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StuckPhysicsStudent
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Homework Statement
Show that the temperature of non-relativistic matter scales as ##a^{-2}## in the absence of interactions. Start from the zero-order part of Eq. (4.68) and assume a form ##f_{dm} \propto e^{-E/T}=e^{-p^2/2mT}##. Note that his argument does not apply to electrons and protons: as long as they are couple to the photons, their temperature scales as ##a^{-1}##.
Homework Equations
Eq. (4.68)
$$\frac {\partial f_{dm}} {\partial t}+\frac{\hat p^i}{a}\frac{p}{E}\frac {\partial f_{dm}} {\partial x^i}-\frac {\partial f_{dm}} {\partial E} \bigg[ \frac{da/dt}{a}\frac{p^2}{E}+\frac{p^2}{E}\frac {\partial \Phi} {\partial t}+\frac{\hat p^i p}{a}\frac {\partial \Psi} {\partial x^i} \bigg]=0$$
The Attempt at a Solution
So I will use overdots to represent partial derivative with respect to time and primes for derivatives with respect to ##x^i##. I'm assuming since ##e^{-E/T}=e^{-p^2/2mT}## that ##E=p^2/2m## and that p is a function of both ##x^i## and ##t##.
Since this is first order I ignore the terms with ##\Phi## and ##\Psi## because they are perturbations to the metric. So 4.68 is just
$$\frac {\partial f_{dm}} {\partial t}+\frac{\hat p^i}{a}\frac{p}{E}\frac {\partial f_{dm}} {\partial x^i}-\frac {\partial f_{dm}} {\partial E} \bigg[ \frac{da/dt}{a}\frac{p^2}{E}\bigg]=0$$
So taking the derivatives of ##f_{dm}## and just sticking with ##e^{-E/T}## for now, we have
$$\frac{\partial f_{dm}} {\partial t}=-\frac{\dot{E}}{T} e^{-E/T}$$
$$\frac{\partial f_{dm}} {\partial x^i}=-\frac{E'}{T} e^{-E/T}$$
$$\frac{\partial f_{dm}} {\partial E}=-\frac{1}{T}e^{-E/T}$$
where all the ##-\frac{1}{T}e^{-E/T}## go away and I am left with no term for temperature. So I messed up somewhere because I need a T to remain.
Thanks for the help in advance.