(i) Given that $g(p) d p$ is the number of eigenstates of a gas particle with momentum between $p$ and $p+d p$, write down the Bose-Einstein distribution $\bar{n}(p)$ for the average number of particles with momentum between $p$ and $p+d p$, as a function of temperature $T$ and chemical potential $\mu$.

Given that $\mu=0$ and $g(p)=8 \pi \frac{V p^{2}}{h^{3}}$ for a gas of photons, obtain a formula for the energy density $\rho_{T}$ at temperature $T$ in the form

$$\rho_{T}=\int_{0}^{\infty} \epsilon_{T}(\nu) d \nu,$$

where $\epsilon_{T}(\nu)$ is a function of the photon frequency $\nu$ that you should determine. Hence show that the value $\nu_{\text {peak }}$ of $\nu$ at the maximum of $\epsilon_{T}(\nu)$ is proportional to $T$.

A thermally isolated photon gas undergoes a slow change of its volume $V$. Why is $\bar{n}(p)$ unaffected by this change? Use this fact to show that $V T^{3}$ remains constant.

(ii) According to the "Hot Big Bang" theory, the Universe evolved by expansion from an earlier state in which it was filled with a gas of electrons, protons and photons (with $n_{e}=n_{p}$ ) at thermal equilibrium at a temperature $T$ such that

$$2 m_{e} c^{2} \gg k T \gg B$$

where $m_{e}$ is the electron mass and $B$ is the binding energy of a hydrogen atom. Why must the composition have been different when $k T \gg 2 m_{e} c^{2}$ ? Why must it change as the temperature falls to $k T \ll B$ ? Why does this lead to a thermal decoupling of radiation from matter?

The baryon number of the Universe can be taken to be the number of protons, either as free particles or as hydrogen atom nuclei. Let $n_{b}$ be the baryon number density and $n_{\gamma}$ the photon number density. Why is the ratio $\eta=n_{b} / n_{\gamma}$ unchanged by the expansion of the universe? Given that $\eta \ll 1$, obtain an estimate of the temperature $T_{D}$ at which decoupling occurs, as a function of $\eta$ and $B$. How does this decoupling lead to the concept of a "surface of last scattering" and a prediction of a Cosmic Microwave Background Radiation (CMBR)?

Part II