Our Mathematical Universe

by Max Tegmark

  In 2006 Angélica and I were invited to Stockholm to help celebrate that the COBE discovery had been awarded the Nobel Prize in Physics. As is so common in science, there had been acrimony within the COBE team about credit attribution. The prize was shared between George Smoot and John Mather, and I was relieved to see them both take a conciliatory approach. They were able to invite the entire COBE team to come and bask in well-deserved glory, and I felt that the unending stream of elegant parties helped bring closure to the rifts by emphasizing the obvious—they’d all accomplished something much more important than helping two guys get prizes: their first baby pictures of our Universe had created a vibrant new research field and ushered in a whole new era in cosmology. I just wish Gamow, Alpher and Herman could have been there, too.

  On March 21, 2013, I got up at five a.m. full of anticipation and tuned in to a live webcast from Paris, where the Planck satellite team were releasing their first microwave background images. ACBAR, ACT, the South Pole Telescope and other experiments had improved our microwave background knowledge over the past decade, but this was the greatest milestone since WMAP. While I was shaving, George Efstathiou described the results, and I felt a wave of nostalgia and excitement sweep over me. I had a flashback to March 1995, when George had invited me to Oxford to work with him on new methods for analyzing Planck data. It was the first time I’d ever been invited anywhere for research collaboration, and I felt most grateful for the opportunity. We developed a novel technique for cleaning out contaminating symbols, which helped bolster the case that the European Space Agency should fund Planck. Now the results would finally be revealed to the eighteen-years-older Max I saw in the bathroom mirror!

  When George showed the new Planck sky map, I just had to put my razor down so I could place our foreground-cleaned WMAP map next to George’s map on my laptop screen. Wow—they agreed beautifully! I thought to myself. And the axis of evil is still there! I’ve placed the two maps together in Figure 3.5 so that you can compare them. As you can see, all the large-scale patterns match up exquisitely, but the Planck map has much more tiny spots. This is because of its greatly superior sensitivity and resolution, which allows it to image tiny patterns that the WMAP satellite blurred out. The Planck map was definitely worth the wait! I’ve projected it as a sphere for you so that you can enjoy it in high-quality color on the front cover of this book. Because of its superb quality, Planck effectively provides the answer sheet for grading the performance of WMAP, and after carefully digesting the Planck results, it’s clear to me that the WMAP team deserves an A+. As does the Planck team. However, I think the greatest surprise with Planck was that there was no surprise: it basically confirmed the cosmological picture we’d already come to believe, with much better precision. The cosmic microwave background had come of age.

  In summary, we’ve now pushed the frontier of knowledge back from about 14 billion to about 400,000 years after our Big Bang, and seen that everything around us came from a hot plasma that filled all space. Back then, there were no people, planets, stars or galaxies—just atoms bouncing around and radiating light. But we still haven’t explored the mystery of where these atoms came from.

  Where Did the Atoms Come From?

  The Cosmic Fusion Reactor

  We saw that Gamow’s audacious extrapolation backward in time successfully predicted the cosmic microwave background, which has now given us stunning baby pictures of our Universe. As if this smashing success wasn’t enough, he pushed his extrapolation even farther back in time and worked out the consequences. The longer ago it was, the hotter it was. We saw that 400,000 years after our Big Bang, the hydrogen that filled space was thousands of degrees, about half as hot as the surface of the Sun, so it did what hydrogen in the Sun’s surface does: glows, producing the cosmic microwave–background radiation. Gamow also realized that a minute after our Big Bang, the hydrogen temperature was about a billion degrees, even hotter than the core of the Sun, so it must have done what hydrogen in the Sun’s core does: fusion, converting hydrogen into helium. However, the expansion and cooling of our Universe soon switched off this cosmic fusion reactor, by making it too cold to function, so it didn’t have time to turn everything into helium. Encouraged by Gamow, his students Alpher and Herman made a detailed calculation of what would happen with the fusion, although since they were working in the late 1940s, their calculations were limited by the lack of modern computers.

  But how can this prediction be tested, given that our Universe wasn’t transparent during its first 400,000 years, with everything that happened back then hidden from view, censored by the cosmic microwave–background plasma screen? Gamow realized that the situation was the same as with the dinosaur theory: you can’t see what happened directly, but you can find fossil evidence! Repeating their calculations with modern data and computers, you predict that back when our entire Universe was a fusion reactor, it fused about 25% of its mass into helium. When you measure the helium fraction of distant intergalactic gas by studying its spectrum with a telescope, you find … 25%! To me, this finding is just as impressive as discovering a fossilized Tyrannosaurus rex femur: direct evidence that crazy things happened in the past, in this case everything being crazy hot like the center of the Sun. And helium isn’t the only fossil. Big Bang nucleosynthesis, as Gamow’s theory became known, also predicts that about one in every 300,000 atoms out there should be deuterium1 and about one in every five billion atoms should be lithium—both of these fractions have now been measured and agree beautifully with the theoretical prediction.

  * * *

  1Deuterium is hydrogen’s big brother, weighing twice as much because it contains not only a proton but also a neutron.

  Big Bang in Trouble

  However, success didn’t come easy: Gamow’s hot Big Bang got a cool reception. Indeed, the name Big Bang was coined by one of its detractors, Fred Hoyle, in an attempt to ridicule it. According to the 1950 scorecard, the theory had made two major predictions, both wrong: the age of our Universe and the abundance of atoms. Hubble’s initial measurement of the cosmic expansion predicted that our Universe was less than two billion years old, and geologists were underwhelmed by the idea that our Universe was younger than some of their rocks. Moreover, Gamow, Alpher and Herman had hoped that Big Bang nucleosynthesis would produce essentially all the atoms around us in the right proportions, but found that it failed to produce even remotely enough carbon, oxygen and other everyday atoms—making merely helium, deuterium and puny amounts of lithium.

  We now know that Hubble had grossly underestimated the distances to his galaxies. Because of this, he mistakenly concluded that our Universe expands seven times faster than it really does, suggesting that our Universe is seven times younger than it really is. When better distance measurements started correcting this error during the 1950s, the unhappy geologists were vindicated and appeased.

  The second Big Bang “failure” also melted away around the same time. Gamow had done pioneering research on fusion reactions in stars, and the work by him and others suggested that stars produce helium and little else, just as our Sun is doing right now. This is why he hoped that Big Bang nucleosynthesis could explain where the rest of the atoms came from. In the 1950s, however, a seemingly surprising nuclear-physics coincidence was discovered that linked nuclear-energy levels of helium, beryllium, carbon and oxygen, facilitating fusion. As Fred Hoyle was the first to realize, this coincidence enabled stars in the late stages of their lives to turn helium into carbon, oxygen and most of the other atoms that you and I are made of. Moreover, it became clear that stars end their lives by blowing apart, recycling many of the atoms that they’ve made into gas clouds that can later form new stars, planets and ultimately you and me. In other words, we’re more connected to the heavens than our ancestors realized: we’re made of star stuff. Just as we are in our Universe, our Universe is in us. This insight transformed Gamow’s Big Bang nucleosynthesis from failure to smashing success: our Universe made helium and a smidgen of
deuterium and lithium during its first few minutes, and stars made the rest of our atoms later on.1 The mystery of where the atoms came from had been solved. And when it rains, it pours: just as the hot Big Bang theory was finally coming in from the cold, the cosmology world was electrified by the 1964 discovery of Gamow’s other prediction: the Big Bang afterglow, the cosmic microwave–background radiation.

  * * *

  1The stars add further to the 25% of helium made by Big Bang nucleosynthesis. We can tell the two sources of helium apart with our telescopes: the farther back in time we look, the less helium we see, bottoming out at 25% when we look back to times before most stars had formed.

  What Is a Big Bang, Really?

  We’ve now pushed the frontier of our knowledge back about 14 billion years, to a time when our entire Universe was a blazing hot fusion reactor. When I say I believe in the Big Bang Hypothesis, I mean that I’m convinced of this but nothing more.

  Big Bang Hypothesis: Everything we can observe was once hotter than the core of the Sun, expanding so fast that it doubled its size in under a second.

  That’s definitely big enough a bang that I feel we can call it Big with a capital B. However, take note that my definition is quite conservative, saying nothing whatsoever about what happened before that. For example, this hypothesis does not imply that our Universe was one second old at the time, or that it was ever infinitely dense or came from some sort of singularity where our math breaks down. The question Do we have evidence for a Big Bang singularity? from the last chapter has a very simple answer: No! Sure, if we extrapolate Friedmann’s equations as far back in time as they’ll go, they break down in an infinitely dense singularity about a second before Big Bang nucleosynthesis, but the theory of quantum mechanics that we’ll explore in Chapter 7 tells us that this extrapolation breaks down before reaching a singularity. I think it’s crucial to distinguish between what we have solid evidence for and what’s highly speculative, and the truth is that although we have some exciting theories and hints about what happened earlier, which we’ll explore in Chapter 5, we frankly don’t yet know. This is the current frontier of our knowledge. Indeed, we don’t even know for sure that our Universe really had a beginning at all, as opposed to spending an eternity doing something we don’t understand prior to Big Bang nucleosynthesis.

  In summary, we humans have now pushed our knowledge frontier remarkably far back in time, revealing the cosmic storyline I’ve tried to illustrate in Figure 3.7. A million years after our Big Bang, space was filled with nearly uniform transparent gas. If we were to watch the cosmic drama running backward in time, we’d see this gas get gradually hotter, with its atoms smashing into each other progressively harder until they break apart into atomic nuclei and free electrons—a plasma. Then we’d see the helium atoms get smashed apart into protons and neutrons. Then these get smashed into their building blocks: quarks. Then we cross our knowledge frontier and enter the realm of scientific speculation—in Chapter 5, we’ll explore what’s labeled “inflation” and “quantum fuzz” in Figure 3.7. If we jump back to a million years after our Big Bang and instead let time run forward, we see gravity amplify the slight clumping of the gas into galaxies, stars and the rich cosmic structure we observe around us today.

  Figure 3.7: Although we know very little about our ultimate origins, we know a great deal about what happened during the subsequent 14 billion years. As our Universe expanded and cooled, quarks assembled into protons (hydrogen nuclei) and neutrons, which in turn fused into helium nuclei. Then these nuclei formed atoms by capturing electrons, and gravity clumped these atoms into the galaxies, stars and planets that we observe today.

  Click here to see a larger image.

  But gravity can only amplify small fluctuations into larger fluctuations—it can’t create fluctuations out of nowhere. If something is perfectly smooth and uniform, gravity will keep it that way forever, unable to create any dense clumps, let alone galaxies. This means that, early on, there must have been small seed fluctuations for gravity to amplify, acting like a form of cosmic blueprints that determined where galaxies would form. Where did these seed fluctuations come from? In other words, we’ve seen where the atoms in our Universe came from, but what about the grand galactic patterns into which they got arranged? Where did the cosmic large-scale structure come from? Of the many questions we’ve asked in cosmology, I feel that this one has turned out to be the most fruitful of all. In the next two chapters, let’s explore why.

  THE BOTTOM LINE

  • Because distant light takes time to reach us, telescopes let us see cosmic history unfold.

  • About 14 billion years ago, everything that we can now observe was hotter than the core of the Sun, expanding so fast that it doubled its size in under a second. This is what I call our Big Bang.

  • Although we don’t know what happened earlier, we know a great deal about what’s happened since then: expansion and clustering.

  • Our Universe spent a few minutes being a giant fusion reactor, like the core of our Sun, converting hydrogen into helium and other light elements, until the cosmic expansion diluted and cooled our Universe enough to stop the fusion.

  • Doing the math, we predict that about 25% of the hydrogen turned into helium; measurements beautifully agree with this prediction and also match the predictions for other light elements.

  • After another 400,000 years of expansion and dilution, this hydrogen-helium plasma had cooled into transparent gas. We see this transition as a distant plasma wall whose faint glow has become known as the cosmic microwave background, triggering two Nobel Prizes.

  • Over the billions of years that followed, gravity transformed our Universe from uniform and boring to clumpy and interesting, amplifying the tiny density fluctuations that we see in the cosmic microwave background into planets, stars, galaxies and the cosmic large-scale structure that we see around us today.

  • The cosmic expansion predicts that distant galaxies should be receding from us according to a simple formula, in good agreement with what we actually observe.

  • This entire history of our Universe is accurately described by simple physical laws that let us predict the future from the past, and the past from the future.

  • These physical laws that govern the history of our Universe are all cast in terms of mathematical equations, so our most accurate description of our cosmic history is a mathematical description.

  4

  Our Universe by Numbers

  Cosmologists are often wrong, but never in doubt.

  —Lev Landau

  In theory, theory and practice are the same. In practice, they are not.

  —Albert Einstein

  “Wow!” My jaw dropped, and I stood there by the roadside utterly speechless. I’d looked at it every day of my life, yet I’d never really seen it before. It was about five a.m., and I’d pulled off the highway through the Arizona desert to check the map. When all of a sudden, it hit me: the sky! This wasn’t the lame light-polluted Stockholm firmament under which I’d grown up, with the Big Dipper and a sparse smattering of other dim stars. It was spectacular and absolutely overwhelming, with thousands of brilliant points of light forming beautiful and intricate patterns, and the Milky Way glowing like a magnificent Galactic highway across the sky.

  My view was enhanced by the dry desert air and being more than two kilometers above sea level, but I suspect that you, too, have at some point gotten far enough from city lights to be awed by the sky. So what exactly was it that we marveled at? Partly the stars themselves, no doubt, and the vastness of it all. But also something else: the patterns. Our ancestors were so intrigued by them that they created myths to explain them, and some cultures imagined them grouped into constellations depicting mythological figures. The stars clearly aren’t uniformly spread across the sky like polka dots, but clustered. The largest stellar clustering pattern I saw that night was our Milky Way Galaxy, and our telescopes have revealed that galaxies, too, are clustered into
intricate patterns, forming groups, galaxy clusters, and enormous filamentary patterns spanning hundreds of millions of light-years. Where did these patterns come from? What’s the origin of this grand cosmic structure?

  At the end of the last chapter, our exploration of the destabilizing effects of gravity also led us to wonder about the origin of the cosmic large-scale structure. In other words, we’re led intellectually to the same question that we’re led to emotionally when awestruck by the sky: whence the structure? This is the key question that we’re going to explore in this chapter.

  Wanted: Precision Cosmology

  As we saw in the last chapter, we humans still don’t understand the ultimate origins of our Universe, specifically what happened before the epoch when our Universe was a giant fusion reactor and doubled its size in under a second. However, we now understand a great deal about what’s happened during the 14 billion years since then: expansion and clustering. These two basic processes, both controlled by gravity, have transformed hot, smooth quark soup into today’s star-studded cosmos. In the last chapter’s fast-forward history of our Universe, we saw that the gradual expansion diluted and cooled the elementary particles, enabling them to cluster into ever-larger structures such as atomic nuclei, atoms, molecules, stars and galaxies. We know of four fundamental forces of nature, and three of them have taken turns driving this clustering process: first the strong nuclear force pulled the nuclei together, then the electromagnetic force made the atoms and the molecules, and finally gravity built the grand structures that adorn our night sky.