Recombination and the Cosmic Microwave Background

For its first 380,000 years the universe was an opaque plasma: free electrons scattered photons so efficiently that light could not travel a meaningful distance before being deflected. When the temperature fell far enough for electrons to bind to protons, the free-electron density collapsed, the scattering stopped, and the photons that had last scattered streamed to us across the entire subsequent history of the universe. Those photons are the cosmic microwave background (CMB), the oldest electromagnetic image available and a near-perfect blackbody. This lesson derives the ionization history through the Saha equation, locates the surface of last scattering, and characterizes the CMB spectrum and its dipole.

Recombination and the Saha equation

The relevant reaction is the capture of an electron by a proton to form neutral hydrogen, balanced by photoionization,

with binding energy $B=13.6\text{eV}$. Define the ionization fraction as the fraction of baryons in free protons,

where charge neutrality sets $n_e=n_p$. In chemical equilibrium the relative abundances of the three species follow the Saha equation, obtained by equating chemical potentials with each equilibrium density given by its non-relativistic phase-space integral:

Dividing by $n_b$ and using $n_e=n_p=X{n}_b$ and $n_{\text{H}}=(1-X)n_b$ gives the Saha equation in the form that determines $X(T)$:

The baryon density $n_b=\eta n_{\gamma }\propto T^3$ is known from the baryon-to-photon ratio. The right side is a steep function of temperature through the exponential, so $X$ falls sharply once $k_{B}T$ drops well below $B$.

The transition happens at a temperature well below the binding energy for the same reason as the deuterium bottleneck: photons outnumber baryons by $\sim 10^9$, so the high-energy tail of the blackbody spectrum keeps hydrogen ionized until $k_{B}T\ll B$. Setting $X=1/2$ in the Saha equation gives the temperature of half-recombination,

roughly forty times cooler than $B/k_B$. The ionization fraction plunges from near unity to below $10^{-3}$ over a redshift interval of only a few hundred.

The Saha equation assumes equilibrium, which fails at the end of recombination. As $X$ drops, the recombination rate $\propto n_e{n}_p\propto X^2$ falls faster than the expansion, so the reaction cannot keep the ionization at its equilibrium value. A full treatment with the recombination rate equations (the Peebles analysis) shows the true ionization freezes out at a residual $X\approx 10^{-4}$ rather than continuing to zero. Recombination also proceeds not directly to the ground state — each such capture emits a photon that immediately reionizes another atom — but through the two-photon decay of the metastable 2s level and the redshifting of Lyman-$\alpha$ photons, both slow. These delays push the completion of recombination and photon decoupling to slightly lower redshift than the Saha estimate.

The surface of last scattering

Photons in the plasma scatter off free electrons by Thomson scattering, with cross section ${\sigma }_T=6.65\times 10^{-29}{\text{m}}^2$. The scattering rate per photon is $\Gamma =n_e{\sigma }_{T}c$, and the mean free path is $1/(n_e{\sigma }_T)$. While $X\approx 1$, the free-electron density is high and the mean free path is tiny; the universe is optically thick and photons random-walk. As $X$ collapses during recombination, $n_e$ drops by orders of magnitude, the mean free path grows past the Hubble length, and photons decouple — they travel freely thereafter.

The optical depth from an observer back to redshift $z$ is

and photon decoupling occurs where $\tau \sim 1$. This happens at

Because decoupling is not instantaneous, the last-scattering event of a given photon is spread over a range of redshift; the visibility function $g(z)=e^{-\tau }\mathbf{d}\tau /\mathbf{d}z$, the probability that a photon last scattered at $z$, is a peaked distribution of width $\Delta z\approx 80$ centered on $z_{\text{dec}}$. The surface of last scattering is therefore a shell of finite thickness surrounding every observer, at a comoving distance equal to the present particle horizon minus a negligible correction.

Before decoupling the photons were tightly coupled to the baryons in a single photon-baryon fluid; after decoupling they free-stream, preserving the temperature they had at last scattering, redshifted by the expansion. The CMB is thus a snapshot of the universe at $z\approx 1090$, and its tiny temperature variations, treated in the next lesson, are a direct image of the density field at that epoch.

The blackbody spectrum

Before decoupling, frequent interactions — Thomson scattering, together with double Compton and bremsstrahlung that create and destroy photons — drove the radiation to a blackbody spectrum. Free expansion after decoupling preserves the blackbody form, because redshifting a Planck spectrum of temperature $T$ produces a Planck spectrum of temperature $T/(1+z)$. The present CMB is therefore predicted to be a blackbody, and it is: the COBE/FIRAS instrument measured

with deviations from a pure Planck spectrum below one part in $10^4$, making the CMB the most perfect blackbody known. The spectral radiance follows the Planck function

peaking near $\nu \approx 160\text{GHz}$ (wavelength $\approx 2\text{mm}$), in the microwave band. The energy density is ${\epsilon }_{\gamma }=a_{\text{rad}}{T}_0^4$ and the number density $n_{\gamma ,0}\approx 411{\text{cm}}^{-3}$, the same photons whose abundance relative to baryons fixed $\eta$ in nucleosynthesis.

The blackbody form is itself a stringent test. A universe that was merely cold and dilute, without a hot dense past, would have no mechanism to thermalize starlight into so perfect a Planck spectrum across the whole sky. The FIRAS spectrum is among the most direct pieces of evidence for the hot Big Bang.¹

The dipole anisotropy

The CMB is isotropic to about one part in $10^3$, but at that level it shows a dipole: one hemisphere of the sky is slightly hotter, the opposite slightly cooler. This is a Doppler effect from the observer's motion through the CMB rest frame. An observer moving with speed $v=\beta c$ sees the temperature vary with angle $\theta$ from the direction of motion as

a dipole of amplitude $\Delta T=\beta T_0$ to first order. The measured dipole amplitude is $\Delta T\approx 3.4\text{mK}$, corresponding to

This is the velocity of the Solar System with respect to the CMB rest frame, compounded of the Sun's orbit in the Galaxy, the Galaxy's motion in the Local Group, and the Local Group's infall toward the Great Attractor. The dipole is a kinematic foreground, subtracted before the primordial anisotropies are analyzed.

Recombination is not the last time the gas changes ionization state. When the first stars and galaxies formed, at $z\sim 6$–$10$, their ultraviolet light reionized the intergalactic hydrogen. Reionization scatters a small fraction of CMB photons on their way to us, quantified by the optical depth to reionization $\tau \approx 0.054$. This scattering slightly damps the small-scale anisotropies and generates a large-scale polarization signal, from which $\tau$ is measured; it is one of the six parameters of the concordance model. Reionization does not erase the last-scattering surface — the universe is far too dilute at $z\sim 8$ to become opaque again — but it is a second, partial scattering screen between us and the CMB.

Once the dipole is removed, the residual temperature fluctuations are at the $10^{-5}$ level and are primordial — the density perturbations at last scattering. Their statistics, encoded in the angular power spectrum, are the subject of the next lesson on CMB anisotropies and cosmological parameters, where the positions and heights of the acoustic peaks measure the geometry and contents of the universe.