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Let $q_{\mathrm{in}}$ be the photoionization rate per neutral hydrogen atom just outside the empty cavity. With an isotropic central source and no attenuation inside that cavity,The cross section is zero below threshold; this permits the lower integration limit zero. Pure hydrogen with hydrogen ionization fraction $\zeta$ has $n_{\mathrm{HI}}=n_H(1-\zeta )$ and $n_e=n_p=n_H\zeta$. Local photoionization equilibrium is thereforeDefine the dimensionless quantity ${\Theta }_{\mathrm{phys}}={\alpha }_r{n}_H/q_{\mathrm{in}}$. Solving the resulting quadratic equation, ${\Theta }_{\mathrm{phys}}{\zeta }_{\mathrm{in}}^2+{\zeta }_{\mathrm{in}}-1=0$, gives the physical rootThe rationalized form avoids numerical cancellation for small ${\Theta }_{\mathrm{phys}}$. Strong illumination gives ${\zeta }_{\mathrm{in}}\simeq 1-{\Theta }_{\mathrm{phys}}$; weak illumination gives ${\zeta }_{\mathrm{in}}\simeq {\Theta }_{\mathrm{phys}}^{-1/2}$.

Eliminating $Q$ with the H-alpha luminosity, $Q=2.2(4\pi D^2{F}_{\mathrm{H}\alpha }){\lambda }_{\mathrm{H}\alpha }/(hc)$, yieldsThe printed definition of $\Theta$ is inconsistent with the defined radii. The source calls $r_{\ast }$ a physical stellar radius, but its factor $(2\pi r_{\ast }D{)}^2$ gives $\Theta$ units of length to the fourth power, rather than a dimensionless value. It also omits the physically necessary cavity radius $r_{\mathrm{in}}$. For a concrete counterexample, double $r_{\mathrm{in}}$ while holding the source, $D$, and $n_H$ fixed: the incident photoionization rate falls by four, so ${\Theta }_{\mathrm{phys}}$ rises by four; the printed expression does not change. The root has the requested form with the corrected ${\Theta }_{\mathrm{phys}}$ above. There is no supplied relation allowing $r_{\mathrm{in}}$ to be replaced by $r_{\ast }$.

Ignoring helium, charge neutrality gives $n_e=n_p$, while baryon conservation gives $n_B=n_p+n_H$. Defining the hydrogen ionization fraction $X_e=n_e/n_B$ yields $n_p=n_e=X_e{n}_B$ and $n_H=(1-X_e)n_B$. Substituting in the abundance relation givesUse $n_B=\eta n_{\gamma }$ and $n_{\gamma }=2\zeta (3)T^3/{\pi }^2$ to obtain the Saha equation:If the positive right side is denoted by $A(T)$, the physical root isThis form avoids subtracting nearly equal numbers when $A$ is small, and makes the limits transparent: $X_e\simeq 1-A$ for $A\ll 1$, whereas $X_e\simeq A^{-1/2}$ for $A\gg 1$. It is an equilibrium prediction; a kinetic cosmological recombination treatment is needed after reactions cease to keep up with expansion.

Write $n_b=n_p+n_H$ for the cosmological baryon density in this hydrogen-only model, and $x_e=n_p/n_b$ for the hydrogen ionization fraction. By charge neutrality, $n_e=n_p$, so the Saha ionization equation becomesThe baryon-to-photon ratio is $\eta =n_b/n_{\gamma }$, and the equilibrium photon number density is $n_{\gamma }=2\zeta (3)T^3/{\pi }^2$. At an order-one hydrogen ionization fraction, neglecting factors of order one therefore givesReplacing $T$ inside the slowly varying logarithm by $B$ givesWith $B=13.6\mathrm{eV}$, $M_e=5\times 10^5\mathrm{eV}$ and $\eta =10^{-10}$, this is $T\simeq 0.351\mathrm{eV}\simeq 4.1\times 10^3\mathrm{K}$. Solving the preceding implicit approximation instead gives $T\simeq 0.306\mathrm{eV}\simeq 3.5\times 10^3\mathrm{K}$. Both are reasonable estimates of the recombination temperature at the accuracy requested. The small baryon abundance delays hydrogen recombination: even when $T\ll B$, the high-energy tail of the abundant photons can still ionize the much rarer hydrogen atoms.

Assume dilute nondegenerate nonrelativistic particles, a common temperature, chemical equilibrium, charge neutrality $n_p=n_e$, and photons with zero chemical potential. The reaction balance gives ${\mu }_p+{\mu }_e={\mu }_H$. Dividing the two charged-species equilibrium densities by the neutral density cancels their chemical-potential factors and leavesFor ground-state hydrogen, $g_p=g_e=2$ and $g_H=4$, and $m_H\simeq m_p$ in the thermal prefactor. The rest-mass difference is $I/c^2$. These approximations give the Saha ionization equationExcited levels or different degeneracy conventions would change the prefactor; the stated assumptions give the printed normalization.

Write the right side as $S(T)$. Since $n_e=X_e{n}_B$ and $n_H=(1-X_e)n_B$, the hydrogen ionization fraction obeys $X_e^2/(1-X_e)=S(T)/n_B$. With $n_B=\eta n_{\gamma }$, the given blackbody photon density then yieldsThere are vastly more photons than baryons. Even when the typical photon energy is below the ionization energy, the high-energy tail can contain enough ionizing photons to keep nearly every baryon ionized. Neutral hydrogen forms only when that tail becomes sufficiently depleted relative to the very small baryon-to-photon ratio. This entropy/number-density effect explains why cosmological recombination occurs at a temperature far below $I$, rather than simply when $kT$ first falls below the binding energy. The Saha ionization equation describes the equilibrium estimate; late kinetic freeze-out is a separate limitation of that approximation.

P. J. E. Peebles made the connection between microscopic processes and cosmological observables quantitative. His 1968 paper treated cosmological recombination as a kinetic problem. Capturing an electron directly into the hydrogen ground state releases a photon that can ionize another atom. Net neutralization depends on escape of resonance radiation through cosmological redshift and on two-photon spontaneous emission from the $2s$ state. An equilibrium Saha equation alone does not give that history.^[ref]

This matters for an acoustic prediction because the hydrogen ionization fraction fixes the supply of free electrons for Thomson scattering. In physical units the scattering rate per photon is ${\Gamma }_T=n_e{\sigma }_{T}c$. The cosmological visibility function describes the distribution of last-scattering times. Thus atomic physics helps determine when the photon-baryon acoustic oscillator is read out, as well as how rapidly tight coupling fails.

In 1970 P. J. E. Peebles and J. T. Yu followed adiabatic cosmological perturbations by numerically integrating the photon collision equation through cosmological recombination. Their calculation connected acoustic evolution and dissipation to residual matter structure and microwave brightness fluctuations. It did not assume that a perfect photon-baryon fluid remained valid throughout last scattering. Their illustrative cosmological models and normalization were historical choices, rather than the present Lambda-CDM model; their predicted brightness variance versus angular resolution should not be relabelled as a modern measured $C_{\ell }$ spectrum.^[ref]

The continuing idea is a forward prediction: specify initial conditions, evolve the coupled matter and radiation, project onto the sky, and compare the result with an instrument's readout. Coherent initial phases are needed for acoustic variance peaks explains why a collection of oscillating modes does not automatically produce a peaked power spectrum. The BOOMERANG acoustic-peak measurement provides a later observational test of this physical programme.

For hydrogen ionization fraction $X_e$ of order one, the hydrogen-only Saha equation gives $1\sim n_b(2\pi /(m_{e}T){)}^{3/2}{e}^{B/T}$. The baryon-to-photon ratio and photon number density imply $n_b\sim \eta T^3$. Therefore $1\sim \eta (T/m_e{)}^{3/2}{e}^{B/T}$ and the displayed logarithmic balance follows. A small $\eta$ supplies many ionizing photons per baryon, so neutral hydrogen forms at a temperature much smaller than its Hydrogen binding energy. The numerical temperature depends on the ionization-fraction convention and on whether equilibrium remains valid.