Annu. Rev. Astron. Astrophys. 1992. 30:
499-542
Copyright © 1992 by . All rights reserved |

**3.2 The Age of the Universe**

By a trivial change of variables in Equation 9, from *a* to
*z* and from
to *t*, we obtain an integral that relates redshift
*z*_{1} to lookback time from the present,

This integral can be solved analytically in some special cases, e.g.
when _{} = 0
(Kolb & Turner 1990,
equations 3.22-3.25;
Sandage 1961a)
or when _{tot} = 1
(Weinberg 1989).
In general, it can be calculated
numerically without difficulty. By inspection of the integrand, one
sees that at fixed
_{M}, increasing
_{} lengthens the lookback time to
any redshift. Eliminating _{} in
favor of _{tot}, one
likewise finds that
at fixed _{tot}, the
lookback time to any *z* lengthens for decreasing
_{M}.
Figure 3 shows the lookback time as a
function of redshift for the five models A-E.

The integral in Equation 16 goes to a finite limit, the age of the
universe, as *z*_{1} ->
.
Kolb & Turner (1990)
give analytic formulas for
the special cases _{} = 0 and
_{tot} = 1 (see also
Sandage 1961a).
Figure 4
shows the results of numerical integration for general cases with
_{},
_{tot} in the same
range that was shown in
Figure 1. One sees that if
_{M},_{}
is bounded to a plausible range, then the age of the universe is
between about 0.5 and 2 Hubble times. In
Section 4.2 we will compare
these ages to observational constraints.

Because the contours in Figure 4 are not too
different from lines of
constant slope, one can readily write a simple approximation that is
valid to within a few percent in the range 0 <
_{M}
1, 0 <
_{tot}
1,and
serviceable anywhere away from the loitering line;

where

and ``sinn^{-1}'' is defined as sinh^{-1} if
_{a}
1 (the usual case) and as
sin^{-1} if
_{a} > 1. (In fact,
Equation 17 is the exact result when
_{tot} = 1.)