This chapter is an introduction to the theory of time series analysis. In Section 4.1 we discuss the estimators of the sample mean and the correlation function of a time series. In Section 4.2 we introduce non-parametric methods of the spectral analysis of time series, including the multitapering method. A detailed discussion of the time series spectral analysis can be found in Refs. [153, 154, 155, 156].

In Sections 4.3–4.5 we discuss useful tests of the time series. One type of test is for the presence of periodicities in the data, which we discuss in Section 4.3. In Section 4.4 we introduce two goodness-of-fit tests describing whether the data come from a given probability distribution: Pearson's χ2 test and Kolmogorov–Smirnov test. Other types of tests are tests for Gaussianity and linearity of the data, which are discussed in Section 4.5. Both tests use higher-order spectra of time series, which are also introduced in Section 4.5.

Sample mean and correlation function

We assume that we have N contiguous data samples xk (k = 1, …, N) of the stochastic process. We also assume that the underlying process is stationary and ergodic (i.e. satisfying the ergodic theorem, see Section 3.2). We immediately see that the N samples of the stochastic process that constitute our observation cannot be considered as a stationary process. They would be a stationary sequence only asymptotically as we extend the number of samples N to infinity. As we shall see this has profound consequences on the statistical properties of the estimators of the spectrum.

Gravitational waves are predicted by Einstein's general theory of relativity. The only potentially detectable sources of gravitational waves are of astrophysical origin. So far the existence of gravitational waves has only been confirmed indirectly from radio observations of binary pulsars, notably the famous Hulse and Taylor pulsar PSR B1913+16 [1]. As gravitational waves are extremely weak, a very careful data analysis is required in order to detect them and extract useful astrophysical information. Any gravitational-wave signal present in the data will be buried in the noise of a detector. Thus the data from a gravitational-wave detector are realizations of a stochastic process. Consequently the problem of detecting gravitational-wave signals is a statistical one.

The purpose of this book is to introduce the reader to the field of gravitational-wave data analysis. This field has grown considerably in the past years as a result of commissioning a world-wide network of long arm interferometric detectors. This network together with an existing network of resonant detectors collects a very large amount of data that is currently being analyzed and interpreted. Plans exist to build more sensitive laser interferometric detectors and plans to build interferometric gravitational-wave detectors in space.

This book is meant both for researchers entering the field of gravitational-wave data analysis and the researchers currently analyzing the data. In our book we describe the basis of the theory of time series analysis, signal detection, and parameter estimation. We show how this theory applies to various cases of gravitational-wave signals. In our applications we usually assume that the noise in the detector is a Gaussian and stationary stochastic process.

In this chapter we very briefly review the theory of gravitational radiation. A detailed exposition of the theory can be found in many textbooks on general relativity, e.g. in Chapters 35–37 of [34], Chapter 9 of [35], or Chapter 7 of [36]. A detailed exposition of the theory of gravitational waves is contained in the recent monograph [37]. Reference [38] is an introductory review of the theory of gravitational radiation and Ref. [16] is an accessible review of different aspects of gravitational-wave research. Some parts of the present chapter closely follow Sections 9.2 and 9.3 of the review article [16].

The chapter begins (in Section 1.1) with a discussion of general relativity theory in the limit of weak gravitational fields. In this limit spacetime geometry is a small perturbation of the flat geometry of Minkowski spacetime. We restrict our considerations to coordinate systems in which the spacetime metric is the sum of the Minkowski metric and a small perturbation. We linearize Einstein field equations with respect to this perturbation and then we study two classes of coordinate transformations that preserve splitting the metric into the sum of Minkowski metric and its small perturbation: global Poincaré transformations and gauge transformations. Finally we discuss the harmonic gauge, which allows one to write the linearized Einstein field equations in the form of inhomogeneous wave equations for the metric perturbation.

In Sections 1.2–1.4 we introduce gravitational waves as time-dependent vacuum solutions of the linearized Einstein equations. In Section 1.2 we study the simplest such solution, namely a monochromatic plane gravitational wave. In Section 1.3 we introduce the TT coordinate system in which description of gravitational waves is especially simple.

Data from a gravitational-wave detector are realizations of a stochastic (or random) process, thus in order to analyze them we need a statistical model. In this chapter we present a theory of the detection of signals in noise and an estimation of the signal's parameters from a statistical point of view. We begin in Section 3.1 with a brief introduction to random variables and in Section 3.2 we present the basic concepts of stochastic processes. A comprehensive introduction to mathematical statistics can be found, for example, in the texts [116, 117]. Our treatment follows the monograph [118]. Other expositions can be found in the texts [119, 120]. A general introduction to stochastic processes is given in [121]. Advanced treatment of the subject can be found in [122, 123].

In Section 3.3 we present the problem of hypothesis testing and in Section 3.4 we discuss its application to the detection of deterministic signals in Gaussian noise. Section 3.5 is devoted to the problem of estimation of stochastic signals. Hypothesis testing is discussed in detail in the monograph [124]. Classical expositions of the theory of signal detection in noise can be found in the monographs [125, 126, 127, 128, 129, 130, 131].

In Section 3.6 we introduce the subject of parameter estimation and present several statistical concepts relevant for this problem. Parameter estimation is discussed in detail in Ref. [132], and Ref. [133] contains a concise account. In Section 3.7 we discuss topics related to the non-stationary stochastic processes.

In this chapter we derive the responses of different detectors to a given gravitational wave described in a TT coordinate system related to the solar system barycenter by wave polarization functions h+ and h×. We start in Section 5.1 by enumerating existing Earth-based gravitational-wave detectors, both laser interferometers and resonant bars. We give their geographical location and orientation with respect to local geographical directions.

In Section 5.2 we obtain a general response of a detector without assuming that the size of the detector is small compared to the wavelength of the gravitational wave. Such an approximation is considered in Section 5.3. In Section 5.4 we specialize our general formulae to the case of currently operating ground-based detectors and to the planned spaceborne detector LISA.

Detectors of gravitational waves

There are two main methods of detecting gravitational waves that have been implemented in the currently working instruments. One method is to measure changes induced by gravitational waves on the distances between freely moving test masses using coherent trains of electromagnetic waves. The other method is to measure the deformation of large masses at their resonance frequencies induced by gravitational waves. The first idea is realized in laser interferometric detectors and Doppler tracking experiments [169, 170, 171, 172], whereas the second idea is implemented in resonant mass detectors [173, 174, 175].

Currently networks of resonant detectors and laser interferometric detectors are working around the globe and collecting data. In Table 5.1 geographical positions and orientations of Earth-based laser interferometric gravitational-wave detectors are given whereas in Table 5.2 resonant detectors are listed.

It is convenient to split the expected astrophysical sources of gravitational waves into three main categories, according to the temporal behavior of the waveforms they produce: burst, periodic, and stochastic sources. In Sections 2.1–2.3 of the present chapter we enumerate some of the most typical examples of gravitational-wave sources from these categories (more detailed reviews can be found in [45], Section 9.4 of [16], and [46, 47]). Many sources of potentially detectable gravitational waves are related to compact astrophysical objects: white dwarfs, neutron stars, and black holes. The physics of compact objects is thoroughly studied in the monograph [48].

In the rest of the chapter we will perform more detailed studies of gravitational waves emitted by several important astrophysical sources. In Section 2.4 we derive gravitational-wave polarization functions h+ and h× for different types of waves emitted by binary systems. As an example of periodic waves we consider, in Section 2.5, gravitational waves coming from a triaxial ellipsoid rotating along a principal axis; we derive the functions h+ and h× for these waves. In Section 2.6 we relate the amplitude of gravitational waves emitted during a supernova explosion with the total energy released in gravitational waves and with the time duration and the frequency bandwidth of the gravitational-wave pulse. Finally in Section 2.7 we express the frequency dependence of the energy density of stationary, isotropic, and unpolarized stochastic background of gravitational waves in terms of their spectral density function.