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SECOND LECTURE - THE EXPANDING UNIVERSE

O ur sun and the nearby stars are all part of a vast collection of stars called the Milky Way galaxy. For a long time it was thought that this was the whole universe. It was only in 1924 that the American astronomer Edwin Hubble demonstrated that ours was not the only galaxy. There were, in fact, many others, with vast tracks of empty space between them. In order to prove this he needed to determine the distances to these other galaxies. We can determine the distance of nearby stars by observing how they change position as the Earth goes around the sun. But other galaxies are so far away that, unlike nearby stars, they really do appear fixed. Hubble was forced, therefore, to use indirect methods to measure the distances.

Now the apparent brightness of a star depends on two factors—luminosity and how far it is from us. For nearby stars we can measure both their apparent brightness and their distance, so we can work out their luminosity. Conversely, if we knew the luminosity of stars in other galaxies, we could work out their distance by measuring their apparent brightness. Hubble argued that there were certain types of stars that always had the same luminosity when they were near enough for us to measure. If, therefore, we found such stars in another galaxy, we could assume that they had the same luminosity. Thus, we could calculate the distance to that galaxy. If we could do this for a number of stars in the same galaxy, and our calculations always gave the same distance, we could be fairly confident of our estimate. In this way, Edwin Hubble worked out the distances to nine different galaxies.

We now know that our galaxy is only one of some hundred thousand million that can be seen using modern telescopes, each galaxy itself containing some hundred thousand million stars. We live in a galaxy that is about one hundred thousand light-years across and is slowly rotating; the stars in its spiral arms orbit around its center about once every hundred million years. Our sun is just an ordinary, average-sized, yellow star, near the outer edge of one of the spiral arms. We have certainly come a long way since Aristotle and Ptolemy, when we thought that the Earth was the center of the universe.

Stars are so far away that they appear to us to be just pinpoints of light. We cannot determine their size or shape. So how can we tell different types of stars apart? For the vast majority of stars, there is only one correct characteristic feature that we can observe—the color of their light. Newton discovered that if light from the sun passes through a prism, it breaks up into its component colors—its spectrum—like in a rainbow. By focusing a telescope on an individual star or galaxy, one can similarly observe the spectrum of the light from that star or galaxy. Different stars have different spectra, but the relative brightness of the different colors is always exactly what one would expect to find in the light emitted by an object that is glowing red hot. This means that we can tell a star’s temperature from the spectrum of its light. Moreover, we find that certain very specific colors are missing from stars’ spectra, and these missing colors may vary from star to star. We know that each chemical element absorbs the characteristic set of very specific colors. Thus, by matching each of those which are missing from a star’s spectrum, we can determine exactly which elements are present in the star’s atmosphere.

In the 1920s, when astronomers began to look at the spectra of stars in other galaxies, they found something most peculiar: There were the same characteristic sets of missing colors as for stars in our own galaxy, but they were all shifted by the same relative amount toward the red end of the spectrum. The only reasonable explanation of this was that the galaxies were moving away from us, and the frequency of the light waves from them was being reduced, or red-shifted, by the Doppler effect. Listen to a car passing on the road. As the car is approaching, its engine sounds at a higher pitch, corresponding to a higher frequency of sound waves; and when it passes and goes away, it sounds at a lower pitch. The behavior of light or radial waves is similar. Indeed, the police made use of the Doppler effect to measure the speed of cars by measuring the frequency of pulses of radio waves reflected off them.

In the years following his proof of the existence of other galaxies, Hubble spent his time cataloging their distances and observing their spectra. At that time most people expected the galaxies to be moving around quite randomly, and so expected to find as many spectra which were blue-shifted as ones which were red–shifted. It was quite a surprise, therefore, to find that the galaxies all appeared red-shifted. Every single one was moving away from us. More surprising still was the result which Hubble published in 1929: Even the size of the galaxy’s red shift was not random, but was directly proportional to the galaxy’s distance from us. Or, in other words, the farther a galaxy was, the faster it was moving away. And that meant that the universe could not be static, as everyone previously thought, but was in fact expanding. The distance between the different galaxies was growing all the time.

The discovery that the universe was expanding was one of the great intellectual revolutions of the twentieth century. With hindsight, it is easy to wonder why no one had thought of it before. Newton and others should have realized that a static universe would soon start to contract under the influence of gravity. But suppose that, instead of being static, the universe was expanding. If it was expanding fairly slowly, the force of gravity would cause it eventually to stop expanding and then to start contracting. However, if it was expanding at more than a certain critical rate, gravity would never be strong enough to stop it, and the universe would continue to expand forever. This is a bit like what happens when one fires a rocket upward from the surface of the Earth. If it has a fairly low speed, gravity will eventually stop the rocket and it will start falling back. On the other hand, if the rocket has more than a certain critical speed–about seven miles a second–gravity will not be strong enough to pull it back, so it will keep going away from the Earth forever.

This behavior of the universe could have been predicted from Newton’s theory of gravity at any time in the nineteenth, the eighteenth, or even the late seventeenth centuries. Yet so strong was the belief in a static universe that it persisted into the early twentieth century. Even when Einstein formulated the general theory of relativity in 1915, he was sure that the universe had to be static. He therefore modified his theory to make this possible, introducing a socalled cosmological constant into his equations. This was a new “antigravity” force, which, unlike other forces, did not come from any particular source, but was built into the very fabric of space-time. His cosmological constant gave space-time an inbuilt tendency to expand, and this could be made to exactly balance the attraction of all the matter in the universe so that a static universe would result.

Only one man, it seems, was willing to take general relativity at face value. While Einstein and other physicists were looking for ways of avoiding general relativity’s prediction of a nonstatic universe, the Russian physicist Alexander Friedmann instead set about explaining it.

THE FRIEDMANN MODELS

The equations of general relativity, which determined how the universe evolves in time, are too complicated to solve in detail. So what Friedmann did, instead, was to make two very simple assumptions about the universe: that the universe looks identical in whichever direction we look, and that this would also be true if we were observing the universe from anywhere else. On the basis of general relativity and these two assumptions, Friedmann showed that we should not expect the universe to be static. In fact, in 1922, several years before Edwin Hubble’s discovery, Friedmann predicted exactly what Hubble found.

The assumption that the universe looks the same in every direction is clearly not true in reality. For example, the other stars in our galaxy form a distinct band of light across the night sky called the Milky Way. But if we look at distant galaxies, there seems to be more or less the same number of them in each direction. So the universe does seem to be roughly the same in every direction, provided one views it on a large scale compared to the distance between galaxies.

For a long time this was sufficient justification for Friedmann’s assumption— as a rough approximation to the real universe. But more recently a lucky accident uncovered the fact that Friedmann’s assumption is in fact a remarkably accurate description of our universe. In 1965, two American physicists, Arno Penzias and Robert Wilson, were working at the Bell Labs in New Jersey on the design of a very sensitive microwave detector for communicating with orbiting satellites. They were worried when they found that their detector was picking up more noise than it ought to, and that the noise did not appear to be coming from any particular direction. First they looked for bird droppings on their detector and checked for other possible malfunctions, but soon ruled these out. They knew that any noise from within the atmosphere would be stronger when the detector is not pointing straight up than when it is, because the atmosphere appears thicker when looking at an angle to the vertical.

The extra noise was the same whichever direction the detector pointed, so it must have come from outside the atmosphere. It was also the same day and night throughout the year, even though the Earth was rotating on its axis and orbiting around the sun. This showed that the radiation must come from beyond the solar system, and even from beyond the galaxy, as otherwise it would vary as the Earth pointed the detector in different directions.

In fact, we know that the radiation must have traveled to us across most of the observable universe. Since it appears to be the same in different directions, the universe must also be the same in every direction, at least on a large scale. We now know that whichever direction we look in, this noise never varies by more than one part in ten thousand. So Penzias and Wilson had unwittingly stumbled across a remarkably accurate confirmation of Friedmann’s first assumption.

At roughly the same time, two American physicists at nearby Princeton University, Bob Dicke and Jim Peebles, were also taking an interest in microwaves. They were working on a suggestion made by George Gamow, once a student of Alexander Friedmann, that the early universe should have been very hot and dense, glowing white hot. Dicke and Peebles argued that we should still be able to see this glowing, because light from very distant parts of the early universe would only just be reaching us now. However, the expansion of the universe meant that this light should be so greatly red-shifted that it would appear to us now as microwave radiation. Dicke and Peebles were looking for this radiation when Penzias and Wilson heard about their work and realized that they had already found it. For this, Penzias and Wilson were awarded the Nobel Prize in 1978, which seems a bit hard on Dicke and Peebles.

Now at first sight, all this evidence that the universe looks the same whichever direction we look in might seem to suggest there is something special about our place in the universe. In particular, it might seem that if we observe all other galaxies to be moving away from us, then we must be at the center of the universe. There is, however, an alternative explanation: The universe might also look the same in every direction as seen from any other galaxy. This, as we have seen, was Friedmann’s second assumption.

We have no scientific evidence for or against this assumption. We believe it only on grounds of modesty. It would be most remarkable if the universe looked the same in every direction around us, but not around other points in the universe. In Friedmann’s model, all the galaxies are moving directly away from each other. The situation is rather like steadily blowing up a balloon which has a number of spots painted on it. As the balloon expands, the distance between any two spots increases, but there is no spot that can be said to be the center of the expansion. Moreover, the farther apart the spots are, the faster they will be moving apart. Similarly, in Friedmann’s model the speed at which any two galaxies are moving apart is proportional to the distance between them. So it predicted that the red shift of a galaxy should be directly proportional to its distance from us, exactly as Hubble found.

Despite the success of his model and his prediction of Hubble’s observations, Friedmann’s work remained largely unknown in the West. It became known only after similar models were discovered in 1935 by the American physicist Howard Robertson and the British mathematician Arthur Walker, in response to Hubble’s discovery of the uniform expansion of the universe.

Although Friedmann found only one, there are in fact three different kinds of models that obey Friedmann’s two fundamental assumptions. In the first kind—which Friedmann found—the universe is expanding so sufficiently slowly that the gravitational attraction between the different galaxies causes the expansion to slow down and eventually to stop. The galaxies then start to move toward each other and the universe contracts. The distance between two neighboring galaxies starts at zero, increases to a maximum, and then decreases back down to zero again.

In the second kind of solution, the universe is expanding so rapidly that the gravitational attraction can never stop it, though it does slow it down a bit. The separation between neighboring galaxies in this model starts at zero, and eventually the galaxies are moving apart at a steady speed.

Finally, there is a third kind of solution, in which the universe is expanding only just fast enough to avoid recollapse. In this case the separation also starts at zero, and increases forever. However, the speed at which the galaxies are moving apart gets smaller and smaller, although it never quite reaches zero.

A remarkable feature of the first kind of Friedmann model is that the universe is not infinite in space, but neither does space have any boundary. Gravity is so strong that space is bent round onto itself, making it rather like the surface of the Earth. If one keeps traveling in a certain direction on the surface of the Earth, one never comes up against an impassable barrier or falls over the edge, but eventually comes back to where one started. Space, in the first Friedmann model, is just like this, but with three dimensions instead of two for the Earth’s surface. The fourth dimension—time—is also finite in extent, but it is like a line with two ends or boundaries, a beginning and an end. We shall see later that when one combines general relativity with the uncertainty principle of quantum mechanics, it is possible for both space and time to be finite without any edges or boundaries. The idea that one could go right around the universe and end up where one started makes good science fiction, but it doesn’t have much practical significance because it can be shown that the universe would recollapse to zero size before one could get round. You would need to travel faster than light in order to end up where you started before the universe came to an end—and that is not allowed.

But which Friedmann model describes our universe? Will the universe eventually stop expanding and start contracting, or will it expand forever? To answer this question we need to know the present rate of expansion of the universe and its present average density. If the density is less than a certain critical value, determined by the rate of expansion, the gravitational attraction will be too weak to halt the expansion. If the density is greater than the critical value, gravity will stop the expansion at some time in the future and cause the universe to recollapse.

We can determine the present rate of expansion by measuring the velocities at which other galaxies are moving away from us, using the Doppler effect. This can be done very accurately. However, the distances to the galaxies are not very well known because we can only measure them indirectly. So all we know is that the universe is expanding by between 5 percent and 10 percent every thousand million years. However, our uncertainty about the present average density of the universe is even greater.

If we add up the masses of all the stars that we can see in our galaxy and other galaxies, the total is less than one-hundredth of the amount required to halt the expansion of the universe, even in the lowest estimate of the rate of expansion. But we know that our galaxy and other galaxies must contain a large amount of dark matter which we cannot see directly, but which we know must be there because of the influence of its gravitational attraction on the orbits of stars and gas in the galaxies. Moreover, most galaxies are found in clusters, and we can similarly infer the presence of yet more dark matter in between the galaxies in these clusters by its effect on the motion of the galaxies. When we add up all this dark matter, we still get only about one-tenth of the amount required to halt the expansion. However, there might be some other form of matter which we have not yet detected and which might still raise the average density of the universe up to the critical value needed to halt the expansion.

The present evidence, therefore, suggests that the universe will probably expand forever. But don’t bank on it. All we can really be sure of is that even if the universe is going to recollapse, it won’t do so for at least another ten thousand million years, since it has already been expanding for at least that long. This should not unduly worry us since by that time, unless we have colonies beyond the solar system, mankind will long since have died out, extinguished along with the death of our sun.

THE BIG BANG

All of the Friedmann solutions have the feature that at some time in the past, between ten and twenty thousand million years ago, the distance between neighboring galaxies must have been zero. At that time, which we call the big bang, the density of the universe and the curvature of space-time would have been infinite. This means that the general theory of relativity— on which Friedmann’s solutions are based—predicts that there is a singular point in the universe.

All our theories of science are formulated on the assumption that space–time is smooth and nearly flat, so they would all break down at the big bang singularity, where the curvature of space–time is infinite. This means that even if there were events before the big bang, one could not use them to determine what would happen afterward, because predictability would break down at the big bang. Correspondingly, if we know only what has happened since the big bang, we could not determine what happened beforehand. As far as we are concerned, events before the big bang can have no consequences, so they should not form part of a scientific model of the universe. We should therefore cut them out of the model and say that time had a beginning at the big bang.

Many people do not like the idea that time has a beginning, probably because it smacks of divine intervention. (The Catholic church, on the other hand, had seized on the big bang model and in 1951 officially pronounced it to be in accordance with the Bible.) There were a number of attempts to avoid the conclusion that there had been a big bang. The proposal that gained widest support was called the steady state theory. It was suggested in 1948 by two refugees from Nazi–occupied Austria, Hermann Bondi and Thomas Gold, together with the Briton Fred Hoyle, who had worked with them on the development of radar during the war. The idea was that as the galaxies moved away from each other, new galaxies were continually forming in the gaps in between, from new matter that was being continually created. The universe would therefore look roughly the same at all times as well as at all points of space.

The steady state theory required a modification of general relativity to allow for the continual creation of matter, but the rate that was involved was so low—about one particle per cubic kilometer per year—that it was not in conflict with experiment. The theory was a good scientific theory, in the sense that it was simple and it made definite predictions that could be tested by observation. One of these predictions was that the number of galaxies or similar objects in any given volume of space should be the same wherever and whenever we look in the universe.

In the late 1950s and early 1960s, a survey of sources of radio waves from outer space was carried out at Cambridge by a group of astronomers led by Martin Ryle. The Cambridge group showed that most of these radio sources must lie outside our galaxy, and also that there were many more weak sources than strong ones. They interpreted the weak sources as being the more distant ones, and the stronger ones as being near. Then there appeared to be fewer sources per unit volume of space for the nearby sources than for the distant ones.

This could have meant that we were at the center of a great region in the universe in which the sources were fewer than elsewhere. Alternatively, it could have meant that the sources were more numerous in the past, at the time that the radio waves left on their journey to us, than they are now. Either explanation contradicted the predictions of the steady state theory. Moreover, the discovery of the microwave radiation by Penzias and Wilson in 1965 also indicated that the universe must have been much denser in the past. The steady state theory therefore had regretfully to be abandoned.

Another attempt to avoid the conclusion that there must have been a big bang and, therefore, a beginning of time, was made by two Russian scientists, Evgenii Lifshitz and Isaac Khalatnikov, in 1963. They suggested that the big bang might be a peculiarity of Friedmann’s models alone, which after all were only approximations to the real universe. Perhaps, of all the models that were roughly like the real universe, only Friedmann’s would contain a big bang singularity. In Friedmann’s models, the galaxies are all moving directly away from each other. So it is not surprising that at some time in the past they were all at the same place. In the real universe, however, the galaxies are not just moving directly away from each other—they also have small sideways velocities. So in reality they need never have been all at exactly the same place, only very close together. Perhaps, then, the current expanding universe resulted not from a big bang singularity, but from an earlier contracting phase; as the universe had collapsed, the particles in it might not have all collided, but they might have flown past and then away from each other, producing the present expansion of the universe. How then could we tell whether the real universe should have started out with a big bang?

What Lifshitz and Khalatnikov did was to study models of the universe which were roughly like Friedmann’s models but which took account of the irregularities and random velocities of galaxies in the real universe. They showed that such models could start with a big bang, even though the galaxies were no longer always moving directly away from each other. But they claimed that this was still only possible in certain exceptional models in which the galaxies were all moving in just the right way. They argued that since there seemed to be infinitely more Friedmann-like models without a big bang singularity than there were with one, we should conclude that it was very unlikely that there had been a big bang. They later realized, however, that there was a much more general class of Friedmann-like models which did have singularities, and in which the galaxies did not have to be moving in any special way. They therefore withdrew their claim in 1970.

The work of Lifshitz and Khalatnikov was valuable because it showed that the universe could have had a singularity—a big bang—if the general theory of relativity was correct. However, it did not resolve the crucial question: Does general relativity predict that our universe should have the big bang, a beginning of time? The answer to this came out of a completely different approach started by a British physicist, Roger Penrose, in 1965. He used the way light cones behave in general relativity, and the fact that gravity is always attractive, to show that a star that collapses under its own gravity is trapped in a region whose boundary eventually shrinks to zero size. This means that all the matter in the star will be compressed into a region of zero volume, so the density of matter and the curvature of space-time become infinite. In other words, one has a singularity contained within a region of space-time known as a black hole.

At first sight, Penrose’s result didn’t have anything to say about the question of whether there was a big bang singularity in the past. However, at the time that Penrose produced his theorem, I was a research student desperately looking for a problem with which to complete my Ph.D. thesis. I realized that if one reversed the direction of time in Penrose’s theorem so that the collapse became an expansion, the conditions of his theorem would still hold, provided the universe were roughly like a Friedmann model on large scales at the present time. Penrose’s theorem had shown that any collapsing star must end in a singularity; the time-reversed argument showed that any Friedmann-like expanding universe must have begun with a singularity. For technical reasons, Penrose’s theorem required that the universe be infinite in space. So I could use it to prove that there should be a singularity only if the universe was expanding fast enough to avoid collapsing again, because only that Friedmann model was infinite in space.

During the next few years I developed new mathematical techniques to remove this and other technical conditions from the theorems that proved that singularities must occur. The final result was a joint paper by Penrose and myself in 1970, which proved that there must have been a big bang singularity provided only that general relativity is correct and that the universe contains as much matter as we observe.

There was a lot of opposition to our work, partly from the Russians, who followed the party line laid down by Lifshitz and Khalatnikov, and partly from people who felt that the whole idea of singularities was repugnant and spoiled the beauty of Einstein’s theory. However, one cannot really argue with the mathematical theorem. So it is now generally accepted that the universe must have a beginning.