We can simplify this classification of solutions (excluding the matter-free ones) by choosing parameters other than and k to classify the shape. Every universe contains either a turning point where H =0, or a point of inflection where the acceleration is zero. Classify the former by the ratio xturn of -to-matter density at the turning point (where = by definition), and the latter by , the density parameter at inflection, (where xi = 1/2 by definition). As the following table shows, xturn varies smoothly from infinity to minus infinity through all the solutions with turning points:
k | rturn | xturn | Type | |
positive, 1 | +1 | (3/)1/2 | 3/2(3/)1/2 1 | Bounce |
1/2 | +1 | 2 | 2 | Bounce |
27/32 | +1 | 4/3 | 1 | Bounce |
1 | +1 | 1 | 1/2 | Loiter |
0 | +1 | 2/3 | 0 | Bang-crunch |
-4 | +1 | 1/2 | -1/4 | Bang-crunch |
- 1 | +1 | (- 2/)1/3 | -1 | Bang-crunch |
negative | 0 | (- 2/)1/3 | -1 | Bang-crunch |
- 1 | -1 | (- 2/)1/3 | -1 | Bang-crunch |
-5 | -1 | 1 | -5/2 | Bang-crunch |
-1 | -1 | 2 | -4 | Bang-crunch |
negative, || 1 | -1 | (- 3/)1/2 | -6(- 3/)1/2 - 1 | Bang-crunch |
Although xturn is continuous, the bounce universes effectively form a separate solution family.
Similarly, varies from zero to positive infinity through all the continuous expansion solutions (in this case we always have ri = ):
k | ri | ||
positive, 1 | -1 | 1 | 1 |
1/8 | -1 | 2 | 1/3 |
1 | -1 | 1 | 1/2 |
8 | -1 | 1/2 | 2/3 |
1 | -1 | 1 | 1 |
positive | 0 | 1 | |
1 | +1 | 1 | 1 |
8 | +1 | 1/2 | 2 |
1 | +1 | 1 | + |
The three families intersect at the loitering solution, where just reaches infinity, so ri = rturn = 1 at the Einstein point. Also the bang-crunch and continously expansion families meet at the k = - 1 cycloid solution, which is equivalent to a bang-crunch solution with xturn = - , and also to a continuous expansion solution with = 0. The final limit, with xturn = + is equivalent to the de Sitter solution.
In terms of and k, the new parameters are given by
xturn = rturn3/2; = = | (17) |
The turnaround radius rturn is given by positive real roots of the cubic equation:
2 - 3kr + r3 = 0 | (18) |
We can convert from these new parameters back to , k using
= ; k = Sign( - 1) | (19) |
= ; k = Sign(1 + xturn) | (20) |