The Fabric of the Cosmos: Space, Time, and the Texture of Reality - Brian Greene (2004)
Part III. SPACETIME AND COSMOLOGY
Chapter 11. Quanta in the Sky with Diamonds
INFLATION, QUANTUM JITTERS, AND THE ARROW OF TIME
The discovery of the inflationary framework launched a new era in cosmological research, and in the decades since, many thousands of papers have been written on the subject. Scientists have explored just about every nook and cranny of the theory you could possibly imagine. While many of these works have focused on details of technical importance, others have gone further and shown how inflation not only solves specific cosmological problems beyond the reach of the standard big bang, but also provides powerful new approaches to a number of age-old questions. Of these, there are three developments—having to do with the formation of clumpy structures such as galaxies; the amount of energy required to spawn the universe we see; and (of prime importance to our story) the origin of time’s arrow—on which inflation has ushered in substantial and, some would say, spectacular progress.
Let’s take a look.
Inflationary cosmology’s solution to the horizon and flatness problems was its initial claim to fame, and rightly so. As we’ve seen, these were major accomplishments. But in the years since, many physicists have come to believe that another of inflation’s achievements shares the top spot on the list of the theory’s most important contributions.
The lauded insight concerns an issue that, to this point, I have encouraged you not to think about: How is it that there are galaxies, stars, planets, and other clumpy things in the universe? In the last three chapters, I asked you to focus on astronomically large scales—scales on which the universe appears homogeneous, scales so large that entire galaxies can be thought of as single H2O molecules, while the universe itself is the whole, uniform glass of water. But sooner or later cosmology has to come to grips with the fact that when you examine the cosmos on “finer” scales you discover clumpy structures such as galaxies. And here, once again, we are faced with a puzzle.
If the universe is indeed smooth, uniform, and homogeneous on large scales—features that are supported by observation and that lie at the heart of all cosmological analyses—where could the smaller-scale lumpiness have come from? The staunch believer in standard big bang cosmology can, once again, shrug off this question by appealing to highly favorable and mysteriously tuned conditions in the early universe: “Near the very beginning,” such a believer can say, “things were, by and large, smooth and uniform, but not perfectly uniform. How conditions got that way, I can’t say. That’s just how it was back then. Over time, this tiny lumpiness grew, since a lump has greater gravitational pull, being denser than its surroundings, and therefore grabs hold of more nearby material, growing larger still. Ultimately, the lumps got big enough to form stars and galaxies.” This would be a convincing story were it not for two deficiencies: the utter lack of an explanation for either the initial overall homogeneity or these important tiny nonuniformities. That’s where inflationary cosmology provides gratifying progress. We’ve already seen that inflation offers an explanation for the large-scale uniformity, and as we’ll now learn, the explanatory power of the theory goes even further. According to inflationary cosmology, the initial nonuniformity that ultimately resulted in the formation of stars and galaxies came from quantum mechanics.
This magnificent idea arises from an interplay between two seemingly disparate areas of physics: the inflationary expansion of space and the quantum uncertainty principle. The uncertainty principle tells us that there are always trade-offs in how sharply various complementary physical features in the cosmos can be determined. The most familiar example (see Chapter 4) involves matter: the more precisely the position of a particle is determined, the less precisely its velocity can be determined. But the uncertainty principle also applies to fields. By essentially the same reasoning we used in its application to particles, the uncertainty principle implies that the more precisely the value of a field is determined at one location in space, the less precisely its rate of change at that location can be determined. (The position of a particle and the rate of change of its position—its velocity—play analogous roles in quantum mechanics to the value of a field and the rate of change of the field value, at a given location in space.)
I like to summarize the uncertainty principle by saying, roughly speaking, that quantum mechanics makes things jittery and turbulent. If the velocity of a particle can’t be delineated with total precision, we also can’t delineate where the particle will be located even a fraction of a second later, since velocity now determines position then. In a sense, the particle is free to take on this or that velocity, or more precisely, to assume a mixture of many different velocities, and hence it will jitter frantically, haphazardly going this way and that. For fields, the situation is similar. If a field’s rate of change can’t be delineated with total precision, then we also can’t delineate what the value of the field will be, at any location, even a moment later. In a sense, the field will undulate up or down at this or that speed, or, more precisely, it will assume a strange mixture of many different rates of change, and hence its value will undergo a frenzied, fuzzy, random jitter.
In daily life we aren’t directly aware of the jitters, either for particles or fields, because they take place on subatomic scales. But that’s where inflation makes a big impact. The sudden burst of inflationary expansion stretched space by such an enormous factor that what initially inhabited the microscopic was drawn out to the macroscopic. As a key example, pioneers1 of inflationary cosmology realized that random differences between the quantum jitters in one spatial location and another would have generated slight inhomogeneities in the microscopic realm; because of the indiscriminate quantum agitation, the amount of energy in one location would have been a bit different from what it was in another. Then, through the subsequent inflationary swelling of space, these tiny variations would have been stretched to scales far larger than the quantum domain, yielding a small amount of lumpiness, much as tiny wiggles drawn on a balloon with a Magic Marker are stretched clear across the balloon’s surface when you blow it up. This, physicists believe, is the origin of the lumpiness that the staunch believer in the standard big bang model simply declares, without justification, to be “how it was back then.” Through the enormous stretching of inevitable quantum fluctuations, inflationary cosmology provides an explanation: inflationary expansion stretches tiny, inhomogeneous quantum jitters and smears them clear across the sky.
Over the few billion years following the end of the brief inflationary phase, these tiny lumps continued to grow through gravitational clumping. Just as in the standard big bang picture, lumps have slightly higher gravitational pull than their surroundings, so they draw in nearby material, growing larger still. In time, the lumps grew large enough to yield the matter making up galaxies and the stars that inhabit them. Certainly, there are numerous steps of detail in going from a little lump to a galaxy, and many still need elucidation. But the overall framework is clear: in a quantum world, nothing is ever perfectly uniform because of the jitteriness inherent to the uncertainty principle. And, in a quantum world that experienced inflationary expansion, such nonuniformity can be stretched from the microworld to far larger scales, providing the seeds for the formation of large astrophysical bodies like galaxies.
That’s the basic idea, so feel free to skip over the next paragraph. But for those who are interested, I’d like to make the discussion a bit more precise. Recall that inflationary expansion came to an end when the inflaton field’s value slid down its potential energy bowl and the field relinquished all its pent-up energy and negative pressure. We described this as happening uniformly throughout space—the inflaton value here, there, and everywhere experienced the same evolution—as that’s what naturally emerges from the governing equations. However, this is strictly true only if we ignore the effects of quantum mechanics. On average, the inflaton field value did indeed slide down the bowl, as we expect from thinking about a simple classical object like a marble rolling down an incline. But just as a frog sliding down the bowl is likely to jump and jiggle along the way, quantum mechanics tells us that the inflaton field experienced quivers and jitters. On its way down, the value may have suddenly jumped up a little bit over there or jiggled down a little bit over there. And because of this jittering, the inflaton reached the value of lowest energy at different places at slightly different moments. In turn, inflationary expansion shut off at slightly different times at different locations in space, so that the amount of spatial expansion at different locations varied slightly, giving rise to inhomogeneities—wrinkles—similar to the kind you see when the pizza maker stretches the dough a bit more in one place than another and creates a little bump. Now the normal intuition is that jitters arising from quantum mechanics would be too small to be relevant on astrophysical scales. But with inflation, space expanded at such a colossal rate, doubling in size every 10−37 seconds, that even a slightly different duration of inflation at nearby locations resulted in a significant wrinkle. In fact, calculations undertaken in specific realizations of inflation have shown that the inhomogeneities produced in this way have a tendency to be too large; researchers often have to adjust details in a given inflationary model (the precise shape of the inflaton field’s potential energy bowl) to ensure that the quantum jitters don’t predict a universe that’s too lumpy. And so inflationary cosmology supplies a ready-made mechanism for understanding how the small-scale nonuniformity responsible for lumpy structures like stars and galaxies emerged in a universe that on the largest of scales appears thoroughly homogeneous.
According to inflation, the more than 100 billion galaxies, sparkling throughout space like heavenly diamonds, are nothing but quantum mechanics writ large across the sky. To me, this realization is one of the greatest wonders of the modern scientific age.
The Golden Age of Cosmology
Dramatic evidence supporting these ideas comes from meticulous satellite-based observations of the microwave background radiation’s temperature. I have emphasized a number of times that the temperature of the radiation in one part of the sky agrees with that in another to high accuracy. But what I have yet to mention is that by the fourth digit after the decimal place, the temperatures in different locations do differ. Precision measurements, first accomplished in 1992 by COBE (the Cosmic Background Explorer satellite) and more recently by WMAP (the Wilkinson Microwave Anisotropy Probe), have determined that while the temperature might be 2.7249 Kelvin in one spot in space, it might be 2.7250 Kelvin in another, and 2.7251 Kelvin in still another.
The wonderful thing is that these extraordinarily small temperature variations follow a pattern on the sky that can be explained by attributing them to the same mechanism that has been suggested for seeding galaxy formation: quantum fluctuations stretched out by inflation. The rough idea is that when tiny quantum jitters are smeared across space, they make it slightly hotter in one region and slightly cooler in another (photons received from a slightly denser region expend more energy overcoming the slightly stronger gravitational field, and hence their energy and temperature are slightly lower than those of photons received from a less dense
Figure 11.1 (a) Inflationary cosmology’s prediction for temperature variations of the microwave background radiation from one point to another on the sky. (b) Comparison of those predictions with satellite-based observations.
region). Physicists have carried out precise calculations based on this proposal, and generated predictions for how the microwave radiation’s temperature should vary from place to place across the sky, as illustrated in Figure 11.1a. (The details are not essential, but the horizontal axis is related to the angular separation of two points on the sky, and the vertical axis is related to their temperature difference.) In Figure 11.1b, these predictions are compared with satellite observations, represented by little diamonds, and as you can see there is extraordinary agreement.
I hope you’re blown away by this concordance of theory and observation, because if not it means I’ve failed to convey the full wonder of the result. So, just in case, let me reemphasize what’s going on here: satellite-borne telescopes have recently measured the temperature of microwave photons that have been traveling toward us, unimpeded, for nearly 14 billion years. They’ve found that photons arriving from different directions in space have nearly identical temperatures, differing by no more than a few ten-thousandths of a degree. Moreover, the observations have shown that these tiny temperature differences fill out a particular pattern on the sky, demonstrated by the orderly progression of diamonds in Figure 11.1b. And marvel of marvels, calculations done today, using the inflationary framework, are able to explain the pattern of these minuscule temperature variations—variations set down nearly 14 billion years ago—and, to top it off, the key to this explanation involves jitters arising from quantum uncertainty. Wow.
This success has convinced many physicists of the inflationary theory’s validity. What is of equal importance, these and other precision astronomical measurements, which have only recently become possible, have allowed cosmology to mature from a field based on speculation and conjecture to one firmly grounded in observation—a coming of age that has inspired many in the field to call our era the golden age of cosmology.
Creating a Universe
With such progress, physicists have been motivated to see how much further inflationary cosmology can go. Can it, for example, resolve the ultimate mystery, encapsulated in Leibniz’s question of why there is a universe at all? Well, at least with our current level of understanding, that’s asking for too much. Even if a cosmological theory were to make headway on this question, we could ask why that particular theory—its assumptions, ingredients, and equations—was relevant, thus merely pushing the question of origin one step further back. If logic alone somehow required the universe to exist and to be governed by a unique set of laws with unique ingredients, then perhaps we’d have a convincing story. But, to date, that’s nothing but a pipe dream.
A related but somewhat less ambitious question, one that has also been asked in various guises through the ages, is: Where did all the mass/energy making up the universe come from? Here, although inflationary cosmology does not provide a complete answer, it has cast the question in an intriguing new light.
To understand how, think of a huge but flexible box filled with many thousands of swarming children, incessantly running and jumping. Imagine that the box is completely impermeable, so no heat or energy can escape, but because it’s flexible, its walls can move outward. As the children relentlessly slam into each of the box’s walls—hundreds at a time, with hundreds more immediately to follow—the box steadily expands. Now, you might expect that because the walls are impermeable, the total energy embodied by the swarming children will stay fully within the expanding box. After all, where else could their energy go? Well, although a reasonable proposition, it’s not quite right. There is some place for it to go. The children expend energy every time they slam into a wall, and much of this energy is transferred to the wall’s motion. The very expansion of the box absorbs, and hence depletes, the children’s energy.
Even though space doesn’t have walls, a similar kind of energy transfer takes place as the universe expands. Just as the fast-moving children work against the inward force exerted by the box’s walls as it expands, the fast-moving particles in our universe work against an inward force as space expands: They work against the inward force of gravity. And just as the total energy embodied by the children drops because it’s continuously transferred to the energy of the walls as the box expands, the total energy carried by ordinary particles of matter and radiation drops becasue it is continually transferred to gravity as the universe expands. In short, by drawing an analogy between the inward force exerted by the box’s walls and the inward force exerted by gravity (an analogy that can be established mathematically), we conclude that gravity depletes the energy in fast-moving particles of matter and radiation as space swells. The loss of energy from fast-moving particles from cosmic expansion has been confirmed by observations of the microwave background radiation.26
Let’s now modify our analogy a bit to gain insight into how an inflaton field impacts our description of energy exchange as space expands. Imagine that a few pranksters among the children hook up a number of enormous rubber bands between each of the opposite, outward-moving walls of the box. The rubber bands exert an inward, negative pressure on the box walls, which has exactly the opposite effect of the children’s outward, positive pressure; rather than transferring energy to the expansion of the box, the rubber bands’ negative pressure “saps” energy from the expansion. As the box expands, the rubber bands get increasingly taut, which means they embody increasing amounts of energy.
This modified scenario is relevant to cosmology because, as we’ve learned, like the pranksters’ rubber bands, a uniform inflaton field exerts a negative pressure within an exanding universe. And so, just as the total energy embodied by the rubber bands increases as the box expands because they extract energy from the box’s walls, the total energy embodied by the inflaton field increases as the universe expands because it extracts energy from gravity.27
To summarize: as the universe expands, matter and radiation lose energy to gravity while an inflaton field gains energy from gravity.
The pivotal nature of these observations becomes clear when we try to explain the origin of the matter and radiation that make up galaxies, stars, and everything else inhabiting the cosmos. In the standard big bang theory, the mass/energy carried by matter and radiation has steadily decreased as the universe has expanded, and so the mass/energy in the early universe greatly exceeded what we see today. Thus, instead of offering an explanation for where all the mass/energy currently inhabiting the universe originated, the standard big bang fights an unending uphill battle: the farther back the theory looks, the more mass/energy it must somehow explain.
In inflationary cosmology, though, much the opposite is true. Recall that the inflationary theory argues that matter and radiation were produced at the end of the inflationary phase as the inflaton field released its pent-up energy by rolling from perch to valley in its potential-energy bowl. The relevant question, therefore, is whether, just as the inflationary phase was drawing to a close, the theory can account for the inflaton field embodying the stupendousquantity of mass/energy necessary to yield the matter and radiation in today’s universe.
The answer to this question is that inflation can, without even breaking a sweat. As just explained, the inflaton field is a gravitational parasite—it feeds on gravity—and so the total energy the inflaton field carried increased as space expanded. More precisely, the mathematical analysis shows that the energy density of the inflaton field remained constant throughout the inflationary phase of rapid expansion, implying that the total energy it embodied grew in direct proportion to the volume of the space it filled. In the previous chapter, we saw that the size of the universe increased by at least a factor of 1030 during inflation, which means the volume of the universe increased by a factor of at least (10 30)3 = 1090. Consequently, the energy embodied in the inflaton field increased by the same huge factor: as the inflationary phase drew to a close, a mere 10−35 or so seconds after it began, the energy in the inflaton field grew by a factor on the order of 1090, if not more. This means that at the onset of inflation, the inflaton field didn’t need to have much energy, since the enormous expansionit was about to spawn would enormously amplify the energy it carried. A simple calculation shows that a tiny nugget, on the order of 10 −26 centimeters across, filled with a uniform inflaton field—and weighing a mere twenty pounds—would, through the ensuing inflationary expansion, acquire enough energy to account for all we see in the universe today.2
Thus, in stark contrast to the standard big bang theory in which the total mass/energy of the early universe was huge beyond words, inflationary cosmology, by “mining” gravity, can produce all the ordinary matter and radiation in the universe from a tiny, twenty-pound speck of inflatonfilled space. By no means does this answer Leibniz’s question of why there is something rather than nothing, since we’ve yet to explain why there is an inflaton or even the space it occupies. But the something in need of explanation weighs a whole lot less than my dog Rocky, and that’s certainly a very different starting point than envisaged in the standard big bang.28
Inflation, Smoothness, and the Arrow of Time
Perhaps my enthusiasm has already betrayed my bias, but of all the progress that science has achieved in our age, advances in cosmology fill me with the greatest awe and humility. I seem never to have lost the rush I initially felt years ago when I first read up on the basics of general relativity and realized that from our tiny little corner of spacetime we can apply Einstein’s theory to learn about the evolution of the entire cosmos. Now, a few decades later, technological progress is subjecting these once abstract proposals for how the universe behaved in its earliest moments to observational tests, and the theories really work.
Recall, though, that besides cosmology’s overall relevance to the story of space and time, Chapters 6 and 7 launched us into a study of the universe’s early history with a specific goal: to find the origin of time’s arrow. Remember from those chapters that the only convincing framework we found for explaining time’s arrow was that the early universe had extremely high order, that is, extremely low entropy, which set the stage for a future in which entropy got ever larger. Just as the pages of War and Peace wouldn’t have had the capacity to get increasingly jumbled if they had not been nice and ordered at some point, so too the universe wouldn’t have had the capacity to get increasingly disordered—milk spilling, eggs breaking, people aging—unless it had been in a highly ordered configuration early on. The puzzle we encountered is to explain how this high-order, low-entropy starting point came to be.
Inflationary cosmology offers substantial progress, but let me first remind you more precisely of the puzzle, in case any of the relevant details have slipped your mind.
There is strong evidence and little doubt that, early in the history of the universe, matter was spread uniformly throughout space. Ordinarily, this would be characterized as a high-entropy configuration—like the carbon dioxide molecules from a bottle of Coke being spread uniformly throughout a room—and hence would be so commonplace that it would hardly require an explanation. But when gravity matters, as it does when considering the entire universe, a uniform distribution of matter is a rare, low-entropy, highly ordered configuration, because gravity drives matter to form clumps. Similarly, a smooth and uniform spatial curvature also has very low entropy; it is highly ordered compared with a wildly bumpy, nonuniform spatial curvature. (Just as there are many ways for the pages of War and Peace to be disordered but only one way for them to be ordered, so there are many ways for space to have a disordered, nonuniform shape, but very few ways in which it can be fully ordered, smooth, and uniform.) So we are left to puzzle: Why did the early universe have a low-entropy (highly ordered) uniform distribution of matter instead of a high-entropy (highly disordered) clumpy distribution of matter such as a diverse population of black holes? And why was the curvature of space smooth, ordered, and uniform to extremely high accuracy rather than being riddled with a variety of huge warps and severe curves, also like those generated by black holes?
As first discussed in detail by Paul Davies and Don Page,3 inflationary cosmology gives important insight into these issues. To see how, bear in mind that an essential assumption of the puzzle is that once a clump forms here or there, its greater gravitational pull attracts yet more material, causing it to grow larger; correspondingly, once a wrinkle in space forms here or there, its greater gravitational pull tends to make the wrinkle yet more severe, leading to a bumpy, highly nonuniform spatial curvature. When gravity matters, ordinary, unremarkable, high-entropy configurations are lumpy and bumpy.
But note the following: this reasoning relies completely on the attractivenature of ordinary gravity. Lumps and bumps grow because they pull strongly on nearby material, coaxing such material to join the lump. During the brief inflationary phase, though, gravity was repulsive and that changed everything. Take the shape of space. The enormous outward push of repulsive gravity drove space to swell so swiftly that initial bumps and warps were stretched smooth, much as fully inflating a shriveled balloon stretches out its creased surface.29 What’s more, since the volume of space increased by a colossal factor during the brief inflationary period, the density of any clumps of matter was completely diluted, much as the density of fish in your aquarium would be diluted if the tank’s volume suddenly increased to that of an Olympic swimming pool. Thus, although attractive gravity causes clumps of matter and creases of space to grow, repulsive gravity does the opposite: it causes them to diminish, leading to an ever smoother, ever more uniform outcome.
Thus, by the end of the inflationary burst, the size of the universe had grown fantastically, any nonuniformity in the curvature of space had been stretched away, and any initial clumps of anything at all had been diluted to the point of irrelevance. Moreover, as the inflaton field slid down to the bottom of its potential-energy bowl, bringing the burst of inflationary expansion to a close, it converted its pent-up energy into a nearly uniform bath of particles of ordinary matter throughout space (uniform up to the tiny but critical inhomogeneities coming from quantum jitters). In total, this all sounds like great progress. The outcome we’ve reached via inflation —a smooth, uniform spatial expansion populated by a nearly uniform distribution of matter—was exactly what we were trying to explain. It’s exactly the low-entropy configuration that we need to explain time’s arrow.
Entropy and Inflation
Indeed, this is significant progress. But two important issues remain.
First, we seem to be concluding that the inflationary burst smooths things out and hence lowers total entropy, embodying a physical mechanism—not just a statistical fluke—that appears to violate the second law of thermodynamics. Were that the case, either our understanding of the second law or our current reasoning would have to be in error. In actuality, though, we don’t have to face either of these options, because total entropy does not go down as a result of inflation. What really happens during the inflationary burst is that the total entropy goes up, but it goes up much less than it might have. You see, by the end of the inflationary phase, space was stretched smooth and so the gravitational contribution to entropy—the entropy associated with the possible bumpy, nonordered, nonuniform shape of space—was minimal. However, when the inflaton field slid down its bowl and relinquished its pent-up energy, it is estimated to have produced about 1080 particles of matter and radiation. Such a huge number of particles, like a book with a huge number of pages, embodies a huge amount of entropy. Thus, even though the gravitational entropy went down, the increase in entropy from the production of all these particles more than compensated. The total entropy increased, just as we expect from the second law.
But, and this is the important point, the inflationary burst, by smoothing out space and ensuring a homogeneous, uniform, low-entropy gravitational field, created a huge gap between what the entropy contribution from gravity was and what it might have been. Overall entropy increased during inflation, but by a paltry amount compared with how much it could have increased. It’s in this sense that inflation generated a low-entropy universe: by the end of inflation, entropy had increased, but by nowhere near the factor by which the spatial expanse had increased. If entropy is likened to property taxes, it would be as if New York City acquired the Sahara Desert. The total property taxes collected would go up, but by a tiny amount compared with the total increase in acreage.
Ever since the end of inflation, gravity has been trying to make up the entropy difference. Every clump—be it a galaxy, or a star in a galaxy, or a planet, or a black hole—that gravity has subsequently coaxed out of the uniformity (seeded by the tiny nonuniformity from quantum jitters) has increased entropy and has brought gravity one step closer to realizing its entropy potential. In this sense, then, inflation is a mechanism that yielded a large universe with relatively low gravitational entropy, and in that way set the stage for the subsequent billions of years of gravitational clumping whose effects we now witness. And so inflationary cosmology gives a direction to time’s arrow by generating a past with exceedingly low gravitational entropy; the future is the direction in which this entropy grows.4
The second issue becomes apparent when we continue down the path to which time’s arrow led us in Chapter 6. From an egg, to the chicken that laid it, to the chicken’s feed, to the plant kingdom, to the sun’s heat and light, to the big bang’s uniformly distributed primordial gas, we followed the universe’s evolution into a past that had ever greater order, at each stage pushing the puzzle of low entropy one step further back in time. We have just now realized that an even earlier stage of inflationary expansion can naturally explain the smooth and uniform aftermath of the bang. But what about inflation itself? Can we explain the initial link in this chain we’ve followed? Can we explain why conditions were right for an inflationary burst to happen at all?
This is an issue of paramount importance. No matter how many puzzles inflationary cosmology resolves in theory, if an era of inflationary expansion never took place, the approach will be rendered irrelevant. Moreover, since we can’t go back to the early universe and determine directly whether inflation occurred, assessing whether we’ve made real progress in setting a direction to time’s arrow requires that we determine the likelihood that the conditions necessary for an inflationary burst were achieved. That is, physicists bristle at the standard big bang’s reliance on finely tuned homogeneous initial conditions that, while observationally motivated, are theoretically unexplained. It feels deeply unsatisfying for the low-entropy state of the early universe simply to be assumed; it feels hollow for time’s arrow to be imposed on the universe, without explanation. At first blush, inflation offers progress by showing that what’s assumed in the standard big bang emerges from inflationary evolution. But if the initiation of inflation requires yet other, highly special, exceedingly low-entropy conditions, we will pretty much find ourselves back at square one. We will merely have traded the big bang’s special conditions for those necessary to ignite inflation, and the puzzle of time’s arrow will remain just as puzzling.
What are the conditions necessary for inflation? We’ve seen that inflation is the inevitable result of the inflaton field’s value getting stuck, for just a moment and within just a tiny region, on the high-energy plateau in its potential energy bowl. Our charge, therefore, is to determine how likely this starting configuration for inflation actually is. If initiating inflation proves easy, we’ll be in great shape. But if the necessary conditions are extraordinarily unlikely to be attained, we will merely have shifted the question of time’s arrow one step further back—to finding the explanation for the low-entropy inflaton field configuration that got the ball rolling.
I’ll first describe current thinking on this issue in the most optimistic light, and then return to essential elements of the story that remain cloudy.
As mentioned in the previous chapter, the inflationary burst is best thought of as an event occurring in a preexisting universe, rather than being thought of as the creation of the universe itself. Although we don’t have an unassailable understanding of what the universe was like during such a preinflationary era, let’s see how far we can get if we assume that things were in a thoroughly ordinary, high-entropy state. Specifically, let’s imagine that primordial, preinflationary space was riddled with warps and bumps, and that the inflaton field was also highly disordered, its value jumping to and fro like the frog in the hot metal bowl.
Now, just as you can expect that if you patiently play a fair slot machine, sooner or later the randomly spinning dials will land on triple diamonds, we expect that sooner or later a chance fluctuation within this highly energetic, turbulent arena of the primordial universe will cause the inflaton field’s value to jump to the correct, uniform value in some small nugget of space, initiating an outward burst of inflationary expansion. As explained in the previous section, calculations show that the nugget of space need only have been tiny—on the order of 10−26 centimeters across—for the ensuing cosmological expansion (inflationary expansion followed by standard big bang expansion) to have stretched it larger than the universe we see today. Thus, rather than assuming or simply declaring that conditions in the early universe were right for inflationary expansion to take place, in this way of thinking about things an ultramicroscopic fluctuation weighing a mere twenty pounds, occurring within an ordinary, unremarkable environment of disorder, gave rise to the necessary conditions.
What’s more, just as the slot machine will also generate a wide variety of nonwinning results, in other regions of primordial space other kinds of inflaton fluctuations would also have happened. In most, the fluctuation wouldn’t have had the right value or have been sufficiently uniform for inflationary expansion to occur. (Even in a region that’s a mere 10−26 centimeters across, a field’s value can vary wildly.) But all that matters to us is that there was one nugget that yielded the space-smoothing inflationary burst that provided the first link in the low-entropy chain, ultimately leading to our familiar cosmos. As we see only our one big universe, we only need the cosmic slot machine to pay out once.5
Since we are tracing the universe back to a statistical fluctuation from primordial chaos, this explanation for time’s arrow shares certain features with Boltzmann’s original proposal. Remember from Chapter 6 that Boltzmann suggested that everything we now see arose as a rare but every so often expectable fluctuation from total disorder. The problem with Boltzmann’s original formulation, though, was that it could not explain why the chance fluctuation had gone so far overboard and produced a universe hugely more ordered than it would need to be even to support life as we know it. Why is the universe so vast, having billions and billions of galaxies, each with billions and billions of stars, when it could have drastically cut corners by having, say, just a few galaxies, or even only one?
From the statistical point of view, a more modest fluctuation that produced some order but not as much as we currently see would be far more likely. Moreover, since on average entropy is on the rise, Boltzmann’s reasoning suggests that it would be much more likely that everything we see today just now arose as a rare statistical jump to lower entropy. Recall the reason: the farther back the fluctuation happened, the lower the entropy it would have had to attain (entropy starts to rise after any dip to low entropy, as in Figure 6.4, so if the fluctuation happened yesterday, it must have dipped down to yesterday’s lower entropy, and if it happened a billion years ago, it must have dipped down to that era’s even lower entropy). Hence, the farther back in time, the more drastic and improbable the required fluctuation. Thus, it is much more likely that the jump just happened. But if we accept this conclusion, we can’t trust memories, records, or even the laws of physics that underlie the discussion itself—a completely intolerable position.
The tremendous advantage of the inflationary incarnation of Boltzmann’s idea is that a small fluctuation early on—a modest jump to the favorable conditions, within a tiny nugget of space—inevitably yields the huge and ordered universe we are aware of. Once inflationary expansion set in, the little nugget was inexorably stretched to scales at least as large as the universe we currently see, and hence there is no mystery as to why the universe didn’t cut corners; there is no mystery why the universe is vast and is populated by a huge number of galaxies. From the get-go, inflation gave the universe an amazing deal. A jump to lower entropy within a tiny nugget of space was leveraged by inflationary expansion into the vast reaches of the cosmos. And, of utmost importance, the inflationary stretching didn’t just yield any old large universe. It yielded our large universe—inflation explains the shape of space, it explains the large-scale uniformity, and it even explains the “smaller”-scale inhomogeneities such as galaxies and temperature variations in the background radiation. Inflation packages a wealth of explanatory and predictive power in a single fluctuation to low entropy.
And so Boltzmann may well have been right. Everything we see may have resulted from a chance fluctuation out of a highly disordered state of primeval chaos. In this realization of his ideas, though, we can trust our records and we can trust our memories: the fluctuation did not happen just now. The past really happened. Our records are records of things that took place. Inflationary expansion amplifies a tiny speck of order in the early universe—it “wound up” the universe to a huge expanse with minimal gravitational entropy—so the 14 billion years of subsequent unwinding, of subsequent clumping into galaxies, stars, and planets, presents no puzzle.
In fact, this approach even tells us a bit more. Just as it’s possible to hit the jackpot on a number of slot machines on the floor of the Bellagio, in the primordial state of high entropy and overall chaos there was no reason why the conditions necessary for inflationary expansion would arise only in a single spatial nugget. Instead, as Andrei Linde has proposed, there could have been many nuggets scattered here and there that underwent space-smoothing inflationary expansion. If that were so, our universe would be but one among many that sprouted—and perhaps continue to sprout—when chance fluctuations made the conditions right for an inflationary burst, as illustrated in Figure 11.2. As these other universes would likely be forever separate from ours, it’s hard to imagine how we would ever establish whether this “multiverse” picture is true. However, as a conceptual framework, it’s both rich and tantalizing. Among other things, it suggests a possible shift in how we think about cosmology: In Chapter 10, I described inflation as a “front end” to the standard big bang theory, in which the bang is identified with a fleeting burst of rapid expansion. But if we think of the inflationary sprouting of each new universe in Figure 11.2 as its own bang, then inflation itself is best viewed as the overarching cosmological framework within which big bang–like evolutions happen, bubble by bubble. Thus, rather than inflation’s being incorporated into the standard big bang theory, in this approach the standard big bang would be incorporated into inflation.
Figure 11.2 Inflation can occur repeatedly, sprouting new universes from older ones.
Inflation and the Egg
So why do you see an egg splatter but not unsplatter? Where does the arrow of time that we all experience come from? Here is where this approach has taken us. Through a chance but every so often expectable fluctuation from an unremarkable primordial state with high entropy, a tiny, twenty-pound nugget of space achieved conditions that led to a brief burst of inflationary expansion. The tremendous outward swelling resulted in space’s being stretched enormously large and extremely smooth, and, as the burst drew to a close, the inflaton field relinquished its hugely amplified energy by filling space nearly uniformly with matter and radiation. As the inflaton’s repulsive gravity diminished, ordinary attractive gravity became dominant. And, as we’ve seen, attractive gravity exploits tiny inhomogeneities caused by quantum jitters to cause matter to clump, forming galaxies and stars and ultimately leading to the formation of the sun, the earth, the rest of the solar system, and the other features of our observed universe. (As discussed, some 7 billion or so years ATB, repulsive gravity once again became dominant, but this is only relevant on the largest of cosmic scales and has no direct impact on smaller entities like individual galaxies or our solar system, where ordinary attractive gravity still reigns.) The sun’s relatively low-entropy energy was used by low-entropy plant and animal life forms on earth to produce yet more low-entropy life forms, slowly raising the total entropy through heat and waste. Ultimately, this chain produced a chicken that produced an egg— and you know the rest of the story: the egg rolled off your kitchen counter and splattered on the floor as part of the universe’s relentless drive to higher entropy. It’s the low-entropy, highly ordered, uniformly smooth nature of the spatial fabric produced by inflationary stretching that is the analog of having the pages of War and Peace all in their proper numerical arrangement; it is this early state of order—the absence of severe bumps or warps or gargantuan black holes—that primed the universe for the subsequent evolution to higher entropy and hence provided the arrow of time we all experience. With our current level of understanding, this is the most complete explanation for time’s arrow that has been given.
The Fly in the Ointment?
To me, this story of inflationary cosmology and time’s arrow is lovely. From a wild and energetic realm of primordial chaos, there emerged an ultramicroscopic fluctuation of uniform inflaton field weighing far less than the limit for carry-on luggage. This initiated inflationary expansion, which set a direction to time’s arrow, and the rest is history.
But in telling this story, we’ve made a pivotal assumption that’s as yet unjustified. To assess the likelihood of inflation’s being initiated, we’ve had to specify the characteristics of the preinflationary realm out of which inflationary expansion is supposed to have emerged. The particular realm we’ve envisioned—wild, chaotic, energetic—sounds reasonable, but delineating this intuitive description with mathematical precision proves challenging. Moreover, it is only a guess. The bottom line is that we don’t know what conditions were like in the supposed preinflationary realm, in the fuzzy patch of Figure 10.3, and without that information we are unable to make a convincing assessment of the likelihood of inflation’s initiating; any calculation of the likelihood depends sensitively on the assumptions we make.6
With this hole in our understanding, the most sensible summary is that inflation offers a powerful explanatory framework that bundles together seemingly disparate problems—the horizon problem, the flatness problem, the origin-of-structure problem, the low-entropy-of-the-early-universe problem—and offers a single solution that addresses them all. This feels right. But to go to the next step, we need a theory that can cope with the extreme conditions characteristic of the fuzzy patch— extremes of heat and colossal density—so that we will stand a chance of gaining sharp, unambiguous insight into the earliest moments of the cosmos.
As we will learn in the next chapter, this requires a theory that can overcome perhaps the greatest obstacle theoretical physics has faced during the last eighty years: a fundamental rift between general relativity and quantum mechanics. Many researchers believe that a relatively new approach called superstring theory may have accomplished this, but if superstring theory is right, the fabric of the cosmos is far stranger than almost anyone ever imagined.