Consider what happens when you observe a particular object, such as a page of this book. Light, either from the sun or a lamp, is continually being reflected off the page, and some of this light will enter your eyes. This will stimulate the retina, thereby causing a signal to be sent via the optic nerve to your brain. The brain then deciphers this signal and ‘reads’ the words that are printed. The key point is that light has to be reflected if you are to see the words. In effect, it must escape from the surface of the page.

Suppose that we are located on the surface of a star and, using a cannon, fire a tennis ball upwards. The distance travelled by the ball is determined by the speed it has when it is released. If the ball is moving sufficiently fast, it can escape completely from the star's influence. In that case, it need never fall back to the surface.

What would happen if the star began to collapse? Its density would gradually increase, and the force of gravity near its surface would become stronger. This means that the ball would have to be released with a greater speed than before in order to escape. If the collapse were to proceed unhindered, the density and gravitational pull of the star would soon become extremely high.

During the past few decades of research a plausible picture of the universe's most distant past has begun to emerge. The current view is that the universe came into existence some ten billion years ago in the form of a huge, exploding ‘fireball’. This was the big bang.

We are going to discuss some key features of the big bang in this book. In particular, we will look at the central question of just how ‘big’ it really was. But before we can begin our journey back towards the origin of the universe, we must work out our present location within it. Let us therefore embark on a brief sight-seeing tour of the universe.

The Earth belongs to a collection of objects known as the solar system. The central and largest object in this system is the sun. Nine planets, including the Earth, orbit the sun. Pluto is the planet most distant from the sun, and Pluto's orbit may be viewed as the edge of the solar system.

The nearest significant object to the Earth, its moon, is some four hundred thousand kilometres away. For comparison, the distance between the Earth and the sun is roughly one hundred and fifty million kilometres, whereas the average distance between the sun and Pluto is approximately six billion kilometres.

What lies beyond the solar system? As we travel past Pluto, the vastness of empty space soon becomes apparent.

The Planck time is the furthest back in time that we can go before quantum gravitational effects become significant. It has been suggested that inflation may have occurred when the universe was just 10−43 seconds old. This is precisely the era when superstring theory is supposed to be important.

More significantly, at least for our discussion, is that all the spatial dimensions of the universe would have had similar sizes. We discussed some of the theoretical arguments as to why there may be higher dimensions in Chapter 6. The existence of these extra dimensions has always been viewed as a problem. We know that the universe today contains only three large space dimensions, so why have the others remained so small that they cannot be seen? In other words, why is it that only three of the dimensions have grown to cosmological sizes? What is it that prevents the universe from having a different number of large dimensions?

Inflation, by its very nature, increases the size of the universe by a huge factor in a very small amount of time. We might therefore expect it to shed some light on the question of why three of the dimensions in the universe are so much larger than the others. The problem of small dimensions would be solved if only three of them were able to inflate to large sizes.

The theory of gravity that describes the large-scale dynamics of the universe was developed by Albert Einstein during the first two decades of the twentieth century. It is referred to as the general theory of relativity. It will be helpful if we now consider some of the ideas behind this theory.

We should begin by considering the speed of light. In 1865 the English physicist James Clerk Maxwell derived equations proving that electromagnetic radiation travels in a vacuum at a constant and finite speed. One of the key assumptions that Einstein later made was to suppose that two observers who are moving at a constant speed relative to one another would measure the same value for the speed of light.

Einstein's assumption goes against our intuition, to say the least. What might we expect? Speed is a relative quantity; we can measure the speed of an object only in terms of its relationship to something else. For example, when we say that a train travels through a station at a constant speed of one hundred kilometres an hour, what we really mean is that the distance between the train and the platform changes at this rate.

Let us consider two trains, A and B, that simultaneously travel through the station at this speed. If the two trains are moving in the same direction, they will appear to be at rest relative to each other.

Our journey has been remarkably successful. We have followed the development of the universe from its earliest moments through to the present. Our picture is that the universe has been expanding ever since it came into existence some ten billion years ago. However, Einstein's theory of general relativity breaks down at extremely early times when the universe was very small. The critical time scale over which the theory does not apply is the Planck time, corresponding to 10−43 seconds. We cannot employ Einstein's theory to determine the nature of the universe before this time.

The question we will address in this concluding chapter is whether we can travel back beyond the Planck time. Are we able to complete the story and cross this final barrier, or does the breakdown of Einstein's theory at this point represent a fundamental limit to our understanding?

If we are going to discuss the origin of the universe, we must first extend Einstein's theory in some suitable way. How might we proceed to modify the theory? We may answer this question by appealing to quantum effects. We have seen in the preceding chapters how these effects become significant on very small scales. The reason why Einstein's theory breaks down before the Planck time is that it fails to take into account the quantum fluctuations that are inherently present in any physical process involving gravity.

These uncertainties are not important on large scales and can be ignored.

Exploring the subject of elementary particles is rather like trying to find our way around an enormous zoo without the help of a guidebook to identify the different species of animal. How are we to make sense of it all? We will begin by summarizing some of the properties exhibited by the elementary particles. It is helpful to picture each particle as a tiny sphere that has three fundamental characteristics: electric charge, mass and spin. The different particles can be described in terms of these three basic quantities.

Electric charge is a familiar concept. Some particles carry it, but others do not. Those that do not are said to be electrically neutral. Likewise some particles have a mass, but others are massless. The mass of a particle contributes to its total energy, because mass is just another form of energy. Mass may be converted into energy and vice versa, and, indeed, a huge amount of energy may be produced from a relatively small mass.

The amount of spin that a particle carries determines its rate of rotation. We can view spinning particles as rotating about an axis. The electron is an example of a spinning particle. The spin of all elementary particles is severely restricted. Those particles that do not rotate have zero spin. Particles that do rotate have a spin that is directly related to that of the electron.

Elementary particles are divided into two main groups depending on the amount of spin that they carry.

A simple derivation of the Robertson–Walker metric

As we saw in the first chapter, the universe appears to be homogeneous and isotropic around us on scales of more than a 100 million light years or so, so that on this scale the density of galaxies is approximately the same and all directions from us appear to be equivalent. From these observations one is led to the Cosmological Principle which states that the universe looks the same from all positions in space at a particular time, and that all directions in space at any point are equivalent. This is an intuitive statement of the Cosmological Principle which needs to be made more precise. For example, what does one mean by ‘a particular time’? In Newtonian physics this concept is unambiguous. In special relativity the concept becomes well-defined if one chooses a particular inertial frame. In general relativity, however, there are no global inertial frames. To define ‘a moment of time’ in general relativity which is valid globally, a particular set of circumstances are necessary, which, in fact, are satisfied by a homo-geneous and isotropic universe.

To define ‘a particular time’ in general relativity which is valid globally in this case, we proceed as follows. Introduce a series of non-intersecting space-like hypersurfaces, that is, surfaces any two points of which can be connected to each other by a curve lying entirely in the hypersurface which is space-like everywhere.

Introduction

In Chapter 4 we saw that all the Friedmann models have singularities in the finite past, that is, at a finite time in the past, which we have called t = 0; the scale factor R(t) goes to zero and correspondingly some physical variables, such as the energy density, go to infinity. Only exceptionally, such as in the de Sitter or the steady state models (see Fig. 6.1), is there no singularity in the finite past. But these latter models have some unphysical or unorthodox feature, such as the continuous creation of matter, which is not generally acceptable. The presence of singularities in the universe, where physical variables such as the mass-energy density or the pressure or the strength of the gravitational field go to infinity seems doubtful to many people, who therefore feel uneasy about this kind of prediction of the equations of general relativity. This was partly the motivation with which Einstein searched for a ‘unified field theory’. In this connection he says (1950):

The material in the earlier edition, to which there appears to have been a favourable response, has been kept intact as far as possible in this new edition except for minor changes. A number of new additions have been made. Some standard topics have been added to the introduction to general relativity, such as Killing vectors. Not all these topics are used later in the book, but some may be of use to the beginning student for mathematical aspects of cosmological studies. Observational aspects have been brought up to date in an extended chapter on the cosmological constant. As this is a book on mathematical cosmology, the treatment of observations is not definitive or exhaustive by any means, but hopefully it is adequate. To clarify the role of the cosmological constant, much discussed in recent years, an exact, somewhat unusual solution with cosmological constant is included. Whether the solution is new is not clear: it is meant to provide a ‘comprehension exercise’. One reviewer of the earlier edition wondered why the Hubble constant and the deceleration parameter were chosen for a separate chapter. I believe these two parameters are among the most important in cosmology; adequate understanding of these helps to assess observations generally. Within the last year or two, through analyses of supernovae in distant galaxies, evidence seems to be emerging that the universe may be accelerating, or at least the deceleration may be not as much as was supposed earlier.

Introduction

In this appendix we consider topics some of which are extensions of material covered in the earlier chapters, and other additional ones which are not necessarily recent developments, but may have relevance for cosmological studies generally. We discuss both observational and theoretical matters.

Neutrino types

A significant discrepancy between theory based on the standard model of particle physics and observation of the flux of solar neutrinos on the surface of the Earth has been noticed for some years. In spite of much effort, an adequate explanation of this discrepancy has not been found.

As discussed in Section 8.8, the number of types of neutrino is of cosmological importance. Among relevant points to emerge at the 14th International Conference on Neutrino Physics and Astrophysics at CERN in 1990 was that there are three neutrino types unless the mass of the fourth one exceeds 45 GeV; the relic abundance of such a heavy neutrino is not sufficient to contribute to dark matter (Griest and Silk, 1990; Salati, 1990). These results come from LEP, the Large Electron Positron collider at CERN.

A large detector has been set up at Mount Ikenoyama in an active zinc mine in Japan, known as the Super-Kamiokande Detector (Kearns, Kajita and Totsuka, 1999). The original experiment was concerned with the detection of proton decay, and was set up at Kamioka, a mining town about 250 km from Tokyo.

In this chapter we present an elementary discussion of some basic concepts in cosmology. Although the mathematical formalism is essential, some of the main ideas underlying the formalism are simple and it helps to have an intuitive and qualitative notion of these ideas.

Cosmology is the study of the large-scale structure and behaviour of the universe, that is, of the universe taken as a whole. The term ‘as a whole’ applied to the universe needs a precise definition, which will emerge in the course of this book. It will be sufficient for the present to note that one of the points that has emerged from cosmological studies in the last few decades is that the universe is not simply a random collection of irregularly distributed matter, but it is a single entity, all parts of which are in some sense in unison with all other parts. This, at any rate, is the view taken in the ‘standard models’ which will be our main concern. We may have to modify these assertions when considering the inflationary models in a later chapter.

When considering the large-scale structure of the universe, the basic constituents can be taken to be galaxies, which are congregations of about 10 stars bound together by their mutual gravitational attraction. Galaxies tend to occur in groups called clusters, each cluster containing anything from a few to a few thousand galaxies.