- #1

StuckPhysicsStudent

- 4

- 0

## 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.