3. Expansion times

The predicted time of expansion from the very early universe to redshift z is

(64)

where E(z) is defined in Eq. (11). If = 0 the present age is t₀ < H₀^-1. In the Einstein-de Sitter model the present age is t₀ = 2 / (3H₀). If the dark energy density is significant and evolving, we may write = ₀f (z), where the function of redshift is normalized to f (0) = 1. Then E(z) generalizes to

(65)

In the XCDM parametrization with constant w_X (Eq. [45]), f (z) = (1 + z)^{3(1 + w_X)}. Olson and Jordan (1987) present the earliest discussion we have found of H₀t₀ in this picture (before it got the name). In scalar field models, f (z) generally must be evaluated numerically; examples are in Peebles and Ratra (1988).

The relativistic correction to the active gravitational mass density (Eq. [8]) is not important at the redshifts at which galaxies can be observed and the ages of their star populations estimated. At moderately high redshift, where the nonrelativistic matter term dominates, Eq. (64) is approximately

(66)

That is, the ages of star populations at high redshift are an interesting probe of _M0 but they are not very sensitive to space curvature or to a near constant dark energy density. ⁽⁷¹⁾

Recent analyses of the ages of old stars ⁽⁷²⁾ indicate the expansion time is in the range

(67)

at 95% confidence, with central value t₀ 13 Gyr. Following Krauss and Chaboyer (2001) these numbers add 0.8 Gyr to the star ages, under the assumption star formation commenced no earlier than z = 6 (Eq. [66]). A naive addition in quadrature to the uncertainty in H₀ (Eq. [6]) indicates the dimensionless age parameter is in the range

(68)

at 95% confidence, with central value H₀ t₀ 0.89. The uncertainty here is dominated by that in t₀. In the spatially-flat CDM model (_K0 = 0), Eq. (68) translates to 0.15 _M0 0.8, with central value _M0 0.4. In the open model with ₀ = 0, the constraint is _M0 0.6 with the central value _M0 0.4. In the inverse power-law scalar field dark energy case (Sec. II.C) with power-law index = 4, the constraint is 0.05 _M0 0.8.

We should pause to admire the unification of the theory and measurements of stellar evolution in our galaxy, which yield the estimate of t₀, and the measurements of the extragalactic distance scale, which yield H₀, in the product in Eq. (68) that agrees with the relativistic cosmology with dimensionless parameters in the range now under discussion. As we indicated in Sec. III, there is a long history of discussion of the expansion time as a constraint on cosmological models. The measurements now are tantalizingly close to a check of consistency with the values of _M0 and ₀ indicated by other cosmological tests.

⁷¹ The predicted maximum age of star populations in galaxies at redshifts z 1 does still depend on ₀ and _K0, and there is the advantage that the predicted maximum age is a lot shorter than today. This variant of the expansion time test is discussed by Nolan et al. (2001), Lima and Alcaniz (2001), and references therein. Back.

⁷² See Carretta et al. (2000), Krauss and Chaboyer (2001), Chaboyer and Krauss (2002), and references therein. Back.