Flatness Problem

Why is Ω = 1 to 60 decimal places? Inflation flattens spacetime geometry exponentially

Ω (today)
1.0000
|Ω-1| at Planck
10⁻⁶⁰
Curvature k
≈0
N e-folds
60
The Flatness Problem: Ω ≡ ρ/ρ_critical is an unstable fixed point — in standard cosmology (Ω-1) grows as ∝a² (matter era). For |Ω-1| ~ 0.01 today, we need |Ω-1| < 10⁻⁶⁰ at the Planck epoch. This extraordinary fine-tuning screams for an explanation. Inflation provides it: during exponential expansion a → a·e^N, the curvature term K/(aH)² ∝ e^(-2N) is driven to zero. With N≥60, any initial Ω is flattened to 1 regardless of starting value.