Recombination (cosmology)

In cosmology, recombination refers to the epoch at which charged electrons and protons in the universe first formed electrically neutral hydrogen atoms. After the theoretical Big Bang, the universe was a hot, dense plasma of photons, electrons, and protons. The interaction of photons with the plasma made the universe effectively opaque to radiation. 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 about 1 part in 10,000.

Shortly after, photons decoupled from matter in the universe, which leads to recombination sometimes being called photon decoupling, although 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 we observe today as cosmic microwave background radiation. Recombination occurred when the universe was roughly 380,000 years old, or at a redshift of z = 1100.

Derivation of recombination epoch

It is possible to find a rough estimate of the redshift of the recombination epoch, starting by considering that during the era preceding recombination, the photons were primarily coupled to matter through the reaction

$H+\gamma \longleftrightarrow p+e^{-}.$

This reaction requires that the photon (γ) have an energy of at least 13.6 electron volts. As long as photons are coupled to matter, this reaction will be in statistical equilibrium, and so the Saha equation can be applied to determine the equilibrium values of the constituents. This results in the equation

${\frac {n_{\text{H}}}{n_{\text{p}}n_{\text{e}}}}=\left({\frac {m_{\text{e}}k_{\text{B}}T}{2\pi \hbar ^{2}}}\right)^{-3/2}\exp \left({\frac {Q}{k_{\text{B}}T}}\right),$

where n represents the number density of the subscripted particle, 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 Q is the binding energy of hydrogen. Noting that charge neutrality requires n_e = n_p, and then defining the fractional ionization as

$X={\frac {n_{\text{p}}}{n_{\text{p}}+n_{\text{H}}}},$

the Saha equation can be rewritten as

${\frac {1-X}{X}}=n_{\text{p}}\left({\frac {m_{\text{e}}k_{\text{B}}T}{2\pi \hbar ^{2}}}\right)^{-3/2}\exp \left({\frac {Q}{k_{\text{B}}T}}\right).$

The final step is to put this equation in terms of the number density of photons, which is related to the number density of baryons through the baryon-to-photon ratio η. The reason for this is that the baryon-to-photon ratio can be measured, and the number density of photons is given by

$n_{\gamma }=0.243\left({\frac {k_{\text{B}}T}{\hbar c}}\right)^{3},$

where c is the speed of light. Then

${\frac {1-X}{X^{2}}}=3.84\eta \left({\frac {k_{\text{B}}T}{m_{\text{e}}c^{2}}}\right)^{3/2}\exp \left({\frac {Q}{k_{\text{B}}T}}\right).$

Solving this equation for a 50 percent ionization yields a recombination temperature of roughly 4000 K. This, in turn, gives the redshift as approximately z = 1500, as the temperature of the radiation in the universe is given by T_r = 2.728 (1 + z). In units of electronvolts, this temperature is roughly 0.3 eV. This is nearly two orders of magnitude lower than the binding energy of hydrogen, which may seem strange, as equilibrium often occurs when the energy scales of two phenomena are roughly equal. The reason for the difference is because photons greatly outnumber baryons; the baryon-to-photon ratio is approximately 10⁻⁹. If there are roughly the same number of photons with an energy greater than the binding energy of hydrogen as there are hydrogen atoms, then the gas will remain ionized. There will be some photons in the Wien region of the black body spectrum with an energy greater than kT, and the number of photons with an energy greater than 13.6 eV does not drop below the number of hydrogen atoms until the temperature is roughly 4000K or 0.3 eV. A different statement of this is that recombination was delayed due to the high entropy of the universe.

This derivation relied on the assumptions of thermodynamic equilibrium and recombination directly to the ground state of hydrogen, each of which simplifies the calculation but also modifies the result. Recombination to an excited state of hydrogen means that recombination proceeds more slowly than that predicted with the Saha equation. A more careful treatment of the physics of recombination yields a value closer to z = 1100.

Impact

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 to directly learn about the universe at larger redshifts. 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. 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. However, these two events are distinct, and that 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.

Bibliography

• Galli, S.; Bean, R.; Melchiorri, A.; Silk, J. (2008). "Delayed recombination and cosmic parameters". Physical Review D. 78 (6): 063532. arXiv:0807.1420. doi:10.1103/PhysRevD.78.063532.{{cite journal}}: CS1 maint: multiple names: authors list (link)
• Longair, Malcolm (2006). Galaxy Formation. Springer. ISBN 9783540734772.
• Padmanabhan, Thanu (1993). Structure formation in the universe. Cambridge University Press. ISBN 0521424860.
• Ryden, Barbara (2003). Introduction to Cosmology. Addison-Wesley. ISBN 0805389121.