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General Relativity - FRW Metric

  1. Apr 16, 2015 #1
    1. The problem statement, all variables and given/known data
    (a) Find the FRW metric, equations and density parameter. Express the density parameter in terms of a and H.
    (b) Express density parameter as a function of a where density dominates and find values of w.
    (c) If curvature is negligible, what values must w be to prevent a singularity? Find a.
    (d) Find an expression for the deceleration parameter and redshift.

    2. Relevant equations

    3. The attempt at a solution

    The metric is given by
    [tex] ds^2 = c^2 dt^2 - a(t)^2 \left[ \frac{dr^2}{1-kr^2} + r^2(d\theta^2 + sin^2 \theta d\phi^2) \right] [/tex]
    The FRW equations are
    [tex] \left( \frac{\dot a}{a} \right)^2 = \frac{8 \pi G \rho_I}{3} + \frac{1}{3} \Lambda c^2 - \frac{kc^2}{a^2 (t)}[/tex]
    [tex] \ddot a(t) = -\frac{4\pi G}{3} \left(\rho_I + \frac{3P}{c^2} \right) a(t) + \frac{1}{3} \Lambda c^2 a(t) [/tex]
    Density parameter is given by
    [tex]\Omega = \frac{8\pi G}{3H^2}\left( \rho_I + \frac{\Lambda c^2}{8 \pi G} - \frac{3 kc^2}{8 \pi G} \right)[/tex]

    How do I express it in terms of ##a## and ##H## only? I know that ##\rho_I \propto a^{-3(1+w_I)} = \rho_I(0) a^{-3(1+w_I)} ##

    This is as far as I can go. Would appreciate any input, many thanks!
  2. jcsd
  3. Apr 17, 2015 #2
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  10. Apr 25, 2015 #9
    Managed some progress with part (a)! I think they are looking for ##H^2 = \frac{8 \pi G}{3}\rho = H_0a^{-3(1+w_I)}##.

    For part (b), I think they want ##\Omega = \frac{8 \pi G}{3 H_0^2} \rho_I a^{-3(1+w_I)}##.

    I suppose the confusion was that I thought ##\rho_I = \rho_{I,0} a^{-3(1+w_I)}## when in fact they use ##\rho_I## as the density of today.
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