So far we have treated galaxies as isolated, non-interacting systems. However, in the hierarchical scenario of structure formation, galaxies and their associated dark matter halos undergo frequent interactions with each other. As we have seen in §7.3.6, a large fraction of dark matter halos are expected to be dynamically young (i.e. to have experienced a merger event in their recent history). In fact, as shown by Li et al. (2007), each halo, independent of its mass, experiences about three major mergers (defined as mergers with a progenitor mass ratio larger than 1/3) after its main progenitor has acquired 1% of its present-day mass. Hence, galaxies and their associated dark matter halos cannot be considered isolated ‘island universes', but are constantly influenced by gravitational interactions with other systems. These interactions may have dramatic impact on the morphologies and star-formation histories of galaxies, making the study of their nature and frequency an important part of galaxy formation and evolution.

Consider a body S which has an encounter with a perturber P with impact parameter b and initial velocity v∞ (in the limit of infinite, initial separation between S and P). Let q be a particle (e.g. a star) in S, at a distance r(t) from the center of S, and let R(t) be the position vector of P from S (see Fig. 12.1 for an illustration).

In the last two chapters we have described various models for the formation of structures in the cosmic density field. In this chapter we focus on how to test these models with observations. Since the cosmic density field is believed to be a random field generated by some random processes, model tests should be based on statistical properties of the cosmic density field, rather than on matching the predicted and observed fields point by point. Our task is therefore twofold. First, we need to develop statistical measures to characterize the cosmic density field. At any given time, the dynamical state of the cosmic density field is given by the mass distribution in space and the velocities of all mass elements. Thus, statistical characterizations of the cosmic density field are mostly based on the density and velocity fields of matter in the Universe. Such statistical characterizations are described in §§6.1-6.3, and models for the time evolution of some of these statistics are presented in §6.4. The second task is to find suitable observational probes of the cosmic density field. For many years, the distribution of galaxies has been used to infer the mass distribution in the Universe, based on the assumption that there is a simple and well-defined relation between the two. We describe various statistical measures of the galaxy distribution in §6.5.

As we have seen in Chapter 2, there is ample evidence that galaxies reside in extended halos of dark matter. According to the current paradigm, these dark matter halos form through gravitational instability. As we have seen in Chapters 4 and 5, density perturbations grow linearly until they reach a critical density, after which they turn around from the expansion of the Universe and collapse to form virialized dark matter halos. These halos continue to grow in mass (and size), either by accreting material from their neighborhood or by merging with other halos. Some of these halos may survive as bound entities after merging into a bigger halo, thus giving rise to a population of subhalos. This process is illustrated in Fig. 7.1, which shows the formation of a dark matter halo in a numerical simulation of structure formation in a CDM cosmology. It shows how a small volume with small perturbations initially expands with the Universe. As time proceeds, small-scale perturbations grow and collapse to form small halos. At a later stage, these small halos merge together to form a single virialized dark matter halo with an ellipsoidal shape, which reveals some substructure in the form of dark matter subhalos.

In Chapter 6 we have described the overall statistical properties of the cosmic density field. In this chapter we focus on the statistical properties of the discrete halos, and on their internal structure.

In the main text we have seen that galaxy formation involves many physical processes. In broad terms these processes can be divided into three main categories: gravitational, gas-dynamical and radiative. Although the physical principles governing most of these processes are well established, the dynamical systems are often so complicated that it is generally difficult to obtain analytical solutions. Thanks to the revolutionary development of powerful computers, it has become possible to tackle some of these problems using numerical simulations. In a numerical simulation, the mass distribution is usually represented by particles or sampled on a grid, and the motion of each mass element is traced numerically by taking into account its interactions with other mass elements. The solutions would be exact if we were able to simulate the motions of all individual atoms or elementary particles. Unfortunately this can never be achieved since the systems of interest (galaxies) contain of the order of 1068 protons. In practice, therefore, the pseudo-particles or mass elements used to represent the mass distribution each have a mass that is typically orders of magnitude larger than that of an actual atom. Such a representation is clearly an approximation, which may impose serious limitations on the reliability of the simulations.

In general, numerical simulations of galaxy formation can be divided in two broad categories: N-body simulations and hydrodynamical simulations. Given the importance of numerical simulations in modern astrophysics, this appendix briefly describes some of the basic numerical methods used.

Many objects in the present-day Universe, including galaxies and clusters of galaxies, have densities orders of magnitude higher than the average density of the Universe. These objects are thus in the highly nonlinear regime, where δ»1. To complete our description of structure formation in the Universe, we therefore need to go beyond perturbation growth in the linear and quasi-linear regimes, discussed in the previous chapter, and address the gravitational collapse of overdensities in the nonlinear regime.

In this chapter, we study the nonlinear gravitational collapse and dynamics of collisionless systems in which non-gravitational effects are negligible. In general, nonlinear gravitational dynamics is difficult to deal with analytically, and so in many applications computer simulations have to be used to follow the evolution in detail. However, if simple assumptions are made about the symmetry of the system, analytical models can still be constructed (§§5.1- 5.3). Although these models are not expected to give accurate descriptions of the true nonlinear problem of gravitational collapse, they provide valuable insight into the complex processes involved. In §5.4 we describe the dynamics of collisionless equilibrium systems. These dynamical models describe the end states of the nonlinear gravitational collapse of a collisionless system, and are applicable to both galaxies and dark matter halos in a steady state. As such, these models are often used to model the observed kinematics of galaxies in an attempt to constrain their masses and their orbital structures.

By and large, a galaxy is observed through, and defined by, its stellar content. Hence, any theory of galaxy formation has to address the question of how stars form. As we have seen in the previous chapter, the baryonic gas in galaxy-sized halos can cool within a time that is shorter than the age of the halo. Consequently the gas is expected to lose pressure support and to flow towards the center of the halo potential well, causing its density to increase. Once its density exceeds that of the dark matter in the central part of the halo, the cooling gas becomes self gravitating and collapses under its own gravity. As we have seen in the previous chapter, in the presence of efficient cooling, self-gravitating gas is unstable and can collapse catastrophically. Ultimately, this cooling process may lead to the formation of dense, cold gas clouds within which star formation can occur. In this chapter we take a closer look at the actual process of star formation. The main questions to be addressed are:

1. (i) When does large-scale star formation occur in a galaxy?

2. (ii) What are the main processes that drive star formation?

3. (iii) What is the rate at which stars form in a gas cloud?

4. (iv) What mass fraction of a gas cloud can be converted into stars?

5. (v) What is the initial mass function (IMF), which describes the mass distribution of stars at birth?