Saha Equation & Recombination

The balance, in one sentence

Take a box of hydrogen and turn up the heat. At low temperature it sits as neutral atoms; at high temperature collisions and energetic photons knock electrons free and it becomes a plasma of bare protons and electrons. Somewhere in between, the two processes — ionization stripping electrons off, recombination capturing them back — reach a balance. The Saha equation is the single relation that tells you exactly where that balance lies: what fraction of the gas is ionized, given its temperature and density.

Meghnad Saha wrote it down in 1920 to explain stellar spectra, but its most dramatic application is cosmological. Run the clock of the expanding universe backward and everything gets hotter and denser. At early times the cosmos was a fully ionized plasma, opaque to light because photons scattered endlessly off free electrons. As it expanded and cooled, the Saha balance tipped: electrons combined with protons, the free-electron fraction crashed, and within a cosmologically brief window the fog cleared. The light set free at that moment has been travelling ever since — we call it the cosmic microwave background, and the Saha equation is what dates its release.

The equation itself

For a single ionization stage — neutral hydrogen H ⇌ proton p + electron e⁻ — the Saha equation in its number-density form is

n_e n_p / n_H = (2π m_e k_B T / h²)^(3/2) · exp(−B / k_B T)

where n_e, n_p, n_H are the number densities of free electrons, protons and neutral hydrogen atoms, m_e is the electron mass, T the temperature, and B = 13.6 eV is the binding (ionization) energy of hydrogen's ground state. The prefactor (2π m_e k_B T / h²)^(3/2) is the inverse cube of the electron thermal de Broglie wavelength — it counts the available quantum phase-space states for the freed electron. The exponential is the Boltzmann penalty for paying the binding energy.

It is cleaner to write it for the ionization fraction x_e ≡ n_e / n_b, where n_b is the total baryon (here, hydrogen) number density. With charge neutrality n_e = n_p, the relation rearranges into the dimensionless Saha form for cosmology:

x_e² / (1 − x_e) = (1 / n_b) · (2π m_e k_B T / h²)^(3/2) · exp(−B / k_B T)

This single algebraic equation, with n_b set by the cosmic baryon density and T set by the photon temperature, is all you need to draw the first-order picture of recombination. Notice what controls it: the exponential of the binding energy in units of temperature, divided by the density. Lower the density — as cosmic expansion relentlessly does — and you make ionization easier, delaying recombination to lower temperature.

Why recombination waits until 3,000 K

Here is the result that surprises almost everyone. The binding energy is 13.6 eV, and the temperature corresponding to that energy is

T(B) = B / k_B = 13.6 eV / (8.617 × 10⁻⁵ eV/K) ≈ 1.58 × 10⁵ K

So you might guess hydrogen stays ionized only until the universe cools below ~158,000 K. In fact recombination does not complete until roughly 3,000 K — more than fifty times cooler. Why the enormous delay?

The culprit is the staggering number of photons per baryon. The baryon-to-photon ratio is

η = n_b / n_γ ≈ 6.1 × 10⁻¹⁰    →    n_γ / n_b ≈ 1.6 × 10⁹

There are about 1.6 billion photons for every proton. A blackbody at temperature T has a high-energy tail; the fraction of photons with energy above 13.6 eV is exponentially tiny, but when you multiply a tiny exponential by a billion-fold abundance, you still have enough hard photons to re-ionize freshly formed atoms. Recombination cannot win until the temperature has dropped far enough that even ~10⁹ photons no longer carry a single ionizing one per baryon. Quantitatively, that suppression factor is exactly the 1/n_b in the Saha formula made large. The balance tips near

k_B T_rec ≈ B / ln[ (m_e k_B T / 2π ℏ²)^{3/2} / n_b ] ≈ 13.6 eV / 40 ≈ 0.34 eV
T_rec ≈ 3,700 K   (where x_e first falls to ~0.5 under pure Saha)

This is the single most important qualitative lesson of cosmic recombination: it is delayed not by atomic physics but by the entropy of the photon bath. The same equation in a dense stellar photosphere (n ≈ 10¹⁷ cm⁻³) tips at ~10,000 K, because there the density term is enormous and the photon dilution argument runs the other way.

Recombination by the numbers

The pure-Saha curve and the corrected (kinetic) results disagree in their tails but agree on the headline epoch. Here are the canonical figures.

Quantity	Value	Note
Ionization energy of H	13.6 eV	Ground-state binding B
T equivalent of B	1.58 × 10⁵ K	Naive (wrong) recombination T
Baryon-to-photon ratio η	6.1 × 10⁻¹⁰	From BBN + CMB
Saha x_e = 0.5	T ≈ 3,700 K, z ≈ 1360	Half-ionized point
Last scattering (visibility peak)	z ≈ 1090	Planck 2018: z* = 1089.9
Temperature at last scattering	≈ 2,970 K ≈ 0.26 eV	T = 2.725 K × (1 + z*)
Cosmic age at recombination	≈ 3.8 × 10⁵ yr	~380,000 years
Width of last-scattering surface	Δz ≈ 80	Δt ≈ 1.15 × 10⁵ yr
Residual (frozen-out) x_e	≈ 2 × 10⁻⁴	Peebles freeze-out floor
Present CMB temperature	2.7255 K	FIRAS / COBE

The redshift and temperature track together by the simple cooling law T(z) = T₀(1 + z) with T₀ = 2.7255 K, because CMB photons redshift exactly as the universe expands. Multiply 2.7255 K by (1 + 1090) and you recover ≈ 2,970 K — the temperature of the surface of last scattering written into the present-day microwave sky.

Where Saha breaks: freeze-out and the Peebles equation

The Saha equation assumes the gas remains in instantaneous equilibrium — that recombination and ionization rates always balance perfectly. In a static box that is fine. In an expanding universe it fails in the tail, and the failure is physically rich.

Recombination directly to the ground state is self-defeating: each such capture emits a 13.6 eV photon that promptly ionizes a neighbouring atom, so it makes no net neutral hydrogen. Atoms must instead cascade down to the first excited state and then reach the ground state by one of two slow channels: emission of a Lyman-alpha photon (which itself is trapped, scattering ~10⁸ times before redshifting out of resonance) or the forbidden two-photon decay of the 2s state (rate Λ₂ₛ ≈ 8.22 s⁻¹). These bottlenecks make real recombination far slower than equilibrium would allow.

Jim Peebles in 1968 — and independently Yakov Zel'dovich, Vladimir Kurt and Rashid Sunyaev the same year — wrote the kinetic rate equation that replaces Saha out of equilibrium:

dx_e/dt = −C [ α n_b x_e² − β (1 − x_e) e^(−B₂/k_B T) ]

C = (Λ₂ₛ + Λ_α) / (Λ₂ₛ + Λ_α + β)   ← Peebles bottleneck factor

Here α is the recombination coefficient (capture to excited states), β the photoionization rate, B₂ = 3.4 eV the binding energy of the n = 2 level, and C is the fraction of excited atoms that successfully reach the ground state before being re-ionized. As the universe expands, x_e cannot keep up with the falling equilibrium value: the reaction rate drops below the Hubble expansion rate H, the chemistry freezes out, and a small residual ionization x_e ≈ 2 × 10⁻⁴ is left frozen into the gas. Modern codes (RECFAST, CosmoRec, HyRec) solve a refined multilevel version of this system to the ~0.1% accuracy that Planck data demand.

Recombination, decoupling, and the last scattering surface

It is worth being precise about three distinct events that all happen near z ≈ 1100 and are easy to conflate.

• Recombination is the chemistry: x_e drops from ~1 toward its frozen-out floor. Half-ionization (Saha) is at z ≈ 1360; the steep fall is centred near z ≈ 1100.
• Photon decoupling / last scattering is the optics: the Thomson-scattering rate Γ = n_e σ_T c falls below the expansion rate H, so a typical photon scatters for the last time. Planck places this at z* = 1089.9.
• The drag epoch z_drag ≈ 1060 is slightly later — the moment baryons stop being dragged by the photon pressure, the redshift that sets the baryon acoustic oscillation sound-horizon ruler.

Because the free-electron fraction does not snap to zero but declines smoothly, the moment of last scattering is spread over a range of redshift. The probability density that a photon last scattered at redshift z is the visibility function

g(z) = e^(−τ(z)) dτ/dz ,    τ(z) = ∫ n_e σ_T c dt

which peaks sharply at z ≈ 1090 with a full width of about Δz ≈ 80. That width corresponds to a comoving slab roughly 19 Mpc thick (about 17 kpc in proper distance at that epoch), crossed in roughly 115,000 years. The finite thickness is not a curiosity: it smooths out temperature fluctuations on angular scales smaller than the slab depth — the photon-diffusion damping known as Silk damping — and it is precisely modelled when fitting the CMB power spectrum.

The same equation that classifies stars

Saha did not have the early universe in mind. In 1920, working at Calcutta and then in London and Berlin, he was trying to understand why stars of different temperatures show such different spectra. A hot O star displays ionized helium absorption lines; the Sun (a G star) shows neutral metals and weak hydrogen; a cool M star shows titanium-oxide molecular bands and almost no hydrogen lines. The puzzle was that all these stars contain essentially the same elements.

The Saha equation resolved it at a stroke: the ionization state of each element in a stellar photosphere is set by temperature and pressure, and it is the ionization state — not the abundance — that controls which absorption lines appear. Cecilia Payne-Gaposchkin used exactly this relation in her 1925 thesis to show that the Sun is overwhelmingly hydrogen and helium, one of the most important results in twentieth-century astrophysics. The familiar OBAFGKM spectral sequence is, at root, a temperature ladder read off through the Saha equation.

The cosmological application differs only in the numbers you feed in. A stellar photosphere has density ~10¹⁷ particles per cm³; the recombining universe had ~200 baryons per cm³ at z = 1100, and the present cosmic mean is ~2.5 × 10⁻⁷ per cm³. Lower density pushes the ionization balance to lower temperature — which is why the cosmos recombined at 3,000 K while a stellar surface ionizes hydrogen at ~10,000 K.

Where it shows up: evidence and missions

• The CMB blackbody. COBE/FIRAS measured the present microwave background as a 2.7255 K blackbody to better than 50 parts per million — the most perfect blackbody known. That spectrum is the redshifted thermal radiation set free at recombination, exactly as the Saha picture predicts.
• The acoustic peaks. WMAP and then Planck measured the CMB temperature power spectrum's acoustic peaks. Their positions and heights depend sensitively on the recombination history — the redshift and width of last scattering — so fitting them with RECFAST/HyRec is a precision test of Saha-plus-Peebles physics. Planck 2018 pins z* = 1089.9 ± 0.4.
• Spectral distortions. The trapped Lyman-alpha and two-photon decays during recombination should imprint faint deviations from a perfect blackbody — the "cosmological recombination lines" — a target for proposed missions such as PIXIE.
• Stellar spectroscopy. Every spectral type assigned to a star, from the hottest Wolf-Rayet down to the coolest brown dwarf, rests on Saha-equation ionization balance in the photosphere.
• Helium recombination. Before hydrogen, doubly ionized helium recombined near z ≈ 6000 and singly ionized helium near z ≈ 2000 — two earlier Saha transitions that modern recombination codes track because they subtly shift the free-electron history seen in the CMB.

Common misconceptions and edge cases

• "Recombination happened when kT equalled 13.6 eV." No — that would be 158,000 K. The billion-to-one photon-to-baryon ratio delays it to ~3,000 K. Forgetting the density term in the Saha equation is the single most common error.
• "Recombination and last scattering are the same instant." They are close but distinct. Recombination is the chemistry; last scattering is when Thomson opacity drops below the expansion rate. Decoupling lags the steepest part of recombination slightly, and the drag epoch (z ≈ 1060) is later still.
• "The Saha equation gives the right answer everywhere." It is correct only in equilibrium. In the expanding universe it overpredicts how completely hydrogen neutralises; the gas freezes out with x_e ≈ 2 × 10⁻⁴, which the Peebles kinetic equation captures and Saha does not. That residual ionization matters for later reionization and for CMB polarization.
• "It's called recombination because electrons re-combine." Electrons and protons had never been bound before, so nothing "re"-combines. The name is borrowed from laboratory plasma physics, where it does describe ions re-capturing electrons. "Primordial recombination" or simply "hydrogen formation" would be more accurate.
• "Direct capture to the ground state drives recombination." It cannot — each ground-state capture emits a 13.6 eV photon that re-ionizes a neighbour. Net neutralisation proceeds only through the slow Lyman-alpha escape and the 2s→1s two-photon decay, which is exactly why Saha (which ignores this bottleneck) is incomplete.
• "The CMB comes from the Big Bang itself." Not directly — the CMB is the light released at recombination, ~380,000 years later, when the Saha balance finally let the plasma neutralise. Earlier than that the universe was opaque; we cannot see the Big Bang in photons, only in (so far undetected) neutrinos and gravitational waves.