Questions on Weinberg Cosmology Book

[wphysics]

I am following up Weinberg Cosmology book, but I have one question.

In chapter 3.1, we have Eq (3.1.3) and (3.1.4)

s(T) = \frac{\rho(T) + p(T)}{T}
T\frac{dp(T)}{dT} = \rho(T) + p(T)

In Eq (3.1.5), we have the Fermi-Dirac or Bose-Einstein distributions.

n(p, T) = \frac{4 \pi g p^2}{(2 \pi \hbar)^3} \frac{1}{exp(\sqrt{p^2 + m^2} / k_B T) \pm 1}.

From using this number distribution, the author said we have the energy density and pressure of a particle mass m are given by Eq (3.1.6) and (3.1.7).

\rho(T) = \int n(p, T) dp \sqrt{p^2 + m^2}
p(T) = \int n(p, T) dp \frac{p^2}{3\sqrt{p^2 + m^2}}

Here, energy density is straightforward by the definition of number density.
But, for pressure, the author said it can be derived from Eq(3.1.4), the second equation on this post.

However, I cannot derive this pressure equation using Eq(3.1.4). Can somebody help me do this?

Thank you.

 

[Bill_K]

My Weinberg seems a lot different from your Weinberg! But tracing through it, here's where the p(T) equation comes from. For a particle (using c = 1), the energy is E = γm and the momentum is p = γmv. Thus p/E = v (Eq.1)

For a system of particles, the energy-momentum tensor is T^μν = ∑_n p_n^μ dx_n^ν/dt δ³(x-x_n), or using Eq.1, T^μν = ∑_n p_n^μ p_n^ν/E_n δ³(x-x_n

For a perfect fluid, the spatial components of T^μν are related to the pressure p by T^ij = p δ^ij.

Thus p = (1/3) Ʃ_j T^jj = (1/3) Ʃ_n p_n²/E_n δ³(x-x_n).

This gets you the p² in the numerator, the E in the denominator, and the 1/3 out front.