The Fabric of the Cosmos: Space, Time, and the Texture of Reality

by Brian Greene

  The insights of Linde and of Albrecht and Steinhardt—now called new inflation— fixed these vexing problems. By changing the shape of the potential energy bowl to that in Figure 10.2, these researchers realized, the inflaton could relax to its zero energy value by "rolling" down the energy hill to the valley, a gradual and graceful process that had no need for the quantum jump of the original proposal. And, as their calculations showed, this somewhat more gradual rolling down the hill sufficiently prolonged the inflationary burst of space so that one single bubble easily grew large enough to encompass the entire observable universe. Thus, in this approach, there is no need to worry about coalescing bubbles. What was of equal importance, rather than converting the inflaton field's energy to that of ordinary particles and radiation through bubble collisions, in the new approach the inflaton gradually accomplished this energy conversion uniformly throughout space by a process akin to friction: as the field rolled down the energy hill—uniformly throughout space—it gave up its energy by "rubbing against" (interacting with) more familiar fields for particles and radiation. New inflation thus retained all the successes of Guth's approach, but patched up the significant problem it had encountered.

  About a year after the important progress offered by new inflation, Andrei Linde had another breakthrough. For new inflation to occur successfully, a number of key elements must all fall into place: the potential energy bowl must have the right shape, the inflaton field's value must begin high up on the bowl (and, somewhat more technically, the inflaton field's value must itself be uniform over a sufficiently large spatial expanse). While it's possible for the universe to achieve such conditions, Linde found a way to generate an inflationary burst in a simpler, far less contrived setting. Linde realized that even with a simple potential energy bowl, such as that in Figure 9.1a, and even without finely arranging the inflaton field's initial value, inflation could still naturally take place. The idea is this. Imagine that in the very early universe, things were "chaotic"—for example, imagine that there was an inflaton field whose value randomly bounced around from one number to another. At some locations in space its value might have been small, at other locations its value might have been medium, and at yet other locations in space its value might have been high. Now, nothing particularly noteworthy would have happened in regions where the field value was small or medium. But Linde realized that something fantastically interesting would have taken place in regions where the inflaton field happened to have attained a high value (even if the region were tiny, a mere 10 -33 centimeters across). When the inflaton field's value is high—when it is high up on the energy bowl in Figure 9.1a— a kind of cosmic friction sets in: the field's value tries to roll down the hill to lower potential energy, but its high value contributes to a resistive drag force, and so it rolls very slowly. Thus, the inflaton field's value would have been nearly constant and (much like an inflaton on the top of the potential energy hill in new inflation) would have contributed a nearly constant energy and a nearly constant negative pressure. As we are now very familiar, these are the conditions required to drive a burst of inflationary expansion. Thus, without invoking a particularly special potential energy bowl, and without setting up the inflaton field in a special configuration, the chaotic environment of the early universe could have naturally given rise to inflationary expansion. Not surprisingly, Linde had called this approach chaotic inflation. Many physicists consider it the most convincing realization of the inflationary paradigm.

  12. Those familiar with the history of this subject will realize that the excitement over Guth's discovery was generated by its solutions to key cosmological problems, such as the horizon and flatness problems, as we describe shortly.

  13. You might wonder whether the electroweak Higgs field, or the grand unified Higgs field, can do double duty—playing the role we described in Chapter 9, while also driving inflationary expansion at earlier times, before forming a Higgs ocean. Models of this sort have been proposed, but they typically suffer from technical problems. The most convincing realizations of inflationary expansion invoke a new Higgs field to play the role of the inflaton.

  14. See note 11, this chapter.

  15. For example, you can think of our horizon as a giant, imaginary sphere, with us at its center, that separates those things with which we could have communicated (the things within the sphere) from those things with which we couldn't have communicated (those things beyond the sphere), in the time since the bang. Today, the radius of our "horizon sphere" is roughly 14 billion light-years; early on in the history of the universe, its radius was much less, since there had been less time for light to travel. See also note 10 from Chapter 8.

  16. While this is the essence of how inflationary cosmology solves the horizon problem, to avoid confusion let me highlight a key element of the solution. If one night you and a friend are standing on a large field happily exchanging light signals by turning flashlights on and off, notice that no matter how fast you then turn and run from each other, you will always be able subsequently to exchange light signals. Why? Well, to avoid receiving the light your friend shines your way, or for your friend to avoid receiving the light you send her way, you'd need to run from each other at faster than light speed, and that's impossible. So, how is it possible for regions of space that were able to exchange light signals early on in the universe's history (and hence come to the same temperature, for example) to now find themselves beyond each other's communicative range? As the flashlight example makes clear, it must be that they've rushed apart at faster than the speed of light. And, indeed, the colossal outward push of repulsive gravity during the inflationary phase did drive every region of space away from every other at much faster than the speed of light. Again, this offers no contradiction with special relativity, since the speed limit set by light refers to motion through space, not motion from the swelling of space itself. So a novel and important feature of inflationary cosmology is that it involves a short period in which there is superluminal expansion of space.

  17. Note that the numerical value of the critical density decreases as the universe expands. But the point is that if the actual mass/energy density of the universe is equal to the critical density at one time, it will decrease in exactly the same way and maintain equality with the critical density at all times.

  18. The mathematically inclined reader should note that during the inflationary phase, the size of our cosmic horizon stayed fixed while space swelled enormously (as can easily be seen by taking an exponential form for the scale factor in note 10 of Chapter 8). That is the sense in which our observable universe is a tiny speck in a gigantic cosmos, in the inflationary framework.

  19. R. Preston, First Light (New York: Random House Trade Paperbacks, 1996), p. 118.

  20. For an excellent general-level account of dark matter, see L. Krauss, Quintes sence:The Mystery of Missing Mass in the Universe (New York: Basic Books, 2000).

  21. The expert reader will recognize that I am not distinguishing between the various dark matter problems that emerge on different scales of observation (galactic, cosmic) as the contribution of dark matter to the cosmic mass density is my only concern here.

  22. There is actually some controversy as to whether this is the mechanism behind all type Ia supernovae (I thank D. Spergel for pointing this out to me), but the uniformity of these events—which is what we need for the discussion—is on a strong observational footing.

  23. It's interesting to note that, years before the supernova results, prescient theoretical works by Jim Peebles at Princeton, and also by Lawrence Krauss of Case Western and Michael Turner of the University of Chicago, and Gary Steigman of Ohio State, had suggested that the universe might have a small nonzero cosmological constant. At the time, most physicists did not take this suggestion too seriously, but now, with the supernova data, the attitude has changed significantly. Also note that earlier in the chapter we saw that the outward push of a cosmological constant can be mimicked by a Higgs field that, like t
he frog on the plateau, is perched above its minimum energy configuration. So, while a cosmological constant fits the data well, a more precise statement is that the supernova researchers concluded that space must be filled with something like a cosmological constant that generates an outward push. (There are ways in which a Higgs field can be made to generate a long-lasting outward push, as opposed to the brief outward burst in the early moments of inflationary cosmology. We will discuss this in Chapter 14, when we consider the question of whether the data do indeed require a cosmological constant, or whether some other entity with similar gravitational consequences can fit the bill.) Researchers often use the term "dark energy" as a catchall phrase for an ingredient in the universe that is invisible to the eye but causes every region of space to push, rather than pull, on every other.

  24. Dark energy is the most widely accepted explanation for the observed accelerated expansion, but other theories have been put forward. For instance, some have suggested that the data can be explained if the force of gravity deviates from the usual strength predicted by Newtonian and Einsteinian physics when the distance scales involved are extremely large—of cosmological size. Others are not yet convinced that the data show cosmic acceleration, and are waiting for more precise measurements to be carried out. It is important to bear these alternative ideas in mind, especially should future observations yield results that strain the current explanations. But currently, there is widespread consensus that the theoretical explanations described in the main text are the most convincing.

  Chapter 11

  1. Among the leaders in the early 1980s in determining how quantum fluctuations would yield inhomogeneities were Stephen Hawking, Alexei Starobinsky, Alan Guth, So-Young Pi, James Bardeen, Paul Steinhardt, Michael Turner, Viatcheslav Mukhanov, and Gennady Chibisov.

  2. Even with the discussion in the main text, you may still be puzzled regarding how a tiny amount of mass/energy in an inflaton nugget can yield the huge amount of mass/energy constituting the observable universe. How can you wind up with more mass/energy than you begin with? Well, as explained in the main text, the inflaton field, by virtue of its negative pressure, "mines" energy from gravity. This means that as the energy in the inflaton field increases, the energy in the gravitational field decreases. The special feature of the gravitational field, known since the days of Newton, is that its energy can become arbitrarily negative. Thus, gravity is like a bank that is willing to lend unlimited amounts of money—gravity embodies an essentially limitless supply of energy, which the inflaton field extracts as space expands.

  The particular mass and size of the initial nugget of uniform inflaton field depend on the details of the model of inflationary cosmology one studies (most notably, on the precise details of the inflaton field's potential energy bowl). In the text, I've imagined that the initial inflaton field's energy density was about 10 82 grams per cubic centimeter, so that a volume of (10 -26 centimeters) 3 = 10 -78 cubic centimeters would have total mass of about 10 kilograms, i.e., about 20 pounds. These values are typical to a fairly conventional class of inflationary models, but are only meant to give you a rough sense of the numbers involved. To give a flavor of the range of possibilities, let me note that in Andrei Linde's chaotic models of inflation (see note 11 of Chapter 10), our observable universe would have emerged from an initial nugget of even smaller size, 10 -33 centimeters across (the so-called Planck length), whose energy density was even higher, about 10 94 grams per cubic centimeter, combining to give a lower total mass of about 10 -5 grams (the so-called Planck mass). In these realizations of inflation, the initial nugget would have weighed about as much as a grain of dust.

  3. See Paul Davies, "Inflation and Time Asymmetry in the Universe," in Nature, vol. 301, p. 398; Don Page, "Inflation Does Not Explain Time Asymmetry," in Nature, vol. 304, p. 39; and Paul Davies, "Inflation in the Universe and Time Asymmetry," in Nature, vol. 312, p. 524.

  4. To explain the essential point, it is convenient to split entropy up into a part due to spacetime and gravity, and a remaining part due to everything else, as this captures intuitively the key ideas. However, I should note that it proves elusive to give a mathematically rigorous treatment in which the gravitational contribution to entropy is cleanly identified, separated off, and accounted for. Nevertheless, this doesn't compromise the qualitative conclusions we reach. In case you find this troublesome, note that the whole discussion can be rephrased largely without reference to gravitational entropy. As we emphasized in Chapter 6, when ordinary attractive gravity is relevant, matter falls together into clumps. In so doing, the matter converts gravitational potential energy into kinetic energy that, subsequently, is partially converted into radiation that emanates from the clump itself. This is an entropy-increasing sequence of events (larger average particle velocities increase the relevant phase space volume; the production of radiation through interactions increases the total number of particles—both of which increase overall entropy). In this way, what we refer to in the text as gravitational entropy can be rephrased as matter entropy generated by the gravitational force. When we say gravitational entropy is low, we mean that the gravitational force has the potential to generate significant quantities of entropy through matter clumping. In realizing such entropy potential, the clumps of matter create a non-uniform, non-homogeneous gravitational field—warps and ripples in spacetime—which, in the text, I've described as having higher entropy. But as this discussion makes clear, it really can be thought of as the clumpy matter (and radiation produced in the process) as having higher entropy (than when uniformly dispersed). This is good since the expert reader will note that if we view a classical gravitational background (a classical spacetime) as a coherent state of gravitons, it is an essentially unique state and hence has low entropy. Only by suitably coarse graining would an entropy assignment be possible. As this note emphasizes, though, this isn't particularly necessary. On the other hand, should the matter clump sufficiently to create black holes, then an unassailable entropy assignment becomes available: the area of the black hole's event horizon (as explained further in Chapter 16) is a measure of the black hole's entropy. And this entropy can unambiguously be called gravitational entropy.

  5. Just as it is possible both for an egg to break and for broken eggshell pieces to reassemble into a pristine egg, it is possible for quantum-induced fluctuations to grow into larger inhomogeneities (as we've described) or for sufficiently correlated inhomogeneities to work in tandem to suppress such growth. Thus, the inflationary contribution to resolving time's arrow also requires sufficiently uncorrelated initial quantum fluctuations. Again, if we think in a Boltzmann-like manner, among all the fluctuations yielding conditions ripe for inflation, sooner or later there will be one that meets this condition as well, allowing the universe as we know it to initiate.

  6. There are some physicists who would claim that the situation is better than described. For example, Andrei Linde argues that in chaotic inflation (see note 11, Chapter 10), the observable universe emerged from a Planck-sized nugget containing a uniform inflaton field with Planck scale energy density. Under certain assumptions, Linde further argues that the entropy of a uniform inflaton field in such a tiny nugget is roughly equal to the entropy of any other inflaton field configuration, and hence the conditions necessary for achieving inflation weren't special. The entropy of the Planck-sized nugget was small but on a par with the possible entropy that the Planck-sized nugget could have had. The ensuing inflationary burst then created, in a flash, a huge universe with an enormously higher entropy—but one that, because of its smooth, uniform distribution of matter, was also enormously far from the entropy that it could have. The arrow of time points in the direction in which this entropy gap is being lessened.

  While I am partial to this optimistic vision, until we have a better grasp on the physics out of which inflation is supposed to have emerged, caution is warranted. For example, the expert reader will note that this approach makes favorable but unjustified assump
tions about the high-energy (transplanckian) field modes—modes that can affect the onset of inflation and play a crucial role in structure formation.

  Chapter 12

  1. The circumstantial evidence I have in mind here relies on the fact that the strengths of all three nongravitational forces depend on the energy and temperature of the environment in which the forces act. At low energies and temperatures, such as those of our everyday environment, the strengths of all three forces are different. But there is indirect theoretical and experimental evidence that at very high temperatures, such as occurred in the earliest moments of the universe, the strengths of all three forces converge, indicating, albeit indirectly, that all three forces themselves may fundamentally be unified, and appear distinct only at low energies and temperatures. For a more detailed discussion see, for example, The Elegant Universe, Chapter 7.