Cosmic Recombination — The Saha Equation & Last Scattering

Cosmic Epoch

Redshift z 1100

Baryon density Ω_bh² 0.0224

Cooling speed 1.0×

Ionization State

ionized 99%neutral 1%

Fully ionized plasma

Temperature T

3003 K

Ionization X_e

0.4%

Scale factor a

1/1101

Photon scatters/s

How it works

As the universe expands, its temperature falls as T(z) = T₀(1+z) with T₀ = 2.725 K. Above ≈4000 K, thermal photons keep hydrogen fully ionized (a proton–electron plasma). The equilibrium ionization fraction X_e = n_e/n_b is set by the Saha equation, solved live every frame from real physical constants:

Xe² / (1 − Xe) = S(T, n_b)

S = (1/n_b) · ( m_e k_B T / 2πℏ² )^3/2 · exp(−B / k_B T)

Xe = [ −S + √(S² + 4S) ] / 2

B = 13.6 eV is the hydrogen binding energy and n_b = n_b,0·(1+z)³ is the baryon number density, set by the Ω_bh² slider. The exp(−B/k_B T) term makes the transition razor-sharp: a swing of only a few hundred kelvin takes X_e from ≈1 to ≈10⁻³ around T ≈ 3000–4000 K (z ≈ 1100–1400) — cosmological recombination.

The bright orbiting particle is a CMB photon undergoing Thomson scattering off free electrons. Its mean free path scales as ℓ ∝ 1/(X_e·n_b), so the model schedules its next bounce after a randomized interval proportional to that mean free path. While X_e is high it scatters constantly (the plasma is opaque); once X_e drops below ≈1%, bounces become rare and the photon effectively free-streams — the moment this happens is the last scattering surface, the origin of the CMB we observe today.

Redshift z — sets the cosmic epoch directly; drag it down through z ≈ 1100–1400 to watch recombination happen.
Ω_bh² — the universe's baryon density; a denser universe recombines at a slightly higher redshift (Saha's n_b dependence).
Cool the universe — animates z decreasing smoothly over time, like watching cosmic history play forward.

Free proton (ionized) Neutral hydrogen atom CMB photon

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