Pulsar timing is a process of iteratively fitting pulse arrival times to constrain the spindown, astrometric, and possibly binary parameters of a pulsar, by enforcing integer numbers of pulsar rotations between the arrival times. Phase connection is the process of unambiguously determining those rotation numbers between the times of arrival (TOAs) while determining a pulsar timing solution. Pulsar timing currently requires a manual process of step-by-step phase connection performed by individuals. In an effort to quantify and streamline this process, we created the Algorithmic Pulsar Timer, APT, an algorithm which can accurately phase connect and time isolated pulsars. Using the statistical F-test and knowledge of parameter uncertainties and covariances, the algorithm decides what new data to include in a fit, when to add additional timing parameters, and which model to attempt in subsequent iterations. Using these tools, the algorithm can phase-connect timing data that previously required substantial manual effort. We tested the algorithm on 100 simulated systems, with a 99% success rate. APT combines statistical tests and techniques with a logical decision-making process, very similar to the manual one used by pulsar astronomers for decades, and some computational brute-force, to automate the often tricky process of isolated pulsar phase connection, setting the foundation for automated fitting of binary pulsar systems.
We report pulsar timing observations carried out in L-band with NTSC's 40-meter Haoping Radio Telescope (HRT), which was constructed in 2014. The observations were carried out using the pulsar machine we developed. Timing observations toward millisecond pulsar J0437-4715 obtains a timing residual (r.m.s) of 397ns in the time span of 284 days. And our observations successfully detected Crab pulsar's glitch that happened on July 23rd, 2019.
Ongoing research on Pulsar Timing Array (PTA) to detect gravitational radiation is reviewed. Here, we discuss the use of millisecond pulsars as a gravitational wave detector, the sources of gravitational radiation detectable by PTAs and the current status of PTA experiments pointing out the future possibilities.
We introduce a method for performing a robust Bayesian analysis of non-Gaussianity present in pulsar timing data, simultaneously with the pulsar timing model, and additional stochastic parameters such as those describing red spin noise and dispersion measure variations. The parameters used to define the presence of non-Gaussianity are zero for Gaussian processes, giving a simple method of defining the strength of non-Gaussian behaviour. We use simulations to show that assuming Gaussian statistics when the noise in the data is drawn from a non-Gaussian distribution can significantly increase th…
A new approach to the problem of gravitational waves detection based on simultaneous timing of several pulsars and subsequent expansion of the post-fit timing data into components of different spectral kind (with different spectral indices) is proposed. Presence of a signal caused by stochastic gravitational waves in spectral components is tested with the two-point angular correlation function as proposed in the pulsar timing array. This new approach was applied to timing data of a few millisecond pulsars and allowed to detect a signature similar to one predicted for gravitational wave backgro…
The 70-m DSN's Deep Space Station antenna 14 (DSS-14) at Goldstone has recently been outfitted with instrumentation to enable pulsar searching and timing operation. Systems capable of similar operations are undergoing installation at DSS-63, and are planned for DSS-43. The Goldstone system is the first of these to become operational, with a 640 MHz bandwidth stretching from 1325-1965 MHz. Initial results from the pulsar timing pipeline show short-term residuals of < 100 ns for pulsar B1937+21. Commissioning obsefvations at DSS-14 to obtain a baseline set of TOA measurements on several millisec…
A new Bayesian method for the analysis of folded pulsar timing data is presented that allows for the simultaneous evaluation of evolution in the pulse profile in either frequency or time, along with the timing model and additional stochastic processes such as red spin noise, or dispersion measure variations. We model the pulse profiles using `shapelets' - a complete ortho-normal set of basis functions that allow us to recreate any physical profile shape. Any evolution in the profiles can then be described as either an arbitrary number of independent profiles, or using some functional form. We …
This paper contains details on the algorithms implemented in the TEMPO2 pulsar timing software package and describes how the software is used. Information is given on how to download and install the software, use the various interfaces, simulate realistic data sets and develop the software. The use of TEMPO2 in predictive mode is also described.
Pulsars are special objects whose positions can be determined independently from timing, radio interferometric, and Gaia astrometry at sub-milliarcsecond (mas) precision; thus, they provide a unique way to monitor the link between dynamical and kinematic reference frames. We aimed to assess the orientation consistency between the dynamical reference frame represented by the planetary ephemeris and the kinematic reference frames constructed by Gaia and VLBI through pulsar positions. We identified 49 pulsars in Gaia Data Release 3 and 62 pulsars with very long baseline interferometry (VLBI) posi…
Pulsar timing observations are usually analysed with least-square-fitting procedures under the assumption that the timing residuals are uncorrelated (statistically "white"). Pulsar observers are well aware that this assumption often breaks down and causes severe errors in estimating the parameters of the timing model and their uncertainties. Ad hoc methods for minimizing these errors have been developed, but we show that they are far from optimal. Compensation for temporal correlation can be done optimally if the covariance matrix of the residuals is known using a linear transformation that wh…
Next generation radio telescopes, namely the Five-hundred-meter Aperture Spherical Telescope (FAST) and the Square Kilometer Array (SKA), will revolutionize the pulsar timing arrays (PTAs) based gravitational wave (GW) searches. We review some of the characteristics of FAST and SKA, and the resulting PTAs, that are pertinent to the detection of gravitational wave signals from individual supermassive black hole binaries.
Here we present a Bayesian method of including discrete measurements of dispersion measure due to the interstellar medium in the direction of a pulsar as prior information in the analysis of that pulsar. We use a simple simulation to show the efficacy of this method, where the inclusion of the additional measurements results in both a significant increase in the precision with which the timing model parameters can be obtained, and an improved upper limit on the amplitude of any red noise in the dataset. We show that this method can be applied where no multi-frequency data exists across much of…
Supermassive black hole binaries are one of the primary targets for gravitational wave searches using pulsar timing arrays. Gravitational wave signals from such systems are well represented by parametrized models, allowing the standard Generalized Likelihood Ratio Test (GLRT) to be used for their detection and estimation. However, there is a dichotomy in how the GLRT can be implemented for pulsar timing arrays: there are two possible ways in which one can split the set of signal parameters for semi-analytical and numerical extremization. The straightforward extension of the method used for con…
We present a principal component analysis method which tracks and compensates for short-timescale variability in pulsar profiles, with a goal of improving pulsar timing precision. We couple this with a fast likelihood technique for determining pulse time of arrival, marginalizing over the principal component amplitudes. This allows accurate estimation of timing errors in the presence of pulsar variability. We apply the algorithm to the slow pulsar PSR J2139+0040 using an archived set of untargeted raster-scan observations at arbitrary epochs across four years, obtaining an improved timing solu…
A pulsar timing array (PTA) refers to a program of regular, high-precision timing observations of a widely distributed array of millisecond pulsars. Here we review the status of the three primary PTA projects and the joint International Pulsar Timing Array project. We discuss current results related to ultra-low-frequency gravitational wave searches and highlight opportunities for the near future.
Resolvable Supermassive Black Hole Binaries are promising sources for Pulsar Timing Array based gravitational wave searches. Search algorithms for such targets must contend with the large number of so-called pulsar phase parameters in the joint log-likelihood function of the data. We compare the localization accuracy for two approaches: Maximization over the pulsar phase parameters (MaxPhase) against marginalization over them (AvPhase). Using simulated data from a pulsar timing array with 17 pulsars, we find that for weak and moderately strong signals, AvPhase outperforms MaxPhase significantl…
Pulsars are amongst the most stable rotators known in the Universe. Over many years some millisecond pulsars rival the stability of atomic clocks. Comparing observations of many such stable pulsars may allow the first direct detection of gravitational waves, improve the Solar System planetary ephemeris and provide a means to study irregularities in terrestrial time scales. Here we review the goals and status of current and future pulsar timing array projects.
To successfully detect gravitational waves with pulsar timing arrays, we need to have a comprehensive understanding of the physical origins and statistical characteristics of the noise in pulse arrival times and identify mitigation methods to reduce the noise. In this paper we will review radiometer noise, phase jitter noise and timing noise in the noise budget of pulsar timing and show various efforts used to reduce them. We will briefly discuss the results of an overall assessment of the components and physical causes of the timing residuals for millisecond pulsars in the North American Nano…
Analysis of pulsar timing data-sets may provide the first direct detection of gravitational waves. This paper, the third in a series describing the mathematical framework implemented into the tempo2 pulsar timing package, reports on using tempo2 to simulate the timing residuals induced by gravitational waves. The tempo2 simulations can be used to provide upper bounds on the amplitude of an isotropic, stochastic, gravitational wave background in our Galaxy and to determine the sensitivity of a given pulsar timing experiment to individual, supermassive, binary black hole systems.
In order to maximize the sensitivity of pulsar timing arrays to a stochastic gravitational wave background, we present computational techniques to optimize observing schedules. The techniques are applicable to both single and multi-telescope experiments. The observing schedule is optimized for each telescope by adjusting the observing time allocated to each pulsar while keeping the total amount of observing time constant. The optimized schedule depends on the timing noise characteristics of each individual pulsar as well as the performance of instrumentation. Several examples are given to illu…
Pulsar Timing Arrays use a set of millisecond pulsars in an attempt to directly detect nanohertz gravitational waves. For this purpose, high precision timing of the pulsars is essential and ultimately a precision of the order of ~100 ns is required. Propagation effects in the interstellar medium cause the radio emission from a pulsar to be dispersed and scattered, introducing time variable delays of the pulses on their way to Earth. If these delays are not properly corrected for, they may cause significant errors in the timing analysis of a pulsar. These proceedings will review the effects of …
In this work we review the application of the theory of Gaussian processes to the modeling of noise in pulsar-timing data analysis, and we derive various useful and optimized representations for the likelihood expressions that are needed in Bayesian inference on pulsar-timing-array datasets. The resulting viewpoint and formalism lead us to two improved parameter-sampling schemes inspired by Gibbs sampling. The new schemes have vastly lower chain autocorrelation lengths than the Markov Chain Monte Carlo methods currently used in pulsar-timing data analysis, potentially speeding up Bayesian infe…
We present timing models for 20 millisecond pulsars in the Parkes Pulsar Timing Array. The precision of the parameter measurements in these models has been improved over earlier results by using longer data sets and modelling the non-stationary noise. We describe a new noise modelling procedure and demonstrate its effectiveness using simulated data. Our methodology includes the addition of annual dispersion measure (DM) variations to the timing models of some pulsars. We present the first significant parallax measurements for PSRs J1024-0719, J1045-4509, J1600-3053, J1603-7202, and J1730-2304,…
Gravitational waves (GWs) are ripples in space-time that are known to exist but have not yet been detected directly. Once they are, a key feature of any viable theory of gravity will be demonstrated and a new window on the Universe opened. GW astronomy was named as one of five key discovery areas in the New Worlds, New Horizons Decadal Report. Pulsar timing probes GW frequencies, and hence source classes, that are inaccessible to any other detection method and can uniquely constrain the nonlinear nature of General Relativity. Pulsar timing is therefore a critical capability with its own discov…
A fully coherent acceleration search algorithm has been applied to the Parkes multi- beam pulsar survey of the Galactic plane to search for previously undiscovered relativistic binary pulsars. The search has resulted in the discovery of 16 pulsars including a binary millisecond pulsar and an intermittent pulsar. Despite a number of promising candidates there have been no new discoveries of relativistic binary pulsars. Here we detail the acceleration search performed in our analysis and present coherent timing solutions for each of pulsars discovered. In light of the lack of discoveries of rela…
We construct realistic sensitivity curves for pulsar timing array searches for gravitational waves, incorporating both red and white noise contributions to individual pulsar noise spectra, as well as the effect of fitting to a pulsar timing model. We demonstrate the method on both simulated pulsars and a realistic array consisting of a subset of NANOGrav pulsars used in recent analyses. A comparison between the results presented here and exact sensitivity curves shows agreement to tens of percent. The resulting sensitivity curves can be used to assess the detectability of predicted gravitation…
A new Bayesian software package for the analysis of pulsar timing data is presented in the form of TempoNest which allows for the robust determination of the non-linear pulsar timing solution simultaneously with a range of additional stochastic parameters. This includes both red spin noise and dispersion measure variations using either power law descriptions of the noise, or through a model-independent method that parameterises the power at individual frequencies in the signal. We use TempoNest to show that at noise levels representative of current datasets in the European Pulsar Timing Array …
Precision astrometry is an integral component of successful pulsar timing campaigns. Astrometric parameters are commonly derived by fitting them as parameters of a timing model to a series of pulse times of arrival (TOAs). TOAs measured to microsecond precision over several-year spans can yield position measurements with sub-milliarcsecond precision. However, timing-based astrometry can become biased if a pulsar displays any red spin noise, which can be compared to the red noise signal produced by the stochastic gravitational wave background. We investigate how noise of different spectral type…
There are two types of timing irregularities seen in pulsars: glitches and timing noise. Both of these phenomena can help us probe the interior of such exotic objects. This article presents a brief overview of the observational and theoretical aspects of pulsar timing irregularities and the main results from the investigations of these phenomena in India. The relevance of such Indian programs for monitoring of young pulsars with the Square Kilometer Array (SKA) is presented, highlighting possible contributions of the Indian neutron star community to the upcoming SKA endeavour.
Pulsar timing is a technique that uses the highly stable spin periods of neutron stars to investigate a wide range of topics in physics and astrophysics. Pulsar timing arrays (PTAs) use sets of extremely well-timed pulsars as a Galaxy-scale detector with arms extending between Earth and each pulsar in the array. These challenging experiments look for correlated deviations in the pulsars' timing that are caused by low-frequency gravitational waves (GWs) traversing our Galaxy. PTAs are particularly sensitive to GWs at nanohertz frequencies, which makes them complementary to other space- and grou…
The Five-hundred-meter Aperture Spherical Telescope (FAST) will become one of the world-leading telescopes for pulsar timing array (PTA) research. The primary goals for PTAs are to detect (and subsequently study) ultra-low-frequency gravitational waves, to develop a pulsar-based time standard and to improve solar system planetary ephemerides. FAST will have the sensitivity to observe known pulsars with significantly improved signal-to-noise ratios and will discover a large number of currently unknown pulsars. We describe how FAST will contribute to PTA research and show that jitter- and timing…
We derive scaling laws for the signal-to-noise ratio of the optimal cross-correlation statistic, and show that the large power-law increase of the signal-to-noise ratio as a function of the the observation time $T$ that is usually assumed holds only at early times. After enough time has elapsed, pulsar timing arrays enter a new regime where the signal to noise only scales as $\sqrt{T}$. In addition, in this regime the quality of the pulsar timing data and the cadence become relatively un-important. This occurs because the lowest frequencies of the pulsar timing residuals become gravitational-w…
At the highest levels of pulsar timing precision achieved to date, experiments are limited by noise intrinsic to the pulsar. This stochastic wideband impulse modulated self-noise (SWIMS) limits pulsar timing precision by randomly biasing the measured times of arrival and thus increasing the root mean square (rms) timing residual. We discuss an improved methodology of removing this bias in the measured times of arrival by including information about polarized radiation. Observations of J0437-4715 made over a one-week interval at the Parkes Observatory are used to demonstrate a nearly 40 per cen…
Gravitational wave bursts produced by supermassive binary black hole mergers will leave a persistent imprint on the space-time metric. Such gravitational wave memory signals are detectable by pulsar timing arrays as a glitch event that would seem to occur simultaneously for all pulsars. In this paper, we describe an initial algorithm which can be used to search for gravitational wave memory signals. We apply this algorithm to the Parkes Pulsar Timing Array data set. No significant gravitational wave memory signal is founded in the data set.
We estimate the sensitivity to nano-Hertz gravitational waves of pulsar timing experiments in which two highly-stable millisecond pulsars are tracked simultaneously with two neighboring radio telescopes that are referenced to the same time-keeping subsystem (i.e. "the clock"). By taking the difference of the two time-of-arrival residual data streams we can exactly cancel the clock noise in the combined data set, thereby enhancing the sensitivity to gravitational waves. We estimate that, in the band ($10^{-9} - 10^{-8}$) Hz, this "interferometric" pulsar timing technique can potentially improve…
Three pulsar timing arrays are now producing high quality data sets. As reviewed in this paper, these data sets are been processed to 1) develop a pulsar-based time standard, 2) search for errors in the solar system planetary ephemeris and 3) detect gravitational waves. It is expected that the data sets will significantly improve in the near future by combining existing observations and by using new telescopes.
We describe the procedure, nuances, issues, and choices involved in creating times-of-arrival (TOAs), residuals and error bars from a set of radio pulsar timing data. We discuss the issue of mis-matched templates, the problem that wide- bandwidth backends introduce, possible solutions to that problem, and correcting for offsets introduced by various observing systems.
High-precision pulsar timing is central to a wide range of astrophysics and fundamental physics applications. When timing an ensemble of millisecond pulsars in different sky positions, known as a pulsar timing array (PTA), one can search for ultra-low-frequency gravitational waves (GWs) through the spatial correlations that spacetime deformations by passing GWs are predicted to induce on the pulses' times-of-arrival (TOAs). A pulsar-timing model, requires the use of a solar-system ephemeris (SSE) to properly predict the position of the solar-system barycentre, the (quasi-)inertial frame where …
We present an analysis of timing residual (noise) of 54 pulsars obtained from 25-m radio telescope at Urumqi Observatory with a time span of 5~8 years, dealing with statistics of the Hurst parameter. The majority of these pulsars were selected to have timing noise that look like white noise rather than smooth curves. The results are compared with artificial series of different constant pairwise covariances. Despite the noise like appearance, many timing residual series showed Hurst parameters significantly deviated from that of independent series. We concluded that Hurst parameter may be capab…
Noise is ubiquitous in pulsar signals where red noise has been attributed to effects arising from the interstellar medium, spin noise, and pulsar mode changes, among others. The red noise, however, has not been detected in all pulsars. Using the dataset from the North American Nanohertz Observatory for Gravitational Waves (NANOGrav), we investigate twenty-three pulsars and show that an evaluation of the mean square deviation and probability distribution of timing residuals can provide a straightforward way of determining the presence of red noise. The results agree with those reported in the l…
Modeling of frequency-dependent effects, contributed by the turbulence in the free electron density of interstellar plasma, is required to enable the detection of the expected imprints from the stochastic gravitational-wave (GW) background in pulsar timing data. In this work, we present an investigation of temporal variations of interstellar medium for a set of millisecond pulsars (MSPs) with the upgraded GMRT aided by large fractional bandwidth at lower observing frequencies. Contrary to the conventional narrow-band analysis using a frequency invariant template profile, we applied $PulsePortr…
Pulsar Timing Array (PTA) experiments exploit the clock-like behaviour of an array of millisecond pulsars, with the goal of detecting low-frequency gravitational waves. PTA experiments have been in operation over the last decade, led by groups in Europe, Australia, and North America. These experiments use the most sensitive radio telescopes in the world, extremely precise pulsar timing models and sophisticated detection algorithms to increase the sensitivity of PTAs. No detection of gravitational waves has been made to date with this technique, but PTA upper limits already contributed to rule …
Complementary to ground-based laser interferometers, pulsar timing array experiments are being carried out to search for nanohertz gravitational waves. Using the world's most powerful radio telescopes, three major international collaborations have collected $\sim$10-year high precision timing data for tens of millisecond pulsars. In this paper we give an overview on pulsar timing experiments, gravitational wave detection in the nanohertz regime, and recent results obtained by various timing array projects.
The error in the Solar system ephemeris will lead to dipolar correlations in the residuals of pulsar timing array for widely separated pulsars. In this paper, we utilize such correlated signals, and construct a Bayesian data-analysis framework to detect the unknown mass in the Solar system and to measure the orbital parameters. The algorithm is designed to calculate the waveform of the induced pulsar-timing residuals due to the unmodelled objects following the Keplerian orbits in the Solar system. The algorithm incorporates a Bayesian-analysis suit used to simultaneously analyse the pulsar-tim…
High-sensitivity radio-frequency observations of millisecond pulsars usually show stochastic, broadband, pulse-shape variations intrinsic to the pulsar emission process. These variations induce jitter noise in pulsar timing observations; understanding the properties of this noise is of particular importance for the effort to detect gravitational waves with pulsar timing arrays. We assess the short-term profile and timing stability of 22 millisecond pulsars that are part of the Parkes Pulsar Timing Array sample by examining intra-observation arrival time variability and single-pulse phenomenolo…
The PSRIX backend is the primary pulsar timing instrument of the Effelsberg 100-m radio telescope since early 2011. This new ROACH-based system enables bandwidths up to 500 MHz to be recorded, significantly more than what was possible with its predecessor, the Effelsberg-Berkeley Pulsar Processor (EBPP). We review the first four years of PSRIX timing data for 33 pulsars collected as part of the monthly European Pulsar Timing Array (EPTA) observations. We describe the automated data analysis pipeline, CoastGuard, that we developed to reduce these observations. We also introduce TOASTER, the EPT…
The European Pulsar Timing Array (EPTA) is a multi-institutional, multi-telescope collaboration, with the goal of using high-precision pulsar timing to directly detect gravitational waves. In this article we discuss the EPTA member telescopes, current achieved timing precision, and near-future goals. We report a preliminary upper limit to the amplitude of a gravitational wave background. We also discuss the Large European Array for Pulsars, in which the five major European telescopes involved in pulsar timing will be combined to provide a coherent array that will give similar sensitivity to th…
A "pulsar timing array" (PTA), in which observations of a large sample of pulsars spread across the celestial sphere are combined, allows investigation of "global" phenomena such as a background of gravitational waves or instabilities in atomic timescales that produce correlated timing residuals in the pulsars of the array. The Parkes Pulsar Timing Array (PPTA) is an implementation of the PTA concept based on observations with the Parkes 64-m radio telescope. A sample of 20 millisecond pulsars is being observed at three radio-frequency bands, 50cm (~700 MHz), 20cm (~1400 MHz) and 10cm (~3100 M…
We propose a novel method for observing the gravitational wave signature of super-massive black hole (SMBH) mergers. This method is based on detection of a specific type of gravitational waves, namely gravitational wave burst with memory (BWM), using pulsar timing. We study the unique signature produced by BWM in anomalous pulsar timing residuals. We show that the present day pulsar timing precision allows one to detect BWM due to SMBH mergers from distances up to 1 Gpc (for case of equal mass 10^8 Msun SMBH). Improvements in precision of pulsar timing together with the increase in number of o…
In the past five years, approximately one third of the 65 pulsars discovered by radio observations of Fermi unassociated sources are black widow pulsars (BWPs). BWPs are binary millisecond pulsars with companion masses ranging from 0.01-0.1 solar masses which often exhibit radio eclipses. The bloated companions in BWP systems exert small torques on the system causing the orbit to change on small but measurable time scales. Because adding parameters to a timing model reduces sensitivity to a gravitational wave (GW) signal, the need to fit many orbital frequency derivatives to the timing data is…
Nancay radiotelescope is involved in high precision timing since 20 years. Since 2004, a coherent dedispersion instrumentation enables numerous routine observations on more than 200 pulsars using half of the time if this 100-meters class radiotelescope. Two main programs are currently conducted. A large set of young and old pulsars is timed for a multi-wavelength approach, complementary to the very successful high energy observations of pulsars done by FERMI. A set of highly stable millisecond pulsars is monitored as our contribution to the European Pulsar Timing Array in order to probe the co…
The sensitivity of Pulsar Timing Arrays to gravitational waves depends on the noise present in the individual pulsar timing data. Noise may be either intrinsic or extrinsic to the pulsar. Intrinsic sources of noise will include rotational instabilities, for example. Extrinsic sources of noise include contributions from physical processes which are not sufficiently well modelled, for example, dispersion and scattering effects, analysis errors and instrumental instabilities. We present the results from a noise analysis for 42 millisecond pulsars (MSPs) observed with the European Pulsar Timing Ar…
Vanderbilt University（范德堡大学）；OzGrav: The ARC Center of Excellence for Gravitational Wave Discovery（ARC引力波发现卓越中心）；Swinburne University of Technology（斯威本科技大学）；LPC2E, OSUC, Univ Orleans, CNRS（奥尔良大学CNRS LPC2E实验室）；Max Planck Institute for Gravitational Physics (Albert Einstein Institute)（马克斯·普朗克引力物理学研究所（爱因斯坦研究所））；Leibniz Universität Hannover（汉诺威莱布尼茨大学）；Jodrell Bank Centre for Astrophysics, Department of Physics and Astronomy, University of Manchester（曼彻斯特大学天体物理学与天文学系乔德雷尔银行天体物理中心）
A low-frequency gravitational-wave background (GWB) from the cosmic merger history of supermassive black holes is expected to be detected in the next few years by pulsar timing arrays. A GWB induces distinctive correlations in the pulsar residuals --- the expected arrival time of the pulse less its actual arrival time. Simplifying assumptions are made in order to write an analytic expression for this correlation function, called the Hellings and Downs curve for an isotropic GWB, which depends on the angular separation of the pulsar pairs, the gravitational-wave frequency considered, and the di…
