One of the classical tests of cosmology is the comparison of
timescales. With a knowledge of H_{0} , the average density of
matter,
, and the value of
the cosmological constant,
,
integration of the Friedmann equation

(7) |

yields a measure of the expansion age of the universe. This expansion
age can be compared with other independent estimates of the age of the
Galaxy and its oldest stars, t_{0}, and thus offers a test of
various
possible cosmological models. For example, the dimensionless product,
H_{0} t_{0}, is 2/3 in the simplest case where
_{m} = 1,
_{} = 0 (the Einstein-de
Sitter model), and the product
is 1 for the case of an empty universe where the matter and energy
density are zero.

An accurate determination of the expansion age requires not only the
value of H_{0} , but also accurate measurements of
_{m} and
_{}. At the time when the
Key Project was begun, the
strong motivation from inflationary theory for a flat universe,
coupled with a strong theoretical preference for
_{} = 0,
favored the Einstein-de Sitter model (e.g.,
Kolb & Turner
1990).
In addition, the ages of globular cluster stars were estimated at that
time to be ~ 15 Gyr
(VandenBerg,
Bolte & Stetson 1996;
Chaboyer et
al. 1996).
However, for a value of H_{0} = 72 km s^{-1}
Mpc^{-1}, as found in
this paper, the Einstein-de Sitter model yields a very young
expansion age of only 9 ± 1 Gyr, significantly younger than the
globular cluster and other age estimates.

Over the past several years, much progress has been made toward
measuring cosmological parameters, and the Einstein-de Sitter model
is not currently favored. For example, estimates of cluster velocity
dispersions, X-ray masses, baryon fractions, and weak lensing studies
all have provided increasingly strong evidence for a
low-matter-density
(_{m}) universe (e.g.,
Bahcall & Fan
1998).
In addition, strong new evidence for a flat universe has emerged from
measurements of the position of the first acoustic peak in recent
cosmic microwave background anisotropy experiments
(de Bernardis et
al. 2000;
Lange et
al. 2000).
Together with evidence for a low matter
density, and with recent data from high-redshift supernovae
(Riess et
al. 1998;
Perlmutter et
al. 1999),
evidence for a non-zero
cosmological constant has been increasing. Moreover, the age
estimates for globular clusters have been revised downward to 12-13
Gyr, based on a new calibration from the Hipparcos satellite
(Chaboyer 1998;
Carretta et
al. 2000).
A non-zero value of the cosmological
constant helps to avoid a discrepancy between the expansion age and
other age estimates. For H_{0} = 72 km s^{-1}
Mpc^{-1},
_{m} = 0.3,
_{} = 0.7, the expansion age
is 13 ± 1 Gyr,
consistent to within the uncertainties, with recent globular cluster
ages. In Table 15, we show expansion ages for
different values of H_{0} and a range of flat models.

In Figure 9 H_{0}t_{0} is
plotted as a function of
. Two curves are shown: the
solid curve is for the case where
= 0, and the dashed curve
allows for non-zero
under the assumption of a flat universe. The ± 1- and
2- limits are plotted for
H_{0} = 72 km s^{-1} Mpc^{-1}, t_{0} =
12.5 Gyr, assuming independent uncertainties of ±10% in each quantity,
and adding the uncertainties in quadrature. These data are consistent
with either a low-density
(_{m} ~ 0.1) open
universe, or a flat universe with
_{m} ~ 0.35,
_{} = 0.65;
however, with these data alone, it is not possible to discriminate
between an open or flat universe. As described above, recent studies favor
_{total} = 1, a
low-matter-density universe
(_{m} ~ 0.3), and a
non-zero value of the cosmological constant. Note,
however, that the open circle at
_{m} = 1,
= 0,
represents the Einstein-de Sitter case, and is inconsistent with the
current values of H_{0} and t_{0} only at a ~
2- level.