3.2. Spatially open and flat cosmological models
The preceding discussion referred to closed universe models for which = 1 and E < 0. For flat and open models ( = 0, - 1) the total energy is non-negative E 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 . For > 0 the universe passes through an inflection point at which the expansion of the universe changes from deceleration ( < 0) to acceleration ( > 0) (from (3) & (4) it can be shown that this usually occurs at a redshift when 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:
which interpolates smoothly between the matter dominated epoch in the past (a t2/3) and an inflationary epoch in the future (a e(/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.