Why is scale factor inversely proprotional to temperature?
There is a nice clear discussion of this in the paper Cooling in the Universe by Sohrab Rahvar. We are not supposed to give link only answers, but I'm not sure what is to be gained by just duplicating the working in Rahvar's paper. Basically you define temperature by the associated random thermal velocities, then write down the geodesic equation for these in an FLRW spacetime. This is realtively straightforward as only a few of the Christoffel symbols are nonzero, and the result falls out that the velocities scale as $a^{1}$. That means the temperature scales as $a^{1}$.
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Cheeku
Updated on August 01, 2022Comments

Cheeku over 1 year
In all lectures, books and papers I have read about, the scale factor of the universe is inversely proportional to temperature
$$a \propto \frac{1}{T}$$
What is the reasoning behind this relation?
People did mention stretching of wavelength proportional to scale factor and wavelength being inversely proportional to temperature by Wein's law, but somehow this argument doesn't seem convincing to me. How did we initially come about to this relation?

Cheeku over 9 yearsIt is time till I touch relativity quantitatively, but at least I now know that a better proof exists, and the fact for nonrelativistic particles. Thanks.