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Density parameter against scale factor for perfect fluids

  1. Sep 5, 2016 #1
    1. The problem statement, all variables and given/known data

    This is a basic cosmology problem.

    The Friedmann equations are

    ##\Big( \frac{\dot{a}}{a}\Big)^{2}+\frac{k}{a^{2}}=\frac{8\pi}{3m_{Pl}^{2}}\rho## and ##\Big( \frac{\ddot{a}}{a} \Big) = - \frac{4\pi}{3m_{Pl}^{2}}(\rho + 3p)##.

    Using the density parameter ##\Omega \equiv \frac{\rho}{\rho_{c}}=\frac{8\pi}{3m_{Pl}^{2}}\frac{\rho}{H^{2}}##, we can write the density parameter as ##\Omega = 1 + \frac{k}{(aH)^{2}}##.

    Furthermore, for perfect fluids, ##p=\omega\rho## so that the continuity equation ##\dot{\rho}+3\Big(\frac{\dot{a}}{a}\Big)(\rho + p)=0## for perfect fluids leads to ##\rho \propto a^{-3(1+w)}##.

    (a) Show that ##\frac{d\Omega}{d\text{ln}a}=(1+3w)\Omega(\Omega -1)##.

    (b) For matter and radiation, ##1+3w>0##. Show that this implies that ##\frac{d|\Omega -1|}{d\text{ln}a}>0##. What does this mean for a flat universe?

    2. Relevant equations

    3. The attempt at a solution

    I have to substitute ##\rho \propto a^{-3(1+w)}## into the Freidmann equation ##\Big( \frac{\dot{a}}{a}\Big)^{2}+\frac{k}{a^{2}}=\frac{8\pi}{3m_{Pl}^{2}}\rho## and find an expression for H in terms of a and k.

    Is this the correct approach?
     
  2. jcsd
  3. Sep 11, 2016 #2
    Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.
     
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