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Notes: Chapter 10

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  • Notes: Chapter 10

    Notes: Chapter 10
    1. As we've seen, the big bang's bang is not an explosion that took place at one location in a preexisting spatial expanse, and that's why we've not also asked where it banged. The playful description of the big bang's deficiency we've used is due to Alan Guth; see, for example, his The Inflationary Universe (Reading, Eng.: Perseus Books, 1997), p. xiii. Return to Text

    2. The term "big bang" is sometimes used to denote the event that happened at time-zero itself, bringing the universe into existence. But since, as we'll discuss in the next chapter, the equations of general relativity break down at time-zero, no one has any understanding of what this event actually was. This omission is what we've meant by saying that the big bang theory leaves out the bang. In this chapter, we are restricting ourselves to realms in which the equations do not break down. Inflationary cosmology makes use of such well-behaved equations to reveal a brief explosive swelling of space that we naturally take to be the bang left out by the big bang theory. Certainly, though, this approach leaves unanswered the question of what happened at the initial moment of the universe's creation — if there actually was such a moment. Return to Text

    3. Abraham Pais, Subtle Is the Lord (Okford: Oxford University Press, 1982), p. 253. Return to Text

    4. For the mathematically inclined reader: Einstein replaced the original equation G„, = 87cT„, by G,„ + Ag,„ = 8:rcTi„ where A is a number denoting the size of the cosmological constant. Return to Text

    5. When I refer to an object's mass in this context, I am referring to the sum total mass of its particulate constituents. If a cube, say, were composed of 1,000 gold atoms, I'd be referring to 1,000 times the mass of a single such atom. This definition jibes with Newton's perspective. Newton's laws say that such a cube would have a mass that is 1,000 times that of a single gold atom, and that it would weigh 1,000 times as much as a single gold atom. According to Einstein, though, the weight of the cube also depends on the kinetic energy of the atoms (as well as all other contributions to the energy of the cube). This follows from E=mc^2: more energy (E), regardless of the source, translates into more mass (m). Thus, an equivalent way of expressing the point is that because Newton didn't know about E=mc^2, his law of gravity uses a definition of mass that misses various contributions to energy, such as energy associated with motion. Return to Text

    6. The discussion here is suggestive of the underlying physics but does not capture it fully. The pressure exerted by the compressed spring does indeed influence how strongly the box is pulled earthward. But this is because the compressed spring affects the total energy of the box and, as discussed in the previous paragraph, according to general relativity, the total energy is what's relevant. However, the point I'm explaining here is that pressure itself — not just through the contribution it makes to total energy — generates gravity, much as mass and energy do. According to general relativity, pressure gravitates. Also note that the repulsive gravity we are referring to is is the internal gravitational field experienced within a region of space suffused by something that has negative rather than positive pressure. In such a situation, negative pressure will contribute a repulsive gravitational field acting within the region. Return to Text

    7. Mathematically, the cosmological constant is represented by a number, usually denoted by A (see note 4). Einstein found that his equations made perfect sense regardless of whether A was chosen to be a positive or a negative number. The discussion in the text focuses on the case of particular interest to modern cosmology (and modern observations, as will be discussed) in which A is positive, since this gives rise to negative pressure and repulsive gravity. A negative value for A yields ordinary attractive gravity. Note, too, that since the pressure exerted by the cosmological constant is uniform, this pressure does not directly exert any force: only pressure differences, like what your ears feel when you're underwater, result in a pressure force. Instead, the force exerted by the cosmological constant is purely a gravitational force. Return to Text

    8. Familiar magnets always have both a north and a south pole. By contrast, grand unified theories suggest that there may be particles that are like a purely north or purely south magnetic pole. Such particles are called monopoles and they could have a major impact on standard big bang cosmology. They have never been observed. Return to Text

    9. Guth and Tye recognized that a supercooled Higgs field would act like a cosmological constant, a realization that had been made earlier by Martinus Veltman and others. In fact, Tye has told me that were it not for a page limit in Physical Review Letters, the journal to which he and Guth submitted their paper, they would not have struck a final sentence noting that their model would entail a period of exponential expansion. But Tye also notes that it was Guth's achievement to realize the important cosmological implications of a period of exponential expansion (to be discussed later in this and in the next chapter), and thereby put inflation front and center on cosmologists' maps.

    In the sometimes convoluted history of discovery, the Russian physicist Alexei Starobinsky had, a few years earlier, found a different means of generating what we now call inflationary expansion, work described in a paper that was not widely known among western scientists. However, Starobinsky did not emphasize that a period of such rapid expansion would solve key cosmological problems (such as the horizon and flatness problems, to be discussed shortly), which explains, in part, why his work did not generate the enthusiastic response that Guth's received. In 1981, the Japanese physicist Katsuhiko Sato also developed a version of inflationary cosmology, and even earlier (in 1978), Russian physicists Gennady Chibisov and Andrei Linde hit upon the idea of inflation, but they realized that — when studied in detail — it suffered from a key problem (discussed in note 11) and hence did not publish their work.

    The mathematically inclined reader should note that it is not difficult to see how accelerated expansion arises. One of Einstein's equations is dt2/a/dt2/a = -4p/3(p + 3p) where a, p, and p are the scale factor of the universe (its "size"), the energy density, and the pressure density, respectively. Notice that if the righthand side of this equation is positive, the scale factor will grow at an increasing rate: the universe's rate of growth will accelerate with time. For a Higgs field perched on a plateau, its pressure density turns out to equal the negative of its energy density (the same is true for a cosmological constant), and so the righthand side is indeed positive. Return to Text

    10. The physics underlying these quantum jumps is the uncertainty principle, covered in Chapter 4. 1 will explicitly discuss the application of quantum uncertainty to fields in both Chapter 11 and Chapter 12, but to presage that material, briefly note the following. The value of a field at a given point in space, and the rate of change of the field's value at that point, play the same role for fields as position and velocity (momentum) play for a particle. Thus, just as we can't ever know both a definite position and a definite velocity for a particle, a field can't have a definite value and a definite rate of change of that value, at any given point in space. The more definite the field's value is at one moment, the more uncertain is the rate of change of that value — that is, the more likely it is that the field's value will change a moment later. And such change, induced by quantum uncertainty, is what I mean when referring to quantum jumps in the field's value. Return to Text

    11. The contribution of Linde and of Albrecht and Steinhardt was absolutely crucial, because Guth's original model — now called old inflation — suffered from a pernicious flaw. Remember that the supercooled Higgs field (or, in the terminology we introduce shortly, the inflaton field) has a value that is perched on the bump in its energy bowl uniformly across space. And so, while I've described how quickly the supercooled inflaton field could take the jump to the lowest energy value, we need to ask whether this quantum-induced jump would happen everywhere in space at the same time. And the answer is that it wouldn't. Instead, as Guth argued, the relaxation of the inflaton field to a zero energy value takes place by a process called bubble nucleation: the inflaton drops to its zero energy value at one point in space, and this sparks an outward-spreading bubble, one whose walls move at light speed, in which the inflaton drops to the zero energy value with the passing of the bubble wall. Guth envisioned that many such bubbles, with random centers, would ultimately coalesce to give a universe with zero-energy inflaton field everywhere. The problem, though, as Guth himself realized, was that the space surrounding the bubbles was still infused with a non-zero-energy inflaton field, and so such regions would continue to undergo rapid inflationary expansion, driving the bubbles apart. Hence, there was no guarantee that the growing bubbles would find one another and coalesce into a large, homogeneous spatial expanse. Moreover, Guth argued that the inflaton field energy was not lost as it relaxed to zero energy, but was converted to ordinary particles of matter and radiation inhabiting the universe. To achieve a model compatible with observations, though, this conversion would have to yield a uniform distribution of matter and energy throughout space. In the mechanism Guth proposed, this conversion would happen through the collision of bubble walls, but calculations — carried out by Guth and Erick Weinberg of Columbia University, and also by Stephen Hawking, Ian Moss, and John Steward of Cambridge University — revealed that the resulting distribution of matter and energy was not uniform. Thus, Guth's original inflationary model ran into significant problems of detail.

    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. Return to Text

    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. Return to Text

    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. Return to Text

    14. See note 11, this chapter. Return to Text

    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. Return to Text

    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. Return to Text

    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. Return to Text

    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. Return to Text

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

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

    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. Return to Text

    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. Return to Text

    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 the 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. Return to Text

    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. Return to Text
    Last edited by Reviewer; 10-15-2012, 07:26 PM.