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Incosmology,recombination refers to theepoch during which chargedelectrons andprotons first becamebound to formelectrically neutralhydrogenatoms. Recombination occurred about378000 years^[1] after theBig Bang (at aredshift ofz = 1100).^[2] The word "recombination" is misleading, since the Big Bang theory does not 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 birth of the universe.

Immediately after theBig Bang, the universe was a hot, denseplasma ofphotons,leptons, andquarks: thequark epoch. At 10⁻⁶ seconds, the Universe had expanded and cooled sufficiently to allow for the formation ofprotons: thehadron epoch. This plasma was effectively opaque to electromagnetic radiation due toThomson scattering by free electrons, as themean 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 universeexpanded, it also cooled. Eventually, the universe cooled to the point that the radiation field could not immediately ionize neutral hydrogen, and atoms became energetically favored.^[3] The fraction of free electrons and protons as compared to neutral hydrogen decreased to a few parts in10000.

Recombination involves electrons binding to protons (hydrogen nuclei) to form neutralhydrogenatoms. Because direct recombinations to theground state (lowest energy) of hydrogen are very inefficient,^{[clarification needed]} these hydrogen atoms generally form with the electrons in a high energy state, and the electrons quickly transition to their low energy state by emittingphotons. Two main pathways exist: from the 2p state by emitting aLyman-a 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.^{[clarification needed]}

This production of photons is known asdecoupling, which leads to recombination sometimes being calledphoton decoupling, but recombination and photon decoupling are distinct events. Once photons decoupled from matter, theytraveled freely through the universe without interacting with matter and constitute what is observed today ascosmic microwave background radiation (in that sense, the cosmic background radiation isinfrared and some redblack-body radiation emitted when the universe was at a temperature of some 3000 K,redshifted by a factor of1100 from the visible spectrum to themicrowave spectrum).

The time frame for recombination can be estimated from the time dependence of the temperature of thecosmic microwave background (CMB).^[4] The microwave background is ablackbody spectrum representing the photons present at recombination, shifted in energy by the expansion of the universe. A blackbody is completely characterized by its temperature; the shift is called theredshift denoted byz:$${\displaystyle T_{\text{CMB}}=2.7\mathrm{K}\times (1+z)}$$where 2.7 K is today's temperature.

The thermal energy at the peak of the blackbody spectrum is theBoltzmann constant,k_B, times the temperature,${\displaystyle k_B{T}_{\text{CMB}}(z)}$ but simply comparing this to the ionization energy of hydrogen atoms will not consider the spectrum of energies. A better estimate evaluates the thermal equilibrium between matter (atoms) and radiation. The density of photons,${\displaystyle n_{\gamma }(E>Q_H)}$ with energyE sufficient to ionize hydrogen is the total density times a factor from the equilibriumBoltzmann distribution:$${\displaystyle n_{\gamma }(E>Q_H)=n_{\gamma }\exp \left(\frac{-Q_H}{k_B{T}_{\text{CMB}}(z)}\right)}$$At equilibrium this will approximately equal the matter (baryon) density. The ratio of baryons to photons,${\displaystyle \eta }$, is known from several sources^[1] including measurements by thePlanck satellite to be around 10^-9. Solving for${\displaystyle z_{\text{rec}}}$ gives value around 1100, which converts to acosmic time value around 400,000 years.

The cosmic ionization history is generally described in terms of the free electron fractionx_e as a function ofredshift. It is the ratio of the abundance of free electrons to the total abundance of hydrogen (both neutral and ionized). Denoting byn_e the number density of free electrons,n_H that of atomic hydrogen andn_p that of ionized hydrogen (i.e. protons),x_e is defined as

${\displaystyle x_{\text{e}}=\frac{n_{\text{e}}}{n_{\text{p}}+n_{\text{H}}}.}$

Since hydrogen only recombines once helium is fully neutral, charge neutrality impliesn_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${\displaystyle p+e^{-}⟷H+\gamma }$ is fast enough that it proceeds near thermal equilibrium. The relative abundance of free electrons, protons and neutral hydrogen is then given by theSaha equation:

${\displaystyle \frac{n_{\text{p}}{n}_{\text{e}}}{n_{\text{H}}}={\left(\frac{m_{\text{e}}{k}_{\text{B}}T}{2\pi {\hslash }^2}\right)}^{\frac{3}{2}}\exp \left(-\frac{E_{\text{I}}}{k_{\text{B}}T}\right),}$

wherem_e is themass of the electron,k_B is theBoltzmann constant,T is the temperature,ħ is thereduced Planck constant, andE_I = 13.6 eV is theionization energy of hydrogen.^[5] Charge neutrality requiresn_e = n_p, and the Saha equation can be rewritten in terms of the free electron fractionx_e:

${\displaystyle \frac{x_{\text{e}}^2}{1-x_{\text{e}}}=(n_{\text{H}}+n_{\text{p}}{)}^{-1}{\left(\frac{m_{\text{e}}{k}_{\text{B}}T}{2\pi {\hslash }^2}\right)}^{\frac{3}{2}}\exp \left(-\frac{E_{\text{I}}}{k_{\text{B}}T}\right).}$

All quantities in the right-hand side are known functions of z, theredshift: the temperature is given byT = (1 +z) × 2.728 K,^[6] and the total density of hydrogen (neutral and ionized) is given byn_p +n_H = (1 +z)³ × 1.6 m⁻³.

Solving this equation for a 50 percent ionization fraction yields a recombination temperature of roughly4000 K, corresponding to redshiftz = 1500.

In 1968, physicistsJim Peebles^[7] in the US andYakov Borisovich Zel'dovich and collaborators^[8] in the USSR independently computed the non-equilibrium recombination history of hydrogen. The basic elements of the model are the following.

• Direct recombinations to the ground state of hydrogen are very inefficient: each such event leads to a photon with energy greater than 13.6 eV, which almost immediately re-ionizes a neighboring hydrogen atom.
• Electrons therefore only efficiently recombine to the excited states of hydrogen, from which they cascade very quickly down to the first excited state, withprincipal quantum numbern = 2.
• From the first excited state, electrons can reach the ground staten = 1 through two pathways:
  • Decay from the 2p state by emitting aLyman-α photon. This photon will almost always be reabsorbed by another hydrogen atom in its ground state. However, cosmological redshifting systematically decreases the photon frequency, and there is a small chance that it escapes reabsorption if it gets redshifted far enough from the Lyman-α line resonant frequency before encountering another hydrogen atom.
  • Decay from the 2s state by emitting two photons. Thistwo-photon decay process is very slow, with a rate^[9] of 8.22 s⁻¹. It is however competitive with the slow rate of Lyman-α escape in producing ground-state hydrogen.
• Atoms in the first excited state may also be re-ionized by the ambientCMB photons before they reach the ground state. When this is the case, it is as if the recombination to the excited state did not happen in the first place. To account for this possibility, Peebles defines the factorC as the probability that an atom in the first excited state reaches the ground state through either of the two pathways described above before being photoionized.

This model is usually described as an "effective three-level 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 inBoltzmann equilibrium with it), and in its ionized state.

Accounting for these processes, the recombination history is then described by thedifferential equation

${\displaystyle \frac{d{x}_{\text{e}}}{dt}=-C\left({\alpha }_{\text{B}}(T)n_{\text{p}}{x}_e-4(1-x_{\text{e}}){\beta }_{\text{B}}(T)e^{-E_{21}/T}\right),}$

whereα_B is the "case B" recombination coefficient to the excited states of hydrogen,β_B is the corresponding photoionization rate andE₂₁ = 10.2 eV is the energy of the first excited state. Note that the second term in the right-hand side of the above equation can be obtained by adetailed balance argument. The equilibrium result given in the previous section would be recovered by setting the left-hand side to zero, i.e. assuming that the net rates of recombination and photoionization are large in comparison to theHubble expansion rate, which sets the overall evolution timescale for the temperature and density. However,C α_Bn_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 atz ≈ 1070.

The simple effective three-level atom model described above accounts for the most important physical processes. However it does rely on approximations that 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 ofcosmic microwave background anisotropies,^[10] 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:

• Accounting for the non-equilibrium populations of the highly excited states of hydrogen. This effectively amounts to modifying the recombination coefficientα_B.
• Accurately computing the rate of Lyman-α escape and the effect of these photons on the 2s–1s transition. This requires solving a time-dependentradiative transfer equation. In addition, one needs to account for higher-orderLyman transitions. These refinements effectively amount to a modification of Peebles'C factor.

Modern recombination theory is believed to be accurate at the level of 0.1%, and is implemented in publicly available fast recombination codes.^[11][12]

Helium nuclei are produced duringBig Bang nucleosynthesis, and make up about 24% of the total mass ofbaryonic matter. Theionization 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,${\displaystyle {\mathrm{H}\mathrm{e}}^{2+}+{\mathrm{e}}^{-}⟶{\mathrm{H}\mathrm{e}}^{+}+\gamma }$ proceeds near Saha equilibrium and takes place around redshiftz ≈ 6000.^[13] The second recombination,${\displaystyle {\mathrm{H}\mathrm{e}}^{+}+{\mathrm{e}}^{-}⟶\mathrm{H}\mathrm{e}+\gamma }$, is slower than what would be predicted from Saha equilibrium and takes place around redshiftz ≈ 2000.^[14] The details of helium recombination are less critical than those of hydrogen recombination for the prediction ofcosmic 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 constantlyscattered 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.^[15] 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 theHubble length, and photons traveled freely without interacting with matter.^[16] 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.^[17] However, these two events are distinct, and in a universe with different values for the baryon-to-photon ratio and matter density, recombination and photon decoupling need not have occurred at the same epoch.^[16]

• Chronology of the universe
• Age of the universe
• Big Bang

• Bromm, Volker (17 April 2014)."AST 376 Cosmology—Lecture Notes: Cosmic Microwave Background (CMB)"(PDF). Austin, Texas: Department of Astronomy,University of Texas at Austin.Archived(PDF) from the original on 23 December 2018. Retrieved1 January 2020.
• Longair, Malcolm (2008).Galaxy Formation.Springer.ISBN 978-3-540-73477-2.
• Maoz, Dan (2016).Astrophysics in a Nutshell (2nd ed.). Princeton, New Jersey; Oxford, UK:Princeton University Press.ISBN 978-0-691-16479-3.LCCN 2015956047.OCLC 950932058.
• Padmanabhan, Thanu (1993).Structure formation in the universe.Cambridge University Press.ISBN 978-0-521-42486-8.
• Peebles, P. J. E. (1968). "Recombination of the Primeval Plasma".The Astrophysical Journal.153: 1.Bibcode:1968ApJ...153....1P.doi:10.1086/149628.
• Ryden, Barbara (2003).Introduction to Cosmology.Addison-Wesley.ISBN 978-0-8053-8912-8.
• Tanabashi, M.; et al. (Particle Data Group) (2018)."Review of Particle Physics".Physical Review D.98 (3). College Park, Maryland:American Physical Society:1–708.Bibcode:2018PhRvD..98c0001T.doi:10.1103/PhysRevD.98.030001.hdl:10044/1/68623.ISSN 1550-7998.OCLC 7814919666.PMID 10020536.
• Zeldovich, Y. B.; Kurt, V. G.; Syunyaev, R. A. (1968). "Recombination of Hydrogen in the Hot Model of the Universe".Zhurnal Eksperimental'noi i Teoreticheskoi Fiziki.55: 278.Bibcode:1968ZhETF..55..278Z.

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