Our Mathematical Universe

by Max Tegmark

  How exactly did gravity do this? If you stop your bike at a red light, you quickly realize that gravity can be destabilizing: you inevitably start tipping sideways and need to put your foot down on the asphalt to avoid falling. The essence of instability is that small fluctuations get amplified. In the stopped-bike example, the farther you get from balance, the stronger gravity will push you in the wrong direction. In the cosmic example, the farther our Universe gets from perfect uniformity, the more forcefully gravity amplifies its clumpiness. If a region of space is slightly denser than its surroundings, then its gravity will pull in neighboring material and make it even denser. Now its gravity is even stronger, making it accrete mass even faster. Just as it’s easier to make money when you have lots of it, it’s easier to accrete mass when you have lots of it. Fourteen billion years was ample time for this gravitational instability to transform our Universe from boring to interesting, amplifying even tiny density fluctuations into huge dense clumps such as galaxies.

  Although this basic picture of expansion and clustering had been worked out during the preceding decades, the details remained sketchy when I started grad school in 1990 and got my first exposure to cosmology. People were still arguing about whether our Universe was 10 billion years old or 20 billion years old, reflecting a long-standing dispute about how fast it was currently expanding, and the harder question of how fast it had expanded in the past was wide open. The clustering story was on even shakier grounds: attempts to get detailed agreement between theory and observation gradually revealed that we had no clue what 96% of our Universe was made of! After the COBE experiment measured 0.002% clumping 400,000 years after our Big Bang, it became clear that gravity wouldn’t have had time to amplify this faint clustering into today’s cosmic large-scale structure unless some invisible form of matter contributed extra gravitational pull.

  This mysterious stuff is known as dark matter, which is really little more than a name for our ignorance. The name invisible matter would be more apt, since it looks transparent rather than dark, and can pass through your hand without your noticing. Indeed, dark matter from space that strikes Earth appears to typically pass unaffected through our entire planet, emerging unscathed on the other side. As if dark matter wasn’t crazy enough, a second mystery substance dubbed dark energy was introduced to make the theoretical predictions match the observed expansion and clustering. It was assumed to affect the cosmic expansion without clustering at all, remaining perfectly uniform at all times.

  Both dark matter and dark energy had had a long and controversial history. The simplest candidate for dark energy was the so-called cosmological constant, the above-mentioned fudge factor that Einstein added to his gravity theory and later called his greatest blunder. Fritz Zwicky postulated dark matter in 1934 to explain the extra gravitational pull that kept galaxy clusters from flying apart, and Vera Rubin discovered in the 1960s that spiral galaxies rotated so fast that they, too, would fly apart unless they contained invisible mass gravitating enough to hold them together. These ideas met with considerable skepticism: if we’re willing to blame unexplained phenomena on entities that are both invisible and can pass through walls, shouldn’t we also start believing in ghosts while we’re at it? Moreover, there was a disturbing precedent: in ancient Greece, when Ptolemy realized that planetary orbits weren’t perfect circles, he cooked up a complicated theory in which they moved on smaller circles (called epicycles) that in turn moved in circles. As we saw earlier, the subsequent discovery of a more accurate law of gravity killed the epicycles, predicting that the orbits aren’t circles but ellipses. Perhaps the need for dark matter and dark energy could be eliminated just as the epicycles were, by discovering a still more accurate law of gravity? Could modern cosmology really be taken seriously?

  Figure 4.1: Both dark matter and dark energy are invisible, which means that they refuse to interact with light and other electromagnetic phenomena. We know of their existence only through their gravitational effects.

  These were the sorts of questions we asked while I was a grad student. Answering them would require much more accurate measurements, to transform cosmology from the data-starved and speculative field that it was into a precision science. Fortunately, that’s exactly what happened.

  Precision Microwave-Background Fluctuations

  As we saw in Figure 3.6, the baby picture of our Universe produced by a cosmic microwave–background experiment can be decomposed as a sum of many different component maps called multipoles which, in essence, contain the contributions from spots of different sizes. Figure 4.2 plots the total amount of fluctuation in each of these multipoles; this curve is called the power spectrum of the microwave background, and encodes the key cosmological information from the map. When you look at a sky map like the one in Figure 3.4 and on the book cover, you see spots of many different sizes just as on a Dalmatian: some spots are about 1 degree across in the sky, others are 2 degrees, and so on. The power spectrum encodes information about how many spots there are of each size.

  What’s so great about the power spectrum is that not only can we measure it, but we can also predict it: for any mathematically defined model of how our Universe expanded and clustered, we can calculate exactly what the power spectrum should be. As you can see in Figure 4.2, the predictions differ wildly between models: indeed, the measurements have now ruled out all but one of the theoretical models in Figure 4.2 beyond any reasonable doubt, even though for each of the killed models, I have at least one respected colleague who used to believe that it was the right one back when I was in grad school. The predicted shape of the power spectrum depends in complicated ways on all the things that affect cosmic clustering (including the density of atoms, the density of dark matter, the density of dark energy and the nature of the seed fluctuations), so if we can adjust our assumptions about all these things so that the prediction matches what we measure, then we’ve not only found a model that works, but also measured these important physical quantities.

  Figure 4.2: Precision measurements of how cosmic microwave–background fluctuations depend on angular scale have totally ruled out many previously popular theoretical models, but agree beautifully with the curve predicted by the current standard model. You can appreciate the most remarkable aspect of modern cosmology here without worrying about any of the details: highly accurate measurements now exist, and they agree with theoretical prediction.

  Telescopes and Computers

  When I first learned about the cosmic microwave background in grad school, there were no measurements whatsoever of the power spectrum. Then the COBE team gave us our first grip on this elusive wiggly curve, measuring that its height on the leftmost side was about 0.001% and that its slope around there was roughly horizontal. There was more information about the power spectrum in the COBE data, but nobody had squeezed it out because this would involve tedious manipulations of a table of numbers called a matrix, which took up 31 megabytes. Although this quantity sounds like a joke these days, being the size of a short video clip on your phone, it was daunting in 1992. So my classmate Ted Bunn and I hatched a sneaky plan: Professor Marc Davis in our department owned a computer called “magicbean” that had more than 32 megabytes of memory, and night after night, I logged on to it in the wee hours of the morning when nobody was paying attention, and let it work on our data analysis. After a few weeks of this clandestine moonshine number crunching, we published a paper with the most accurate measurement so far of the power spectrum shape.

  This project made me realize that, just as telescopes had once transformed astronomy, dramatic improvements in computer technology had the power to take it to yet another level. Indeed, your own computer today is so much better that it could repeat all my calculations with Ted in minutes. I decided that if experimentalists were putting so much hard work into collecting this data about our Universe, people like me owed it to them to milk their data for all it was worth. This became a central theme of my work for the next decade.

  One que
stion I obsessed about was how to best measure the power spectrum. There were quick methods that suffered from inaccuracies and other problems. Then there was the optimal method that my friend Andrew Hamilton had worked out, which unfortunately required an amount of computer time that grew as the sixth power of the number of pixels in the sky map, so measuring the power spectrum from the COBE map would take longer than the age of our Universe.

  It’s November 21, 1996, and it’s dark and quiet at the Institute for Advanced Study in Princeton, New Jersey, where I’m having another crazy night in the office. I’m excited about an idea for replacing the sixth power in Andrew Hamilton’s method by the third power, enabling me to optimally measure the COBE power spectrum in under an hour, and I’m scrambling to finish my paper in time for a Princeton conference the next day. In the physics community, we post all our papers to this free website http://arXiv.org as soon as we’ve finished them, so that our colleagues can read them before they get bogged down in the refereeing and publishing process. The problem was that I had this terrible habit of submitting my papers before I’d finished them, right after the day’s submission deadline ended. This way I could be first in the next daily paper listing. The downside was that, if I failed to finish within twenty-four hours, I’d be publicly humiliated by having an unfinished draft displayed to the world as a permanent monument to my stupidity. This time, my strategy finally backfired, with early birds in Europe getting access to the incomplete mess that was my discussion section before I finally finished it around four a.m. At the conference, my friend Lloyd Knox presented a similar method that he’d developed with Andrew Jaffe and Dick Bond in Toronto, but hadn’t yet written up for publication. When I presented my results, Lloyd smiled and said to Dick: “Fast fingers Tegmark!” Our methods turned out to be quite useful, and have been used for essentially all microwave-background power-spectrum measurements since then. Lloyd and I seem to follow parallel paths through life: we have the same ideas at the same time (indeed, he’d scooped me earlier on a cool formula for noise in microwave-background maps), we got two sons at the same time, and we even got divorced at the same time.

  Gold in the Hills

  With improved experiments, computers and methods, the measurements of the power-spectrum curve in Figure 4.2 kept getting better. As you can see in the figure, the curve was predicted to look a bit like the rolling hills of California, with a series of distinct peaks. If you measure lots of Great Danes, poodles and Chihuahuas and plot their distribution of sizes, you’ll get a curve with three peaks. Similarly, if you measure lots of cosmic microwave–background spots as shown in Figure 3.4 and plot their distribution of sizes, you’ll find that there are certain characteristic spot sizes that are particularly common. The most prominent peak in Figure 4.2 corresponds to spots that are about one degree in angular size. Why? Well, these spots were caused by sound waves rippling through the cosmic plasma near the speed of light, so since the plasma existed for 400,000 years after our Big Bang, the spots grew to be about 400,000 light-years in size. If you calculate how large an angle such a 400,000 light-year blob will cover in the sky today, 14 billion years later, you get about a degree. Unless space is curved, that is.…

  As we discussed in Chapter 2, there is more than one kind of uniform three-dimensional space: in addition to the flat kind that Euclid axiomatized and we all learned about in school, there are curved spaces where the angles obey different rules. I learned in school that the angles in a triangle on a flat sheet of paper add up to 180 degrees. But if you draw it on the curved surface of an orange, they’ll add up to more than 180 degrees, and if you draw it on a saddle, they’ll add up to less (see Figure 2.7 for examples). Similarly, if our physical space were curved like a spherical surface, the angle covered by each microwave-background spot would be bigger, shifting the peaks in the power-spectrum curve to the left, and if space had saddlelike curvature, the spots would look smaller and shift the peaks to the right.

  To me, one of the most beautiful ideas in Einstein’s gravity theory is that geometry isn’t just mathematics: it’s also physics. Specifically, Einstein’s equations show that the more matter space contains, the more curved space gets. This curvature of space causes things to move not in straight lines, but in a motion that curves toward massive objects—thus explaining gravity as a manifestation of geometry. This opens up a totally new way of weighing our Universe: just measure the first peak in the cosmic microwave–background power spectrum! If its position shows that space is flat, then Einstein’s equations tell us that our average cosmic density is about 10−26kg/m3, corresponding to about ten milligrams per Earth volume or about six hydrogen atoms per cubic meter. If the peak is farther to the left, the density is higher, and vice versa. Given all the confusion about dark matter and dark energy, measuring this total density was hugely important, and experimental teams around the world raced for the first peak—which was expected to be the easiest peak to detect because bigger spots are easier to measure than smaller ones.

  I caught my first glimpse of the peak in 1996, in a paper spearheaded by Lyman Page’s student Barth Netterfield using Saskatoon data. “Wow!” I thought, and had to put down my spoonful of Munich müsli to really take it in. At the cerebral level, the theory behind the power-spectrum peaks was very elegant and all, but in my gut, I still felt that our human extrapolations couldn’t work this well. Three years later, Lyman Page’s student Amber Miller spearheaded a more accurate measurement of the first peak, and found it to be roughly in the right place for a flat universe, but somehow, it all still felt too good to be true. Finally, in April 2000, I just had to accept it. A microwave telescope called Boomerang, which had spent eleven days circumnavigating Antarctica, dangling under a high-altitude balloon the size of a football field, had produced by far the most accurate power-spectrum measurements to date, showing a beautiful first peak at exactly the place for a flat universe. So now we knew the total density of our Universe (averaged over space).

  Dark Energy

  This measurement presents an interesting situation when accounting for the cosmic-matter budget. As you can see in Figure 4.3, we know the size of the total budget from the first peak position, but we also know the density of ordinary matter, and of dark matter from measuring its gravitational effects on cosmic clustering. But all this matter makes up only 30% of the total budget, which means that the remaining 70% must be some form of matter that doesn’t cluster—so-called dark energy.

  The most impressive thing I just said is what I didn’t say: the word supernova. Because completely independent evidence for dark energy suggested exactly the same 70% number, based on cosmic expansion rather than clustering. Earlier, we talked about using Cepheid variable stars as standard candles to measure cosmic distances. We cosmologists now have another standard candle in our toolbox that’s even more luminous, so that it can be seen not only millions but even billions of light-years away. These are huge cosmic explosions known as Type Ia supernovae, which during a few seconds can release more energy than a hundred million billion suns.

  Do you remember the rest of the first verse of “Twinkle, Twinkle, Little Star”? When Jane Taylor wrote the lines “Up above the world so high, / Like a diamond in the sky”, she had no idea how right she was: when our Sun eventually dies in about 5 billion years, it will end its days as a so-called white dwarf, which is a giant ball that—like a diamond—is made mostly of carbon atoms. Our Universe is teeming with white dwarfs today, created by stars past. Many of them are continually gaining weight by gobbling up gas from dying companion stars that they’re orbiting. Once they become officially overweight (which happens when they reach 1.4 times the mass of the Sun), they suffer the stellar equivalent of a heart attack: they become unstable and detonate in a gigantic thermonuclear explosion—a Type Ia supernova. Since all these cosmic bombs therefore have the same mass, it comes as no surprise that they’re roughly equally powerful.

  Figure 4.3: The cosmic matter budget. The horizontal positions of the
microwave-background power-spectrum peaks tell us that space is flat and the total cosmic density is about a million trillion trillion (1030) times lower than that of water (averaged across our Universe). The peak heights tell us that ordinary and dark matter make up only about 30% of this density, so there must be 70% of something else—dark energy.

  Moreover, the slight variations in explosive power have been shown to be linked both to the spectrum of the explosion and to how fast it brightens and dims, all of which can be measured, allowing astronomers to turn supernovae Ia into accurate standard candles.

  This technique was used by Saul Perlmutter, Adam Riess, Brian Schmidt, Robert Kirshner and their collaborators to accurately measure the distances to lots of supernovae Ia and also how fast they were receding from us based on their redshift. From these measurements, they made the most accurate reconstruction to date of how fast our Universe has expanded at various times in the past, and in 1998, they announced a startling discovery that earned them the 2011 Nobel Prize in physics: after spending its first 7 billion years slowing down, the cosmic expansion started speeding up again and has accelerated ever since! If you throw a rock up in the air, Earth’s gravitational pull will decelerate its motion away from Earth, so the cosmic acceleration revealed a strange gravitational force that’s repulsive rather than attractive. As I’ll explain in the next chapter, Einstein’s gravity theory predicts that dark energy has exactly this antigravity effect, and the supernova teams found that a cosmic-matter budget with 70% dark energy beautifully explained what they saw.

  A 50% Batting Average

  One of my favorite things about being a scientist is getting to work with such cool people. The person I’ve coauthored the most papers with is a friendly Argentinian named Matias Zaldarriaga. My ex-wife and I secretly nicknamed him “the Great Zalda,” and we agreed that the only thing that topped his talent was his sense of humor. He’d cowritten the computer program everyone used to predict power-spectrum curves like those in Figure 4.2, and he once bet an air ticket to Argentina that his predictions were all wrong and there were no peaks. In preparation for the Boomerang results, we sped up these calculations and computed a huge database of models against which we could compare measurements. So when the Boomerang data became available, I again posted an unfinished paper to http://arXiv.org, and then we had fun working around the clock to finish it before it went public on Sunday evening. Ordinary matter (atoms) can bump into stuff that dark matter simply sails through, and therefore ends up moving differently through space. This means that ordinary and dark matter affects cosmic clustering and the microwave-background power-spectrum curve (see Figure 4.2) in different ways. In particular, adding more atoms to the matter budget lowers the second peak. The Boomerang team reported a really puny second peak, and Matias and I found that this required atoms to make up at least 6% of the cosmic-matter budget. But Big Bang nucleosynthesis, the cosmic fusion–reactor story we discussed in Chapter 3, only works if atoms make up 5%, so something was wrong! I spent these crazy days in Albuquerque where I’d gone to give a talk, and it felt really exhilarating to get to tell the audience about these brand-new clues that our Universe had revealed. Matias and I just barely made our deadline, and our paper hit the Web even before the Boomerang team’s own analysis paper, which was delayed by a pedantic computer on the ridiculous grounds that a figure caption was one word too long.