The numerical answer will depend on the particular way in which inflation happens, and in particular on the vacuum energy during inflation. But the differences between one model and another aren’t that significant, so it suffices to pick an example and stick to it. Let’s say that the energy scale during inflation is 1 percent of the Planck scale; pretty high, but low enough that we’re safely avoiding complications from quantum gravity. In that case, the estimated entropy of our comoving patch at the beginning of inflation is:
Sinflation ≈ 1012.
That’s an incredibly small value, compared either to the 10120 it could have been or even the 1088 it would soon become. It reflects the fact that every single degree of freedom that goes into describing our current universe must have been delicately packed into an extremely smooth, small patch of space, in order for inflation to get going.
The secret of inflation is thereby revealed: It explains why our observable universe was in such an apparently low-entropy, finely tuned early state by assuming that it started in an even lower-entropy state before that. That’s hardly surprising, if we believe the Second Law and expect entropy to grow with time, but it doesn’t seem to address the real issue. Taken at face value, it would seem very surprising indeed that we would find our comoving patch of universe in the kind of lo w-entropy configuration necessary to start inflation. You can’t solve a fine-tuning problem by appealing to an even greater fine-tuning.
OUR COMOVING PATCH REVISITED
Let’s think this through, because we’re deviating from orthodoxy here and it behooves us to be careful.
We have been making two crucial assumptions about the evolution of the observable universe—our comoving patch of space and all of the stuff within it. First, we’re assuming that the observable universe is essentially autonomous—that is, it evolves as a closed system, free from outside influences. Inflation does not violate this assumption; once inflation begins, our comoving patch rapidly turns into a smooth configuration, and that configuration evolves independently of the rest of the universe. This assumption can obviously break down before inflation, and play a role in setting up the initial conditions; but inflation itself does not take advantage of any hypothetical external influences in attempting to explain what we currently see.
The other assumption is that the dynamics of our observable universe are reversible—they conserve information. This seemingly innocuous point implies a great deal. There is a space of states that is fixed once and for all—in particular, it is the same at early times as at late times—and the evolution within that space takes different starting states to different ending states (in the same amount of time). The early universe looks very different from the late universe—it’s smaller, denser, expanding more rapidly, and so on. But (under our assumptions of reversible dynamics) that doesn’t mean the space of states has changed, only that the particular kind of state the universe is in has changed.
The early universe, to belabor the point, is the same physical system as the late universe, just in a very different configuration. And the entropy of any given microstate of that system reflects how many other microstates look similar from a macroscopic point of view. If we were to randomly choose a configuration of the physical system we call the observable universe, it would be overwhelmingly likely to be a state of very high entropy—that is, close to empty space.269
To be honest, however, people tend not to think that way, even among professional cosmologists. We tend to reason that the early universe is a small, dense place, so that when we imagine what states it might be in, we can restrict our attention to small, dense configurations that are sufficiently smooth and well behaved so that the rules of quantum field theory apply. But there is absolutely no justification for doing so, at least within the dynamics itself. When we imagine what possible states the early universe could have been in, we need to include unknown states that are outside the realm of validity of quantum field theory. For that matter, we should include all of the possible states of the current universe, as they are simply different configurations of the same system.
The size of the universe is not conserved—it evolves into something else. When we consider statistical mechanics of gas molecules in a box, it’s okay to keep the number of molecules fixed, because that reflects the reality of the underlying dynamics. But in a theory with gravity, the “size of the universe” isn’t fixed. So it makes no sense—again, just based on the known laws of physics, without recourse to some new principles outside those laws—to assume from the start that the early universe must be small and dense. That’s something we need to explain.
All of which is somewhat problematic for the conventional justification that we put forward for the inflationary universe scenario. According to the previous story, we admit that we don’t know what the early universe was like, but we imagine that it was characterized by wild fluctuations. (The current universe, of course, is not characterized by such fluctuations, so already there is something to be explained.) Among those fluctuations, every once in a while a region will come into existence that is dominated by dark super-energy, and the conventional inflation story follows. After all, how hard can it be to randomly fluctuate into the right conditions to start inflation?
The answer is that it can be incredibly hard. If we truly randomly chose a configuration for the degrees of freedom within that region, we would be overwhelmingly likely to get a high-entropy state: a large, empty universe.270 Indeed, simply by comparing entropies, we’d be much more likely to get our current universe, with a hundred billion galaxies and all the rest, than we would be to get a patch ready to inflate. And if we’re not randomly choosing configurations of those degrees of freedom—well, then, what are we doing? That’s beyond the scope of the conventional inflationary story.
These problems are not specific to the idea of inflation. They would plague any possible scenario that claimed to provide a dynamical explanation for the apparent fine-tuning of our early universe, while remaining consistent with our two assumptions (our comoving patch is a closed system, and its dynamics are reversible). The problem is that the early universe has a low entropy, which means that there are a relatively small number of ways for the universe to look like that. And, while information is conserved, there is no possible dynamical mechanism that can take a very large number of states and evolve them into a smaller number of states. If there were, it would be easy to violate the Second Law.
SETTING THE STAGE
This discussion has intentionally emphasized the hidden skeletons in the closet of the inflationary universe scenario—there are plenty of other books that will emphasize the arguments in its favor.271 But let’s be clear: The problem isn’t really with inflation; it’s with how the theory is usually marketed. We often hear that inflation removes the pressing need for a theory of initial conditions, as inflation will begin under fairly generic circumstances, and once it begins all our problems are solved.
The truth is almost the converse: Inflation has a lot going for it, but it makes the need for a theory of initial conditions much more pressing. Hopefully I’ve made the case that neither inflation nor any other mechanism can, by itself, explain our lo w-entropy early universe under the assumptions of reversibility and autonomous evolution. It’s possible, of course, that reversibility should be the thing to go; perhaps the fundamental laws of physics violate reversibility at a fundamental level. Even though that’s intellectually conceivable, I’ll argue that it’s hard to make such an idea match what we actually see in the world.
A less drastic strategy would be to move beyond the assumption of autonomous evolution. We knew all along that treating our comoving patch as a closed system was, at best, an approximation. It seems like an extremely good approximation right now, or even at any time in the history of the universe about which we actually have empirical data. But surely it breaks down at the very beginning. Inflation could play a crucial role in explaining the universe we see, but only if we can discard the idea that “
we just randomly fluctuated into it,” and come up with a particular reason why the conditions necessary for inflation ever came to pass.
In other words, it seems like the most straightforward way out of our conundrum is to abandon the goal of explaining the unnatural early universe purely in terms of the autonomous evolution of our comoving patch, and instead try to embed our observable universe into a bigger picture. This brings us back to the idea of the multiverse—a larger structure in which the universe we observe is just a tiny part. If something like that is true, we are at least able to contemplate the idea that the evolution of the multiverse naturally gives rise to conditions under which inflation can begin; after that, the story proceeds as above.
So we want to ask, not what our the physical system making up our observable universe should look like, but what a multiverse should look like, and whether it would naturally give rise to regions that look like the universe we see. Ideally, we’d want it to happen without putting in time asymmetry by hand at any step along the way. We want to explain not only how we can get the right conditions to start inflation, but why it might be natural to have a large swath of spacetime (our observable universe) that features those conditions at one end of time and empty space at the other. This is a program that is far from complete, although we have some ideas. We’re deep into speculative territory by now, but if we keep our wits about us we should be able to take a safe journey without being devoured by dragons.
15
THE PAST THROUGH TOMORROW
The eternal silence of these infinite spaces fills me with dread.
—Blaise Pascal, Pensées272
Over the course of this book, we’ve explored the meaning of the arrow of time, as embodied in the Second Law of Thermodynamics, and its relationship with cosmology and the origin of the universe. Finally we have enough background to put it all together and address the question once and for all: Why was the entropy of our observable universe low at early times? (Or even better, so as not to succumb to asymmetric language right from the start: Why do we live in the temporal vicinity of an extremely low-entropy state?)
We’ll address the question, but we don’t know the answer. There are ideas, and some ideas seem more promising than others, but all of them are somewhat vague, and we certainly haven’t yet put the final pieces together. That’s science for you. In fact, that’s the exciting part of science—when you have some clues assembled, and some promising ideas, but are still in the process of nailing down the ultimate answers. Hopefully the prospects sketched in this chapter will serve as a useful guide to wherever cosmologists go next in their attempts to address these deep issues.273
At the risk of being repetitive, let’s review the conundrum one last time, so that we can establish what would count as an acceptable solution to the problem.
All of the macroscopic manifestations of the arrow of time—our ability to turn eggs into omelets but not vice versa, the tendency of milk to mix into coffee but never spontaneously unmix, the fact that we can remember the past but not the future—can be traced to the tendency of entropy to increase, in accordance with the Second Law of Thermodynamics. In the 1870s, Boltzmann explained the microscopic under pinnings of the Second Law: Entropy counts the number of microstates corresponding to each macrostate, so if we start (for whatever reason) in a relatively low-entropy state, it’s overwhelmingly likely that the entropy will increase toward the future. However, due to the fundamental reversibility of the laws of physics, if the only thing we have to go on is the fact that the current state is low entropy, we would with equal legitimacy expect the entropy to have been larger in the past. The real world doesn’t seem to work that way, so we need something else to go on. That something else is the Past Hypothesis: the assumption that the very early universe found itself in an extremely low-entropy state, and we are currently witnessing its relaxation to a state of high entropy. The question of why the Past Hypothesis is true belongs to the realm of cosmology. The anthropic principle is woefully inadequate for the task, since we could easily find ourselves constituted as random fluctuations (Boltzmann brains) in an otherwise empty de Sitter space. Likewise, inflation by itself doesn’t address the question, as it requires an even lower-entropy starting point than the conventional Big Bang cosmology. So the question remains: Why does the Past Hypothesis hold within our observable patch of the universe?
Let’s see if we can’t make some headway on this.
EVOLVING THE SPACE OF STATES
Start with the most obvious hypothesis: Deep down, the fundamental laws of physics simply aren’t reversible. I’ve tried to be careful to allude to the existence of this possibility all along, but have always spoken as if it’s a long shot, or not worthy of our serious attention. There are good reasons for that, though they are not airtight.
A reversible system is one that has a space of states, fixed once and for all, and a rule for evolving those states forward in time that conserves information. Two different states, beginning at some initial time, will evolve predictably into two different states at some specific later time—never into the same future state. That way we can reverse the evolution, since every state the system could currently be in has a unique predecessor at every moment in time.
One way to violate reversibility would be to let the space of states itself actually evolve with time. Perhaps there were simply fewer possible states the universe could have been in at early times, so the small entropy is not so surprising. In that case, many possible microstates that live in the same macrostate as the current universe simply have no possible past state from which they could have come.
Indeed, this is how many cosmologists implicitly speak about what happens in an expanding universe. If we restrict ourselves to “states that look like gentle vibrations of quantum fields around a smooth background,” it’s certainly true that this particular part of the space of states grows with time, as space itself (in the old-fashioned three-dimensional sense of “space”) becomes larger. But that’s very different from imagining that the entire space of states is actually changing with time. Almost nobody would claim to support such a position, if they sat down and thought through what it really meant. I explicitly rejected this possibility when I argued that the early universe was finely tuned—among all the states it could have been in, we included states that look like the universe today, as well as various choices with even higher entropy.
The weirdest thing about the idea that the space of states changes with time is that it requires an external time parameter—a concept of “time” that lives outside the actual universe, and through which the universe evolves. Ordinarily, we think of time as part of the universe—a coordinate on spacetime, measured by various sorts of predictably repetitive clocks. The question “What time is it?” is answered by reference to things going on within the universe—that is to say, to features of the state the universe is currently in. (“The little hand is on the three, and the big hand is on the twelve.”) But if the space of states truly changes with time, that conception becomes insufficient. At any one moment, the universe is actually in one specific state. It makes no sense to say, “The space of states is smaller when the universe is in state X than when it is in state Y.” The space of states, by definition, includes all of the states the universe could hypothetically be in.
Figure 81: On the left we have reversible laws of physics: The system evolves within a fixed space of states, such that different initial states evolve uniquely to different final states. The middle example is irreversible, because the space of states grows with respect to some external time parameter; some states at later times have no predecessors from which they could have come. On the right we have another form of irreversibility, where the space of states remains fixed, but different initial states evolve into the same final state.
So for the space of states to change with time, we would have to posit a notion of time that is not merely measured by features of the state of the universe, but exists outside the universe
as we conventionally understand it. Then it would make sense to say, “When this external time parameter was equal to a certain value, the space of states of the universe was relatively small, and when it had progressed to some other value the space of states had grown larger.”
There’s not much to say about this idea. It’s possible, but very few people advocate it as an approach to the arrow-of-time problem.274 It would require a dramatic rethinking of the way we currently understand the laws of physics; nothing about our current framework suggests the existence of a time parameter lurking outside the universe itself. So for now, we can’t rule it out, but it doesn’t give us a warm and fuzzy feeling.
IRREVERSIBLE MOTIONS
The other way to invent laws of physics that are intrinsically irreversible is to stick once and for all with some space of states, but posit dynamical laws that don’t conserve information. That’s what we saw with checkerboard D back in Chapter Seven; when diagonal lines of gray squares bumped into the vertical line, they simply ceased to exist. There was no way of knowing, from the state at one particular moment in time, precisely what state it came from in the past, since there was no way of reconstructing what diagonals had been lurking around before running afoul of a vertical column.
It’s not hard to come up with slightly more realistic versions of the same idea. In Chapter Eight we contemplated an irreversible game of billiards: a conventional billiards table, where the balls moved forever without losing any energy through friction, except that whenever a ball hit a particular one of the walls of the table, it came perfectly to rest and stayed there forever. The space of states of this system never changes; it always consists of all the possible positions and momenta of the balls on the table. The entropy is defined in the completely conventional way, as the logarithm of the number of states with certain macroscopic properties. But the dynamics are irreversible: Given any one ball stuck to the special wall, we have no way of knowing how long it’s been there. And the entropy of this system flouts the Second Law with impunity; gradually, as more balls get stuck, the system takes up a smaller and smaller portion of the space of states, and the entropy decreases without any intervention from the outside world.
From Eternity to Here: The Quest for the Ultimate Theory of Time Page 43