2. The Closed Model
The Closed Model is characterized
by a positive curvature, k = 1. Thus, the spatial structure is
that of the 3-sphere, similar to the surface of a sphere, but in 3
dimensions instead of 2. This model corresponds to a universe that
begins at a `Big-Bang' and continues to expand until gravity
finally halts the expansion. The universe will then collapse into
a `Big-Crunch', which will resemble the reverse process of the
`Big-Bang'. The ability of the matter (or energy) in the universe
to halt the expansion obviously depends on the density. If the
matter-energy density is too low, the universe will have enough
momentum from the `bang' to escape the pull of gravity. In the
Closed Model the density of the universe is great enough to halt
the expansion and start a contraction. This corresponds to a value
of 
> 1, which is evident from
the use of the Friedmann
equations with k = 1. Plugging this k value into the Friedmann
equations (29), (28) and using
=
-qH2a one gets,
This can be expressed as
Equation (28) takes the form,
Combining (35) and (36) gives,
or
Thus,
Comparing (37) with the critical density
(32) and the value of q > 1/2 in
Table 2, it is evident that
The advantage of equation (37) above, is that the
density is expressed all in quantities that can be measured. In
that, if 2 of the 3 quantities are known the third may be found.
>
c for the
universe to be closed. In terms of
, this gives
