3.2. Expansion Rate and Time

Since we are considering what the Friedmann-Lemaître model does and does not predict we should note that the model allows solutions without a Big Bang, that trace back through a bounce to contraction from arbitrarily low density. This requires > 0 and positive space curvature, and, if the universe is going to contract by a substantial factor before bouncing, very large space curvature and small matter density: the redshift at the bounce is z_max ~ |_R| / _m. The bounce case is seldom mentioned, and I suspect rightly so, for apart from the bizarre initial conditions the redshift z_max required for light element production requires quite unacceptable density parameters. If this assessment is valid we are left with Friedmann-Lemaître solutions that trace back to infinite density, which is bizarre enough but maybe can be finessed by inflation and resolved by better gravity physics.

A Friedmann-Lemaître model that expands from exceedingly high density predicts that stellar evolution ages and radioactive decay ages are less than the cosmological expansion time t₀. Numerical examples are

(10)

The Hubble Space Telescope Key Project (Freedman et al. 1998; Madore et al. 1998) reports

(11)

The systematic error includes length scale calibrations common to most measurements of H₀. A recent survey of evolution ages of the oldest globular cluster stars yields 11.5 ± 1.3 Gyr. (Chaboyer et al. 1998). We have to add the time for expansion from very high redshift to the onset of star formation; a commonly used nominal value is 1 Gyr. If the universe is 14 Gyr old this would put the onset of star formation at z ~ 5 in the Einstein-de Sitter model, z ~ 6 if _m = 0.25 and = 0.75. Since star forming galaxies are observed in abundance at z ~ 3 (Pettini et al. 1998 and references therein) this is conservative. These numbers give

(12)

where the standard deviations have been added in quadrature.

The result agrees with the low density models in equation (10). The Einstein-de Sitter case is off by 1.6 standard deviations, not a serious discrepancy. It could be worse: Pont et al. (1998) put the minimum stellar evolution age at 13 Gyr. With 1 Gyr for star formation this would make the Einstein-de Sitter model > 2.6 off. It could go the other way: an analysis of the distance scale implied by the geometry of the multiply lensed system PG 1115+090 by Keeton & Kochanek (1997) puts the Hubble parameter at h = 0.51 ± 0.14. At t₀ = 14 Gyr this says H₀t₀ = 0.73 ± 0.20, nearly centered on the Einstein-de Sitter value. An elegant argument based on the globular cluster distance to the Coma Cluster of galaxies leads to a similar conclusion (Baum 1998). Most estimates of H₀ are larger, however, and the correction to t₀ for the time to abundant star formation is conservative, so in line 1b of Table 2 I give the Einstein-de Sitter model a modest demerit for its expansion time.

The low density cases pass the time-scale constraint at the accuracy of the present measurements. Since a satisfactory and it is to be hoped feasible measurement would distinguish between the _m ~ 0.25 open and flat cases I lower their grades from this test to ?.