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Rewriting the Friedmann Equation

  1. Dec 7, 2013 #1
    1. The problem statement, all variables and given/known data

    Consider a universe with Cold Matter, Radiation, and Dark Energy satisfying an equation of
    state [itex]p = w\rho.[/itex]

    Variables: p is the pressure of the universe, w is a constant, and [itex]\rho[/itex] is the density of the universe.

    a) Show that the Friedmann equation can be rewritten as
    [itex] H^2(t) = H_0^2(\Omega_{DE}(\frac{a_0}{a(t)})^{3(1+w)} + \Omega_K(\frac{a_0}{a(t)})^2 + \Omega_M(\frac{a_0}{a(t)})^3 + \Omega_R(\frac{a_0}{a(t)})^4)[/itex]

    Variables: [itex]H(t)= \frac{\dot{a}}{a}[/itex] is the Hubble constant, [itex]H_0[/itex] is today's value of H, a(t) and [itex]a_0[/itex] are the scale factor at time t and today, respectively, and the Omegas are fractions of the critical density for Dark Energy (DE), Curvature (K), Matter(M), and Radiation (R).

    b) For the case [itex]w < \frac{-1}{3},[/itex] which component of energy density is likely to dominate the early universe? How about the late universe?

    2. Relevant equations

    (1) [itex]\dot{a}^2 + K = \frac{8 \pi G \rho a^2}{3}[/itex]
    (2) [itex]\dot{\rho} = \frac{-3\dot{a}}{a}(p+\rho)[/itex]
    (3) [itex]\Omega_K = \frac{-K}{a_0^2 H_0^2}[/itex]

    3. The attempt at a solution

    For Part a),

    Putting in the given equation for pressure, (2) can be solved to obtain

    (4) [itex]\rho = \rho_0 \frac{a_0}{a}^{3(1+w)}[/itex]

    where [itex] \rho_0[/itex] is the current density, which itself is just the sum of the densities of dark energy, matter, and radiation at today's time.

    (5) [itex]\rho_0 = \rho_{DE0} + \rho_{M0} + \rho_{R0}[/itex]

    Plugging (5) into (4), (4) into (1), and dividing (1) by [itex]a^2[/itex] yields

    (6) [itex]H^2 = \frac{8 \pi G}{3} (\rho_{DE0} + \rho_{M0} + \rho_{R0} ) (\frac{a_0}{a})^{3(1+w)} - \frac{K}{a^2}[/itex]

    The current densities are related to their fraction of the current critical densities by

    (7) [itex]\rho_{i0} = \frac{3H_0^2 \Omega_i}{8 \pi G}[/itex]
    where i is Dark Energy, Matter, or Radiation

    Putting (7) into each of the densities in (6), and pulling out a factor of [itex]\frac{3H_0^2}{8 \pi G}[/itex] gives

    (8)[itex]H^2 = H_0^2 (\Omega_{DE} + \Omega_{M} + \Omega_{R} ) (\frac{a_0}{a})^{3(1+w)} - \frac{K}{a^2}[/itex]

    which, solving (3) for K and putting into (8) gives an answer very close to the solution

    (9) [itex]H^2 = H_0^2 (\Omega_{DE} + \Omega_{M} + \Omega_{R} ) (\frac{a_0}{a})^{3(1+w)} - \Omega_K (\frac{a_0}{a})^2[/itex]

    This would get me the desired result if I could choose w as [itex]\frac{1}{3}[/itex] for the radiation term, and 0 for the matter term, which are the values of w that solve radiation and matter dominated universes, respectively, but I don't see how I would be allowed to just simply say "on this term, w is 0, and on this one, it's [itex]\frac{1}{3}[/itex], and for this one, I'll leave it as w".

    My Solution to Part b)

    When [itex]w < \frac{-1}{3},[/itex], using the solution in part a), in the early universe, when [itex]\frac{a_0}{a}[/itex] is large, the Radiation term (largest exponent) will dominate. When [itex]\frac{a_0}{a}[/itex] is approximately 1, the Dark Energy term (which has the largest value of the density fractions), should dominate.

    Explicitly, I haven't been given values of the Omegas, which means I've just tailored an answer to what we know about our universe: radiation dominated while young, dark energy dominated now.
  2. jcsd
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