**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
*H*_{0}*t*_{0}
(= 1.05*ht*, where *H*_{0}
100*h* *km*
*s*^{-1}*Mpc*^{-1} and
*t*_{0} 10*tGyr*),
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)

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]

where *S*_{a
1}^{-1}
sinh^{-1}
and
*S*_{a >
1}^{-1}
*sin*^{-1}. A useful crude approximation near
*H*_{0}*t*_{0} ~ 2/3 is

*New Developments:*
Progress is being made in the HST key project to measure
*H*_{0} 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.