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Boltzmann Redux

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  • Boltzmann Redux

    THE FABRIC of the COSMOS, Brian Greene, 2004
    ```(annotated and with added bold highlights by Epsilon=One)
    Chapter 11 - Quanta in the Sky with Diamonds
    Boltzmann Redux
    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.

    Figure 11.2 Inflation can occur repeatedly, sprouting new universes from older ones.

    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.
    Last edited by Reviewer; 10-13-2012, 06:51 AM.