Many Worlds in One: The Search for Other Universes (16 page)
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Authors: Alex Vilenkin
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If there has ever been a problem calling for measures of last resort, it is the cosmological constant problem. Different contributions to the vacuum energy density conspire to cancel one another with an accuracy of one part in 10
120
. This is the most notorious and perplexing case of fine-tuning in
physics. Andrei Linde was one of the first brave souls to apply anthropic reasoning to this problem. He was not satisfied with the vague talk about “other universes” and suggested a specific model of how the cosmological constant could be variable and what could make it change from one place to another.
120
. This is the most notorious and perplexing case of fine-tuning in
physics. Andrei Linde was one of the first brave souls to apply anthropic reasoning to this problem. He was not satisfied with the vague talk about “other universes” and suggested a specific model of how the cosmological constant could be variable and what could make it change from one place to another.
Linde used an idea that had worked for him before. Remember the little ball rolling down in the energy landscape? The ball represented a scalar field; and its elevation, the energy density of the field. As the field rolled downhill, its energy drove the inflationary expansion of the universe.
The feature Linde took from this model of inflation was that different elevations in the landscape correspond to different energy densities. He assumed the existence of another scalar field with an energy landscape of its own. To avoid confusion with the field responsible for inflation, we shall call the latter field “the inflaton”—its usual name in the physics literature. In our neighborhood, the inflaton has already rolled to the bottom of its energy hill. (This happened 14 billion years ago, at the end of inflation.) To prevent his new field from rolling downhill too quickly, Linde had to require that the slope should be exceedingly gentle, much more so than in the model of inflation. Any slope, no matter how small, will eventually cause the field to roll down. With smaller slopes it will take longer “to get the ball rolling.” Linde assumed the slope to be so small that the field would not move much in the 14 billion years that elapsed since the big bang. But if the slope extends to a great length in both directions, the energy density can reach very large positive or negative values. (See
Figure 13.2
.)
Figure 13.2
.)
Figure 13.2
.
A scalar field on a very gentle slope of the energy landscape.
.
A scalar field on a very gentle slope of the energy landscape.
The full energy density of the vacuum—the cosmological constant—is obtained by adding the energy density of the scalar field to the vacuum energy densities of fermions and bosons calculated from particle physics. Even if there are no miraculous cancellations and the particle physics contribution is huge, there will be a spot on the slope where the scalar field contribution has an equal magnitude and opposite sign, so the total vacuum energy density is zero. The scalar field is presumably very close to that spot in our part of the universe.
If the scalar field were to vary from one part of the universe to another, the cosmological “constant” would also be variable, and that is all one needs to apply the anthropic principle. But what could cause the scalar field to vary? Linde had a good answer to this one as well!
Prior to the big bang, in the course of eternal inflation, the field experienced random quantum kicks. As before, we can represent the behavior of the field by that of a party of random walkers (see Chapter 8). The slope of the hill is too small to be of any importance in this case, so the walkers step left and right with nearly equal probability. Even if they start at the same place, the walkers will gradually drift apart and, given enough time, will spread along the entire slope. (Recall, there is no shortage of time in eternal inflation.) Since the walkers represent scalar field values in different regions of space, we conclude that quantum processes during inflation necessarily generate a distribution of regions with all possible values of the field—and therefore all possible values of the cosmological constant.
While the walkers are wandering on the slope, the distances between the regions they represent are being stretched by the exponential inflationary expansion. As a result, the spatial variation of the vacuum energy density is extremely small.
au
You would have to travel googles of miles before you noticed the slightest change.
au
You would have to travel googles of miles before you noticed the slightest change.
Linde’s model can be extended to include more scalar fields and make other constants of nature variable.
av
And if the fundamental particle physics allows the constants to vary, then quantum processes during eternal inflation inescapably generate vast regions of space with all possible values of
the constants. Eternal inflation thus provides a natural arena for applications of the anthropic principle.
av
And if the fundamental particle physics allows the constants to vary, then quantum processes during eternal inflation inescapably generate vast regions of space with all possible values of
the constants. Eternal inflation thus provides a natural arena for applications of the anthropic principle.
Now that we have an ensemble of regions with different values of the cosmological constant, what value should we expect to observe? In regions where the mass density of the vacuum is greater than the density of water (1 gram per cubic centimeter), stars would be torn apart by repulsive gravity. It turns out, however, that a much smaller vacuum density would do enough damage to make observers impossible. This was shown by Steven Weinberg, in a paper that later became a classic of anthropic reasoning.
Figure 13.3
.
Steven Weinberg. (Photo by Frank Curry, Studio Penumbra)
.
Steven Weinberg. (Photo by Frank Curry, Studio Penumbra)
As the universe expands, the density of matter is diluted, and inevitably there comes a time when it drops below that of the vacuum. Weinberg found that once this happens, matter can no longer clump into galaxies; instead it is dispersed by the repulsive vacuum gravity. The larger the cosmological constant is, the earlier is the time of vacuum dominance. And regions
where it dominates before any galaxies have had a chance to form will have no cosmologists to worry about the cosmological constant problem.
where it dominates before any galaxies have had a chance to form will have no cosmologists to worry about the cosmological constant problem.
The effect of a negative cosmological constant is even more devastating. In this case, the vacuum gravity is attractive, and vacuum domination leads to a rapid contraction and collapse of the corresponding regions. The anthropic principle requires that the collapse should not occur before galaxies and observers have had time to evolve.
According to Weinberg’s analysis, the largest mass density of the vacuum that still allows some galaxies to form is about the mass of a few hundred hydrogen atoms per cubic meter—10
27
times smaller than the density of water. This was a great improvement over the googles of tons per cubic centimeter suggested by particle physicists’ calculations.
27
times smaller than the density of water. This was a great improvement over the googles of tons per cubic centimeter suggested by particle physicists’ calculations.
If indeed the smallness of the cosmological constant is due to anthropic selection, then, small though it is, the constant does not have to be exactly zero. In fact, there seems to be no reason why it should be much smaller than required by the anthropic principle. In the late 1980s, the observational accuracy was already reaching the level necessary to detect such values of the constant, and Weinberg made a prediction that it would soon show up in astronomical observations. Indeed, almost a decade later the first hints of cosmological constant appeared in supernova data.
Mediocrity Raised to a Principle
I consider myself an average man, except for the fact that I consider myself an average man.
—MICHEL DE MONTAIGNE
T
he most scathing criticism raised against the anthropic principle is that it does not yield any testable predictions. All it says is that we can observe only those values of the constants that allow observers to exist. This can hardly be regarded as a prediction, since it is guaranteed to be true. The question is, Can we do any better? Is it possible to extract some nontrivial predictions from anthropic arguments?
he most scathing criticism raised against the anthropic principle is that it does not yield any testable predictions. All it says is that we can observe only those values of the constants that allow observers to exist. This can hardly be regarded as a prediction, since it is guaranteed to be true. The question is, Can we do any better? Is it possible to extract some nontrivial predictions from anthropic arguments?
If the quantity I am going to measure can take a range of values, determined largely by chance, then I cannot predict the result of the measurement with certainty. But I can still try to make a statistical prediction. Suppose, for example, I want to predict the height of the first man I am going to see when I walk out into the street. According to the
Guinness Book of Records
, the tallest man in medical history was the American Robert Pershing Wadlow, whose height was 2.72 meters (8 feet 11 inches). The shortest adult man, the Indian Gul Mohammed, was just 56 centimeters tall (about 22 inches). If I want to play it really safe, I should predict that the first man I see will be somewhere between these two extremes. Barring the possibility
of breaking the Guinness records, this prediction is guaranteed to be correct.
Guinness Book of Records
, the tallest man in medical history was the American Robert Pershing Wadlow, whose height was 2.72 meters (8 feet 11 inches). The shortest adult man, the Indian Gul Mohammed, was just 56 centimeters tall (about 22 inches). If I want to play it really safe, I should predict that the first man I see will be somewhere between these two extremes. Barring the possibility
of breaking the Guinness records, this prediction is guaranteed to be correct.
To make a more meaningful prediction, I could consult the statistical data on the height of men in the United States. The height distribution follows a bell curve, shown in
Figure 14.1
, with a median value at 1.77 meters (about 5 feet 9½ inches). (That is, 50 percent of men are shorter and 50 percent are taller than this value.) The first man I meet is not likely to be a giant or a dwarf, so I expect his height to be in the mid-range of the distribution. To make the prediction more quantitative, I can assume that he will not be among the tallest 2.5 percent or shortest 2.5 percent of men in the United States. The remaining 95 percent have heights between 1.63 meters (5 feet 4 inches) and 1.90 meters (6 feet 3 inches). If I predict that the man I meet will be within this range of heights and then perform the experiment a large number of times, I can expect to be right 95 percent of the time. This is known as a prediction at 95 percent confidence level.
Figure 14.1
, with a median value at 1.77 meters (about 5 feet 9½ inches). (That is, 50 percent of men are shorter and 50 percent are taller than this value.) The first man I meet is not likely to be a giant or a dwarf, so I expect his height to be in the mid-range of the distribution. To make the prediction more quantitative, I can assume that he will not be among the tallest 2.5 percent or shortest 2.5 percent of men in the United States. The remaining 95 percent have heights between 1.63 meters (5 feet 4 inches) and 1.90 meters (6 feet 3 inches). If I predict that the man I meet will be within this range of heights and then perform the experiment a large number of times, I can expect to be right 95 percent of the time. This is known as a prediction at 95 percent confidence level.
In order to make a 99 percent confidence level prediction, I would have to discard 0.5 percent at both ends of the distribution. As the confidence level is increased, my chances of being wrong get smaller, but the predicted range of heights gets wider and the prediction less interesting.
Figure 14.1
.
Height distribution of men in the United States. The number of men whose height is within a given interval is proportional to the area under the corresponding portion of the curve. The shaded “tails” of the bell curve mark 2.5 percent at low and high ends of the distribution. The range between the marked areas is predicted at 95 percent confidence level.
.
Height distribution of men in the United States. The number of men whose height is within a given interval is proportional to the area under the corresponding portion of the curve. The shaded “tails” of the bell curve mark 2.5 percent at low and high ends of the distribution. The range between the marked areas is predicted at 95 percent confidence level.
Can a similar technique be applied to make predictions for the constants of nature? I was trying to find the answer to this question in the summer of 1994, when I visited my friend Thibault Damour at the Institut des Hautes Études Scientifiques in France. The institute is located in a small village, Bures-sur-Yvette, a thirty-minute train ride from Paris. I love the French countryside and, despite the calories, French food and wine. The famous Russian physicist Lev Landau used to say that a single alcoholic drink was enough to kill his inspiration for a week. Luckily, this has not been my experience. In the evenings, with my spirits up after a very enjoyable dinner, I would take a walk in the meadows along the little river Yvette, and my thoughts would gradually return to the problem of anthropic predictions.
Suppose some constant of nature, call it
X
, varies from one region of the universe to another. In some of the regions observers are disallowed, while in others observers can exist and the value of
X
will be measured. Suppose further that some Statistical Bureau of the Universe collected and published the results of these measurements. The distribution of values measured by different observers would most likely have the shape of a bell curve, similar to the one in
Figure 14.1
. We could then discard 2.5 percent at both ends of the distribution and predict the value of
X
at a 95 percent confidence level.
X
, varies from one region of the universe to another. In some of the regions observers are disallowed, while in others observers can exist and the value of
X
will be measured. Suppose further that some Statistical Bureau of the Universe collected and published the results of these measurements. The distribution of values measured by different observers would most likely have the shape of a bell curve, similar to the one in
Figure 14.1
. We could then discard 2.5 percent at both ends of the distribution and predict the value of
X
at a 95 percent confidence level.
Figure 14.2
.
An observer randomly picked in the universe. The values of the constants measured by this observer can be predicted from a statistical distribution.
.
An observer randomly picked in the universe. The values of the constants measured by this observer can be predicted from a statistical distribution.
What would be the meaning of such a prediction? If we randomly picked observers in the universe, their observed values of
X
would be in the predicted interval 95 percent of the time. Unfortunately, we cannot test this kind of prediction, because all regions with different values of
X
are beyond our horizon. We can only measure
X
in our local region. What we can do, though, is to think of ourselves as having been randomly picked. We are just one out of a multitude of civilizations scattered throughout the universe. We have no reason to believe a priori that the value of
X
in our region is very rare, or otherwise very special compared with the values measured by other observers. Hence, we can predict, at 95 percent confidence level, that our measurements will yield a value in the specified range. The assumption of being unexceptional is crucial in this approach; I called it “the principle of mediocrity.”
X
would be in the predicted interval 95 percent of the time. Unfortunately, we cannot test this kind of prediction, because all regions with different values of
X
are beyond our horizon. We can only measure
X
in our local region. What we can do, though, is to think of ourselves as having been randomly picked. We are just one out of a multitude of civilizations scattered throughout the universe. We have no reason to believe a priori that the value of
X
in our region is very rare, or otherwise very special compared with the values measured by other observers. Hence, we can predict, at 95 percent confidence level, that our measurements will yield a value in the specified range. The assumption of being unexceptional is crucial in this approach; I called it “the principle of mediocrity.”
Some of my colleagues objected to this name. They suggested “the principle of democracy” instead. Of course, nobody wants to be mediocre, but the name expresses nostalgia for the times when humans were at the center of the world. It is tempting to believe that we are special, but in cosmology, time and again, the assumption of being mediocre proved to be a very fruitful hypothesis.
The same kind of reasoning can be applied to predicting the height of people. Imagine for a moment that you don’t know your own height. Then you can use statistical data for your country and gender to predict it. If, for example, you are an adult man living in the United States and have no reason to think that you are unusually tall or short, you can expect, at 95 percent confidence, to be between 1.63 and 1.90 meters tall.
I later learned that similar ideas had been suggested earlier by the philosopher John Leslie and, independently, by the Princeton astrophysicist Richard Gott. The main interest of these authors was in predicting the longevity of the human race. They argued that humanity is not likely to last much longer than it has already existed, since otherwise we would find ourselves to be born surprisingly early in its history. This is what’s called the “doomsday argument.” It dates back to Brandon Carter, the inventor of the anthropic principle, who presented the argument in a 1983 lecture, but never in print (it appears that Carter already had enough controversy on his hands).
1
Gott also used a similar argument to predict the fall of the Berlin Wall and the lifetime of the British journal
Nature
, where he published his first paper
on this topic. (The latter prediction, that
Nature
will go out of print by the year 6800, is yet to be verified.)
1
Gott also used a similar argument to predict the fall of the Berlin Wall and the lifetime of the British journal
Nature
, where he published his first paper
on this topic. (The latter prediction, that
Nature
will go out of print by the year 6800, is yet to be verified.)
If we have a statistical distribution for the constants of nature measured by all the observers in the universe, we can use the principle of mediocrity for making predictions at a specified confidence level. But where are we going to get the distribution? In lieu of the data from the Statistical Bureau of the Universe, we will have to derive it from theoretical calculations.
The statistical distribution cannot be found without a theory describing the multiverse with variable constants. At present, our best candidate for such a theory is the theory of eternal inflation. As we discussed in the preceding chapter, quantum processes in the inflating spacetime spawn a multitude of domains with all possible values of the constants. We can try to calculate the distribution for the constants from the theory of eternal inflation, and then—perhaps!—we could check the results against the experimental data. This opens an exciting possibility that eternal inflation can, after all, be subjected to observational tests. Of course, I felt this opportunity was not to be missed.
Consider a large volume of space, so large that it includes regions with all possible values of the constants. Some of these regions are densely populated with intelligent observers. Other regions, less favorable to life, are greater in volume, but more sparsely populated. Most of the volume will be occupied by huge barren domains, where observers cannot exist.
The number of observers who will measure certain values of the constants is determined by two factors: the volume of those regions where the constants have the specified values (in cubic light-years, for example), and the number of observers per cubic light-year. The volume factor can be calculated from the theory of inflation, combined with a particle physics model for variable constants (like the scalar field model for the cosmological constant).
2
But the second factor, the population density of observers, is much more problematic. We know very little about the origin of life, let alone intelligence. How, then, can we hope to calculate the number of observers?
2
But the second factor, the population density of observers, is much more problematic. We know very little about the origin of life, let alone intelligence. How, then, can we hope to calculate the number of observers?
What comes to the rescue is that some of the constants do not directly affect the physics and chemistry of life. Examples are the cosmological constant, the neutrino mass, and the parameter, usually denoted by
Q
, that characterizes the magnitude of primordial density perturbations. Variation of such life-neutral constants may influence the formation of galaxies, but not the chances for life to evolve within a given galaxy. In contrast, constants such as the electron mass or Newton’s gravitational constant have a direct impact on life processes. Our ignorance about life and intelligence can be factored out if we focus on those regions where the life-altering constants have the same values as in our neighborhood and only the life-neutral constants are different. All galaxies in such regions will have about the same number of observers, so the density of observers will simply be proportional to the density of galaxies.
3
Q
, that characterizes the magnitude of primordial density perturbations. Variation of such life-neutral constants may influence the formation of galaxies, but not the chances for life to evolve within a given galaxy. In contrast, constants such as the electron mass or Newton’s gravitational constant have a direct impact on life processes. Our ignorance about life and intelligence can be factored out if we focus on those regions where the life-altering constants have the same values as in our neighborhood and only the life-neutral constants are different. All galaxies in such regions will have about the same number of observers, so the density of observers will simply be proportional to the density of galaxies.
3
Thus, the strategy is to restrict the analysis to life-neutral constants. The problem then reduces to the calculation of how many galaxies will form per given volume of space—a well-studied astrophysical problem. The result of this calculation, together with the volume factor derived from the theory of inflation, will yield the statistical distribution we are looking for.
As I was thinking about observers in remote domains with different constants of nature, it was hard to believe that the equations I scribbled in my notepad had much to do with reality. But having gone this far, I bravely pressed ahead: I wanted to see if the principle of mediocrity could shed some new light on the cosmological constant problem.
The first step was already taken by Steven Weinberg. He studied how the cosmological constant affects galaxy formation and found the anthropic bound on the constant—the value above which the vacuum energy would dominate the universe too soon for any galaxies to form. Moreover, as I already mentioned, Weinberg realized that there was a
prediction
implicit in his analysis. If you pick a random value between zero and the anthropic bound, this value is not likely to be much smaller than the bound, for the same reason that the first man you meet is not likely to be a dwarf. Weinberg
argued, therefore, that the cosmological constant in our part of the universe should be comparable to the anthropic bound.
aw
prediction
implicit in his analysis. If you pick a random value between zero and the anthropic bound, this value is not likely to be much smaller than the bound, for the same reason that the first man you meet is not likely to be a dwarf. Weinberg
argued, therefore, that the cosmological constant in our part of the universe should be comparable to the anthropic bound.
aw
The argument sounded convincing, but I had my reservations. In regions where the cosmological constant is comparable to the anthropic bound, galaxy formation is barely possible and the density of observers is very low. Most observers are to be found in regions teeming with galaxies, where the cosmological constant is well below the bound—small enough to dominate the universe only after the process of galaxy formation is more or less complete. The principle of mediocrity says that we are most likely to find ourselves among these observers.
I made a rough estimate, which suggested that the cosmological constant measured by a typical observer should not be much greater than ten times the average density of matter. A much smaller value is also improbable—like the chance of meeting a dwarf. I published this analysis in 1995, predicting that we should measure a value of about ten times the matter density in our local region.
4
More detailed calculations, also based on the principle of mediocrity, were later performed by the Oxford astrophysicist George Efstathiou
5
and by Steven Weinberg, who was now joined by his University of Texas colleagues Hugo Martel and Paul Shapiro. They arrived at similar conclusions.
4
More detailed calculations, also based on the principle of mediocrity, were later performed by the Oxford astrophysicist George Efstathiou
5
and by Steven Weinberg, who was now joined by his University of Texas colleagues Hugo Martel and Paul Shapiro. They arrived at similar conclusions.
I was very excited about this newly discovered possibility of turning anthropic arguments into testable predictions. But very few people shared my enthusiasm. One of the leading superstring theorists, Joseph Polchinski, once said that he would quit physics if a nonzero cosmological constant were discovered.
ax
Polchinski realized that the only explanation for a small cosmological constant would be the anthropic one, and he just could not stand the thought. My talks about anthropic predictions were sometimes followed by an embarrassed silence. After one of my seminars, a prominent Princeton cosmologist rose from his seat and said, “Anyone who wants to work on the anthropic principle—should.” The tone of his remark left little doubt that he believed all such people would be wasting their time.
ax
Polchinski realized that the only explanation for a small cosmological constant would be the anthropic one, and he just could not stand the thought. My talks about anthropic predictions were sometimes followed by an embarrassed silence. After one of my seminars, a prominent Princeton cosmologist rose from his seat and said, “Anyone who wants to work on the anthropic principle—should.” The tone of his remark left little doubt that he believed all such people would be wasting their time.
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