Photon-to-baryon ratio ($s_{\gamma }=n_{\gamma }/n_B=1/\eta$)

The reciprocal of the baryon-to-photon ratio, not a synonym for it. In a thermal photon gas this number ratio is proportional to, but differs from, the thermodynamic entropy per baryon. The exact photon entropy per photon is $4u_{\gamma }/(3k_{B}T{n}_{\gamma })\simeq 3.60$, so the photon-sector entropy per baryon in units of $k_B$ is about $3.60s_{\gamma }$. The entropy sector must be specified when other relativistic species contribute.

With $n_{\gamma }=0.24Q^3$, $u_{\gamma }=0.66\hslash c{Q}^4$, $Q=k_{B}T/(\hslash c)$ and $n_B={\Omega }_B{\rho }_c/m_p$, eliminate $Q$ using $u_{\gamma }={\Omega }_{\gamma }{\rho }_c{c}^2$. The prefactor is $0.24/0.66^{3/4}=0.328$. Here ${\rho }_c$ is the mass-equivalent cosmological critical density; inserting an energy-density critical value without its $c^2$ conversion would give incorrect dimensions. A numerical estimate needs photon and baryon density parameters as additional inputs.