The stochastic background: scaling laws and time to detection for pulsar timing arrays

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-wave dominated. Pulsar timing arrays enter this regime more quickly than one might naively suspect. For T=10 yr observations and typical stochastic background amplitudes, pulsars with residual RMSs of less than about $1\,\mu$s are already in that regime. The best strategy to increase the detectability of the background in this regime is to increase the number of pulsars in the array. We also perform realistic simulations of the NANOGrav pulsar timing array, which through an aggressive pulsar survey campaign adds new millisecond pulsars regularly to its array, and show that a detection is possible within a decade, and could occur as early as 2016.Comment: Submitted to Classical and Quantum Gravity for Focus Issue on Pulsar Timing Arrays. 15 pages, 5 figure

In an expanding universe, what doesn't expand?

The expansion of the universe is often viewed as a uniform stretching of space that would affect compact objects, atoms and stars, as well as the separation of galaxies. One usually hears that bound systems do not take part in the general expansion, but a much more subtle question is whether bound systems expand partially. In this paper, a very definitive answer is given for a very simple system: a classical "atom" bound by electrical attraction. With a mathemical description appropriate for undergraduate physics majors, we show that this bound system either completely follows the cosmological expansion, or -- after initial transients -- completely ignores it. This "all or nothing" behavior can be understood with techniques of junior-level mechanics. Lastly, the simple description is shown to be a justifiable approximation of the relativistically correct formulation of the problem.Comment: 8 pages, 9 eps figure

Robust statistics for deterministic and stochastic gravitational waves in non-Gaussian noise I: Frequentist analyses

Gravitational wave detectors will need optimal signal-processing algorithms to extract weak signals from the detector noise. Most algorithms designed to date are based on the unrealistic assumption that the detector noise may be modeled as a stationary Gaussian process. However most experiments exhibit a non-Gaussian ``tail'' in the probability distribution. This ``excess'' of large signals can be a troublesome source of false alarms. This article derives an optimal (in the Neyman-Pearson sense, for weak signals) signal processing strategy when the detector noise is non-Gaussian and exhibits tail terms. This strategy is robust, meaning that it is close to optimal for Gaussian noise but far less sensitive than conventional methods to the excess large events that form the tail of the distribution. The method is analyzed for two different signal analysis problems: (i) a known waveform (e.g., a binary inspiral chirp) and (ii) a stochastic background, which requires a multi-detector signal processing algorithm. The methods should be easy to implement: they amount to truncation or clipping of sample values which lie in the outlier part of the probability distribution.Comment: RevTeX 4, 17 pages, 8 figures, typos corrected from first version

Sensitivity curves for searches for gravitational-wave backgrounds

We propose a graphical representation of detector sensitivity curves for stochastic gravitational-wave backgrounds that takes into account the increase in sensitivity that comes from integrating over frequency in addition to integrating over time. This method is valid for backgrounds that have a power-law spectrum in the analysis band. We call these graphs “power-law integrated curves.” For simplicity, we consider cross-correlation searches for unpolarized and isotropic stochastic backgrounds using two or more detectors. We apply our method to construct power-law integrated sensitivity curves for second-generation ground-based detectors such as Advanced LIGO, space-based detectors such as LISA and the Big Bang Observer, and timing residuals from a pulsar timing array. The code used to produce these plots is available at https://dcc.ligo.org/LIGO-P1300115/public for researchers interested in constructing similar sensitivity curves

Hellings and Downs correlation of an arbitrary set of pulsars

Pulsar timing arrays (PTAs) detect gravitational waves (GWs) via the correlations they induce in the arrival times of pulses from different pulsars. We assume that the GWs are described by a Gaussian ensemble, which models the confusion noise produced by expected PTA sources. The mean correlation h2μu(γ) as a function of the angle γ between the directions to two pulsars was predicted by Hellings and Downs in 1983. The variance σtot2(γ) in this correlation was recently calculated [B. Allen, Variance of the Hellings-Downs correlation, Phys. Rev. D 107, 043018 (2023)PRVDAQ2470-001010.1103/PhysRevD.107.043018] for a single noise-free pulsar pair at angle γ, which shows that after averaging over many pairs, the variance reduces to an intrinsic cosmic variance σcos2(γ). Here, we extend this to an arbitrary set of pulsars at specific sky locations, with pulsar pairs binned by γ. We derive the linear combination of pulsar-pair correlations which is the optimal estimator of the Hellings and Downs correlation for each bin, illustrating our methods with plots of the expected range of variation away from the Hellings and Downs curve, for the sets of pulsars monitored by three active PTA collaborations. We compute the variance of and the covariance between these binned estimates, and show that these reduce to the cosmic variance and covariance s(γ,γ′) respectively, in the many-pulsar limit. The likely fluctuations away from the Hellings and Downs curve μu(γ) are strongly correlated/anticorrelated in the three angular regions where μu(γ) is successively positive, negative, and positive. We also construct the optimal estimator of the squared strain h2 from pulsar-pair correlation data. Remarkably, when there are very many pulsar pairs, this determines h2 with arbitrary precision because (in contrast to LIGO-like GW detectors) PTAs probe an infinite set of GW modes. To assess if observed deviations away from the Hellings and Downs curve are consistent with predictions, we propose and characterize several χ2 goodness-of-fit statistics. While our main focus is ideal noise-free data, we also show how pulsar noise and measurement noise can be included. Our methods can also be applied to future PTAs, where the improved telescopes will provide larger pulsar populations and higher-precision timing

The Hellings and Downs correlation of an arbitrary set of pulsars

Pulsar timing arrays (PTAs) detect gravitational waves (GWs) via the correlations they induce in the arrival times of pulses from different pulsars. We assume that the GWs are described by a Gaussian ensemble. The mean correlation $h^2 \mu_{\rm u}(\gamma)$ as a function of the angle $\gamma$ between the directions to two pulsars was predicted by Hellings and Downs (HD) in 1983. The variance $\sigma^2_{\rm tot}(\gamma)$ in this correlation was recently calculated by Allen[1] for a single noise-free pulsar pair at angle $\gamma$, which shows that after averaging over many pairs, the variance reduces to an intrinsic cosmic variance $\sigma^2_{\rm cos}(\gamma)$. Here, we extend this to an $arbitrary$ set of pulsars at specific sky locations, with pulsar pairs binned by $\gamma$. We derive the linear combination of pulsar-pair correlations which is the optimal estimator of the HD correlation for each bin, illustrating our methods with plots of the expected range of variation away from the HD curve, for the sets of pulsars monitored by three active PTA collaborations. We compute the variance of and the covariance between these binned estimates, and show that these reduce to the cosmic variance and covariance $s(\gamma,\gamma')$ respectively, in the many-pulsar limit. The likely fluctuations away from the HD curve $\mu_{\rm u}(\gamma)$ are strongly correlated/anticorrelated in the three angular regions where $\mu_{\rm u}(\gamma)$ is successively positive, negative, and positive. We also construct the optimal estimator of the squared strain $h^2$. When there are very many pulsar pairs, this determines $h^2$ with arbitrary precision because PTAs probe an infinite set of GW modes. To assess observed deviations away from the HD curve, we characterize several $\chi^2$ goodness-of-fit statistics. We also show how pulsar noise and measurement noise can be included.Comment: 77 pages, 23 figures, 4 tables. Now includes a complete treatment of the many-pulsar limits both with and without autocorrelations included, as well as construction of h-squared and chi-squared bounds and limit