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

TL;DR

The study provides compelling evidence for a stochastic gravitational-wave background in the NANOGrav 15-year data set, demonstrating Hellings–Downs inter-pulsar correlations through both Bayesian and frequentist analyses. The inferred GWB amplitude and spectral slope, compatible with SMBHB populations, are supported by robust cross-checks, alternative noise models, and extensive validation; however, some spectral features and a marginal monopole component warrant caution and further data. The findings mark a significant milestone in low-frequency gravitational-wave astronomy and set the stage for enhanced discovery potential via the International Pulsar Timing Array in coming years.

Abstract

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. $3σ$) 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. $3.5 - 4σ$). 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.

Figures (18)

• Figure 1: Summary of the main Bayesian and optimal-statistic analyses presented in this paper, which establish multiple lines of evidence for the presence of Hellings--Downs correlations in the 15-year NANOGrav data set. Throughout we refer to the $68.3\%$, $95.4\%$, and $99.7\%$ regions of distributions as $1/2/3\sigma$ regions, even in two dimensions. (a): Bayesian "free-spectrum" analysis, showing posteriors (gray violins) of independent variance parameters for a Hellings--Downs-correlated stochastic process at frequencies $i/T$, with $T$ the total data set time span. The blue represents the posterior median and $1/2\sigma$ posterior bandsa for a power-law model; the dashed black line corresponds to a $\gamma=13/3$ (SMBHB-like) power-law, plotted with the median posterior amplitude. See §\ref{['sec:bayes']} for more details. (b): Posterior probability distribution of GWB amplitude and spectral exponent in a HD power-law model, showing $1/2/3\sigma$ credible regions. The value $\gamma_\mathrm{GWB}=13/3$ (dashed black line) is included in the $99\%$ credible region. The amplitude is referenced to $f_\mathrm{ref}=1\,\mathrm{yr}^{-1}$ (blue) and $0.1\,\mathrm{yr}^{-1}$ (orange). The dashed blue and orange curves in the $\log_{10} A_\mathrm{GWB}$ subpanel shows its marginal posterior density for a $\gamma = 13/3$ model, with $f_\mathrm{ref}=1\,\mathrm{yr}^{-1}$ and $f_\mathrm{ref}=0.1\,\mathrm{yr}^{-1}$, respectively. See §\ref{['sec:bayes']} for more details. (c): Angular-separation--binned inter-pulsar correlations, measured from 2,211 distinct pairings in our 67-pulsar array using the frequentist optimal statistic, assuming maximum-a-posteriori pulsar noise parameters and $\gamma=13/3$ common-process amplitude from a Bayesian inference analysis. The bin widths are chosen so that each includes approximately the same number of pulsar pairs, and central bin locations avoid zeros of the Hellings--Downs curve. This binned reconstruction accounts for correlations between pulsar pairs romano+20212022arXiv220807230A. The dashed black line shows the Hellings--Downs correlation pattern, and the binned points are normalized by the amplitude of the $\gamma=13/3$ common process to be on the same scale. Note that we do not employ binning of inter-pulsar correlations in our detection statistics; this panel serves as a visual consistency check only. See §\ref{['sec:optimal']} for more frequentist results. (d): Bayesian reconstruction of normalized inter-pulsar correlations, modeled as a cubic spline within a variable-exponent power-law model. The violins plot the marginal posterior densities (plus median and 68% credible values) of the correlations at the knots. The knot positions are fixed, and are chosen on the basis of features of the Hellings--Downs curve (also shown as a dashed black line for reference): they include the maximum and minimum angular separations, the two zero crossings of the Hellings--Downs curve, and the position of minimum correlation. See §\ref{['sec:bayes']} for more details.
• Figure 2: Bayes factors between models of correlated red noise in the NANOGrav 15-year data set (see §\ref{['subsec:correlation']} and App. \ref{['sec:bayesapp']}). All models feature variable-$\gamma$ power laws. curn${}^\gamma$ is vastly favored over irn (i.e., we find very strong evidence for common-spectrum excess noise over pulsar intrinsic red-noise alone); hd${}^\gamma$ is favored over curn${}^\gamma$ (i.e., we find positive evidence for Hellings--Downs correlations in the common-spectrum process); dipole and monopole processes are strongly disfavored with respect to curn${}^\gamma$; adding correlated processes to hd${}^\gamma$ is disfavored. While the interpretation of "raw" Bayes factors is somewhat subjective, they can be given a statistical significance within the hypothesis-testing framework by computing their background distributions and deriving the $p$-values of the observed factors, e.g., \ref{['fig:background']}.
• Figure 3: Empirical background distribution of hd${}^\gamma$-to-curn${}^\gamma$ Bayes factor (left, see §\ref{['sec:bayes']}) and noise-marginalized optimal statistic (right, see §\ref{['sec:optimal']}), as computed by the phase-shift technique tlb+17 to remove inter-pulsar correlations. We only compute 5,000 Bayesian phase shifts, compared to 400,000 optimal statistic phase shifts, because of the huge computational resources needed to perform the Bayesian analyses. For the optimal statistic, we also compute the background distribution using 27,000 simulations (orange line) and compare to an analytic calculation (green line). Dotted lines indicate Gaussian-equivalent 2$\sigma$, 3$\sigma$, and 4$\sigma$ thresholds. The dashed vertical lines indicate the values of the detection statistics for the unshifted data sets. For the Bayesian analyses, we find $p=10^{-3}$ (approx. $3\sigma$); for the optimal statistic analyses, we find $p = 5 \times 10^{-5}$--$1.9 \times 10^{-4}$ (approx. $3.5$--$4\sigma$).
• Figure 4: Optimal statistic S/N for HD correlations, distributed over curn${}^\gamma$ (solid lines) and curn${}^{13/3}$ (dashed lines) noise-parameter posteriors. The vertical lines indicate the mean S/Ns. We find S/Ns of $5 \pm 1$ and $4 \pm 1$ for curn${}^\gamma$ and curn${}^{13/3}$, respectively.
• Figure 5: curn${}^\gamma$ posterior distributions using DMGP (red) and DMX (blue) to model DM variations. The dashed line marks $\gamma_\mathrm{CURN} = 13/3$. While the posteriors are broadly consistent, DMGP shifts the $\gamma_\mathrm{CURN}$ posterior to higher values, making it more consistent with $\gamma_\mathrm{CURN}=13/3$.