H(a)² = H0²·[Ωm/a³ + ΩΛ]
ä/a = −(4πG/3)(ρ+3p) + Λc²/3
Λ dominates ⇒ ä > 0 (acceleration)
Galaxies are modeled as points fixed in comoving space; the scale factor a(t) stretches their physical separation over time following the Friedmann equations. When dark energy (ΩΛ, modeled as a cosmological constant) dominates over matter, the second derivative of a(t) becomes positive - the expansion accelerates, exactly as observed via Type Ia supernovae in 1998.
- Ω_Λ (dark energy density) — higher values increase the accelerating push, visibly speeding up late-time expansion.
- Ω_m (matter density) — higher values increase early deceleration from gravitational attraction.
- Time slider — scrub through cosmic history from before today to the far future.
- Auto-advance — animates time forward continuously to watch the expansion history play out.
Real-world application: this same Friedmann-equation framework, fit to supernova and CMB data, is how cosmologists determined dark energy makes up about 68% of the universe's energy budget.