We report multiple lines of evidence for a stochastic signal that is correlated among 67 pulsars from the 15 yr 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 10¹⁴, 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 the latter Bayes factors using a method that removes interpulsar correlations from our data set, finding p = 10⁻³ (≈3σ) for the observed Bayes factors in the null no-correlation scenario. A frequentist test statistic built directly as a weighted sum of interpulsar correlations yields p = 5 × 10⁻⁵ to 1.9 × 10⁻⁴ (≈3.5σ–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 − 0.6 + 0.7 × 10 − 15 (median + 90% credible interval) at a reference frequency of 1 yr⁻¹. 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.

Neutron stars are not only of astrophysical interest, but are also of great interest to nuclear physicists because their attributes can be used to determine the properties of the dense matter in their cores. One of the most informative approaches for determining the equation of state (EoS) of this dense matter is to measure both a star’s equatorial circumferential radius R _e and its gravitational mass M. Here we report estimates of the mass and radius of the isolated 205.53 Hz millisecond pulsar PSR J0030+0451 obtained using a Bayesian inference approach to analyze its energy-dependent thermal X-ray waveform, which was observed using the Neutron Star Interior Composition Explorer (NICER). This approach is thought to be less subject to systematic errors than other approaches for estimating neutron star radii. We explored a variety of emission patterns on the stellar surface. Our best-fit model has three oval, uniform-temperature emitting spots and provides an excellent description of the pulse waveform observed using NICER. The radius and mass estimates given by this model are km and (68%). The independent analysis reported in the companion paper by Riley et al. explores different emitting spot models, but finds spot shapes and locations and estimates of R _e and M that are consistent with those found in this work. We show that our measurements of R _e and M for PSR J0030+0451 improve the astrophysical constraints on the EoS of cold, catalyzed matter above nuclear saturation density.

We report on Bayesian parameter estimation of the mass and equatorial radius of the millisecond pulsar PSR J0030+0451, conditional on pulse-profile modeling of Neutron Star Interior Composition Explorer X-ray spectral-timing event data. We perform relativistic ray-tracing of thermal emission from hot regions of the pulsar’s surface. We assume two distinct hot regions based on two clear pulsed components in the phase-folded pulse-profile data; we explore a number of forms (morphologies and topologies) for each hot region, inferring their parameters in addition to the stellar mass and radius. For the family of models considered, the evidence (prior predictive probability of the data) strongly favors a model that permits both hot regions to be located in the same rotational hemisphere. Models wherein both hot regions are assumed to be simply connected circular single-temperature spots, in particular those where the spots are assumed to be reflection-symmetric with respect to the stellar origin, are strongly disfavored. For the inferred configuration, one hot region subtends an angular extent of only a few degrees (in spherical coordinates with origin at the stellar center) and we are insensitive to other structural details; the second hot region is far more azimuthally extended in the form of a narrow arc, thus requiring a larger number of parameters to describe. The inferred mass M and equatorial radius R _eq are, respectively, 1.34 − 0.16 + 0.15 M ⊙ and 12.71 − 1.19 + 1.14 km , while the compactness GM / R eq c 2 = 0.156 − 0.010 + 0.008 is more tightly constrained; the credible interval bounds reported here are approximately the 16% and 84% quantiles in marginal posterior mass.

PSR J0740+6620 has a gravitational mass of 2.08 ± 0.07 M _⊙, which is the highest reliably determined mass of any neutron star. As a result, a measurement of its radius will provide unique insight into the properties of neutron star core matter at high densities. Here we report a radius measurement based on fits of rotating hot spot patterns to Neutron Star Interior Composition Explorer (NICER) and X-ray Multi-Mirror (XMM-Newton) X-ray observations. We find that the equatorial circumferential radius of PSR J0740+6620 is 13.7 − 1.5 + 2.6 km (68%). We apply our measurement, combined with the previous NICER mass and radius measurement of PSR J0030+0451, the masses of two other ∼2 M _⊙ pulsars, and the tidal deformability constraints from two gravitational wave events, to three different frameworks for equation-of-state modeling, and find consistent results at ∼1.5–5 times nuclear saturation density. For a given framework, when all measurements are included, the radius of a 1.4 M _⊙ neutron star is known to ±4% (68% credibility) and the radius of a 2.08 M _⊙ neutron star is known to ±5%. The full radius range that spans the ±1σ credible intervals of all the radius estimates in the three frameworks is 12.45 ± 0.65 km for a 1.4 M _⊙ neutron star and 12.35 ± 0.75 km for a 2.08 M _⊙ neutron star.

We report on Bayesian estimation of the radius, mass, and hot surface regions of the massive millisecond pulsar PSR J0740+6620, conditional on pulse-profile modeling of Neutron Star Interior Composition Explorer X-ray Timing Instrument event data. We condition on informative pulsar mass, distance, and orbital inclination priors derived from the joint North American Nanohertz Observatory for Gravitational Waves and Canadian Hydrogen Intensity Mapping Experiment/Pulsar wideband radio timing measurements of Fonseca et al. We use XMM-Newton European Photon Imaging Camera spectroscopic event data to inform our X-ray likelihood function. The prior support of the pulsar radius is truncated at 16 km to ensure coverage of current dense matter models. We assume conservative priors on instrument calibration uncertainty. We constrain the equatorial radius and mass of PSR J0740+6620 to be 12.39 − 0.98 + 1.30 km and 2.072 − 0.066 + 0.067 M _⊙ respectively, each reported as the posterior credible interval bounded by the 16% and 84% quantiles, conditional on surface hot regions that are non-overlapping spherical caps of fully ionized hydrogen atmosphere with uniform effective temperature; a posteriori, the temperature is log 10 ( T [ K ] ) = 5.99 − 0.06 + 0.05 for each hot region. All software for the X-ray modeling framework is open-source and all data, model, and sample information is publicly available, including analysis notebooks and model modules in the Python language. Our marginal likelihood function of mass and equatorial radius is proportional to the marginal joint posterior density of those parameters (within the prior support) and can thus be computed from the posterior samples.

We report results from continued timing observations of PSR J0740+6620, a high-mass, 2.8 ms radio pulsar in orbit with a likely ultracool white dwarf companion. Our data set consists of combined pulse arrival-time measurements made with the 100 m Green Bank Telescope and the Canadian Hydrogen Intensity Mapping Experiment telescope. We explore the significance of timing-based phenomena arising from general relativistic dynamics and variations in pulse dispersion. When using various statistical methods, we find that combining ∼1.5 yr of additional, high-cadence timing data with previous measurements confirms and improves on previous estimates of relativistic effects within the PSR J0740+6620 system, with the pulsar mass m p = 2.08 − 0.07 + 0.07 M ⊙ (68.3% credibility) determined by the relativistic Shapiro time delay. For the first time, we measure secular variation in the orbital period and argue that this effect arises from apparent acceleration due to significant transverse motion. After incorporating contributions from Galactic differential rotation and off-plane acceleration in the Galactic potential, we obtain a model-dependent distance of d = 1.14 − 0.15 + 0.17 kpc (68.3% credibility). This improved distance confirms the ultracool nature of the white dwarf companion determined from recent optical observations. We discuss the prospects for future observations with next-generation facilities, which will likely improve the precision on m _p for J0740+6620 by an order of magnitude within the next few years.