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Authors: Alex Vilenkin
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Too Good to Be Wrong
Truth comes out of error more readily than out of confusion.
—FRANCIS BACON
E
very 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.
very 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.
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.
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.
. 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
.
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
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
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
.
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.
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.
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.
.
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.)
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?
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.
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.
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.
.
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.
.
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)
.
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.
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.)
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.)
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