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Rapid progress is being made in measuring the cosmological parameters that describe the dynamical evolution and the geometry of the Universe. In essence, this is the first conclusion of this review. The second conclusion is that despite the considerable advances, the accuracy of cosmological parameters is not yet sufficiently high to discriminate amongst, or to rule out with confidence, many existing, competing, world models. We as observers still need to do better. Fortunately, there are a number of opportunities on the horizon that will allow us to do so.

In the context of the general theory of relativity, and assumptions of large-scale homogeneity and isotropy, the dynamical evolution of the Universe is specified by the Friedmann equation

Equation 1

where a (t) is the scale factor, H = adot / a is the Hubble parameter (and H0 is the Hubble ``constant'' at the present epoch), rhom is the average mass density, k is a curvature term, and Lambda is the cosmological constant, a term which represents the energy density of the vacuum. It is common practice to define the matter density (Omegam = 8 pi G rhom / 3H02), the vacuum energy density (OmegaLambda = Lambda / 3H02), and the curvature term (Omegak = -k / a02 H02) so that Omegam + OmegaLambda = 1 for the case of a flat universe where k = 0. The simplest case is the Einstein-de Sitter model with Omegam = 1 and OmegaLambda = 0. The dimensionless product H0t0 (where t0 is the age of the Universe) is a function of both Omegam and OmegaLambda. In the case of the Einstein-de Sitter Universe

Equation 2

Figure 1

Figure 1. Omegam versus H0 showing current observational limits on cosmological parameters. Solid lines denote expansion ages for an open (OmegaLambda = 0) Universe and the dashed line denotes an expansion age of 15 Gyr in the case of a flat (OmegaLambda neq 0) Universe. See text for details.

Bounds on several cosmological parameters are summarized in Figure 1 in a plot of the matter density as a function of the Hubble constant, following Carroll, Press & Turner (1992). Solid lines represent the expansion ages for 10, 15, and 20 Gyr in an open (Lambda = 0) model. The grey box is defined by values of H0 in the range of 40 to 90 km/sec/Mpc and 0.15 < Omegam < 0.4. The solid arrow denotes the same range in H0 for Omegam = 1. This plot illustrates the well-known ``age'' problem; namely that for an Einstein-de Sitter Universe (Omega = 1, Lambda = 0), H0 must be less than ~ 45 km/sec/Mpc if the ages of globular clusters (t0) are indeed ~ 15 billion years old. This discrepancy is less severe if the matter density of the Universe is less than the critical density, or if a non-zero value of the cosmological constant is allowed. For example, the dashed line indicates an expansion age of 15 Gyr in the case of a flat (Omegam + OmegaLambda = 1) model for Lambda neq 0.

A number of issues that require knowledge of the cosmological parameters remain unresolved at present. First is the question of timescales (H0t0) discussed above; possibly a related issue is the observation of red (if they are indeed old) galaxies at high redshift. Second is the amount of dark matter in the Universe. As discussed below, many dynamical estimates of the mass over a wide range of scale sizes are currently favoring values of Omegam ~ 0.25 ± 0.10, lower than the critical Einstein-de Sitter density. And third is the origin of large-scale structure in the Universe. Accounting for the observed power spectrum of galaxy clustering has turned out to be a challenge to the best current structure formation models.

Taking the current data at face value, there appears to be a conflict with the standard Einstein-de Sitter model. In fact, it is precisely the resolution of these problems that has led to a recent resurgence of interest in a non-zero value of Lambda (e.g. Ostriker & Steinhardt 1995; Krauss & Turner 1995). Another means of addressing these issues (e.g. Bartlett et al. 1995) requires being in conflict with essentially all of the current observational measurements of H0; from purely theoretical considerations, a very low value of H0 (leq 30) could also resolve these issues.

Ultimately we will have to defer to measurement as the arbiter amongst the wide range of cosmological models (and their very different implications) still being discussed in the literature. A wealth of new data is becoming available and progress is being made in the measurement of all of the cosmological parameters discussed below: the matter density, Omegam, the vacuum energy density, OmegaLambda, the expansion rate H0, and age of the oldest stars t0. The central, critical issues now are (and in fact have always been) testing for and eliminating sources of significant systematic error.

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