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**Boltzmann Equation--where'd the 3 come from?**

Greetings, I'm a little bit confused about the derivation for the Boltzmann equation for a particle in thermal equilibrium in the Friedman-Robertson-Walker metric. I've been following the exposition in Kolb and Turner,

*The Early Universe*p. 116. I reproduce all the relevant results here.

In particular:

We are given (K&T, eq 5.5) that for a phase space distribution [tex]f[/tex] the form of the Liouville operator in the FRW model is given by:

[tex]\hat{\mathbf{L}}[f(E,t)] = E\frac{\partial f}{\partial t}-\frac{\dot{R}}{R}|\mathbf{p}|^2\frac{\partial f}{\partial E}[/tex]

Further, the number density [tex]n[/tex] is given by an integral over momenta (K&T eq. 5.6):

[tex]n(t) = \frac{g}{(2\pi)^3}\int d^3p f(E,t)[/tex]

where [tex]g[/tex] is the number of internal degrees of freedom.

The Boltzmann equation, [tex]\hat{\mathbf{L}}[f]= \mathbf{C}[f][/tex], can then be written out by plugging in the above equation for the Liouville operator on the left hand side.

We can then divide by [tex]E[/tex], multiply by [tex]\frac{g}{(2\pi)^3}[/tex], and perform a momentum space integral to express the Boltzmann equation in terms of [tex]n[/tex].

Kolb and Turner write the result as:

[tex]\frac{dn}{dt} + 3\frac{\dot{R}}{R}n = \frac{g}{(2\pi)^3}\int\textbf{C}[f]\frac{d^3p}{E}[/tex]

I'm confused by the factor of 3 in the seccond term and am not sure how this is resolved. I'm also not sure how to treat the energy in the momentum integral--I assume that since [tex]E^2=\mathbf{p}^2+m^2[/tex], one can rewrite the momentum integral in spherical coordinates where the function [tex]f[/tex] is a function of the radial coordinate alone. I assume some integration by parts is necessary, but this still does not account for the factor of 3.

Any help would be appreciated,

Best,

Flip

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