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FIFTH LECTURE - THE ORIGIN AND FATE OF THE UNIVERSE

Throughout the 1970s I had been working mainly on black holes. However, in 1981 my interest in questions about the origin of the universe was reawakened when I attended a conference on cosmology in the Vatican. The Catholic church had made a bad mistake with Galileo when it tried to lay down the law on a question of science, declaring that the sun went around the Earth. Now, centuries later, it had decided it would be better to invite a number of experts to advise it on cosmology.

At the end of the conference the participants were granted an audience with the pope. He told us that it was okay to study the evolution of the universe after the big bang, but we should not inquire into the big bang itself because that was the moment of creation and therefore the work of God.

I was glad then that he did not know the subject of the talk I had just given at the conference. I had no desire to share the fate of Galileo; I have a lot of sympathy with Galileo, partly because I was born exactly three hundred years after his death.

THE HOT BIG BANG MODEL

In order to explain what my paper was about, I shall first describe the generally accepted history of the universe, according to what is known as the “hot big bang model.” This assumes that the universe is described by a Friedmann model, right back to the big bang. In such models one finds that as the universe expands, the temperature of the matter and radiation in it will go down. Since temperature is simply a measure of the average energy of the particles, this cooling of the universe will have a major effect on the matter in it. At very high temperatures, particles will be moving around so fast that they can escape any attraction toward each other caused by the nuclear or electromagnetic forces. But as they cooled off, one would expect particles that attract each other to start to clump together.

At the big bang itself, the universe had zero size and so must have been infinitely hot. But as the universe expanded, the temperature of the radiation would have decreased. One second after the big bang it would have fallen to about ten thousand million degrees. This is about a thousand times the temperature at the center of the sun, but temperatures as high as this are reached in H-bomb explosions. At this time the universe would have contained mostly photons, electrons, and neutrinos and their antiparticles, together with some protons and neutrons.

As the universe continued to expand and the temperature to drop, the rate at which electrons and the electron pairs were being produced in collisions would have fallen below the rate at which they were being destroyed by annihilation.

So most of the electrons and antielectrons would have annihilated each other to produce more photons, leaving behind only a few electrons.

About one hundred seconds after the big bang, the temperature would have fallen to one thousand million degrees, the temperature inside the hottest stars. At this temperature, protons and neutrons would no longer have sufficient energy to escape the attraction of the strong nuclear force. They would start to combine together to produce the nuclei of atoms of deuterium, or heavy hydrogen, which contain one proton and one neutron. The deuterium nuclei would then have combined with more protons and neutrons to make helium nuclei, which contained two protons and two neutrons. There would also be small amounts of a couple of heavier elements, lithium and beryllium.

One can calculate that in the hot big bang model about a quarter of the protons and neutrons would have been converted into helium nuclei, along with a small amount of heavy hydrogen and other elements. The remaining neutrons would have decayed into protons, which are the nuclei of ordinary hydrogen atoms. These predictions agree very well with what is observed.

The hot big bang model also predicts that we should be able to observe the radiation left over from the hot early stages. However, the temperature would have been reduced to a few degrees above absolute zero by the expansion of the universe. This is the explanation of the microwave background of radiation that was discovered by Penzias and Wilson in 1965. We are therefore thoroughly confident that we have the right picture, at least back to about one second after the big bang. Within only a few hours of the big bang, the production of helium and other elements would have stopped. And after that, for the next million years or so, the universe would have just continued expanding, without anything much happening. Eventually, once the temperature had dropped to a few thousand degrees, the electrons and nuclei would no longer have had enough energy to overcome the electromagnetic attraction between them. They would then have started combining to form atoms.

The universe as a whole would have continued expanding and cooling. However, in regions that were slightly denser than average, the expansion would have been slowed down by extra gravitational attraction. This would eventually stop expansion in some regions and cause them to start to recollapse. As they were collapsing, the gravitational pull of matter outside these regions might start them rotating slightly. As the collapsing region got smaller, it would spin faster—just as skaters spinning on ice spin faster as the draw in their arms. Eventually, when the region got small enough, it would be spinning fast enough to balance the attraction of gravity. In this way, disklike rotating galaxies were born.

As time went on, the gas in the galaxies would break up into smaller clouds that would collapse under their own gravity. As these contracted, the temperature of the gas would increase until it became hot enough to start nuclear reactions. These would convert the hydrogen into more helium, and the heat given off would raise the pressure, and so stop the clouds from contracting any further. They would remain in this state for a long time as stars like our sun, burning hydrogen into helium and radiating the energy as heat and light.

More massive stars would need to be hotter to balance their stronger gravitational attraction. This would make the nuclear fusion reactions proceed so much more rapidly that they would use up their hydrogen in as little as a hundred million years. They would then contract slightly and, as they heated up further, would start to convert helium into heavier elements like carbon or oxygen. This, however, would not release much more energy, so a crisis would occur, as I described in my lecture on black holes.

What happens next is not completely clear, but it seems likely that the central regions of the star would collapse to a very dense state, such as a neutron star or black hole. The outer regions of the star may get blown off in a tremendous explosion called a supernova, which would outshine all the other stars in the galaxy. Some of the heavier elements produced near the end of the star’s life would be flung back into the gas in the galaxy. They would provide some of the raw material for the next generation of stars.

Our own sun contains about 2 percent of these heavier elements because it is a second– or third–generation star. It was formed some five thousand million years ago out of a cloud of rotating gas containing the debris of earlier supernovas. Most of the gas in that cloud went to form the sun or got blown away. However, a small amount of the heavier elements collected together to form the bodies that now orbit the sun as planets like the Earth.

OPEN QUESTIONS

This picture of a universe that started off very hot and cooled as it expanded is in agreement with all the observational evidence that we have today. Nevertheless, it leaves a number of important questions unanswered. First, why was the early universe so hot? Second, why is the universe so uniform on a large scale—why does it look the same at all points of space and in all directions?

Third, why did the universe start out with so nearly the critical rate of expansion to just avoid recollapse? If the rate of expansion one second after the big bang had been smaller by even one part in a hundred thousand million million, the universe would have recollapsed before it ever reached its present size. On the other hand, if the expansion rate at one second had been larger by the same amount, the universe would have expanded so much that it would be effectively empty now.

Fourth, despite the fact that the universe is so uniform and homogenous on a large scale, it contains local lumps such as stars and galaxies. These are thought to have developed from small differences in the density of the early universe from one region to another. What was the origin of these density fluctuations?

The general theory of relativity, on its own, cannot explain these features or answer these questions. This is because it predicts that the universe started off with infinite density at the big bang singularity. At the singularity, general relativity and all other physical laws would break down. One cannot predict what would come out of the singularity. As I explained before, this means that one might as well cut any events before the big bang out of the theory, because they can have no effect on what we observe. Space–time would have a boundary— a beginning at the big bang. Why should the universe have started off at the big bang in just such a way as to lead to the state we observe today? Why is the universe so uniform, and expanding at just the critical rate to avoid recollapse? One would feel happier about this if one could show that quite a number of different initial configurations for the universe would have evolved to produce a universe like the one we observe.

If this is the case, a universe that developed from some sort of random initial conditions should contain a number of regions that are like what we observe. There might also be regions that were very different. However, these regions would probably not be suitable for the formation of galaxies and stars. These are essential prerequisites for the development of intelligent life, at least as we know it. Thus, these regions would not contain any beings to observe that they were different.

When one considers cosmology, one has to take into account the selection principle that we live in a region of the universe that is suitable for intelligent life. This fairly obvious and elementary consideration is sometimes called the anthropic principle. Suppose, on the other hand, that the initial state of the universe had to be chosen extremely carefully to lead to something like what we see around us. Then the universe would be unlikely to contain any region in which life would appear.

In the hot big bang model that I described earlier, there was not enough time in the early universe for heat to have flowed from one region to another. This means that different regions of the universe would have had to have started out with exactly the same temperature in order to account for the fact that the microwave background has the same temperature in every direction we look. Also, the initial rate of expansion would have had to be chosen very precisely for the universe not to have recollapsed before now. This means that the initial state of the universe must have been very carefully chosen indeed if the hot big bang model was correct right back to the beginning of time. It would be very difficult to explain why the universe should have begun in just this way, except as the act of a God who intended to create beings like us.

THE INFLATIONARY MODEL

In order to avoid this difficulty with the very early stages of the hot big bang model, Alan Guth at the Massachusetts Institute of Technology put forward a new model. In this, many different initial configurations could have evolved to something like the present universe. He suggested that the early universe might have had a period of very rapid, or exponential, expansion. This expansion is said to be inflationary—an analogy with the inflation in prices that occurs to a greater or lesser degree in every country. The world record for price inflation was probably in Germany after the first war, when the price of a loaf of bread went from under a mark to millions of marks in a few months. But the inflation we think may have occurred in the size of the universe was much greater even than that—a million million million million million times in only a tiny fraction of a second. Of course, that was before the present government.

Guth suggested that the universe started out from the big bang very hot. One would expect that at such high temperatures, the strong and weak nuclear forces and the electromagnetic force would all be unified into a single force. As the universe expanded, it would cool, and particle energies would go down. Eventually there would be what is called a phase transition, and the symmetry between the forces would be broken. The strong force would become different from the weak and electromagnetic forces. One common example of a phase transition is the freezing of water when you cool it down. Liquid water is symmetrical, the same at every point and in every direction. However, when ice crystals form, they will have definite positions and will be lined up in some direction. This breaks the symmetry of the water.

In the case of water, if one is careful, one can “supercool” it. That is, one can reduce the temperature below the freezing point—0 degrees centigrade—without ice forming. Guth suggested that the universe might behave in a similar way: The temperature might drop below the critical value without the symmetry between the forces being broken. If this happened, the universe would be in an unstable state, with more energy than if the symmetry had been broken. This special extra energy can be shown to have an antigravitational effect. It would act just like a cosmological constant.

Einstein introduced the cosmological constant into general relativity when he was trying to construct a static model of the universe. However,in this case, the universe would already be expanding. The repulsive effect of this cosmological constant would therefore have made the universe expand at an everincreasing rate. Even in regions where there were more matter particles than average, the gravitational attraction of the matter would have been outweighed by the repulsion of the effective cosmological constant. Thus, these regions would also expand in an accelerating inflationary manner.

As the universe expanded, the matter particles got farther apart. One would be left with an expanding universe that contained hardly any particles. It would still be in the supercooled state, in which the symmetry between the forces is not broken. Any irregularities in the universe would simply have been smoothed out by the expansion, as the wrinkles in a balloon are smoothed away when you blow it up. Thus, the present smooth and uniform state of the universe could have evolved from many different nonuniform initial states. The rate of expansion would also tend toward just the critical rate needed to avoid recollapse.

Moreover, the idea of inflation could also explain why there is so much matter in the universe. There are something like 1,080 particles in the region of the universe that we can observe. Where did they all come from? The answer is that, in quantum theory, particles can be created out of energy in the form of particle/antiparticle pairs. But that just raises the question of where the energy came from. The answer is that the total energy of the universe is exactly zero.

The matter in the universe is made out of positive energy. However, the matter is all attracting itself by gravity. Two pieces of matter that are close to each other have less energy than the same two pieces a long way apart. This is because you have to expend energy to separate them. You have to pull against the gravitational force attracting them together. Thus, in a sense, the gravitational field has negative energy. In the case of the whole universe, one can show that this negative gravitational energy exactly cancels the positive energy of the matter. So the total energy of the universe is zero.

Now, twice zero is also zero. Thus, the universe can double the amount of positive matter energy and also double the negative gravitational energy without violation of the conservation of energy. This does not happen in the normal expansion of the universe in which the matter energy density goes down as the universe gets bigger. It does happen, however, in the inflationary expansion, because the energy density of the supercooled state remains constant while the universe expands. When the universe doubles in size, the positive matter energy and the negative gravitational energy both double, so the total energy remains zero. During the inflationary phase, the universe increases its size by a very large amount. Thus, the total amount of energy available to make particles becomes very large. As Guth has remarked, “It is said that there is no such thing as a free lunch. But the universe is the ultimate free lunch.” THE END OF INFLATION

The universe is not expanding in an inflationary way today. Thus, there had to be some mechanism that would eliminate the very large effective cosmological constant. This would change the rate of expansion from an accelerated one to one that is slowed down by gravity, as we have today. As the universe expanded and cooled, one might expect that eventually the symmetry between the forces would be broken, just as supercooled water always freezes in the end. The extra energy of the unbroken symmetry state would then be released and would reheat the universe. The universe would then go on to expand and cool, just like the hot big bang model. However, there would now be an explanation of why the universe was expanding at exactly the critical rate and why different regions had the same temperature.

In Guth’s original proposal, the transition to broken symmetry was supposed to occur suddenly, rather like the appearance of ice crystals in very cold water. The idea was that “bubbles” of the new phase of broken symmetry would have formed in the old phase, like bubbles of steam surrounded by boiling water.

The bubbles were supposed to expand and meet up with each other until the whole universe was in the new phase. The trouble was, as I and several other people pointed out, the universe was expanding so fast that the bubbles would be moving away from each other too rapidly to join up. The universe would be left in a very nonuniform state, with some regions having symmetry between the different forces. Such a model of the universe would not correspond to what we see.

In October 1981 I went to Moscow for a conference on quantum gravity. After the conference, I gave a seminar on the inflationary model and its problems at the Sternberg Astronomical Institute. In the audience was a young Russian, Andrei Linde. He said that the difficulty with the bubbles not joining up could be avoided if the bubbles were very big. In this case, our region of the universe could be contained inside a single bubble. In order for this to work, the change from symmetry to broken symmetry must have taken place very slowly inside the bubble, but this is quite possible according to grand unified theories.

Linde’s idea of a slow breaking of symmetry was very good, but I pointed out that his bubbles would have been bigger than the size of the universe at the time. I showed that instead the symmetry would have broken everywhere at the same time, rather than just inside bubbles. This would lead to a uniform universe, like we observe. The slow symmetry breaking model was a good attempt to explain why the universe is the way it is. However, I and several other people showed that it predicted much greater variations in the microwave background radiation than are observed. Also, later work cast doubt on whether there would have been the right kind of phase transition in the early universe. A better model, called the chaotic inflationary model, was introduced by Linde in 1983. This doesn’t depend on phase transitions, and it can give us the right size of variations of the microwave background. The inflationary model showed that the present state of the universe could have arisen from quite a large number of different initial configurations. It cannot be the case, however, that every initial configuration would have led to a universe like the one we observe. So even the inflationary model does not tell us why the initial configuration was such as to produce what we observe. Must we turn to the anthropic principle for an explanation? Was it all just a lucky chance? That would seem a counsel of despair, a negation of all our hopes of understanding the underlying order of the universe.

QUANTUM GRAVITY

In order to predict how the universe should have started off, one needs laws that hold at the beginning of time. If the classical theory of general relativity was correct, the singularity theorem showed that the beginning of time would have been a point of infinite density and curvature. All the known laws of science would break down at such a point. One might suppose that there were new laws that held at singularities, but it would be very difficult even to formulate laws at such badly behaved points and we would have no guide from observations as to what those laws might be. However, what the singularity theorems really indicate is that the gravitational field becomes so strong that quantum gravitational effects become important: Classical theory is no longer a good description of the universe. So one has to use a quantum theory of gravity to discuss the very early stages of the universe. As we shall see, it is possible in the quantum theory for the ordinary laws of science to hold everywhere, including at the beginning of time. It is not necessary to postulate new laws for singularities, because there need not be any singularities in the quantum theory.

We don’t yet have a complete and consistent theory that combines quantum mechanics and gravity. However, we are thoroughly certain of some features that such a unified theory should have. One is that it should incorporate Feynman’s proposal to formulate quantum theory in terms of a sum over histories. In this approach, a particle going from A to B does not have just a single history as it would in a classical theory. Instead, it is supposed to follow every possible path in space–time. With each of these histories, there are associated a couple of numbers, one representing the size of a wave and the other representing its position in the cycle—its phase.

The probability that the particle, say, passes through some particular point is found by adding up the waves associated with every possible history that passes through that point. When one actually tries to perform these sums, however, one runs into severe technical problems. The only way around these is the following peculiar prescription: One must add up the waves for particle histories that are not in the real time that you and I experience but take place in imaginary time.

Imaginary time may sound like science fiction, but it is in fact a well–defined mathematical concept. To avoid the technical difficulties with Feynman’s sum over histories, one must use imaginary time. This has an interesting effect on space–time: The distinction between time and space disappears completely. A space–time in which events have imaginary values of the time coordinate is said to be Euclidean because the metric is positive definite.

In Euclidean space-time there is no difference between the time direction and directions in space. On the other hand, in real space-time, in which events are labeled by real values of the time coordinate, it is easy to tell the difference. The time direction lies within the light cone, and space directions lie outside. One can regard the use of imaginary time as merely a mathematical device—or trick—to calculate answers about real space-time. However, there may be more to it than that. It may be that Euclidean space-time is the fundamental concept and what we think of as real space-time is just a figment of our imagination.

When we apply Feynman’s sum over histories to the universe, the analogue of the history of a particle is now a complete curved space–time which represents the history of the whole universe. For the technical reasons mentioned above, these curved space–times must be taken to be Euclidean. That is, time is imaginary and is indistinguishable from directions in space. To calculate the probability of finding a real space-time with some certain property, one adds up the waves associated with all the histories in imaginary time that have that property. One can then work out what the probable history of the universe would be in real time.

THE NO BOUNDARY CONDITION

In the classical theory of gravity, which is based on real space-time, there are only two possible ways the universe can behave. Either it has existed for an infinite time, or else it had a beginning at a singularity at some finite time in the past. In fact, the singularity theorems show it must be the second possibility. In the quantum theory of gravity, on the other hand, a third possibility arises. Because one is using Euclidean space-times, in which the time direction is on the same footing as directions in space, it is possible for space-time to be finite in extent and yet to have no singularities that formed a boundary or edge. Space-time would be like the surface of the Earth, only with two more dimensions. The surface of the Earth is finite in extent but it doesn’t have a boundary or edge. If you sail off into the sunset, you don’t fall off the edge or run into a singularity. I know, because I have been around the world.

If Euclidean space-times direct back to infinite imaginary time or else started at a singularity, we would have the same problem as in the classical theory of specifying the initial state of the universe. God may know how the universe began, but we cannot give any particular reason for thinking it began one way rather than another. On the other hand, the quantum theory of gravity has opened up a new possibility. In this, there would be no boundary to space–time. Thus, there would be no need to specify the behavior at the boundary. There would be no singularities at which the laws of science broke down and no edge of space–time at which one would have to appeal to God or some new law to set the boundary conditions for space-time. One could say: “The boundary condition of the universe is that it has no boundary.” The universe would be completely self-contained and not affected by anything outside itself. It would be neither created nor destroyed. It would just be.

It was at the conference in the Vatican that I first put forward the suggestion that maybe time and space together formed a surface that was finite in size but did not have any boundary or edge. My paper was rather mathematical, however, so its implications for the role of God in the creation of the universe were not noticed at the time–just as well for me. At the time of the Vatican conference, I did not know how to use a no boundary idea to make predictions about the universe. However, I spent the following summer at the University of California, Santa Barbara. There, a friend and colleague of mine, Jim Hartle, worked out with me what conditions the universe must satisfy if space–time had no boundary.

I should emphasize that this idea that time and space should be finite without boundary is just a proposal. It cannot be deduced from some other principle. Like any other scientific theory, it may initially be put forward for aesthetic or metaphysical reasons, but the real test is whether it makes predictions that agree with observation. This, however, is difficult to determine in the case of quantum gravity, for two reasons. First, we are not yet sure exactly which theory successfully combines general relativity and quantum mechanics, though we know quite a lot about the form such a theory must have. Second, any model that described the whole universe in detail would be much too complicated mathematically for us to be able to calculate exact predictions. One therefore has to make approximations—and even then, the problem of extracting predictions remains a difficult one.

One finds, under the no boundary proposal, that the chance of the universe being found to be following most of the possible histories is negligible. But there is a particular family of histories that are much more probable than the others. These histories may be pictured as being like the surface of the Earth, with a distance from the North Pole representing imaginary time; the size of a circle of latitude would represent the spatial size of the universe. The universe starts at the North Pole as a single point. As one moves south, the circle of latitude get bigger, corresponding to the universe expanding with imaginary time. The universe would reach a maximum size at the equator and would contract again to a single point at the South Pole. Even though the universe would have zero size at the North and South poles, these points would not be singularities any more than the North and South poles on the Earth are singular. The laws of science will hold at the beginning of the universe, just as they do at the North and South poles on the Earth.

The history of the universe in real time, however, would look very different. It would appear to start at some minimum size, equal to the maximum size of the history in imaginary time. The universe would then expand in real time like the inflationary model. However, one would not now have to assume that the universe was created somehow in the right sort of state. The universe would expand to a very large size, but eventually it would collapse again into what looks like a singularity in real time. Thus, in a sense, we are still all doomed, even if we keep away from black holes. Only if we could picture the universe in terms of imaginary time would there be no singularities.

The singularity theorems of classical general relativity showed that the universe must have a beginning, and that this beginning must be described in terms of quantum theory. This in turn led to the idea that the universe could be finite in imaginary time, but without boundaries or singularities. When one goes back to the real time in which we live, however, there will still appear to be singularities. The poor astronaut who falls into a black hole will still come to a sticky end. It is only if he could live in imaginary time that he would encounter no singularities.

This might suggest that the so–called imaginary time is really the fundamental time, and that what we call real time is something we create just in our minds. In real time, the universe has a beginning and an end at singularities that form a boundary to space-time and at which the laws of science break down. But in imaginary time, there are no singularities or boundaries. So maybe what we call imaginary time is really more basic, and what we call real time is just an idea that we invent to help us describe what we think the universe is like. But according to the approach I described in the first lecture, a scientific theory is just a mathematical model we make to describe our observations. It exists only in our minds. So it does not have any meaning to ask: Which is real, “real” or “imaginary” time? It is simply a matter of which is a more useful description.

The no boundary proposal seems to predict that, in real time, the universe should behave like the inflationary models. A particularly interesting problem is the size of the small departures from uniform density in the early universe. These are thought to have led to the formation first of the galaxies, then of stars, and finally of beings like us. The uncertainty principle implies that the early universe cannot have been completely uniform. Instead, there must have been some uncertainties or fluctuations in the positions and velocities of the particles. Using the no boundary condition, one finds that the universe must have started off with just the minimum possible nonuniformity allowed by the uncertainty principle.

The universe would have then undergone a period of rapid expansion, like in the inflationary models. During this period, the initial nonuniformities would have been amplified until they could have been big enough to explain the origin of galaxies. Thus, all the complicated structures that we see in the universe might be explained by the no boundary condition for the universe and the uncertainty principle of quantum mechanics.

The idea that space and time may form a closed surface without boundary also has profound implications for the role of God in the affairs of the universe. With the success of scientific theories in describing events, most people have come to believe that God allows the universe to evolve according to a set of laws. He does not seem to intervene in the universe to break these laws. However, the laws do not tell us what the universe should have looked like when it started. It would still be up to God to wind up the clockwork and choose how to start it off. So long as the universe had a beginning that was a singularity, one could suppose that it was created by an outside agency. But if the universe is really completely self-contained, having no boundary or edge, it would be neither created nor destroyed. It would simply be. What place, then, for a creator?