From Eternity to Here: The Quest for the Ultimate Theory of Time - Sean Carroll (2010)

Part I. TIME, EXPERIENCE, AND THE UNIVERSE

Chapter 3. THE BEGINNING AND END OF TIME

What has the universe got to do with it? You’re here in Brooklyn! Brooklyn is not expanding!

—Alvy Singer’s mom, Annie Hall

Imagine that you are wandering around in the textbook section of your local university bookstore. Approaching the physics books, you decide to leaf through some volumes on thermodynamics and statistical mechanics, wondering what they have to say about entropy and the arrow of time. To your surprise (having been indoctrinated by the book you’re currently reading, or at least the first two chapters and the jacket copy), there is nothing there about cosmology. Nothing about the Big Bang, nothing about how the ultimate explanation for the arrow of time is to be found in the low-entropy boundary condition at the beginning of our observable universe.

There is no real contradiction, nor is there a nefarious conspiracy on the part of textbook writers to keep the central role of cosmology hidden from students of statistical mechanics. For the most part, people interested in statistical mechanics care about experimental situations in laboratories or kitchens here on Earth. In an experiment, we can control the conditions before us; in particular, we can arrange systems so that the entropy is much lower than it could be, and watch what happens. You don’t need to know anything about cosmology and the wider universe to understand how that works.

But our aims are more grandiose. The arrow of time is much more than a feature of some particular laboratory experiments; it’s a feature of the entire world around us. Conventional statistical mechanics can account for why it’s easy to turn an egg into an omelet but hard to turn an omelet into an egg. What it can’t account for is why, when we open our refrigerator, we are able to find an egg in the first place. Why are we surrounded by exquisitely ordered objects such as eggs and pianos and science books, rather than by featureless chaos?

Part of the answer is straightforward: The objects that populate our everyday experience are not closed systems. Of course an egg is not a randomly chosen configuration of atoms; it’s a carefully constructed system, the assembly of which required a certain set of resources and available energy, not to mention a chicken. But we could ask the same question about the Solar System, or about the Milky Way galaxy. In each case, we have systems that are for all practical purposes isolated, but nevertheless have a much lower entropy than they could.

The answer, as we know, is that the Solar System hasn’t always been a closed system; it evolved out of a protostellar cloud that had an even lower entropy. And that cloud came from the earlier galaxy, which had an even lower entropy. And the galaxy was formed out of the primordial plasma, which had an even lower entropy. And that plasma originated in the very early universe, which had an even lower entropy still.

And the early universe came out of the Big Bang. The truth is, we don’t know much about why the early universe was in the configuration it was; that’s one of the questions motivating us in this book. The ultimate explanation for the arrow of time as it manifests itself in our kitchens and laboratories and memories relies crucially on the very low entropy of the early universe.

You won’t usually find any discussion of this story in conventional textbooks on statistical mechanics. They assume that we are interested in systems that start with relatively low entropy, and take it from there. But we want more—why did our universe have such a small entropy at one end of time, thereby setting the stage for the subsequent arrow of time? It makes sense to start by considering what we do know about how the universe has evolved from its beginning up to today.

THE VISIBLE UNIVERSE

Our universe is expanding, filled with galaxies gradually moving apart from one another. We experience only a small part of the universe directly, and in trying to comprehend the bigger picture it’s tempting to reach for analogies. The universe, we are told, is like the surface of a balloon, on which small dots have been drawn to represent individual galaxies. Or the universe is like a loaf of raisin bread rising in the oven, with each galaxy represented by one of the raisins.

These analogies are terrible. And not only because it seems demeaning to have something as majestic as a galaxy be represented by a tiny, wrinkled raisin. The real problem is that any such analogy brings along with it associations that do not apply to the actual universe. A balloon, for example, has an inside and an outside, as well as a larger space into which it is expanding; the universe has none of those things. Raisin bread has an edge, and is situated inside an oven, and smells yummy; there are no corresponding concepts in the case of the universe.

So let’s take another tack. To understand the universe around us, let’s consider the real thing. Imagine standing outside on a clear, cloudless night, far away from the lights of the city. What do we see when we look into the sky? For the purposes of this thought experiment, we can grant ourselves perfect vision, infinitely sensitive to all the different forms of electromagnetic radiation.

We see stars, of course. To the unaided eye they appear as points of light, but we have long since figured out that each star is a massive ball of plasma, glowing through the energy of internal nuclear reactions, and that our Sun is a star in its own right. One problem is that we don’t have a sense of depth—it’s hard to tell how far away any of those stars are. But astronomers have invented clever ways to determine the distances to nearby stars, and the answers are impressively large. The closest star, Proxima Centauri, is about 40 trillion kilometers away; traveling at the speed of light, it would take about four years to get there.

Stars are not distributed uniformly in every direction. On our hypothetical clear night, we could not help but notice the Milky Way—a fuzzy band of white stretching across the sky, from one horizon to the other. What we’re seeing is actually a collection of many closely packed stars; the ancient Greeks suspected as much, and Galileo verified that idea when he turned his telescope on the heavens. In fact, the Milky Way is a giant spiral galaxy—a collection of hundreds of billions of stars, arranged in the shape of a disk with a bulge in the center, with our Solar System located as one of the distant suburbs on one edge of the disk.

For a long time, astronomers thought that “the galaxy” and “the universe” were the same thing. One could easily imagine that the Milky Way constituted an isolated collection of stars in an otherwise empty void. But it was well known that, in addition to pointlike stars, the night sky featured fuzzy blobs known as “nebulae,” which some argued were giant collections of stars in their own right. After fierce debates between astronomers in the early years of the twentieth century,34 Edwin Hubble was eventually able to measure the distance to the nebula M33 (the thirty-third object in Charles Messier’s catalog of fuzzy celestial objects not to be confused by when one was searching for comets), and found that it is much farther away than any star. M33, the Triangulum Galaxy, is in fact a collection of stars comparable in size to the Milky Way.

Upon further inspection, the universe turns out to be teeming with galaxies. Just as there are hundreds of billions of stars in the Milky Way, there are hundreds of billions of galaxies in the observable universe. Some galaxies (including ours) are members of groups or clusters, which in turn describe sheets and filaments of large-scale structure. On average, however, galaxies are uniformly distributed through space. In every direction we look, and at every different distance from us, the number of galaxies is roughly equal. The observable universe looks pretty much the same everywhere.

BIG AND GETTING BIGGER

Hubble was undoubtedly one of the greatest astronomers of history, but he was also in the right place at the right time. He bounced around a bit after graduating from college, spending time variously as a Rhodes scholar, high school teacher, lawyer, soldier in World War I, and for a while as a basketball coach. But ultimately he earned a Ph.D. in astronomy from the University of Chicago in 1917 and moved to California to take up a position at the Mount Wilson Observatory outside Los Angeles. He arrived to find the brand-new Hooker telescope, featuring a mirror 100 inches across, at the time the world’s largest. It was at the 100-inch that Hubble made the observations of variable stars in other galaxies, establishing for the first time their great distance from the Milky Way.

Meanwhile other astronomers, led by Vesto Slipher, had been measuring the velocity of spiral nebulae using the Doppler effect.35 If an object is moving with respect to you, any wave it emits (such as light or sound) will get compressed if it’s moving toward you, and stretched if it’s moving away. In the case of sound, we experience the Doppler effect as a raising of the pitch of objects that are coming toward us, and a lowering of the pitch as they move away. Similarly, we see the light from objects moving toward us shifted toward the blue (shorter wavelengths) than we would expect, and light from objects moving away is shifted toward the red (longer wavelengths). So an approaching object is blueshifted, while a receding object is redshifted.

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Figure 7: Edwin Hubble, surveyor of the universe, smoking a pipe.

What Slipher found was that the vast majority of nebulae were redshifted. If these objects were moving randomly through the universe, we would expect about as many blueshifts as redshifts, so this pattern came as a surprise. If the nebulae were small clouds of gas and dust, we might have concluded that they had been forcibly ejected from our galaxy by some unknown mechanism. But Hubble’s result, announced in 1925, scotched that possibility—what we were seeing was a collection of galaxies the size of our own, all running away from us as if they were afraid or something.

Hubble’s next discovery made it all snap into place. In 1929 he and his collaborator Milton Humason compared the redshifts of galaxies to the distances he had measured, and found a striking correlation: The farther the galaxies were, the faster they were receding. This is now known as Hubble’s Law: The apparent recession velocity of a galaxy is proportional to its distance from us, and the constant of proportionality is known as the Hubble constant.36

Hidden within this simple fact—the farther away things are, the faster they are receding—lies a deep consequence: We are not at the center of some giant cosmic migration. You might get the impression that we are somehow special, what with all of these galaxies moving way from us. But put yourself in the place of an alien astronomer within one of those other galaxies. If that astronomer looks back at us, of course they would see the Milky Way receding from them. But if they look in the opposite direction in the sky, they will also see galaxies moving away from them—because, from our perspective, those more distant galaxies are moving even faster. This is a very profound feature of the universe in which we live. There isn’t any particular special place, or central point away from which everything is moving. All of the galaxies are moving away from all of the other galaxies, and each of them sees the same kind of behavior. It’s almost as if the galaxies aren’t moving at all, but rather that the galaxies are staying put and space itself is expanding in between them.

Which is, indeed, precisely what’s going on, from the modern way of looking at things. These days we think of space not as some fixed and absolute stage through which matter moves, but as a dynamical and lively entity in its own right, according to Einstein’s general theory of relativity. When we say space is expanding, we mean that more space is coming into existence in between galaxies. Galaxies themselves are not expanding, nor are you, nor are individual atoms; anything that is held together by some local forces will maintain its size, even in an expanding universe. (Maybe you are expanding, but you can’t blame the universe.) A light wave, which is not bound together by any forces, will be stretched, leading to the cosmological redshift. And, of course, galaxies that are sufficiently far apart not to be bound by their mutual gravitational attraction will be moving away from one another.

This is a magnificent and provocative picture of the universe. Subsequent observations have confirmed the idea that, on the very largest scales, the universe is homogeneous: It’s more or less the same everywhere. Clearly the universe is “lumpy” on smaller scales (here’s a galaxy, there’s a void of empty space next to it), but if you consider a sufficiently large volume of space, the number of galaxies and the amount of matter within it will be essentially the same, no matter which volume you pick. And the whole shebang is gradually getting bigger; in about 14 billion years, every distant galaxy we observe will be twice as far away as it is today.

We find ourselves in the midst of an overall smooth distribution of galaxies, the space between them expanding so that every galaxy is moving away from every other.37 If the universe is expanding, what’s it expanding into? Nothing. When we’re talking about the universe, there’s no need to invoke something for it to expand into—it’s the universe—it doesn’t have to be embedded in anything else; it might very well be all there is. We’re not used to thinking like this, because the objects we experience in our everyday lives are all situated within space; but the universe is space, and there’s no reason for there to be any such thing as “outside.”

Likewise, there doesn’t have to be an edge—the universe could just continue on infinitely far in space. Or, for that matter, it could be finite, by wrapping back on itself, like the surface of a sphere. There is a good reason to believe we will never know, on the basis of actual observations. Light has a finite speed (1 light-year per year, or 300,000 kilometers per second), and there is only a finite time since the Big Bang. As we look out into space, we are also looking backward in time. Since the Big Bang occurred approximately 14 billion years ago, there is an absolute limit to how far we can peer in the universe.38 What we see is a relatively homogeneous collection of galaxies, about 100 billion of them all told, steadily expanding away from one another. But outside our observable patch, things could be very different.

THE BIG BANG

I’ve been casually throwing around the phrase the Big Bang. It’s a bit of physics lingo that has long since entered the popular lexicon. But of all the confusing aspects of modern cosmology, probably none has been the subject of more misleading or simply untrue statements—including by professional cosmologists who really should know better—than “the Big Bang.” Let’s take a moment to separate what we know from what we don’t.

The universe is smooth on large scales, and it’s expanding; the space between galaxies is growing. Assuming that the number of atoms in the universe stays the same,39 matter becomes increasingly dilute as time goes by. Meanwhile, photons get redshifted to longer wavelengths and lower energies, which means that the temperature of the universe decreases. The future of our universe is dilute, cold, and lonely.

Now let’s run the movie backward. If the universe is expanding and cooling now, it was denser and hotter in the past. Generally speaking (apart from some niceties concerning dark energy, more about which later), the force of gravity acts to pull things together. So if we extrapolate the universe backward in time to a state that was denser than it is today, we would expect such an extrapolation to continue to be good; in other words, there’s no reason to expect any sort of “bounce.” The universe should simply have been more and more dense further and further in the past. We might imagine that there would be some moment, only a finite amount of time ago, when the universe was infinitely dense—a “singularity.” It’s that hypo thetical singularity that we call “the Big Bang.”

Note that we are referring to the Big Bang as a moment in the history of the universe, not as a place in space. Just as there is no special point in the current universe that defines a center of the expansion, there is no special point corresponding to “where the Bang happened.” General relativity says that the universe can be squeezed into zero size at the moment of the singularity, but be infinitely big at every moment after the singularity.

So what happened before the Big Bang? Here is where many discussions of modern cosmology run off the rails. You will often read something like the following: “Before the Big Bang, time and space did not exist. The universe did not come into being at some moment in time, because time itself came into being. Asking what happened before the Big Bang is like asking what lies north of the North Pole.”

That all sounds very profound, and it might even be right. But it might not. The truth is, we just don’t know. The rules of general relativity are unambiguous: Given certain kinds of stuff in the universe, there must have been a singularity in the past. But that’s not really an internally consistent conclusion. The singularity itself would be a moment when the curvature of spacetime and the density of matter were infinite, and the rules of general relativity simply would not apply. The correct deduction is not that general relativity predicts a singularity, but that general relativity predicts that the universe evolves into a configuration where general relativity itself breaks down. The theory cannot be considered to be complete; something happens where general relativity predicts singularities, but we don’t know what.

Possibly general relativity is not the correct theory of gravity, at least in the context of the extremely early universe. Most physicists suspect that a quantum theory of gravity, reconciling the framework of quantum mechanics with Einstein’s ideas about curved spacetime, will ultimately be required to make sense of what happens at the very earliest times. So if someone asks you what really happened at the moment of the purported Big Bang, the only honest answer would be: “I don’t know.” Once we have a reliable theoretical framework in which we can ask questions about what happens in the extreme conditions characteristic of the early universe, we should be able to figure out the answer, but we don’t yet have such a theory.

It might be that the universe didn’t exist before the Big Bang, just as conventional general relativity seems to imply. Or it might very well be—as I tend to believe, for reasons that will become clear—that space and time did exist before the Big Bang; what we call the Bang is a kind of transition from one phase to another. Our quest to understand the arrow of time, anchored in the low entropy of the early universe, will ultimately put this issue front and center. I’ll continue to use the phrase “the Big Bang” for “that moment in the history of the very early universe just before conventional cosmology becomes relevant,” whatever that moment might actually be like in a more complete theory, and whether or not there is some kind of singularity or boundary to the universe.

HOT, SMOOTH BEGINNINGS

While we don’t know what happened at the very beginning of the universe, there’s a tremendous amount that we do know about what happened after that. The universe started out in an incredibly hot, dense state. Subsequently, space expanded and matter diluted and cooled, passing through a variety of transitions. A suite of observational evidence indicates that it’s been about 14 billion years from the Big Bang to the present day. Even if we don’t claim to know the details of what happened at the earliest moments, it all happened within a very short period of time; most of the history of the universe has occurred long since its mysterious beginnings, so it’s okay to talk about how many years a given event occurred after the Big Bang. This broad-stroke picture is known as the “Big Bang model” and is well understood theoretically and supported by mountains of observational data, in contrast with the hypothetical “Big Bang singularity,” which remains somewhat mysterious.

Our picture of the early universe is not based simply on theoretical extrapolation; we can use our theories to make testable predictions. For example, when the universe was about 1 minute old, it was a nuclear reactor, fusing protons and neutrons into helium and other light elements in a process known as “primordial nucleosynthesis.” We can observe the abundance of such elements today and obtain spectacular agreement with the predictions of the Big Bang model.

We also observe cosmic microwave background radiation. The early universe was hot as well as dense, and hot things give off radiation. The theory behind night-vision goggles is that human beings (or other warm things) give off infrared radiation that can be detected by an appropriate sensor. The hotter something is, the more energetic (short wavelength, high frequency) is the radiation it emits. The early universe was extremely hot and gave off a lot of energetic radiation.

What is more, the early universe was opaque. It was sufficiently hot that electrons could not stay bound to atomic nuclei, but flew freely through space; photons frequently bounced off the free electrons, so that (had you been around) you wouldn’t have been able to see your hand in front of your face. But eventually the temperature cooled to a point where electrons could get stuck to nuclei and stay there—a process called recombination, about 400,000 years after the Big Bang. Once that happened, the universe was transparent, so light could travel essentially unimpeded from that moment until today. Of course, it still gets redshifted by the cosmological expansion, so the hot radiation from the period of recombination has been stretched into microwaves, about 1 centimeter in wavelength, reaching a current temperature of 2.7 Kelvin (-270.4 degrees Celsius).

The story of the evolution of the universe according to the Big Bang model (as distinguished from the mysterious moment of the Big Bang itself) therefore makes a strong prediction: Our universe should be suffused with microwave radiation from all directions, a relic from an earlier time when the universe was hot and dense. This radiation was finally detected by Arno Penzias and Robert Wilson in 1965 at Bell Labs in Holmdel, New Jersey. And they weren’t even looking for it—they were radio astronomers who became somewhat annoyed at this mysterious background radiation they couldn’t get rid of. Their annoyance was somewhat mollified when they won the Nobel Prize in 1978.41 It was the discovery of the microwave background that converted most of the remaining holdouts for the Steady State theory of cosmology (in which the temperature of the universe would be constant through time, and new matter is continually created) over to the Big Bang point of view.

TURNING UP THE CONTRAST KNOB ON THE UNIVERSE

The universe is a simple place. True, it contains complicated things like galaxies and sea otters and federal governments, but if we average out the local idiosyncrasies, on very large scales the universe looks pretty much the same everywhere. Nowhere is this more evident than in the cosmic microwave background. Every direction we look in the sky, we see microwave background radiation that looks exactly like that from an object glowing serenely at some fixed temperature—what physicists call “blackbody” radiation. However, the temperature is ever so slightly different from point to point on the sky; typically, the temperature in one direction differs from that in some other direction by about 1 part in 100,000. These fluctuations are called anisotropies—tiny departures from the otherwise perfectly smooth temperature of the background radiation in every direction.

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Figure 8: Temperature anisotropies in the cosmic microwave background, as measured by NASA’s Wilkinson Microwave Anisotropy Probe. Dark regions are slightly colder than average, light regions are slightly hotter than average. The differences have been dramatically enhanced for clarity.

These variations in temperature reflect slight differences in the density of matter from place to place in the early universe. Saying that the early universe was smooth is not just a simplifying assumption; it’s a testable hypothesis that is strongly supported by the data. On very large scales, the universe is still smooth today. But the scales have to be pretty large—over 300 million light-years or so. On smaller scales, like the size of a galaxy or the Solar System or your kitchen, the universe is quite lumpy. It wasn’t always like that; at early times, even small scales were very smooth. How did we get here from there?

The answer lies in gravity, which acts to turn up the contrast knob on the universe. In a region that has slightly more matter than average, there is a gravitational force that pulls things together; in regions that are slightly underdense, matter tends to flow outward to the denser regions. By this process—the evolution of structure in the universe—the tiny primordial fluctuations revealed in the microwave background anisotropies grow into the galaxies and structures we see today.

Imagine that we lived in a universe much like our current one, with the same kind of distribution of galaxies and clusters, but that was contracting rather than expanding. Would we expect that the galaxies would smooth out toward the future as the universe contracted, creating a homogeneous plasma such as we see in the past of our real (expanding) universe? Not at all. We would expect the contrast knob to continue to be turned up, even as the universe contracted—black holes and other massive objects would gather matter from the surrounding regions. Growth of structure is an irreversible process that naturally happens toward the future, whether the universe is expanding or contracting: It represents an increase in entropy. So the relative smoothness of the early universe, illustrated in the image of the cosmic microwave background, reflects the very low entropy of those early times.

THE UNIVERSE IS NOT STEADY

The Big Bang model seems like a fairly natural picture, once you believe in an approximately uniform universe that is expanding in time. Just wind the clock backward, and you get a hot, dense beginning. Indeed, the basic framework was put together in the late 1920s by Georges Lemaître, a Catholic priest from Belgium who had studied at Cambridge and Harvard before eventually earning his doctorate from MIT.42 (Lemaître, who dubbed the beginning of the universe the “Primeval Atom,” refrained from drawing any theological conclusions from his cosmological model, despite the obvious temptation.)

But there is a curious asymmetry in the Big Bang model, one that should come as no surprise to us by now: the difference between time and space. The idea that matter is smooth on large scales can be elevated into the “Cosmological Principle”: There is no such thing as a special place in the universe. But it seems clear that there is a special time in the universe: the moment of the Big Bang.

Some mid-century cosmologists found this stark distinction between smoothness in space and variety in time to be a serious shortcoming of the Big Bang model, so they set about developing an alternative. In 1948, three leading astrophysicists—Hermann Bondi, Thomas Gold, and Fred Hoyle—suggested the Steady State model of the universe.43 They based this model on the “Perfect Cosmological Principle”—there is no special place and no special time in the universe. In particular, they suggested that the universe wasn’t any hotter or denser in the past than it is today.

The pioneers of the Steady State theory (unlike some of their later followers) were not crackpots. They understood that Hubble had discovered the expansion of the universe, and they respected the data. So how can the universe be expanding without diluting and cooling down? The answer they suggested was that matter was continually being created in between the galaxies, precisely balancing the dilution due to the expansion of the universe. (You don’t need to make much: about one hydrogen atom per cubic meter every billion years. It’s not like your living room will start filling up.) Creation of matter wouldn’t happen all by itself; Hoyle invented a new kind of field, called the C-field, which he hoped would do the trick, but the idea never really caught on among physicists.

From our jaded modern perspective, the Steady State model seems like a lot of superstructure constructed on the basis of some fairly insubstantial philosophical presuppositions. But many great theories begin that way, before they are confronted with the harsh realities of data; Einstein certainly leaned on his own philosophical preferences during the construction of general relativity. But unlike relativity, when the data ultimately confronted the Steady State model, the result was not pretty.44 The last thing you would expect from a model in which the temperature of the universe remains constant is a relic background radiation that indicates a hot beginning. After Penzias and Wilson discovered the microwave background, support for the Steady State theory crumbled, although there remains to this day a small cadre of true believers who invent ingenious ways of avoiding the most straightforward interpretation of the data.

Nevertheless, thinking about the Steady State model brings home the perplexing nature of time in the Big Bang model. In the Steady State cosmology, there was still unmistakably an arrow of time: Entropy increased, without limit, in the same direction, forever and ever. In a very legitimate sense, the problem of explaining the low-entropy initial conditions of the universe would be infinitely bad in a Steady State universe; whatever those conditions were, they were infinitely far in the past, and the entropy of any finite-sized system today would have been infinitesimally small. One could imagine that considerations of this form might have undermined the Steady State model from the start, if cosmologists had taken the need to explain the low entropy of the early universe seriously.

In the Big Bang picture, things don’t seem quite as hopeless. We still don’t know why the early universe had a low entropy, but at least we know when the early universe was: It was 14 billion years ago, and its entropy was small but not strictly zero. Unlike in the Steady State model, in the context of the Big Bang you can at least put your finger directly on where (really “when”) the problem is located. Whether or not this is really an improvement can’t be decided until we understand cosmology within a more comprehensive framework.

BUT IT IS ACCELERATING

We know a good deal about the evolution of the universe over the last 14 billion years. What’s going to happen in the future?

Right now the universe is expanding, becoming increasingly colder and ever more dilute. For many years the big question in cosmology had been, “Will expansion continue forever, or will the universe eventually reach a maximum size and begin to contract toward a Big Crunch at the end of time?” Debating the relative merits of these alternatives was a favorite parlor game among cosmologists ever since the early days of general relativity. Einstein himself favored a universe that was finite in both space and time, so he liked the idea of an eventual re-collapse. Lemaître, in contrast, preferred the idea that the universe would continue to cool off and expand forever: ice, rather than fire.

Performing measurements that would decide the question empirically turned out to be more difficult. General relativity would seem to make a clear prediction: As the universe expands, the gravitational force between galaxies pulls all of them together, working to slow the expansion down. The question was simply whether there was enough matter in the universe to actually cause a collapse, or whether it would expand ever more gradually but for all eternity. For a long time it was a hard question to answer, as observations seemed to indicate that there was almost enough matter to reverse the expansion of the universe—but not quite enough.

The breakthrough occurred in 1998, from a completely different method. Rather than measuring the total amount of mass in the universe, and comparing with theory to determine whether there was enough to eventually reverse the universe’s expansion, one could go out and directly measure the rate at which the expansion was slowing down. Easier said than done, of course. Basically what one had to do was what Hubble had done years before—measure both distances and apparent velocities of galaxies, and look at the relationship between them—but to enormously higher precision and at much greater distances. The technique eventually used was to search for Type Ia supernovae, exploding stars that not only have the virtue of being very bright (and therefore visible over cosmological distances), but also have almost the same brightness in every event (so that the apparent brightness can be used to deduce the distance to the supernova).45

The hard work was done by two teams: one led by Saul Perlmutter of Lawrence Berkeley National Laboratory, and one led by Brian Schmidt of Mount Stromlo Observatory in Australia. Perlmutter’s group, which contained a number of particle physicists converted to the cause of cosmology, started earlier, and had championed the supernova technique in the face of considerable skepticism. Schmidt’s group, which included a number of experts on supernova astronomy, started later but managed to catch up. The teams maintained a rivalry that was often friendly and occasionally less so, but they both made crucial contributions, and rightfully share the credit for the ultimate discovery.

As it happens, Brian Schmidt and I were office mates in graduate school at Harvard in the early 1990s. I was the idealistic theorist, and he was the no-nonsense observer. In those days, when the technology of large-scale surveys in astronomy was just in its infancy, it was a commonplace belief that measuring cosmological parameters was a fool’s errand, doomed to be plagued by enormous uncertainties that would prevent us from determining the size and shape of the universe with anything like the precision we desired. Brian and I made a bet concerning whether we would be able to accurately measure the total matter density of the universe within twenty years. I said we would; Brian was sure we wouldn’t. We were poor graduate students at the time, but purchased a small bottle of vintage port, to be secreted away for two decades before we knew who had won. Happily for both of us, we learned the answer long before then; I won the bet, due in large part to the efforts of Brian himself. We split the bottle of port on the roof of Harvard’s Quincy House in 2005.

And the answer is: The universe isn’t decelerating at all; it’s actually accelerating! If you were to measure the apparent recession velocity of a galaxy, and (hypothetically) came back a billion years later to measure it again, you would find that the velocity was now higher.46 How can that be reconciled with the supposed prediction of general relativity that the universe should be slowing down? Like most such predictions of general relativity, there are hidden assumptions: in this case, that the primary source of energy in the universe consists of matter.

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Figure 9: The accelerating universe.

To a cosmologist, matter is shorthand for “any collection of particles, each of which is moving much more slowly than the speed of light.” (If particles are moving at or close to the speed of light, cosmologists refer to them as “radiation,” whether or not they are actually electromagnetic radiation in the usual sense.) Einstein taught us long ago that particles have energy, even when they’re not moving at all: E = mc2 means that the energy of a perfectly stationary massive particle is given by its mass times the speed of light squared. For our present purposes, the crucial aspect of matter is that it dilutes away as the universe expands.47 What general relativity actually predicts is that the expansion should be decelerating, as long as the energy is diluting away. If it’s not—if the density of energy, the amount of energy in each cubic centimeter or cubic light-year of space, is approximately constant—then that energy provides a perpetual impulse to the expansion of space, and the universe will actually be accelerating.

It’s possible, of course, that general relativity is not the correct theory of gravity on cosmological scales, and that possibility is one that physicists take very seriously. It seems more likely, however, that general relativity is correct, and the observations are telling us that most of the energy in the universe is not in the form of “matter” at all, but rather in the form of some stubbornly persistent stuff that sticks around even as space expands. We’ve dubbed that mysterious stuff “dark energy,” and the nature of the dark energy is very much a favorite research topic for modern cosmologists, both theorists and observers.

We don’t know much about dark energy, but we do know two very crucial things: It’s nearly constant throughout space (the same amount of energy from place to place), and also nearly constant in density through time (the same amount of energy per cubic centimeter at different times). So the simplest possible model for dark energy would be one featuring an absolutely constant density of energy through all space and time. And in fact, that’s an old idea, dating back to Einstein: He called it “the cosmological constant,” and these days we often call it “vacuum energy.” (Some people may try to convince you that there is some difference between vacuum energy and the cosmological constant—don’t fall for it. The only difference is which side of the equation you put it on, and that’s no difference at all.)

What we’re suggesting is that every cubic centimeter of space—out in the desolate cold between the galaxies, or at the center of the Sun, or right in front of your face—there is a certain amount of energy, over and above whatever comes from the particles and photons and other things that are actually located in that little cube. It’s called “vacuum energy” because it’s present even in a vacuum, in a perfectly empty space—a minimum amount of energy inherent in the fabric of spacetime itself.48 You can’t feel it, you can’t see it, you can’t do anything with it, but it is there. And we know it is there because it exerts a crucial influence on the universe, imparting a gentle push that causes distant galaxies to accelerate away from us.

Unlike the gravity caused by ordinary matter, the effect of vacuum energy is to push things apart rather than pull them together. When Einstein first proposed the cosmological constant in 1917, his motivation was to explain a static universe, one that wasn’t expanding or contracting. This wasn’t a misguided philosophical stance—it was the best understanding according to the astronomy of the day; Hubble wouldn’t discover the expansion of the universe until 1929. So Einstein imagined a universe in delicate balance between the pull of gravity among galaxies and the push of the cosmological constant. Once he learned of Hubble’s discovery, he regretted ever introducing the cosmological constant—had he resisted the temptation, he might have predicted the expansion of the universe before it was discovered.

THE MYSTERY OF VACUUM ENERGY

In theoretical physics, it’s not easy to un-invent a concept. The cosmological constant is the same as the idea of vacuum energy, the energy of empty space itself. The question is not “Is vacuum energy a valid concept?”—it’s “What value should we expect the vacuum energy to have?”

Modern quantum mechanics implies that the vacuum is not a boring place; it’s alive with virtual particles. A crucial consequence of quantum mechanics is Werner Heisenberg’s uncertainty principle: It’s impossible to pin down the observable features of any system into one unique state with perfect precision, and that includes the state of empty space. So if we were to look closely enough at empty space, we would see particles flashing into and out of existence, representing quantum fluctuations of the vacuum itself. These virtual particles are not especially mysterious or hypothetical—they are definitely there, and they have measurable effects in particle physics that have been observed many times over.

Virtual particles carry energy, and that energy contributes to the cosmological constant. We can add up the effects of all such particles to obtain an estimate for how large the cosmological constant should be. But it wouldn’t be right to include the effects of particles with arbitrarily high energies. We don’t believe that our conventional understanding of particle physics is adequate for very high-energy events—at some point, we have to take account of the effects of quantum gravity, the marriage of general relativity with quantum mechanics, which remains an incomplete theory at the moment.

So instead of appealing to the correct theory of quantum gravity, which we still don’t have, we can simply examine the contributions to the vacuum energy of virtual particles at energies below where quantum gravity becomes important. That’s the Planck energy, named after German physicist Max Planck, one of the pioneers of quantum theory, and it turns out to be about 2 billion joules (a conventional unit of energy).49 We can add up the contributions to the vacuum energy from virtual particles with energies ranging from zero up to the Planck energy, and then cross our fingers and compare with what we actually observe.

The result is a complete fiasco. Our simple estimate of what the vacuum energy should be comes out to about 10105 joules per cubic centimeter. That’s a lot of vacuum energy. What we actually observe is about 10-15 joules per cubic centimeter. So our estimate is larger than the experimental value by a factor of 10120—a 1 followed by 120 zeroes. Not something we can attribute to experimental error. This has been called the biggest disagreement between theoretical expectation and experimental reality in all of science. For comparison, the total number of particles in the observable universe is only about 1088; the number of grains of sand on all the Earth’s beaches is only about 1020.

The fact that the vacuum energy is so much smaller than it should be is a serious problem: the “cosmological constant problem.” But there is also another problem: the “coincidence problem.” Remember that vacuum energy maintains a constant density (amount of energy per cubic centimeter) as the universe expands, while the density of matter dilutes away. Today, they aren’t all that different: Matter makes up about 25 percent of the energy of the universe, while vacuum energy makes up the other 75 percent. But they are changing appreciably with respect to each other, as the matter density dilutes away with the expansion and the vacuum energy does not. At the time of recombination, for example, the energy density in matter was a billion times larger than that in vacuum energy. So the fact that they are somewhat comparable today, uniquely in the history of the universe, seems like a remarkable coincidence indeed. Nobody knows why.

These are serious problems with our theoretical understanding of vacuum energy. But if we put aside our worries concerning why the vacuum energy is so small, and why it’s comparable in density to the energy in matter, we are left with a phenomenological model that does a remarkable job of fitting the data. (Just like Carnot and Clausius didn’t need to know about atoms to say useful things about entropy, we don’t need to understand the origin of the vacuum energy to understand what it does to the expansion of the universe.) The first direct evidence for dark energy came from observations of supernovae in 1998, but since then a wide variety of methods have independently confirmed the basic picture. Either the universe is accelerating under the gentle influence of vacuum energy, or something even more dramatic and mysterious is going on.

THE DEEPEST FUTURE

As far as we can tell, the density of vacuum energy is unchanging as the universe expands. (It could be changing very slowly, and we just haven’t been able to measure the changes yet—that’s a major goal of modern observational cosmology.) We don’t know enough about vacuum energy to say for sure what will happen to it indefinitely into the future, but the obvious first guess is that it will simply stay at its current value forever.

If that’s true, and the vacuum energy is here to stay, it’s straightforward to predict the very far future of our universe. The details get complicated in an interesting way, but the outline is relatively simple.50 The universe will continue to expand, cool off, and become increasingly dilute. Distant galaxies will accelerate away from us, becoming more and more redshifted as they go. Eventually they will fade from view, as the time between photons that could possibly reach us becomes longer and longer. The entirety of the observable universe will just be our local group of gravitationally bound galaxies.

Galaxies don’t last forever. The stars in them burn their nuclear fuel and die. Out of the remnant gas and dust more stars can form, but a point of diminishing returns is reached, after which all of the stars in the galaxy are dead. We are left with white dwarfs (stars that once burned, and ran out of fuel), brown dwarfs (stars that never burned in the first place), and neutron stars (stars that used to be white dwarfs but collapsed further under the pull of gravity). These objects may or may not be stable in their own right; our best current theoretical guess is that the protons and neutrons that make them up aren’t perfectly stable themselves but will eventually decay into lighter particles. If that’s true (and admittedly, we’re not sure), the various forms of dead stars will eventually dissipate into a thin gas of particles that disperse into the void. It won’t be quick; a reasonable estimate is 1040 years from now. For comparison, the current universe is about 1010 years old.

Besides stars, there are also black holes. Most large galaxies, including our own, have giant black holes at the center. In a galaxy the size of the Milky Way, with about 100 billion stars, the black hole might be a few million times as massive as the Sun—big compared to any individual star, but still small compared to the galaxy as a whole. But it will continue to grow, sweeping up whatever unfortunate stars happen to fall into it. Ultimately, however, all of the stars will have been used up. At that point, the black hole itself begins to evaporate into elementary particles. That’s the remarkable discovery of Stephen Hawking from 1976, which we’ll discuss in detail in Chapter Twelve: “black holes ain’t so black.” Due once again to quantum fluctuations, a black hole can’t help but gradually radiate out into the space around it, slowly losing energy in the process. If we wait long enough—and now we’re talking 10100 years or so—even the supermassive black holes at the centers of galaxies will evaporate away to nothing.

Regardless of how the details play out, we are left with the same long-term picture. Other galaxies move away from us and disappear; our own galaxy will evolve through various stages, but the end result is a thin gruel of particles dissipating into the void. In the very far future, the universe becomes once again a very simple place: It will be completely empty, as empty as space can be. That’s the diametric opposite of the hot, dense state in which the universe began; a vivid cosmological manifestation of the arrow of time.

THE ENTROPY OF THE UNIVERSE

An impressive number of brain-hours on the part of theoretical physicists have been devoted to the question of why the universe evolved in this particular fashion, rather than in some other way. It’s certainly possible that this question simply has no answer; perhaps the universe is what it is, and the best we can do is to accept it. But we are hopeful, not without reason, that we can do more than accept it—we can explain it.

Given perfect knowledge of the laws of physics, the question “Why has the universe evolved in the fashion it has?” is equivalent to “Why were the initial conditions of the universe arranged in the way they were?” But that latter formulation is already sneaking in an implicit notion of time asymmetry, by privileging past conditions over future conditions. If our understanding of the fundamental, microscopic laws of nature is correct, we can specify the state of the universe at any time, and from there derive both the past and the future. It would be better to characterize our task as that of understanding what would count as a natural history of the universe as a whole.51

There is some irony in the fact that cosmologists have underappreciated the importance of the arrow of time, since it is arguably the single most blatant fact about the evolution of the universe. Boltzmann was able to argue (correctly) for the need for a low-entropy boundary condition in the past, without knowing anything about general relativity, quantum mechanics, or even the existence of other galaxies. Taking the problem of entropy seriously helps us look at cosmology in a new light, which might suggest some resolutions to long-standing puzzles.

But first, we need to be a little more clear about what exactly we mean about “the entropy of the universe.” In Chapter Thirteen we will discuss the evolution of entropy in our observable universe in great detail, but the basic story goes as follows:

1. In the early universe, before structure forms, gravity has little effect on the entropy. The universe is similar to a box full of gas, and we can use the conventional formulas of thermodynamics to calculate its entropy. The total entropy within the space corresponding to our observable universe turns out to be about 1088 at early times.

2. By the time we reach our current stage of evolution, gravity has become very important. In this regime we don’t have an ironclad formula, but we can make a good estimate of the total entropy just by adding up the contributions from black holes (which carry an enormous amount of entropy). A single supermassive black hole has an entropy of order 1090, and there are approximately 1011 such black holes in the observable universe; our total entropy today is therefore something like 10101.

3. But there is a long way to go. If we took all of the matter in the observable universe and collected it into a single black hole, it would have an entropy of 10120. That can be thought of as the maximum possible entropy obtainable by rearranging the matter in the universe, and that’s the direction in which we’re evolving.52

Our challenge is to explain this history. In particular, why was the early entropy, 1088, so much lower than the maximum possible entropy, 10120? Note that the former number is much, much, much smaller than the latter; appearances to the contrary are due to the miracle of compact notation.

The good news is, at least the Big Bang model provides a context in which we can sensibly address this question. In Boltzmann’s time, before we knew about general relativity or the expansion of the universe, the puzzle of entropy was even harder, simply because there was no such event as “the beginning of the universe” (or even “the beginning of the observable universe”). In contrast, we are able to pinpoint exactly when the entropy was small, and the particular form that low-entropy state took; that’s a crucial step in trying to explain why it was like that.

It’s possible, of course, that the fundamental laws of physics simply aren’t reversible (although we’ll give arguments against that later on). But if they are, the low entropy of our universe near the Big Bang leaves us with two basic possibilities:

1. The Big Bang was truly the beginning of the universe, the moment when time began. That may be because the true laws of physics allow spacetime to have a boundary, or because what we call “time” is just an approximation, and that approximation ceases to be valid near the Big Bang. In either case, the universe began in a low-entropy state, for reasons over and above the dynamical laws of nature—we need a new, independent principle to explain the initial state.

2. There is no such thing as an initial state, because time is eternal. In this case, we are imagining that the Big Bang isn’t the beginning of the entire universe, although it’s obviously an important event in the history of our local region. Somehow our observable patch of spacetime must fit into a bigger picture. And the way it fits must explain why the entropy was small at one end of time, without imposing any special conditions on the larger framework.

As to which of these is the correct description of the real world, the only answer is that we don’t know. I will confess to a personal preference for Option 2, as I think it would be more elegant if the world were described as a nearly inevitable result of a set of dynamical laws, without needing an extra principle to explain why it appears precisely this way. Turning this vague scenario into an honest cosmological model will require that we actually take advantage of the mysterious vacuum energy that dominates our universe. Getting there from here requires a deeper understanding of curved spacetime and relativity, to which we now turn.