Pulsar Timing Array Data Research Articles

Multiwavelength constraints on the inflationary consistency relation

We present the first attempt to use a combination of CMB, LIGO, and Pulsar Timing Array (PPTA) data to constrain both the tilt and the running of primordial tensor power spectrum through constraints on the gravitational wave energy density generated in the early universe. Combining measurements at different cosmological scales highlights how complementary data can be used to test the predictions of early universe models including the inflationary consistency relation. Current data prefer a slightly positive tilt (${n}_{t}=0.0{6}_{\ensuremath{-}0.89}^{+0.63}$) and a negative running (${n}_{t,\text{run}}<\ensuremath{-}0.22$) for the tensor power spectrum spectrum. Interestingly, the addition of direct gravitational wave detector data alone puts strong bounds on the tensor-to-scalar ratio $r<0.2$ since the large positive tensor tilt preferred by the Planck temperature power spectrum is no longer allowed. Adding the recently released BICEP2/KECK and Planck 353 GHz polarization cross-correlation data gives an even stronger bound $r<0.1$. We comment on possible effects of a large positive tilt on the background expansion and show that depending on the assumptions regarding the UV cutoff (${k}_{\mathrm{UV}}/{k}_{*}=1{0}^{24}$) of the primordial spectrum of gravitational waves, the strongest bounds on ${n}_{t}=0.0{7}_{\ensuremath{-}0.80}^{+0.52}$ are derived from this effect.

Time-domain implementation of the optimal cross-correlation statistic for stochastic gravitational-wave background searches in pulsar timing data

Supermassive black hole binaries, cosmic strings, relic gravitational waves from inflation, and first order phase transitions in the early universe are expected to contribute to a stochastic background of gravitational waves in the 10^(-9) Hz-10^(-7) Hz frequency band. Pulsar timing arrays (PTAs) exploit the high precision timing of radio pulsars to detect signals at such frequencies. Here we present a time-domain implementation of the optimal cross-correlation statistic for stochastic background searches in PTA data. Due to the irregular sampling typical of PTA data as well as the use of a timing model to predict the times-of-arrival of radio pulses, time-domain methods are better suited for gravitational wave data analysis of such data. We present a derivation of the optimal cross-correlation statistic starting from the likelihood function, a method to produce simulated stochastic background signals, and a rigorous derivation of the scaling laws for the signal-to-noise ratio of the cross-correlation statistic in the two relevant PTA regimes: the weak signal limit where instrumental noise dominates over the gravitational wave signal at all frequencies, and a second regime where the gravitational wave signal dominates at the lowest frequencies.

An all-sky search for continuous gravitational waves in the Parkes Pulsar Timing Array data set

We present results of an all-sky search in the Parkes Pulsar Timing Array (PPTA) Data Release 1 data set for continuous gravitational waves (GWs) in the frequency range from $5\times 10^{-9}$ to $2\times 10^{-7}$ Hz. Such signals could be produced by individual supermassive binary black hole systems in the early stage of coalescence. We phase up the pulsar timing array data set to form, for each position on the sky, two data streams that correspond to the two GW polarizations and then carry out an optimal search for GW signals on these data streams. Since no statistically significant GWs were detected, we place upper limits on the intrinsic GW strain amplitude $h_0$ for a range of GW frequencies. For example, at $10^{-8}$ Hz our analysis has excluded with $95\%$ confidence the presence of signals with $h_0\geqslant 1.7\times 10^{-14}$. Our new limits are about a factor of four more stringent than those of Yardley et al. (2010) based on an earlier PPTA data set and a factor of two better than those reported in the recent Arzoumanian et al. (2014) paper. We also present PPTA directional sensitivity curves and find that for the most sensitive region on the sky, the current data set is sensitive to GWs from circular supermassive binary black holes with chirp masses of $10^{9} M_{\odot}$ out to a luminosity distance of about 100 Mpc. Finally, we set an upper limit of $4 \times 10^{-3} {\rm{Mpc}}^{-3} {\rm{Gyr}}^{-1}$ at $95\%$ confidence on the coalescence rate of nearby ($z \lesssim 0.1$) supermassive binary black holes in circular orbits with chirp masses of $10^{10}M_{\odot}$.

Bayesian estimation of non-Gaussianity in pulsar timing analysis

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 the uncertainties associated with the pulsar timing model parameters. We then apply the method to the publicly available 15 year Parkes Pulsar Timing Array data release 1 dataset for the binary pulsar J0437$-$4715. In this analysis we present a significant detection of non-Gaussianity in the uncorrelated non-thermal noise, but we find that it does not yet impact the timing model or stochastic parameter estimates significantly compared to analysis performed assuming Gaussian statistics. The methods presented are, however, shown to be of immediate practical use for current European Pulsar Timing Array (EPTA) and International Pulsar Timing Array (IPTA) datasets.

Searching for Continuous Gravitational Waves with Pulsar Timing Arrays: Detection and Characterization

Gravitational Waves (GWs) are tiny ripples in the fabric of space-time predicted by Einsteins theory of General Relativity. Pulsar timing arrays (PTAs) offer a unique opportunity to detect low frequency GWs in the near future. Such a detection would be complementary to both LISA and LIGO GW efforts. In this frequency band, the expected source of GWs are Supermassive Black Hole Binaries (SMBHBs) and they will most likely form in an ensemble creating a stochastic GW background with the possibility of a few nearby/massive sources that will be individually resolvable. A direct detection of GWs will open a new window into the fields of astronomy and astrophysics by allowing us to constrain the coalescence rate of SMBHBs, providing further tests on the theory of General Relativity, and giving us access to properties of black holes not accessible by current astronomical techniques. Here we will discuss the development of a robust detection pipeline for single resolvable GW sources that is fully tailored to the unique aspects of PTA data analysis.Â

Using pulsar timing arrays and the quantum normalization condition to constrain relic gravitational waves

In the non-standard model of relic gravitational waves (RGWs) generated in the early universe, the theoretical spectrum is mainly described by an amplitude r and a spectral index β, the latter usually being determined by the slope of the inflaton potential. Pulsar timing arrays (PTAs) data have imposed constraints on the amplitude of strain spectrum for a power-law form as a phenomenological model. Applying these constraints to a generic, theoretical spectrum with r and β as independent parameters, we convert the PTAs constraint into an upper bound on the index β, which turns out to be less stringent than those upper bounds from the Big Bang nucleosynthesis, cosmic microwave background and LIGO/VIRGO, respectively. Moreover, it is found that PTAs constrain the non-standard RGWs more stringently than the standard RGWs. If the condition of the quantum normalization is imposed upon a theoretical spectrum of RGWs, r and β become related. With this condition, a minimum requirement of the horizon size during inflation is greater than the Planck length that results in an upper bound on β, which is comparable in magnitude to that by PTAs. When both PTAs and the quantum normalization are applied to a theoretical spectrum of RGWs, constraints can be obtained for other cosmic processes of the early universe, such as the reheating, a process less understood observationally so far. The resulting constraint is consistent with the slow-roll, massive scalar inflation model. The future square kilometer array will be able to constrain RGWs further and might even detect RGWs, rendering an important probe to the very early universe.

Pulsar timing array analysis for black hole backgrounds

An astrophysical population of supermassive black hole binaries is thought to be the strongest source of gravitational waves in the frequency range covered by pulsar timing arrays (PTAs). A potential cause for concern is that the standard cross-correlation method used in PTA data analysis assumes that the signals are isotropically distributed and Gaussian random, while the signals from a black hole population are likely to be anisotropic and deterministic. Here we show that while the conventional analysis is not optimal for detecting signals from black hole binaries, the technique still works as the standard Hellings–Downs correlation curve turns out to hold for point sources. Moreover, the small effective number of signal samples blurs the distinction between Gaussian and deterministic signals. Possible improvements to the standard cross-correlation analysis that account for the anisotropy of the signal are discussed.

The European Pulsar Timing Array and the Large European Array for Pulsars

The European Pulsar Timing Array (EPTA) and the Large European Array for Pulsars (LEAP) play crucial roles in the global effort to detect gravitational waves (GWs) with a Pulsar Timing Array (PTA) experiment. While the EPTA uses five of the world’s largest cm-radio telescopes, LEAP harvests their combined power to synthesize a 194 m equivalent dish to provide high-precision PTA data for most of the sky. The EPTA has already produced a large variety of results, including astrophysical studies of individual pulsars, tests of theories of gravity, stringent limits on a GW background produced by super-massive binary black holes or the vibration of cosmic strings. It has also undertaken the development of new analysis methods and techniques, and studies of the astrophysics and population of expected GW background sources. This review gives an overview of the EPTA and LEAP set-ups and corresponding activities.

Resolving multiple supermassive black hole binaries with pulsar timing arrays. II. Genetic algorithm implementation

Pulsar timing arrays might detect gravitational waves from massive black hole binaries within this decade. The signal is expected to be an incoherent superposition of several nearly monochromatic waves of different strengths. The brightest sources might be individually resolved, and the overall deconvolved, at least partially, in their individual components. In this paper we extend the maximum-likelihood-based method developed in Babak and Sesana [Phys. Rev. D 85, 044034 (2012)] to search for individual massive black hole binaries in pulsar timing array data. We model the signal as a collection of circular monochromatic binaries, each characterized by three free parameters: two angles defining the sky location and the frequency. We marginalize over all other source parameters, and we apply an efficient multisearch genetic algorithm to maximize the likelihood function and look for sources in synthetic data sets. On data sets characterized by white Gaussian noise plus few injected sources with signal-to-noise ratio in the range 10--60, our search algorithm performs well, recovering all the injections with no false positives. Individual source signal-to-noise ratios are estimated within a few percent of the injected values, sky locations are recovered within a few degrees, and frequencies are determined with sub-Fourier-bin precision.

Weighing the evidence for a gravitational-wave background in the first International Pulsar Timing Array data challenge

We describe an analysis of the First International Pulsar Timing Array Data Challenge, which was designed to test the ability of new and existing algorithms to constrain the properties of a stochastic gravitational-wave background influencing the arrival times of pulsar signals. We employ a robust, unbiased Bayesian framework developed by van Haasteren to study the three Open and Closed data sets of the International Pulsar Timing Array data challenge. We test various models for each data set and use MultiNest to recover the evidence for the purposes of Bayesian model selection. The parameter constraints of the favored model are confirmed using an adaptive Markov chain Monte Carlo technique. Our results for Closed1 favor a gravitational-wave background with strain amplitude at $f=1\text{ }\text{ }{\mathrm{yr}}^{\ensuremath{-}1}$, $A$, of $(1.1\ifmmode\pm\else\textpm\fi{}0.1)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}14}$, power spectral index $\ensuremath{\gamma}=4.30\ifmmode\pm\else\textpm\fi{}0.15$ and no evidence for red-timing noise or single sources. The evidence for Closed2 favors a gravitational-wave background with $A=(6.1\ifmmode\pm\else\textpm\fi{}0.3)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}14}$, $\ensuremath{\gamma}=4.34\ifmmode\pm\else\textpm\fi{}0.09$, with no red-timing noise or single sources. Finally, the evidence for Closed3 favors the presence of red-timing noise and a gravitational-wave background, with no single sources. The properties of the background are $A=(5\ifmmode\pm\else\textpm\fi{}1)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}15}$ and $\ensuremath{\gamma}=4.23\ifmmode\pm\else\textpm\fi{}0.35$, while the properties of the red noise are ${N}_{\mathrm{red}}=(12\ifmmode\pm\else\textpm\fi{}4)\text{ }\text{ }\mathrm{ns}$ and ${\ensuremath{\gamma}}_{\mathrm{red}}=1.5\ifmmode\pm\else\textpm\fi{}0.3$. In all cases the redness of the recovered background is consistent with a source population of inspiraling supermassive black-hole binaries. We also investigate the effect that down-sampling of the data sets has on parameter constraints and run time. Finally, we provide a proof-of-principle study of the ability of the Bayesian framework used in this paper to reconstruct the angular correlation of gravitational-wave background induced timing residuals, comparing this to the Hellings and Downs curve.

DOES A “STOCHASTIC” BACKGROUND OF GRAVITATIONAL WAVES EXIST IN THE PULSAR TIMING BAND?

We investigate the effects of gravitational waves (GWs) from a simulated population of binary super-massive black holes (SMBHs) on pulsar timing array datasets. We construct a distribution describing the binary SMBH population from an existing semi-analytic galaxy formation model. Using realizations of the binary SMBH population generated from this distribution, we simulate pulsar timing datasets with GW-induced variations. We find that the statistics of these variations do not correspond to an isotropic, stochastic GW background. The "Hellings & Downs" correlations between simulated datasets for different pulsars are recovered on average, though the scatter of the correlation estimates is greater than expected for an isotropic, stochastic GW background. These results are attributable to the fact that just a few GW sources dominate the GW-induced variations in every Fourier frequency bin of a 5-year dataset. Current constraints on the amplitude of the GW signal from binary SMBHs will be biased. Individual binary systems are likely to be detectable in 5-year pulsar timing array datasets where the noise is dominated by GW-induced variations. Searches for GWs in pulsar timing array data therefore need to account for the effects of individual sources of GWs.

OPTIMAL STRATEGIES FOR CONTINUOUS GRAVITATIONAL WAVE DETECTION IN PULSAR TIMING ARRAYS

Supermassive black hole binaries (SMBHBs) are expected to emit continuous gravitational waves in the pulsar timing array (PTA) frequency band ($10^{-9}$--$10^{-7}$ Hz). The development of data analysis techniques aimed at efficient detection and characterization of these signals is critical to the gravitational wave detection effort. In this paper we leverage methods developed for LIGO continuous wave gravitational searches, and explore the use of the $\mathcal{F}$-statistic for such searches in pulsar timing data. Babak & Sesana 2012 have already used this approach in the context of PTAs to show that one can resolve multiple SMBHB sources in the sky. Our work improves on several aspects of prior continuous wave search methods developed for PTA data analysis. The algorithm is implemented fully in the time domain, which naturally deals with the irregular sampling typical of PTA data and avoids spectral leakage problems associated with frequency domain methods. We take into account the fitting of the timing model, and have generalized our approach to deal with both correlated and uncorrelated colored noise sources. We also develop an incoherent detection statistic that maximizes over all pulsar dependent contributions to the likelihood. To test the effectiveness and sensitivity of our detection statistics, we perform a number of monte-carlo simulations. We produce sensitivity curves for PTAs of various configurations, and outline an implementation of a fully functional data analysis pipeline. Finally, we present a derivation of the likelihood maximized over the gravitational wave phases at the pulsar locations, which results in a vast reduction of the search parameter space.

A data analysis library for gravitational wave detection

AbstractOne of the main goals of Pulsar Timing Arrays (PTAs) is the direct detection of gravitational waves (GWs). A first detection will be a major leap for astronomy and substantial effort is currently going into timing as many pulsars as possible, with the highest possible accuracy. As part of the individual PTA projects, several groups are developing data analysis methods for the final stage of a gravitational-waves search pipeline: the analysis of the timing residuals. Here we report the progress of on-going work to develop, within a Bayesian framework, a comprehensive and user friendly analysis library to search for gravitational waves in PTA data.

Tracking dispersion measure variations of timing array pulsars with the GMRT

AbstractWe present the results from nearly three years of monitoring of the variations in dispersion measure (DM) along the line-of-sight to 11 millisecond pulsars using the Giant Metrewave Radio Telescope (GMRT). These results demonstrate accuracies of single epoch DM estimates of the order of 5 × 10−4 cm−3 pc. A preliminary comparison with the Parkes Pulsar Timing Array (PPTA) data shows that the measured DM fluctuations are comparable. We show effects of DM variations due to the solar wind and solar corona and compare with the existing models.

Stochastic and continuous gravitational wave analysis pipelines for pulsar timing array data

AbstractThe Nanohertz Observatory for Gravitational Waves (NANOGrav) collaboration aims to detect gravitational waves (GWs) through the precise timing of millisecond pulsars. GWs will come in the form of a stochastic background, continuous sources and burst sources. Here we will review recent progress on the development of data analysis pipelines aimed at the detection of a stochastic background as well as continuous sources. We will introduce the Optimal Statistic and F-Statistic methods that are used in the stochastic and continuous pipelines, respectively. Both pipelines are fully functional on real pulsar timing data and take into account the timing models for each pulsar. Finally, we will present the efficacy of each pipeline on locally simulated data as well as data from the 2012 IPTA data challenge.

Search for the gravitational wave memory effect with the Parkes Pulsar Timing Array

AbstractGravitational 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.

CONSTRAINING THE COALESCENCE RATE OF SUPERMASSIVE BLACK-HOLE BINARIES USING PULSAR TIMING

Pulsar timing observations are used to place constraints on the rate of coalescence of supermassive black-hole (SMBH) binaries as a function of mass and redshift. In contrast to the indirect constraints obtained from other techniques, pulsar timing observations provide a direct constraint on the black-hole merger rate. This is possible since pulsar timing is sensitive to the gravitational waves (GWs) emitted by these sources in the final stages of their evolution. We find that upper bounds calculated from the recently published Parkes Pulsar Timing Array data are just above theoretical predictions for redshifts below 10. In the future, with improved timing precision and longer data spans, we show that a non-detection of GWs will rule out some of the available parameter space in a particular class of SMBH binary merger models. We also show that if we can time a set of pulsars to 10ns timing accuracy, for example, using the proposed Square Kilometre Array, it should be possible to detect one or more individual SMBH binary systems.

DETECTION, LOCALIZATION, AND CHARACTERIZATION OF GRAVITATIONAL WAVE BURSTS IN A PULSAR TIMING ARRAY

Efforts to detect gravitational waves by timing an array of pulsars have focused traditionally on stationary gravitational waves: e.g., stochastic or periodic signals. Gravitational wave bursts --- signals whose duration is much shorter than the observation period --- will also arise in the pulsar timing array waveband. Sources that give rise to detectable bursts include the formation or coalescence of supermassive black holes (SMBHs), the periapsis passage of compact objects in highly elliptic or unbound orbits about a SMBH, or cusps on cosmic strings. Here we describe how pulsar timing array data may be analyzed to detect and characterize these bursts. Our analysis addresses, in a mutually consistent manner, a hierarchy of three questions: \emph{i}) What are the odds that a dataset includes the signal from a gravitational wave burst? \emph{ii}) Assuming the presence of a burst, what is the direction to its source? and \emph{iii}) Assuming the burst propagation direction, what is the burst waveform's time dependence in each of its polarization states? Applying our analysis to synthetic data sets we find that we can \emph{detect} gravitational waves even when the radiation is too weak to either localize the source of infer the waveform, and \emph{detect} and \emph{localize} sources even when the radiation amplitude is too weak to permit the waveform to be determined. While the context of our discussion is gravitational wave detection via pulsar timing arrays, the analysis itself is directly applicable to gravitational wave detection using either ground or space-based detector data.

On measuring the gravitational-wave background 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 inference method for measuring the stochastic gravitational-wave background (GWB) from the PTA data. Our method 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 spindown, annual modulations and jumps due to a change of equipment, (3) It measures simultaneously both the amplitude and the slope of the GWB spectrum, (4) It can deal with unevenly sampled data and coloured pulsar noise spectra. We sample the likelihood function using Markov Chain Monte Carlo (MCMC) simulations. We extensively test our approach on mock PTA datasets, and find that the Bayesian inference method has significant benefits over currently proposed counterparts. We show the importance of characterising all red noise components in pulsar timing noise by demonstrating that the presence of a red component would significantly hinder a detection of the GWB. Lastly, we explore the dependence of the signal-to-noise ratio on the duration of the experiment, number of monitored pulsars, and the magnitude of the pulsar timing noise. These parameter studies will help formulate observing strategies for the PTA experiments.