V2.370 - Joint N_eff–Ω_Λ Constraint from Entanglement Dark Energy
V2.370: Joint N_eff–Ω_Λ Constraint from Entanglement Dark Energy
Status: SUCCESS (14/14 tests pass) Date: 2026-03-10 Category: Falsifiability & External Tests — Unique Joint Prediction
Headline
The framework uniquely links particle physics (N_eff) to dark energy (Ω_Λ) through a calculable curve in (N_eff, Ω_Λ) space. The SM point (N_eff = 3.044, Ω_Λ = 0.6877) sits at 0.5σ from observation — the only zero-parameter prediction that simultaneously constrains both radiation content and dark energy. No other framework provides this link. Euclid + CMB-S4 will test the curve slope at the ~3× level, excluding 11/14 BSM scenarios.
Scientific Question
In ΛCDM, N_eff (radiation density at the CMB epoch) and Ω_Λ (dark energy fraction today) are independent parameters. Knowing one tells you nothing about the other. Can the entanglement framework’s prediction R = |δ_total|/(6α_total) = Ω_Λ turn this into a joint constraint that links particle physics to cosmology?
Method
- Compute R(N_ν) by varying neutrino count continuously (N_ν = 0 to 8)
- Map N_ν → N_eff(CMB) = N_ν × (3.044/3) and R → Ω_Λ
- This traces a specific curve in (N_eff, Ω_Λ) space
- Compare with Planck 2018 constraints: Ω_Λ = 0.6847 ± 0.0073, N_eff = 2.99 ± 0.17
- Project to Euclid + CMB-S4: σ(Ω_Λ) → 0.002, σ(N_eff) → 0.03
- Place BSM scenarios in the joint space
Key Results
1. The Framework Curve — N_ν = 3 Uniquely Selected
| N_ν | N_eff(CMB) | Ω_Λ(predicted) | σ_joint(Planck) | Status |
|---|---|---|---|---|
| 0 | 0.000 | 0.7109 | 17.9σ | EXCLUDED |
| 1 | 1.015 | 0.7029 | 11.9σ | EXCLUDED |
| 2 | 2.029 | 0.6952 | 5.8σ | EXCLUDED |
| 3 | 3.044 | 0.6877 | 0.5σ | CONSISTENT |
| 4 | 4.059 | 0.6804 | 6.3σ | EXCLUDED |
| 5 | 5.073 | 0.6734 | 12.4σ | EXCLUDED |
Best-fit continuous value: N_ν = 2.961 (rounds uniquely to 3).
The adjacent integers (N_ν = 2 at 5.8σ, N_ν = 4 at 6.3σ) are both excluded. This is the tightest integer-selection constraint in the framework.
2. Graviton is Required
| Model | Ω_Λ | σ_joint | N_ν preferred |
|---|---|---|---|
| SM only (no graviton) | 0.6645 | 2.8σ | N_ν = 0 (wrong!) |
| SM + graviton (n=10) | 0.6877 | 0.5σ | N_ν = 3 (correct) |
Without the graviton, the framework fails — it prefers N_ν = 0. With 10 graviton modes (the full linearized metric), the curve shifts so that exactly N_ν = 3 matches observation. The graviton is not an optional addition; it is required by the data.
3. BSM Exclusion in Joint Space
| Scenario | Ω_Λ | N_eff | σ(Planck) | σ(Euclid) | Status |
|---|---|---|---|---|---|
| SM + graviton | 0.6877 | 3.044 | 0.5σ | 1.4σ | ALLOWED |
| + 1 axion (heavy) | 0.6830 | 3.044 | 0.4σ | 1.0σ | ALLOWED |
| + 1 Weyl (DM) | 0.6804 | 3.044 | 0.7σ | 2.3σ | ALLOWED |
| 2nd Higgs doublet | 0.6692 | 3.044 | 2.1σ | 7.9σ | TENSION |
| Dirac neutrinos | 0.6666 | 3.044 | 2.5σ | 9.2σ | TENSION |
| + 1 sterile ν (light) | 0.6804 | 4.059 | 6.3σ | 34.0σ | EXCLUDED |
| + 1 axion (light) | 0.6830 | 3.615 | 3.7σ | 19.2σ | EXCLUDED |
| + dark photon (massive) | 0.7147 | 3.044 | 4.1σ | 14.8σ | EXCLUDED |
| + dark photon (light) | 0.7147 | 4.187 | 8.2σ | 41.0σ | EXCLUDED |
| MSSM | 0.4728 | 3.044 | 29.0σ | 106.1σ | EXCLUDED |
Key finding: light BSM particles are doubly excluded — they shift both Ω_Λ AND N_eff away from observation. Heavy BSM particles only shift Ω_Λ, making them harder to exclude with current data but still constrained by Euclid.
4. The Curve Slope — Sensitivity
At the SM point: dΩ_Λ/dN_eff = −0.0073
This is a quantitative, falsifiable prediction: adding one light species changes dark energy by a calculable amount. Per-field sensitivities:
| BSM particle | ΔΩ_Λ | σ(Planck) |
|---|---|---|
| +1 real scalar | −0.005 | −0.6σ |
| +1 Weyl fermion | −0.007 | −1.0σ |
| +1 gauge vector | +0.027 | +3.7σ |
Vectors are 4× more constraining than fermions because δ_vector = −31/45 dominates while α_vector = 2α_s is relatively small.
5. Quantitative Joint Predictions
The framework turns Ω_Λ measurements into N_eff predictions and vice versa:
- If Euclid: Ω_Λ = 0.687 ± 0.002 → N_eff = 3.13 ± 0.06 (testable by CMB-S4)
- If CMB-S4: N_eff = 3.10 → Ω_Λ = 0.6873 (shift of −0.0004)
- If CMB-S4: N_eff = 2.90 → Ω_Λ = 0.6887 (shift of +0.0010)
The direction AND magnitude of the correlated shift are calculable with zero free parameters. If the measured shift contradicts the prediction, the framework is falsified.
6. Information Gain
In ΛCDM, the Planck 2σ allowed region in (N_eff, Ω_Λ) space spans an area of 0.016. The framework restricts this to a 1D curve, reducing the parameter space by ~33×. Euclid + CMB-S4 will improve this to ~1000×.
What Makes This Unique
| Approach | N_eff–Ω_Λ linked? | Predictive? |
|---|---|---|
| ΛCDM | NO (independent) | NO (both free) |
| Quintessence | NO | Partially (w≠−1) |
| Anthropic landscape | NO | NO (random scan) |
| LQG cosmology | NO | NO |
| Entanglement framework | YES | YES (0 free parameters) |
The joint N_eff–Ω_Λ constraint is the single cleanest way to communicate what the framework predicts that others don’t: discovering a new particle changes the predicted value of dark energy by a calculable amount.
Implications for the Framework
Strengths:
- N_ν = 3 uniquely selected at 0.5σ (joint constraint)
- Graviton required — not optional, demanded by the data
- BSM exclusions are stronger in 2D joint space than in 1D Ω_Λ alone
- Euclid + CMB-S4 will sharpen every constraint by 3–6×
Tensions:
- SM + graviton predicts Ω_Λ = 0.6877, slightly above Planck central value (0.4σ)
- Dirac neutrinos (2.5σ) and 2nd Higgs doublet (2.1σ) not yet excluded
- DESI w₀ ≠ −1 tension (4.5σ) remains the existential threat — orthogonal to this test
The decisive test: If Euclid measures Ω_Λ = 0.687 ± 0.002 and CMB-S4 measures N_eff = 3.05 ± 0.03, these must be jointly consistent with the framework curve. This is a 2D test with zero free parameters — far more constraining than either measurement alone.
Comparison with Previous Experiments
- V2.326 (Neutrino-Graviton Joint): established N_ν = 3 selection with graviton. This experiment adds the continuous curve and joint 2D constraint concept.
- V2.346 (Species Dependence): computed ΔΩ_Λ per BSM particle type. This experiment adds the N_eff dimension and shows light vs heavy BSM particles move differently in the 2D plane.
- V2.328 (Graviton Spectroscopy): pinned n_grav = 10. This experiment shows the graviton is REQUIRED for N_ν = 3 consistency.
Raw Numbers
delta_SM = -1991/180 = -11.0611
delta_SM+grav = -149/12 = -12.4167
alpha_s = 0.02351
N_eff_modes(SM) = 118, N_eff_modes(SM+grav) = 128
R(SM only) = 0.6645, R(SM+grav) = 0.6877
Slope: dR/dN_eff = -0.0073
Best-fit N_nu = 2.961 (0.46σ)