Further Simplification

We can simplify this classification of solutions (excluding the matter-free ones) by choosing parameters other than $\lambda$ 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 x_turn of $\Lambda$-to-matter density at the turning point (where $\Omega_{\rm turn}^{}$ = $\infty$ by definition), and the latter by $\Omega_{i}^{}$, the density parameter at inflection, (where x_i = 1/2 by definition). As the following table shows, x_turn varies smoothly from infinity to minus infinity through all the solutions with turning points:

$\lambda$	k	rturn	xturn	Type
positive, $\ll$ 1	+1	(3/$\lambda$)1/2	3/2(3/$\lambda$)1/2 $\gg$ 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
$\ll$ - 1	+1	(- 2/$\lambda$)1/3	-1	Bang-crunch
negative	0	(- 2/$\lambda$)1/3	-1	Bang-crunch
$\ll$ - 1	-1	(- 2/$\lambda$)1/3	-1	Bang-crunch
-5	-1	1	-5/2	Bang-crunch
-1	-1	2	-4	Bang-crunch
negative, |$\lambda$| $\ll$ 1	-1	(- 3/$\lambda$)1/2	-6(- 3/$\lambda$)1/2 $\ll$ - 1	Bang-crunch

Although x_turn is continuous, the bounce universes effectively form a separate solution family.

Similarly, $\Omega_{i}^{}$ varies from zero to positive infinity through all the continuous expansion solutions (in this case we always have r_i = $\lambda^{-1/3}_{}$):

$\lambda$	k	ri	$\Omega_{i}^{}$
positive, $\ll$ 1	-1	$\gg$ 1	$\lambda^{1/3}_{}$ $\ll$ 1
1/8	-1	2	1/3
1	-1	1	1/2
8	-1	1/2	2/3
$\gg$ 1	-1	$\ll$ 1	1
positive	0	$\lambda^{-1/3}_{}$	1
$\gg$ 1	+1	$\ll$ 1	1
8	+1	1/2	2
1	+1	1	+ $\infty$

The three families intersect at the loitering solution, where $\Omega_{i}^{}$ just reaches infinity, so r_i = r_turn = 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 x_turn = - $\infty$, and also to a continuous expansion solution with $\Omega_{i}^{}$ = 0. The final limit, with x_turn = + $\infty$ is equivalent to the de Sitter solution.

In terms of $\lambda$ and k, the new parameters are given by

$$\lambda \Omega_{i}^{} {1 \over 1 - k r_i} {1 \over 1 - k \lambda^{-1/3}}$$ (17)

The turnaround radius r_turn is given by positive real roots of the cubic equation:

$$\lambda$$ (18)

If there are no such roots the universe is in the continuous expansion family; if there are two (only for 0 < $\lambda$ < 1) the smaller and larger correspond to the bang-crunch and bounce turning points respectively; there are never three different positive real solutions.

We can convert from these new parameters back to $\lambda$, k using

$$\lambda \left(\vphantom{ { \Omega_i \over \vert \Omega_i - 1 \vert}}\right. {\Omega_i \over \vert \Omega_i - 1 \vert} \left.\vphantom{ { \Omega_i \over \vert \Omega_i - 1 \vert}}\right)^{3}_{} \Omega_{i}^{}$$ (19)

$$\lambda {k x_{\rm turn} \over 4} \left(\vphantom{ {3 \over 1 + x_{\rm turn} } }\right. {3 \over 1 + x_{\rm turn}} \left.\vphantom{ {3 \over 1 + x_{\rm turn} } }\right)^{3}_{}$$ (20)

Conversion back to $\Omega_{m}^{}$, $\Omega_{\Lambda}^{}$ is best done via $\lambda$ and k.