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Authors: Alex Vilenkin

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Let us now briefly summarize what we have learned about eternal inflation. If we were somehow able to observe the eternally inflating universe from outside, as the surface of the Earth can be observed from outer space, we would see a multitude of island universes scattered in the vast inflating sea of false vacuum. If the universe is closed, then the view that would open in front of us might in fact resemble a picture of the globe, with continents and archipelagos surrounded by the ocean.
aa
This globe is expanding at a staggering speed, the island universes are also growing exceedingly fast, and tiny new islands constantly appear and immediately start to expand. The number of island universes rapidly grows with time. It grows without bound and becomes infinite in the limit of infinite future.
The inhabitants of island universes, like us, see a very different picture. They do not perceive their universe as a finite island. For them it appears as a self-contained, infinite universe. The boundary between this universe and the inflating part of spacetime is the big bang, a moment in their past. We cannot travel to the inflating sea, simply because it is impossible to travel to the past.
Remarkably, the entire “master” universe, which contains all the infinite island universes, may be closed and finite. The apparent contradiction is resolved by the fact that the internal notion of time in island universes is different from the “global” time that one has to use to describe the entire spacetime. In the global time, the outer parts of island universes are not yet formed and will complete their formation in the infinite future, while in the interior time the island universe forms all at once. The spacetime structure of a closed, eternally inflating universe is illustrated in
Figure 10.3
.
The surprising feature of island universes—that they are infinite when viewed from the inside—turned out to be important. It later led me to what perhaps is the most striking consequence of eternal inflation.
Figure 10.3
. Spacetime of a one-dimensional, closed, eternally inflating universe. This universe is filled with false vacuum at the initial moment (bottom of the figure) and has three island universes by the time corresponding to the top of the figure.
The King Lives!
Must not all things that
can
happen
have
already happened, been done, run past?
—NIETZSCHE
T
he first glimpse of an idea came to me in the summer of 2000. As it often happens, my immediate impulse was to share it with someone. You may get more credit if you work alone, but working together is so much more fun! And if you are blessed to have a good collaborator, it can be a real joy. By a stroke of luck, my old friend Jaume Garriga happened to be in town. When I told him about my thought, he understood it instantly.
Jaume is a soft-spoken, quiet fellow. He does not say much, but he does speak his mind. On that occasion, he only said, “This is a very marketable idea.” That was not exactly an endorsement. He meant that it was the kind of idea that would be more attractive to mass media than to physicists. But I could tell that Jaume was hooked. He was about to leave for his native Catalonia, and we agreed to resume the discussion during my visit to the University of Barcelona, where he worked.
Two months later, Jaume met me and my wife at the Barcelona airport. We arrived on a weekend and had two days free before the beginning of my “official” visit. I could not wait to get back to our physics discussion, but it turned out that our program had already been fixed. As we drove out onto
the highway, Jaume told us that we were going to his father’s farm: “They are expecting us for dinner.” We passed the formidable mountain mass of Montserrat, which suddenly rises out of flat, reddish terrain, and continued north, into a greener, hilly countryside. In an hour or so we drove up to the Garriga family farm.
Figure 11.1
.
Jaume Garriga. (Courtesy of Takahiro Tanaka)
Amazingly, the same family has worked this land for more than 750 years. The farmhouse was an impressive Catalan
masia
, looking like a small fortress, complete with a tower. I was totally blown away and forgot all about physics.
The dinner was served in a spacious hall, where the Garriga family had gathered. As a guest of honor, I was seated next to Jaume’s father, who enchanted us with tales of the ancient history of the land and made sure that my wineglass was never empty. Toward the end of the dinner he excused himself and walked out of the hall. Jaume explained: “He went to tell the cows to go home.” The cows did not have to be shepherded; they just needed a friendly reminder.
Figure 11.2
. Jaume in his younger years at the family farm. (Courtesy of Jaume Garriga)
After the dinner, Jaume’s older brother led us up the winding stairway to the top of the tower. It was used as a watchtower in times of danger. When the enemy was sighted, the guardsmen could signal with a torch to similar towers at the neighboring farms, all the way to the duke’s garrison at the castle of Cardona, about five miles away. We looked out of the small square windows of the tower—to check if there were any villains in sight. The sun was already setting over the hills. In the distance, we saw the cows coming home, by themselves, from the pasture.
We left the farm in the morning and headed north, toward the mountains. Our destination was the little coastal village of Cadaqués, the home of Salvador Dalí. My wife is fascinated by Dalí’s art, and she wanted to see his house and the village where he spent most of his life. She wanted to go there every time we visited Barcelona, but once I got to the university, I was invariably distracted by physics discussions and other equally important matters, so in the end there was never any time left for the trip. Now she said this was it: we were going to Cadaqués
before
Barcelona.
The narrow winding road climbed up the mountains, clinging dangerously
to the slopes, and then weaved its way down to the cliffs and secluded blue coves of Costa Brava. We entered the village in the early afternoon, when the sun was in its full Mediterranean splendor. The whitewashed houses of Cadaqués crowded the hillsides, tumbling down and stopping right at the water’s edge. Higher up the slope stood a rustic white church, austere and beautiful.
Our visit to Dalí’s house did not go as planned. Jaume’s wife, Julie, who decided to join us at the last minute, took along their baby daughter, Clara. As we were entering the museum, Clara protested loudly, so the ladies went in, while Jaume and I were left to babysit. Soon we were deep in the discussion of our physics problem. By the time our wives returned, the museum was already closing. So I did not get to see the so-much-talked-about Casa Dalí.
We spent the rest of the afternoon wandering about the village. As we strolled the narrow cobblestone streets of Cadaqués, Jaume and I continued our discussion, and a new picture of the universe was gradually taking shape. It was bizarre and disturbing.
Figure 11.3
.
Port Alguer (Cadaqués)
by Salvador Dalí. (© 2005 Salvador Dalí, Gala-Salvador Dalí Foundation/Artists Rights Society [ARS], New York)
The conversation revolved around the remote regions of the universe and how they could differ from our local
cosmic
neighborhood. Since each island universe is infinite from the viewpoint of its inhabitants, it can be divided into an infinite number of regions having the same size as our observable region. For short, we called them “O-regions.”
Imagine an infinite space packed with gigantic spheres, 80 billion light-years in diameter. Each sphere is an O-region. The spheres expand with the expansion of the universe; consequently, they were smaller at earlier times. All of these O-regions looked pretty much the same at the big bang—that is, at the end of inflation. But they were different in detail. Small density perturbations, brought about by random quantum processes during inflation, differed from one region to another. As these perturbations were amplified by gravity, the macroscopic properties of O-regions began to diverge. By the time of galaxy formation, the details of how galaxies are distributed in different O-regions varied considerably, although statistically the regions were still very similar. Later on, the evolution of life and intelligence was influenced by chance, leading to further divergence of properties. We can thus expect the histories of different O-regions to be rather different.
The key observation was that the number of distinct configurations of matter that can possibly be realized in any O-region—or, for that matter, in any finite system—is finite. One might think that arbitrarily small changes could be made in the system, thus creating an infinite number of possibilities. But this is not the case.
If I move my chair by 1 centimeter, I change the state of our O-region. I could instead move it by 0.9, 0.99, 0.999, etc., centimeter—an infinite sequence of possible displacements, which more and more closely approach the limit of 1 centimeter. The problem, however, is that displacements too close to one another cannot even in principle be distinguished, because of the quantum-mechanical uncertainty.
In classical, Newtonian physics, the state of a physical system can be described by specifying the positions and velocities of all its constituent particles. We now know that such a description can only be used for macroscopic,
massive objects, and even then only approximately. In the quantum world, particles are inherently fuzzy and cannot be precisely localized.
At the core of quantum physics is the
uncertainty principle
, discovered by Werner Heisenberg in 1927. It says that the position and the velocity of a particle cannot both be accurately measured. The more precisely we measure the position, the greater is the uncertainty in the velocity. If the position is measured exactly, then the velocity is completely undetermined, and vice versa—if we measure velocity exactly, we have no idea where the particle is.
Heisenberg offered the following intuitive explanation for the uncertainty. A simple way to determine the location of a particle is to shine light on it. The light waves will be scattered by the particle in all directions. Some of them will be registered by our eyes, or by our measuring apparatus, and we will see where the particle is. The image of the particle obtained in this way cannot be absolutely sharp: the details smaller than the wavelength of light are necessarily blurred, so the position cannot be measured more accurately than the light’s wavelength. To deal with this problem, we could use light of shorter and shorter wavelengths, but that’s where the quantum nature of light comes into play. Light consists of photons whose energy is inversely proportional to the wavelength. When the particle is illuminated with very-short-wavelength light, it is being bombarded by highly energetic photons. The particle recoils from the impact, and thus its velocity is altered. This recoil is the origin of the uncertainty: the greater the accuracy we want to achieve in position measurement, the shorter-wavelength light we should use, and the greater will be the impact on the velocity of the particle we are observing.
Even if we are not interested in the particle’s velocity and want to know only its position, Heisenberg’s argument indicates that in order to localize the particle more and more accurately, we would have to supply larger and larger amounts of energy. In any realistic physical system with limited energy, the localization accuracy is also limited.
Since we cannot pin down precise positions of particles, we can instead use what is called the coarse-grained description. Suppose the volume of our O-region is divided into cubic cells of a certain size, say, 1 cubic centimeter each. A coarse-grained state is given by indicating the cell occupied by
each particle in the region. A more refined description is obtained by making the cells smaller. But there is a limit to this refinement, since the energy cost of localizing particles to small cells will eventually exceed the available energy in the O-region.
Clearly, the number of ways in which a finite number of particles can be distributed into a finite number of cells is also finite. Hence, the material content of our O-region can only be in a finite number of distinct states. A very rough estimate of this number gives 10 to the power 10
90
, or 1 followed by 10
90
zeros—far too many zeros to fit in the pages of this book. This is a fantastically huge number, but the important point is that the number is finite.
So far, so good. One problem though is that some distant regions may contain more matter and energy than ours. Rare, large quantum fluctuations during inflation may produce some strongly over-dense regions, full of high-energy particles. As the number of particles and their energy grow, the number of possible states is also increased. But only up to a point. If more and more energy is packed into a region, its gravity gets stronger, and eventually the whole region turns into a black hole. Thus gravity puts an absolute bound on the number of states that can possibly exist in a region of a given size, regardless of its contents.
The precise value of the bound is still a subject of investigation. It was originated by Jacob Bekenstein in the 1980s and was taken up more recently, in the context of superstring theory, by Gerard ’t Hooft, Leonard Susskind, and others. This work suggests that the maximum number of states in a region depends only on the surface area of its boundary. For an O-region, this number is 10 to the power 10
123
(1 followed by more than a google zeros!).
ab

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