Both quasars and radio galaxies have large radio luminosities, and the most luminous of these can be detected at very large redshifts. Indeed, one of the complications encountered in determining the distribution of their intrinsic luminosities (the radio luminosity function) is that the objects are so faint optically that the largest telescopes are needed to obtain the spectra from which redshifts can be derived. At the same time, this raises the expectation that radio sources can serve as probes of the large-scale geometry of the Universe. The first indication of cosmological evolution was provided by the statistical relation between numbers and flux density of radio sources, the source counts; a similar test of cosmologies, the luminosity-volume testwas applied to visible quasars. These tests immediately ruled out the steady-state model of the Universe, but did not contribute to the precision cosmology which later emerged from the WMAP measurements of the CMB. The extension of the source counts in surveys of extreme sensitivity has, however, contributed dramatically to the astrophysics of radio-source evolution.

Another cosmological test, the relation between apparent source diameter and luminosity, emerges from the geometry outlined in Chapter 14. This again proves to be more of interest regarding source evolution than for cosmology itself. A further observational field opened by the geometrical theory is gravitational lensing, which was discovered as a radio phenomenon and is now observed through most of the electromagnetic spectrum.

Astronomy makes use of more than 20 decades of the electromagnetic spectrum, from radio to gamma rays. The observing techniques vary so much over this enormous range that there are distinct disciplines of gamma-ray, X-ray, ultraviolet, optical, infrared, millimetre and radio astronomy, often concentrated in individual observatories. Modern astrophysics depends on a synthesis of observations from the whole wavelength range, and the concentration on radio in this text needs some rationale. Apart from the history of the subject, which developed from radio communications rather than as a deliberate extension of conventional astronomy, there are two outstanding characteristics that call for a special exposition. First, the astrophysics: long-wavelength radio waves are most often observed as a continuum in which the interaction with matter follows classical electrodynamics. High-energy electrons are involved; they are created in a variety of circumstances, and their radiation as they circulate in magnetic fields gives evidence of new phenomena, often showing a close link to the phenomena observed in X-rays and gamma-rays. At the shorter wavelengths the low quantum energy gives access to spectral lines from atomic and molecular species at comparatively low temperatures. Second, the techniques: radio astronomy takes account of the phase as well as the intensity of incoming radio waves, allowing the development of interferometers of astonishingly high angular resolution and sensitivity

The electromagnetic signals that give information about the Universe have the characteristics of random noise. More specifically, in the radio part of the spectrum, the signals are composed of Rayleigh, or Gaussian, noise, the result of an assemblage of many random oscillators with random frequency and phase. As one moves to shorter wavelengths, through the infrared and into the optical, ultraviolet and X-ray bands, the discrete character of photons becomes increasingly dominant, and the random noise obeys Poisson statistics, sometimes called shot noise. Throughout the spectrum, the process of detecting and measuring the signals gathered by a telescope is almost always electronic; for the optical astronomer the eye and the photographic plate are not sensitive enough, and at both the radio and X-ray ends of the spectrum electronic means have always been essential. The device that measures the power of the incoming signal is a radiometer; when it measures power as a function of frequency, it is a spectrometer.

At wavelengths shorter than about 100 μm, immediate detection of the received power is almost always forced on the observer because the laws of quantum mechanics require any amplifier to add extraneous noise. For the radio astronomer, the incoming signal is amplified before its power is measured in a detector, and the construction of low-noise amplifiers has become an art.

Cygnus A (Figure 13.1) is one of the strongest radio sources in the sky. It was discovered by Hey in his early survey of the radio sky (Hey et al. 1946a, 1946b); its trace can even be discerned on Reber's 1944 map (see Appendix 3 for a brief history). It is, however, an inconspicuous object optically, and it was not identified until its position was known to an accuracy of 1 arcmin (Smith 1951; Baade and Minkowski 1954). Its optical counterpart was found to be an eighteenth-magnitude galaxy with a recession velocity of 17 000 km s-1, that is, with redshift z= 0.06, indicating a distance of almost 1000 million light years. The source was shown to be double, with an overall size of more than a minute of arc, through the pioneering interferometry observations of Jennison and das Gupta (1953). Furthermore, the large angular size and double-lobed shape of Cygnus A were such distinctive features that similar radio sources might well be recognizable at very much greater distances; this was confirmed in 1960 when the radio source 3C 295 was identified by Minkowski with another similar galaxy, this time with a redshift of 0.45. Another galaxy, NGC 5128, had already been identified as a radio source known as Centaurus A (Bolton et al.1949); this again showed the characteristic double-lobed shape, but with an angular diameter of 4° it was obviously much closer.

In astrophysical contexts, the propagation of radio waves is governed, as for other parts of the electromagnetic spectrum, by the laws of radiative transfer and refraction. In radio astronomy, however, there is an emphasis on classical (non-quantized) radiative and refractive processes. Synchrotron radiation is the dominant radiation process at the longer wavelengths; spectral-line emission is observed mainly at shorter wavelengths. Maser action, the microwave equivalent of lasers, is encountered in several astrophysical contexts: this is due to the low energy of radio photons which can be significantly amplified by small population inversions in rotational and vibrational energy levels. Refraction is important in astrophysical plasmas; even though these are usually electrically neutral, protons have a negligible effect and the electron gas can have a significant effect on the velocity of radio waves. In the presence of a magnetic field, birefringence can lead to Faraday rotation of the plane of polarization.

In this chapter we set out the basic theories of radiative transfer, and outline the processes of radiation that are of particular importance in radio astronomy: free-free emission, line emission (and particularly maser emission) in dilute gas and synchrotron radiation. Free- free emission, or bremsstrahlung, is the main source in ionized hydrogen clouds, whereas synchrotron radiation is responsible for the background radiation in our Galaxy (Chapter 8) and is also practically universal in discrete radio sources from supernova remnants to quasars.

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.