Recombination and Photon Decoupling

Cosmological recombination is the nonequilibrium formation of neutral atoms as the expanding plasma cools. Photon decoupling is related but distinct: it is the loss of Thomson opacity after the free-electron density falls. The first process determines the ionization history $x_e(z)$; the second turns that history into a finite probability distribution for the last scattering of a CMB photon.

Required background. States, Processes, and the Laws of Thermodynamics supplies Thermodynamics and Statistical Mechanics for chemical equilibrium and detailed balance.

Helpful background. The Einstein Equation supplies the gravitational convention, while The Friedmann Equations supplies $H(z)$. Relativistic Matter and Compact Objects supplies Relativistic Matter and Compact Objects thermodynamics. Classical Field Modes and Canonical Variables supplies the Flat-Spacetime Quantum Field Theory mode convention used for photons, and Free Fields, Creation Operators, and Fock Space supplies occupation-state counting.

The Saha equilibrium estimate

Set $a_0=1$, so $1+z=a^{-1}$ and $T_\gamma=T_0(1+z)$. Let $n_{\rm H}$ be the total number density of hydrogen nuclei, ionized or neutral, and define the hydrogen free-electron fraction

$$x_e\equiv\frac{n_e}{n_{\rm H}}.$$

This analytic section excludes helium, so $0\leq x_e\leq1$; a precision code adds the helium electrons and specifies whether $x_e$ is normalized to hydrogen nuclei or all baryons. In natural units $\hbar=c=k_B=1$, charge neutrality and chemical equilibrium for $e^-+p\leftrightarrow {\rm H}+\gamma$ give

$$\boxed{ \frac{x_e^2}{1-x_e} =\frac{1}{n_{\rm H}} \left(\frac{m_eT_\gamma}{2\pi}\right)^{3/2} e^{-B_1/T_\gamma} },$$

where $B_1=13.6\ {\rm eV}$ is the ground-state binding energy. The spin degeneracies cancel for ground-state hydrogen. Because photons outnumber baryons enormously, the high-energy blackbody tail maintains ionization until $T_\gamma$ is far below $B_1$; a characteristic Saha value is $T_\gamma\sim0.3\ {\rm eV}$. Tong derives the equilibrium relation and its photon-to-baryon interpretation in Tong 2019, §2.3.3, printed pp. 99–102, eqs. (2.26)–(2.31) (Official PDF).

For an SI calculation, the phase-space factor in this equation is

$$\left(\frac{m_e k_BT}{2\pi\hbar^2}\right)^{3/2} e^{-B_1/(k_BT)}, \qquad n_{\rm H}(z)=n_{{\rm H}0}(1+z)^3.$$

The benchmark below uses the spectroscopic value $B_1=13.5984346\ {\rm eV}$; writing $13.6\ {\rm eV}$ in the analytic discussion is harmless at the displayed precision but shifts the numerical peak by about $0.2$ in redshift.

The Saha equation is an algebraic equilibrium curve, not a dynamical recombination history. It becomes unreliable once the expansion time is too short for the atomic populations to track that curve.

The Lyman bottleneck and the Peebles equation

Direct recombination to the ground state emits an ionizing photon that is usually absorbed by another atom. A net neutral atom instead reaches the ground state mainly by two slow routes: cosmological redshifting lets a Lyman-$\alpha$ photon escape resonance, or the metastable $2s$ state decays through a two-photon transition. The high Lyman optical depth is therefore a kinetic bottleneck.

An effective three-level hydrogen atom packages this physics into a case-B recombination coefficient $\alpha_B$, which excludes direct capture to the ground state. In the analytic approximation set the matter and radiation temperatures equal, $T_m=T_\gamma=T$, and define the photoionization coefficient from the $n=2$ manifold by detailed balance,

$$\beta_B(T)=\alpha_B(T) \left(\frac{m_eT}{2\pi}\right)^{3/2}e^{-B_2/T}, \qquad B_2=3.4\ {\rm eV}.$$

In SI units the factor multiplying $\alpha_B$ is

$$\left(\frac{m_ek_BT}{2\pi\hbar^2}\right)^{3/2} e^{-B_2/(k_BT)}.$$

If $E_\alpha=B_1-B_2$ and $n_{1s}\simeq n_{\rm H}(1-x_e)$, the Peebles equation is

$$\boxed{ \frac{dx_e}{dt} =-C\left[ \alpha_Bn_{\rm H}x_e^2 -\beta_B(1-x_e)e^{-E_\alpha/T} \right] },$$

with

$$C=\frac{1+K_{\rm esc}\Lambda_{2s,1s}n_{1s}} {1+K_{\rm esc}(\Lambda_{2s,1s}+\beta_B)n_{1s}}, \qquad K_{\rm esc}=\frac{\lambda_\alpha^3}{8\pi H}.$$

Here $\lambda_\alpha$ is the Lyman-$\alpha$ rest wavelength, $\Lambda_{2s,1s}$ is the two-photon decay rate, and $K_{\rm esc}$ encodes Lyman-$\alpha$ escape by cosmological redshifting on an expanding branch. Its units are volume times time; it is distinct from the spatial-curvature constant $K$ used elsewhere in this volume. This reduction assumes quasi-equilibrium within the excited manifold and treats resonant escape in the Sobolev approximation. The factor $0<C\leq1$ is the probability that an atom in the excited manifold reaches the ground state before photoionization. In equilibrium the two exponentials combine as $B_2+E_\alpha=B_1$, recovering the Saha relation. During recombination the bracket is positive, so $dx_e/dt\lt0$ as required. Equivalently, $dx_e/dz$ has the opposite sign because $dt/dz=-[(1+z)H]^{-1}$.

Seager, Sasselov, and Scott give the hydrogen and helium effective equations, atomic inputs, and their ranges of validity in Seager, Sasselov, and Scott 1999, §3, printed pp. L3–L4, eqs. (1)–(7). The assumptions here are narrower: homogeneous hydrogen, a blackbody radiation bath, charge neutrality, negligible collisional excitation, and no late reionization.

Optical depth and the visibility function

Let $\eta$ be conformal time and $\eta_0$ its value today. With the Thomson cross section $\sigma_T$, define the optical depth from $\eta$ to the observer by

$$\tau(\eta)=\int_\eta^{\eta_0} a(\tilde\eta)n_e(\tilde\eta)\sigma_T\,d\tilde\eta.$$

This convention gives $\tau'=-an_e\sigma_T$. A photon last scattered in $d\eta$ around $\eta$ with probability

$$\boxed{ g(\eta)\,d\eta =-\tau'(\eta)e^{-\tau(\eta)}d\eta },$$

where $g\geq0$ is the visibility density per unit conformal time. Its normalization is not an extra assumption:

$$\int_{\eta_i}^{\eta_0}g(\eta)\,d\eta =\left.e^{-\tau}\right|_{\eta_i}^{\eta_0} =1-e^{-\tau(\eta_i)}\simeq1$$

when the chosen early surface is optically thick. In redshift variables the same probability measure has a different coordinate density,

$$p(z)=e^{-\tau(z)}\frac{d\tau}{dz}, \qquad \frac{d\tau}{dz} =\frac{c\,\sigma_Tn_{\rm H}(z)x_e(z)}{(1+z)H(z)},$$

where $c$ has been restored for an SI evaluation. The peak of $g$ defines a useful last-scattering epoch, but its nonzero width is physical. Recombination does not happen on one time slice, and the local criterion $n_e\sigma_Tc\sim H$ does not replace the integrated survival probability $e^{-\tau}$.

Thus $g(\eta)d\eta=p(z)dz$ after reversing the orientation of the redshift integral. Neither a peak position nor a full width at half maximum is invariant under a nonlinear change of coordinate; the diagnostic below reports the redshift density $p(z)$.

A Saha-only response check

A small calculation isolates the direct equilibrium and opacity effects. Take $T_0=2.7255\ {\rm K}$, $Y_p=0.2458$, $\omega_b\equiv\Omega_bh^2=0.02237$, $\omega_m=0.143$, and $\omega_r=7.08\times10^{-5}$. Here $h\equiv H_0/(100\,\mathrm{km\,s^{-1}\,Mpc^{-1}})$; these are fixed diagnostic inputs, not a fitted cosmology. Use

$$n_{{\rm H}0} =\frac{(1-Y_p)\omega_b\rho_{c,100}}{m_p}, \qquad H(z)=100\frac{{\rm km}}{{\rm s\,Mpc}} \sqrt{\omega_r(1+z)^4+\omega_m(1+z)^3},$$

where

$$\rho_{c,100} =\frac{3(100\ {\rm km\,s^{-1}Mpc^{-1}})^2}{8\pi G}.$$

The SI constants used here give $n_{{\rm H}0}=0.189465137\,\mathrm{m^{-3}}$ and $100\,\mathrm{km\,s^{-1}\,Mpc^{-1}}=3.2407792894\times10^{-18}\,\mathrm{s^{-1}}$, with $\sigma_T=6.6524587321\times10^{-29}\,\mathrm{m^2}$.

Solve the Saha quadratic at each redshift and integrate $p(z)$ by the trapezoid rule from $z=0$ to $3000$ with $\Delta z=0.1$. Linear interpolation between the two half-maximum crossings gives the widths below. At the upper boundary $\tau(3000)\simeq162.0187$, so the exact truncated normalization $1-e^{-\tau(3000)}$ is unity to the shown precision. The trapezoid result is $0.999999821$; its $1.8\times10^{-7}$ deficit is discretization error, not leakage through the early boundary. Halving the step gives $0.999999955$ and changes the FWHM from $141.86199$ to $141.86195$; the peak is then limited by the redshift grid.

Input change	Peak redshift	FWHM in redshift
baseline	$1288.2$	$141.9$
$\omega_b\times0.9$	$1290.9$	$143.2$
$\omega_b\times1.1$	$1285.6$	$140.7$
$H(z)\times0.9$	$1282.6$	$140.0$
$H(z)\times1.1$	$1293.2$	$143.6$

In this controlled Saha diagnostic, more baryons increase the opacity and move last scattering slightly later while narrowing the distribution; faster expansion reduces optical depth per redshift and moves it earlier while broadening it. The $H$ variation leaves the Saha $x_e$ curve fixed, so it tests only the direct opacity channel. A Peebles or multilevel calculation must also recompute the kinetic lag, which can compete with these trends.

The baseline peak is intentionally not a precision prediction: Saha equilibrium releases photons too early. Modern calculations evolve hydrogen and helium multilevel populations, two-photon channels, line feedback, continuum opacity, Lyman-$\alpha$ frequency diffusion, the matter temperature, and radiative transfer. HyRec summarizes that precision boundary in Ali-Haïmoud and Hirata 2011, official abstract. A numerical value from such a code requires a pinned release and input file; none is being claimed by the table above.

Interpretation and onward use

The visibility function supplies the time window through which acoustic temperature and polarization sources are observed. Its peak, width, and tails affect peak phases, diffusion damping, and polarization generation, but those perturbative consequences belong to Einstein–Boltzmann Evolution and Line-of-Sight CMB Anisotropies. Late reionization adds a separate low-redshift optical-depth contribution; it must not be folded into the recombination peak without saying so.

Exercises

At fixed ionization fraction, use the Saha equation to find the response of the recombination temperature to the hydrogen density. Evaluate it at $T=0.3\ {\rm eV}$ and explain why its sign need not match the baryon-density shift of the visibility peak in the table.

Solution

Holding $x_e$ fixed holds the Saha right-hand side fixed, so

$$0=d\ln\left(\frac{T^{3/2}e^{-B_1/T}}{n_{\rm H}}\right) =\left(\frac32+\frac{B_1}{T}\right)d\ln T-d\ln n_{\rm H}.$$

Therefore

$$\frac{d\ln T}{d\ln n_{\rm H}} =\frac{1}{3/2+B_1/T} \simeq0.0214$$

at $B_1/T=13.6/0.3$. Increasing the density moves a fixed-$x_e$ recombination marker to slightly higher temperature and redshift. The derivative treats the local physical $n_{\rm H}$ and $T$ as independent variables. In the table it is instead the present normalization $n_{{\rm H}0}$ that varies while $n_{\rm H}=n_{{\rm H}0}(T/T_0)^3$. Holding $x_e$ fixed there gives

$$\frac{d\ln T}{d\ln n_{{\rm H}0}} =\frac{1}{B_1/T-3/2}\simeq0.0228.$$

The visibility peak depends on the opacity $c\sigma_Tn_{\rm H}x_e/[(1+z)H]$ integrated along the photon path. In the Saha-only table, the direct increase in opacity wins and delays last scattering. The distinction is precisely why recombination and photon decoupling should not be identified.

References

• Ali-Haïmoud, Y., and C. M. Hirata, “HyRec: A Fast and Highly Accurate Primordial Hydrogen and Helium Recombination Code,” Physical Review D 83, 043513 (2011), DOI.
• Seager, S., D. D. Sasselov, and D. Scott, “A New Calculation of the Recombination Epoch,” Astrophysical Journal 523, L1–L5 (1999), DOI, Author-hosted PDF.
• Tong, D., Cosmology, University of Cambridge Part II Mathematical Tripos lecture notes, Michaelmas 2019, Official PDF.