|Annu. Rev. Astron. Astrophys. 1988. 26:
Copyright © 1988 by . All rights reserved
4.4. The Look-Back Time as a Function of A and q0
To use Equation 35 we must change E(t) into E(z) by the relation between the look-back time = t0 - t1 and the redshift as a function of q0. 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 q0 = 0 and q0 = 1/2 for empty space and for flat space-time, respectively.
Recall that R(t) ~ t for q0 = 0 and R(t) ~ t2/3 for q0 = 1/2. Using these dependencies and the Lemaitre equation of R0/R1 = 1 + z gives the following relations for the look-back time:
The general case for any q0 is found by combining the age equations (Sandage 1961a, Equations 61 and 65) of T0 = f (q0, H0) with the R0/R1 = q(z, q0) Friedmann solution, together with R0/R1 = 1 + z. Tables are given in Sandage (1961b).