V2.102 - Corrected Graviton Entanglement Entropy
V2.102: Corrected Graviton Entanglement Entropy
Headline
The observed cosmological constant is bracketed by SM-only and SM+graviton predictions. Using the correct graviton entanglement entropy (Benedetti-Casini 2020: δ_grav = -61/45, not the erroneous -212/45 from Christensen-Duff’s trace anomaly), and the f=6 self-consistency factor from V2.101:
| Scenario | Λ_pred/Λ_obs | Deviation |
|---|---|---|
| SM only (f=6, HK α) | 0.959 | 4.1% low |
| SM + graviton | 1.059 | 5.9% high |
| Target | 1.000 | 0% |
The observed value Ω_Λ = 0.685 lies between the SM-only prediction (R = 0.657) and the SM+graviton prediction (R = 0.725).
Critical Correction
V2.100 used δ_graviton = -212/45 from Christensen-Duff (1978). This is the heat kernel B_4 coefficient (trace anomaly), NOT the entanglement entropy log coefficient.
Benedetti-Casini (2020, Phys. Rev. D 101, 045004) computed the actual entanglement entropy:
- c_log(graviton) = -61/45 = -1.356 (3.48× smaller than -212/45)
General formula for bosonic spin h: c_log(h) = -2(1 + 15h²)/90
- h=0 (complex scalar): -1/45
- h=1 (photon, free): -16/45
- h=2 (graviton, free): -61/45
Free vs Full Algebra
Edge modes at the entangling surface modify the log coefficient:
- Photon: δ_free = -16/45, δ_full = -31/45 (edge adds -1/3)
- Graviton: δ_free = -61/45, δ_full = unknown (graviton edge modes not computed)
Our SM prediction uses full algebra for gauge fields (-31/45 per vector).
Key Finding
The alpha_graviton needed for exact agreement is 9.1 × alpha_scalar — much larger than the 2 × alpha_scalar estimate from the spin decomposition. This suggests either:
- The graviton contributes less to δ than Benedetti-Casini predicts (unlikely — it’s exact)
- The graviton’s α is larger than 2 × α_scalar
- The SM-only prediction (4% off) is the correct comparison point, and the graviton contribution is partially cancelled by some other effect
Sensitivity
| α_graviton estimate | R_SM+grav | Λ/Λ_obs | Deviation |
|---|---|---|---|
| 0 (no area) | 0.738 | 1.077 | 7.7% |
| 2×α_scalar | 0.725 | 1.059 | 5.9% |
| 1×α_vector | 0.725 | 1.059 | 5.9% |
| 9.1×α_scalar (exact fit) | 0.685 | 1.000 | 0% |
Bottom Line
The prediction is already within 4-6% of observation — 121 orders of magnitude closer than the naive QFT estimate. The remaining uncertainty is dominated by the unknown α_graviton and the question of whether graviton edge modes exist.