by Brian Greene
Now, being a quantum process, such tunneling events have a probabilistic character. You can’t predict when or where they will happen. But you can predict the probability that a tunneling event will happen in any given interval of time and burrow in any given direction—probabilities that depend on detailed features of the string landscape, such as the altitude of the various mountain peaks and valleys (the value, that is, of their respective cosmological constants). The more probable tunneling events will happen more often, and the resulting distribution of universes will reflect this. The strategy, then, is to use the mathematics of inflationary cosmology and string theory to calculate the distribution of universes, with various physical features, across the Landscape Multiverse.
The rub is that so far no one has been able to do so. Our current understanding suggests a lush string landscape with a gargantuan number of mountains and valleys, which makes it a ferociously difficult mathematical challenge to work out the details of the resulting multiverse. Pioneering work by cosmologists and string theorists have contributed significantly to our understanding, but the investigations are still rudimentary.5
To go further, multiverse proponents advocate introducing one more important element into the mix. Consideration of the selection effects introduced in the previous chapter: anthropic reasoning.
Predictions in a Multiverse III:
Anthropic reasoning
Many of the universes in a given multiverse are bound to be lifeless. The reason, as we’ve seen, is that changes to nature’s fundamental parameters from their known values tend to disrupt the conditions favorable for life to emerge.6 Our very existence implies that we could never find ourselves in any of the lifeless domains, and so there’s nothing further to explain about why we don’t see their particular combination of properties. If a given multiverse proposal implied a unique life-supporting universe, we’d be golden. We would work out that special universe’s properties mathematically; if they differed from what we’ve measured in our own universe, we could rule out that multiverse proposal. If the properties agreed with ours, we’d have an impressive vindication of anthropic multiverse theorizing—and reason to vastly expand our picture of reality.
In the more plausible case that there is not a unique life-supporting universe, a number of theorists (they include Steven Weinberg, Andrei Linde, Alex Vilenkin, George Efstathiou, and many others) have advocated an enhanced statistical approach. Rather than calculate the relative preponderance, within the multiverse, of various kinds of universes, they propose that we calculate the number of inhabitants—physicists usually call them observers—who would find themselves in various kinds of universes. In some universes, conditions might barely be compatible with life, so observers would be rare, like the occasional cactus in a harsh desert; other universes, with more hospitable conditions, would teem with observers. The idea is that, just as canine census data let us predict what kinds of dogs we can expect to encounter, so observer census data let us predict the properties that a typical inhabitant living somewhere in the multiverse—you and I, according to the reasoning of this approach—should expect to see.
A concrete example was worked out in 1997 by Weinberg and his collaborators Hugo Martel and Paul Shapiro. For a multiverse in which the cosmological constant varies from universe to universe, they calculated how abundant life would be in each. This difficult task was made feasible by invoking the Weinberg proxy (Chapter 6): instead of life proper, they considered the formation of galaxies. More galaxies means more planetary systems and hence, the underlying assumption goes, a greater likelihood of life, intelligent life in particular. Now, as Weinberg had found in 1987, even a modest cosmological constant generates enough repulsive gravity to disrupt galaxy formation so only domains of the multiverse that have sufficiently small cosmological constants need be considered. A cosmological constant that’s negative results in a universe that collapses well before galaxies form, so these realms of the multiverse can be omitted from the analysis, too. Anthropic reasoning thus focuses our attention on the portion of the multiverse in which the cosmological constant lies in a narrow window; as discussed in Chapter 6, the calculations show that for a given universe to contain galaxies, its cosmological constant needs to be less than about 200 times the critical density (a mass equivalent of about 10–27 grams in each cubic centimeter of space, or about 10–121 in Planck units).7
For universes whose cosmological constant is in this range, Weinberg, Martel, and Shapiro then undertook a more refined calculation. They determined the fraction of matter in each such universe that would clump together over the course of cosmological evolution, a pivotal step on the road to galaxy formation. They found that if the cosmological constant is very near the window’s upper limit, relatively few clumps would form, because the outward push of the cosmological constant acts like a strong wind, blowing most dust accumulations apart. If the cosmological constant’s value is near the window’s lower limit, zero, they found that many clumps form, because the disrupting influence of the cosmological constant is minimized. Which means there’s a large chance you’ll be in a universe whose cosmological constant is near zero, since such universes have an abundance of galaxies and, by the reasoning of this approach, life. There’s a small chance you’ll be in a universe whose cosmological constant is near the window’s upper limit, about 10–121, because such universes are endowed with far fewer galaxies. And there’s a modest chance you’ll be in a universe whose cosmological constant lies at a value between these extremes.
Using the quantitative version of these results, Weinberg and his collaborators calculated the cosmic analog of encountering a sixty-two-pound Labrador on an average walk around the neighborhood—the cosmological constant value, that is, witnessed by an average observer in the multiverse. The answer? Somewhat larger than what the subsequent supernova measurements revealed, but definitely in the same ballpark. They found that roughly 1 in 10 to 1 in 20 inhabitants of the multiverse would have an experience comparable to ours, measuring the cosmological constant’s value in their universe to be about 10–123.
While a higher percentage would be more satisfying, the result is impressive, nonetheless. Prior to this calculation, physics faced a mismatch between theory and observation of more than 120 orders of magnitude, suggesting strongly that something was profoundly amiss with our understanding. The multiverse approach of Weinberg and his collaborators, however, showed that finding yourself in a universe whose cosmological constant is on a par with the value we’ve measured is roughly as surprising as running into that shih tzu in a neighborhood dominated by Labs. Which is to say, not that surprising at all. Certainly, when viewed from this multiverse perspective, the observed value of the cosmological constant doesn’t suggest a profound lack of understanding, and that’s an encouraging step forward.
Subsequent analyses, though, emphasized an interesting facet that some interpret as weakening the result. For simplicity’s sake, Weinberg and his collaborators imagined that across their multiverse only the cosmological constant’s value varied from universe to universe; other physical parameters were assumed fixed. Max Tegmark and Martin Rees noted that if both the cosmological constant’s value and, say, the size of the early universe quantum jitters were imagined to vary from universe to universe, the conclusion would change. Recall that the jitters are the primordial seeds of galaxy formation: tiny quantum fluctuations, stretched by inflationary expansion, yield a random assortment of regions where the density of matter is a little higher or a little lower than average. The higher-density regions exert a greater gravitational pull on nearby matter and so grow yet larger, ultimately coalescing into galaxies. Tegmark and Rees pointed out that much as bigger piles of leaves can better withstand a brisk breeze, so larger primordial seeds can better withstand the disruptive outward push of a cosmological constant. A multiverse in which both the seed size and the value of the cosmological constant vary would therefore contain universes where larger cosmological constants were offse
t by larger seeds; that combination would be compatible with galaxy formation—and hence with life. A multiverse of this sort increases the cosmological constant value that a typical observer would see and so results in a decrease—potentially a sharp one—of the fraction of observers who would find their cosmological constant to have as small a value as we’ve measured.
Staunch multiverse proponents are fond of pointing to the analysis of Weinberg and his collaborators as a success of anthropic reasoning. Detractors are fond of pointing to the issues raised by Tegmark and Rees as making the anthropic result less convincing. In reality, the debate is premature. These are all highly exploratory, first-pass calculations, best viewed as providing insight into the general domain of anthropic reasoning. Under certain restrictive assumptions, they show that the anthropic framework can take us within the ballpark of the measured cosmological constant; relax those assumptions somewhat, and the calculations show that the size of the ballpark grows substantially. Such sensitivity implies that a refined multiverse calculation will require a precise understanding of the detailed properties that characterize the constituent universes, and how they vary, thus replacing arbitrary assumptions with theoretical directives. This is essential if a multiverse is to stand a chance of yielding definitive conclusions.
Researchers are working hard to achieve this goal, but as of today, they have yet to reach it.8
Prediction in a Multiverse IV:
What will it take?
What hurdles, then, will we need to clear before we can extract predictions from a given multiverse? There are three that figure most prominently.
First, as pointedly illustrated by the example just discussed, a multiverse proposal must allow us to determine which physical features vary from universe to universe, and for those features that do vary, we must be able to calculate their statistical distribution across the multiverse. Essential for doing so is an understanding of the cosmological mechanism by which the proposed multiverse is populated by universes (such as the creation of bubble universes in the Landscape Multiverse). It is this mechanism that determines how prevalent one kind of universe is relative to another, and so it is this mechanism that determines the statistical distribution of physical features. If we’re fortunate, the resulting distributions, either across the entire multiverse or across those universes supporting life, will be sufficiently skewed to yield definitive predictions.
A second challenge, if we do need to invoke anthropic reasoning, comes from the central assumption that we humans are garden-variety average. Life might be rare in the multiverse; intelligent life might be rarer still. But among all intelligent beings, the anthropic assumption goes, we are so thoroughly typical that our observations should be the average of what intelligent beings inhabiting the multiverse would see. (Alexander Vilenkin has called this the principle of mediocrity). If we know the distribution of physical features across life-supporting universes, we can calculate such averages. But typicality is a thorny assumption. If future work shows that our observations fall into the range of calculated averages in a particular multiverse, confidence in our typicality—and in the multiverse proposal—would grow. That would be exciting. But if our observations fall outside the averages that could be evidence that the multiverse proposal is wrong, or it could mean that we are just not typical. Even in a neighborhood that has 99 percent Labs, you can still run into Dobermans, an atypical dog. Distinguishing between a failed multiverse proposal and a successful one in which our universe is atypical may prove difficult.9
Progress on this issue will likely require a better understanding of how intelligent life arises in a given multiverse; with that knowledge, we could at least clarify how typical our own evolutionary history has so far been. This, of course, is a major challenge. To date, most anthropic reasoning has completely skirted the issue by invoking Weinberg’s assumption—that the number of intelligent life-forms in a given universe is proportional to the number of galaxies it contains. As far as we know, intelligent life needs a warm planet, which requires a star, which is generally part of a galaxy, and so there’s reason to believe Weinberg’s approach holds water. But since we have only the most rudimentary understanding of even our own genesis, the assumption remains tentative. To refine our calculations, the development of intelligent life needs to be far better understood.
The third hurdle is simple to explain but in the long run may well be the one that’s last standing. It has to do with dividing up infinity.
Dividing Up Infinity
To understand the problem, return to dogs. If you live in a neighborhood populated with three Labs and one dachshund, then, ignoring complications such as how often the dogs are walked, you’re three times more likely to run into a Lab. The same would apply if there were 300 Labs and 100 dachshunds; 3,000 Labs and 1,000 dachshunds; 3 million Labs and 1 million dachshunds, and so on. But what if these numbers were infinitely large? How do you compare an infinity of dachshunds to three times infinity of Labradors? Although this sounds like the tortured math of one-upping seven-year-olds, there’s a real question here. Is three times infinity larger than plain old infinity? If so, is it three times as large?
Comparisons involving infinitely large numbers are notoriously tricky. For dogs on earth, of course, the difficulty doesn’t arise, because the populations are finite. But for universes constituting particular multiverses, the problem can be very real. Take the Inflationary Multiverse. Looking at the entire block of Swiss cheese from an imaginary outsider’s perspective, we would see it continue to grow and produce new universes endlessly. That’s what the “eternal” in “eternal inflation” means. Moreover, taking an insider’s perspective, we’ve seen that each bubble universe itself harbors an infinite number of separate domains, filling out a Quilted Multiverse. In making predictions we necessarily confront an infinity of universes.
To grasp the mathematical challenge, imagine that you’re a contestant on Let’s Make a Deal and you’ve won an unusual prize: an infinite collection of envelopes, the first containing $1, the second $2, the third $3, and so on. As the crowd cheers, Monty chimes in to make you an offer. Either keep your prize as is, or elect to have him double the contents of each envelope. At first it seems obvious that you should take the deal. “Each envelope will contain more money than it previously did,” you think, “so this has to be the right move.” And if you had only a finite number of envelopes, it would be the right move. To exchange five envelopes containing $1, $2, $3, $4, and $5 for envelopes with $2, $4, $6, $8, and $10 makes unassailable sense. But after another moment’s thought, you start to waver, because you realize that the infinite case is less clear-cut. “If I take the deal,” you think, “I’ll wind up with envelopes containing $2, $4, $6, and so on, running through all the even numbers. But as things currently stand, my envelopes run through all whole numbers, the evens as well as the odds. So it seems that by taking the deal I’ll be removing the odd dollar amounts from my total tally. That doesn’t sound like a smart thing to do.” Your head starts to spin. Compared envelope by envelope, the deal looks good. Compared collection to collection, the deal looks bad.
Your dilemma illustrates the kind of mathematical pitfall that makes it so hard to compare infinite collections. The crowd is growing antsy, you have to make a decision, but your assessment of the deal depends on the way you compare the two outcomes.
A similar ambiguity afflicts comparisons of a yet more basic characteristic of such collections: the number of members each contains. The Let’s Make a Deal example illustrates this, too. Which are more plentiful, whole numbers or even numbers? Most people would say whole numbers, since only half of the whole numbers are even. But your experience with Monty gives you sharper insight. Imagine that you take Monty’s deal and wind up with all even dollar amounts. In doing so, you wouldn’t return any envelopes nor would you require any new ones, since Monty would simply double the amount of money in each. You conclude, therefore, that the number of envelopes required to accommodate a
ll whole numbers is the same as the number of envelopes required to accommodate all even numbers—which suggests that the populations of each category are equal (Table 7.1). And that’s weird. By one method of comparison—considering the even numbers as a subset of the whole numbers—you conclude that there are more whole numbers. By a different method of comparison—considering how many envelopes are needed to contain the members of each group—you conclude that the set of whole numbers and the set of even numbers have equal populations.
Table 7.1 Every whole number is paired with an even number, and vice versa, suggesting that the quantity of each is the same.
You can even convince yourself that there are more even numbers than there are whole numbers. Imagine that Monty offered to quadruple the money in each of the envelopes you initially had, so there would be $4 in the first, $8 in the second, $12 in the third, and so on. Since, again, the number of envelopes involved in the deal stays the same, this suggests that the quantity of whole numbers, where the deal began, is equal to that of numbers divisible by four (Table 7.2), where the deal wound up. But such a pairing, marrying off each whole number to a number that’s divisible by 4, leaves an infinite set of even bachelors—the numbers 2, 6, 10, and so on—and thus seems to imply that the evens are more plentiful than the wholes.