Follow ionization fraction as the universe cools—see when photons last scatter to form the CMB
Recombination is one of the most important events in cosmic history. At a redshift of z ≈ 1100 (about 380,000 years after the Big Bang), the universe cooled enough for electrons and protons to combine into neutral hydrogen atoms. Before recombination, the universe was ionized plasma—photons constantly scattered off free electrons via Thomson scattering, making the universe opaque. After recombination, photons could travel freely, and the universe became transparent. The photons we observe as the cosmic microwave background (CMB) were emitted during this brief transition period, forming what we call the "last scattering surface."
The ionization fraction xe (the fraction of free electrons) is determined by the Saha equation, which balances the ionization and recombination rates in thermal equilibrium:
xe² / (1 - xe) = (me T / 2π)^(3/2) * exp(-13.6 eV / T) / nb
where me is the electron mass, T is the temperature, 13.6 eV is the hydrogen ionization energy, and nb is the baryon number density. Naively, we might expect recombination when T ~ 13.6 eV (about z ~ 6000). However, recombination actually occurs much later, at T ~ 0.3 eV (z ~ 1100). This delay is caused by the huge photon-to-baryon ratio η ≈ 10⁹: even when T drops below 13.6 eV, there are still enough high-energy photons in the tail of the blackbody distribution to ionize hydrogen. Recombination must wait until T ~ 13.6 eV / 30 before the ionization fraction drops significantly.
The full recombination process is more subtle than the Saha equation suggests. Direct recombination to the ground state (p + e⁻ → H + 13.6 eV photon) produces a Lyman-α photon that can immediately ionize another hydrogen atom, trapping the system. The key path is recombination to the n=2 level, followed by 2-photon or Lyman-α decay. This creates a "bottleneck" at the n=2 level, slowing recombination. The effective recombination rate is set by the cosmological redshift rate, not the atomic physics rate. This feedback between expansion and atomic transitions is what sets the precise redshift of recombination.
The visibility function g(z) describes the probability that a CMB photon we observe today last scattered at redshift z. It is the product of two factors:
Mathematically: g(z) = τ'(z) * exp(-τ(z)), where τ(z) is the optical depth to Thomson scattering. The visibility function peaks sharply at z ≈ 1100 (the "last scattering surface") and has a width Δz ≈ 80. This narrow peak is why we can think of the CMB as coming from a well-defined "surface" in the early universe, even though it is actually a finite-thickness shell. The width of the visibility function sets the Silk damping scale, which suppresses CMB fluctuations on small angular scales.
The mean free path of photons λmfp = 1 / (ne σT), where σT is the Thomson cross section and ne is the free electron density, grows dramatically during recombination. Before recombination (z > 1200), λmfp is tiny compared to the horizon, so photons are tightly coupled to the plasma. Around z ~ 1100, λmfp becomes comparable to the horizon, and photons begin to free-stream. After recombination, λmfp exceeds the horizon size, and photons travel unimpeded for billions of years until they reach our detectors as the CMB. This transition from diffusion to free-streaming is crucial for understanding the physics of CMB anisotropies and acoustic oscillations.
Before recombination, photons and baryons are tightly coupled via Compton scattering. The photon temperature Tγ ∝ (1+z) and the baryon temperature Tb are essentially equal: Tb ≈ Tγ. However, after photon decoupling at z ~ 1100, the two fluids evolve independently. Photons continue to cool as Tγ ∝ (1+z) due to cosmological redshift (today Tγ = 2.725 K). Baryons cool faster because the expanding gas does adiabatic work: Tb ∝ (1+z)², assuming negligible heating from gravitational collapse and star formation. At z = 0, the gas would be at ~50 K if it stayed neutral, but in practice reionization (z ~ 10) and astrophysical heating bring it to ~10⁴ K in the intergalactic medium.
The CMB temperature map from Planck shows fluctuations ΔT ~ 100 μK on a 2.725 K background. These fluctuations are sourced by density perturbations (Sachs-Wolfe effect), Doppler shifts from bulk velocities, and gravitational redshift (integrated Sachs-Wolfe effect). The angular power spectrum Cℓ has a series of acoustic peaks at ℓ ~ 220, 540, 800, ... corresponding to modes that were at maximum compression (peaks) or rarefaction (troughs) at recombination. The peak positions depend on the curvature and matter/dark energy content; the peak heights depend on baryon density and other parameters.
Thomson scattering during recombination produces linear polarization (E-modes) from the local quadrupole anisotropy. The polarization pattern traces the velocity field at last scattering and provides independent information about the baryon-photon fluid. Gravitational waves from inflation would produce B-mode polarization, but this has not yet been detected at the recombination surface. The reionization bump at ℓ ~ 10 in the EE spectrum comes from scattering at z ~ 10, not z ~ 1100, and constrains the reionization optical depth τreion ≈ 0.05.
Vary baryon density Ωb h²: Increase Ωb h² and watch recombination happen earlier (higher z) and faster (narrower visibility function). Decrease it and recombination happens later and more gradually. This shift in the recombination redshift changes the sound horizon angle, moving the CMB acoustic peaks. CMB observations pin down Ωb h² to sub-percent precision.
Watch the ionization fraction drop: The ionization fraction xe stays near 1 until z ~ 1200, then drops rapidly to xe ~ 10⁻⁴ by z ~ 1000. The transition is very sharp—most of recombination happens in Δz ~ 100. This is much faster than naively expected because the Saha equation has exponential sensitivity to T. Small residual ionization (xe ~ 10⁻⁴) persists to low redshift.
Visibility function peak: The visibility function g(z) peaks sharply at z ≈ 1090 with width Δz ≈ 80. About 90% of CMB photons last scattered between z = 1050 and z = 1130. This narrow peak justifies treating the CMB as coming from a 2D "surface" rather than a 3D volume. The finite thickness sets the damping scale via photon diffusion.
Mean free path evolution: Watch the mean free path λmfp grow from sub-horizon scales at z > 1200 to super-horizon scales at z < 1000. The transition λmfp ~ horizon happens right at the visibility function peak. Before this, photons diffuse; after this, they free-stream. The diffusion length (Silk scale) is the integral of λmfp over the visibility function.
Compare with Hubble rate: The recombination rate (set by the Hubble expansion rate) competes with the ionization rate (set by blackbody photons in the high-energy tail). When the universe expands slowly (low H₀), recombination happens later because there's more time for ionizing photons to maintain equilibrium. This H₀ dependence is subdominant compared to the Ωb h² dependence.
The theory of recombination was first worked out by Yakov Zel'dovich, Rashid Sunyaev, and Jim Peebles in the late 1960s. Peebles' 1968 paper "Recombination of the Primeval Plasma" laid out the multi-level atom calculation and predicted the redshift z ≈ 1100 for hydrogen recombination. The visibility function formalism was developed in the 1990s with the advent of precision CMB cosmology codes like CMBFAST and later CAMB and CLASS. These codes solve the full Boltzmann equations for photons, baryons, dark matter, and neutrinos to compute the CMB power spectrum. Today, we can measure recombination parameters with percent-level precision thanks to Planck and other CMB experiments.
For more, see: Peebles (1968), "Recombination of the Primeval Plasma" for the classic treatment, Hu & Sugiyama (1996), "Small-Scale Cosmological Perturbations: An Analytic Approach" for the visibility function and acoustic oscillations, or Dodelson, "Modern Cosmology" (Chapter 8) for a comprehensive textbook discussion of recombination, photon decoupling, and CMB physics.