3.2. Hubble Age
As alluded to earlier, in a Friedman-Robertson-Walker Universe, the age of the Universe is directly related to both the overall density of energy, and to the equation of state of the dominant component of this energy density. The equation of state is parameterized by the ratio = p / , where p stands for pressure and for energy density. It is this ratio which enters into the second order Friedman equation describing the change in Hubble parameter with time, which in turn determines the age of the Universe for a specific net total energy density.
The fact that this depends on two independent parameters has meant that one could reconcile possible conflicts with globular cluster age estimates by altering either the energy density, or the equation of state. An open universe, for example, is older for a given Hubble Constant, than is a flat universe, while a flat universe dominated by a cosmological constant can be older than an open matter dominated universe.
If, however, we incorporate the recent geometric determination which suggests we live in a flat Universe into our analysis, then our constraints on the possible equation of state on the dominant energy density of the universe become more severe. If, for existence, we allow for a diffuse component to the total energy density with the equation of state of a cosmological constant ( = - 1), then the age of the Universe for various combinations of matter and cosmological constant are shown below.
_{M} | _{x} | t_{0} |
1 | 0 | 9.7 ± 1 |
0.2 | 0.8 | 15.3 ± 1.5 |
0.3 | 0.7 | 13.7 ± 1.4 |
0.35 | 0.65 | 12.9 ± 1.3 |
Clearly, a matter-dominated flat universe is in trouble if one wants to reconcile the inferred Hubble age with the lower limit on the age of the universe inferred from globular clusters. In fact, if one took the above constraints at face value, such a Universe is ruled out on the basis of age estimates and the Hubble constant estimates. However, I am old enough to know that systematic uncertainties in cosmology often shift parameters well outside their formal two sigma, or even three sigma limits. In order to definitely rule out a flat matter dominated universe using a comparison of stellar and Hubble ages, uncertainties in both would have to be reduced by at least a factor of two.