*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
=
-*qH*^{2}*a* 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
>
_{c} for the
universe to be closed. In terms of
, this gives

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.