Fear of Physics (21 page)

 

 

The ultimate testing ground of Einstein’s ideas is the universe itself. General relativity not only tells us about the curvature of space around local masses but implies that the geometry of the entire universe is governed by the matter within it. If there is sufficient average density of mass, the average curvature of space will be large enough so that space will eventually curve back on itself in the three-dimensional analogue of the two-dimensional surface of a sphere. More important, it turns out that in this case
the universe would have to stop expanding eventually and recollapse in a “big crunch,” the reverse of the big bang.

There is something fascinating about a “closed” universe, as such a high-density universe is called. I remember first learning about it as a high school student when I heard the astrophysicist Thomas Gold lecture, and it has stayed with me ever since. In a universe that closes back upon itself, light rays—which, of course, travel in straight lines in such a space—will eventually return back to where they started, just like latitudes and longitudes on the surface of the Earth. Thus, light can never escape out to infinity. Now, when such a thing happens on a smaller scale, that is, when an object has such a high density that not even light can escape from its surface, we call it a black hole. If our universe is closed, we are actually living inside a black hole! Not at all like you thought it would be, or like the Disney movie! That is because, as systems get larger and larger, the average density needed to produce a black hole gets smaller and smaller. A black hole with the mass of the sun would be about a kilometer in size when it first formed, with an average density of many tons per cubic centimeter. A black hole with the mass of our visible universe, however, would first form with a size comparable to the visible universe and an average density of only about 10
^–29
grams per cubic centimeter!

The current wisdom, though, suggests that we do not live inside a black hole. Since the early 1980s most theorists, at least, believed the average density of space was such that the universe is just on the borderline between a closed universe, which closes in on itself and that in general will recollapse, and an open universe, which is infinite and will in general continue to expand unabated forever. This borderline case, called a “flat” universe, is also infinite in spatial extent, and if the dominant gravitational attraction comes from matter, it will continue to expand forever,
with the expansion slowing down but never quite stopping. Since a flat universe requires about 100 times as much matter than is visible in all the stars and galaxies we can see, theorists had come to expect that as much as 99 percent of the universe is composed of dark matter, invisible to telescopes, as I described in chapter 3, even though the dark matter that had been inferred by weighing galaxies and clusters was about a factor of three smaller than the amount needed for a flat universe.

How can we tell if this supposition is correct? One way is to try to determine the total density of matter around galaxies and clusters of galaxies, as I described in chapter 3, which gave no direct indication of a flat universe. There is another way, at least in principle, which is essentially the same as an intelligent bug in Kansas trying to determine whether the Earth is round without going around it and without leaving the surface. Even if such a bug could not picture a sphere in his mind, just as we cannot picture a curved three-dimensional space in ours, he might envisage it by generalizing a flat two-dimensional experience. There are geometrical measurements that can be done at the Earth’s surface that are consistent only if this surface is a sphere. For example, Euclid told us over twenty centuries ago that the sum of the three angles in any triangle drawn on a piece of paper is 180°. If I draw a triangle with one 90° angle, the sum of the other two angles must be 90°. So each of these angles must be less than 90°, as shown in the two cases below:

This is true only on a flat piece of paper, however. On the surface of the sphere, I can draw a triangle for which
each
of the angles is 90°! Simply draw a line along the equator, then go up a longitude line to the North Pole, and then make a 90° angle and go back down another longitude to the equator:

Similarly, you may remember that the circumference of a circle of radius
r
is 2πr. However, on a sphere, if you travel out a distance
r
in all directions from, say, the North Pole, and then draw a circle connecting these points, you will find that the circumference of this circle is smaller than 2πr. This is easier to understand if you look at the sphere from outside:

If we were to lay out big triangles, or big circles, on the Earth’s surface, we could use the deviations from Euclid’s predictions to measure its curvature and we would find that it is a sphere. As can be seen from the drawings, however, in order to deviate significantly from the flat-surface predictions of Euclid, we would have to produce extremely large figures, comparable to the size of the Earth. We can similarly probe the geometry of our three-dimensional space. Instead of using the circumference of circles, which is a good way to map out the curvature of a two-dimensional surface, we could instead use the area or volume of spheres. If we consider a large enough sphere of radius
r
centered on the position of the Earth, the volume inside this sphere should deviate from the prediction of Euclid if our three-dimensional space is curved.

How can we measure the volume of a sphere whose size is a significant fraction of the visible universe? Well, if we assume that the density of galaxies in the universe is roughly constant over space at any time, then we can assume that the volume of any region will be directly proportional to the total number of galaxies inside that region. All we have to do, in principle, is
count
galaxies as a function of distance. If space were curved, then we should be able to detect a deviation from Euclid’s prediction. This was, in fact, tried in 1986 by two young astronomers who were then at Princeton, E. Loh and E. Spillar, and the results they announced purported to give evidence for the first time that the universe is flat, as we theorists expect it is. Unfortunately, it was shown shortly after they announced their results that uncertainties, due primarily to the fact that galaxies evolve in time and merge, made it impossible to use the data then at hand to draw any definitive conclusion.

Another way to probe the geometry of the universe is to measure the angle spanned by a known object such as a ruler held at some distance from your eye. On a flat plane, for example, it is
clear that this angle continues to decrease as the ruler gets farther and farther away:

However, on a sphere, this need not be the case:

In the early 1990s a study was performed of the angle spanned by very compact objects in the center of distant galaxies measured with radio telescopes out to distances of almost half the size of the visible universe. The behavior of this angle with increasing distance is again almost exactly what one would predict for a flat universe. A colleague of mine and I have shown that this test too has possible ambiguities due to the possible cosmic evolution over time of the objects examined.

In 1998 a totally different and unexpected opportunity arose for using just this procedure to determine the geometry of the universe, based on measurements of what is known as the Cosmic Microwave Background (CMB) radiation, the afterglow of the big bang.

This radiation, first measured in 1965, is coming at us from all directions and has been traveling through space largely unimpeded for almost 14 billion years. The last time this radiation interacted significantly with matter was when the universe was only 100,000 years old and had a temperature of about 3,000 degrees above absolute zero on the Kelvin scale. As a result, when we observe this radiation today it gives us a “picture” of what the distribution of matter and radiation was at that very early time. Since the radiation we measure today has been traveling at us from all directions for almost the entire age of the universe, measuring this radiation today across the whole sky provides a “picture” of a spherical surface, when the radiation last interacted with matter, located more than 14 billion light years away.

This spherical surface provides a perfect way to use the geometric method described earlier, if one could just find some kind of “ruler” on the surface whose angular size we could measure. Fortunately nature has provided just such a ruler because of gravity. Since for matter gravity is always attractive, any lump of matter will tend to collapse inward unless some form of pressure keeps it from collapsing. Before the universe had cooled below about 3,000 degrees it turned out that matter, primarily in the form of hydrogen, was completely ionized and interacted strongly with radiation. This provided a form of pressure that would have stopped any lump of matter up to a certain size from collapsing.

Why only up to a certain size? Well, when the universe was only 100,000 years old, any light ray could have traveled only about 100,000 light years. Since no information can travel faster than light, neither pressure nor even local gravitational attraction could have acted to affect initial lumps of matter larger than about 100,000 light years across. Once the universe cooled below 3,000 degrees, and matter became neutral, the radiation pressure on matter dropped to close to zero, and lumps could begin to collapse,
with the first lumps collapsing being those lumps that had previously not been affected by the pressure of radiation, i.e., those lumps that were approximately 100,000 light years across.