The Hidden Reality: Parallel Universes and the Deep Laws of the Cosmos - Brian Greene (2011)
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.
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 are 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 nowreceived 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
So much for measuring distances to faraway galaxies containing brilliant Type Ia supernovae. How do we learn about the rate of the universe’s expansion ages ago, when each of those cosmic beacons momentarily ignited? The physics involved isn’t much more complex than that at work in neon signs.
A neon sign glows red because when a current runs through the sign’s gaseous interior, orbiting electrons in the neon atoms are momentarily knocked into higher-energy states. Then, as the neon atoms calm, the excited electrons jump down to their normal state of motion, relinquishing the extra energy by emitting photons. The color of the photons—their wavelength—is determined by the energy they carry. A key discovery, fully established by quantum mechanics in the early decades of the twentieth century, is that atoms of a given element have a unique collection of possible electron energy jumps; this translates into a unique collection of colors for released photons. For neon atoms, a dominant color is red (or, really, reddish orange), which accounts for the appearance of neon signs. Other elements—helium, oxygen, chlorine, and so on—exhibit similar behavior, the main difference being the wavelengths of the photons emitted. A “neon” sign of a color other than red is more than likely filled with mercury (if it’s blue) or helium (if it’s gold), or is made from glass tubes coated with substances, typically phosphors, whose atoms can emit light of yet other wavelengths.
Much of observational astronomy relies on the very same considerations. Astronomers use telescopes to gather light from distant objects, and from the colors they find—the particular wavelengths of light they measure—they can identify the chemical composition of the sources. An early demonstration occurred during the solar eclipse of 1868, when the French astronomer Pierre Janssen and, independently, the English astronomer Joseph Norman Lockyer examined light from the outermost shell of the sun, peeking just beyond the moon’s rim, and found a mysterious bright emission with a wavelength that no one could reproduce in the laboratory using known substances. This led to the bold—and correct—suggestion that the light was emitted by a new, hitherto unknown element. The unknown substance was helium, which thus claims the singular distinction of being the only element discovered in the sun before it was found on earth. Such work established convincingly that, much as you can be uniquely identified by the pattern of lines making up your fingerprint, so an atomic species is uniquely identified by the pattern of wavelengths of the light it emits (and also absorbs).
In the decades that followed, astronomers who examined the wavelengths of light gathered from more and more distant astrophysical sources became aware of a peculiar feature. Although the collection of wavelengths resembled those familiar from laboratory experiments with well-known atoms such as hydrogen and helium, they were all somewhat longer. From one distant source, the wavelengths might be 3 percent longer; from another source, 12 percent longer; from a third 21 percent longer. Astronomers named this effect redshift, in recognition that ever longer wavelengths of light, at least in the visible part of the spectrum, become ever redder.
Naming is a good start, but what causes the wavelengths to stretch? The well-known answer, which emerged most clearly from the observations of Vesto Slipher and Edwin Hubble, is that the universe is expanding. The static map framework introduced earlier is tailor-made for providing an intuitive explanation.
Picture a light wave undulating its way from the Noa Galaxy toward earth. As we plot the light’s progress across our unchanging map, we see a uniform succession of wave crests, one following another, as the undisturbed wave train heads toward our telescope. The uniformity of the waves might lead you to think that the wavelength of the light when emitted (the distance between successive wave crests) will be the same as when it’s received. But the delightfully interesting part of the story comes into focus when we use the map’s legend to convert map distances into real distances. Because the universe is expanding, the map’s conversion factor is larger when the light concludes its journey than it was at inception. The implication is that although the light’s wavelength as measured on the map is unchanging, when converted to real distances, the wavelength grows. When we finally receive the light, its wavelength is longer than when it was emitted. It’s as if light waves are threads stitched through a piece of spandex. Just as stretching the spandex stretches the stitching, so expanding the spatial fabric stretches the light waves.
We can be quantitative. If the wavelength appears stretched by 3 percent, then the universe is 3 percent larger now than it was when the light was emitted; if the light appears 21 percent longer, then the universe has stretched 21 percent since the light began its journey. Redshift measurements thus tell us about the size of the universe when the light we’re now examining was emitted, as compared with the size of the universe today.* It’s a straightforward final step to parlay a series of such redshift measurements into a determination of the universe’s expansion profile over time.
A pencil mark drawn long ago on your child’s wall records how tall she was at the date specified. A series of pencil marks gives her height at a series of dates. Given enough marks, you can determine how quickly she was growing at various times in the past. A growth spurt at nine, a slower period until eleven, another rapid spurt at thirteen, and so on. When astronomers measure a Type Ia supernova’s redshift, they’re determining an analogous “pencil mark” for space. Much like your child’s height marks, a series of such redshift measurements of various Type Ia supernovae would enable them to calculate how quickly the universe was growing over various intervals in the past. With those data, in turn, the astronomers could determine the rate at which the expansion of space has been slowing. That was the plan of attack laid out by the research teams.
To execute it, they would have to complete one remaining step: dating the universe’s pencil marks. The teams needed to determine when the light from a given supernova was emitted. This is a straightforward task. Since the difference between a supernova’s apparent and intrinsic brightness reveals its distance, and since we know light’s speed, we should be able to immediately calculate how long ago the supernova’s light was emitted. The reasoning is right, but there is one essential subtlety, to do with the “post-facto” stretching of light’s trajectory mentioned above, that’s worth emphasizing.
When light travels in an expanding universe, it covers a given distance partly because of its intrinsic speed through space, but partly also because of the stretching of space itself. You can compare this with what happens on an airport’s moving walkway. Without increasing your intrinsic speed, you travel farther than you otherwise would because the moving walkway augments your motion. Similarly, without increasing its intrinsic speed, light from a distant supernova travels farther than it otherwise would because during its journey the stretching space augments its motion. To judge correctly when the light we now see was emitted, we must take account of both contributions to the distance it covers. The math gets a little involved (see the notes if you are curious), but it is by now thoroughly understood.7
Being careful about this point, as well as numerous other theoretical and observational details, both groups were able to work out the size of the universe’s scale factor at various identifiable times in the past. They were able, that is, to find a series of dated pencil marks delineating the universe’s size, and therefore to determine how the expansion rate has been changing over the history of the cosmos.
After checking, and rechecking, and checking again, both teams released their conclusions. For the last 7 billion years, contrary to long-held expectations, the expansion of space has not been slowing down. It’s been speeding up.
A summary of this pioneering work, together with subsequent observations that cinched the case even more tightly, is given in Figure 6.2. The observations revealed that until about 7 billion years ago, the scale factor did indeed behave as expected: its growth gradually slowed down. Had this continued, the graph would have leveled off or even turned downward. But the data show that at about the 7-billion-year mark, something dramatic happened. The graph turned upward, which means that the growth rate of the scale factor began to increase. The universe kicked into high gear as the expansion of space started to accelerate.
Figure 6.2 The scale factor of the universe over time, showing that cosmic expansion slowed down until about 7 billion years ago, when it began to speed up.
Our cosmic destiny turns on the shape of this graph. With accelerated expansion, space will continue to spread indefinitely, dragging away distant galaxies ever farther and ever faster. A hundred billion years from now, any galaxies not now resident in our neighborhood (a gravitationally bound cluster of about a dozen galaxies called our “local group”) will exit our cosmic horizon and enter a realm permanently beyond our capacity to see. Unless future astronomers have records handed down to them from an earlier era, their cosmological theories will seek explanations for an island universe, with galaxies numbering no more than students in a backwoods school, floating in a static sea of darkness. We live in a privileged age. Insights the universe giveth, accelerated expansion will taketh away.
As we will see in the pages that follow, the limited view on offer for future astronomers is all the more striking when compared with the enormity of the cosmic expanse to which our generation has been led in attempting to explain the accelerated expansion.
The Cosmological Constant
If you saw a ball’s speed increase after someone threw it upward, you’d conclude that something was pushing it away from the earth’s surface. The supernova researchers similarly concluded that the unexpected speeding up of the cosmic exodus required something to push outward, something to overwhelm the inward pull of attractive gravity. As we’re now amply familiar, this is the very job description which makes the cosmological constant, and the repulsive gravity to which it gives rise, the ideal candidate. The supernova observations thus ushered the cosmological constant back into the limelight, not through the “bad judgment of conviction” to which Einstein had alluded in his letter decades earlier, but through the raw power of data.
The data also allowed the researchers to fix the numerical value of the cosmological constant—the amount of dark energy suffusing space. Expressing the result in terms of an equivalent amount of mass, as is conventional among physicists (using E = mc2 in the less familiar form, m = E/c2), the researchers showed that the supernova data required a cosmological constant of just under 10–29 grams in every cubic centimeter.8 The outward push of such a small cosmological constant would have been trumped for the first 7 billion years by the inward pull of ordinary matter and energy, in keeping with the observational data. But the expansion of space would have diluted ordinary matter and energy, ultimately allowing the cosmological constant to gain the upper hand. Remember, the cosmological constant does not dilute; the repulsive gravity supplied by a cosmological constant is an intrinsic feature of space—every cubic meter of space contributes the same outward push, dictated by the cosmological constant’s value. And so the more space there is between any two objects, arising from cosmic expansion, the stronger the force driving them apart. By about the 7-billion-year mark, the cosmological contant’s repulsive gravity would have carried the day; the universe’s expansion has been speeding up ever since, just as the data in Figure 6.2 attest.
To conform more fully to convention, I should re-express the cosmological constant’s value in the units physicists more typically use. Much as it would be strange to ask a grocer for 1015 picograms of potatoes (instead, you’d ask for 1 kilogram, an equivalent measure in more sensible units), or tell a waiting friend that you’ll be with her in 109 nanoseconds (instead, you’d say 1 second, an equivalent measure in more sensible units); it is similarly odd for a physicist to quote the energy of the cosmological constant in grams per cubic centimeter. Instead, for reasons that will shortly become apparent, the natural choice is to express the cosmological constant’s value as a multiple of the so-called Planck mass (about 10–5 grams) per cubic Planck length (a cube that measures about 10–33 centimeters on each side and so has a volume of 10–99 cubic centimeters). In these units, the cosmological constant’s measured value is about 10–123, the tiny number that opened this chapter.9
How sure are we of this result? The data establishing accelerated expansion have only become more conclusive in the years since the first measurements were made. Moreover, complementary measurements (focusing on, for example, detailed features of the microwave background radiation; see Fabric of the Cosmos, Chapter 14) dovetail spectacularly well with the supernova results. If there’s room for maneuvering, it lies in what we accept as an explanation for the accelerated expansion. Taking general relativity as the mathematical description of gravity, the only option is indeed the antigravity of a cosmological constant. Other explanations emerge if we modify this picture by including additional exotic quantum fields (which, much as we found in inflationary cosmology, can for periods of time masquerade as a cosmological constant),10 or alter the equations of general relativity (so that attractive gravity drops off in strength with separation more precipitously than it does according to Newton’s or Einstein’s mathematics, thus allowing distant regions to rush away more quickly, without requiring a cosmological constant). But to date, the simplest and most convincing explanation for the observations of accelerated expansion is that the cosmological constant doesn’t vanish, and so space is suffused with dark energy.
To many researchers, the discovery of a nonzero cosmological constant is the single most surprising observational result to have emerged in their lifetimes.
When I first caught wind of the supernova results suggesting a nonzero cosmological constant, my reaction was typical of many physicists. “It just can’t be.” Most (but not all) theoreticians had concluded decades before that the value of the cosmological constant was zero. This view initially arose from the “Einstein’s greatest blunder” lore, but, over time, a variety of compelling arguments emerged to support it. The most potent came from considerations of quantum uncertainty.
Because of quantum uncertainty and the attendant jitters experienced by all quantum fields, even empty space is home to frenetic microscopic activity. And much like atoms bouncing around a box or kids jumping around a playground, quantum jitters harbor energy. But unlike atoms or kids, quantum jitters are ubiquitous and inevitable. You can’t declare a region of space closed and send the quantum jitters home; the energy supplied by quantum jitters permeates space and can’t be removed. Since the cosmological constant is nothing but energy that permeates space, quantum field jitters provide a microscopic mechanism that generates a cosmological constant. That’s a pivotal insight. You’ll recall that when Einstein introduced the notion of a cosmological constant, he did so abstractly—he didn’t specify what it might be, where it might come from, or how it might arise. The link to quantum jitters makes it inevitable that had Einstein not dreamed up the cosmological constant, someone engaged with quantum physics subsequently would have. Once quantum mechanics is taken into account, you are forced to confront an energy contribution provided by fields that’s uniformly spread through space, and so you are led directly to the notion of a cosmological constant.
The question this raises is one of numerical detail. How much energy is contained in these omnipresent quantum jitters? When theorists calculated the answer, they got a well-nigh ridiculous result: there should be an infinite amount of energy in every volume of space. To see why, think of a field jittering inside an empty box of any size. Figure 6.3 shows some sample shapes the jitters can assume. Every such jitter contributes to the field’s energy content (in fact, the shorter the wavelength, the more rapid the jitter and hence the greater the energy). And since there are infinitely many possible wave shapes, each with a shorter wavelength than the previous, the total energy contained in the jitters is infinite.11
Although clearly unacceptable, the result did not engender fits of apoplexy because researchers recognized it as a symptom of the larger, well-recognized problem that we discussed earlier: the hostility between gravity and quantum mechanics. Everyone knew that you can’t trust quantum field theory on super-small distance scales. Jitters with wavelengths as small as the Planck scale, 10–33 centimeters, and smaller, have energy (and from m = E/c2, mass equivalent) so large that the gravitational force matters. To describe them properly requires a framework that melds quantum mechanics and general relativity. Conceptually, this shifts the discussion to string theory, or to any other proposed quantum theory that includes gravity. But the immediate and more pragmatic response among researchers was simply to declare that the calculations should disregard jitters on scales smaller than the Planck length. Failure to implement this exclusion would extend a quantum field theory calculation into a realm clearly beyond its range of validity. The expectation was that we will one day understand string theory or quantum gravity well enough to deal with the super-small jitters quantitatively, but the interim stopgap was to mathematically quarantine the most pernicious fluctuations. The import of the directive is clear: if you ignore jitters shorter than the Planck length, you’re left with only a finite number, so the total energy they contribute to a region of empty space is also finite.
Figure 6.3 There are infinitely many wave shapes in any volume and hence infinitely many distinct quantum jitters. This yields the problematic result of an infinite energy contribution.
That’s progress. Or, at the very least, it shifts the burden to future insights that would, fingers crossed, tame the super-small-wavelength quantum fluctuations. But even so, researchers found that the resulting answer for the energy jitters, while finite, was still gargantuan, about 1094 grams per cubic centimeter. This is far larger than what you’d get from compressing all the stars in all the known galaxies into a thimble. Focusing on an infinitesimal cube, one that measures a Planck length on each side, this stupendous density amounts to 10–5 grams per cubic Planck length, or 1 Planck mass per Planck volume (which is why these units, like kilos for potatoes and seconds for waiting, are the natural and sensible choice). A cosmological constant of this magnitude would drive such an enormously fast outward burst that everything from galaxies to atoms would be ripped apart. More quantitatively, astronomical observations had established a tight limit on how large a cosmological constant could be, if there were one at all, and the theoretical results exceeded the limit by a staggering factor of more than a hundred orders of magnitude. While a large finite number for the energy that suffuses space is better than an infinite one, physicists realized the dire need for dramatically reducing the result from their calculations.
Here’s where theoretical prejudice came to the fore. Assume for the moment that the cosmological constant is not just small. Assume it’s zero. Zero is a favorite number of theoreticians because there’s a tried and true way for it to emerge from calculations: symmetry. For example, imagine that Archie has enrolled in a continuing education course and for homework has to add together the sixty-third power of each of the first ten positive numbers, 163 + 263 + 363 + 463 + 563 + 663 + 763 + 863 + 963 + 1063, and then add the result to the sum of the sixty-third power of each of the first ten negative numbers, (–1)63 + (–2)63 + (–3)63 + (–4)63 + (–5)63 + (–6)63 + (–7)63 + (–8)63 + (–9)63 + (–10)63. What’s the final tally? As he laboriously calculates, getting ever-more frustrated, multiplying and then adding together numbers with more than five dozen digits, Edith chimes in: “Use symmetry, Archie.” “Huh?” What she means is that each term in the first collection has a symmetric balancing term in the second: 163 and (–1)63 sum to 0 (a negative raised to an odd power remains negative); 263 and (–2)63 sum to 0, and so on. The symmetry between the expressions results in a total cancellation, as if they were children of equal weight balancing on opposite sides of a seesaw. Needing no calculations at all, Edith shows that the answer is 0.
Many physicists believed—or, I should really say, hoped—that a similar total cancellation due to an as yet unidentified symmetry in the laws of physics would rescue the calculation of the energy contained in quantum jitters. Physicists surmised that the huge energies from quantum jitters would cancel against some as yet unidentified huge balancing contributions, once the physics was sufficiently well understood. This was about the only strategy physicists could come up with for tamping down the unruly results of the rough calculations. And that’s why many theorists concluded that the cosmological constant had to be zero.
Supersymmetry provides a concrete example of how this could play out. Recall from Chapter 4 (Table 4.1) that supersymmetry entails a pairing of species of particles, and hence species of fields: electrons are paired with species of particles called supersymmetric electrons, or selectrons for short; quarks with squarks; neutrinos with sneutrinos, and so on. All of these “sparticle” species are currently hypothetical, but experiments in the next few years at the Large Hadron Collider may change that. In any event, an intriguing fact came to light when theoreticians examined mathematically the quantum jitters associated with each of the paired fields. For every jitter of the first field, there’s a corresponding jitter of its partner that has the same size but opposite sign, much as in Archie’s math homework. And just as in that example, when we add together all such contributions pair by pair, they cancel out, yielding a final result of zero.12
The catch, and it’s a big one, is that the total cancellation occurs only if both members of a pair have not only the same electric and nuclear charges (which they do), but also the same mass. Experimental data have ruled this out. Even if nature makes use of supersymmetry, the data show that it can’t be realized in its most potent form. The as yet unknown particles (selectrons, squarks, sneutrinos, and so on) must be much heavier than their known counterparts—only this can explain why they haven’t been seen in accelerator experiments. When the different particle masses are accounted for, the symmetry is disturbed, the balancing is unbalanced, and the cancellations are imperfect; the result is once again huge.
Over the years, many analogous proposals were put forward, invoking a range of additional symmetry principles and cancellation mechanisms, but none achieved the goal of establishing theoretically that the cosmological constant should vanish. Even so, most researchers took this merely as a sign of our incomplete understanding of physics, not as a clue that belief in a vanishing cosmological constant was misguided.
One physicist who challenged the orthodoxy was the Nobel laureate Steven Weinberg.* In a paper published in 1987, more than a decade before the revolutionary supernova measurements, Weinberg suggested an alternative theoretical scheme that yielded a decidedly different outcome: a cosmological constant that is small but not zero. Weinberg’s calculations were based on one of the most polarizing concepts to have gripped the physics community in decades—a principle some revere and others vilify, a principle some call profound and others call silly. Its official, if misleading, name is the anthropic principle.
Nicolaus Copernicus’ heliocentric model of the solar system is acknowledged as the first convincing scientific demonstration that we humans are not the focal point of the cosmos. Modern discoveries have reinforced the lesson with a vengeance. We now realize that Copernicus’ result is but one of a series of nested demotions overthrowing long-held assumptions regarding humanity’s special status: we’re not located at the center of the solar system, we’re not located at the center of the galaxy, we’re not located at the center of the universe, we’re not even made of the dark ingredients constituting the vast majority of the universe’s mass. Such cosmic downgrading, from headliner to extra, exemplifies what scientists now call the Copernican principle: in the grand scheme of things, everything we know points toward human beings not occupying a privileged position.
Nearly five hundred years after Copernicus’ work, at a commemorative conference in Kraków, one presentation in particular—given by the Australian physicist Brandon Carter—provided a tantalizing twist to the Copernican principle. Carter expounded his belief that an overadherence to the Copernican perspective might, in certain circumstances, divert researchers from significant opportunities for making progress. Yes, Carter agreed, we humans are not central to the cosmic order. Yet, he continued, aligning with similar insights articulated by scientists such as Alfred Russel Wallace, Abraham Zelmanov, and Robert Dicke, there is one arena in which we do play an absolutely indispensable role: our own observations. However far we have been demoted by Copernicus and his legacy, we top the bill when credits are conferred for the gathering and analyzing of the data that mold our beliefs. Because of this unavoidable position, we must take account of what statisticians call selection bias.
It’s a simple and widely applicable idea. If you are investigating trout populations but only canvass the Sahara Desert, your data will be biased by your focusing on an environment particularly inhospitable to your subject. If you are studying the general public’s interest in opera, but send your survey solely to the database collected by the journal Can’t Live Without Opera, your results won’t be accurate because the respondents are not representative of the population as a whole. If you are interviewing a group of refugees who have endured astoundingly harsh conditions during their trek to safety, you might conclude that they are among the hardiest ethnicities on the planet. Yet, when you learn the devastating fact that you are speaking with less than 1 percent of those who started out, you realize that such a deduction is biased because only the phenomenally strong survived the journey.
Addressing these biases is vital for getting meaningful results and for avoiding the futile search to explain conclusions based on unrepresentative data. Why are trout extinct? What’s the cause of the public’s surging interest in opera? Why is it that a particular ethnicity is so astoundingly resilient? Biased observations can launch you on meaningless quests to explain things that a broader, more representative view renders moot.
In most cases, these types of biases are easily identified and corrected. But there’s a related variety of bias that’s more subtle, one so basic it can easily be overlooked. It’s the kind in which limitations on when and where we are able to live can have a profound impact on what we are able to see. If we fail to take proper account of the impact such intrinsic limitations have on our observations, then, as in the examples above, we can draw wildly erroneous conclusions, including some that may impel us on fruitless journeys to explain meaningless MacGuffins.
For instance, imagine that you’re intent on understanding (as was the great scientist Johannes Kepler) why the earth is 93 million miles from the sun. You want to find, deep within the laws of physics, something that will explain this observational fact. For years you struggle mightily but are unable to synthesize a convincing explanation. Should you keep trying? Well, if you reflect on your efforts, taking account of selection bias, you will soon realize that you’re on a wild goose chase.
The laws of gravity, Newton’s as well as Einstein’s, allow a planet to orbit a star at any distance. If you were to grab hold of the earth, move it to some arbitrary distance from the sun, and then set it in motion again at the right velocity (a velocity easy to work out with basic physics), it would happily go into orbit. The only thing special about being 93 million miles from the sun is that it yields a temperature range on earth conducive to our being here. If earth were much closer or much farther away from the sun, the temperature would be much hotter or colder, eliminating an essential ingredient for our form of life: liquid water. This reveals the inbuilt bias. The very fact that we measure the distance from our planet to the sun mandates that the result we find must be within the limited range compatible with our own existence. Otherwise, we wouldn’t be here to contemplate the earth’s distance from the sun.
If earth were the only planet in the solar system, or the only planet in the universe, you still might feel compelled to carry your investigations further. Yes, you might say, I understand that my own existence is tied to the earth’s distance from the sun, yet this only heightens my urge to explain why the earth happens to be situated at such a cozy, life-compatible position. Is it just a lucky coincidence? Is there a deeper explanation?
But the earth is not the only planet in the universe, let alone in the solar system. There are many others. And this fact casts such questions in a very different light. To see what I mean, imagine that you mistakenly think a particular shop carries only a single shoe size, and so are gleefully surprised when the salesman brings you a pair that fits perfectly. “Of all possible shoe sizes,” you reflect, “it’s amazing that the single one they carry is mine. Is that just a lucky coincidence? Is there a deeper explanation?” But when you learn that the shop actually carries a wide range of sizes, the questions evaporate. A universe with many planets, situated at a range of distances from their host stars, provides a similar situation. Just as it’s no big surprise that among all the shoes in the shop there’s at least one pair that fits, so it’s no big surprise that among all the planets in all the solar systems in all the galaxies there’s at least one at the right distance from its host star to yield a climate conducive to our form of life. And it’s on one of those planets, of course, that we live. We simply couldn’t evolve or survive on the others.
So there is no fundamental reason why the earth is 93 million miles from the sun. A planet’s orbital distance from its host star is due to the vagaries of historical happenstance, the innumerable detailed features of the swirling gas cloud from which a particular solar system coalesced; it’s a contingent fact that’s unavailable for fundamental explanation. Indeed, these astrophysical processes have produced planets throughout the cosmos, orbiting their respective suns at a vast assortment of distances. We find ourselves on one such planet situated 93 million miles from our sun because that’s a planet on which our form of life could evolve. Failure to take account of this selection bias would lead one to search for a deeper answer. But that’s a fool’s errand.
Carter’s paper emphasized the importance of paying heed to such bias, an accounting he called the anthropic principle (an unfortunate name, because the idea would apply equally well to any form of intelligent life that makes and analyzes observations, not just to humans). No one took exception to this element of Carter’s argument. The controversial part was his suggestion that the anthropic principle might cast its net not just over things in the universe, like planetary distances, but over the universe itself.
What would that mean?
Imagine you’re puzzling over some fundamental feature of the universe, say the mass of an electron, .00054 (expressed as a fraction of the proton’s mass), or the strength of the electromagnetic force, .0073 (expressed by its coupling constant), or, of primary interest to us here, the value of the cosmological constant, 1.38 × 10–123 (expressed in Planck units). Your intention is to explain why these constants have the particular values they do. You try and try but come up emptyhanded. Take a step back, Carter says. Maybe you’re failing for the same reason you’d fail to explain the earth-sun distance: there is no fundamental explanation. Just as there are many planets at many distances and we necessarily inhabit one whose orbit yields hospitable conditions, maybe there are many universes with many different values for the “constants” and we necessarily inhabit the one in which the values are conducive to our existence.
In this way of thinking, to ask why the constants have their particular values is to ask the wrong kind of question. There is no law dictating their values; their values can and do vary across the multiverse. Our intrinsic selection bias ensures that we find ourselves in that part of the multiverse in which the constants have the values with which we’re familiar simply because we’re unable to exist in the parts of the multiverse where the values are different.
Note that the reasoning would fall flat if our universe were unique because you could still ask the “lucky coincidence” or “deeper explanation” questions. Much as a potent explanation for why the shop has your shoe size requires that the shelves be stocked with many different sizes, and much as a potent explanation for why there’s a planet situated at a bio-friendly distance from its host star requires planets orbiting their stars at many different distances, so a potent explanation of nature’s constants requires a vast assortment of universes endowed with many different values for those constants. Only in this setting—a multiverse, and a robust one at that—does anthropic reasoning have the capacity to make the mysterious mundane.*
Clearly, then, the degree to which you are swayed by the anthropic approach depends on the degree to which you are convinced of its three essential assumptions: (1) our universe is part of a multiverse; (2) from universe to universe in the multiverse, the constants take on a broad range of possible values; and (3) for most variations of the constants away from the values we measure, life as we know it would fail to take hold.
In the 1970s, when Carter put forward these ideas, the notion of parallel universes was anathema to many physicists. Certainly, there’s still ample reason to be skeptical. But we’ve seen in the previous chapters that although the case for any particular version of the multiverse is surely tentative, there’s reason for giving this new view of reality serious consideration, Assumption 1. Many scientists now are. Regarding Assumption 2, we’ve also seen that, for example, in the Inflationary and Brane Multiverses, we would indeed expect physical features, such as the constants of nature, to vary from universe to universe. Later in this chapter we’ll look at this point more closely.
But what about Assumption 3, concerning life and the constants?
Life, Galaxies, and Nature’s Numbers
For many of nature’s constants, even modest variations would render life as we know it impossible. Make the gravitational constant stronger, and stars burn up too quickly for life on nearby planets to evolve. Make it weaker and galaxies don’t hold together. Make the electromagnetic force stronger, and hydrogen atoms repel each other too strongly to fuse and supply power to stars.13 But what about the cosmological constant? Does life’s existence depend on its value? This is the issue Steven Weinberg took up in his 1987 paper.
Because the formation of life is a complex process about which our understanding is in its earliest stages, Weinberg recognized that it was hopeless to determine how one or another value of the cosmological constant directly impacts the myriad steps that breathe life into matter. But rather than give up, Weinberg introduced a clever proxy for the formation of life: the formation of galaxies. Without galaxies, he reasoned, the formation of stars and planets would be thoroughly compromised, with a devastating impact on the chance that life might emerge. This approach was not only eminently reasonable but also useful: it shifted the focus to determining the impact that cosmological constants of various sizes would have on galaxy formation, and that was a problem Weinberg could solve.
The essential physics is elementary. While precise details of galaxy formation are an active area of research, the broad-brush process involves a kind of astrophysical snowball effect. A clump of matter forms here or there, and by virtue of being more dense than its surroundings, it exerts a greater gravitational pull on nearby matter and thus grows larger still. The cycle continues feeding on itself to ultimately produce a swirling mass of gas and dust, from which stars and planets coalesce. Weinberg’s realization was that a cosmological constant with a value large enough would disrupt the clumping process. The repulsive gravity it would generate, if sufficiently strong, would thwart galactic formation by making the initial clumps—which were small and fragile—stream apart before they had time to become robust by attracting surrounding matter.
Weinberg worked out the idea mathematically and found that a cosmological constant any larger than a few hundred times the current cosmological density of matter, a few protons per cubic meter, would disrupt the formation of galaxies. (Weinberg also considered the impact of a negative cosmological constant. The constraints in that case are even tighter, because a negative value increases the attractive pull of gravity and makes the whole universe collapse before stars even have time to ignite.). If you imagine, then, that we’re part of a multiverse and that the cosmological constant’s value varies over a wide range from universe to universe, much as planet-star distances vary over a wide range from solar system to solar system—the only universes that could have galaxies, and hence the only universes we could inhabit, are ones in which the cosmological constant is no larger than Weinberg’s limit, which in Planck units is about 10–121.
After years of failed efforts by the community of physicists, this was the first theoretical calculation to result in a value for the cosmological constant that was not absurdly larger than limits inferred from observational astronomy. Nor did it contradict a belief widely held at the time of Weinberg’s work, that the cosmological constant vanished. Weinberg took this apparent progress one step further by encouraging an even more aggressive interpretation of his result. He suggested that we should expect to find ourselves in a universe with a cosmological constant whose value is as small as it needs to be for us to exist, but not a whole lot smaller. A much smaller constant, he reasoned, would call for an explanation that goes beyond mere compatibility with our existence. That is, it would require precisely the kind of explanation that physics had valiantly sought but so far failed to find. This led Weinberg to suggest that more refined measurements might one day reveal that the cosmological constant doesn’t vanish but, instead, has a value near or at the upper limit that he’d calculated. As we’ve seen, within a decade of Weinberg’s paper, the observations of the Supernova Cosmology Project and the High-Z Supernova Search Team proved this suggestion prophetic.
But to assess fully this unconventional explanatory framework, we need to examine Weinberg’s reasoning more closely. Weinberg is imagining a sprawling multiverse so diverse in population that it just has to contain at least one universe with the cosmological constant we’ve observed. But what kind of multiverse will guarantee, or at least make it highly likely, that this is the case?
To think this through, consider first an analogous problem with simpler numbers. Imagine you work for the notorious film producer Harvey W. Einstein, who has asked you to put out a casting call for the lead in his new indie, Pulp Friction. “How tall do you want him?” you ask. “I dunno. Taller than a meter, less than two. But you better make sure whatever height I decide, there’s someone who fits the bill.” You’re tempted to correct your boss, noting that because of quantum uncertainty he really doesn’t need to have every height represented but, thinking back on what happened to the surly little talking fly who tried that, you refrain.
Now you face a decision. How many actors should you have at the audition? You reason: If W. measures heights to a centimeter’s accuracy, there are a hundred different possibilities between one and two meters. So you need at least a hundred actors. But since some actors who show up may have the same height, leaving other heights unrepresented, you’d better gather more than a hundred. To be safe, maybe you should put out a call for a few hundred actors. That’s a lot, but fewer than what you’d need if W. measured heights to a millimeter’s accuracy. In that case, there’d be a thousand different heights between one and two meters, so to be safe you’d need to gather a few thousand actors.
The same reasoning is relevant for the case of universes with different cosmological constants. Assume that all the universes in a multiverse have cosmological constant values between zero and one (in the usual Planck units); smaller values lead to universes that collapse, larger values would strain the applicability of our mathematical formulations, compromising all understanding. So just as the actors’ heights had a range of one (in meters), the universes’ cosmological constants have a range of one (in Planck units). As for accuracy, the analog of W. using centimeter ticks, or millimeter ticks, is the precision with which we can measure the cosmological constant. Today’s accuracy is about 10–124 (in Planck units). In the future, our accuracy will no doubt improve, but as we’ll see, that will hardly affect our conclusions. Then just as there are 102 different possible heights spaced at least 10–2 meters apart (1 centimeter) in a one-meter range, and 103 different possible heights spaced at least 10–3 meters apart (1 millimeter), so there are 10124different values of the cosmological constant spaced at least 10–124 apart between the values 0 and 1.
To ensure that every possible cosmological constant is realized, we’d therefore need a multiverse with at least 10124 different universes. But as with the actors, we need to account for possible duplicates, universes that may have the same cosmological constant value. And so to play it safe and make it highly likely that every possible cosmological constant value is realized, we should have a multiverse with far more than 10124 universes, say a million times more, bringing it to a nice even 10130 universes. I’m being cavalier because when we’re talking about numbers this large, the exact values hardly matter. No familiar example of anything—not the number of cells in your body (1013); not the number of seconds since the big bang (1018); not the number of photons in the observable part of the universe (1088)—comes even remotely close to the number of universes we’re contemplating. The bottom line is that Weinberg’s approach for explaining the cosmological constant works only if we’re part of a multiverse in which there are a huge number of different universes; their cosmological constants must fill out some 10124 distinct values. Only with that many different universes is there a high likelihood that there’s one with a cosmological constant that matches ours.
Are there theoretical frameworks that naturally yield such a spectacular profusion of universes with different cosmological constants?14
From Vice to Virtue
There are. We encountered such a framework in the previous chapter. A count of the different possible forms for the extra dimensions in string theory, when including fluxes that can thread through them, came to about 10500. This dwarfs 10124. Multiply 10124 by a few hundred orders of magnitude and 10500 still dwarfs it. Subtract 10124 from 10500, and then subtract it again, and again, and do so a billion times over, and you’d barely make a dent. The result would still be nearly 10500.
Critically, the cosmological constant does indeed vary from one such universe to another. Just as magnetic flux carries energy (it can move things), so the fluxes threading holes in Calabi-Yau shapes also have energy, whose quantity is quite sensitive to the shape’s geometrical details. If you have two different Calabi-Yau shapes with different fluxes penetrating different holes, their energies will generally be different too. And since a given Calabi-Yau shape is attached to every point in the familiar three large dimensions of space, much as circular loops of pile attach to every point on the large extended base of a carpet, the energy the shape contains would uniformly fill the three large dimensions, much as soaking the individual fibers in a carpet’s pile would make the entire carpet backing uniformly heavy. Thus, should one or another of the 10500 different dressed-up Calabi-Yau shapes constitute the requisite extra dimensions, the energy it contains would contribute to the cosmological constant. Results obtained by Raphael Bousso and Joe Polchinski made this observation quantitative. They argued that the various cosmological constants supplied by the 10500 or so different possible forms for the extra dimensions are distributed uniformly across a broad range of values.
This is just what the doctor ordered. Having 10500 tick marks distributed across a range from 0 to 1 ensures that many of them lie extremely close to the value of the cosmological constant astronomers have measured during the past decade. It may be hard to find the explicit examples among the 10500 possibilities, because even if today’s fastest computers took a single second to analyze each form for the extra dimensions, after a billion years only a paltry 1032 examples would have been examined. But this reasoning suggests strongly that they exist.
Certainly, a collection of 10500 different possible forms for the extra dimensions is about as far from a unique universe as anyone imagined string theory research would ever take us. And for those who’ve held strongly to Einstein’s dream of finding a unified theory describing one single universe—ours—these developments came with significant discomfort. But analysis of the cosmological constant casts the situation in a different light. Rather than despair because a unique universe seems not to emerge, we are encouraged to celebrate: string theory makes the least plausible part of Weinberg’s explanation of the cosmological constant—the requirement that there be many more than 10124 different universes—suddenly seem plausible.
The Final Step, in Brief
The elements of a tantalizing story seem to be coming together. But a gap remains in the reasoning. It’s one thing for string theory to allow for a huge number of possible distinct universes. It’s another to claim that string theory ensures that all of the possible universes to which it can give rise are actually out there, parallel worlds populating a vast multiverse. As emphasized most emphatically by Leonard Susskind—who was inspired by the pioneering work of Shamit Kachru, Renata Kallosh, Andrei Linde, and Sandip Trivedi—if we weave eternal inflation into the tapestry, the gap can be filled.15
I’ll now explain this final step, but in case you’re reaching saturation and just want the punch line, here’s a three-sentence summary. The Inflationary Multiverse—the ever-expanding Swiss cheese cosmos—contains a vast, ever-increasing number of bubble universes. The idea is that when inflationary cosmology and string theory are melded, the process of eternal inflation sprinkles string theory’s 10500 possible forms for the extra dimensions across the bubbles—one form for the extra dimensions per bubble universe—providing a cosmological framework that realizes all possibilities. By this reasoning, we live in that bubble whose extra dimensions yield a universe, cosmological constant and all, that’s hospitable to our form of life and whose properties agree with observations.
In the remainder of the chapter, I will flesh out the details, but if you’re ready to move on, feel free to jump ahead to the chapter’s last section.
The String Landscape
In explaining inflationary cosmology back in Chapter 3, I used a variation on a common metaphor. A mountain’s peak represents the highest value of energy contained in an inflaton field suffusing space. The act of rolling down the mountain and coming to rest at a low point in the terrain represents the inflaton shedding this energy, which in the process is converted to particles of matter and radiation.
Let’s revisit three aspects of the metaphor, updating them with insights we’ve since acquired. First, we’ve learned that the inflaton is only one source of the energy that may fill space; other contributions come from the quantum jitters of any and all fields—electromagnetic, nuclear, and so on. To revise the metaphor accordingly, altitude will now reflect the combined energy uniformly suffusing space contributed by all sources.
Second, the original metaphor envisioned the base of the mountain, where the inflaton finally comes to rest, as being at “sea level,” altitude zero, meaning the inflaton has shed all its energy (and pressure). But with our revised metaphor, the height of the mountain’s base should represent the combined energy suffusing space from all sources after inflation has drawn to a close. This is another name for that bubble universe’s cosmological constant. The mystery in explaining our cosmological constant thus translates into the mystery of explaining the altitude of our mountain’s base—why is it so close to, but not exactly at, sea level?
Finally, we initially considered the simplest of mountainous terrains, a peak leading smoothly to a base, where the inflaton would ultimately settle (see Figure 3.1). We then went a step further, taking account of other ingredients (Higgs fields) whose evolution and final resting places would influence the physical features manifest in the bubble universes (see Figure 3.6). In string theory, the range of possible universes is richer still. The shape of the extra dimensions determines the physical features within a given bubble universe, and so the possible “resting places,” the various valleys in Figure 3.6b, now represent the possible shapes the extra dimensions can take. To accommodate the 10500 possible forms for these dimensions, the mountain terrain therefore needs a lush assortment of valleys, ledges, and outcroppings, as represented in Figure 6.4. Any such feature in the terrain where a ball could come to rest represents a possible shape into which the extra dimensions could relax; the altitude at that location represents the cosmological constant of the corresponding bubble universe. Figure 6.4 illustrates what’s called the string landscape.
With this more refined understanding of the mountain—or landscape—metaphor, we now consider how quantum processes affect the form of the extra dimensions in this setting. As we will see, quantum mechanics lights up the landscape.
Figure 6.4 The string landscape can be visualized schematically as a mountainous terrain in which different valleys represent different forms for the extra dimensions, and altitude represents the cosmological constant’s value.
Quantum Tunneling in the Landscape
While Figure 6.4 is necessarily schematic (each of the different Higgs fields in Figure 3.6 has its own axis; similarly each of the roughly 500 different field fluxes that can thread a Calabi-Yau shape should also have its own axis—but sketching mountains in a 500-dimensional space is a challenge), it correctly suggests that universes with different forms for the extra dimensions are part of a connected terrain.16 And when quantum physics is taken into account, using results discovered independently of string theory by the legendary physicist Sidney Coleman in collaboration with Frank De Luccia, the connections between the universes allow for dramatic transmutations.
The core physics relies on a process known as quantum tunneling. Imagine a particle, an electron for instance, encountering a solid barrier, say a slab of steel ten feet thick, that classical physics predicts it can’t penetrate. A hallmark of quantum mechanics is that the rigid classical notion of “can’t penetrate” often translates into the softer quantum declaration of “has a small but nonzero probability of penetrating.” The reason is that the quantum jitters of a particle allow it, every so often, to suddenly materialize on the other side of an otherwise impervious barrier. The moment at which such quantum tunneling happens is random; the best we can do is predict the likelihood that it will take place during one interval or another. But the math says that if you wait long enough, penetration through just about any barrier will happen. And it does happen. If it didn’t, the sun wouldn’t shine: for hydrogen nuclei to get close enough to fuse, they must tunnel through the barrier created by the electromagnetic repulsion of their protons.
Coleman and De Luccia, and many who have since followed their lead, scaled quantum tunneling up from single particles to an entire universe that’s faced with a similar “impenetrable” barrier separating its current configuration from another that’s possible. To get a feel for their result, imagine two possible universes that are otherwise identical save for a field, uniformly suffusing each, whose energy is higher in one, lower in the other. In the absence of a barrier, the higher energy-field value rolls to the lower, like a ball rolling down a hill as we’ve seen in the discussion of inflationary cosmology. But what happens if the field’s energy curve has a “mountainous bump” separating its current value from the one it seeks, as in Figure 6.5? Coleman and De Luccia found that much as is the case for a single particle, a universe can do what classical physics forbids: it can jitter its way—it can quantum tunnel—through the barrier and reach the lower energy configuration.
Figure 6.5 An example of a field’s energy curve that has two values—two troughs or valleys—where the field naturally comes to rest. A universe suffused with the higher-energy field value can quantum tunnel to the lower value. The process involves a small randomly located region of space in the original universe acquiring the lower field value; the region then expands, converting an ever-wider domain from the higher to the lower energy.
But because we are talking about a universe and not just a single particle, the tunneling process is more involved. It’s not that the field’s value throughout all of space tunnels simultaneously through the barrier, Coleman and De Luccia argued; rather, a “seed” tunneling event would create a small, randomly located bubble suffused with the smaller field energy. The bubble would then grow, much like Vonnegut’s ice-nine, ever enlarging the domain in which the field had tunneled to the lower energy.
These ideas can be applied directly to the string landscape. Imagine that the universe has a particular form for the extra dimensions, which corresponds to the left valley in Figure 6.6a. Because of this valley’s high altitude, the three familiar spatial dimensions are permeated by a large cosmological constant—yielding strong repulsive gravity—and so are rapidly inflating. This expanding universe, together with its extra dimensions, is illustrated on the left side of Figure 6.6b. Then, at some random location and moment, a tiny region of space tunnels through the intervening mountain to the valley on the right side of Figure 6.6a. Not that the tiny region of space moves (whatever that would mean); rather, the form of the extra dimensions (its shape, size, fluxes it carries) in this little region changes. The extra dimensions in the tiny region transmute, acquiring the form associated to the right valley in Figure 6.6a. This new bubble universe lies within the original, as illustrated in Figure 6.6b.
The new universe will rapidly expand and continue to transform the extra dimensions as it spreads. But since the new universe’s cosmological constant has decreased—its altitude in the landscape is lower than the original—the repulsive gravity it experiences is weaker, and so it won’t expand as fast as the original universe. We thus have an expanding bubble universe, with the new form for the extra dimensions, contained in even faster expanding bubble universe, with the original form for the extra dimensions.17
The process can repeat. At other locations inside the original universe as well as inside the new one, further tunneling events cause additional bubbles to open up, creating regions with yet different forms for the extra dimensions (Figure 6.7). In due course, the expanse of space will be riddled with bubbles inside of bubbles inside of bubbles—each undergoing inflationary expansion, each with a different form for the extra dimensions, and each with a smaller cosmological constant than the larger bubble universe within which it formed.
The result is a more intricate version of the Swiss cheese multiverse we found in our earlier encounter with eternal inflation. In that version, we had two types of regions: the “cheesy” ones that were undergoing inflationary expansion and the “holes” that weren’t. This was a direct reflection of the simplified landscape with a single mountain whose base we assumed to be at sea level. The richer string theory landscape, with its sundry peaks and valleys corresponding to different values of the cosmological constant, gives rise to the many different regions in Figure 6.7—bubbles inside of bubbles inside of bubbles, like a sequence of Matryosta dolls, each painted by a different artist. Ultimately, the relentless series of quantum tunnelings through the mountainous string landscape realizes every possible form for the extra dimensions in one or another bubble universe. This is the Landscape Multiverse.
Figure 6.6 (a) A quantum tunneling event, within the string landscape. (b) The tunneling creates a small region of space—represented by the smaller and darker bubble—within which the form of the extra dimensions has changed.
Figure 6.7 The tunneling process can repeat, yielding a vast nested sequence of expanding bubble universes, each with a different form for the extra dimensions.
The Landscape Multiverse is just what we need for Weinberg’s explanation of the cosmological constant. We’ve argued that the string landscape ensures that there are, in principle, possible forms for the extra dimensions that would have a cosmological constant in the ballpark of the observed value: there are valleys in the string landscape whose tiny altitude is on par with the tiny but nonzero cosmological constant that the supernovae observations revealed. When the string landscape combines with eternal inflation, all possible forms for the extra dimensions, including those with such a small cosmological constant, are brought to life. Somewhere within the vast nested sequence of bubbles constituting the Landscape Multiverse, there are universes whose cosmological constant is about 10–123, the minuscule number that launched this chapter. And according to this line of thought, it is in one of those bubbles that we live.
The Rest of Physics?
The cosmological constant is but one feature of the universe we inhabit. It is arguably among the most puzzling, since its small measured value is so famously at odds with the numbers that emerge from the most straightforward estimates using established theory. This chasm draws singular focus to the cosmological constant and underlies the urgency of finding a framework, however exotic, with the capacity to explain it. Proponents of the interlocking set of ideas laid out above argue that the string multiverse does just that.
But what about all the other features of our universe—the existence of three kinds of neutrinos, the particular mass of the electron, the strength of the weak nuclear force, and so on? While we can at least imagine calculating these numbers, no one has as yet managed to do so. You might wonder whether their values, too, are ripe for a multiverse-based explanation. Indeed, researchers surveying the string landscape have found that these numbers, like the cosmological constant, also vary from place to place, and hence—at least in our current understanding of string theory—are not uniquely determined. This leads to a perspective very different from what dominated in the early days of research on the subject. It suggests that trying to calculate the properties of the fundamental particles, like trying to explain the distance between the earth and the sun, may be misguided. Like planetary distances, some or all of the properties would vary from one universe to the next.
For this line of thinking to be credible, though, we need at a bare minimum to know not only that there are bubble universes in which the cosmological constant has the right value, but also that in at least one such bubble the forces and the particles agree with what scientists in our universe have measured. We need to be sure that our universe, in all its detail, is somewhere in the landscape. This is the goal of a vibrant field called string model building. The research program amounts to hunting around the string landscape and examining possible forms for the extra dimensions mathematically, in search of universes that most resemble ours. It’s a formidable challenge, because the landscape is too large and intricate to be fully studied in any systematic way. Progress requires sharp calculational skills as well as intuition regarding which pieces to assemble—the extra-dimensional shape, its size, the field fluxes cycling its holes, the presence of various branes, and so on. Those who lead this charge combine the best of rigorous science with an artistic sensibility. To date, no one has found an example that reproduces the features of our universe exactly. But with some 10500 possibilities awaiting exploration, the consensus is that our universe has a home somewhere in the landscape.
Is This Science?
In this chapter we’ve turned a logical corner. Until now, we’ve been exploring the implications for reality, writ large, of various developments in fundamental physics and cosmology research. I delight in the possibility that copies of the earth exist in the far reaches of space, or that our universe is one of many bubbles in an inflating cosmos, or that we live on one of many braneworlds constituting a giant cosmic loaf. These are undeniably provocative and alluring ideas.
But with the Landscape Multiverse, we’ve invoked parallel universes in a different way. In the approach we’ve just followed, the Landscape Multiverse is not merely broadening our view of what might be out there. Instead, an array of parallel universes, worlds that may be beyond our ability to visit or see or test or influence, now and perhaps always, are directly invoked to provide insight into observations we make here, in this universe.
Which raises an essential question: Is this science?
*One point of language. For the most part, I use the terms “cosmological constant” and “dark energy” interchangeably. When I need a little more precision, I take the value of the cosmological constant to denote the amount of dark energy suffusing space. As noted earlier, physicists often use the term “dark energy” a bit more liberally, to mean anything that can look like or masquerade as a cosmological constant over reasonably long time scales, but might slowly change and hence not truly be constant.
*It’s also how 3D movie technology works: by suitably choosing the spatial offsets on the screen of nearly duplicate images, the filmmaker causes your brain to interpret the resulting parallaxes as different distances, creating the illusion of a 3D environment.
*If space is infinitely big, you might wonder what it means to say that the universe is larger now than it was in the past. The answer is that “larger” refers to the distances between galaxies today compared with the distances between those same galaxies in the past. The expansion of the universe means the galaxies are now farther apart, which is reflected mathematically in the universe’s scale factor being larger. In the case of an infinite universe, “larger” does not refer to the overall size of space, since once infinite always infinite. But for ease of language, I will continue to refer to the changing size of the universe, even in the case of infinite space, with the understanding that I’m referring to the changing distances between galaxies.
*The Cambridge astrophysicist George Efstathiou was also one of the early pioneers who argued strongly and convincingly for a nonzero cosmological constant.
*In Chapter 7, we will examine more thoroughly and more generally the challenges of testing theories that involve a multiverse; we will also more closely analyze the role of anthropic reasoning in yielding potentially testable outcomes.