Our Mathematical Universe

by Max Tegmark

  What are these different phases of space like? Imagine that you get a car as a birthday present, with the key in the ignition, but you have never heard of cars before and have absolutely no information about how they work. Being an inquisitive person, you get inside and start messing with the various buttons, knobs and levers. Eventually, you figure out how to use it and get quite good at driving. But unbeknownst to you, somebody has removed the letter R by the gearshift and messed with the transmission so that you need to apply a crazy amount of force to shift into Reverse. This means that unless someone tells you, you’ll probably never figure out that the car can drive backwards as well. If asked to describe how the car worked, you’d incorrectly assert that, without exception, as long as the engine is running, the harder you push on the accelerator pedal, the faster the car moves forward. If in a parallel universe, the car had instead required huge force to shift into forward drive mode, you’d have concluded that this strange machine worked differently and only moved backwards.

  Our Universe is very much like this car. As illustrated in Figure 6.6, it has a bunch of “knobs” that control how it works: the laws according to which things move when you do various things to them and so forth—what we’re told in school are the laws of physics, including so-called constants of nature. Each setting of the knobs corresponds to one of the phases of space, so if there are 500 knobs with 10 possible settings each, there are 10500 different phases.

  When I was in high school, I was incorrectly taught that these laws and constants were always valid, and never changed either from place to place or from time to time. Why this mistake? Because an enormous amount of energy—much more than we have at our disposal—is required to change the settings of these knobs, just as the gearshift on that car, so we didn’t realize that the settings could be changed. Nor that there even were any settings to change: unlike gearshifts, nature’s knobs are well hidden. They come in the form of so-called high-mass fields and other obscure entities, and huge energy is required not only to alter them, but even to detect that they exist in the first place.

  Figure 6.6: The very fabric of space and time seems to have various built-in knobs that can be dialed to different settings in different parts of the Level II multiverse. Our actual Universe seems to have thirty-two knobs that can be dialed continuously, as we’ll see in Chapter 10, as well as additional ones with a discrete number of settings that control what kinds of particles that can exist.

  Click here to see a larger image.

  So how then have physicists figured out that these knobs probably do exist, and that we could actually make our Universe work differently if we had enough energy? In the same way that you could, if you were really inquisitive, figure out that your car could in principle drive backwards: by examining in detail how its parts work! You could figure it out by carefully examining the transmission gearbox. In the same way, detailed study of the smallest building blocks of nature suggests to us that, with enough energy, they could be rearranged in a way such that our Universe would operate differently—we’ll explore the workings of these building blocks in the next chapter. Eternal inflation would have provided enough energy for the quantum fluctuations to actually make all such possible rearrangements in different Level I multiverses. It acted like an extremely strong gorilla that randomly messed with all the knobs and gearshifts in a whole parking lot full of cars: by the time it was done, some fraction of them would be in Reverse.

  In summary, the Level II multiverse fundamentally changes our notion of physical laws. Many of the regularities that we used to view as fundamental laws, which by definition hold anywhere and anytime, have turned out to be merely effective laws, local bylaws that can vary from place to place, corresponding to different knob settings defining space in different phases. Table 6.1 summarizes these notions and how they’re related to parallel universes. This change continues an old trend: whereas Copernicus thought that it was a fundamental law that planets orbit in perfect circles, we now know that more general orbits are allowed, and that the level of non-circularity (which astronomers call “eccentricity”) of an orbit is effectively a knob that can be changed only slowly and with difficulty once a solar system has formed. The Level II multiverse takes this concept to a new level by downgrading many more fundamental laws to effective laws, as we’ll explore next.

  * * *

  1For detailed step-by-step accounts of how the Level II multiverse was discovered and developed by Andrei Linde, Alex Vilenkin, Alan Guth, Sidney Coleman, Frank de Luccia, Raphael Bousso, Joe Polchinski, Lenny Susskind, Shamit Kachru, Renata Kallosh, Sandip Trivedi and others, I recommend the recent books by Brian Greene, Lenny Susskind and Alexander Vilenkin in the “Further Reading” section at the end of this book. The Greene and Susskind books provide good introductions to string theory by two of its pioneers.

  Fine-Tuning as Evidence for the Level II Multiverse

  So does the Level II multiverse really exist? As we’ve seen, evidence for eternal inflation (of which there’s plenty) is evidence for the Level II multiverse, because the former predicts the latter. We also saw that if there are laws or constants of nature that can in principle vary from place to place, then eternal inflation will make them do so across the Level II multiverse. But is there any more direct evidence that doesn’t hinge so crucially on theoretical arguments?

  Multiverse Terminology That We Use in This Book

  Physical reality Everything that exists; Chapter 12 argues that this equals the Level IV multiverse

  Space The part of physical reality that’s continuously connected to what we can observe; with eternal inflation, this equals the Level II multiverse

  Our Universe The part of physical reality we can in principle observe; quantum complications aside, this is the spherical region of space from which light has had time to reach us during the 14 billion years since our Big Bang

  Parallel universe A part of physical reality that can in principle be observed from somewhere else but not from here—parallel universes are not a theory, but a prediction of certain theories

  Multiverse A collection of universes

  Level I multiverse Distant regions of space that are currently but not forever unobservable; they have the same effective laws of physics but may have different histories

  Level II multiverse Distant regions of space that are forever unobservable because space between here and there keeps inflating; they obey the same fundamental laws of physics, but their effective laws of physics may differ

  Level III multiverse Different parts of quantum Hilbert space (Chapter 8); same diversity as Level II

  Level IV multiverse All mathematical structures (Chapter 12), corresponding to different fundamental laws of physics

  Fundamental laws The mathematical equations that govern physics

  Effective laws Particular solution to the mathematical equations that describe physics; can be mistaken for fundamental laws if the same solution is implemented throughout universe

  Fine-tuning Physical constants in the effective laws having values in a very narrow range allowing life; observed fine-tuning is arguable evidence for the Level II multiverse

  Table 6.1: Summary of key multiverse concepts and how they’re interrelated

  I’m going to argue that there is: the fact that our Universe appears highly fine-tuned for life. Basically, we’ve discovered that many of those knobs that we discussed appear tuned to very special values, and if we could change them even by quite small amounts, then life as we know it would become impossible. Tweak the dark-energy knob and galaxies never form, tweak another knob and atoms become unstable, and so on. Lacking pilot training, I’d feel terrified to mess with any of the knobs in an airplane cockpit, but if I could randomly mess with the knobs of our Universe, my survival odds would be even worse.

  I’ve seen three main reactions to this observed fine-tuning:

  1. Fluke: It’s just a fluke coincidence and there’s nothing more to it.

  2. Design: It’s ev
idence that our Universe was designed by some entity (perhaps a deity or an advanced universe-simulating life form) with the knobs deliberately fine-tuned to allow life.

  3. Multiverse: It’s evidence for the Level II multiverse, since if the knobs have all settings somewhere, it’s natural that we’ll exist and find ourselves in a habitable region.

  We’ll explore the fluke and multiverse interpretations below and the simulation interpretation in Chapter 12. But first, let’s explore the fine-tuning evidence to see what all the fuss is about.

  Fine-Tuned Dark Energy

  As we saw in Chapter 4, our cosmic history has been a gravitational tug-of-war between dark matter trying to pull things together and dark energy trying to push them apart. Because galaxy formation is all about pulling things together, I think of dark matter as our friend and dark energy as our enemy. The cosmic density used to be dominated by dark matter, and its friendly gravitational attraction helped assemble galaxies such as our own. However, because the cosmic expansion diluted the dark matter but not the dark energy, the unkind gravitational repulsion of dark energy eventually gained the upper hand, sabotaging further galaxy formation. This means that if the dark energy had had significantly larger density, it would have started gaining the upper hand much sooner, before any galaxies had had time to form. The result would be a stillborn universe, remaining forever dark and lifeless, containing nothing more complex or interesting than nearly uniform gas. If, on the other hand, the dark-energy density were reduced enough to be significantly negative (which is allowed by Einstein’s gravity theory), then our Universe would have stopped expanding, recollapsing in a cataclysmic Big Crunch before any life had had time to evolve. In summary, if you actually figure out how to change the dark-energy density by turning the dark-energy knob in Figure 6.6, then please don’t turn it too far in either direction, because this would be just as bad for life as pressing the Off button.

  How far could you rotate the dark-energy knob before the “Oops!” moment? The current setting of the knob, corresponding to the dark-energy density we’ve actually measured, is about 10−27 kilograms per cubic meter, which is almost ridiculously close to zero compared to the available range: the natural maximum value for the dial is a dark-energy density around 1097 kilograms per cubic meter, which is when the quantum fluctuations fill space with tiny black holes, and the minimum value is the same with a minus sign in front. If rotating the dark-energy knob in Figure 6.6 by a full turn would vary the density across the full range, then the actual knob setting for our Universe is about 10−123 of a turn away from the halfway point. That means that if you want to tune the knob to allow galaxies to form, you have to get the angle by which you rotate it right to over 120 decimal places! Although this sounds like an impossible fine-tuning task, some mechanism appears to have done precisely this for our Universe.

  Fine-Tuned Particles

  In the next chapter, we’ll explore the microworld of elementary particles. There are many knobs there, too, determining particles’ masses and how strongly particles interact with each other, and the science community has gradually come to realize that many of these knobs are fine-tuned as well.

  For instance, if the electromagnetic force were weakened by a mere 4%, then the Sun would immediately explode as its hydrogen fused into so-called diprotons, an otherwise nonexistent kind of neutron-free helium. If it were significantly strengthened, previously stable atoms such as carbon and oxygen would radioactively decay away.

  If the so-called weak nuclear force were substantially weaker, there would be no hydrogen around, since it would all have been converted to helium shortly after our Big Bang. If it were either much stronger or much weaker, the neutrinos from a supernova explosion would fail to blow away the outer parts of the star, and it’s doubtful whether life-supporting heavier elements such as iron would ever be able to leave the stars where they were produced and end up in planets such as Earth.

  If electrons were much lighter, there could be no stable stars, and if they were much heavier, there could be no ordered structures such as crystals and DNA molecules. If protons were 0.2% heavier, they’d decay into neutrons unable to hold on to electrons, so there would be no atoms. If they were instead much lighter, then neutrons inside of atoms would decay into protons, so there would be no stable atoms except for hydrogen. Indeed, the proton mass depends on another knob that has a very wide range of variation and needs to be fine-tuned to thirty-three decimal places to get any stable atoms other than hydrogen.

  Fine-Tuned Cosmology

  Many of these fine-tuning examples were discovered in the seventies and eighties by Paul Davies, Brandon Carter, Bernard Carr, Martin Rees, John Barrow, Frank Tipler, Steven Weinberg and other physicists. And more examples just kept turning up. My first foray into this was with Martin Rees, a white-haired astronomer with impeccable British manners who’s one of my science heroes. I haven’t seen anybody else look as happy and excited when they give a talk, and it’s as if his eyes beam out enthusiasm. He was the first member of the scientific establishment to encourage me to follow my heart and pursue non-mainstream topics. In the last chapter, we saw that the cosmic seed–fluctuation amplitude was about 0.002%. Martin and I calculated that if it were much smaller, galaxies wouldn’t have formed, and if it were much larger, frequent asteroid impacts and other difficulties would ensue.

  This is what I was talking about when I put Alan Guth to sleep. My talk host, Alex Vilenkin, stayed awake, however, and we later teamed up to study neutrinos, ghostlike particles that our Big Bang created in abundance. We found that they, too, appeared somewhat fine-tuned, in that making them significantly heavier would sabotage galaxy formation. My MIT colleague Frank Wilczek had an idea for how the dark-matter density could vary from universe to universe, and together with Martin Rees and my friend Anthony Aguirre, we calculated that turning the dark-matter knob far from its observed value is also bad for our health.

  The Fluke Explanation

  So what are we to make of this fine-tuning? First of all, why can’t we just dismiss it all as a bunch of fluke coincidences? Because the scientific method doesn’t tolerate unexplained coincidences: saying, “My theory requires an unexplained coincidence to agree with observation” is equivalent to saying, “My theory is ruled out.” For example, we’ve seen how inflation predicts that space is flat and the spots in the cosmic microwave background should have an average size around a degree, and that the experiments described in Chapter 4 confirmed this. Suppose that the Planck team had observed a much smaller average spot size, prompting them to announce that they’d ruled out inflation with 99.999% confidence. This would mean that random fluctuations in a flat universe could have caused spots to appear as unusually small as they measured, tricking them into an incorrect conclusion, but that with 99.999% probability, this wouldn’t happen. In other words, inflation would require a 1-in-100,000 unexplained coincidence in order to agree with the measurement. If Alan Guth and Andrei Linde now held a joint press conference, insisting that there was no evidence against inflation because they had a gut feeling that the Planck measurements were just a fluke coincidence that should be dismissed, they’d be violating the scientific method.

  In other words, random fluctuations mean that we can never be 100% sure of anything in science—there’s always a tiny probability that you got really unlucky with random measurement noise, that your detector malfunctioned, or even that the whole experiment was just a hallucination. In practice, however, a theory that’s ruled out at 99.999% confidence is normally considered dead as a doornail by the scientific community. Yet the theory that the dark-energy fine-tuning is a fluke requires us to believe in a much more unlikely unexplained coincidence, and is therefore ruled out at about 99.999999 … percent confidence, where there are about 120 nines after the decimal point.

  The A Word

  What about the Level II multiverse explanation of fine-tuning? A theory where the knobs of nature take essentially all possible values somewhere
will predict with 100% certainty that a habitable universe like ours exists, and since we can only live in a habitable universe, we shouldn’t be surprised to find ourselves in one.

  Although this explanation is logical, it’s quite controversial. After all the silly historical attempts to keep Earth as the center of our Universe, the opposite viewpoint has gotten deeply entrenched. Known as the Copernican principle, it holds that there’s nothing special about our place in space and time. Brandon Carter proposed a direct competitor that he called the weak anthropic principle: “We must be prepared to take account of the fact that our location in the universe is necessarily privileged to the extent of being compatible with our existence as observers.” Some of my colleagues view this as an objectionable step backwards, reminiscent of geocentrism. When taking fine-tuning into account, the Level II–multiverse picture does indeed violate the Copernican principle big time, as illustrated in Figure 6.7: the vast majority of all universes are stone dead, and our own is extremely atypical: it contains way less dark energy than most other ones, and also has highly unusual settings of many other “knobs.”

  Explaining things we can observe by introducing parallel universes that we can’t observe also rubs some of my colleagues the wrong way. I remember a 1998 talk at Fermilab, home of the famous particle accelerator outside Chicago, where the audience erupted in an audible hiss when a speaker mentioned the “A word,” anthropic. Indeed, to sneak under the radar and past the referee, Martin Rees and I went out of our way to avoid using the A word anywhere in the abstract of that first anthropic paper we wrote together.…