Recombination (Cosmology)

Recombination in cosmology is the process by which free electrons became bound to atomic nuclei as the early universe expanded and cooled, forming neutral atoms. It includes successive stages of helium recombination and the later, especially important formation of neutral hydrogen. The resulting decrease in free-electron density allowed radiation to cease scattering frequently and begin travelling largely freely. In the standard cosmological model, the main epoch of photon last scattering occurred approximately 380,000 years after the Big Bang, at a temperature near 3,000 kelvin. The radiation observed from this epoch is the cosmic microwave background (CMB). (esa.int)

Physical setting and chronology

Before recombination, ordinary matter consisted predominantly of an ionized plasma of hydrogen and helium nuclei and free electrons. Radiation and ordinary matter were tightly coupled through interactions involving the electrons. Photons scattered repeatedly rather than propagating over cosmological distances without interruption. Expansion lowered the radiation temperature until neutral atoms could survive in substantial numbers. (esa.int)

Recombination was not a single instantaneous event. Its principal stages occurred at different redshifts, conventionally denoted by $z$:

These values characterize broad recombination eras, not sharply defined boundaries. Hydrogen recombination overlaps the main period of photon last scattering, whose characteristic redshift is about $1{,}100$. Different definitions of a “recombination redshift”—such as a specified neutral fraction or the peak probability of last scattering—therefore yield different values. (arxiv.org)

The name “recombination” does not imply that these nuclei and electrons had previously existed together as neutral atoms. It is the conventional atomic-physics term applied to their combination as the universe cooled. (background.uchicago.edu)

Atomic physics and equilibrium

The elementary hydrogen process can be represented schematically as

where $p$ is a proton and $\gamma$ a photon. The forward direction is radiative recombination; the reverse is photoionization. Hydrogen’s ground-state ionization energy is approximately 13.6 electronvolts, much greater than the typical thermal energy of the radiation near last scattering. Nevertheless, photons in the high-energy tail of the radiation distribution can ionize hydrogen, so comparing the binding energy with the mean photon energy alone does not determine when recombination occurs. (background.uchicago.edu)

When recombination and photoionization maintain chemical equilibrium, the ionization fraction can be estimated using the Saha ionization equation. For an idealized hydrogen-only gas,

Here $x_e=n_e/n_{\mathrm H}$, $n_{\mathrm H}$ is the total number density of hydrogen nuclei, $m_e$ is the electron mass, $T$ the common matter and radiation temperature, $k_{\mathrm B}$ the Boltzmann constant, $h$ the Planck constant, and $\chi_{\mathrm H}$ the hydrogen ionization energy. The exponential dependence explains why cooling produces a rapid change in the equilibrium neutral fraction. This equation is useful early in recombination, but it does not describe the entire subsequent evolution. (background.uchicago.edu)

Why recombination departs from equilibrium

Capturing an electron does not necessarily produce a lasting increase in neutral hydrogen. A direct capture into the ground state emits a photon energetic enough to ionize another atom. Much of the effective net recombination therefore proceeds through excited states, followed by transitions that allow atoms to reach the ground state without immediately reversing the process. (articles.adsabs.harvard.edu)

Two routes are particularly important:

• Lyman-alpha escape. The $2p\rightarrow1s$ transition emits a Lyman-alpha photon. Such photons are readily reabsorbed by other hydrogen atoms. Cosmic expansion shifts their frequencies away from resonance, permitting a small net escape from this repeated absorption and emission.
• Two-photon decay. The metastable $2s$ state reaches the ground state through $2s\rightarrow1s+2\gamma$. This provides an additional route around the strong trapping of Lyman-alpha radiation.

The net rates of these processes, rather than electron capture alone, largely control hydrogen recombination near last scattering. (academic.oup.com)

As expansion continues and density falls, the reactions become too slow to maintain equilibrium. A small residual ionized fraction consequently survives: recombination does not remove every free electron. Accurate calculations evolve atomic populations, the radiation field, and the electron abundance together, including higher excited states and radiative transfer effects. (background.uchicago.edu)

Helium introduces additional complications. Its line photons interact with other helium atoms and with the developing neutral-hydrogen population. Absorption by neutral hydrogen helps helium resonance photons escape their original transitions and accelerates the formation of neutral helium. (arxiv.org)

Recombination, decoupling, and last scattering

Three closely related concepts describe different aspects of this period:

• Recombination concerns the binding of electrons into atoms.
• Photon decoupling concerns the weakening of interactions between radiation and matter.
• Last scattering concerns the final scattering events experienced by the photons subsequently observed as the CMB.

The free-electron scattering rate is approximately

where $\sigma_{\mathrm T}$ is the Thomson scattering cross section and $c$ the speed of light. Recombination rapidly reduces $n_e$, allowing photons to propagate increasingly freely. These processes overlap, but their physical definitions are not interchangeable. (articles.adsabs.harvard.edu)

Last scattering is spread over a finite interval rather than confined to an infinitely thin boundary. Its distribution is described by a visibility function, which combines the probability of scattering at a given epoch with the probability of avoiding subsequent scattering. The “surface of last scattering” is therefore a shell of finite thickness in an observer’s past view of the universe. (academic.oup.com)

Recombination did not create the bulk of the CMB photons. It allowed the pre-existing thermal radiation to escape frequent scattering. Continued expansion subsequently stretched its wavelengths into the microwave range and lowered its effective temperature to about 2.7 kelvin. Atomic recombination emission is a much smaller additional component. (esa.int)

Cosmological significance

Before decoupling, the coupled photon–baryon fluid supported acoustic oscillations. Last scattering preserved information about their phases in the CMB temperature pattern. Modes near maximal compression or rarefaction contribute to the acoustic peaks used to determine cosmological parameters within models such as the Lambda–cold dark matter model. (background.uchicago.edu)

Photons also diffused through the plasma before last scattering. This mixed radiation from hotter and colder regions and suppressed fluctuations on small scales, an effect called Silk damping. The detailed recombination history therefore affects both the acoustic pattern and its small-scale damping. Errors in that history can bias inferred cosmological parameters. (background.uchicago.edu)

Atomic captures and transitions additionally produce cosmological recombination radiation: weak spectral features superimposed on the nearly blackbody CMB spectrum. Hydrogen and the two helium recombination stages contribute distinct, redshift-broadened features. Their predicted shapes depend on atomic processes, primordial abundances, and the expansion history. (arxiv.org)

Development of the theory and modelling limits

In 1968, P. J. E. Peebles and, independently, Yakov Zel’dovich, Vladimir Kurt, and Rashid Sunyaev developed non-equilibrium treatments of primordial recombination. Their effective three-level description incorporated Lyman-alpha escape and two-photon decay. Later multilevel calculations followed many excited states of hydrogen and helium and identified corrections important for precision CMB analysis. (academic.oup.com)

Numerical implementations include RECFAST, HyRec, and CosmoRec. These balance computational speed against detailed atomic and radiation physics; effective multilevel methods allow highly excited states to be incorporated without evolving every state separately during each cosmological calculation. (arxiv.org)

The approximate age and redshift assigned to recombination depend on the assumed cosmological parameters and on the chosen definition of the epoch. Recombination calculations also have distinct accuracy requirements for predicting the free-electron history and for predicting the much weaker recombination spectrum. Uncertainties in collisional processes and neutral-helium atomic modelling can matter more for the latter. (aanda.org)

References

1. Planck and the cosmic microwave backgroundesa.int
2. The cosmic microwave background and inflationesa.int
3. CosmoSpec: Fast and detailed computation of the cosmological recombination radiation from hydrogen and heliumarxiv.org
4. CMB Introduction: Recombinationbackground.uchicago.edu
5. Wandering in the Background: A Cosmic Microwave Background Explorerbackground.uchicago.edu
6. Recombination of the Primeval Plasmaarticles.adsabs.harvard.edu
7. Recombinationbackground.uchicago.edu
8. HyRec: A fast and highly accurate primordial hydrogen and helium recombination codearxiv.org
9. Primordial helium recombination I: feedback, line transfer, and continuum opacityarxiv.org
10. HYREC-2: a highly accurate sub-millisecond recombination codearxiv.org