by Brian Greene
Well, almost. Because your crime was particularly devious, at the last minute the parole board decides that you must solve a second puzzle. You are given two identical old-time Jack-in-the-box toys, and your new challenge is to find a way to make each have a different weight. But in this go-around, not only are you forbidden to change the amount of mass in either object, you are also required to keep both at exactly the same temperature. Again, were Newton given this puzzle, he would immediately resign himself to life in the Hall. Since the toys have identical masses, he would conclude that their weights are identical, and so the puzzle is insoluble. But once again, your knowledge of general relativity comes to the rescue: On one of the toys you compress the spring, tightly squeezing Jack under the closed lid, while on the other you leave Jack in his popped-up posture. Why? Well, a compressed spring has more energy than an uncompressed one; you had to exert energy to squeeze the spring down and you can see evidence of your labor because the compressed spring exerts pressure, causing the toy's lid to strain slightly outward. And, again, according to Einstein, any additional energy affects gravity, resulting in additional weight. Thus, the closed Jack-in-the-box, with its compressed spring exerting an outward pressure, weighs a touch more than the open Jack-in-the-box, with its uncompressed spring. This is a realization that would have escaped Newton, and with it you finally do earn back your freedom.
The solution to that second puzzle hints at the subtle but critical feature of general relativity that we're after. In his paper presenting general relativity, Einstein showed mathematically that the gravitational force depends not only on mass, and not only on energy (such as heat), but also on any pressures that may be exerted. And this is the essential physics we need if we are to understand the cosmological constant. Here's why. Outward-directed pressure, like that exerted by a compressed spring, is called positive pressure. Naturally enough, positive pressure makes a positive contribution to gravity. But, and this is the critical point, there are situations in which the pressure in a region, unlike mass and total energy, can be negative, meaning that the pressure sucks inward instead of pushing outward. And although that may not sound particularly exotic, negative pressure can result in something extraordinary from the point of view of general relativity: whereas positive pressure contributes to ordinary attractive gravity, negative pressure contributes to "negative" gravity, that is, to repulsive gravity! 6
With this stunning realization, Einstein's general relativity exposed a loophole in the more than two-hundred-year-old belief that gravity is always an attractive force. Planets, stars, and galaxies, as Newton correctly showed, certainly do exert an attractive gravitational pull. But when pressure becomes important (for ordinary matter under everyday conditions, the gravitational contribution of pressure is negligible) and, in particular, when pressure is negative (for ordinary matter like protons and electrons, pressure is positive, which is why the cosmological constant can't be composed of anything familiar) there is a contribution to gravity that would have shocked Newton. It's repulsive.
This result is central to much of what follows and is easily misunderstood, so let me emphasize one essential point. Gravity and pressure are two related but separate characters in this story. Pressures, or more precisely, pressure differences, can exert their own, nongravitational forces. When you dive underwater, your eardrums can sense the pressure difference between the water pushing on them from the outside and the air pushing on them from the inside. That's all true. But the point we're now making about pressure and gravity is completely different. According to general relativity, pressure can indirectly exert another force—it can exert a gravitational force—because pressure contributes to the gravitational field. Pressure, like mass and energy, is a source of gravity. And remarkably, if the pressure in a region is negative, it contributes a gravitational push to the gravitational field permeating the region, not a gravitational pull.
This means that when pressure is negative, there is competition between ordinary attractive gravity, arising from ordinary mass and energy, and exotic repulsive gravity, arising from the negative pressure. If the negative pressure in a region is negative enough, repulsive gravity will dominate; gravity will push things apart rather than draw them together. Here is where the cosmological constant comes into the story. The cosmological term Einstein added to the equations of general relativity would mean that space is uniformly suffused with energy but, crucially, the equations show that this energy has a uniform, negative pressure. What's more, the gravitational repulsion of the cosmological constant's negative pressure overwhelms the gravitational attraction coming from its positive energy, and so repulsive gravity wins the competition: a cosmo logical constant exerts an overall repulsive gravitational force. 7
For Einstein, this was just what the doctor ordered. Ordinary matter and radiation, spread throughout the universe, exert an attractive gravitational force, causing every region of space to pull on every other. The new cosmological term, which he envisioned as also being spread uniformly throughout the universe, exerts a repulsive gravitational force, causing every region of space to push on every other. By carefully choosing the size of the new term, Einstein found that he could precisely balance the usual attractive gravitational force with the newly discovered repulsive gravitational force, and produce a static universe.
Moreover, because the new repulsive gravitational force arises from the energy and pressure in space itself, Einstein found that its strength is cumulative; the force becomes stronger over larger spatial separations, since more intervening space means more outward pushing. On the distance scales of the earth or the entire solar system, Einstein showed that the new repulsive gravitational force is immeasurably tiny. It becomes important only over vastly larger cosmological expanses, thereby preserving all the successes of both Newton's theory and his own general relativity when they are applied closer to home. In short, Einstein found he could have his cake and eat it too: he could maintain all the appealing, experimentally confirmed features of general relativity while basking in the eternal serenity of an unchanging cosmos, one that was neither expanding nor contracting.
With this result, Einstein no doubt breathed a sigh of relief. How heart-wrenching it would have been if the decade of grueling research he had devoted to formulating general relativity resulted in a theory that was incompatible with the static universe apparent to anyone who gazed up at the night sky. But, as we have seen, a dozen years later the story took a sharp turn. In 1929, Hubble showed that cursory skyward gazes can be misleading. His systematic observations revealed that the universe is not static. It is expanding. Had Einstein trusted the original equations of general relativity, he would have predicted the expansion of the universe more than a decade before it was discovered observationally. That would certainly have ranked among the greatest discoveries—it might have been the greatest discovery—of all time. After learning of Hubble's results, Einstein rued the day he had thought of the cosmological constant, and he carefully erased it from the equations of general relativity. He wanted everyone to forget the whole sorry episode, and for many decades everyone did.
In the 1980s, however, the cosmological constant resurfaced in a surprising new form and ushered in one of the most dramatic upheavals in cosmological thinking since our species first engaged in cosmological thought.
Of Jumping Frogs and Supercooling
If you caught sight of a baseball flying upward, you could use Newton's law of gravity (or Einstein's more refined equations) to figure out its subsequent trajectory. And if you carried out the required calculations, you'd have a solid understanding of the ball's motion. But there would still be an unanswered question: Who or what threw the ball upward in the first place? How did the ball acquire the initial upward motion whose subsequent unfolding you've evaluated mathematically? In this example, a little further investigation is all it generally takes to find the answer (unless, of course, the aspiring big-leaguers realize that the ball just hit is on a collision course wi
th the windshield of a parked Mercedes). But a more difficult version of a similar question dogs general relativity's explanation of the expansion of the universe.
The equations of general relativity, as originally shown by Einstein, the Dutch physicist Willem de Sitter, and, subsequently, Friedmann and Lemaître, allow for an expanding universe. But, just as Newton's equations tell us nothing about how a ball's upward journey got started, Einstein's equations tell us nothing about how the expansion of the universe got started. For many years, cosmologists took the initial outward expansion of space as an unexplained given, and simply worked the equations forward from there. This is what I meant earlier when I said that the big bang is silent on the bang.
Such was the case until one fateful night in December 1979, when Alan Guth, a physics postdoctoral fellow working at the Stanford Linear Accelerator Center (he is now a professor at MIT), showed that we can do better. Much better. Although there are details that today, more than two decades later, have yet to be resolved fully, Guth made a discovery that finally filled the cosmological silence by providing the big bang with a bang, and one that was bigger than anyone expected.
Guth was not trained as a cosmologist. His specialty was particle physics, and in the late 1970s, together with Henry Tye from Cornell University, he was studying various aspects of Higgs fields in grand unified theories. Remember from the last chapter's discussion of spontaneous symmetry breaking that a Higgs field contributes the least possible energy it can to a region of space when its value settles down to a particular nonzero number (a number that depends on the detailed shape of its potential energy bowl). In the early universe, when the temperature was extraordinarily high, we discussed how the value of a Higgs field would wildly fluctuate from one number to another, like the frog in the hot metal bowl whose legs were being singed, but as the universe cooled, the Higgs would roll down the bowl to a value that would minimize its energy.
Guth and Tye studied reasons why the Higgs field might be delayed in reaching the least energetic configuration (the bowl's valley in Figure 9.1c). If we apply the frog analogy to the question Guth and Tye asked, it was this: what if the frog, in one of its earlier jumps when the bowl was starting to cool, just happened to land on the central plateau? And what if, as the bowl continued to cool, the frog hung out on the central plateau (leisurely eating worms), rather than sliding down to the bowl's valley? Or, in physics terms, what if a fluctuating Higgs field's value should land on the energy bowl's central plateau and remain there as the universe continues to cool? If this happens, physicists say that the Higgs field has supercooled, indicating that even though the temperature of the universe has dropped to the point where you'd expect the Higgs value to approach the low-energy valley, it remains trapped in a higher-energy configuration. (This is analogous to highly purified water, which can be supercooled below 0 degrees Celsius, the temperature at which you'd expect it to turn into ice, and yet remain liquid because the formation of ice requires small impurities around which the crystals can grow.)
Guth and Tye were interested in this possibility because their calculations suggested it might be relevant to a problem (the magnetic monopole problem 8 ) researchers had encountered with various attempts at grand unification. But Guth and Tye realized that there might be another implication and, in retrospect, that's why their work proved pivotal. They suspected that the energy associated with a supercooled Higgs field— remember, the height of the field represents its energy, so the field has zero energy only if its value lies in the bowl's valley—might have an effect on the expansion of the universe. In early December 1979, Guth followed up on this hunch, and here's what he found.
A Higgs field that has gotten caught on a plateau not only suffuses space with energy, but, of crucial importance, Guth realized that it also contributes a uniform negative pressure. In fact, he found that as far as energy and pressure are concerned, a Higgs field that's caught on a plateau has the same properties as a cosmological constant: it suffuses space with energy and negative pressure, and in exactly the same proportions as a cosmological constant. So Guth discovered that a supercooled Higgs field does have an important effect on the expansion of space: like a cosmological constant, it exerts a repulsive gravitational force that drives space to expand. 9
At this point, since you are already familiar with negative pressure and repulsive gravity, you may be thinking, All right, it's nice that Guth found a specific physical mechanism for realizing Einstein's idea of a cosmological constant, but so what? What's the big deal? The concept of a cosmological constant had long been abandoned. Its introduction into physics was nothing but an embarrassment for Einstein. Why get excited over rediscovering something that had been discredited more than six decades earlier?
Inflation
Well, here's why. Although a supercooled Higgs field shares certain features with a cosmological constant, Guth realized that they are not completely identical. Instead, there are two key differences—differences that make all the difference.
First, whereas a cosmological constant is constant—it does not vary with time, so it provides a constant, unchanging outward push—a supercooled Higgs field need not be constant. Think of a frog perched on the bump in Figure 10.1a. It may hang out there for a while, but sooner or later a random jump this way or that—a jump taken not because the bowl is hot (it no longer is), but merely because the frog gets restless— will propel the frog beyond the bump, after which it will slide down to the bowl's lowest point, as in Figure 10.1b. A Higgs field can behave similarly. Its value throughout all of space may get stuck on its energy bowl's central bump while the temperature drops too low to drive significant thermal agitation. But quantum processes will inject random jumps into the Higgs field's value, and a large enough jump will propel it off the plateau, allowing its energy and pressure to relax to zero. 10 Guth's calculations showed that, depending on the precise shape of the bowl's bump, this jump could have happened rapidly, perhaps in as short a time as .00000000000000000000000000000001 (10 -35 ) seconds. Subsequently, Andrei Linde, then working at the Lebedev Physical Institute in Moscow, and Paul Steinhardt, then working with his student Andreas Albrecht at the University of Pennsylvania, discovered a way for the Higgs field's relaxation to zero energy and pressure throughout all of space to happen even more efficiently and significantly more uniformly (thereby curing certain technical problems inherent to Guth's original proposal 11 ). They showed that if the potential energy bowl had been smoother and more gradually sloping, as in Figure 10.2, no quantum jumps would have been necessary: the Higgs field's value would quickly roll down to the valley, much like a ball rolling down a hill. The upshot is that if a Higgs field acted like a cosmological constant, it did so only for a brief moment.
Figure 10.1 ( a ) A supercooled Higgs field is one whose value gets trapped on the energy bowl's high-energy plateau, like the frog on a bump. ( b ) Typically, a supercooled Higgs field will quickly find its way off the plateau and drop to a value with lower energy, like the frog's jumping off the bump.
The second difference is that whereas Einstein carefully and arbitrarily chose the value of the cosmological constant—the amount of energy and negative pressure it contributed to each volume of space—so that its outward repulsive force would precisely balance the inward attractive force arising from the ordinary matter and radiation in the cosmos, Guth was able to estimate the energy and negative pressure contributed by the Higgs fields he and Tye had been studying. And the answer he found was more than 1000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000 (10 100 ) times larger than the value Einstein had chosen. This number is huge, obviously, and so the outward push supplied by the Higgs field's repulsive gravity is monumental compared with what Einstein envisioned originally with the cosmological constant.
Figure 10.2 A smoother and more gradually sloping bump allows the Higgs field value to roll down to the zero-energy valley more easily and more uniformly throughout space.
Now, if we combine these two observations—that the Higgs field will stay on the plateau, in the high-energy, negative-pressure state, only for the briefest of instants, and that while it is on the plateau, the repulsive outward push it generates is enormous—what do we have? Well, as Guth realized, we have a phenomenal, short-lived, outward burst. In other words, we have exactly what the big bang theory was missing: a bang, and a big one at that. That's why Guth's discovery is something to get excited about. 12
The cosmological picture emerging from Guth's breakthrough is thus the following. A long time ago, when the universe was enormously dense, its energy was carried by a Higgs field perched at a value far from the lowest point on its potential energy bowl. To distinguish this particular Higgs field from others (such as the electroweak Higgs field responsible for giving mass to the familiar particle species, or the Higgs field that arises in grand unified theories 13 ) it is usually called the inflaton field. 25 Because of its negative pressure, the inflaton field generated a gigantic gravitational repulsion that drove every region of space to rush away from every other; in Guth's language, the inflaton drove the universe to inflate. The repulsion lasted only about 10 -35 seconds, but it was so powerful that even in that brief moment the universe swelled by a huge factor. Depending on details such as the precise shape of the inflaton field's potential energy, the universe could easily have expanded by a factor of 10 30 , 10 50 , or 10 100 or more.