At what redshift does energy density of matter = energy density of radiation?

[zeion]

Hello. This is question for my course work, I was wondering if I could get some insight, here is the question:

 

Assume that the vast majority of the photons in the present Universe are cosmic microwave radiation photons that are a relic of the big bang. For simplicity, also assume that all the photons have the energy corresponding to the wavelength of the peak of a 2.73K black-body radiation curve. At Approximately what redshift will the energy density in radiation be equal to the energy density in matter?

 

(hint: work out the energy density in photons at the present time. Then work it out for baryons, assuming a proton for a typical baryon. Remember how the two quantities scale with redshift to work out when the energy density is the same.)

 

[math]

\rho_M \propto a^{-3}

[/math]

[math]

\rho_\gamma \propto a^{-4}

[/math]

[math]

T \propto a^{-1}

[/math]

[math]

1 + z = \frac{v}{v_0} = \frac{\lambda_0}{\lambda} = \frac{a(t_0)}{a(t)}

[/math]

 

How do I calculate the energy density of photons and protons at the present time? Do I use E = mc^2?

[Radical Edward]

what are your variables there? It's been a while since I worked with that sort of stuff. lambda I guess is wavelength, v, velocity, t and t_o are time, rhos are density, but what about a and z?

[zeion]

a is the scale factor for the size of the universe where the a = 1 at the present time.. and 0.5 sometime in the past when it was half the size, etc.. and z is the redshift.