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Density of a univrse filled with x-matter

  1. Apr 1, 2007 #1
    1. The problem statement, all variables and given/known data
    Consider a flat (k=0) universe filled up with X-matter, with an equation of state: [TEX]p_X/c^2=w\rho_X[/TEX], where -1<w<1. Find the expressions for the evolution of density [TEX]\rho_X(a),a(t) and \rho(t)[/TEX], where a is the evolving scale factor

    2. Relevant equations
    [TEX]\left ( \frac{da}{dt} \frac{1}{a} \right ) ^2 = \frac{8 G \pi \rho}{3}-\frac{k}{a^2} \\
    \dot{\rho}}=-3 \frac{\dot{a}}{a}(\rho + p)[/TEX]

    3. The attempt at a solution
    The big question is, am I on the right track to solve this? I started by solving [TEX]\left ( \frac{da}{dt} \frac{1}{a} \right ) ^2 = \frac{8 G \pi \rho}{3}-\frac{k}{a^2}[/TEX], which gives me: [TEX]a=Ce^{\pm \sqrt{\frac{8 G \pi \rho}{3}}t}[/TEX]
    Now I have a(t), and should be able to solve [TEX]\rho(t)[/TEX] by solving the next equation. But this one has the two a's in it, which are time dependant. This gives me a pretty tricky equation to solve. So, is this the right approach, or is there a better way? I only want hints, not a solution.

    Thanks/ Johan

    I hope the latex-script comes out right...
  2. jcsd
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