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.