by Brian Greene
Weinberg worked out the idea mathematically and found that a cosmological constant any larger than a few hundred times the current cosmological density of matter, a few protons per cubic meter, would disrupt the formation of galaxies. (Weinberg also considered the impact of a negative cosmological constant. The constraints in that case are even tighter, because a negative value increases the attractive pull of gravity and makes the whole universe collapse before stars even have time to ignite.). If you imagine, then, that we’re part of a multiverse and that the cosmological constant’s value varies over a wide range from universe to universe, much as planet-star distances vary over a wide range from solar system to solar system—the only universes that could have galaxies, and hence the only universes we could inhabit, are ones in which the cosmological constant is no larger than Weinberg’s limit, which in Planck units is about 10–121.
After years of failed efforts by the community of physicists, this was the first theoretical calculation to result in a value for the cosmological constant that was not absurdly larger than limits inferred from observational astronomy. Nor did it contradict a belief widely held at the time of Weinberg’s work, that the cosmological constant vanished. Weinberg took this apparent progress one step further by encouraging an even more aggressive interpretation of his result. He suggested that we should expect to find ourselves in a universe with a cosmological constant whose value is as small as it needs to be for us to exist, but not a whole lot smaller. A much smaller constant, he reasoned, would call for an explanation that goes beyond mere compatibility with our existence. That is, it would require precisely the kind of explanation that physics had valiantly sought but so far failed to find. This led Weinberg to suggest that more refined measurements might one day reveal that the cosmological constant doesn’t vanish but, instead, has a value near or at the upper limit that he’d calculated. As we’ve seen, within a decade of Weinberg’s paper, the observations of the Supernova Cosmology Project and the High-Z Supernova Search Team proved this suggestion prophetic.
But to assess fully this unconventional explanatory framework, we need to examine Weinberg’s reasoning more closely. Weinberg is imagining a sprawling multiverse so diverse in population that it just has to contain at least one universe with the cosmological constant we’ve observed. But what kind of multiverse will guarantee, or at least make it highly likely, that this is the case?
To think this through, consider first an analogous problem with simpler numbers. Imagine you work for the notorious film producer Harvey W. Einstein, who has asked you to put out a casting call for the lead in his new indie, Pulp Friction. “How tall do you want him?” you ask. “I dunno. Taller than a meter, less than two. But you better make sure whatever height I decide, there’s someone who fits the bill.” You’re tempted to correct your boss, noting that because of quantum uncertainty he really doesn’t need to have every height represented but, thinking back on what happened to the surly little talking fly who tried that, you refrain.
Now you face a decision. How many actors should you have at the audition? You reason: If W. measures heights to a centimeter’s accuracy, there are a hundred different possibilities between one and two meters. So you need at least a hundred actors. But since some actors who show up may have the same height, leaving other heights unrepresented, you’d better gather more than a hundred. To be safe, maybe you should put out a call for a few hundred actors. That’s a lot, but fewer than what you’d need if W. measured heights to a millimeter’s accuracy. In that case, there’d be a thousand different heights between one and two meters, so to be safe you’d need to gather a few thousand actors.
The same reasoning is relevant for the case of universes with different cosmological constants. Assume that all the universes in a multiverse have cosmological constant values between zero and one (in the usual Planck units); smaller values lead to universes that collapse, larger values would strain the applicability of our mathematical formulations, compromising all understanding. So just as the actors’ heights had a range of one (in meters), the universes’ cosmological constants have a range of one (in Planck units). As for accuracy, the analog of W. using centimeter ticks, or millimeter ticks, is the precision with which we can measure the cosmological constant. Today’s accuracy is about 10–124 (in Planck units). In the future, our accuracy will no doubt improve, but as we’ll see, that will hardly affect our conclusions. Then just as there are 102 different possible heights spaced at least 10–2 meters apart (1 centimeter) in a one-meter range, and 103 different possible heights spaced at least 10–3 meters apart (1 millimeter), so there are 10124 different values of the cosmological constant spaced at least 10–124 apart between the values 0 and 1.
To ensure that every possible cosmological constant is realized, we’d therefore need a multiverse with at least 10124 different universes. But as with the actors, we need to account for possible duplicates, universes that may have the same cosmological constant value. And so to play it safe and make it highly likely that every possible cosmological constant value is realized, we should have a multiverse with far more than 10124 universes, say a million times more, bringing it to a nice even 10130 universes. I’m being cavalier because when we’re talking about numbers this large, the exact values hardly matter. No familiar example of anything—not the number of cells in your body (1013); not the number of seconds since the big bang (1018); not the number of photons in the observable part of the universe (1088)—comes even remotely close to the number of universes we’re contemplating. The bottom line is that Weinberg’s approach for explaining the cosmological constant works only if we’re part of a multiverse in which there are a huge number of different universes; their cosmological constants must fill out some 10124 distinct values. Only with that many different universes is there a high likelihood that there’s one with a cosmological constant that matches ours.
Are there theoretical frameworks that naturally yield such a spectacular profusion of universes with different cosmological constants?14
From Vice to Virtue
There are. We encountered such a framework in the previous chapter. A count of the different possible forms for the extra dimensions in string theory, when including fluxes that can thread through them, came to about 10500. This dwarfs 10124. Multiply 10124 by a few hundred orders of magnitude and 10500 still dwarfs it. Subtract 10124 from 10500, and then subtract it again, and again, and do so a billion times over, and you’d barely make a dent. The result would still be nearly 10500.
Critically, the cosmological constant does indeed vary from one such universe to another. Just as magnetic flux carries energy (it can move things), so the fluxes threading holes in Calabi-Yau shapes also have energy, whose quantity is quite sensitive to the shape’s geometrical details. If you have two different Calabi-Yau shapes with different fluxes penetrating different holes, their energies will generally be different too. And since a given Calabi-Yau shape is attached to every point in the familiar three large dimensions of space, much as circular loops of pile attach to every point on the large extended base of a carpet, the energy the shape contains would uniformly fill the three large dimensions, much as soaking the individual fibers in a carpet’s pile would make the entire carpet backing uniformly heavy. Thus, should one or another of the 10500 different dressed-up Calabi-Yau shapes constitute the requisite extra dimensions, the energy it contains would contribute to the cosmological constant. Results obtained by Raphael Bousso and Joe Polchinski made this observation quantitative. They argued that the various cosmological constants supplied by the 10500 or so different possible forms for the extra dimensions are distributed uniformly across a broad range of values.
This is just what the doctor ordered. Having 10500 tick marks distributed across a range from 0 to 1 ensures that many of them lie extremely close to the value of the cosmological constant astronomers have measured during the past decade. It may be hard to find the explicit examples among the 10500 possibilities, because even if today’
s fastest computers took a single second to analyze each form for the extra dimensions, after a billion years only a paltry 1032 examples would have been examined. But this reasoning suggests strongly that they exist.
Certainly, a collection of 10500 different possible forms for the extra dimensions is about as far from a unique universe as anyone imagined string theory research would ever take us. And for those who’ve held strongly to Einstein’s dream of finding a unified theory describing one single universe—ours—these developments came with significant discomfort. But analysis of the cosmological constant casts the situation in a different light. Rather than despair because a unique universe seems not to emerge, we are encouraged to celebrate: string theory makes the least plausible part of Weinberg’s explanation of the cosmological constant—the requirement that there be many more than 10124 different universes—suddenly seem plausible.
The Final Step, in Brief
The elements of a tantalizing story seem to be coming together. But a gap remains in the reasoning. It’s one thing for string theory to allow for a huge number of possible distinct universes. It’s another to claim that string theory ensures that all of the possible universes to which it can give rise are actually out there, parallel worlds populating a vast multiverse. As emphasized most emphatically by Leonard Susskind—who was inspired by the pioneering work of Shamit Kachru, Renata Kallosh, Andrei Linde, and Sandip Trivedi—if we weave eternal inflation into the tapestry, the gap can be filled.15
I’ll now explain this final step, but in case you’re reaching saturation and just want the punch line, here’s a three-sentence summary. The Inflationary Multiverse—the ever-expanding Swiss cheese cosmos—contains a vast, ever-increasing number of bubble universes. The idea is that when inflationary cosmology and string theory are melded, the process of eternal inflation sprinkles string theory’s 10500 possible forms for the extra dimensions across the bubbles—one form for the extra dimensions per bubble universe—providing a cosmological framework that realizes all possibilities. By this reasoning, we live in that bubble whose extra dimensions yield a universe, cosmological constant and all, that’s hospitable to our form of life and whose properties agree with observations.
In the remainder of the chapter, I will flesh out the details, but if you’re ready to move on, feel free to jump ahead to the chapter’s last section.
The String Landscape
In explaining inflationary cosmology back in Chapter 3, I used a variation on a common metaphor. A mountain’s peak represents the highest value of energy contained in an inflaton field suffusing space. The act of rolling down the mountain and coming to rest at a low point in the terrain represents the inflaton shedding this energy, which in the process is converted to particles of matter and radiation.
Let’s revisit three aspects of the metaphor, updating them with insights we’ve since acquired. First, we’ve learned that the inflaton is only one source of the energy that may fill space; other contributions come from the quantum jitters of any and all fields—electromagnetic, nuclear, and so on. To revise the metaphor accordingly, altitude will now reflect the combined energy uniformly suffusing space contributed by all sources.
Second, the original metaphor envisioned the base of the mountain, where the inflaton finally comes to rest, as being at “sea level,” altitude zero, meaning the inflaton has shed all its energy (and pressure). But with our revised metaphor, the height of the mountain’s base should represent the combined energy suffusing space from all sources after inflation has drawn to a close. This is another name for that bubble universe’s cosmological constant. The mystery in explaining our cosmological constant thus translates into the mystery of explaining the altitude of our mountain’s base—why is it so close to, but not exactly at, sea level?
Finally, we initially considered the simplest of mountainous terrains, a peak leading smoothly to a base, where the inflaton would ultimately settle (see Figure 3.1). We then went a step further, taking account of other ingredients (Higgs fields) whose evolution and final resting places would influence the physical features manifest in the bubble universes (see Figure 3.6). In string theory, the range of possible universes is richer still. The shape of the extra dimensions determines the physical features within a given bubble universe, and so the possible “resting places,” the various valleys in Figure 3.6b, now represent the possible shapes the extra dimensions can take. To accommodate the 10500 possible forms for these dimensions, the mountain terrain therefore needs a lush assortment of valleys, ledges, and outcroppings, as represented in Figure 6.4. Any such feature in the terrain where a ball could come to rest represents a possible shape into which the extra dimensions could relax; the altitude at that location represents the cosmological constant of the corresponding bubble universe. Figure 6.4 illustrates what’s called the string landscape.
With this more refined understanding of the mountain—or landscape—metaphor, we now consider how quantum processes affect the form of the extra dimensions in this setting. As we will see, quantum mechanics lights up the landscape.
Figure 6.4 The string landscape can be visualized schematically as a mountainous terrain in which different valleys represent different forms for the extra dimensions, and altitude represents the cosmological constant’s value.
Quantum Tunneling in the Landscape
While Figure 6.4 is necessarily schematic (each of the different Higgs fields in Figure 3.6 has its own axis; similarly each of the roughly 500 different field fluxes that can thread a Calabi-Yau shape should also have its own axis—but sketching mountains in a 500-dimensional space is a challenge), it correctly suggests that universes with different forms for the extra dimensions are part of a connected terrain.16 And when quantum physics is taken into account, using results discovered independently of string theory by the legendary physicist Sidney Coleman in collaboration with Frank De Luccia, the connections between the universes allow for dramatic transmutations.
The core physics relies on a process known as quantum tunneling. Imagine a particle, an electron for instance, encountering a solid barrier, say a slab of steel ten feet thick, that classical physics predicts it can’t penetrate. A hallmark of quantum mechanics is that the rigid classical notion of “can’t penetrate” often translates into the softer quantum declaration of “has a small but nonzero probability of penetrating.” The reason is that the quantum jitters of a particle allow it, every so often, to suddenly materialize on the other side of an otherwise impervious barrier. The moment at which such quantum tunneling happens is random; the best we can do is predict the likelihood that it will take place during one interval or another. But the math says that if you wait long enough, penetration through just about any barrier will happen. And it does happen. If it didn’t, the sun wouldn’t shine: for hydrogen nuclei to get close enough to fuse, they must tunnel through the barrier created by the electromagnetic repulsion of their protons.
Coleman and De Luccia, and many who have since followed their lead, scaled quantum tunneling up from single particles to an entire universe that’s faced with a similar “impenetrable” barrier separating its current configuration from another that’s possible. To get a feel for their result, imagine two possible universes that are otherwise identical save for a field, uniformly suffusing each, whose energy is higher in one, lower in the other. In the absence of a barrier, the higher energy-field value rolls to the lower, like a ball rolling down a hill as we’ve seen in the discussion of inflationary cosmology. But what happens if the field’s energy curve has a “mountainous bump” separating its current value from the one it seeks, as in Figure 6.5? Coleman and De Luccia found that much as is the case for a single particle, a universe can do what classical physics forbids: it can jitter its way—it can quantum tunnel—through the barrier and reach the lower energy configuration.
Figure 6.5 An example of a field’s energy curve that has two values—two troughs or valleys—where the field naturally comes to rest. A universe suffused with th
e higher-energy field value can quantum tunnel to the lower value. The process involves a small randomly located region of space in the original universe acquiring the lower field value; the region then expands, converting an ever-wider domain from the higher to the lower energy.
But because we are talking about a universe and not just a single particle, the tunneling process is more involved. It’s not that the field’s value throughout all of space tunnels simultaneously through the barrier, Coleman and De Luccia argued; rather, a “seed” tunneling event would create a small, randomly located bubble suffused with the smaller field energy. The bubble would then grow, much like Vonnegut’s ice-nine, ever enlarging the domain in which the field had tunneled to the lower energy.
These ideas can be applied directly to the string landscape. Imagine that the universe has a particular form for the extra dimensions, which corresponds to the left valley in Figure 6.6a. Because of this valley’s high altitude, the three familiar spatial dimensions are permeated by a large cosmological constant—yielding strong repulsive gravity—and so are rapidly inflating. This expanding universe, together with its extra dimensions, is illustrated on the left side of Figure 6.6b. Then, at some random location and moment, a tiny region of space tunnels through the intervening mountain to the valley on the right side of Figure 6.6a. Not that the tiny region of space moves (whatever that would mean); rather, the form of the extra dimensions (its shape, size, fluxes it carries) in this little region changes. The extra dimensions in the tiny region transmute, acquiring the form associated to the right valley in Figure 6.6a. This new bubble universe lies within the original, as illustrated in Figure 6.6b.