by Brian Greene
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.
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 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 d
ata 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.
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 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 becau
se 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.