Our Mathematical Universe

by Max Tegmark

  Alex found that the question of where and when inflation ends is quite subtle and interesting. We know that inflation ends in at least some places, since 14 billion years ago, it ended in the part of space that we now inhabit. This means that there must be some physical process which can get rid of the inflating substance, causing it to decay into ordinary non-inflating matter, which then keeps expanding, clustering, and ultimately forming galaxies, stars and planets as we described in the last chapter. Radioactivity famously makes unstable substances decay into others, so let’s suppose that the inflating substance is similarly unstable. This means that there’s some time scale called the half-life during which half of the inflating substance will decay. As illustrated in Figure 5.7, we now have an interesting tug-of-war between the doubling caused by inflation and the halving caused by decay. For inflation to work, the former has to win so that the total inflating volume grows over time. This means that the doubling time of the inflating substance has to be shorter than its half-life. The figure illustrates such an example, where inflation triples the size of space while one third of the inflating substance decays away, over and over again. As you can see, the total volume of space that’s still inflating keeps doubling forever. In parallel, non-inflating regions of space are continuously being produced by the decay of inflating space, so the amount of non-inflating volume, where inflation has ended and galaxies can form, keeps doubling, too.

  Figure 5.7: Schematic illustration of eternal inflation. For each volume of inflating substance (symbolized by a cube) that decays into a non-inflating Big Bang universe like ours, two other inflating volumes don’t decay, instead tripling their volume. The result is a never-ending process where the number of Big Bang universes increases as 1, 2, 4, etc., doubling at each step. So what we call our Big Bang (one of the flashes) isn’t the beginning of everything, but the end of inflation in our part of space.

  This perpetual property of inflation turned out to be much more general than originally expected. Andrei Linde, who coined the term “eternal inflation,” discovered that even the very simplest inflation model that he’d proposed, which we talked about above, inflated eternally through an elegant mechanism related to the quantum fluctuations that generated our cosmological seed fluctuations.

  By now, a very large class of inflation models have been analyzed in detail by researchers around the globe, and it’s been found that almost all of them lead to eternal inflation. Although most of these calculations are rather complicated, the schematic illustration of Figure 5.7 captures the essence of why inflation is generally eternal: for inflation to work in the first place, the inflating substance needs to expand faster than it decays, and this automatically makes the total amount of inflating stuff grow without limit.

  The discovery of eternal inflation has radically transformed our understanding of what’s out there in space on the largest scales. Now I can’t help but feel that our old story sounds like a fairy tale, with its single narrative in a simple sequence: “Once upon a time, there was inflation. Inflation made our Big Bang. Our Big Bang made galaxies.” Figure 5.7 illustrates why this story is too naive: it yet again repeats our human mistake of assuming that all we know of so far is all that exists. We see that even our Big Bang is just a small part of something much grander, a treelike structure that’s still growing. In other words, what we’ve called our Big Bang wasn’t the ultimate beginning, but rather the end—of inflation in our part of space.

  How to Make an Infinite Space in a Finite Volume

  That kindergartner in Chapter 2 asked whether space goes on forever. Eternal inflation gives a clear answer: space isn’t just huge—it’s infinite. With infinitely many galaxies, stars and planets.

  Let’s explore this notion more carefully. Although the schematic nature of Figure 5.7 doesn’t make this clear, we’re still talking about just one single connected space. Right now (we’ll return to what “right now” means below), some parts of this space are expanding very fast because they contain inflating matter, other parts are expanding more slowly because inflation has ended there, and yet other parts, like the region that’s inside our Galaxy, are no longer expanding at all. So does inflation end? The detailed inflation research we mentioned above shows that the answer is: yes and no. It ends and it doesn’t end, in the following sense:

  1. In almost all parts of space, inflation will eventually end in a Big Bang like ours.

  2. There will nonetheless be some points in space where inflation never ends.

  3. The total inflating volume increases forever, doubling at regular intervals.

  4. The total post-inflationary volume containing galaxies also increases forever, doubling at regular intervals.

  But does this really mean that space is infinite already? This brings us back to another one of our questions from Chapter 2: How could an infinite space get created in a finite time? It sounds impossible. But as I mentioned, inflation is like a magic show where seemingly impossible tricks happen through creative use of the laws of physics. Indeed, inflation can do something even better, which I think is its most amazing trick of all: it can create an infinite volume inside a finite volume! Specifically, it can start with something smaller than an atom and create an infinite space inside of it, containing infinitely many galaxies, without affecting the exterior space.

  Figure 5.8 illustrates how inflation does this trick. It shows a slice through space and time, where the left and right edges correspond to two points where inflation never ends, and the bottom edge corresponds to a time when the entire region between these two points is inflating. It’s hard to draw an expanding three-dimensional space, so I’ve ignored both the expansion and two of the three space dimensions in the picture, because neither of these two complications affect the basic argument. Eventually, inflation will end everywhere except at the left and right edges; the curved boundary shows the exact time when it ends at different places. Once inflation ends in a given region, the traditional Big Bang story from the last two chapters starts unfolding there, with a hot cosmic fusion reactor eventually cooling to form atoms, galaxies, and perhaps observers like us.

  Here’s the key part of the trick: according to Einstein’s theory of general relativity, an observer living in one of these galaxies will perceive space and time differently than I’ve defined them with my axes in the drawing. Our physical space doesn’t come with centimeter marks built in the way a ruler does, nor does our Universe come with a bunch of clocks pre-installed. Instead, any observer needs to define her own measurement rods and clocks, which in turn define her notion of space and time. This idea can lead to one of Einstein’s core insights, immortalized by the slogan “It’s all relative”: that different observers can perceive space and time in different ways. In particular, simultaneity can be relative. Suppose you email an astronaut friend on Mars:

  Figure 5.8: As described in the text, inflation can create an infinite universe inside of what looks like a subatomic volume from the outside. An observer inside will view A as simultaneous with B, C as simultaneous with D, the infinite U-shaped surface where inflation ends as her time zero, the infinite U-shaped surface where atoms form as her time 400,000 years, etc. For simplicity, this cartoon ignores both the expansion of space and two of the three space dimensions.

  Click here to see a larger image.

  Hey, how are things over there?

  Ten minutes later, she gets your message, which was transmitted at the speed of light using radio waves. While you’re waiting, you receive an email from Nigeria, offering cheap Rolex watches. Another ten minutes later, you get her reply:

  Good, but I miss Earth!

  Now which event happened first, you receiving the spam or your astronaut friend sending her message? Amazingly, Einstein discovered that this simple question has no simple answer. Instead, the correct answer depends on the velocity of the person answering it! For example, if I’m zooming past Earth toward Mars in a spaceship, intercept all three emails, and analyze the situation, I�
�ll determine that according to my onboard clock, your friend on Mars sent the message before you got the spam. If I’m flying in the opposite direction, I’ll determine that you got the spam first. Confusing? That’s what most of Einstein’s colleagues thought as well when he presented his relativity theory, but countless experiments have since confirmed that this is how time works. The only circumstance when we can definitely say that an event on Mars happened before an event on Earth is if we can send a message from Mars after the Mars event that reaches Earth before the Earth event.

  Now let’s apply this to the situation in Figure 5.8. For someone outside of this region, it might make sense to define space and time as the horizontal and vertical directions, respectively, just as I’ve drawn the figure, so that the four events I’ve circled happened in the order A, B, C, D. Moreover, B definitely happened before D because you could imagine sending a message from B to D, and similarly, A definitely happened before C. But can we really be sure that A happened before B, given that the two events are too far apart for light to have time to reach one from the other? Einstein’s answer is no. Indeed, for an observer living in one of these galaxies, it makes more sense to define inflation as having ended at a particular fixed time, since the end of inflation corresponds to her Big Bang, so according to her, the events A and B are simultaneous! As you can see, the “Inflation ends” surface is not horizontal. In fact, it’s infinite, since it bends up like the letter U toward the left and right edges of the plot where we agreed that inflation never ends. This means that as far as she’s concerned, her Big Bang occurred at a single instant in a truly infinite space! Where did the infinity come sneaking in from? You can see that it snuck in via the infinite future time available, by her space direction being curved progressively more upward.

  She’ll similarly conclude that her space is infinite at later times. For example, if she builds a cosmic microwave–background experiment to take baby pictures of her 400,000-year-old universe, the plasma surface she’s imaging corresponds to the surface in the picture where protons and electrons combine into transparent (invisible) hydrogen atoms. Since you can see that this is also an infinite U-shaped surface, she’ll perceive her 400,000-year-old universe as having been infinite. She’ll also consider events C and D simultaneous, since they lie on the U-shaped surface where the first galaxies form, and so on. Because you can stack an infinite number of these U-shapes inside each other, she’ll feel that her universe is infinite in both space and future time—even though it all neatly fits into an initially subatomic region according to the outside observer. The fact that space expands inside doesn’t necessarily increase the amount of room it all takes as seen from outside: remember that Einstein allows space to stretch and produce more volume from nothing, without taking it from someplace else. In practice, this infinite universe might look something like a subatomic black hole from the outside. In fact, Alan Guth and collaborators even explored the speculative possibility of doing this trick yourself for real: creating in your laboratory something that looks like a small black hole from the outside and that looks like an infinite universe from the inside—as to whether this is really possible, the jury is still out. If you’re harboring demiurgic urges, I highly recommend Brian Greene’s instructions for “aspiring universe creators” in his book The Hidden Reality.

  We began our exploration of inflation earlier in this chapter by lamenting the unsatisfactory answers that Friedmann’s classic Big Bang theory gave to some basic questions, so let’s conclude our exploration by reviewing how inflation answers them:

  Q: What caused our Big Bang?

  A: The repeated doubling in size of an explosive subatomic speck of inflating material.

  Q: Did our Big Bang happen at a single point?

  A: Almost: it began in a region of space much smaller than an atom.

  Q: Where in space did our Big Bang explosion happen?

  A: In that tiny region—but inflation stretched it out to about the size of a grapefruit growing so fast that the subsequent expansion made it larger than all the space that we see today.

  Q: How could our Big Bang create an infinite space in a finite time?

  A: Inflation produces an infinite number of galaxies by continuing forever. According to general relativity, an observer in one of these galaxies will view space and time differently, perceiving space as having been infinite already when inflation ended.

  In summary, inflation has radically transformed our understanding of our cosmic origins, replacing the awkward unanswered questions of Friedmann’s Big Bang model by a simple mechanism that creates our Big Bang from almost nothing. It has also given us more than we asked for: a space that isn’t just huge but truly infinite, with infinite numbers of galaxies, stars and planets. And as we’ll see in the next chapter, that’s just the tip of the iceberg.

  THE BOTTOM LINE

  • There are serious problems with the earliest stages of Friedmann’s Big Bang model.

  • Inflation theory solves them all, and explains the mechanism that caused the Big Bang.

  • Inflation explains why space is so flat, which we’ve measured to about 1% accuracy.

  • It explains why, on average, our distant Universe looks the same in all directions, with only 0.002% fluctuations from place to place.

  • It explains the origins of these 0.002% fluctuations as quantum fluctuations stretched by inflation from microscopic to macroscopic scales, then amplified by gravity into today’s galaxies and cosmic large-scale structure.

  • Inflation even explains cosmic acceleration, which nabbed a 2011 Nobel Prize, as inflation restarting, in slow motion, doubling the size of our Universe not every split second but every 8 billion years.

  • Inflation theory says that our Universe grew much like a human baby: an accelerating growth phase, in which the size doubled at regular intervals, was followed by a more leisurely decelerating growth phase.

  • What we call our Big Bang wasn’t the beginning but the end—of inflation in our part of space—and inflation typically continues forever in other places.

  • Inflation generically predicts that our space isn’t just huge, but infinite, filled with infinite galaxies, stars and planets, with initial conditions generated randomly by quantum fluctuations.

  6

  Welcome to the Multiverse

  If the doors of perception were cleansed every thing would appear to man as it is, Infinite.

  For man has closed himself up, till he sees all things thro’ narrow chinks of his cavern.

  —William Blake, The Marriage of Heaven and Hell

  Two things are infinite: the universe and human stupidity; and I’m not sure about the universe.

  —attributed to Albert Einstein

  Are you ready for controversy? The science we’ve explored so far in this book has by now become mostly mainstream and well accepted. We now enter the controversial, which many of my physics colleagues will argue passionately either for or against.

  The Level I Multiverse

  Is there another copy of you reading this book, deciding to put it aside without finishing this sentence, while you’re reading on? A person living on a planet called Earth, with misty mountains, fertile fields and sprawling cities, in a solar system with seven other planets? The life of this person has been identical to yours in every respect—until now, that is, when your decision to read on signals that your two lives are diverging.

  You probably find this idea strange and implausible, and I must confess that this is my gut reaction, too. Yet it looks like we might just have to live with it, since the simplest and most popular cosmological model today predicts that this person actually exists in a galaxy about 101029 meters from here. This proposition doesn’t even assume speculative modern physics, but merely that space is infinite and rather uniformly filled with matter. Your alter ego is simply a prediction of eternal inflation, which, as we’ve seen in the last chapter, agrees with all current observational evidence and is implicitly used a
s the basis for most calculations and simulations presented at cosmology conferences.

  What’s a Universe?

  Before we start talking in earnest about other universes, it’s crucial that we’re clear on what we mean by our own Universe. This is the terminology we’ll use in this book:

  Term Definition

  Physical reality Everything that exists

  Our Universe Everything that exists

  The part of physical reality we can in principle observe

  If we ignore the quantum complications of the next chapter, the following universe definition is equivalent.

  In the past chapters, we also referred to this region as our observable Universe. Geekier-sounding synonyms that are popular with astronomers are our horizon volume, or the region within our particle horizon. Astronomers also like to talk about our Hubble volume, whose size is in the same ballpark, defined as the region within which galaxies are receding slower than light.

  Given that other universes may exist, I find it a bit arrogant referring to our own as the universe, so I try to avoid using that term altogether. But this is clearly a matter of taste, since New Yorkers refer to their town as “the city,” and Americans and Canadians refer to their joint baseball championship as “the World Series.”