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By Root 1982 0

different possibilities between one and two meters. So you need at least a hundred actors. But since some actors who show up may have the same height, leaving other heights unrepresented, you’d better gather more than a hundred. To be safe, maybe you should put out a call for a few hundred actors. That’s a lot, but fewer than what you’d need if W. measured heights to a millimeter’s accuracy. In that case, there’d be a thousand different heights between one and two meters, so to be safe you’d need to gather a few thousand actors.

The same reasoning is relevant for the case of universes with different cosmological constants. Assume that all the universes in a multiverse have cosmological constant values between zero and one (in the usual Planck units); smaller values lead to universes that collapse, larger values would strain the applicability of our mathematical formulations, compromising all understanding. So just as the actors’ heights had a range of one (in meters), the universes’ cosmological constants have a range of one (in Planck units). As for accuracy, the analog of W. using centimeter ticks, or millimeter ticks, is the precision with which we can measure the cosmological constant. Today’s accuracy is about 10–124 (in Planck units). In the future, our accuracy will no doubt improve, but as we’ll see, that will hardly affect our conclusions. Then just as there are 102 different possible heights spaced at least 10–2 meters apart (1 centimeter) in a one-meter range, and 103 different possible heights spaced at least 10–3 meters apart (1 millimeter), so there are 10124 different values of the cosmological constant spaced at least 10–124 apart between the values 0 and 1.

To ensure that every possible cosmological constant is realized, we’d therefore need a multiverse with at least 10124 different universes. But as with the actors, we need to account for possible duplicates, universes that may have the same cosmological constant value. And so to play it safe and make it highly likely that every possible cosmological constant value is realized, we should have a multiverse with far more than 10124 universes, say a million times more, bringing it to a nice even 10130 universes. I’m being cavalier because when we’re talking about numbers this large, the exact values hardly matter. No familiar example of anything—not the number of cells in your body (1013); not the number of seconds since the big bang (1018); not the number of photons in the observable part of the universe (1088)—comes even remotely close to the number of universes we’re contemplating. The bottom line is that Weinberg’s approach for explaining the cosmological constant works only if we’re part of a multiverse in which there are a huge number of different universes; their cosmological constants must fill out some 10124 distinct values. Only with that many different universes is there a high likelihood that there’s one with a cosmological constant that matches ours.

Are there theoretical frameworks that naturally yield such a spectacular profusion of universes with different cosmological constants?14

From Vice to Virtue

There are. We encountered such a framework in the previous chapter. A count of the different possible forms for the extra dimensions in string theory, when including fluxes that can thread through them, came to about 10500. This dwarfs 10124. Multiply 10124 by a few hundred orders of magnitude and 10500 still dwarfs it. Subtract 10124 from 10500, and then subtract it again, and again, and do so a billion times over, and you’d barely make a dent. The result would still be nearly 10500.

Critically, the cosmological constant does indeed vary from one such universe to another. Just as magnetic flux carries energy (it can move things), so the fluxes threading holes in Calabi-Yau shapes also have energy, whose quantity is quite sensitive to the shape’s geometrical details. If you have two different Calabi-Yau shapes with different fluxes penetrating different holes, their energies will generally be different too. And since a given Calabi-Yau