 Solution

The official notes, PDF p. 27, handwritten p. 20 derive the early-time photon count, its vanishing limit, and the failure of thermal equilibrium when the count is of order one. Two details need supplementing.

First, their constant $N_{\gamma 0}$ is the normalization obtained by extrapolating the early-time particle horizon formula to $z=0$. It is not the count within today's actual particle horizon. Let $N_0=N_{\gamma }(t_0)$ and let $r_{h0}$ be that horizon's physical radius. Using the paper's early-time approximation givesFor a present matter-dominated approximation, $r_{h0}\simeq 2c/H_0$. Alternatively, retaining both components in the flat model exactly, ${\Omega }_R=1/(2+z_{\mathrm{eq}})$, and integrating the particle horizon givesThe second boxed formula uses the exact two-component normalization and the radiation-dominated early-time limit. Its large-$z_{\mathrm{eq}}$ leading term agrees with the first expression and $r_{h0}\simeq 2c/H_0$.

Second, a particle horizon containing only order-one photons cannot support a smooth, locally representative radiation density: discrete-particle fluctuations are of the same order as the mean, and a single photon's direction is strongly anisotropic. Thus the fluid description assuming spatial homogeneity and cosmological isotropy is not self-consistent on that causal scale. This does not by itself disprove an ensemble-averaged cosmological symmetry; it limits the extrapolation of the smooth thermal equilibrium description.