Annu. Rev. Astron. Astrophys. 1988. 26: 561-630 Copyright © 1988 by Annual Reviews. All rights reserved

1. THE ELEMENTS OF PRACTICAL COSMOLOGY

The standard model of cosmology, based on what has come to be called the Friedmann-Lemaitre-Robertson-Walker (FLRW) model (hereinafter simply the Friedmann model), is now part of scientific culture. The most popular current version leads to the hot big bang (HBB) description of events near the beginning of the cosmic expansion, which has often been called a creation ⁽²⁾ moment at the beginning of physical time. In this review a prejudice in favor of the HBB (in contrast to cold beginnings discussed, for example, by Layser (1987) in his remarkable book on the growth of order in the Universe] can hardly be suppressed, successful as the model has become in providing an understanding of the abundance of He⁴ and the 3-K radiation. Nevertheless, if a description of beginnings in this sense is to be confined within the methods of science rather than to be colored by teleological metaphysics, the model must pass the tests normal to science rather than to be accepted as revealed truth. The purpose of this review is to discuss the direct tests of observation that lead to the view that a hot beginning to a current universe of finite age did occur.

Reviews of the theoretical aspects of the FLRW standard model from various viewpoints have appeared previously in this series. Novikov & Zeldovich (1967) surveyed the physical aspects of the HBB early Universe. Harrison (1973) summarized and discussed the various chemical eras, starting from a presumed initial singularity of very high temperature to the time of decoupling of matter and radiation, with the consequent formation of atoms some 30,000 yr after the Creation. Steigman (1976) reviewed the evidence and the reason(s) for the present matter-antimatter asymmetry. Boesgaard & Steigman (1985) discussed the theory and compared its predictions with observations of big bang nucleosynthesis. This comparison of the observed abundances of H, D, He³, He⁴, and Li⁷ with the calculations provides one of the two most powerful proofs of the HBB model. The other, of course, is the 3-K microwave background (MWB) radiation itself, discussed in these Reviews by Sunyaev & Zeldovich (1980) from the theoretical standpoint, and by Thaddeus (1972) and Weiss (1980) from the observational. The spectrum of the radiation resembles closely that of a blackbody. This is an important argument supporting a relic origin for the radiation, although alternate explanations have been proposed (Hoyle et al. 1968, Layzer & Hively 1973, Rana 1981, and references therein).

Other theoretical aspects of the standard model have been developed in these pages by Gould (1968) and Field (1972) in their reviews of the intergalactic medium, by Gott (1977) in his discussion of galaxy formation, and by Ellis (1984) in his survey of alternatives to the HBB standard model.

Particularly useful among the many workshop and conference proceedings that give entrance to the extensive archive literature are Physical Cosmology (Balian et al. 1979), Astrophysical Cosmology (Bruck et al. 1982), Progress in Cosmology (Wolfendale 1981), Cosmology and Fundamental Physics (Setti & Van Hove 1983), and Inner Space/Outer Space (Kolb et al. 1986).

Most of these discussions center on theoretical consequences of the HBB model. There have been only a few systematic reviews of results of the several direct (mostly geometrical) tests of the model. To be sure, important expositions of the principles of some of the classical tests are contained in discussions of the general properties of the models, such as the foundational reviews by Robertson (1933, 1955), the comprehensive summary by Zeldovich (1965), the lectures by Gunn (1978), and the textbooks by McVittie (1965), Peebles (1971), Weinberg (1972), Rowan-Robinson (1981), Narlikar (1983), and Zeldovich & Novikov (1983). But in all of these accounts, details of the practical methods of the subject are kept as a black art, taken to be known, and therefore not set out in detail.

The present review is concerned with the observational aspects of the subject. This is because no textbook now exists on what every student should know if practical cosmology - the linchpin of the laboratory part of the subject - is to become their way of life. The emphasis is on the details of the calculations (i.e. the equations) that are necessary to make comparisons between the models and the data. My aim is to assess critically if the model does in fact have experimental verification beyond the admittedly very powerful tests of the Gamow, Alpher, and Herman 3-K radiation, and the consequent predictions of baryon and nucleosynthesis out of the HBB.

The most satisfactory outcome of any such test would be some direct verification of the curvature of space by an experimental geometrical measurement similar to those proposed by Gauss and by Karl Schwarzschild. Spatial curvature is required by the foundation of the theory, deeply buried as it is in the covering theory of general relativity (Section 2). Barring such a test (none has yet been successful), a direct verification that the redshift is due to a true expansion of the geometrical manifold would be most helpful, but again such a demonstration is not quite available yet (see Section 8).

One precise prediction of the theory is that the form of the redshift-distance relation [observed at fixed cosmic time - i.e. found by reducing the observed World picture to the World map (in the language of Milne) to account for the light travel time] be strictly linear, not exponential as in a "tired light" theory, nor in any other form as in some nonstandard models. Tests of the linearity of the redshift vector field are singularly robust and are featured later in this review.

One of the central requirements of the standard model is that the time since the Creation (defined here as the beginning of physical time) be related to the observed Hubble expansion rate H₀^-1 by a factor that depends on the density parameter ₀ (in principle observable) via the connection (dictated by relativity) between the matter density and the space-time curvature. This relation is kc²/R² = H₀²(₀ - 1) if the cosmological constant is zero. Otherwise, the curvature has the additional term of c²/3 added. This test of the time scale, made by comparing the theoretical value of the age of the Universe, T₀ = H₀^-1 f(₀, ), with other clocks set ticking at the singularity, must work if the standard model is to be an adequate description. Because the test is so powerful, it has a chance to give a bona fide scientific judgment, if the relevant times can be accurately measured. We consider this test at length later in this review.

Finally, it is a commonplace that if any model is correct, it must have verifiable predictive power. The HBB ideas would seem to have already passed the high hurdle of the 3-K MWB radiation first predicted by Gamow (1946, 1948) and his fellow "originalists" (Alpher 1948, Alpher et al. 1948, 1953, Alpher & Herman 1948, 1950) and later required by Peebles (1986), Wagoner et al. (1967), Wagoner (1973), and now so many others. The radiation was subsequently discovered by Penzias & Wilson (1965). At the time, Dicke et al. (1965) were engaged in a search whose purpose was in fact to verify their independent prediction as a requirement of a hot big bang.

The most elementary prediction of any model that does not postulate continuous creation is that the mean contents of the Universe (suitably spatially averaged) were once younger than they are now. Verification of this required evolution in the look-back time is yet nascent. The variety of observational tests using galaxies at different redshifts, i.e. at different look-back times, give suggestive but not yet quite overwhelming evidence for evolution with time (see Section 7).

In the sections that follow I assume no detailed familiarity with the theoretical literature, nor familiarity at all with the observations. We develop the necessary apparatus for the tests as we need them so as to lay bare the assumptions upon which they rest. The level is aimed at first-year graduate students to provide them entrance to the literature for the necessary data, equations, and correction tables.

The menu for this journey through the test maze begins with the simplest geometrical predictions of curved space. Here the galaxy number count-distance test is set out in its most direct form of the volume V(r) enclosed within the "distance" l between us and coordinate point r in the comoving manifold (defined in the next section). Because the l distances are needed but are not themselves measured by rigid rods (suitably defined), we introduce next the distance-redshift relation that follows from the requirements of homogeneity and isotropy of the Robertson-Walker spaces. This leads naturally to the line element with its magic of accounting for the light-travel-time effects by using the null geodesic equation for light rays. This, in turn, permits direct entry to the redshift-distance equations, and therefrom to the redshift-luminosity (Mattig) relations for standard candles. This is the Hubble diagram straightaway. The practical details of corrections for aperture effect, K dimming, cluster richness, and the Bautz-Morgan contrast correlation are then set out. It is from the Hubble diagram for cluster galaxies that the linearity test for the form of the velocity field is most directly made.

Armed now with the m(z, q₀) Mattig equation, the N(z, q₀) count-redshift relation (as a function of space curvature) can be transformed to the observed N(m, q₀) predictions, integrating over the luminosity function. It is the comparison of this prediction with the observed N(m) relation that motivated Hubble to claim a geometrical measurement of the space curvature (Section 5).

These developments dispose, then, of the N(m, q₀) and the m(z, q₀) tests, which are half of the four classical hopes (Sandage 1961a) to find the one World model.

The remaining two tests are the angular size-redshift relation, and the time-scale comparison. The angular size variation with redshift should be the most direct way to sample the geometry (Hoyle 1959). The theory of this test leads to the surface brightness ~ (1 + z)^-4 relation, which must be valid if the expansion is real. Success in performing the experiment centers about the use of metric rather than isophotal galaxy diameters. The search for a suitable measure of a metric size is the present stumbling block, one yet to be adequately dislodged, but progress has been made (Section 8).

The time-scale test depends on the value of the Hubble expansion rate H₀. The problems of its determination in the presence of observational bias in the data samples are set out in Section 9, where evidence favoring the long distance scale, which requires a low value of H₀, is discussed.

² Creation is a flammable word that triggers responses often not intended by writers who use it. Gamow, in reply to a critic who complained about the title of his famous book "The Creation of the Universe," advised his reader to interpret creation as something similar to a lady's fashion rather than to misinterpret it as a theological statement. If it were the latter, the inquiry would be removed from the possibility of using the scientific method to discover, rather than some other method to reveal. When creation is used in this review, its meaning is in the Gamow sense. Nevertheless, the subject is possibly as close as science can come to the questions of origins - hence its enormous appeal. Back.