Estimating Redshift of Photon Gas in Universe Transition

[roya]

Homework Statement

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]

Homework 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]

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

 

[mark726]

Thank you for your post. Estimating the redshift of the photon gas in the universe at the time of transition from radiation to matter domination is a challenging problem, but I will do my best to guide you through the steps.

Firstly, your approach of finding the scale factor R(t) in terms of R0 and t0 and then plugging it into the redshift equation is a good start. However, there are a few things that need to be corrected.

To begin with, in a flat universe (k=0), the sum of the density parameters \Omega_m and \Omega_\Lambda should be equal to 1, not the individual parameters themselves. This means that \Omega_m + \Omega_\Lambda = 1. From this, you can find that \Omega_m = 0.3 and \Omega_\Lambda = 0.7, which are the given cosmological parameters.

Next, the equation \rho = \rho_{matter} + \rho_{radiation} is correct, but it is not the equation you should use to find the relation between \rho and R. Instead, you should use the equations given in the problem statement: \rho_{matter} \propto R^{-3} and \rho_{radiation} \propto R^{-4}. This means that \rho_{matter} = c_1 R^{-3} and \rho_{radiation} = c_2 R^{-4}, where c_1 and c_2 are constants. You can find these constants by using the fact that at the time of transition, \rho_{matter} = \rho_{radiation}. This will give you the relation between \rho and R.

Now, you can use the Friedmann equation to find the relation between H and \rho. This will give you a differential equation for R, which you can solve to find R(t). Once you have R(t), you can use the redshift equation 1+z_{eq}=\frac{R_0}{R(t)} to find the redshift at the time of transition.

I hope this helps clarify your confusion. If you have any further questions, please do not hesitate to ask.