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 xturn 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 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:

$ \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 xturn 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 ri = $ \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 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 = - $ \infty$, and also to a continuous expansion solution with $ \Omega_{i}^{}$ = 0. The final limit, with xturn = + $ \infty$ is equivalent to the de Sitter solution.

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

xturn = $\displaystyle \lambda$rturn3/2;      $\displaystyle \Omega_{i}^{}$ = $\displaystyle {1 \over 1 - k r_i}$ = $\displaystyle {1 \over 1 - k \lambda^{-1/3}}$ (17)

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

2 - 3kr + $\displaystyle \lambda$r3 = 0 (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

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

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

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

Patrick Leahy 2000-04-27