V2.451: CMB Low-Multipole Horizon Imprint — Does the Framework Predict the Low Quadrupole?

Question

The framework derives Λ from entanglement entropy at the cosmological (de Sitter) horizon. If this horizon is physically significant as an entangling surface, it might leave observable imprints on the CMB at scales comparable to the horizon.

The CMB quadrupole (l=2) is anomalously low: D₂^{obs} ≈ 150 μK² vs D₂^{ΛCDM} ≈ 1100 μK² (only 14% of the theoretical prediction). Could the framework’s horizon entanglement explain this through a natural IR cutoff?

Method

1. Compute the Sachs-Wolfe (SW) power spectrum C_l at l=2..30 analytically
2. Apply IR cutoffs at four physically motivated horizon scales:
   • Hubble radius: c/H₀ = 4451 Mpc
   • de Sitter horizon: c/(H₀√Ω_Λ) = 5379 Mpc
   • Event horizon: ∫dt/a (future) = 5091 Mpc
   • Particle horizon: η₀ = 14695 Mpc
3. Compute the suppression factor W(l) = C_l^{cut}/C_l^{no cut} for each
4. Compare suppressed D_l with Planck observed values
5. Find the best-fit cutoff and compare with framework horizons

Key Results

The Anomaly

l	D_l^{obs} (μK²)	D_l^{ΛCDM} (μK²)	Ratio	Status
2	150	1100	0.14	Anomalously low
3	801	1080	0.74	Low but within 1σ CV
5	1376	1060	1.30	Normal
10	1084	1060	1.02	Normal
20	971	1060	0.92	Normal

The anomaly is concentrated at l=2 ONLY. Higher multipoles are consistent with ΛCDM.

Suppression Factors by Horizon Scale

Horizon	W(l=2)	W(l=3)	W(l=5)	W(l=10)	W(l=20)
Hubble radius (x=19.8)	0.007	0.014	0.037	0.157	0.943
de Sitter (x=16.4)	0.011	0.020	0.055	0.247	0.999
Event horizon (x=17.3)	0.009	0.020	0.052	0.198	0.997
Particle horizon (x=6.0)	0.086	0.146	0.698	1.000	1.000
Needed to match D₂	0.136	~0.74	~1.0	~1.0	~1.0

Critical finding: ALL framework horizons over-suppress the CMB.

• The Hubble/dS/EH cutoffs suppress l=2 to <2% (need 14%) AND devastate l=3-10 where no anomaly exists (D₃ suppressed to 2% vs observed 74%)
• The particle horizon is the least bad but still over-suppresses l=3 (15% vs needed ~74%)

The Fundamental Problem: Pattern Mismatch

The observed anomaly is ONLY at l=2. Any smooth IR cutoff suppresses a RANGE of multipoles (l=2 through l~x_cut). The observed data require:

• Suppress l=2 by 86%
• Leave l=3 at 74%
• Leave l=5+ untouched

No smooth cutoff can produce this pattern. The l=2 “anomaly” is an isolated outlier, consistent with cosmic variance (which gives σ(D₂)/D₂ = √(2/5) = 63%).

Best-Fit Cutoff

Property	Value
x_cut = k·η*	4.5
k_cut	3.2 × 10⁻⁴ Mpc⁻¹
R_cut	19.5 Gpc
Δχ² improvement	2.9 (for 1 parameter)
BIC with cutoff	25.5
BIC without cutoff	25.0

The best-fit cutoff scale (19.5 Gpc) is LARGER than any physical horizon in the framework (max: particle horizon at 14.7 Gpc). It matches the literature value (k_c ≈ 2-5 × 10⁻⁴ Mpc⁻¹, R = 13-31 Gpc), but this is phenomenology, not physics.

BIC disfavors the cutoff: 25.5 > 25.0 (the Δχ² = 2.9 doesn’t justify the extra parameter).

Model Comparison

Model	χ²	χ²/dof	BIC	Verdict
ΛCDM (no cutoff)	25.0	0.86	25.0	PREFERRED
Framework Ω_Λ=0.688	25.0	0.86	25.0	Same as ΛCDM
Particle horizon cutoff	23.6	0.84	27.0	Overfitting
Best-fit cutoff	22.1	0.79	25.5	Marginal
de Sitter cutoff	88.5	3.16	91.9	REJECTED
Hubble cutoff	116.4	4.16	119.8	REJECTED

The framework’s de Sitter horizon cutoff makes the fit 3.5× worse than no cutoff.

Physical Interpretation

The framework’s entanglement is a vacuum property, not a modification of primordial perturbations.

• Entanglement entropy S = α·A + δ·ln(A) is computed in the vacuum state of quantum fields. This determines the BACKGROUND values of G and Λ.
• CMB anisotropies are PERTURBATIONS on top of this vacuum, created by inflation and evolved through standard cosmological dynamics.
• These are different things. The vacuum entanglement constrains background geometry; it does not modify the statistics of perturbations.

A horizon IR cutoff would require the entanglement structure to MODIFY the primordial power spectrum — a much stronger claim than the framework currently makes, and one that is NOT supported by the framework’s theoretical structure.

Honest Assessment

The Verdict

The framework does NOT predict CMB anomalies. This closes strategic prediction #6 (CMB large-scale anomalies) with a clear negative result.

The low CMB quadrupole is:

• An isolated outlier at l=2 only (not a systematic suppression)
• Within 1.5σ of cosmic variance (5 independent samples at l=2)
• Inconsistent with any smooth IR cutoff (which would suppress l=3-10 too)
• NOT predicted by the framework’s horizon entanglement

What This Means

1. The framework = ΛCDM at the perturbation level. Its unique predictions are in the BACKGROUND quantities (Ω_Λ, w, BH entropy), not in the CMB power spectrum.

2. No smoking gun in the CMB. The framework cannot be confirmed or falsified by low-l CMB observations. This narrows the confirmation path to: Ω_Λ precision (Euclid), w = -1 (DESI), species-dependence (new particles), and neutrino type (0νββ experiments).

3. The horizon cutoff idea is dead. All physically motivated cutoffs either over-suppress (de Sitter, Hubble) or don’t match the data pattern (particle horizon). The BIC disfavors even the best-fit cutoff.

Strengths

• Clean, honest negative result — rules out a speculative extension
• Quantitative comparison of all horizon scales
• Demonstrates that the observed anomaly pattern (l=2 only) is inconsistent with any smooth cutoff
• Closes a gap in the project’s exploration of unique predictions

Weaknesses

• Uses Sachs-Wolfe approximation (adequate for low l, but not precision cosmology)
• ISW computation had normalization issues (simplified kernel)
• Planck low-l data are approximate (should use full Commander likelihood)
• Did not explore non-sharp cutoff forms (exponential, modular Hamiltonian window) — but the pattern mismatch argument applies to ALL smooth suppressions

Files

• src/cmb_horizon.py — Full computation (SW + cutoffs + model comparison)
• tests/test_cmb.py — 13/13 tests passing
• run_experiment.py — Experiment driver
• results.json — Machine-readable results