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8. H0 t0

One of the most powerful tests for a non-zero cosmological constant is provided by a comparison of the expansion and oldest-star ages. To quote Carroll, Press and Turner (1990), ``A high value of H0 (> 80 km/s/Mpc, say), combined with no loss of confidence in a value 12-14 Gyr as a minimum age for some globular clusters, would effectively prove the existence of a significant OmegaLambda term. Given such observational results, we know of no convincing alternative hypotheses.''

In Figure 3, the dimensionless product of H0t0 is plotted as a function of Omega. Two different cases are illustrated: an open OmegaLambda = 0 Universe, and a flat Universe with OmegaLambda + Omegam = 1. Suppose that both H0 and t0 are both known to ± 10% (1-sigma, including systematic errors). The dashed and dot-dashed lines indicate 1-sigma and 2-sigma limits, respectively for values of H0 = 70 km/sec/Mpc and t0 = 15 Gyr. Since the two quantities H0 and t0 are completely independent, the two errors have been added in quadrature, yielding a total uncertainty on the product of H_0t0 of ± 14% rms. These values of H0 and t0 are consistent with a Universe where OmegaLambda = 0.8, Omegam = 0.2. The Einstein-de Sitter model (Omegam = 1, OmegaLambda = 0) is excluded (at 2.5sigma).

Despite the enormous progress recently in the measurements of H0 and t0, Figure 3 demonstrates that significant further improvements are still needed. First, in the opinion of this author, total (including both statistical and systematic) uncertainties of ± 10% have yet to be achieved for either H0 or t0. Second, assuming that such accuracies will be forthcoming in the near future for H0 (as the Key Project, supernova programs and other surveys near completion), and for t0 (as HIPPARCHOS provides an improved calibration both for RR Lyraes and subdwarfs), it is clear from this figure that if H0 is as high as 70 km/sec/Mpc, then accuracies of significantly better than ± 10% will be required to rule in or out a non-zero value for Lambda. (If H0 were larger (or smaller), this discrimination would be simplified!)

Figure 3

Figure 3. The product of H0t0 as a function of Omega. The dashed curve indicates the case of a flat Universe with OmegaLambda + Omegam = 1. The abscissa in this case corresponds to OmegaLambda. The solid curve represents a Universe with OmegaLambda = 0. In this case, the abscissa should be read as Omegam. The dashed and dot-dashed lines indicate 1-sigma and 2-sigma limits, respectively for values of H0 = 70 km/sec/Mpc and t0 = 15 Gyr in the case where both quantities are known to ± 10% (1-sigma). The large open circle denotes values of H0t0 = 2/3 and Omegam = 1 (i.e., those predicted by the standard Einstein-de Sitter model). Also shown for comparison is a solid line for the case H0 = 50 km/sec/Mpc, t0 = 15 Gyr.

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