Calculate Redshift Where Radiation and Matter Energy Densities Equal

AI Thread Summary
To determine the redshift at which the energy densities of radiation and matter are equal, one must calculate the current energy density of cosmic microwave background radiation using the Stefan-Boltzmann law, considering its black-body spectrum at 2.73K. The energy density of matter can be derived from the matter density parameter, using the Hubble constant to find physical matter density. Both densities scale differently with redshift, with radiation scaling as a^-4 and matter as a^-3. By equating these densities and converting the scale factor to redshift, the desired value can be found. This approach combines principles of cosmology and thermodynamics to solve the problem effectively.
zeion
Messages
455
Reaction score
1
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.)

<br /> \rho_M \propto a^{-3}<br />

<br /> \rho_\gamma \propto a^{-4}<br />

<br /> <br /> T \propto a^{-1}<br /> <br />

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

How can I calculate the energy density of photons and protons at the present time? Do I use E = mc^2?
 
Space news on Phys.org
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).
 
I'm confused about this same question, can anyone else clarify please?

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.
 
https://en.wikipedia.org/wiki/Recombination_(cosmology) Was a matter density right after the decoupling low enough to consider the vacuum as the actual vacuum, and not the medium through which the light propagates with the speed lower than ##({\epsilon_0\mu_0})^{-1/2}##? I'm asking this in context of the calculation of the observable universe radius, where the time integral of the inverse of the scale factor is multiplied by the constant speed of light ##c##.
Why was the Hubble constant assumed to be decreasing and slowing down (decelerating) the expansion rate of the Universe, while at the same time Dark Energy is presumably accelerating the expansion? And to thicken the plot. recent news from NASA indicates that the Hubble constant is now increasing. Can you clarify this enigma? Also., if the Hubble constant eventually decreases, why is there a lower limit to its value?

Similar threads

Back
Top