# At what redshift does energy density in matter equal energy density in radiation?

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.)

$$\rho_M \propto a^{-3}$$

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

$$T \propto a^{-1}$$

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

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

Wallace
You need to first consider how to calculate the energy density in radiation, given that you are told that it follows a black-body spectrum of a given temperature. That's the hardest part of this question. You shouldn't need to worry about the energy of a proton, if you assume a reasonable Hubble's constant value of
$$H_0$$ ~ $$72 Mpc/Km/s$$
and a matter density today of
$$\Omega_m$$ ~ $$0.3$$
that will give you the matter density today to compare with the radiation density today, then you need to scale these back as function of a the scale factor $$a(t)$$ to find the point at which they are equal. Then convert that scale factor to a redshift.

Do I use the Stefan-Boltzmann law to calculate radiation density?
How do I use Hubble's constant to solve this?
What unit is that matter density measured in?

Wallace
Do I use the Stefan-Boltzmann law to calculate radiation density?

Yes

How do I use Hubble's constant to solve this?
What unit is that matter density measured in?

Write down the definition of the matter density parameter $$\Omega_m$$. You should be able to find this in any textbook on the subject. From that definition you should see that if you specify the Hubble constant and the matter density parameter, then you will have a number for the physical matter density $$\rho_m$$ as a result (there are some physical constants in the expression as well, but they also have known values that you can plug in).