figure 5.6 Einstein visits the Mount Wilson Observatory offices. In 1931. Einstein visited the Pasadena offices of the Mount Wilson Observatory. George Ellery Hale, builder of the 100-inch telescope and founder of the observatory, looks down from his portrait in the library. Hubble {apparently being patted on the head by Hale) is at the left. Einstein, holding chalk, is in front of the blackboard. Courtesy of The Observatories of the Carnegie Institution of Washington will he the last stronghold to collapse. To drop the cosmic constant would drop the bottom out of space."11

While the theory of relativity has gone from triumph to triumph with the discovery of black holes, images of gravitational lenses, and precision tests of its predictions in the feeble gravity of the solar system and the more powerful tests from neutron stars locked in a closc orbit, the cosmological constant acquired a special status as theoretical poison ivy—an idea to be avoided.12 Eddington wandered farther and farther from the mainstream of theoretical developments in this and in other areas, following his own path into the wilderness.

"The biggest blunder of my life" is Einstein's anathema (whether he said it or not!). From time to time A has been picked out of Einstein's trash basket for further examination, but overall, the cos-

mological constant acquired a very bad reputation and was, for the most part, kept out of the discussion of practical cosmology. After all, if it had embarrassed Einstein, what would it do to the rest of us? But, as we will see, the cosmological coastant, or something that resembles it very closely, is back again, but this time with evidence Eddington may yet get the last laugh as we all go diving in Einstein's dumpster.

In 1932, Einstein and dc Sitter wrote a paper in which they swore off using the cosmological constant until "an increase in the precision of data derived from observations will enable us in the future to fix its sign and determine its value

Eddington wasn't ready to give up the cosmological constant, and chided Einstein and dc Sitter:

Einstein came to stay with me shortly afterwards, and I took him to task about [the paper] I ie replied- "I did not think the paper (abandoning A] important myself, hut deSitter was keen on it "Just after Einstein had gone, deSitter wrote to me announcing a visit He added "You will have seen the paper by Einstein and myself. I do not myself consider the result of much importance, but Einstein seemed to think that it was.' "ll

The application of Einstein's general relativity to the expansion of the universe was worked our in 1922 by the Russian meteorologist Alcxandr Friedmann, reinvented by the Belgian Abbe Georges Le-maitre in 1927, and discovered for the third time by physicist Howard P. Robertson. Even before Hubbie s discovery, the connection between the expanding universe and gravitation was reasonably well understood. Gravitation slows cosmic expansion.

If, for the moment, you follow Einstein's example (but not Eddington s) after the discovery of cosmic expansion and put the cosmological constant on the shelf, the possibilities are limited. In that case, the expansion of the universe is completely governed by the competition between motion, as expressed by the Hubble constant, and gravitation, given by the density of gravitating mass-energy. We have a shorthand for talking about the average density of the universe. Wc compare the observed density with a "critical density" that divides expansion forever from contraction at some far-off time in the future. The ratio of these two densities is just a number: to give it a ring of the ultimate and a whiff of cschatol-ogy, wc use the last letter of the Greek alphabet, il, (omega) as the symbol for that ratio.

The simplest picture is one where there is no matter. Or, anyway, not enough matter to matter. If £1, the density of matter divided by the critical density, is near zero, and if the universe starts off expanding, later expansion will not be significantly slowed by gravity Cosmic expansion would continue without limit, neither decelerating nor accelerating, bur coasting on indefinitely. If you start with a Big Bang everywhere, you get an expanding universe with Hubble's law for every observer.

If the universe has an appreciable mass density, with Qn, of 0.3, or 0.6 or 0.9, or any value smaller than one, the universe will still continue to expand without limit as in the Q = 0 ease. Here, I've written where the subscript "m" is to remind us we're discussing the effects of matter, without including the cosmological constant.

Friedmann's solutions for general relativity predict the course of cosmic expansion if you start with an expanding universe. In the presence of significant amounts of gravitating matter, gravitation slows expansion. A universe with Qm less than one will grow more dilute as it continues to expand—when you work out the physical details, the expansion will never stop. Even though it is always slowing down, an expanding universe with Q. below one will keep on keeping on, expanding forever.

The critical density itself is the amazingly small number of about 10 2(1 kilograms per cubic meter, or about 6 hydrogen atoms in a typical cubic meter of the universe.Our common sense world of everyday things does not give us a feeling for these numbers. In the room where you're sitting, the air has about lO" particles in every cubic meter. A very good laboratory "vacuum," say in the beam line of particle accelerator or the aluminizing tank at an observatory, might have 101"5 atoms in a cubic meter. What we think of as "empty" is a million billion times above the cosmic average for the universe. One path to forecasting the future of cosmic expansion would tx; lo take the average number of galaxies per cubic megaparsec from a big sample like the Las Campanas Redshifl Survey and multiply by the mass of each galaxy. When we did this, we found for matter that clusters with galaxies is about 0.3 ± 0.1.

There is also a simple mapping between density and the geometry of the universe. If A is part of the picture, you have to include its effect by computing the mass equivalent of that vacuum energy, which we call Oa. General relativity is a thoroughly tested theory of gravitation based on Einstein's idea that matter (and energy) curve space. It turns out that Q = Qm + QA= 1 corresponds to flat space, of the type we all learned about in high school where parallel lines don't meet; ii greater than 1 corresponds to the geometry of a sphere, like the geometry of the surface of the Earth, where lines of longitude, which look parallel at the equator, intersect at the poles. And a low-density universe, with Q less than 1, has the geometry of a saddle, in which the relations between distances and angles are the opposite from those seen on a sphere.

The geometry of space is not just an abstraction. If there are objects of constant brightness (".standard candles" in astronomical jargon), or objects of constant size ("standard rulers") then astronomers can make measurements to determine the geometry of the universe. In 196l, Allan Sandage, who was Bubble's only student and his heir in carrying forward the program of observational cosmology, wrote a paper in 'Ihe AstrophysicalJournal that set out the program to measure the geometry of the universe and to determine its fate by astronomical observation. The article, "The Ability of the 200-Inch Telescope to Discriminate Between Selected World Models/' described how the Hale telescope at Palomar Mountain could be used to measure the shape of the universe and to see the deceleration caused by mass in the universe.16 Sandage showed that the best method was to measure the relation between redshift and distance for objects in an expanding universe. You determine which of the "selected world models" represents the universe we live in by measuring the present expansion rate and the present rate of deceleration from observations. Most of the discussion in Sandage's classic paper is for the case of A = 0. For completeness, Sandage included a brief section near the end of this long paper that showrs how to detect a cosmological constant that would produce an accelerating universe, but the discussion for the next 35 years centered on finding just two numbers: the present expansion rate, the Hubble constant, //f>, and the present rate of deceleration, which (for A = 0) gives Q.17

Sandage's program for the Hale telescope was to make a Hubble diagram that extends over a large enough distance so that the cosmological effects of geometry and deceleration would make a measurable difference in the apparent brightness of an object at a given redshift. For the Earth, with a diameter of 12,000 kilometers, the effects of curvature get noticeable when you travel distances of thousands of kilometers. When you fly across the Atlantic, you definitely want the pilot to take curvature into account, flying over Newfoundland on the way to Paris from New York. That's why a glotx? is so helpful for understanding big distances, even though a flat, foldable road map will do fine for getting lost in Boston. For the universe, the natural time scale is the expansion time, about 14 billion years, and the natural distance scale is 14 billion light-years. So you need to look back several billion light-years for the global effects predicted by general relativity to make a significant difference. Technical difficulties mount as you push out to great distances where cosmology matters: the objects are exceedingly faint, and you arc looking at them when they were very young. Just at distances where the effects of cosmology begin to lie important, the uncertainties in the measurements begin to grow large.

For decades, Sandage pursued this program at Palomar with the 200-inch telescope, using the brightest galaxies as standard candles, because you can see them halfway across the universe and they seemed to have a small scatter in their intrinsic brightness. A big galaxy has the brightness of 50 billion stars. Hut galaxies are funny things. 'Iliey arc not really single "things" at all, but collections of stars, and the stars themselves change their brightness as they age over times of a few billion years. Also, galaxies are not so small compared to the separation between them, so in a few billion years, galaxies, especially galaxies in clusters, collide, merge, and grow. These changes in galactic properties can mask the subtle changes in brightness with redshift that cosmology produces. 'Ihe 25-year enterprise of determining the shape and fate of the universe by observing galaxies did not produce a conclusive result.1* But applying the same ideas to better-behaved standard candles, type la supernovae, with the more powerful telescopes that have superceded the 200-inch has given a strong and unexpected indication of the history of cosmic expansion. The cosmological constant is back: only this time, with evidence.

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