V2.361 - Sound Horizon & BAO Confrontation
V2.361: Sound Horizon & BAO Confrontation
Question
V2.354 found a 4.4σ tension between the framework’s CMB distance scale and Planck. Is this real physics or a computational artifact? And does the framework survive confrontation with the full DESI DR2 BAO dataset (13 data points)?
The Resolution
The 4.4σ tension was entirely an artifact of the Eisenstein-Hu (1998) fitting formula, which gives r_d ≈ 150 Mpc instead of the correct ~147 Mpc.
The error chain:
EH formula → z_d ≈ 1020 (should be 1060) → r_d ≈ 150 Mpc (should be 147)
→ D_M/r_d too low → 4.4σ "tension"With CAMB-calibrated r_d = 147.09 Mpc, the CMB tension drops to +0.7σ.
Why r_d Is Identical for Framework and Planck
The sound horizon r_d depends only on pre-recombination physics:
- omega_m * h^2 = 0.1430 (fixed by Planck CMB)
- omega_b * h^2 = 0.02237 (fixed by Planck CMB)
- N_eff = 3.044 (standard neutrino decoupling)
The framework changes Omega_Lambda (and hence H_0), but this only affects post-recombination distances. The sound horizon is set at z ~ 1060, long before dark energy matters. Therefore r_d is identical for both:
| Scenario | r_d (Mpc) | r_d * h | z_drag |
|---|---|---|---|
| Planck LCDM | 147.09 | 99.06 | 1059.9 |
| Framework (1-loop) | 147.09 | 99.23 | 1059.9 |
DESI DR2 BAO Confrontation
Full confrontation with 13 DESI DR2 data points across 7 redshift bins:
| Scenario | BAO chi2 | CMB chi2 | Total chi2/14 |
|---|---|---|---|
| Planck LCDM | 15.03 | 0.73 | 1.13 |
| Framework (tree) | 14.49 | 0.30 | 1.06 |
| Framework (1-loop) | 14.74 | 0.55 | 1.09 |
The framework achieves chi2/pt = 1.13 for BAO alone, marginally better than Planck LCDM’s 1.16. The difference (Delta chi2 = -0.3) is not significant — BAO cannot distinguish the two with current data.
Largest Individual Tension
The LRG1 D_H/r_d at z = 0.51 shows +2.85σ for the framework (and +2.88σ for Planck LCDM). This is a known tension in the DESI data that affects both models equally — it is not framework-specific.
BAO Constraint on Graviton Modes
The BAO chi2 as a function of n_grav:
| n_grav | Omega_Lambda | BAO chi2/pt | CMB tension |
|---|---|---|---|
| 8 | 0.699 | 1.50 | -0.58σ |
| 9 | 0.693 | 1.20 | -0.00σ |
| 10 | 0.688 | 1.11 | +0.55σ |
| 11 | 0.682 | 1.23 | +1.08σ |
| 12 | 0.677 | 1.52 | +1.59σ |
n_grav = 10 gives the best BAO fit, consistent with V2.350 and V2.360.
Key Results
- 4.4σ RESOLVED: The V2.354 tension was a fitting formula artifact, not physics
- BAO passed: chi2/pt = 1.13 across 13 DESI DR2 points
- r_d identical: Framework and Planck share r_d = 147.09 Mpc (same pre-recombination physics)
- n_grav = 10 confirmed: Best BAO fit at the framework’s predicted graviton mode count
- Indistinguishable from LCDM: Delta chi2 = -0.3 (framework marginally better, not significant)
Honest Assessment
Strength: The framework survives the most precise BAO dataset ever compiled. The 4.4σ scare was entirely computational. With proper r_d, every distance measure is within 1σ of DESI (except the known LRG1 D_H outlier).
Limitation: BAO cannot distinguish the framework from Planck LCDM. The Omega_Lambda difference (0.6858 vs 0.6847) is too small for current BAO precision. This is expected — the distinguishing power comes from other observables (graviton modes, neutrino nature, BH log corrections).
What would kill the framework: If future BAO data (DESI Year 3+) shows chi2/pt >> 2 for Omega_Lambda = 0.6858 while Omega_Lambda = 0.6847 fits well, the framework would be in tension. Current data shows no hint of this.
Files
src/sound_horizon.py: Core calculation (sound horizon, BAO distances, DESI data)tests/test_sound_horizon.py: 14 tests, all passingresults.json: Full numerical output