Footnote on page 151 of Weinberg's Cosmology book

[jouvelot]

Hi all,

In this footnote, it is mentioned that Eq. 3.1.7, giving the pressure p(T) of a particle, can be derived from the law of conservation of energy (Eq. 3.1.4)

Tdp(T)/dT = ρ(T)+p(T)

and a previous definition (Eq. 3.1.6) of the energy density ρ(T) based on Fermi-Dirac or Bose-Einstein distributions (Eq. 3.1.5).

Just as a sanity check, I mentally plugged the provided definition of p(T) in the conservation equation and cannot see from the top of my head how this is going to work. Indeed, the derivative dp(T)/dT will introduce, among other things, the Boltzmann constant in the lhs of the equation, and I see no way to eliminate it, since it doesn't seem to occur in a similar manner in the equation rhs. Any hint?

Thanks in advance, and Happy New Year to all.

Bye,

Pierre

 

[George Jones]

Just as a sanity check, I mentally plugged the provided definition of p(T) in the conservation equation and cannot see from the top of my head how this is going to work. Indeed, the derivative dp(T)/dT will introduce, among other things, the Boltzmann constant in the lhs of the equation, and I see no way to eliminate it, since it doesn't seem to occur in a similar manner in the equation rhs. Any hint?

Look at equations (3.61) and (3.62) on the page that I have attached from Baumann's cosmology lecture notes. On the next page, Baumann writes "Integrating by parts, we find
$$\frac{dP}{dT}=\frac{\rho + P}{T}"$$

Do you see what happens?

 

[jouvelot]

Hello George,

I have no problem deriving Eq. 3.1.4. My issue has to do with the comment in the footnote on Page 151 that states that Eq. 3.1.7 can be derived _from_ Eq. 3.1.4 and also Eq. 3.1.6 (the derivation of which is simple too). The sheet you provided doesn't seem to help in that regard.

Thanks a lot for your help anyway :)

Bye,

Pierre