Changes for Recombination

The recombination history of hydrogen

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

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

Since hydrogen only recombines once helium is fully neutral, charge neutrality implies n_e = n_p, i.e. x_e is also the fraction of ionized hydrogen.

Rough estimate from equilibrium theory

It is possible to find a rough estimate of the redshift of the recombination epoch assuming the recombination reaction ${\displaystyle p+e^{-}\longleftrightarrow H+\gamma }$ is fast enough that it proceeds near thermal equilibrium. The relative abundance of free electrons, protons and neutral hydrogen is then given by the Saha equation:

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

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.^[2] Charge neutrality requires n_e = n_p, and the Saha equation can be rewritten in terms of the free electron fraction x_e:

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

All quantities in the right-hand side are known functions of redshift: the temperature is given by T = 2.728 (1 + z) K,^[3] 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.

The effective three-level atom (TLA)

In 1968, physicists Jim Peebles^[4] in the US and Yakov Borisovich Zel'dovich and collaborators^[5] 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, with principal quantum number n = 2.
• From the first excited state, electrons can reach the ground state n =1 through two pathways:
  • Decay from the 2p state by emitting a Lyman-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. This two-photon decay process is very slow, with a rate^[6] 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 ambient CMB 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 factor C 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 in Boltzmann equilibrium with it), and in its ionized state.

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

${\displaystyle {\frac {dx_{\text{e}}}{dt}}=-C\left(\alpha _{\text{B}}(T)n_{\text{p}}x_{e}-4(1-x_{\text{e}})\beta _{\text{B}}(T)\exp \left(-{\frac {E_{21}}{k_{B}T}}\right)\right),}$

where E₂₁ = 10.2 eV is the energy of the first excited state, α_B is the "case B" recombination coefficient to the excited states of hydrogen, and β_B is the corresponding photoionization rate which is related to α_B by:

${\displaystyle \mathbf {\beta _{B}(T)=\alpha _{B}(T)\left({\frac {m_{e}k_{B}T}{2\pi \hbar ^{2}}}\right)^{\frac {3}{2}}\exp \left(-{\frac {E_{I}-E_{21}}{k_{B}T}}\right).} }$

Note that the second term in the right-hand side of the differential equation can be obtained by a detailed 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 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.

Modern developments

The simple effective three-level atom model described above accounts for the most important physical processes. However, it does rely on approximations which lead to errors in the predicted recombination history at the level of 10% or so. Due to the rapid development of observations for the CMB temperature anisotropies (especially on small scales), more details about the recombination are needed. For example, the damping scale of diffusion damping is related to the free-electron number density n_e determined by recombination. Inaccurate estimation of the damping scale will result in a biased prediction of the CMB radiation power on small scales, which in response affects our understanding of the matter, baryonic matter fraction, etc.^[7] Several research groups have revisited the details of the picture for recombination over the last two decades.

The refinements of the theory can be divided into two categories:^[1]

• Accurately computing the rate of Lyman-α escape, higher-order Lyman transitions and the effect of these photons on the 2s-1s transition. This requires solving a time-dependent radiative transfer equation, which includes influences from the two-photon transitions and the frequency diffusion in the Lyman-α line. These refinements effectively amount to modifying the Peebles' C factor, and are most prominent at the early stage of recombination (z ≳ 1000). At this stage, the recombination rate strongly depended on the small net decay rate from n = 2 to n = 1 state.

• Accounting for the non-equilibrium populations of the highly excited states of hydrogen. This was achieved by the multilevel atom method (MLA) that is able to consider an arbitrarily large number of excited states of hydrogen.^[8] This refinement effectively amounts to modifying the recombination coefficient α_B, and is most prominent at the late stage of recombination. It introduces a 10% difference in x_e when z ≲ 800, and causes the hydrogen recombination to finish earlier.

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], modified from the MLA method.

Primordial helium recombination

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, ${\displaystyle \mathrm {He} ^{2+}+\mathrm {e} ^{-}\longrightarrow \mathrm {He} ^{+}+\gamma }$ proceeds near Saha equilibrium and takes place around redshift z ≈ 6000.^[11] The second recombination, ${\displaystyle \mathrm {He} ^{+}+\mathrm {e} ^{-}\longrightarrow \mathrm {He} +\gamma }$, is slower than what would be predicted from Saha equilibrium due to the low net decay rate from He I to the ground state.^[12] However, at z < 2200, the H I population increased rapidly, keep absorbing more and more photons created in He I recombination. This phenomenon speeded up the He I recombination, and caused the He I recombination to finish around z ≈ 1800.^[12] A faster helium recombination results in smaller x_e, which will lead to a lower small scale CMB radiation power, for both the temperature and the polarization.

Since the helium recombination happened much earlier than the hydrogen recombination, the details of it are less critical than those of the hydrogen for the prediction of cosmic microwave background anisotropies (at most 3% change on x_e).^[12] The universe was still very optically thick after helium had recombined and before hydrogen started its recombination.

Primordial light barrier

Prior to recombination, photons were not able to freely travel through the universe, as they constantly scattered off the free electrons and protons. The scattering rate can be expressed as

${\displaystyle \mathbf {n_{e}\sigma _{t}=(n_{p}+n_{H})x_{e}\sigma _{t}} }$

where σ_t is the Thomson cross-section. 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] When this scattering rate is smaller than the expansion rate of the universe, photons will decouple from the matter, and start to travel freely without interacting with matter (free streaming). An order of magnitude evaluation gives^[14]

${\displaystyle \mathbf {{\frac {n_{e}\sigma _{t}}{H}}\simeq 10^{2}x_{e},} }$

where H, stands for the expansion rate, is the Hubble parameter. From this equation we see when x_e was of order 10^-2, photons started free streaming. Because the free electron fraction x_e dropped rapidly from 1 to 10^-3 during recombination, the 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 baryon-to-photon ratio and matter density, recombination and photon decoupling need not have occurred at the same epoch.^[16] Detailed analysis shows that the decoupling happened approximately at T = 3000 K (z ≈ 1100) in our universe, earlier than that using the simple n_eσ_t = H criterion.^[17]

Changes of Big Bang nucleosynthesis

Measurements and status of theory

The theory of BBN gives a detailed mathematical description of the production of the light "elements" deuterium, helium-3, helium-4, and lithium-7. Modern BBN theory can reach a precision of 10^-4 for the mass fraction of helium-4.^[18] Specifically, the theory yields precise quantitative predictions for the mixture of these elements, that is, the primordial abundances at the end of the big-bang.

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References