Our Mathematical Universe

by Max Tegmark

  Personally, my only objection to Carter’s anthropic principle is that it contains the word principle, suggesting that it’s somehow optional. But no, the use of correct logic when confronting a theory with observation isn’t optional. If most of space is uninhabitable, then we should clearly expect to find ourselves in a place that’s special in the sense of being habitable. Indeed, most of space seems rather uninhabitable even if we limit ourselves to our own Universe: good luck surviving in an intergalactic void or inside a star! For example, only a thousandth of a trillionth of a trillionth of a trillionth of our Universe lies within a kilometer of a planetary surface, so that’s quite a special place, yet that’s where we find ourselves and it’s hardly surprising.

  Figure 6.7: If the dark-energy density (here represented by darkness of shading) varies from universe to universe, then galaxies, planets and life will only emerge in those universes where it’s the lowest. In this illustration, the habitable least-dark fraction is 20% of the universes, but the real fraction may be closer to 10−120.

  As a more interesting example, consider M, the mass of our Sun. M affects the luminosity of the Sun, and using basic physics, one can compute that life as we know it on Earth is only possible if M is in the narrow range between 1.6 × 1030kg and 2.4 × 1030kg—otherwise Earth’s climate would be colder than on Mars or hotter than on Venus. The measured value is M ~ 2.0 × 1030kg. This apparently unexplained coincidence of the habitable and observed M values may appear disturbing given that calculations show that stars in the much broader mass range from M ~ 1029kg to 1032kg can exist: the mass of our Sun appears fine-tuned for life. However, we can explain this apparent coincidence because there’s an ensemble of many such systems with different “knob settings”: we now know that there are many solar systems with a range of sizes of the central star and the planetary orbits, and we should obviously expect to find ourselves living in one of the inhabitable solar systems.

  The interesting point here is that we could have used this fine-tuning of our Solar System to argue that different solar systems should exist even before any were discovered. Using the exact same logic, we can use the observed fine-tunings of our Universe to argue for the existence of different universes. The only difference is whether the other predicted entities are observable or not, but this difference doesn’t weaken the argument, since it never enters into the logic.

  What Can We Ever Hope to Predict?

  We physicists like measuring numbers. Such as these, for example:

  Parameter Observed Value

  Mass of Earth 5.9742 × 1024kg

  Mass of electron 9.10938188 × 10−31kg

  Radius of Earth’s orbit in Solar System 149,597,870,691 × 1024m

  Radius of electron’s orbit in hydrogen atom 5.29177211 × 10−11m

  We also like trying to predict such numbers from first principles. But will we ever succeed, or is this merely wishful thinking? Before making his famous discovery that planetary orbits are ellipses, Johannes Kepler had an elegant theory related to the third number in the table above: he proposed that the orbits of Mercury, Venus, Earth, Mars, Jupiter and Saturn had exactly the same proportions to one another as six spheres nested like Russian Dolls that had between them an octahedron, icosahedron, dodecahedron, tetrahedron and cube, respectively (see Figure 7.2). Aside from the fact that his theory was soon ruled out by better measurements, its entire premise seems silly now that we know that there are other solar systems: the particular orbits we measure in our Solar System don’t tell us anything fundamental about our Universe, merely something about our location in it, in this case which particular solar system we live in. In this sense, we can think of these digits as just part of our address in space, as part of our cosmic postal code. For example, to explain to an extraterrestrial mailman which solar system in our Galactic neighborhood we wanted our package delivered to, we could tell him to come to the one with eight planets whose orbits were 1.84, 2.51, 4.33, 12.7, 24.7, 51.1 and 76.5 times larger than that of the innermost planet, and he might say: “Oh, I know which solar system you’re talking about!” In the same vein, we’ve permanently given up on predicting Earth’s mass or radius from first principles because we know that many planets with different sizes exist.

  But what about the mass and orbital size for an electron? These numbers are the same for all electrons in our Universe that we’ve checked, so we’ve gotten our hopes up that they may be truly fundamental properties of our physical world that we’ll one day be able to compute from theory alone, in the spirit of Kepler’s orbit model. Indeed, as recently as 1997, the famous string theorist Ed Witten told me that he thought string theory would one day predict how many times lighter an electron is than a proton. Yet when I last saw him at Andrei Linde’s sixtieth birthday party, he confessed after some wine that he’d given up on ever predicting all constants of nature.

  Why this new pessimism? Because history is repeating itself. The Level II multiverse does to the electron’s mass what other planets did to Earth’s mass, demoting it from being a fundamental property of nature to being merely part of our cosmic address. For any number that varies across the Level II multiverse, measuring its value simply narrows down the options for what particular universe we happen to be in.

  As we’ll see in Chapter 10, we’ve so far discovered thirty-two independent numbers built into our Universe that we’re trying to measure to as many decimal places as possible. Do they all vary across the Level II multiverse, or can any of them be computed from first principles (or from some other shorter list of numbers)? We still lack a successful fundamental theory of physics that can answer this question, so until we do, it’s interesting to look at the measurements for some hints. Numbers that vary across the multiverse should look random to us if we’re living in a random universe. Do the measured numbers look random? Well, you can judge for yourself in Figure 6.8, where I’ve plotted the masses of the nine fundamental particles called fermions in particle physics. Aside from the funny scale I’ve used, where the mass increases tenfold for every few centimeters you go to the right, it looks to me like nine randomly thrown darts. In fact, these nine numbers have passed some stringent statistical randomness tests with flying colors, consistent with being randomly generated from what statisticians call a uniform distribution with a slope below 10%.

  Figure 6.8: The nine masses that we’ve managed to measure for so-called fermion particles look rather random, as some multiverse models predict, suggesting that we’ll never manage to calculate them from first principles. The scale shows how many times heavier than an electron each particle is.

  Click here to see a larger image.

  All Isn’t Lost

  If we’re living in a random habitable universe, the numbers should still look random, but with a probability distribution that favors habitability. By combining predictions about how the numbers vary across the multiverse with the relevant physics of galaxy formation and so on, we can make statistical predictions for what we should actually observe, and such predictions have so far agreed fairly well with data for dark energy, dark matter and neutrinos (Figure 6.9). Indeed, Steven Weinberg’s first prediction of a non-zero dark-energy density was made this way.

  I’ve had fun going through the full list of known “knobs” of our “universal controller,” pondering what would happen if they were set differently. For example, please don’t rotate the Figure 6.6 knobs for the number of space and time dimensions, since it would be lethal. If you increase the number of space dimensions beyond three, there can be neither stable solar systems nor stable atoms. For instance, going to a four-dimensional space changes Newton’s inverse-square law for the gravitational force to an inverse-cube law, for which there are no stable orbits whatsoever. I got quite excited when I figured this out, and then realized that I’d just broken my personal scooping record: the Austrian physicist Paul Ehrenfest had discovered this already back in 1917…. Spaces with less than three dimensions don’t allow solar systems because gravity ceases
to be attractive, and they’re probably too simple to contain observers also for other reasons—for example, two neurons can’t cross. Changing the number of time dimensions isn’t as absurd as you might think, and Einstein’s theory of general relativity can handle this just fine. However, I once wrote a paper showing that doing that would eliminate the key mathematical property of physics that allows us to make predictions, thus making it pointless to evolve a brain. As Figure 6.10 illustrates, this leaves three space dimensions and one time dimension as the only viable option. In other words, an infinitely intelligent baby could in principle, before making any observations at all, calculate from first principles that there’s a Level II multiverse with different combinations of space and time dimensions, and that 3 + 1 is the only option supporting life. Paraphrasing Descartes, it could then think, Cogito, ergo three space dimensions and one time dimension, before opening its eyes for the very first time and verifying its predictions.

  Figure 6.9: If the densities of dark energy, dark matter and neutrinos vary dramatically across a Level II multiverse, then most universes will be devoid of galaxies and lifeless, and a random observer should expect to measure values in a fairly narrow range quantified by the probability distributions shown. We should expect the measured values to fall in the central gray region with 90% probability, and indeed we do.

  Click here to see a larger image.

  The entire Level II multiverse exists in a single space, so how can the dimensionality vary within it? Well, according to the most popular string-theory models, it’s only the apparent dimensionality that varies: the true space always has nine dimensions, but we don’t notice six of them because they’re microscopically curled up in the spirit of the cylinder from Figure 2.7: if you travel a tiny distance along one of these six hidden dimensions, you find yourself back where you started. Supposedly, all nine dimensions started out curled up, and then in our patch of space, inflation stretched three of them out to astronomical size while leaving six of them tiny and invisible. Elsewhere in the Level II multiverse, inflation stretched out different numbers of dimensions, creating worlds that seem anywhere from zero-dimensional to nine-dimensional.

  Figure 6.10: With more than three space dimensions, there are no stable atoms or solar systems. With fewer, there’s no gravitational attraction. With more or less than one time dimension, physics loses all predictive power, and there would be no point in evolving a brain. In a Level II multiverse where the number of space and time dimensions varies from universe to universe, we should therefore expect to find ourselves in a universe with three dimensions of space and one of time, since all other universes are probably uninhabited.

  Mathematicians have identified many different ways in which these extra dimensions can be curled up and filled with energy (for example, generalized magnetic fields can loop around inside the hidden dimensions), and in string theory, these many options correspond to the changeable knobs that we explored earlier. Different options can correspond not only to different physical constants in the dimensions that aren’t curled up, but also to different rules for what elementary particles can exist and the effective equations that describe them. There might be Level II parallel universes where there are, say, ten rather than six kinds of quarks.

  In summary, this means that although the fundamental equations of physics (those of string theory, perhaps) remain valid throughout the Level II multiverse, the apparent laws of physics that observers will uncover can change from one Level I multiverse to another. In other words, these apparent laws are universal not in the dictionary sense of “always applicable,” but only in the literal sense of “applicable in our Universe.” They’re multiversal only at Level I, not at Level II. The fundamental equations, however, are multiversal even at Level II—they won’t change until we get to Chapter 12 and the Level IV multiverse.…

  Multiverse Halftime Roundup

  We’ve explored lots of crazy-sounding ideas in this chapter, so let’s end it by taking a step back and looking at the big picture. I think of inflation as the explanation that doesn’t stop—inflating or explaining. Just as cell division didn’t make merely one baby and stop, but a huge and diverse population of humans, it looks like inflation didn’t make merely one universe and stop, but a huge and diverse population of parallel universes, perhaps realizing all possible options for what we used to think of as physical constants. Which would explain yet another mystery: the fact that our Universe is so fine-tuned for life. Even though most of the parallel universes created by inflation are stillborn, there will be some where conditions are just right for life, and it’s not surprising that this is where we find ourselves.

  My colleague Eddie Farhi likes to call Alan Guth “The Enabler,” because eternal inflation enables everything that can happen to actually happen: inflation produces space for it to take place and creates initial conditions allowing the story to play itself out. In other words, inflation is a process converting potentiality into reality.

  If you feel uncomfortable talking about our Level II multiverse, just say “space” instead, remembering that all of our Level I and Level II parallel universes are simply distant regions of one and the same infinite space. It’s just that the structure of this space is much richer than Euclid imagined: it’s expanding so that we can only see the small part of it that we call our Universe, and its faraway properties are more diverse than what we see in our telescopes. The Chapter 3 notion that our Universe is homogeneous and looks the same everywhere is just an interlude, valid only on intermediate scales: gravity makes things clumpy and interesting on smaller scales, and inflation makes things diverse and interesting on larger scales.

  If you’re still struggling to make inner peace with parallel universes, here’s another way of thinking about them that might help. Alan Guth mentioned it in a recent MIT talk, but it has nothing to do with inflation. When we discover an object in nature, the scientific thing to do is look for a mechanism that created it. Cars are created by car factories, rabbits are created by rabbit parents and solar systems are created from gravitational collapse in giant molecular clouds. So it’s quite reasonable to assume that our Universe was created by some sort of universe-creation mechanism (perhaps inflation, perhaps something totally different). Now here’s the thing: all the other mechanisms we mentioned naturally produce many copies of whatever they create; a cosmos containing only one car, one rabbit, and one solar system would seem quite contrived. In the same vein, it’s arguably more natural for the correct universe-creation mechanism, whatever it is, to create many universes rather than just the one we inhabit.

  If we apply this same argument to whatever mechanism started inflation and ultimately produced our Level II multiverse, we conclude that it probably produced many separate Level II multiverses that are completely disconnected. However, this variant appears to be untestable, since it would neither add any qualitatively different worlds nor alter the probability distribution for their properties—all possible Level I multiverses are already realized within each of these Level II multiverses.

  Inflation aside, there might be other mechanisms that create universes. An idea proposed by Richard Tolman and John Wheeler and recently elaborated on by Paul Steinhardt and Neil Turok is that our cosmic history is cyclic, going through an infinite series of Big Bangs. If it exists, the ensemble of such incarnations would also form a multiverse, perhaps with a diversity similar to that of Level II.

  Another universe-creation mechanism, proposed by Lee Smolin, involves mutating and sprouting new universes through black holes rather than through inflation. This would produce a Level II multiverse as well, with natural selection favoring universes with maximal black-hole production. My friend Andrew Hamilton from Chapter 4 may have uncovered such a universe-creation mechanism: he’s investigated an instability that occurs inside black holes shortly after they form, and it may be violent enough to trigger inflation that would create a Level I multiverse—which would be entirely contained inside the original black hol
e, but its inhabitants would probably neither know nor care about this fact.

  In so-called braneworld scenarios, another three-dimensional world could be quite literally parallel to ours, a short distance away in a higher dimension. However, I don’t think that such a world (brane) deserves to be called a parallel universe separate from our own, since it can interact with it gravitationally much as we do with dark matter.

  Parallel universes remain highly controversial. However, there’s been a striking shift in the scientific community during the past decade, where multiverses have gone from having lunatic-fringe status to being discussed openly at physics conferences and in peer-reviewed papers. I think that the success of precision cosmology and inflation has played a major role in this shift, as has the discovery of dark energy and the failure to explain its fine-tuning by other means. Even those of my colleagues who dislike the multiverse idea now tend to grudgingly acknowledge that the basic arguments for it are reasonable. The main critique has shifted from “This makes no sense and I hate it” to “I hate it.”

  In my opinion, our job as scientists isn’t to tell our Universe how to work in order to conform to our human prejudice, but to look at it with open minds and try to figure out how it actually works.

  We humans have a well-documented tendency toward hubris, arrogantly imagining ourselves at center stage, with everything revolving around us. We’ve gradually learned that it’s instead we who are revolving around the Sun, which is itself revolving around one galaxy among countless others. Thanks to breakthroughs in physics, we may be gaining still deeper insights into the very nature of reality—indeed, in this book, we’re still only two multiverse levels down, with two to go, and will start exploring the Level III multiverse in the next chapter. The price we have to pay is becoming more humble—which will probably do us good—but in return we may find ourselves inhabiting a reality grander than our ancestors imagined in their wildest dreams.