Recombination (cosmology)

Cosmological recombination is the epoch in which cooling allowed free electrons and ions to form neutral atoms, sharply reducing the free-electron density.

Approximating decoupling by a single time and retaining only a photon monopole and Doppler dipole omits the finite cosmological visibility function width, diffusion or Silk damping, quadrupole anisotropy and polarization generated as tight coupling fails. The photon-baryon acoustic oscillator also make the emission density and velocity strongly scale dependent. They can be included in the initial monopole and dipole, so their existence alone does not invalidate the subsequent collisionless line-of-sight solution for free-streaming photons. Smaller angles correspond to larger angular multipole $\ell$.

When the effective recombination rate per Electron becomes smaller than the expansion rate, the small remaining ionized fraction can no longer track chemical equilibrium. A nonzero Electron fraction remains, exceeding the exponentially small Saha equation prediction. The effective rate includes atomic radiative bottlenecks; chemical freeze-out is distinct from photon decoupling by scattering.

In hydrogen-only cosmology, charge neutrality gives $n_e=n_p$ and baryon conservation gives $n_b=n_p+n_H$. The fraction $X_e$ lies between zero and one, with $n_H=(1-X_e)n_b$. With helium present, Electron abundance per hydrogen nucleus and per total baryon are different conventions and must be specified. The hydrogen-only Saha equation gives the equilibrium fraction.

Photon decoupling occurs when the Thomson-scattering rate drops below the Hubble expansion rate. Photons then free stream and form the cosmic microwave background.

The physical Thomson scattering rate equals the inverse photon mean scattering time. Comparing it with the Hubble parameter gives a local approximate decoupling criterion. A more precise last-scattering temperature is obtained from optical depth and the visibility maximum; these conventions need not give exactly the same temperature.

The recombination temperature is the photon temperature at which the free-electron fraction falls sharply as neutral atoms form. For hydrogen it is about $0.3\mathrm{eV}$ because the photon-to-baryon ratio is very large.

For hydrogen ionization fraction $X_e$ of order one, the hydrogen-only Saha equation gives $1\sim n_b(2\pi /(m_{e}T){)}^{3/2}{e}^{B/T}$. The baryon-to-photon ratio and photon number density imply $n_b\sim \eta T^3$. Therefore $1\sim \eta (T/m_e{)}^{3/2}{e}^{B/T}$ and the displayed logarithmic balance follows. A small $\eta$ supplies many ionizing photons per baryon, so neutral hydrogen forms at a temperature much smaller than its Hydrogen binding energy. The numerical temperature depends on the ionization-fraction convention and on whether equilibrium remains valid.

With the rounded hydrogen Saha equation coefficient, the condition $X_e=1/2$ gives the displayed equation for $y=\mathcal{B}/T$. Its low-temperature root is about $42$, hence $T\simeq 0.32$ eV for hydrogen binding energy $13.6$ eV. Dropping the slowly varying logarithm gives the rough leading-log estimate $T\sim 0.4$ eV. This equilibrium convention is distinct from photon decoupling and from a full kinetic recombination calculation.

Chemical equilibrium for $A^{+}+e^{-}\leftrightarrow A^0+\gamma$ givesafter neglecting the ion--atom mass difference in the translational prefactor.

For classical nondegenerate Electrons, the free-particle Boltzmann occupancy has the form $z_e\exp [-p^2/(2mkT)]$, with chemical potential measured relative to its rest energy. Counting the two spin states in phase space gives $n_e=2z_e(2\pi mkT/h^2{)}^{3/2}$. A bound level of energy $-I_n$ has relative occupancy $(g_n/g_{+})z_e{e}^{I_n/kT}$ against its parent ion. Eliminating $z_e$ gives the level-specific Saha ionization equation.

Because equilibrium reactions balance chemical potentials and photons have zero chemical potential,The rest-mass difference in the Maxwell--Boltzmann densities then supplies the Boltzmann factor $e^{-I_A/(k_{B}T)}$.

For $p+e^{-}\leftrightarrow H+\gamma$, the hydrogen binding energy is the rest-energy defect

With singly ionized helium and helium mass fraction $Y_p$, charge neutrality gives

Writing $\mathcal{F}=n_e/n_B$ couples the two equilibrium equations:

When $Y_p=0$, $\mathcal{F}=X_{H^{+}}$, and

For a hydrogen-only gas with $g_e=g_p=2$, $g_H=4$ and $m_H\simeq m_p$, the Saha equation is $(1-X_e)/X_e^2=n_B[2\pi /(m_{e}T){]}^{3/2}{e}^{I/T}$. Defining ${\eta }_{\gamma }=n_B/n_{\gamma }$ and using $n_{\gamma }=2\zeta (3)T^3/{\pi }^2$ gives coefficient $2\zeta (3){\eta }_{\gamma }/{\pi }^2$ multiplying $(2\pi T/m_e{)}^{3/2}{e}^{I/T}$. If instead ${\eta }_s=n_B/s$, with entropy density $s=(2{\pi }^2/45)g_{\ast s}{T}^3$, that coefficient is $(2{\pi }^2/45)g_{\ast s}{\eta }_s$. The two expressions agree because $s/n_{\gamma }={\pi }^4{g}_{\ast s}/[45\zeta (3)]$. The reference entropy sector must be specified, especially after particles have decoupled.