What Redshift Equates Cosmic Microwave Radiation and Matter Energy Densities?

[zeion]

Homework Statement

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.)

Homework Equations

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



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


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

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

The Attempt at a Solution

Not sure where to start.. how do I work out the energy density for photons and protons at the present time? Do I use E = mc^2?

 

[mark726]

Thank you for your question. It is an interesting one that requires some understanding of cosmology and the properties of radiation and matter. Let me guide you through the steps to find the approximate redshift at which the energy density in radiation and matter are equal.

Firstly, let's define some important quantities:

- Energy density of radiation (\rho_\gamma): This is the amount of energy per unit volume of space occupied by radiation.
- Energy density of matter (\rho_M): This is the amount of energy per unit volume of space occupied by matter.
- Redshift (z): This is a measure of the expansion of the universe, where higher values of z correspond to earlier times in the universe.
- Scale factor (a): This is a measure of the size of the universe, where larger values of a correspond to larger sizes of the universe.

Now, let's work out the energy density of photons at the present time. We can use the Stefan-Boltzmann law, which states that the energy density of black-body radiation is given by:

\rho_\gamma = \frac{\pi^2}{15} \frac{k_B^4}{\hbar^3 c^3} T^4

Where k_B is the Boltzmann constant, \hbar is the reduced Planck constant, c is the speed of light, and T is the temperature of the radiation. Plugging in the values for these constants and the temperature of 2.73K, we get:

\rho_\gamma = 4.17 \times 10^{-14} \text{J/m}^3

Next, let's work out the energy density of matter. We can use the equation you provided, \rho_M \propto a^{-3}, to relate the energy density of matter to the scale factor. However, we need to know the energy density of matter at a specific time or redshift. For simplicity, let's assume that the energy density of matter at the present time is equal to the energy density of baryonic matter (protons and neutrons). We can use the equation E=mc^2 to relate the energy density to the mass density of baryonic matter. Assuming a proton for a typical baryon, we get:

\rho_M = m_p n_b

Where m_p is the mass of a proton and n_b is the number density of baryons. We can also use the