Baryon-to-photon ratio ($\eta$)

The baryon-to-photon ratio is the number density of baryons divided by the number density of photons. In the present universe it is of order $10^{-9}$.

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.

The cosmological baryon density is the mass density in baryonic matter, including hydrogen and helium. Up to nuclear binding-energy and composition corrections it is the proton mass times the baryon number density. The baryon-to-photon ratio links that number density to the photon number density. Larger baryon density increases nuclear reaction rates and permits the deuterium bottleneck to end earlier.

Thermal photon number density is proportional to $T^3$. A sudden photon heating factor $(1+r{)}^{1/4}$ therefore raises photon number by $(1+r{)}^{3/4}$ and dilutes the baryon-to-photon ratio by its inverse when baryon number is conserved. Comparing Big Bang nucleosynthesis and cosmic microwave background baryon-density determinations can test entropy injection between the two epochs, subject to changes in the expansion rate and nuclear abundances.