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Homework Help: Cosmology Question that I really don't have a clue on.

  1. Jan 5, 2009 #1
    1. The problem statement, all variables and given/known data
    a) Using the Robertson-Walker metric, write down an expression for the proper distance in terms of the coordinate r. For objects moving with the expansion of the universe, show that the proper distance increases with time in the manner described by the Hubble Law.

    b) Use the Friedmann equations for a matter-dominated universe Universe with [itex]\Lambda = 0[/itex] to show that:
    [tex] H(z) = H_0 (1+z)(1 + \Omega_0 z)^{1/2}[/itex]

    c) Define the angular diameter distance [itex] d_A [/itex] of an object and relate this to the coordinate r appearing in the Robertson-Walker metric. Show that in a matter dominated universe with [itex]\Omega_0 = 1 [/itex],
    [tex] d_A = \frac{2c}{H}\lbrace(1+z)^{-1} - (1+z)^{-3/2}\rbrace [/tex]

    2. Relevant equations
    [tex] ds^2 = c^2 dt^2 - a^2(t)\lbrace \frac{dr^2}{1-kr^2} \rbrace + r^2(d\theta^2 + sin^2\theta d\phi) [/tex]

    [tex] \lbrace\frac{da}{dt}\rbrace^2 + kc^2 = \frac{8 \pi G \rho^2}{3} + \frac{\Lambda c^2 a^2}{3} [/tex]

    3. The attempt at a solution
    a) If ds=0, then let [itex] l_p = cdt = adr[/itex] and so by considering
    [tex] \frac{dr}{dt} = Hr [/tex]
    [tex] H = \frac{ \dot{a}}{a} [/tex]
    it means that:
    [tex] \dot{l_p} = \dot{a}r [/tex]

    I don't even have the slightest idea where to start for parts b and c. Any help on those would be greatly appreciated.
  2. jcsd
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