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Figure 5 4 The very first Hubble diagram In 1929. Edwin Hubble plotted the velocities of galaxies, determined from their redshifts* against their distances, measured from cepheids and other methods This diagram shows that the velocity is proportional to the distance, although individual galaxies depart noticeably from this relation and a few very nearby galaxies (like M3I) are approaching lb The slope of the Hubble diagram is the Hubble constant, measured in kilometers per second per megaparsec Hubble's original work showed a slope of 528 kilometers per second per megaparsec. over seven times larger than the modern value near 70 kilometers per second per megaparsec Courtesy of Publications of the National Academy of Sciences.

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Figure 5 4 The very first Hubble diagram In 1929. Edwin Hubble plotted the velocities of galaxies, determined from their redshifts* against their distances, measured from cepheids and other methods This diagram shows that the velocity is proportional to the distance, although individual galaxies depart noticeably from this relation and a few very nearby galaxies (like M3I) are approaching lb The slope of the Hubble diagram is the Hubble constant, measured in kilometers per second per megaparsec Hubble's original work showed a slope of 528 kilometers per second per megaparsec. over seven times larger than the modern value near 70 kilometers per second per megaparsec Courtesy of Publications of the National Academy of Sciences.

the brightest stars in galaxies. The next rung of his ladder of distances resorted to properties of the galaxies themselves to judge still larger distances. The precision of this chain of reasoning was not great, but the early results, though riddled with errors, were enough to show something very profound about the universe.

When scientists have two lists of things—a list of redshifts and a list of distances—you know what they will do. They will plot a graph. That's because we seek the mathematical relation that underpins the observations. The book of nature is written in the language of mathematics, and a graph is the easiest way to see how two quantities are related.

As plotted by Hubble in 1929, the relation between redshift and distance shows that we live in an expanding universe. As Eddington had astutely noted from very fragmentary data 6 years earlier, almost all galaxies are redshifted—moving away from us—and Hub-

hie showed the velocity is proportional to the distance. When you observe a galaxy that is twice as far, you find it is moving away twice as fast. A simple equation connects the measured velocity with the measured distance:

Velocity = (Some number) X Distance V - Htt x D

We call that equation Hubble's law, and the number, the Hubble constant, is the slope of the line in a Hubble diagram of velocity versus distance. We use the symbol Hti for the present-day value of the Hubble constant. The H is for "Hubble" (though he modestly used K, a usage I would like to bring back). //„ is pronounced "aitch-nought" where the "nought" means the I kibble constant measured here and now in the nearby universe. Despite its name, the Hubble constant was different in the distant cosmic past. H,t is measured in astronomers' units of kilometers per second per megaparsec, where a megaparsec is about 3 million light-years. This peculiar form of units keeps the physicists at a respectful distance to avoid contamination.^

Hubble's law is definitely not common sense—but it is the essential observation that shows we live in an expanding universe. Most of the undergraduates (and 1 suppose most of the faculty) at my institution are quite self-centered. If they think of Hubble's law at all, they think it confirms their belief that the universe is organized with themselves at the center and everyone else moving away from them. This is the egocentric universe.

But if there is any lesson to be learned from our location in the universe, or any lesson from the history of astronomy, it's that we humans are probably not the central pivot of the universe. The Earth isn't the center of the solar system, the sun isn't the centcr of our galaxy, and we would be slow learners to insist that our galaxy occupies the central position in the universe.

Instead of assuming that we are al a special place with a unique view of the universe, astronomy today takes the opposite approach. We assume our view is completely typical and the general layout of the universe as viewed from any other location would be the same. To get started, we assume that the universe is the same in all directions and the same from place to place. Of course, we know that isn't true in every detail. All galaxies are not identical, so the viewr from M31 can't be exactly the same as the view from our galaxy. Hut if you take a large enough piece of the universe, on average, one region is like another.

Now this is a simple and appealing assumption, but it is also subject to observational test. Unlike political theory, wc don't hold scientific truths to be self-evident. We test them by measurement. We can see whether one volume in the universe is like another by making maps of the locations of galaxies and determining empirically how big a patch you need to measure to get a fair sample of the universe. Measuring galaxy redshifts enables you to measure how far away they arc, at least for galaxies that arc far enough for the cosmic expansion to be larger than the individual motions of galaxies.

In 19H3, a group of us glimpsed the biggest structures in the universe from redshift samples of a few hundred galaxies. We were lucky, and detected the Great Void in Bootes, a big region without galaxies about 100 megaparsecs across.7 Since we only knew this was the biggest structure in our own survey, and it was about the biggest thing wc could have seen, wc didn't know quire what to make of it. Subsequent redshift surveys led by my colleagues at the Harvard-Smithsonian Center for Astrophysics, Margaret Gcllcr and John Huchra, showed that galaxies form a filamentary structure of great voids and great walls, with features of about the size of the Bootes Void seen in all directions. In the biggest redshift survey of the early 1990s, we showed that once you get to this scale, things seem to even out. We reached the end of greatness and the beginning of homogeneity.*

Today, redshift surveys arc big enterprises, with redshifts for hundreds of thousands of galaxies being systematically measured by highly automated systems. This field has changed from a cottage industry into assembly-line work. The observed scale of voids and filaments requires that you take a cube of at least a few hundred million light years on a side to gel the average properties of the local universe. Once you blur your view to this scale, one piece of the universe is like another. Swiss cheese has a well-determined

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Figure 5.5. The Las Campanas Redshift Survey. The redshrfts of 23.697 galaxies were measured by a single Harvard graduate student, Huan Lin, as pare of this collaboration. The galaxies were selected by their apparent brightness in six thin slices across the sky. This plot, with Las Campanas at the center, uses the redshift and position on the sky to show where the galaxies are located in space. They are dumped, with great voids, great sheets, and great dusters, all on scales less than about 7000 kilometer per second (about 100 megaparsees For a Hubble constant of 70). On larger scales, the structure seems to even out—this survey was the first that was large enough to see the end of greatness and the beginnir^g of cosmic homogeneity. Courtesy of Huan Lin and the Las Campanas Redshift Survey.

Figure 5.5. The Las Campanas Redshift Survey. The redshrfts of 23.697 galaxies were measured by a single Harvard graduate student, Huan Lin, as pare of this collaboration. The galaxies were selected by their apparent brightness in six thin slices across the sky. This plot, with Las Campanas at the center, uses the redshift and position on the sky to show where the galaxies are located in space. They are dumped, with great voids, great sheets, and great dusters, all on scales less than about 7000 kilometer per second (about 100 megaparsees For a Hubble constant of 70). On larger scales, the structure seems to even out—this survey was the first that was large enough to see the end of greatness and the beginnir^g of cosmic homogeneity. Courtesy of Huan Lin and the Las Campanas Redshift Survey.

average density once you take a big enough hunk to include both holes and the tasty solids. That's how they can sell it by the pound. A few hundred million light years sounds big for the size of voids, but the observable universe contains over 10,000 cells of this size, so it, too, can have a reasonably well-determined density.

If everyone sees the same universe we do, just from different locations, that's sufficient to make Hubhle's law into a recipe for an expanding universe. Start with one dimension—a long, stretchy rubber band. If you glue a little button on the rubber band every centimeter, and then stretch it out, the buttons will move away from one another. If you stretch the rubber band to twice its length, the buttons will each be twice as far apart. If you think about it from the point of view of an ant on each of the buttons, every ant sees all her neighbors moving away, and the more distant ones moving away more rapidly. In fact, this simple stretching produces a displacement, a rate of expansion, which is just proportional to distance. This cxactly echoes Hubble's law. It is Hubhle's law.

But Hubble's law is not just a demonstration. Hubble's law is measured in the real universe in which we live. The hard part is imagining all of this in two or three or four dimensions. Two dimensions would be something like the stretching surface of a balloon as it is inflated. Ants on the surfacc would sec Hubble's law. In three dimensions, try to imagine a giant jungle gym made of growing bamboo. If you were to hang on to one of the intersections, you'd see all your neighbors receding slowly and distant playmates receding rapidly. You'd see Hubble's law.

When the problem shifts up to three dimensions, it is our own common sense that makes it hard for us to understand these ideas. We can see the weather balloon growing with time, and sec the two-dimensional surface stretching. But we're not so good at imagining a space that is expanding in three Cor four!) dimensions. A homely, but nourishing, metaphor is to imagine you are a raisin in a baking loaf of raisin bread. As the bread expands in all directions while baking, all the other raisins move away from you, obeying Hubble's law. Cosmic expansion does not depend on an edge and does not need to have a centcr—to each observer it seems as though the local space is stretching away from you and it seems that you're at the center of your own nutshell.

But that's the same as the view you'd have from any other galaxy. An observer on M31 could invent a common-sense egocentric universe based on observations from M3I, another observer from a galaxy in the Virgo Cluster could do the same, and so could an observer in a galaxy deep in a Hubble Space Telescope field. You could say that cach of them is equally justified in considering themselves the center of the universe. Which is to say, not at all. Everybody's common sense is just slightly askew because we don't lcam the properties of an expanding space from our everyday experiences Maybe we should plant jungle gyms of live bamboo.

While Doppler's trumpets on a train are a vivid way to see the connection between motion and pitch, the cosmological rcdshifr is not precisely the same thing. It is more helpful to think of the red-shift as the effect of the universe stretching out while light travels from a distant galaxy. Light is emitted from a star in a distant galaxy with a particular wavelength set by quantum mechanics. This wavelength gets stretched out by cosmic expansion while the light is in flight. The longer the trip, the greater the rcdshift. That's Hubbie s law. Formally, the redshift is just a number: we u^e the symbol z for redshift:

^ _ Wavelength observed ^ Wavelength emitted

For small redshifts, the speed of light (c) times the redshift (z) gives a velocity. Although we often express redshift as a velocity, it is not exactly a common-sense velocity. The redshift ctoesn't tell us how fast galaxies are moving through a grid of space; it measures the expansion of space that has taken place while the light from a galaxy is in flight to us.

This distinction makes a difference when wre measure the velocities of galaxies that are zooming around in clusters of galaxies. There, all the galaxies are essentially at the same distance, and have the same cosmological redshift, but in addition, they have an extra velocity toward or away from us that is due to their own motion in the grip of the local gravitational field. The velocities of galaxies in clusters reveal the amount of matter in the universe by giving a quantitative measure of its gravitational effect. This is how Zwicky first detected the dark matter.

Does the expanding universe means that everything around us is growing in size5 No.

This is the question correctly answered by Alvy Singer's mother in the first few minutes (if Woody Allen's movie Annie Hall. Alvy's mother rakes him to see the family doctor, Dr. Flicker, because young Alvy, depressed by the meaninglessness of homework in an expanding universe, won't do his. As Alvy explains his angst, his mother interjects, yelling: "What's it to you? Brooklyn is not expanding."

She's right about this. Objccts like the Earth (and by extension, Brooklyn) whose structure is determined by electrical repulsion between the electrons in atoms, or by local gravity, do not share the overall expansion of space.

In the 1920s, Einstein's immense prestige, based on the success of general relativity, plus the correct formulation of the problem of an expanding universe by Alcxandr Fricdmann and others, made the cosmological constant a ccntral element in understanding the universe. The way we usually tell the story, Hubble's 1929 result cut the legs out from under this quest. If the universe is expanding, not static, then there is no need for a cosmological constant. You stan with an expanding universe, and it keeps on coasting outward.

By 1931, Einstein had abandoned the cosmological term, noting 1 lubble's observations "which the theory of general relativity can account for in a natural way, namely, without a lambda term." And he sent it on its way with the cursc of sour grapes, saying it was "theoretically unsatisfactory anyway." The legend, promulgated by the physicist George Gamow in his autobiography (but which appears nowhere in Einstein's own writings), is that Einstein called this "perhaps the biggest blunder of my lifc.'"J I suppose what Einstein (or perhaps Gamow) meant was that if Einstein had ignored the astronomers, stuck with the mathematically beautiful form of his equations, and not introduced A, he would have predicted cosmic expansion a decade ahead of its astronomical discovery, which would have been yet another feat of thcorctical brilliancc. Of course, he might just as well have predicted cosmic contraction due to gravity, perhaps noting the approach of M31 to the Milky Way, as observed by Slipher in 1912. Then that would have been his biggest blunder, revealed when the galaxies beyond M31 did not show blueshifts

Curiously, Arthur Stanley Eddington, Einstein's great promoter among scientists and envoy to the public, was not so quick to recant. He had noted the evidence of Slipher's velocity measurements in his 1923 book on relativity and he thought the expansion of the universe as observed by Hubble might be the clue to the role of the cosmological constant, which can do more than just balance gravitation: it can cause the expansion and produce an accelerating universe. He did not abandon the cosmological constant in 1929. Eddington explained his ideas in a vivid public talk at the International Astronomical Union's meeting in Cambridge, Massachusetts, in September 1932. He extended the metaphor of the astronomer as sleuth to the breaking point:

I am a detective in search of a criminal—the cosmological constant. 1 know he exists, but I do not know bis appearance; for instance I do not know if he is a little man or a tall man . . The first move was to search for footprints at the scene of the crime The search has revealed foot prints, cr what look like footprints—the recession of the spiral nebulae.10

Eddington thought the origin of the expansion measured by Hubble might lie in the repulsive effcet of the cosmological constant. Perhaps the galaxies had slowly started to expand from rest a very long time ago, and the expansion we see today is just the accumulated effect of A accelerating the universe over the eons. So, unlike Einstein. Eddington did not abandon A, invented to make a static universe, once Hubble had shown the universe was expanding. Instead, he looked to A as the source of the observed expansion. If there was an increase in expansion speed over time caused by steady repulsion this would show up in the Hubble diagram, with distant galaxies rcccding more slowly than you'd expcct in a universe that was coasting out from a Big Bang. The measurements of Eddington's time did not extend over large enough distances to look deep into the cosmic past for this accelerating effect. So, although Einstein was done with the cosmological constant, Eddington was not. With a rhetorical flourish that has seemed extravagant, bordering on silly, for most of 60 years, Eddington proclaimed: "If ever the theory of relativity falls into disrepute the cosmical constant

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