2.2. The Flatness Feature
The ``critical density'', _{c}, is defined by
A universe with k = 0 has = _{c} and is said to be ``flat''. It is useful to define the dimensionless density parameter
If is close to unity the term dominates in the Friedmann equation and the Universe is nearly flat. If deviates significantly from unity the k term (the ``curvature'') is dominant.
The Flatness feature stems from the fact that = 1 is an unstable point in the evolution of the Universe. Because a^{-3} or a^{-4} throughout the history of the Universe, the term in the Friedmann equation falls away much more quickly than the k / a^{2} term as the Universe expands, and the k / a^{2} comes to dominate. This behavior is illustrated in Fig. 1.
Figure 1. In the SBB (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 today is remarkably close to unity even after 15 Billion years of evolution. Today the value of is within an order of magnitude of unity, and that means that at early times must have taken values that were set extremely closely to _{c}. For example at the epoch of Grand Unified Theories (GUTs) (T 10^{16} GeV), has to equal _{c} to around 55 decimal places. This is simply an important property 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.