We obtain most of the information about the universe from light. Over the last century, the development of x-ray, radio and infrared detectors has given us new windows on the universe. Understanding the propagation of light in an expanding universe is therefore critical to the interpretation of observations.

Problem 2.1 Estimate the total amount of energy received by all optical telescopes over the course of the last century and compare this energy to that needed to return this book to your bookshelf.

There is a fundamental limit to how far we can see, since no particles can travel faster than light. The finite speed of light leads to “horizons” and sets an absolute constraint on our ability to comprehend the entire universe. The term “horizon” is used in different contexts in the literature, often without clear definition, and one of the purposes of this chapter is to carefully delineate the various usages. We will study in detail conformal diagrams, which are a useful pictorial way of representing horizons and the causal global structure of spacetime. Finally, we discuss the basic kinematical tests which aim to measure the distance, angular size, speed and acceleration of distant objects. Using these tests, one can obtain information about the expansion rate and deceleration parameter at earlier times, and thus probe the evolutionary history of the universe.

After recombination, the primordial radiation freely streams through the universe without any further scattering. An observer today detects the photons that last interacted with matter at redshift z ≈ 1000, far beyond the stars and galaxies. The pattern of the angular temperature fluctuations gives us a direct snapshot of the distribution of radiation and energy at the moment of recombination, which is representative of what the universe looked like when it was a thousand times smaller and a hundred thousand times younger than today.

The first striking feature is that the variations in intensity across the sky are tiny, less than 0.01% on average. We can conclude from this that the universe was extremely homogeneous at that time, in contrast to the lumpy, highly inhomogeneous distribution of matter seen today. The second striking feature is that the average amplitude of the inhomogeneities is just what is required in a universe composed of cold dark matter and ordinary matter to explain the formation of galaxies and large-scale structure. Moreover, the temperature autocorrelation function indicates that the inhomogeneities have statistical properties in perfect accordance with what is predicted by inflationary models of the universe.

In a map showing the microwave background temperature across the sky, the features subtending a given angle are associated with physics on a spatial scale that can be computed from the angle and the angular diameter distance to the last scattering surface.

In the previous chapters we studied the geometrical properties of the universe. Now we turn to its thermal history. This history can be subdivided into several periods. Here we focus mainly on the period between neutrino decoupling and recombination. This period is characterized by a sequence of important departures from thermal and chemical equilibrium that shaped the present state of the universe.

We begin with an overview of the main thermal events and then turn to their detailed description. In particular, in this chapter we study the decoupling of neutrinos, primordial nucleosynthesis and recombination. Our considerations are based on well understood and tested laws of particle, nuclear and atomic physics below a few MeV and, as such, are not likely to be a rich source of future research. However, this is important background material which underlies the concept of the hot expanding universe.

The composition of the universe

According to the Friedmann equations, the expansion rate of the universe is determined by the energy density and equation of state of its constituents. The main components of the matter composition that played an important role at temperatures below a few MeV are primordial radiation, baryons, electrons, neutrinos, dark matter and dark energy.

Primordial radiation The cosmic microwave background (CMB) radiation has temperature Tγ0 ≃ 2.73 K. Its current energy density is about εγ0 ≃ 10−34 g cm−3 and constitutes only 10−5 of the total energy density.

The most important feature of our universe is its large scale homogeneity and isotropy. This feature ensures that observations made from our single vantage point are representative of the universe as a whole and can therefore be legitimately used to test cosmological models.

For most of the twentieth century, the homogeneity and isotropy of the universe had to be taken as an assumption, known as the “Cosmological Principle.” Physicists often use the word “principle” to designate what are at the time wild, intuitive guesses in contrast to “laws,” which refer to experimentally established facts.

The Cosmological Principle remained an intelligent guess until firm empirical data, confirming large scale homogeneity and isotropy, were finally obtained at the end of the twentieth century. The nature of the homogeneity is certainly curious. The observable patch of the universe is of order 3000 Mpc (1 Mpc ≃ 3.26 × 106 light years ≃ 3.08 × 1024 cm). Redshift surveys suggest that the universe is homogeneous and isotropic only when coarse grained on 100 Mpc scales; on smaller scales there exist large inhomogeneities, such as galaxies, clusters and superclusters. Hence, the Cosmological Principle is only valid within a limited range of scales, spanning a few orders of magnitude.

Moreover, theory suggests that this may not be the end of the story. According to inflationary theory, the universe continues to be homogeneous and isotropic over distances larger than 3000 Mpc, but it becomes highly inhomogeneous when viewed on scales much much larger than the observable patch.

This textbook is designed both for serious students of physics and astrophysics and for those with a particular interest in learning about theoretical cosmology. There are already many books that survey current observations and describe theoretical results; my goal is to complement the existing literature and to show where the theoretical results come from. Cosmology uses methods from nearly all fields of theoretical physics, among which are General Relativity, thermodynamics and statistical physics, nuclear physics, atomic physics, kinetic theory, particle physics and field theory. I wanted to make the book useful for undergraduate students and, therefore, decided not to assume preliminary knowledge in any specialized field. With very few exceptions, the derivation of every formula in the book begins with basic physical principles from undergraduate courses. Every chapter starts with a general elementary introduction. For example, I have tried to make such a geometrical topic as conformal diagrams understandable even to those who have only a vague idea about General Relativity. The derivations of the renormalization group equation, the effective potential, the non-conservation of fermion number, and quantum cosmological perturbations should also, in principle, require no prior knowledge of quantum field theory. All elements of the Standard Model of particle physics needed in cosmological applications are derived from the initial idea of gauge invariance of the electromagnetic field. Of course, some knowledge of general relativity and particle physics would be helpful, but this is not a necessary condition for understanding the book.

The laws of particle interactions are well established only below the energy currently reached by accelerators, which is about a few hundred GeV. The next generation of accelerators will allow us to go a couple of orders of magnitude further, but even in the remote future it will be impossible to overcome the existing gap of about seventeen orders of magnitude to reach the Planckian scale. Therefore, the only “laboratories” for testing particle theories at very high energies are the very early universe and astrophysical sources of highly energetic particles. The quality of cosmological information is much worse than that gained from accelerators. However, given the lack of choice, we can still hope to learn essential features of high-energy physics based on cosmological and astrophysical observations.

The particle theory describing interactions below the TeV scale is called the Standard Model and it comprises the unified electroweak theory and quantum chromodynamics, both based on the idea of local gauge symmetry. Attempts to incorporate the electroweak and strong interactions in some larger symmetry group and thus unify them have not yet met with success. Unfortunately, there are too many ways to extend the theory beyond the Standard Model while remaining in agreement with available experimental data. Only further experiments can help us in selecting the “correct theory of nature.”

Measurements of the cosmic microwave background tell us that the universe was very homogeneous and isotropic at the time of recombination. Today, however, the universe has a well developed nonlinear structure. This structure takes the form of galaxies, clusters and superclusters of galaxies, and, on larger scales, of voids, sheets and filaments of galaxies. Deep redshift surveys show, however, that when averaged over a few hundred megaparsecs, the inhomogeneities in the density distribution remain small. The simple explanation as to how nonlinear structure could develop from small initial perturbations is based on the fact of gravitational instability.

Gravitational instability is a natural property of gravity. Matter is attracted to high-density regions, thus amplifying already existing inhomogeneities. To ensure that the small initial inhomogeneities present at recombination produce the nonlinear structure observed today, we have to study how fast they grow in an expanding universe. The complete general relativistic analysis of gravitational instability is rather involved and the physical interpretation of the results is not always straightforward. For this reason we develop the theory of gravitational instability in several steps.

In this chapter we consider gravitational instability in the Newtonian theory of gravity. The results derived in this theory are applicable only to nonrelativistic matter on scales not exceeding the Hubble horizon. First, we find out how small inhomogeneities grow in a nonexpanding universe (Jeans theory). The main purpose here is to determine which types of perturbations can exist in homogeneous, isotropic media, and to introduce methods to analyze them.

Matter is distributed very homogeneously and isotropically on scales larger than a few hundred megaparsecs. The CMB gives us a “photograph” of the early universe, which shows that at recombination the universe was extremely homogeneous and isotropic (with accuracy ∼ 10−4) on all scales up to the present horizon. Given that the universe evolves according to the Hubble law, it is natural to ask which initial conditions could lead to such homogeneity and isotropy.

To obtain an exhaustive answer to this question we have to know the exact physical laws which govern the evolution of the very early universe. However, as long as we are interested only in the general features of the initial conditions it suffices to know a few simple properties of these laws. We will assume that inhomogeneity cannot be dissolved by expansion. This natural surmise is supported by General Relativity (see Part II of this book for details). We will also assume that nonperturbative quantum gravity does not play an essential role at sub-Planckian curvatures. On the other hand, we are nearly certain that nonperturbative quantum gravity effects become very important when the curvature reaches Planckian values and the notion of classical spacetime breaks down. Therefore we address the initial conditions at the Planckian time ti = tPl ∼ 10−43 s.

In this chapter we discuss the initial conditions problem we face in a decelerating universe and show how this problem can be solved if the universe undergoes a stage of the accelerated expansion known as inflation.

Since the beginning of the 1970s, we have witnessed spectacular progress in the development of cosmology, which started with a breakthrough in the theoretical understanding of the physical processes in the early universe and culminated in a series of observational discoveries. The time is ripe for a textbook which summarizes the new knowledge in a rigorous and yet accessible form.

The beginning of the new era in theoretical cosmology can be associated with the development of the gauge theories of weak, electromagnetic and strong interactions. Until that time, we had no idea of properties of matter at densities much greater than nuclear density ∼ 1014 g/cm3, and everybody thought that the main thing we need to know about the early universe is the equation of state of superdense matter. In the beginning of the 1970s we learned that not only the size and the temperature of our universe, but also the properties of elementary particles in the early universe were quite different from what we see now. According to the theory of the cosmological phase transitions, during the first 10−10 seconds after the big bang there was not much difference between weak and electromagnetic interactions. The discovery of the asymptotic freedom for the first time allowed us to investigate the properties of matter even closer to the big bang, at densities almost 80 orders of magnitude higher than the nuclear density.

One of the central issues of contemporary cosmology is the explanation of the origin of primordial inhomogeneities, which serve as the seeds for structure formation. Before the advent of inflationary cosmology the initial perturbations were postulated and their spectrum was designed to fit observational data. In this way practically any observation could be “explained”, or more accurately described, by arranging the appropriate initial conditions. In contrast, inflationary cosmology truly explains the origin of primordial inhomogeneities and predicts their spectrum. Thus it becomes possible to test this theory by comparing its predictions with observations.

According to cosmic inflation, primordial perturbations originated from quantum fluctuations. These fluctuations have substantial amplitudes only on scales close to the Planckian length, but during the inflationary stage they are stretched to galactic scales with nearly unchanged amplitudes. Thus, inflation links the large-scale structure of the universe to its microphysics. The resulting spectrum of inhomogeneities is not very sensitive to the details of any particular inflationary scenario and has nearly universal shape. This leads to concrete predictions for the spectrum of cosmic microwave background anisotropies.

In the previous chapter we studied gravitational instability in a universe filled with hydrodynamical matter. To understand the generation of primordial fluctuations we have to extend our analysis to the case of a scalar field condensate and quantize the cosmological perturbations. In this chapter we study the behavior of perturbations during an inflationary stage and calculate their resulting spectrum.