2.5. The Age of the Universe

Measured independent lower bounds on the Hubble constant and on the age of the oldest globular clusters provide a lower bound on H₀t₀ (= 1.05ht, where H₀ 100h km s^-1Mpc^-1 and t₀ 10tGyr), and thus an interesting constraint in the _m - plane. The exact expressions are computable in the various regions of parameter space. For example, for = 0, the relation is (e.g., [15], Eq. 2.79)

(5)

where C₊₁^-1 cos^-1 and C_-1^-1 cosh^-1. A very useful approximation in the presence of a cosmological constant, that is an exact solution for a flat universe, is [1]

(6)

where S_{a 1}^-1 sinh^-1 and S_{a > 1}^-1 sin^-1. A useful crude approximation near H₀t₀ ~ 2/3 is

(7)

New Developments: Progress is being made in the HST key project to measure H₀ based on Cepheids in Virgo and Fornax.

Pro: The method does not depend on fluctuations, GI, biasing, etc. The current error of ~ 20% in the Hubble constant will hopefully be reduced soon to the level of 10% percent. This method is likely to provide the most stringent upper bound on _m and lower bound on .

Con: The errors in the determination of the age of the universe based on globular clusters are uncertain. The major source of error are the distances to Pop-II stars in the globular clusters, and complex stellar evolution issues.

Current Results: The most likely estimates are of h 0.6 - 0.7 [16] and t 1.5 [17], corresponding to ht 1. They seem to favor a possible deviation from the Einstein deSitter model towards low _m or high or both. However, one only needs to appeal to the current ~ 1 lower bounds (say h 0.53 and t 1.2) in order to accommodate the Einstein deSitter model.

Figure 1 displays in the _m- plane the ~ two-sigma constraints from the global measures discussed above. Superposed is the main constraint from cosmic flows (Section 4 below). The joint permitted range for _m is thus roughly 0.4 to 1.1. Low _m models of _m 0.3 are significantly ruled out.