A Spin-weighted spherical harmonics

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


We describe an alternative approach to the analysis of gravitational-wave backgrounds, based on the formalism used to characterise the polarisation 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, unpolarised 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.

04.80.Nn, 04.30.Db, 07.05.Kf, 95.55.Ym

I Introduction

Near-future detections of gravitational waves (GWs) will open a new window onto the cosmos by allowing astrophysical and cosmological phenomena that generate only weak or difficult-to-detect electromagnetic signatures to be probed for the first time and with an unprecedented precision. Within the next several years a global network of advanced kilometre-scale laser interferometers will come online, providing insights into stellar-mass compact binary systems and stochastic gravitational-wave backgrounds in the kHz band Harry et al. (2010); Adv (2009); Somiya (2012); Unnikrishnan (2013). In years, the launch of a m arm-length space-based laser interferometer will allow precision tests of fundamental physics, and perform detailed demographic studies of massive black-holes throughout the Universe Amaro-Seoane et al. (2012).

Concurrently with these efforts are dedicated programs observing the regular pulsed-emission from ensembles of Galactic millisecond pulsars with the aim of detecting and characterising nanohertz gravitational waves van Haasteren et al. (2011); Demorest et al. (2013); Shannon et al. (2013); Manchester and IPTA (2013). The long-term stability of integrated pulse profiles allows incredibly accurate models of the time-of-arrival (TOA) of pulses to be constructed, and enables these pulsars to be used as standard clocks in the sky. Potential gravitational-wave targets in the nHz band are single resolvable sources (e.g., chirping supermassive black-hole (SMBH) binaries Sesana et al. (2009); Lee et al. (2011); Ellis et al. (2012) or cosmic-string bursts Damour and Vilenkin (2001); Leblond et al. (2009); Key and Cornish (2009)) and stochastic backgrounds from the superposition of many inspiraling SMBH binary systems Rajagopal and Romani (1995); Jaffe and Backer (2003); Wyithe and Loeb (2003), decaying cosmic-string networks Vilenkin (1981a, b); Ölmez et al. (2010); Sanidas et al. (2012), or even backgrounds of primordial origin Grishchuk (1976, 2005).

A pulsar timing array (PTA) can be thought of as a galactic-scale gravitational-wave detector Foster and Backer (1990). When a gravitational wave transits the Earth-pulsar line-of-sight it creates a perturbation in the intervening metric, causing a change in the proper separation, which manifests as a redshift in the pulse frequency Sazhin (1978); Detweiler (1979); Estabrook and Wahlquist (1975); Burke (1975). Standard timing-models only factor in deterministic influences to the TOAs, such that a subtraction of modelled TOAs from the raw observations will result in a stream of timing-residuals, which encode the influence of gravitational waves along with stochastic noise processes. A PTA allows one to cross-correlate the residuals from many pulsars, leveraging the common influence of a gravitational-wave background against undesirable, uncorrelated noise processes.

In fact, for a Gaussian-stationary, isotropic, unpolarised stochastic background composed of plus/cross gravitational-wave polarisation states, the cross-correlation of timing-residuals is a unique smoking-gun signature of the background’s presence, and depends only on the angular separation between pulsars on the sky: this is the famous Hellings and Downs curve Hellings and Downs (1983). Backgrounds composed of non-Einsteinian polarisation states Lee et al. (2008); Chamberlin and Siemens (2012), or influenced by non-zero graviton mass Lee (2013), will induce different correlation signatures, as will anisotropy in the background’s energy density Mingarelli et al. (2013); Taylor and Gair (2013), where the signature will contain rich information on the distribution of gravitational-wave power with respect to the position of pulsars on the sky.

Each of these standard analyses assumes a model describing the nature of the background and then tries to measure a small number of model parameters. For an isotropic, unpolarised and uncorrelated background there is just one measurable parameter, which is the amplitude of the background. While such analyses are optimal for the type of background being modelled, they will not be as sensitive to alternative models and will not indicate whether the model is correct. In this paper, we describe how pulsar timing residuals can be used instead to construct a map of the gravitational-wave background that makes no assumptions about its nature. The properties of the observed background can be checked for consistency with any particular model, e.g., to what extent it is isotropic, unpolarised and uncorrelated, but this approach has the potential to reveal much more, since we will extract all of the information that can be determined about the background by a given pulsar timing array. This information will not only tell us which out of our current models provides the best description of the background, but will clearly indicate if none of those models are accurate and therefore that new physical models of the background are required.

The polarisation of the cosmic microwave background (CMB), which has two independent components, can be represented as a transverse traceless tensor field on the sky Kamionkowski et al. (1997). In the analysis of CMB data, the polarisation field is represented as a superposition of gradients and curls of spherical harmonics, and CMB measurements attempt to determine the individual components of those modes. A gravitational-wave background is also a transverse traceless tensor field on the sky and so the same formalism can be applied to the analysis of a gravitational-wave background. It is this that we describe in this paper. That the CMB approach can be readily applied to gravitational waves is most easily seen from the fact that the gradients and curls of spherical harmonics can also be written as the real and imaginary parts of spin- spin-weighted spherical harmonics, which are widely used to decompose the gravitational-wave emission from a source Thorne (1980).

Any gravitational-wave background can be decomposed as a sum of gradient and curl modes. The components of this decomposition are the expansion coefficients of the metric perturbation in terms of the gradient and curl spherical harmonics, see Eq. (10). The signature that arises in the cross-correlation of the timing residuals of pairs of pulsars in a PTA can therefore be computed as a sum of the cross-correlation curves (overlap reduction functions) of each mode. For an unpolarised statistically isotropic background the overlap reduction functions for the individual models are just Legendre polynomials and the Hellings and Downs curve can be recovered straightforwardly as a superposition of these. Three modes are sufficient to represent the Hellings and Downs correlation for reasonable assumptions about the PTA, so applying this formalism to an isotropic, unpolarised and uncorrelated background will not be much more computationally challenging than the standard analysis.

The overlap reduction functions for individual modes can also be computed for anisotropic backgrounds. For pulsar timing arrays, the resulting expression is relatively simple since the response of a pulsar to curl modes is identically zero, while the response to a gradient mode is proportional to the corresponding spherical harmonic evaluated at the direction to the pulsar. For anisotropic, unpolarised and uncorrelated backgrounds, the integral expressions for the spherical harmonic components of the overlap reduction function can be evaluated analytically, allowing us to extend the results given for quadrupole and lower backgrounds in Mingarelli et al. (2013).

It is also relatively straightforward to reconstruct a map of the gravitational-wave sky for that part of the background spanned by the gradient modes visible to a PTA, and we describe how this can be done. For a PTA consisting of pulsars, at any given frequency we make two measurements—an amplitude and a phase—with each pulsar. Since PTAs are static, the response function is frequency-independent and we would therefore not expect to be able to measure more than real components of the background. The fact that PTAs are sensitive to only components of the background is consistent with recent unpublished results by Cornish and van Haasteren (private communication). We describe how we can recover these complex combinations of gradient mode components and which components we expect to measure most accurately (those for the low- modes). In practice, we can either restrict our mapping search to fewer than low- modes or use singular-value decomposition (SVD) of the mapping matrix to determine the linear combinations to which the array is sensitive. Since we make no assumptions about the properties of the underlying background in this analysis, we can interpret the map that we obtain in terms of its implications for fundamental physics, as described below. To characterise an isotropic, unpolarised and uncorrelated background we need to reach an angular resolution of , which requires pulsars, well within reach of current PTA efforts. To reach the angular resolution at which we expect to resolve individual sources with a PTA we must probe , which will require pulsars. This should be achievable with the Square-Kilometre Array (SKA) Smits et al. (2009).

In our approach, each individual mode used in the decomposition describes a background that is correlated between different points on the sky. By this we will mean a correlation in the gravitational radiation coming from different angular directions, which is different from the correlation between the pulsar responses, present for all types of background. It is also different from spatial correlations that may exist between the metric perturbations evaluated at different locations in space. A background that is spatially homogeneous and isotropic can have spatial correlations provided the correlations depend only on the distance between any two points and , and any background of this form will be uncorrelated in Fourier (angle) space. We focus on angular correlations because we will measure the gravitational-wave background at a single point only and therefore cannot compute spatial correlations from our data. Assumptions about the presence or absence of spatial correlations are needed to compute the statistical properties of a background in any particular physical model, but here we will focus only on a measurement of the background and so the angular correlation properties are the most important.

The gravitational-wave background in the pulsar timing band is most likely to be generated by a superposition of emission from many individual astrophysical sources. Such a background will not show angular correlations between different sky locations, but would show anisotropy indicative of the spatial distribution of sources contributing to the background. A background of cosmological origin could in principle show angular correlations on some scale, and the spectrum of modes present will be characteristic of the quantum fluctuations that produced it. However, there are no mechanisms currently known that would generate such correlations in the nanohertz frequency band. Nonetheless, the power of the analysis described here is that it can represent any background and it makes no assumptions about the correlation properties or isotropy. It will allow us to derive a map of the background, free from model assumptions, that will encode all of the details about the underlying physical processes that produced the background and that are possible to deduce from our observations. If the map indicates the presence of correlated emission or significant anisotropy, it will be a startling and profound result, pointing either to unmodelled physics in the early Universe or an unknown systematic affecting the timing data. In either case, the result would be of great significance.

This paper is organised as follows: In Section II we describe the general formalism, which is based on that used to characterise CMB polarisation and can be used to describe arbitrary gravitational-wave backgrounds. We include a description of the basis functions used to expand the backgrounds, and we give definitions of the response functions and overlap reduction functions for arbitrary gravitational-wave detectors. In Section III we specialise to the case of PTAs, deriving the overlap reduction function for an unpolarised statistically isotropic background, and show how the Hellings and Downs curve for an isotropic, unpolarised and uncorrelated background is well-approximated by a combination of the first three modes, . We show that the response of a pulsar to the curl modes of a gravitational-wave background is identically zero, while the response to an individual gradient mode is simply proportional to the corresponding spherical harmonic evaluated at the direction to the pulsar. We also demonstrate how the formalism can be used by recovering the coefficients of the expansion from a simulated pulsar-timing data set. In Section IV we compute the overlap reduction functions needed to represent arbitrary anisotropic backgrounds, giving explicit expressions for a PTA. In Section V we discuss how one can reconstruct a map of the gravitational-wave sky in terms of the gradient components visible to a PTA. We show that an -pulsar array can measure (complex) combinations of the gradient components of the background, but is blind to the curl component, irrespective of the value of . Finally, in Section VI we summarise the results and discuss some of the implications if a measurement of these parameters is made that is indicative of significant correlations in the background.

We also include several appendices: Appendices A and B contain useful definitions and identities for spin-weighted spherical harmonics and associated Legendre functions and Legendre polynomials, respectively. In Appendix C, we calculate the oscillatory behavior of the pulsar term for an isotropic, unpolarised and uncorrelated stochastic background, and show that it is negligible. In Appendix D, we derive the grad and curl response for a static interferometer, and find that the curl response is zero, similar to that for a PTA. In Appendix E, we derive analytic expressions for the spherical harmonic components of the overlap reduction function for anisotropic, unpolarised and uncorrelated backgrounds for all values of and , extending the analytical results of Mingarelli et al. (2013).

Ii General formalism

The gravitational-wave field is a symmetric transverse-traceless tensor field, with two independent polarisation states, and , which transform under rotations of the polarisation axes defined at each point on the sky Maggiore (2008). In the analysis of the CMB, polarisation is characterised by a two dimensional, symmetric and trace-free matrix, which is analogous to the symmetric transverse-traceless metric perturbations describing a general gravitational-wave field. Therefore, our analysis will closely parallel the treatment of polarisation in analyses of the CMB, see e.g. Kamionkowski et al. (1997); Challinor et al. (2000).

ii.1 Gradient and curl spherical harmonics

Any symmetric trace-free rank-two tensor field on the two-sphere can be written as the sum of the “gradient” of a scalar field


plus the “curl” of another scalar field


where a semi-colon denotes covariant differentiation, is the metric tensor on the sphere, and is the Levi-Civita anti-symmetric tensor


Following standard practice, we use the metric tensor and its inverse to “lower” and “raise” tensor indices—e.g., . In standard spherical coordinates ,


Since any scalar field on the two-sphere can be written as a sum of spherical harmonics, , it follows that any symmetric trace-free rank-two tensor field can be written as a sum of gradients and curls of spherical harmonics Kamionkowski et al. (1997); Zerilli (1970).

Defining the gradient and curl spherical harmonics for by:




it follows that


Note that we have adopted the notational convention used in the CMB literature, e.g., Kamionkowski et al. (1997), by putting parentheses around multipole moment indices and to distinguish these indices from spatial tensor indices , , etc.

ii.2 Expanding the metric perturbations

In transverse-traceless coordinates, the metric perturbations associated with a gravitational wave are transverse to the direction of propagation and hence define a symmetric trace-free tensor field on the two-sphere. The Fourier components of the field can therefore be decomposed as




Note that the summation over starts at and not at , as would be the case if we were expanding a scalar function on the sphere in terms of ordinary (i.e., undifferentiated) spherical harmonics . In what follows we will use the shorthand notation for . From the above definitions it follows that




with respect to complex conjugation and parity (i.e., ) transformations. Note that the gradient modes have “electric-type” parity, while the curl modes have “magnetic-type” parity. These are sometimes referred to as “ modes” and “ modes”, respectively, in the CMB literature.

A general stochastic gravitational-wave background can be written as a superposition of plane waves having frequency and propagation direction . We assume that gravitational waves of different frequencies are uncorrelated with one another, which follows if the background is stationary with respect to time. Using the preceding decomposition, we can therefore write the metric perturbation induced by an arbitrary stochastic background in transverse-traceless coordinates as


Introducing the usual orthogonal coordinate axes on the sky


and defining two polarization tensors by


the gradient and curl spherical harmonics can be written explicitly as Hu and White (1997):




These functions are related to spin-2 spherical harmonics Newman and Penrose (1966); Goldberg et al. (1967) through the equation


and can be written in terms of associated Legendre functions as


Using this explicit form for the gradient and curl spherical harmonics, Eq. (14) becomes


In terms of the more traditional “plus” and “cross” decomposition of the Fourier components,


we see that


and, conversely,


Finally, in terms of spin-weighted spherical harmonics:




These latter expressions for , , , and are convenient when one can make use of relations derived for the spin-weighted spherical harmonics (see, e.g., Appendix A).

ii.3 Statistical properties of the background

The statistical properties of a Gaussian-stationary background are encoded in the quadratic expectation values or, equivalently, , for and . For a statistically unpolarised and uncorrelated isotropic background


where . The factor of has been included so that is the two-sided gravitational-wave strain power, when summed over both polarizations. Using Eq. (28) and assuming the above expectation values, it follows that


where the last line follows from the orthogonality relation


which is a consequence of Eqs. (7) and (8). In a similar way, one can show that


where the zero expectation values follow from


which is a consequence of Eq. (9). Thus, if we define


where the correlation functions have the form


we deduce that an isotropic, unpolarised and uncorrelated background may be described by Eqs. (37) and (38) with


for .

ii.4 Statistically isotropic backgrounds

Stochastic backgrounds described by expectation values of the form given in Eq. (37) are said to be statistically isotropic. This means that there is no preferred direction on the sky, even though there can be non-trivial angular dependence in the distribution of gravitational-wave power via the . The fact that the quadratic expectation values in Eq. (37) depend only on and not on is equivalent to the statement that the angular distribution is independent of the orientation of the reference frame in which it is evaluated. In Sec. IV.1, we will extend our analysis to include more general (i.e., statistically anisotropic) backgrounds, allowing expectation values that can also depend on , cf. Eq. (106). In principle, the correlation functions for a statistically isotropic background are arbitrary, but if we impose additional physicality constraints the forms are restricted, as we shall discuss in Sec. VI.1. Requiring the background to be statistically unpolarised imposes the restrictions


which follow from invariance of the expectation values under rotations about a point on the sky. In addition, invariance of the expectation values under a parity transformation () further requires


To see that this is indeed the case, recall that under a parity transformation, cf. Eq. (13),


for which


Thus, invariance under a parity transformation requires


so . Similarly, one can show . Hence, a statistically isotropic, unpolarised and parity-invariant background is completely characterised by the single correlation function .

ii.5 Detector response functions

The response of a detector to a passing gravitational wave is given by the convolution of the metric perturbations with the impulse response of the detector:


If we expand the metric perturbations in terms of the plus and cross Fourier modes , where , we can write the response as




Alternatively, if we expand the metric perturbations in terms of the gradient and curl spherical harmonic modes , where , we have




The detector response functions implicitly depend on the assumptions made about the choice of polarisation axes, but we will assume these are consistent with the definitions used in Eqs. (15) and (16) above. Note that the response functions for the two different mode decompositions are related by:


and, conversely,


which follow from Eq. (17).

ii.6 Overlap reduction function

Using Eqs. (37) and (38) for a statistically isotropic background, and assuming and , the expectation value of the correlation between two detectors, labeled by and , can be written as


where is the overlap reduction function (see, e.g., Christensen (1992); Flanagan (1993); Finn et al. (2009)), and is given by




where are the gradient and curl response functions for the two detectors, . In Sec. IV.1, we will extend our analysis to compute overlap reduction functions for general anisotropic backgrounds.

Iii Application to pulsar timing arrays

In this section, we apply the above formalism to PTAs, deriving the overlap reduction function for statistically isotropic backgrounds, and showing how one can recover the Hellings and Downs curve. The same approach can also be used to characterise gravitational-wave backgrounds in other frequency bands, relevant to ground-based or space-based detectors. Although the overlap reduction functions in those cases will be different due to the different detector response functions, they can be calculated in a similar way to the pulsar timing response derived here.

iii.1 Detector response functions

As a plane gravitational wave transits the Earth-pulsar line-of-sight, it creates a perturbation in the intervening metric, causing a change in the proper separation, which is manifested as a redshift in the pulse frequency Sazhin (1978); Detweiler (1979); Estabrook and Wahlquist (1975); Burke (1975):