One of the main goals of QSO research is to use these objects as a probe of the history of the Universe. Two specific aims are first, to determine the characteristics of the QSO population as a function of redshift, and second, to find the lookback time at which QSOs first appeared, as this provides some measure of the time scale for galaxy formation in the early Universe. Both of these important aims require large and preferably unbiased samples of QSOs. In this chapter, we consider how large samples of QSOs might be obtained through various survey techniques, and how the samples we obtain might be affected by various biases.

The measurable quantity that will result from surveys is the QSO ‘surface density’ dN(F,z)/dΩ i.e., number of QSOs per unit solid angle (square degree) as a function of flux F and redshift z. From this, we can compute the ‘luminosity function’, which is the relative number of AGNs at a given luminosity, and the ‘space density’, which is the total number of sources per unit comoving volume! over some specified luminosity range – when the luminosity function is correctly normalized, the total space density is simply the integral of the luminosity function over its entire range.

The primary goal of QSO surveys then is to determine dN(F,z)/dΩ in an accurate and unbiased fashion. This is a difficult and complicated undertaking because QSOs are faint and their surface density is low; the total surface density of QSOs brighter than B = 21 mag is only ∼40deg-2.

As we pointed out in Chapter 3, the main problem with sustaining an active nucleus by gravitational accretion over its lifetime of at least 108 years is funneling enough mass into the nucleus. Removing a sufficient amount of angular momentum from the gas flowing into the nucleus requires breaking the azimuthal symmetry of the galaxy's gravitational potential. A clear way to do that is by gravitational interactions with other systems, as was originally suggested by Toomre and Toomre (1972) and by Gunn (1979). This provides motivation for examining the nearby environment of AGNs to see if indeed there is evidence for interactions with nearby galaxies. The two specific questions that we want to consider are:

1. What kinds of galaxies harbor AGNs? Are there any discernible differences between galaxies with active nuclei and those without them?

2. Does the presence or absence of companion galaxies have anything to do with whether or not a galaxy harbors an AGN?

We will consider these issues separately, although they are clearly related.

Host Galaxies

The study of the ‘host galaxies’, those galaxies that contain active nuclei, is a very difficult undertaking. The major problems were alluded to at the beginning of this book: the light from the AGN itself often dominates the total light from the galaxy, particularly in the case of the highest-luminosity AGNs, which are spatially rare and thus typically found only at great distances. Consequently the work on the lowerluminosity end of the AGN distribution, i.e., Seyfert galaxies, has tended to yield less ambiguous results.

Like many textbooks, this one arose out of the author's frustration. While I believe that there are many excellent journal articles, scholarly reviews, conference proceedings, and even a few advanced monographs on active galactic nuclei (AGNs), there is no single place where a beginning student can get the very basic background necessary to get the most out of the more research-oriented material. The aims of this book are thus actually twofold: first, I wanted to summarize our basic, if marginal, understanding of AGNs at what I believe is a level of familiarity that should be expected of doctoral-level students in astronomy, and second, I wanted to provide a fairly comprehensive introduction to AGNs that would serve as a gateway to the more specialized review articles and research literature for students who have research ambitions in the field. The intended audience is thus advanced undergraduate and beginning graduate students in astronomy and astrophysics. Fairly complete undergraduate preparation in physics is assumed, as is some basic understanding of extragalactic astronomy.

I have tried to focus on basic issues and avoid minutiae and arcane issues, even though some of these undoubtedly will turn out to be tremendously important in the future. I have attempted to compile the basic background material that is by and- large familiar to researchers in AGNs, although I caution that it is by no means complete: research-level competence in the field of AGNs will require a good deal more background than is given here.

This book had its origins in a workshop held in Cape Town from June 27 to 2 July 1994, with participants from South Africa, USA, Canada, UK, Sweden, Germany, and India. The meeting considered in depth recent progress in analyzing the evolution and structure of cosmological models from a dynamical systems viewpoint, and the relation of this work to various other approaches (particularly Hamiltonian methods). This book is however not a conference report. It was written by some of the conference participants, based on what they presented at the workshop but altered and extended after reflection on what was learned there, and then extensively edited so as to form a coherent whole. This process has been very useful: a considerable increase in understanding has resulted, particularly through the emphasis on relating the results of the qualitative analysis to possible observational tests. Apart from describing the development of the subject and what is presently known, the book serves to delineate many areas where the answers are still unknown. The intended readers are graduate students or research workers from either discipline (cosmological modeling or dynamical systems theory) who wish to engage in research in the area, tackling some of these unsolved problems.

The role of the two editors has been somewhat different.

In Section 5.1 we give an overview of the use of qualitative methods in analyzing Bianchi cosmologies, expanding on the brief remarks in the Introduction to the book. Section 5.2 provides an introduction to the use of expansion–normalized variables in conjunction with the orthonormal frame formalism, thereby laying the foundation for the detailed analysis of the Bianchi models with non–tilted perfect fluid source in Chapters 6 and 7. In Section 5.3 we discuss, from a general perspective, the use of dynamical systems methods in analyzing the evolution of Bianchi cosmologies, referring to the background material in Chapter 4.

Overview

As explained in Section 1.4.2 there are two main approaches to formulating the field equations for Bianchi cosmologies:

1. the metric approach,

2. the orthonormal frame approach.

In the metric approach the basic variables are the metric components gαβ(t) relative to a group–invariant, time–independent frame (see (1.89)). This approach was initiated by Taub (1951) in a major paper. After a number of years researchers became aware that the Bianchi models admitted additional structure, namely the automorphism group, which plays an important role in identifying the physically significant variables (also referred to as gauge–invariant variables, or the true degrees of freedom). This group is defined to be the set of time–dependent linear transformations (1.87) of the spatial frame vectors that preserve the structure equations (1.88).

In this chapter we give a brief overview of some aspects of the theory of dynamical systems. We assume that the reader is familiar with the theory of systems of linear differential equations, and with the elementary stability analysis of equilibrium points of systems of non–linear differential equations (e.g. Perko 1991). We emphasize instead the fundamental concept of the flow and various other geometrical concepts such as α– and ω–limit sets, attractors and stable/unstable manifolds, which have proved useful in applications in cosmology. In the interest of readability we have stated some of the definitions and theorems in a simplified form; full details may be found in the references cited. One important aspect of the theory that we do not discuss due to limitations of space is structural stability and bifurcations. We refer to Perko (1991, chapter 4) for an introduction to these matters. We also note that the discussion of chaotic dynamical systems is deferred until Chapter 11.

To date, applications of the theory of dynamical systems in cosmology have been confined to the finite dimensional case, corresponding to systems of ordinary differential equations, although in Chapter 13 we obtain a glimpse of the potential for using infinite dimensional dynamical systems. We restrict our discussion to the finite dimensional case, referring the interested reader to books such as Hale (1988), Temam (1988a,b) and Vishik (1992) for an introduction to the infinite dimensional case.

The cosmological models proposed by A. Einstein and W. de Sitter in 1917, based on Einstein's theory of general relativity, initiated the modern study of cosmology. The concept of an expanding universe was introduced by A. Friedmann and G. Lemaître in the 1920s, and gained credence in the 1930s because of Hubble's observations of galaxies showing a systematic increase of redshift with distance, together with Eddington's proof of the instability of the Einstein static model. Since the 1940s the implications of following an expanding universe back in time have been systematically investigated, with an emphasis on four distinct epochs in the history of the universe:

(1) The galactic epoch, which is the period of time extending from galaxy formation to the present. This is the epoch that is most accessible to observation. During this period, matter in a cosmological model is usually idealized as a pressure–free perfect fluid, with galaxy clusters or galaxies acting as the particles of the fluid. The cosmic background radiation has negligible dynamic effect in this period.

(2) The pre–galactic epoch, during which matter is idealized as a gas, with the particles being the gas molecules, atoms, nuclei, or elementary particles at different times. The epoch is divided into a post–decoupling period, when matter and radiation evolve essentially independently, and a pre–decoupling period, when matter is ionized and is strongly interacting with radiation through Thomson scattering. The observed cosmic microwave background radiation is interpreted as evidence for the existence of this pre–decoupling period.