In our previous paper \cite{PTA1} we derived a generic expression for the pulse redshift the main observable for the Pulsar Timing Array (PTA) experiment for detection of gravitational waves for all possible polarizations induced by modifications of general relativity (GR). In this work we provide a generic expression of the overlap reduction function for PTA without using the short wavelength approximation for tensorial polarization. We are convinced, that the short wavelength approximation is not applicable to the overlap reduction function for PTA's, since the removal of the exponential terms in the integrand would lead to poles for $x, y$ and $l$ polarizations and discontinuities for $+$ and $\times$. In this work we provide a series expansion to calculate the integral exactly and investigate the behaviour of the series for short wavelength values via numerical evaluation of the analytical series. We find a disagreement for the limit of co-located pulsars with the Hellings \& Downs curve.
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…
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…
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 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…
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…
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…
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.
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…
We provide comments on the article by Shannon et al. (Sep 2015) entitled "Gravitational waves from binary supermassive black holes missing in pulsar observations". The purpose of this letter is to address several misconceptions of the public and other scientists regarding the conclusions of that work.
Pulsar-timing has become a celebrated tool for probing modifications to General Relativity in the strong-field surroundings of neutron stars. Here we investigate whether scalar-tensor theories that incorporate a nonminimally coupled scalar degree of freedom may pass pulsar-timing tests, by computing the scalar charges entering such observables. In particular we show that for positive values of the nonminimal coupling $ξ$, pulsar-timing constraints may be evaded even in the presence of spontaneous scalarization.
In the last decade, the use of an ensemble of radio pulsars to constrain the characteristic strain caused by a stochastic gravitational wave background has advanced the cause of detection of very low frequency gravitational waves significantly. This electromagnetic means of gravitational wave detection, called Pulsar Timing Array(PTA), is reviewed in this article. The principle of operation of PTA, the current operating PTAs and their status is presented along-with a discussion of the main challenges in the detection of gravitational waves using PTA.
The maximum frequency of gravitational waves (GWs) detectable with traditional pulsar timing methods is set by the Nyquist frequency ($f_{\rm{Ny}}$) of the observation. Beyond this frequency, GWs leave no temporal-correlated signals; instead, they appear as white noise in the timing residuals. The variance of the GW-induced white noise is a function of the position of the pulsars relative to the GW source. By observing this unique functional form in the timing data, we propose that we can detect GWs of frequency $>$ $f_{\rm{Ny}}$ (super-Nyquist frequency GWs; SNFGWs). We demonstrate the feasib…
The sensitivity of pulsar timing arrays to gravitational waves is, at some level, limited by timing noise. Red timing noise - the stochastic wandering of pulse arrival times with a red spectrum - is prevalent in slow-spinning pulsars and has been identified in many millisecond pulsars. Phenomenological models of timing noise, such as from superfluid turbulence, suggest that the timing noise spectrum plateaus below some critical frequency, $f_c$, potentially aiding the hunt for gravitational waves. We examine this effect for individual pulsars by calculating minimum observation times, $T_{\rm m…
The extremely regular, periodic radio emission from millisecond pulsars makes them useful tools for studying neutron star astrophysics, general relativity, and low-frequency gravitational waves. These studies require that the observed pulse times of arrival be fit to complex timing models that describe numerous effects such as the astrometry of the source, the evolution of the pulsar's spin, the presence of a binary companion, and the propagation of the pulses through the interstellar medium. In this paper, we discuss the benefits of using Bayesian inference to obtain pulsar timing solutions. …
We have begun an exciting era for gravitational wave detection, as several world-leading experiments are breaching the threshold of anticipated signal strengths. Pulsar timing arrays (PTAs) are pan-Galactic gravitational wave detectors that are already cutting into the expected strength of gravitational waves from cosmic strings and binary supermassive black holes in the nHz-$μ$Hz gravitational wave band. These limits are leading to constraints on the evolutionary state of the Universe. Here, we provide a broad review of this field, from how pulsars are used as tools for detection, to astrophy…
We constrain the cosmological density of cosmic string loops using two observational signatures -- gravitational microlensing and the Kaiser-Stebbins effect. Photometry from RXTE and CoRoT space missions and pulsar timing from Parkes Pulsar Timing Array, Arecibo and Green Bank radio telescopes allow us to probe cosmic strings in a wide range of tensions $Gμ/c^2=10^{-16}\div10^{-10}$. We find that pulsar timing data provide the most stringent constraints on the abundance of light strings at the level $Ω_s \sim 10^{-3}$. Future observational facilities such as the Square Kilometer Array will all…
We provide an observation method for gravitational waves using a pulsar timing array to extend the observational frequency range up to the rotational frequency of pulsars. For this purpose, we perform an analysis of a perturbed electromagnetic wave in perturbed spacetime from the field perspective. We apply the analysis to the received electromagnetic waves in a radio telescope, which partially composes the periodic electromagnetic pulse emitted by a pulsar. For simple observation, two frequency windows are considered. For each window, we propose gauge-invariant quantities and discuss their ob…
A low frequency stochastic background of gravitational waves may be detected by pulsar timing experiments in the next five to ten years. Using methods developed to analyze interferometric gravitational wave data, in this paper we lay out the optimal techniques to detect a background of gravitational waves using a pulsar timing array. We show that for pulsar distances and gravitational wave frequencies typical of pulsar timing experiments, neglecting the effect of the metric perturbation at the pulsar does not result in a significant deviation from optimality. We discuss methods for setting upp…
Searches for stochastic gravitational-wave backgrounds using pulsar timing arrays look for correlations in the timing residuals induced by the background across the pulsars in the array. The correlation signature of an isotropic, unpolarized gravitational-wave background predicted by general relativity follows the so-called Hellings and Downs curve, which is a relatively simple function of the angle between a pair of Earth-pulsar baselines. In this paper, we give a pedagogical discussion of the Hellings and Downs curve for pulsar timing arrays, considering simpler analogous scenarios involving…
We study the observation of stochastic gravitational-wave background (SGWB) made by pulsar-timing arrays in the spherical harmonic space. Instead of using the Shapiro time delay, we keep the Sachs-Wolfe line-of-sight integral for the timing residual of an observed pulsar. We derive the power spectrum of the timing residual, from which the overlap reduction functions and the bipolar spherical harmonics coefficients are constructed for the SGWB intensity and polarization anisotropies. We have reproduced the previous results, noting that we have developed a fast algorithm for computing accurate o…
Of the current known pulsars, the Crab pulsar (B0531+21) is one of the most promising sources of gravitational waves. The relatively large timing noise of the Crab causes its phase evolution to depart from a simple spin-down model. This effect needs to be taken in to account when performing time domain searches for the Crab pulsar in order to avoid severely degrading the search efficiency. The Jodrell Bank Crab pulsar ephemeris is examined to see if it can be used for tracking the phase evolution of any gravitational wave signal from the pulsar, and we present a method of heterodyning the data…
Pulsar timing arrays act to detect gravitational waves by observing the small, correlated effect the waves have on pulse arrival times at Earth. This effect has conventionally been evaluated assuming the gravitational wave phasefronts are planar across the array, an assumption that is valid only for sources at distances $R\gg2πL^2/λ$, where $L$ is physical extent of the array and $λ$ the radiation wavelength. In the case of pulsar timing arrays (PTAs) the array size is of order the pulsar-Earth distance (kpc) and $λ$ is of order pc. Correspondingly, for point gravitational wave sources closer …
We study the observation of polarized stochastic gravitational-wave background (SGWB) in pulsar-timing-array experiments. The time residual for an observed pulsar is formulated as a line-of-sight integral that incorporates the effects of the pulsar term, from which we construct the correlation function of the time residual between a pair of pulsars in terms of the overlap reduction functions (ORFs) for the SGWB intensity and polarization anisotropies. Our formulation provides a numerical scheme for computing the ORFs for high multipole moments and the lowest-moment ORFs for the SGWB linear pol…
The majority of fast millisecond pulsars are in binary systems, so that any periodic signal they emit is modulated by both Doppler and relativistic effects. Here we show how well-established binary models can be used to account for these effects in searches for gravitational waves from known pulsars within binary systems. A seperate issue affecting certain pulsar signals is that of timing noise and we show how this, with particular reference to the Crab pulsar, can be compensated for by using regularly updated timing ephemerides.
We extend our previous work on applying CMB techniques to the mapping of gravitational-wave backgrounds to backgrounds which have non-GR polarisations. Our analysis and results are presented in the context of pulsar-timing array observations, but the overarching methods are general, and can be easily applied to LIGO or eLISA observations using appropriately modified response functions. Analytic expressions for the pulsar-timing response to gravitational waves with non-GR polarisation are given for each mode of a spin-weighted spherical-harmonic decomposition of the background, which permit the…
Relic gravitational waves (RGWs) , a background originated during inflation, would give imprints on the pulsar timing residuals. This makes RGWs be one of important sources for detection using the method of pulsar timing. In this paper, we discuss the effects of RGWs on the single pulsar timing, and give quantitively the timing residuals caused by RGWs with different model parameters. In principle, if the RGWs are strong enough today, they can be detected by timing a single millisecond pulsar with high precision after the intrinsic red noise in pulsar timing residuals were understood, even tho…
Over the last few years, a set of new results from pulsar timing has introduced much tighter constraints on violations of the strong equivalence principle (SEP), either via a direct verification of the universality of free fall for a pulsar in a triple star system, or from tests of the nature of gravitational waves, in particular a search for dipolar gravitational wave emission in a variety of binary pulsars with different masses. No deviations from the SEP have been detected in our experiments. These results introduce some of the most stringent constraints on several classes of alternative th…
The opening of the gravitational wave window by ground-based laser interferometers has made possible many new tests of gravity, including the first constraints on polarization. It is hoped that within the next decade pulsar timing will extend the window by making the first detections in the nano-Hertz frequency regime. Pulsar timing offers several advantages over ground-based interferometers for constraining the polarization of gravitational waves due to the many projections of the polarization pattern provided by the different lines of sight to the pulsars, and the enhanced response to longit…
Pulsar timing experiments are currently searching for gravitational waves, and this dissertation focuses on the development and study of the pulsar timing residual models used for continuous wave searches. The first goal of this work is to re-present much of the fundamental physics and mathematics concepts behind the calculations and theory used in pulsar timing. While there exist many reference sources in the literature, I try to offer a fully self-contained explanation of the fundamentals of this research which I hope the reader will find helpful. The next goal broadly speaking has been to f…
This paper discusses spectral sensitivity of solitary and binary pulsars to a low-frequency (colored) noise. It is a third paper in a series of papers devoted to analysis of the low-frequency noise in pulsar timing observations and its impact on observable fitting parameters of timing model.
Pulsar timing-array correlation measurements offer an exciting opportunity to test the nature of gravity in the cosmologically novel nanohertz gravitational wave regime. The stochastic gravitational wave background is assumed Gaussian and random, while there are limited pulsar pairs in the sky. This brings theoretical uncertainties to the correlation measurements, namely the pulsar variance due to pulsar samplings and the cosmic variance due to Gaussian signals. We demonstrate a straightforward calculation of the mean and the variances on the Hellings-Downs correlation relying on a power spect…
Individual supermassive black hole binaries in non-circular orbits are possible nanohertz gravitational wave sources for the rapidly maturing Pulsar Timing Array experiments. We develop an accurate and efficient approach to compute Pulsar Timing Array signals due to gravitational waves from inspiraling supermassive black hole binaries in relativistic eccentric orbits. Our approach employs a Keplerian-type parametric solution to model third post-Newtonian accurate precessing eccentric orbits while a novel semi-analytic prescription is provided to model the effects of quadrupolar order gravitati…
We study the ultralight vector dark matter with a mass around $10^{-23}\,\mathrm{eV}$. The vector field oscillating coherently on galactic scales induces oscillations of the spacetime metric with a frequency around nHz, which is detectable by pulsar timing arrays. We find that the pulsar timing signal due to the vector dark matter has nontrivial angular dependence unlike the scalar dark matter and the maximal amplitude is three times larger than that of the scalar dark matter.
We open the discussion into how the Laser Interferometer Space Antenna (LISA) observations of supermassive black-hole (SMBH) mergers (in the mass range ~10^6-10^8 Msun) may be complementary to pulsar timing-based gravitational wave searches. We consider the toy model of determining pulsar distances by exploiting the fact that LISA SMBH inspiral observations can place tight parameter constraints on the signal present in pulsar timing observations. We also suggest, as a future path of research, the use of LISA ring-down observations from the most massive (>~ a few 10^7 Msun) black-hole mergers, …
If the cosmological QCD phase transition is strongly first order and lasts sufficiently long, it generates a background of gravitational waves which may be detected via pulsar timing experiments. We estimate the amplitude and the spectral shape of such a background and we discuss its detectability prospects.
A Realistic Pulsar -- Supermassive Black Hole Timing Model
Peking University（北京大学）；Kavli Institute for Astronomy and Astrophysics, Peking University（北京大学科维理天文与天体物理研究所）；National Astronomical Observatories, Chinese Academy of Sciences（中国科学院国家天文台）
General relativity has predicted the existence of gravitational waves (GW), which are waves of the distortions of space-time with two degrees of polarization and the propagation speed of light. Alternative theories predict more polarizations, up to a maximum of six, and possible deviation of propagation speed from the light speed. The present paper reviews recent proposals to test the gravity theories in the radiation regime by observing GWs using pulsar timing arrays.
Long-term precise timing of Galactic millisecond pulsars holds great promise for measuring the long-period (months-to-years) astrophysical gravitational waves. Several gravitational-wave observational programs, called Pulsar Timing Arrays (PTA), are being pursued around the world. Here we develop a Bayesian algorithm for measuring the stochastic gravitational-wave background (GWB) from the PTA data. Our algorithm has several strengths: (1) It analyses the data without any loss of information, (2) It trivially removes systematic errors of known functional form, including quadratic pulsar spin-d…
Non-Gaussian timing residual skewness in pulsar timing arrays
Max Planck Institute for Gravitational Physics (Albert Einstein Institute), Leibnitz Universität Hannover（马克斯·普朗克引力物理研究所（爱因斯坦研究所），莱布尼茨汉诺威大学）；Department of Physics, National Central University（国立中央大学物理系）；Institute of Physics, Academia Sinica（台湾中央研究院物理研究所）；Institute of Astronomy and Astrophysics, Academia Sinica（台湾中央研究院天文及天文物理研究所）
Precision timing of large arrays (>50) of millisecond pulsars will detect the nanohertz gravitational-wave emission from supermassive binary black holes within the next ~3-7 years. We review the scientific opportunities of these detections, the requirements for success, and the synergies with electromagnetic instruments operating in the 2020s.
Pulsar Timing Arrays (PTAs) use high accuracy timing of a collection of low timing noise pulsars to search for gravitational waves in the microhertz to nanohertz frequency band. The sensitivity of such a PTA depends on (a) the direction of the gravitational wave source, (b) the timing accuracy of the pulsars in the array and (c) how the available observing time is allocated among those pulsars. Here, we present a simple way to calculate the sensitivity of the PTA as a function of direction of a single GW source, based only on the location and root-mean-square residual of the pulsars in the arr…
Pulsar Timing Arrays are a prime tool to study unexplored astrophysical regimes with gravitational waves. Here we show that the detection of gravitational radiation from individually resolvable super-massive black hole binary systems can yield direct information about the masses and spins of the black holes, provided that the gravitational-wave induced timing fluctuations both at the pulsar and at the Earth are detected. This in turn provides a map of the non-linear dynamics of the gravitational field and a new avenue to tackle open problems in astrophysics connected to the formation and evolu…
The pulsar timing residuals induced by gravitational waves from non-evolving single binary sources are affected by many parameters related to the relative positions of the pulsar and the gravitational wave sources. We will fully analyze the effects due to different parameters one by one. The standard deviations of the timing residuals will be calculated with a variable parameter fixing a set of other parameters. The orbits of the binary sources will be generally assumed to be elliptical. The influences of different eccentricities on the pulsar timing residuals will also studied in detail. We f…
Gravitational waves provide a new probe of the Universe which can reveal a number of cosmological and astrophysical phenomena that cannot be observed by electromagnetic waves. Different frequencies of gravitational waves are detected by different means. Among them, precision measurements of pulsar timing provides a natural detector for gravitational waves with light-year scale wavelengths. In this review, first a basic framework to detect a stochastic gravitational wave background using pulsar timing array is introduced, and then possible interpretations of the latest observational result of 1…
The effect of a narrow-band isotropic stochastic GW background on pulsar timing and astrometric measurements is studied. Such a background appears in some theories of gravity. We show that the existing millisecond pulsar timing accuracy ($\sim 0.2 \rm{μs}$) strongly constrains possible observational consequences of theory of massive gravity with spontaneous Lorentz braking \cite{dtt:2005}, essentially ruling out significant contribution of massive gravitons to the local dark halo density. The present-day accuracy of astrometrical measurements ($\sim 100 \rm{μas}$) sets less stringent constrain…
We study how to probe bispectra of stochastic gravitational waves with pulsar timing arrays. The bispectrum is a key to probe the origin of stochastic gravitational waves. In particular, the shape of the bispectrum carries valuable information of inflation models. We show that an appropriate filter function for three point correlations enables us to extract a specific configuration of momentum triangles in bispectra. We also calculate the overlap reduction functions and discuss strategy for detecting the bispectrum with multiple pulsars.
The coherent oscillation of ultralight dark matter in the mass regime around $10^{-23}$ eV induces changes in gravitational potential with the frequency in the nanohertz range. This effect is known to produce a monochromatic signal in the pulsar timing residuals. Here we discuss a multifield scenario that produces a wide spectrum of frequencies, such that the ultralight particle oscillation can mimic the pulsar timing signal of stochastic common spectrum process. We discuss how ultralight dark matter with various spins produces such a wide band spectrum on pulsar timing residuals and perform t…
The detection of a stochastic background of low-frequency gravitational waves by pulsar-timing and astrometric surveys will enable tests of gravitational theories beyond general relativity. These theories generally permit gravitational waves with non-Einsteinian polarization modes, which may propagate slower than the speed of light. We use the total-angular-momentum wave formalism to derive the angular correlation patterns of observables relevant for pulsar timing arrays and astrometry that arise from a background of subluminal gravitational waves with scalar, vector, or tensor polarizations. …
Pulsar timing arrays (PTAs) will be sensitive to a finite number of gravitational wave (GW) "point" sources (e.g. supermassive black hole binaries). N quiet pulsars with accurately known distances d_{pulsar} can characterize up to 2N/7 distant chirping sources per frequency bin Δf_{gw}=1/T, and localize them with "diffraction limited" precision δθ\gtrsim (1/SNR)(λ_{gw}/d_{pulsar}). Even if the pulsar distances are poorly known, a PTA with F frequency bins can still characterize up to (2N/7)[1-(1/2F)] sources per bin, and the quasi-singular pattern of timing residuals in the vicinity of a GW so…
