Beyond the Hellings–Downs curve: Non-Einsteinian gravitational waves in pulsar timing array correlations

We present observations and timing analyses of 68 millisecond pulsars (MSPs) comprising the 15 yr data set of the North American Nanohertz Observatory for Gravitational Waves (NANOGrav). NANOGrav is a pulsar timing array (PTA) experiment that is sensitive to low-frequency gravitational waves (GWs). This is NANOGrav’s fifth public data release, including both “narrowband” and “wideband” time-of-arrival (TOA) measurements and corresponding pulsar timing models. We have added 21 MSPs and extended our timing baselines by 3 yr, now spanning nearly 16 yr for some of our sources. The data were collected using the Arecibo Observatory, the Green Bank Telescope, and the Very Large Array between frequencies of 327 MHz and 3 GHz, with most sources observed approximately monthly. A number of notable methodological and procedural changes were made compared to our previous data sets. These improve the overall quality of the TOA data set and are part of the transition to new pulsar timing and PTA analysis software packages. For the first time, our data products are accompanied by a full suite of software to reproduce data reduction, analysis, and results. Our timing models include a variety of newly detected astrometric and binary pulsar parameters, including several significant improvements to pulsar mass constraints. We find that the time series of 23 pulsars contain detectable levels of red noise, 10 of which are new measurements. In this data set, we find evidence for a stochastic GW background.

We search for an isotropic stochastic gravitational-wave background (GWB) in the newly released 11 year data set from the North American Nanohertz Observatory for Gravitational Waves (NANOGrav). While we find no evidence for a GWB, we place constraints on a population of inspiraling supermassive black hole (SMBH) binaries, a network of decaying cosmic strings, and a primordial GWB. For the first time, we find that the GWB constraints are sensitive to the solar system ephemeris (SSE) model used and that SSE errors can mimic a GWB signal. We developed an approach that bridges systematic SSE differences, producing the first pulsar-timing array (PTA) constraints that are robust against SSE errors. We thus place a 95% upper limit on the GW-strain amplitude of A GWB < 1.45 × 10−15 at a frequency of f = 1 yr−1 for a fiducial f −2/3 power-law spectrum and with interpulsar correlations modeled. This is a factor of ∼2 improvement over the NANOGrav nine-year limit calculated using the same procedure. Previous PTA upper limits on the GWB (as well as their astrophysical and cosmological interpretations) will need revision in light of SSE systematic errors. We use our constraints to characterize the combined influence on the GWB of the stellar mass density in galactic cores, the eccentricity of SMBH binaries, and SMBH–galactic-bulge scaling relationships. We constrain the cosmic-string tension using recent simulations, yielding an SSE-marginalized 95% upper limit of Gμ < 5.3 × 10−11—a factor of ∼2 better than the published NANOGrav nine-year constraints. Our SSE-marginalized 95% upper limit on the energy density of a primordial GWB (for a radiation-dominated post-inflation universe) is ΩGWB(f) h 2 < 3.4 × 10−10.

The observations from pulsar timing arrays (PTAs), led by the North American Nanohertz Observatory for Gravitational Waves (NANOGrav), have provided opportunities to constrain primordial gravitational waves at low frequencies. In this paper, we analyze the best-fit parameter values for different Gauss-Bonnet Inflationary Gravitational Wave (GB-IGW) models with the PTArcade program, and we compare the results with the observations of NANOGrav, European Pulsar Timing Array (EPTA), Parkes Pulsar Timing Array (PPTA), and International Pulsar Timing Array (IPTA). We find the potential parameter n derived from the GB-IGW model is not necessarily positive. Instead, PTA data suggest the possibility of new parameter ranges. Meanwhile, the other parameters' results, such as the spectral indices of the scalar and tensor perturbation also have differences with traditional Cosmic Microwave Background analyses from Planck 2018, showing the reason for the different phenomena of Gravitational Wave power spectra expected. Additionally, the GB-IGW models we calculated are closer to the central values of multiple PTA datasets compared to the standard inflation model, making the GB-IGW model more likely to be detected by future space-based gravitational wave observatories.

We study the detectability of circular polarization in a stochastic gravitational wave background from various sources such as supermassive black hole binaries, cosmic strings, and inflation in the early universe with pulsar timing arrays. We calculate generalized overlap reduction functions for the circularly polarized stochastic gravitational wave background. We find that the circular polarization can not be detected for an isotropic background. However, there is a chance to observe the circular polarization for an anisotropic gravitational wave background. We also show how to separate polarized gravitational waves from unpolarized gravitational waves.

Primordial black holes (PBHs) can constitute the predominant fraction of dark matter (DM) if PBHs reside in the currently unconstrained “sublunar” mass range. PBHs originating from scalar perturbations generated during inflation can naturally appear with a broad spectrum in a class of models. The resulting stochastic gravitational wave (GW) background generated from such PBH production can account for the recently reported North American Nanohertz Observatory for Gravitational Waves (NANOGrav) pulsar timing array data signal, and will be testable in future GW observations by interferometer-type experiments such as Laser Interferometer Space Antenna (LISA). We show that the broad mass function of such PBH DM is already being probed by Subaru Hyper Suprime-Cam (HSC) microlensing data and is consistent with a detected candidate event. Upcoming observations of HSC will be able to provide an independent definitive test of the stochastic GW signals originating from such PBH DM production scenarios.

The gravitational waves (GWs) from supermassive binary black holes (BBHs) have long been sought by pulsar timing array (PTA) experiments, in the forms of both a stochastic GW background (GWB) and individual sources. Evidence for a GWB was reported recently by several PTAs with origins to be determined. Here we use a BBH population synthesis model to investigate the detection probability of individual BBHs by the Chinese PTA (CPTA) and the constraint on the GWB origin that may be obtained by PTA observations of both GWB and individual BBHs. If the detected GWB signal is entirely due to BBHs, a significantly positive redshift evolution (∝ (1 + z)2.07) of the mass scaling relation between supermassive black holes and their host galaxies is required. In this case, we find that the detection probability of individual BBHs is ∼85% or 64% if using a period of 3.4 yr of CPTA observation data, with an expectation of ∼1.9 or 1.0 BBHs detectable with a signal-to-noise ratio ≥3 or 5, and it is expected to increase to >95% if the observation period is extended to 5 yr or longer. Even if the contribution from BBHs to the GWB power signal is as small as ∼10%, a positive detection of individual BBHs can still be expected within an observation period of ∼10 yr. A nondetection of individual BBHs within several years from now jointly with the detected GWB signal can put a strong constraint on the upper limit of the BBH contribution to the GWB signal and help identify/falsify a cosmological origin.

Gravitational wave was directly detected for the first time in 2015 after a century of its prediction by Einstein’s General Theory of Relativity in 1916. But, the continuous, low frequency Gravitational-wave Background (GWB) that is thought to be present throughout the universe has not been detected directly so far. The researchers at North American Nanohertz Observatory for Gravitational Waves (NANOGrav) have recently reported detection of a low-frequency signal that could be ‘Gravitational-wave Background (GWB)’.

The subtle influence of gravitational waves on the apparent positioning of celestial bodies offers novel observational windows [1,2,3,4]. We calculate the expected astrometric signal induced by an isotropic Stochastic Gravitational Wave Background (SGWB) in the short distance limit. Our focus is on the resultant proper motion of Solar System objects, a signal on the same time scales addressed by Pulsar Timing Arrays (PTA). We derive the corresponding astrometric deflection patterns, finding that they manifest as distinctive dipole and quadrupole correlations or, in some cases, may not be present. Our analysis encompasses both Einsteinian and non-Einsteinian polarisations. We estimate the upper limits for the amplitude of SGWBs that could be obtained by tracking the proper motions of large numbers of solar system objects such as asteroids. We find that for SGWBs with negative spectral indices, such as that generated by Super Massive Black Hole Binaries (SMBHB), the constraints from these observations could rival those from PTAs. With the Gaia satellite and the Vera C. Rubin Observatory poised to track an extensive sample of asteroids — ranging from 𝒪(105) to 𝒪(106), we highlight the significant future potential for similar surveys to contribute to our understanding of the SGWB.

Observations of low-frequency gravitational waves (GWs) will require the highest possible timing precision from an array of the most spin-stable pulsars. We can improve the sensitivity of a pulsar timing array (PTA) to different GW sources by observing pulsars with low timing noise over years to decades and distributed across the sky. We discuss observing strategies for a PTA focused on a stochastic GW background such as from unresolved supermassive black hole binaries as well as focused on single continuous-wave sources. First, we describe the method to calculate a PTA’s sensitivity to different GW-source classes. We then apply our method to the 45 pulsars presented in the North American Nanohertz Observatory for the GW 11 year data set. For expected amplitudes of the stochastic background, we find that all pulsars contribute significantly over the timescale of decades; the exception is for very pessimistic values of the stochastic-background amplitude. For individual single sources, we find that a number of pulsars contribute to the sensitivity of a given source, but that which pulsars contribute is different depending on the source, or versus an all-sky metric. Our results seem robust to the presence of red noise in pulsar arrival times. It is critical to obtain more robust pulsar-noise parameters as they heavily affect our results. Our results show that it is also imperative to locate and time as many high-precision pulsars as possible, as quickly as possible, to maximize the sensitivity of next-generation PTA detectors.

A population of supermassive black hole (SMBH) binaries is expected to generate a stochastic gravitational wave background (SGWB) in the pulsar timing array (PTA) frequency range of 10−9 to $10^{-7}\, {\rm Hz}$. Detection of this signal is a current observational goal and so predictions of its characteristics are of significant interest. In this work, we use SMBH binary mergers from the MassiveBlackII simulation to estimate the characteristic strain of the stochastic background. We examine both a gravitational wave (GW) driven model of binary evolution and a model which also includes the effects of stellar scattering and a circumbinary gas disc. Results are consistent with PTA upper limits and similar to estimates in the literature. The characteristic strain at a reference frequency of $1\, {\rm yr}^{-1}$ is found to be $A_{\rm {yr}^{-1}}= 6.9 \times 10^{-16}$ and $A_{\rm {yr}^{-1}}= 6.4 \times 10^{-16}$ in the GW-driven and stellar scattering/gas disc cases, respectively. Using the latter approach, our models show that the SGWB is mildly suppressed compared to the purely GW-driven model as frequency decreases inside the PTA frequency band.

The null energy condition (NEC) is a fundamental principle in general relativity, and its violation could leave discernible signatures in gravitational waves (GWs). A violation of the NEC during the primordial era would imprint a blue-tilted spectrum on the stochastic gravitational wave background (SGWB) at nanohertz frequencies, potentially accounting for the recently detected signal by pulsar timing arrays. Remarkably, models of NEC violation during inflation also predict a nearly scale-invariant GW spectrum in the millihertz frequency range, which could be detectable by upcoming space-based GW detectors such as Taiji. The observation of this distinctive spectrum would provide compelling evidence for new physics beyond the standard cosmological paradigm. In this study, we explore Taiji’s ability to detect an SGWB arising from NEC violation during inflation, considering various foregrounds and noise sources, including an extragalactic foreground from binary black hole mergers throughout the Universe, a galactic foreground from white dwarf binaries, and the intrinsic noise of the Taiji detector. Employing comprehensive Bayesian parameter estimation techniques to analyze simulated Taiji data, we demonstrate a remarkable improvement in precision of three orders of magnitude compared to the NANOGrav 15-year data set for measuring the tensor power spectrum amplitude, PT,2, during the second inflationary stage. This substantial enhancement in measurement capabilities underscores Taiji’s potential as a powerful probe for investigating the NEC violation in the early Universe.

In the summer of 2023, the pulsar timing arrays (PTAs) announced a compelling evidence for the existence of a nanohertz stochastic gravitational wave background (SGWB). Despite this breakthrough, however, several critical questions remain unanswered: What is the source of the signal? How can cosmic variance be accounted for? To what extent can we constrain nanohertz gravity? When will individual supermassive black hole binaries become observable? And how can we achieve a stronger detection? These open questions have spurred significant interests in PTA science, making this an opportune moment to revisit the astronomical and theoretical foundations of the field, as well as the data analysis techniques employed. In this review, we focus on the theoretical aspects of the SGWB as detected by PTAs. We provide a comprehensive derivation of the expected signal and its correlation, presented in a pedagogical manner, while also addressing current constraints. Looking ahead, we explore future milestones in the field, with detailed discussions on emerging theoretical considerations such as cosmic variance, the cumulants of the one- and two-point functions, subluminal gravitational waves, and the anisotropy and polarization of the SGWB.

Observing and timing a group of millisecond pulsars with high rotational stability enables the direct detection of gravitational waves (GWs). The GW signals can be identified from the spatial correlations encoded in the times-of-arrival of widely spaced pulsar-pairs. The Chinese Pulsar Timing Array (CPTA) is a collaboration aiming at the direct GW detection with observations carried out using Chinese radio telescopes. This short article serves as a “table of contents” for a forthcoming series of papers related to the CPTA Data Release 1 (CPTA DR1) which uses observations from the Five-hundred-meter Aperture Spherical radio Telescope. Here, after summarizing the time span and accuracy of CPTA DR1, we report the key results of our statistical inference finding a correlated signal with amplitude logAc=−14.4−2.8+1.0 for spectral index in the range of α ∈ [ − 1.8, 1.5] assuming a GW background (GWB) induced quadrupolar correlation. The search for the Hellings–Downs (HD) correlation curve is also presented, where some evidence for the HD correlation has been found that a 4.6σ statistical significance is achieved using the discrete frequency method around the frequency of 14 nHz. We expect that the future International Pulsar Timing Array data analysis and the next CPTA data release will be more sensitive to the nHz GWB, which could verify the current results.

The stochastic gravitational wave (GW) background provides a fascinating window to the physics of the very early universe. Beyond the nearly scale-invariant primordial GW spectrum produced during inflation, a spectrum with a much richer structure is typically generated during the preheating phase after inflation (or after some other phase transition at lower energies). This raises the question of what one can learn from a future observation of the stochastic gravitational wave background spectrum about the underlying physics during preheating. Recently, it has been shown that during preheating non-perturbative quasi-stable objects like oscillons can act as strong sources for GW, leading to characteristic features such as distinct peaks in the spectrum. In this paper, we study the GW production from oscillons using semi-analytical techniques. In particular, we discuss how the GW spectrum is affected by the parameters that characterise a given oscillon system, e.g. by the background cosmology, the asymmetry of the oscillons and the evolution of the number density of the oscillons. We compare our semi-analytic results with numerical lattice simulations for a hilltop inflation model and a KKLT scenario, which differ strongly in some of these characteristics, and find very good agreement.

We study the co-evolution of supermassive black holes (SMBHs) with galaxies by means of semi-analytic model (SAM) of galaxy formation based on sub-halo merger trees built from Millennium and Millennium-II simulation. We utilize the simulation results from Guo 2013 and Henriques 2015 to study two aspects of the co-evolution, \emph{i.e.} the stochastic gravitational wave (GW) background generated by SMBH merger and the SMBH/galaxy clustering. The characteristic strain amplitude of GW background predicted by Guo 2013 and Henriques 2015 models are $A_{yr^{-1}}=5.00\times10^{-16}$ and $A_{yr^{-1}}=9.42\times10^{-17}$, respectively. We find the GW amplitude is very sensitive to the galaxy merger rate. The difference in the galaxy merger rate between Guo 2013 and Henriques 2015, results in a factor $5$ deviation in the GW strain amplitude. For clusterings, we calculate the spatially isotropic two point auto- and cross-correlation functions (2PCFs) for both SMBHs and galaxies by using the mock catalogs generated from Guo 2013 model. We find that all 2PCFs have positive dependence on both SMBH and galaxy mass. And there exist a significant time evolution in 2PCFs, namely, the clustering effect is enhanced at lower redshifts. Interestingly, this result is not reported in the active galactic nuclei samples in SDSS. Our analysis also shows that, roughly, SMBHs and galaxies, with galaxy mass $10^2\sim10^3$ larger than SMBH mass, have similar pattern of clustering, which is a reflection of the co-evolution of SMBH and galaxy. Finally, we calculate the first ten multiples of the angular power spectrum of the energy density of GW background. We find the amplitude of angular power spectrum of the first ten multiples is about $10\%$ to $60\%$ of the monopole component in the whole frequency range.