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How do I find the scale factor of cosmological constant?

  1. Apr 25, 2015 #1
    1. The problem statement, all variables and given/known data

    (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?

    a4rlts.png

    2. Relevant equations


    3. The attempt at a solution

    Part(a)

    For dust/matter: ##\rho \propto a^{-3}##. For Curvature: ##\rho \propto a^{-2}##. For Vacuum: ##\rho = const.##.
    313sopy.png

    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?
     
  2. jcsd
  3. May 25, 2015 #2
    anyone tried part (d) yet? I'm sure my parts (a)-(c) are right.
     
  4. May 31, 2015 #3
    bumpp on part (d)
     
  5. Jun 1, 2015 #4
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