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 wX (Eq. [45]), f (z) = (1 + z)3(1 + wX). 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.