From Eternity to Here

by Sean Carroll

The existence of a positive cosmological constant allows us to actually prove a somewhat rigorous result, rather than just spinning through a collection of thought experiments. The cosmic no-hair theorem states that, under the familiar set of “reasonable assumptions,” a universe with a positive vacuum energy plus some matter fields will, if it lasts long enough for the vacuum energy to take over, eventually evolve into empty universe with nothing but vacuum energy. The cosmological constant always wins, in other words.

The resulting universe—empty space with a positive vacuum energy—is known as de Sitter space, after Dutch physicist Willem de Sitter, one of the first after Einstein to study cosmology within the framework of general relativity.

As we mentioned back in Chapter Three, empty space with zero vacuum energy is known as Minkowski space, while empty space with a negative vacuum energy is anti- de Sitter space.

Three different versions of “empty space,” with different values of the vacuum energy: Minkowski space when the vacuum energy vanishes, de Sitter when it is positive, and anti-de Sitter when it is negative.

In Minkowski space, two particles initially at rest will stay motionless with respect to each other; in de Sitter space they are pushed apart, while in anti-de Sitter space they are pulled together.

Everything we’ve been arguing points to the idea that de Sitter space is the ultimate endpoint of cosmological evolution when the vacuum energy is positive, and hence the highest-entropy state we can think of in the presence of gravity. That’s not a definitive statement—the state of the art isn’t sufficiently advanced to allow for definitive statements along these lines—but it’s suggestive.

Indeed, if we believe in the holographic principle, we can assign a definite value to the entropy contained within any observable patch of de Sitter space. The answer is a huge number, and the entropy is larger when the vacuum energy is smaller.

Cosmologists talk about the “true vacuum,” in which the vacuum energy takes on its lowest possible value, but also various possible “false vacua,” in which the effective vacuum energy is higher.

The idea that “high entropy” means “empty space” becomes a lot more complicated when empty space can take on different forms, corresponding to different values of the vacuum energy.

WHY DON’T WE LIVE IN EMPTY SPACE?

If we allow the universe to expand until all of the matter and cosmic background radiation has diluted away, leaving only those particles that are produced out of de Sitter space by quantum effects, the temperature will be about 10—29 Kelvin. Cold by anyone’s standards.

When we take quantum effects in de Sitter space into account, the universe acts like a box of gas at a fixed temperature, and that situation will last forever.

If we wait long enough, our universe will empty out until it looks like de Sitter space with a tiny temperature, and stay that way forever. There will be random fluctuations in the thermal radiation that lead to all sorts of unlikely events—including the spontaneous generation of galaxies, planets, and Boltzmann brains. The chance that any one such thing happens at any particular time is small, but we have an eternity to wait, so every allowed thing will happen.

And it’s worth emphasizing that this puzzle makes the arrow-of-time problem enormously more pressing. Before this issue was appreciated, we had something of a fine-tuning problem: Why did the early universe have such a low entropy? But we were at least allowed to shrug our shoulders and say, “Well, maybe it just did, and there is no deeper explanation.” But now that’s no longer good enough. In de Sitter space, we can reliably predict the number of times in the history of the universe (including the infinite future) that observers will appear surrounded by cold and forbidding emptiness, and compare them to the observers who will find themselves in comfortable surroundings full of stars and galaxies, and the cold and forbidding emptiness is overwhelmingly likely. This is more than just an uncomfortable fine-tuning; it’s a direct disagreement between theory and observation, and a sign that we have to do better.

14

INFLATION AND THE MULTIVERSE

But there is a danger of begging the important question—why was it ever dominated by dark energy in that way? Inflation doesn’t provide any sort of answer by itself to the riddle of why entropy was low in the early universe, other than to assume that it started even lower (which is arguably a bit of a cheat).

It seems likely, in the judgment of most working cosmologists, that some version of inflation is correct—the question is, why did inflation ever happen?

THE CURVATURE OF SPACE

Therefore: If there is any noticeable amount of curvature whatsoever in the early universe, the universe today should be very obviously curved. A flat universe is like a pencil balanced exactly on its tip; if there were any deviation to the left or right, the pencil would tend to fall pretty quickly onto its side. Similarly, any tiny deviation from perfect flatness at early times should have become progressively more noticeable as time went on. But as a matter of observational fact, the universe looks very flat. As far as anyone can tell, there is no measurable curvature in the universe today at all.

This state of affairs is known as the flatness problem. Because the universe is so flat today, it had to be incredibly flat in the past. But why?

MAGNETIC MONOPOLES

Grand Unified Theories, or GUTs for short, attempt to provide a unified account of all the forces of nature other than gravity.

GUTs also predicted the existence of a new kind of particle, the magnetic monopole.

But according to GUTs, monopoles should be able to exist. In fact, in the late 1970s people realized that you could sit down and calculate the number of monopoles that should be created in the aftermath of the Big Bang. And the answer is: way too many. The total amount of mass in monopoles, according to these calculations, should be much higher than the total mass in ordinary protons, neutrons, and electrons. Magnetic monopoles should be passing through your body all the time.

INFLATION

THE HORIZON PROBLEM

The horizon problem is this: How did those widely separated points know to have almost the same conditions? Even though they are all within our cosmological horizon, their own cosmological horizons are much smaller, since they are much closer to the Big Bang. These days it’s a standard exercise for graduate students studying cosmology to calculate the size of the cosmological horizons for such points, under the assumptions of the standard Big Bang model; the answer is that points separated by more than about one degree on the sky have horizons that don’t overlap at all. In other words, there is no event in spacetime that is in the past of all these different points, and there is no way that any signal could be communicated to each of them.

Nevertheless, they all share nearly identical physical conditions. How did they know?

But how? That’s the horizon problem. As you can see, it’s closely connected to the entropy problem. Having the entire early universe share very similar conditions is a low-entropy configuration, as there are only a limited number of ways it can happen.

Inflation seems to provide a neat solution to the horizon problem. During the era of inflation, space expands by an enormous amount; points that were initially quite close get pushed very far apart. In particular, points that were widely separated when the microwave background was formed were right next to each other before inflation began—thereby answering the “How did they know to have similar conditions?” question. More important, during inflation the universe is dominated by dark super-energy, which—like any form of dark energy—has essentially the same density everywhere. There might be other forms of energy in the patch of space where inflation begins, but they are quickly diluted away; inflation stretches space flat, like pulling at the edges of a wrinkled bedsheet. The natural outcome of inflation is a universe that is very uniform on large scales.

TRUE AND FALS...

You may wonder what it is that actually creates this dark super-energy that drives inflation. The answer is a quantum field, just like the fields whose vibrations show up as the particles around us. Unfortunately, none of the fields we know—the neutrino field, the electromagnetic field, and so on—are right for the job. So cosmologists simply propose that there is a brand new field, imaginatively dubbed the “inflaton,” whose task it is to drive inflation. Inventing new fields out of whole cloth like this is not quite as disreputable as it sounds; the truth is, inflation supposedly takes place at energies far higher than we can directly re-create in laboratories here on Earth. There are undoubtedly any number of new fields that become relevant at such energies, even if we don’t know what they are; the question is whether any of them have the right properties to be the inflaton (i.e., give rise to a temporary phase of dark super-energy that expands the universe by a tremendous amount before decaying away).

That kind of energy, associated with the value of the field itself rather than changes in the field from place to place or time to time, is known as “potential energy.”

Looking at the potential energy curve for some field, the bottom of every valley defines a distinct vacuum state.)

Rather than the inflaton being stuck in a false vacuum “valley,” imagine that it starts out on an elevated plateau—a long stretch that is nearly flat. The field then slowly rolls down the plateau, keeping the energy almost constant but not quite, before ultimately falling off a cliff (the phase transition). This is called “new inflation” and is the most popular implementation of the inflationary universe idea among cosmologists today.

Inflation does its best to make the universe as smooth as possible, but there is a fundamental limit imposed by quantum mechanics. Things can’t become too smooth, or we would violate the Heisenberg Uncertainty Principle by pinpointing the state of the universe too precisely. The inevitable quantum fuzziness in the energy density from place to place during inflation gets imprinted on the amount of matter and radiation the inflaton converts into, and that translates into a very specific prediction for what kinds of perturbations in density we should see in the early universe. It’s those primordial perturbations that imprint temperature fluctuations in the cosmic microwave background, and eventually grow into stars, galaxies, and clusters. So far, the kinds of perturbations predicted by inflation match the observations very well.

ETERNAL INFLATION

In fact, there is a way to “reheat” the interior of the true-vacuum bubble, to create the conditions of the Big Bang model: an episode of new inflation inside the bubble. We imagine that the inflaton field inside the bubble doesn’t land directly at the bottom of its potential, corresponding to the true vacuum; instead, it lands on an intermediate plateau, from which the field slowly rolls toward that minimum. In this way, there can be a phase of new inflation within each bubble; the energy density from the inflaton potential while it’s on the plateau can later be converted into matter and radiation, and we end up with a perfectly plausible universe.

So old inflation, once it starts, never ends. You can make bubbles of true vacuum that look like our universe, but the region of false vacuum outside always keeps growing. More bubbles keep appearing, and the process never terminates. That’s the idea of “eternal inflation.”

THE MULTIVERSE

WHAT GOOD IS INFLATION?

We don’t know what conditions in the extremely early universe were really like. Let’s assume it was dense and crowded, but not necessarily smooth; there may have been wild fluctuations from place to place. These may have included black holes, oscillating fields, and even somewhat empty patches. Now imagine that at least one small region of space within this mess is relatively quiet, with its energy density consisting mostly of dark super-energy from the inflaton field. While the rest of space goes on its chaotic way, this particular patch begins to inflate; its volume increases by an enormous amount, while any preexisting perturbations get wiped clean by the inflationary stretching. At the end of the day, that particular patch evolves into a region of space that looks like our universe as described by the standard Big Bang model, regardless of what happens to the rest of the initially fluctuating primordial soup. Therefore, it doesn’t require any delicate, unnatural fine-tuning of initial conditions to get a universe that is spatially flat and uniform over large distances; it arises robustly from generic, randomly fluctuating initial conditions.

OUR COMOVING PATCH REVISITED

SETTING THE STAGE

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 low-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.

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.

15

THE PAST THROUGH TOMORROW

EVOLVING THE SPACE OF STATES