H0 TENSION RESOLUTION

The Hubble Tension Problem

The Hubble constant H₀ describes the current expansion rate of the universe. Two primary measurement methods yield discrepant values:

Early Universe (CMB-based):

• Planck 2018: H₀ = 67.4 ± 0.5 km/s/Mpc
• Method: Sound horizon scale from baryon acoustic oscillations
• Assumes: ΛCDM cosmology with static Λ

Late Universe (Local measurements):

• SH0ES 2022: H₀ = 73.04 ± 1.04 km/s/Mpc
• Method: Cepheid-calibrated Type Ia supernovae distance ladder
• Direct measurement: No cosmological model assumptions

The Tension:

• Discrepancy: ΔH₀ ≈ 5.6 km/s/Mpc
• Statistical significance: 5.0σ (excluding systematics)
• Probability of chance: p < 3 × 10⁻⁷

Formal Framework

Definition [[109_PRED14.1_H0-Tension-Resolution|PRED14.1]].1: The H₀ tension resolution function R(G) maps grace functions to Hubble tension reduction:

R: G(t, Ψ) → [0, 1]

Where R = 0 indicates full tension and R = 1 indicates complete resolution.

Theorem [[109_PRED14.1_H0-Tension-Resolution|PRED14.1]].1: For the grace function G(t, Ψ) = Λ₀ + α·Ψ(t), there exists a unique α* such that R(G) = 1.

Proof:

1. Define tension metric: T = |H₀_local - H₀_CMB|/σ_combined

2. Current tension: T₀ = 5.6/1.15 ≈ 4.9σ

3. Grace-modified H₀_local = H₀_CMB × √(G(t₀)/Λ₀)

4. Setting H₀_local = 73.0: √(G(t₀)/Λ₀) = 73.0/67.4 = 1.083

5. Therefore: G(t₀)/Λ₀ = 1.173

6. α* = 0.173 × Λ₀/Ψ₀

7. Uniqueness: monotonic relationship between α and H₀_local ∎

Bridge Count: 7