Rh

8k Gp0 3 R

8k gp0 3 R3

where we have used equation (20.16) in the last step. Substituting equation (20.25) into the equation for q (equation 20.24), we have q = 1/2

k > 0 In this case, q can be arbitrarily large. It even approaches infinity as R approaches Rmax (since R = 0 at Rmax ). This means that any value of q in the range q > 1/2

will produce a closed universe.

k < 0 We have already seen that q must be greater than zero, so the range of q given by

will produce an open universe.

The deceleration will depend on the density of matter in the universe. We can define a critical density, pcrit, such that the universe is closed if P > Pcrit and open if p < pcrit. If p = pcrit, we will have k = 0, and the universe is on the boundary between open and closed. This last point allows us to find pcrit, since it is the density for k = 0, or q = 1/2. If we set q = 1/2 in equation (20.24) and solve for the density, p, we have pcrit

H2 8k G

It is convenient to define a density parameter O, which is the ratio of the true density to the critical density. That is

0 0