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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)
[tex]D_C = \int_0^X \frac{1}{\sqrt{1-kr^2}} [/tex]
[tex]D_C = \frac{1}{\sqrt k} sin^{-1}\left( \sqrt k X \right) [/tex]
[tex]X = \frac{1}{\sqrt k} sin \left( \sqrt k D_C\right) [/tex]
Furthest distance is simply ##\frac{1}{\sqrt k}##.
Part(d)
Starting with the Raychauduri Equation:
[tex]\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}[/tex]
For a static solution,
[tex]0 = -\frac{4\pi G}{3}\left( \rho + \frac{3P}{c^2} \right) + \frac{1}{3} \Lambda c^2 - \frac{kc^2}{a^2}[/tex]
[tex]0 = -\frac{4\pi G}{3} \rho + \frac{1}{3} \Lambda c^2 - \frac{kc^2}{a^2}[/tex]
[tex] 0 = -\rho + \frac{\Lambda c^2}{4 \pi G} - \frac{3 kc^2}{4 \pi G a^2} [/tex]
Thus ##\rho_E = \frac{\Lambda c^2}{4 \pi G}##.
How do I find the scale factor for this density?