Our Mathematical Universe
Page 16
Figure 6.1: If you fly to the right, past the circle in the toroidal universe, you immediately reenter at the corresponding point on the left circle—leave at A, reenter at A, etc.; the two As are actually the same physical point. This means that the cosmic microwave–background patterns along the two circles should look similar to us, since they’re actually one and the same.
Click here to see a larger image.
Enough about infinite space. What about the infinite-matter part of the assumption? Before inflation, it was often justified by appealing to the so-called Copernican principle, that we humans don’t occupy any special place in the cosmos: if there are galaxies around here, there should be galaxies everywhere.
What do recent observations have to say about it? Specifically, how uniform is the matter distribution on large scales? In an “island universe” model where space is infinite but all the matter is confined to a finite region, almost all members of the Level I multiverse would be dead, consisting of nothing but empty space. Such models have been popular historically, originally with the island being Earth and the celestial objects visible to the naked eye, and in the early twentieth century, with the island being the known part of the Milky Way Galaxy. The island-universe model has been demolished by recent observations. The 3-D galaxy maps from the last chapter have shown that the spectacular large-scale structure observed (galaxy groups, clusters, superclusters, walls) gives way to dull uniformity on large scales, with no coherent structures larger than about a billion light-years.
The larger the scale we observe, the more uniformly filled with matter our Universe looks (Figure 4.6). Barring conspiracy theories where our Universe is designed to fool us, the observations thus speak loud and clear: space as we know it appears to continue far beyond the edge of our Universe, teeming with galaxies, stars and planets.
* * *
1If the cosmic expansion continues to accelerate (currently an open question), the observable Universe will eventually stop growing: all galaxies beyond a certain distance will eventually recede faster than light and be forever invisible to us.
Where Are the Level I Parallel Universes?
We’ve seen that if they exist, then Level I parallel universes are simply universe-sized parts of our space that are so far away that light from them hasn’t yet had time to reach us. Does the fact that we’re at the center of our Universe mean that we’re somehow in a special place in space? Well, if you’re walking on a large field when fog has cut the visibility to 50 meters, you’ll feel like you’re at the center of a fog sphere, beyond which (akin to the edge of our Universe) you can’t see anything. But that doesn’t mean that you’re in any sort of special place, at the center of anything fundamental, because everyone else on that field will find themselves at the center of their own fog spheres. In the same way, any observers anywhere in space will find themselves at the centers of their universes. Also, there are no physical boundaries between neighboring universes, just as there’s no special boundary 50 meters into the fog—the field and the fog have the same properties over there as where you are. Moreover, universes can overlap just as fog spheres can: just as someone 30 meters away on the field can see both you and regions that you can’t see, the universe of someone in a galaxy 5 billion light-years from us would contain both Earth and regions of space that lie outside of our Universe.
If eternal inflation or something else created an infinite number of such parallel universes, then how far away is the nearest identical copy of our own? According to classical physics, a universe can be arranged in infinitely many different ways, so there’s no guarantee that you’d ever find an exactly identical one. Classically, there are infinitely many options even for the distance between two particles, since it requires infinitely many decimal places to specify. However, there’s clearly only a finite number of universe possibilities that our collective human civilization can ever distinguish between in practice, since our brains and computers can store only a finite amount of information. Moreover, we can only measure things with finite accuracy—our current record in physics is measuring a quantity to about sixteen decimal places.
Quantum mechanics limits the variety even at a fundamental level. As we’ll explore in the next two chapters, quantum mechanics adds a sort of intrinsic fuzziness to nature that makes it meaningless to talk about where things are beyond a certain level of precision. The result of this limitation is that the total number of ways in which our Universe can be arranged is finite. A conservative estimate, erring on the high side, is that there are at most 1010118 possible ways in which a universe the size of ours can be arranged.1 An even more conservative bound, known as the holographic principle, says that a volume the size of our Universe can be arranged in, at most, 1010124 ways.2 Otherwise, you’d have to pack so much stuff into it that it would form a black hole larger than itself.
These are huuuuuuge numbers, even larger than the famed googolplex. Little boys tend to obsess about big things, and I’ve overheard my sons and their friends try to outdo each other by naming ever bigger numbers. After trillions, octillions and so on, someone inevitably drops the G-bomb: googolplex. After which, a moment of awed silence ensues. As you may know, a googolplex is one followed by a googol zeros, where a googol is one followed by a hundred zeros. So it’s 1010100, which isn’t one followed by a hundred zeros, but one followed by 10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 zeros! This number is so large that you couldn’t write it out even in principle, since it contains more digits than there are atoms in our Universe. I always suspected that Google was an ambitious company. When I visited them for a conference, I discovered that they call their corporate campus the Googleplex.
Although 1010118 is huge beyond astronomical, it’s still nothing compared with infinity. This means that if eternal inflation made a space containing infinitely many Level I parallel universes, then you’ll find it containing all possibilities. Specifically, you’ll have to check on average about 1010118 universes until you find a copy of any particular kind of universe, as illustrated in Figure 6.2. So if you could travel in a straight line until you reached the closest identical copy of our own Universe, you’d need to journey about 1010118 universe diameters. If you’re willing to look in all directions to find our closest copy, the distance to the closest one comes out be about the same, which is also about the same as 1010118 meters, given the funny mathematical behavior of double exponents (powers of powers).3
Figure 6.2: In a toy universe where four different locations can each hold one of two kinds of particles, there are only 24 possible arrangements (top left). This means that in a Level I multiverse of such universes, on average you have to check 16 universes until you find a repeat of a particular universe. If our Universe can similarly contain 1010118 particles arranged in 1010118 different ways, then you’ll have to travel past about 1010118 parallel universes before reaching an identical copy.
Click here to see a larger image.
Closer by, about ~ 101091 meters away, there should be a sphere of radius 100 light-years identical to the one centered here, so all perceptions that we have during the next century will be identical to those of our counterparts over there. About ~ 101029 meters away, there should be an identical copy of you. Indeed, there are probably copies of you much closer than that, since the planet formation and evolutionary processes that have tipped the odds in your favor are at work everywhere. There are probably at least 1020 habitable planets in our own Universe volume alone.
* * *
1This is an extremely conservative estimate, simply counting all possible quantum states that a universe (horizon volume) can have that are no hotter than 108 degrees. Although the actual calculation requires quantum-mechanical details, the number 10118 can be roughly understood as the number of protons that the so-called Pauli exclusion principle would allow you to pack into a universe at this temperature (our own Universe contains only about 1080 pr
otons). If each of these 10118 slots can be either occupied or unoccupied, there are 210118 ~ 1010118 possibilities.
2That’s two to the power of the surface area of our Universe measured in so-called Planck units. The books by Lenny Susskind and Brian Greene in the “Suggestions for Further Reading” section describe the holographic principle in detail and how it was developed from ideas of Gerard t’Hooft, Lenny Susskind, Charles Thorn, Raphael Bousso, Jacob Bekenstein, Stephen Hawking, Juan Maldacena and others.
3If you’re a math buff, note that 1010118 universe diameters ≈ 1010118 × 1027 m = 1010118+27 m ≈ 1010118 m. If you’re willing to look in all directions to find our closest copy, then you need to explore a spherical volume around us containing about 1010118 universes, whose radius exceeds that of our Universe by a factor (1010118)1/3 = 1010118/3 ≈ 1010117.53 ≈ 1010118.
The Level II Multiverse
Earlier, I called inflation the gift that keeps on giving, because every time you think it can’t possibly predict something more radical than it already has, it does. If you felt that the Level I multiverse was large and hard to stomach, try imagining an infinite set of distinct ones, some perhaps with apparently different laws of physics. Andrei Linde, Alex Vilenkin, Alan Guth and their colleagues have shown that this is what inflation typically predicts, and we’ll refer to it as the Level II multiverse.
Many Universes in One Space
How can physics possibly allow such craziness? Well, we saw in Figure 5.8 how inflation could create an infinite volume inside of a finite volume. As Figure 6.3 illustrates, there’s no reason why inflation can’t do this in several adjacent volumes, ending up with several infinite regions (Level I multiverses), as long as inflation is eternal and never ends at the boundaries between them. This means that if you live in one of these Level I multiverses, it’s impossible for you to visit a neighboring one: inflation keeps creating intervening space faster than you can travel through it. I imagine trying this with my kids in the backseat of my rocket:
Figure 6.3: If eternal inflation creates three infinite regions using the mechanism from Figure 5.8, then travel between them is impossible because inflation keeps creating new space between you and your destination faster than you can travel through it.
Click here to see a larger image.
“Dad, are we there yet?”
“We have one light-year left to go.”
“Dad, are we there yet?”
“We have two light-years left to go.”
In other words, although these other parts of the Level II multiverse are in the same space as we are, they’re more than infinitely far away in the sense that we’d never reach them even if we traveled at the speed of light forever. In contrast, you can in principle travel to an arbitrarily distant part of our Level I multiverse if you’re patient enough and the cosmic expansion decelerates.1
I’ve simplified things in Figure 6.3 by ignoring the fact that space is expanding. The eternally inflating regions in the figure, which I’ve drawn as thin vertical bars separating the U-shaped Level I multiverses, will in fact expand rapidly, and eventually, parts within them will stop inflating, giving rise to additional U-shaped regions. This makes things even more interesting, giving the Level II multiverse a treelike structure as illustrated in Figure 6.4. Any inflating region keeps expanding rapidly, but inflation eventually ends in various parts of it, forming U-shaped regions that each constitute an infinite Level I multiverse. This tree continues growing forever, creating an infinite number of such U-shaped regions—all of them together form the Level II multiverse. Within each such region, the end of inflation transforms the inflating substance into particles that eventually cluster into atoms, stars and galaxies. Alan Guth likes to call each Level I multiverse a “pocket universe,” because it conveniently fits into a small part of the tree.
Figure 6.4: The expansion of space and the fact that inflation keeps ending in certain places gives the Level II multiverse a treelike structure. Inflation continues in the tree-shaped gray part of space and time, and each U-shaped region where inflation has ended is an infinite Level I multiverse.
* * *
1If the dark energy sticks around so that our cosmic acceleration continues, then even most Level I parallel universes will remain forever separate, with the intervening space stretching faster than light can travel through it. We don’t yet understand dark energy well enough to know whether this will happen.
Diversity!
Earlier in this chapter, I mentioned that the Level II multiverse can contain infinite regions with apparently different laws of physics. But this sounds absurd: how can the laws of physics allow different laws of physics? As we’ll now see, the key idea is that fundamental laws of physics, which by definition hold anywhere and anytime, can give rise to a complicated physical state of affairs where the effective laws of physics inferred by self-aware observers vary from place to place.
If you were a fish who’d lived your entire life in the ocean, you might make the mistake of thinking of water not as a substance, but as empty space. What a human would think of as a property of water, say, the friction when swimming through it, you might misinterpret as a fundamental law of physics: “a fish in uniform motion ends up at rest—unless flapping its fins.” You’d probably have no idea that water can exist in three different phases—solid, liquid and gaseous—and that your “empty space” was simply the liquid phase, a particular solution to the equations describing water.
This example may sound silly, and if a real fish were to think this, we might be tempted to laugh at it. But could it be that what we humans think of as empty space is also some form of medium? Then the last laugh would be on us! As a matter of fact, there’s mounting evidence that this is exactly how things are. Not only does our “empty space” seem to be a sort of medium, but it appears to have way more than three phases—perhaps about 10500, and perhaps even infinitely many, which opens up the possibility that, in addition to curving, stretching and vibrating, our space may even be able to do something analogous to freezing and evaporating!
Figure 6.5: Can space freeze? A fish might think of water as empty space, because it’s the only medium it knows. But if a clever fish figured out the physical laws governing water molecules, it could realize that they have three different solutions, “phases,” corresponding to the liquid water it knows and also to steam and ice, which it’s never seen. In the same way, what we’ve thought of as empty space may be a medium with 10500 or more different phases, of which we’ve experienced only one.
How did physicists reach this conclusion? Well, if a fish were sufficiently intelligent, it could build experiments and determine that its “space” is made of water molecules obeying certain mathematical equations. By studying these equations, it could, as illustrated in Figure 6.5, determine that they have three different solutions corresponding to the three phases of solid ice, liquid water and gaseous steam, even if it had never seen either an iceberg or a steam-producing underwater volcanic vent. In exactly the same way, we physicists are searching for equations describing our own space and its contents. We haven’t yet found the final answer, but the approximate answers we’ve found so far tend to share a key feature: they have more than one solution (phase) that describes a uniform space. String theory, which is a leading candidate for a final answer, has been found to have perhaps 10500 or more solutions, and there’s no indication that competing theories such as loop quantum gravity have a single unique solution either. Physicists like to call the collection of all possible solutions the landscape of the theory.1 However, this pessimistic conclusion rests on a rather dubious premise: that the way inflation occurred in our region of space is the only way that inflation occurred anywhere. These solutions, whose properties constitute effective laws of physics, all correspond to different possibilities allowed by the same fundamental laws of physics.
What does this have to do with inflation? Remarkably, eternal inflation has the property that it creates all possible kinds of
space! It realizes the entire landscape. In fact, for each phase that space can have, it creates infinitely many Level I multiverses full of that phase. This means that we observers are easily tricked into making the same mistake as the fish: because we observe space to have the same properties everywhere in our Universe, we’re tempted to mistakenly conclude that space is like that everywhere else as well.
How does inflation do this? Well, it requires lots of energy to change the phase of space, so the everyday processes that we can observe are unable to do it. But back during inflation, there was an enormous amount of energy in each small volume, enough for the previously mentioned quantum fluctuations to occasionally cause a phase change in some tiny region, which would then inflate to become an enormous region containing only that same phase. Moreover, a given region of space has to be in a definite phase in order to stop inflating. This ensures that boundary regions between two phases keep inflating forever, so that each phase fills an entire infinite Level I multiverse.