Lab 18 — Cosmology

The Friedmann Equation — One Line That Sets a Universe’s Whole Life

Assume the universe is homogeneous and isotropic and Einstein’s equations collapse to a single line. All that is left is to say what is inside — radiation Ω_r, matter Ω_m, curvature Ω_k, a cosmological constant Ω_Λ — and the entire history follows. Each component dilutes at its own rate (radiation fastest, then matter, while Λ never dilutes at all), so which one dominates keeps changing, and that ordering *is* the history of the universe. Above is the scale factor a(t); below is the same solution drawn as space itself. Radiation is pinned by the measured quantity Ω_r h² = 4.15×10⁻⁵, with Ω_r derived from H₀. Because radiation is always present, every setting starts radiation-dominated — and exact Einstein–de Sitter (Ω_k = Ω_Λ = 0) is not on the slider: hold flat at Ω_m = 1 and Ω_Λ becomes −9×10⁻⁵, which recollapses at a ≈ 10⁴.

The numbers are real. Feed in the measured values (H₀ = 67.4 km/s/Mpc, Ω_m = 0.315, Ω_Λ = 0.685) and the age comes out 13.79 Gyr, against Planck 2018’s 13.797 ± 0.023 Gyr. The deceleration parameter q₀ = −0.527 is negative — that negative sign is exactly what the 1998 supernova surveys found when they discovered the expansion is speeding up. Λ overtook matter only at z ≈ 0.3, geologically recently; matter and radiation were equal at z ≈ 3400, just before the universe became transparent. The integrator is checked against every case with a closed form: Einstein–de Sitter (2/3H₀), the empty universe (1/H₀), radiation only (1/2H₀), the asinh solution for flat ΛCDM, the closed form for matter + radiation, and the recollapse time of a closed matter universe — to 1e-8 or better for the first five, 2e-4 for the recollapse. Two caveats: the galaxies drawn at small a are comoving markers — for the first few hundred million years there were no galaxies yet — and neutrinos are non-relativistic today, so Ω_r is the effective early-time radiation extrapolated back as a⁻⁴, which is standard and accurate where it matters (a ≪ 1).