Recombination (cosmology)

Recombination is the cosmological epoch when free electrons and protons first bonded into electrically neutral hydrogen atoms. The name is a historical misnomer: the Big Bang theory does not claim that protons and electrons had previously been combined, but the term was coined before the Big Bang became the leading explanation for the universe's origin.

Just before recombination, the universe contained almost no neutral hydrogen. Although atoms could theoretically form, any newly created neutral hydrogen was instantly ionized by the abundant high-energy photons. As the universe expanded and cooled, the radiation field eventually lost enough energy to ionize neutral hydrogen, making atoms energetically stable.

Photon decoupling—when photons began traveling freely through space without interacting with matter—occurred around the same time as recombination, but the two are distinct events driven by different physical processes. Recombination happened when expansion lowered the temperature, allowing atoms to form and releasing some photons. Photon decoupling happened when expansion diluted the electron density, enabling the far more numerous photons to propagate without scattering; this produced the cosmic microwave background radiation.

The timing of recombination can be estimated from the cosmic microwave background's temperature evolution. The CMB is a blackbody spectrum, shifted in energy by cosmic expansion (redshift, denoted z). Today's CMB temperature is about 2.7 K. Simply comparing the blackbody peak thermal energy (Boltzmann constant times temperature) to hydrogen's ionization energy (13.6 eV) is insufficient, because the photon energy distribution matters. A better estimate considers thermal equilibrium between matter and radiation. The density of photons with enough energy to ionize hydrogen equals the total photon density multiplied by a Boltzmann factor. At equilibrium, this roughly equals the baryon density. The baryon-to-photon ratio, known from Planck satellite measurements and other sources, is about 10⁻⁹. Solving for the redshift gives a value near 1100, corresponding to roughly 400,000 years after the Big Bang.

The cosmic ionization history is described by the free electron fraction, xₑ, as a function of redshift. This is the ratio of free electron density to total hydrogen density (neutral plus ionized). Since hydrogen recombines only after helium is fully neutral, charge neutrality implies that the free electron density equals the proton density, so xₑ also represents the fraction of ionized hydrogen.

A rough estimate of the recombination redshift comes from assuming the reaction H⁺ + e⁻ ⇌ H + γ proceeds near thermal equilibrium. The Saha equation then gives the relative abundances of free electrons, protons, and neutral hydrogen. With charge neutrality, the equation becomes an expression for xₑ. All quantities on the right side are known functions of redshift: temperature scales as T = 2.7 K × (1 + z), and the total hydrogen density is given by n_H = (1 + z)³ × (baryon density today). Solving for 50% ionization yields a recombination temperature of roughly 3000 K, corresponding to z ≈ 1100.

In 1968, Jim Peebles in the US and Yakov Borisovich Zel'dovich and collaborators in the USSR independently calculated hydrogen's non-equilibrium recombination history. Their model's key points are: Direct recombinations to the ground state are inefficient because each such event produces a photon above 13.6 eV, which almost immediately re-ionizes a neighboring hydrogen atom. Instead, electrons efficiently recombine to excited states, then quickly cascade to the first excited state (n = 2). From there, two pathways lead to the ground state (n = 1): decay from the 2p state via a Lyman-α photon, which is usually reabsorbed but may escape if cosmological redshift moves it far enough from the Lyman-α resonance before encountering another hydrogen atom; and decay from the 2s state via two-photon emission, a slow process (rate 8.22 s⁻¹) that nevertheless competes with the slow Lyman-α escape. Atoms in the first excited state can also be re-ionized by ambient CMB photons before reaching the ground state, effectively undoing the recombination. To account for this, Peebles defined a factor C as the probability that an atom in the first excited state reaches the ground state.