Detection methods for stochastic gravitationalwave 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 groundbased and spacebased laser interferometers, spacecraft Doppler tracking, and pulsar timing arrays; and we allow for anisotropy, nonGaussianity, and nonstandard 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 selfcontained and comprehensive as possible, targeting graduate students and new researchers looking to enter this field.
Keywords
Gravitational waves Data analysis Stochastic backgrounds1 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 gravitationalwave 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 Gravitationalwave Observatory (LIGO) (Aasi et al. 2015). [LIGO consists of two 4 kmlong 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 postmerger 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 gravitationalwave window onto the universe.
GW150914 is just the first of many gravitationalwave 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 lightyears 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 signaltonoise ratio than was observed for GW150914 (which was 24). In addition, the Advanced Virgo detector (Acernese et al. 2015) (a 3 kmlong laser interferometer in Cascina, Italy) and KAGRA (Aso et al. 2013) (a 3 kmlong 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 templatebased searches for compact binary inspirals and searches for generic gravitationalwave transients in the two LIGO detectors (Abbott et al. 2016e, d). The network matchedfilter signaltonoise 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 socalled “goldplated” 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 signaltonoise ratios are too low.
The total rate of merger events from the population of stellarmass 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 LIGOlike 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 “popcornlike”, with the majority of the individual signals being too weak to individually detect. Since the arrival times of the merger signals are randomlydistributed, 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 gravitationalwave signal produced by a large number of weak, independent, and unresolved sources. The background doesn’t have to be popcornlike, 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 gravitationalwave 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 gravitationalwave 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 gravitationalwave 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 gravitationalwave “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?
Crosscorrelation methods can be used whenever one has multiple detectors that respond to the common gravitationalwave 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 gravitationalwave 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 crosscorrelation search for a “vanilla” (Gaussianstationary, unpolarized, isotropic) background, one can search for nonGaussian backgrounds, anisotropic backgrounds, circularlypolarized backgrounds, and backgrounds with polarization components predicted by alternative (nongeneralrelativity) theories of gravity. These searches are discussed in Sects. 7 and 8.
Overview of analysis methods for stochastic gravitationalwave backgrounds
Early analyses (before 2000)  More recent analyses 

Used frequentist statistics  Use both frequentist and Bayesian inference 
Used crosscorrelation methods  Use crosscorrelation methods and stochastic templates; use null channels or knowledge about instrumental noise when crosscorrelation is not available 
Assumed Gaussian noise  Have allowed nonGaussian noise 
Assumed stationary, Gaussian, unpolarized, and isotropic gravitationalwave backgrounds  Have allowed nonGaussian, polarized, and anisotropic gravitationalwave backgrounds 
Were done primarily in the context of groundbased detectors (e.g., resonant bars and LIGOlike interferometers) where the smallantenna (i.e., longwavelength) approximation was valid  Have been done in the context of spacebased detectors (e.g., spacecraft tracking, LISA) and pulsar timing arrays for which the smallantenna 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 spacebased interferometer like LISA is guaranteed to detect the gravitationalwave 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 gravitationalwave spectrum
1.2.1 Cosmic microwave background experiments
At the extreme lowfrequency end of the spectrum, corresponding to gravitationalwave 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 Bmode 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 Bmode 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 gravitationalwave background with any of the higherfrequency 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 gravitationalwave background is a “holy grail” of gravitationalwave astronomy.
1.2.2 Pulsar timing arrays
At frequencies between \({\sim }10^{9}~\mathrm{Hz}\) and \(10^{7}~\mathrm{Hz}\), corresponding to gravitationalwave 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 mostlikely gravitationalwave source for PTAs is a gravitationalwave 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 backends 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 gravitationalwave 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 Spacebased interferometers
At frequencies between \({\sim }10^{4}~\mathrm{Hz}\) and \(10^{1}~\mathrm{Hz}\), corresponding to gravitationalwave periods of order hours to minutes, proposed spacebased 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 stellarmass objects around supermassive black holes, and (iii) the stochastic confusion noise produced by compact whitedwarf 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 multiband gravitationalwave astronomy (Sesana 2016).
The basic spacebased interferometer configuration consists of three satellites (each housing two lasers, two telescopes, and two test masses) that fly in an equilateraltriangle 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 technologydemonstration 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 Earthbased detectors (Harms et al. 2013) and also secondgeneration spacebased interferometers—the BigBang Observer (BBO) (Phinney et al. 2004) and the DECIhertz interferometer Gravitationalwave 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 intermediatemass (\(10^3\)–\(10^4~M_\odot \)) binary black holes, galactic and extragalactic neutron star binaries, and a cosmologicallygenerated 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 gravitationalwave 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 nonGaussian, polarized or unpolarized, etc.). These searches are necessarily implemented differently for different detectors, since, for example, groundbased detectors like LIGO and Virgo operate in the smallantenna (or longwavelength) limit, while pulsar timing arrays operate in the shortwavelength 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 groundbased, spacebased, and pulsar timing observations. Moreover, reviews of gravitationalwave 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 gravitationalwave 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 timeofflight calculation underpins the detector response functions, and the choice of prior for the gravitationalwave template defines the search. A matchedfilter search for binary mergers and a crosscorrelation 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 gravitationalwave 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 gravitationalwave backgrounds, and show how these techniques arise naturally from the standard templatebased approach. We derive the frequentist crosscorrelation 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 singledetector response functions and the associated antenna patterns for groundbased and spacebased laser interferometers, spacecraft Doppler tracking, and pulsar timing measurements. (We do not discuss resonant bar detectors or CMBbased 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 Gaussianstationary, 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 gravitationalwave sky: (i) a frequentist gravitationalwave radiometer search, which is optimal for point sources, (ii) searches that decompose the gravitationalwave power on the sky in terms of spherical harmonics, and (iii) a phasecoherent search that can map both the amplitude and phase of a gravitationalwave background at each location on the sky. In Sect. 8, we discuss searches for: (i) nonGaussian backgrounds, (ii) circularlypolarized backgrounds, and (iii) backgrounds having nonstandard (i.e., nongeneralrelativity) polarization modes. We also briefly describe extensions of the crosscorrelation search method to look for nonstochasticbackgroundtype signals—in particular, longduration unmodelled transients and continuous (nearlymonochromatic) gravitationalwave signals from sources like Sco X1.
In Sect. 9, we discuss realworld complications introduced by irregular sampling, nonstationary and nonGaussian 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 gravitationalwaves. 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 nonstationarity and nonGaussianity. In Appendix D we describe the relationship between continuous functions of time and frequency and their discretelysampled counterparts. Appendices E, F, G are adapted from Gair et al. (2015), with details regarding spinweighted scalar, vector, and tensor spherical harmonics. Finally, Appendix H gives a “Rosetta stone” for translating back and forth between different response function conventions for gravitationalwave backgrounds.
2 Characterizing a stochastic gravitationalwave 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 gravitationalwave 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 planewave 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 gravitationalwave signal stochastic?
The standard “textbook” definition of a stochastic background of gravitational radiation is a random gravitationalwave 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 planewave 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 higherorder moments (Appendix B). For nonGaussian backgrounds, third and/or higherorder 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 gravitationalwave 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 gravitationalwave 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 signalplusnoise models, given the observed data. The signalplusnoise 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., nonoverlapping 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 loweramplitude background, leaving behind a residual nondeterministic 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 nondeterministic 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 nonGaussian backgrounds in more detail.
2.2 Planewave expansions
2.2.1 Polarization basis
2.2.2 Tensor spherical harmonic basis
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 2sphere 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
2.3 Statistical properties
For Gaussian backgrounds we need only consider quadratic expectation values, since all higherorder moments are either zero or can be written in terms of the quadratic moments (Appendix B). For nonGaussian 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 gravitationalwave signals from the early Universe is expected to be Gaussian (via the central limit theorem), as well as isotropicallydistributed on the sky. Contrast this with the superposition of gravitational waves produced by unresolved Galactic whitedwarf binaries radiating in the LISA band (\(10^{4}~\mathrm{Hz}\) to \(10^{1}~\mathrm{Hz}\)). Although this confusionlimited 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 Gaussianstationary backgrounds
More general Gaussianstationary backgrounds (e.g., polarized, statistically isotropic but with correlated radiation, etc.) can be represented by appropriately changing the righthandside 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
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 gravitationalwave 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 gravitationalwave 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.
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.
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 pvalues for parameter estimation and hypothesis testing  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 “longrun 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(dH), 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 maximumlikelihood statistic for a particular parameter, or the maximumlikelihood ratio to compare a signalplusnoise model to a noiseonly 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 pvalues 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 socalled 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 gravitationalwave 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 gravitationalwave 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.
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 gravitationalwave data analysis, the gravitationalwave 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
3.2.3 Frequentist upper limits
3.2.4 Frequentist parameter estimation
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 gravitationalwave 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 twosided intervals for nonnull results. This procedure, developed by Feldman and Cousins (1998), also solves the problem that the choice of an upper limit or twosided 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 twosided confidence interval is based on the data and not decided upon before performing the experiment.
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
3.3.2 Bayesian upper limits
3.3.3 Bayesian model selection
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
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 frequentistBayesian analysis, doing Monte Carlo simulations: (i) over different noiseonly 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 nullhypothesis 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 pvalue 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
3.5.1 Simulated data
3.5.2 Frequentist analysis
3.5.3 Bayesian analysis
Finally, the Bayes factor for the signalplusnoise model \(\mathcal{M}_1\) relative to the noiseonly 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  – 
pvalue  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 gravitationalwave 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 gravitationalwave signals using a network of gravitationalwave 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 gravitationalwave detectors comes in a variety of forms. For groundbased interferometers such as LIGO and Virgo, the data comes from the error signal in the differential armlength control system, which is nonlinearly related to the laser phase difference, which in turn is linearly related to the gravitationalwave 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 spindown 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 spacebased gravitationalwave 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 gravitationalwave strain.
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. Crosscorrelation 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 crosscorrelation 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 crosscorrelation methods, e.g., using a null channel as a noise calibrator.
The basic idea of using crosscorrelation to search for stochastic gravitationalwaves 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
We have assumed here that there is no crosscorrelated 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
4.3.1 White noise and signal
4.3.2 Colored noise and signal
We do not bother to write down the maximumlikelihood 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 optimallyfiltered crosscorrelation statistic for isotropic stochastic backgrounds. There one assumes a particular spectral shape for the gravitationalwave power spectral density, and then simply estimates its overall amplitude. That simplifies the analysis considerably.
4.4 Maximumlikelihood detection statistic
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
Correlations enter the Bayesian analysis via the covariance matrix C that appears in the likelihood function \(p(dS_{n_1},S_{n_2}, S_h,\mathcal{M}_1)\). The covariance matrix for the data includes the crossdetector signal correlations, as we saw in (4.15). So although one does not explicitly construct a crosscorrelation statistic in the Bayesian framework, cross correlations do play an important role in the calculations.
4.6 Comparing frequentist and Bayesian crosscorrelation 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 \(\Lambda \).
4.6.2 Bayesian analysis
Figure 18 shows the marginalized posterior \(p(S_hd,\mathcal{M}_1)\) for the stochastic signal variance given the data d and signalplusnoise model \(\mathcal{M}_1\). The peak of the posterior lies close the frequentist maximumlikelihood 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 signalplus noise model \(\mathcal{M}_1\) (blue curves) and the noiseonly model \(\mathcal{M}_0\) (green curves). For comparison, the frequentist maximumlikelihood 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 signalplusnoise 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 crosscorrelation methods aren’t available
Crosscorrelation methods can be used whenever one has two or more detectors that respond to a common gravitationalwave 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 gravitationalwave signal, or the modulation of an anisotropic signal due to the motion of the detector (as is expected for the confusionnoise from galactic compact white dwarf binaries for LISA). Without some way of differentiating instrumental noise from gravitationalwave “noise”, there is no hope of detecting a stochastic background.
4.7.1 Singledetector excess power statistic
4.7.2 Null channel method
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 nearoptimal) data analysis algorithms to search for gravitational waves. To specify the likelihood function, for example, requires models not only for the gravitationalwave 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 nontrivial 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 groundbased and spacebased laser interferometers, spacecraft Doppler tracking, and pulsar timing arrays. We conclude this section by discussing how the motion of a detector relative to the gravitationalwave 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 gravitationalwave 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 groundbased laser interferometers.
5.1 Detector response
Gravitational waves are timevarying perturbations to the background geometry of spacetime. Since gravitational waves induce timevarying changes in the separation between two freelyfalling objects (socalled test masses), gravitationalwave 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 timevarying 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 freelyfalling objects. Spacecraft Doppler tracking, pulsar timing arrays, and ground and spacebased laser interferometers (e.g., LIGOlike and LISAlike 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
5.1.2 Pulsar timing
5.1.3 Laser interferometers
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  L (km)  \(f_*\) (Hz)  f (Hz)  \(f/f_*\)  Relation 

Groundbased interferometer  \({\sim } 1\)  \({\sim } 10^5\)  \(10\,\,\hbox {to}\,10^4\)  \(10^{4}\,\hbox {to}\,10^{1}\)  \(f\ll f_*\) 
Spacebased 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
5.2.1 Oneway tracking
5.2.2 Twoway tracking
5.2.3 Michelson interferometer
5.3 Overlap functions
As mentioned in Sect. 4, a stochastic gravitationalwave background manifests itself as a nonvanishing 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 gravitationalwave signal and other noise sources. In this section, we calculate the expected correlation due to a gravitationalwave background, allowing for nontrivial detector response functions and nontrivial 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
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 nontrivial 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 groundbased interferometers, which operate in the smallantenna 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 largeantenna, smallseparation 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
5.3.4 Autocorrelated response
5.4 Examples of overlap functions
5.4.1 LHOLLO overlap function
5.4.2 Bigbang observer overlap function
5.4.3 Pulsar timing overlap function (Hellings and Downs curve)
5.5 Moving detectors
5.5.1 Monochromatic plane waves
5.5.2 Stochastic backgrounds
Figure 40 shows the expected timedomain output of a particular Michelson combination, X(t), of the LISA data over a 5year period. The combined signal (red) consists of both detector noise (black) and the confusionlimited gravitationalwave signal from the galactic population of compact whitedwarf 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).
5.5.3 Rotational and orbital motion of Earthbased detectors
As mentioned above, given the broadband 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 quasistatic. So, for example, the motion of a single Earthbased 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 gravitationalwave background from a different spatial location and with a different orientation.
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 phasecoherent approach for mapping anisotropic gravitationalwave 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 signaltonoise 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 optimallyfiltered crosscorrelation statistic (Allen 1997; Allen and Romano 1999) for an Gaussianstationary, unpolarized, isotropic gravitationalwave background can be derived as a matchedfilter statistic for the expected crosscorrelation. This derivation of the optimallyfiltered crosscorrelation 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
6.2 Correlated measurements
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
6.4 Optimal filtering for a stochastic background
6.4.1 Optimal estimators for individual frequency bins
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 whitedwarf 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 gravitationalwave power on the sky can be used to discriminate between cosmologicallygenerated backgrounds, produced in the very early Universe, and astrophysicallygenerated backgrounds, produced by more recent populations of astrophysical sources. In addition, an anisotropic distribution of power may allow us to detect the gravitationalwave 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 groundbased detectors like LIGO, Virgo, etc., or yearly orbital motion for spacebased detectors like LISA). This approach assumes a known distribution of gravitationalwave power \(\mathcal{P}(\hat{n})\), and filters the data so as to maximize the signaltonoise ratio of the harmonics of the correlated signal. The second approach (Sect. 7.3) constructs maximumlikelihood estimates of the gravitationalwave power on the sky based on crosscorrelated 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 gravitationalwave power on the sky. The fourth and final approach we describe (Sect. 7.5) attempts to measure both the amplitude and phase of the gravitationalwave 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 crosscorrelation 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 maximumlikelihood estimates of gravitationalwave power; Thrane et al. (2009) and Talukder et al. (2011) for maximumlikelihood ratio detection statistics; and Gair et al. (2014), Cornish and van Haasteren (2014) and Romano et al. (2015) regarding phasecoherent 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 spacebased interferometers like LISA or the BigBang 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 crosscorrelating data from two independent LISAtype 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
7.1.2 Shortterm Fourier transforms
7.1.3 Crosscorrelations
Figure 43 shows maps of the real and imaginary parts of \(\gamma (t; f, \hat{n})\) (appropriately normalized) for the strain response of the 4km 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 Earthfixed 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 \(\gamma (t; f, \hat{n})\)
Example: Earthbased interferometers
Example: Pulsar timing arrays
In Fig. 46, we show plots of the spherical harmonic components of \(\gamma (t;f, \hat{n})\) calculated using the Earthtermonly Dopplerfrequency response functions (5.21) for pulsar timing. Since there is no frequency or timedependence 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 zaxis and the other is in the xzplane, 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
7.2.1 Timedependent crosscorrelation
7.2.2 Calculation of the optimal filter
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 gravitationalwave 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 crosscorrelated 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 groundbased and spacebased 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 maximumlikelihood estimators for the \(\mathcal{P}_{lm}\), using singularvalue decomposition to ‘invert’ the Fisher matrix (or point spread function), which maps the true gravitationalwave power distribution to the measured distribution on the sky.
7.3 Maximumlikelihood estimates of gravitationalwave power
In this section, we describe an approach for constructing maximumlikehood estimates of the gravitationalwave 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 gravitationalwave detectors typically does not have perfect coverage of the sky, the inversion requires some form of regularization, which we describe below. The gravitationalwave 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 maximumlikelihood estimators
7.3.2 Extension to a network of detectors
7.3.3 Error estimates
7.3.4 Point spread functions
Example: Pulsar timing arrays
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: Earthbased interferometers
In Fig. 49 we plot point spread functions for gravitationalwave power for the LIGO HanfordLIGO Livingston pair of detectors. The lefthand plot is for a point source located at the center of the map, \((\theta ,\phi )=(90^\circ , 0^\circ )\), while the righthand plot is for a point source located at \((\theta ,\phi )=(60^\circ , 0^\circ )\) (indicated by black dots). We assumed equal whitenoise 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 figureeight with a bright region at the center of the figureeight 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
7.3.5 Singularvalue decomposition
Expression (7.33) for the maximumlikelihood 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 singularvalue 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.
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 maximumlikelihood estimates, which is dominated by the modes to which we are least sensitive.
7.3.6 Radiometer and spherical harmonic decomposition methods
The gravitationalwave 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 maximumlikelihood approach for mapping gravitationalwave 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.
Gravitationalwave radiometer
Spherical harmonic decomposition
7.4 Frequentist detection statistics
Such a matchedfilter 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 Phasecoherent mapping
Phasecoherent mapping is an approach that constructs estimates of both the amplitude and phase of the gravitationalwave 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 gravitationalwave power \(\mathcal{P}(\hat{n}) = h_+^2 + h_\times ^2\). The gravitationalwave 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 Maximumlikelihood estimators and Fisher matrix
7.5.2 Point spread functions
To illustrate the above procedure, we calculate point spread functions for phasecoherent 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 equalnoise 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 gravitationalwave power, we need to do the same for the estimators \(\hat{a}\) for the phasecoherent mapping approach. Although we could use singularvalue decomposition for the Fisher matrix F given by (7.69), we will first whiten the data, which leads us directly to pseudoinverse 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 skymap basis vectors, which we will describe in more detail in Sect. 7.5.4. An alternative approach involving the pseudoinverse of the unwhitened response matrix is given in Gair et al. (2014) and Appendix B of Romano et al. (2015).
7.5.4 Basis skies
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 2N (the factor of 2 corresponding to the two polarizations, \(+\) and \(\times \)). This means that one can extract at most 2N real pieces of information about the gravitationalwave background with an Npulsar array. This is typically fewer than the number of modes of the background that we would like to recover.
7.5.5 Underdetermined reconstructions
7.5.6 Pulsar timing arrays
To explicitly demonstrate that a pulsar timing array is insensitive to the curlcomponent of a gravitationalwave background, Gair et al. (2014) constructed maximumlikelihood 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 maximumlikelihood 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 maximumlikelihood recovered map from the gradient component and the total simulated background, respectively. Note that the maximumlikelihood 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 gravitationalwave 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 Groundbased interferometers
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
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 nonGaussian 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 NonGaussian backgrounds
In Sect. 2.1, we asked the question “when is a gravitationalwave 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 timefrequency space, again making them individually unresolvable, but producing a confusion noise that can be detected by crosscorrelation 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 dutycycle is small, then the resulting stochastic signal will be nonGaussian and “popcornlike”, 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 toymodel signal consisting of simulated sineGaussian 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 lowerlefthand 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 Gaussiandistributed, as expected from the central limit theorem.
8.1.1 NonGaussian search methods: overview
8.1.2 Likelihood function approach for nonGaussian backgrounds
Fundamentally, searching for nonGaussian 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.
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).
Generic likelihood for unresolvable signals
8.1.3 Frequentist detection statistic for nonGaussian backgrounds
We can perform exactly the same calculations, making the same assumptions, for the likelihood functions constructed from any of the nonGaussian 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 maximumlikelihood ratios \(\Lambda _\mathrm{ML}^\mathrm{NG}(d)\) for their nonGaussian 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.
8.1.4 Bayesian model selection

\(\mathcal{M}_0\): noiseonly 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 Gaussianstochastic signal model defined by (8.6).

\(\mathcal{M}_2\): noise plus the mixtureGaussian 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 Gaussianstochastic signal model. This is a hybrid signal model that allows for both stochastic and deterministic components for the signal.
8.1.5 Fourthorder correlation approach for nonGaussian backgrounds
In this section, we briefly describe a fourthorder correlation approach for detecting nonGaussian 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 nonGaussian contribution to the background from any Gaussiandistributed 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 gravitationalwave 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 nonisotropic response functions can also be done (Seto 2009).
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 (4thorder) overlap functions, which are skyaverages of the product of the response functions for four different detectors. The expected value of the 4thorder detection statistic for this more general analysis involves generalized overlap functions for both the (squared) overall intensity and circular polarization components of the nonGaussian background. We will discuss circular polarization in the following section, but in the simpler context of Gaussianstationary isotropic backgrounds. Readers should see Seto (2009) for more details regarding circular polarization in the context of nonGaussian 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 gravitationalwave 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 crosscorrelation methods to look for evidence of circular polarization in a stochastic background.
8.2.1 Polarization correlation matrix
8.2.2 Overlap functions
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 higherorder 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 maximumlikelihood estimators of gravitationalwave power for unpolarized anisotropic backgrounds (Sect. 7.3). For a general discussion of component separation for isotropic backgrounds, see Parida et al. (2016).
8.2.4 Example: component separation for two baselines
8.2.5 Effective overlap functions for I and V for multiple baselines
8.3 NonGR polarization modes: preliminaries
8.3.1 Transformation of the polarization tensors under a rotation about \(\hat{n}\)
8.3.2 Polarization and spherical harmonic basis expansions
8.3.3 Detector response
8.3.4 Searches for nonGR polarizations using different detectors
Evidence for nonGR polarization modes can show up in searches for either deterministic or stochastic gravitationalwave 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 coaligned (and hence see the same polarization). Adding Virgo, KAGRA, and LIGOIndia 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 mostlikely first come from stochastic background observations.
In the following sections, we describe stochastic background search methods for nonGR polarization modes using both groundbased 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 groundbased 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 nonGR polarizations using groundbased detectors
We now describe crosscorrelation searches for nonGR polarization modes using a network of groundbased laser interferometers. For additional details, see Nishizawa et al. (2009).
8.4.1 Response functions
8.4.2 Overlap functions
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.
8.4.4 Effective overlap functions for multiple baselines
8.5 Searches for nonGR polarizations using pulsar timing arrays
As discussed in Sect. 8.3.4 it is also possible to search for nonGR polarizations using a pulsar timing array. Although the general concepts are the same as those for groundbased 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 crosscorrelating 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
8.5.2 Spherical harmonic basis response functions
8.5.3 Overlap functions
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 antidirectional (\(\zeta _{IJ}=\pi \)) pulsars (Lee et al. 2008; Chamberlin and Siemens 2012; Gair et al. 2015). The pulsar terms need to be included in the scalarlongitudinal response functions for all cases to obtain a finite result, which again depend on the frequency f via the distances to the pulsars. A semianalytic 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 semianalytic 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
Relative uncertainties for the tensor, breathing, scalarlongitudinal, and vectorlongitudinal polarization modes searched for separately or in various combinations for \(l_\mathrm{max}=2\) and \(N_p=30\) pulsars
(l, m) mode  

(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 nonzero only for \(l\ge 2\), while the response to the breathing modes is nonzero only for \(l=0,1\). Adding the scalarlongitudinal modes to the analysis worsens the recovery of the tensor and breathing modes by about an order of magnitude, as the scalarlongitudinal modes can also have nonzero 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 scalarlongitudinal modes depends strongly on the distances to the pulsars. The uncertainity in the recovery of the scalarlongitudinal 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 scalarlongitudinal modes is much larger for small angular separations between the pulsars (Sect. 8.5.3; Fig. 68). Finally, adding the vectorlongitudinal modes to the analysis weakens the recovery of the scalarlongitudinal 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 vectorlongitudinal response, as indicated in (8.114), but it is much weaker than the frequency dependence of the scalarlongitudinal 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 crosscorrelation techniques described in Sect. 4 to search for signals that don’t really constitute a stochastic gravitationalwave background. Using a stochasticbased crosscorrelation 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 crosscorrelation method can be used to search for (i) longduration unmodelled transients and (ii) persistent (or continuous) gravitational waves from targeted sources.
8.6.1 Searches for longduration unmodelled transients
The Stochastic Transient Analysis Multidetector Pipeline (Thrane et al. 2011) (STAMP for short) is a crosscorrelation search for unmodelled longduration 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 quasimonochromatic signals that one expects from e.g., rotating (nonaxisymmetric) neutron stars. STAMP was developed in the context of groundbased interferometers, but the general method, which we briefly describe below, is also valid for other types of gravitationalwave detectors.
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 timefrequency map, which requires using some form of patternrecognition 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 timeshifting 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 followup investigations using auxiliary instrumental and environmental channels, and data quality indicators. This provides additional confidence that the gravitationalwave candidate is not some spurious instrumental or environmental artefact.)
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 allsky search for longduration gravitationalwave transients; the second, a triggeredsearch for longduration gravitationaltransients coincident with long duration gammaray bursts.
8.6.2 Searches for targetedsources of continuous gravitational waves
The gravitationalwave 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 lowmass Xray binaries like Sco X1 (Abadie et al. 2011; Messenger et al. 2015; Abbott et al. 2016a). For example, Sco X1 is expected to emit gravitational waves from the (suspected) rotating neutron star at its core, having nonaxisymmetric distortions produced by the accretion of matter from the lowmass companion. The parameters of this system that determine the phase evolution of the gravitational radiation are not wellconstrained: (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 lowmass 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 lowmass companion star. Both of these features translate into a large parameter space volume over which to search, making fullycoherent matchedfilter searches for the gravitationalwave signal computationally challenging (Messenger et al. 2015).
9 Realworld complications
Experience with realworld 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 realworld data sets.
9.1 Observatoryspecific challenges
To begin the discussion, we highlight some of the challenges associated with realworld data, which are specific to the different observational domains—e.g., groundbased detectors, spacebased 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 Groundbased interferometers
Analysis of data from the first and second generation groundbased interferometers have shown that the data are neither perfectly stationary nor Gaussian. The nonstationarity can be broadly categorized as having two components: slow, adiabatic drifts in the noise spectrum with time; and shortduration noise transients, referred to as glitches (Blackburn et al. 2008), which have compact support in timefrequency. These glitches are also the dominant cause of nonGaussianity in the noise distributions, giving rise to long “tails” (large amplitude events with nonnegligible 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 datataking mode due to an environmental disturbance or instrumental malfunction), scheduled maintenance, etc.
An analysis of LIGOVirgo data that assumes the noise spectrum is constant over days or weeks would produce biased results. In practice, the data is analyzed using \({\sim } 1\) minlong 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 nonstationary. 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 gravitationalwave 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 gravitationalwave signal, subtracting the timing model unfortunately removes part of the signal as well. Subtraction of the timing model also introduces nonstationarity into the data (Haasteren and Levin 2013), again making timedomain analyses the only possibility (van Haasteren et al. 2009).
9.1.3 Spacebased 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 repointing of the communication antennae and orbit adjustments. Possible sources of nonstationarity 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 noiseorthogonal signal channels can be synthesized, but these combinations are also signal orthogonal, and so crosscorrelation cannot be used to detect a signal.
9.2 Nonstationary noise
Data from existing gravitationalwave detectors, including bars, interferometers, and pulsar timing, exhibit various degrees of nonstationarity. Here we give examples relevant to groundbased interferometers, but the situation is similar for the other detection techniques.
Nonstationary behavior can manifest itself in many forms, and there are no doubt many factors that contribute to the nonstationarity seen in interferometer data. Nonetheless, a simple twopart model does a good job of capturing the bulk of the nonstationary features. The twopart model consists of a slowlyvarying noise spectral density \(S_n(t{;}f)\), and localized noise transients or “glitches”. The slow drift in the spectrum can be modeled as a locallystationary 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 nonstationarity poses more of a challenge, as the nonstationarity persists even for short data segments.
9.2.1 Local stationarity
9.2.2 Glitches
9.3 NonGaussian noise
Gaussian noise processes are ubiquitous in nature, and provide a remarkably good model for the data seen in gravitationalwave detectors. Properly whitened gravitationalwave 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.
Since gravitationalwave data typically have highlycolored 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 squareroot 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 Fourierdomain and timedomain samples for the simulated data shown in Fig. 73. By eye, the frequencydomain samples appear fairly Gaussian, while the timedomain 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 pvalue of \(p=2.6\times 10^{5}\) for the Fourierdomain samples and \(p<10^{20}\) for the timedomain samples. Applying the same analysis to the locallystationary \(\text {AR}(1)\) 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 AndersonDarling test, while the whitened timedomain samples do not. Overall, glitches cause much larger departures from Gaussianity than adiabatic variation in the noise levels.
Todate, there have been no detailed studies of the effects of nonstationary and nonGaussian 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 LIGOVirgo analyses using timeshifted 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 crosscorrelation 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 nonstationarity 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 nonstationary, 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).
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 timeshifted data and simulated signals. An alternative approach is to develop more flexible noise models that can account for various types of nonstationarity and nonGaussianity.
Additional models for nonstationary and nonGaussian 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 nonGaussian noise models, including exponential and twocomponent Gaussian models. The twocomponent Gaussian model combined with a nonstationary glitch model was studied in Littenberg and Cornish (2010). Student’s tdistribution was considered in Rover (2011). A nonstationary and nonGaussian noise model was derived in Principe and Pinto (2008) based on a Poisson distribution of sineGaussian glitches.
9.6 Correlated noise
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 cosmicmicrowavebackground 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, spectrumbased 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 narrowband 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 autocorrelated noise level in each detector, and thus very hard to detect outside of the crosscorrelation 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 gravitationalwave signals that will differ from the spatial correlation pattern of the noise (unless we are incredibly unlucky!). In principle, the difference in the frequencydependent 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 gravitationalwave signal using a single instrument.
Singledetector detection techniques will be put to the test when the first spacebased gravitationalwave 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 lowfrequency gravitational waves, forming an approximate null channel (see Sect. 4.7 for a discussion of null channels). Other combinations, such as the socalled A and E channels (Prince et al. 2002), are much more sensitive to gravitationalwave 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).
LISAtype observatories operate as synthetic interferometers by forming gravitationalwave observables in postprocessing using different combinations of the phasemeter readouts from each interspacecraft laser link. The combinations synthesize effective equalpathlength interferometers to cancel the otherwise overwhelming laser frequency noise. These combinations have to account for the unequal and timevarying distances between the spacecraft.
An additional complication for space interferometers operating in the mHz frequency range are the millions of astrophysical signals that can drownout a cosmologicallygenerated 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 whitedwarf binaries in our galaxy. The annual modulation of the whitedwarf 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 nottodistant 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 inband for a fraction of the observation time), while the reverse is true for longlived signals, such as the slowly evolving supermassive blackhole 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 gravitationalwave 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 gravitationalwave 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.
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 tidallysqueezed 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 groundbased and ballonborne 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 singlefield “slowroll” inflation model, and direct the reader to more extensive CMBfocused reviews, e.g., Kamionkowski and Kovetz (2015), that cover more exotic models.
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 blackbody 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 curlfree Emodes and divergencefree Bmodes, and using measurements made on a wide range of angular scales. For a more indepth description, see Chapter 27 of the Review of Particle Physics (Olive et al. 2014).
Coherent motion in the primordial plasma can polarize the CMB photons through Thomson scattering. Scalar perturbations source curlfree Emode polarization anisotropies, while the tensor perturbations source divergencefree Bmode polarization anisotropies, in addition to Emodes. 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 (nanoKelvin or smaller polarization fluctuations as compared to microKelvin temperature fluctuations) and foreground noise. The main noise contributions come from gravitational lensing, which converts a fraction of the much larger Emode anisotropy into Bmodes, 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 Bmode 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 Bmode contribution is very challenging, the payoff 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 onboard clock and radio telemetry replacing the regular lighthouselike radio emission of a pulsar. The \({\sim }1\)–10 AU Earthspacecraft separation places spacecraft Doppler tracking between pulsar timing and future LISAlike missions in terms of baseline and gravitationalwave frequency coverage. In principle, a fleet of spacecraft each equipped with accurate clocks and highpower radio transmitters could be used to perform the same type of crosscorrelation analysis used in pulsar timing, but todate 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 \(201726\) 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 stellarremnant 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
Footnotes
 1.
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.
 2.
Actually, even if the gravitygradient and seismic noise were zero, one couldn’t go below \({\sim }1~\mathrm {Hz}\) with the current generation of groundbased laser interferometers, since the suspended mirrors (i.e., the test masses) are no longer freely floating when you go below their resonant frequencies: \({\sim }1~\mathrm {Hz}\).
 3.
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.
 4.
The gravitationalwave propagation direction, which we will denote by \(\hat{k}\), is given by \(\hat{k}=\hat{n}\).
 5.
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.
 6.
Since the model \(\bar{h}(t)\) will differ from the actual h(t), we use an overbar for the model to distinguish the two.
 7.
Our convention for Fourier transform is \(\tilde{h}(f)= \int _{\infty }^\infty dt \>e^{i2\pi f t} h(t)\).
 8.
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.
 9.
 10.
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., \(R^{ab}(f,\hat{n}) = e^{i 2\pi f\hat{n}\cdot {\vec {x}}/c}\bar{R}^{ab}(f,\hat{n})\).
 11.
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 gravitationalwave source, \(\hat{n}\), as opposed to the direction of gravitationalwave propagation, \(\hat{k}=\hat{n}\). Appendix H provides expressions relating the response functions calculated using these two different conventions.
 12.
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.
 13.Recall from Footnote 10 that the phase factors \(e^{i 2\pi f\hat{n}\cdot {\vec {x}}_{I,J}/c}\) are already contained in our definition of the response functions \(R^A_{I,J}(f,k)\). If we explicitly display this dependence thenwhere \(\bar{R}^A_{I,J}(f,\hat{n}) \equiv \bar{R}^{ab}_{I,J}(f,\hat{n})e^A_{ab}(\hat{n})\). One often sees this latter expression for \(\Gamma _{IJ}(f)\) in the literature.$$\begin{aligned} \Gamma _{IJ}(f) \equiv \frac{1}{8\pi } \int d^2\Omega _{\hat{n}} \sum _A \bar{R}_I^A(f,\hat{n})\bar{R}_J^{A}{}^*(f,\hat{n}) e^{i 2\pi f\hat{n}\cdot ({\vec {x}}_I {\vec {x}}_J)/c}, \nonumber \end{aligned}$$
 14.
This is needed, for example, to calculate the overlap functions for an array of seismometers in the smallantenna 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 Ith seismometer.
 15.
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.
 16.
In equatorial coordinates, the galactic center is located at \((\mathrm{ra},\mathrm{dec}) = (6^\mathrm{h} 15^\mathrm{m}, 29^\circ )\).
 17.
For correlated measurements, \(Q_i = \sum _j(\bar{C}^{1})_{ij}/h_i\) where \(\bar{C}^{1}\) is the inverse of the rescaled covariance matrix \(\bar{C}_{ij}\equiv C_{ij}/(h_i h_j)\).
 18.
 19.
The number of terms in the expansion is given by \(2+\mathrm{floor}(1+l/2)\).
 20.
 21.
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.
 22.
 23.
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 \(\mathcal{S}\)).
 24.
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).
 25.
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.
 26.
One should think of a radiometer upperlimit map as a convenient way of representing upper limits for a collection of individual pointsource 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 sidestepping the deconvolution problem associated with a nontrivial point spread function for the search. In other words, each pixel of a radiometer map corresponds to a separate analysis.
 27.
The normalized leakage of a window w(t) is defined as \(\tilde{w}(f)/\tilde{w}(0)\).
 28.
If the rectangular window is defined to be nonzero for \(t\in [T/2,T/2]\) instead of [0, T], then \(\tilde{w}_d(f)=T\mathcal {D}_N(f\Delta t)\), which does not include the phase factor on the righthand side of (D.22).
Notes
Acknowledgements
JDR acknowledges support from National Science Foundation Awards PHY1205585, CREST HRD1242090, PHY1505861. NJC acknowledges support from National Science Foundation Awards PHY1306702 and PHY1607343, and NASA award NNX16AB98G. JDR and NJC acknowledge support from the National Science Foundation NANOGrav Physics Frontier Center, NSF PFC1430284. We also thank members of the LIGOVirgo 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 LIGOP1600242.
Supplementary material
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