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3.2. Spatially open and flat cosmological models

The preceding discussion referred to closed universe models for which kappa = 1 and E < 0. For flat and open models (kappa = 0, - 1) the total energy is non-negative E geq 0 and motion in the potential V(a) becomes unbounded, since a particle always has sufficient energy to surmount the potential barrier in figure (2). As a result the expansion factor a(t) shows monotonic behaviour, starting from the singular point at a = 0, t = 0 and increasing without bound as t rightarrow infty. For Lambda > 0 the universe passes through an inflection point at which the expansion of the universe changes from deceleration (addot < 0) to acceleration (addot > 0) (from (3) & (4) it can be shown that this usually occurs at a redshift when Lambda is still not dominating the expansion dynamics of the universe; see section 4.3).

In the important case when the universe is spatially flat and contains pressureless matter (dust) and a positive cosmological constant, the expansion factor has the exact analytical form:

Equation 12 (12)

which interpolates smoothly between the matter dominated epoch in the past (a propto t2/3) and an inflationary epoch in the future (a propto e(Lambda/3)1/2t). Equation (12) will be used later, when we examine some observational aspects of a universe with a cosmological constant in Section 4.

Finally, oscillating, bouncing and loitering models, as well as the static Einstein universe, are clearly absent in flat and open FRW models.

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