Annu. Rev. Astron. Astrophys. 1988. 26:
561-630
Copyright © 1988 by . All rights reserved |

**4.4. The Look-Back Time as a Function of A and q _{0}**

To use Equation 35 we must change *E*(*t*) into
*E*(*z*) by the relation between the look-back time
= *t*_{0} -
*t*_{1} and the redshift as a function
of *q*_{0}. The general case requires the closed solution
of *R*(*t*) from the
Friedmann equation. Before setting down this general solution, it is
instructive to consider again the simple cases of *q*_{0} =
0 and *q*_{0} = 1/2
for empty space and for flat space-time, respectively.

Recall that *R*(*t*) ~ *t* for *q*_{0} = 0 and
*R*(*t*) ~ *t*^{2/3} for
*q*_{0} = 1/2. Using
these dependencies and the Lemaitre equation of
*R*_{0}/*R*_{1} = 1 + *z* gives the
following relations for the look-back time:

(36) |

The general case for any *q*_{0} is found by combining the
age equations
(Sandage 1961a,
Equations 61 and 65) of
*T*_{0} = *f* (*q*_{0},
*H*_{0}) with the
*R*_{0}/*R*_{1} = *q*(*z*,
*q*_{0}) Friedmann solution, together with
*R*_{0}/*R*_{1} = 1 + *z*. Tables are
given in
Sandage (1961b).