How do I find the scale factor of cosmological constant?

[unscientific]

Homework Statement

(a)Sketch how the contributions change with time
(b)For no cosmological constant, how long will this universe exist?
(c)How far would a photon travel in this metric?
(d)Find particular density $\rho_E$ and scale factor
(e)How would this universe evolve?[/B]

Homework Equations

The Attempt at a Solution

Part(a)[/B]
For dust/matter: $\rho \propto a^{-3}$. For Curvature: $\rho \propto a^{-2}$. For Vacuum: $\rho = const.$.

In early times, dust dominated. Late times, curvature dominated.Part(b)
For dust: $a \propto t^{\frac{2}{3}}$. For Curvature: $a \propto t$. The universe first expands a little then reaches the big crunch where $\dot a = 0$ then starts to contract. In late times, curvature dominates.
I suppose a rough time would be of order $\propto t_0$.

Part(c)
$$D_C = \int_0^X \frac{1}{\sqrt{1-kr^2}}$$
$$D_C = \frac{1}{\sqrt k} sin^{-1}\left( \sqrt k X \right)$$
$$X = \frac{1}{\sqrt k} sin \left( \sqrt k D_C\right)$$
Furthest distance is simply $\frac{1}{\sqrt k}$.

Part(d)
Starting with the Raychauduri Equation:
$$\frac{\ddot a}{a} = -\frac{4\pi G}{3}\left( \rho + \frac{3P}{c^2} \right) + \frac{1}{3} \Lambda c^2 - \frac{kc^2}{a^2}$$
For a static solution,
$$0 = -\frac{4\pi G}{3}\left( \rho + \frac{3P}{c^2} \right) + \frac{1}{3} \Lambda c^2 - \frac{kc^2}{a^2}$$
$$0 = -\frac{4\pi G}{3} \rho + \frac{1}{3} \Lambda c^2 - \frac{kc^2}{a^2}$$
$$0 = -\rho + \frac{\Lambda c^2}{4 \pi G} - \frac{3 kc^2}{4 \pi G a^2}$$
Thus $\rho_E = \frac{\Lambda c^2}{4 \pi G}$.

How do I find the scale factor for this density?

 

[unscientific]

anyone tried part (d) yet? I'm sure my parts (a)-(c) are right.

 

[unscientific]

bumpp on part (d)

 

[unscientific]

bumpp