The Hidden Reality Part 5

Part of the appeal of a cyclical cosmology is its apparent ability to avoid the knotty issue of how the universe began. If the universe goes through cycle after cycle, and if the cycles have always happened (and perhaps always will), then the problem of an ultimate beginning is sidestepped. Each cycle has its own beginning, but the theory provides a concrete physical cause: the termination of the previous cycle. And if you ask about the beginning of the entire cycle of universes, the answer is simply that there was no such beginning, because the cycles have been repeating for eternity.

In a sense, then, cyclical models are an attempt to have your cosmological cake and eat it too. Back in the early days of scientific cosmology, the steady state steady state theory provided its own end run around the question of cosmic origin by suggesting that although the universe is expanding, it did not have a beginning: as the universe expands, new matter is continually created to fill the additional s.p.a.ce, ensuring that constant conditions are maintained throughout the cosmos for all eternity. But the steady state theory ran afoul of astronomical observations pointing strongly toward earlier epochs whose conditions differed markedly from those we experience today. Most pointed of all were observations zeroing in on an earliest cosmological phase that was far from steady and stately, being instead chaotic and combustible. A big bang undermines dreams of steady state, bringing the question of origin back to center stage. It's here that cyclical cosmologies offer a compelling alternative. Each cycle theory provided its own end run around the question of cosmic origin by suggesting that although the universe is expanding, it did not have a beginning: as the universe expands, new matter is continually created to fill the additional s.p.a.ce, ensuring that constant conditions are maintained throughout the cosmos for all eternity. But the steady state theory ran afoul of astronomical observations pointing strongly toward earlier epochs whose conditions differed markedly from those we experience today. Most pointed of all were observations zeroing in on an earliest cosmological phase that was far from steady and stately, being instead chaotic and combustible. A big bang undermines dreams of steady state, bringing the question of origin back to center stage. It's here that cyclical cosmologies offer a compelling alternative. Each cycle can can incorporate a big-bang-like past, in alignment with the astronomical data. But by stringing together an infinite number of cycles the theory still avoids having to supply an ultimate beginning. Cyclical cosmologies, so it would seem, thereby meld the most attractive features of the steady state and big bang models. incorporate a big-bang-like past, in alignment with the astronomical data. But by stringing together an infinite number of cycles the theory still avoids having to supply an ultimate beginning. Cyclical cosmologies, so it would seem, thereby meld the most attractive features of the steady state and big bang models.

Then in the 1950s, the Dutch astrophysicist Herman Zanstra called attention to a problematic feature of cyclical models, one that was implicit in Tolman's a.n.a.lysis a couple of decades earlier. Zanstra showed that there couldn't have been an infinite number of cycles preceding our own. The wrench in the cosmological works was the Second Law of Thermodynamics. This law, which we'll discuss more fully in Chapter 9 Chapter 9, establishes that disorder-entropy-increases over time. It's something we routinely experience. Kitchens, however ordered in the morning, have a way of becoming disordered by nightfall; the same goes for laundry bins, desktops, and playrooms. In these everyday settings, the increase in entropy is a mere nuisance; in cyclic cosmology, the increase in entropy is pivotal. As Tolman himself had realized, the equations of general relativity link the entropy content of the universe with the duration of a given cycle. More entropy means more disordered particles squeezed together when the universe shrinks; that generates a more powerful rebound, s.p.a.ce expands further, and so the cycle lasts longer. Looking back from today, the Second Law then implies that ever-earlier cycles would have had ever-less entropy (because the Second Law says that entropy increases toward the future, it must decrease toward the past),* and would thus have had ever-shorter durations. Working this out mathematically, Zanstra showed that sufficiently far back in time the cycles would have been so short that they would have ceased. They and would thus have had ever-shorter durations. Working this out mathematically, Zanstra showed that sufficiently far back in time the cycles would have been so short that they would have ceased. They would would have had a beginning. have had a beginning.

Steinhardt and company claim that their new version of cyclical cosmology avoids this pitfall. In their approach, the cycles arise not from a universe expanding, contracting, and expanding again but rather from the separation separation between braneworlds expanding, contracting, and expanding again. The branes themselves continually expand-and they do so throughout each and every cycle. Entropy builds from one cycle to the next, just as the Second Law requires, but because the branes expand the entropy is spread over ever-larger spatial volumes. The total entropy goes up, but the entropy between braneworlds expanding, contracting, and expanding again. The branes themselves continually expand-and they do so throughout each and every cycle. Entropy builds from one cycle to the next, just as the Second Law requires, but because the branes expand the entropy is spread over ever-larger spatial volumes. The total entropy goes up, but the entropy density density goes down. By the end of each cycle, the entropy is so diluted that its density is driven very nearly to zero-a full reset. And so, unlike what happens in the a.n.a.lysis of Tolman and Zanstra, the cycles can continue indefinitely toward the future as well as the past. The braneworld Cyclic Multiverse has no need for a beginning to time. goes down. By the end of each cycle, the entropy is so diluted that its density is driven very nearly to zero-a full reset. And so, unlike what happens in the a.n.a.lysis of Tolman and Zanstra, the cycles can continue indefinitely toward the future as well as the past. The braneworld Cyclic Multiverse has no need for a beginning to time.8 Sidestepping an age-old conundrum is a feather in the Cyclic Multiverse's cap. But as its proponents note, the Cyclic Multiverse goes beyond offering resolution to cosmological conundra-it makes a specific prediction that distinguishes it from the widely accepted inflationary paradigm. In inflationary cosmology, the violent burst of expansion in the early universe would have so thoroughly disturbed the spatial fabric that substantial gravitational waves would have been produced. These ripples would have left trace imprints on the cosmic microwave background radiation, and highly sensitive observations are now seeking them out. A brane collision, by contrast, creates a momentary maelstrom-but without the spectacular inflationary stretching of s.p.a.ce, any gravitational waves produced would almost certainly be too weak to create a lasting signal. So evidence of gravitational waves produced in the early universe would be strong evidence against the Cyclic Multiverse. On the other hand, failure to observe any evidence of these gravitational waves would severely challenge a great many inflationary models and make the cyclic framework all the more attractive.

The Cyclic Multiverse is widely known within the physics community but is viewed, almost as widely, with much skepticism. Observations have the capacity to change this. If evidence for braneworlds emerges from the Large Hadron Collider, and if signs of gravitational waves from the early universe remain elusive, the Cyclic Multiverse will likely garner increased support.

In Flux.

The mathematical realization that string theory is not just a theory of strings but also includes branes has had a major impact on research in the field. The braneworld scenario, and the multiverses to which it gives rise, is one resulting area of investigation with the capacity to profoundly remake our perspective on reality. Without the more exact mathematical methods developed over the last decade and a half, most of these insights would have remained beyond reach. Nevertheless, the main problem physicists hoped the more exact methods would address-the need to pick one form for the extra dimensions out of the many candidates that theoretical a.n.a.lyses have uncovered-has not yet been solved. Far from it. The new methods have actually made the problem all the more challenging. They've resulted in the discovery of vast new troves of possible forms for the extra dimensions, increasing the candidate pool enormously while not providing an iota of insight into how to single out one as ours.

Pivotal to these developments is a property of branes called flux flux. Just as an electron gives rise to an electric field, an electric "mist" that permeates the area around it, and a magnet gives rise to a magnetic field, a magnetic "mist" that permeates its region, so a brane gives rise to a brane field brane field, a brane "mist" that permeates its region, as in Figure 5.5 Figure 5.5. When Faraday was performing the first experiments with electric and magnetic fields, in the early 1800s, he imagined quantifying their strength by delineating the density of field lines at a given distance from the source, a measure he called the field's flux flux. The word has since become ensconced in the physics lexicon. The strength of a brane's field is also delineated by the flux it generates.

String theorists, including Raphael Bousso, Polchinski, Steven Giddings, Shamit Kachru, and many others realized that a full description of string theory's extra dimensions requires not only specifying their shape and size-which researchers in this area, including me, had focused on more or less exclusively in the 1980s and early 1990s-but also also specifying the brane fluxes that permeate them. Let me take a moment to flesh this out. specifying the brane fluxes that permeate them. Let me take a moment to flesh this out.

Figure 5.5 Electric flux produced by an electron; magnetic flux produced by a bar magnet; brane flux produced by a brane Electric flux produced by an electron; magnetic flux produced by a bar magnet; brane flux produced by a brane.

Since the earliest mathematical work investigating string theory's extra dimensions, researchers have known that Calabi-Yau shapes typically contain a great many open regions, like the s.p.a.ce inside a beach ball, a doughnut's hole, or within a blown gla.s.s sculpture. But it wasn't until the early years of the new millennium that theorists realized that these open regions needn't be completely empty. They can be wrapped by one or another brane, and threaded by flux piercing through them, as in Figure 5.6 Figure 5.6. Previous research (as summarized, for instance, in The Elegant The Elegant Universe Universe) had for the most part considered only "naked" Calabi-Yau shapes, from which all such adornments were absent. When researchers realized that a given Calabi-Yau shape could be "dressed up" with these additional features, they uncovered a gargantuan collection of modified forms for the extra dimensions.

Figure 5.6 Parts of the extra dimensions in string theory can be wrapped by branes and threaded by fluxes, yielding "dressed-up" Calabi-Yau shapes. (The figure uses a simplified version of a Calabi-Yau shape-a "three-hole doughnut"-and represents wrapped branes and flux lines schematically with glowing bands encircling portions of the s.p.a.ce. Parts of the extra dimensions in string theory can be wrapped by branes and threaded by fluxes, yielding "dressed-up" Calabi-Yau shapes. (The figure uses a simplified version of a Calabi-Yau shape-a "three-hole doughnut"-and represents wrapped branes and flux lines schematically with glowing bands encircling portions of the s.p.a.ce.) A rough count gives a sense of scale. Focus on fluxes. Just as quantum mechanics establishes that photons and electrons come in discrete units-you can have 3 photons and 7 electrons, but not 1.2 photons or 6.4 electrons-so quantum mechanics shows that flux lines also come in discrete bundles. They can penetrate a surrounding surface once, twice, three times, and so on. But apart from this restriction to whole numbers, there's in principle no other limit. In practice, when the amount of flux is large, it tends to distort the surrounding Calabi-Yau shape, rendering previously reliable mathematical methods inaccurate. To avoid venturing into these more turbulent mathematical waters, researchers typically consider only flux numbers that are about 10 or less.9 This means that if a given Calabi-Yau shape contains one open region, we can dress it up with flux in ten different ways, yielding ten new forms for the extra dimensions. If a given Calabi-Yau has two such regions, there are 10 10=100 different flux dressings (10 possible fluxes through the first paired with 10 through the second); with three open regions there are 103 different flux dressings, and so on. How large can the number of these dressings get? Some Calabi-Yau shapes have on the order of five hundred open regions. different flux dressings, and so on. How large can the number of these dressings get? Some Calabi-Yau shapes have on the order of five hundred open regions. The same reasoning yields on the order of 10 The same reasoning yields on the order of 10500 different forms for the extra dimensions different forms for the extra dimensions.

In this way, rather then winnowing the candidates to a few specific shapes for the extra dimensions, the more refined mathematical methods have led to a cornucopia of new possibilities. All of a sudden, Calabi-Yau s.p.a.ces can clothe themselves with far more outfits than there are particles in the observable universe. For some string theorists, this caused great distress. As emphasized in the previous chapter, without a means of choosing the exact form for the extra dimensions-which we now realize means also selecting the flux outfit that shape wears-the mathematics of string theory loses its predictive power. Much hope had been placed on mathematical methods that could go beyond the limitations of perturbation theory. Yet, when some of those methods materialized, the problem of fixing the form for the extra dimensions only got worse. Some string theorists lost heart.

Others, more sanguine, believe it's too early to give up hope. One day-perhaps a day that's just around the corner, perhaps a day that's far off-we will discover the missing principle that determines what the extra dimensions look like, including the fluxes the shape may be sporting.

Others still have taken a more radical tack. Maybe, they suggest, the decades of fruitless attempts to pin down the form for the extra dimensions are telling us something. Maybe, these radicals brazenly continue, we need to take seriously all all of the possible shapes and fluxes emerging from string theory's mathematics. Maybe, they urge, the reason the mathematics contains all these possibilities is of the possible shapes and fluxes emerging from string theory's mathematics. Maybe, they urge, the reason the mathematics contains all these possibilities is that they're all real that they're all real, each shape being the extra-dimensional part of its own separate universe. And maybe, grounding a seemingly wild flight of fancy in observational data, this is just what's needed to address perhaps the th.o.r.n.i.e.s.t problem of all: the cosmological constant.

*You can think of this as a grand generalization of the results touched on in Chapter 4 Chapter 4, in which different forms for the extra dimensions can give rise to identical physical models.

This wasn't the result of a mysterious mathematical coincidence. Instead, in a precise mathematical sense, strings are highly symmetric shapes, and it was this symmetry that wiped away the inconsistencies. See note 4 for details.

*The first revolution was the 1984 results of John Schwarz and Michael Green, which launched the modern version of the subject.

*If you're being careful, you'll note that a slice of bread is really three-dimensional (width and height on the slice's surface, but also depth from the slice's thickness), but don't let that trouble you. The thickness of the bread will remind us that our slices are visual stand-ins for large three-branes.

*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 a.s.suming, in keeping with observations, that entropy decreases toward the past. See The Fabric of the Cosmos 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 .[image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image] 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. 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 s.p.a.ce.

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 Many universes. universes.

The Return of the Cosmological Constant Remember that the cosmological constant, if it exists, fills s.p.a.ce 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 s.p.a.ce 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." 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 2020; 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: But hindsight is not always 2020; 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 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. 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 spectral lines-light emitted by distant stars-and, just as Einstein had antic.i.p.ated, 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 s.p.a.ce 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 s.p.a.ce 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 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, s.p.a.ce 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 s.p.a.ce 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 s.p.a.ce 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 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 s.p.a.ced 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 quant.i.tative, 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 compet.i.tion 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 apparent brightness (how bright it appears when observed from earth) with its brightness (how bright it appears when observed from earth) with its intrinsic intrinsic brightness (how bright it would appear if observed from close by), you can thus work out its distance. 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 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...o...b..ting. Well-developed physics of stellar structure establishes that if the white dwarf pulls away enough material (so that its total ma.s.s 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 ma.s.s 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-s.p.a.ce 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 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; scale factor; in an expanding universe, the scale factor increases with time. 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 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, s.p.a.ce 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 now received the supernova's light, since it has just received the supernova's light, since it has just now now completed its journey, we must use the scale factor that's just completed its journey, we must use the scale factor that's just now now written in the map's legend to translate the separation on the map-the trajectory from the supernova to us, traced in 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 Figure 6.1d-into the physical distance traveled. The procedure makes clear that the result is the distance now now between us and the current location of the Noa Galaxy: the third of our multiple-choice options. 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 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) (b) During the light's journey, the universe expands, which is reflected in the map's legend During the light's journey, the universe expands, which is reflected in the map's legend. (c) (c) When we receive the light, its intensity has been diluted through the spreading When we receive the light, its intensity has been diluted through the spreading. (d) (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 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 s.p.a.ce that traced its path, the length of that line would increase as s.p.a.ce 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 now spreads-and this sphere's radius is the length of the light's trajectory spreads-and this sphere's radius is the length of the light's trajectory now 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 gla.s.s 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 sh.e.l.l 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 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 grows. When we finally receive the light, its wavelength is longer than when it was emitted. It's as if light waves are threads st.i.tched through a piece of spandex. Just as stretching the spandex stretches the st.i.tching, so expanding the spatial fabric stretches the light waves.

We can be quant.i.tative. 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 size of the universe when the light we're now examining was emitted, as compared with the size of the universe today. 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 It's a straightforward final step to parlay a series series of such redshift measurements into a determination of the universe's expansion profile over time. 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 a.n.a.logous "pencil mark" for s.p.a.ce. 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 s.p.a.ce 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 s.p.a.ce, but partly also because of the stretching of s.p.a.ce 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 s.p.a.ce 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.

Cosmic Acceleration.

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 s.p.a.ce has not been slowing down. It's been speeding up 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 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 increase. The universe kicked into high gear as the expansion of s.p.a.ce 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 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, s.p.a.ce 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 cl.u.s.ter 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 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. 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 s.p.a.ce. Expressing the result in terms of an equivalent amount of ma.s.s, as is conventional among physicists (using E=mc2 in the less familiar form, m=E/c in the less familiar form, m=E/c2), the researchers showed that the supernova data required a cosmological constant of just under 1029 grams in every cubic centimeter. 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 s.p.a.ce 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 s.p.a.ce-every cubic meter of s.p.a.ce contributes the same outward push, dictated by the cosmological constant's value. And so the more s.p.a.ce 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 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 s.p.a.ce 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 s.p.a.ce-every cubic meter of s.p.a.ce contributes the same outward push, dictated by the cosmological constant's value. And so the more s.p.a.ce 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 Figure 6.2 attest. 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 10 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 ma.s.s (about 10 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 ma.s.s (about 105 grams) per cubic Planck length (a cube that measures about 10 grams) per cubic Planck length (a cube that measures about 1033 centimeters on each side and so has a volume of 10 centimeters on each side and so has a volume of 1099 cubic centimeters). In these units, the cosmological constant's measured value is about 10 cubic centimeters). In these units, the cosmological constant's measured value is about 10123, 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 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 s.p.a.ce is suffused with dark energy. 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 s.p.a.ce 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.

Explaining Zero.

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 s.p.a.ce 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 s.p.a.ce closed and send the quantum jitters home; the energy supplied by quantum jitters permeates s.p.a.ce and can't be removed. Since the cosmological constant is nothing but energy that permeates s.p.a.ce, quantum field jitters provide a microscopic mechanism that generates 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 s.p.a.ce, and so you are led directly to the notion of a cosmological constant. 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 s.p.a.ce, 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 How much energy is contained in these omnipresent quantum jitters? contained in these omnipresent quantum jitters? When theorists calculated the answer, they got a well-nigh ridiculous result: there should be an When theorists calculated the answer, they got a well-nigh ridiculous result: there should be an infinite infinite amount of energy in every volume of s.p.a.ce. To see why, think of a field jittering inside an empty box of any size. amount of energy in every volume of s.p.a.ce. To see why, think of a field jittering inside an empty box of any size. Figure 6.3 Figure 6.3 shows some sample shapes the jitters can a.s.sume. 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. shows some sample shapes the jitters can a.s.sume. 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, 1033 centimeters, and smaller, have energy (and from m=E/c centimeters, and smaller, have energy (and from m=E/c2, ma.s.s 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 quant.i.tatively, 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 s.p.a.ce 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 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 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 105 grams per cubic Planck length, or 1 Planck ma.s.s 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 quant.i.tatively, 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 s.p.a.ce is better than an infinite one, physicists realized the dire need for dramatically reducing the result from their calculations. grams per cubic Planck length, or 1 Planck ma.s.s 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 quant.i.tatively, astronomical observations had established a tight limit