We quantify pulsar timing array (PTA) sensitivity to anisotropy in the gravitational wave background using the cross-correlation based Fisher information matrix in the pixel and spherical harmonic bases. We use a set of simulations to empirically determine scaling relations of a PTA's sensitivity to anisotropy with the number of pulsars $N_\mathrm{psr}$ in the array, the error $δt$ on the times of arrival, the frequency $f_\mathrm{GW}$ of the gravitational waves, and the angular scale $ΔΩ$ of the anisotropy. The sensitivity scales approximately as $N_\mathrm{psr}^{0.8}$, $δt^{-0.08}$, and $ΔΩ^{1.6}-ΔΩ^{2.1}$ (depending on the ranges of $\ell$ and $m$ under consideration). In addition, we use realistic simulations to project the NANOGrav PTA sensitivity to a 30-year baseline and quantify the growth in sensitivity at several timeslices. Except at the lowest frequencies, we find negligible effect on sensitivity through increasing the observation duration only. Finally, we introduce a multi-resolution pixel basis motivated by the large dependence of the sensitivity on sky location, and demonstrate the operation of the basis through a set of injections and recoveries.

The observed gravitational-wave background (GWB) spectrum is higher in amplitude than model predictions by a factor of 2–3. Using a semi-analytic model, we evaluate the effect of a high-scatter supermassive black hole (SMBH) scaling relation ( M _BH – M _bulge ) on models of the nanohertz GWB. By implementing an intrinsic scatter of the M _BH – M _bulge relation, which is larger at higher redshift, but matches local observations, we find that the amplitude of GWB models increases to be consistent with the low-frequency end of the GWB spectrum. This amplitude increase is not uniform across frequencies, a strongly evolving scatter preferentially increases the number density of the most massive SMBHs which, in the GWB spectrum, minimizes the strength of the low-frequency turnover. Our models with positively evolving intrinsic scatter can reproduce the electromagnetically observed overmassive SMBHs at 4 < z < 6 without changing the M _BH – M _bulge normalization though we find that including moderate normalization evolution marginally improves fits to the GWB data. We conclude that the M _BH – M _bulge relation which best describes the available GWB and electromagnetic data sets has intrinsic scatter that evolves as $\varepsilon (z)={\varepsilon }_{0}+(0.56\pm 0.4){\mathrm{log}}_{10}(1+z)$ and normalization that evolves as α ( z ) = α _0 (1 + z ) ^0.84±0.35 . The results of this work imply that the M _BH – M _bulge relation we see today is not universal throughout cosmic time and that a diversity of seeding models and growth mechanisms may be at play in the early stages of SMBH–galaxy evolution.

We present the first targeted searches for continuous gravitational waves (CWs) from 114 active galactic nuclei that may host supermassive black hole binaries, using the NANOGrav 15 yr dataset. By incorporating electromagnetic priors on sky location, distance, redshift, and CW frequency, our strain and chirp-mass upper limits are typically improved by a factor of ∼2 (median 2.2) relative to all-sky limits at the same frequency. Bayesian comparisons against a model including only a Hellings–Downs-correlated background disfavors a CW signal for all targets, with a mean Bayes factor of 0.73 ± 0.32. Two targets have Bayes factors slightly above unity, but coherence tests, random-targeting experiments, and a conservative accounting of the 114-target trials factor all indicate that they are consistent with noise. We use these two candidates as worked examples to illustrate an end-to-end targeted CW search analysis and a suite of follow-up tests that future promising candidates would need to pass. We find that the electromagnetic interpretations of both candidates are ambiguous, and we update the constraints on a putative binary in 3C 66B, ruling out part of its previously allowed parameter space. Ultimately, our results demonstrate the current sensitivity of targeted pulsar timing array searches for CWs and define a road map for future multimessenger CW detections.

We test the impact of an evolving supermassive black hole mass scaling relation (M-BH-M-bulge) on the predictions for the gravitational-wave background (GWB). The observed GWB amplitude is 2-3 times higher than predicted by astrophysically informed models, which suggests the need to revise the assumptions in those models. We compare a semi-analytic model's ability to reproduce the observed GWB spectrum with a static versus evolving-amplitude M-BH-M-bulge relation. We additionally consider the influence of the choice of galaxy stellar mass function (GSMF) on the modeled GWB spectra. Our models are able to reproduce the GWB amplitude with either a large number density of massive galaxies or a positively evolving M-BH-M-bulge amplitude (i.e., the MBH/Mbulge ratio was higher in the past). If we assume that the M-BH-M-bulge amplitude does not evolve, our models require a GSMF that implies an undetected population of massive galaxies (M-star >= 10(11)M circle dot at z > 1). When the M-BH-M-bulge amplitude is allowed to evolve, we can model the GWB spectrum with all fiducial values and an M-BH-M-bulge amplitude that evolves as alpha(z) = alpha 0(1 + z)(1.04 +/- 0.5).

We report updated nHz gravitational wave (GW) significance, characterization, and interpretations using the customized chromatic-noise models (CNMs) developed in Larsen, Baier et al. (2026). for the NANOGrav 15-year data set. We find increased evidence for the Hellings-Downs (HD) correlation signature of the stochastic gravitational wave background (GWB), with a Bayes factor of 1571±14 for HD-correlations over a common uncorrelated red-noise process using a power-law model with 14 Fourier modes. We find this ∼8× increase in Bayes factor from Agazie et al. (2023a) is a result of improved noise mitigation. Assuming an analytic null distribution for the frequentist interpulsar correlation statistic, this corresponds to a slightly more significant measurement from 3.16σ to 3.32σ against the no-correlation scenario. Spectral inference with CNMs brings the power-law GWB amplitude down to A_ GWB = 2.1^+0.6_-0.5×10^-15 at fixed γ_ GWB = 13/3. In a varied-γ analysis, the spectral index increases to γ_ GWB=3.5^+0.7_-0.6. We report updates on an all-sky continuous gravitational wave (CW) search as well as select targeted searches and calculate a 3.2× larger detection volume for the NANOGrav detector. With CNMs, we find reduced evidence for a non-Einsteinian, scalar-transverse mode of gravity. Finally, we reinterpret the GWB first with the assumption of an astrophysical background sourced by SMBHBs and then assuming the more exotic origins of cosmic inflation, a first-order cosmological phase transition, and stable cosmic strings. Under both the SMBHB hypothesis and the cosmological hypotheses, we see only marginal shifts in model parameter posteriors which are consistent with the slightly quieter and steeper power-law GWB spectrum.

Abstract The observed gravitational-wave background (GWB) spectrum is higher in amplitude than model predictions by a factor of 2–3. Using a semi-analytic model, we evaluate the effect of a high-scatter supermassive black hole (SMBH) scaling relation ( M BH – M bulge ) on models of the nanohertz GWB. By implementing an intrinsic scatter of the M BH – M bulge relation, which is larger at higher redshift, but matches local observations, we find that the amplitude of GWB models increases to be consistent with the low-frequency end of the GWB spectrum. This amplitude increase is not uniform across frequencies, a strongly evolving scatter preferentially increases the number density of the most massive SMBHs which, in the GWB spectrum, minimizes the strength of the low-frequency turnover. Our models with positively evolving intrinsic scatter can reproduce the electromagnetically observed overmassive SMBHs at 4 < z < 6 without changing the M BH – M bulge normalization though we find that including moderate normalization evolution marginally improves fits to the GWB data. We conclude that the M BH – M bulge relation which best describes the available GWB and electromagnetic data sets has intrinsic scatter that evolves as ε ( z ) = ε 0 + ( 0.56 ± 0.4 ) log 10 ( 1 + z ) and normalization that evolves as α ( z ) = α 0 (1 + z ) 0.84±0.35 . The results of this work imply that the M BH – M bulge relation we see today is not universal throughout cosmic time and that a diversity of seeding models and growth mechanisms may be at play in the early stages of SMBH–galaxy evolution.

The observed GWB spectrum is higher in amplitude than model predictions by a factor of 2-3. Using a semi-analytic model, we evaluate the effect of a high-scatter supermassive black hole (SMBH) scaling relation (M_BH-M_bulge) on models of the nanohertz gravitational wave background (GWB). By implementing an intrinsic scatter of the M_BH-M_bulge relation, which is larger at higher redshift, but matches local observations, we find that the amplitude of GWB models increases to be consistent with the low-frequency end of the GWB spectrum. This amplitude increase is not uniform across frequencies, a strongly evolving scatter preferentially increases the number density of the most massive SMBHs which, in the GWB spectrum, minimizes the strength of the low-frequency turnover. Our models with positively evolving intrinsic scatter can reproduce the electromagnetically observed overmassive SMBHs at 4 < z < 6 without changing the M_BH-M_bulge normalization though we find that including moderate normalization evolution marginally improves fits to the GWB data. We conclude that the M_BH-M_bulge relation which best describes the available GWB and electromagnetic data sets has intrinsic scatter that evolves as ε(z) = ε_0 + (0.56 ± 0.4) log_10(1 + z) and normalization that evolves as α(z) = α_0 (1 + z)^0.84 ± 0.35. The results of this work imply that the M_BH-M_bulge relation we see today is not universal throughout cosmic time and that a diversity of seeding models and growth mechanisms may be at play in the early stages of SMBH-galaxy evolution.

In the published article, we presented inferences about the population of supermassive black hole (SMBH) binaries emitting at nanohertz gravitational-wave (GW) frequencies based on the 15 yr dataset from the North American Nanohertz Observatory for Gravitational Waves (NANOGrav). Here, we report two bugs in the astrophysical analysis software used for the published article. While parameter posteriors of our inference can be seen to have slight differences on close inspection, none of the conclusions from the original publication have changed, including the finding that the measured GW background is dominated by the most massive, high-mass-ratio SMBH binaries. Our corrected results show slightly less evidence for environmental interactions with SMBH binaries and indicate that the population of SMBH binaries contributing to the GW background occurs at slightly higher mass ratios and redshifts than indicated in the published article. Correcting these errors does not change the central conclusion that astrophysically motivated models of SMBH binary populations are able to reproduce both the amplitude and shape of the observed low-frequency GW spectrum. These errors did not affect any other analyses of the NANOGrav 15 yr dataset, including the evidence for a GW background presented in another of our published articles.

The NANOGrav 15 yr (NG15) data set provides evidence for a gravitational-wave background (GWB) signal at nHz frequencies, which is expected to originate either from a cosmic population of inspiraling supermassive black hole binaries or new particle physics in the early Universe. A firm identification of the source of the NG15 signal requires an accurate reconstruction of its frequency spectrum. In this Letter, we provide such a spectral characterization of the NG15 signal based on a piecewise power-law (PPL) ansatz that strikes a balance between existing alternatives in the literature. Our PPL reconstruction is more flexible than the standard constant power-law model, which describes the GWB spectrum in terms of only two parameters: an amplitude A and a spectral index gamma. Concurrently, it better approximates physically realistic GWB spectra-especially those of cosmological origin-than the free spectral model, since the latter allows for arbitrary variations in the GWB amplitude from one frequency bin to the next. Our PPL reconstruction of the NG15 signal relies on individual PPL models with a fixed number of internal nodes (i.e., constant power law, broken power law, doubly broken power law, etc.), which are ultimately combined in a Bayesian model average. The data products resulting from our analysis provide the basis for fast refits of spectral GWB models.

Evidence has emerged for a stochastic signal correlated among 67 pulsars within the 15 yr pulsar-timing data set compiled by the NANOGrav collaboration. Similar signals have been found in data from the European, Indian, Parkes, and Chinese pulsar timing arrays. This signal has been interpreted as indicative of the presence of a nanohertz stochastic gravitational-wave background (GWB). To explore the internal consistency of this result, we investigate how the recovered signal strength changes as we remove the pulsars one by one from the data set. We calculate the signal strength using the (noise-marginalized) optimal statistic, a frequentist metric designed to measure the correlated excess power in the residuals of the arrival times of the radio pulses. We identify several features emerging from this analysis that were initially unexpected. The significance of these features, however, can only be assessed by comparing the real data to synthetic data sets. After conducting identical analyses on simulated data sets, we do not find anything inconsistent with the presence of a stochastic GWB in the NANOGrav 15 yr data. The methodologies developed here can offer additional tools for application to future, more sensitive data sets. While this analysis provides an internal consistency check of the NANOGrav results, it does not eliminate the necessity for additional investigations that could identify potential systematics or uncover unmodeled physical phenomena in the data.

The cosmic merger history of supermassive black hole binaries (SMBHBs) is expected to produce a low-frequency gravitational wave background (GWB). Here we investigate how signs of the discrete nature of this GWB can manifest in pulsar timing arrays (PTAs) through excursions from, and breaks in, the expected fGW-2/3 power law of the GWB strain spectrum. To do this, we create a semianalytic SMBHB population model, fit to North American Nanohertz Observatory for Gravitational Waves (NANOGrav's) 15 yr GWB amplitude, and with 1000 realizations, we study the populations' characteristic strain and residual spectra. Comparing our models to the NANOGrav 15 yr spectrum, we find two interesting excursions from the power law. The first, at 2 nHz, is below our GWB realizations with a p-value significance p = 0.05-0.06 (approximate to 1.8 sigma-1.9 sigma). The second, at 16 nHz, is above our GWB realizations with p = 0.04-0.15 (approximate to 1.4 sigma-2.1 sigma). We explore the properties of a loud SMBHB that could cause such an excursion. Our simulations also show that the expected number of SMBHBs decreases by 3 orders of magnitude, from similar to 106 to similar to 103, between 2 and 20 nHz. This causes a break in the strain spectrum as the stochasticity of the background breaks down at 26-19+28nHz , consistent with predictions pre-dating GWB measurements. The diminished GWB signal from SMBHBs at frequencies above the 26 nHz break opens a window for PTAs to detect continuous GWs from individual SMBHBs or GWs from the early Universe.

We conducted high-precision timing of PSR J1946+2052 to determine the masses of the two neutron stars in the system, test general relativity (GR) and assessed the system's potential for future measurement of the moment of inertia of the pulsar. We analysed seven years of timing data from the Arecibo 305-m radio telescope, the Green Bank Telescope (GBT), and the Five-hundred-meter Aperture Spherical radio Telescope (FAST). The data processing accounted for dispersion measure variations and relativistic spin precession-induced profile evolution. We employed both DDFWHE and DDGR binary models to measure the spin parameters, kinematic parameters and orbital parameters. The timing campaign has resulted in the precise measurement of five post-Keplerian parameters, which yield very precise masses for the system and three tests of general relativity. One of these is the second most precise test of the radiative properties of gravity to date: the intrinsic orbital decay, Ṗ_ b,int=-1.8288(16)×10^-12 s s^-1, represents 1.00005(91) of the GR prediction, indicating that the theory has passed this stringent test. The other two tests, of the Shapiro delay parameters, have precisions of 6% and 5% respectively; this is caused by the moderate orbital inclination of the system, ∼ 74^∘; the measurements of the Shapiro delay parameters also agree with the GR predictions. Additionally, we analysed the higher-order contributions of , including the Lense-Thirring contribution. Both the second post-Newtonian and the Lense-Thirring contributions are larger than the current uncertainty of (δ=4×10^-4 deg yr^-1), leading to the higher-order correction for the total mass.

Pulsar timing array observations have found evidence for an isotropic gravitational-wave background with the Hellings–Downs angular correlations between pulsar pairs. This interpretation hinges on the measured shape of the angular correlations, which is predominantly 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 yr pulsar timing data set collected by the North American Nanohertz Observatory for Gravitational Waves (NANOGrav). When including Legendre polynomials with multipoles ℓ ≥ 2, we only find a significant signal in the quadrupole with an amplitude consistent with general relativity and nonzero at the ∼95% confidence level and a Bayes factor of 200. When we include multipoles ℓ ≤ 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 yr 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.

Aims. Double neutron star (DNS) systems are superb laboratories for testing theories of gravity and constraining the equation of state of ultra-dense matter. PSR J1946+2052 is a particularly intriguing DNS system due to its orbital period (1h 53 m), as it is the shortest among all DNS systems known in our Galaxy. Methods. We aim to conduct high-precision timing of PSR J1946+2052 to determine the masses of the two neutron stars in the system, test general relativity (GR), and assess the system's potential for future measurement of the moment of inertia of the pulsar. Results. We analysed seven years of timing data from the Arecibo 305-m radio telescope, the Green Bank Telescope, and the Five-hundred-meter Aperture Spherical radio Telescope. The data processing accounted for dispersion measure variations and relativistic spin precession-induced profile evolution. We employed both theory-independent (DDFWHE) and GR-dependent (DDGR) binary models to measure the spin parameters, kinematic parameters, and orbital parameters. Results. The timing campaign resulted in the precise measurement of five post-Keplerian parameters, which yield very precise masses for the system (total mass M = 2.531858(60) M-circle dot, companion mass M-c = 1.2480(21) M-circle dot, and pulsar mass M-p = 1.2838(21) M-circle dot), and three tests of GR. One of these tests is the second-most precise test of the radiative properties of gravity to date. The intrinsic orbital decay, P-b,P-int = -1.8288(16) x 10(-12), s s(-1), represents 1.00005(91) of the GR prediction, indicating that the theory has passed this stringent test. The other two tests of the Shapiro delay parameters have precisions of 6% and 5%, respectively. This is caused by the moderate orbital inclination of the system, similar to 74 degrees. The measurements of the Shapiro delay parameters also agree with the GR predictions. Additionally, we analysed the higher-order contributions of omega(center dot), including the Lense-Thirring contribution. Both the second post-Newtonian and the Lense-Thirring contributions are larger than the current uncertainty of omega(center dot)( delta omega(center dot) = 4 x 10(-4) deg yr(-1)), leading to the higher-order correction for the total mass.

The NANOGrav 15 yr data provide compelling evidence for a stochastic gravitational-wave (GW) background at nanohertz frequencies. The simplest model-independent approach to characterizing the frequency spectrum of this signal consists of a simple power-law fit involving two parameters: an amplitude A and a spectral index gamma. In this Letter, we consider the next logical step beyond this minimal spectral model, allowing for a running (i.e., logarithmic frequency dependence) of the spectral index, gamma(run)(f) = gamma + beta ln (f/f(ref)). We fit this running-power-law (RPL) model to the NANOGrav 15 yr data and perform a Bayesian model comparison with the minimal constant-power-law (CPL) model, which results in a 95% credible interval for the parameter beta consistent with no running, beta is an element of[-0.80, 2.96], and an inconclusive Bayes factor, B(RPL versus CPL) = 0.69 +/- 0.01. We thus conclude that, at present, the minimal CPL model still suffices to adequately describe the NANOGrav signal; however, future data sets may well lead to a measurement of nonzero beta. Finally, we interpret the RPL model as a description of primordial GWs generated during cosmic inflation, which allows us to combine our results with upper limits from Big Bang nucleosynthesis, the cosmic microwave background, and LIGO-Virgo-KAGRA.

Pulsar timing array experiments have reported evidence for a stochastic background of nanohertz gravitational waves consistent with the signal expected from a population of supermassive black hole binaries. Their analyses assume power-law spectra for intrinsic pulsar noise and for the background, as well as a Hellings-Downs cross-correlation pattern among the gravitational-wave- induced residuals across pulsars. These assumptions may not be realized in actuality. We test them in the NANOGrav 15 yr dataset using Bayesian posterior predictive checks. After fitting our fiducial model to real data, we generate a population of simulated dataset replications. We use the replications to assess whether the optimal statistic significance, interpulsar correlations, and spectral coefficients are extreme. We recover Hellings-Downs correlations in simulated datasets at significance levels consistent with the correlations measured in the NANOGrav 15 yr dataset. A similar test on spectral coefficients shows that their values in real data are not extreme compared to their distributions across replications. We also evaluate the evidence for the stochastic background using posterior predictive versions of the frequentist optimal statistic and of Bayesian model comparison and find comparable significance (3.26 and 36 respectively) to what was previously reported for the standard statistics. We conclude with novel visualizations of the reconstructed gravitational waveforms that enter the residuals for each pulsar. Our analysis strengthens confidence in the identification and characterization of the gravitational-wave background.

Aims. Double neutron star (DNS) systems are superb laboratories for testing theories of gravity and constraining the equation of state of ultra-dense matter. PSR J1946+2052 is a particularly intriguing DNS system due to its orbital period (1h 53 m), as it is the shortest among all DNS systems known in our Galaxy. Methods. We aim to conduct high-precision timing of PSR J1946+2052 to determine the masses of the two neutron stars in the system, test general relativity (GR), and assess the system’s potential for future measurement of the moment of inertia of the pulsar. Results. We analysed seven years of timing data from the Arecibo 305-m radio telescope, the Green Bank Telescope, and the Five-hundred-meter Aperture Spherical radio Telescope. The data processing accounted for dispersion measure variations and relativistic spin precession-induced profile evolution. We employed both theory-independent (DDFWHE) and GR-dependent (DDGR) binary models to measure the spin parameters, kinematic parameters, and orbital parameters. Results. The timing campaign resulted in the precise measurement of five post-Keplerian parameters, which yield very precise masses for the system (total mass M = 2.531858(60) M ⊙ , companion mass M c = 1.2480(21) M ⊙ , and pulsar mass M p = 1.2838(21) M ⊙ ), and three tests of GR. One of these tests is the second-most precise test of the radiative properties of gravity to date. The intrinsic orbital decay, Ṗ b,int = −1.8288(16) × 10 −12 , s s −1 , represents 1.00005(91) of the GR prediction, indicating that the theory has passed this stringent test. The other two tests of the Shapiro delay parameters have precisions of 6% and 5%, respectively. This is caused by the moderate orbital inclination of the system, ∼74°. The measurements of the Shapiro delay parameters also agree with the GR predictions. Additionally, we analysed the higher-order contributions of ω ˙, including the Lense-Thirring contribution. Both the second post-Newtonian and the Lense-Thirring contributions are larger than the current uncertainty of ω ˙ ( δω ˙ = 4 × 10 −4 deg yr −1 ), leading to the higher-order correction for the total mass.

We present the results of a search for nonlinear gravitational-wave (GW) memory in the NANOGrav 15 yr data set. We find no significant evidence for memory signals in the data set, with a maximum Bayes factor of 3.1 in favor of a model including memory. We therefore place upper limits on the strain of potential GW memory events as a function of sky location and observing epoch. We find upper limits that are not always more constraining than previous NANOGrav results. We show that it is likely due to the increase in common red noise between the 12.5 and 15 yr NANOGrav data sets.

Recently we found compelling evidence for a gravitational wave background with Hellings and Downs (HD) correlations in our 15-year data set. These correlations describe gravitational waves as predicted by general relativity, which has two transverse polarization modes. However, more general metric theories of gravity can have additional polarization modes which produce different interpulsar correlations. In this work we search the NANOGrav 15-year data set for evidence of a gravitational wave background with quadrupolar Hellings and Downs (HD) and Scalar Transverse (ST) correlations. We find that HD correlations are the best fit to the data, and no significant evidence in favor of ST correlations. While Bayes factors show strong evidence for a correlated signal, the data does not strongly prefer either correlation signature, with Bayes factors $\sim 2$ when comparing HD to ST correlations, and $\sim 1$ for HD plus ST correlations to HD correlations alone. However, when modeled alongside HD correlations, the amplitude and spectral index posteriors for ST correlations are uninformative, with the HD process accounting for the vast majority of the total signal. Using the optimal statistic, a frequentist technique that focuses on the pulsar-pair cross-correlations, we find median signal-to-noise-ratios of 5.0 for HD and 4.6 for ST correlations when fit for separately, and median signal-to-noise-ratios of 3.5 for HD and 3.0 for ST correlations when fit for simultaneously. While the signal-to-noise-ratios for each of the correlations are comparable, the estimated amplitude and spectral index for HD are a significantly better fit to the total signal, in agreement with our Bayesian analysis.