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2.2. The Flatness Feature

The ``critical density'', rhoc, is defined by

Equation 4 (4)

A universe with k = 0 has rho = rhoc and is said to be ``flat''. It is useful to define the dimensionless density parameter

Equation 5 (5)

If Omega is close to unity the rho term dominates in the Friedmann equation and the Universe is nearly flat. If Omega deviates significantly from unity the k term (the ``curvature'') is dominant.

The Flatness feature stems from the fact that Omega = 1 is an unstable point in the evolution of the Universe. Because rho propto a-3 or a-4 throughout the history of the Universe, the rho term in the Friedmann equation falls away much more quickly than the k / a2 term as the Universe expands, and the k / a2 comes to dominate. This behavior is illustrated in Fig. 1.

Figure 1

Figure 1. In the SBB Omega(a) tends to evolve away from unity as the Universe expands.

Despite the strong tendency for the equations to drive the Universe away from critical density, the value of Omega today is remarkably close to unity even after 15 Billion years of evolution. Today the value of Omega is within an order of magnitude of unity, and that means that at early times rho must have taken values that were set extremely closely to rhoc. For example at the epoch of Grand Unified Theories (GUTs) (T approx 1016 GeV), rho has to equal rhoc to around 55 decimal places. This is simply an important property rho must have for the SBB to fit the current observations, and at this point we merely take it as a feature (the ``Flatness Feature'') of the SBB.

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