Ever since I wrote my semi-popular book The Ultimate Fate of the Universe I have been meaning to write a technical version of it. There are of course many good books on cosmology and it seemed doubtful to me whether the inclusion of a chapter on the distant future of the universe would itself justify another book. However, in recent years there have been two interesting developments in cosmology, namely inflationary models and quantum cosmology, with their connection with particle physics and quantum mechanics, and I believe the time is ripe for a book containing these topics. Accordingly, this book has a chapter each on inflationary models, quantum cosmology and the distant future of the universe (as well as a chapter on singularities not usually contained in the standard texts).

This is essentially an introductory book. None of the topics dealt with have been treated exhaustively. However, I have tried to include enough introductory material and references so that the reader can pursue the topic of his interest further.

A knowledge of general relativity is helpful; I have included a brief exposition of it in Chapter 2 for those who are not familiar with it. This material is very standard; the form given here is taken essentially from my book Rotating Fields in General Relativity.

In the process of writing this book, I discovered two exact cosmological solutions, one connecting radiation and matter dominated eras and the other representing an inflationary model for a sixth degree potential. These have been included in Sections 4.5 and 9.4 respectively as I believe they are new and have some physical relevance.

Introduction

In the previous chapters we have considered in some detail the ‘standard’ model of the universe. It is pertinent to ask what the prediction of the standard model is for the distant future of the universe. The future of the universe has been the subject of much speculation, in one form or another, from time immemorial. It is only in the last few decades that enough progress has been achieved in cosmology to study this question scientifically. In this chapter we shall attempt to provide an account of – or at any rate limit the possibilities for – the distant future of the universe, on the basis of the present state of knowledge. We refer the reader to Rees (1969), Davies (1973), Islam (1977, 1979a,b, 1983a,b), Barrow and Tipler (1978) and Dyson (1979) for more material on this topic. This chapter is based mostly on the papers by Islam and Dyson.

The distant future of the universe is dramatically different depending on whether it expands forever, or it stops expanding at some future time and recollapses. In the earlier chapters we have considered in detail the conditions under which these possibilities are likely to arise. As galaxies are the basic constituents of the universe, to examine the distant future of the universe we must consider the long term evolution of a typical galaxy. We will first assume that we are in an open universe, or, at any rate, that an indefinite time in the future is available.

Introduction

We saw in the previous chapters that the standard model predicts a singularity sometime in the past history of the universe where the density tends to infinity. In Chapter 7 we also saw there is reason to believe that the existence of singularities may not be a feature peculiar to the highly symmetric Friedmann models, but may exist in any general solution of Einstein's equations representing a cosmological situation. Many physicists think that the existence of singularities in general relativity is unphysical and points to the breakdown of the theory in the extreme situations that singularities purport to represent. Indeed, in these extreme conditions the quantum nature of space-time may come into play, and there have been suggestions that when the quantum theory of gravitation is taken into account, singularities may not arise. However, the quantization of gravitation is notoriously difficult – there does not, at present, exist any satisfactory quantum theory of gravitation, whether the gravitation theory is general relativity or any other reasonable theory of gravity. However, there have been some approximate schemes to try and answer at least partially some of the questions that a quantum theory of gravitation is supposed to answer. One of these schemes is quantum cosmology. We shall only give a brief and incomplete account of quantum cosmology in this chapter, as the technicalities are mostly beyond the scope of this book. This chapter is based mainly on Hartle and Hawking (1983), Hartle (1984, 1986), Narlikar and Padmanabhan (1983), and Islam (1993, 1994).

Introduction

As mentioned in Chapter 1, the ‘cosmic background radiation’ discovered originally by Penzias and Wilson in 1965 provides evidence that the universe must have gone through a hot dense phase. We have also seen that the Friedmann models (described in Chapter 4), if they are regarded as physically valid, predict that the density of mass-energy must have been very high in the early epochs of the universe. In fact, of course, the Friedmann models imply that the mass-energy density goes to infinity as the time t approaches the ‘initial moment’ or ‘the initial singularity’, at t = 0. This is what is referred to as the ‘big bang’, meaning an explosion at every point of the universe in which matter was thrown asunder violently, from an infinite or near infinite density. However, the precise nature of the physical situation at t = 0, or the situation before t = 0 (or whether it is physically meaningful to talk about any time before t = 0) – these sorts of questions are entirely unclear. In this and the following chapter we shall try to deal partly with some questions of this kind. In the present chapter we simply assume that there was a catastrophic event at t = 0, and try to describe the state of the universe from about t = 0.01 s until about t = one million years. This will be our definition of the ‘early universe’, which specifically excludes the first hundredth of a second or so, during which, as we shall see in the next chapter, and as speculations go, events occurred which are of a very different nature from those occurring in the ‘early universe’ according to the definition given here.

The Bronze Age (2000 BC–1000 BC) was a period of major advances and changes in the riverine cultures of Mesopotamia, Egypt, India and China. Exceptional developments in irrigation and agriculture had led to the establishment of large urban civilisations in which arts and sciences were encouraged and patronised. Efficient tax collection and revenue management had freed funds to support a class of people who could devote their time to study, observation and contemplation. The alphabet and numbers were formalised and the practice of keeping records, both civil and military, was well established. Concepts of space and measurement and concerns with heavenly bodies and physical structures led to the development of arithmetic and geometry.

The development of predictive and exact sciences followed from the study of motions of the Sun, Moon and the five visible planets. The periodicity of the motion of these bodies was utilised to establish a quantitative measure of time, and the correlation between the rising and setting of groups of stars and seasons was developed into a calendar, which we use today (in slightly modified form). The systematic study of the motion of the Sun and Moon by the Mesopotamian priest-astronomers enabled them to identify the cause of eclipses and also to predict future eclipses. The discipline of record keeping had been extended to note the occurrences of unusual astronomical events and irregularities in the movements of planets. The belief-systems developed by these cultures defined man's position in the cosmos and his relationship with nature.

Gravity is the only truly universal force. It moulded the universe and it almost certainly will overwhelmingly determine the end of the universe. Today we also know (certainly believe) that gravitation is primarily responsible for the formation of the large structures we see around us: the Earth, the solar system, the stars, and the galaxies. Gravity has fashioned the beautiful and awe-inspiring sights in the sky, which have inspired both philosophers and mystics. Without gravity the sky would have been a very boring sight. Even more exotic objects, only visible at radio, X-ray or other energies, are present in the sky and these have also been fashioned by gravity. After the formation of the solar system, gravity has played a benign role in the evolution of life in the solar system. The strong gravity of Jupiter has shielded Earth from destructive impacts by comets and asteroids, and it is reasonable to say that life on Earth would not have survived without this ‘gravity shield’. It is for this reason that there is such excitement at the discovery of large planets around other stars. Life, as we know it, may not (almost certainly does not) exist on these large planets, but without such a large planet and its gravity shield, life certainly would not survive and flourish on an inner planet if there is one. In this chapter the central role of gravity in shaping the universe and even the climate on Earth is described.

FIGURES OF PLANETS

The sizes and the figures of the planets have been subjects of inquiry and discussion since antiquity. The Greeks accepted the Platonic belief in perfect shapes and believed that the planets, and the Earth in particular, were perfect spheres.

Gravity is beautifully successful in describing the universe at macroscopic level but at microscopic level it is totally inadequate. By contrast, quantum theory is highly successful in describing matter on small scales. The detailed understanding of chemical reactions, lasers, microchips and nuclear weapons is based entirely on quantum physics. The concept of an atom originated in Greek antiquity, but it was suppressed under the baleful influence of Plato and Aristotle. In the early nineteenth century John Dalton resurrected ‘atomism’. Dalton was a Quaker and was born in 1766 in Eaglesfield, Cumberland, in England. He recognised that atoms could help to understand the data being accumulated by chemical and physical experiments. Dalton's atoms were the smallest indivisible unit of a substance and they retained their chemical properties. He maintained that chemical reactions were just the rearrangement of these basic units of matter. Dalton presented his work in a two-volume thesis called New System of Chemical Philosophy, published between 1808 and 1827. This laid the basis of modern chemistry. Dalton's atoms were not accepted enthusiastically by all chemists of the early nineteenth century. However, it was soon realised that the chemists and physicists of the day had independently accumulated data which suggested the atomic nature of matter. In particular, physicists such as Maxwell and Boltzmann maintained that the pressure exerted by gas in a container could be explained if the gas was assumed to be a collection of hard spheres, like billiard balls, which bounced off the walls of a container in accordance with Newtonian mechanics.

For 200 years, from the beginning of the eighteenth century to the beginning of the twentieth century, Newtonian mechanics reigned supreme. By the late nineteenth century, the few simple laws of Newtonian physics could explain, with uncanny accuracy, most of the disparate phenomena of the natural world. Everything in the heavens and on Earth appeared to obey the laws, and the mastery of these laws was bringing mankind the mastery of the environment. These laws dominated the way both scientists and laypersons thought. Newtonian mechanics has a deterministic framework for the cosmos and this was deeply satisfying to the Judeo-Christian culture of Western Europe. There were those who questioned Newton's assumptions of absolute space and time, independent of man, but anyone doubting the validity of the laws of motion or gravitation was not taken seriously by the scientific community.

Newton had applied his theory of gravitation to two-body systems, such as the Sun and a planet. In the eighteenth century various attempts were made to extend it to three gravitating bodies. In 1682 Halley had claimed that the comet then observed in the sky had also appeared in 1531 and 1607; given, then, that the period of the comet was about 75 years, he predicted that it would reappear in 1758. Months before its appearance, the French mathematician Alexis Clairaut used tedious and brute-force mathematics to calculate the gravitational perturbations of Jupiter and Saturn on the otherwise elliptical orbit of Comet Halley.

Our world is ruled by two sets of laws: the laws of gravity and the laws of quantum mechanics. The laws of gravity describe the large structures in the universe such as the Earth, the solar system, stars, galaxies and the universe itself. These laws allow us to predict the path and motion of spacecraft and asteroids and also the evolution of the universe. The laws of quantum mechanics, on the other hand, describe the very small structures such as molecules, atoms and subatomic particles. They enable us to understand the three subatomic forces, lasers, CD players and nuclear weapons. One of the great puzzles of the twentieth century is that these two sets of laws, each employing a different set of mathematics and each making astonishingly accurate predictions in its own regime, should be so profoundly different and incompatible.

Quantum mechanics is a child of the twentieth century. Its origins can be traced back to the year 1900, when Planck proposed the particle nature of electromagnetic radiation to explain the black-body spectrum. The character of the laws of motion and the laws of gravity, on the other hand, has unfolded over a considerably longer period. Today, the concepts of mass, force and gravity are very familiar, but they are also deeply mysterious and are intimately linked to our understanding of motion. Historically motion was perhaps the first natural phenomenon to be investigated scientifically.

Gunpowder, an invention imported from China, proved immensely popular with the warring princes of fifteenth-century Europe. These princes were using gunpowder in their frequent wars, to hurl large projectiles at or over the walls of towns and cities they were attacking. By the middle of the sixteenth century the casting and boring of cannons had progressed to a stage where serious consideration had to be given to aiming and firing of guns. All over Europe gunners began to look at ways of increasing the range and aim of their artillery. But the path of the cannon ball made no sense within the context of Aristotelian doctrine. The Aristotelian laws of motion stated that the natural state of all ‘earthly’ objects was to be at rest. Motion away from the centre of the Earth was only possible with a ‘mover’ which had to be in contact with the object being moved. When the mover was removed the object should fall straight down to Earth. But cannon balls (or projectiles generally) did not fall straight down to Earth after they left the muzzle of the gun – they followed a curved path. Even the most ardent supporter of Aristotle could see that there was a flaw in the Aristotelian laws of motion. An alternative to the Aristotelian attempts to explain the motion of projectiles was the concept of the impressed force. According to this view, there is an incorporeal motive force that is imparted to the projectile, causing it to continue moving. This view was proposed in the seventh century by John Philoponus, by the eleventh-century Persian philosopher Avicenna and the twelfth-century Arab philosopher Abu al Barakat al-Baghdadi.

The Arab instrumentation, observations, astronomical tables and maps retained their superiority at least until the middle of the thirteenth century. But the diffusion of Greek literature from the Middle East and Spain gradually rekindled the spirit of inquiry in Europe. In the twelfth and the thirteenth centuries the writings of Aristotle on physics, metaphysics and ethics became available in Latin, translated from either Greek or Arab sources. These were crucial for the greatest of the medieval Christian thinkers, St Thomas Aquinas (1225–1274 AD). Aquinas studied Aristotle in great detail and wrote numerous commentaries on a variety of Aristotle's works. One of the Aristotelian themes that influenced Aquinas was that knowledge is not innate but is gained from the senses and from logical inference of self-evident truths. To this Aquinas added divine revelation as an additional basis for inference. Aquinas used Aristotle's dictum that everything is moved by something else to argue that the observable order of cause and effect is not self-explanatory. It can only be explained by existence of the ‘First Cause’ or God. The concept of the ‘First Cause’ can be traced back to the Greek thinkers and had become an underlying assumption in the Judeo-Christian world-view. The argument for the existence of God inferred from motion was given doctrinal status in the first two ‘proofs of God’ of Aquinas:

• Things are in motion, hence there is a first mover

• Things are caused, hence there is a first cause

But where Aristotle was concerned with understanding how the world functions, Aquinas was concerned more fundamentally with explaining why it exists.