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Homework Help: Cosmology: estimating the redshift of the photon gas at matter-radiation equality

  1. Sep 11, 2010 #1
    1. The problem statement, all variables and given/known data
    The problem is translated from a different language, so I hope I am not missing anything.
    I need to estimate the redshift of the photon gas in the universe, at the time of transition from radiation to matter domination.
    Cosmological parameters:
    [tex] k=0 [/tex] (meaning a flat universe)
    [tex] \Omega_\Lambda=0.7 [/tex]
    [tex] H_0=71\frac{km}{s\cdot{Mpc}} [/tex]
    also it is mentioned that [tex]\rho_{matter} \propto R^{-3} ~,~ \rho_{radiation} \propto R^{-4}[/tex]

    2. Relevant equations
    Friedmann equation:
    [tex]H^2=\left(\frac{\dot{R}}{R}\right)^2 = \frac{8}{3}\pi G\rho - \frac{kc^2}{R^2} + \frac{\Lambda}{3}[/tex]
    redshift in terms of universe scale factor
    [tex]1+z_{eq}=\frac{R_0}{R(t)}[/tex]
    and also the definition of the omegas:
    [tex]\Omega_m=\frac{\rho}{\frac{8}{3} \pi GH^2}[/tex]
    [tex]\Omega_\Lambda=\frac{\Lambda}{3H^2}[/tex]


    3. The attempt at a solution
    I am not completely sure that this is the right approach, but it's the only thing that comes to mind. Basically what I am trying to do is find the scale factor R(t) in terms of R0 and t0, and just plug it into the redshift equation. In a flat universe (k=0) [tex]\Omega_m + \Omega_\Lambda = 1 [/tex] (which is derived easily using the friedmann equation), so [tex]\Omega_m=0.3[/tex]

    This gives me [tex]H=\frac{\dot{R}}{R}[/tex] in terms of [tex]\rho[/tex].
    but this is where i get confused. If I knew the proportionality relation between [tex]\rho (t)[/tex] and [tex]R(t)[/tex] then I could have solved the differential equation and find R..
    [tex]\rho=\rho_{matter} + \rho_{radiation}[/tex], so if either matter or radiation density were dominant, I would use the dominant parameter, but the solution is for a time where they are equal... and if i try to equate them and find the total density that way, I get very awkward results... I am pretty sure that this is wrong and that I am lacking some basic understanding... I really hope someone can help me out here.


    One more thing that really confuses me, is if I try to find R(t) using [tex]\Omega_\Lambda[/tex].
    This gives me a differential equation for R, which I can solve
    [tex]\frac{\dot{R}}{R}=\sqrt{\frac{\Lambda}{3\Omega_\Lambda}}[/tex]
    [tex]R(t)=R_0 e^{\sqrt{\frac{\Lambda}{3\Omega_\Lambda}}(t-t_0)}[/tex]

    I guess t0 I can find using the hubble constant [tex]t_0=H_0^{-1}[/tex]
    but this doesn't make sense.. why is the proportional relation between the densities and the scale factor is given if not used... and of course matter-radiation equality is not taken into consideration ..
    Very confused :uhh:
     
    Last edited: Sep 11, 2010
  2. jcsd
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