence for a narrow-band monopolar feature near 4 nHz.Source: §6 · p. 16
  • The signal appears persistent across longer data slices, multiple pulsars, and independent single-telescope data sets, and is inconsistent with an individual circular SMBHB.These checks support a stochastic background interpretation rather than a single resolvable binary.Source: §6 · p. 16
  • Under purely GW-driven evolution, SMBHBs predict a characteristic-strain spectrum proportional to f^-2/3, but population amplitudes vary by more than an order of magnitude.The observed amplitude is consistent with SMBH and galaxy-evolution models but lies toward the upper end of predictions.Source: §6 · p. 16
  • The data provide little support for a low-frequency turnover, while exotic cosmological sources can produce similar nanohertz backgrounds and remain difficult to distinguish spectrally.Future longer data sets and cross-PTA comparisons are expected to clarify the signal’s astrophysical or cosmological origin.Source: §6 · pp. 16–17

A. ADDITIONAL DATA SET DETAILS

The 15-year data set combines observations from Arecibo, GBT, and the VLA for 67 pulsars. Telescope visibility and frequency coverage determine the available pulsars, while split-telescope sets provide independent cross-checks.

A. ADDITIONAL DATA SET DETAILS

  • 67 pulsars were observed between July 2004 and August 2020 using Arecibo, GBT, and the VLA.The data set includes sky-location and angular-separation coverage for the pulsar pairs.Source: A. ADDITIONAL DATA SET DETAILS · p. 23
  • Arecibo observed 33 pulsars, GBT observed 36, and the VLA observed seven, with some overlap among telescopes.Pulsar visibility depended primarily on declination and telescope access.Source: A. ADDITIONAL DATA SET DETAILS · p. 23
  • Observations used multiple radio-frequency bands to measure interstellar-medium dispersion and included bandwidths up to 800 MHz.The analysis also formed split-telescope data sets for Arecibo and GBT, but not the shorter, less sensitive VLA-only set.Source: A. ADDITIONAL DATA SET DETAILS · p. 23

B. BAYESIAN METHODS & DIAGNOSTICS

The analysis uses MCMC posterior sampling, convergence checks, bootstrap uncertainty estimation, and several Bayesian model-comparison methods. These methods support comparisons among nested and disjoint noise and correlation models.

B. BAYESIAN METHODS & DIAGNOSTICS

  • MCMC samples the joint posterior, with convergence required at R-hat < 1.01 for all parameters.Most analyses use PTMCMC, with posterior samples postprocessed using chainconsumer.Source: B. BAYESIAN METHODS & DIAGNOSTICS · p. 23
  • Bootstrap resampling of MCMC samples estimates uncertainty distributions for quantiles and marginalized posterior summaries.The approach replaces the earlier analytic approximation for marginalized-posterior uncertainty.Source: B. BAYESIAN METHODS & DIAGNOSTICS · p. 23
  • Thermodynamic integration estimates Bayesian evidence by integrating likelihood expectations across parallel-tempered chains.The chains use geometrically spaced inverse temperatures.Source: B. BAYESIAN METHODS & DIAGNOSTICS · p. 24
  • The Savage–Dickey density ratio compares nested models by evaluating posterior-to-prior density at the parameter value fixed by the simpler model.This applies when one model freezes a subset of parameters, such as θ0 = 0.Source: B. BAYESIAN METHODS & DIAGNOSTICS · p. 24
  • Product-space sampling compares disjoint models by sampling a shared parameter space with a model-indexing parameter and counting activation frequencies.Approximate posterior reweighting is also used when direct sampling of an expensive model is difficult.Source: B. BAYESIAN METHODS & DIAGNOSTICS · p. 24

C. BROKEN POWER-LAW MODEL

The analysis tests spectral deviations from a standard power law using a broken power-law model and a t-process spectrum. The broken model sets the frequency basis, while the t-process accommodates outlying frequency bins and broadens spectral uncertainty.

C. BROKEN POWER-LAW MODEL

  • The broken-power-law model allows separate low- and high-frequency slopes, with fbreak setting the transition and ℓ = 0.1 controlling smoothness.The cutoff is selected from inference using a modified common red-noise model.Source: C. BROKEN POWER-LAW MODEL · p. 25
  • A broken power law places the spectral break at the MAP fbreak = 2.75 × 10^-8 Hz, motivating 14 common-spectrum frequencies.The posterior median and 90% credible region for fbreak are 3.2+5.4−1.2 × 10^-8 Hz.Source: C. BROKEN POWER-LAW MODEL · p. 25

D. T-PROCESS SPECTRUM MODEL

  • The t-process spectrum allows individual frequency powers to vary more freely around the power law, targeting bins f1, f6, f7, and f8 that are in tension with it.It uses an inverse-gamma mixture that produces Student’s-t-distributed spectral powers.Source: D. T-PROCESS SPECTRUM MODEL · p. 25
  • The t-process posterior is more consistent with steeper power laws and includes γ = 13/3 within 1σ credibility.This model broadens the spectrum specifically at the frequency bins identified as anomalous.Source: D. T-PROCESS SPECTRUM MODEL · p. 25

E. TURNOVER MODEL

The turnover model describes environmentally driven changes in SMBHB evolution that can alter the gravitational-wave spectrum. The data hint at a turnover below 10 nHz, but the comparison with a pure power law is inconclusive and parameter estimates depend on DM modeling.

E. TURNOVER MODEL

  • Environmental interactions can accelerate binary evolution through the GW spectrum, reducing emission relative to a GW-driven inspiral.The turnover model represents the transition from environmentally dominated evolution at lower frequencies to GW-dominated evolution at higher frequencies.Source: E. TURNOVER MODEL · p. 25
  • 65% credibility places the transition frequency f0 below 10 nHz, while 75% credibility places the bend frequency below 10 nHz.These results provide hints of departures from a pure power law.Source: E. TURNOVER MODEL · p. 25
  • The t-process power-law model permits local spectral deviations and yields wider constraints that are more consistent with γ = 13/3.Figure 13 compares its spectral posteriors with the standard power-law model and free-spectral bin-variance posteriors.Source: Figure 13 · p. 26
  • A Bayes factor of 1.46±0.02 provides inconclusive support for curnturnover over curnγ.The turnover parameters are sensitive to whether DM fluctuations are modeled with DMX or DMGP.Source: E. TURNOVER MODEL · p. 26

F. SKY SCRAMBLES

Sky scrambles construct null distributions by preserving measured inter-pulsar correlation strength while removing its dependence on pulsar angular separation. Applied to the 15-year data, they yield significant Bayesian and optimal-statistic results that are stable across match thresholds.

F. SKY SCRAMBLES

  • Sky scrambles test whether inter-pulsar correlations depend on angular separation, whereas phase shifts test whether inter-pulsar correlations are absent.The two procedures therefore correspond to different null hypotheses when a correlated signal is present.Source: F. SKY SCRAMBLES · p. 26
  • The analysis generated 10,000 independent scrambled ORFs but computed detection statistics for 5,000 because obtaining qualifying scrambles and evaluating them was computationally limited.The match thresholds were below 0.1 relative to the unscrambled ORF and below 0.17 between scrambled ORFs.Source: F. SKY SCRAMBLES · p. 26
  • Only 8 of 5,000 sky scrambles exceeded the observed Bayes factor, giving p = 1.6 × 10^-3, while none exceeded the observed noise-marginalized mean S/N, giving p < 10^-4.Figure 14 summarizes both empirical detection-statistic distributions.Source: F. SKY SCRAMBLES · p. 26, Figure 14 · p. 27
  • The null distributions changed little for match thresholds below about 0.2, although stricter thresholds reduced the number of usable scrambles and p-value precision.The comparison includes thresholds of 0.17, 0.1, and 0.08.Source: F. SKY SCRAMBLES · p. 26

G. MULTIPLE-CORRELATION OPTIMAL STATISTIC

The MCOS fits multiple inter-pulsar correlation patterns sharing one spectral shape and finds that models containing Hellings–Downs and monopole components are preferred. Simulations qualify the monopole indication because a Hellings–Downs signal alone can produce it.

G. MULTIPLE-CORRELATION OPTIMAL STATISTIC

  • The MCOS fits correlation coefficients with a linear model whose coefficients are squared component amplitudes, and reports each component’s coefficient-to-error ratio as S/N.Pulsar-noise posterior uncertainty induces a distribution of MCOS statistics.Source: G. MULTIPLE-CORRELATION OPTIMAL… · p. 27
  • Models with both HD and monopole correlations are preferred among all models, with S/N 3.4 ± 0.8 for HD and 2.9 ± 0.8 for monopole correlations.Adding a dipole gives S/N 2.9 ± 0.6 for HD, 2.4 ± 0.6 for monopole, and 0.6 ± 0.4 for dipole correlations.Source: G. MULTIPLE-CORRELATION OPTIMAL… · pp. 27–28, Figure 16 · p. 28
  • Across simulations without spatially correlated injections, the observed mean HD and monopole statistics have p-values below 10^-2 and approximately 4 × 10^-2, respectively.These empirical p-values quantify how often the observed statistics arise in the simulated no-correlated-signal setting.Source: G. MULTIPLE-CORRELATION OPTIMAL… · p. 28
  • A real HD signal can excite the monopole statistic, so MCOS S/N measures apparent correlation patterns but has limited ability to identify their physical source.The quoted S/Ns quantify compatibility with uncorrelated noise, not confusion between different correlated-noise patterns.Source: G. MULTIPLE-CORRELATION OPTIMAL… · p. 28

H. MULTIPLE-CORRELATION OPTIMAL STATISTIC SIMULATIONS

  • Simulations find that the observed monopole S/N can occur when only an HD-correlated stochastic process is present, so the monopole indication is possible but weaker than the HD indication.The MCOS analysis therefore does not by itself establish a distinct monopolar source.Source: G. MULTIPLE-CORRELATION OPTIMAL… · p. 28

H.1. Astrophysical simulations

Astrophysical and model-checking simulations test whether the observed MCOS statistics arise from noise, an HD-correlated GWB, or modeling deficiencies. They support the observed HD statistic as consistent with an HD GWB while leaving the monopole statistic compatible with several explanations.

H.1. Astrophysical simulations

  • The simulations include 100 realizations each with no correlated signal, an HD GWB of amplitude 2.7 × 10^-15 and spectral index 13/3, or omitted additional pulsar-noise power.The third case tests whether excess common-spectrum noise is mistaken for spatially correlated GWB power.Source: §H.1 · p. 29
  • The observed HD S/N is unlikely under noise-only or uncorrelated common-spectrum simulations, with p < 10^-2, but is fully consistent with an injected HD GWB, with p = 0.64.The observed monopole S/N is compatible with intrinsic pulsar noise or a real HD signal.Source: §H.1 · p. 29, Figure 18 · p. 29

H. MULTIPLE-CORRELATION OPTIMAL STATISTIC SIMULATIONS

  • The simulations compare MCOS statistics from replicated data generated using pulsar-noise and GW parameters inferred from the 15-year data set.The astrophysical simulations use MAP parameters, whereas model checking draws parameters from the hd13/3 posterior.Source: §H.2 · pp. 29–30

H.2. Model-checking simulations

  • In posterior-based model checking, the replicated monopole S/N exceeded the observed value in 11% of 500 draws, making the observed monopole plausible under an HD-only GWB model.The result does not by itself indicate that the hd13/3 model is misspecified.Source: §H.2 · p. 30
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