In several papers about Recombination, (for instance Peebles, Astrophysical Journal, vol. 153, eq. 39) it is claimed that the evolution of matter temperature follows the equation \begin{equation} \frac{dT}{dt} = -2\frac{\dot{a}}{a}T + \frac{2}{3 K_B n} \epsilon \end{equation} where the first term in the right hand side accounts for the expansion of the universe and $\epsilon$ is the energy injected into the gas per second per unit physical volume. How can I derive this equation? I suppose that this equation can be derived applying the First Principle of Thermodynamics to a comoving volume in the expanding universe or maybe taking the second moment of a Boltzmann equation. Any help or reference in which this equation is discussed would be very helpful.
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$\begingroup$ Are you talking about this paper? If so, the equation you reference in your question is not (as far as I can tell) the same equation as in the paper. What's more, that paper pretty much derives the equation throughout. $\endgroup$– zephyrCommented Sep 21, 2018 at 14:47
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$\begingroup$ The difference between the equation I wrote and the one in the paper is that in the latter what I called $\epsilon$ is explicitly calculated for Compton Scattering. Anyway I can't see where this equation is derived. $\endgroup$– JohnNNNYCommented Sep 21, 2018 at 19:28
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$\begingroup$ I've voted for this question to be reopened - see meta discussion $\endgroup$– Chappo Hasn't ForgottenCommented Sep 28, 2018 at 22:17
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$\begingroup$ I found the answer anyway. The equation can be derived using the first principle of thermodynamics to a comoving volume and the equation of state for a perfect gas. $\endgroup$– JohnNNNYCommented Sep 29, 2018 at 7:32
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