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3.2. Age of the universe and cosmological constant

From equation (24) we can also determine the current age of the universe by the integral

Equation 34 (34)

Since most of the contribution to this integral comes from late times, we can ignore the radiation term and set OmegaR approx 0. When both OmegaNR and OmegaLambda are present and are arbitrary, the age of the universe is determined by the integral

Equation 35 (35)

The integral, which cannot be expressed in terms of elementary functions, is well approximated by the numerical fit given in the second line. Contours of constant H0t0 based on the (exact) integral are shown in figure 11. It is obvious that, for a given OmegaNR, the age is higher for models with OmegaLambda neq 0.

Figure 11

Figure 11. Lines of constant H0t0 in the OmegaNR - OmegaLambda plane. The eight lines are for H0 t0 = (1.08, 0.94, 0.9, 0.85, 0.82, 0.8, 0.7, 0.67) as shown. The diagonal line is the contour for models with OmegaNR + OmegaLambda = 1.

Observationally, there is a consensus [49, 50] that h approx 0.72 ± 0.07 and t0 approx 13.5 ± 1.5 Gyr [83]. This will give H0 t0 = 0.94 ± 0.14. Comparing this result with the fit in (35), one can immediately draw several conclusions:

If the universe is populated by dust-like matter (with w = 0) and another component with an equation of state parameter wX, then the age of the universe will again be given by an integral similar to the one in equation (35) with OmegaLambda replaced by OmegaX(1 + z)3(1+wX). This will give

Equation 36 (36)

The integrand varies from 0 to (OmegaNR + OmegaX)-1/2 in the range of integration for w < 0 with the rapidity of variation decided by w. As a result, H0 t0 increases rapidly as w changes from 0 to -3 or so and then saturates to a plateau. Even an absurdly negative value for w like w = - 100 gives H0 t0 of only about 1.48. This shows that even if some exotic dark energy is present in the universe with a constant, negative w, it cannot increase the age of the universe beyond about H0 t0 approx 1.48.

The comments made above pertain to the current age of the universe. It is also possible to obtain an expression similar to (34) for the age of the universe at any given redshift z

Equation 37 (37)

and use it to constrain OmegaLambda. For example, the existence of high redshift galaxies with evolved stellar population, high redshift radio galaxies and age dating of high redshift QSOs can all be used in conjunction with this equation to put constrains on OmegaLambda [84, 85, 86, 87, 88, 89]. Most of these observations require either OmegaLambda neq 0 or Omegatot < 1 if they have to be consistent with h gtapprox 0.6. Unfortunately, the interpretation of these observations at present requires fairly complex modeling and hence the results are not water tight.

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