De Sitter Horizon Entanglement Entropy
Experiment V2.90: De Sitter Horizon Entanglement Entropy
Status: COMPLETE
Summary
Computes the entanglement entropy of a free scalar field across the cosmological horizon in static de Sitter spacetime, and compares with flat-space values to determine whether spacetime curvature modifies alpha and delta.
The de Sitter static patch has metric ds^2 = -(1-H^2 r^2)dt^2 + dr^2/(1-H^2 r^2)
- r^2 dOmega^2 with horizon at r_H = 1/H. On the lattice, the lapse function f(j) = 1 - H^2 j^2 modifies the radial coupling matrix.
Key Result: Curvature Reduces Alpha by ~6%, Delta Extraction Fails
- alpha_dS (d2S) = 0.02052 vs alpha_flat (d2S) = 0.02180: -5.9% curvature correction
- Delta extraction via d3S fails in de Sitter (curvature terms swamp the signal)
- This is consistent with V2.68’s finding that d3S fails on de Sitter backgrounds
The curvature correction to alpha is small and in the wrong direction for closing the self-consistency gap: lower alpha increases R = |delta|/(12*alpha), but only by ~6%.
Phase 1: Flat-Space Baseline (Horizon Entropy)
Flat-space S(n) for n = 5..60 with proportional cutoff C=6, N=500. This IS the horizon entanglement entropy for H = 1/n in the Bunch-Davies vacuum.
- alpha (d2S, proportional cutoff) = 0.02180 +/- 4e-7
- delta (d3S) = -0.00732 (34% error vs theory -0.01111; known: need N=1000+ for <2%)
Phase 2: De Sitter Comparison
For each n_H (horizon radius), compute entropy at n_sub = n_H/4 (sub-horizon fraction = 0.25) using the de Sitter Hamiltonian with lapse regularization.
Per-point curvature effects (S_dS/S_flat - 1):
| n_H | n_sub | H*n_sub | Rel. diff |
|---|---|---|---|
| 10 | 2 | 0.200 | -2.8% |
| 20 | 5 | 0.250 | -6.0% |
| 40 | 10 | 0.250 | -6.1% |
| 60 | 15 | 0.250 | -6.1% |
The curvature effect saturates at ~6% for H*n_sub ~ 0.25 (where the lapse is f ~ 0.94 at the partition surface).
Phase 3: Coefficient Extraction
| Method | alpha (flat) | alpha (dS) | Change |
|---|---|---|---|
| d2S mean | 0.02180 | 0.02052 | -5.9% |
| 4-param fit | 0.02193 | 0.02067 | -5.7% |
| Method | delta (flat) | delta (dS) | Notes |
|---|---|---|---|
| d3S mean | -0.00732 | +0.02570 | dS extraction completely fails |
| 4-param fit | +6.38 | +4.45 | Fit delta unreliable (proportional cutoff, small N) |
Delta extraction requires N >= 1000 and is known to fail on curved backgrounds (V2.68). The reliable result is the alpha comparison via d2S.
Phase 4: Self-Consistency Implications
Using the reliable d2S alpha and theoretical delta = -1/90:
| Quantity | Flat | De Sitter | Change |
|---|---|---|---|
| alpha (d2S) | 0.02180 | 0.02052 | -5.9% |
| delta (theory) | -0.01111 | -0.01111 | 0% (conformal anomaly) |
| R = |delta|/(12*alpha) | 0.04248 | 0.04511 | +6.2% |
The curvature correction moves R in the right direction (toward 1) but the effect is negligible: from 0.042 to 0.045 for a single scalar. For the full SM, V2.69 found alpha(H) ~ alpha(0) to <1% at similar H*n_sub values.
Phase 5: Conclusions
- De Sitter curvature reduces alpha by ~6% at H*n_sub ~ 0.25
- Delta is a conformal anomaly and should be robust (though lattice extraction fails)
- The curvature effect is too small to significantly change R_SM = 0.329
- This confirms V2.69’s finding that the self-consistency gap is not a curvature artifact
Files
| File | Description |
|---|---|
| src/desitter_horizon_entropy.py | De Sitter Hamiltonian, entropy computation, coefficient extraction |
| src/horizon_self_consistency.py | Self-consistency analysis and Lambda prediction |
| tests/test_horizon_entropy.py | 21 tests (all pass) |
| run_experiment.py | 5-phase driver |