Scalefactor calculations for special cases using the Friedmann Equation

In summary, to obtain the scale factor and redshift when matter and radiation energy densities were equal, use the conservation of energy-momentum and the fact that radiation energy-momentum tensor is traceless. Equate the expressions for dust and radiation energy densities and integrate Friedmann equations to find the value of scale factor at matter-radiation equality. When using only radiation, a \propto t1/2 while with matter only, a \propto t2/3. Remember to consider the average energy per blackbody photon and assume zero curvature and critical density. The approximation using only matter is closer to the correct answer.
  • #1
karan9
3
0
(i) Obtain the scale factor a(t) and redshift z when the energy density of matter and radiation were equal.
(ii) Next use the a(t) relation for a matter-only universe to estimate the time of matter-radiation equality.
(iii) Repeat (ii) but using the a(t) relation for a radiation-only universe. Which approximation, (ii) or (iii), is closer to the correct answer?

Can someone help me?

Thanks!
 
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  • #2
How far have you gotten? You should start by figuring out how dust and radiation energy densities depend on the scale factor. You can find this out using the conservation of energy-momentum, and the fact that radiation energy-momentum tensor is traceless.

After that you just equate the expressions you got, and integrate Friedmann equations
 
  • #3
Ive understood the concept of how I'm supposed to do it. But, I am confused on how to start the radiation equation. As in what formula do I use to find the scale factor for a radiation dominated universe?
 
  • #4
karan9 said:
Ive understood the concept of how I'm supposed to do it. But, I am confused on how to start the radiation equation. As in what formula do I use to find the scale factor for a radiation dominated universe?

The way I understand the question is that you should use the scaling laws to find the value of scale factor at matter-radiation equality as a function of their densities today.
 
  • #5
Just remember that the average energy/photon is proportional to 1/a. I'm guessing that you should assume zero curvature, that the density is always at critical, and that the kinetic energy of the matter can be neglected. You'll need to assume some value for the amount of matter per photon (about 10-35kg/photon).
 
  • #6
Oh that clears up part i for me. Thanks guys! My answer for part i comes to a=2.8*10^-4
Is that right?

I am confused about the method we have been asked to use for part ii.
Do i just plug in the value of a into the (da)^2 = Ho^2/a^2 equation and solve for 't' ?
Am I missing something?

Thanks Again.
 
  • #7
Also remember that the average energy per blackbody photon is ~2.7kT, which is a little higher than you might guess.
Just to give a few clues, with radiation only, you'll find that a [itex]\propto[/itex] t1/2 whereas with matter only, a [itex]\propto[/itex] t2/3. Then [itex]\rho[/itex] [itex]\propto[/itex] k/tn where n is a certain positive integer that I won't disclose (same integer for radiation and matter), and k is a constant, although kradiation is slightly larger than kmatter.
 

1. What is the Friedmann Equation and how is it used in scalefactor calculations for special cases?

The Friedmann Equation is a fundamental equation in cosmology that describes the evolution of the universe. It is used in scalefactor calculations for special cases, such as when considering the expansion of the universe or the matter content of the universe.

2. How does the Friedmann Equation account for the expansion of the universe?

The Friedmann Equation includes a term for the expansion of the universe, known as the scalefactor. This term accounts for the change in size of the universe over time and is dependent on the matter and energy content of the universe.

3. Can the Friedmann Equation be used to calculate the scalefactor for all cases?

No, the Friedmann Equation is only applicable for special cases such as homogeneous and isotropic universes. It does not account for more complex scenarios such as the presence of dark energy or the effects of gravitational waves.

4. How does the Friedmann Equation relate to the Hubble Constant?

The Hubble Constant is the proportionality constant between the recession velocity of distant galaxies and their distance. The Friedmann Equation includes this constant and uses it to calculate the expansion rate of the universe.

5. Are there any limitations to using the Friedmann Equation for scalefactor calculations?

Yes, the Friedmann Equation is based on certain assumptions about the universe and may not accurately describe all scenarios. It also does not take into account the effects of dark matter and dark energy, which are important components of the universe's energy density.

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