13.4 Recombination and Decoupling

Recombination and Decoupling

The Process of Recombination

Recombination is the era when free protons and electrons in the primordial plasma combined to form neutral hydrogen atoms. Despite the name, these particles had never been combined before, so "recombination" is a historical misnomer that has stuck.

This transition occurred roughly 380,000 years after the Big Bang, once the universe cooled to about 3000 K. Above that temperature, photons carried enough energy to immediately re-ionize any hydrogen atom that formed. Below it, electrons could finally bind to protons and stay bound.

As neutral hydrogen accumulated, the number of free electrons dropped sharply. Fewer free electrons meant fewer targets for photon scattering, which set the stage for decoupling. The process was gradual rather than instantaneous, unfolding over a redshift range of roughly $z \approx 1400$ to $z \approx 1100$.

The Saha equation describes the ionization equilibrium during this epoch. It gives the balance between ionized and neutral hydrogen as a function of temperature. The ionization fraction $x_e = n_e / (n_H + n_p)$ quantifies what fraction of hydrogen remains ionized at a given temperature. During recombination, $x_e$ drops from near unity to a small residual value.

Decoupling and the Cosmic Microwave Background

Decoupling is the direct consequence of recombination. As free electrons vanished into neutral atoms, photons lost their scattering partners and began streaming freely through the universe. This is the moment the universe became transparent to radiation.

The photons released at decoupling have been propagating ever since, redshifting with the expansion of the universe. Today they form the Cosmic Microwave Background (CMB), observed at a nearly uniform temperature of about 2.725 K.

The CMB is not perfectly uniform. Tiny temperature fluctuations at the level of $\Delta T / T \sim 10^{-5}$ are imprinted across the sky. These anisotropies reflect density variations in the photon-baryon fluid at the time of decoupling. Overdense regions were slightly hotter (or cooler, depending on competing effects like gravitational redshift), and these density contrasts eventually grew under gravity into the galaxies, clusters, and large-scale structure we observe today.

Mathematical Description of Recombination

The Saha equation governs the ionization equilibrium under the assumption that recombination and ionization reactions remain in thermal equilibrium:

$\frac{n_e n_p}{n_H} = \left(\frac{2\pi m_e k_B T}{h^2}\right)^{3/2} e^{-E_I / k_B T}$

where:

• $n_e$, $n_p$, $n_H$ are the number densities of electrons, protons, and neutral hydrogen
• $m_e$ is the electron mass
• $k_B$ is Boltzmann's constant, $h$ is Planck's constant
• $T$ is the radiation temperature
• $E_I = 13.6 \text{ eV}$ is the ionization energy of hydrogen

The exponential factor $e^{-E_I / k_B T}$ dominates the temperature dependence. As $T$ drops below $\sim 0.3 \text{ eV}$ (roughly 3000 K), this factor plummets and the equilibrium shifts heavily toward neutral hydrogen.

The Saha equation predicts a very sharp transition in $x_e$. In reality, recombination proceeds more slowly than the Saha prediction because the process falls out of equilibrium. Direct recombination to the ground state produces a Lyman-continuum photon energetic enough to immediately ionize another atom, so net recombination proceeds mainly through excited states (especially via two-photon decay from the $2s$ state). The Peebles three-level atom model captures this non-equilibrium behavior and yields a residual ionization fraction of $x_e \sim 10^{-3}$ that freezes out rather than dropping to zero.

Last Scattering Surface

Concept and Significance of the Last Scattering Surface

The last scattering surface is the spherical shell in spacetime from which CMB photons had their final scattering interaction before reaching us. It represents the earliest epoch we can observe directly with electromagnetic radiation.

Think of it as a cosmic boundary: photons arriving from this surface carry information about conditions at $z \approx 1100$, but anything happening at earlier times (higher redshift) is hidden behind the opaque plasma. The last scattering surface is not infinitely thin. Because recombination took a finite amount of time, it has a characteristic thickness of $\Delta z \sim 80$, corresponding to a comoving depth of roughly 10 Mpc. This finite thickness slightly smears out CMB anisotropies on the smallest angular scales.

Physics of Photon-Matter Interactions

Before recombination, the dominant photon-matter interaction was Thomson scattering: the elastic scattering of photons off free electrons. The Thomson cross-section is:

$\sigma_T = \frac{8\pi}{3}\left(\frac{e^2}{m_e c^2}\right)^2 \approx 6.65 \times 10^{-25} \text{ cm}^2$

where $e$ is the electron charge, $m_e$ is the electron mass, and $c$ is the speed of light. This cross-section is independent of photon frequency (valid as long as photon energies are well below $m_e c^2$, which holds throughout recombination).

The optical depth $\tau$ of the universe depends on both $\sigma_T$ and the free electron density $n_e$. Before recombination, $n_e$ was high and the universe was optically thick ($\tau \gg 1$), meaning photons scattered many times over short distances. As recombination progressed and $n_e$ plummeted, $\tau$ dropped below unity and photons could escape. The last scattering surface corresponds to the epoch where $\tau \approx 1$.

Photon Mean Free Path and Universe Transparency

The photon mean free path is the average distance a photon travels between successive Thomson scatterings:

$\lambda_{\text{mfp}} = \frac{1}{n_e \, \sigma_T}$

Before recombination, $n_e$ was large and $\lambda_{\text{mfp}}$ was tiny compared to cosmological scales. As $n_e$ dropped during recombination, $\lambda_{\text{mfp}}$ grew rapidly.

The universe became effectively transparent when the mean free path exceeded the Hubble radius:

$\lambda_{\text{mfp}} > c \, H^{-1}$

where $H$ is the Hubble parameter at that epoch. Once this condition was satisfied, a typical photon would never scatter again within a Hubble time, and it could free-stream to us as part of the CMB.

Because the transition in $x_e$ was gradual, the shift from opaque to transparent was not a sharp boundary. The finite duration of this transition gives the last scattering surface its thickness, which in turn sets a damping scale for CMB anisotropies. Fluctuations on angular scales smaller than this thickness are washed out, an effect known as Silk damping (or diffusion damping), which suppresses power in the CMB at high multipoles.