At what redshift does energy density of matter equal 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]

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?

[modest]

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

 

You can get the energy density of photons from the Stefan-Boltzmann law:

 

[math]u = \frac{4}{c} \sigma T^4= \frac {4 \times 5.7 \times 10^{-8}Wm^{-2}K^{-4} \times (2.7K)^4}{3 \times 10^8m/s}=?[/math]

 

2.7 bing the temp of CMBR. You can get the matter energy density by noting that the density is 10^11 solar masses per cubic megaparsec and, as you say, use e=mc^2.

 

If you want to cheat, both are given here:

 

http://hyperphysics.phy-astr.gsu.edu/hbase/Astro/expand.html

 

Then use:

[math]\rho_m = a^{-3}[/math]

and

[math]\rho_r = a^{-3}[/math]

 

and you should get an answer that roughly agrees with equation 137 here:

http://www.astro.uu.se/~nisse/courses/kos2006/lnotes/ln6.pdf

 

~modest

 

EDIT: Sorry, that's [math]\rho_m \propto a^{-3}[/math] and [math]\rho_r \propto a^{-4}[/math]. I wrote that in a rush.

[zeion]

thx this was very helpful

[Little Bang]

If you want the energy density of a proton you could use f = MC^2/h

where M is equal to the mass of the proton.Then E = fh

[modest]

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

If you want the energy density of a proton you could use f = MC^2/h

where M is equal to the mass of the proton.Then E = fh

 

Multiply both sides of your first equation (f = MC^2/h) by h then substitute the second equation into the result. What do you get?

 

~modest

[Little Bang]

I see your point. Does that mean the f = MC^2/h is an invalid equation?

[modest]

E = MC^2

 

Correct.

 

I see your point. Does that mean the f = MC^2/h is an invalid equation?

 

Not at all, it's very useful for finding frequency. It's just that finding energy if you know mass is much easier with e=mc^2. Your method would have worked, just more complicated with more terms.

 

~modest

[Little Bang]

I agree.