The Epoch of Recombination

The day the universe became transparent

For its first 378,000 years the universe was opaque — a glowing, featureless fog. Not because it was dusty, but because it was a plasma: every electron was free, unattached to any nucleus, and free electrons are superb scatterers of light. A photon could travel only a tiny fraction of the horizon before bouncing off an electron, like sunlight trying to cross the interior of the Sun. Matter and radiation were welded into a single hot fluid, sloshing together, with no clear sky and no straight-line light.

Then the universe cooled to about 3000 K. Electrons slowed enough to be captured by the bare protons and helium nuclei, forming the first neutral atoms. Neutral hydrogen barely scatters the cool radiation around it, so as the free electrons vanished the fog lifted. Photons that had been ricocheting for hundreds of thousands of years suddenly found nothing to scatter off, and streamed freely across the cosmos. That release is the single most important light source in cosmology: the cosmic microwave background. The epoch of recombination is, quite literally, the surface we see when we look as far back as light allows.

One naming caveat up front, because it confuses everyone: the electrons and protons were never combined before this epoch, so "recombination" is a misnomer borrowed from lab plasma physics. The universe started ionised and combined for the first time. The word stuck regardless.

The Saha equation and why 3000 K, not 158,000 K

Hydrogen's ground-state binding energy is 13.6 eV. Set k_B T = 13.6 eV and you get a temperature of about 158,000 K. Naively you would expect hydrogen to go neutral when the universe was that hot. It did not — recombination waited until the universe was roughly 50 times colder. The reason is the staggering number of photons per baryon.

In thermal equilibrium the balance between ionisation and recombination is governed by the Saha equation. Written for the free-electron fraction x_e = n_e / n_H of a pure-hydrogen plasma, it is

  x_e²        1    ( m_e k_B T )^(3/2)
─────────  = ───  ( ───────── )      · exp( −13.6 eV / k_B T )
 1 − x_e      n_b  (  2π ℏ²    )

The key feature is the prefactor (m_e k_B T / 2πℏ²)^(3/2) / n_b, which is enormous because the baryon number density n_b is tiny. Equivalently, there are about η⁻¹ ≈ 1.6 × 10⁹ photons for every baryon. With a billion-plus photons competing to re-ionise each atom, the rare high-energy photons in the Wien tail of the blackbody spectrum keep hydrogen ionised long after the average photon energy has dropped well below 13.6 eV.

Solving Saha for the actual baryon density gives ionisation fractions of:

  x_e ≈ 0.9   at  T ≈ 4500 K   (z ≈ 1650)
  x_e ≈ 0.5   at  T ≈ 3700 K   (z ≈ 1360)
  x_e ≈ 0.1   at  T ≈ 3100 K   (z ≈ 1130)
  x_e ≈ 0.01  at  T ≈ 2800 K   (z ≈ 1030)

So the transition is sharp but it is anchored near 3000 K, not 158,000 K — purely a counting effect. This delay matters: it places recombination after matter-radiation equality and at a redshift where the universe is already matter-dominated, which shapes the acoustic peaks of the CMB.

Beyond equilibrium: the Peebles bottleneck

The Saha equation assumes the gas stays in equilibrium, but recombination is too fast for that to remain true. The subtlety, worked out independently by Jim Peebles and by Yakov Zeldovich, Vladimir Kurt and Rashid Sunyaev in 1968, is that recombination directly to the ground state is self-defeating: each such capture emits a 13.6 eV photon that immediately ionises a neighbouring atom. Net progress can only happen through the excited states.

An electron must therefore cascade down to the first excited level (n = 2) and then reach the ground state by one of two slow channels:

• Lyman-α escape. A 2p → 1s transition emits a Lyman-α photon at 121.6 nm. In the dense early gas this photon is almost instantly re-absorbed by another atom; net escape only happens as cosmic expansion redshifts it out of the line. This is a slow leak.
• 2s two-photon decay. The metastable 2s state cannot decay by a single photon (it is dipole-forbidden), so it decays by emitting two photons simultaneously, at a rate of only about Λ₂ₛ ≈ 8.22 s⁻¹. Slow, but it never gets reabsorbed because two soft photons rarely re-excite an atom.

These two bottlenecks throttle recombination so that the real ionisation history lags the Saha prediction and freezes out at a residual x_e ≈ 2 × 10⁻⁴ rather than going all the way to zero. Peebles captured this in an ordinary differential equation for x_e(z):

dx_e/dt = −C [ α_B n_b x_e²  −  β_B (1 − x_e) e^(−B_α/k_B T) ]

where α_B is the case-B recombination coefficient (summing captures to n ≥ 2), β_B the corresponding photoionisation rate, B_α the binding energy of the n = 2 level, and C the Peebles factor — the fraction of excited atoms that reach the ground state before being re-ionised. Modern CMB analysis uses high-precision multilevel codes — RECFAST (Seager, Sasselov & Scott 1999), CosmoRec, and HyRec — that track hundreds of levels and percent-level corrections, because the recombination history feeds directly into the predicted CMB power spectrum.

The key numbers

Quantity	Value	Note
Time after Big Bang	≈ 378,000 yr	at z ≈ 1090 (Planck 2018)
Redshift of recombination	z ≈ 1090	peak of the visibility function
Redshift of last scattering	z* = 1089.80 ± 0.21	Planck 2018 best fit
Photon temperature then	≈ 2970 K ≈ 0.256 eV	= 2.725 K × (1 + z*)
Hydrogen ionisation energy	13.6 eV (≈ 158,000 K)	note the ~50× mismatch
Photon-to-baryon ratio η⁻¹	≈ 1.6 × 10⁹	η = 6.1 × 10⁻¹⁰
Thickness of last-scattering shell	Δz ≈ 80 (≈ ±115,000 yr)	finite, not a sharp wall
Residual free-electron fraction	x_e ≈ 2 × 10⁻⁴	frozen-out leftover
CMB temperature today	2.72548 ± 0.00057 K	COBE/FIRAS, Fixsen 2009
Comoving distance to LSS	≈ 45.6 Gly (≈ 14.0 Gpc)	the most distant light we see; lookback time ≈ 13.8 Gyr

Note the distinction between the He and H transitions: doubly ionised helium recombines near z ≈ 6000, singly ionised helium near z ≈ 2500, and hydrogen near z ≈ 1100. Helium goes first because its ionisation energies (54.4 eV and 24.6 eV) are higher.

Decoupling and the last-scattering surface

Recombination is the cause; decoupling is the consequence. Photons stay glued to matter as long as the Thomson scattering rate exceeds the cosmic expansion rate. The scattering rate per photon is

Γ = n_e σ_T c     with   σ_T = 6.652 × 10⁻²⁵ cm²  (Thomson cross-section)

As x_e plummets during recombination, the free-electron density n_e = x_e n_b collapses, Γ drops below the Hubble rate H(z), and photons decouple. The probability that a CMB photon last scattered at redshift z is described by the visibility function g(z) = (dτ/dz) e^−τ, where τ is the Thomson optical depth back to z. The visibility function is a peaked curve centred at z ≈ 1090 with a width Δz ≈ 80.

That finite width is why the last-scattering surface is not an infinitely thin wall but a shell roughly 115,000 years deep in time. The finite thickness smears small-scale CMB anisotropies — this is part of the Silk damping tail that suppresses the high-multipole acoustic peaks. The geometry matters too: because the universe is transparent after recombination, we look out in every direction and the LSS appears as a sphere centred on us, at a comoving distance of about 45.6 billion light-years (≈ 14.0 Gpc) — even though the light itself has been travelling for only about 13.8 billion years, because the intervening space has expanded.

Worked example: when does Thomson scattering switch off?

Let's estimate the decoupling redshift by asking when the scattering rate equals the expansion rate, Γ = H. We need the free-electron density and the Hubble rate as functions of redshift.

Step 1 — baryon density. Today the baryon number density is about n_{b,0} ≈ 0.25 m⁻³. It scales as (1 + z)³, so at z = 1090:

n_b(z) = 0.25 × (1091)³ m⁻³ ≈ 3.2 × 10⁸ m⁻³

Step 2 — free electrons. From the Saha/Peebles history, x_e ≈ 0.1 near z ≈ 1100, so

n_e = x_e n_b ≈ 0.1 × 3.2 × 10⁸ ≈ 3.2 × 10⁷ m⁻³

Step 3 — scattering rate. With σ_T = 6.652 × 10⁻²⁹ m² and c = 3 × 10⁸ m/s:

Γ = n_e σ_T c
  = (3.2 × 10⁷)(6.652 × 10⁻²⁹)(3 × 10⁸)
  ≈ 6.4 × 10⁻¹³ s⁻¹

Step 4 — Hubble rate. In the matter-dominated era H(z) ≈ H₀ √(Ω_m) (1 + z)^(3/2). With H₀ ≈ 2.2 × 10⁻¹⁸ s⁻¹ and Ω_m ≈ 0.31:

H(1090) ≈ 2.2 × 10⁻¹⁸ × √0.31 × (1091)^(3/2)
        ≈ 2.2 × 10⁻¹⁸ × 0.557 × 3.6 × 10⁴
        ≈ 4.4 × 10⁻¹⁴ s⁻¹

Result. At z ≈ 1090, Γ ≈ 6 × 10⁻¹³ s⁻¹ is still about 15 times larger than H ≈ 4 × 10⁻¹⁴ s⁻¹ — photons are just on the cusp. Drop x_e by another factor of ten (which happens within Δz ≈ 80 as recombination races on) and Γ falls below H: decoupling. This back-of-envelope reproduces the precise answer that detailed codes give, z* ≈ 1090, and shows why decoupling is so abrupt — both n_e and the scattering rate are falling exponentially fast.

Discovery: predicting, then hearing, the relic light

The physics of recombination and the relic radiation it releases were predicted before they were observed:

• 1948 — Gamow, Alpher & Herman. Working out Big Bang nucleosynthesis, Ralph Alpher and Robert Herman predicted a relic radiation field with a present temperature of about 5 K. The prediction was largely forgotten for over a decade.
• 1965 — Penzias & Wilson. Arno Penzias and Robert Wilson, using a horn antenna at Bell Labs, found an isotropic 3.5 K excess they could not eliminate. Robert Dicke's Princeton group (Peebles, Roll, Wilkinson) immediately identified it as the relic of recombination. Penzias and Wilson won the 1978 Nobel Prize.
• 1968 — Peebles; Zeldovich, Kurt & Sunyaev. Both groups independently solved the non-equilibrium recombination problem, identifying the Lyman-α and 2s two-photon bottlenecks that delay recombination beyond the Saha prediction.
• 1990 — COBE/FIRAS. John Mather's instrument measured the CMB spectrum to be a blackbody at 2.725 K with deviations under 50 parts per million — the most perfect blackbody ever measured. Mather and George Smoot shared the 2006 Nobel Prize.
• 2003–2013 — WMAP and Planck. WMAP and then ESA's Planck satellite mapped the temperature anisotropies to exquisite precision, pinning the last-scattering redshift at z* = 1089.80 ± 0.21 and the age at recombination to ≈ 378,000 years.

Related transitions and what came after

Recombination sits in a sequence of cosmic milestones, and it is easy to confuse it with its neighbours:

Event	Redshift	What happens	Effect on light
Matter-radiation equality	z ≈ 3400	matter density overtakes radiation	growth of structure begins
Helium recombination	z ≈ 6000 & 2500	He III→II, then He II→I	minor optical-depth steps
Hydrogen recombination	z ≈ 1090	H⁺ + e⁻ → H	universe turns transparent
Dark ages	z ≈ 1090 → 20	neutral gas, no stars yet	only the 21 cm line
Cosmic dawn	z ≈ 20 → 15	first stars (Pop III) ignite	first starlight
Reionization	z ≈ 10 → 6	starlight re-ionises hydrogen	partial re-scattering of CMB

Note the irony: recombination made the universe transparent, but reionization — when the first stars and quasars stripped the electrons off hydrogen again — made it partly opaque to scattering once more. That late re-scattering imprints a measurable optical depth τ ≈ 0.054 on the CMB, which is how Planck constrains the timing of cosmic dawn.

Common misconceptions and subtleties

• "Recombination created the CMB spectrum." No — the blackbody spectrum already existed, maintained by tight coupling before recombination. Recombination merely released photons that were already thermalised. That is why the CMB is such a perfect blackbody even though it was emitted by recombining atoms.
• "The universe went neutral instantly at 13.6 eV." No — the photon-to-baryon ratio of ~1.6 × 10⁹ delays recombination to ~3000 K (0.26 eV), about 50 times colder than the binding energy, and the transition still takes Δz ≈ 80 to complete.
• "Recombination, decoupling, and last scattering are the same event." They are nearly simultaneous but physically distinct: the chemistry (recombination), the dynamical uncoupling of photons from baryons (decoupling), and the geometric surface of final scatters (last scattering). Their redshifts differ by tens.
• "The CMB shows us the Big Bang." It shows the universe at 378,000 years, not t = 0. Everything before recombination is hidden behind the opaque last-scattering wall; to see earlier we need neutrinos or gravitational waves.
• "Recombination went to completion." It froze out at a residual ionisation x_e ≈ 2 × 10⁻⁴ because the Lyman-α and 2s bottlenecks could not keep up with the expansion. A tiny free-electron population survived all the way to reionization.
• "It's called recombination because atoms had combined before." They had not. The universe was born ionised; this was the first combination. The name is borrowed from lab plasma jargon and is technically a misnomer.