Omega_K Curvature ~10^{-3}

The curvature parameter Ω_K measures the deviation of the universe's total energy density from the critical value required for flat geometry. Standard single-field slow-roll inflation robustly predicts Ω_K ≈ 0 to exquisite precision — perhaps one part in 10⁵ or smaller. Current CMB observations (Planck 2018 full data) find a mild but persistent preference for a slightly closed universe: Ω_K = −0.044 ± 0.015 (using CMB alone), a ~2.8σ deviation from flatness. While individually modest, this result is inconsistent with the predictions of the simplest inflationary models and has not gone away with improved data.

Within ΛCDM+inflation, a measured Ω_K at the 10⁻² level would be deeply problematic, as inflation should have reduced any primordial curvature by at least 60 e-folds. If Planck's CMB-only preference for negative curvature is real and not a systematic artifact, it would falsify standard single-field inflation. The tension is partially resolved when BAO data is combined with CMB data, pulling Ω_K closer to zero — but this creates a new internal inconsistency within ΛCDM, because the CMB and BAO should not require different curvature values. The parameter Ω_K thus sits in an uncomfortable zone: too large for inflation, too model-dependent for a clear verdict.