by Brian Greene
*You could still ask whether the entire higher-dimensional spatial expanse can move, but however interesting to contemplate, it’s not relevant to the discussion here.
*For readers familiar with the puzzle of time’s arrow, note that I am assuming, in keeping with observations, that entropy decreases toward the past. See The Fabric of the Cosmos, Chapter 6, for a detailed discussion.
CHAPTER 6
New Thinking About an Old Constant
The Landscape Multiverse
The difference between 0 and . might not seem like much. And by any familiar measure it’s not. Yet there’s growing suspicion that this tiny difference may be responsible for a radical shift in how we envision the landscape of reality.
The tiny number printed above was first measured in 1998 by two teams of astronomers making meticulous observations of exploding stars in distant galaxies. Since then, the work of many has corroborated the teams’ result. What is the number, and why such a fuss? Evidence is mounting that it’s what I referred to earlier as the entry on the third line of the general relativity tax form: Einstein’s cosmological constant, which specifies the amount of invisible dark energy permeating the fabric of space.
As the result continues to hold up under intense scrutiny, physicists are becoming increasingly confident that decades of previous observations and theoretical deductions, which had convinced the vast majority of researchers that the cosmological constant was 0, have been overthrown. Theorists scurried to figure out where they’d gone wrong. But not all had. Years earlier, a contentious line of thought had suggested that a nonzero cosmological constant might one day be found. The key supposition? We’re living in one of many universes. Many universes.
The Return of the Cosmological Constant
Remember that the cosmological constant, if it exists, fills space with a uniform invisible energy—dark energy—whose iconic feature would be its repulsive gravitational force. Einstein latched on to the idea in 1917, invoking the cosmological constant’s antigravity to balance the otherwise attractive gravitational pull of the universe’s ordinary matter, and thus allow for a cosmos that neither expanded nor contracted.*
Many have reported that upon learning of Hubble’s 1929 observations, which established that space is expanding, Einstein called the cosmological constant his “greatest blunder.” George Gamow recounted a conversation in which Einstein is purported to have said this, but given Gamow’s penchant for playful hyperbole, some have questioned the accuracy of the story.1 What’s certain is that Einstein dropped the cosmological constant from his equations when the observations showed that his belief in a static universe was misguided, noting years later that had “Hubble’s expansion been discovered at the time of the creation of the general theory of relativity, the cosmological constant would never have been introduced.”2 But hindsight is not always 20–20; it can sometimes blur earlier clarity. In 1917, in a letter he wrote to the physicist Willem de Sitter, Einstein expressed a more nuanced perspective:
In any case, one thing stands. The general theory of relativity allows the inclusion of the cosmological constant in the field equations. One day, our actual knowledge of the composition of the fixed star sky, the apparent motions of fixed stars, and the position of spectral lines as a function of distance, will probably have come far enough for us to be able to decide empirically the question of whether or not the cosmological constant vanishes. Conviction is a good motive, but a bad judge.3
Some eight decades later, the Supernova Cosmology Project, led by Saul Perlmutter, and the High-Z Supernova Search Team, led by Brian Schmidt, took this very approach. They carefully studied an abundance of spectral lines—light emitted by distant stars—and, just as Einstein had anticipated, they were able to address empirically the question of whether the cosmological constant vanishes.
To the shock of many, they found strong evidence that it doesn’t.
Cosmic Destiny
When these astronomers began their work, neither group was focused on measuring the cosmological constant. Instead, the teams had set their sights on measuring another cosmological feature, the rate at which the expansion of space is slowing. Ordinary attractive gravity acts to pull every object closer to every other, so it causes the expansion speed to decrease. The precise rate of slowdown is central to predicting what the universe will be like in the far future. A big slowdown would mean that the expansion of space would diminish all the way to zero and then reverse its motion, leading to a period of spatial contraction. Unabated, this might result in a big crunch—a reverse of the big bang—or perhaps a bounce, as in the cyclical models introduced in the previous chapter. A small slowdown would yield a very different outcome. Much as a ball with a high speed can escape the earth’s gravity and head ever farther outward, if the speed of spatial expansion were high enough, and the rate of its slowdown sufficiently meager, space could expand forever. By measuring the cosmic slowdown, the two groups sought the ultimate fate of the cosmos.
The approach of each team was straightforward: measure how fast space was expanding at various times in the past, and by comparing those speeds determine the rate at which the expansion has been slowing over the course of cosmic history. Okay. But how would you do this? As with many questions in astronomy, the answer comes down to careful measurements of light. Galaxies are luminous beacons whose motion traces the spatial expansion. If we could determine how fast galaxies at a range of distances were receding from us when, long ago, they emitted the light we now see, we could determine how fast space was expanding at a variety of moments in the past. By comparing those speeds, we’d learn the rate of cosmic slowdown. That’s the essential idea.
To fill in the details, we need to address two primary questions. From today’s observations of faraway galaxies, how can we determine their distances, and how can we determine their speeds? Begin with distance.
Distance and Brightness
One of the oldest and most important problems in astronomy is to determine the distances to celestial objects. And one of the first techniques for doing so, parallax, is an approach with which five-year-olds routinely experiment. Children can be fascinated (momentarily) by looking at an object while alternately closing their left and right eyes because the object appears to jump from side to side. If you haven’t been five for some time, try the experiment by holding up this book and looking at one of its corners. The jump occurs because your left and right eyes, being spaced apart, have to point at different angles to focus on the same spot. For objects that are farther away, the jumping is less noticeable, because the difference in angle gets smaller. This simple observation can be made quantitative, providing a precise correlation between the difference in angle between the lines of sight of your two eyes—the parallax—and the distance of the object you’re viewing. But don’t worry about working out the details; your visual system does it automatically. It’s why you see the world in 3D.*
When you look at stars in the night sky, the parallax is too small to be reliably measured; your eyes are just too close together to yield a significant difference in angle. But there’s a clever way around this: measure the position of a star on two occasions, some six months apart, thus using the two locations of the earth in place of the two locations of your eyes. The larger separation of the observing locations increases the parallax; it’s still small, but in some cases is big enough to be measured. Back in the early 1800s there was an intense competition among a group of scientists to be the first to measure such stellar parallax; in 1838, the German astronomer and mathematician Friedrich Bessel won the bragging rights, successfully measuring the parallax to a star called 61 Cygni, in the constellation Cygnus. The angular difference turned out to be .000084 degrees, placing the star about 10 light-years away.
Since then, the technique has been steadily refined and is now undertaken by satellites that can measure parallax angles far smaller than what Bessel achieved. Such advances have allowed for accurate distance measurements of stars that a
re up to a few thousand light-years away, but much beyond that the angular differences again become too small, and the method is thwarted.
Another approach, which has the capacity to measure yet greater celestial distances, is based on an even simpler idea: the farther away you move a light-emitting object, be it a car’s headlights or a blazing star, the more the emitted light will spread out during its journey toward you, and so the dimmer it will appear. By comparing an object’s apparent brightness (how bright it appears when observed from earth) with its intrinsic brightness (how bright it would appear if observed from close by), you can thus work out its distance.
The hitch, and it’s not a small one, lies in establishing the intrinsic brightness of astrophysical objects. Is a star dim because it’s especially distant or because it just doesn’t give off much light? This makes clear why a long-standing effort has been to find a relatively common astronomical species whose intrinsic brightness can be reliably determined without the need to stand right next to it. If you could find such so-called standard candles, you’d have a uniform benchmark for judging distances. The degree to which one standard candle appeared dimmer than another would tell you directly how much farther away it is.
For over a century, a variety of standard candles have been proposed and used, with varying success. In recent times, the most fruitful method has made use of a kind of stellar explosion called a Type Ia supernova. A Type Ia supernova occurs when a white dwarf star pulls material from the surface of a companion, typically a nearby red giant that it’s orbiting. Well-developed physics of stellar structure establishes that if the white dwarf pulls away enough material (so that its total mass increases to about 1.4 times that of the sun), it can no longer support its own weight. The bloated dwarf star collapses, setting off an explosion so violent that the light generated rivals the combined output of the other 100 billion or so stars residing in the galaxy it inhabits.
These supernovae are ideal standard candles. Because the explosions are so powerful, we can see them out to fantastically large distances. And, crucially, because the explosions are all the result of the same physical process—a white dwarf’s mass increasing to about 1.4 times that of the sun’s, resulting in stellar collapse—the ensuing supernovae flare to a very similar peak intrinsic brightness. The challenge in using Type Ia supernovae, however, is that in a typical galaxy they take place only once every few hundred years: How do you catch them in the act? Both the Supernova Cosmology Project and the High-Z Supernova Search Team tackled this obstacle in a manner reminiscent of epidemiological studies: accurate information about even relatively rare conditions can be gained if you study large populations. Similarly, by using telescopes equipped with wide-field-of-view detectors capable of simultaneously examining thousands of galaxies, the researchers were able to locate dozens of Type Ia supernovae, which could then be closely observed with more conventional telescopes. On the basis of how bright each appeared, the teams were able to calculate the distance to dozens of galaxies situated billions of light-years away—thus accomplishing the first step in the task they’d set for themselves.
Whose Distance Is It, Anyway?
Before moving on to the next step, the determination of how fast the universe was expanding when each of these distant supernovae happened, let me briefly untangle a potential knot of confusion. When we’re talking about distances on such fantastically large scales, and in the context of a universe that’s continually expanding, the question inevitably arises of which distance the astronomers are actually measuring. Is it the distance between the locations we and a given galaxy each occupied eons ago, when the galaxy emitted the light we’re just now seeing? Is it the distance between our current location and the location the galaxy occupied eons ago, when it emitted the light we’re just now seeing? Or is it the distance between our current location and the galaxy’s current location?
Here’s what I consider the most insightful way of thinking about these and a whole slew of similarly confusing cosmological questions.
Imagine you want to know the distances, as the crow flies, among three cities, New York, Los Angeles, and Austin, so you measure their separation on a map of the United States. You find that New York is 39 centimeters from Los Angeles; Los Angeles is 19 centimeters from Austin; and Austin is 24 centimeters from New York. You then convert these measurements into real-world distances by looking at the map’s legend, which provides a conversion factor—1 centimeter = 100 kilometers—which allows you to conclude that the three cities are about 3,900 kilometers, 1,900 kilometers, and 2,400 kilometers apart, respectively.
Now imagine that the earth’s surface swells uniformly, doubling all separations. This would certainly be a radical transformation, but even so your map of the United States would continue to be perfectly valid as long as you made one important change. You’d need to modify the legend so that the conversion factor read “1 centimeter = 200 kilometers.” Thirty-nine centimeters, 19 centimeters, and 24 centimeters on the map would now correspond to 7,800 kilometers, 3,800 kilometers, and 4,800 kilometers across the expanded United States. Were the expansion of the earth to continue, your static, unchanging map would remain accurate, as long as you continually updated its legend with the conversion factor relevant at each moment—1 centimeter = 200 kilometers at noon; 1 centimeter = 300 kilometers at two p.m.; 1 centimeter = 400 kilometers at four p.m.—to reflect how locations were being dragged apart by the expanding surface.
The expanding earth proves a useful conceit because similar considerations apply to the expanding cosmos. Galaxies don’t move under their own power. Rather, like the cities on our expanding earth, they race apart because the substrate in which they’re embedded—space itself—is swelling. This means that had some cosmic cartographer mapped galaxy locations billions of years ago, the map would be as valid today as it was then.4 But, like the legend for the map of an expanding earth, the cosmic map’s legend must be updated to ensure that the conversion factor, from map distances to real distances, remains accurate. The cosmological conversion factor is called the universe’s scale factor; in an expanding universe, the scale factor increases with time.
Whenever you think about the expanding universe, I urge you to picture an unchanging cosmic map. Think of it as if it were any ordinary map lying flat on a table, and account for the cosmic expansion by updating the map’s legend over time. With a little practice, you’ll see that this approach vastly simplifies conceptual hurdles.
As a case in point, consider light from a supernova explosion in the distant Noa Galaxy. When we compare the supernova’s apparent brightness with its intrinsic brightness, we are measuring the dilution of the light’s intensity between emission (Figure 6.1a) and reception (Figure 6.1c), arising from its having spread out on a large sphere (drawn as a circle in Figure 6.1d) during the journey. By measuring the dilution, we determine the size of the sphere—its surface area—and then, with a little high school geometry, we can determine the sphere’s radius. This radius traces the light’s entire trajectory, and so its length equals the distance the light has traveled. Now the question that initiated this section pops up: To which of the three candidate distances, if any, does the measurement correspond?
During the light’s journey, space has continually expanded. But the only change this requires to the static cosmic map is a regular updating of the scale factor recorded in the legend. And since we have just now received the supernova’s light, since it has just now completed its journey, we must use the scale factor that’s just now written in the map’s legend to translate the separation on the map—the trajectory from the supernova to us, traced in Figure 6.1d—into the physical distance traveled. The procedure makes clear that the result is the distance now between us and the current location of the Noa Galaxy: the third of our multiple-choice options.
Figure 6.1 (a) Light from a distant supernova spreads as it travels toward us (we are situated in the galaxy on the map’s right-hand side). (b) During the light’s journey,
the universe expands, which is reflected in the map’s legend. (c) When we receive the light, its intensity has been diluted through the spreading. (d) When we compare the supernova’s apparent brightness to its intrinsic brightness, we are measuring the area of the sphere on which it has spread (drawn as a circle), and hence also its radius. The radius of the sphere traces the light’s trajectory. Its length is the distance now between us and the galaxy that contained the supernova, so that’s what the observations determine.
Notice, too, that because the universe is continually expanding, earlier segments of a photon’s journey continue to stretch long after the photon has sped past. If a photo painted a line on space that traced its path, the length of that line would increase as space expanded. By applying the map’s scale factor at the time of reception to the light’s entire journey, the third answer directly incorporates all such expansion. This is the right approach, because the amount by which the light’s intensity is diluted depends on the size of the sphere over which the light now spreads—and this sphere’s radius is the length of the light’s trajectory now, including all post facto stretching.5
When we compare the intrinsic brightness of a supernova with its apparent brightness, we are therefore determining the distance now between us and the galaxy it occupied. Those are the distances the two groups of astronomers measured.6
The Colors of Cosmology