How does a particle's kinetic energy vary with a(t)?

[George Jones]

http://arxiv.org/abs/0808.1552
Note on the thermal history of decoupled massive particles
Hongbao Zhang
(Submitted on 11 Aug 2008)
"This note provides an alternative approach to the momentum decay and thermal evolution of decoupled massive particles. Although the ingredients in our results have been addressed in [Weinberg's new Cosmology text], the strategies employed here are simpler, and the results obtained here are more general."

I hope someone can help me understand this. I read trhe Zhang article recommended by marcus.

This seems to say that the temperature of a collection of particles not in equilibrium varies inversly with a. Near the end of the article, the following is said:

... although the spectrum has still kept the form of the Fermi-Dirac and Bose-Einstein distributions
since decoupling, it is not the thermal spectrum with the effective temperature and
chemical potential since the effective mass is not equal to the static mass.

What I think all this comes to, although it is not said explicitly (and I am not at all clear in my mind that I have it right) is the Fermi-Dirac and Bose-Einstein distributions form are preserved assuming an artificial temperature, while the real, Maxwell-Boltzmann distibution of the energy (in terms of the velocity-squared of the collection of particles) would correspond to a temperature that varied inversely with a².

That's not quite correct. The correct statement is that temperature varies inversely with $a$ for relativistic particles (which includes radiation), and inversely as $a^2$ for non-relativistic particles (which includes all the ordinary matter and dark matter in our present universe). At the time of neutrino decoupling, the neutrinos are still highly relativistic, so their temperature will still vary as $1 / a$. Whether or not the neutrinos become non-relativistic at some point after decoupling depends on the neutrino masses; for small enough neutrino masses, they could still be relativistic even today.

There seems to be a misunderstanding here. The temperature scaling of a decoupled species is determined by properties at the time of decoupling. If a species was non-relativistic at the time of decoupling, then, after decoupling, its temperature is inversely proportional to a^2. If a species was (ultra)relativistic at the time of decoupling, then, after decoupling, its temperature is inversely proportional to a. The latter is true even when a species that was relativistic at the time of decoupling is cooled by expansion to non-relativistic speeds.

I think this is what Chalnoth means by

Also, let me add one other thing. This statement isn't true:

Temperature varies inversely as $1/a(t)$ as long as the neutrinos remain relativistic. They don't need to interact with anything to maintain this temperature scaling because they'll retain their thermal distribution without interactions.

I spent a pleasant morning looking at the quantitative details of Zhang's paper, a couple of whose points seem to be somewhat well-known before publication of the paper. Right now, I have to pick up my daughter from school, and take her to a piano lesson and soccer practice. I will try to make an effort tomorrow to fill in the quantitative details of how Zhang's paper relates to the above.

 

[PeterDonis]

The latter is true even when a species that was relativistic at the time of decoupling is cooled by expansion to non-relativistic speeds.

Hm, I wasn't aware of this. Is there a quick heuristic explanation of why it's true? I'm not getting anything from the Zhang paper that is helpful in that regard.

 

[Chalnoth]

There seems to be a misunderstanding here. The temperature scaling of a decoupled species is determined by properties at the time of decoupling. If a species was non-relativistic at the time of decoupling, then, after decoupling, its temperature is inversely proportional to a^2. If a species was (ultra)relativistic at the time of decoupling, then, after decoupling, its temperature is inversely proportional to a. The latter is true even when a species that was relativistic at the time of decoupling is cooled by expansion to non-relativistic speeds.

I think this is what Chalnoth means by

Also, let me add one other thing. This statement isn't true:

Temperature varies inversely as 1/a(t) as long as the neutrinos remain relativistic. They don't need to interact with anything to maintain this temperature scaling because they'll retain their thermal distribution without interactions.

No, that's not what I meant. What I was getting at is that the thermal distribution that they had at the time of decoupling remains thermal from there on. That is, the expansion reduces their temperature but doesn't make them non-thermal. This is certainly a true statement as long as the neutrinos remain relativistic. I am not quite certain it's true when the neutrinos transition from relativistic to non-relativistic.

The calculations I was trying to do later were using the assumption that you could deal with all particles of a given energy independently, and track their behavior forward in time.

 

[Mordred]

There is a decent coverage on entropy conservation and neutrinos dropping out of thermal equilibrium in this article.

http://www.google.ca/url?sa=t&sourc...geOUuN9MWcq5b0Raw&sig2=U50khye_5kdEY9U2yXd-mg