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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 Omega > 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 addot = -qH2a one gets,


This can be expressed as

Equation 35 (35)

Equation (28) takes the form,

Equation 36 (36)

Combining (35) and (36) gives,





Equation 37 (37)

Comparing (37) with the critical density (32) and the value of q > 1/2 in Table 2, it is evident that rho > rhoc for the universe to be closed. In terms of Omega, this gives

Equation 38 (38)

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.

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