# Calculate Redshift Where Radiation and Matter Energy Densities Equal

• zeion
In summary: First, you need 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 ofH_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
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.)

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

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

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

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

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

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?

zeion said:
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).

For energy density of radiation, how would i use the stefan boltzman law?

Write down the Stefan Boltzmann law. Think about the terms in the equation. Which one do you need to calculate, and which ones are you already given?

Note that because this is a homework question, I'm following the guidelines for answering homework from the Homework Help forum, rather than just stating the answer.

## 1. How do you calculate redshift where radiation and matter energy densities equal?

The redshift where radiation and matter energy densities equal can be calculated using the equation z = (Ωmr)1/3 - 1, where z is the redshift and Ωm and Ωr are the matter and radiation energy densities, respectively.

## 2. What is the significance of calculating redshift where radiation and matter energy densities equal?

Calculating the redshift where radiation and matter energy densities equal can provide insight into the evolution of the universe and the relative contributions of radiation and matter to the total energy density. It can also help in determining the age of the universe and the expansion rate of the universe.

## 3. How is the redshift where radiation and matter energy densities equal related to the cosmic microwave background (CMB) radiation?

The redshift where radiation and matter energy densities equal, also known as the matter-radiation equality, is closely related to the CMB radiation. The CMB is a remnant of the radiation from the early universe, and the redshift of the matter-radiation equality marks the time when the universe transitioned from being dominated by radiation to being dominated by matter.

## 4. Are there any other factors that need to be considered when calculating the redshift where radiation and matter energy densities equal?

Yes, there are other factors that can affect the calculation of the redshift where radiation and matter energy densities equal. These include the presence of dark energy, the curvature of the universe, and the presence of other types of matter such as dark matter.

## 5. How is the redshift where radiation and matter energy densities equal related to the expansion of the universe?

The redshift where radiation and matter energy densities equal is related to the expansion of the universe through the Hubble parameter, which describes the rate of expansion of the universe. The redshift of the matter-radiation equality can provide valuable information about the expansion rate of the universe and how it has changed over time.

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