Putting the ice crystal into the supercooled water is very similar to giving the rolling ball a shove over the nearby hill. It’s the event that pushes the system “over the edge.” In the case of the rolling ball, the shove has to be strong enough to push it over the barrier. A tiny push won’t do it. The ball will just roll back to the starting position. The same is true for the supercooled water. If the ice crystal is smaller than a certain critical size, it will just melt back into the surrounding liquid. For example, an ice crystal just a few molecules in diameter will not grow and take over.
But even without someone adding a chip of ice, the supercooled water will not last forever. The reason is that the molecules in the liquid are constantly fluctuating, bouncing off one another and rearranging themselves. This motion is due to both the thermal jitters and the quantum jitters. Every once in a while, by accident, a group of molecules will arrange themselves into a small crystal. Most of the time the crystal will be so small that it will quickly melt into its surroundings.
Very rarely, however, a larger crystal will spontaneously form by random accident. Then the crystal will explosively grow, and everything will freeze. The phenomenon is called bubble nucleation, the growing ice crystal being thought of as an expanding bubble. A very similar thing will happen to water that is superheated above the boiling point. The only difference is that a bubble of steam will spontaneously nucleate and grow.
The boundary between the solid ice and the liquid water (or the steam and water) is called a domain wall. It is like a membrane between the two different phases.2 In fact the domain wall has properties of its own, for example, surface tension that tries to shrink the bubble. Another example of a domain wall is the boundary between ordinary water and air. As a child I was fascinated by the trick of floating a steel sewing needle on the surface of a cup of water. The domain boundary separating air and water is like a skin stretched over the liquid. It has surface tension, which actually has to be pierced in order for an object to penetrate it.
A vacuum with a positive cosmological constant is a lot like a supercooled or superheated liquid. It is metastable and can decay by nucleating bubbles. Every vacuum corresponds to a valley in the Landscape with a particular altitude or energy density. However, although the vacuum may seem quiet and featureless to our coarse senses, quantum fluctuations continually create tiny bubbles of space whose properties correspond to neighboring valleys. Usually the bubbles quickly shrink and disappear. But if the neighboring valley has lower altitude, then every so often a bubble will appear that is large enough to start growing. Will it take over everything? We will see shortly.
The domain walls that separate a bubble from its environment are two-dimensional surfaces that resemble membranes. These are not the first membranes we have encountered. In chapter 10 we learned about Polchinski’s D-branes. In many cases the domain walls are nothing but the membranelike D2-branes.
Cloning Space
One thing is missing in the analogy between the cosmic bubbling of pocket universes and the bubbling of ice crystals in supercooled fluids: namely, the tendency for space to expand. Each point on the Landscape has a cosmological constant. Recall that a positive cosmological constant means a universal repulsion, which causes matter to separate. A modern general relativist would say that space itself is expanding, or inflating, and that matter is simply being carried along for the ride.
Long ago, when Einstein was still experimenting with the cosmological constant, the Dutch astronomer Willem de Sitter began the study of inflating space. The space or, more accurately, space-time that de Sitter discovered, and which carries his name, is the solution to Einstein’s equations when there is no energy or gravitating matter other than the ubiquitous vacuum energy of empty space, i.e., a cosmological constant. Like Einstein, de Sitter assumed the cosmological constant was positive. What he found was an inflating space that grows exponentially with time. Exponential expansion means that in a certain time interval space doubles; then, in the next time interval, it doubles again; and then again. It grows to two times—four times—eight times—sixteen times its original size, in the same way that compound interest expands money. At a rate of 5 percent interest, your capital would double in fourteen years. The cosmological constant is like the interest rate: the bigger the cosmological constant, the faster the space inflates. Like any expanding space, de Sitter space satisfies Hubble’s Law—velocity proportional to distance.
We have used the analogy of an expanding rubber balloon to visualize a growing universe. But de Sitter space is different from the rubber of an exponentially expanding balloon in an important way. In the case of a balloon, the rubber—the fabric of the balloon—becomes increasingly stretched, stressed, and thinned out by the expansion. Eventually it reaches its limits, and the balloon pops. But the fabric of de Sitter space never changes. It is as though the rubber molecules were continually giving birth to new rubber molecules in order to fill the spaces created by the expansion. Think of the rubber molecules as cloning themselves to fill in the gaps.
Of course no real rubber molecules are being continuously created. Space itself is reproducing to fill the gaps. One might say that space is cloning itself—each small volume giving birth to offspring, thereby growing exponentially.
Suppose an observer in de Sitter space, moving with the general expansion, looked around at her surroundings: what would she see? You might expect that she would see the universe changing with time, getting bigger and bigger. Surprisingly, that’s not the case. All around her she would see space flowing away according to Hubble’s Law: the close things moving slowly, the distant things moving faster. At some distance the fluid of space would be rushing away so rapidly that the recessional velocity would become equal to the speed of light. At even farther distances the outgoing points would recede with an even greater velocity! Space in those regions would be flowing away so fast that even light signals, emitted straight toward the observer, would be swept away. Because no signal can travel faster than light, contact with these distant regions is completely cut off. The farthest points that can be observed, i.e., the point where the recessional velocity is the speed of light, is called the horizon, or more properly, the event horizon.
The concept of a cosmic event horizon—an ultimate barrier to our observations or a point of no return—is one of the most fascinating consequences of an accelerating universe. Like the horizon of the earth, it is by no means an end of space. It is merely the end to what we can see. When an object crosses the horizon, it says good-bye forever. Some objects may even have initially formed beyond the horizon. The observer can never have any knowledge of them. But if such objects are permanently beyond the limits of our knowledge, do they matter at all? Is there any reason to include the regions outside the horizon in a scientific theory? Some philosophers would argue that they are metaphysical constructions that have no more business in a scientific theory than the concepts of heaven, hell, and purgatory. Their existence is a sign that the theory has unverifiable and, therefore, unscientific elements to it—or so they say.
The trouble with that view is it does not permit us to appeal to a vast and diverse megaverse of pocket universes, an idea which does have explanatory power: most importantly here, the power to explain the anthropic fine-tuning of our region of space. We will see shortly that all the other pockets are in the mysterious ghostly portions of space out beyond our horizon. Without the idea of a megaverse of pockets, there is no natural way to formulate a sensible Anthropic Principle. My own view of this dilemma will be explained in the next chapter, but I will state it briefly here. I believe this whole discussion is based on a fallacy. In a universe governed by quantum mechanics, the apparent ultimate barriers are not so ultimate. In principle, objects behind horizons are quite within our grasp, but only in principle. More on this in the next chapter.
Curiously, in a universe accelerating under the influence of a cosmological constant, the distance to the event horizon never changes. It is fixed b
y the value of the cosmological constant—the larger the cosmological constant, the smaller the distance to the horizon. The observer lives in an unchanging world of finite radius bounded by her horizon, but in exactly the way that the earth’s horizon eludes anyone who tries to approach it, the cosmic horizon of de Sitter space can never be reached. It’s always a finite distance away, but when you approach it, there’s nothing there! However, if we could get outside the de Sitter space—watch it from a distance, so to speak—we would see the whole space exponentially growing with time.
Metastable de Sitter Space
I want to return to the topic of metastable substances, but with a new twist—suppose the substance in question is inflating. To help visualize the expanding metastable substance, imagine an infinite shallow lake of supercooled water. To simulate the cloning of space, the bottom of the lake could be filled with small feeder pipes that continuously provide new supercooled water. In order to make room for the new fluid, the water spreads out horizontally—any two molecules getting farther apart because new molecules come to fill in the growing space between them. If boats were floating on the lake, they would separate and lose contact. The lake inflates just like de Sitter space.
In that inflating body of supercooled water, crystals of ice will randomly nucleate from time to time. If they are large enough, they will grow and become expanding ice islands. But because they are being carried along with the spreading fluid, the growing islands may separate so fast that they never meet one another. The regions between the islands inflate and prevent the entire lake from becoming solid ice. The space between islands eternally grows, remaining liquid, even though the islands of ice also grow indefinitely. Nevertheless, any observer floating along with the flow will end up surrounded by ice: given enough time, a tiny crystal of ice will eventually nucleate in the person’s vicinity and swallow her up. This is a somewhat paradoxical but, nonetheless, correct conclusion: there is always plenty of liquid water but any given bit of it is sooner or later engulfed in ice.
What I have described is a precise analogy for the phenomenon called Eternal Inflation: growing islands of alternate vacuum in a sea of eternally inflating space. It is by no means a new idea. My colleague at Stanford, Andrei Linde, is one of the great thinkers who have pioneered many of the modern ideas of cosmology. For as long as I’ve known Andrei—certainly since he came to the United States from Russia, about fifteen years ago—he has been preaching the doctrine of an eternally inflating universe, constantly spinning off bubbles of many kinds.3 Alexander Vilenkin is another Russian-American cosmologist who has determinedly tried to push cosmology in the direction of a superinflating megaverse of enormous diversity. But, for the most part, physicists have ignored these ideas, at least until very recently. What is shaking up the field right now is the realization that String Theory—our best guess for a theory of nature—has features that mesh very well with these older ideas.
The combination of general relativity, quantum mechanics, and an initial high-density universe, together with the Landscape of String Theory, suggest that an eternally inflating, metastable universe may be inevitable.
Eternal Inflation
If you purchased this book hoping to find the ultimate answer to how the universe began, I am afraid you will be disappointed. Neither I nor anyone else knows. Some think it began with a singularity, an infinitely violent state of infinite energy density. Others, notably Stephen Hawking and his followers, believe in a quantum tunneling from nothing. But however it began, we know one thing. At some time in the past, the universe existed in a state of very large energy density, probably trapped in an inflationary expansion. Almost all cosmologists will agree that a history of rapid exponential inflation is very likely the explanation of many puzzles of cosmology. In chapter 5 we learned about the observational basis for this belief. It seems all but certain that the observable history of our universe began about fourteen billion years ago at a point in the Landscape with enough energy density to inflate our patch of space by at least 1020 times. That is probably an enormous underestimate. The energy density during this period was very large—how large we don’t know for sure but vastly larger than anything we can make in the laboratory, even during the most violent collisions of elementary particles in the largest accelerators. It appears that at that time, the universe was not quite trapped in a valley of the Landscape but was resting on a slightly tilted plateau. As it inflated, our pocket of space (the observable universe) slowly rolled down the shallow tilt, toward a sudden, steep ledge, and when it got to the edge of the ledge, it quickly descended, converting potential energy to heat and particles. This event, which created the material of the universe, is called reheating. Finally, the universe rolled down to our present valley, with its tiny anthropic cosmological constant. That’s it: cosmology as we know it was a brief roll from one value of the vacuum energy to another. All the interesting things happened during this transitory period.
How did our pocket universe get up on the ledge? That’s what we don’t know. But it is mighty convenient that it started where it did. Without the Inflation caused by the energy density on the ledge, the universe could not have evolved to the large, matter-filled universe we see around us: a universe big enough, smooth enough, and with density contrasts just right for our own existence.
The problem with a theory that places us on the ledge at the very beginning is that this is just one of a stupendous number of starting points. Its only distinction is that it provides a potentially successful beginning for a universe with a chance for life to evolve. Arbitrarily placing the universe at such a lucky spot on the Landscape would defeat the goal of explaining the world without an intelligent designer. But as I will explain, a theory with an enormous Landscape has no choice. To my mind it is completely inevitable—mathematically certain—that some parts of space will evolve to find themselves at the lucky spot. But not everyone agrees.
The Princeton cosmologist Paul Steinhardt, in a critique of the Anthropic Principle, says: “The Anthropic Principle makes an enormous number of assumptions—regarding the existence of multiple universes.… Why do we need to postulate an infinite number of universes with all sorts of different properties just to explain our one?” The answer is that we don’t need to postulate them. They are unavoidable consequences of well-tested, conventional principles of general relativity and quantum mechanics.
It is ironic that Steinhardt’s own work contained the original germ of the Eternal Inflation idea, including the arguments that I find so inevitable. The bubbling up of an infinity of pocket universes is as certain as the bubbling of an opened bottle of champagne. There are only two assumptions: the existence of a Landscape and the fact that the universe started with a very high density of energy, i.e., that it started at high altitude. The first may prove to be no assumption at all. The mathematics of String Theory seems to make the Landscape unavoidable. And the second—high-energy density—is a feature of every scientific cosmology that begins with the Big Bang. Let me explain why I, together with most other cosmologists, find Eternal Inflation to be such a compelling idea.
The ideas that I am going to tell you about are not my own. They were pioneered by the cosmologists Alan Guth, Andrei Linde, Paul Steinhardt, and Alexander Vilenkin and owe a large debt to the seminal work of one of the great physicists of my generation, Sidney Coleman. Let us begin with a universe, or perhaps just a patch of space, located at an arbitrary point in the Landscape with only the simple requirement that the energy density is rather large. Like any mechanical system it will begin to evolve toward regions of lower potential energy. Think of rolling a bowling ball from the top of Mount Everest. What is the likelihood that it will roll all the way to sea level without getting stuck somewhere? Not too good. Far more likely it will come to rest in some local valley not far from the mountain. The initial conditions—exactly where it began and with what velocity it started to roll—hardly matter.
As with bowling balls, so it goes with the patch
of space we are following: it will most likely plop into some valley, where it will begin to inflate. A stupendous volume of space will be cloned, all located in the same valley. There are of course lower valleys, but to get to them the universe would have to climb over mountain passes at elevations higher than the starting valley, and it cannot do so because it doesn’t have the energy. So it just sits there and inflates forever.
But we’ve forgotten one thing. The vacuum has the quantum jitters. Just like the thermal jitters of supercooled water, the quantum jitters cause small bubbles to form and disappear. The interior of these bubbles may lie in a neighboring valley, with smaller altitude. This bubbling is constantly going on, but most of the bubbles are too small to grow. The surface tension on the domain walls separating the bubble from the rest of the vacuum squeezes them away. But, as in the supersaturated case, every now and then a bubble forms that is big enough to start growing.
The mathematics describing this bubble formation in an inflating universe has been known for many years. In 1977 Sidney Coleman and Frank De Luccia wrote a paper that was to become a classic. In their paper they calculated the rate at which such bubbles would appear in an inflating universe, and although the rate could be very small—very few bubbles per unit volume—it most certainly is not zero. The calculations used only the most trustworthy, well-tested methods of quantum field theory and are considered by modern physicists to be rock solid. Thus, unless there is something terribly wrong, the inflating vacuum will spin off growing bubbles located in neighboring valleys.
Do bubbles collide, eventually coalescing so that all of space winds up in some new valley? Or does the space between the bubbles expand too quickly to allow the islands to merge? The answer depends on the competition between two rates—the rate of bubble formation and the rate that space reproduces, or the rate of cloning. If the bubbles form very quickly, they will rapidly collide and merge, the whole space moving to some new point on the Landscape. But if the rate at which space reproduces is greater than the rate at which bubbles form, cloning wins, and the bubbles never catch up with one another. Like the islands of ice in the inflating supercooled lake, the bubbles evolve in isolation, eventually moving beyond one another’s horizons. The majority of space continues to eternally inflate.
The Cosmic Landscape Page 32
