COSMOLOGY, OBSERVATIONAL TESTS VIRGINIA TRIMBLE AND STEPHEN P. MARAN Astronomers make observational tests of cosmology to determine the large-scale properties of the universe so as to distinguish between alternative concepts, or models, that attempt to account for its nature, history, and future. The comparison process is often described as observational testing of the models, which are generally based on Albert Einstein's theory of general relativity. Because it appears that the universe is expanding, these observational tests include measurements to determine the rate at which the universe is now expanding and the rate at which this expansion changes with time. Other observational tests include attempts to determine the density of matter in the universe, the abundances of light elements and isotopes thought to be created in the Big Bang, measurements of the microwave background radiation, and studies of the large-scale distribution of galaxies in space. These tests are intended to yield estimates of the age and size of the universe and the curvature of space, to provide information on the distribution and properties of unseen "dark matter," and to elucidate how the universe may have come to take its present form, and how that form will change with time. The first, and for years the only, discovery in modern observational cosmology was Edwin Hubble's recognition in the 1920s of a linear relationship between redshifts seen in the spectra of galaxies outside our own cluster (the Local Group) and the distances of those galaxies from us. The relationship ****************** is called Hubble's law. Here z is the redshift (*****, a measure of the amount that light of wavelength * is shifted toward longer wavelengths), c is the speed of light, and d is distance. **, the proportionality factor, is called Hubble's constant, although it can vary with time. ** has units of velocity divided by distance, or reciprocal time, and is typically given in kilometers per second per megaparsec. One megaparsec (Mpc) equals approximately ************ cm, or about 3.26 million light-years. Most scientists quickly interpreted Hubble's law to mean overall, uniform expansion of the observable universe. The redshifts result from the expansion of space-time, rather than from the motion of objects through space, and so are not quite the same as ordinary Doppler shifts of spectral lines. In particular, there is no unique relationship between redshift and relative velocity, as there is in Doppler shifts. Hubble's law does not, as it might first seem, imply the existence of a center for the expansion, let alone our presence at that center. Rather, an observer anywhere in the universe will see redshifts proportional to distances from himself, and the same value of ** at a given time. Since Hubble's work, the majority of cosmologists has thought in terms of a universe expanding out of a hot, dense initial state over a finite (but long) period of time. Such an expansion is describable by Einstein's equations in the form written down, also in the 1920s, by Alexander Friedmann in the USSR, Georges Lemaitre in Belgium, and Howard P. Robertson and Arthur Walker in the US. A model for such a universe is called a Big Bang model or evolutionary cosmology. Wholly different concepts include the steady-state cosmology and sometimes involve claims that the redshifts of galaxies have a noncosmological origin. For about the next 35 years, observational cosmology consisted largely of a search for accurate values of two numbers, the present expansion rate **, and its acceleration or deceleration, denoted ** During that period, the best estimates of ** gradually dropped from about 500 to about ************* (corresponding to a characteristic time scale or "age of the universe" of 2-20 billion years, respectively), while ** has been uncertain even in sign. The situation changed rapidly after, and largely because of, the 1965 discovery by Arno Penzias and Robert Wilson of a universal background of microwave radiation at a temperature of about ***. This radiation, predicted in 1948-49 by Ralph Alpher and Robert Herman, working with George Gamow, is interpreted as a first-hand relic of the hot, dense early universe. Its properties allow us to ask, and partially to answer, a wide range of questions about processes that occurred after the BigBang 10-20 billion years ago. In particular, the agreement between the observed amounts of helium, deuterium, and lithium in the universe and calculations based on a Hot Big Bang model is an impressive indicator of the correctness of our standard picture. PARAMETERS It is convenient to label the Positions of galaxies by coordinates that remain attached to them (comoving coordinates), multiplied by a factor **** that describes the expansion of the universe. Hubble's constant H is then simply *********************, and *** is its present value. The "dot" notation, for example, ****, for time derivatives is standard. The acceleration or deceleration of the expansion can be stated in dimensionless form as ********************************* Again, *** is the present value, positive if the expansion is slowing down, negative if the expansion is speeding up. During the expansion, the density of rest-mass energy drops as *****, and that of radiation energy as ****. The age of the universe, that is, the time elapsed since densities were arbitrarily (extremely) high, is ***** for **** and ***** for *****. In addition, if ordinary gravitation is the only important force on a cosmological scale, a universe with **** will expand and cool forever, while one with ******* will eventually recontract, heating us all out of existence as the temperature of the background radiation increases, long before we are crushed by the increasing density. The above considerations come just from application of calculus and conservation of mass energy. To go further, we apply general relativity to express how R(t) responds to the presence throughout the universe of an average mass-energy density p and a uniform pressure P: *********************** The new parameter, *, is Einstein's famous cosmological constant, which he is said, late in life, to have deeply regretted including in his equations. Originally introduced to permit a static universe, it may seem unnecessary in an expanding one, but is nevertheless part of the most general equation. * can be either positive or negative. Thus it can increase or decrease the effect of gravitating matter. But, since p decreases as R grows and * does not, *, if nonzero, will eventually dominate universal dynamics over sufficiently long times and distances. The pressure term can be dropped at this point as it is negligible after the first *** yr or so of expansion. Then the equation of motion has a first integral of the form ********************, where the intergration constant k can have values ***, *** or 0. fork***, the four-dimensional space-time is curved like a sphere (*****) or saddle (hypersphere, ******), and R(t) represents the radius of curvature of the space. For k=0, the universe is the flat Euclidean space of high school geometry Thus we have the five parameters H, q, p, *, and k to describe the behavior of R(t). Mercifully, they are not all independent. If we express the density in the convenient dimensionless form ********************** then ********************** and ********************** If *= 0, then ********, and die dynamical future of the universe can be found from its present density, with the dividing point between eternal expansion and eventual recontraction occurring at ***********, called the closure or critical density Pc. For this critical case, k=0. Thus a universe that just barely expands forever has Euclidean (flat) geometry. It is important to realize that p includes all forms of mass energy present in the universe, and thus must include invisible matter, not just the stars, galaxies, and radiation we easily detect. All these equations and considerations apply only to a homogeneous (same in all places) and isotropic (same in all directions) universe. This is clearly a bad description on small scales, since the distributions of planets, stars, galaxies, and clusters of galaxies are all inhomogeneous. But the uniformity around the sky both of the 3K radiation and of Hubble's law (for sufficiently large distances) suggests that homogeneity and isotropy are reasonable assumptions on very large scales of at least 100-6000 Mpc. (There are theorems within general relativity that guarantee that local measurements of, for instance, density will be cosmologically meaningful under these circumstances.) Additional parameters are needed to describe a universe in which the expansion is accompanied by overall rotation or shear. Such models have not been explored in as much detail as the purely expansionary ones, and can be severely constrained by the isotropy of the 3K radiation. If we expand the idea of observational cosmology to cover processes occurring in the early universe, then a number of additional measurable quantities become of interest. These include the precise temperature of the microwave radiation; its deviations either from a blackbody spectrum or from exact isotropy; the abundances of helium, deuterium, and lithium (before nuclear reactions in stars began modifying them); the properties of the largest structures (superclusters of galaxies and voids between them) in the universe and of the largest deviations from smooth Hubble expansion; and the nature of the primordial density fluctuations that eventually formed galaxies. OBSERVATIONS Distance Scale and H From Hubble's time to the present, cosmology has been bedeviled by the difficulty of establishing distances to astronomical objects beyond the range of parallax measurements. The general approach has been for nearby stars and star clusters, where geometrical methods are applicable, to be used to calibrate more distant, brighter ones (for which apparent brightness or apparent angular size plus known brightness or size imply distance), and for these, in turn, to be used to calibrate whole galaxies and clusters of galaxies. The large drop in the estimated value of ** from 500 to 50-100 km s**Mpc** between 1930 and 1965 occurred in several steps as astronomers recognized, first, that a bright sort of Cepheid variable star had been mistaken for a fainter sort, and second, that small star clusters and the gas they illuminate had been mistaken for single stars. The redshifts of galaxies, also required for Hubble's constant **, are readily measured to much higher accuracy than are the distances. Since about 1965, two lines of investigation, associated closely with the names of Allan Sandage and Gerard de Vaucouleurs, have persisted in yielding two different values of **, near 50 and 100 km s***Mpc***1, respectively. The two distance scales differ even within our own galaxy and are separated by a full factor of 2 when they reach out to where the universal expansion dominates over local, peculiar motions of galaxies (there being also some disagreement about whether this occurs at about 10 or 100 Mpc). Thus, there is immediately a factor of at least 2 uncertainty in the estimated age of the universe and of 4 in the brightness and other properties of distant galaxies and quasars. Hopes have been expressed that study of Cepheids in the Virgo cluster of galaxies with the Hubble Space Telescope (after its optics are corrected), or sufficiently careful modeling of the behavior of supernovae in still more distant galaxies, may resolve the prolonged discrepancy. Other Parameters The distance scale enters into all the other parameters in one way or another, to the detriment of precise measurement. The least-well- determined parameter is *, which does not dominate any directly measurable effect. Thus it is constrained only by its relationship to the other parameters. The geometrical parameters, * and *, are not much better off. To determine them directly, we need to measure lengths, surface areas, or volumes at known, very large distances and to compare them with Euclidean values. However, this requires that accurate distances be known and that the existence of objects of known linear size at large distance be established, and neither requirement is satisfied. Nevertheless, we know that the current radius of curvature of the universe, **, must be quite large compared with **** (3000-6000 Mpc), and it could be infinite. If *****, then universes with k=0 or k=-1 are infinite in volume and the part we can observe is infinitesimal. *** ***** universe has finite volume proportional to R*(t), and we might be observing at least a few percent of it. The deceleration parameter ** can be measured by looking at the value of H in the past. This means extending the Hubble relation out far enough so that the linear form breaks down. But the only distance indicator we have, apart from the redshift itself, is the apparent brightness of galaxies. At the redshifts required to see deviations from linearity, galaxies are half or less of their present age. Thus apparent brightness can only be turned into a distance if we understand clearly how the real brightnesses vary with time. As Fig.1 indicates, the differences in cz (the product of the speed of light and the redshift) versus apparent brightness due to plausible models of brightness evolution are larger than those due to the likely range of **. Rapid improvement in this situation is not expected. The density parameter ** is rather better known, partly because the value of H cancels in the ratio of ** to p contributed by any particular kind of object. Measured values of ** increase with the size of the system whose dynamics are investigated, from less than 0.01 for single galaxies to 0.1-0.3 for large clusters and superclusters. Significant material may well exist between these structures, however, and ***=1, favored by some theories, cannot be excluded observationally. The density contributed by ordinary (baryonic) matter, made of protons, neutrons, and electrons, is separately limited to roughly **** 0.015-0.15 if the helium, deuterium, and lithium we now see are relics of a homogeneous Hot Big Bang. Thus a density as large as *** requires either the existence of a nonbaryonic component or a highly inhomogeneous early universe. If, on the other hand, ** (or **) were very much larger than 1, its effects would show in a Hubble diagram like Fig.1, unless a carefully tuned value of * compensated. Finally, limits can be set to the present age of the universe both from the oldest stars (in globular clusters) in our galaxy and from the abundances of radioactive nuclides, including ****, ****, *****, ****, and Sr**. Values found are invariably in excess of ********* years, ruling out, for instance, the combination H=100 kms** Mpc****1, *******, whose age is only 6.7x*** years, and marginally ruling out any combination of large H and *=0. Just how much larger than ******* years the age must be depends on how much time is required for the formation of galaxies and the first stars, on the precise chemical compositions of globular cluster stars, and (yet again!) on precise distances. Values from at least ************* years are currently defensible and defended. STATUS OF OBSERVATIONAL COSMOLOGY Table 1 summarizes our present understanding of the measurable quantities. From the point of view of a professional astronomer, the lingering factor of two uncertainties are a persistent frustration, particularly because they bracket the dividing line between open (infinite volume, expands forever) and closed (finite volume, eventually recontracts) cosmological models. From a philosophical point of view, however, it should probably be regarded as a triumph that we have a sell-consistent model of what has happened over the past ********* years in a volume larger by a factor of **** or more than our own solar system. Additional Reading Gott, J.R., Gunn, J.E., Schramm, D.N., and Tinsley,B.M., (1976). Will the Universe expand forever? Scientific American 234 (No.3)62. Harrison, E.R.(1981). Cosmology: The Science of the Universe. Cambridge University Press, Cambridge. Sandage, A.R.(1988). Observational tests of world models. Ann. Rev. Astron. Ap.26 561. Silk, J.(1989). The Big Bang, rev. ed. W.H. Freeman and Company, New York. Trimble, V.(1987). Existence and nature of dark matter in the Universe. Ann. Rev. Aston. Ap.25 425. Zeldovich, Ya.B. and Novikov, I.D.(1983). Relativistic Astrophysics 2. The Structure and Evolution of the Universe. University of Chicago Press, Chicago. See also Background Radiation, Microwave; Cosmology, Big Bang Theory; Cosmology, Galaxy Formation; Cosmology, Inflationary Universe; Dark Matter, Cormological; Hubble Constant; Superclusters, Dynamics and Models.