In this section, a personal selection of landmarks in cosmology are described, which also serves as an overview of the lecture course.

The isotropic Universe

This chapter will describe homogeneous and isotropic world models, whereas inhomogeneities in the Universe will be discussed in Chap.8. We discuss why it is meaningful to consider homogeneous world models, despite the fact that the Universe around us is inhomogeneous, filled with stars, galaxies and galaxy clusters. A conceptually simple way of describing homogeneous world models will be presented, and it will be argued that a Newtonian description of these models covers all the essential features of relativistic models. Indeed, (nearly) all equations derived in the context of Newtonian cosmology are valid also in relativistic models, the so-called Friedmann-Lemaitre models.

Introduction to General Relativity

After Einstein completed his Special Relativity, he soon noticed that there was no simply way to include gravitational forces into that theory. He worked from 1907 until 1915 to obtain a theory which combines Special Relativity with gravity -- and found the theory of General Relativity. Rightfully, many physicists think that this is the largest achievement in physics in the last century. Even those who do not fully share this view admit that it is one of the great discoveries of mankind. This lecture course is not the proper place to do justice to this beautiful theory, but we shall give a very quick overview on what it contains, and how the concepts and equations of General Relativity can be obtained. It must be stressed that this brief chapter cannot be a substitute for a proper course on General Relativity.

Cosmological solutions of Einstein's equations

We shall now consider homogeneous and isotropic world models in the framework of General Relativity. For that, we first need to define more precisely what homogeneity and isotropy means in General Relativity. We shall then derive the expanasion equations of cosmology, interpret them and derive relations between observables (redshift) and cosmic time, scale factor, distances, volumes, etc.

Thermal history of the Universe

The CMB photons that we see today have been more energetic in the past, due to the effect of redshift. As we will demonstrate below, cosmic expansion preserves the blackbody property of a photon distribution; hence, at earlier epochs the photons also had a Planck spectrum, but with a higher temperature. This implies that the Universe was hot in the past. Since the discussion in the previous section showed that at some point in the past, the scale factor must have been extremely close to zero, this means that shortly after the Big Bang, the Universe must have been very hot. In these early phases, one can expect that highly energetic processes occurred; for example, when the temperature was above 1 MeV, electron-positron pairs could be created. Furthermore, the energy of nucleons in the early phases of the cosmic evolution was high enough to enable the formation of atomic nuclei, similar to the nucleosynthesis occurring in the central regions of stars. In this section we shall consider aspects of the thermal history of the Universe.

Gravitational Lensing

Gravitational lensing has become one of the most interesting tools to study the mass distribution in the Universe. Since gravitational light deflection is independent of the nature and state of the matter, it is ideally suited to investigate the distribution of all (and thus also of dark) matter in the Universe. Lensing results have now become available over a wide range of scales, from the search for MACHOs in the Galactic halo, to the mass distribution in galaxies and clusters of galaxies, and the statistical properties of the large-scale matter distribution in the Universe. Here and in Chaps.7 and 11, after introducing the concepts of strong and weak lensing, several applications are outlined, from strong lensing by galaxies, to strong and weak lensing by clusters and the lensing properties of the large-scale structure.

Weak Gravitational Lensing

Multiple images, microlensing (with appreciable magnifications) and arcs in clusters are phenomena of strong lensing. In weak gravitational lensing, the Jacobi matrix of the lens equation is very close to the unit matrix, which implies weak distortions and small magnifications. Those cannot be identified in individual sources, but only in a statistical sense; the basics of these effects will be described in this chapter, together with several applications.

Structure Formation in the Universe

In Chap.4, we have considered a homogeneous Universe -- which is a truly boring place to be in. The real Universe shows structure on various scales up to 200 Mpc such as is seen in the form of `Great Walls', first detected in the CfA galaxy redshift survey, but now found to occur more frequently in the Sloan Digital Sky Survey. Therefore, we need to understand how the structure observed in the Universe today has been formed and how it has evolved in the course of cosmic evolution. A highly abbreviated scenario is that tiny perturbations are assumed to be present at high redshift, e.g., inflated quantum fluctuations. They develop through self-gravity, that is, they are amplified due to gravitational instability. Furthermore, other effects like pressure, free-streaming of particles etc. affect the evolution of structure. The earliest traces of the inhomogeneities are visible in the small anisotropies on the CMB, which we will discuss in more detail in Chap.9. These inhomogeneities seem to be present at the time when the photons decouple, at redshifts of 1100, and have been amplified by gravitational instability to yield the large-scale structure in the current Universe. In this chapter, we shall describe the physics of this structure growth.

CMB anisotropies

The cosmic microwave background consists of photons that last interacted with matter at z=1100. Since the Universe was already inhomogeneous at this time, it is expected that these spatial inhomogeneities are reflected in a (small) anisotropy of the CMB: the angular distribution of the CMB temperature reflects the matter inhomogeneities at the redshift of decoupling of radiation and matter. Since the discovery of the CMB in 1965, such anisotropies have been searched for. Under the assumption that the matter in the Universe only consists of baryons, the expectation was that we would find relative fluctuations in the CMB temperature of order 1/1000 on scales of a few arcminutes. This expectation is based on the theory of gravitational instability for structure growth: to account for the density fluctuations observed today, one needs relative density fluctuations at z=1000 at least of order 1/1000. Despite increasingly more sensitive observations, these fluctuations were not detected, except for the dipole anisotropy: we have a peculiar velocity relative to the frame in which CMB is isotropic. The upper limits of temperature fluctuations resulting from these searches for anisotropies provided one of the arguments that, in the mid-1980s, caused the idea of the existence of dark matter on cosmic scales to increasingly enter the minds of cosmologists. As we will see soon, in a Universe which is dominated by dark matter the expected CMB fluctuations on small angular scales are considerably smaller than in a purely baryonic Universe. Only with the COBE satellite were temperature fluctuations in the CMB finally observed in 1992. Over the past years, sensitive and significant measurements of the CMB anisotropy have also been carried out using balloons and ground-based telescopes. We will first describe the physics of CMB anisotropies, before turning to the observational results and their interpretation. As we will see, the CMB anisotropies depend on nearly all cosmological parameters. Therefore, from an accurate mapping of the angular distribution of the CMB and by comparison with theoretical expectations, all these parameters can, in principle, be determined.

Inflation

We have seen before that the standard expansion history of the Universe, as described by the Friedmann-Lemaitre models, leads to a number of problems, in particular the horizon and the flatness problem. We summarize the basic problems here:

• Horizon problem. Why is the Universe so homogeneous? For example, why is the CMB temperature the same within 1/100000 from all directions, when causally connected regions subtend an angle of 1 degree on the surface of last scattering?
• Flatness problem. Why is the total density of the Universe so close to the critical density today? As we have demonstrated in Chap.4, this requires very fine tuning of the density at some early epoch.
• Origin of fluctuations. Through which process can density fluctuations be generated which at the early epochs have been outside the horizon and thus not in causal contact?
• Matter-antimatter asymmetry. When the Universe had a temperature above 1 GeV, protons, neutrons and electrons had a similar number density as photons in the Universe. The same had been true for the number density of antiprotons, antineutrons and positrons. However, today the ratio of the number densities of protons, neutrons and electrons to that of photons are only a billionth, and the abundance of antimatter is essentially zero. What has caused this preference of matter over antimatter? This problem is called baryogenesis.
• Relics of the early Universe. Unified gauge theories predict the presence of a variety of superheavy stable particle species produced in the early Universe; they should have survived annihilation and should have a density much higher than the critical today. Example for this are magnetic monopoles and other topological defects.