Our Mathematical Universe

by Max Tegmark

  Figure 3.4: It looks like we’re in the middle of a giant plasma sphere, because we see this plasma wall from the previous figure in every direction that we look.

  Click here to see a larger image.

  The discovery was a sensation, and earned them the 1978 Nobel Prize in physics. From the calculations of Gamow and his students, it followed that the plasma sphere in Figure 3.4 must have been about half as hot as the Sun’s surface, and that as the radiation from its hot glow traveled through space for 14 billion years to reach us, it cooled down thousandfold to the observed three degrees above absolute zero as space expanded thousandfold. In other words, our entire Universe was once as hot as a star, and the wild thousandfold extrapolation of Gamow’s hot Big Bang theory had been tested and vindicated.

  Baby Pictures of Our Universe

  Now that the plasma sphere had been detected, the race was on to take the first photos of it. Because the temperature of the radiation was basically the same in all directions, the images Penzias and Wilson could make looked like one of those joke postcards labeled “San Francisco in the Fog,” where all you see is uniform white. To get interesting photos that would qualify as the first baby pictures of our Universe, one would need to increase the contrast to detect slight variations from place to place. These variations had to exist, because if the conditions had been identical everywhere in the past, then the laws of physics would have kept them identical everywhere today, in stark contrast to the clumpy Universe we now observe with galaxies in some places and not in others.

  However, taking these cosmic baby pictures proved so difficult that it took almost three decades of technological development. To suppress measurement noise, Penzias and Wilson had to use liquid helium to cool their detector down to near the temperature of the cosmic microwave background. The temperature fluctuations from place to place in the sky turned out to be tiny, around a thousandth of a percent, so producing baby pictures required a hundred thousand times more sensitivity than that of Penzias and Wilson’s measurements. Experimentalists around the world took on the challenge and failed. Some said it was hopeless, but others refused to give up. On May 1, 1992, when I was halfway through grad school, the fledgling Internet was abuzz with rumors: George Smoot was going to announce the results from the most ambitious microwave-background experiment to date, performed from the cold darkness of space by a NASA satellite called COBE, the Cosmic Background Explorer. My Ph.D.-thesis advisor, Joe Silk, was scheduled to introduce George’s talk, and before he flew out to Washington, D.C., I asked him what he thought the odds were of a discovery. Joe guessed that they hadn’t seen the cosmic fluctuations, just a radio noise from our own Galaxy.

  But instead of delivering an anticlimactic lecture, George Smoot dropped a bombshell that transformed not only my own career, but the entire field of cosmology: he and his team members had found the fluctuations! Stephen Hawking hailed this as “the most important discovery of the century, if not of all time,” because as we’ll see below, these baby pictures of when our Universe was “only” 400,000 years old contained crucial clues to our cosmic origins.

  The Gold Rush

  Now that COBE had found gold, there was a wild rush to mine more of it. As you can see in Figure 3.5, the COBE sky map was pretty fuzzy, because of low-resolution imaging that smoothed out features smaller than about 7 degrees—the natural next step was therefore to zoom in on a small part of the sky with higher resolution and less noise. As I’ll explain below, such high-resolution maps encode the answer to some key cosmological questions. I’d loved photography ever since I saved up for my first camera at age twelve by delivering junk mail in Stockholm, so imaging our Universe instinctively appealed to me. I’d also enjoyed messing with images and computer graphics, whether it was for my high school’s newspaper, Curare, or for the shareware computer game FRAC, a 3-D Tetris clone that paid for my 1991 around-the-world trek. I therefore felt very fortunate when various experimentalists let me team up with them on converting their data into sky maps.

  Figure 3.5: When showing maps of the whole sky, it’s convenient to project them onto a flat page just as we do with Earth maps (top), simply interpreting them as looking up toward the sky rather than down toward the ground. The “baby picture of our Universe” from COBE (bottom left) was quite fuzzy, motivating many experiments to zoom in on small sky patches with higher resolution (middle left) before the WMAP and Planck satellites delivered high-resolution maps of the entire sky (right) with three megapixels and fifty megapixels, respectively. These sky maps are rotated relative to the Earth map so that map midplane corresponds not to the plane of Earth’s equator but to the plane of our Galaxy (bottom left gray stripe); Earth’s North Pole points toward the center of the Saskatoon map. (Earth map credit: Patrick Dineen)

  Click here to see a larger image.

  My first stroke of luck was meeting Lyman Page, a young professor from Princeton. I liked his playful, boyish smile, and worked up the courage to ask him about possible collaboration after a conference talk he gave. I liked him even more after learning that he’d spent years sailing the Atlantic before grad school. He ended up entrusting me with data from a microwave telescope in the Canadian town Saskatoon, with which he and his group had spent three years scanning the sky patch directly above the North Pole.

  Converting this into a map was surprisingly hard because the data didn’t consist of sky photos, merely of long tables of numbers encoding how many volts had been measured by adding and subtracting different parts of the sky in various complicated ways. But I also found it surprisingly exciting, requiring my utmost efforts with information theory and computational number-crunching, and after many müsli-fueled evenings in the Munich office where I was doing my postdoc, I was able to finish the Saskatoon map in Figure 3.5 just in time for my talk at a big cosmology conference in the French Alps. Although I’ve given hundreds of talks by now, there are a few that stand out in my memory as magic moments that infallibly make me smile every time I remember them. This was one of those magic ones. My heart pounded as I walked up to the podium and looked around the room. It was packed with people, many of whom I knew from reading their work and most of whom had no idea who I was. They’d come to the conference more for the great skiing than for hearing total beginners like me. But I didn’t just feel my heart race—I also felt a great energy in the room. People were really excited about all the new cosmic microwave–background developments, and I felt honored and thrilled to get to be a small part of this. The year 1996 was back in the pre-Cambrian era when we still gave our talks using plastic transparencies, and I ended with the ace in my deck: a slide showing the Saskatoon map just as in Figure 3.5, as a zoom of the COBE map. I could feel a ripple of excitement spreading through the room, and a bunch of people stood around the overhead projector for most of the ensuing coffee break requesting to look at it again and asking questions. Dick Bond, one of the founders of cosmic microwave–background cosmology, came over and said, smiling: “I can’t believe Lyman gave you the data!”

  I felt that cosmology had entered a golden age, where new discoveries were bringing new people and new funding into the field, which led to new discoveries in a virtuous circle. The very next month, in April 1996, funding was approved for two new satellites with radically better resolution and sensitivity than COBE. One was the NASA mission WMAP, spearheaded by Lyman Page and a close-knit group of colleagues, and the other was the European-led mission Planck, for which I’d had lots of fun making calculations and forecasts for the grant proposal. Since space missions involve many years of planning, smaller teams around the world raced to steal the thunder from WMAP and Planck, or at least grab some of the lowest-hanging fruit before they launched. As a result, the Saskatoon project ended up being just the first of many fun data collaborations for me. I got to work with the builders of experiments with exotic names such as HACME, QMAP, Tenerife, POLAR, PIQ and Boomerang to make baby pictures of our Universe from their data or figure out what they taught us abou
t our cosmos. My basic game plan was to be the middleman between theory and experiment: I felt that cosmology was transforming from a data-starved field into one with more data than people knew how to handle, so I decided to develop tools for taking full advantage of this data avalanche. Specifically, my strategy was to use a branch of mathematics known as information theory to figure out how much relevant information about our Universe was contained in a given data set. Typically, the megabytes, gigabytes or terabytes measured would contain only a modest number of bits of cosmological information, scrambled and hidden in some complicated way among vast amounts of noise from detector electronics, atmospheric emission, Galactic radiation, and other sources. Although there was a known mathematically perfect method for extracting this needle from the haystack, it was usually too complex to do in practice, requiring millions of years of computer calculations. I published various data-analysis methods that weren’t necessarily perfect, but were able to extract almost all the information quickly enough to be useful in practice.

  I love the cosmic microwave background for many reasons. For example, I can thank it for my first marriage and for the existence of my sons, Philip and Alexander: I got together with my ex-wife, Angélica de Oliveira Costa, because she came from Brazil to Berkeley as a grad student to work with George Smoot, and we ended up collaborating closely not only on diaper changing, but also on many of the data-analysis projects I mentioned. One such project was QMAP, a telescope flown by Lyman Page, Mark Devlin and collaborators on a high-altitude balloon to avoid most of the microwave noise caused by Earth’s atmosphere.

  · · ·

  Oh, no! It’s about two a.m. on May 1, 1998, and things look grim. There are only seven hours left before our flight will depart to Chicago, where I’m supposed to present the new QMAP results at a cosmology conference, but Angélica and I are still in my office at the Institute for Advanced Study in Princeton, shaking our heads. So far, all cosmic microwave–background experiments had required you to believe that no mistakes had been made and nothing important had been overlooked. A key to believability in science is having an independent experiment confirm your results, but because people had looked in different directions with different resolutions, it was never possible to compare the sky images made by two different experiments and check whether they agreed with each other. Up until this moment, that is: the Saskatoon and QMAP sky maps have a major overlap in a banana-shaped sky patch that you can see in Figure 3.5. Angélica and I are staring with dismay at my computer screen and feeling our hearts sink: there are the Saskatoon and QMAP maps side by side, and they don’t agree at all! We squint and try to imagine that the discrepancies are just due to instrumental noise. No, wishful thinking goes only so far. All this work just to realize that at least one of the maps is all wrong. And how can I possibly give a talk about this? It would be total humiliation not just for us, but for all the people who built and ran the experiments.

  Suddenly Angélica, who’s been poring over our computer program, discovers a suspicious minus sign, which would, crudely speaking, cause the QMAP map to come out upside down. We fix it, rerun the code, and look at each other with disbelief as the new map appears on the screen: now the agreement between the two maps is stunning! A clutch play! We sleep for a few hours, fly to Chicago, I whip my talk together on pure adrenaline, and I run all the way from our rental car to the Fermilab auditorium to arrive just in time for my talk. I’m so excited that I don’t even realize my transgression until the evening, when our car is mysteriously missing.

  “Where did you park it?” the guard asks.

  “Oh, right outside, in front of the fire hydrant,” I reply—and suddenly find myself thinking Doh!!! for the second time that day.…

  The Cosmic Beach Ball

  The great gold rush to mine the microwave sky continued for years, with over twenty different experiments spurring each other on—I’ll tell you more about some of them below. And then there was WMAP. At two p.m. on March 11, 2003, the room was packed: we were all glued to the screen where the WMAP team members were announcing their results live on NASA-TV. Whereas ground- and balloon-based experiments could only map parts of the sky, the WMAP satellite had mapped the whole sky just as COBE had, with dramatically better sensitivity and resolution. I felt like when I was a little kid on Christmas Eve and Santa Claus finally arrived—except that I’d been eagerly awaiting this moment not for months but for years. It was worth the wait: the resulting images were stunning. As was their work ethic and sleep deprivation: they’d gone from funding to construction, launch, data analysis, and results in under six years, three times faster than COBE. Indeed, the WMAP project leader, Chuck Bennett, almost killed himself keeping it on schedule: David Spergel, another key contributor to the project, told me that Chuck collapsed and had to be hospitalized for three weeks after launch.

  Moreover, they made all their data publicly available online, so that cosmologists around the world could take a crack at reanalyzing it themselves. Cosmologists like me. Now it was my turn to work like crazy while they caught up on sleep. Their measurements were superb, but contaminated with radio noise from our own Galaxy, which you can see in Figure 3.5 as the horizontal band in the COBE map. The bad news is that such microwave contamination from our Galaxy and others exists everywhere in the sky, even if the level is too low to be easily seen. The good news is that the contamination has a different color than the signal (it depends on frequency in a different way), and that WMAP imaged the sky at five separate frequencies. The WMAP team had used this extra information to clean out the contamination, but I was excited about an even better method for doing this, based on information theory, which produced a cleaner map with higher-resolution (Figure 3.5, bottom right). After working all out on this for a month with Angélica and my old friend Andrew Hamilton, we submitted our paper and my life started returning to normal. I had fun making the ball-like image of the microwave background in Figure 3.4 and on the front cover of this book, and the WMAP team liked it so much that they made their own version and printed it on a plastic beach ball, which to this day adorns my office. I call it my “universe,” because it’s the iconic image of what bounds everything we can in principle observe.

  The Axis of Evil

  As I’ll explain further on, key cosmic clues lie encoded in the sizes of the spots you see in the cosmic microwave background. Just as we can decompose sounds and colors into different frequencies, we can decompose two-dimensional microwave-background maps as a sum of many different component maps (see Figure 3.6) that go by the geeky-sounding name of multipoles. These multipole maps, in essence, contain the contributions from spots of different sizes, and ever since COBE, something had seemed to be fishy with the second multipole, called the quadrupole: the largest spots in the map appeared weaker than expected. Yet nobody had ever been able to make a map of the quadrupole to see what was going on with it: this required a map of the entire sky, but microwaves from our Galaxy contaminated part of the sky beyond repair.

  Until now: our map appeared so clean that perhaps it was usable across the whole sky. It was late at night, shortly before we submitted our map paper. Angélica and the kids were asleep, and I was tempted to hit the sack. But I was really curious to see what that pesky quadrupole looked like, and decided to write a computer program making a picture of it. When it finally popped up on my screen (Figure 3.6, left), I got intrigued: it wasn’t just weak as expected (the temperature fluctuations in the hot and cold spots were really close to zero), but the pattern formed a funny-looking one-dimensional band across the sky rather than being a random mess as theory predicted. I was really sleepy now, but decided to reward myself for all this late-night programming and debugging with one more image, so I changed 2 to 3 in my program and reran it to get a plot of the third multipole, known as the octupole. Whoa! What the …? Up popped another one-dimensional band (Figure 3.6, middle), seemingly aligned with the quadrupole. This was not how our Universe was supposed to be! As opposed to photos of y
ou, photos of our Universe weren’t supposed to have any special direction, such as “up”: they should look similar no matter how you rotate them. Yet these baby-universe images on my computer screen had these bands of zebra-like stripes aligned in only one particular direction. Suspecting a bug in my code, I changed 3 to 4 and reran, but the plot of the fourth multipole (Figure 3.6, right) looked just as expected: a random mess with no special direction.

  After Angélica had double-checked everything, we mentioned this surprising discovery in our map paper. I was amazed by how it caught on. It got mentioned in the New York Times, which sent a photographer to take mug shots of us. We and many other groups looked into it in more detail, one of which dubbed the special direction “the axis of evil.” Some argued that it was a statistical fluke or galaxy contamination, while others argued that it was even more puzzling than we’d claimed, finding additional anomalies even for multipoles 4 and 5 using a different method. Some exotic explanations, such as our living in a small “bagel universe” where space connects back on itself (see this page), were ruled out by further analysis, and to this day, I’m as puzzled by the axis of evil as I was that first night.

  Figure 3.6: When decomposing the WMAP map from Figure 3.5 into a sum of multipoles showing spots of progressively smaller sizes, the first two (left and middle maps) show a mysterious alignment around what’s been dubbed “the axis of evil.” The different colors show how much warmer or colder than average the sky is in different directions; the bar shows the scale in μK, millionths of degrees.

  A Microwave Background Comes of Age