S21: Equilibrium Particle Abundances

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Exercise 21.1.1

Answer

We start from

\[n_\chi = \frac{g}{h^3}\int d^3p \left[\exp\left(\frac{E(p)}{k_B T}\right) +1 \right]^{-1}\]

and \(E(p) = pc\).

Because the integrand only depends on the magnitude of the momentum, \(p = \sqrt{p_x^2+p_y^2+p_z^2}\), we are motivated to transform from the Cartesian \(p_x,p_y,p_z\) to spherical coordinates \(p,\theta_p,\phi_p\), just like we have done before in configuration space (\(x,y,z \rightarrow r,\theta,\phi\)). The area of a sphere in momentum space of radius p is \( 4\pi p^2 \) and the volume contained in a spherical shell that extends from \(p\) to \(p+dp\) is \(4\pi p^2 dp\). So we can replace \(\int d^3p\) with \(\int 4\pi p^2 dp\). So we have

\[n_\chi = \frac{4\pi g}{h^3}\int p^2 dp \left[\exp\left(\frac{pc}{k_B T}\right) +1 \right]^{-1}.\]

Setting \(x = pc/(k_B T)\) so \(p=xk_BT/c\) and \(dp = dx k_BT/c\) we get

\[n_\chi = \frac{4\pi g}{h^3}\left(\frac{k_B T}{c}\right)^3\int x^2 dx \left[e^x +1 \right]^{-1}.\]

Exercise 21.2.1

Answer

The parameter \(x = mc^2/(k_B T)\) determines whether the particles are relativistic or non-relativistic. The typical kinetic energy is given roughly by \(k_B T\) and the rest mass energy is \(mc^2\). For \(x<<1\) kinetic energy is much greater than rest mass energy, which means the particles are relativistic. For \(x>>1\) we have the reverse so the particles are non-relativistic.

In the relativistic limit we saw above that \(n_\chi \propto T^3\). Since \(a \propto 1/T\) we have \(a^3 n_\chi \propto T^{-3}T^3 = T^0\) is indepdent of \(T\) and therefore independent of \(x\). Therefore the graph of \(a^3 n_\chi\) vs. \(x\) at low \(x\) is a horizontal line.

In the non-relativistic limit we have

\[a^3 n_\chi \propto e^{-x}x^{3/2}. \]

An exponential always eventually wins over a power-law so at large enough \(x\), \(a^3 n_\chi\) will be a decreasing function of \(x\).

If we smoothly connect a decreasing function of \(x\) at high \(x\) with the horizontal line at low \(x\) we'll get something qualitatively similar to the graph shown in the next chapter.