From Eternity to Here (59 page)

One potential problem in discussing the curvature of space by itself is that general relativity gives us the freedom to slice spacetime into three-dimensional copies of space evolving through time in a multitude of ways; the definition of “space” is not unique. Fortunately, in our observed universe there is a natural way to do the slicing: We define “time” such that the density of matter is approximately constant through space on large scales, but diminishing as the universe expands. The distribution of matter, in other words, defines a natural rest frame for the universe. This doesn’t violate the precepts of relativity in any way, because it’s a feature of a particular configuration of matter, not of the underlying laws of physics.

In general, space could curve in arbitrary ways from place to place, and the discipline of differential geometry was developed to handle the mathematics of curvature. But in cosmology we’re lucky in that space is uniform over large distances, and looks the same in every direction. In that case, all you have to do is specify a single number—the “curvature of space”—to tell me everything there is to know about the geometry of three-dimensional space.

The curvature of space can be a positive number, or a negative number, or zero. If the curvature is zero, we naturally say that space is “flat,” and it has all the characteristics of geometry as we usually understand it. These characteristics were first set out by Euclid, and include properties like “initially parallel lines stay parallel forever,” and “the angles inside a triangle add up to precisely 180 degrees.” If the curvature is positive, space is like the surface of a sphere—except that it’s three-dimensional. Initially parallel lines do eventually intersect, and angles inside a triangle add up to
greater
than 180 degrees. If the curvature is negative, space is like the surface of a saddle, or of a potato chip. Initially parallel lines grow apart, and angles inside a triangle—well, you can probably guess.
²⁵⁷

According to the rules of general relativity, if the universe starts flat, it stays flat. If it starts curved, the curvature gradually diminishes away as the universe expands. However, as we know, the density of matter and radiation also dilutes away. (For right now, forget you’ve ever heard about dark energy, which changes everything.) When you plug in the equations, the density of matter or radiation decreases
faster
than the amount of curvature. Relative to matter and radiation, curvature becomes more relevant to the evolution of the universe as space expands.

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.
²⁵⁸

Figure 74:
Ways that space can have a uniform curvature. From top to bottom: positive curvature, as on a sphere; negative curvature, as on a saddle; zero curvature, as on a flat plane.

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?

The flatness problem bears a family resemblance to the entropy problem we discussed in the last chapter. In both cases, it’s not that there is some blatant disagreement between theory and observation—all we have to do is posit that the early universe had some particular form, and everything follows nicely from there. The problem is that the “particular form” seems to be incredibly unnatural and finely tuned, for no obvious reason. We could say that both the entropy and the spatial curvature of the early universe just were small, and there’s no explanation beyond that. But these apparently unnatural features of the universe might be a clue to something important, so it behooves us to take them seriously.

MAGNETIC MONOPOLES

Alan Guth wasn’t trying to solve the flatness problem when he hit upon the idea of inflation. He was interested in a very different puzzle, known as the
monopole problem
.

Guth, for that matter, wasn’t especially interested in cosmology. In 1979, he was in his ninth year of being a postdoctoral researcher—the phase of a scientist’s career in between graduate school and becoming a faculty member, when they concentrate on research without having to worry about teaching duties or other academic responsibilities. (And without the benefit of any job security whatsoever; most postdocs never succeed in getting a faculty job, and eventually leave the field.) Nine years is past the time when a talented postdoc would normally have moved on to become an assistant professor somewhere, but Guth’s publication record at this point in his career didn’t really reflect the ability that others saw in him. He had labored for a while on a theory of quarks that had fallen out of favor, and was now trying to understand an obscure prediction of the newly popular “Grand Unified Theories”: the prediction of magnetic monopoles.

Grand Unified Theories, or GUTs for short, attempt to provide a unified account of all the forces of nature other than gravity. They became very popular in the 1970s, both for their inherent simplicity, and because they made an intriguing prediction: that the proton, the stalwart elementary particle that (along with the electron and the neutron) forms the basis for all the matter around us, would ultimately decay into lighter particles. Giant laboratories were built to search for proton decay, but it hasn’t yet been discovered. That doesn’t mean that GUTs aren’t right; they are still quite popular, but the failure to detect proton decay has left physicists at a loss over how these theories should be tested.

GUTs also predicted the existence of a new kind of particle, the magnetic monopole. Ordinary charged particles are electric monopoles—that is, they have either a positive charge or a negative charge, and that’s all there is to it. No one has ever discovered an isolated “magnetic charge” in Nature. Magnets as we know them are always dipoles—they come with a north pole and a south pole. Cut a magnet in half between the poles, and two new poles pop into existence where you made the cut. As far as experimenters can tell, looking for an isolated magnetic pole—a monopole—is a lot like looking for a piece of string with only one end.

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.

There is an easy way out of this, of course: GUTs might not be right. And that still might be the correct solution. But Guth, while thinking about the problem, hit on a more interesting one: inflation.

INFLATION

Dark energy—a source of energy density that is approximately (or exactly) constant throughout space and time, not diluting away as the universe expands—makes the universe accelerate, by imparting a perpetual impulse to the expansion. We believe that most of the energy in the universe, between 70 percent and 75 percent of the total, is currently in the form of dark energy. But in the past, when matter and radiation were denser, dark energy presumably had about the same density it has today, so it would have been relatively unimportant.

Now imagine that, at some other time in the very early universe, there was dark energy with an extraordinarily larger energy density—call it “dark super-energy.”
²⁵⁹
It dominated the universe and caused space to accelerate at a terrific rate. Then—for reasons to be specified later—this dark super-energy suddenly decayed into matter and radiation, which formed the hot plasma making up the early universe we usually think about. The decay was almost complete, but not quite, leaving behind the relatively minuscule amount of dark energy that has just recently become important to the dynamics of the universe.

That’s the scenario of inflation. Basically, inflation takes a tiny region of space and blows it up to an enormous size. You might wonder what the big deal is—who cares about a temporary phase of dark super-energy, if it just decays into matter and radiation? The reason why inflation is so popular is because it’s like confession—it wipes away prior sins.

Figure 75:
Inflation takes a tiny patch of space and expands it rapidly to a tremendous size. This figure is not at all to scale; inflation occurs in a tiny fraction of a second, and expands space by more than a factor of 10
²⁶
.

Consider the monopole problem. Monopoles are (if GUTs are correct) produced in copious amounts in the extremely early universe. So imagine that inflation happens pretty early, but later than the production of monopoles. In that case, as long as inflation lasts long enough, space expands by such a tremendous amount that all the monopoles are diluted away practically to nothing. As long as the decay of the dark super-energy into matter and radiation doesn’t make any more monopoles (which it won’t, if it’s not too energetic), voilà—no more monopole problem.

Likewise with spatial curvature. The problem there was that curvature dilutes away more gradually than matter or radiation, so if there were any curvature at all early on it should be extremely noticeable today. But dark energy dilutes away even more gradually than curvature—indeed, it hardly dilutes away at all. So again, if inflation goes on long enough, curvature can get diluted to practically nothing, before matter and radiation are re-created in the decay of the dark super-energy. No more flatness problem.

You can see why Guth was excited about the idea of inflation. He had been thinking about the monopole problem, but from the other side—not trying to solve it, but using it as an argument against GUTs. In his original work on the problem, with Cornell physicist Henry Tye, they had ignored the possible role of dark energy and established that the monopole problem was very hard to solve. But once Guth sat down to study the effects that an early period of dark energy could have, a solution to the monopole problem dropped right into his lap—that’s worth at least a single box, right there.

The double-box-worthiness came when Guth understood that his idea would also solve the flatness problem, which he hadn’t even been thinking about. Completely coincidentally, Guth had gone to a lecture some time earlier by Princeton physicist Robert Dicke, one of the first people to study the cosmic microwave background. Dicke’s lecture, held at a Cornell event called “Einstein Day,” pointed out several loose ends in the conventional cosmological model. One of them was the flatness problem, which stuck with Guth, even though his research at the time wasn’t especially oriented toward cosmology.

So when he realized that inflation solved not only the monopole problem but also the flatness problem, Guth knew he was onto something big. And indeed he was; almost overnight, he went from being a struggling postdoc to being a hot property on the faculty job market. He chose to return to MIT, where he had been a graduate student, and he’s still teaching there today.

THE HORIZON PROBLEM

In working out the consequences of inflation, Guth realized that the scenario offered a solution to yet another cosmological fine-tuning puzzle: the
horizon problem
. Indeed, the horizon problem is arguably the most insistent and perplexing issue in standard Big Bang cosmology.

The problem arises from the simple fact that the early universe looks more or less the same at widely separated points. In the last chapter, we noted that a “typical” state of the early universe, even if we insisted that it be highly dense and rapidly expanding, would tend to be wildly fluctuating and inhomogeneous—it should resemble the time-reverse of a collapsing universe. So the fact that the universe was so smooth is a feature that seems to warrant an explanation. Indeed, it’s fair to say that the horizon problem is really a reflection of the entropy problem as we’ve presented it, although it’s usually justified in a different way.