# Detection methods for stochastic gravitational-wave backgrounds: a unified treatment

## Abstract

We review detection methods that are currently in use or have been proposed to search for a stochastic background of gravitational radiation. We consider both Bayesian and frequentist searches using ground-based and space-based laser interferometers, spacecraft Doppler tracking, and pulsar timing arrays; and we allow for anisotropy, non-Gaussianity, and non-standard polarization states. Our focus is on relevant data analysis issues, and not on the particular astrophysical or early Universe sources that might give rise to such backgrounds. We provide a unified treatment of these searches at the level of detector response functions, detection sensitivity curves, and, more generally, at the level of the likelihood function, since the choice of signal and noise models and prior probability distributions are actually what define the search. Pedagogical examples are given whenever possible to compare and contrast different approaches. We have tried to make the article as self-contained and comprehensive as possible, targeting graduate students and new researchers looking to enter this field.

### Keywords

Gravitational waves Data analysis Stochastic backgrounds## 1 Introduction

The real voyage of discovery consists not in seeking new landscapes, but in having new eyes.

Marcel Proust

It is an exciting time for the field of gravitational-wave astronomy. The observation, on September 14th, 2015, of gravitational waves from the inspiral and merger of a pair of black holes (Abbott et al. 2016e) has opened a radically new way of observing the Universe. The event, denoted GW150914, was observed simultaneously by the two detectors of the Laser Interferometer Gravitational-wave Observatory (LIGO) (Aasi et al. 2015). [LIGO consists of two 4 km-long laser interferometers, one located in Hanford, Washington, the other in Livingston, LA.] The merger event that produced the gravitational waves occured in a distant galaxy roughly 1.3 billion light years from Earth. The initial masses of the two black holes were estimated to be \(36^{+5}_{-4}\ \mathrm{M}_\odot \) and \(29^{+4}_{-4}\ \mathrm{M}_\odot \), and that of the post-merger black hole as \(62^{+4}_{-4}\ \mathrm{M}_\odot \) (Abbott et al. 2016f). The difference between the initial and final masses corresponds to \(3.0^{+0.5}_{-0.5}\ \mathrm{M}_\odot c^2\) of energy radiated in gravitational waves, with a peak luminosity of *more than ten times the combined luminosity of all the stars in all the galaxies in the visible universe*! The fact that this event was observed *only* in gravitational waves—and not in electromagnetic waves—illustrates the complementarity and potential for new discoveries that comes with the opening of the gravitational-wave window onto the universe.

GW150914 is just the first of many gravitational-wave signals that we expect to observe over the next several years. Indeed, roughly 3 months after the detection of GW150914, a second event, GW151226, was observed by the two LIGO detectors (Abbott et al. 2016d). This event also involved the inspiral and merger of a pair of stellar mass black holes, with initial component masses \(14.2^{+8.3}_{-3.7}\ \mathrm{M}_\odot \) and \(7.5^{+2.3}_{-2.3}\ \mathrm{M}_\odot \), and a final black hole mass of \(20.8^{+6.1}_{-1.7}\ \mathrm{M}_\odot \). The source was at a distance of roughly 1.4 billion light-years from Earth, comparable to that of GW150914. Advanced LIGO will continue interleaving observation runs and commissioning activities to reach design sensivity around 2020 (Aasi et al. 2015), which will allow detections of signals like GW150914 and GW151226 with more than three times the signal-to-noise ratio than was observed for GW150914 (which was 24). In addition, the Advanced Virgo detector (Acernese et al. 2015) (a 3 km-long laser interferometer in Cascina, Italy) and KAGRA (Aso et al. 2013) (a 3 km-long cryogenic laser interferometer in Kamioka mine in Japan) should both be taking data by the end of 2016. There are also plans for a third LIGO detector in India (Iyer et al. 2011). A global network of detectors such as this will allow for much improved position reconstruction and parameter estimation of the sources (Abbott et al. 2016i).

### 1.1 Motivation and context

GW150914 and GW151226 were single events—binary black hole mergers that were observed with both template-based searches for compact binary inspirals and searches for generic gravitational-wave transients in the two LIGO detectors (Abbott et al. 2016e, d). The network matched-filter signal-to-noise ratio (Owen and Sathyaprakash 1999) for these two events, using relativitistic waveform models for binary black holes, was 24 and 13, respectively. The probability that these detections were due to noise alone is \({<} 2\times 10^{-7}\), corresponding to a significance greater than \(5\sigma \)—the standard for so-called “gold-plated” detections. But for every loud event like GW150914 or GW151226, we expect many more quiet events that are too distant to be individually detected, since the associated signal-to-noise ratios are too low.

The total rate of merger events from the population of stellar-mass binary black holes of which GW150914 and GW151226 are members can be estimated^{1} by multiplying the local rate estimate of 9–240 \(\mathrm{Gpc}^{-3}\, \mathrm{year}^{-1}\) (Abbott et al. 2016g) by the comoving volume out to some large redshift, e.g., \(z\sim 6\). This yields a total rate of binary black hole mergers between \({\sim }1\) per minute and a few per hour. Since the duration of each merger signal in the sensitive band of a LIGO-like detector is of order a few tenths of a second to \({\sim } 1\) s, the *duty cycle* (the fraction of time that the signal is “on” in the data) is \({\ll } 1\). This means that the combined signal from such a population of binary black holes will be “popcorn-like”, with the majority of the individual signals being too weak to individually detect. Since the arrival times of the merger signals are randomly-distributed, the combined signal from the population of binary black holes is itself random—it is an example of a *stochastic background* of gravitational radiation.

More generally, a stochastic background of gravitational radiation is *any* random gravitational-wave signal produced by a large number of weak, independent, and unresolved sources. The background doesn’t have to be popcorn-like, like the expected signal from the population of binary black holes which gave rise to GW150914 and GW151226. It can be composed of individual deterministic signals that overlap in time (or in frequency) producing a “confusion” noise analogous to conversations at a cocktail party. Such a confusion noise is produced by the galactic population of compact white dwarf binaries. (For this case, the stochastic signal is so strong that it becomes a *foreground*, acting as an additional source of noise when trying to detect *other* weak gravitational-wave signals in the same frequency band). Alternatively, the signal can be *intrinsically* random, associated with stochastic processes in the early Universe or with unmodeled sources, like supernovae, which produce signals that are not described by deterministic waveforms.

The focus of this review article is on data analysis strategies (i.e., detection methods) that can be used to detect and ultimately characterize a stochastic gravitational-wave background. To introduce this topic and to set the stage for the more detailed discussions to follow in later sections, we ask (and start to answer) the following questions:

#### 1.1.1 Why do we care about detecting a stochastic background?

Detecting a stochastic background of gravitational radiation can provide information about astrophysical source populations and processes in the very early Universe, which are inaccessible by any other means. For example, electromagnetic radiation cannot provide a picture of the Universe any earlier than the time of last of scattering (roughly 400,000 years after the Big Bang). Gravitational waves, on the other hand, can give us information all the way back to the onset of inflation, a mere \({\sim } 10^{-32}~\mathrm{s}\) after the Big Bang. (See Maggiore 2000 for a detailed discussion of both cosmological and astrophysical sources of a stochastic gravitational-wave background).

#### 1.1.2 Why is detection challenging?

Stochastic signals are effectively another source of noise in a single detector. So the fundamental problem is how to distinguish between gravitational-wave “noise” and instrumental noise. It turns out that there are several ways to do this, as we will discuss in the later sections of this article.

#### 1.1.3 What detection methods can one use?

Cross-correlation methods can be used whenever one has multiple detectors that respond to the common gravitational-wave background. For single detector analyses e.g., for the Laser Space Interferometer Antenna (LISA), one needs to take advantage of null combinations of the data (which act as instrument noise monitors) or use instrument noise modeling to try to distinguish the gravitational-wave signal from instrumental noise. Over the past 15 years or so, the number of detection methods for stochastic backgrounds has increased considerably. So now, in addition to the standard cross-correlation search for a “vanilla” (Gaussian-stationary, unpolarized, isotropic) background, one can search for non-Gaussian backgrounds, anisotropic backgrounds, circularly-polarized backgrounds, and backgrounds with polarization components predicted by alternative (non-general-relativity) theories of gravity. These searches are discussed in Sects. 7 and 8.

*all*analyses use a likelihood function, e.g., for defining frequentist statistics or for calculating posterior distributions for Bayesian inference (as will be described in more detail in Sect. 3), and take advantage of cross-correlations if multiple detectors are available (as will be described in more detail in Sect. 4).

Overview of analysis methods for stochastic gravitational-wave backgrounds

Early analyses (before 2000) | More recent analyses |
---|---|

Used frequentist statistics | Use both frequentist and Bayesian inference |

Used cross-correlation methods | Use cross-correlation methods and stochastic templates; use null channels or knowledge about instrumental noise when cross-correlation is not available |

Assumed Gaussian noise | Have allowed non-Gaussian noise |

Assumed stationary, Gaussian, unpolarized, and isotropic gravitational-wave backgrounds | Have allowed non-Gaussian, polarized, and anisotropic gravitational-wave backgrounds |

Were done primarily in the context of ground-based detectors (e.g., resonant bars and LIGO-like interferometers) where the small-antenna (i.e., long-wavelength) approximation was valid | Have been done in the context of space-based detectors (e.g., spacecraft tracking, LISA) and pulsar timing arrays for which the small-antenna approximation is not valid |

#### 1.1.4 What are the prospects for detection?

The prospects for detection depend on the source of the background (i.e., astrophysical or cosmological) and the type of detector being used. For example, a space-based interferometer like LISA is *guaranteed* to detect the gravitational-wave confusion noise produced by the galactic population of compact white dwarf binaries. Pulsar timing arrays, on the other hand, should be able to detect the confusion noise from supermassive black hole binaries (SMBHBs) at the centers of merging galaxies, provided the binaries are not affected by their environments in a way that severely diminishes the strength of the background (Shannon et al. 2015). Detection sensitivity curves are a very convenient way of comparing theoretical predictions of source strengths to the sensivity levels of the various detectors (as we will discuss in Sect. 10).

### 1.2 Searches across the gravitational-wave spectrum

^{2}noise sources below 10 Hz, and photon shot noise above a couple of kHz). Outside this band there are several other experiments—both currently operating and planned—that should also be able to detect gravitational waves. An illustration of the gravitational-wave spectrum, together with potential sources and relevant detectors, is shown in Fig. 1. We highlight a few of these experiments below.

#### 1.2.1 Cosmic microwave background experiments

At the extreme low-frequency end of the spectrum, corresponding to gravitational-wave periods of order the age of the Universe, the Planck satellite (ESA 2016c) and other cosmic microwave background (CMB) experiments, e.g., BICEP and Keck (BICEP/Keck 2016) are looking for evidence of relic gravitational waves from the Big Bang in the *B*-mode component of CMB polarization maps (Kamionkowski et al. 1997; Hu and White 1997; Ade et al. 2015a). In 2014, BICEP2 announced the detection of relic gravitational waves (Ade et al. 2014), but it was later shown that the observed *B*-mode signal was due to contamination by intervening dust in the galaxy (Flauger et al. 2014; Mortonson and Seljak 2014). So at present, these experiments have been able to only *constrain* (i.e., set upper limits on) the amount of gravitational waves in the very early Universe (Ade et al. 2015a). But these constraints severely limit the possibility of detecting the relic gravitational-wave background with any of the higher-frequency detection methods, unless its spectrum increases with frequency. [Note that standard models of inflation predict a relic background whose energy density is almost constant in frequency, leading to a strain spectral density that decreases with frequency.] Needless to say, the detection of a primordial gravitational-wave background is a “holy grail” of gravitational-wave astronomy.

#### 1.2.2 Pulsar timing arrays

At frequencies between \({\sim }10^{-9}~\mathrm{Hz}\) and \(10^{-7}~\mathrm{Hz}\), corresponding to gravitational-wave periods of order decades to years, pulsar timing arrays (PTAs) can be used to search for gravitational waves. This is done by carefully monitoring the arrival times of radio pulses from an array of galactic millisecond pulsars, looking for *correlated* modulations in the arrival times induced by a passing gravitational wave (Detweiler 1979; Hellings and Downs 1983). The most-likely gravitational-wave source for PTAs is a gravitational-wave background formed from the incoherent superposition of signals produced by the inspirals and mergers of SMBHBs in the centers of distant galaxies (Jaffe and Backer 2003). These searches continue to improve their sensitivity by upgrading instrument back-ends and discovering more millisecond pulsars that can be added to the array. These improvements have led to more constraining upper limits on the amplitude of the gravitational-wave background (Shannon et al. 2015; Arzoumanian et al. 2016), with a detection being likely before the end of this decade (Siemens et al. 2013; Taylor et al. 2016b).

#### 1.2.3 Space-based interferometers

At frequencies between \({\sim }10^{-4}~\mathrm{Hz}\) and \(10^{-1}~\mathrm{Hz}\), corresponding to gravitational-wave periods of order hours to minutes, proposed space-based interferometers like LISA can search for gravitational waves from a wide variety of sources (Gair et al. 2013). These include: (i) inspirals and mergers of SMBHBs with masses of order \(10^6~\mathrm{M}_\odot \), (ii) captures of compact stellar-mass objects around supermassive black holes, and (iii) the stochastic confusion noise produced by compact white-dwarf binaries in our galaxy. In fact, hundreds of binary black holes that are individually resolvable by LISA will coalesce in the aLIGO band within a 10 year period, opening up the possibility of doing *multi-band* gravitational-wave astronomy (Sesana 2016).

The basic space-based interferometer configuration consists of three satellites (each housing two lasers, two telescopes, and two test masses) that fly in an equilateral-triangle formation, with arm lengths of order several million km. A variant of the original LISA design was selected in February 2017 by the European Space Agency (ESA) as the 3rd large mission in its Cosmic Vision Program (ESA 2016a). The earliest launch date for LISA would be around 2030. A technology-demonstration mission, called LISA Pathfinder (ESA 2016b), was launched in December 2015, meeting or exceeding all of the requirements for an important subset of the LISA technologies (Armano et al. 2016).

#### 1.2.4 Other detectors

Finally, in the frequency band between \({\sim }0.1~\mathrm {Hz}\) and \(10~\mathrm {Hz}\), there are proposals for both Earth-based detectors (Harms et al. 2013) and also second-generation space-based interferometers—the Big-Bang Observer (BBO) (Phinney et al. 2004) and the DECI-hertz interferometer Gravitational-wave Observatory (DECIGO) (Ando et al. 2010). Such detectors would be sensitive to gravitational waves with periods between \({\sim }10~\mathrm {s}\) and \(0.1~\mathrm {s}\). The primary sources in this band are intermediate-mass (\(10^3\)–\(10^4~M_\odot \)) binary black holes, galactic and extra-galactic neutron star binaries, and a cosmologically-generated stochastic background.

### 1.3 Goal of this article

Starting with the pioneering work of Grishchuk (1976), Detweiler (1979), Hellings and Downs (1983), and Michelson (1987), detection methods for gravitational-wave backgrounds have increased in scope and sophistication over the years, with several new developments occuring rather recently. As mentioned above, we have search methods now that target different properties of the background (e.g., isotropic or anisotropic, Gaussian or non-Gaussian, polarized or unpolarized, etc.). These searches are necessarily implemented differently for different detectors, since, for example, ground-based detectors like LIGO and Virgo operate in the *small-antenna* (or *long-wavelength*) limit, while pulsar timing arrays operate in the *short-wavelength* limit. Moreover, each of these searches can be formulated in terms of either Bayesian or frequentist statistics. *The goal of this review article is to discuss these different detection methods from a perspective that attempts to *unify* the different treatments, emphasizing the similarities that exist when viewed from this broader perspective.*

### 1.4 Unification

The extensive literature describing stochastic background analyses leaves the reader with the impression that highly specialized techniques are needed for ground-based, space-based, and pulsar timing observations. Moreover, reviews of gravitational-wave data analysis leave the impression that the analysis of stochastic signals is somehow fundamentally different from that of any other signal type. Both of these impressions are misleading. The apparent differences are due to differences in terminology and perspective. By adopting a common analysis framework and notation, we are able to present a *unified* treatment of gravitational-wave data analysis across source classes and observation techniques.

We will provide a unified treatment of the various methods at the level of detector response functions, detection sensitivity curves, and, more generally, at the level of the likelihood function, since the choice of signal and noise models and prior probability distributions are actually what define the search. The same photon time-of-flight calculation underpins the detector response functions, and the choice of prior for the gravitational-wave template defines the search. A *matched-filter* search for binary mergers and a *cross-correlation* search for stochastic signals are both derived from the same likelihood function, the difference being that the former uses a parameterized, deterministic template, while the latter uses a stochastic template. Hopefully, by the end of this article, the reader will see that the plethora of searches for different types of backgrounds, using different types of detectors, and using different statistical inference frameworks are not all that different after all.

### 1.5 Outline

The rest of the article is organized as follows: We begin in Sect. 2 by specifying the quantities that one uses to characterize a stochastic gravitational-wave background. In Sect. 3, we give an overview of statistical inference by comparing and contrasting how the Bayesian and frequentist formalisms address issues related to hypothesis testing, model selection, setting upper limits, parameter estimation, etc. We then illustrate these concepts in the context of a very simple toy problem. In Sect. 4, we introduce the key concept of correlation, which forms the basis for the majority of detection methods used for gravitational-wave backgrounds, and show how these techniques arise naturally from the standard template-based approach. We derive the frequentist cross-correlation statistic for a simple example. We also describe how a null channel is useful when correlation methods are not possible.

In Sect. 5, we go into more detail regarding the different types of detectors. In particular, we calculate single-detector response functions and the associated antenna patterns for ground-based and space-based laser interferometers, spacecraft Doppler tracking, and pulsar timing measurements. (We do not discuss resonant bar detectors or CMB-based detection methods in this review article. However, current bounds from CMB observations will be reviewed in Sect. 10). By correlating the outputs of two such detectors, we obtain expressions for the correlation coefficient (or *overlap reduction function*) for a Gaussian-stationary, unpolarized, isotropic background as a function of the separation and orientation of the two detectors. In Sect. 6, we discuss optimal filtering. Section 7 extends the analysis of the previous sections to *anisotropic* backgrounds. Here we describe several different analyses that produce maps of the gravitational-wave sky: (i) a frequentist gravitational-wave radiometer search, which is optimal for point sources, (ii) searches that decompose the gravitational-wave power on the sky in terms of spherical harmonics, and (iii) a phase-coherent search that can map both the amplitude and phase of a gravitational-wave background at each location on the sky. In Sect. 8, we discuss searches for: (i) non-Gaussian backgrounds, (ii) circularly-polarized backgrounds, and (iii) backgrounds having non-standard (i.e., non-general-relativity) polarization modes. We also briefly describe extensions of the cross-correlation search method to look for *non-stochastic-background-type* signals—in particular, long-duration unmodelled transients and continuous (nearly-monochromatic) gravitational-wave signals from sources like Sco X-1.

In Sect. 9, we discuss real-world complications introduced by irregular sampling, non-stationary and non-Gaussian detector noise, and correlated environmental noise (e.g., Schumann resonances). We also describe what one can do if one has only a single detector, as is the case for LISA. Finally, we conclude in Sect. 10 by discussing prospects for detection, including detection sensitivity curves and current observational results.

We also include several appendices: In Appendix A we discuss different polarization basis tensors, and a Stokes’ parameter characterization of gravitational-waves. In Appendices B and C, we summarize some standard statistical results for a Gaussian random variable, and then discuss how to define and test for non-stationarity and non-Gaussianity. In Appendix D we describe the relationship between continuous functions of time and frequency and their discretely-sampled counterparts. Appendices E, F, G are adapted from Gair et al. (2015), with details regarding spin-weighted scalar, vector, and tensor spherical harmonics. Finally, Appendix H gives a “Rosetta stone” for translating back and forth between different response function conventions for gravitational-wave backgrounds.

## 2 Characterizing a stochastic gravitational-wave background

When you can measure what you are speaking about, and express it in numbers, you know something about it, when you cannot express it in numbers, your knowledge is of a meager and unsatisfactory kind; it may be the beginning of knowledge, but you have scarely, in your thoughts, advanced to the stage of science.

William Thomson, Baron Kelvin of Largs

In this section, we define several key quantities (e.g., fractional energy density spectrum, characteristic strain, distribution of gravitational-wave power on the sky), which are used to characterize a stochastic background of gravitational radiation. The definitions are appropriate for both isotropic and anisotropic backgrounds. Our approach is similar to that found in Allen and Romano (1999) for isotropic backgrounds and for the standard polarization basis. For the plane-wave decomposition in terms of tensor spherical harmonics, we follow Gair et al. (2014, 2015). Detailed derivations can be found in those papers.

### 2.1 When is a gravitational-wave signal stochastic?

The standard “textbook” definition of a stochastic background of gravitational radiation is *a random gravitational-wave signal produced by a large number of weak, independent, and unresolved sources*. To say that it is random means that it can be characterized only statistically, in terms of expectation values of the field variables or, equivalently, in terms of the Fourier components of a plane-wave expansion of the metric perturbations (Sect. 2.3.1). If the number of independent sources is sufficiently large, the background will be Gaussian by the central limit theorem. Knowledge of the first two moments of the distribution will then suffice to determine all higher-order moments (Appendix B). For non-Gaussian backgrounds, third and/or higher-order moments will also be needed.

Although there is general agreement with the above definition, there has been some confusion and disagreement in the literature (Rosado 2011; Regimbau and Mandic 2008; Regimbau and Hughes 2009; Regimbau 2011) regarding some of the defining properties of a stochastic background. This is because terms like *weak* and *unresolved* depend on details of the observation (e.g., the sensitivity of the detector, the total observation time, etc.), which are not intrinsic properties of the background. So the answer to the question “When is a gravitational-wave signal stochastic?” is not as simple or obvious as it might initially seem.

In Cornish and Romano (2015), we addressed this question in the context of searches for gravitational-wave backgrounds produced by a population of astrophysical sources. We found that it is best to give *operational* definitions for these properties, framed in the context of Bayesian inference. We will discuss Bayesian inference in more detail in Sect. 3, but for now the most important thing to know is that by using Bayesian inference we can calculate the probabilities of different signal-plus-noise models, given the observed data. The signal-plus-noise model with the largest probability is the preferred model, i.e., the one that is most consistent with the data. This is the essence of Bayesian model selection.

So we define a signal to be *stochastic* if a Bayesian model selection calculation prefers a stochastic signal model over any deterministic signal model. We also define a signal to be *resolvable* if it can be decomposed into *separate* (e.g., non-overlapping in either time or frequency) and *individually detectable* signals, again in a Bayesian model selection sense.^{3} If the background is associated with the superposition of signals from many astrophysical sources—as we expect for the population of binary black holes which gave rise to GW150914 and GW151226—then we should *subtract out* any bright deterministic signals that standout above the lower-amplitude background, leaving behind a residual non-deterministic signal whose statistical properties we would like to determine. In the context of Bayesian inference, this ‘subtraction’ is done by allowing *hybrid* signal models, which consist of both parametrized deterministic signals and non-deterministic backgrounds. By using such hybrid models we can investigate the statistical properties of the residual background without the influence of the resolvable signals.

We will return to these ideas in Sect. 8.1, when we discuss searches for non-Gaussian backgrounds in more detail.

### 2.2 Plane-wave expansions

*f*, and coming from different directions \(\hat{n}\) on the sky:

^{4}

#### 2.2.1 Polarization basis

#### 2.2.2 Tensor spherical harmonic basis

*gradient*and

*curl*tensor spherical harmonics (Gair et al. 2014):

*l*and

*m*) on the 2-sphere. Appendix G contains additional details regarding gradient and curl spherical harmonics.

Note that we have adopted the notational convention used in the CMB literature, e.g., Kamionkowski et al. (1997), by putting parentheses around the *lm* indices to distinguish them from the spatial tensor indices *a*, *b*, etc. In addition, summations over *l* and *m* start at \(l=2\), and not \(l=0\) as would be the case for the expansion of a scalar field on the 2-sphere in terms of ordinary (i.e., undifferentiated) spherical harmonics. In what follows, we will use \(\sum _{(lm)}\) as shorthand notation for \(\sum _{l=2}^\infty \sum _{m=-l}^l\) unless indicated otherwise.

#### 2.2.3 Relating the two expansions

*E*-modes and

*B*-modes (corresponding to the gradient and curl spherical harmonics). The most relevant property of the gradient and curl spherical harmonics is that they transform like combinations of spin-weight \(\pm 2\) fields with respect to rotations of an orthonormal basis at points on the 2-sphere. Explicitly,

### 2.3 Statistical properties

*moments*(Appendix B) of the metric perturbations:

*lm*) label the multipole components for the gradient and curl tensor spherical harmonic decomposition. Without loss of generality we can assume that the background has zero mean:

*stationary*(Appendix C). This means that all statistical quantities constructed from the metric perturbations at times

*t*, \(t'\), etc., depend only on the difference between times, e.g., \(t-t'\), and not on the choice of time origin. We expect this to be true given that the age of the universe is roughly 9 orders of magnitude larger than realistic observation times, \({\sim }10~\mathrm {year}\). It is thus unlikely that a stochastic gravitational-wave background has statistical properties that vary over the time scale of the observation.

For Gaussian backgrounds we need only consider quadratic expectation values, since all higher-order moments are either zero or can be written in terms of the quadratic moments (Appendix B). For non-Gaussian backgrounds (Sect. 8.1), third and/or higher order moments will also be needed.

Beyond our assumption of stationarity, the specific form of the expectation values will depend, in general, on the source of the background. For example, a cosmological background produced by the superposition of a large number of independent gravitational-wave signals from the early Universe is expected to be Gaussian (via the central limit theorem), as well as isotropically-distributed on the sky. Contrast this with the superposition of gravitational waves produced by unresolved Galactic white-dwarf binaries radiating in the LISA band (\(10^{-4}~\mathrm{Hz}\) to \(10^{-1}~\mathrm{Hz}\)). Although this confusion-limited astrophysical foreground is also expected to be Gaussian and stationary, it will have an *anisotropic distribution*, following the spatial distribution of the Milky Way. The anistropy will be encoded as a modulation in the LISA output, due to the changing antenna pattern of the LISA constellation in its yearly orbit around the Sun. Hence, different sources will give rise to different statistical distributions, which we will need to consider when formulating our data analysis strategies.

#### 2.3.1 Quadratic expectation values for Gaussian-stationary backgrounds

*strain power spectral density*function (units of \(\mathrm{strain}^2/\mathrm{Hz}\)), summed over both polarizations and integrated over the sky. The factor of \(\delta (f-f')\) arises due to our assumption of stationarity; the factor of \(\delta _{AA'}\) (or \(\delta ^{PP'}\)) is due to our assumption that the polarization modes are statistically independent of one another and have no preferred component; and the factor of \(\delta ^2(\hat{n},\hat{n}')\) (or \(\delta _{ll'}\delta _{mm'}\)) is due to our assumption of spatial homogeneity and isotropy.

*f*. It is related to \(S_h(f)\) via

More general Gaussian-stationary backgrounds (e.g., polarized, statistically isotropic but with correlated radiation, etc.) can be represented by appropriately changing the right-hand-side of the quadratic expectation values. However, for the remainder of this section and for most of the article, we will consider “vanilla” isotropic backgrounds, whose quadratic expectation values (2.14) or (2.15) are completely specified by the power spectral density \(S_h(f)\).

### 2.4 Fractional energy density spectrum

*f*to \(f+df\), and \(\rho _c\equiv 3c^2 H_0^2/8\pi G\) is the critical energy density need to close the universe. The

*total*energy density in gravitational waves normalized by the critical energy density is thus

*independent*of the value of the Hubble constant. However, since recent measurements by Planck (Ade et al. 2015b; ESA 2016c) have shown that \(h_0=0.68\) to a high degree of precision, we have assumed this value in this review article and quote limits directly on \(\Omega _\mathrm {gw}(f)\) (Sect. 10). The specific functional form for \(\Omega _\mathrm {gw}(f)\) depends on the source of the background, as we shall see explicitly below.

### 2.5 Characteristic strain

## 3 Statistical inference

If your experiment needs statistics, you ought to have done a better experiment.

Ernest Rutherford

In this section, we review statistical inference from both the Bayesian and frequentist perspectives. Our discussion of frequentist and Bayesian upper limits, and the example given in Sect. 3.5 comparing Bayesian and frequentist analyses is modelled in part after Röver et al. (2011). Readers interested in more details about Bayesian statistical inference should see, e.g., Howson and Urbach (1991), Howson and Urbach (2006), Jaynes (2003), Gregory (2005) and Sivia and Skilling (2006). For a description of frequentist statistics, we recommend Helstrom (1968), Wainstein and Zubakov (1971) and Feldman and Cousins (1998).

### 3.1 Introduction to Bayesian and frequentist inference

Statistical inference can be used to answer questions such as “Is a gravitational-wave signal present in the data?” and, if so, “What are the physical characteristics of the source?” These questions are addressed using the techniques of classical (also known as *frequentist*) inference and *Bayesian* inference. Many of the early theoretical studies and observational papers in gravitational-wave astronomy followed the frequentist approach, but the use of Bayesian inference is growing in popularity. Moreover, many contemporary analyses cannot be classified as purely frequentist or Bayesian.

*d*, given that the star has mass

*m*. This probability distribution is the

*likelihood*, denoted \(p(d\vert m)\). In contrast, in the Bayesian interpretation the data are known (after all, it is what is measured!), and the mass of the star is what we are uncertain about,

^{5}so the relevant probability is that the mass has a certain value, given the data. This probability distribution is the

*posterior*, \(p(m \vert d)\). The likelihood and posterior are related via Bayes’ theorem:

*p*(

*m*) is the prior probability distribution for

*m*, and the normalization constant,

*marginalized likelihood*, or

*evidence*. For uniform (flat) priors the frequentist confidence intervals for the parameters will coincide with the Bayesian credible intervals, but the interpretation remains quiet distinct.

The choice of prior probability distributions is a source of much consternation and debate, and is often cited as a weakness of the Bayesian approach. But the choice of probability distribution for the likelihood (which is also important for the frequentist approach) is often no less fraught. The prior quantifies what we know about the range and distribution of the parameters in our model, while the likelihood quantifies what we know about our measurement apparatus, and, in particular, the nature of the measurement noise. The choice of prior is especially problematic in a new field where there is little to guide the choice. For example, electromagnetic observations and population synthesis models give some guidance about black hole masses, but the mass range and distribution is currently not well constrained. The choice of likelihood can also be challenging when the measurement noise deviates from the stationary, Gaussian ideal. More details related to the choice of likelihood and choice of prior will be given in Sect. 3.6.

*are*already present in existing data sets, but most are at levels where we are unable to distinguish them from noise processes. For detection we demand that a model for the data that includes a gravitational-wave signal be favored over a model having no gravitational-wave signal. In Bayesian inference a detection might be announced when the odds ratio between models with and without gravitational-wave signals gets sufficiently large, while in frequentist inference a detection might be announced when the

*p*-value for some test statistic is less than some prescribed threshold. These different approaches to deciding whether or not to claim a detection (e.g., Bayesian model selection or frequentist hypothesis testing), as well as differences in regard to parameter estimation, are described in the following subsections. Table 2 provides an overview of the key similarities and differences between frequentist and Bayesian inference, to be described in detail below.

Comparison of frequentist and Bayesian approaches to statistical inference

Frequentist | Bayesian |
---|---|

Probabilities assigned only to propositions about outcomes of repeatable experiments (i.e., random variables), not to hypotheses or parameters which have fixed but unknown values | Probabilities can be assigned to hypotheses and parameters since probability is degree of belief (or confidence, plausibility) in any proposition |

Assumes measured data are drawn from an underlying probability distribution, which assumes the truth of a particular hypothesis or model (likelihood function) | Same |

Constructs a statistic to estimate a parameter or to decide whether or not to claim a detection | Needs to specify prior degree of belief in a particular hypothesis or parameter |

Calculates the probability distribution of the statistic (sampling distribution) | Uses Bayes’ theorem to update the prior degree of belief in light of new data (i.e., likelihood “plus” prior yields posterior) |

Constructs confidence intervals and | Constructs posteriors and odds ratios for parameter estimation and hypothesis testing/model comparison |

### 3.2 Frequentist statistics

As mentioned above, classical or *frequentist* statistics is a branch of statistical inference that interprets probability as the “long-run relative occurrence of an event in a set of identical experiments.” Thus, for a frequentist, probabilities can only be assigned to propositions about outcomes of (in principle) repeated experiments (i.e., *random variables*) and not to hypotheses or parameters describing the state of nature, which have fixed but unknown values. In this interpretation, the measured data are drawn from an underlying probability distribution, which assumes the truth of a particular hypothesis or model. The probability distribution for the data is just the likelihood function, which we can write as *p*(*d*|*H*), where *d* denotes the data and *H* denotes an hypothesis.

Statistics play an important role in the frequentist framework. These are random variables constructed from the data, which typically estimate a signal parameter or indicate how well the data fit a particular hypothesis. Although it is common to construct statistics from the likelihood function (e.g., the maximum-likelihood statistic for a particular parameter, or the maximum-likelihood ratio to compare a signal-plus-noise model to a noise-only model), there is no a priori restriction on the form of a statistic other than it be *some* function of the data. Ultimately, it is the goal of the analysis and the cleverness of the data analyst that dictate which statistic (or statistics) to use.

To make statistical inferences in the frequentist framework requires knowledge of the probability distribution (also called the *sampling distribution*) of the statistic. The sampling distribution can either be calculated analytically (if the statistic is sufficiently simple) or via Monte Carlo simulations, which effectively construct a histogram of the values of the statistic by simulating many independent realizations of the data. Given a statistic and its sampling distribution, one can then calculate either *confidence intervals* for parameter estimation or *p*-values for hypothesis testing. (These will be discussed in more detail below). Note that a potential problem with frequentist statistical inference is that the sampling distribution depends on data values that were *not* actually observed, which is related to how the experiment was carried out *or might have been* carried out. The so-called *stopping problem* of frequentist statistics is an example of such a problem (Howson and Urbach 2006).

#### 3.2.1 Frequentist hypothesis testing

Suppose, as a frequentist, you want to test the hypothesis \(H_1\) that a gravitational-wave signal, having some fixed but unknown amplitude \(a>0\), is present in the data. Since you cannot assign probabilities to hypotheses or to parameters like *a* as a frequentist, you need to introduce instead an alternative (or *null*) hypothesis \(H_0\), which, for this example, is the hypothesis that there is no gravitational-wave signal in the data (i.e., that \(a=0\)). You then argue for \(H_1\) by arguing *against*\(H_0\), similar to proof by contradiction in mathematics. Note that \(H_1\) is a *composite* hypothesis since it depends on a range of values of the unknown parameter *a*. It can be written as the union, \(H_1=\cup _{a>0} H_a\), of a set of simple hypotheses \(H_a\) each corresponding to a single fixed value of the parameter *a*.

*test*or

*detection statistic*, on which the statistical test will be based. As mentioned above, you will need to calculate analytically or via Monte Carlo simulations the sampling distribution for \(\Lambda \) under the assumption that the null hypothesis is true, \(p(\Lambda |H_0)\). If the observed value of \(\Lambda \) lies far out in the tails of the distribution, then the data are most likely not consistent with the assumption of the null hypothesis, so you reject \(H_0\) (and thus accept \(H_1\)) at the \(p*100\)% level, where

*p*-value (or

*significance*) of the test; it is illustrated graphically in Fig. 3. The

*p*-value required to reject the null hypothesis determines a

*threshold*\(\Lambda _*\), above which you reject \(H_0\) and accept \(H_1\) (e.g., claim a detection). It is related to the

*false alarm probability*for the test as we explain below.

*false alarm*errors, which arise if the data are such that you reject the null hypothesis (i.e., \(\Lambda _\mathrm{obs}>\Lambda _*\)) when it is actually true, and (ii) type II or

*false dismissal*errors, which arise if the data are such that you accept the null hypothesis (i.e., \(\Lambda _\mathrm{obs}<\Lambda _*\)) when it is actually false. The false alarm probability \(\alpha \) and false dismissal probability \(\beta (a)\) are given explicitly by

*a*is the amplitude of the gravitational-wave signal, assumed to be present under the assumption that \(H_1\) is true. To calculate the false dismissal probability \(\beta (a)\), one needs the sampling distribution of the test statistic assuming the presence of a signal with amplitude

*a*.

Different test statistics are judged according to their false alarm and false dismissal probabilities. Ideally, you would like your statistical test to have false alarm and false dismissal probabilities that are both as small as possible. But these two properties compete with one another as setting a larger threshold value to minimize the false alarm probability will increase the false dismissal probability. Conversely, setting a smaller threshold value to minimize the false dismissal probability will increase the false alarm probability.

In the context of gravitational-wave data analysis, the gravitational-wave community is (at least initially) reluctant to falsely claim detections. Hence the false alarm probability is set to some very low value. The best statistic then is the one that minimizes the false dismissal probability (i.e., maximizes detection probability) for fixed false alarm. This is the *Neyman*–*Pearson criterion*. For medical diagnosis, on the other hand, a doctor is very reluctant to falsely dismiss an illness. Hence the false dismissal probability will be set to some very low value. The best statistic then is the one which minimizes the false alarm probability for fixed false dismissal.

#### 3.2.2 Frequentist detection probability

*detection probability*or

*power*of the test. It is the fraction of times that the test statistic \(\Lambda \) correctly identifies the presence of a signal of amplitude

*a*in the data, for a fixed false alarm probability \(\alpha \) (which sets the threshold \(\Lambda _*\)). A plot of detection probability versus signal strength is often used to show how strong a signal has to be in order to detect it with a certain probability. Since detection probability does not depend on the observed data—it depends only on the sampling distribution of the test statistic and a choice for the false alarm probability—detection probability curves are often used as a

*figure-of-merit*for proposed search methods for a signal. Figure 4 shows a detection probability curve, with the value of

*a*needed to be detectable with 90% frequentist probability indicated by the dashed vertical line. We will denote this value of

*a*by \(a^{90\%,\mathrm{DP}}\). Note that as the signal amplitude goes to zero, the detection probability reduces to the false alarm probability \(\alpha \), which for this example was chosen to be 0.10.

#### 3.2.3 Frequentist upper limits

*upper limit*) on the strength of the signal that one was trying to detect. The upper limit depends on the observed value of the test statistic, \(\Lambda _\mathrm{obs}\), and a choice of confidence level, \(\mathrm {CL}\), interpreted in the frequentist framework as the long-run relative occurrence for a set of repeated identical experiments. For example, one defines the 90% confidence-level upper limit \(a^{90\%,\mathrm{UL}}\) as the minimum value of

*a*for which \(\Lambda \ge \Lambda _\mathrm{obs}\) at least 90% of the time:

#### 3.2.4 Frequentist parameter estimation

*a*, like the amplitude of a gravitational-wave signal, is slightly different than the method used to claim a detection. You need to first construct a statistic (called an

*estimator*) \(\hat{a}\) of the parameter

*a*you are interested in. (This might be a maximum-likelihood estimator of

*a*, but other estimators can also be used). You then calculate its sampling distribution \(p(\hat{a}|a, H_a)\). Note that statements like

*not*be interpreted as a statement about the probability of

*a*lying within a particular interval \([\hat{a}-\Delta ,\hat{a}+\Delta ]\), since

*a*is not a random variable. Rather, it should be interpreted as a probabilistic statement about the

*set of intervals*\(\{[\hat{a}-\Delta ,\hat{a}+\Delta ]\}\) for all possible values of \(\hat{a}\). Namely, in a set of many repeated experiments, 0.95 is the fraction of the intervals that will contain the true value of the parameter

*a*. Such an interval is called a \(95\%\)

*frequentist confidence interval*. This is illustrated graphically in Fig. 6.

It is important to point out that an estimator can sometimes take on a value of the parameter that is *not physically allowed*. For example, if the parameter *a* denotes the amplitude of a gravitational-wave signal (so physically \(a\ge 0\)), it is possible for \(\hat{a} <0\) for a particular realization of the data. Note that there is nothing mathematically wrong with this result. Indeed, the sampling distribution for \(\hat{a}\) specifies the probability of obtaining such values of \(\hat{a}\). It is even possible to have a confidence interval \([\hat{a}-\Delta , \hat{a}+\Delta ]\) all of whose values are unphysical, especially if one is trying to detect a weak signal in noise. Again, this is mathematically allowed, but it is a little awkward to report a frequentist confidence interval that is completely unphysical. We shall see that within the Bayesian framework unphysical intervals and unphysical posteriors never arise, as a simple consequence of including a prior distribution on the parameter that requires \(a > 0\).

#### 3.2.5 Unified approach for frequentist upper limits and confidence intervals

Frequentists also have a way of avoiding unphysical or empty confidence intervals, which at the same time *unifies* the treatment of upper limits for null results and two-sided intervals for non-null results. This procedure, developed by Feldman and Cousins (1998), also solves the problem that the choice of an upper limit or two-sided confidence interval leads to intervals that do not have the proper coverage (i.e., the probability that an interval contains the true value of a parameter does not match the stated confidence level) if the choice of reporting an upper limit or two-sided confidence interval is *based on the data* and not decided upon before performing the experiment.

*ordering*) of the values of the random variable to include in the acceptance intervals for an unknown parameter. If we let

*a*denote the parameter whose value we are trying to determine, and \(\hat{a}\) be an estimator of

*a*with sampling distribution \(p(\hat{a}|a,H_a)\), then the choice of acceptance intervals becomes, for each value of

*a*, how do we choose \([\hat{a}_1, \hat{a}_2]\) such that

*a*that maximizes the sampling distribution \(p(\hat{a}|a,H_a)\) for a given value of \(\hat{a}\). The prescription then for constructing the acceptance intervals is to find, for each allowed value of

*a*, values of \(\hat{a}_1\) and \(\hat{a}_2\) such that \(R(\hat{a}_1|a)=R(\hat{a}_2|a)\) and for which (3.9) is satisfied. The set of all such acceptance intervals for different values of

*a*forms a

*confidence belt*in the \(\hat{a}a\)-plane, which is then used to construct an upper limit or a two-sided confidence interval for a particular observed value of the estimator \(\hat{a}\), as explained below and illustrated in Fig. 7.

*a*with variance \(\sigma ^2\):

*a*represents the amplitude of a signal, so that \(a > 0\). (Recall that it is possible, however, for the estimator \(\hat{a}\) to take on negative values). Then \(a_\mathrm{best}=\hat{a}\) if \(\hat{a} > 0\), while \(a_\mathrm{best} = 0\) if \(\hat{a} \le 0\), for which

### 3.3 Bayesian inference

In the following subsections, we again describe parameter estimation and hypothesis testing, but this time from the perspective of Bayesian inference.

#### 3.3.1 Bayesian parameter estimation

*a*, is estimated in terms of its posterior distribution,

*p*(

*a*|

*d*), in light of the observed data

*d*. As discussed in the introduction to this section, the posterior

*p*(

*a*|

*d*) can be calculated from the likelihood

*p*(

*d*|

*a*) and the prior probability distribution

*p*(

*a*) using Bayes’ theorem

*p*(

*a*|

*d*), a Bayesian confidence interval (often called a

*credible interval*given the Bayesian interpretation of probability as degree of belief, or state of knowledge, about an event) is simply defined in terms of the area under the posterior between one parameter value and another. This is illustrated graphically in Fig. 8, for the case of a 95% symmetric credible interval, centered on the mode of the distribution \(a_\mathrm{mode}\). If the posterior distribution depends on two parameters

*a*and

*b*, but you really only care about

*a*, then you can obtain the posterior distribution for

*a*by marginalizing the joint distribution

*p*(

*a*,

*b*|

*d*) over

*b*:

*nuisance parameters*. Although nuisance parameters can be handled in a straight-forward manner using Bayesian inference, they are problematic to deal with (i.e., they are a nuisance!) in the context of frequentist statistics. The problem is that marginalization doesn’t make sense to a frequentist, for whom parameters cannot be assigned probability distributions.

*nats*, while using a base 2 logarithm gives the information in

*bits*. If the data tells us nothing about the parameter, then \(p(d\vert a) = \mathrm{constant}\), which implies \(p(a\vert d)=p(a)\) and thus \(I=0\).

#### 3.3.2 Bayesian upper limits

*a*has a value in the indicated range. One usually sets an upper limit on a parameter when the mode of the distribution for the parameter being estimated is not sufficiently displaced from zero, as shown in Fig. 9.

#### 3.3.3 Bayesian model selection

*p*(

*d*) involves a sum over all possible models:

*prior*odds ratio for models \(\alpha ,\beta \), while the second term is the evidence ratio, or

*Bayes factor*,

While the foundations of Bayesian inference were laid out by Laplace in the 1700s, it did not see widespread use until the late twentieth century with the advent of practical implementation schemes and the development of fast electronic computers. Today, Monte Carlo sampling techniques, such as Markov Chain Monte Carlo (MCMC) and Nested Sampling, are used to sample the posterior and estimate the evidence (Skilling 2006; Gair et al. 2010). Successfully applying these techniques is something of an art, but in principle, once the likelihood and prior have been written down, the implementation of Bayesian inference is purely mechanical. Calculating the likelihood and choosing a prior will be discussed in some detail in Sect. 3.6.

### 3.4 Relating Bayesian and frequentist detection statements

*maximum-likelihood ratio*. For concreteness, let us assume that we have two models \(\mathcal{M}_0\) (noise-only) and \(\mathcal{M}_1\) (noise plus gravitational-wave signal), with parameters \(\mathbf {\theta }_n\) and \(\{\mathbf {\theta }_n,\mathbf {\theta }_h\}\), respectively. The frequentist detection statistic will be defined in terms of the ratio of the maxima of the likelihood functions for the two models:

*informative*—i.e., when the likelihood functions are peaked relative to the joint prior probability distributions of the parameters. For an arbitrary model \(\mathcal{M}\) with parameters \(\mathbf {\theta }\), the Laplace approximation yields:

*d*; \(\Delta V_\mathcal{M}\) is the characteristic spread of the likelihood function around its maximum (the volume of the uncertainty ellipsoid for the parameters); and \(V_\mathcal{M}\) is the total parameter space volume of the model parameters. Applying this approximation to models \(\mathcal{M}_0\) and \(\mathcal{M}_1\) in (3.25), we obtain

*Occam penalty factor*, which prefers the simpler of two models that fit the data equally well. The first term has the interpretation of being the squared signal-to-noise ratio of the data, assuming an additive signal in Gaussian-stationary noise, and it can be used as an alternative frequentist detection statistic in place of \(\Lambda _\mathrm{ML}\).

*sky*and

*phase scrambles*to effectively destroy signal-induced spatial correlations between pulsars while retaining the statistical properties of each individual dataset. This is similar to doing time-slides for LIGO analyses, which are used to assess the significance of a detection.

Bayes factors and their interpretation in terms of the strength of the evidence in favor of one model relative to the other

\(\mathcal{B}_{\alpha \beta }(d)\) | \(2\ln \mathcal{B}_{\alpha \beta }(d)\) | Evidence for model \(\mathcal{M}_\alpha \) relative to \(\mathcal{M}_\beta \) |
---|---|---|

\({<}1\) | \({<}0\) | Negative (supports model \(\mathcal{M}_\beta \)) |

1–3 | 0–2 | Not worth more than a bare mention |

3–20 | 2–6 | Positive |

20–150 | 6–10 | Strong |

\({>}150\) | \({>}10\) | Very strong |

Taylor et al. (2016a) even go so far as to perform a *hybrid* frequentist-Bayesian analysis, doing Monte Carlo simulations: (i) over different noise-only realizations, and (ii) over different sky and phase scrambles, which null the correlated signal. These simulations produce different null *distributions* for the Bayes factor, similar to a null-hypothesis distribution for a frequentist detection statistic (in this case, the log of the Bayes factor). The significance of the measured Bayes factor is then its corresponding *p*-value with respect to one of these null distributions. The utility of such a hybrid analysis is its ability to better assess the significance of a detection claim, especially when there might be questions about the suitability of one of the models (e.g., the noise model) used in the construction of a likelihood function.

### 3.5 Simple example comparing Bayesian and frequentist analyses

*i*labels the individual samples of the data. The likelihood functions for the noise-only and signal-plus-noise models \(\mathcal{M}_0\) and \(\mathcal{M}_1\) are thus simple Gaussians:

*a*to be flat over the interval \((0,a_\mathrm{max}]\), so \(p(a)=1/a_\mathrm{max}\).

*a*is given by the sample mean of the data:

*a*and has variance \(\sigma _{\hat{a}}^2 =\sigma ^2/N\) (the familiar variance of the sample mean). Thus, the sampling distribution of \(\hat{a}\) is simply

*truncated*Gaussian on the interval \(a\in (0,a_\mathrm{max}]\), with mean \(\hat{a}\) and variance \(\sigma _{\hat{a}}^2\).

*sufficient statistic*for

*a*. This means that the posterior distribution for

*a*can be written simply in terms of \(\hat{a}\), in lieu of the individual samples \(d\equiv \{d_1, d_2, \ldots , d_N\}\). The Bayes factor

*square*of a (single) Gaussian random variable \(\rho =\bar{d}\sqrt{N}/\sigma \). Moreover, since \(\rho \) has mean \(\mu \equiv a\sqrt{N}/\sigma \) and unit variance, the sampling distribution for \(\Lambda \) in the presence of a signal is a

*noncentral chi-squared*distribution with one degree of freedom and non-centrality parameter \(\lambda \equiv \mu ^2 = a^2 N/\sigma ^2\):

*a*and hence \(\lambda \) are equal to zero), \(\Lambda \) is given by an (ordinary) chi-squared distribution with one degree of freedom:

*analytic*expressions for the sampling distributions for the detection statistic \(\Lambda \) is due to the simplicity of the signal and noise models. For more complicated real-world problems, these distributions would need to be generated

*numerically*using fake signal injections and time-shifts to produce many different realizations of the data (signal plus noise) from which one can build up the distributions.

*not*a sufficient statistic for

*a*, due to the fact that \(\Lambda \) involves the

*square*of the maximum-likelihood estimate \(\hat{a}\)—i.e., \(\Lambda = \hat{a}^2 N/\sigma ^2\). Thus, we cannot take \(p(\Lambda |a,\mathcal{M}_1)\) conditioned on \(\Lambda \) (assuming a flat prior on

*a*from \([0,a_\mathrm{max}]\)) to get the posterior distribution for

*a*given

*d*, since we would be missing out on data samples that give negative values for \(\hat{a}\). Another way to see this is to start with \(p(\Lambda |a,\mathcal{M}_1)\) given by (3.45), and then make a change of variables from \(\Lambda \) to \(\hat{a}\) using the general transformation relation

*p*(

*a*|

*d*) from (3.33)—and

*not*from (3.47)—if we want the posterior to have the proper dependence on

*a*.

#### 3.5.1 Simulated data

*weak*and (moderately)

*strong*signals. Single realizations of the data for the two different injections are shown in Fig. 11. The noise realization is the same for the two injections.

#### 3.5.2 Frequentist analysis

*N*, \(\sigma \), and the probability distributions (3.44) and (3.45) for the frequentist detection statistic \(\Lambda \), we can calculate the detection threshold for fixed false alarm probability \(\alpha \) (which we will take to equal 10%), and the corresponding detection probability as a function of the amplitude

*a*. The detection threshold turns out to equal \(\Lambda _* = 2.9\) (so 10% of the area under the probability distribution \(p(\Lambda |\mathcal{M}_0)\) is for \(\Lambda \ge \Lambda _*\)). The value of the amplitude

*a*needed for 90% confidence detection probability with 10% false alarm probability is given by \(a^{90\%,\mathrm{DP}}=0.30\). (These results for the detection threshold and detection probability do

*not*depend on the particular realizations of the simulated data). The corresponding curves are shown in Fig. 12.

*a*. The corresponding values of the detection statistic are \(\Lambda _\mathrm{obs} = 0.72\) and 11.2 for the two injections, and have

*p*-values equal to 0.45 and \(9.0\times 10^{-4}\), as shown in Fig. 13. The 95% frequentist confidence interval is given simply by \([\hat{a}-2\sigma _{\hat{a}},\hat{a}+2\sigma _{\hat{a}}]\), since \(\hat{a}\) is Gaussian-distributed, and has values \([-0.11,0.29]\) and [0.14, 0.54], respectively. These intervals contain the true value of the amplitudes for the two injections, \(a_0=0.05\) and 0.3.

#### 3.5.3 Bayesian analysis

*a*given the value of the maximum-likelihood estimator \(\hat{a}\), which (as we discussed earlier) is a sufficient statistic for the data

*d*. Recall that the posterior for

*a*for this example is simply a truncated Gaussian from 0 to \(a_\mathrm{max}\) centered on \(\hat{a}\), which could be negative, see (3.36). The left two panels show the graphical construction of the Bayesian 90% upper limit and 95% credible interval for the amplitude

*a*for the weak injection, \(a^{90\%,\mathrm{UL}}=0.23\) and [0, 0.26]. The right two panels show similar plots for the strong injection, \(a^{90\%,\mathrm{UL}}=0.46\) and [0.14, 0.54].

Finally, the Bayes factor for the signal-plus-noise model \(\mathcal{M}_1\) relative to the noise-only model \(\mathcal{M}_0\) can be calculated by taking the ratio of the marginalized likelihood \(p(d|\mathcal{M}_1)\) given by (3.35) to \(p(d|\mathcal{M}_0)\) given by (3.30). Doing this, we find 2 ln \(B_{10} = -2.2\) and 9.2 for the weak and strong signal injections, respectively. The Laplace approximation to this quantity is given by (3.41), with values \(-2.0\) and 8.5, respectively.

#### 3.5.4 Comparison summary

Tabular summary of the frequentist and Bayesian analysis results for the simulated data (both weak and strong injections)

(Weak injection, \(a_0=-0.05\)) | (Strong injection, \(a_0=0.3\)) | |||
---|---|---|---|---|

Frequentist | Bayesian | Frequentist | Bayesian | |

Detection threshold (\(\Lambda _*\)) | 2.9 | – | 2.9 | – |

Detection statistic (\(\Lambda _\mathrm{obs}\)) | 0.72 | – | 11.2 | – |

| 0.45 | – | \(9.0\times 10^{-4}\) | – |

90% upper limit | 0.20 | 0.23 | 0.46 | 0.46 |

95% interval | \([-0.11,0.29]\) | [0, 0.26] | [0.14, 0.54] | [0.14, 0.54] |

ML estimator (\(\hat{a}\)) | 0.085 | 0.085 | 0.335 | 0.335 |

Bayes factor (\(2\ln \mathcal{B}_{10}\)) | – | \(-2.2\) | – | 9.2 |

Laplace approximation | – | \(-2.0\) | – | 8.5 |

### 3.6 Likelihoods and priors for gravitational-wave searches

To conclude this section on statistical inference, we discuss some issues related to calculating the likelihood and choosing a prior in the context of searches for gravitational-wave signals using a network of gravitational-wave detectors.

#### 3.6.1 Calculating the likelihood

Defining the likelihood function (for either a frequentist or Bayesian analysis) involves understanding the instrument response and the instrument noise. The data collected by gravitational-wave detectors comes in a variety of forms. For ground-based interferometers such as LIGO and Virgo, the data comes from the error signal in the differential arm-length control system, which is non-linearly related to the laser phase difference, which in turn is linearly related to the gravitational-wave strain. For pulsar timing arrays, the data comes from the arrival times of radio pulses (derived from the folded pulse profiles), which must be corrected using a complicated timing model that takes into account the relative motion of the telescopes and the pulsars, along with the spin-down of the pulsars, in addition to a variety of propagation effects. The timing residuals formed by subtracting the timing model from the raw arrival times contain perturbations due to gravitational waves integrated along the line of sight to the pulsar. For future space-based gravitational-wave detectors such as LISA, the data will be directly read out from phase meters that perform a heterodyne measurement of the laser phase. Synthetic combinations of these phase read outs (chosen to cancel laser phase noise) are then linearly proportional to the gravitational-wave strain.

*t*can be written as

*h*(

*t*) is shorthand for the gravitational-wave metric perturbations \(h_{ab}(t,{\vec {x}})\) convolved with the instrument response function and converted into the appropriate quantity—phase shift, time delay, differential arm length error, etc. (A detailed calculation of

*h*(

*t*) and the associated detector response functions will be given in Sect. 5.2). As mentioned above, the data

*d*(

*t*) may be the quantity that is measured directly, or, more commonly, some quantity that is derived from the measurements such as timing residuals or calibrated strain. In any analysis, it is important to marginalize over the model parameters used to make the conversion from the raw data.

*d*(

*t*) is found by demanding that the residual

^{6}for the gravitational-wave signal. The likelihood of observing a collection of discretely-sampled data \(d \equiv \{ d_1, d_2, \ldots , d_N\}\), where \(d_i\equiv d(t_i)\), is then given by \(p( d\vert \bar{h}) = p_n(r)\), where \(r\equiv \{r_1, r_2,\ldots , r_N\}\) with \(r_i\equiv r(t_i)\). Since instrument noise is due to a large number of small disturbances combined with counting noise in the large-number limit, the central limit theorem suggests that the noise distribution can be approximated by a multi-variate normal (Gaussian) distribution:

*I*,

*J*labels the detector, and

*i*,

*j*labels the discrete time or frequency sample for the corresponding detector. Note here that the parameters \(\mathbf {\theta }\) appearing in (3.18) are the individual time or frequency samples \(\bar{h}_i\).

#### 3.6.2 Choosing a prior

## 4 Correlations

Correlation is not cause, it is just a ‘music of chance’.

Siri Hustvedt

Stochastic gravitational waves are indistinguishable from unidentified instrumental noise in a single detector, but are correlated between pairs of detectors in ways that differ, in general, from instrumental noise. Cross-correlation methods basically use the random output of one detector as a template for the other, taking into account the physical separation and relative orientation of the two detectors. In this section, we introduce cross-correlation methods in the context of both frequentist and Bayesian inference, analyzing in detail a simple toy problem (the data are “white” and we ignore complications that come from the separation and relative orientation of the detectors—this we discuss in detail in Sect. 5). We also briefly discuss possible alternatives to cross-correlation methods, e.g., using a null channel as a noise calibrator.

The basic idea of using cross-correlation to search for stochastic gravitational-waves can be found in several early papers (Grishchuk 1976; Hellings and Downs 1983; Michelson 1987; Christensen 1990, 1992; Flanagan 1993). The derivation of the likelihood function in Sect. 4.2 follows that of Cornish and Romano (2013); parts of Sect. 4.4 are also discussed in Allen et al. (2003) and Drasco and Flanagan (2003).

### 4.1 Basic idea

*h*denotes the common gravitational-wave signal and \(n_1\), \(n_2\) the noise in the two detectors. To cross correlate the data, we simply form the product of the two samples, \(\hat{C}_{12}\equiv d_1 d_2\). The expected value of the correlation is then

We have assumed here that there is no cross-correlated noise (instrumental or environmental). If there is correlated noise, then the simple procedure describe above needs to be augmented. This will be discussed in more detail in Sect. 9.6.

### 4.2 Relating correlations and likelihoods

### 4.3 Extension to multiple data samples

*white*(i.e., the data are uncorrelated between time samples) or (ii) both

*colored*(i.e., allowing for correlations in time). The white noise example will be analyzed in more detail in Sects. 4.4–4.6.

#### 4.3.1 White noise and signal

*products*of the likelihoods (4.5) and (4.8) for the individual data samples. We can write these product likelihoods as single multivariate Gaussian distributions:

*N*independent data samples.

*known*parameters, then the frequentist estimator of \(S_h\) would also include

*auto-correlation*terms for each detector:

#### 4.3.2 Colored noise and signal

*independent*of one another. (This assumes that the data are

*stationary*, so that there is no preferred origin of time). Assuming multivariate Gaussian distributions as before, the variances \(S_{n_1}\), \(S_{n_2}\), and \(S_h\) generalize to

*power spectral densities*, which are functions of frequency defined by

^{7}The factor of 1 / 2 in (4.20) is for

*one-sided*power spectra, for which the integral of the power spectrum over

*positive*frequencies equals the variance of the data:

*Parseval’s theorem*, see e.g., (D.40). For

*N*samples of discretely-sampled data from each of two detectors \(I=1,2\) (total duration

*T*), the likelihood function for a Gaussian stochastic signal template becomes (Allen et al. 2002; Cornish and Romano 2013):

We do not bother to write down the maximum-likelihood estimators of the signal and noise power spectral densities for this particular example. We will return to this problem in Sect. 6, where we discuss the *optimally-filtered* cross-correlation statistic for isotropic stochastic backgrounds. There one assumes a particular spectral *shape* for the gravitational-wave power spectral density, and then simply estimates its overall amplitude. That simplifies the analysis considerably.

### 4.4 Maximum-likelihood detection statistic

*N*samples of data in each of two detectors, having uncorrelated white noise and a common white stochastic signal. As described in Sect. 3.4, one can calculate a frequentist detection statistic based on the

*maximum-likelihood ratio*:

*autocorrelations*of the data. In the absence of a signal, they are maximum-likelihood estimators of the noise variances \(S_{n_1}\) and \(S_{n_2}\). But in the presence of a signal, they are maximum-likelihood estimators of the

*combined*variances \(S_1\equiv S_{n_1}+S_h\) and \(S_2\equiv S_{n_2}+S_h\).

*weak*compared to the detector noise—i.e., \(S_h\ll S_{n_I}\), for \(I=1,2\)—the above expression reduces to

*coherence*) of the data from the two detectors. It is a measure of how well the data in detector 2

*matches*that in detector 1.

From (4.17), we see that \(\Lambda (d)\) is a ratio of the square of a sum of products of Gaussian random variables to the product of a sum of squares of Gaussian random variables. This is a sufficiently complicated expression that we will estimate the distribution of \(\Lambda (d)\)*numerically*, doing fake signal injections into many realizations of simulated noise to build up the sampling distribution. We do this explicitly in Sect. 4.6, when we compare the frequentist and Bayesian correlation methods for this example.

### 4.5 Bayesian correlation analysis

*statistically independent*of one another so that the joint prior distributions factorize into a product of priors for the individual parameters. We use Jeffrey’s priors for the individual noise variances:

^{8}prior for the signal variance:

Correlations enter the Bayesian analysis via the covariance matrix *C* that appears in the likelihood function \(p(d|S_{n_1},S_{n_2}, S_h,\mathcal{M}_1)\). The covariance matrix for the data includes the cross-detector signal correlations, as we saw in (4.15). So although one does not explicitly construct a cross-correlation statistic in the Bayesian framework, cross correlations do play an important role in the calculations.

### 4.6 Comparing frequentist and Bayesian cross-correlation methods

#### 4.6.1 Frequentist analysis

As mentioned previously, the form (4.27) of the detection statistic \(\Lambda (d)\) is sufficiently complicated that it was simplest to resort to numerical simulations to estimate its sampling distribution, \(p(\Lambda |S_{n_1}, S_{n_2}, S_h, \mathcal{M}_1)\). We took 50 values for each of \(S_{n_1}\), \(S_{n_2}\), and \(S_h\) in the interval [0, 3], and then for each of the corresponding \(50^3\) points in parameter space, we generated \(10^4\) realizations of the data, yielding \(10^4\) values of \(\Lambda (d)\). By histogramming these values for each point in parameter space, we were able to estimate the probability density function (and also the cumulative distribution function) for \(\Lambda \).

*exclusion*and

*inclusion*regions for our simulated data with \(\Lambda _\mathrm{obs}=7.6\). The 90% confidence-level exclusion region \(\mathcal{E}_{90\%}\) lies

*above*the red surface; it consists of points \((S_{n_1},S_{n_2},S_h)\) satisfying

*below*the red surface is the 90% confidence-level inclusion region \(\mathcal{I}_{90\%}\). Note that construction of these regions is such that the

*true*values of the parameters \(S_{n_1}\), \(S_{n_2}\), and \(S_h\) have a 90% frequentist probability of lying in \(\mathcal{I}_{90\%}\). This generalizes, to multiple parameters, the definition of the frequentist 90% confidence-level upper-limit for a single parameter, which was discussed in detail in Sect. 3.2.3. Note that it is not correct to simply “cut” the surface using the maximum-likelihood point estimates \(\hat{S}_{n_1}= 0.78\) and \(\hat{S}_{n_2}=1.46\) to obtain a single value for \(S_h^{90\%, \mathrm{UL}}\). One needs to include the whole region in order to get the correct frequentist coverage.

*regions*for the given point estimates. For example, \((S_{n_1}, S_{n_2}, S_h)\in \mathcal{I}_{95\%}\) for the observed point estimate \(\hat{S}_{h,\mathrm{obs}}\) if and only if \(\hat{S}_{h,\mathrm{obs}}\) is contained in the symmetric 95% confidence-level interval centered on the mode of the probability distribution \(p(\hat{S}_h|S_{n_1}, S_{n_2}, S_h, \mathcal{M}_1)\). These regions again generalize to multiple parameters the definition of a frequentist confidence

*interval*for a single parameter, which was discussed in detail in Sect. 3.2.4. They will be different, in general, for the different maximum-likelihood estimators. But in order to move on to the Bayesian analysis for this example, we will leave the explicit construction of these regions to the interested reader.

#### 4.6.2 Bayesian analysis

*positive*evidence (see Table 3) in favor of a correlated stochastic signal in the data.

Figure 18 shows the marginalized posterior \(p(S_h|d,\mathcal{M}_1)\) for the stochastic signal variance given the data *d* and signal-plus-noise model \(\mathcal{M}_1\). The peak of the posterior lies close the frequentist maximum-likelihood estimator \(\hat{S}_h=0.40\) (blue dotted vertical line), and easily contains the injected value in its 95% Bayesian credible interval (grey shaded region). Figure 19 shows similar plots for the marginalized posteriors for the noise variances \(S_{n_1}\) and \(S_{n_2}\) for both the signal-plus noise model \(\mathcal{M}_1\) (blue curves) and the noise-only model \(\mathcal{M}_0\) (green curves). For comparison, the frequentist maximum-likelihood estimators \(\hat{S}_{n_1}, \hat{S}_{n_2}=0.78, 1.46\) and 1.18, 1.86 for the two models are shown by the corresponding (blue and green) dotted vertical lines. Again, the peaks of the Bayesian posterior distributions lie close to these values. The 95% Bayesian credible intervals for \(S_{n_1}\) and \(S_{n_2}\) for the signal-plus-noise model \(\mathcal{M}_1\) are also shown (grey shaded region). These intervals easily contain the injected values for these two parameters.

### 4.7 What to do when cross-correlation methods aren’t available

Cross-correlation methods can be used whenever one has two or more detectors that respond to a common gravitational-wave signal. The beauty of such methods is that even though a stochastic background is another source of “noise” in a single detector, the common signal components in multiple detectors combine coherently when the data from pairs of detectors are multiplied together and summed, as described in Sect. 4.1. But with only a single detector, searches for a stochastic background need some other way to distinguish the signal from the noise—e.g., a difference between the spectra of the noise and the gravitational-wave signal, or the modulation of an anisotropic signal due to the motion of the detector (as is expected for the confusion-noise from galactic compact white dwarf binaries for LISA). Without some way of differentiating instrumental noise from gravitational-wave “noise”, there is no hope of detecting a stochastic background.

*N*samples of data from each of two detectors \(I=1,2\) (which we will call

*channels*in what follows), but let’s assume that the second channel is

*insensitive*to the gravitational-wave signal:

#### 4.7.1 Single-detector excess power statistic

*excess power*statistic from the autocorrelated data to estimate the signal variance:

*foreground*signal from galactic white-dwarf binaries in the LISA band. For frequencies below a few mHz, the gravitational-wave confusion noise from these binaries is expected to dominate the LISA instrument noise (Hils et al. 1990; Bender and Hils 1997; Hils and Bender 2000; Nelemans et al. 2001).

#### 4.7.2 Null channel method

*off-source*measurement using detector 1, then we could estimate the noise variance \(S_{n_1}\) directly from the detector output, free of contamination from gravitational waves. Using this noise estimate, \(\hat{S}_{n_1}\), we could then define our excess power statistic as

*combination*of the data (called a

*null channel*) for which the response to gravitational waves is strongly suppressed. The

*symmetrized Sagnac*combination of the data for LISA (Tinto et al. 2001; Hogan and Bender 2001) is one such example.

*a*would be replaced by a function of frequency—i.e., a

*transfer function*relating the noise in the two channels). To begin with, we will also assume that

*a*is

*known*. Then the data from the second channel can be used as a

*noise calibrator*for the first channel. The frequentist estimators for this scenario are:

*a*is not known a priori, but is described by its own prior probability distribution \(p_a(a)\), we have

*a*.

*Jeffrey’s prior*for \(S_{n_2}\):

*log-normal*prior for

*a*:

*a*. Note that \(A=0.67a_0\) and \(1.5a_0\) correspond to priors for

*a*that are

*biased*away from its true value \(a=a_0\). Note also that 68% of the prior distribution is contained in the region \(a\in [A/\Sigma , A\Sigma ]\) (so \(\Sigma =1\) corresponds to a delta-function prior—i.e., no uncertainty in

*a*). The particular realization that we used consisted of \(N=100\) samples of data (4.37) with \(S_h=1\), \(S_{n_2}=1\), and \(S_{n_1} = a_0 S_{n_2}\) with \(a_0=1\). Note that for the biased priors for

*a*(associated with the dashed and dotted curves in Fig. 20), an under (over) estimate in

*a*corresponds to over (under) estimate in \(S_h\), as \(S_h\) is effectively the difference between the estimated variance in channel 1 and

*a*times the estimated variance in channel 2. For this particular realization of the data, the mode of the “0%, unbiased” posterior for \(S_h\) is about 20% less than the injected value, \(S_h=1\). On average, they would agree.

## 5 Geometrical factors

There is geometry in the humming of the strings, there is music in the spacing of the spheres.

Pythagoras

In the previous sections, we ignored many details regarding detector response and detector geometry. We basically assumed that the detectors were *isotropic*, responding equally well to all gravitational waves, regardless of the waves’ directions of propagation, frequency content, and polarization. We also ignored any loss in sensitivity in the correlations between data from two or more detectors, due to the separation and relative orientation of the detectors. But these details *are* important if we want to design optimal (or near-optimal) data analysis algorithms to search for gravitational waves. To specify the likelihood function, for example, requires models not only for the gravitational-wave signal and instrument noise, but also for the response of the detectors to the waves that a source produces.

In this section, we fill in these details. We first discuss the response of a single detector to an incident gravitational wave. We then show how these non-trivial detector responses manifest themselves in the correlation between data from two or more detectors. The results are first derived in a general setting making no assumption, for example, about the wavelength of a gravitational wave to the characteristic size of a detector. The general results are then specialized, as appropriate, to the case of ground-based and space-based laser interferometers, spacecraft Doppler tracking, and pulsar timing arrays. We conclude this section by discussing how the motion of a detector relative to the gravitational-wave source affects the detector response.

The approach we take in this section is similar in spirit to that of Hellings (1991), attempting to unify the treatment of detector response functions and correlation functions across different gravitational-wave detectors. Readers interested in more details about the effect of detector geometry on the correlation of data from two or more detectors should see the original papers by Hellings and Downs (1983) for pulsar timing arrays, and Flanagan (1993) and Christensen (1990, 1992) for ground-based laser interferometers.

### 5.1 Detector response

Gravitational waves are time-varying perturbations to the background geometry of spacetime. Since gravitational waves induce time-varying changes in the separation between two freely-falling objects (so-called test masses), gravitational-wave detectors are designed to be as sensitive as possible to this changing separation. For example, a resonant bar detector acts like a giant tuning fork, which is set into oscillation when a gravitational wave of the natural frequency of the bar is incident upon it. These oscillations produce a stress against the equilibrium electromagnetic forces that exist within the bar. The stress (or oscillation) is measured by a strain gauge (or accelerometer), indicating the presence of a gravitational wave. The response for a bar detector is thus the fractional change in length of the bar, \(h(t) = \Delta l(t)/l\), induced by the wave. Since the length of the bar is typically much smaller than the wavelength of a gravitational wave at the bar’s resonant frequency, the response is most easily computed using the geodesic deviation equation (Misner et al. 1973) for the time-varying tidal field.

In this article, we will focus our attention on *beam* detectors, which use electromagnetic radiation to monitor the separation of two or more freely-falling objects. Spacecraft Doppler tracking, pulsar timing arrays, and ground- and space-based laser interferometers (e.g., LIGO-like and LISA-like detectors) are all examples of beam detectors, which can be used to search for gravitational waves (see, e.g., Section 4.2 in Sathyaprakash and Schutz 2009).

#### 5.1.1 Spacecraft Doppler tracking

*t*in the absence of the gravitational wave. A schematic representation of \(\Delta T(t)\) for spacecraft Doppler tracking is given in Fig. 21.

#### 5.1.2 Pulsar timing

*one-way*transmission of electromagnetic radiation (i.e., radio pulses are emitted by a pulsar and received by a radio antenna on Earth). The response for such a system is simply the timing residual

#### 5.1.3 Laser interferometers

*strain*measurement in the two arms

*L*), and \(\Delta T(t)\) is the difference in round-trip travel times as before. Thus, interferometer phase and strain response are simply related to one another.

^{9}(Misner et al. 1973; Schutz 1985; Hartle 2003) since the unperturbed separation

*L*of the two test masses can be larger than or comparable to the wavelength \(\lambda \equiv c/f\) of an incident gravitational wave having frequency

*f*. This is definitely the case for pulsar timing where

*L*is of order a few kpc, and for spacecraft Doppler tracking where

*L*is of order tens of AU. It is also the case for space-based detectors like LISA (\(L=5\times 10^6 \mathrm{\ km}\)) for gravitational waves with frequencies around a tenth of a Hz. On the other hand, for Earth-based detectors like LIGO (\(L=4~\mathrm{km}\)), \(L\ll \lambda \) is a good approximation below a few kHz. Thus, the approach that we will take in the following subsections is to calculate the detector response in general, not making any approximation a priori regarding the relative sizes of \(\lambda =c/f\) and

*L*. To recover the standard expressions (i.e., in the long-wavelength or small-antenna limit) for Earth-based detectors like LIGO will be a simple matter of taking the limit

*fL*/

*c*to zero. For reference, Table 5 summarizes the characteristic properties (i.e., size, characteristic frequency, sensitivity band, etc.) of different beam detectors.

Characteristic properties of different beam detectors: column 2 is the arm length or characteristic size of the detector (tens of AU for spacecraft Doppler tracking; a few kpc for pulsar timing); column 3 is the frequency corresponding to the characteristic size of the detector, \(f_*\equiv c/L\); columns 4 and 5 are the frequencies at which the detector is sensitive in units of Hz and units of \(f_*\), respectively; and column 6 is the relationship between *f* and \(f_*\)

Beam detector |
| \(f_*\) (Hz) |
| \(f/f_*\) | Relation |
---|---|---|---|---|---|

Ground-based interferometer | \({\sim } 1\) | \({\sim } 10^5\) | \(10\,\,\hbox {to}\,10^4\) | \(10^{-4}\,\hbox {to}\,10^{-1}\) | \(f\ll f_*\) |

Space-based interferometer | \({\sim } 10^6\) | \({\sim } 10^{-1}\) | \(10^{-4}\,\hbox {to}\,10^{-1}\) | \(10^{-3}\,\hbox {to}\, 1\) | \(f\lesssim f_*\) |

Spacecraft Doppler tracking | \({\sim } 10^9\) | \({\sim } 10^{-4}\) | \(10^{-6}\,\hbox {to}\,10^{-3}\) | \(10^{-2}\,\hbox {to}\,10\) | \(f\sim f_*\) |

Pulsar timing | \({\sim } 10^{17}\) | \({\sim } 10^{-12}\) | \(10^{-9}\,\hbox {to}\,10^{-7}\) | \(10^3\,\hbox {to}\,10^5\) | \(f\gg f_*\) |

### 5.2 Calculation of response functions and antenna patterns

*linear*in the metric perturbations \(h_{ab}(t,{\vec {x}})\) describing the wave, and can be written as the convolution of the metric perturbations \(h_{ab}(t,{\vec {x}})\) with the

*impulse response*\(R^{ab}(t,{\vec {x}})\) of the detector:

*t*. In terms of a plane-wave expansion (2.1) of the metric perturbations, we have

^{10}

*f*, coming from direction \(\hat{n}\), and having polarization \(A=+,\times \). Plots of \(|R^A(f,\hat{n})|\) for fixed frequency

*f*are

*antenna beam patterns*for gravitational waves with polarization

*A*. A plot of

*f*is the beam pattern for an

*unpolarized*gravitational wave—i.e., a wave having statistically equivalent \(+\) and \(\times \) polarization components.

#### 5.2.1 One-way tracking

*t*is given by Estabrook and Wahlquist (1975):

*Earth term*and

*pulsar term*, respectively.

*timing transfer function*for one-way photon propagation along \({\vec {u}}=L\hat{u}\). (Here \(\mathrm{sinc}\,x \equiv \sin x/x\)). If we choose \(\vec {r}_2\) to be the origin of coordinates, then \(\mathcal{T}_{{\vec {u}}}(f,\hat{n}\cdot \hat{u})\) contains all the frequency-dependence of the timing response. For example, for normal incidence of the gravitational wave (\(\hat{n}\cdot \hat{u}=0\)), \(|\mathcal{T}_{{\vec {u}}}(f,0)| = (L/c)\,|\mathrm{sinc}(\pi fL/c)|\). Figure 25 is a plot of \(|\mathcal{T}_{{\vec {u}}}(f,0)|\) versus frequency on a logarithmic frequency scale.

*t*as indicated in (5.1). This simply pulls-down a factor of \(i2\pi f\) from the exponential in \(\Delta T(t)\), leading to

*one-way*spacecraft Doppler tracking.

^{11}

*identically zero*. We will discuss the consequences of this result in more detail in Sect. 7.5.6, in the context of phase-coherent mapping of anisotropic gravitational-wave backgrounds.

#### 5.2.2 Two-way tracking

*two-way*spacecraft Doppler tracking, we need to generalize the calculation of the previous subsection to include a return trip of the photon from \(\vec {r}_2\) back to \(\vec {r}_1\). This can be done by simply summing the expressions for the one-way timing residuals:

*three*terms corresponding to the final reception of the photon at \(\vec {r}_1\) at time

*t*, the reflection of the photon at \(\vec {r}_2\) at time \(t-L/c\), and the emission of the photon at \(\vec {r}_1\) at time \(t-2L/c\). The timing response function is given by

*two-way*(or roundtrip) photon propagation along \({\vec {u}}\) and back. For normal incidence, the magnitude of the timing transfer function is given by \(|\mathcal{T}_{{{\vec {u}}},\mathrm{rt}}(f,0)|= (2L/c)|\mathrm{sinc}(2\pi fL/c)|\), which is identical to the expression for one-way tracking with

*L*/

*c*replaced by 2

*L*/

*c*. We also note that if we choose the origin of coordinates to be at \(\vec {r}_1\) (which we can always do for a single detector), and if the frequency

*f*is such that \(fL/c\ll 1\), then the timing response simplifies to

*small-antenna limit*(instead of

*long-wavelength limit*) for this type of limit, since it avoids an ambiguity that might arise if we want to compare three or more length scales. For example, if we have two detectors that are physically separated and the wavelength of a gravitational wave is

*large*compared to the size of each detector but

*small*compared to the separation of the detectors, we would be in the long-wavelength limit with respect to detector size but in the short-wavelength limit with respect to detector separation. (This is actually the case for the current network of ground-based interferometers). The terminology

*small-antenna, large-separation limit*is more appropriate for this case.

#### 5.2.3 Michelson interferometer

^{12}

*free-spectral range*of the interferometer, \(f = f_\mathrm{fsr}\equiv c/(2L)\). Similar plots of the antenna patterns for unpolarized gravitational waves are given in Fig. 30. In Fig. 31 we show colorbar plots of the antenna patterns for the strain response to unpolarized gravitational waves for the LIGO Hanford and Virgo interferometers (located in Hanford, WA and Cascina, Italy, respectively), again evaluated in the small-antenna limit.

### 5.3 Overlap functions

As mentioned in Sect. 4, a stochastic gravitational-wave background manifests itself as a non-vanishing correlation between the data taken by two or more detectors. This correlation differs, in general, from that due to instrumental noise, allowing us to distinguish between a stochastic gravitational-wave signal and other noise sources. In this section, we calculate the expected correlation due to a gravitational-wave background, allowing for non-trivial detector response functions and non-trivial detector geometry. Interested readers can find more details in Hellings and Downs (1983), Christensen (1990, 1992), Flanagan (1993), and Finn et al. (2009).

#### 5.3.1 Definition

*I*and

*J*. In the presence of a gravitational wave, these data will have the form

*I*,

*J*to the gravitational wave, and \(n_{I,J}\) denote the contribution from instrumental noise. If the instrumental noise in the two detectors are uncorrelated with one another, it follows that the expected correlation of the data is just the expected correlation of the detector responses, \(\langle d_I d_J\rangle = \langle h_I h_J\rangle \). If we also assume that the gravitational wave is due to a stationary, Gaussian, isotropic, and unpolarized stochastic background, then

*overlap function*for the two detectors

*I*,

*J*written in terms of the polarization-basis response function \(R^A_{I,J}(f,\hat{n})\),

^{13}where \(A=\{+,\times \}\). In terms of the tensor spherical harmonic-basis response functions \(R^P_{I,J(lm)}(f)\), we would have

#### 5.3.2 Interpretation

Expression (5.36) for the overlap function involves four length scales: the lengths of the two detectors, \(L_I\) and \(L_J\), which appear in the response functions \(R^A_{I,J}(f,\hat{n})\); the separation of the detectors, \(s\equiv |{\vec {x}}_I-{\vec {x}}_J|\), which appears in the exponential factor; and the wavelength of the gravitational waves, \(\lambda = c/f\). In general, one has to evaluate the integral in (5.36) *numerically*, due to the non-trivial frequency dependence of the response functions. However, as we shall see in Sect. 5.4, in certain limiting cases of the ratio of these length scales, we can do the integral *analytically* and obtain relatively simple expressions for the overlap function in terms of spherical Bessel or trigonometric functions. This is the case for ground-based interferometers, which operate in the *small-antenna limit*—i.e., \(fL/c\ll 1\) for both detectors, even though the separation can be large compared to the wavelength, \(fs/c \gtrsim 1\). It is also the case for pulsar timing arrays, which operate in the *large-antenna, small-separation limit*, since \(fL/c\gg 1\) for each pulsar and \(fs/c\ll 1\) for different radio receivers on Earth. (The Earth effectively resides at the solar system barycenter relative to the wavelength of the gravitational waves relevant for pulsar timing).

#### 5.3.3 Normalization

*normalized*overlap function \(\gamma _{IJ}(f)\propto \Gamma _{IJ}(f)\) by requiring that \(\gamma _{IJ}(0)=1\) for two detectors that are co-located and co-aligned. For the strain response of two identical equal-arm Michelson interferometers, this leads to the relation

#### 5.3.4 Auto-correlated response

*auto-correlated*response of a

*single*detector, we can simply set \(I=J\) in the previous expressions. This means that the gravitational-wave strain power \(S_h(f)\) and the detector response power \(P_{h_I}(f)\) in detector

*I*are related by

*integrated over the whole sky*. A plot of the normalized transfer function \(\gamma _{II}(f)\) for the strain response of an equal-arm Michelson interferometer is shown in Fig. 32. Compared to Fig. 25 for the timing transfer function \(|\mathcal{T}_{{\vec {u}}}(f,0)|\) for one-way photon propagation evaluated at normal incidence of the gravitational wave, we see that the relevant frequency scale for an equal-arm Michelson is

*c*/ (2

*L*) (as opposed to

*c*/

*L*) due to the round-trip motion of the photons. Also, the hard nulls in Fig. 25 have been softened into

*dips*due to averaging of the waves over the whole sky. The high-frequency ‘bumps’ for \(\gamma _{II}(f)\) are lower than those for \(|\mathcal{T}_{{\vec {u}}}(f,0)|\) due to the squaring of \(|R^A_I(f,\hat{n})|\) which enters into the definition of \(\Gamma _{II}(f)\) (and \(\gamma _{II}(f)\)). Figure 33 is an extended version of Fig. 32, with the appropriate frequency ranges for ground-based interferometers (like LIGO), space-based interferometers (like LISA), spacecraft Doppler tracking, and pulsar timing searches indicated on the plot. See also Table 5 for more details.

### 5.4 Examples of overlap functions

#### 5.4.1 LHO-LLO overlap function

*detector tensors*; they are symmetric and trace-free with respect to their

*ab*indices. In terms of the detector tensors, the overlap function becomes

*ab*and

*cd*index pairs. The most general expression that we construct for \(\Gamma _{abcd}(\Delta {\vec {x}})\) given \(\delta _{ab}\), \(s_a\), and its symmetry properties is:

*analytically*, leading to

^{14}then all terms contribute (Coughlin and Harms 2014):

*trace-free*detector tensors, as is the case for ground-based interferometers, there is no contribution from the

*A*and

*C*terms. Thus, in the small-antenna limit, the overlap function for the strain response of two equal-arm Michelson interferometers can be written as a sum of the first three spherical Bessel functions with coefficients that depend on the product of the frequency and separation of the two detectors. (The analytic expression for the overlap function can also be derived using (5.37), which involves the tensor spherical harmonic response functions. A detailed derivation using these response functions is given in Romano et al., 2015).

#### 5.4.2 Big-bang observer overlap function

#### 5.4.3 Pulsar timing overlap function (Hellings and Downs curve)

*N*pulsars, labeled by index \(I=1,2,\ldots , N\). Each pulsar defines a one-way tracking beam detector with the position of pulsar

*I*at \(\vec {p}_{I}\) and the position of detector

*I*(i.e., a radio receiver on Earth) by \({\vec {x}}_{I}\). For convenience, we will take the origin of coordinates to lie at the solar system barycenter. Since the diameter of the Earth (\({\sim } 10^4~\mathrm{km}\)) and its distance from the Sun (\({\sim } 10^8~\mathrm{km}\)) are both small compared to the wavelength of gravitational waves relevant for pulsar timing (\(\lambda = c/f\sim 10^{13}~\mathrm{km}\)), we can effectively set \({\vec {x}}_{I}\approx {\vec {x}}_J\approx {\vec {0}}\) in the argument of the exponential term that enters expression (5.36) for the overlap function. Thus,

*I*. But since \({\vec {x}}_I\approx {\vec {0}}\), it follows that \(\hat{u}_I\) and \(\hat{u}_J\) are just unit vectors pointing from the location of pulsars

*I*and

*J*

*toward*the solar system barycenter. For distinct pulsars (\(I\ne J\)), we can ignore the exponential terms in the square brackets, since \(fL/c\gg 1\) for \(L\sim 1~\mathrm{kpc}\ ({=}3\times 10^{16}~\mathrm{km})\) implies that \(e^{-i 2\pi fL_I(1+\hat{n}\cdot \hat{u}_I)/c}\) and its product with the corresponding term for pulsar

*J*are rapidly varying functions of \(\hat{n}\) and do not contribute significantly when integrated over the whole sky (Hellings and Downs 1983; Anholm et al. 2009). (For a single pulsar (\(I=J\)), the product of the two exponential terms equals 1 and hence cannot be ignored). With these simplifications, the integral can be done analytically (Hellings and Downs 1983; Anholm et al. 2009; Jenet and Romano 2015). The result is

*I*and

*J*relative to the solar system barycenter. (For Doppler frequency measurements, the overlap function is

*independent*of frequency, \(\Gamma _{IJ}=\chi (\zeta _{IJ})/3\)). \(\chi (\zeta )\) is the

*Hellings and Downs*function (Hellings and Downs 1983); it depends only on the angular separation of a pair of pulsars. The normalization was chosen so that for a single pulsar, \(\chi (0)=1\) (for two

*distinct*pulsars occupying the same angular position on the sky, \(\chi (0)=0.5\)). A plot of the Hellings and Downs curve is given in Fig. 37.

*independent*of frequency; it is a function of the

*angle*\(\zeta \) between different pulsar pairs. This contrasts with the overlap functions for the two LIGO interferometers and for BBO given in Figs. 34 and 36. These overlap functions were calculated for a fixed pair of detectors; they are functions instead of the

*frequency*of the gravitational wave. (ii) The value of the Hellings and Downs function \(\chi (\zeta _{IJ})\) for a pair of pulsars

*I*,

*J*can be written as a Legendre series in the cosine of the angle between the two pulsars. This follows immediately if one uses (5.37) for the overlap function and (5.23) for the pulsar timing response functions in the tensor spherical harmonic basis. As shown in Gair et al. (2014):

### 5.5 Moving detectors

*modulation*in both the

*amplitude*and the

*phase*of the response of a detector to a monochromatic, plane-fronted gravitational wave (Cutler 1998). For Earth-based interferometers like LIGO, the modulation is due to both the Earth’s daily rotation and yearly orbital motion around the Sun. For space-based interferometers like LISA, the modulation is due to the motion of the individual spacecraft as they orbit the Sun with a period of 1 year. For example, for the original LISA design, three spacecraft fly in an equilateral-triangle configuration around the Sun. The center-of-mass (or guiding center) of the configuration moves in a circular orbit of radius 1 AU, at an angle of \(20^\circ \) behind Earth, while the configuration ‘cartwheels’ in retrograde motion about the guiding center, also with a period of 1 year (see Fig. 39).

#### 5.5.1 Monochromatic plane waves

*orientation*of the detector as well as the detector’s

*translational*motion relative the source. The time-varying orientation leads to changes in the response of the detector to the \(+\) and \(\times \) polarization components of the wave, \(|R^+ h_+|\) and \(|R^\times h_\times |\). The translational motion leads to a Doppler shift in the observed frequency of the wave, which is proportional to

*v*/

*c*times the nominal frequency, where

*v*is velocity of the detector relative to the source:

#### 5.5.2 Stochastic backgrounds

*broad-band*, the Doppler shift associated with the phase modulation of the individual component plane waves is not important, as the gravitational-wave signal power is (at worst) shuffled into nearby bins.

^{15}On the other hand, the amplitude modulation of the signal, due to the time-varying orientation of a detector,

*can*be significant if the background is

*anisotropic*—i.e., stronger coming from certain directions on the sky than from others. (We will discuss searches for anisotropic backgrounds in detail in Sect. 7). As the lobes of the antenna pattern sweep through the “hot” and “cold” spots of the anisotropic background, the amplitude of the signal is modulated in time.

Figure 40 shows the expected time-domain output of a particular Michelson combination, *X*(*t*), of the LISA data over a 5-year period. The combined signal (red) consists of both detector noise (black) and the confusion-limited gravitational-wave signal from the galactic population of compact white-dwarf binaries. At frequencies \({\sim } 10^{-4} - 10^{-3}~\mathrm{Hz}\), which corresponds to the lower end of LISA’s sensitivity band, the contribution from these binaries dominates the detector noise. The modulation of the detector output is clearly visible in the figure. The peaks in amplitude are more than 50% larger than the minimima; they repeat on a 6 month time scale, as expected from LISA’s yearly orbital motion around the Sun (Fig. 39).

^{16}The motion of the LISA constellation was taken from Cutler (1998), and the antenna pattern was calculated for the

*X*-Michelson combination of the LISA data, assuming the small-antenna approximation for the interferometer response functions. The full animation corresponds to LISA’s orbital period of 1 year. Go to http://dx.doi.org/10.1007/s41114-017-0004-1 to view the animation.

#### 5.5.3 Rotational and orbital motion of Earth-based detectors

As mentioned above, given the broad-band nature of a stochastic signal, the Doppler shift associated with the motion of a detector does not play an important role for stochastic background searches. This means that we can effectively ignore the velocity of a detector, and treat its motion as *quasi-static*. So, for example, the motion of a single Earth-based detector like LIGO can be thought of as synthesizing a *set* of *static* virtual detectors located along an approximately circular ring 1 AU from the solar system barycenter (Romano et al. 2015). Each virtual detector in this set observes the gravitational-wave background from a different spatial location and with a different orientation.

*correlated*with one another. Basically, we want two neighboring virtual detectors to be spaced far enough apart that they provide

*independent*information about the background. For a gravitational wave of frequency

*f*, the minimal separation corresponds to \(|\Delta {\vec {x}}|\approx \lambda /2\), where \(\lambda = c/f\) is the wavelength of the gravitational wave. For smaller separations, the two detectors will be driven in coincidence (on average), as discussed in item (iii) at the very end of Sect. 5.4.1. Writing \(|\Delta {\vec {x}}|= v\Delta t\) and solving for \(\Delta t\) yields

We will return to this idea of using the motion of a detector to synthesize a set of static virtual detectors when we discuss a *phase-coherent* approach for mapping anisotropic gravitational-wave backgrounds in Sect. 7.5.

## 6 Optimal filtering

Filters are for cigarettes and coffee.

Cassandra Clare

Optimal filtering, in its most simple form, is a method of combining data so as to extremize some quantity of interest. The optimality criterion depends on the particular application, but for signal processing, one typically wants to: (i) maximize the detection probability for a fixed rate of false alarms, (ii) maximize the signal-to-noise ratio of some test statistic, or (iii) find the minimal variance, unbiased estimator of some quantity. Finding such optimal combinations plays a key role in both Bayesian and frequentist approaches to statistical inference (Sect. 3), and it is an important tool for every data analyst. For a Bayesian, the optimal combinations are often implicitly contained in the likelihood function, while for a frequentist, optimal filtering is usually more explicit, as there is much more freedom in the construction of a statistic.

In this section, we give several simple examples of optimal (or matched) filtering for deterministic signals, and we then show how the standard optimally-filtered cross-correlation statistic (Allen 1997; Allen and Romano 1999) for an Gaussian-stationary, unpolarized, isotropic gravitational-wave background can be derived as a matched-filter statistic for the expected cross-correlation. This derivation of the optimally-filtered cross-correlation statistic differs from the standard derivation given, e.g., in Allen (1997), but it illustrates a connection between searches for deterministic and stochastic signals, which is one of the goals of this review article.

### 6.1 Optimal combination of independent measurements

*N*

*independent*measurements

*a*is some astrophysical parameter that we want to estimate and \(n_i\) are (independent) noise terms. Assuming the noise has zero mean and known variance \(\sigma _i^2\) (which can be different from measurement to measurement), it follows that

*unbiased, minimal variance*estimator of

*a*. Unbiased (i.e., \(\langle \hat{a}\rangle =a\)) implies

*weighted average*that gives less weight to the noiser measurements (i.e., those with large variance \(\sigma _i^2\)). The variance of the optimal combination is

*N*independent and identically-distributed measurements as we saw in Sect. 3.5.

*a*, assuming that the noise terms \(n_i\) are Gaussian-distributed and independent of one another. In fact, similar to what we showed in Sect. 3.5, one can rewrite the argument of the exponential so that

*a*, if the prior for

*a*is flat.

### 6.2 Correlated measurements

*N*measurements \(d_i\) are

*correlated*, so that the covariance matrix

*C*has non-zero elements

Note that although (6.14) shows how to optimally combine data that are correlated with one another, it turns out that for most practical purposes one can get by using expressions like (6.8) and (6.18) below, which are valid for *uncorrelated* data. This is because the values of the Fourier transform of a stationary random process are uncorrelated for different frequency bins. Basically, the Fourier transform is a rotation in data space to a basis in which the covariance matrix is diagonal; this is called a *Karhunen–Loeve transformation*. (See also Appendix D.6). This is one of the reasons why much of signal processing is done in the frequency domain.

### 6.3 Matched filter

*a*. We will also assume that the different measurements are independent, as will be the case for a stationary random process in the frequency domain. Since \(\langle d_i\rangle = a h_i\) is not a constant, the analysis of the previous subsection does not immediately apply. However, if we simply rescale \(d_i\) by \(h_i\), we obtain a new set of measurements

*is*now valid. Thus,

*a*.

*matched filter*(Wainstein and Zubakov 1971) since the data \(d_i\) are projected onto the expected signal shape \(h_i\) (as well as weighted by the inverse of the noise variance \(\sigma _i^2\)). The particular combination

*optimal filter*for this analysis.

^{17}When there are many possible candidate signal shapes, one constructs a

*template bank*—i.e., a collection of possible shapes against which the data compared. By normalizing each of the templates so that \(\sum _i (h_i^2/\sigma _i^2)=1\), the signal-to-noise ratio of the matched filter

### 6.4 Optimal filtering for a stochastic background

*T*be the total observation time of the measurement. In the frequency domain, the measurements are given by the values of the complex-valued cross-correlation

*x*(

*f*) for different frequencies correspond to the measurements \(d_i\) of the previous subsections. Since we are assuming uncorrelated detector noise,

^{18}In the weak-signal limit, the covariance matrix is dominated by the diagonal terms:

*S*in Allen (1997) and Allen and Romano (1999), is given by \(S=\hat{\Omega }_0 T\).

#### 6.4.1 Optimal estimators for individual frequency bins

*individual*frequency bins, of width \(\Delta f\), centered at

*each*(positive) frequency

*f*:

*T*is the duration of the data segments used in calculating the Fourier transforms \(\tilde{d}_1(f)\), \(\tilde{d}_2(f)\); and \(\Gamma _{12}(f)\) is the overlap function for the two detectors.

#### 6.4.2 More general parameter estimation

The analyses in the previous two subsections take as given the spectral shape of an isotropic stochastic background, and then construct estimators of its overall amplitude. But it is also possible to construct estimators of *both* the amplitude and spectral index of the background. One simply treats these as free parameters in the signal model e.g., when constructing the likelihood function. Interested readers should see Mandic et al. (2012) for details.

## 7 Anisotropic backgrounds

Sameness is the mother of disgust, variety the cure.

Francesco Petrarch

An anisotropic background of gravitational radiation has *preferred* directions on the sky—the associated signal is stronger coming from certain directions (“hot” spots) than from others (“cold” spots). The anisotropy is produced primarily by sources that follow the local distribution of matter in the universe (e.g., compact white-dwarf binaries in our galaxy), as opposed to sources at *cosmological* distances (e.g., cosmic strings or quantum fluctuations in the gravitational field amplified by inflation Allen, 1997; Maggiore, 2000), which would produce an *isotropic* background. This means that the measured distribution of gravitational-wave power on the sky can be used to discriminate between cosmologically-generated backgrounds, produced in the very early Universe, and astrophysically-generated backgrounds, produced by more recent populations of astrophysical sources. In addition, an anisotropic distribution of power may allow us to detect the gravitational-wave signal in the first place; as the lobes of the antenna pattern of a detector sweep across the “hot” and “cold” spots of the anisotropic distribution, the amplitude of the signal is modulated in time, while the detector noise remains unaffected (Adams and Cornish 2010).

In this section, we describe several different approaches for searching for anisotropic backgrounds of gravitational waves: The first approach (described in Sect. 7.2) looks for modulations in the correlated output of a pair of detectors, at harmonics of the rotational or orbital frequency of the detectors (e.g., daily rotational motion for ground-based detectors like LIGO, Virgo, etc., or yearly orbital motion for space-based detectors like LISA). This approach assumes a known distribution of gravitational-wave power \(\mathcal{P}(\hat{n})\), and filters the data so as to maximize the signal-to-noise ratio of the harmonics of the correlated signal. The second approach (Sect. 7.3) constructs maximum-likelihood estimates of the gravitational-wave power on the sky based on cross-correlated data from a network of detectors. This approach produces sky maps of \(\mathcal{P}(\hat{n})\), analogous to sky maps of temperature anisotropy in the cosmic microwave background radiation. The third approach (Sect. 7.4) constructs frequentist detection statistics for either an unknown or an assumed distribution of gravitational-wave power on the sky. The fourth and final approach we describe (Sect. 7.5) attempts to measure both the amplitude *and* phase of the gravitational-wave background at each point on the sky, making minimal assumptions about the statistical properties of the signal. This latter approach produces sky maps of the real and imaginary parts of the random fields \(h_+(f,\hat{n})\) and \(h_\times (f,\hat{n})\), from which the power in the background \(\mathcal{P}(\hat{n}) = |h_+|^2 + |h_\times |^2\) is just one of many quantities that can be estimated from the measured data.

Numerous papers have been written over the last \({\approx }20\) years on the problem of detecting anisotropic stochastic backgrounds, starting with the seminal paper by Allen and Ottewill (1997), which laid the foundation for much of the work that followed. Readers interested in more details should see Allen and Ottewill (1997) regarding modulations of the cross-correlation statistic at harmonics of the Earth’s rotational frequency; Ballmer (2006a, b), Mitra et al. (2008), Thrane et al. (2009), Mingarelli et al. (2013) and Taylor and Gair (2013) for maximum-likelihood estimates of gravitational-wave power; Thrane et al. (2009) and Talukder et al. (2011) for maximum-likelihood ratio detection statistics; and Gair et al. (2014), Cornish and van Haasteren (2014) and Romano et al. (2015) regarding phase-coherent mapping. For results of actual analyses of initial LIGO data and pulsar timing data for anisotropic backgrounds, see Abadie et al. (2011) and Taylor et al. (2015) and Sect. 10.2.5.

Note that we will not discuss in any detail methods to detect anisotropic backgrounds using space-based interferometers like LISA or the Big-Bang Observer (BBO). As mentioned in Sect. 5.5.2, the confusion noise from the galactic population of compact white dwarf binaries is a guaranteed source of anisotropy for such detectors. At low frequencies, measurements made using a single LISA will be sensitive to only the \(l=0,2,4\) components of the background, while cross-correlating data from two independent LISA-type detectors (as in BBO) will allow for extraction of the full range of multipole moments. The proposed data analysis methods are similar to those that we will discuss in Sects. 7.2 and 7.3, but using the synthesized *A*, *E*, and *T* data channels for a single LISA (see Sect. 9.7). Readers should see Giampieri and Polnarev (1997), Cornish (2001), Ungarelli and Vecchio (2001), Seto (2004), Seto and Cooray (2004), Kudoh and Taruya (2005), Edlund et al. (2005) and Taruya and Kudoh (2005) for details.

### 7.1 Preliminaries

#### 7.1.1 Quadratic expectation values

*isotropic*component of the background, and sets the overall normalization of the strain power spectral density \(S_h(f)\).

#### 7.1.2 Short-term Fourier transforms

*short-term*Fourier transform. Note that, in this notation,

*t*labels a

*particular*time chunk, and is not a variable that is subsequently Fourier transformed.

#### 7.1.3 Cross-correlations

*t*and frequency

*f*:

*IJ*from both \(\hat{C}_{IJ}(t;f)\) and \(\gamma _{IJ}(t;f,\hat{n})\) when there is no chance for confusion.

Figure 43 shows maps of the real and imaginary parts of \(\gamma (t; f, \hat{n})\) (appropriately normalized) for the strain response of the 4-km LIGO Hanford and LIGO Livingston interferometers evaluated at \(f=0~\mathrm{Hz}\) (top two plots) and \(f=200~\mathrm{Hz}\) (bottom two plots). (In the Earth-fixed frame, the detectors don’t move so there is no time dependence to worry about). Note the presence of oscillations or ‘lobes’ for the \(f=200~\mathrm{Hz}\) plots, which come from the exponential factor \(e^{-i2\pi f\hat{n}\cdot \Delta {\vec {x}}/c}\) of the product of the two response functions (5.43). For \(f=0\), this factor is unity.

#### 7.1.4 Spherical harmonic components of \(\gamma (t; f, \hat{n})\)

*t*and for fixed frequency

*f*, these functions are scalar fields on the unit 2-sphere and hence can be expanded in terms of the ordinary spherical harmonics \(Y_{lm}(\hat{n})\):

*Example: Earth-based interferometers*

*analytic*expressions for \(\gamma _{lm}(t;f)\) for a pair of Earth-based interferometers in the short-antenna limit. If we set \(t=0\), then \(\gamma _{lm}(0;f)\) can be written as a linear combination

^{19}involving spherical Bessel functions, \(j_n(x)/x^n\) (for

*l*even) and \(j_n(x)/x^{n-1}\) (for

*l*odd), where

*x*depends on the relative separation of the two detectors, \(x \equiv 2\pi f|\Delta {\vec {x}}|/c\). The coefficients of the expansions are complex numbers that depend on the relative orientation of the detectors. Explicit expression for the first few spherical harmonic components for the LIGO Hanford–LIGO Livingston pair are given below:

*Example: Pulsar timing arrays*

In Fig. 46, we show plots of the spherical harmonic components of \(\gamma (t;f, \hat{n})\) calculated using the Earth-term-only Doppler-frequency response functions (5.21) for pulsar timing. Since there is no frequency or time-dependence for these response functions, the spherical harmonic components of \(\gamma (\hat{n})\) depend only of the angular separation \(\zeta \) between the two pulsars that define the detector pair. As shown in Mingarelli et al. (2013) and Gair et al. (2014), these functions can be calculated analytically for *all* values of *l* and *m*. A detailed derivation with all the relevant formulae can be found in Appendix E of Gair et al. (2014); there the calculation is done in a ‘computational’ frame, where one of the pulsars is located along the *z*-axis and the other is in the *xz*-plane, making an angle \(\zeta \) with respect to the first. In this computational frame, all of the components \(\gamma _{lm}(\zeta )\) are real. Note that up to an overall normalization factor^{20} of \(3/\sqrt{4\pi }\), the function \(\gamma _{00}(\zeta )\) is just the Hellings and Downs function for an unpolarized, isotropic stochastic background, shown in Fig. 37.

### 7.2 Modulations in the correlated output of two detectors

*m*th harmonic of the correlation has a frequency dependence proportional to

*z*-axis points along the Earth’s rotational axis). In this section, we derive the above result following the presentation in Allen and Ottewill (1997) and construct an optimal filter for the cross-correlation that maximizes the signal-to-noise ratio for the

*m*th harmonic. This was the first concrete approach that was proposed for detecting an anisotropic stochastic background.

#### 7.2.1 Time-dependent cross-correlation

*t*, and where we have included a filter function \(\tilde{Q}(t;f)\), whose specific form we will specify later. Since the cross-correlation is periodic with a period \(T_\mathrm{mod} = 1\) sidereal day (due to the motion of the detectors attached to the surface of the Earth), we can expand \(\hat{C}(t)\) as a Fourier series:

*T*is the total observation time, e.g., 1 sidereal year, which we will assume for simplicity is an integer multiple of \(T_\mathrm{mod}\).

*t*. These two results can now be cast in terms of the Fourier components \(\hat{C}_m\) using (7.17). Since (7.12) implies

#### 7.2.2 Calculation of the optimal filter

*m*th harmonic \(\hat{C}_m\), we

*maximize*the (squared) signal-to-noise:

*inner product*on the space of complex-valued functions (Allen 1997):

*fixed*vector

*A*. But since this ratio is proportional to the squared cosine of the angle between \(\tilde{Q}\) and

*A*, it is maximized by choosing \(\tilde{Q}\)

*proportional*to

*A*. Thus,

*m*th harmonic.

Note that this expression reduces to the standard form of the optimal filter (6.35) for an isotropic background, \(\mathcal{P}_{lm} = \delta _{l0}\delta _{m0}\mathcal{P}_{00}\). Note also that the optimal filter assumes knowledge of both the spectral shape \(\bar{H}(f)\)*and* the angular distribution of gravitational-wave power on the sky, \(\mathcal{P}_{lm}\). So if one has some model for the expected anisotropy (e.g., a dipole in the same direction as the cosmic microwave background), then one can filter the cross-correlated data to be optimally sensitive to the harmonics \(\hat{C}_m\) induced by that anisotropy.

#### 7.2.3 Inverse problem

In Allen and Ottewill (1997), there was no attempt to solve the *inverse problem*—that is, given the *measured values* of the correlation harmonics, how can one *infer* (or *estimate*) the components \(\mathcal{P}_{lm}\)? The first attempt to solve the inverse problem was given in Cornish (2001), in the context of correlation measurements for both ground-based and space-based interferometers. Further developments in solving the inverse problem were given in subsequent papers, e.g., Ballmer (2006a, b), Mitra et al. (2008) and Thrane et al. (2009), which we explain in more detail in the following subsections. Basically, these latter methods constructed frequentist maximum-likelihood estimators for the \(\mathcal{P}_{lm}\), using singular-value decomposition to ‘invert’ the Fisher matrix (or point spread function), which maps the true gravitational-wave power distribution to the measured distribution on the sky.

### 7.3 Maximum-likelihood estimates of gravitational-wave power

In this section, we describe an approach for constructing maximum-likehood estimates of the gravitational-wave power distribution \(\mathcal{P}(\hat{n})\). It is a solution to the inverse problem discussed at the end of the previous subsection. But since a network of gravitational-wave detectors typically does not have perfect coverage of the sky, the inversion requires some form of regularization, which we describe below. The gravitational-wave radiometer and spherical harmonic decomposition methods (Sect. 7.3.6) are the two main implementations of this approach, and have been used to analyze LIGO science data (Abadie et al. 2011; Abbott et al. 2016a).

#### 7.3.1 Likelihood function and maximum-likelihood estimators

*IJ*indices for notational convenience.

^{21}Since the gravitational-wave power distribution \(\mathcal{P}\) enters quadratically in the exponential of the likelihood, we can immediately write down the maximum-likelihood estimators of \(\mathcal{P}\):

*F*is called the

*Fisher information matrix*. It is typically a singular matrix, since the response matrix \(M=\bar{H}\gamma \) usually has

*null*directions (i.e., anisotropic distributions of gravitational-wave power that are mapped to zero by the detector response). Inverting

*F*therefore requires some sort of regularization, such as singular-value decomposition (Press et al. 1992) (Sect. 7.3.5). The vector

*X*is the so-called

*dirty map*, as it represents the gravitational-wave sky as ‘seen’ by a pair of detectors. If the spectral shape \(\bar{H}(f)\) that we used for our signal model exactly matches that of the observed background, then

*point spread function*, which is a characteristic feature of any imaging system. We give plots of point spread functions for both pulsar timing arrays and ground-based interferometers in Sect. 7.3.4.

#### 7.3.2 Extension to a network of detectors

*network*of detectors. One simply replaces

*X*and

*F*in (7.33) by their network expressions, which are simply sums of the dirty maps and Fisher matrices for each distinct detector pair:

#### 7.3.3 Error estimates

*F*is the covariance matrix for the dirty map

*X*, while \(F^{-1}\) is the covariance matrix of the clean map \(\hat{\mathcal{P}}\). We will see below (Sect. 7.3.5) that regularization necessarily changes these results as one cannot recover modes of \(\mathcal{P}\) to which the detector network is insensitive. This introduces a bias in \(\hat{\mathcal{P}}\), and changes the corresponding elements of the covariance matrix for \(\hat{\mathcal{P}}\).

#### 7.3.4 Point spread functions

*Example: Pulsar timing arrays*

*actual-noise*weighting, based on the timing noise values given in the second column of Table 6. Note that this latter plot is similar to the small-

*N*plots in Fig. 47, being dominated by pulsars with low timing noise—in this particular case, J0437−4715 and J2124−3358, which have the lowest and third-lowest timing noise.

Actual pulsar locations and timing noise

Pulsar name | Timing noise (\(\upmu \)s) | Pulsar name | Timing noise (\(\upmu \)s) |
---|---|---|---|

J0437−4715 | 0.14 | J1730−2304 | 0.51 |

J0613−0200 | 2.19 | J1732−5049 | 1.81 |

J0711−6830 | 1.04 | J1744−1134 | 0.17 |

J1022\(+\)1001 | 0.60 | J1824−2452 | 3.62 |

J1024−0719 | 0.35 | J1909−3744 | 0.56 |

J1045−4509 | 3.24 | J1939\(+\)2134 | 3.58 |

J1600−3053 | 2.67 | J2124−3358 | 0.25 |

J1603−7202 | 1.64 | J2129−5721 | 2.55 |

J1643−1224 | 4.86 | J2145−0750 | 0.50 |

J1713\(+\)0747 | 0.89 | B1855\(+\)0900 | 0.70 |

*Example: Earth-based interferometers*

In Fig. 49 we plot point spread functions for gravitational-wave power for the LIGO Hanford-LIGO Livingston pair of detectors. The left-hand plot is for a point source located at the center of the map, \((\theta ,\phi )=(90^\circ , 0^\circ )\), while the right-hand plot is for a point source located at \((\theta ,\phi )=(60^\circ , 0^\circ )\) (indicated by black dots). We assumed equal white-noise power spectra for the two detectors, and we combined the contributions from 100 discrete frequencies between 0 and 100 Hz, and 100 discrete time chunks over the course of one sidereal day. The point spread functions for the two different point source locations are shaped, respectively, like a *figure-eight* with a bright region at the center of the figure-eight pattern, and a *tear drop* with a bright region near the top of the drop. These results are in agreement with Mitra et al. (2008) (see e.g., Fig. 1 in that paper). Provided one combines data over a full sidereal day, the point spread function is independent of the right ascension (i.e., azimuthal) angle of the source. Readers should see Mitra et al. (2008) for more details, including a stationary phase approximation for calculating the point spread function.

*Angular resolution estimates*

*f*is gravitational-wave frequency and

*D*is separation between a pair of detectors. Thus, the larger the separation between detectors and the higher frequencies searched for, the better the angular resolution. For a pulsar timing array consisting of

*N*pulsars, the corresponding estimate is given by

*l*for a spherical harmonic decomposition of the background having angular features of size \(\Delta \theta \). The last approximate equality follows from the fact that, at each frequency, one can extract at most

*N*(complex) pieces of information about the gravitational-wave background using an

*N*-pulsar array (Boyle and Pen 2012; Cornish and van Haasteren 2014; Gair et al. 2014); and those

*N*pieces of information correspond to the number of spherical harmonic components (

*lm*) out to \(l_\mathrm{max}\), so \(N\sim l_\mathrm{max}^2\). (We will discuss this again in Sect. 7.5.4, in the context of

*basis skies*for a phase-coherent search for anisotropic backgrounds). Note that if we knew the distances to the pulsars in the array and used information from the pulsar-term contribution to the timing residuals (5.17), then \(\Delta \theta \) for a pulsar timing array would have the same form as (7.42), but with

*D*now representing the Earth-pulsar distance. See Boyle and Pen (2012) for details.

#### 7.3.5 Singular-value decomposition

Expression (7.33) for the maximum-likelihood estimator \(\hat{\mathcal{P}}\) involves the inverse of the Fisher matrix *F*. But this is just a *formal* expression, as *F* is typically a singular matrix, requiring some sort of regularization to invert. Here we describe how *singular-value decomposition* (Press et al. 1992) can be used to ‘invert’ *F*. Since this a general procedure, we will frame our discussion in terms of an arbitrary matrix *S*.

*S*into the product of three matrices:

*U*and

*V*are \(n\times n\) and \(m\times m\) unitary matrices, and \(\Sigma \) is an \(n\times m\) rectangular matrix with (real, non-negative) singular values \(\sigma _k\) along its diagonal, and with zeros everywhere else. We will assume, without loss of generality, that the singular values are arranged from largest to smallest along the diagonal. We define the

*pseudo-inverse*\(S^+\) of

*S*as

*S*is a square matrix with non-zero determinant, then the pseudo-inverse \(S^+\) is identical to the ordinary matrix inverse \(S^{-1}\). Thus, the pseudo-inverse of a matrix generalizes the notion of ordinary inverse to non-square or singular matrices.

As a practical matter, it is important to note that if the nonzero singular values of \(\Sigma \) vary over several orders of magnitude, it is usually necessary to first set to zero (by hand) all nonzero singular values \(\le \) some minimum threshold value \(\sigma _\mathrm{min}\) (e.g., \(10^{-5}\) times that of the largest singular value). Alternatively, we can set those very small singular values equal to the threshold value \(\sigma _\mathrm{min}\). This procedure helps to reduce the noise in the maximum-likelihood estimates, which is dominated by the modes to which we are least sensitive.

*biased*estimator of \(\mathcal{P}\) if \(F^+\ne F^{-1}\), as was discussed in Thrane et al. (2009).

*F*, hence making the matrix less singular without any external form of regularization.

#### 7.3.6 Radiometer and spherical harmonic decomposition methods

The gravitational-wave radiometer (Ballmer 2006a, b; Mitra et al. 2008) and spherical harmonic decomposition methods (Thrane et al. 2009; Abadie et al. 2011) are two different ways of implementing the maximum-likelihood approach for mapping gravitational-wave power \(\mathcal{P}(\hat{n})\). They differ primarily in their choice of signal model, and their approach for deconvolving the detector response from the underlying (true) distribution of power on the sky.

*Gravitational-wave radiometer*

*diagonal element*\(F_{\hat{n}\hat{n}}\) to obtain an estimate of the point-source amplitude at \(\hat{n}\):

*X*is the dirty map (7.34). Thus, the radiometer method estimates the strength of point sources at different points on the sky,

*ignoring*any correlations between neighboring pixels.

*IJ*the above estimator (7.49) is equivalent to an appropriately normalized cross-correlation statistic:

*Spherical harmonic decomposition*

*f*is the maximum gravitational-wave frequency and

*D*is the separation between a pair of detectors, sets an upper limit on the size of \(l_\mathrm{max}\), since the detector network is not able to resolve features having smaller angular scales. For example, for the LIGO Hanford–LIGO Livingston detector pair (\(D= 3000\ \mathrm{km}\)) and a stochastic background having contributions out to \(f\sim 500~\mathrm{Hz}\), we find \(l_\mathrm{max} \lesssim 30\). Alternatively, one can use Bayesian model selection to determine the value of \(l_\mathrm{max}\) that is most consistent with the data.

*X*.

### 7.4 Frequentist detection statistics

*X*is the ‘dirty’ map, which is related to \(\hat{\mathcal{P}}\) via \(\hat{\mathcal{P}} = F^{-1} X\). The last form suggests a standard matched-filter statistic:

*assumed*distribution of gravitational-wave power on the sky, normalized such that

Such a matched-filter statistic was proposed in Appendix C of Thrane et al. (2009) and studied in detail in Talukder et al. (2011). One nice property of this statistic is that it does not require inverting the Fisher matrix. Hence it avoids the inherent bias (7.47) and introduction of other uncertainties associated with the deconvolution process. Indeed, if we are *given* a model of the expected anisotropy, \(\lambda (d)\) is the *optimal* statistic for detecting its presence. Thus, \(\lambda (d)\) is especially good at detecting weak anisotropic signals. See Talukder et al. (2011) for more details.

### 7.5 Phase-coherent mapping

Phase-coherent mapping is an approach that constructs estimates of both the amplitude and phase of the gravitational-wave background at each point of the sky (Cornish and van Haasteren 2014; Gair et al. 2014; Romano et al. 2015). In some sense, it can be thought of as the “square root” of the approaches described in the previous subsections, which attempt to measure the distribution of gravitational-wave *power*\(\mathcal{P}(\hat{n}) = |h_+|^2 + |h_\times ^2|\). The gravitational-wave signal can be characterized in terms of either the standard polarization basis components \(\{h_+(f,\hat{n}), h_\times (f,\hat{n})\}\) or the tensor spherical harmonic components \(\{a^G_{(lm)}(f), a^C_{(lm)}(f)\}\). In what follows we will restrict our attention the polarization basis components, although a similar analysis can be carried out in terms of the spherical harmonic components (Gair et al. 2014).

#### 7.5.1 Maximum-likelihood estimators and Fisher matrix

*I*labels the different detectors, and \(\tilde{n}_I(t;f)\) denotes the corresponding detector noise. Given our assumption (7.3) that the spectral and angular dependence of the background factorize with known spectral function \(\bar{H}(f)\), we can rewrite the above equation as

*A*and directions \(\hat{n}\) on the sky.

*I*at time

*t*. Thus, we can write down a likelihood function for the data \(d\equiv \{\tilde{d}_I(t;f)\}\) given

*a*:

*I*, times

*t*, and frequencies

*f*, or summations over polarizations

*A*and directions \(\hat{n}\) on the sky. Note that (7.65) has exactly the same form as (7.32), so the same general remarks made in Sect. 7.3.1 apply here as well. Namely, the maximum-likelihood estimators of

*a*are

*M*,

*N*here are different, of course, from those in Sect. 7.3.1). Explicit expression for

*X*and

*F*are given below:

*A*, compared to the corresponding expressions, (7.37) and (7.38), for gravitational-wave power.

#### 7.5.2 Point spread functions

*A*and \(\hat{n}\) vary. Since there are two polarization modes (\(+\) and \(\times \)), there are actually

*four*different point spread functions for each direction \(\hat{n}'\) on the sky:

To illustrate the above procedure, we calculate point spread functions for phase-coherent mapping, for pulsar timing arrays consisting of \(N=1\), 2, 5, 10, 25, 50, 100 pulsars. Figure 53 show plots of these point spread functions. The pulsars are randomly distributed over the sky (indicated by white stars), and the point source is located at the center of the maps (indicated by a black dot). For simplicity, we assumed a single frequency bin, and used equal-noise weighting for calculating the point spread functions. (In addition, there is no time dependence as the directions to the pulsars are fixed on the sky). Different rows in the figure correspond to different numbers of pulsars in the array. Different columns correspond to different choices for *A* and \(A'\): columns 1, 2 correspond to the \(A=+,\times \) response of the pulsar timing array to an \(A'=+\)-polarized point source; columns 3, 4 correspond to the \(A=+,\times \) response of the pulsar timing array to an \(A'=\times \)-polarized point source. Note that for \(N=1\), the point spread functions are proportional to either \(R_I^+(\hat{n})\) or \(R_I^\times (\hat{n})\) for that pulsar, producing maps similar to those shown in Fig. 27. As *N* increases the \(++\) and \(\times \times \) point spread functions (columns 1 and 4) become tighter around the location of the point source, which is at the center of the maps. But since the \(+\) and \(\times \) polarizations are orthogonal, the \(\times +\) and \(+\times \) point spread functions (columns 2 and 3) have values close to zero around the location of the point source.

#### 7.5.3 Singular value decomposition

Just as we had to deconvolve the detector response in order to obtain the estimators \(\hat{\mathcal{P}}\) for gravitational-wave power, we need to do the same for the estimators \(\hat{a}\) for the phase-coherent mapping approach. Although we could use singular-value decomposition for the Fisher matrix *F* given by (7.69), we will first *whiten* the data, which leads us directly to pseudo-inverse of the whitened response matrix *M*, (7.63). This is the approach followed in Cornish and van Haasteren (2014) and Romano et al. (2015), and it leads to some interesting results regarding *sky-map basis vectors*, which we will describe in more detail in Sect. 7.5.4. An alternative approach involving the pseudo-inverse of the unwhitened response matrix is given in Gair et al. (2014) and Appendix B of Romano et al. (2015).

*L*is a lower triangular matrix. The whitened data are then given by \(\bar{d} = L^\dagger d\) (since this has unit covariance matrix), and the whitened response matrix is given by \(\bar{M} = L^\dagger M\). In terms of these whitened quantities,

*always*possible to define the pseudo-inverse of a matrix in terms of its singular-value decomposition. Thus, given the singular-value decomposition:

#### 7.5.4 Basis skies

*V*corresponding to the non-zero singular values of \(\Sigma \) are

*basis vectors*(which we will call

*basis skies*) in terms of which \(\hat{a}\) can be written as a linear combination. Similarly, if write the whitened response to the gravitational-wave background as

*U*corresponding to the non-zero singular values of \(\Sigma \) can be interpreted as

*range vectors*for the response. To be more explicit, let \(u_{(k)}\) and \(v_{(k)}\) denote the

*k*th columns of

*U*and

*V*, and let

*r*be the number of non-zero singular values of \(\Sigma \). Then

*a*and

*b*is defined as \(a\cdot b = a^\dagger b\). If we further expand \(\bar{d} = \bar{M} a + \bar{n}\) in the first of these equations, then

*a*onto the basis skies \(v_{(k)}\) for only the non-zero singular values of \(\Sigma \).

In Fig. 54, we show plots of the real parts of the \(+\) and \(\times \)-polarization basis skies for a pulsar timing array consisting of \(N=5\) pulsars randomly distributed on the sky. The imaginary components of the basis skies are identically zero, and hence are not shown in the figure. The basis skies are shown in decreasing size of their singular values, from top to bottom. In general, if *N* is the number of pulsars in the array, then the number of basis skies is 2*N* (the factor of 2 corresponding to the two polarizations, \(+\) and \(\times \)). This means that one can extract at most 2*N* real pieces of information about the gravitational-wave background with an *N*-pulsar array. This is typically fewer than the number of modes of the background that we would like to recover.

#### 7.5.5 Underdetermined reconstructions

*n*is less than the number of modes

*m*that we are trying to recover (so \(n<m\)), or where there are certain modes of the gravitational-wave background (e.g.,

*null skies*) that our detector network is simply insensitive to. Then, for both of these cases, the linear system of equations that we are trying to solve, \(\bar{d} = \bar{M} a\), is

*underdetermined*—i.e., there exist multiple solutions for

*a*, which differ from (7.75) by terms of the form

*arbitrary*gravitational-wave background. (Note that \(a_\mathrm{null}\) is an element of the

*null space*of \(\bar{M}\) as it maps to zero under the action of \(\bar{M}\)). Our solution for \(\hat{a}\) given in (7.75) sets to zero those modes that we are insensitive to. Our solution also sets to zero the variance of these modes.

#### 7.5.6 Pulsar timing arrays

*half*of all possible modes of a gravitational-wave background, regardless of how many pulsars there are in the array. Note that this statement is not restricted to the tensor spherical harmonic analysis; it is also true in terms of the standard \((+,\times )\) polarization components, since \(a^G_{(lm)}(f)\) and \(a^C_{(lm)}(f)\) are linear combinations of \(h_+(f,\hat{n})\) and \(h_\times (f,\hat{n})\), see (2.11). It is just that the insensivity of a pulsar timing array to half of the gravitational-wave modes is

*manifest*in the gradient and curl spherical harmonic basis for which (7.80) is valid.

To explicitly demonstrate that a pulsar timing array is insensitive to the curl-component of a gravitational-wave background, Gair et al. (2014) constructed maximum-likelihood estimates of a simulated background containing both gradient and curl modes. The total simulated background and its gradient and curl components are shown in the top row (panels a–c) of Fig. 55. (Note that this is for a noiseless simulation so as not to confuse the lack of reconstructing the curl component with the presence of detector noise). Panel e shows the maximum-likelihood recovered map for a pulsar timing array consisting of \(N=100\) pulsars randomly distributed on the sky. Panels d and f are residual maps obtained by subtracting the maximum-likelihood recovered map from the gradient component and the total simulated background, respectively. Note that the maximum-likelihood recovered map resembles the gradient component of the background, consistent with the fact that a pulsar timing array is insenstive to the curl component of a gravitational-wave background. The residual map for the gradient component (panel d) is much cleaner than the residual map for the total simulated background (panel f), which has angular structure that closely resembles the curl component of the background.

#### 7.5.7 Ground-based interferometers

*at the origin*of coordinates. Since a translation mixes gradient and curl components, the response functions for an interferometer displaced from the origin by \(\hat{x}_0\) are given by Romano et al. (2015):

*L*. Here

*j*symbols (see, e.g., Wigner 1959; Messiah 1962). Note that the curl response is now non-zero, and both response functions depend on frequency via the quantity \(\alpha \), which equals \(2\pi \) times the number of radiation wavelengths between the origin and the vertex of the interferometer. These expressions are valid in an arbitrary translated and rotated coordinate system, provided the angles for \(\hat{u}\), \(\hat{v}\), and \(\hat{x}_0\) are calculated in the rotated frame.

*and*curl components of a gravitational-wave background. This is in contrast to a pulsar timing array, which is insensitive to the curl component, because one vertex of all the pulsar baselines are ‘pinned’ to the solar system barycenter. To illustrate this difference, we show in Fig. 56, maximum-likelihood recovered sky maps for simulated grad-only and curl-only anistropic backgrounds injected into noise for a 3-detector network of ground-based interferometers (Hanford–Livingston–Virgo). The grad-only and curl-only backgrounds are the same as those used for the simulated maps in Fig. 55. In contrast to the recovered maps shown in that figure for the pulsar timing array, the maximum-likelihood maps (bottom row) for the network of ground-based interferometers reproduce the general angular structure of both the grad-only

*and*curl-only injected maps (shown in the top row). (The noise for these injections degrades the recovery compared to the noiseless injections in Fig. 55). See Romano et al. (2015) for more details and related simulations.

## 8 Searches for other types of backgrounds/signals

No idea is so outlandish that it should not be considered with a searching but at the same time a steady eye.

Winston Churchill

*anisotropic*signals, which are stronger coming from certain directions on the sky than from others. In this section, we discuss search methods for non-Gaussian signals (Sect. 8.1), circularly polarized backgrounds (Sect. 8.2), and additional polarization modes predicted by alternative (non-general-relativity) metric theories of gravity (Sects. 8.3, 8.4, 8.5). In Sect. 8.6, we also briefly mention searches for other types of gravitational-wave signals, which are not really stochastic backgrounds, but nonetheless can be searched for using the basic idea of cross-correlation, which we developed in Sect. 4. The majority of the search methods that we will describe here have been implemented “across the band”—i.e., for ground-based interferometers, space-based interferometers, and pulsar timing arrays. For these methods, we will highlight any significant differences in the implementations for the different detectors, if there are any.

Of course, we do not have enough time or space in this section to do justice for all of these methods. As such, readers are strongly encouraged to read the original papers for more details. For non-Gaussian backgrounds, see Drasco and Flanagan (2003), Seto (2009), Thrane (2013), Martellini and Regimbau (2014) and Cornish and Romano (2015); for circular polarization, see Seto and Taruya (2007, 2008) and Kato and Soda (2016); for polarization modes in alternative theories of gravity, see Lee et al. (2008), Nishizawa et al. (2009), Chamberlin and Siemens (2012) and Gair et al. (2015); and for the other types of signals, see Thrane et al. (2011) and Messenger et al. (2015).

### 8.1 Non-Gaussian backgrounds

In Sect. 2.1, we asked the question “when is a gravitational-wave signal stochastic” to highlight the practical distinction between searches for deterministic and stochastic signals. From an operational perspective, a signal is stochastic if it is best searched for using a stochastic signal model (i.e., one defined in terms of probability distributions), even if the signal is *intrinsically* deterministic, e.g., a superposition of sinusoids. This turns out to be the case if the signals are: (i) *sufficiently weak* that they are individually unresolvable in a single detector, and hence can only be detected by integrating their correlated contribution across multiple detectors over an extended period of time, or (ii) they are *sufficiently numerous* that they overlap in time-frequency space, again making them individually unresolvable, but producing a *confusion noise* that can be detected by cross-correlation methods. If the rate of signals is large enough, the confusion noise will be Gaussian thanks to the central limit theorem. But if the rate or duty-cycle is small, then the resulting stochastic signal will be non-Gaussian and “popcorn-like”, as we discussed in Sect. 1.1. This is the type of signal that we expect from the population of binary black holes that produced GW150914 and GW151226; and it is the type of signal that we will focus on in the following few subsections.

Figure 57 illustrates the above statements in the context of a simple toy-model signal consisting of simulated sine-Gaussian bursts (each with a width \(\sigma _t= 1~\mathrm{s}\)) having different rates or duty cycles. The left two panels correspond to the case where there is 1 burst every 10 seconds (on average). The probability distribution of the signal samples *h* (estimated by the histogram in the lower-left-hand panel) is far from Gaussian for this case. The right two panels correspond to 100 bursts every second (on average), for which the probability distribution is approximately Gaussian-distributed, as expected from the central limit theorem.

#### 8.1.1 Non-Gaussian search methods: overview

*skewness*and (excess)

*kurtosis*of the distribution, which are the third and fourth-order

*cumulants*, defined as follows: If

*X*is a random variable with probability distribution \(p_X(x)\), then the

*moments*are defined by (Appendix B):

*cumulants*by

*mean*\(\mu \) and

*variance*\(\sigma ^2\) of the distribution. For a Gaussian distribution, \(c_3=0, c_4=0, \ldots \). For a distribution with zero mean, the above formulas simplify to \(c_1=0\), \(c_2=\mu _2\), \(c_3=\mu _3\), and \(c_4 = \mu _4 - 3\mu _2^2\). The higher-order-moment approach requires 3rd or 4th-order correlation measurements (Sect. 8.1.5).

#### 8.1.2 Likelihood function approach for non-Gaussian backgrounds

Fundamentally, searching for non-Gaussian stochastic signals is no different than searching for a Gaussian stochastic signal. In both cases one must: (i) specify a signal model, (ii) incorporate that signal model into a likelihood function or frequentist detection statistic/estimator, and (iii) then analyze the data to determine how likely it is that a signal is present. It is the choice of signal model, of course, that determines what type of signal is being searched for.

^{22}with covariance matrix \(C_n\), the probability of observing data

*d*in a network of detectors given signal model \(\bar{h}\) is (3.53):

*I*. (The subscript

*i*labels either a time or frequency sample for the analysis, whichever is being used). Since one is often not interested in the particular values of \(\bar{h}\), but rather the values of the parameters \(\mathbf {\theta }_h\) that describe the signal, one marginalizes over \(\bar{h}\):

Several different signal priors, which have been proposed in the literature, are given below. For simplicity, we will consider the case where the detectors are colocated and coaligned, and have isotropic antenna patterns, so that the contribution from the signal is the same in each detector, and is independent of direction on the sky. For real analyses, these simplifications will need to be dropped, as is done e.g., in Thrane (2013).

*Gaussian signal prior*

*non-Gaussian signal prior*

*Mixture-Gaussian signal prior*

*non-Gaussian signal prior*

*n*). The Edgeworth expansion is referenced off a Gaussian probability distribution, and is thus said to be a

*semi-parametric*representation of a non-Gaussian distribution. This prior reduces to the Drasco and Flanagan signal prior when \(c_3=0\), \(c_4=0\).

*Multi-sinusoid signal prior*

*deterministic*signal prior, corresponding to the superposition of

*M*sinusoids with unknown amplitudes, frequencies, and phases, \(\mathbf {\theta }_h = \{A_I, f_I, \varphi _I\vert I=1,2,\ldots , M\}\). This was one of the signal models used in Cornish and Romano (2015) to investigate the question of when is a signal stochastic.

*Superposition of finite-duration deterministic signals*

*I*th signal and \(t_I\) is its arrival time. Note that these signal waveforms can be

*extended*in time, having a characteristic duration \(\tau \). Thus, this signal model is intermediate between the single-sample burst and multi-sinusoid signal models.

*Generic likelihood for unresolvable signals*

*I*,

*J*, and takes as its fundamental data vector estimates of the signal-to-noise ratio of the cross-correlated power in the two detectors:

*i*is over time-frequency pixels

*tf*. The functions

*S*and

*B*are probability distributions for \(\hat{\rho }_i\) for the signal and noise models, respectively. These distributions are generic in the sense that they are to be estimated using Monte Carlo simulations with injected signals for the signal model

*S*, and via time-slides on real data for the noise model

*B*. They need not be Gaussian for either the signal or the detector noise. The vectors \(\mathbf {\theta }_h\) and \(\mathbf {\theta }_n\) denote parameters specific to the signal and noise models. Although the above likelihood function was not obtained by explicitly marginalizing over \(\bar{h}\), mathematically there is some signal prior and noise model which yields this likelihood upon marginalization.

#### 8.1.3 Frequentist detection statistic for non-Gaussian backgrounds

*N*is the number of samples, and where the last approximate equality assumes that the gravitational-wave signal is weak compared to the detector noise. We have added the superscript G to indicate that this is for a Gaussian-stochastic signal model.

We can perform exactly the same calculations, making the same assumptions, for the likelihood functions constructed from *any* of the non-Gaussian signal priors given above (in Sect. 8.1.2). These calculations have already been done for the Drasco–Flanagan and Martellini–Regimbau signal priors (Drasco and Flanagan 2003; Martellini and Regimbau 2014). The expressions that they find for the maximum-likelihood ratios \(\Lambda _\mathrm{ML}^\mathrm{NG}(d)\) for their non-Gaussian signal models are rather long and not particularly informative, so we do not bother to write them down here (interested readers should see (1.8) in Drasco and Flanagan 2003, and the last equation in Martellini and Regimbau 2014). The values of the parameters that maximize the likelihood ratio are estimators of \(\xi \), \(\alpha \), \(S_{n_1}\), \(S_{n_2}\) for the Drasco and Flanagan signal model, and estimators of \(\xi \), \(\alpha \), \(c_3\), \(c_4\), \(S_{n_1}\), \(S_{n_2}\) for the Martellini and Regimbau signal model.

*burst*statistic, which is just the maximum of the absolute value of the data samples in e.g., detector 1: \(\Lambda ^\mathrm{B}(d) = \max _{i} |d_{1i}|\). The false alarm and false dismissal probabilities were both chosen to equal 0.01 for this calculation. From the figure one sees that for \(\xi \,{\gtrsim }10^{-3}\), the Gaussian-stochastic cross-correlation statistic performs best. For smaller values of \(\xi \), the non-Gaussian statistic is better. In particular, for \(\xi \sim 10^{-4}\). there is a factor of \({\sim }2\) improvement in the minimum detectable signal amplitude if one uses the non-Gaussian maximum-likelihood detection statistic.

#### 8.1.4 Bayesian model selection

*prior*odds for the two models, while the second term is the

*Bayes factor*:

*frequency-domain version*of the short-duration time-domain bursts discussed in the previous subsections. Five different models were considered:

\(\mathcal{M}_0\): noise-only model, consisting of uncorrelated white Gaussian noise in two detectors with unknown variances \(\sigma _1^2\), \(\sigma _2^2\).

\(\mathcal{M}_1\): noise plus the Gaussian-stochastic signal model defined by (8.6).

\(\mathcal{M}_2\): noise plus the mixture-Gaussian stochastic signal model defined by (8.8).

\(\mathcal{M}_3\): noise plus the deterministic multisinusoid model defined by (8.11).

\(\mathcal{M}_4\): noise plus the deterministic multisinusoid signal model plus the Gaussian-stochastic signal model. This is a

*hybrid*signal model that allows for both stochastic and deterministic components for the signal.

*different*realizations of the data; they are not uncertainties in the Bayes factors associated with different Monte Carlo simulations for a

*single*realization, which were \({\lesssim }10\%\).

#### 8.1.5 Fourth-order correlation approach for non-Gaussian backgrounds

In this section, we briefly describe a fourth-order correlation approach for detecting non-Gaussian stochastic signals, originally proposed in Seto (2009). The key idea is that by forming a particular combination of data from 4 detectors (the *excess kurtosis*), one can separate the non-Gaussian contribution to the background from any Gaussian-distributed component. This approach requires that the noise in the four detectors be uncorrelated with one another, but it does not require that the noise be Gaussian. Here we sketch out the calculation for 4 colocated and coaligned detectors, which we will assume have isotropic antenna patterns, so that the contribution from the gravitational-wave signal is the same in each detector, and is independent of direction on the sky. These simplifying assumptions are not essential for this approach; the calculation for separated and misalinged detectors with non-isotropic response functions can also be done (Seto 2009).

*I*and \(\tilde{h}\) denotes the total gravitational-wave contribution, which has a Gaussian-stochastic component \(\tilde{g}\), and a non-Gaussian component formed from the superposition of short-duration burst signals \(\tilde{b}_i\), \(i=1,2,\ldots , n\). We assume that the noise in the detectors are uncorrelated with one another and with the gravitational-wave signals, and that the individual gravitational-wave signals are also uncorrelated amongst themselves. The (random) number of bursts present in a particular segment of data is determined by a Poisson distribution

*estimator*of the expected correlations. Since the noise in the detectors are uncorrelated with everything, the only contributions to \(\mathcal{K}\) will come from expectation values of products of \(\tilde{h} = \tilde{g} +\sum _i \tilde{b}_i\) with itself. Calculating the quadratic terms that enter (8.23), we find:

As mentioned already, the above calculation can be extended to the case of separated and misaligned detectors (Seto 2009). In so doing, one obtains expressions for *generalized* (4th-order) overlap functions, which are sky-averages of the product of the response functions for four different detectors. The expected value of the 4th-order detection statistic for this more general analysis involves generalized overlap functions for both the (squared) overall intensity and circular polarization components of the non-Gaussian background. We will discuss circular polarization in the following section, but in the simpler context of Gaussian-stationary isotropic backgrounds. Readers should see Seto (2009) for more details regarding circular polarization in the context of non-Gaussian stochastic signals discussed above.

### 8.2 Circular polarization

Up until now, we have only considered *unpolarized* stochastic backgrounds. That is, we have assumed that the gravitational-wave power in the \(+\) and \(\times \) polarization modes are equal (on average) and are statistically independent of one another (i.e., there are no correlations between the \(+\) and \(\times \) polarization modes). It is possible, however, for some processes in the early Universe to give rise to *parity violations* (Alexander et al. 2006), which would manifest themselves as an asymmetry in the amount of right and left *circularly* polarized gravitational waves. Following Seto and Taruya (2007, 2008), we now describe how to generalize our cross-correlation methods to look for evidence of circular polarization in a stochastic background.

#### 8.2.1 Polarization correlation matrix

*polarized anisotropic*Gaussian-stationary background. (We will restrict attention to isotropic backgrounds later on). It turns out that these expectation values can be written in terms of the

*Stokes’ parameters*

*I*,

*Q*,

*U*, and

*V*, which are defined for a monochromatic plane gravitational wave in Appendix A. If we expand \(h_{ab}(f,\hat{n})\) in terms of the

*linear*polarization basis tensors \(e^A_{ab}(\hat{n})\), where \(A=\{+,\times \}\), we have

*circular*polarization basis tensors \(e^C_{ab}(\hat{n})\), where \(C=\{R,L\}\), then

*V*is a measure of a possible asymmetry between the right and left circular polarization components:

*I*and

*V*are ordinary scalar (spin 0) fields on the sphere, while

*Q*and

*U*transform like spin 4 fields under a rotation of the unit vectors \(\{\hat{l},\hat{m}\}\) tangent to the sphere. Thus,

*I*and

*V*can be written as linear combinations of the ordinary spherical harmonics \(Y_{lm}(\hat{n})\):

*Q*,

*U*components of the polarization correlation matrix vanish if the background is isotropic (i.e., has only a contribution from the monopole \(l=0\), \(m=0\)). So for simplicity, we will restrict our attention to polarized

*isotropic backgrounds*, for which the circular polarization correlation matrix becomes diagonal and the quadratic exprectation values reduce to:

#### 8.2.2 Overlap functions

*I*and

*J*to such a background. Similar to (5.9), we can write the response of detector

*I*as

*R*,

*L*label the right and left circular polarization states for both the Fourier components and the detector response functions. Writing down a similar expression for the response of detector

*J*, and using (8.36) to evaluate the expected value of the product of the responses, we find

*I*and

*V*Stokes parameters for a polarized isotropic stochastic background. Using

*I*and

*V*overlap functions for the LIGO-Virgo detector pairs, using the small-antenna limit for the strain response functions. The overlap functions have been normalized (5.40) so that \(\gamma ^{(I)}_{IJ}(f)=1\) for colocated and coaligned detectors. Similar plots can be made for other interferometer pairs, by simply using the appropriate response functions for those detectors.

Note that for pulsar timing, one can show that \(\Gamma ^{(V)}_{IJ}(f) = 0\) for any pair of pulsars. This means that one cannot detect the presence of a circularly polarized stochastic background using a pulsar timing array if one restricts attention to just the isotropic component of the background. One must include higher-order multipoles in the analysis—i.e., do an *anisotropic* search as discussed in Sect. 7. Such an analysis for anisotropic polarized backgrounds using pulsar timing arrays is given in Kato and Soda (2016). In that paper, they extend the analysis of Mingarelli et al. (2013) to include circular polarization. See Kato and Soda (2016) for additional details.

#### 8.2.3 Component separation: ML estimates of *I* and *V*

As shown in Seto and Taruya (2007, 2008), in order to separate the *I*(*f*) and *V*(*f*) contributions to a polarized isotropic background at each frequency *f*, we will need to analyze data from at least two independent baselines (so three or more detectors). In what follows, we will use the notation \(\alpha =1,2,\ldots , N_b\) to denote the individual baselines (detector pairs) and \(\alpha _1\), \(\alpha _2\) to denote the two detectors that constitute that baseline. The formalism we adopt here is similar to that for constructing maximum-likelihood estimators of gravitational-wave power for unpolarized anisotropic backgrounds (Sect. 7.3). For a general discussion of component separation for isotropic backgrounds, see Parida et al. (2016).

*T*is the duration of the data. Assuming that the noise in the individual detectors are uncorrelated with one another, we can easily calculate the expected value of \(\hat{C}_\alpha (f)\) using our previous result (8.40). The result is

*M*is an \(N_f N_b\times 2 N_f\) detector network response matrix, and \(\mathcal{S}\) is an \(2N_f\times 1\) vector containing the unknown Stokes’ parameters, which we want to estimate from the data.

^{23}

*M*and \(\mathcal{N}\) given above. As before, inverting

*F*may require some sort of regularization, e.g., using singular-value decomposition (Sect. 7.3.5). If that’s the case then \(F^{-1}\) should be replaced in the above formula by its pseudo-inverse \(F^+\). The uncertainty in the maximum likelihood recovered values is given by the covariance matrix

#### 8.2.4 Example: component separation for two baselines

*M*is a square \(2 N_f\times 2 N_f\) matrix, which we assume has non-zero determinant. Then it follows simply from the definitions (8.52) of

*F*and

*X*that

#### 8.2.5 Effective overlap functions for *I* and *V* for multiple baselines

*I*and

*V*can easily be extended to the case of an arbitrary number of baselines \(\alpha = 1,2,\ldots , N_b\). For multiple baselines with noise spectra \(N_\alpha (f)\equiv P_{n_{\alpha _1}}(f) P_{n_{\alpha _2}}(f)\), one can show that

*effective*overlap functions for

*I*and

*V*associated with a multibaseline detector network by basically inverting the above uncertainties. For simplicity, we will assume that the noise power spectra for the detectors are equal to one another so that \(N_\alpha \equiv N\) can be factored out of the above expressions. We then define

*I*and

*V*components of the background. Plots of \(\Gamma _\mathrm{eff}^{(I)}(f)\) and \(\Gamma _\mathrm{eff}^{(V)}(f)\) are shown in Fig. 62 for the multibaseline network formed from the LIGO Hanford, LIGO Livingston, and Virgo detectors. Recall that the overlap functions for the individual detectors pairs are shown in Fig. 61. Dips in sensitivity correspond to frequencies where the determinant of \(\bar{F}\) is zero (or close to zero).

### 8.3 Non-GR polarization modes: preliminaries

*tensor*modes predicted by general relativity (GR); two

*vector*(or “shear”) modes, which we will denote by

*X*and

*Y*; and two

*scalar*modes: a “breathing” mode

*B*and a pure longitudinal mode

*L*(see, e.g., Nishizawa et al. 2009). The tensor and breathing modes are

*transverse*to the direction of propagation, while the two vector modes and the scalar longitudinal mode have

*longitudinal*components (parallel to the direction of propagation). See Fig. 63.

*z*-axis, and \(\hat{l}\) and \(\hat{m}\) point along the

*x*and

*y*axes, the polarization tensors can be represented by the following \(3\times 3\) matrices:

#### 8.3.1 Transformation of the polarization tensors under a rotation about \(\hat{n}\)

*X*,

*Y*, and the spin 0 nature of the scalar modes

*B*,

*L*.

#### 8.3.2 Polarization and spherical harmonic basis expansions

*X*,

*Y*, we need to use the

*vector-gradient*and

*vector-curl*spherical harmonics \(Y^{V_G}\), \(Y^{V_C}\), which are defined in terms of spin-weight \(\pm 1\) spherical harmonics (Appendices F and E). For the scalar modes, we can use

#### 8.3.3 Detector response

#### 8.3.4 Searches for non-GR polarizations using different detectors

Evidence for non-GR polarization modes can show up in searches for *either* deterministic or stochastic gravitational-wave signals. Whether these alternative polarization modes are first discovered from the observation of gravitational waves from a resolvable source (like a binary black hole merger) or from a stochastic background depends in part on the type and number of detectors making the observations. For example, individual binary black holes mergers (GW150914 and GW151226) have already been observed by advanced LIGO. But it was not possible to extract information about the polarization of the waves, since the two LIGO interferometers are effectively co-aligned (and hence see the *same* polarization). Adding Virgo, KAGRA, and LIGO-India to the global network will eventually allow for the extraction of this polarization information. Pulsar timing arrays, on the other hand, are expected to first detect a stochastic background from the inspirals of SMBHBs in the centers of distant galaxies (Rosado et al. 2015). So if evidence of alternative polarization modes are discovered by pulsar timing, it will most-likely first come from stochastic background observations.

In the following sections, we describe stochastic background search methods for non-GR polarization modes using both ground-based interferometers (Sect. 8.4) and pulsar timing arrays (Sect. 8.5). We will calculate antenna patterns, overlap functions, and discuss component separation for the tensor, vector, and scalar polarization modes. For ground-based interferometers, our discussion will be based on Nishizawa et al. (2009). For pulsar timing arrays, see Lee et al. (2008), Chamberlin and Siemens (2012) and Gair et al. (2015).

### 8.4 Searches for non-GR polarizations using ground-based detectors

We now describe cross-correlation searches for non-GR polarization modes using a network of ground-based laser interferometers. For additional details, see Nishizawa et al. (2009).

#### 8.4.1 Response functions

*X*and

*Y*polarizations for the vector modes, and equal power in the

*B*and

*L*polarizations for the scalar modes.

#### 8.4.2 Overlap functions

*independently polarized*, but is otherwise Gaussian-stationary and isotropic. This means that the quadratic expectation values of the Fourier components of the metric perturbations can be written as

*I*as

#### 8.4.3 Component separation: ML estimates of \(S_h^{(T)}\), \(S_h^{(V)}\), and \(S_h^{(S)}\)

Proceeding along the same lines as in Sect. 8.2.3, we now describe a method for separating the tensor, vector, and scalar contributions to the total strain spectral density. As shown in Nishizawa et al. (2009), we will need to analyze data from at least three independent baselines (so at least three detectors) to separate the tensor, vector, and scalar contributions at each frequency *f*. As before, we will adopt the notation \(\alpha =1,2,\ldots , N_b\) to denote the individual baselines (detector pairs) and \(\alpha _1\), \(\alpha _2\) to denote the two detectors that constitute that baseline.

*M*and \(\mathcal{N}\) given above, and with the standard proviso about possibly having to use singular-value decomposition to invert

*F*. The uncertainty in the maximum-likelihood recovered values is given by the covariance matrix

*effective*overlap functions for the tensor, vector, and scalar modes for a multibaseline network of detectors.

#### 8.4.4 Effective overlap functions for multiple baselines

*effective*overlap functions for the tensor, vector, and scalar modes, associated with a multibaseline detector network. As we did in Sect. 8.2.5, we will assume for simplicity that the noise power spectra for the detectors are equal to one another so that \(N_\alpha \equiv N\) can be factored out of the above expressions. We then define

### 8.5 Searches for non-GR polarizations using pulsar timing arrays

As discussed in Sect. 8.3.4 it is also possible to search for non-GR polarizations using a pulsar timing array. Although the general concepts are the same as those for ground-based interferometers, there are some important differences, as the vector and scalar longitudinal polarization modes require keeping the pulsar term in the response functions to avoid possible singularities. We shall see below that the sensitivity to the vector and scalar longitudinal modes increases dramatically when cross-correlating data from pairs of pulsars with small angular separations. For additional details, see Lee et al. (2008), Chamberlin and Siemens (2012) and Gair et al. (2015).

#### 8.5.1 Polarization basis response functions

*L*is its distance from Earth (see Sect. 5.2.1 with \(\hat{p}=-\hat{u}\)). Without loss of generality, we have assumed that the location of the measurement is at the origin of coordinates. Note that we have kept the

*pulsar term*(the second term in the square brackets) since, as we shall see below, it is needed to get finite expressions for the response and overlap functions for the vector and scalar longitudinal modes.

*B*and the tensor \(+\) mode have the same form for our particular choice of \(\{\hat{l},\hat{m}\}\). This is not a problem, however, as we can still distinguish these modes due to their different behavior under rotations. The difference between the breathing and tensor modes becomes more apparent in terms of the spherical harmonic basis response functions \(R^B_{(lm)}(f)\) and \(R^G_{(lm)}(f)\), which are given in (8.98).

^{24}The factor of \(\sin \theta \) in the numerator for \(R^X(f,\hat{n})\) “softens” the \((1-\cos \theta )^{-1}\) singularity to \((1-\cos \theta )^{-1/2}\), so that it becomes integrable when calculating the vector longitudinal overlap functions (Lee et al. 2008; Chamberlin and Siemens 2012; Gair et al. 2015). (We will discuss this in more detail in Sect. 8.5.3). By keeping the pulsar term we remove these singularities as can be seen by expanding the full expressions in (8.95) for \(\theta \ll 1\):

#### 8.5.2 Spherical harmonic basis response functions

*Thus, pulsar timing arrays are also insensitive to the curl component of the vector-longitudinal modes.*(iii) In the limit \(y\gg 1\), only the response to the scalar-longitudinal mode has frequency dependence (via

*y*). (iv) The response to the breathing mode has non-zero contributions only from \(l=0\) and \(l=1\). In terms of power (which is effectively the square of the response), this means that pulsar timing observations will be insensitive to anisotropies in power in the breathing mode beyond quadrupole (i.e., \(l=2\)).

#### 8.5.3 Overlap functions

*anisotropic*backgrounds will be briefly mentioned in Sect. 8.5.4. Details can be found in Gair et al. 2015). Making these assumptions, the quadratic expectation values of the Fourier coefficients \(h_A(f,\hat{n})\) take the form

*A*on the right-hand side of this expression.

*f*via the distances to the pulsars, \(2\pi fL_I/c\) and \(2\pi fL_J/c\). (See Appendix J of Gair et al. (2015) for an analytic expression for \(\Gamma ^{(V)}_{IJ}(f)\) in the limit \(\zeta _{IJ}\rightarrow 0\).)

Finally, for the scalar longitudinal overlap function \(\Gamma ^L_{IJ}(f)\), there is no known analytic expression for the integral in (8.105), except in the limit of codirectional (\(\zeta _{IJ}=0\)) and anti-directional (\(\zeta _{IJ}=\pi \)) pulsars (Lee et al. 2008; Chamberlin and Siemens 2012; Gair et al. 2015). The pulsar terms need to be included in the scalar-longitudinal response functions for all cases to obtain a finite result, which again depend on the frequency *f* via the distances to the pulsars. A semi-analytic expression for \(\Gamma ^L_{IJ}(f)\) is derived in Gair et al. (2015), which is valid in the \(2\pi fL/c \gg 1\) limit. The semi-analytic expression effectively replaces the double integral over directions on the sky \(\hat{n}=(\theta ,\phi )\) with just a single numerical integration over \(\theta \). See Gair et al. (2015) for additional details regarding that calculation.

#### 8.5.4 Component separation and anisotropic backgrounds

*anisotropic*stochastic backgrounds. The spherical harmonic components of the overlap functions

*analytically*for the tensor, vector, and breathing polarization modes for all values of

*l*and

*m*, while the components of the scalar longitudinal overlap function admit only semi-analytic expressions. (This is similar to what we described in the previous section in the context of an isotropic background). Plots of the first few spherical harmonics components, as a function of the angular separation \(\zeta _{IJ}\) between a pair of pulsars, are given in Figures 1, 5, 2, and 3 of Gair et al. (2015).

*lm*) indices here correspond to an expansion of the Fourier components of the metric perturbations in terms of tensor (spin 2), vector (spin 1), and scalar (spin 0) spherical harmonics:

*I*to the background. The expansion coefficients \(a_{(lm)}^P(f)\) give the contributions of the different polarization modes to the background, and \(R^P_{I(lm)}(f)\) are the response functions for those particular coefficients. For an angular resolution of order \(180^\circ /l_\mathrm{max}\), the total number of modes that are (in principle) accessible to a pulsar timing array with a sufficient number of pulsars is

*l*and \(y\equiv 2\pi fL/c\). If we take the \(y\gg 1\) limit of these equations, we recover the approximate expressions given in (8.98). But to separate the various components of the background, we need to use these more complicated expressions to break the angular-direction degeneracy.

Relative uncertainties for the tensor, breathing, scalar-longitudinal, and vector-longitudinal polarization modes searched for separately or in various combinations for \(l_\mathrm{max}=2\) and \(N_p=30\) pulsars

( | |||||||||
---|---|---|---|---|---|---|---|---|---|

(0, 0) | \((1,-1)\) | (1, 0) | (1, 1) | \((2,-2)\) | \((2,-1)\) | (2, 0) | (2, 1) | (2, 2) | |

Tensor | − | − | − | − | 0.44 | 0.38 | 0.32 | 0.38 | 0.44 |

Tensor | − | − | − | − | 0.49 | 0.39 | 0.37 | 0.39 | 0.49 |

Breathing | 0.16 | 0.53 | 0.46 | 0.53 | − | − | − | − | − |

Tensor | − | − | − | − | 16.2 | 10.5 | 11.4 | 10.5 | 16.2 |

Breathing | 4.36 | 16.1 | 14.1 | 16.1 | − | − | − | − | − |

Longitudinal | 0.71 | 0.96 | 0.84 | 0.96 | 1.21 | 0.78 | 0.86 | 0.78 | 1.21 |

Tensor | − | − | − | − | 1.4e5 | 5.4e4 | 8.0e4 | 5.4e4 | 1.4e5 |

Breathing | 18.4 | 9.4e4 | 6.2e4 | 9.4e4 | − | − | − | − | − |

Longitudinal | 3.08 | 11.5 | 8.68 | 11.5 | 20.9 | 7.51 | 11.9 | 7.52 | 20.9 |

Vector | − | 6.6e4 | 4.4e4 | 6.6e4 | 7.0e4 | 2.7e4 | 4.0e4 | 2.7e4 | 7.0e4 |

The entries in the table reflect our expectations for recovering the different modes of the background. Namely, there is little change in our ability to recover the tensor modes when the breathing modes are also included in the analysis. This is because the tensor modes are non-zero only for \(l\ge 2\), while the response to the breathing modes is non-zero only for \(l=0,1\). Adding the scalar-longitudinal modes to the analysis worsens the recovery of the tensor and breathing modes by about an order of magnitude, as the scalar-longitudinal modes can also have non-zero values for all values of *l*. (There are simply more parameters to recover). But one is still able to break the degeneracy as the response to the scalar-longitudinal modes depends *strongly* on the distances to the pulsars. The uncertainity in the recovery of the scalar-longitudinal modes is about an order of magnitude less than that for the tensor and breathing modes, since the analysis assumes equal intrinsic amplitudes for all the modes, while the correlated response to the scalar-longitudinal modes is much larger for small angular separations between the pulsars (Sect. 8.5.3; Fig. 68). Finally, adding the vector-longitudinal modes to the analysis weakens the recovery of the scalar-longitudinal modes by about an order of magnitude, again because more parameters need to be recovered. However, it *severely worsens* the recovery of all the other modes, because of the degeneracy in the response on the angular direction to the pulsars. There is some dependence on frequency for the vector-longitudinal response, as indicated in (8.114), but it is much weaker than the frequency dependence of the scalar-longitudinal modes. So the degeneracy is not broken nearly as strongly for these modes. See Gair et al. (2015) for more details.

### 8.6 Other searches

It is also possible to use the general cross-correlation techniques described in Sect. 4 to search for signals that don’t really constitute a stochastic gravitational-wave background. Using a stochastic-based cross-correlation method to search for such signals is not optimal, but it still gives valid results for detection statistics or estimators of signal parameters, with error bars that properly reflect the uncertainty in these quantities. It is just that these error bars are *larger* than those for an optimal (minimum variance) search, which is better “tuned” for the signal. Below we briefly describe how the general cross-correlation method can be used to search for (i) long-duration unmodelled transients and (ii) persistent (or continuous) gravitational waves from targeted sources.

#### 8.6.1 Searches for long-duration unmodelled transients

The Stochastic Transient Analysis Multi-detector Pipeline (Thrane et al. 2011) (STAMP for short) is a cross-correlation search for unmodelled long-duration transient signals (“bursts”) that last on order a few seconds to several hours or longer. The duration of these transients are long compared to the typical merger signal from inspiralling binaries (tens of milliseconds to a few seconds), but short compared to the persistent quasi-monochromatic signals that one expects from e.g., rotating (non-axisymmetric) neutron stars. STAMP was developed in the context of ground-based interferometers, but the general method, which we briefly describe below, is also valid for other types of gravitational-wave detectors.

*inverse*of the integrand of the overlap function \(\gamma _{IJ}(t;f,\hat{n})\) for a particular direction \(\hat{n}\) on the sky:

*t*;

*f*) for a point source in direction \(\hat{n}\), which follows from (7.7). The data \(\tilde{s}_{IJ}(t;f,\hat{n})\) for a single direction \(\hat{n}\) define a

*time-frequency map*. For a typical analysis using the LIGO Hanford and LIGO Livingston interferometer, a single map has a frequency range from about 50 to \({\sim }1000~\mathrm{Hz}\), and a time duration of a couple hundred seconds (or whatever the expected duration of the transient might be). A strong burst signal shows up as

*cluster*or

*track*of bright pixels in the time-frequency map, which stands out above the noise. The data analysis problem thus becomes a

*pattern recognition*problem.

The procedure for deciding whether or not a signal is present in the data can be broken down into three steps: (i) determine if a statistically significant clump or track of bright pixels is present in a time-frequency map, which requires using some form of pattern-recognition or clustering algorithm (see Thrane et al. 2011 and relevant references cited therein); (ii) calculate the value of the detection statistic \(\Lambda \), obtained from a weighted sum of the power in the pixels for each cluster determined by the previous step; (iii) compare the observed value of the detection statistic to a threshold value \(\Lambda _*\), which depends on the desired false alarm rate. This threshold is typically calculated by time-shifting the data to empirically determine the sampling distribution of \(\Lambda \) in the absence of a signal. If \(\Lambda _\mathrm{obs}>\Lambda _*\), then reject the null hypothesis and claim detection as discussed in 3.2.1. (Actually, in practice, this last step is a bit more complicated, as one typically does follow-up investigations using auxiliary instrumental and environmental channels, and data quality indicators. This provides additional confidence that the gravitational-wave candidate is not some spurious instrumental or environmental artefact.)

*accretion disk instability waveform*, based on a model by van Putten (2001); van Putten and Levinson (2003); van Putten (2008). The signal is a (inverse) “chirp” in gravitational radiation having an exponentially decaying frequency. (The magnitude of the signal increases with time as the frequency

*decreases*.) The injected signal is strong enough to be seen by eye in the raw time-frequency map (left panel). After applying a clustering algorithm, the fluctuations in the detector noise have been noticeably reduced (right panel).

Readers should see Thrane et al. (2011) for many more details regarding STAMP, and Abbott et al. (2016c) and Aasi et al. (2013) for results from analyses of LIGO data taken during their 5th and 6th science runs—the first paper describes an all-sky search for long-duration gravitational-wave transients; the second, a triggered-search for long-duration gravitational-transients coincident with long duration gamma-ray bursts.

#### 8.6.2 Searches for targeted-sources of continuous gravitational waves

The gravitational-wave radiometer method (Sect. 7.3.6) can also be used to look for gravitational waves from persistent (continuous) sources at known locations on the sky, e.g., the galactic center, the location of SN 1987A, or from low-mass X-ray binaries like Sco X-1 (Abadie et al. 2011; Messenger et al. 2015; Abbott et al. 2016a). For example, Sco X-1 is expected to emit gravitational waves from the (suspected) rotating neutron star at its core, having non-axisymmetric distortions produced by the accretion of matter from the low-mass companion. The parameters of this system that determine the phase evolution of the gravitational radiation are not well-constrained: (i) Since the neutron star at the core has not been observed to emit pulsations in the radio or any electromagnetic band, the orbital parameters of the binary are estimated instead from optical observations of the low-mass companion (Steeghs and Casares 2002; Galloway et al. 2014). These observations do not constrain the orbital parameters as tightly as being able to directly monitor the spin frequency of the neutron star. (ii) The intrinsic spin evolution of the neutron star also has large uncertainties due to the high rate of accretion from the low-mass companion star. Both of these features translate into a *large* parameter space volume over which to search, making fully-coherent matched-filter searches for the gravitational-wave signal computationally challenging (Messenger et al. 2015).

*narrow-band, targeted*radiometer search, cross-correlating data from a pair of detectors with a filter function proportional to the integrand \(\gamma _{IJ}(t;f,\hat{n}_0)\) of the overlap function evaluated at the direction \(\hat{n}_0\) to the source on the sky:

*robust*in the sense that it makes minimal assumptions about the source. The detection efficiency of the search could be improved if one had additional information about the signal (e.g., if one knew that the radiation was circularly polarized), which could then be included in the stochastic signal model.

## 9 Real-world complications

Experience with real-world data, however, soon convinces one that both stationarity and Gaussianity are fairy tales invented for the amusement of undergraduates.

D.J. Thompson(Thomson 1994)

The analyses described in the previous sections assumed that the instrument noise is stationary, Gaussian distributed, and uncorrelated between detectors. The analyses also implicitly assumed that the data were regularly sampled and devoid of gaps, facilitating an easy transition between the recorded time series and the frequency domain where many of the analyses are performed. In practice, all of these assumptions are violated to varying degrees, and the analyses of real data require additional care. Analyses that assume stationary, Gaussian noise can produced biased results when applied to more complicated real-world data sets.

### 9.1 Observatory-specific challenges

To begin the discussion, we highlight some of the challenges associated with real-world data, which are specific to the different observational domains—e.g., ground-based detectors, space-based detectors, and pulsar timing. Then, in the following subsections, we discuss the complications in more detail, and suggest ways to deal with or mitigate these problems.

#### 9.1.1 Ground-based interferometers

Analysis of data from the first and second generation ground-based interferometers have shown that the data are neither perfectly stationary nor Gaussian. The non-stationarity can be broadly categorized as having two components: slow, adiabatic drifts in the noise spectrum with time; and short-duration noise transients, referred to as *glitches* (Blackburn et al. 2008), which have compact support in time-frequency. These glitches are also the dominant cause of non-Gaussianity in the noise distributions, giving rise to long “tails” (large amplitude events with non-negligible probability), which extend past a core distribution that is well described as Gaussian. The data are evenly sampled by design, though there are often large gaps between data segments due to “loss of lock” (the interferometer being knocked out of data-taking mode due to an environmental disturbance or instrumental malfunction), scheduled maintenance, etc.

An analysis of LIGO-Virgo data that assumes the noise spectrum is constant over days or weeks would produce biased results. In practice, the data is analyzed using \({\sim } 1\) min-long segments. Glitches, on the other hand, do not pose a significant problem for stochastic searches as they are rarely coherent between detectors. Glitches are a more serious problem for searches that target short duration, deterministic signals.

#### 9.1.2 Pulsar timing arrays

Pulsar timing data are, in many ways, far more challenging to analyze (Haasteren and Levin 2010). The lack of dedicated telescope facilities, and the practical constraints associated with making the observations, result in data that are irregularly sampled. Moreover, the very long observation timelines (years to decades) and the mixture of facilities yield data sets that have been collected using a variety of receivers, data recorders, and pulse folding schemes. The heterogeneity of the observations causes the data to be non-stationary. In addition, the characteristic period of the gravitational waves searched for is of order the duration of the observations. Thus, Fourier domain methods for pulsar timing analyses have, at best, limited formal utility.

An additional complication for pulsar timing analyses is that a complicated deterministic timing model that predicts the time of arrival of each pulse has to be subtracted from the data to produce the timing residuals used in the gravitational-wave analyses. The timing model includes a pulsar spindown model and a detailed pulse propagation model that accounts for the relative motion of the Earth and pulsar. Many of the pulsars are in binary systems, so the timing model has to include relativistic orbital motion, and propagation effects such as the Shapiro time delay. Since errors in the timing model are strongly correlated with the gravitational-wave signal, subtracting the timing model unfortunately removes part of the signal as well. Subtraction of the timing model also introduces non-stationarity into the data (Haasteren and Levin 2013), again making time-domain analyses the only possibility (van Haasteren et al. 2009).

#### 9.1.3 Space-based detectors

For future space detectors we can only guess at the nature of the noise. Results from the LISA Pathfinder mission provide some insight (Armano et al. 2016), but only for a subset of the detector components, and for somewhat different flight hardware. The data will be regularly sampled, but data gaps are expected due to re-pointing of the communication antennae and orbit adjustments. Possible sources of non-stationarity include variations in the solar wind, thermal variations, and tidal perturbations from the Earth and other solar system bodies. The plans for the first space interferometers envision a single array of 3 spacecraft with 6 laser links. From these links three noise-orthogonal signal channels can be synthesized, but these combinations are also signal orthogonal, and so cross-correlation cannot be used to detect a signal.

### 9.2 Non-stationary noise

Data from existing gravitational-wave detectors, including bars, interferometers, and pulsar timing, exhibit various degrees of non-stationarity. Here we give examples relevant to ground-based interferometers, but the situation is similar for the other detection techniques.

Non-stationary behavior can manifest itself in many forms, and there are no doubt many factors that contribute to the non-stationarity seen in interferometer data. Nonetheless, a simple two-part model does a good job of capturing the bulk of the non-stationary features. The two-part model consists of a slowly-varying noise spectral density \(S_n(t{;}f)\), and localized noise transients or “glitches”. The slow drift in the spectrum can be modeled as a locally-stationary noise process (Dahlhaus 2011), which has the nice feature that for small enough time segments, the data in each segment can be treated as stationary. The glitch contribution to the non-stationarity poses more of a challenge, as the non-stationarity persists even for short data segments.

#### 9.2.1 Local stationarity

*q*(

*t*) and \(\epsilon (t)\) are slowly-varying functions of time. The local power spectrum

*S*(

*t*;

*f*) for this process has the form

*T*with

*N*samples, \(f_{\mathrm{N}}=N/(2T)=1/(2 \Delta t)\). Figure 70 shows the average and local spectra for \(T=1024\) seconds of data sampled at 1024 Hz with

*k*is defined by

*N*limit,

*c*(

*k*) for \(k > 0\) is Gaussian distributed with zero mean and variance \(\sigma ^2=1/N\) (Dwivedi and Subba Rao 2009). It is convenient to use the scaled auto-covariance \(C(k) \equiv \sqrt{N} c(k)\), which has unit variance for stationary, Gaussian noise. Figure 71 compares

*C*(

*k*) computed for the locally-stationary \(\text {AR}(1)\) model shown in Fig. 70, and a stationary \(\text {AR}(1)\) model with \(q(t)=q_0\) and \(\epsilon (t) = \epsilon _0\). The locally-stationary model shows clear departures from stationarity when the auto-correlation is computed using the full data set (as evidenced by the large autocorrelations for small lags), while the data in each of the 32 sub-segments shows no signs of non-stationarity.

*C*(

*k*) as an indicator of non-stationarity is that any window that is applied to the time-domain data to lessen spectral leakage in the Fourier transform necessarily makes the data non-stationary. Choosing a window function (Appendix D.2) that is unity across most of the samples, such as a Tukey window (D.14), lessens the taper-induced non-stationarity, but does not eliminate the effect. The solution is to apply a correction to the autocorrelation that accounts for the window. Figure 72 shows the impact that a Tukey window has on the mean and variance of the Fourier autocorrelation

*C*(

*k*). In this simulation \(N= 32768\) samples were used with a Tukey window that is constant across the central 90% of the samples. By subtracting the mean and scaling by the square-root of the variance caused by the Tukey window, the non-stationarity caused by the filter can be corrected for.

#### 9.2.2 Glitches

### 9.3 Non-Gaussian noise

Gaussian noise processes are ubiquitous in nature, and provide a remarkably good model for the data seen in gravitational-wave detectors. Properly whitened gravitational-wave data typically have a Gaussian core that accounts for the bulk of the samples, along with a small number of outliers in the tails of the distribution. Even these small departures can severely impact analyses that assume perfectly Gaussian distributions.

*least-squares*(maximum-likelihood) data analysis technique in an effort to determine the orbit of the newly discovered dwarf planet Ceres. Gauss showed that if measurement errors are: (i) more likely small than large, (ii) symmetric, and (iii) have zero mean, then they follow a normal distribution (first described by de Moivre in 1733). Gauss’ proof relied on the law of large numbers: he assumed that under repeated measurements the most-likely value of a quantity is given by the mean of the measured values. The assumptions used in Gauss’ derivation were placed on a firmer footing by Laplace, who derived the central-limit theorem, which states that the arithmetic mean of a sufficiently large number of independent random deviates will be approximately normally distributed, regardless of the underlying distributions the deviates are drawn from, so long as the distributions have finite first and second moments. The central limit theorem is often invoked to explain the ubiquity of Gaussian measurement errors. While the classic central limit theorem applies to noise contributions that are fundamentally stochastic (such as those with a quantum origin), a variant of the central limit theorem also applies to the sum of a large number of

*deterministic*effects, so long as the deterministic processes obey certain conditions (Imkeller and von Storch 2001).

Since gravitational-wave data typically have highly-colored spectra, one cannot simply compare the distribution of samples in time or frequency to a Gaussian distribution. The data first have to be whitened. This can be done by dividing the Fourier coefficients by the square-root of an estimate of the power spectra, and inverse Fourier transforming the result to arrive at a whitened time series. Figure 74 shows histograms of the whitened Fourier-domain and time-domain samples for the simulated data shown in Fig. 73. By eye, the frequency-domain samples appear fairly Gaussian, while the time-domain samples show clear departures from Gaussianity. Applying the Anderson–Darling test (Anderson and Darling 1954) to both sets of samples indicates that the Gaussian hypothesis is rejected in both cases, with a *p*-value of \(p=2.6\times 10^{-5}\) for the Fourier-domain samples and \(p<10^{-20}\) for the time-domain samples. Applying the same analysis to the locally-stationary \(\text {AR}(1)\) model generated using 32 seconds of data (i.e., setting \(T=32\) s in the model for *q*(*t*) and \(\epsilon (t)\)), we find that the whitened Fourier coefficients generally pass the Anderson-Darling test, while the whitened time-domain samples do not. Overall, glitches cause much larger departures from Gaussianity than adiabatic variation in the noise levels.

To-date, there have been no detailed studies of the effects of non-stationary and non-Gaussian noise on stochastic background analyses beyond the theoretical investigations in Allen et al. (2002), Allen et al. (2003) and Himemoto et al. (2007). However, a variety of checks have been applied to the LIGO-Virgo analyses using time-shifted data and hardware and software signal injections, and the results were found to be consistent with the performance expected for stationary, Gaussian noise (Abbott et al. 2005, 2007). In particular, the distribution of the residuals of the cross-correlation detection statistic, formed by subtracting the mean and scaling by the square root of the variance, have been shown to be Gaussian distributed (Abbott et al. 2007).

### 9.4 Gaps and irregular sampling

Data gaps and irregular sampling do not significantly impact the analyses of interferometer data, but pose a major challenge to pulsar timing analyses.

#### 9.4.1 Interferometer data

Interferometer data are regularly sampled, and gaps in the data pose no great challenge since the non-stationarity already demands that the analysis be performed on short segments of coincident data. The main difficulty working with short segments of data is accounting for the filters that need to be applied to suppress spectral leakage (Abbott et al. 2005; Lazzarini and Romano 2004).

#### 9.4.2 Pulsar timing data

The collection of pulsar timing data is constrained by telescope, funding, and personnel availability. A large number of pulsars are now observed fairly regularly, with observations occurring every 2–3 weeks. Older data sets are less regularly sampled, and often have gaps of months or even years (Arzoumanian et al. 2015). Moreover, the sensitivity of the instruments varies significantly over time, making the data highly non-stationary, thus obviating the benefit of performing the analyses in the frequency domain. For these reasons, modern pulsar timing analyses are conducted directly in the time domain (van Haasteren et al. 2009).

*i*and

*j*, and (ii) a stationary red noise component \(S_{ij}\) that depends on the lag \(\vert i - j\vert \) (van Haasteren et al. 2009). These contribute to the time-domain noise correlation matrix \(C_n\), which appears in the Gaussian likelihood (3.51):

### 9.5 Advanced noise modeling

The traditional approach to noise modeling has been to assume a simple model, such as the noise being stationary and Gaussian, and then measure the consequences this has on the analyses using Monte Carlo studies of time-shifted data and simulated signals. An alternative approach is to develop more flexible noise models that can account for various types of non-stationarity and non-Gaussianity.

*BayesWave/BayesLine*algorithm, which uses a two-part noise model composed of a stationary, Gaussian component

*S*(

*f*), and short duration glitches,

*g*(

*t*), modeled as Gaussian-enveloped sinusoids (Cornish and Littenberg 2015; Littenberg and Cornish 2015). The spectral model

*S*(

*f*) is based on a cubic-spline with a variable number of control points to model the smoothly-varying part of the spectrum, and a collection of truncated Lorentzians to model sharp line features. The optimal number and placement of the control points and Lorentizians is determined from the data using a trans-dimensional Markov Chain Monte Carlo technique. The same technique is used to determine the number of sine-Gaussian glitches and their parameters (central time and frequency, duration, etc.). This approach has been applied to both LIGO data (Cornish and Littenberg 2015; Littenberg and Cornish 2015) and pulsar timing data (Ellis and Cornish 2016). Figure 75 demonstrates the application of the

*BayesWave*and

*BayesLine*algorithms to data from the LIGO Hanford detector during the S6 science run of the initial LIGO detectors. Removing the glitches has a significant impact on the inferred power spectra. Figure 76 displays histograms of the whitened Fourier coefficients for the data shown in Fig. 75 with and without glitch removal.

Additional models for non-stationary and non-Gaussian noise have been considered by several authors. The detection of deterministic and stochastic signals was considered in Allen et al. (2002), Allen et al. (2003) and Himemoto et al. (2007) for a variety of non-Gaussian noise models, including exponential and two-component Gaussian models. The two-component Gaussian model combined with a non-stationary glitch model was studied in Littenberg and Cornish (2010). Student’s *t*-distribution was considered in Rover (2011). A non-stationary and non-Gaussian noise model was derived in Principe and Pinto (2008) based on a Poisson distribution of sine-Gaussian glitches.

### 9.6 Correlated noise

*fundamental*limit to the detection of stochastic signals.

If the spectral shape of either, or preferably both, the signal and the correlated noise are known, then it is possible to separate the contributions using techniques similar to those that are used to separate the primordial cosmic-microwave-background signal from foreground contamination (Bennett et al. 2003). When the cause of the correlated noise is not fully understood, or when searching for signals with arbitrary spectral shapes, spectrum-based component separation will not be possible.

Several sources of correlated noise have been hypothesized, and in some cases observed, for both interferometer and pulsar timing analyses. Some of the correlations are due to the electronics (Abbott et al. 2005), such as correlations between harmonics of the 60 Hz AC power lines between the LIGO Hanford and LIGO Livingston detectors, and correlations at multiples of 16 Hz from the data sampling referenced to clocks on the Global Positioning System satellites. These narrow-band correlations are easily removed using notch filters. Correlations in the global time standard can also impact pulsar timing observations, as can errors in the ephemeris used in the timing model.

#### 9.6.1 Schumann resonances

Perhaps the greatest challenge comes from correlated noise sources of unknown origin. Such noise sources may be well below the auto-correlated noise level in each detector, and thus very hard to detect outside of the cross-correlation analysis. One way of separating these noise sources from a stochastic signal is to build a large number of interferometers at many locations around the world. Each pair of detectors will then have a unique overlap function for gravitational-wave signals that will differ from the spatial correlation pattern of the noise (unless we are incredibly unlucky!). In principle, the difference in the frequency-dependent spatial correlation patterns of the signal and the noise will allow the two components to be separated.

### 9.7 What can one do with a single detector (e.g., LISA)?

The discovery of the cosmic microwave background was described in a paper with the unassuming title “A Measurement of Excess Antenna Temperature at 4080 Mc/s” (Penzias and Wilson 1965). Penzias and Wilson used a single microwave horn, and announced the result after convincing themselves that no instrumental noise sources, including pigeon droppings, could be responsible for the excess noise seen in the data. In principle, the same approach could be used to detect a stochastic gravitational-wave signal using a single instrument.

Single-detector detection techniques will be put to the test when the first space-based gravitational-wave interferometer is launched, since (unless the funding landscape changes dramatically) the instrument will be a single array of 3 spacecraft. Assuming that pairs of laser links operate between each pair of spacecraft, it will be possible to synthesize multiple interferometry signals from the phase readouts (Estabrook et al. 2000). One particular combination of the phase readouts, called the *T* channel, corresponds to a Sagnac interferometer, and is relatively insensitive to low-frequency gravitational waves, forming an approximate null channel (see Sect. 4.7 for a discussion of null channels). Other combinations, such as the so-called *A* and *E* channels (Prince et al. 2002), are much more sensitive to gravitational-wave signals. Using the Sagnac *T* to measure the instrument noise, the relative power levels in the \(\{A,E,T\}\) channels can be used to separate a stochastic signal from instrument noise (Tinto et al. 2001).

LISA-type observatories operate as synthetic interferometers by forming gravitational-wave observables in post-processing using different combinations of the phasemeter readouts from each inter-spacecraft laser link. The combinations synthesize effective equal-path-length interferometers to cancel the otherwise overwhelming laser frequency noise. These combinations have to account for the unequal and time-varying distances between the spacecraft.

*j*that receives light from spacecraft

*i*. Permuting the spacecraft labels \(\{1,2,3\}\) yields equivalent expressions for the Michelson observables

*Y*and

*Z*, as shown in panel (a) of Fig. 78. The phasemeter readouts \(\Phi _{ij}(t)\) are impacted by acceleration noise \(S^a_{ij}\) and position noise \(S^p_{ij}\). When the noise levels in each spacecraft are equal, there exist noise-orthogonal combinations (Prince et al. 2002; Adams and Cornish 2010):

*A*,

*E*are rotated by 45 degrees with respect to each other, and provide instantaneous measurements of the \(+\) and \(\times \) polarization states. The Sagnac-like

*T*channel is relatively insensitive to gravitational waves for frequencies below the transfer frequency \(f_*\equiv c/(2\pi L)\). The

*T*channel can be used to infer the instrument noise level, so that any excess in the

*A*,

*E*channels can then be confidently attributed to gravitational waves (Tinto et al. 2001). For frequencies \(f \ll f_*\) the \(\{A,E,T\}\) channels have uncorrelated responses to unpolarized, isotropic stochastic gravitational-wave signals.

*T*channel as a noise reference as the noise combinations in

*T*differ from those in

*A*,

*E*. For example, the acceleration noise appears in

*T*as (Adams and Cornish 2010):

*A*and

*E*as

*T*provides a measurement of the average noise, which can then be used as an estimator for the noise in

*A*,

*E*. An analysis that assumes common noise levels will overstate the sensitivity to a signal. A more conservative approach is allow for unequal noise levels and to infer the individual contributions from the data. For example, if one link is particularly noisy, it will dominate the noise in

*T*, and enter unequally in

*A*and

*E*, making it possible to identify the bad link and account for it in the analysis.

*T*channel (Adams and Cornish 2010). The sensitivity decreases for less informative priors. In the limit that the priors allow for arbitrarily complicated functional forms for the noise and signal spectra—forms so

*contrived*that they can compensate for the differences in the transfer functions—it becomes impossible to separate signal from noise. In practice, a combination of pre-flight and on-board testing, combined with physical modeling, will hopefully constrain the noise model sufficiently to inform the analysis and allow for component separation.

An additional complication for space interferometers operating in the mHz frequency range are the millions of astrophysical signals that can drown-out a cosmologically-generated stochastic background. While the brightest signals from massive black hole mergers, stellar captures, and galactic binaries can be identified and subtracted, a large number of weaker overlapping signals will remain, creating a residual *confusion noise*. The largest source of confusion noise is expected to come from millions of compact white-dwarf binaries in our galaxy. The annual modulation of the white-dwarf confusion noise due to the motion of the LISA spacecraft (see Fig. 40) will allow for this component to be separated from an isotropic stochastic background, though at the cost of reduced sensitivity to the background (Adams and Cornish 2014).

## 10 Prospects for detection

It’s tough to make predictions, especially about the future.

Yogi Berra

The detection of the binary black hole merger signals GW150914 and GW151226 give us confidence that stochastic gravitational waves will be detected in the not-to-distant future. Not only do they show that our basic measurement principles are sound, they also point to the existence of a much larger population of weaker signals from more distant sources that will combine to form a stochastic background that may be detected by 2020 (Abbott et al. 2016h). Indeed, a confusion background from the superposition of weaker signals eventually becomes the limiting noise source for detecting individual systems (Barack and Cutler 2004). As a general rule of thumb, individual bright systems will be detected before the background for transient signals (those that are in-band for a fraction of the observation time), while the reverse is true for long-lived signals, such as the slowly evolving supermassive black-hole binaries targeted by pulsar timing arrays (Rosado et al. 2015). The prospects for detecting more exotic stochastic signals, such as those from phase transitions in the early Universe or inflation, are much less certain, but are worth pursing for their high scientific value. In this section we begin with a brief review of detection sensitivities curves across the gravitational-wave spectrum, followed by a review of the current limits and prospects for detection in each observational window.

### 10.1 Detection sensitivity curves

Detector sensitivity curves provide a useful visual indicator of the sensitivity of an instrument to potential gravitational-wave sources. A good pedagogical description of the various types of sensitivity curve in common use can be found in Moore et al. (2015a). Here we provide a more condensed summary.

*fT*describes the boost that we get by coherently integrating the signal over many cycles. For deterministic signals the amplitude signal-to-noise ratio grows as \(T^{1/2}\). Since sensitivity curves are usually plotted in terms of the amplitude spectral density \(h_\mathrm{eff}(f) = \sqrt{S_n(f)}\), it is natural to plot signals in terms of the square-root of the numerator of (10.4). Representative LISA sources are represented in this way in panel (d) of Fig. 79. An alternative choice is to plot both of these quantities multiplied by the square-root of the frequency, which yield the characteristic strain for the signal, \(h_c(f)\), as well as for the noise, \(h_n(f)\). Examples of characteristic strain sensitivity curves are shown in Fig. 80.

*power*signal-to-noise ratio. Similar to the amplitude signal-to-noise ratio for deterministic signals, the power signal-to-noise ratio for stochastic signals grows as \(T^{1/2}\). (This assumes we are in the weak-signal limit, and that the effective low-frequency cutoff does not change with time. See Siemens et al. (2013) for a more complicated scaling that occurs for pulsar timing arrays). Following the same logic as was applied to deterministic signals, it would be natural to plot \((2f T)^{1/4} \sqrt{S_h(f)}\) against sensitivity curves defined by \(\sqrt{S_\mathrm{net}(f)}\). Unfortunately, such conventions are not uniformly applied, and the factor of \((2f T)^{1/4}\) is often applied to \(\sqrt{S_\mathrm{net}(f)}\) instead:

### 10.2 Current observational results

#### 10.2.1 CMB isotropy

The cosmic microwave background (CMB) provides a snapshot of the Universe \({\approx }400,000\) years after the big bang. During this epoch, the dense, hot plasma that filled the early Universe dilutes and cools to the point where electrons and ions combine to form a neutral gas that is transparent to photons. Maps of the CMB contain a record of the conditions when the CMB photons were last scattered.

Gravitational waves propagating through the early Universe, referred to as tensor perturbations in the CMB literature, can leave an imprint in the temperature and polarization pattern when CMB photons scatter off the tidally-squeezed plasma. The challenge is to separate out the contributions from primordial scalar, vector, and tensor perturbations, and to separate these primordial contributions from subsequent scattering by dust grains and hot gas.

Observations by the *COBE, WMAP* and *Planck* missions, along with a host of ground-based and ballon-borne experiments, have provided strong evidence in support for the inflation paradigm, where the Universe undergoes a short period of extremely rapid expansion driven by some, as yet unknown, *inflaton* field. To keep the discussion brief, we focus our review on the standard single-field “slow-roll” inflation model, and direct the reader to more extensive CMB-focused reviews, e.g., Kamionkowski and Kovetz (2015), that cover more exotic models.

*a*(

*t*). The Einstein equations for a FLRW Universe containing an inflaton field \(\phi \) with potential \(V(\phi )\) are given by

^{25}

*V*, and the slow-roll parameters \(\epsilon _V\) and \(\eta _V\):

*V*, and the two leading terms in the Taylor-series expansion of the inflaton potential, \(V_{,\phi }\) and \(V_{,\phi \phi }\). Additionally measuring \(n_t\) would provide a consistency check for the slow-roll model.

One challenge in measuring \(P_s(k)\) and \(P_t(k)\) is that the scalar and tensor perturbations both source temperature and polarization anisotropies in the CMB radiation. Another challenge is that foreground gas and dust can also contribute to the temperature and polarization anisotropies. The various components can be teased apart by observing a wide range of CMB energies across a wide range of angular scales.

The primordial contribution to the CMB follows a black-body spectrum, while the dominant foreground contribution from gas and dust have very different spectra. By observing at multiple CMB wavelengths the primordial and foreground contributions can be separated. Separating the scalar and tensor contributions to the primordial component of the temperature anisotropies can be achieved by making maps that cover a wide range of angular scales, while separating their contributions to the polarization anisotropies can be achieved by decomposing the signal into curl-free *E*-modes and divergence-free *B*-modes, and using measurements made on a wide range of angular scales. For a more in-depth description, see Chapter 27 of the Review of Particle Physics (Olive et al. 2014).

*TT*) power spectra using the best fit \(\Lambda \)CDM model from

*Planck*are shown in panel (a) of Fig. 83. By comparing the CMB anisotropy at very large scales (\(\ell \sim 2\)–10) and degree scales (\(\ell \sim 200\)), it is possible to constrain the

*tensor-to-scalar ratio*(Knox and Turner 1994):

*Planck*temperature map, combined with weak lensing data, provide a precise measurement for the amplitude and spectral index of the scalar perturbations:

*Planck*bound on

*r*is the most stringent possible using CMB temperature data (Knox and Turner 1994). (In fact, it beats the theoretical limit slightly since the analysis also used weak lensing and

*WMAP*polarization data). In order to improve on this bound, or to detect the tensor contribution, CMB polarization data must also be used.

*Planck*bound on

*r*can be mapped into constraints on the gravitational-wave energy density via (Turner et al. 1993; Lasky et al. 2016):

*Planck*bound from the

*B*-mode power spectrum, along with existing and projected bounds from pulsar timing and aLIGO are shown in Fig. 84, which is taken from Lasky et al. (2016). Also shown are curves for theoretical models with a large tensor-to-scalar ratio (\(r=0.11\)) and a range of spectral tilts \(n_t\).

Coherent motion in the primordial plasma can polarize the CMB photons through Thomson scattering. Scalar perturbations source curl-free *E*-mode polarization anisotropies, while the tensor perturbations source divergence-free *B*-mode polarization anisotropies, in addition to *E*-modes. In principle, by decomposing the polarization into *E* and *B* components, and using observations across a range of angular scales, it should be possible to separate the scalar and tensor contributions. In practice, the measurements are extremely challenging due to the weakness of the signals (nano-Kelvin or smaller polarization fluctuations as compared to micro-Kelvin temperature fluctuations) and foreground noise. The main noise contributions come from gravitational lensing, which converts a fraction of the much larger *E*-mode anisotropy into *B*-modes, and scattering by dust grains, which can convert unpolarized CMB radiation into *E* and *B* modes. Both of these potential noise sources have recently been detected (Hanson et al. 2013; Ade et al. 2015c). The detection of *B*-mode polarization on large angular scales by *BICEP2* was originally interpreted as having a primordial origin (Ade et al. 2014), but a joint analysis using *Planck* dust maps (Ade et al. 2015c) showed the signal to be consistent with foreground noise.

While detecting the primordial *B*-mode contribution is very challenging, the pay-off is very large, as measuring the amplitude of the tensor perturbations, \(A_t\), fixes the energy scale of inflation, and can be used to strongly constrain models of inflation.

#### 10.2.2 Pulsar timing

Pulsar timing observations have made tremendous progress in the past 10 years and are now producing limits that seriously constrain astrophysical models for supermassive black hole mergers. The current observations are most sensitive at \(f\sim 10^{-8}\ \mathrm {Hz}\), so we choose a reference frequency of \(f_\mathrm{ref} = 10^{-8}\ \mathrm {Hz}\), and quote the latest bounds on \(\Omega _\mathrm{gw}(f) = \Omega _\beta (f/f_\mathrm{ref})^\beta \) in terms of bounds on \(\Omega _\beta \) for a Hubble constant value of \(H_0 = 70~\mathrm{km}\,\mathrm{s}^{-1}\,\mathrm{Mpc}^{-1}\).

#### 10.2.3 Spacecraft Doppler tracking

Spacecraft Doppler tracking (Armstrong 2006) operates on the same principles as pulsar timing, with a precision on-board clock and radio telemetry replacing the regular lighthouse-like radio emission of a pulsar. The \({\sim }1\)–10 AU Earth-spacecraft separation places spacecraft Doppler tracking between pulsar timing and future LISA-like missions in terms of baseline and gravitational-wave frequency coverage. In principle, a fleet of spacecraft each equipped with accurate clocks and high-power radio transmitters could be used to perform the same type of cross-correlation analysis used in pulsar timing, but to-date the analyses have been limited to single spacecraft studies.

#### 10.2.4 Interferometer bounds

Data from the initial LIGO and Virgo observation runs, and more recently, from advanced LIGO’s first observing run (O1), have been used to place constraints on the fractional energy density of isotropic stochastic backgrounds across multiple frequency bands between \(20-1726\) Hz. The bounds are quoted in terms of \(\Omega _\mathrm{gw}(f) =\Omega _\beta (f/f_\mathrm{ref})^\beta \) for \(\beta =0\) (flat in energy density), \(\beta =3\) (flat in strain spectral density), and \(\beta =2/3\) (appropriate for a stochastic signal from a population of inspiralling binaries). The \(\beta =0\) bounds are quoted for the lower frequency bands, where the sensitivity is greatest for signals with this slope, while the \(\beta =3\) bounds are quoted for the higher frequency bands. The \(\beta =2/3\) bound is motivated by the detection of multiple binary black hole mergers during O1, which implies that stellar-remnant black holes may produce a detectable stochastic signal from the superposition of many individually undetected sources (Abbott et al. 2016h). The bounds assume a Hubble constant value of \(H_0 = 68~\mathrm{km}\,\mathrm{s}^{-1}\,\mathrm{Mpc}^{-1}\).

*Initial LIGO and Virgo data*

*Advanced LIGO’s first observing run O1*

#### 10.2.5 Bounds on anisotropic backgrounds

*G*is Newton’s gravitational constant, and \(\mathcal{P}_{\hat{n}_0}\) is the signal power of a single point source in direction \(\hat{n}_0\) (which is the radiometer signal model).

^{26}The reference frequency for all the maps is \(f_\mathrm{ref}=25~\mathrm{Hz}\), corresponding to the most sensitive part of the frequency band for a stochastic search at advanced LIGO design sensitivity. All the searches include frequencies \(20< f < 500~\mathrm{Hz}\), which more than cover the regions of 99% sensitivity for each spectral index.

The coalescence rate is expected to vary significantly with redshift *z*, so this simple calculation, which assumes a constant coalescence rate, provides only a rough estimate.

Actually, even if the gravity-gradient and seismic noise were zero, one couldn’t go below \({\sim }1~\mathrm {Hz}\) with the current generation of ground-based laser interferometers, since the suspended mirrors (i.e., the test masses) are no longer freely floating when you go below their resonant frequencies: \({\sim }1~\mathrm {Hz}\).

Signals may be separable even when overlapping in time and frequency if the detector has good sky resolution, or if the signals have additional complexities due to effects such as orbital evolution and precession.

The gravitational-wave propagation direction, which we will denote by \(\hat{k}\), is given by \(\hat{k}=-\hat{n}\).

In some treatments, the Bayesian interpretation is equated to philosophical schools such as Berkeley’s empiricist idealism, or subjectivism, which holds that things only exist to the extent that they are perceived, while the frequentist interpretation is equated to Platonic realism, or metaphysical objectivism, holding that things exist objectively and independently of observation. These equivalences are false. A physical object can have a definite, Platonic existence, and Bayesians can still assign probabilities to its attributes since our ability to measure is limited by imperfect equipment.

Since the model \(\bar{h}(t)\) will differ from the actual *h*(*t*), we use an overbar for the model to distinguish the two.

Our convention for Fourier transform is \(\tilde{h}(f)= \int _{-\infty }^\infty dt \>e^{-i2\pi f t} h(t)\).

A flat prior for \(S_h\) yields more conservative (i.e., larger) upper limits for \(S_h\) than a Jeffrey’s prior, since there is more prior weight at larger values of \(S_h\) for a flat prior than for a Jeffrey’s prior.

See Creighton et al. (2009) and Koop and Finn (2014) for an alternative derivation of the response of a detector to gravitational waves, which is done in terms of the curvature tensor and not the metric perturbations.

Some authors (Christensen 1990, 1992; Flanagan 1993; Allen and Romano 1999; Cornish and Larson 2001; Finn et al. 2009), including us in the past, have defined the response function \(R^{ab}(f,\hat{n})\)*without* the factor of \(e^{i 2\pi f\hat{n}\cdot {\vec {x}}/c}\). If one chooses coordinates so that the measurement is made at \({\vec {x}}={\vec {0}}\), then these two definitions agree. Just be aware of this possible difference when reading the literature. To distinguish the two definitions, we will use the symbol \(\bar{R}^{ab}(f,\hat{n})\) to denote the expression without the exponential term, i.e., \(R^{ab}(f,\hat{n}) = e^{i 2\pi f\hat{n}\cdot {\vec {x}}/c}\bar{R}^{ab}(f,\hat{n})\).

There is a factor of \((-1)^l\) difference between \(R^G_{(lm)}(f)\) in (5.23) and (92) in Gair et al. (2014). The difference is due to the change in expressing the response functions in terms of the direction to the gravitational-wave source, \(\hat{n}\), as opposed to the direction of gravitational-wave propagation, \(\hat{k}=-\hat{n}\). Appendix H provides expressions relating the response functions calculated using these two different conventions.

Although Fig. 28 shows \(\hat{u}\) and \(\hat{v}\) making right angles with one another, the following calculation is valid for \(\hat{u}\) and \(\hat{v}\) separated by an *arbitrary* angle.

This is needed, for example, to calculate the overlap functions for an array of seismometers in the small-antenna limit (Coughlin and Harms 2014). For this case, the detector tensors are simply \(D_I^{ab} \equiv u_I^a u_I^b\), where \(\hat{u}_I\) is a unit vector pointing along the sensitive direction of the *I*th seismometer.

Actually, the bin size for a typical LIGO search for a stochastic background is *larger* than the \({\sim } 10^{-2}~\mathrm{Hz}\) Doppler shift due to the Earth’s orbital motion around the Sun.

In equatorial coordinates, the galactic center is located at \((\mathrm{ra},\mathrm{dec}) = (-6^\mathrm{h} 15^\mathrm{m}, -29^\circ )\).

For correlated measurements, \(Q_i = \sum _j(\bar{C}^{-1})_{ij}/h_i\) where \(\bar{C}^{-1}\) is the inverse of the re-scaled covariance matrix \(\bar{C}_{ij}\equiv C_{ij}/(h_i h_j)\).

The last equality in (6.23) follows from (5.38) with the Dirac delta function \(\delta (f-f')\) replaced by its finite-time version \(\delta _T(f-f') = T\mathrm {sinc}[\pi (f-f')T]\), which equals *T* when \(f=f'\).

The functions here are a factor of 1 / 2 smaller than those in Fig. 8 in Gair et al. (2014), due to different definitions of \(\gamma (t; f, \hat{n})\). Compare (115) in that paper to (7.8) and (7.10) above.

The multiplications inside the exponential are *matrix* multiplications—either summations over sky directions \(\hat{n}\) or summations over discrete times and frequencies, *t* and *f*.

At times it will be convenient to think of *M* as an \(N_f\times N_f\) block diagonal matrix with \(N_b\times 2\) blocks, one for each frequency. At other times, it will be convenient to think of *M* as an \(N_b\times 2\) block matrix with diagonal \(N_f\times N_f\) blocks. The calculations we need to do usually determine which representation is most appropriate. (Similar statements can be made for the vectors \(\hat{C}\) and \(\mathcal{S}\)).

This corresponds to the direction to the pulsar and the direction to the source of the gravitational wave being the same. For this case, the radio pulse from the pulsar and the gravitational wave travel in phase with one another from the pulsar to Earth. It is as if the radio pulse “surfs” the gravitational wave (Chamberlin and Siemens 2012).

For our discussion of inflation, we will work in *particle physics units* where both \(c=1\) and \(\hbar =1\). In place of using Newton’s gravitational constant *G*, we will use the *reduced* Planck mass \(M_\mathrm{Pl} \equiv (\hbar c/8\pi G)^{1/2} = 2.435\times 10^{18}~\mathrm{GeV}/c^2\). In these units \(M_\mathrm{Pl}^{2} = 1/8\pi G\), which simplifies several of the formulae. If you want to reinstate all of the relevant factors of \(\hbar \) and *c*, note that the inflaton field \(\phi \) has dimensions of energy and the inflaton potential \(V(\phi )\) has dimensions of energy density.

One should think of a radiometer upper-limit map as a convenient way of representing upper limits for a *collection* of individual point-source signal models, one for each point on the sky. As described in Sect. 7.3.6, the radiometer analysis ignores correlations between neighboring pixels on the sky, completely side-stepping the deconvolution problem associated with a non-trivial point spread function for the search. In other words, each pixel of a radiometer map corresponds to a separate analysis.

## Acknowledgements

JDR acknowledges support from National Science Foundation Awards PHY-1205585, CREST HRD-1242090, PHY-1505861. NJC acknowledges support from National Science Foundation Awards PHY-1306702 and PHY-1607343, and NASA award NNX16AB98G. JDR and NJC acknowledge support from the National Science Foundation NANOGrav Physics Frontier Center, NSF PFC-1430284. We also thank members of the LIGO-Virgo stochastic working group and members of NANOGrav for countless discussions related to all things stochastic. Special thanks go out to Bruce Allen, Matt Benacquista, Nelson Christensen, Gwynne Crowder, Yuri Levin, Tyson Littenberg, Chris Messenger, Soumya Mohanty, Tanner Prestegard, Eric Thrane, and Michele Vallisneri, who either commented on parts of the text or provided figures for us to use. Special thanks also go out to an anonymous referee for many comments and useful suggestions for improving parts of the text. This research made use of Python and its standard libraries: numpy and matplotlib. We also made use of MEALPix (a Matlab implementation of HEALPix, Górski et al., 2005), developed by the GWAstro Research Group and available from http://gwastro.psu.edu. Finally, we thank the editors of *Living Reviews in Relativity* (especially Bala Iyer and Frank Schulz) for their incredible patience while this article was being written. This document has been assigned LIGO Document Control Center number LIGO-P1600242.