The Cosmic Landscape (57 page)

As I have repeatedly emphasized, there is no known explanation of the special properties of our pocket other than the populated Landscape—no explanation that does not require supernatural forces. But there are real problems with our current understanding of the populated Landscape, and some are potentially very serious. In my own view the biggest challenges have to do with Eternal Inflation—the mechanism that may populate the Landscape. The cloning of space is not seriously questioned by anyone, and neither is the spinning off of bubbles by the metastable vacuum. Both ideas are based on some of the most trusted principles of general relativity and quantum mechanics. But no one has a clear understanding of how these observations are to be turned into predictions—even statistical guesses—about our universe.

Given a megaverse, endlessly filled with pocket universes, the Anthropic Principle is an effective tool to weed out and eliminate most of them as candidates for our universe. Those that don’t support our kind of life can be tossed in the trash. That provides marvelous explanatory power for questions like, why is the cosmological constant small? But much of the controversy over the Anthropic Principle has to do with a more ambitious agenda, the hope that it can substitute for the silver bullet in predicting all of nature.

This is an unreasonable expectation. There is no reason why every feature of nature should be determined by the existence of life. Some features will be determined by mathematical reasoning of the traditional kind, some by anthropic consideration, and some features may just be accidental environmental facts.

As always, the world of the big-brained fish (chapter 5) is a good place to get some perspective. Let’s follow the fish as they learn more about their world:

In time, with the help of the codmologists, the fish came to the realization that they inhabited a planet revolving around a hot glowing nuclear reactor—a star—that provided the heat that warmed their water. The question that had obsessed their best minds would take on an entirely new complexion. Realizing that the temperature depends on how distant the star is, the puzzle would be restated: “Why is the orbital distance of our planet from the source of heat so finely tuned?” But the answer of the codmologists is the same. The universe is big. It has many stars and planets, and some small fraction are just the right distance for liquid water and for fish.

But some fyshicists are unhappy with the answer. They correctly claim that the temperature depends on something else besides the orbital distance. The luminosity of the star—the rate that it radiates energy—comes into the equation. “We could be close to a small, dim star or far from a bright giant. There is a whole range of possibilities. The Ickthropic Principle is a failure. There is no way that it can explain the distance to our star.”

But it was never the intention of the codmologists to explain every feature of nature. Their claim that the universe is big and contains a wide variety of environments is as valid as ever. The criticism that the Ickthropic Principle can’t explain everything is a straw man, set up by the fyshicists just to knock it down.

There are very close parallels between this story and the case of the Anthropic Principle. One example involves both the cosmological constant and the lumpiness in the early universe. In chapter 2 I related how Weinberg explained the fact that the cosmological constant is so incredibly small—if it were much bigger, the very small density contrasts (lumpiness) in the universe could not have grown into galaxies. But suppose the initial density contrasts were a bit stronger. Then a somewhat larger cosmological constant could be tolerated. As in the case of the distance and luminosity of the star, there is a range of possible values for the cosmological constant and lumpiness which permit life, or at least galaxies. The Anthropic Principle by itself is powerless to choose between them. Some physicists take this as evidence against the Anthropic Principle. Once again I regard it as a straw man.

But it is possible that with further input both the fyshicists and we could do better. Let’s bring in the astro-fyshicists: the experts on how stars form and evolve. These fishy scientists have studied the formation of stars from giant gas clouds, and as expected, they find that a range of luminosities is possible. There is no way to be certain of the stellar luminosity without getting above the surface and observing the star, but still it seems that some values for the luminosity are more likely than others. Indeed, the astro-fyshicists find that the majority of long-lived stars should have a luminosity of between 10
²⁶
and 10
²⁷
watts. Their star is probably in this range.

Now the codmologists are in business. With such luminosity the planet would have to be about 100,000,000 miles from the star in order to have a climate temperate enough for liquid water. That prediction is not as absolute as they might like. Like all probabilistic claims, it could be wrong. But still, it’s better than no prediction.

What the two situations—one involving liquid water and the other the formation of galaxies—have in common is that anthropic (or ickthropic) considerations alone are not enough to determine or predict everything. This is inevitable if there is more than one valley in the Landscape that can support our kind of life. With 10
⁵⁰⁰
valleys it seems certain that this will be the case. Let’s call such vacuums anthropically acceptable. The usual physics and chemistry may be very similar in many of these—electrons, nuclei, gravity, galaxies, stars, and planets much like we know them in our own world. The differences may be in those things that only a high-energy particle physicist would be interested in. For example, there are many particles in nature—the top-quark, the tau lepton, the bottom-quark, and others—whose detailed properties hardly matter at all to the ordinary world. They are too heavy to make any difference except in high-energy collisions in giant accelerator laboratories. Some of these vacuums (including our own) may have many new types of particles that make little or no difference to ordinary physics. Is there any way to explain in which of these anthropically acceptable vacuums we live? Obviously, the Anthropic Principle cannot help us predict which one we live in—any of these vacuums is acceptable.

This conclusion is frustrating. It leaves the theory open to the serious criticism that it has no predictive power, something that scientists are very sensitive about. To address this deficiency many cosmologists have tried to supplement the Anthropic Principle with additional probabilistic assumptions. For example, instead of asking precisely what the value is of the top-quark mass, we might try to ask what the probability is that the top-quark has a mass in a particular range.

Here is one such proposal: eventually we will know enough about the Landscape to know just how many valleys exist for each range of the top-quark mass. Some values of the mass may correspond to a huge number of valleys—some may correspond to a much smaller number. The proposal is simple enough—the values of the top-quark mass that correspond to a great many valleys are more probable than values that correspond to few valleys. To carry out this kind of program, we would have to know far more about the Landscape than we do now. But let’s put ourselves in the future, when the details of the Landscape have been mapped out by String Theory, and we know the number of vacuums with any conceivable set of properties. Then the natural proposal would be that the relative probability for two distinct values of some constant would be the ratio of the number of appropriate vacuums. For example, if there were twice as many vacuums with mass value M
₁
as mass value M
₂
, it would follow that M
₁
was twice as likely as M
₂
. If we are lucky we might find that some value of the top-quark mass corresponds to an exceptionally large number of valleys. Then we might move forward by assuming this value to be true for our world.

No single prediction of this kind, based as it is on probability, can make or break the theory, but many successful statistical predictions would add great weight to our confidence.

The idea I have just outlined is tempting, but there are serious reasons to question the logic. Remember that the Landscape is merely the space of possibilities. If we were fyshicists thinking about the Landscape of possible planets, we might count all sorts of bizarre possibilities as long as they were solutions to the equations of physics: planets with cores of pure gold among others. The equations of physics have just as many solutions corresponding to huge golden balls as iron balls.
⁹
The logic of counting possibilities would say that an iron-cored planet is no more likely to be the fyshicists’ home than a golden planet—obviously a mistake.
¹⁰

What we really want to know is not how many
possibilities
of each kind there are: what we want to know is how many
planets
of each kind there are. For this we need much more than the abstract counting of possibilities. We need to know how iron and gold are produced during the slow nuclear burning inside stars.

Iron is the most stable of all the elements. It is more difficult to dislodge a proton or neutron from an iron nucleus than from any other. Consequently, nuclear burning proceeds down the periodic table, hydrogen to helium to lithium, until finally it ends with iron. As a result iron is far more common in the universe than any of the elements with a higher atomic number, including gold. That’s why iron is cheap and gold costs almost five hundred dollars an ounce. Iron is ubiquitous in the universe: gold, by contrast, is very rare. Almost all solid planets will have far more iron in their cores than gold. By comparison with iron planets, the number of solid-gold planets in the universe is minute, very possibly zero. We want to count
actualities
rather than possibilities.

The same logic as applies to planets ought to apply to pocket universes. But now we encounter a horrific problem with Eternal Inflation. Because it goes on forever, Eternal Inflation (as it is now conceived) will create an infinite number of pockets—in fact an infinite number of every kind of pocket universe. Thus, we face an age-old mathematical problem of comparing infinite numbers. Which infinity is bigger than which and by how much?

The problem of comparing infinite numbers goes back to Georg Cantor, who in the late nineteenth century asked exactly that question—how do you compare the size of two sets, each of which has an infinite number of elements? First he began by asking how to compare ordinary numbers. Suppose, for example, we have a bunch of apples and a bunch of oranges. The obvious answer is to count both bunches, but if all we want to know is which is bigger, then there is a more primitive thing we can do—something that doesn’t even require any knowledge of numbers: line up the apples and next to them line up the oranges, matching each orange to an apple. If some apples are left over, then there are more apples. If oranges are left over, then there are more oranges. If the oranges and apples match, their number is the same.

Cantor said the same thing could be done with infinite (or what he called transfinite) sets. Take, for example, the even integers and the odd integers. There is an infinite number of each, but is the infinite number the same? Line them up and see if they can be made to match in such a way that there is an even for every odd. Mathematicians call this a
one-to-one correspondence.

1 3 5 7 9   11 13 . . .

2 4 6 8 10 12 14 . . .

Note that the two lists eventually contain every odd integer and every even integer—none is left out. Moreover, they match exactly, so Cantor concluded that the number of evens and odds are the same even though they are both infinite.

What about the total number of integers, even and odd? That is obviously larger than the number of even integers—twice as large. But Cantor disagreed. The even integers can be matched exactly with the list of all integers.

1 2 3 4 5 6 7 . . .

2 4 6 8 10 12 14 . . .

According to the only mathematical theory of infinite numbers, the theory that Cantor constructed, the number of even integers is the same as the number of all integers! What is more, the set of numbers divisible by 10—10, 20, 30, 40, etc.—is exactly the same infinite size. The integers, the even or odd integers, the integers that are divisible by ten—they are all examples of what mathematicians call countably infinite sets, and they are all equally large.
¹¹

Let’s do a thought experiment involving infinite numbers. Imagine an infinite bag filled with all the integers written on scraps of paper. Here is the experiment: first shake the bag to thoroughly mix up the scraps. Now reach in and draw out a single integer. The question is—what is the probability that you’ve pulled out an even integer?

The naive answer is simple. Since half the integers are even, the probability must be one-half—50 percent. But we can’t really do this experiment because no one can make an infinite bag of integers. So to test the theory, we can cheat a little and use a finite bag, containing, let’s say, the first thousand integers. Sure enough if we do the experiment over and over we will, indeed, find the probability to pull an even integer is one-half. Next we can do the same experiment with a bag filled with the first ten thousand integers. Again, since half the scraps are even and half are odd, we will find the probability for an even integer is one-half. Do it again with the first one hundred thousand integers, the first million integers, the first billion, and so on. Each time the probability is one-half. It is reasonable to extrapolate from this that if the bag had an infinite number of scraps, the probability would remain one-half.