Scalefactor calculations for special cases using the Friedmann Equation

karan9

(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!!

clamtrox

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

karan9

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?

clamtrox

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.

BillSaltLake

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).

karan9

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.

BillSaltLake

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] t^1/2 whereas with matter only, a [itex]\propto[/itex] t^2/3. 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 k_radiation is slightly larger than k_matter.