From Eternity to Here: The Quest for the Ultimate Theory of Time - Sean Carroll (2010)
Part IV. FROM THE KITCHEN TO THE MULTIVERSE
Chapter 13. THE LIFE OF THE UNIVERSE
Time is a great teacher, but unfortunately it kills all its pupils.
What should the universe look like?
This might not be a sensible question. The universe is a unique entity; it’s different in kind from the things we typically think about, all of which exist in the universe. Objects within the universe belong to larger collections of objects, all of which share common properties. By observing these properties we can get a feel for what to expect from that kind of thing. We expect that cats usually have four legs, ice cream is usually sweet, and supermassive black holes lurk at the centers of spiral galaxies. None of these expectations is absolute; we’re talking about tendencies, not laws of nature. But our experience teaches us to expect that certain kinds of things usually have certain properties, and in those unusual circumstances where our expectations are not met, we might naturally be moved to look for some sort of explanation. When we see a three-legged cat, we wonder what happened to its other leg.
The universe is different. It’s all by itself, not a member of a larger class. (Other universes might exist, at least for certain definitions of “universe”; but we certainly haven’t observed any.) So we can’t use the same kind of inductive, empirical reasoning—looking at many examples of something, and identifying common features—to justify any expectations for what the universe should be like.232
Nevertheless, scientists declare that certain properties of the universe are “natural” all the time. In particular, I’m going to suggest that the low entropy of the early universe is surprising, and argue that there is likely to be an underlying explanation. When we notice that an unbroken egg is in a low-entropy configuration compared to an omelet, we have recourse to a straightforward explanation: The egg is not a closed system. It came out of a chicken, which in turn is part of an ecosystem here on Earth, which in turn is embedded in a universe that has a low-entropy past. But the universe, at least at first glance, does seem to be a closed system—it was not hatched out of a Universal Chicken or anything along those lines. A truly closed physical system with a very low entropy is surprising and suggests that something bigger is going on.233
The right attitude toward any apparently surprising feature of the observed universe, such as the low early entropy or the small vacuum energy, is to treat it as a potential clue to a deeper understanding. Observations like this aren’t anywhere near as definitive as a straightforward experimental disagreement with a favored theory; they are merely suggestive. In the backs of our minds, we’re thinking that if the configuration of the universe were chosen randomly from all possible configurations, it would be in a very high-entropy state. It’s not, so therefore the state of the universe isn’t just chosen randomly. Then how is it chosen? Is there some process, some dynamical chain of events, that leads inevitably to the seemingly non-random configuration of our universe?
OUR HOT, SMOOTH EARLY DAYS
If we think of the universe as a physical system in a randomly chosen configuration, the question “What should the universe look like?” is answered by “It should be in a high-entropy state.” We therefore need to understand what a high-entropy state of the universe would look like.
Even this formulation of the question is not quite right. We don’t actually care about the particular state of the universe right this moment; after all, yesterday it was different, and tomorrow it will be different again. What we really care about is the history of the universe, its evolution through time. But understanding what would constitute a natural history presupposes that we understand something about the space of states, including what high-entropy states look like.
Cosmologists have traditionally done a very sloppy job of addressing this issue. There are a couple of reasons for this. One is that the expansion of the universe from a hot, dense early state is such an undeniable brute fact that, once you’ve become accustomed to the idea, it seems hard to imagine any alternative. You begin to see your task as a theoretical cosmologist as one of explaining why our universe began in the particular hot, dense early state that it did, rather than some different hot, dense early state. This is temporal chauvinism at its most dangerous—unthinkingly trading in the question “Why does the universe evolve in the way it does?” for “Why were the initial conditions of the universe set up the way they were?”
The other thing standing in the way of more productive work on the entropy of the universe is the inescapable role of gravity. By “gravity” we mean everything having to do with general relativity and curved spacetime—everyday stuff like apples falling and planets orbiting stars, but also black holes and the expansion of the universe. In the last chapter, we focused on the one example where we think we know the entropy of an object with a strong gravitational field: a black hole. That example does not seem immediately helpful when thinking about the whole universe, which is not a black hole; it bears a superficial resemblance to a white hole (since there is a singularity in the past), but even that is of little help, since we are inside it rather than outside. Gravity is certainly important to the universe, and that’s especially true at early times when space was expanding very rapidly. But appreciating that it’s important doesn’t help us address the problem, so most people simply put it aside.
There is one other strategy, which appears innocent at first but really hides a potentially crucial mistake. That’s to simply separate out gravity from everything else, and calculate the entropy of the matter and radiation within spacetime while forgetting that of spacetime itself. Of course, it’s hard to be a cosmologist and ignore the fact that space is expanding; however, we can take the expansion of space as a given, and simply consider the state of the “stuff ” (particles of ordinary matter, dark matter, radiation) within such a background. The expanding universe acts to dilute away the matter and cool off the radiation, just as if the particles were all contained in a piston that was gradually being pulled out to create more room for them to breathe. It’s possible to calculate the entropy of the stuff in that particular background, exactly as it’s possible to calculate the entropy of a collection of molecules inside an expanding piston.
At any one time in the early universe, we have a gas of particles at a nearly constant temperature and nearly constant density from place to place. In other words, a configuration that looks pretty much like thermal equilibrium. It’s not exactly thermal equilibrium, because in equilibrium nothing changes, and in the expanding universe things are cooling off and diluting away. But compared to the rate at which particles are bumping into one another, the expansion of space is relatively slow, so the cooling off is quite gradual. If we just consider matter and radiation in the early universe, and neglect any effects of gravity other than the overall expansion, what we find is a sequence of configurations that are very close to thermal equilibrium at a gradually declining density and temperature.234
But that’s a woefully incomplete story, of course. The Second Law of Thermodynamics says, “The entropy of a closed system either increases or remains constant”; it doesn’t say, “The entropy of a closed system, ignoring gravity, either increases or remains constant.” There’s nothing in the laws of physics that allows us to neglect gravity in situations where it’s important—and in cosmology it’s of paramount importance.
By ignoring the effects of gravity on the entropy, and just considering the matter and radiation, we are led to nonsensical conclusions. The matter and radiation in the early universe was close to thermal equilibrium, which means (neglecting gravity) that it was in its maximum entropy state. But today, in the late universe, we’re clearly not in thermal equilibrium (if we were, we’d be surrounded by nothing but gas at constant temperature), so we are clearly notin a configuration of maximum entropy. But the entropy didn’t go down—that would violate the Second Law. So what is going on?
What’s going on is that it’s not okay to ignore gravity. Unfortunately, including gravity is not so easy, as there is still a lot we don’t understand about how entropy works when gravity is included. But as we’ll see, we know enough to make a great deal of progress.
WHAT WE MEAN BY OUR UNIVERSE
For the most part, up until now I have stuck to well-established ground: either reviewing things that all good working physicists agree are correct, or explaining things that are certainly true that all good working physicists shouldagree are correct. In the few genuinely controversial exceptions (such as the interpretation of quantum mechanics), I tried to label them clearly as unsettled. But at this point in the book, we start becoming more speculative and heterodox—I have my own favorite point of view, but there is no settled wisdom on these questions. I’ll try to continue distinguishing between certainly true things and more provisional ideas, but it’s important to be as careful as possible in making the case.
First, we have to be precise about what we mean by “our universe.” We don’t see all of the universe; light travels at a finite speed, and there is a barrier past which we can’t see—in principle given by the Big Bang, in practice given by the moment when the universe became transparent about 380,000 years after the Big Bang. Within the part that we do see, the universe is homogenous on large scales; it looks pretty much the same everywhere. There is a corresponding strong temptation to take what we see and extrapolate it shamelessly to the parts we can’t see, and imagine that the entire universe is homogenous throughout its extent—either through a volume of finite size, if the universe is “closed,” or an infinitely big volume, if the universe is “open.”
But there’s really no good reason to believe that the universe we don’t see matches so precisely with the universe we do see. It might be a simple starting assumption, but it’s nothing more than that. We should be open to the possibility that the universe eventually looks completely different somewhere beyond the part we can see (even if it keeps looking uniform for quite a while before we get to the different parts).
So let’s forget about the rest of the universe, and concentrate on the part we can see—what we’ve been calling “the observable universe.” It stretches about 40 billion light-years around us.235 But since the universe is expanding, the stuff within what we now call the observable universe was packed into a smaller region in the past. What we do is erect a kind of imaginary fence around the stuff within our currently observable universe, and keep track of what’s inside the fence, allowing the fence itself to expand along with the universe (and be smaller in the past). This is known as our comoving patch of space, and it’s what we have in mind when we say “our observable universe.”
Figure 65: What we call “the observable universe” is a patch of space that is “comoving”—it expands along with the universe. We trace back along our light cones to the Big Bang to define the part of the universe we can observe, and allow that region to grow as the universe expands.
Our comoving patch of space is certainly not, strictly speaking, a closed system. If an observer were located at the imaginary fence, they would notice various particles passing in and out of our patch. But on average, the same number and kind of particles would come in and go out, and in the aggregate they would be basically indistinguishable. (The smoothness of the cosmic microwave background convinces us that the uniformity of our universe extends out to the boundary of our patch, even if we’re not sure how far it continues beyond.) So for all practical purposes, it’s okay to think of our comoving patch as a closed system. It’s not really closed, but it evolves just as if it were—there aren’t any important influences from the outside that are affecting what goes on inside.
CONSERVATION OF INFORMATION IN AN EXPANDING SPACE TIME
If our comoving patch defines an approximately closed system, the next step is to think about its space of states. General relativity tells us that space itself, the stage on which particles and matter move and interact, evolves over time. Because of this, the definition of the space of states becomes more subtle than it would have been if spacetime were absolute. Most physicists would agree that information is conserved as the universe evolves, but the way that works is quite unclear in a cosmological context. The essential problem is that more and more things can fit into the universe as it expands, so—naїvely, anyway—it looks as if the space of states is getting bigger. That would be in flagrant contradiction to the usual rules of reversible, information-conserving physics, where the space of states is fixed once and for all.
To grapple with this issue, it makes sense to start with the best understanding we currently have of the fundamental nature of matter, which comes from quantum field theory. Fields vibrate in various ways, and we perceive the vibrations as particles. So when we ask, “What is the space of states in a particular quantum field theory?” we need to know all the different ways that the fields can vibrate.
Any possible vibration of a quantum field can be thought of as a combination of vibrations with different specific wavelengths—just as any particular sound can be decomposed into a combination of various notes with specific frequencies. At first you might think that any possible wavelength is allowed, but actually there are restrictions. The Planck length—the tiny distance of 10-33 centimeters at which quantum gravity becomes important—provides a lower limit on what wavelengths are allowed. At smaller distances than that, spacetime itself loses its conventional meaning, and the energy of the wave (which is larger when the wavelength is shorter) becomes so large that it would just collapse to a black hole.
Likewise, there is an upper limit on what wavelengths are allowed, given by the size of our comoving patch. It’s not that vibrations with longer wavelengths can’t exist; it’s that they just don’t matter. Wavelengths larger than the size of our patch are effectively constant throughout the observable universe.
So it’s tempting to take “the space of states of the observable universe” as consisting of “vibrations in all the various quantum fields, with wavelengths larger than the Planck length and smaller than the size of our comoving patch.” The problem is, that’s a space of states that changes as the universe expands. Our patch grows with time, while the Planck length remains fixed. At extremely early times, the universe was very young and expanding very rapidly, and our patch was relatively small. (Exactly how small depends on details of the evolution of the early universe that we don’t know.) There weren’t that many vibrations you could squeeze into the universe at that time. Today, the Hubble length is enormously larger—about 1060 times larger than the Planck length—and there are a huge number of allowed vibrations. Under this way of thinking, it’s not so surprising that the entropy of the early universe was small, because the maximum allowed entropy of the universe at that time was small—the maximum allowed entropy increases as the universe expands and the space of states grows.
Figure 66: As the universe expands, it can accommodate more kinds of waves. More things can happen, so the space of states would appear to be growing.
But if a space of states changes with time, the evolution clearly can’t be information conserving and reversible. If there are more possible states today than there were yesterday, and two distinct initial states always evolve into two distinct final states, there must be some states today that didn’t come from anywhere. That means the evolution can’t be reversed, in general. All of the conventional reversible laws of physics we are used to dealing with feature spaces of states that are fixed once and for all, not changing with time. The configuration within that space will evolve, but the space of states itself never changes.
We seem to have something of a dilemma. The rules of thumb of quantum field theory in curved spacetime would seem to imply that the space of states grows as the universe expands, but the ideas on which all this is based—quantum mechanics and general relativity—conform proudly to the principle of information conservation. Clearly, something has to give.
The situation is reminiscent of the puzzle of information loss in black holes. There, we (or Stephen Hawking, more accurately) used quantum field theory in curved spacetime to derive a result—the evaporation of black holes into Hawking radiation—that seemed to destroy information, or at least scramble it. Now in the case of cosmology, the rules of quantum field theory in an expanding universe seem to imply fundamentally irreversible evolution.
I am going to imagine that this puzzle will someday be resolved in favor of information conservation, just as Hawking (although not everyone) now believes is true for black holes. The early universe and the late universe are simply two different configurations of the same physical system, evolving according to reversible fundamental laws within precisely the same space of possible states. The right thing to do, when characterizing the entropy of a system as “large” or “small,” is to compare it to the largest possible entropy—not the largest entropy compatible with some properties the system happens to have at the time. If we were to look at a box of gas and find that all of the gas was packed tightly into one corner, we wouldn’t say, “That’s a high-entropy configuration, as long as we restrict our attention to configurations that are packed tightly into that corner.” We would say, “That’s a very low-entropy configuration, and there’s probably some explanation for it.”
All of this confusion arises because we don’t have a complete theory of quantum gravity, and have to make reasonable guesses on the basis of the theories we think we do understand. When those guesses lead to crazy results, something has to give. We gave a sensible argument that the number of states described by vibrating quantum fields changes with time as the universe expands. If the total space of states remains fixed, it must be the case that many of the possible states of the early universe have an irreducibly quantum-gravitational character, and simply can’t be described in terms of quantum fields on a smooth background. Presumably, a better theory of quantum gravity would help us understand what those states might be, but even without that understanding, the basic principle of information conservation assures us that they must be there, so it seems logical to accept that and try to explain why the early universe had such an apparently low-entropy configuration.
Not everyone agrees.236 A certain perfectly respectable school of thought goes something like this: “Sure, information might be conserved at a fundamental level, and there might be some fixed space of states for the whole universe. But who cares? We don’t know what that space of states is, and we live in a universe that started out small and relatively smooth. Our best strategy is to use the rules suggested by quantum field theory, allowing only a very small set of configurations at very early times, and a much larger set at later times.” That may be right. Until we have the final answers, the best we can do is follow our intuitions and try to come up with testable predictions that we can compare against data. When it comes to the origin of the universe, we’re not there yet, so it pays to keep an open mind.
Because we don’t have quantum gravity all figured out, it’s hard to make definitive statements about the entropy of the universe. But we do have some basic tools at our disposal—the idea that entropy has been increasing since the Big Bang, the principle of information conservation, the predictions of classical general relativity, the Bekenstein-Hawking formula for black-hole entropy—that we can use to draw some reliable conclusions.
One obvious question is: What does a high-entropy state look like when gravity is important? If gravity is not important, high-entropy states are states of thermal equilibrium—stuff tends to be distributed uniformly at a constant temperature. (Details may vary in particular systems, such as oil and vinegar.) There is a general impression that high-entropy states are smooth, while lower-entropy states can be lumpy. Clearly that’s just a shorthand way of thinking about a subtle phenomenon, but it’s a useful guide in many circumstances.237 Notice that the early universe is, indeed, smooth, in accordance with the let’s-ignore-gravity philosophy we just examined.
But in the later universe, when stars and galaxies and clusters form, it becomes simply impossible to ignore the effects of gravity. And then we see something interesting: The casual association of “high-entropy” with “smooth” begins to fail, rather spectacularly.
For many years now, Sir Roger Penrose has been going around trying to convince people that this feature of gravity—things get lumpier as entropy increases in the late universe—is crucially important and should play a prominent role in discussions of cosmology. Penrose became famous in the late 1960s and early 1970s through his work with Hawking to understand black holes and singularities in general relativity, and he is an accomplished mathematician as well as physicist. He is also a bit of a gadfly, and takes great joy in exploring positions that run decidedly contrary to the conventional wisdom in various fields, from quantum mechanics to the study of consciousness.
Figure 67: Roger Penrose, who has done more than anyone to emphasize the puzzle of the low-entropy early universe.
One of the fields in which Penrose likes to poke holes in cherished beliefs is theoretical cosmology. When I was a graduate student in the late 1980s, theoretical particle physicists and cosmologists mostly took it for granted that some version of inflationary cosmology (discussed in the next chapter) must be true; astronomers tended to be more cautious. Today, this belief is even more common, as evidence from the cosmic microwave background has shown that the small variations in density from place to place in the early universe match very well with what inflation would predict. But Penrose has been consistently skeptical, primarily on the basis of inflation’s failure to explain the low entropy of the early universe. I remember reading one of his papers as a student, and appreciating that Penrose was saying something important but feeling that he had missed the point. It took two decades of thinking about entropy before I became convinced that he has mostly been right all along.
We don’t have a full picture of the space of microstates in quantum gravity, and correspondingly lack a rigorous understanding of entropy. But there is a simple strategy for dealing with this obstacle: We consider what actually happens in the universe. Most of us believe that the evolution of the observable universe has always been compatible with the Second Law, and entropy has been increasing since the Big Bang, even if we’re hazy on the details. If entropy tends to go up, and if there is some process that happens all the time in the universe, but its time-reverse never happens, it probably represents an increase in entropy.
An example of this is “gravitational instability” in the late universe. We’ve been tossing around the notion of “when gravity is important” and “when gravity is not important,” but what is the criterion to decide whether gravity is important? Generally, given some collection of particles, their mutual gravitational forces will always act to pull the particles together—the gravitational force between particles is universally attractive. (In contrast, for example, with electricity and magnetism, which can be either attractive or repulsive depending on what kinds of electric charges you are dealing with.238) But there are other forces, generally collected together under the rubric of “pressure,” that prevent everything from collapsing to a black hole. The Earth or the Sun or an egg doesn’t collapse under its own gravitational pull, because each is supported by the pressure of the material inside it. As a rule of thumb, “gravity is important” means “the gravitational pull of a collection of particles overwhelms the pressure that tries to keep them from collapsing.”
In the very early universe, the temperature is high and the pressure is enormous. 239 The local gravity between nearby particles is too weak to bring them together, and the initial smoothness of the matter and radiation is preserved. But as the universe expands and cools, the pressure drops, and gravity begins to take over. This is the era of “structure formation,” where the initially smooth distribution of matter gradually begins to condense into stars, galaxies, and larger groups of galaxies. The initial distribution was not perfectly featureless; there were small deviations in density from place to place. In the denser regions, gravity pulled particles even closer together, while the less dense regions lost particles to their denser neighbors and became even emptier. Through gravity’s persistent efforts, what was a highly uniform distribution of matter becomes increasingly lumpy.
Penrose’s point is this: As structure forms in the universe, entropy increases. He puts it this way:
Gravitation is somewhat confusing, in relation to entropy, because of its universally attractive nature. We are used to thinking about entropy in terms of an ordinary gas, where having the gas concentrated in small regions represents low entropy . . . and where in the high-entropy state of thermal equilibrium, the gas is spread uniformly. But with gravity, things tend to be the other way about. A uniformly spread system of gravitating bodies would represent relatively low entropy (unless the velocities of the bodies are enormously high and/or the bodies are very small and/or greatly spread out, so that the gravitational contributions become insignificant), whereas high entropy is achieved when the gravitating bodies clump together.240
All of that is completely correct, and represents an important insight. Under certain conditions, such as those that pertain in the universe on large scales today, even though we don’t have a cut-and-dried formula for the entropy of a system including gravity, we can say with confidence that the entropy increases as structure forms and the universe becomes lumpier.
There is another way of reaching a similar conclusion, through the magic of thought experiments. Take the current macrostate of the universe—some collection of galaxies, dark matter, and so forth, distributed in a certain way through space. But now let’s make a single change: Imagine that the universe is contracting rather than expanding. What should happen?
It should be clear that what won’t happen is a simple time-reversal of the actual history of the universe from its smooth initial state to its lumpy present—at least, not for the overwhelming majority of microstates within our present macrostate. (If we precisely time-reversed the specific microstate of our present universe, then of course that is exactly what would happen.) If the distribution of matter in our present universe were to start contracting together, individual stars and galaxies would not begin to disperse and smooth out. Instead, the gravitational force between heavy objects would draw them together, and the amount of lumpy structure would actually increase, even as the universe contracted. Black holes would form, and coalesce together to create bigger black holes. There would ultimately be a sort of Big Crunch, but (as Penrose emphasizes) it wouldn’t look anything like the smooth Big Bang from which our universe came. Places where the density was high and black holes formed would crash into a future singularity relatively quickly, while places that were emptier would survive for longer.
Figure 68: When gravity is unimportant, increasing entropy tends to smooth things out; when gravity does become important, matter tends to clump together as entropy increases.
This story fits in well with the idea that the space of states within our comoving patch remains fixed, but when the universe is small most of the states cannot be described as vibrating quantum fields in an otherwise smooth space. Such a picture would be completely inadequate to describe the chaotic black-hole-filled mess that we would generically expect from a collapsing universe. But such a messy configuration is just as much an allowed state of the universe as the relatively smooth backgrounds we traditionally deal with in cosmology. Indeed, such a configuration has a higher entropy than a smooth universe (which we know because a collapsing universe would generically evolve into something messy), which means that there are many more microstates of that form than of the form where everything is relatively smooth. Why our actual universe is so atypical is, of course, at the heart of our mystery.
THE EVOLUTION OF ENTROPY
We’ve now assembled enough background knowledge to follow Penrose and take a stab at quantifying how the entropy of our universe changes from early times to today. We know the basic story of how our comoving patch evolves—at early times it was small, and full of hot, dense gas that was very close to perfectly smooth, and at later times it is larger and cooler and more dilute, and contains a distribution of stars and galaxies that is quite lumpy on small scales, although it’s still basically smooth over very large distances. So what is its entropy?
At early times, when things were smooth, we can calculate the entropy by simply ignoring the influence of gravity. This might seem to go against the philosophy I’ve thus far been advocating, but we’re not saying that gravity is irrelevant in principle—we’re simply taking advantage of the fact that, in practice, our early universe was in a configuration where the gravitational forces between particles didn’t play a significant role in the dynamics. Basically, it was just a box of hot gas. And a box of hot gas is something whose entropy we know how to calculate.
The entropy of our comoving patch of space when it was young and smooth is:
Searly ≈ 1088.
The “≈” sign means “approximately equal to,” as we want to emphasize that this is a rough estimate, not a rigorous calculation. This number comes from simply treating the contents of the universe as a conventional gas in thermal equilibrium, and plugging in the formulas worked out by thermodynamicists in the nineteenth century, with one additional feature: Most of the particles in the early universe are photons and neutrinos, moving at or close to the speed of light, so relativity is important. Up to some numerical factors that don’t change the answer very much, the entropy of a hot gas of relativistic particles is simply equal to the total number of such particles. There are about 1088particles within our comoving patch of universe, so that’s what the entropy was at early times. (It increases a bit along the way, but not by much, so treating the entropy as approximately constant at early times is a good approximation.)
Today, gravity has become important. It is not very accurate to think of the matter in the universe as a gas in thermal equilibrium with negligible gravity; ordinary matter and dark matter have condensed into galaxies and other structures, and the entropy has increased considerably. Unfortunately, we don’t have a reliable formula that tracks the change in entropy during the formation of a galaxy.
What we do have is a formula for the circumstance in which gravity is at its most important: in a black hole. As far as we know, very little of the total mass of the universe is contained in the form of black holes.241 In a galaxy like the Milky Way, there are a number of stellar-sized black holes (with maybe 10 times the mass of the Sun each), but the majority of the total black hole mass is in the form of a single supermassive black hole at the galactic center. While supermassive black holes are certainly big—often over a million times the mass of the Sun—that’s nothing compared to an entire galaxy, where the total mass might be 100 billion times the mass of the Sun.
But while only a tiny fraction of the mass of the universe appears to be in black holes, they contain a huge amount of entropy. A single supermassive black hole, a million times the mass of the Sun, has an entropy according to the Bekenstein-Hawking formula of 1090. That’s a hundred times larger than all of the nongravitational entropy in all the matter and radiation in the observable universe.242
Even though we don’t have a good understanding of the space of states of gravitating matter, it’s safe to say that the total entropy of the universe today is mostly in the form of these supermassive black holes. Since there are about 100 billion (1011 ) galaxies in the universe, it’s reasonable to approximate the total entropy by assuming 100 billion such black holes. (They might be missing from some galaxies, but other galaxies will have larger black holes, so this is not a bad approximation.) With an entropy of 1090 per million-solar-mass black hole, that gives us a total entropy within our comoving patch today of
Stoday ≈ 10101.
Mathematician Edward Kasner coined the term googol to stand for 10100, a number he used to convey the idea of an unimaginably big quantity. The entropy of the universe today is about ten googols. (The folks from Google used this number as an inspiration for the name of their search engine; now it’s impossible to refer to a googol without being misunderstood.)
When we write the current entropy of our comoving patch as 10101, it doesn’t seem all that much larger than its value in the early universe, 1088. But that’s just the miracle of compact notation. In fact, 10101 is ten trillion (1013) times bigger than 1088. The entropy of the universe has increased by an enormous amount since the days when everything was smooth and featureless.
Still, it’s not as big as it could be. What is the maximum value the entropy of the observable universe could have? Again, we don’t know enough to say for sure what the right answer is. But we can say that the maximum entropy must be at least a certain number, simply by imagining that all of the matter in the universe were rearranged into one giant black hole. That’s an allowed configuration for the physical system corresponding to our comoving patch of universe, so it’s certainly possible that the entropy could be that large. Using what we know about the total mass contained in the universe, and plugging once again into the Bekenstein-Hawking entropy formula for black holes, we find that the maximum entropy of the observable universe is at least
Smax ≈ 10120.
That’s a fantastically big number. A hundred quintillion googols! The maximum entropy the observable universe could have is at least that large.
These numbers drive home the puzzle of entropy that modern cosmology presents to us. If Boltzmann is right, and entropy characterizes the number of possible microstates of a system that are macroscopically indistinguishable, it’s clear that the early universe was in an extremely special state. Remember that the entropy is the logarithm of the number of equivalent states, so a state with entropy S is one of 10S indistinguishable states. So the early universe was in one of
different states. But it could have been in any of the
possible states that are accessible to the universe. Again, the miracle of typography makes these numbers look superficially similar, but in fact the latter number is enormously, inconceivably larger than the former. If the state of the early universe were simply “chosen randomly” from among all possible states, the chance that it would have looked like it actually did are ridiculously tiny.
The conclusion is perfectly clear: The state of the early universe was not chosen randomly among all possible states. Everyone in the world who has thought about the problem agrees with that. What they don’t agree on is why the early universe was so special—what is the mechanism that put it in that state? And, since we shouldn’t be temporal chauvinists about it, why doesn’t the same mechanism put the late universe in a similar state? That’s what we’re here to figure out.
We’ve established that the early universe was in a very unusual state, which we think is something that demands explanation. What about the question we started this chapter with: What should the universe look like? What is the maximum-entropy state into which we can arrange our comoving patch?
Roger Penrose thinks the answer is a black hole.
What about the maximum-entropy state? Whereas with a gas, the maximum entropy of thermal equilibrium has the gas uniformly spread throughout the region in question, with large gravitating bodies, maximum entropy is achieved when all the mass is concentrated in one place—in the form of an entity known as a black hole.243
You can see why this is a tempting answer. As we’ve seen, in the presence of gravity, entropy increases when things cluster together, rather than smoothing out. A black hole is certainly as densely packed as things can possibly get. As we discussed in the last chapter, a black hole represents the most entropy we can squeeze into a region of spacetime with any fixed size; that was the inspiration behind the holographic principle. And the resulting entropy is undoubtedly a big number, as we’ve seen in the case of a supermassive black hole.
But in the final analysis, that’s not the best way to think about it.244 A black hole doesn’t maximize the total entropy a system can have—it only maximizes the entropy that can be packed into a region of fixed size. Just as the Second Law doesn’t say “entropy tends to increase, not including gravity,” it also doesn’t say “entropy per volume tends to increase.” It just says “entropy tends to increase,” and if that requires a big region of space, then so be it. One of the wonders of general relativity—and a crucial distinction with the absolute spacetime of Newtonian mechanics—is that sizes are never fixed. Even without a complete understanding of entropy, we can get a handle on the answer by following in Penrose’s footsteps, and simply examining the natural evolution of systems toward higher-entropy states.
Consider a simple example: a collection of matter gathered in one region of an otherwise empty universe, without even any vacuum energy. In other words, a spacetime that is empty almost everywhere, except for some particular place where some matter particles are congregated. Because most of space has no energy in it at all, the universe won’t be expanding or contracting, so nothing really happens outside the region where the matter is located. The particles will contract together under their own gravitational force.
Let’s imagine that they collapse all the way to a black hole. Along the way, there’s no question that the entropy increases. However, the black hole doesn’t just sit there for all of eternity—it gives off Hawking radiation, gradually shrinking as it loses energy, and eventually evaporating away completely.
Figure 69: A black hole has a lot of entropy, but it evaporates into radiation that has even more entropy.
The natural behavior of black holes in an otherwise empty universe is to radiate away into a dilute gas of particles. Because such behavior is natural, we expect it to represent an increase in entropy—and it does. We can explicitly compare the entropy of the black hole to the entropy of the radiation into which it evaporates—and the entropy of the radiation is higher. By about 33 percent, to be specific. 245
Now, the density of entropy has clearly gone down—when we had a black hole, all that entropy was packed into a small volume, whereas the Hawking radiation is emitted gradually and gets spread out over a huge region of space. But again, the density of entropy isn’t what we care about; it’s just the total amount.
The lesson of this thought experiment is that the rule of thumb “when we take gravity into consideration, higher-entropy states are lumpy rather than smooth” is not an absolute law; it’s merely valid under certain circumstances. The black hole is lumpier (higher contrast) than the initial collection of particles, but the eventual dispersing radiation isn’t lumpy at all. In fact, as the radiation scurries off to the ends of the universe, we approach a configuration that grows ever smoother, as the density everywhere approaches zero.
So the answer to the question “What does a high-entropy state look like, when we take gravity into account?” isn’t “a lumpy, chaotic maelstrom of black holes,” nor is it even “one single giant black hole.” The highest-entropy states look like empty space, with at most a few particles here and there, gradually diluting away.
That’s a counterintuitive claim, worth examining from different angles.246 The case of a collection of matter that all falls together to form a black hole is a relatively straightforward one, where we can actually plug in numbers and verify that the entropy increases when the black hole evaporates away. But that’s a far cry from proving that the result (an increasingly dilute gas of particles moving through empty space) is really the highest-entropy configuration possible. We should try to contemplate other possible answers. The guiding principles are that we want a configuration that other kinds of configurations naturally evolve into, and that itself persists forever.
What if, for example, we had an array of many black holes? We might imagine that black holes filled the universe, so that the radiation from one black hole eventually fell into another one, which kept them from evaporating away. However, general relativity tells us that such a configuration can’t last. By sprinkling objects throughout the universe, we’ve created a situation where space has to either expand or contract. If it expands, the distance between the black holes will continually grow, and eventually they will simply evaporate away. As before, the long-term future of such a universe simply looks like empty space.
If space is contracting, we have a different story. When the entire universe is shrinking, the future is likely to end in a Big Crunch singularity. That’s a unique case; on the one hand, the singularity doesn’t really last forever (since time ends there, at least as far as we know), but it doesn’t evolve into something else, either. We can’t rule out the possibility that the future evolution of some hypothetical universe ends in a Big Crunch, but our lack of understanding of singularities in quantum gravity makes it difficult to say very much useful about that case. (And our real world doesn’t seem to be behaving that way.)
One clue is provided by considering a collapsing collection of matter (black holes or otherwise) that looks exactly like a contracting universe, but one where the matter only fills a finite region of space, rather than extending throughout it. If the rest of the universe is empty, this local region is exactly like the situation we considered before, where a group of particles collapsed to make a black hole. So what looks from the inside like a universe collapsing to a Big Crunch looks from the outside like the formation of a giant black hole. In that case, we know what the far future will bring: It might take a while, but that black hole will radiate away into nothing. The ultimate state is, once again, empty space.
Figure 70: An array of black holes cannot remain static. It will either expand and allow the black holes to evaporate away, approaching empty space (top right), or collapse to make a Big Crunch or a single larger black hole (bottom right).
We can be a little more systematic about this. Cosmologists are used to thinking of universes that are doing the same thing all throughout space, because the observable part of our own universe seems to be like that. But let’s not take that for granted; let’s ask what could be going on throughout the universe, in perfect generality.
The notion that the space is “expanding” or “contracting” doesn’t have to be an absolute property of the entire universe. If the matter in some particular region of space is moving apart and diluting away, it will look locally like an expanding universe, and likewise for contraction when matter moves together. So if we imagine sprinkling particles throughout an infinitely big space, most of the time we will find that some regions are expanding and diluting, while other regions are contracting and growing more dense.
But if that’s true, a remarkable thing happens: Despite the apparent symmetry between “expanding” and “contracting,” pretty soon the expanding regions begin to win. And the reason is simple: The expanding regions are growing in volume, while the contracting ones are shrinking. Furthermore, the contracting regions don’t stay contracted forever. In the extreme case where matter collapses all the way to a black hole, eventually the black holes just radiate away. So starting from initial conditions that contain both expanding and contracting regions, if we wait long enough we’ll end up with empty space—entropy increasing all the while.247
Figure 71: Initial conditions (at bottom) in a universe with both expanding and contracting regions. The expanding regions grow in size and become increasingly dilute. The contracting regions grow denser at first, but at some point will begin to evaporate into the surrounding emptiness.
In each of these examples, the crucial underlying feature is the dynamical nature of spacetime in general relativity. In a fixed, absolute spacetime (such as Boltzmann would have presumed), it makes sense to imagine a universe filled with gas at a uniform temperature and density—thermal equilibrium everywhere. That’s a high-entropy state, and a natural guess for what the universe “should” look like. It’s no surprise that Boltzmann suggested that our observable universe could be a statistical fluctuation within such a configuration.
But general relativity changes everything. A gas of uniform density in a static spacetime is not a solution to Einstein’s equation—the universe would have to be either expanding or contracting. Before Einstein came along it made sense to start your thought experiments by fixing the average density of matter, or the total volume of the region under consideration. But in general relativity, these aren’t things you’re allowed to simply keep fixed; they will tend to change with time. One way of thinking about it is to realize that general relativity always gives you a way to increase the entropy of any particular configuration: Make the universe bigger, and let the stuff expand to fill the new volume. The ultimate end of such a process is, of course, empty space. That’s what counts as a “high-entropy” state once we take gravity into account.
None of these arguments is airtight, of course. They are suggestive of a result that seems to hang together and make sense, once you think it through, but that’s far short of a definitive demonstration of anything. The claim that the entropy of some system within the universe can increase by scattering its elements across a vast expanse of space seems pretty safe. But the conclusion that empty space is therefore the highest-entropy state is more tentative. Gravity is tricky, and there’s a lot we don’t understand about it, so it’s not a good idea to become too emotionally invested in any particular speculative scenario.
THE REAL WORLD
Let’s apply these ideas to the real world. If high-entropy states are those that look like empty space, presumably our actual observable universe should be evolving toward such a state. (It is.)
We have casually been assuming that when things collapse under the force of gravity, they end up as a black hole before ultimately evaporating away. It’s far from obvious that this holds true in the real world, where we see lots of objects held together by gravity, but that are very far from being black holes—planets, stars, and even galaxies.
But the reality is, all of these things will eventually “evaporate” if we wait long enough. We can see this most clearly in the case of a galaxy, which can be thought of as a collection of stars orbiting under their mutual gravitational pull. As individual stars pass by other stars, they interact much like molecules in a box of gas, except that the interaction is solely through gravity (apart from those very rare instances when the stars smack right into one another). These interactions can exchange energy from one star to the other.248 Over the course of many such encounters, stars will occasionally pick up so much energy that they reach escape velocity and fly away from the galaxy altogether. The rest of the galaxy has now lost some of its energy, and as a consequence it shrinks, so that its stars are clustered more tightly together. Eventually, the remaining stars are going to be packed so closely that they all fall into a black hole at the center. From that point, we return to our previous story.
Similar logic works for any other object in the universe, even if the details might differ. The basic point is that, given some rock or star or planet or what have you, that particular physical system wants to be in the highest-entropy arrangement of the constituents from which it is made. That’s a little poetic, as inanimate objects don’t really have desires, but it reflects the reality that an unfettered evolution of the system would naturally bring it into a higher-entropy configuration.
You might think that the evolution is, in fact, fettered: A planet, for example, might have a higher entropy if its entire mass collapsed into a black hole, but the pressure inside keeps it stable. Here’s where the miracle of quantum mechanics comes in. Remember that a planet isn’t really a collection of classical particles; it’s described by a wave function, just like everything else. That wave function characterizes the probability that we will find the constituents of the planet in any of their possible configurations. One of those possible configurations, inevitably, will be a black hole. In other words, from the point of view of someone observing the planet (or anything else), there is a tiny chance they will find that it has spontaneously collapsed into a black hole. That’s the process known as “quantum tunneling.”
Do not be alarmed. Yes, it’s true, just about everything in the universe—the Earth, the Sun, you, your cat—has a chance of quantum-tunneling into the form of a black hole at any moment. But the chance is very small. It would be many, many times the age of the universe before there were a decent chance of it happening. But in a universe that lasts for all eternity, that means the chances are quite good that it will eventually happen—it’s inevitable, in fact. No collection of particles can simply sit undisturbed in the universe forever. The lesson is that matter will find a way to transform into a higher-entropy configuration, if one exists; it might be via tunneling into the form of a black hole, or through much more mundane channels. No matter what kind of lump of matter you have in the universe, it can increase in entropy by evaporating into a thin gruel of particles moving away into empty space.
As we discussed back in Chapter Three, there’s more than matter and radiation in the universe—there’s also dark energy, responsible for making the universe accelerate. We don’t know for sure what the dark energy is, but the leading candidate is “vacuum energy,” also known as the cosmological constant. Vacuum energy is simply a constant amount of energy inherent in every cubic centimeter of space, one that remains fixed throughout space and time.
The existence of dark energy both simplifies and complicates our ideas about high-entropy states in the presence of gravity. I’ve been suggesting that the natural behavior of matter is to disperse away into empty space, which is therefore the best candidate for a maximum-entropy state. In a universe like ours, with a vacuum energy that is small but greater than zero, this conclusion becomes even more robust. A positive vacuum energy imparts a perpetual impulse to the expansion of the universe, which helps the general tendency of matter and radiation to dilute away. If, within the next few years, human beings perfect an immortality machine and/or drug, cosmologists who live forever will have to content themselves with observing an increasingly empty universe. Stars will die out, black holes will evaporate, and everything will be pushed away by the accelerating effects of vacuum energy.
In particular, if the dark energy is really a cosmological constant (rather than something that will ultimately fade away), we can be sure that the universe will never re-collapse into a Big Crunch of any sort. After all, the universe is not only expanding but also accelerating, and that acceleration will continue forever. This scenario—which, let’s not forget, is the most popular prognosis for the real world according to contemporary cosmologists—vividly highlights the bizarre nature of our low-entropy beginnings. We’re contemplating a universe that has existed for a finite time in the past but will persist forever into the future. The first few tens of billions of years of its existence are a hot, busy, complex and interesting mess, which will be followed by an infinite stretch of cold, empty quietness. (Apart from the occasional statistical fluctuation; see next section.) Although it’s not much more than a gut feeling, it just seems like a waste to face the prospect of an endless duration of dark loneliness after a relatively exciting few years in our universe’s past.
The existence of a positive cosmological constant allows us to actually prove a somewhat rigorous result, rather than just spinning through a collection of thought experiments. The cosmic no-hair theorem states that, under the familiar set of “reasonable assumptions,” a universe with a positive vacuum energy plus some matter fields will, if it lasts long enough for the vacuum energy to take over, eventually evolve into empty universe with nothing but vacuum energy. The cosmological constant always wins, in other words.249
The resulting universe—empty space with a positive vacuum energy—is known as de Sitter space, after Dutch physicist Willem de Sitter, one of the first after Einstein to study cosmology within the framework of general relativity. As we mentioned back in Chapter Three, empty space with zero vacuum energy is known as Minkowski space, while empty space with a negative vacuum energy is anti- de Sitter space. Even though spacetime is empty in de Sitter space, it’s still curved, because of the positive vacuum energy. The vacuum energy, as we know, imparts a perpetual impulse to the expansion of space. If we consider two particles initially at rest in de Sitter space, they will gradually be pulled apart by the expansion. Likewise, if we trace their motion into the past, they would have been coming toward each other, but ever more slowly as the space between them was pushed apart. Anti-de Sitter space is the reverse; particles are pulled toward each other.
Figure 72: Three different versions of “empty space,” with different values of the vacuum energy: Minkowski space when the vacuum energy vanishes, de Sitter when it is positive, and anti-de Sitter when it is negative. In Minkowski space, two particles initially at rest will stay motionless with respect to each other; in de Sitter space they are pushed apart, while in anti-de Sitter space they are pulled together. The larger the magnitude of the vacuum energy, the stronger the pushing or pulling.
Everything we’ve been arguing points to the idea that de Sitter space is the ultimate endpoint of cosmological evolution when the vacuum energy is positive, and hence the highest-entropy state we can think of in the presence of gravity. That’s not a definitive statement—the state of the art isn’t sufficiently advanced to allow for definitive statements along these lines—but it’s suggestive.
You might wonder how empty space can have a large entropy—entropy is supposed to count the number of ways we can rearrange microstates, and what is there to rearrange if all we have is empty space? But this is just the same puzzle that faced us with black holes. The answer must be that there are a large number of microstates describing the quantum states of space itself, even when it’s empty. Indeed, if we believe in the holographic principle, we can assign a definite value to the entropy contained within any observable patch of de Sitter space. The answer is a huge number, and the entropy is larger when the vacuum energy is smaller.250 Our own universe is evolving toward de Sitter space, and the entropy of each observable patch will be about 10120. (The fact that this is the same as the entropy we would get by collapsing all the matter in the observable universe into a black hole is a coincidence—it’s the same coincidence as the fact that the matter density and vacuum energy are approximately equal at the present time, even though the matter dominated in the past and the vacuum energy will dominate in the future.)
While de Sitter space provides a sensible candidate for a high-entropy state, the idea of vacuum energy complicates our attempts to understand entropy in the context of quantum gravity. The basic problem is that the effective vacuum energy—what you would actually measure as the energy of the vacuum at any particular event in spacetime—can certainly change, at least temporarily. Cosmologists talk about the “true vacuum,” in which the vacuum energy takes on its lowest possible value, but also various possible “false vacua,” in which the effective vacuum energy is higher. Indeed, it’s possible that we might be in a false vacuum right now. The idea that “high entropy” means “empty space” becomes a lot more complicated when empty space can take on different forms, corresponding to different values of the vacuum energy.
That’s a good thing—we don’t want empty space to be the highest-entropy state possible, because we don’t live there. In the next couple of chapters we’re going to see whether we can’t take advantage of different possible values of vacuum energy to somehow make sense of the universe. But first we need to assure ourselves that, without some strategy along those lines, it really would be extremely surprising that we don’t live in a universe that is otherwise empty. And that calls for another visit with some of the giants on whose shoulders we are standing, Boltzmann and Lucretius.
WHY DON’T WE LIVE IN EMPTY SPACE?
We began this chapter by asking what the universe should look like. It’s not obvious that this is even a sensible question to ask, but if it is, a logical answer would be “it should look like it’s in a high-entropy state,” because there are a lot more high-entropy states than low-entropy ones. Then we argued that truly high-entropy states basically look like empty space; in a world with a positive cosmological constant, that means de Sitter space, a universe with vacuum energy and nothing else.
So the major question facing modern cosmology is: “Why don’t we live in de Sitter space?” Why do we live in a universe that is alive with stars and galaxies? Why do we live in the aftermath of our Big Bang, an enormous conflagration of matter and energy with an extraordinarily low entropy? Why is there so much stuff in the universe, and why was it packed so smoothly at early times?
One possible answer would be to appeal to the anthropic principle. We can’t live in empty space because, well, it’s empty. There’s nothing there to do the living. That sounds like a perfectly reasonable assumption, although it doesn’t exactly answer the question. Even if we couldn’t live precisely in empty de Sitter space, that doesn’t explain why our early universe is nowhere near to being empty. Our actual universe seems to be an enormously more dramatic departure from emptiness than would be required by any anthropic criterion.
You might find these thoughts reminiscent of our discussion of the Boltzmann-Lucretius scenario from Chapter Ten. There, we imagined a static universe with an infinite number of atoms, so that there was an average density of atoms throughout space. Statistical fluctuations in the arrangements of those atoms, it was supposed, could lead to temporary low-entropy configurations that might resemble our universe. But there was a problem: That scenario makes a strong prediction, namely, that we (under any possible definition of we) should be the smallest possible fluctuation away from thermal equilibrium consistent with our existence. In the most extreme version, we should be disembodied Boltzmann brains, surrounded by a gas with uniform temperature and density. But we’re not, and further experiments continue to reveal more evidence that the rest of the universe is not anywhere near equilibrium, so this scenario seems to be ruled out experimentally.
The straightforward scenario Boltzmann had in mind would doubtless be dramatically altered by general relativity. The most important new ingredient is that it’s impossible to have a static universe filled with gas molecules. According to Einstein, space filled with matter isn’t going to just sit there; it’s going to either expand or contract. And if the matter is sprinkled uniformly throughout the universe, and is made of normal particles (which don’t have negative energy or pressure), there will inevitably be a singularity in the direction of time where things are getting more dense—a past Big Bang if the universe is expanding, or a future Big Crunch if the universe is contracting. (Or both, if the universe expands for a while and then starts to re-contract.) So this carefree Newtonian picture where molecules persist forever in a happy static equilibrium is not going to be relevant once general relativity comes into the game.
Instead, we should contemplate life in de Sitter space, which replaces a gas of thermal particles as the highest-entropy state. If all you knew about was classical physics, de Sitter space would be truly empty. (Vacuum energy is a feature of spacetime itself; there are no particles associated with it.) But classical physics isn’t the whole story; the real world is quantum mechanical. And quantum field theory says that particles can be created “out of nothing” in an appropriate curved spacetime background. Hawking radiation is the most obvious example.
It turns out, following very similar reasoning to that used by Hawking to investigate black holes, that purportedly empty de Sitter space is actually alive with particles popping into existence. Not a lot of them, we should emphasize—we’re talking about an extremely subtle effect. (There are a lot of virtual particles in empty space, but not many real, detectable ones.) Let’s imagine that you were sitting in de Sitter space with an exquisitely sensitive experimental apparatus, capable of detecting any particles that happened to be passing your way. What you would discover is that you were actually surrounded by a gas of particles at a constant temperature, just as if you were in a box in thermal equilibrium. And the temperature wouldn’t go away as the universe continued to expand—it is a feature of de Sitter space that persists for all eternity.251
Admittedly, you wouldn’t detect very many particles; the temperature is quite low. If someone asked you what the “temperature of the universe” is right now, you might say 2.7 Kelvin, the temperature of the cosmic microwave background radiation. That’s pretty cold; 0 Kelvin is the lowest possible temperature, room temperature is about 300 Kelvin, and the lowest temperature ever achieved in a laboratory on Earth is about 10-10 Kelvin. If we allow the universe to expand until all of the matter and cosmic background radiation has diluted away, leaving only those particles that are produced out of de Sitter space by quantum effects, the temperature will be about 10-29 Kelvin. Cold by anyone’s standards.
Still, a temperature is a temperature, and any temperature above zero allows for fluctuations. When we take quantum effects in de Sitter space into account, the universe acts like a box of gas at a fixed temperature, and that situation will last forever. Even if we have a past that features a dramatic Big Bang, the future is an eternity of ultra-cold temperature that never drops to zero. Hence, we should expect an endless future of thermal fluctuations—including Boltzmann brains and any other sort of thermodynamically unlikely configuration we might have worried about in an eternal box of gas.
And that would seem to imply that all of the troublesome aspects of the Boltzmann-Lucretius scenario are troublesome aspects of the real world. If we wait long enough, our universe will empty out until it looks like de Sitter space with a tiny temperature, and stay that way forever. There will be random fluctuations in the thermal radiation that lead to all sorts of unlikely events—including the spontaneous generation of galaxies, planets, and Boltzmann brains. The chance that any one such thing happens at any particular time is small, but we have an eternity to wait, so every allowed thing will happen. In that universe—our universe, as far as we can tell—the overwhelming majority of mathematical physicists (or any other kind of conscious observer) will pop out of the surrounding chaos and find themselves drifting alone through space.252
The acceleration of the universe was discovered in 1998. Theorists chewed over this surprising result for a while before the problem with Boltzmann brains became clear. It was first broached in a 2002 paper by Lisa Dyson, Matthew Kleban, and Leonard Susskind, with the ominous title “Disturbing Implications of a Cosmological Constant,” and amplified in a follow-up paper by Andreas Albrecht and Lorenzo Sorbo in 2004.253 The resolution to the puzzle is still far from clear. The simplest way out is to imagine that the dark energy is not a cosmological constant that lasts forever, but an ephemeral source of energy that will fade away long before we hit the Poincaré recurrence time. But it’s not clear exactly how this would work, and compelling models of decaying dark energy turn out to be difficult to construct.
So the Boltzmann-brain problem—“Why do we find ourselves in a universe evolving gradually from a state of incredibly low entropy, rather than being isolated creatures that recently fluctuated from the surrounding chaos?”—does not yet have a clear solution. And it’s worth emphasizing that this puzzle makes the arrow-of-time problem enormously more pressing. Before this issue was appreciated, we had something of a fine-tuning problem: Why did the early universe have such a low entropy? But we were at least allowed to shrug our shoulders and say, “Well, maybe it just did, and there is no deeper explanation.” But now that’s no longer good enough. In de Sitter space, we can reliably predict the number of times in the history of the universe (including the infinite future) that observers will appear surrounded by cold and forbidding emptiness, and compare them to the observers who will find themselves in comfortable surroundings full of stars and galaxies, and the cold and forbidding emptiness is overwhelmingly likely. This is more than just an uncomfortable fine-tuning; it’s a direct disagreement between theory and observation, and a sign that we have to do better.