Our Mathematical Universe

by Max Tegmark

  Although you might find these definitions reasonable, please beware that some people use these words differently, which can cause confusion. In particular, some people use the phrase I eschew, “the universe,” to mean everything that exists, in which case, by definition, there can’t be any parallel universes.

  Now that we’ve defined our Universe, how big is it? As we discussed, our Universe is a spherical region with Earth at the center. The stuff near the edges of our Universe, from which light has only now reached us after a 14-billion-year space journey, is currently about 5 × 1026 meters away from us.1 As far as we currently know, our Universe contains about 1011 galaxies, 1023 stars, 1080 protons and 1089 photons (particles of light).

  This is certainly a lot of stuff, but could there exist even more, farther away in space? As we saw, inflation predicts that there is. Your doppelgänger’s universe (this page), if it exists, is a sphere of the same size centered over there, none of which we can see or have any contact with yet, because light or other information from there hasn’t had time to reach us. This is the simplest (but far from the only) example of parallel universes. I like to call this kind, a distant region of space the size of our Universe, a Level I parallel universe. All the Level I parallel universes together form the Level I multiverse. Table 6.1 defines all the different types of multiverses we explore in this book and how they’re interrelated.

  By our very definition of universe, one might expect the notion that our observable Universe is merely a small part of a larger multiverse to be forever in the domain of metaphysics. Yet the epistemological borderline between physics and metaphysics is defined by whether a theory is experimentally testable, not by whether it’s weird or involves unobservable entities. Technology-powered experimental breakthroughs have therefore expanded the frontiers of physics to incorporate ever more abstract (and at the time counterintuitive) concepts such as a round rotating Earth, an electromagnetic field, time slowdown at high speeds, quantum superpositions, curved space and black holes. As we’ll see below, it’s becoming increasingly clear that theories grounded in modern physics can in fact be empirically testable, predictive and falsifiable even if they involve a multiverse. Indeed, in the rest of this book, we’ll be exploring as many as four distinct levels of parallel universes, so that, to me, the most interesting question isn’t whether there’s a multiverse (since Level I isn’t that controversial), but rather how many levels it has.

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  1As we saw in Chapter 3, this is more than 14 billion light-years because light gets helped along by the expansion of space.

  What Are Level I Parallel Universes Like?

  Suppose inflation really happened and made our space infinite. Then there are infinitely many Level I parallel universes. Moreover, as Figure 5.8 illustrates, all of the infinite space was created full of matter which, much like here in our own Universe, gradually formed atoms, galaxies, stars and planets. This means that most of the Level I parallel universes shared our own cosmic history in broad brushstrokes. However, most of them differ from our Universe in the details, because they started out slightly differently. The reason they did is that, as we saw in the previous chapter, the seed fluctuations responsible for all cosmic structure were generated by quantum fluctuations which are for all practical purposes random (see this page).

  Our physics description of the world is traditionally split into two parts: how things start out and how things change. In other words, we have initial conditions and we have laws of physics specifying how the initial conditions evolve over time. Observers living in parallel universes at Level I observe the exact same laws of physics as we do, but with different initial conditions than those in our Universe. For example, the particles start out in slightly different places, moving with slightly different speeds. It’s these slight differences that ultimately determine what happens in their universes: which regions turn into galaxies, which regions become intergalactic voids, which stars get planets, which planets get dinosaurs, which planets get their dinosaurs killed by an asteroid collision, and so on. In other words, the quantum-induced differences between parallel universes get amplified over time into very different histories. In summary, students in Level I parallel universes would learn the same thing in physics class but different things in history class.

  But would those students exist in the first place? It feels extremely unlikely that your life turned out exactly as it did, since it required so many things to happen: Earth had to form, life had to evolve, the dinosaurs had to go extinct, your parents had to meet, you had to get the idea to read this book, etc. But the probability of all these outcomes happening clearly isn’t zero, since it in fact happened right here in our Universe. And if you roll the dice enough times, even the most unlikely things are guaranteed to happen. With infinitely many Level I parallel universes created by inflation, quantum fluctuations effectively rolled the dice infinitely many times, guaranteeing with 100% certainty that your life would occur in one of them. Indeed, in infinitely many of them, since even a tiny fraction of an infinite number is still an infinite number.

  And an infinite space doesn’t contain only exact copies of you. It contains many more people who are almost like you, yet slightly different. So if you were able to go meet the closest person out there in space who looked like your spitting image, this person would probably speak an alien language you couldn’t understand and would have experienced a life quite different from yours. But out of all your infinitely many look-alikes out there on other planets, there’s also one who speaks English, lives on a planet identical to Earth, and has experienced a life completely indistinguishable from yours in all ways. This person subjectively feels exactly like you feel. Yet there may be some very minor difference in how the particles move in your alter ego’s brain that’s too subtle to make a perceptible difference now, but which in a few seconds will make your alter ego put this book aside while you read on, causing your two lives to start diverging.

  This raises an interesting philosophical point that will come back and haunt us in Chapter 11: if there are indeed many copies of “you,” with identical past lives and memories, this kills the traditional notion of determinism: you can’t predict your own future—even if you have complete knowledge of the entire past and future history of the cosmos! The reason you can’t is that there’s no way for you to determine which of these copies is “you” (they all feel that they are). Yet their lives will typically begin to differ eventually, so the best you can do is predict probabilities for what you’ll experience from now on.

  In summary, in an infinite space created by inflation, everything that can happen according to the laws of physics does happen. And it happens an infinite number of times. This means that there are parallel universes where you never get a parking ticket, where you have a different name, where you’ve won a million-dollar lottery, where Germany won World War II, where dinosaurs still roam Earth, and where Earth never formed in the first place. Although each of these outcomes occur in an infinite number of universes, some occur in a larger fraction than others, and making sense of this raises a host of intriguing issues that we’ll tackle in Chapter 11.

  Are Parallel Universes Unscientific?

  Hold on!!! Did I just go bananas??? I mean, so far in this book, I’ve mostly written about stuff that I hope you found pretty reasonable. Sure, some of the scientific discoveries I wrote about were controversial at the time, but at least they’re accepted by the scientific mainstream today. But then things started going kind of crazy in this chapter. And this last business about infinite copies of us doing everything we can imagine—this just sounds nuts. Totally nuts. So before going any farther down this rabbit hole, we need to pause for a sanity check. First of all, is it really science to talk about such crazy things that we can’t even observe, or have I crossed the line into pure philosophical speculation?

  Let’s be more specific. The influential Austro-British philosopher Karl Popper popularized the now widely accepted adage “If
it’s not falsifiable, then it’s not scientific.” Physics is all about testing mathematical theories against observation: if a theory can’t be tested even in principle, then it’s logically impossible to ever falsify it, which, by Popper’s definition, means that it’s unscientific. It follows then that the only thing that can have any hope of being scientific is a theory. Which brings us to a very important point:

  Parallel universes are not a theory, but a prediction of certain theories.

  Of theories such as inflation. Parallel universes (if they exist) are things, and things can’t be scientific, so a parallel universe can’t be scientific any more than a banana can.

  Therefore, we must reformulate our question about philosophical speculation in terms of theories, which leads to the following crucial question:

  Are theories predicting the existence of unobservable entities unfalsifiable and therefore unscientific? This is where I think it gets really interesting, because this question has a clear answer: For a theory to be falsifiable, we need not be able to observe and test all its predictions, merely at least one of them. Consider the following analogy:

  Theory Prediction

  General relativity Black-hole interiors

  Inflation (Chapter 5) Level I parallel universes

  Inflation + landscape (Chapter 6) Level II parallel universes

  Collapse-free quantum mechanics (Chapter 8) Level III parallel universes

  External reality hypothesis (Chapter 10) Level IV parallel universes

  Because Einstein’s theory of general relativity has successfully predicted many things that we can observe, such as the detailed motion of Mercury around the Sun, the bending of light by gravity, and the gravitational slowing of clocks, we consider it a successful scientific theory and take seriously also its predictions for things we can’t observe—for example, that space continues inside black-hole event horizons1 and that (contrary to early misconceptions) nothing funny happens right at the horizon. Analogously, the successful predictions of inflation that we’ve described in the last two chapters make inflation a scientific theory, which makes it reasonable to take seriously its other predictions as well—both testable predictions such as what future cosmic microwave–background experiments should measure and seemingly untestable predictions such as the existence of parallel universes. The last three examples in the table on this page involve theories we’ll describe later in the book that predict additional types of parallel universes.

  Another important thing about physics theories is that if you like one, you have to buy the whole package. You’re not allowed to say: “Well, I like how general relativity explains Mercury’s orbit, but I don’t like black holes, so I’m going to opt out of that feature.” You can’t buy general relativity with the black holes removed the way you can buy coffee with the caffeine removed. General relativity is a rigid mathematical theory with no adjustments possible; you have to either accept all its predictions, or you have to start over from scratch and invent a different mathematical theory that agrees with all of general relativity’s successful predictions while simultaneously predicting that black holes can’t exist. This turns out to be extremely difficult, and so far, all such attempts have failed.

  In the same way, parallel universes aren’t optional in eternal inflation. They come as part of the package, and if you don’t like them, then you have to find a different mathematical theory that solves the bang problem, the horizon problem and the flatness problem, that generates the cosmic seed fluctuations—and doesn’t predict parallel universes. This, too, has proven difficult, which is why more and more of my colleagues are—often grudgingly—beginning to take parallel universes seriously.

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  1Although you can, in principle, enter a black hole and observe what happens inside (if its tidal forces don’t “spaghettify” you first), you won’t be able to publish your findings in a scientific journal, since you effectively went there with a one-way ticket.

  Evidence for Level I Parallel Universes

  Okay, so we’ve settled one thing: we don’t need to feel guilty for talking about parallel universes in this book, even though it’s supposed to be a scientific book. But just because something is scientific, it doesn’t have to be correct, so let’s take a closer look at the evidence for parallel universes.

  Earlier in this chapter, we saw that the Level I multiverse, including your doppelgängers, is a logical consequence of eternal inflation. We’ve also seen that inflation is currently the most popular early-universe theory in the scientific community, and that inflation is typically eternal, thus producing the Level I multiverse. In other words, the best evidence for the Level I multiverse is the evidence we have for inflation. Does this prove that your doppelgängers exist? Certainly not! At this point, we can’t be 100% certain that inflation is eternal, or even that it happened at all. Fortunately, inflation research is now a very active field both theoretically and experimentally, so we’re likely to gain more evidence for or against eternal inflation (and consequently for or against the Level I multiverse) in the years ahead.

  So far, our entire discussion in this chapter has been in the context of inflation. But does the Level I multiverse stand and fall with inflation? No! For there to be no Level I parallel universes at all, there must be no space whatsoever beyond the region we can see. I don’t have a single science colleague who’s argued for such a small space, and someone arguing for it could be likened to an ostrich with its head in the sand, claiming that only what it can see can exist. We all accept the existence of things that we can’t see but could see if we moved or waited, like ships beyond the horizon. Objects beyond our cosmic horizon have similar status, since our observable Universe grows by roughly a light-year every year as light from farther away has time to reach us.1

  What about evidence for our doppelgängers? If we tease apart our arguments above, we see that the “everything that can happen does happen” property of the Level I multiverse follows from two logically distinct assumptions, both of which could conceivably be correct even without inflation:

  1. Infinite space and matter: Early on, there was an infinite space filled with hot expanding plasma.

  2. Random seeds: Early on, a mechanism operated such that any region could receive any possible seed fluctuations, seemingly at random.

  Let’s explore these two assumptions in turn. I think the second one is a pretty reasonable assumption, regardless of inflation. We’ve observed that these random-looking seed fluctuations exist, so we know that some mechanism made them. We’ve measured their statistical properties carefully using cosmic microwave–background and galaxy maps, and their random properties are consistent with what’s known to statisticians as a “Gaussian random field,” which satisfies assumption 2. Moreover, if inflation didn’t happen and distant spatial regions were never able to communicate with each other (Figure 5.2), then this mechanism would be guaranteed to roll the dice independently in each region.

  What about the assumption of infinite space and matter? Well, an infinite space rather uniformly filled with matter used to be the standard assumption in mainstream cosmology even long before inflation was invented, and is now part of what’s known as the cosmological standard model. Yet this assumption and its Level I multiverse implications used to be controversial; indeed, an assertion along these lines was one of the heresies for which the Vatican had Giordano Bruno burned at the stake in 1600. Those of us who have published on this topic more recently, including George Ellis, Geoff Brundrit, Jaume Garriga and Alex Vilenkin, have thus far avoided the stake, but let’s nonetheless take a critical look at the infinite space and infinite matter assumption.

  We saw in Chapter 2 that although the simplest model of space (dating back to Euclid) is infinite, Einstein’s general relativity allows various elegant ways in which space can be finite. If space curves back on itself like a hypersphere (Figure 2.7), then the total volume of this hypersphere must be at least a hundred times larger than the part of it th
at we can observe (our Universe) in order to explain why our visible part of space is so flat that cosmic microwave–background experiments haven’t detected any curvature. In other words, even if we live in a finite space of the hypersphere kind, then there are at least a hundred Level I parallel universes.

  What about a finite space of the torus (bagel) kind that we explored in Chapter 2, where space is flat but you nonetheless return to your starting point if you travel some distance? Such a space is like that of one of those computer games where you can fly off the screen and instantly reenter on the other side, so if you could see far enough in front of you, you’d see the back of your own head—and infinitely many regularly spaced copies of you in all directions, much like if you were standing in a mirror-covered room. If our space has this property, what’s the smallest size it could have? It clearly has to be much larger than our Galaxy, since our telescopes don’t show infinite copies of the Milky Way lined up in tidy rows. But if the size were, say, 10 billion light-years, this test would fail: we wouldn’t see the nearest copy of our Galaxy because it didn’t exist 10 billion light-years ago. Fortunately, there’s an even more sensitive test: we can look for a recognizable object such as a bright galaxy 5 billion light-years away, and then look for the same object 5 billion light-years in the opposite direction. Such searches have also come up empty-handed. The most sensitive test of all is to use the most distant thing we can see, the cosmic microwave background, and look for matching patterns in opposite directions as in Figure 6.1—many research teams, Angélica and I included, tried this and found nothing. Also, if space has a finite volume, only certain perturbation frequencies are allowed, just as the air in a flute can only vibrate at certain special frequencies. This distorts the microwave-background power spectrum in a particular way that Angélica and others have looked for without finding anything. In summary, it’s still possible that space is finite, but finite space models have been severely constrained by observations in recent years, so the only spaces still allowed have a volume that’s comparable to or greater than our Universe. This makes it really tough to avoid at least a handful of parallel universes. Moreover, having exactly one universe right now would require a strange unexplained “Why now?” coincidence, since there would have been more than one universe when light had only had time to reach us from a smaller fraction of space.