Many Worlds in One: The Search for Other Universes

by Vilenkin, Alex

  COSMIC INFLATION

  What would happen if, at some early epoch, the space of the universe were in a false-vacuum state? If the matter density at that epoch were less than needed to balance the universe, then the repulsive gravity of the vacuum would have prevailed. This would cause the universe to expand—even if it did not expand to begin with.

  To have a definite picture in mind, we shall assume that the universe is closed. Then it swells like an expanding balloon, as shown in Figure 3.1 in Chapter 3. As the volume of the universe grows, the matter is diluted and its mass density is reduced. But the mass density of false vacuum is a fixed constant; it always remains the same. Hence, very quickly the matter density becomes negligible, and we are left with a uniform, expanding sea of false vacuum.

  Figure 5.2. Alan Guth in his office at MIT. Guth is the proud winner of the 2005 contest for the messiest office, organized by The Boston Globe. (Photo by Larry Fink)

  The expansion is driven by the false-vacuum tension, overcoming the attractive force of the vacuum mass density. Since neither of these quantities changes with time, the expansion rate remains constant as well. The expansion rate tells by what fraction the universe grows in a chosen unit of time (say, in 1 second). Its meaning is very similar to that of the inflation rate in economics: the percentage increase of prices in a year. In 1980, when Guth gave his seminar at Harvard, the rate of economic inflation in the United States was 14 percent. If it remained fixed at that value, prices would double every 5.3 years. Similarly, a constant expansion rate of the universe implies that there is a fixed time in which the size of the universe doubles.

  The growth pattern with a constant doubling time is called exponential. It is known to build up very quickly to gigantic numbers. If a slice of pizza costs $1 now, then after 10 doubling cycles (53 years in our example) its price will be $1,024, and after 330 cycles it will be $10100. This stupendous number, 1 followed by 100 zeros, has a special name: a googol. Guth suggested that we adopt the term “inflation” in cosmology, using it to describe an exponential expansion of the universe.

  The doubling time in a false-vacuum universe is unbelievably short. The higher the vacuum energy, the shorter the time. For the electroweak vacuum, the universe would expand by a googol in one-thirtieth of a microsecond, and for the grand-unified vacuum this expansion would happen 1026 times faster. In this tiny fraction of a second, a region the size of an atom would be blown to dimensions much greater than the entire currently observable universe.

  Since false vacuum is unstable, it eventually decays, and its energy ignites a hot fireball of particles. This event signals the end of inflation and the starting point of the usual cosmological evolution. We thus get an enormous, hot, expanding universe from a tiny initial seed. As an extra bonus, amazingly, the horizon and flatness problems of big bang cosmology disappear in this scenario.

  The essence of the horizon problem is that the distances between some parts of the observable universe appear to have always been greater than the distance traveled by light since the big bang. This implies that these regions have never interacted with one another, and then it is hard to explain how they could reach nearly identical temperatures and densities. In the standard big bang theory, the distance traveled by light grows proportionally to the age of the universe, while the separation between the regions increases more slowly, because cosmic expansion is being slowed down by gravity. Regions that cannot interact now will be able to do so in the future, when the light-travel distance finally catches up with their separation. But at earlier times the light-travel distance fell even shorter of the mark, so that if the regions cannot interact at the present epoch, they were surely unable to do so in the past. The root of the problem can thus be traced to the attractive nature of gravity, which causes the expansion to slow down with time.

  In a false-vacuum universe, however, gravity is repulsive; so instead of slowing down, expansion accelerates. Then the situation is reversed: regions that can exchange light signals will lose their ability to interact in the future. And more important, regions that are out of each other’s reach must have interacted in the past. The horizon problem has disappeared!

  The flatness problem dissolves just as easily. It turns out that the universe is driven away from the critical density only if its expansion is slowing down. In the case of accelerated, inflationary expansion, the opposite is true: the universe is driven toward the critical density, and thus to flatness. Since inflation enlarges the universe by an enormous factor, we can see only a tiny part of it. This observable region appears to be flat, just as the surface of the Earth appears flat when you look at it from close by.

  In summary, a brief period of inflation makes the universe large, hot, uniform, and flat, setting just the right initial conditions for the standard big bang cosmology.

  The theory of inflation was about to begin its conquest of the world. As for Guth himself, his days as a postdoc were over. He accepted a job offer from his alma mater, the Massachusetts Institute of Technology, where he has remained ever since.

  This would be a nice happy ending to the story of inflation, except for one unfortunate problem: the theory did not work.

  6

  Too Good to Be Wrong

  Truth comes out of error more readily than out of confusion.

  —FRANCIS BACON

  THE GRACEFUL EXIT PROBLEM

  Every physicist knows the sinking sensation of discovering a fatal flaw in the beautiful theory that you thought up a few days ago. Alas, this is the fate of most beautiful theories. And so it was with inflation. As usual, the devil was in the details. On a closer examination, the false vacuum did not decay as smoothly as anticipated.

  The process of vacuum decay is similar to the boiling of water. Small bubbles of true vacuum pop out randomly and expand in the midst of false vacuum (Figure 6.1). As the bubbles grow, their interiors remain nearly empty and all the energy released from converting false vacuum into true is concentrated in the expanding bubble walls. When bubbles collide and merge, their walls disintegrate into elementary particles. The end result is true vacuum filled with a hot fireball of matter.

  This is indeed what happens if bubbles pop out at a feverish rate, so that the whole decay process is complete in less than one doubling time. That would mean, however, that inflation ends too soon, way before the universe becomes homogeneous and flat. We are interested in the opposite case, when the bubble-formation rate is low, so that the universe can expand by a large factor before bubbles begin to collide. But, as the Swiss physicist Paul Ehrenfest used to say, that’s where the frog jumps into the water.

  Figure 6.1. Small bubbles of true vacuum pop out randomly and expand. Bubbles that formed earlier have grown to a bigger size.

  The trouble is that the space between the bubbles is filled with false vacuum and is therefore rapidly expanding. The bubbles grow very fast, at speeds approaching the speed of light, but even that is no match for the exponential expansion of false vacuum. If the bubbles do not collide within one doubling time after they are formed, then at later times their separation will only grow, so they will never collide.

  The conclusion is that inflation cannot possibly end. The bubbles grow to unlimited sizes, and new, small bubbles keep popping out in the expanding gaps between them. As a result, the wonderful uniformity created by inflation is completely destroyed. The absence of a suitable ending to inflationary expansion has become known as the graceful exit problem.

  Guth realized that there was a problem a few months after he went public with his new theory. His paper on inflation was not yet written at that time, and for a very simple reason: Alan Guth is the biggest procrastinator in the world. (I learned this firsthand after working with him on a number of research projects.) Of course, Guth was disappointed to find a serious flaw in his theory. But still he felt the idea was too good to be wrong. When he finally got around to writing the paper in August of 1980, he concluded it with these words: “I am publishing this paper in
the hope that it will … encourage others to find some way to avoid the undesirable features of the inflationary scenario.”1

  SCALAR FIELD

  To get to the root of the problem, let us now discuss the false-vacuum decay in more detail. The decay process was studied by the Harvard physicist Sidney Coleman, who described it in terms of scalar fields.

  A field is a quantity that has some value at every point in space. The values may vary from one point to another and can also change with time. A simple example of a field is the temperature. The North Pole, the tip of Cape Cod, the center of the Sun—all points in the universe have a certain value of temperature. Another familiar example is the magnetic field. In addition to its magnitude, this field also has a direction. We don’t feel the magnetic field, but its presence becomes evident when we examine a compass. The compass needle will point in the direction of the field, and the field strength can be judged by how forcefully it causes the needle to swing in that direction.

  Fields like the temperature, which do not have any direction, are called scalar fields. They are characterized by a single number: their magnitude. Scalar fields play an important role in elementary particle physics. According to modern particle theories, the space of the universe is pervaded by a number of scalar fields, whose values determine the vacuum energy, as well as the particle masses and their interactions. In other words, these fields determine what vacuum we live in. At present, the scalar fields are at their true vacuum values, but things could have been different at earlier epochs.

  To illustrate the physics of vacuum decay, we shall consider a single scalar field and focus on how it affects the vacuum energy. Each cubic centimeter of space has energy, which depends on the magnitude of the field. The exact dependence is not currently known, but its general form is expected to resemble a hilly landscape, as in Figure 6.2, with hilltops (maxima) at some values of the field and valleys (minima) at others. The behavior of the scalar field is very similar to that of a ball rolling along the terrain depicted in this energy landscape. The value of the field is represented by the location of the ball along the horizontal axis. Depending on the initial position of the ball, it will roll down to one or the other energy minimum in the figure. The lower minimum has almost zero energy density; it corresponds to the true vacuum. The upper minimum corresponds to a high-energy false vacuum.

  Suppose now that we start with a false vacuum at every point in space. This situation is represented by the ball lying in the upper minimum. It will lie there for a very long time, unless someone kicks it upward, supplying the energy needed to go over the barrier and into the lower minimum. But according to the quantum theory, objects can “tunnel” through energy barriers. If you were to observe such an event, you would see the ball disappear and instantly materialize on the other side of the barrier.

  Figure 6.2. The energy landscape of a scalar field with a false and a true vacuum. The field can tunnel through the barrier separating the two vacua.

  Quantum tunneling is a probabilistic process. You cannot predict exactly when it will happen, but you can calculate the probability for it to happen in a given interval of time. For a macroscopic object, like a ball, the tunneling probability is extremely low. If, for example, you want a can of Coke to tunnel out of a vending machine, you will have to wait for much longer than the present age of the universe. But in the microscopic world of elementary particles, quantum tunneling is much more common. As I mentioned in Chapter 4, George Gamow used the tunneling effect to explain the decay of radioactive atomic nuclei. In the case of a false vacuum, the probability that a large region of space will tunnel to the true vacuum is completely negligible. The tunneling occurs in a tiny, microscopic region, resulting in a small true-vacuum bubble. This is the bubble-formation process that we discussed in the preceding section of this chapter. The tunneling probability may be large or small, depending on the shape of the energy function. (The probability is large for low and narrow energy barriers.)

  Despite the similarity between the tunneling of a ball and that of a scalar field, there is also an important difference. The ball tunnels between two different points in space, while for a scalar field the tunneling occurs between two different values of the field at the same location.

  What has transpired from this analysis is that if there is an energy barrier between the two vacua, then false-vacuum decay can proceed only through quantum tunneling. The tunneling results in a haphazard pattern of bubbles that never merge, so the decay process is never complete. But what would happen if we remove the barrier?

  SLOW DOES IT

  Andrei Linde, a young Russian cosmologist, was the first to consider unorthodox scalar field models that had no barrier between the false and true vacua.

  As before, suppose we start with a small closed universe and a scalar field in the false-vacuum state. If there is no barrier, the ball representing the field simply rolls down toward the true vacuum (see Figure 6.3). There are no bubbles, and the field remains uniform in the entire space as it rolls downhill. When it gets to the bottom, the scalar field starts oscillating back and forth. The energy of the oscillations is then quickly dissipated into a fireball of particles, while the field settles at the energy minimum.

  The problem is, however, that in the absence of a barrier the field would roll down very fast and inflation would be cut off too early. Having recognized this danger, Linde made a crucial step. He suggested that the energy function should have the form of a hill with a very gentle slope, as shown in Figure 6.4. The flat region near the hilltop in the figure plays the role of the false vacuum. If the ball is placed somewhere in that region, it will start rolling very slowly. And since the slope is so flat, the ball will remain at about the same height. Now, remember, the height in this figure stands for the energy density of the scalar field, and keeping it constant is all that is needed to sustain the constant rate of inflation.

  Figure 6.3. The energy landscape without a barrier. The scalar field quickly rolls down to the true vacuum.

  Figure 6.4. A “flattened hill” energy landscape. The scalar field slowly rolls downhill, while inflation continues.

  Linde’s key observation was that in the flat region near the hilltop, the scalar field rolls very slowly, and therefore it takes a long time to traverse that region. In the meantime, the universe expands exponentially, resulting in a huge expansion factor. When the field gets to the steeper part of the energy slope, it rolls down faster, and when it finally reaches the minimum, it oscillates and dumps its energy into a hot fireball of particles. At this point we have an enormous, hot, expanding universe, which is also homogeneous and nearly flat. The graceful exit problem has been solved!

  Figure 6.5. Andrei Linde (left) with Slava Mukhanov (of Ludwig-Maximilians University in Munich). (Courtesy of Sugumi Kanno)

  All that is needed is a scalar field whose energy function has the form of a flattened hill, as in Figure 6.4. You may be wondering how the scalar field got on the top of the hill to begin with. Good question. But wait until Chapter 17.

  Linde’s paper appeared in February of 1982, and a few months later essentially the same idea was published independently by the American physicists Andreas Albrecht and Paul Steinhardt. The theory of inflation had been rescued.

  Another important question is whether or not such scalar fields really exist in nature. Unfortunately, we don’t know. There is no direct evidence for their existence. Scalar fields appearing in the simplest electroweak and grand-unified theories have energy functions that are too steep for the purposes of inflation. But there is a class of supersymmetric theories, which include plenty of scalar fields with flat energy functions. The superstring theory, which is now the leading candidate for the fundamental theory of nature, belongs to this class. (We shall have more to say about superstrings in Chapter 15.)

  THE NUFFIELD WORKSHOP

  The next act of the drama was set in the medieval university town of Cambridge. By invitation of Stephen Hawking, some thirty cosmologist
s from all over the world converged there in the summer of 1982. They gathered for a three-week workshop on the very early universe, funded by the Nuffield Foundation. I was thrilled to be among the participants: Hawking asked me to talk about my recent work on cosmic strings.

  I instantly fell in love with Cambridge. Early in the morning I would get up to stroll through the grounds of old colleges. Gothic chapels, clock towers, austere walled courts with their perfect rectangular lawns and bright splashes of flowers—vestiges of another, more contemplative age. By nine o’clock I would be back to modernity, sitting in the conference hall and waiting for the talks to begin. Thankfully, there were only two talks a day, one in the morning and one in the afternoon, with plenty of time left for informal discussions. British food was not among the highlights of the trip, but British beer is quite a different story, and I spent many evening hours discussing physics and other matters over a pint of lager.

  The program of the meeting emphasized recent developments in cosmology, and inevitably the theory of inflation took the center stage. The graceful exit problem was now out of the way, but still there was another major concern.

  It is true that inflation makes the universe flat and smooth, but perhaps it does too good a job of that. No galaxies or stars would ever form in a perfectly homogeneous universe. As we discussed in Chapter 4, galaxies have evolved from small variations in the density. The origin of these primordial inhomogeneities, or density perturbations, became the central issue of the workshop.