**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:

(12) |

which interpolates smoothly between the matter dominated epoch in the past
(*a*
*t*^{2/3}) 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.