Our Mathematical Universe

by Max Tegmark

  To me, Friedmann is one of the great unsung heroes of cosmology. While writing this, I couldn’t resist reading his original 1922 paper, and noticed that it ends by giving an intriguing example of a vast universe containing five billion trillion suns’ worth of mass, from which he calculates a lifetime of about ten billion years—in the same ballpark as the accepted modern value for the age of our Universe. He doesn’t explain where he got this from, years before galaxies were discovered, but it was certainly a fitting ending to a remarkable paper by a remarkable person.

  Our Universe Is Expanding

  Five years later, history repeated itself: an MIT graduate student, the Belgian priest and astrophysicist Georges Lemaître, again published Friedmann’s Big Bang solution, which he had been unaware of and had rediscovered. And once again, it was largely ignored by the scientific community.

  What finally made people take note of the Big Bang idea wasn’t new theoretical work, but new measurements. Now that Edwin Hubble had established that galaxies existed, an obvious next step for him was to start mapping out how they were distributed in space and how they moved. As I mentioned in the previous chapter, it’s often easy to measure how fast something is moving toward or away from us, since this motion shifts the lines of its spectrum. Red light has the lowest frequency of all the colors in the rainbow, so if a galaxy is moving away from us, the colors of all its spectral lines will be redshifted, shifted toward redder colors, and the higher its speed, the greater its redshift. If the galaxy is moving toward us, its colors will instead be blueshifted toward higher frequencies.

  If galaxies were just moving around at random, we’d expect about half of them to be redshifted and the rest blueshifted. Surprisingly, almost all the galaxies that Hubble studied were redshifted. Why were they all receding from us? Didn’t they like us? Did we say something wrong? Moreover, Hubble discovered that the greater the distance d to the galaxy, the higher the velocity v with which it receded from us, according to the formula:

  v = Hd

  which we now know as Hubble’s law. Here H is the so-called Hubble parameter, which Hubble modestly called K in his seminal 1929 paper on the subject, so as not to appear too conceited. Interestingly, Georges Lemaître had shown in his ignored 1927 paper that the expanding universe solution predicted Hubble’s law: if everything was expanding away from everything else, then we’d see the distant galaxies expand away from us like this.

  If a galaxy is moving straight away from us, this suggests that it was very close to us in the past. How long ago? If you see a car speeding away after a bank robbery, you can estimate how long ago it left the bank by dividing its distance by its speed. If we do this for the receding galaxies, Hubble’s law gives the same answer d/v = 1/H for all of them! This answer is 1/H ≈ 14 billion years, using modern measurements, so Hubble’s discovery suggests that something rather dramatic happened about 14 billion years ago, involving lots of matter squeezed together here at high density. To get a more exact answer, we need to factor in the extent to which the car/universe has been accelerating/decelerating/cruising at constant speed since leaving the crime scene. When we do this today, using Friedmann’s equations and modern measurements, we find that the required correction is quite small, at the percent level: after its Big Bang, our Universe spent about the first half of its time decelerating, then the rest of the time accelerating, so the corrections roughly cancel out.

  Making Sense of an Expanding Universe

  After Hubble’s measurements were announced, even Einstein was convinced, and now our Universe was expanding even officially. But what does it mean that our Universe is expanding? We’re now ready to tackle four more of the questions from the beginning of Chapter 2.

  First of all, are galaxies really moving away from us, or is space just expanding? Conveniently, Einstein’s theory of gravity (general relativity) says that these are two equivalent viewpoints that are equally valid, as illustrated in Figure 3.2, so you’re free to think about it in whichever way you find more intuitive.1 From the first viewpoint (left), space isn’t changing but the galaxies are moving through space like the chocolate chips in a muffin that’s rising because of the baking powder you put in the batter. All galaxies/chocolate chips move farther apart from all others, and more widely separated pairs get separated faster. In particular, if you’re standing on a specific chocolate chip/galaxy, you’ll see that the motion of all the others relative to you obeys Hubble’s law: they’re all receding straight away from you, and one twice as far recedes at twice the speed. Remarkably, things will look the same whichever chocolate chip or galaxy you’re observing from, so if the distribution of galaxies has no end, then the expansion has no center—it looks the same from everywhere.

  From the second viewpoint, space is like the muffin dough: it expands, so just as the chocolate chips aren’t moving relative to the dough, the galaxies aren’t moving through space. Instead, we can think of the galaxies as being at rest in space (Figure 3.2, right) while all the distances between them get redefined. It’s as if the tick marks on imaginary rulers connecting the galaxies get relabeled so that their spacing corresponds not to a millimeter but to a centimeter—now all intergalactic distances are ten times larger than they used to be.

  This answers another one of our questions: Don’t galaxies receding faster than the speed of light violate relativity theory? Hubble’s law v = Hd implies that galaxies will move away from us faster than the speed of light c if their distance from us is greater than c/H ≈ 14 billion light-years, and we have no reason to doubt that such galaxies exist, so doesn’t this violate Einstein’s claim that nothing can go faster than light? The answer is yes and no: it violates Einstein’s special relativity theory from 1905 but not his general relativity theory from 1915, and the latter is Einstein’s final word on the subject, so we’re okay. General relativity liberalizes the speed limit: whereas special relativity says that no two objects can move faster than light relative to one another under any circumstances, general relativity merely insists that they can’t move faster than light relative to one another when they’re in the same place—in contrast, the galaxies speeding away from us superluminally are all very far from us. If we think of space as expanding, then we can rephrase this by saying that nothing is allowed to move faster than light through space, but space itself is free to stretch however fast it wants to.

  Speaking of distant galaxies, I’ve seen newspaper articles talking about ones as far as about 30 billion light-years away from us. If our Universe is only 14 billion years old, how can we see objects that are 30 billion light-years away? How did their light have time to reach us? Moreover, we just figured out that they’re receding from us faster than the speed of light, which makes the notion that we can see them sound even weirder. Here the answer is that we’re not seeing these distant galaxies where they are now, but where they were when they emitted the light that reaches us now. Just as we see the Sun the way it looked eight minutes ago at the position where it was eight minutes ago, we might see a distant galaxy the way it looked 13 billion years ago, at the position where it was back then—which was about eight times closer to Earth than it is now! So the light from this galaxy never needed to travel more than 13 billion light-years through space to reach us, because the stretching of space made up for the difference—it’s as if you walk up an escalator and move twenty meters while taking only ten one-meter steps.

  * * *

  1Mathematically, the different viewpoints correspond to different choices of space coordinates, and Einstein’s theory allows you to pick whichever coordinate system you want for space and time.

  What’s Our Universe Expanding Into?

  Won’t there be a cosmic traffic accident somewhere far away where galaxies expanding away from us crash into whatever they’re expanding into? If our Universe expands according to Friedmann’s equations, there are no such problems: as Figure 3.2 illustrated, the expansion looks the same from everywhere in space, so there can’t be any such troub
le spots. If we take the viewpoint that distant galaxies really are receding through a static space, then the reason they never collide with more distant galaxies is that those are receding even faster: you can’t rear-end a speeding Porsche if you’re driving a Model T Ford. If you instead take the viewpoint that space is expanding, the explanation is simply that volume isn’t conserved. From hearing about the Middle East on the news, we’re used to the idea that you can’t get more space without taking it away from someone else. However, general relativity says the exact opposite: more volume can be created in a particular region between some galaxies without this new volume expanding into other regions—the new volume simply stays between those same galaxies (Figure 3.2, right).

  The Cosmic Classroom

  In other words, as crazy and counterintuitive as it sounds, the expanding universe is both logical and supported by astronomical observations. In fact, the observational evidence has grown dramatically stronger since the days of Edwin Hubble, thanks to modern technology and new discoveries that we’ll explore below. The most basic conclusion is simply that even our Universe itself is changing: when we push our knowledge frontier back many billions of years, we discover a universe that hadn’t expanded as much, and was therefore denser and more crowded. This means that the space we inhabit isn’t the boring static space once axiomatized by Euclid, but a dynamic evolving space that once had some sort of childhood—and perhaps some sort of birth about 14 billion years ago.

  Dramatically better telescopes have now improved our vision to the point that we can see our evolving cosmos quite directly. Imagine that you’re giving a presentation in a large auditorium. Suddenly you notice something funny about the audience. The rows of chairs closest to you are all occupied by people around your own age. But about ten rows back, you see only teenagers. Behind them are a bunch of younger kids, and behind them a row of toddlers. Behind them, near the very back of the room, you see only babies. The very last row is completely empty, as far as you can see. When we gaze out into our Universe with our best telescopes, we see something similar: nearby are lots of large and mature galaxies like our own, but very far away, we see mostly small baby galaxies that don’t yet look fully developed. Beyond them we see no galaxies at all, merely darkness. Since it takes light longer to reach us from farther away, gazing into the distance is equivalent to observing the past. The darkness behind the galaxies is the epoch before the first galaxies had time to form. Back then, space was filled with hydrogen and helium gas that gravity hadn’t yet had time to clump into galaxies, and since this gas is transparent like helium in balloons at birthday parties, it’s invisible to our telescopes.

  But there’s a mystery: during your presentation, you suddenly realize that there’s energy coming from beyond that empty last row: the rear wall of the auditorium isn’t completely dark, but gives off a faint glow of microwaves! Why? Bizarre as it sounds, this is what we see when we peer into the most distant depths of our Universe. To understand this, we need to continue our quest to push our knowledge frontier even farther back in time.

  Where Did the Mysterious Microwaves Come From?

  To me, a key lesson from both Newton and Friedmann is this simple mantra: “Dare to extrapolate!” Specifically, take your current understanding of the laws of physics, apply them in a new uncharted situation, and ask whether they predict something interesting that we can observe. Newton took the laws of motion that Galileo had established on Earth and extrapolated them to the Moon and beyond. Friedmann took the laws of motion and gravity that Einstein had established in our Solar System and extrapolated them to our entire Universe. Given how successful this mantra was, you might think that it would catch on as a meme in the scientific community. In particular, you might think that after 1929 when Friedmann’s expanding-universe idea gained acceptance, scientists around the world would race against each other to systematically explore what happened if you extrapolated it backward in time. Well, if you’d have thought this, you’d have been wrong.… No matter how emphatically we scientists claim to be rational seekers of truth, we’re as prone as anyone to human foibles such as prejudice, peer pressure and herd mentality. Overcoming these shortcomings clearly takes more than just talent for calculating.

  To me, the next cosmological superhero who had what it took was another Russian: George Gamow. His Ph.D. advisor in Leningrad was none less than Alexander Friedmann, and although Friedmann died two years into Gamow’s studies, both his ideas and his intellectual boldness lived on in Gamow.

  The Cosmic Plasma Screen

  Given that our Universe is currently expanding, it must have been denser and more crowded in the past. But has it always been expanding? Perhaps not: Friedmann’s work allows for the possibility that our Universe was once contracting, and that all the material moving toward us gently slowed down, stopped and started accelerating away from us. Such a cosmic bounce could only have happened if the density of matter were much lower than we now know it to be. Gamow decided to systematically explore the other option, which was more generic and more radical: expansion ever since the beginning. As he explained in a 1946 book, this proposition implies that if we imagine the cosmic drama to be a movie and rewind it, playing it backward, we’ll see the density of our Universe increase without limit. Since intergalactic space is filled with hydrogen, this gas will get more and more compressed and therefore hotter and hotter the farther back in time we look. If you keep heating an ice cube, it melts. If you keep heating liquid water, it transforms into gas: steam. Similarly, if you keep heating hydrogen gas, it turns into a fourth phase: plasma. Why? Well, a hydrogen atom is simply an electron orbiting around a proton, and hydrogen gas is just a bunch of such atoms bouncing against each other. If the temperature rises, the atoms move faster and bump each other harder. If it gets hot enough, the bumps get so violent that the atoms break apart and the electrons and protons go their separate ways—a hydrogen plasma is simply such a soup of free electrons and protons.

  In other words, Gamow predicted that our Universe began with a hot Big Bang, and that plasma once filled all of space. What’s exceptionally interesting about this is that the prediction is testable: whereas cold hydrogen gas is transparent and invisible, hot hydrogen plasma is opaque and glows brightly, like the surface of the Sun. This means that when we gaze ever farther into space as in Figure 3.3, we should encounter old galaxies nearby, then young galaxies beyond them, then transparent hydrogen gas, then a wall of glowing hydrogen plasma. We can’t see beyond this wall, because it’s opaque and therefore obstructs what came before it like a cosmic censor. Moreover, as illustrated in Figure 3.4, this is what we should see in all directions, since wherever we look, we’re also looking back in time. It therefore looks to us like we’re surrounded by a gigantic plasma sphere.

  In his 1946 book, Gamow’s Big Bang theory predicted that we should be able to observe this plasma sphere. He got his students Ralph Alpher and Robert Herman to work things out in more detail, and a few years later, they published a paper predicting that it would glow with a temperature of about five degrees above absolute zero, meaning that it would mainly give off microwaves rather than visible light. They unfortunately failed to convince any astronomers to search for this cosmic microwave–background radiation in the sky, and their work was largely forgotten just as Friedmann’s expanding-universe discovery was.

  Figure 3.3: Since it takes time for distant light to reach us, looking farther away means looking farther back in time. Beyond the most distant galaxies, we see an opaque wall of glowing hydrogen plasma, whose glow has taken about 14 billion years to reach us. This is because the same hydrogen that fills space today was hot enough to be plasma about 14 billion years ago, when our Universe was only about 400,000 years old. (Credit: Adapted from NASA/WMAP team)

  Seeing the Afterglow

  By 1964, a group at Princeton University had realized that this observable microwave signal should exist and planned an observational search for it, but they were beaten to the punch
. The same year, Arno Penzias and Robert Wilson were testing a new state-of-the-art microwave telescope at Bell Labs in New Jersey and discovered something puzzling: their telescope detected a signal they couldn’t explain, and this signal remained the same regardless of where they pointed the telescope. Weird! They were expecting to detect signals only when they pointed at particular objects in the sky, such as the Sun or a satellite transmitting microwaves. Instead, it was as if the whole sky was glowing, with a temperature of three degrees above absolute zero—close to the five degrees that Gamow’s group had predicted. They carefully checked for local sources of noise, and for a while, suspicion fell on pigeons that were nesting in the telescope and leaving droppings there. I got to have lunch with Arno a while back, and he told me that they put the pigeons in a wooden box with food and sent it to another Bell Labs campus far away with instructions to release the birds. Unfortunately, they were homing pigeons.… Although his book merely states that they “eliminated” the pigeons when they returned, I got him to reveal the grim truth after some wine: it involved a shotgun.… Although the pigeons were gone, the mysterious signal remained: they had discovered the cosmic microwave background, the faint afterglow of our Big Bang.