Observing gravitational waves with solar system astrometry

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.

ABSTRACTWe searched for an isotropic stochastic gravitational wave background in the second data release of the International Pulsar Timing Array, a global collaboration synthesizing decadal-length pulsar-timing campaigns in North America, Europe, and Australia. In our reference search for a power-law strain spectrum of the form $h_c = A(f/1\, \mathrm{yr}^{-1})^{\alpha }$, we found strong evidence for a spectrally similar low-frequency stochastic process of amplitude $A = 3.8^{+6.3}_{-2.5}\times 10^{-15}$ and spectral index α = −0.5 ± 0.5, where the uncertainties represent 95 per cent credible regions, using information from the auto- and cross-correlation terms between the pulsars in the array. For a spectral index of α = −2/3, as expected from a population of inspiralling supermassive black hole binaries, the recovered amplitude is $A = 2.8^{+1.2}_{-0.8}\times 10^{-15}$. None the less, no significant evidence of the Hellings–Downs correlations that would indicate a gravitational-wave origin was found. We also analysed the constituent data from the individual pulsar timing arrays in a consistent way, and clearly demonstrate that the combined international data set is more sensitive. Furthermore, we demonstrate that this combined data set produces comparable constraints to recent single-array data sets which have more data than the constituent parts of the combination. Future international data releases will deliver increased sensitivity to gravitational wave radiation, and significantly increase the detection probability.

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.

Large-area sky surveys show that massive galaxies undergo at least one major merger in a Hubble time. Ongoing pulsar timing array (PTA) experiments are aimed at measuring the gravitational-wave (GW) emission from binary supermassive black holes (SMBHs) at the centres of galaxy merger remnants. In this paper, using the latest observational estimates for a range of galaxy properties and scaling relations, we predict the amplitude of the GW background generated by the binary SMBH population. We also predict the numbers of individual binary SMBH GW sources. We predict the characteristic strain amplitude of the GW background to lie in the range 5.1 × 10−16 < Ayr < 2.4 × 10−15 at a frequency of (1 yr)−1, with 95 per cent confidence. Higher values within this range, which correspond to the more commonly preferred choice of galaxy merger time-scale, will fall within the expected sensitivity ranges of existing PTA projects in the next few years. In contrast, we find that a PTA consisting of at least 100 pulsars observed with next-generation radio telescopes will be required to detect continuous-wave GWs from binary SMBHs. We further suggest that GW memory bursts from coalescing SMBH pairs are not viable sources for PTAs. Both the GW background and individual GW source counts are dominated by binaries formed in mergers between early-type galaxies of masses ≳5 × 1010 M⊙ at redshifts ≲1.5. Uncertainties in the galaxy merger time-scale and the SMBH mass–galaxy bulge mass relation dominate the uncertainty in our predictions.

In this Letter, we carry out the first systematic investigation of the expected gravitational wave (GW) background generated by supermassive black hole (SMBH) binaries in the nHz frequency band accessible to pulsar timing arrays (PTAs). We take from the literature several estimates of the redshift-dependent galaxy mass function and of the fraction of close galaxy pairs to derive a wide range of galaxy merger rates. We then exploit empirical black hole–host relations to populate merging galaxies with SMBHs. The result of our procedure is a collection of a large number of phenomenological SMBH binary merger rates consistent with current observational constraints on the galaxy assembly at z &amp;lt; 1.5. For each merger rate we compute the associated GW signal, eventually producing a large set of estimates of the nHz GW background that we use to infer confidence intervals of its expected amplitude. When considering the most recent SMBH–host relations, accounting for overmassive black holes in brightest cluster galaxies, we find that the nominal 1σ interval of the expected GW signal is only a factor of 3–10 below current PTA limits, implying a non-negligible chance of detection in the next few years.

We present new limits on an isotropic stochastic gravitational-wave background (GWB) using a six pulsar dataset spanning 18 yr of observations from the 2015 European Pulsar Timing Array data release. Performing a Bayesian analysis, we fit simultaneously for the intrinsic noise parameters for each pulsar, along with common correlated signals including clock, and Solar System ephemeris errors, obtaining a robust 95$\%$ upper limit on the dimensionless strain amplitude $A$ of the background of $A<3.0\times 10^{-15}$ at a reference frequency of $1\mathrm{yr^{-1}}$ and a spectral index of $13/3$, corresponding to a background from inspiralling super-massive black hole binaries, constraining the GW energy density to $\Omega_\mathrm{gw}(f)h^2 < 1.1\times10^{-9}$ at 2.8 nHz. We also present limits on the correlated power spectrum at a series of discrete frequencies, and show that our sensitivity to a fiducial isotropic GWB is highest at a frequency of $\sim 5\times10^{-9}$~Hz. Finally we discuss the implications of our analysis for the astrophysics of supermassive black hole binaries, and present 95$\%$ upper limits on the string tension, $G\mu/c^2$, characterising a background produced by a cosmic string network for a set of possible scenarios, and for a stochastic relic GWB. For a Nambu-Goto field theory cosmic string network, we set a limit $G\mu/c^2<1.3\times10^{-7}$, identical to that set by the {\it Planck} Collaboration, when combining {\it Planck} and high-$\ell$ Cosmic Microwave Background data from other experiments. For a stochastic relic background we set a limit of $\Omega^\mathrm{relic}_\mathrm{gw}(f)h^2<1.2 \times10^{-9}$, a factor of 9 improvement over the most stringent limits previously set by a pulsar timing array.

The steadily improving sensitivity of pulsar timing arrays (PTAs) suggests that gravitational waves (GWs) from supermassive black hole binary (SMBHB) systems in the nearby universe will be de- tectable sometime during the next decade. Currently, PTAs assume an equal probability of detection from every sky position, but as evidence grows for a non-isotropic distribution of sources, is there a most likely sky position for a detectable single source of GWs? In this paper, a collection of galactic catalogs is used to calculate various metrics related to the detectability of a single GW source resolv- able above a GW background, assuming that every galaxy has the same probability of containing a SMBHB. Our analyses of these data reveal small probabilities that one of these sources is currently in the PTA band, but as sensitivity is improved regions of consistent probability density are found in predictable locations, specifically around local galaxy clusters.

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 &lt; 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 &lt; 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 &lt; 3.4 × 10−10.

The direct detection of gravitational waves (GWs) by pulsar timing arrays (PTAs) is very likely within the next decade. While the stochastic GW background is a promising candidate for detection it is also possible that single resolvable sources may be detectable as well. In this work we will focus on the detection and characterization of single GW sources from supermassive black hole binaries. We introduce a fully Bayesian data analysis pipeline that is meant to carry out a search, characterization, and evaluation phase. This will allow us to rapidly locate the global maxima in parameter space, map out the posterior, and finally weigh the evidence of a GW detection through a Bayes factor. Here we will make use of an adaptive metropolis algorithm and parallel tempering. We test this algorithm on realistic simulated data that are representative of modern PTAs.

While supermassive black hole (SMBH) binaries are not the only viable source for the low-frequency gravitational wave background (GWB) signal evidenced by the most recent pulsar timing array (PTA) data sets, they are expected to be the most likely. Thus, connecting the measured PTA GWB spectrum and the underlying physics governing the demographics and dynamics of SMBH binaries is extremely important. Previously, Gaussian processes (GPs) and dense neural networks have been used to make such a connection by being built as conditional emulators; their input is some selected evolution or environmental SMBH binary parameters and their output is the emulated mean and standard deviation of the GWB strain ensemble distribution over many Universes. In this paper, we use a normalizing flow (NF) emulator that is trained on the entirety of the GWB strain ensemble distribution, rather than only mean and standard deviation. As a result, we can predict strain distributions that mirror underlying simulations very closely while also capturing frequency covariances in the strain distributions as well as statistical complexities such as tails, non-Gaussianities, and multimodalities that are otherwise not learnable by existing techniques. In particular, we feature various comparisons between the NF-based emulator and the GP approach used extensively in past efforts. Our analyses conclude that the NF-based emulator not only outperforms GPs in the ease and computational cost of training but also outperforms in the fidelity of the emulated GWB strain ensemble distributions.

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.

We assess the effects of super-massive black hole (SMBH) environments on the gravitational-wave (GW) signal from binary SMBHs. To date, searches with pulsar timing arrays for GWs from binary SMBHs, in the frequency band $\sim1-100$\,nHz, include the assumptions that all binaries are circular and evolve only through GW emission. However, dynamical studies have shown that the only way that binary SMBH orbits can decay to separations where GW emission dominates the evolution is through interactions with their environments. We augment an existing galaxy and SMBH formation and evolution model with calculations of binary SMBH evolution in stellar environments, accounting for non-zero binary eccentricities. We find that coupling between binaries and their environments causes the expected GW spectral energy distribution to be reduced with respect to the standard assumption of circular, GW-driven binaries, for frequencies up to $\sim20$\,nHz. Larger eccentricities at binary formation further reduce the signal in this regime. We also find that GW bursts from individual eccentric binary SMBHs are unlikely to be detectable with current pulsar timing arrays. The uncertainties in these predictions are large, owing to observational uncertainty in SMBH-galaxy scaling relations and the galaxy stellar mass function, uncertainty in the nature of binary-environment coupling, and uncertainty in the numbers of the most massive binary SMBHs. We conclude, however, that low-frequency GWs from binary SMBHs may be more difficult to detect with pulsar timing arrays than currently thought.

The continuous nanohertz gravitational waves (GWs) from individual supermassive binary black holes (SMBBHs) can be encoded in the timing residuals of pulsar timing arrays (PTAs). For each pulsar, the residuals actually contain an Earth term and a pulsar term, but usually only the Earth term is considered as a signal and the pulsar term is dropped. This leads to parameter-estimation biases (PEBs) for the SMBBHs, and currently there are no convenient evaluations of the PEBs. In this article, we formulate the PEBs for a SMBBH with an eccentric orbit. In our analyses, the unknown phases of pulsar terms are treated as random variables obeying the uniform distribution U[0, 2π), due to the fact that pulsar distances are generally poorly measured. Our analytical results are in accordance with the numerical work by Zhu et al. at 1.5σ level, which implies that our formulae are effective in estimating magnitudes of the PEBs. Additionally, we find that the biases Δφ E and Δe/e for two parameters—that is, Earth-term phase φ E and orbital eccentricity e—monotonically decrease as e increases, which partly confirms a hypothesis in our previous work. Furthermore, we also calculate the PEBs caused by the recently observed common-spectrum process (CSP). We find that if the strain amplitude of the continuous GW is significantly stronger (three times larger, in our cases) than the stochastic GW background, then the PEBs from pulsar terms are larger than those from the CSP. Our formulae of the PEBs can be conveniently applied in the future PTA data analyses.

Supermassive black hole binaries (SMBHBs) are binary systems formed by black holes with masses exceeding millions of solar masses, and are expected to form and evolve in the nuclei of galaxies. The extremely compact nature of these objects leads to the intense and efficient emission of gravitational waves (GWs), which can be detected by the Pulsar Timing Array (PTA) experiment in the form of a gravitational wave background (GWB); that is, a superposition of GW signals coming from different sources. The modelling of the GWB requires some assumptions as to the binary population, and exploration of the whole parameter space involved is hindered by the great computational cost involved. We trained two neural networks (NN) on a semi-analytical modelling of the GWB generated by an eccentric population of MBHBs that interact with the stellar environment. We then used the NN to predict the characteristics of the GW signal in regions of the parameter space that we did not sample analytically. The developed framework allows us to quickly predict the amplitude, shape, and variance of the GWB signals produced in different realisations of the universe.

Supermassive binary black holes (SMBBHs) in galactic centers may radiate gravitational waves (GW) in the nano-Hertz frequency band, which are expected to be detected by pulsar timing arrays (PTAs) in the near future. GW signals from individual SMBBHs at cosmic distances, if detected by PTAs, are potentially powerful standard sirens that can be used to independently measure distances and thus put constraints on cosmological parameters. In this paper, we investigate the constraint that may be obtained on the equation of state (w) of dark energy by using those SMBBHs, expected to be detected by the PTAs in the Square Kilometre Array (SKA) era. By considering both the currently available SMBBH candidates and mock SMBBHs in the universe resulting from a simple galaxy major merger model, we find that ∼200–3000 SMBBHs with chirp mass &gt;109 M ⊙ are expected to be detected with a signal-to-noise ratio &gt;10 by SKA–PTA with conservative and optimistic settings and they can be used to put a constraint on w to an uncertainty of Δw ∼ 0.02–0.1. If further information on the mass and mass ratio of those SMBBHs can be provided by electromagnetic observations (e.g., chirp mass uncertainty ≲50%), the constraint may be further improved to a ≲0.01 level, as many more SMBBHs will be detected by SKA–PTA with relatively better distance measurements and can be used as the standard sirens.