1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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]

    Problem
    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.

    Problem
    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
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?