Our Mathematical Universe

by Max Tegmark

  As if this weren’t enough of a success, Alan realized that inflation solves the flatness problem as well. Suppose you’re the ant on the sphere in Figure 2.7 and can only see a small area of the curved surface that you live on. If inflation suddenly makes the sphere vastly larger, that small area that you can see will look much flatter; a square centimeter on a ping-pong ball is noticeably curved, whereas a square centimeter on the surface of Earth is almost perfectly flat. Similarly, when inflation dramatically expands our own 3-D space, the space within any given cubic centimeter becomes almost perfectly flat. Alan proved that as long as inflation continues long enough to make our observable Universe, it makes space flat enough to last until the present day without a Big Crunch or Big Chill.

  In fact, inflation typically continues a lot longer than that, ensuring that space remains essentially perfectly flat until the present day. In other words, inflation theory made a testable prediction back in the eighties: our space should be flat. As we saw in the last two chapters, we’ve now performed this test to better than 1% precision, and inflation passed the test with flying colors!

  Who Paid for the Ultimate Free Lunch?

  Inflation is like a great magic show—my gut reaction is: This can’t possibly obey the laws of physics! Yet under close enough scrutiny, it does.

  First of all, how can one gram of inflating matter turn into two grams when it expands? Surely, mass can’t just be created from nothing? Interestingly, Einstein offered us a loophole through his special relativity theory, which says that energy E and mass m are related according to the famous formula E = mc2. Here c = 299,792,458 meters per second is the speed of light, and because it’s such a large number, a tiny amount of mass corresponds to a huge amount of energy: less than a kilogram of mass released the energy of the Hiroshima nuclear blast. This means that you can increase the mass of something by adding energy to it. For example, you can make a rubber band very slightly heavier by stretching it: you need to apply energy to stretch it, and this energy goes into the rubber band and increases its mass.

  A rubber band has negative pressure because you need to work to expand it. For a substance with positive pressure, like air, it’s the other way around: you need to do work to compress it. In summary, the inflating substance has to have negative pressure in order to obey the laws of physics, and this negative pressure has to be so huge that the energy required to expand it to twice its volume is exactly enough to double its mass.

  Another puzzling feature of inflation is that it causes accelerated expansion. In high school, I was taught that gravity is an attractive force, so if I have a bunch of expanding stuff, then shouldn’t gravity instead decelerate the expansion, trying to ultimately reverse the motion and pull things back together? Again Einstein comes to the rescue with a loophole, this time from his general relativity theory, which says that gravity is caused not only by mass, but also by pressure. Since mass can’t be negative, the gravity from mass is always attractive. But positive pressure also causes attractive gravity, which means that negative pressure causes repulsive gravity! We just saw that an inflating substance has huge negative pressure. Alan Guth calculated that the repulsive gravitational force caused by its negative pressure is three times stronger than the attractive gravitational force caused by its mass, so the gravity of an inflating substance will blow it apart!

  In summary, an inflating substance produces an antigravity force that blows it apart, and the energy that this antigravity force expends to expand the substance creates enough new mass for the substance to retain constant density. This process is self-sustaining, and the inflating substance keeps doubling its size over and over again. In this way, inflation creates everything we can observe with our telescopes from almost nothing. This prompted Alan Guth to refer to our Universe as “the ultimate free lunch”: inflation predicts that its total energy is very close to zero!

  But according to the Nobel Prize–winning economist Milton Friedman, “there’s no such thing as a free lunch,” so who paid the energy bill for all that galactic grandeur that we observe around us in our Universe? The answer is that gravity did, because the gravitational force injected energy into the inflating matter by stretching it out. But if the total energy of everything can’t change and heavy objects have loads of positive energy according to Einstein’s E = mc2 formula, then this means that gravity must have gotten stuck with a corresponding amount of negative energy! That’s in fact exactly what’s happened. The gravitational field, which is responsible for all gravitational forces, has negative energy. And it gets more negative energy every time gravity accelerates something. Consider, for example, a distant asteroid. If it’s moving only slowly, it has very little motion energy. If it’s far from Earth’s gravitational pull, it also has very little gravitational energy (so-called potential energy). If it gradually falls toward Earth, it will pick up great speed and motion energy—perhaps enough to create a huge crater on impact. Since the gravitational field started with almost no energy and then released all this positive energy, it now has negative energy left.

  We’ve now tackled another question from our list at the beginning of Chapter 2: Doesn’t creation of the matter around us from almost nothing by inflation violate energy conservation? We’ve seen that the answer is no: all the required energy was borrowed from the gravitational field.

  I have to confess that, although this process doesn’t violate the laws of physics, it makes me nervous. I just can’t shake the uneasy feeling that I’m living in a Ponzi scheme of cosmic proportions. If you’d visited Bernie Madoff before his 2008 arrest for embezzling $65 billion, you’d have thought that he was surrounded by real wealth that he actually owned. Yet on closer scrutiny, it turned out that he’d effectively purchased it with borrowed money. Over the years, he doubled the scale of his operation over and over again by cleverly leveraging what he had to borrow even more from naive investors. An inflating universe does exactly the same thing: it doubles its size over and over again by leveraging the energy that it already has to borrow even more energy from the gravitational field. Just like Madoff, the inflating universe exploits an inherent instability in the system to create apparent grandeur out of nothing. I just hope that our Universe proves less unstable than Madoff’s.…

  The Gift That Keeps on Giving

  Inflation Encore

  Like many successful scientific theories, inflation got off to a rough start. Its first firm prediction, that space was flat, seemed inconsistent with mounting observational evidence. As we saw in the last chapter, Einstein’s gravity theory says that space can only be flat if the cosmic density equals a particular critical value. We use the symbol Ωtotal (or just Ω or “Omega” for short) to denote how many times denser our Universe is than this critical density, so inflation predicted that Ω = 1. While I was a grad student, however, our measurements of the cosmic density from galaxy surveys and other data kept getting better, suggesting the much lower value Ω ≈ 0.25, and it became increasingly embarrassing for Alan Guth to travel from conference to conference stubbornly insisting that Ω = 1 despite what his experimental colleagues told him. But Alan stuck to his guns, and history proved him right. As we saw in the last chapter, the discovery of dark energy revealed that we’d only been counting about a quarter of the density, and when we counted dark energy, too, we measured Ω = 1 to better than 1% precision (see Table 4.1).

  The discovery of dark energy gave a huge credibility boost to inflation also for another reason: Now you could no longer dismiss the assumption of a nondiluting substance as nutty and unphysical, because dark energy is precisely such a substance! So the epoch of inflation that created our Big Bang ended 14 billion years ago, but a new epoch of inflation has begun. This new phase of inflation driven by dark energy is just like the old one but in slow motion, doubling the size of our Universe not every split second but every 8 billion years. So the interesting debate is no longer about whether inflation happened or not, but about whether it happened once or twice.

>   Sowing the Seed Fluctuations

  The hallmark of a successful scientific theory is that you get more out of it than you put into it. Alan Guth showed that, with one single assumption (a tiny speck of a hard-to-dilute substance), you could solve three separate cosmological conundrums: the Bang problem, the horizon problem and the flatness problem. Above we saw how inflation did more: it predicted Ω = 1, which was accurately confirmed about two decades later. However, that wasn’t all.

  We ended the last chapter by asking where the galaxies and the large-scale cosmic structure ultimately came from, and much to everybody’s surprise, inflation answered this question, too! And what an answer it gave! The idea was first proposed by two Russian physicists, Gennady Chibisov and Viatcheslav Mukhanov, and when I first heard it, I thought it sounded absurd. Now I think it’s a leading candidate for the most radical and beautiful synthesis of ideas in scientific history.

  In short, the answer is that the cosmic seed fluctuations came from quantum mechanics, the theory of the microworld that we’ll explore in Chapters 7 and 8. But I learned in college that quantum effects are important only for the very smallest things we study, such as atoms, so how can they possibly have any relevance to the very largest things we study, such as galaxies? Well, one of the beauties of inflation is that it connects the smallest and largest scales: during the early stages of inflation, the region of space that now contains our Milky Way Galaxy was much smaller than an atom, so quantum effects could have been important. And indeed they were: as we’ll see in Chapter 7, the so-called Heisenberg uncertainty principle of quantum mechanics prevents any substance, including the inflating material, from being completely uniform. If you try to make it uniform, quantum effects force it to start wiggling around, spoiling the uniformity. When inflation stretched a subatomic region into what became our entire observable Universe, the density fluctuations that quantum mechanics had imprinted were stretched as well, to sizes of galaxies and beyond. As we saw in the last chapter, gravitational instability took care of the rest, amplifying these fluctuations from the tiny 0.002%-level amplitudes with which quantum mechanics had endowed them into the spectacular galaxies, galaxy clusters and superclusters that now adorn our night sky.

  Figure 5.6: This so-called snowflake fractal, invented by the Swedish mathematician Helge von Koch, has the remarkable property that it’s identical to a magnified piece of itself. Inflation predicts that our baby Universe was similarly indistinguishable from a magnified piece of itself, at least in an approximate statistical sense.

  Click here to see a larger image.

  The best part is that this isn’t just qualitative blah blah, but a rigorous quantitative story where everything can be accurately calculated. The power-spectrum curve I’ve plotted in Figure 4.2 is a theoretical prediction for one of the very simplest inflation models, and I find it remarkable how well it matches all the measurements. Inflation models can also predict three of the measured cosmological parameters that I listed in Table 4.1. I’ve already mentioned one of these predictions: Ω = 1. The other two involve the nature of the cosmic-clustering patterns that we explored in the last chapter. In the simplest inflation models, the amplitude of the seed clustering (called Q in the table) depends on how fast the inflating region doubles its size, and with a doubling time around 10−38 seconds, the prediction matches the observed value Q ≈ 0.002%.

  Inflation also makes an interesting prediction for the seed clustering “tilt” parameter (called n in the table). To understand this, we need to look at the jagged curve in Figure 5.6, which is what mathematicians call self-similar, fractal or scale-invariant. All of these words basically mean that if I replace the image by a magnified piece of it, you can’t tell the difference. Since I can repeat this zoom trick as many times as I want, it’s clear that even a trillionth of the curve must look identical to the whole thing. Interestingly, inflation predicts that to a good approximation, our baby Universe was scale-invariant, too, in the sense that you couldn’t tell the difference between a random cubic centimeter of it and a greatly magnified piece of it. Why? Well, during the inflation epoch, magnifying our Universe was basically equivalent to waiting a little, until everything doubled in size yet again. So if you could have time-traveled back to the inflation epoch, seeing that the statistical properties of the fluctuations were scale-invariant would have been equivalent to seeing that these properties didn’t change over time. But inflation predicts that these properties hardly change over time for a very simple reason: the local physical conditions that generate the quantum fluctuations hardly change over time either, since the inflating substance isn’t noticeably changing its density or other properties.

  The tilt parameter n in Table 4.1 measures how close the inflating universe was to scale-invariant. It contrasts the amount of clustering on large and small scales, and is defined so that n = 1 means perfectly scale-invariant (the same clustering on all scales), n < 1 means more clustering on large scales, and n > 1 means more clustering on small scales. Mukhanov and other inflation pioneers had predicted that n would be quite close to 1. When my friend Ted and I moonlighted on the magicbean computer back in Chapter 4, it was to make the most accurate measurement to date of n. Our result was n = 1.15 ± 0.29, confirming that yet another prediction from inflation was looking good.

  The n business gets even more interesting. Because inflation eventually has to end, the inflating substance has to gradually dilute ever so slightly during inflation—otherwise nothing would change and inflation would continue forever. In the simplest inflation models, this decrease in the density causes the amplitude of generated fluctuations to decrease as well. This means that the fluctuations generated later on have lower amplitude. But fluctuations generated later didn’t get stretched as much before inflation ended, so they correspond to fluctuations on smaller scales today. The upshot of all this is the prediction that n < 1. To predict something more specific, you need a model for what the inflating substance is made of. The simplest such model of all, pioneered by Andrei Linde (Figure 5.1), is known in geek-speak as a “scalar field with quadratic potential” (it’s basically a hypothetical cousin of a magnetic field), and it predicts that n = 0.96. Now take another look at Table 4.1. You’ll see that the n measurement has now gotten about 60 times more accurate since those wild magicbean days, and that the latest measurement is n = 0.96 ± 0.005, tantalizingly close to what was predicted!

  Andrei Linde is one of inflation’s pioneers, and has inspired me a lot. I’ll hear someone explain something and think it’s complicated. Then I’ll hear Andrei’s explanation of the same thing and realize that it’s simple when I think about it in the right way—his way. He has a dark but warm sense of humor that undoubtedly helped him survive back in the Soviet Union, and has a mischievous glint in the eye regardless of whether he’s discussing personal things or cutting-edge science.

  All these measurements will keep getting more accurate in the years to come. We also have the potential to measure several additional numbers that inflation models make predictions for. For example, in addition to intensity and color, light has a property called polarization—bees can see it and use it to navigate, and although our human eyes don’t notice it, our polarized sunglasses let light through only if it’s polarized in a particular way. Many popular inflation models predict a rather unique signature in the polarization of the cosmic microwave–background radiation: quantum fluctuations during inflation generate what’s known as gravitational waves, vibrations in the very fabric of spacetime, and these in turn distort the cosmic microwave–background pattern in a characteristic way. If these distortions are detected by a future experiment, I think it will be hailed as smoking-gun evidence that inflation really happened.

  In summary, it’s too early to say for sure whether our Big Bang really was caused by inflation. However, I feel that it’s fair to say that the inflation theory has been way more successful than Alan Guth imagined when he invented it, giving good agreement with precision measurem
ents and emerging as the theory of our cosmic origins that’s taken most seriously by the cosmology community.

  Eternal Inflation

  Our discussion of inflation so far might sound like the typical life cycle of a successful physics idea: new theory solves old problems. Further predictions. Experimental confirmation. Widespread acceptance. Textbooks rewritten. It sounds as though it’s time to give inflation the traditional scientific retirement speech: “Thank you, inflation theory, for your loyal service in tying up some loose ends regarding the ultimate origins of our Universe. Now please go off and retire in neatly compartmentalized textbook chapters, and leave us alone so that we can work on other newer and more exciting problems that aren’t yet solved.” But like a tenacious aging professor, inflation refuses to retire! In addition to being the gift that keeps on giving within its compartmentalized subject area of early-universe cosmology, as we saw above, inflation has given us more radical surprises that were quite unexpected—and to some of my colleagues, also quite unwelcome.

  Unstoppable

  The first shocker is that inflation generally refuses to stop, forever producing more space. This was discovered for specific models by Andrei Linde and Paul Steinhardt. An elegant proof of the existence of this effect was given by Alex Vilenkin, a friendly soft-spoken professor at Tufts University, and the one who invited me to give that talk that put Alan Guth to sleep. While he was a student back in his native Ukraine, he refused a request from the KGB to testify against a fellow student who was critical of the authorities, despite being warned of “consequences.” Although he’d been admitted to physics grad school at Moscow State University, the most prestigious physics program in the Soviet Union, the permission he required for moving to Moscow was never granted. Nor was he able to get any normal jobs. He spent a year struggling as a night watchman at a zoo before finally managing to leave the country. Whenever I get annoyed by a bureaucrat, thinking of Alex’s story transforms my frustration into grateful realization of how small my problems are. Perhaps his disposition to stick with what he believes is right despite authority pressure helps explain why he persisted and discovered things that other great scientists dismissed.