What you may also notice if you are able to watch the dots is that the farther away they are from one another, the faster they separate. In fact the dots would do exactly what Hubble’s galaxies did. Hubble’s Law is the law of dots on the surface of an expanding balloon. Unfortunately Friedmann died in 1925, before he could learn of Hubble’s discovery or the fact that his—Friedmann’s—work had laid the foundation for all future cosmology.
Let’s review some of that cosmology.
The Cosmological Principle and the Three Geometries
“Only two things are infinite, the universe and human stupidity, and I’m not sure about the former.”
— ALBERT EINSTEIN
A couple of years ago I had the good fortune to be invited to South Africa in order to give some lectures in one of the universities. While I was there my wife and I made a trip to the Krueger National Park. The park is an enormous expanse of African veldt and home to all the large mammals of the continent. It was a fabulous experience. In the mornings and evenings we would go out in a Land Rover to view and photograph wildlife. We saw hippos, a rhino, Cape buffalo, a pride of lions devouring antelope, and most impressive of all, an angry rogue male elephant. But for me the most powerful sight of all was the southern sky on a dark, moonless night. The southern sky is much richer than the northern sky that I’m used to, and Krueger is almost completely free of light pollution. The sight of the Milky Way stretched clear across the sky is truly awe-inspiring. But the humbling sense of immensity is deceptive. The entire Milky Way together with all the visible stars is an infinitesimal corner of a much vaster space, homogeneously filled with a hundred billion galaxies, which can be seen only through a large telescope. And even that is a tiny portion of a much bigger cosmos.
According to my dictionary the word homogeneous means “uniform in structure or composition throughout.” When applied to oatmeal it means nice and smooth, i.e., without lumps. Of course if you look at the oatmeal with a magnifying glass, it hardly looks homogeneous. The point is that when you say something is homogeneous, you must qualify the statement by adding “on scales larger than some specified size.” Well-stirred oatmeal is homogeneous on scales larger than an eighth of an inch. Farmer Brown’s wheat field in the middle of Kansas is homogeneous on scales larger than a few feet.
Well, not quite. The oatmeal is only homogeneous on scales from an eighth of an inch to the size of the bowl. Farmer Brown’s field is homogeneous on scales bigger than ten feet but smaller than a mile. On scales of a mile or more, the countryside looks like a crazy quilt of rectangular fields. The right thing to say is that Farmer Brown’s field is homogeneous on scales between ten feet and a fraction of a mile.
Through the naked eye the African night sky is very inhomogeneous. The Milky Way is a bright, narrow band of light that divides a much darker background. But a look through a large telescope reveals billions of galaxies that are, on the whole, homogeneously distributed through the observable universe. According to astronomers the universe appears to be homogeneous and isotropic on scales larger than a hundred million light-years up to at least fifteen billion light-years. The fifteen billion light-year limit is certainly an underestimate that just represents our inability to see farther.
Returning to my dictionary, I found the following definition for the term isotropic: “identical in all directions; invariant with respect to direction.” Isotropic is not the same as homogeneous. Here’s an example. Once while I was diving near a coral reef in the Red Sea, I saw a huge school of closely spaced, small, thin fish that filled quite a large volume homogeneously. For some strange reason, until I got too close, they all faced the same direction. The school appeared homogeneous over some range of scales, but definitely not isotropic. Every place within the school was like every other place, but every direction was not at all like every other direction. The direction the fish faced was special.
Cosmologists and astronomers almost always assume that the universe is homogeneous and isotropic; no matter where you are in the universe and which direction you are facing, you see the same thing. I don’t mean the nearby details but the overall, large-scale features of the universe. Cosmologists call this assumption the cosmological principle. Of course calling it a principle does not make it right. Originally it was just a guess, but gradually better and better observations of several kinds have convinced astronomers and cosmologists that the universe is indeed homogeneous and isotropic over scales ranging from a few hundred million light-years to at least a few tens of billions of light-years. Beyond that we don’t know for sure because there is a limit to our observations. It doesn’t matter how big our telescope, objects farther than fourteen billion light-years are impossible to observe. The reason is the simple fact that the universe is only about fourteen billion years old. In that time light could not have traveled more than fourteen billion light-years; light from more distant places just hasn’t reached us yet. In fact it’s a pretty safe bet that the universe is homogeneous and isotropic out to distance scales much larger than the observable part of the universe. But like Farmer Brown’s field, the universe may become a crazy quilt at a large enough distance: a patchwork of pocket universes.
For now, let’s adopt the very conventional point of view that the cosmological principle is correct out to the largest scales. This raises an interesting question: what kind of overall spatial geometry is compatible with the cosmological principle? By spatial geometry I mean the shape of space. Let’s begin with two-dimensional examples. A 2-sphere is a particular geometry. So are ellipsoids, pear shapes, and banana shapes.4
Among this list, only a sphere is homogeneous and isotropic. Like a circle, a sphere has perfect symmetry; every point is exactly like every other point. An ellipsoid, while not as symmetric as a sphere, still has a good deal of symmetry. For example, its mirror image looks just like itself. But not every place on the ellipsoid is the same as every other. The pear and banana are even less symmetric.
One way to describe the properties of a surface is by its curvature. The curvature of the sphere is absolutely uniform. Mathematically speaking it is a space of uniform positive curvature. The ellipsoid is also a positively curved space, but it is more curved in some places than others. For example, the prolate ellipsoid, which is shaped somewhat like a submarine, is curved more near its ends than at its waist. Of all these examples only the sphere is uniformly curved and homogeneous.
Spheres, ellipsoids, and the surfaces of fruits are closed-and-bounded geometries, meaning that they are finite in extent but without edges. But the truth is that no one knows if the universe is finite in extent. No cosmic Magellan has ever circumnavigated it. It’s entirely possible that the universe goes on forever, in which case it is unbounded, or infinite.
If we allow the possibility that the universe is infinite, then there are two more homogeneous, isotropic geometries. The first is obvious: the infinite flat plane. Think of it as a sheet of paper that goes on and on forever. There are no landmarks on the infinite plane to tell you where you are or which way you are facing. And unlike the surface of the sphere, the plane is not curved: mathematically it has zero curvature. Positive curvature for the sphere, zero curvature for the plane, and finally, the last homogeneous isotropic geometry, the negatively curved “hyperbolic geometry.” To help visualize this, think of a piece of duct pipe bent to a right angle. At the outer “elbow,” the sheet metal is positively curved like the sphere. The inner curved surface is the place where the curvature is negative.
But of course the duct-pipe elbow is not homogeneous. The inner curved region is not at all the same as the outer positive curvature region. A better example is the surface of a saddle. Imagine continuing the saddle shape forever to form an unbounded negatively curved surface. It’s not easy to visualize, but it’s perfectly possible.
All three of these surfaces—sphere, plane, and hyperbolic geometry—are homogeneous. Moreover, all three of them have three-dimensional analogs: the 3-sphere, ordinary Euclidean
three-dimensional space, and the most difficult to visualize, the hyperbolic three-dimensional space.
To envision the three standard types of cosmologies, think of each surface as a rubber sheet (or balloon in the case of the sphere) and fill the surface with dots to represent galaxies. Then start stretching the surface so that the dots begin to separate and the distance between any two grows with time. That’s all there is to it. You now have a rough idea of the three homogeneous, isotropic cosmologies. Cosmologists refer to these three cases as k = 1, k = 0, and k = -1. It’s just shorthand for positive curvature (the sphere), zero curvature (flat space or the plane), and negative curvature (hyperbolic space).
Is the universe finite and bounded as Einstein thought, or is it unbounded, filled with an endless infinity of stars and galaxies? The question fascinated cosmologists throughout the twentieth century and since, but the answer has proved elusive. In the rest of this chapter, I will tell you what has been discovered in the recent past and how it bears on the answer.
The Three Fates
About a month ago I was home working on this book when I was disturbed by a knock on my front door. When I answered it, three very neatly dressed young people handed me a leaflet. I don’t usually bother arguing with proselytizers, but when I saw the title of the booklet—Are You Prepared for the End of the Universe?—I couldn’t resist asking them a few questions. When I asked them how they knew anything about the end of the universe, they told me that modern-day scientists had confirmed the biblical account of Armageddon and that the end of the universe was a scientific certainty.
They were probably right. Modern-day scientists do predict that the universe—at least the universe as we know it—will come to an end. Every reasonable cosmological theory says so. When and how it will happen varies from one set of assumptions to another, but all agree that it won’t happen for some tens of billions of years.
Broadly speaking there are two “end-of-the-world” scenarios. To understand them, think of a stone being thrown vertically into the air. Actually, I want to forget the air. Let’s throw the stone from an airless asteroid. One of two things can happen. The gravitational pull of the asteroid may be sufficient to pull the stone back, or it may not be. In the first case the stone will reverse its outward motion and come crashing back down, but in the second case it will overcome the gravitational attraction and fly off forever. It all depends on whether the initial velocity of the stone is faster than the escape velocity. The escape velocity depends on the mass of the asteroid: the bigger the mass, the larger the escape velocity.
According to the General Theory of Relativity, the fate of the universe is much like the fate of that stone.5 The galaxies (and other material in the universe) have been shot out of the Big Bang explosion and are now flying away from each other. Meanwhile, gravity is working to pull them back. To put it another way, the balloonlike universe is growing, but gravitating matter is slowing down the expansion. Will the expansion keep going, or will gravity reverse it and eventually cause the universe to start shrinking? The answer is quite similar to the case of the asteroid and the stone. If there is enough mass in the universe, it will reverse direction and eventually collapse in a dreadful, big, superheated crunch. On the other hand, if there is not enough mass, the universe will keep expanding indefinitely. In this case the end may be gentler, but eventually the universe will become so thinned out that it will die a cold death.
For both the stone and the universe there is a third possibility. The stone might be precisely at the escape velocity. This would require a perfect balance between gravitational attraction and outward velocity. If you do the math in this case, you will find that the stone keeps going but at an ever-decreasing velocity. The same is true of the universe. If there is an exact balance between mass density and outward expansion, the universe will eternally expand but at an ever-decreasing rate.
Geometry Is Fate
Three possible geometries and three possible fates; is there a connection? Indeed there is. Einstein’s theory of gravity (without a cosmological constant) relates geometry to the presence of mass; mass affects geometry. The Newtonian dictum that “mass is the source of the gravitational field” is replaced by “mass warps and bends space.” That’s the link that relates the three geometries to the three fates. The details are in the difficult mathematics (tensor calculus and Riemannian geometry) of general relativity, but the result (with no cosmological constant) is easy to understand:
If the mass density in the universe is large enough to reverse the outgoing expansion, it will distort space into a sphere, a 3-sphere that is. This is the case of a closed-and-bounded universe. And its fate is a final crunch or, in the technical jargon, a singularity. This case is called the closed universe, or the k = 1 universe.
If the mass density is less than the minimum amount needed to close the universe, then it is also insufficient to reverse the motion. In this case it distorts space into a hyperbolic geometry. The hyperbolic universe expands forever. It’s called the open universe, or the k = –1 case.
If the universe is right on the knife-edge, between open and closed, then the geometry of space is flat, uncurved, Euclidean space, but the universe endlessly expands, albeit at an ever-diminishing rate. This is called the flat universe, and it is labeled k = 0.
So, which is it?
Some say the world will end in fire,
Some say in ice.
From what I’ve tasted of desire
I hold with those who favor fire.
But if it had to perish twice,
I think I know enough of hate
To know that for destruction ice
Is also great
And would suffice.
— Robert Frost, “Fire and Ice”
When I asked the three young missionaries if it would be the hot death or the cold death, they said that it all depended on me. Very probably it would be the hot death unless I changed my ways.
Physicists and cosmologists are less certain of the final reckoning. For decades they have tried to determine which of the three fates will rule the last days. The first way to find out is very direct: use telescopes to look out into the distant reaches of space and count all the mass that can be seen—stars, galaxies, giant clouds of dust, and everything else that can be seen or deduced. Is the gravitational pull of all that material enough to turn the expansion around?
We know how fast the universe is expanding today. Hubble determined that the velocity of a distant galaxy is proportional to its distance—the factor of proportionality being the Hubble constant. This quantity is the best measure of the expansion rate: the larger the Hubble constant, the faster every galaxy is moving away from us. The units of the Hubble constant are velocity per unit distance. Astronomers usually quote it as “kilometers per second per megaparsec.” Everyone will recognize kilometers per second as a unit of velocity. One kilometer per second is about three times the speed of sound, i.e., Mach 3. The megaparsec is less familiar. It’s a unit of length, convenient for the study of cosmology. One megaparsec is about three million light-years, or thirty million trillion kilometers, a little more than the distance to our neighboring galaxy, Andromeda.
The value of the Hubble constant has been repeatedly measured over the years and has been the subject of a lively debate. Astronomers agreed it was somewhere between fifty and one hundred kilometers per second per megaparsec, but only in the recent past has the answer been resolved as about seventy-five in these units. The implication is that at a distance of one megaparsec, the galaxies are receding with a velocity of 75 km/ sec. At two megaparsecs their velocity is 150 km/ sec.
Now 75 km/ sec sounds awfully fast by terrestrial standards. At that rate it would take about ten minutes to circumnavigate the earth. But it’s not at all fast from the viewpoint of a physicist or astronomer. For example, the pinwheel motion of the Milky Way imparts a velocity to the earth that’s ten times faster. And by comparison with the speed of light, it’s a snail’s pace.
Nevertheless, the expansion is slow, and it would take very little mass density to turn it around.
Knowing the expansion rate, it is a straightforward application of Einstein’s equations to compute how large a mass density would be required to prevent the universe from eternally growing. The answer? Just 10–25 kilograms per cubic meter would be the knife-edge value: just barely enough to eventually reverse the outward flow of the galaxies. That’s not much. It’s roughly the mass of fifty protons in a cubic meter. A tiny bit more would be enough to curve the universe into a 3-sphere and turn the Big Bang into a disastrous big crunch. If the density were exactly this critical value, the universe would be flat (i.e., k = 0).
The Cosmic Landscape Page 15
