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

1. Sep 11, 2010

### roya

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:
$$k=0$$ (meaning a flat universe)
$$\Omega_\Lambda=0.7$$
$$H_0=71\frac{km}{s\cdot{Mpc}}$$
also it is mentioned that $$\rho_{matter} \propto R^{-3} ~,~ \rho_{radiation} \propto R^{-4}$$

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

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) $$\Omega_m + \Omega_\Lambda = 1$$ (which is derived easily using the friedmann equation), so $$\Omega_m=0.3$$

This gives me $$H=\frac{\dot{R}}{R}$$ in terms of $$\rho$$.
but this is where i get confused. If I knew the proportionality relation between $$\rho (t)$$ and $$R(t)$$ then I could have solved the differential equation and find R..
$$\rho=\rho_{matter} + \rho_{radiation}$$, 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 $$\Omega_\Lambda$$.
This gives me a differential equation for R, which I can solve
$$\frac{\dot{R}}{R}=\sqrt{\frac{\Lambda}{3\Omega_\Lambda}}$$
$$R(t)=R_0 e^{\sqrt{\frac{\Lambda}{3\Omega_\Lambda}}(t-t_0)}$$

I guess t0 I can find using the hubble constant $$t_0=H_0^{-1}$$
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