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In cosmology, recombination refers to the epoch at which charged electrons and protons first became bound to form electrically neutral hydrogen atoms. Recombination occurred about 370,000 years^{[1]}^{[notes 1]} after the Big Bang (at a redshift of z = 1100^{[2]}). The word "recombination" is misleading, since the Big Bang theory doesn't posit that protons and electrons had been combined before, but the name exists for historical reasons since it was named before the Big Bang hypothesis became the primary theory of the creation of the universe.
Immediately after the Big Bang, the universe was a hot, dense plasma of photons, leptons, and quarks: the quark epoch. At 10^{−6} seconds, the Universe had expanded and cooled sufficiently to allow for the formation of protons: the hadron epoch. This plasma was effectively opaque to electromagnetic radiation due to Thomson scattering by free electrons, as the mean free path each photon could travel before encountering an electron was very short. This is the current state of the interior of the Sun. As the universe expanded, it also cooled. Eventually, the universe cooled to the point that the formation of neutral hydrogen was energetically favored, and the fraction of free electrons and protons as compared to neutral hydrogen decreased to a few parts in 10,000.
Recombination involves electrons binding to protons (hydrogen nuclei) to form neutral hydrogen atoms. Because direct recombinations to the ground state (lowest energy) of hydrogen are very inefficient, these hydrogen atoms generally form with the electrons in a high energy state, and the electrons quickly transition to their low energy state by emitting photons. Two main pathways exist: from the 2p state by emitting a Lymana photon  these photons will almost always be reabsorbed by another hydrogen atom in its ground state  or from the 2s state by emitting two photons, which is very slow.
This production of photons is known as decoupling, which leads to recombination sometimes being called photon decoupling, but recombination and photon decoupling are distinct events. Once photons decoupled from matter, they traveled freely through the universe without interacting with matter and constitute what is observed today as cosmic microwave background radiation (in that sense, the cosmic background radiation is infrared [and some red] blackbody radiation emitted when the universe was at a temperature of some 3000 K, redshifted by a factor of 1100 from the visible spectrum to the microwave spectrum).
The cosmic ionization history is generally described in terms of the free electron fraction x_{e} as a function of redshift. It is the ratio of the abundance of free electrons to the total abundance of hydrogen (both neutral and ionized). Denoting by n_{e} the number density of free electrons, n_{H} that of atomic hydrogen and n_{p} that of ionized hydrogen (i.e. protons), x_{e} is defined as
Since hydrogen only recombines once helium is fully neutral, charge neutrality implies n_{e} = n_{p}, i.e. x_{e} is also the fraction of ionized hydrogen.
It is possible to find a rough estimate of the redshift of the recombination epoch assuming the recombination reaction is fast enough that it proceeds near thermal equilibrium. The relative abundance of free electrons, protons and neutral hydrogen is then given by the Saha equation:
where m_{e} is the mass of the electron, k_{B} is Boltzmann's constant, T is the temperature, ħ is the reduced Planck's constant, and E_{I} = 13.6 eV is the ionization energy of hydrogen.^{[3]}</ref> Charge neutrality requires n_{e} = n_{p}, and the Saha equation can be rewritten in terms of the free electron fraction x_{e}:
All quantities in the righthand side are known functions of redshift: the temperature is given by T = 2.728 (1 + z) K,^{[4]}</ref> and the total density of hydrogen (neutral and ionized) is given by n_{p} + n_{H} = 1.6 (1+z)^{3} m^{−3}.
Solving this equation for a 50 percent ionization fraction yields a recombination temperature of roughly 4000 K, corresponding to redshift z = 1500.
In 1968, physicists Jim Peebles^{[5]} in the US and Yakov Borisovich Zel'dovich and collaborators^{[6]} in the USSR independently computed the nonequilibrium recombination history of hydrogen. The basic elements of the model are the following.
This model is usually described as an "effective threelevel atom" as it requires keeping track of hydrogen under three forms: in its ground state, in its first excited state (assuming all the higher excited states are in Boltzmann equilibrium with it), and in its ionized state.
Accounting for these processes, the recombination history is then described by the differential equation
where α_{B} is the "case B" recombination coefficient to the excited states of hydrogen, β_{B} is the corresponding photoionization rate and E_{21} = 10.2 eV is the energy of the first excited state. Note that the second term in the righthand side of the above equation can be obtained by a detailed balance argument. The equilibrium result given in the previous section would be recovered by setting the lefthand side to zero, i.e. assuming that the net rates of recombination and photoionization are large in comparison to the Hubble expansion rate, which sets the overall evolution timescale for the temperature and density. However, C α_{B} n_{p} is comparable to the Hubble expansion rate, and even gets significantly lower at low redshifts, leading to an evolution of the free electron fraction much slower than what one would obtain from the Saha equilibrium calculation. With modern values of cosmological parameters, one finds that the universe is 90% neutral at z ≈ 1070.
The simple effective threelevel atom model described above accounts for the most important physical processes. However it does rely on approximations which lead to errors on the predicted recombination history at the level of 10% or so. Due to the importance of recombination for the precise prediction of cosmic microwave background anisotropies,^{[8]} several research groups have revisited the details of this picture over the last two decades.
The refinements to the theory can be divided into two categories:
Modern recombination theory is believed to be accurate at the level of 0.1%, and is implemented in publicly available fast recombination codes.^{[9]}^{[10]}
Helium nuclei are produced during Big Bang nucleosynthesis, and make up about 24% of the total mass of baryonic matter. The ionization energy of helium is larger than that of hydrogen and it therefore recombines earlier. Because neutral helium carries two electrons, its recombination proceeds in two steps. The first recombination, proceeds near Saha equilibrium and takes place around redshift z≈ 6000.^{[11]} The second recombination, , is slower than what would be predicted from Saha equilibrium and takes place around redshift z≈ 2000.^{[12]} The details of helium recombination are less critical than those of hydrogen recombination for the prediction of cosmic microwave background anisotropies, since the universe is still very optically thick after helium has recombined and before hydrogen has started its recombination.
Prior to recombination, photons were not able to freely travel through the universe, as they constantly scattered off the free electrons and protons. This scattering causes a loss of information, and "there is therefore a photon barrier at a redshift" near that of recombination that prevents us from using photons directly to learn about the universe at larger redshifts.^{[13]} Once recombination had occurred, however, the mean free path of photons greatly increased due to the lower number of free electrons. Shortly after recombination, the photon mean free path became larger than the Hubble length, and photons traveled freely without interacting with matter.^{[14]} For this reason, recombination is closely associated with the last scattering surface, which is the name for the last time at which the photons in the cosmic microwave background interacted with matter.^{[15]} However, these two events are distinct, and in a universe with different values for the baryontophoton ratio and matter density, recombination and photon decoupling need not have occurred at the same epoch.^{[14]}