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

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3.2 The Age of the Universe

By a trivial change of variables in Equation 9, from a to z and from tau to t, we obtain an integral that relates redshift z1 to lookback time from the present,

Equation 16 16.

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

Figure 3

Figure 3. Lookback time as a function of redshift for the five models A-E. Even at moderate redshifts (z approx 1), the OmegaLambda-dominated models separate cleanly from the open models. However, the absolute differences are small.

The integral in Equation 16 goes to a finite limit, the age of the universe, as z1 -> infty. Kolb & Turner (1990) give analytic formulas for the special cases OmegaLambda = 0 and Omegatot = 1 (see also Sandage 1961a). Figure 4 shows the results of numerical integration for general cases with OmegaLambda, Omegatot in the same range that was shown in Figure 1. One sees that if OmegaM,OmegaLambda 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.

Figure 4

Figure 4. Contours of the age of the universe are shown in the OmegaM, Omegatot) plane. Away from the infinity line, the contours are close to straight lines, and Equation 17 is a good analytic approximation.

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 < OmegaM leq 1, 0 < Omegatot leq 1,and serviceable anywhere away from the loitering line;

Equation 17 17.

where

Equation 18 18.

and ``sinn-1'' is defined as sinh-1 if Omegaa leq 1 (the usual case) and as sin-1 if Omegaa > 1. (In fact, Equation 17 is the exact result when Omegatot = 1.)

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