From Eternity to Here: The Quest for the Ultimate Theory of Time

by Sean M. Carroll

  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 thems
elves 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.

  14

  INFLATION AND THE MULTIVERSE

  Those who think of metaphysics as the most unconstrained or speculative of disciplines are misinformed; compared with cosmology, metaphysics is pedestrian and unimaginative.

  —Stephen Toulmin254

  On a cool Palo Alto morning in December 1979, Alan Guth pedaled his bike as fast as he could to his office in the theoretical physics group at SLAC, the Stanford Linear Accelerator Center. Upon reaching his desk, he opened his notebook to a new page and wrote:

  SPECTACULAR REALIZATION: this kind of supercooling can explain why the universe today is so incredibly flat—and therefore resolve the fine-tuning paradox pointed out by Bob Dicke in his Einstein Day lectures.

  He carefully drew a rectangular box around the words. Then he drew another one.255

  As a scientist, you live for the day when you hit upon a result—a theoretical insight, or an experimental discovery—so marvelous that it deserves a box around it. The rare double-box-worthy results tend to change one’s life, as well as the course of science; as Guth notes, he doesn’t have any other double-boxed results anywhere in his notebooks. The one from his days at SLAC is now on display at the Adler Planetarium in Chicago, open to the page with the words above.

  Guth had hit on the scenario now known as inflation—the idea that the early universe was suffused with a temporary form of dark energy at an ultra-high density, which caused space to accelerate at an incredible rate (the “supercooling” mentioned above). That simple suggestion can explain more or less everything there is to explain about the conditions we observe in our early universe, from the geometry of space to the pattern of density perturbations observed in the cosmic microwave background. Although we do not yet have definitive proof that inflation occurred, it has been arguably the most influential idea in cosmology over the last several decades.

  Figure 73: Alan Guth, whose inflationary universe scenario may help explain why our observed universe is close to smooth and flat.

  Which doesn’t mean inflation is right, of course. If the early universe was temporarily dominated by dark energy at an ultra-high scale, then we can understand why the universe would evolve into just the state it was apparently in at early times. But there is a danger of begging the important question—why was it ever dominated by dark energy in that way? Inflation doesn’t provide any sort of answer by itself to the riddle of why entropy was low in the early universe, other than to assume that it started even lower (which is arguably a bit of a cheat).

  Nevertheless, inflation is an extraordinarily compelling idea, which really does seem to match well with the observed features of our early universe. And it leads to some surprising consequences that Guth himself never foresaw when he first suggested the scenario—including, as we’ll see, a way to make the idea of a “multiverse” become realistic. It seems likely, in the judgment of most working cosmologists, that some version of inflation is correct—the question is, why did inflation ever happen?

  THE CURVATURE OF SPACE

  Imagine you take a pencil and balance it vertically on its tip. Obviously its natural tendency will be to fall over. But you could imagine that if you had an extremely stable surface, and you were a real expert at balancing, you could arrange things so that the pencil remained vertical for a very long time. Like, more than 14 billion years.

  The universe is somewhat like that, where the pencil represents the curvature of space. This can be a more confusing concept than it really should be, because cosmologists sometimes speak about the “curvature of spacetime,” and other times about the “curvature of space,” and those things are different; you’re supposed to figure out from context which one is meant. Just as spacetime can have curvature, space all by itself can as well—and the question of whether space is curved is completely independent of whether spacetime is curved.256

  One potential problem in discussing the curvature of space by itself is that general relativity gives us the freedom to slice spacetime into three-dimensional copies of space evolving through time in a multitude of ways; the definition of “space” is not unique. Fortunately, in our observed universe there is a natural way to do the slicing: We define “time” such that the density of matter is approximately constant through space on large scales, but diminishing as the universe expands. The distribution of matter, in other words, defines a natural rest frame for the universe. This doesn’t violate the precepts of relativity in any way, because it’s a feature of a particular configuration of matter, not of the underlying laws of physics.

  In general, space could curve in arbitrary ways from place to place, and the discipline of differential geometry was developed to handle the mathematics of curvature. But in cosmology we’re lucky in that space is uniform over large distances, and looks the same in every direction. In that case, all you have to do is specify a single number—the “curvature of space”—to tell me everything there is to know about the geometry of three-dimensional space.

  The curvature of space can be a positive number, or a negative number, or zero. If the curvature is zero, we naturally say that space is “flat,” and it has all the characteristics of geometry as we usually understand it. These characteristics were first set out by Euclid, and include properties like “initially parallel lines stay parallel forever,” and “the angles inside a triangle add up to precisely 180 degrees.” If the curvature is positive, space is like the surface of a sphere—except that it’s three-dimensional. Initially parallel lines do eventually intersect, and angles inside a triangle add up to greater than 180 degrees. If the curvature is negative, space is like the surface of a saddle, or of a potato chip. Initially parallel lines grow apart, and angles inside a triangle—well, you can probably guess.257

  According to the rules of general relativity, if the universe starts flat, it stays flat. If it starts curved, the curvature gradually diminishes away as the universe expands. However, as we know, the density of matter and radiation also dilutes away. (For right now, forget you’ve ever heard about dark energy, which changes everything.) When you plug in the equations, the density of matter or radiation decreases faster than the amount of curvature. Relative to matter and radiation, curvature becomes more relevant to the evolution of the universe as space expands.

  Therefore: If there is any noticeable amount of curvature whatsoever in the early universe, the universe today should be very obviously curved. A flat universe is like a pencil balanced exactly on its tip; if there were any deviation to the left or right, the pencil would tend to fall pretty quickly onto its side. Similarly, any tiny deviation from perfect flatness at early times should have become progressively more noticeable as time went on. But as a matter of observational fact, the universe looks very flat. As far as anyone can tell, there is no measurable curvature in the universe today at all.258

  Figure 74: Ways that space can have a uniform curvature. From top to bottom: positive curvature, as on a sphere; negative curvature, as on a saddle; zero curvature, as on a flat plane.

  This state of affairs is known as the flatness problem. Because the universe is so flat today, it had to be incredibly flat in the past. But why?

  The flatness problem bears a family resemblance to the entropy problem we discussed in the last chapter. In both cases, it’s not that there is some blatant disagreement between theory and observation—all we have to do is posit that the early universe had some particular form, and everything follows nicely from there. The problem is that the “particular form” seems to be incredibly unnatural and finely tuned, for no obvious reason. We could say that both the entropy and the spatial curvature of the early universe just were small, and there’s no explanation beyond that. But these apparently unnatural features of the universe might be a
clue to something important, so it behooves us to take them seriously.

  MAGNETIC MONOPOLES

  Alan Guth wasn’t trying to solve the flatness problem when he hit upon the idea of inflation. He was interested in a very different puzzle, known as the monopole problem.

  Guth, for that matter, wasn’t especially interested in cosmology. In 1979, he was in his ninth year of being a postdoctoral researcher—the phase of a scientist’s career in between graduate school and becoming a faculty member, when they concentrate on research without having to worry about teaching duties or other academic responsibilities. (And without the benefit of any job security whatsoever; most postdocs never succeed in getting a faculty job, and eventually leave the field.) Nine years is past the time when a talented postdoc would normally have moved on to become an assistant professor somewhere, but Guth’s publication record at this point in his career didn’t really reflect the ability that others saw in him. He had labored for a while on a theory of quarks that had fallen out of favor, and was now trying to understand an obscure prediction of the newly popular “Grand Unified Theories”: the prediction of magnetic monopoles.

  Grand Unified Theories, or GUTs for short, attempt to provide a unified account of all the forces of nature other than gravity. They became very popular in the 1970s, both for their inherent simplicity, and because they made an intriguing prediction: that the proton, the stalwart elementary particle that (along with the electron and the neutron) forms the basis for all the matter around us, would ultimately decay into lighter particles. Giant laboratories were built to search for proton decay, but it hasn’t yet been discovered. That doesn’t mean that GUTs aren’t right; they are still quite popular, but the failure to detect proton decay has left physicists at a loss over how these theories should be tested.