The Age of the Universe: first attempt

For many years, it was believed that the most likely universe was a flat (or "critical"), matter dominated universe. Let's work out the age of that universe.

We had from last time, the
Friedman equation, the dynamical equation that governs the expansion of the Universe. Let's adopt it for the simple case of no cosmological constant (Lambda=0):

We said that if the Universe was not decelerating -- if there was no gravity to slow the expansion -- the age of the universe would simply be 1/H0. This is the age that corresponds to a Omega=0 (or Omega very small) universe.

What about the age of a critical universe, one that has just enough mass to halt the expansion (Omega=1)? We can integrate the Friedman equation (remembering that rho ~ R-3) to get

Now we use our definition of the critical density:

to substitute in for the density and get

Finally, remember that we defined R such that now, R=1. So to get the age of the universe we set R=1 and solve for t, to get

So a critical universe is 2/3 as old as an empty (or "very open") universe. So the age of the Universe depends both on the Hubble constant and the density parameter. For H 0=72 km/s/Mpc, an open (empty) universe is 13.9 Gyr old, a critical universe is 9.3 Gyr old.

Problem: Globular clusters are old: ~ 9-12 Gyr old. How can the universe be younger than the stars it contains? It can't, driving the "Cosmological Crisis" of the 1990s.