Pulsar timing arrays (PTAs) have recently entered the detection era, quickly moving beyond the goal of simply improving sensitivity at the lowest frequencies for the sake of observing the stochastic gravitational wave background (GWB), and focusing on its accurate spectral characterization. While all PTA collaborations around the world use Fourier-domain Gaussian processes to model the GWB and intrinsic long time-correlated (red) noise, techniques to model the time-correlated radio-frequency-dependent (chromatic) processes have varied from collaboration to collaboration. Here we test a new class of models for PTA data, Gaussian processes based on time-domain kernels that model the statistics of the chromatic processes starting from the covariance matrix. As we will show, these models can be effectively equivalent to Fourier-domain models in mitigating chromatic noise. This work presents a method for Bayesian model selection across the various choices of kernel as well as deterministic chromatic models for nonstationary chromatic events and the solar wind. As PTAs turn toward high frequency (>1 yr(-1)) sensitivity, the size of the basis used to model these processes will need to increase, and these time-domain models present some computational efficiencies compared to Fourier-domain models.

X-ray pulsars are highly magnetized (B∼ 10^12 G) neutron stars accreting from a donor star. Their characteristic X-ray emission arises from accreted material decelerated from relativistic velocities near the magnetic poles of the neutron star. As our line of sight onto the magnetic poles changes with the rotation of the neutron star, the X-rays are periodically modulated, resulting in X-ray pulsations. The shape of the pulse profiles depends on the physics of the interaction between the bright X-rays from the magnetic poles with the infalling matter, the location of the magnetic poles on the neutron star with respect to its spin axis, and on the properties of the space-time around the neutron star. In this review we give a pedagogical introduction to the accretion mechanisms operating in the various types of accreting neutron star systems and the observational techniques used to characterize the pulse profiles. We summarize how the pulse profiles depend on X-ray luminosity and energy and discuss the attempts to connect theoretically these observables with the physical accretion mechanisms. We conclude with an outline of future observational needs and further developments for theoretical models of magnetic accretion.

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.

We present timing solutions spanning nearly two decades for five redback (RB) systems found in globular clusters (GCs), created using a novel technique that effectively "isolates" the pulsar. By accurately measuring the time of passage through periastron (T0) at points over the timing baseline, we use a piecewise-continuous binary model to get local solutions of the orbital variations that we pair with long-term orbital information to remove the orbital timing delays. The isolated pulse times of arrival can then be fit to describe the spin behavior of the millisecond pulsar (MSP). The results of our timing analyses via this method are consistent with those of conventional timing methods for binaries in GCs as demonstrated by analyses of NGC 6440D. We also investigate the observed orbital phase variations for these systems. Quasiperiodic oscillations in Terzan 5P's orbit may be the result of changes to the gravitational quadruple moment of the companion as prescribed by the Applegate model. We find a striking correlation between the standard deviation of the phase variations as a fraction of a system's orbit ( sigma Delta T0 ) and the MSP's spin frequency, as well as a potential correlation between sigma Delta T0 and the binary's projected semimajor axis. While long-term RB timing is fraught with large systematics, our work provides a needed alternative for studying systems with significant orbital variations, especially when high-cadence monitoring observations are unavailable.

Evidence for a low-frequency gravitational-wave background using pulsar timing arrays has generated recent interest into its underlying contributing sources. However, multiple investigations have seen that the significance of the evidence does not change with choice of pulsar modeling techniques but the resulting parameters from the gravitational wave searches do. PSR J1455-3330 is one of the longest-observed pulsars in the array monitored by the North American Nanohertz Observatory for Gravitational Waves (NANOGrav) but showed no evidence for long-timescale red noise, either intrinsic or the common signal found among many pulsars in the array. In this work, we argue that NANOGrav's piecewise-constant function used to model variations in radio-frequency-dependent dispersive delay should not be used for this pulsar, and a much simpler physical model of a fixed solar wind density plus a linear trend in dispersion measure is preferred. When the original model is replaced, (i) the pulsar's timing parallax signal changes from an upper limit to a significant detection, (ii) red noise becomes significant, and (iii) the red noise is consistent with the common signal found for the other pulsars. Neither of these signals are radio-frequency dependent. While the same physical motivation will not apply to many of the pulsars currently used in pulsar timing arrays, we argue for careful physically-motivated timing and noise modeling of pulsars used in precision timing experiments.

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.

Wideband timing of the North American Nanohertz Observatory for Gravitational Waves (NANOGrav) datasets was first done for the 12.5 yr dataset. This method, where a single time of arrival and a single dispersion measure (DM) are measured using the entire bandwidth of each observation, proved to be invaluable for characterizing the time-varying DM, improving handling of frequency-dependent profile variability, as well as data volume reduction. The Canadian Hydrogen Intensity Mapping Experiment (CHIME) Telescope has been observing most NANOGrav millisecond pulsars (MSPs) at nearly daily cadence (compared to roughly monthly cadence for other NANOGrav observations) since 2019, with the objective of integration into future pulsar timing array datasets. In this paper, we show the results of integration of high-cadence, low-observing-frequency CHIME data with data from the NANOGrav experiment for an isolated MSP PSR J0645+5158 and three binary MSPs: PSR J1012+5307, PSR J2145-0750, and PSR J2302+4442. Using a wideband timing pipeline, which we also describe, we present updated timing results for all four sources, including improved relativistic post-Keplerian measurements for the three binary pulsars in this analysis. For PSR J2302+4442, we report an updated strong detection of Shapiro delay from which we measured a companion mass of 0.35-0.04+0.05M circle dot , a pulsar mass of 1.8-0.3+0.3M circle dot , and an orbital inclination of 80 degrees-2+1 . We also report updated constraints on the reflex motion for PSR J2145-0750 using a combination of Very Long Baseline Array astrometry and our updated measurement of the time derivative of the projected semimajor axis of the pulsar orbit as a prior.

In this study, we use the Bayesian methods in the Enterprise package to examine the fully general parameterization of pulsar timing models in tandem with noise. We investigate four pulsars, PSR J1600-3053, PSR J2043+1711, PSR J0740+6620, and PSR J1640+2224, through the lens of Bayesian timing. These four are selected as they are well-studied, but exhibit interesting characteristics under the lens of Bayesian timing. Our new pulsar mass constraints (medians and 68% confidence intervals) for our fully general non-linear Bayesian timing models are m_p=1.6(1) M_⊙ for PSR J2043+1711 and m_p=2.3^+0.9_-0.7 M_⊙ for PSR J1600-3053 both using the NANOGrav 12.5-yr data release, and m_p=2.06(6) M_⊙ for PSR J0740+6620 using the data from Fonseca, et al., 2021. We investigate the effects on placing physical priors on timing model parameters, including restricting the upper limit on the pulsar mass for PSR J1640+2224, which has a mass often estimated to be greater than 3 M_⊙. We find that restricting the allowed sampling space of the pulsar mass for PSR J1640+2224 to m_p<3 M_⊙ results in a pulsar mass of m_p=2.2(5) M_⊙ for PSR J1640+2224 using the NANOGrav 12.5-yr data release. For the first time, we find evidence for intrinsic red noise in PSR J2043+1711. We show how fully general Bayesian timing can better model the interplay of the intrinsic noise and the timing parameters.

Accurate pulsar astrometric estimates are essential to almost all high-precision pulsar timing experiments. Traditional pulsar timing techniques refine these estimates by including them as free parameters when fitting a model to observed pulse time-of-arrival measurements. However, reliable submilliarcsecond astrometric estimations require years of observations. Even then, power from red noise can be inadvertently absorbed into astrometric parameter fits. This effect biases the resulting estimates and reduces the sensitivity to red noise processes, including gravitational waves (GWs). In this work, we seek to mitigate these shortcomings by using pulsar astrometric estimates derived from very long baseline interferometry (VLBI) as priors for the timing fit. First, we used VLBI and timing astrometric estimates of 18 millisecond pulsars to calibrate a rotation between the reference frames used in timing and VLBI, with a precision of similar to 0.7 mas. Through this frame tie, we combined timing- and VLBI-based probabilities to obtain a maximum-posterior astrometric solution. We found offsets between our results and the timing-based astrometric solutions, which, if real, would lead to the absorption of spectral power at the frequencies of interest for single-source GW searches. However, we do not find significant power absorption due to astrometric fitting at the low-frequency domain of the GW background.

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.

Pulsar timing arrays search for stochastic processes such as gravitational waves by comparing pulse time of arrival data for millisecond pulsars to expectations from a background with a given power spectral density (PSD). To make the analysis computationally tractable, the Bayesian likelihood is usually computed using an approximation in which the signal is taken to be a sum of Fourier modes appropriate to the total time of observation, even though the true signal is not periodic. We study the difference between likelihoods computed with this Fourier approximation method for power law spectra and those computed exactly (or using more-closely spaced frequencies as a proxy for the exact result) in the NANOGrav 15-year dataset. We find that the true marginal likelihoods for power-law PSDs are on average about half as large as the likelihoods computed using the Fourier approximation. This could lead to an error of a factor of two in model comparison. However, in the important comparison of uncorrelated vs. Hellings-Downs correlated models, a very similar correction appears in both, so the model comparison is essentially unaffected. We also compare parameter estimation results for power law PSDs, finding little difference between the methods. We briefly discuss spectra with sharper features, for which the approximation could be much worse.

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).

Free-floating objects (FFOs) in interstellar space—rogue planets, brown dwarfs, and large asteroids that are not gravitationally bound to any star—are expected to be ubiquitous throughout the Milky Way. Recent microlensing surveys have discovered several free-floating planets that are not bound to any known stellar systems. Additionally, three interstellar objects, namely 1I/’Oumuamua, 2I/Borisov, and 3I/ATLAS, have been detected passing through our solar system on hyperbolic trajectories. In this work, we search for FFOs on hyperbolic orbits that pass near millisecond pulsars (MSPs), where their gravitational influence can induce detectable perturbations in pulse arrival times. Using the NANOGrav 15 yr narrow band dataset, which contains high-precision timing data for 68 MSPs, we conduct a search for such hyperbolic scattering events between FFOs and pulsars. Although no statistically significant events were detected, this nondetection enables us to place upper limits (ULs) on the number density of FFOs as a function of their mass within our local region of the Galaxy. For example, the UL on the number density for Jupiter-mass FFOs (∼10 ^−2.5 –10 ^−3.5 M _⊙ ) obtained from different pulsars ranges from 5.25 × 10 ^6 pc ^−3 to 5.37 × 10 ^9 pc ^−3 , while the UL calculated by combining results from all the pulsars is 6.03 × 10 ^5 pc ^−3 . These results represent the first constraints on FFO population derived from pulsar timing data.

Pulsar timing arrays (PTAs) conduct low-frequency gravitational-wave (GW) searches, which require comprehensive accounting of various noise sources to achieve robust results. Interstellar propagation effects (e.g., dispersion and scattering) are especially complex noise sources, introducing chromatic delays that can reduce sensitivity to GWs and bias their inference if left unmodeled. These delays also strongly depend on the line-of-sight properties to each individual pulsar. To address this, we present customized chromatic noise models for 67 pulsars in the NANOGrav 15 yr dataset. These models are selected from an expanded suite of Gaussian processes to simultaneously characterize multiple types of chromatic delays and are tailored to each pulsar's dataset. Alongside probing the interstellar medium, we use these models to infer the solar wind electron density over the course of similar to 1.5 solar cycles. We also find evidence for nondispersive chromatic delays in 21 out of 67 NANOGrav pulsars. After applying our chromatic models, we observe significant impacts on the inference of achromatic noise in 19 out of 67 pulsars, finding in several cases that a previously significant achromatic noise process can be partially or entirely described as chromatic. These results demonstrate that refined noise modeling is essential to enhance the sensitivity and accuracy of low-frequency GW searches with PTAs.

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 North American Nanohertz Observatory for Gravitational Waves (NANOGrav) achieves sub-microsecond timing precision for several millisecond pulsars in its pulsar timing array (PTA) with the objective of detecting and characterizing nanohertz gravitational waves. PSR J1713+0747 is one of the most precisely timed pulsars in the array, achieving sub-microsecond timing precision. However, in April 2021, PSR J1713+0747 underwent a sudden and unusual change in pulse shape that disrupted its timing stability. As PSR J1713+0747 is a key contributor to PTA sensitivity, variations in its pulse profile significantly affect the array's sensitivity to nanohertz gravitational waves. We apply frequency-dependent Gaussian component models to decompose the pulse profile and track the evolution of individual components through the event. This component-level method maintains phase-connected timing across the shape-change event. At L-band, the recovered TOAs have a median uncertainty of 0.47 microseconds compared to 0.69 microseconds for standard template matching. At 820 MHz, where profile evolution is stronger, the recovered TOAs have a median uncertainty of 1.63 microseconds compared to 0.67 microseconds for standard template matching. The recovered TOAs achieve timing uncertainties comparable to conventional template matching while allowing data affected by profile variability to be retained in PTA gravitational-wave analyses. These results represent an initial step toward profile-domain timing methods capable of accounting for pulse-profile evolution while reducing the need for additional timing model parameters.

Trojan asteroids are found in the equilateral triangle Lagrange points of the Sun–Jupiter system in a great number, although they also exist less prolifically in other parts of the solar system. Despite up to planetary mass Trojans being predicted in extrasolar systems (i.e., exotrojans), they remain unconfirmed, although strong candidate evidence has emerged recently. For the first time, we extend the search for exotrojans to radio pulsars with low-mass (∼0.01 M _⊙ ) companions using accurately measured pulse times of arrival. With techniques developed for detecting the reflex motion of a star due to a librating Trojan, we place ∼1 M _⊕ upper mass constraints on potential exotrojans around eight pulsars observed in the NANOGrav 15 yr dataset. We find weak evidence consistent with ∼2–4 M _J exotrojans in the PSR J0023+0923 and PSR J1705−1903 binary systems, although the signals likely have a different, unknown source. We also place a libration-independent upper mass constraint of ∼8 M _J on exotrojans in the PSR J1641+8049 system by looking for an inconsistency between the times of superior conjunction as measured by optical light curves and those predicted by radio timing. These results offer initial observational constraints on the existence of exotrojans around pulsars, while their possible formation mechanisms remain unexplored.

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.