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Register a new userUnderstanding the gravitational-wave Hellings and Downs curve for pulsar timing arrays in terms of sound and electromagnetic waves
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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 sound and electromagnetic waves. We calculate Hellings-and-Downs-type functions for these two scenarios and develop a framework suitable for doing more general correlation calculations.
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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. We find that the pulsar timing observables for the scalar longitudinal mode, which diverge with source distance in the luminal limit, are finite in the subluminal case. Furthermore, we apply our results to $f(R)$ gravity, which contains a massive scalar degree of freedom in addition to the standard transverse-traceless modes. The scalar mode in this $f(R)$ theory is a linear combination of the scalar-longitudinal and scalar-transverse modes, exciting only the monopole and dipole for pulsar timing arrays and only the dipole for astrometric surveys.
Recent years have seen a burgeoning interest in using pulsar timing arrays (PTAs) as gravitational-wave (GW) detectors. To date, that interest has focused mainly on three particularly promising source types: supermassive--black-hole binaries, cosmic strings, and the stochastic background from early-Universe phase transitions. In this paper, by contrast, our aim is to investigate the PTA potential for discovering unanticipated sources. We derive significant constraints on the available discovery space based solely on energetic and statistical considerations: we show that a PTA detection of GWs at frequencies above ~3.e-5 Hz would either be an extraordinary coincidence or violate cherished beliefs; we show that for PTAs GW memory can be more detectable than direct GWs, and that, as we consider events at ever higher redshift, the memory effect increasingly dominates an events total signal-to-noise ratio. The paper includes also a simple analysis of the effects of pulsar red noise in PTA searches, and a demonstration that the effects of periodic GWs in the 10^-8 -- 10^-4.5 Hz band would not be degenerate with small errors in standard pulsar parameters (except in a few narrow bands).
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.
We discuss the theory of pulsar-timing and astrometry probes of a stochastic gravitational-wave background with a recently developed total-angular-momentum (TAM) formalism for cosmological perturbations. We review the formalism, emphasizing in particular the features relevant for this work and describe the observables we consider (i.e. the pulsar redshift and stellar angular displacement). Using the TAM approach, we calculate the angular power spectra for the observables and from them derive angular auto- and cross-correlation functions. We provide the full set of power spectra and correlation functions not only for the standard transverse-traceless propagating degrees of freedom in general relativity, but also for the four additional non-Einsteinian polarizations that may arise in alternative-gravity theories. We discuss how pulsar-timing and astrometry surveys can complement and serve as cross checks to one another and comment on the importance of testing the chirality of the gravitational-wave background as a tool to understand the nature of its sources. A simple rederivation of the power spectra from the plane-wave formalism is provided in an Appendix.
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 signal to be mapped across the sky to any desired resolution. We also derive the pulsar-timing overlap reduction functions for the various non-GR polarisations, finding analytic forms for anisotropic backgrounds with scalar-transverse (breathing) and vector-longitudinal polarisations, and a semi-analytic form for scalar-longitudinal backgrounds. Our results indicate that pulsar-timing observations will be completely insensitive to scalar-transverse mode anisotropies in the polarisation amplitude beyond dipole, and anisotropies in the power beyond quadrupole. Analogously to our previous findings that pulsar-timing observations lack sensitivity to tensor-curl modes for a transverse-traceless tensor background, we also find insensitivity to vector-curl modes for a vector-longitudinal background.
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