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In hydrogen-only cosmology, charge neutrality gives $n_e=n_p$ and baryon conservation gives $n_b=n_p+n_H$. The fraction $X_e$ lies between zero and one, with $n_H=(1-X_e)n_b$. With helium present, electron abundance per hydrogen nucleus and per total baryon are different conventions and must be specified. The hydrogen-only Saha equation gives the equilibrium fraction.

Two stages of the thermal history. Big Bang nucleosynthesis is nuclear assembly in an almost fully ionized, radiation-dominated plasma. Around $t\sim 1\mathrm{s}$, at temperatures of order $1\mathrm{MeV}/k_B$, the weak neutron–proton conversion rate falls below the expansion rate. The neutron-to-proton abundance freezes out at order $1/5$ and then decreases by neutron decay. The deuterium bottleneck delays effective nuclear assembly until $k_{B}T\sim 0.07$–$0.1\mathrm{MeV}$, at ages of a few minutes: the high photon-to-baryon ratio permits photodissociation despite the deuteron binding energy exceeding the mean photon energy. Once deuterium survives, reactions rapidly put most remaining neutrons into helium-4. A neutron-to-proton ratio near $1/7$ gives a helium mass fraction $Y_p\simeq 2(n/p)/(1+n/p)\simeq 1/4$, with hydrogen dominant by number and small residual deuterium, helium-3 and lithium-7 abundances. Expansion and low density prevent substantial heavy-element production. Nuclear binding is involved; neutral atoms do not yet survive.

Photon decoupling occurs much later, near a few thousand kelvin and a few hundred thousand years, after matter has overtaken radiation in the expansion dynamics. Electrons and nuclei form neutral atoms during cosmological recombination. The resulting decline of free-electron density reduces Thomson opacity, and photons begin to free stream over cosmological distances. Those photons form the observed cosmic microwave background. Atomic binding and scattering opacity, rather than new nuclear reactions, control this stage. Helium forms neutral atoms earlier than hydrogen. Recombination, photon last scattering, baryon drag release and the later thermal decoupling of the gas are related but not identical events.

The temperature falls through expansion: after the relevant entropy releases have ended, $T_{\gamma }\propto a^{-1}$ and number densities scale as $a^{-3}$. Big Bang nucleosynthesis lies in the radiation-dominated universe; photon decoupling lies after matter–radiation equality, when a matter-dominated approximation is useful but radiation is not exactly negligible.

Photon density and baryon abundance. The cosmic microwave background has a black-body radiation spectrum obeying Planck's law:where $u_{\nu }$ is spectral energy density, not a photon-number density. The measured temperature is about $2.72548\mathrm{K}$.^[ref] The mean photon energy in a black body isUsing the supplied rounded energy density givesThe exact black-body number integral at $2.725\mathrm{K}$ gives $4.1\times 10^8{\mathrm{m}}^{-3}$; the difference reflects rounding of the stated energy density. Either estimate is adequate.

A numerical photon-to-baryon ratio additionally needs a baryon density, which is not printed in this question. Make that observational input explicit rather than infer baryons from radiation alone. For illustration take the standard rounded physical density ${\Omega }_b{h}_H^2\simeq 0.022$, with $H_0=100h_H\mathrm{km}{\mathrm{s}}^{-1}\mathrm{Mp}{\mathrm{c}}^{-1}$. This is consistent with the cosmological parameter fits available around the paper's date.^[ref] It givesCounting baryons, including those bound in helium, rather than counting nuclei then givesBoth populations dilute as $a^{-3}$ after electron–positron annihilation, so this ratio also applies at hydrogen recombination in the standard adiabatic history.

Estimating decoupling. The large photon excess explains why setting $k_{B}T$ equal to the Hydrogen binding energy ${\chi }_H=13.6\mathrm{eV}$ is wrong. Even a small ionizing tail can oppose neutral-atom formation. The crude exponential balance $e^{-{\chi }_H/(k_{B}T)}\sim \eta$ gives ${\chi }_H/(k_{B}T)\sim \ln (1/\eta )\sim 21$, or $T\sim 7\times 10^3\mathrm{K}$. This only establishes the scale and the large suppression below ${\chi }_H/k_B$; it neglects phase-space factors and the opacity criterion.

A more controlled elementary Saha-Thomson decoupling estimate uses the hydrogen-only Saha equation. With $X_e=n_e/n_H$ and ground-state statistical weights combined in the usual way,The electron translational phase-space factor is significant inside the logarithm. This equilibrium relation places the sharp neutralization interval around $T\sim (3\text{–}4)\times 10^3\mathrm{K}$. Combine it with the approximate Thomson decoupling rate criterionFor $Y_p\simeq 0.25$, $n_{b0}\simeq 0.25{\mathrm{m}}^{-3}$, $H_0\simeq 68\mathrm{km}{\mathrm{s}}^{-1}\mathrm{Mp}{\mathrm{c}}^{-1}$, and ${\Omega }_m\simeq 0.31$, solving these two approximate relations gives $T\simeq 3.1\times 10^3\mathrm{K}$, with equilibrium $X_e$ of order $10^{-2}$. Thus the appropriate estimate isThis is not a precision recombination calculation. Hydrogen-only equilibrium omits the kinetic bottleneck in escaping Lyman photons, two-photon processes, the detailed helium history and freeze-out; $\Gamma /H\sim 1$ is not identical to the maximum of the last-scattering visibility function. These approximations must be improved to predict an accurate free-electron fraction or a percent-level last-scattering redshift. The high photon-to-baryon ratio alone does not fix that detailed history.

The electrons are highly nonrelativistic: $k_{B}T/(m_e{c}^2)\sim 5\times 10^{-7}$, and their thermal root-mean-square speed is only about $10^{-3}c$. Photon coupling does not make their kinetic energies relativistic.

Age–temperature relation. During a nearly flat pressureless matter-dominated phase, with radiation, curvature and vacuum energy neglected in the expansion equation,Here the Big Bang time is chosen as $t=0$. Conserved photon entropy gives $T\propto a^{-1}$, soTo normalize this relation, use the matter-era expansion rate expressed in present-day measured parameters, rather than assume the late universe is still matter dominated:The same illustrative parameters give $t_{\mathrm{dec}}\sim 4.5\text{–}5\times 10^5\mathrm{yr}$ for $T_{\mathrm{dec}}\sim 3.0\text{–}3.1\times 10^3\mathrm{K}$. Radiation was still a meaningful correction; retaining it reduces the age, and a more complete recombination calculation gives the familiar scale of roughly $4\times 10^5\mathrm{yr}$.^[ref] A numerical age cannot follow from a proportionality $t\propto T^{-3/2}$ alone: an expansion or density normalization is essential. The small amount of dark energy at decoupling can be neglected even though it affects the present age.

Work in natural units $k_B=\hslash =c=1$, initially neglecting helium and excited hydrogen states. The reaction $p+e^{-}\leftrightarrow H+\gamma$ is in chemical equilibrium when ${\mu }_p+{\mu }_e={\mu }_H$, since equilibrium photons have zero chemical potential. Substitution of the nonrelativistic Maxwell-Boltzmann distribution for the three densities givesFor ground-state hydrogen use $g_e=g_p=2$, $g_H=4$, and $m_H\simeq m_p$ in the translational prefactor. Writing its Hydrogen binding energy as $I$, this becomes $(m_{e}T/(2\pi ){)}^{3/2}{e}^{-I/T}$. Charge neutrality gives $n_p=n_e=X_e{n}_B$, while baryon conservation gives $n_H=(1-X_e)n_B$. Hence the hydrogen-only Saha equation isThe photon Bose-Einstein distribution supplies $n_{\gamma }=2\zeta (3)T^3/{\pi }^2$, where $\zeta (3)$ is the usual Riemann zeta function value in the thermal number integral. With baryon-to-photon ratio ${\eta }_{\gamma }=n_B/n_{\gamma }$, substitution yieldsThis reproduces the requested numerical coefficient, with the printed $\xi (3)$ interpreted as the thermal $\zeta (3)$ value. There is a normalization ambiguity in the PDF: it writes $\eta =n_B/s$ without defining $s$. To reproduce this coefficient, $s$ must mean photon number density. If $s$ has its usual entropy density meaning, the Saha baryon-abundance normalization instead givesThe two forms agree through ${\eta }_{\gamma }=[{\pi }^4{g}_{\ast s}/(45\zeta (3))]{\eta }_s$, with the entropy sector specified. A baryon-to-entropy ratio cannot simply be inserted into the photon-normalized expression.

As the universe cools, the Boltzmann factor favors neutral atoms and cosmological recombination sharply lowers $X_e$. For an order-one ionized fraction the photon-normalized equation gives the balanceup to an order-one ionization-fraction factor. The small baryon abundance means that there are very many photons per baryon: even a small high-energy photon tail can ionize most atoms. Thus the recombination temperature is far below the microscopic binding scale $I\simeq 13.6\mathrm{eV}$, typically a few tenths of an eV for the standard abundance, rather than of order $I$.

Photon decoupling is a related but distinct kinetic event. The Thomson scattering rate $n_e{\sigma }_T$ falls relative to $H$ as free electrons disappear, and photons begin free streaming. The precise last-scattering epoch depends on integrated optical depth, not only the local equality of rates. At still lower density the effective recombination rate per electron also drops below the expansion rate. Residual electron freeze-out leaves a small nonzero ionized fraction, larger than the vanishingly small equilibrium Saha prediction; radiative atomic bottlenecks mean the entire history cannot be obtained from equilibrium alone.

The temperature comparison in the last request needs an interpretation. If “reionisation temperature” means the microscopic thermal ionization scale, the large photon-to-baryon ratio explains why cosmological recombination requires $T\ll I$. If it means later cosmic reionization, there is no universal thermal threshold: stars or other sources supply ionizing photons and can reheat gas. The cosmological photon background is actually colder at that later epoch, because $T_{\gamma }\propto a^{-1}$. Its temperature therefore cannot literally be higher than the background temperature at recombination merely because hydrogen is reionized.

For the reaction $p+e\leftrightarrow H+\gamma$, chemical equilibrium requires ${\mu }_p+{\mu }_e={\mu }_H$ because thermal photons have zero chemical potential. Insert the nonrelativistic Maxwell-Boltzmann distribution number densities and use $m_H=m_p+m_e-\mathcal{B}$:For ground-state hydrogen, including its spin states, $g_e=g_p=2$, $g_H=4$; hence the degeneracy factor is one. Since $m_p/m_H\simeq 1$, the right side is $(m_{e}T/2\pi {)}^{3/2}{e}^{-\mathcal{B}/T}$.

Charge neutrality gives $n_p=n_e$, while baryon conservation gives $n_b=n_p+n_H$. Consequently $n_e=n_p=X_e{n}_b$ and $n_H=(1-X_e)n_b$. Inverting the preceding ratio and inserting the photon number density with $n_b=\eta n_{\gamma }$ gives the hydrogen-only Saha equationWriting $y=\mathcal{B}/T$, its coefficient is $(2\zeta (3)/{\pi }^2)\eta (2\pi \mathcal{B}/m_e{)}^{3/2}\simeq 3.2\times 10^{-16}$, consistent with the rounded $3\times 10^{-16}$. The negligible mass-ratio correction and ground-state approximation are the assumptions behind this form.

DefineUsing $n_e=n_B\mathcal{F}$, $n_{j^{+}}=X_{j^{+}}{n}_j$, and $n_{j^0}=(1-X_{j^{+}})n_j$, the two Saha relations become the closed systemwhereThese are two equations for the two ionization fractions.

When $Y_p\to 0$, $\mathcal{F}=X_{H^{+}}$, and the hydrogen equation reduces tothe usual hydrogen-only Saha equation.

Combining the hydrogen-only Saha equation with the Thomson decoupling rate criterion gives an elementary estimate of photon decoupling. With present baryon density about $0.25{\mathrm{m}}^{-3}$, hydrogen fraction about $0.75$, present photon temperature $2.725\mathrm{K}$ and a matter-era normalization $H_0\sqrt{{\Omega }_m}\simeq 38\mathrm{km}{\mathrm{s}}^{-1}\mathrm{Mp}{\mathrm{c}}^{-1}$, the coupled estimate gives a temperature near $3100\mathrm{K}$. Hydrogen chemical equilibrium fails during the actual kinetic recombination history, and the local scattering-rate criterion differs from the maximum of the visibility function; the estimate is not a precision prediction.

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.