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Homework Help: Friedmann's 1st equation and density parameters

  1. Jan 9, 2014 #1
    1. The problem statement, all variables and given/known data
    I'm trying to work out how the expression:
    $$H^2 = H_0^2 \left [ \Omega_0 \frac{a}{a_0} + 1 - \Omega_0\right]$$
    can be deduced from Friedmann's first equation:
    $$H^2 = \frac{8\pi G \rho}{3} - \frac{kc^2}{R^2}$$
    And I have a number of questions.

    Firstly, I've often seen the 1st Friedmann equation written with a ##\frac{\Lambda c^2}{3}## term in it, but my textbook gives it as above. I guess they are equivalent, but I can't see how immediately. I'd like to know how you get the first expression above from Friedmann's equation so that I can work out whether it is valid for a spatially flat universe (k=0).
    I also note that there is no ##\Omega(t)## term, so the ##a## in the square brackets provides the time-dependent element. Is it possible to write the ##\Omega_0\frac{a}{a_0}## in terms of ##\Omega## instead?

    Thanks in advance!
    Last edited: Jan 9, 2014
  2. jcsd
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