Mapping gravitational-wave backgrounds using methods from CMB analysis: Application to pulsar timing arrays

Abstract

We describe an alternative approach to the analysis of gravitational-wave backgrounds, based on the formalism used to characterize the polarization of the cosmic microwave background. In contrast to standard analyses, this approach makes no assumptions about the nature of the background and so has the potential to reveal much more about the physical processes that generated it. An arbitrary background can be decomposed into modes whose angular dependence on the sky is given by gradients and curls of spherical harmonics. We derive the pulsar timing overlap reduction functions for the individual modes, which are given by simple combinations of spherical harmonics evaluated at the pulsar locations. We show how these can be used to recover the components of an arbitrary background, giving explicit results for both isotropic and anisotropic uncorrelated backgrounds. We also find that the response of a pulsar timing array to curl modes is identically zero, so half of the gravitational-wave sky will never be observed using pulsar timing, no matter how many pulsars are included in the array. An isotropic, unpolarized and uncorrelated background can be accurately represented using only three modes, and so a search of this type will be only slightly more complicated than the standard cross-correlation search using the Hellings and Downs overlap reduction function. However, by measuring the components of individual modes of the background and checking for consistency with isotropy, this approach has the potential to reveal much more information. Each individual mode on its own describes a background that is correlated between different points on the sky. A measurement of the components that indicates the presence of correlations in the background on large angular scales would suggest startling new physics.

Expected properties of the first gravitational wave signal detected with pulsar timing arrays

“…Second, throughout the paper we adopted a conservative definition of GWB (isotropic, stochastic, Gaussian, unpolarised, and stationary); the true signal produced by the superposition of GWs coming from the ensemble of SBHBs might well be dominated by a handful of signals, therefore significantly departing from isotropy and/or Gaussianity. The development of detection algorithms targeting multiple individual sources (Babak & Sesana 2012;Petiteau et al 2013) as well as certain types of anisotropic signals (Gair et al 2014) might prove to be a 'more optimal' strategy than searching for a GWB with the aforementioned properties.…”

“…Using an isotropic search when the background is anisotropic is sub-optimal and will have a lower DP. Techniques for searching for and characterising anisotropic backgrounds have been developed (Gair et al 2014), which could out-perform isotropic searches in certain regimes, although this has not yet been investigated fully. The DP of an optimal anisotropic background search is likely to be lower than an optimal isotropic search for an isotropic background of the same net amplitude, due to the larger number of model parameters in the anisotropic case.…”

The Polarizations of Gravitational Waves

“…A second method to detect GWs is to use pulsar timing arrays (PTAs) [54][55][56][57][58][59][60]. The propagation of radial pulses emanating from pulsars is affected by the stochastic GW background.…”

Limits on Anisotropy in the Nanohertz Stochastic Gravitational Wave Background

“…Hence the angular resolution, and thus l max , is a function of the number of pulsars, N psr , which significantly contribute to a single-source detection, and the SNR of that detection [43]. Sesana and Vecchio [43] find that the angular resolution of a PTA for a resolvable GW source is ΔΩ ∝ 50ð50=N psr Þ 1=2 ð10=SNRÞ 2 deg 2 , and this resolution sets an upper bound on l via l ¼ 180=θ, where θ ¼ ffiffiffiffiffiffiffi ffi ΔΩ p [29]. We analyze a subset of the six best pulsars in the EPTA [19] that encapsulate ∼95% of the full-array SNR in simulated continuous GW searches [44]: PSRs J0613 − 0200, J1012 þ 5307, J1600 − 3053, J1713 þ 0747, J1744 − 1134, J1909 − 3744, where T max ¼ 17.7 years and the GW frequencies with which we characterize red-noise components are ∈ ½1=T max ¼ 1.79; 50=T max ¼ 89.7 nHz .…”