Recombination

Recombination

Recombination is one of the most important events in cosmic history. At redshift z ≈ 1089 (about 380,000 years after the Big Bang), the universe had cooled enough that free electrons could combine with protons to form neutral hydrogen atoms.

Why Recombination Matters

Before recombination the universe was opaque: photons were constantly scattered by free electrons. The mean free path of a photon was tiny. At recombination, the electron density dropped dramatically and the universe became transparent. Photons could travel freely for the first time — these are the CMB photons we observe today.

Temperature at Recombination

$T_{rec} = T_0 \cdot (1 + z_{rec}) = 2.725 \times 1090 \approx 2970 \text{ K}$

Thomson Scattering

Before recombination, photons scattered off free electrons via Thomson scattering. The optical depth through a column of ionised gas:

$\tau = \sigma_T \cdot x_e \cdot n_b \cdot L$

where σ_T = 6.652 × 10⁻²⁹ m² is the Thomson cross section, x_e is the ionisation fraction, n_b is the baryon number density, and L is the path length.

Baryon Number Density

The present-day baryon number density:

$n_b = \frac{\Omega_b \cdot \rho_c}{m_p}$

where ρ_c = 3H²/(8πG) is the critical density and m_p = 1.673 × 10⁻²⁷ kg is the proton mass.

Your Task

Implement four functions. All constants must be defined inside each function body.

• recombination_redshift() — returns the standard value 1089.0
• recombination_temperature_K() — returns T₀ × 1090 with T₀ = 2.725 K inside the function
• thomson_optical_depth(x_e, n_b_m3, path_length_m) — returns τ = σ_T × x_e × n_b × L
• baryon_number_density_m3(Omega_b, H0_km_s_Mpc) — returns n_b = Ω_b × ρ_c / m_p