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Integrating Friedmann Equation of Multi-component universe respect to a and t

  1. Dec 28, 2012 #1
    I am having a trouble finding relationship between 'a' and 't' by integrating friedmann equation in a multi-component universe.

    It would be very helpful if you can help me with just
    matter-curvature only universe and matter-lambda only universe.

    The two integrals looks like following.

    Matter-curvature only:

    [tex]H_0 t = ∫_0^a \frac{da}{[Ω_0/a + (1-Ω_0)]^{1/2}} [/tex]

    Matter-Lambda only:

    [tex]H_0 t = ∫_0^a \frac{da}{[Ω_0/a + (1-Ω_0)a^2]^{1/2}} [/tex]

    Thank you for your help.
     
  2. jcsd
  3. Dec 28, 2012 #2

    BillSaltLake

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    Gold Member

    Try substituting x = 1/a and then use a table of integrals.
     
  4. Dec 31, 2012 #3
    with matter lambda the result is
    a(t)=(ro_matter/ro_lambda)^(1/3)*[sinh[(6*Pi*ro_lambda*G)^(1/2)*t]^(2/3)
    Where ro_x/ro_critical=omega_0x
     
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