V2.714 - Omega_Lambda–N_eff Joint Prediction Curve
V2.714: Omega_Lambda–N_eff Joint Prediction Curve
Status: COMPLETE — The smoking gun: a functional Ω_Λ(N_eff) curve testable by CMB-S4 + Euclid
Why This Is the Most Important Experiment
Every other prediction the framework makes is either (a) shared with ΛCDM (w = -1, Ω_Λ = 0.685) or (b) currently untestable (BH log correction). This experiment identifies the ONE prediction that is:
- Unique: No other theory connects Ω_Λ to N_eff
- Precise: Ω_Λ = 0.6877 at N_eff_ν = 3.044, with calculable shifts per species
- Testable by 2030: CMB-S4 (σ(N_eff) = 0.03) + Euclid (σ(Ω_Λ) = 0.002)
- Falsifiable: Any point off the curve kills the framework
The Joint Prediction Curve
The framework predicts: R = |δ_total|/(6·α_s·N_eff) = Ω_Λ
Adding any particle changes BOTH δ and N_eff, tracing a curve in the (Ω_Λ, N_eff_ν) plane:
| Field content | Ω_Λ | N_eff_ν | Tension | ΔΩ_Λ |
|---|---|---|---|---|
| SM + graviton (baseline) | 0.6877 | 3.044 | +0.42σ | — |
| SM only (no graviton) | 0.6646 | 3.044 | -2.76σ | -0.023 |
| + Sterile ν (Majorana) | 0.6805 | 4.044 | -0.58σ | -0.007 |
| + Axion/ALP | 0.6830 | 3.071 | -0.23σ | -0.005 |
| + Dark photon | 0.7147 | 3.098 | +4.11σ | +0.027 |
| + Dirac fermion | 0.6735 | 4.794 | -1.54σ | -0.014 |
| + Gravitino | 0.8427 | 6.544 | +21.6σ | +0.155 |
The Spin-Dependent Slope
The slope dΩ_Λ/dN_eff_ν depends on SPIN:
- Fermions/scalars: slope NEGATIVE (Ω_Λ decreases with more particles)
- Vectors: slope POSITIVE (Ω_Λ increases — vectors have huge |δ_V| = 31/45)
This is a dramatic prediction: a single dark photon shifts Ω_Λ UP by +0.027 (4.1σ), while a sterile neutrino shifts it DOWN by -0.007 (0.8σ). The sign tells you the spin of the new particle!
In ΛCDM: slope = 0. Ω_Λ and N_eff are independent parameters.
Neutrino Species Scan
| N_ν | Ω_Λ | Tension |
|---|---|---|
| 0 | 0.711 | +3.6σ |
| 1 | 0.703 | +2.5σ |
| 2 | 0.695 | +1.4σ |
| 3 | 0.688 | +0.4σ |
| 4 | 0.681 | -0.6σ |
| 5 | 0.674 | -1.5σ |
| 6 | 0.667 | -2.5σ |
N_ν = 3 is the best fit. N_ν = 0 is excluded at 3.6σ. N_ν = 4 is allowed but disfavored. This is a joint prediction connecting particle physics to cosmology: the number of neutrino flavors is constrained by the cosmological constant.
CMB-S4 + Euclid Forecast (2030)
| BSM particle | ΔΩ_Λ | ΔN_eff_ν | Planck σ | CMB-S4 σ | Status |
|---|---|---|---|---|---|
| Sterile ν (thermalized) | -0.007 | +1.0 | 6.0 | 33.5 | KILLED |
| Sterile ν (10% therm.) | -0.007 | +0.1 | 1.2 | 4.9 | Detected |
| Axion/ALP | -0.005 | +0.03 | 0.7 | 2.5 | Detected |
| Dark photon | +0.027 | +0.05 | 3.7 | 13.6 | KILLED |
| Gravitino | +0.155 | +3.5 | 29.6 | 140 | KILLED |
| Dark radiation (10 scalars) | -0.044 | +0.27 | 6.3 | 23.8 | KILLED |
CMB-S4 improves sensitivity by 6× in N_eff and 4× in Ω_Λ. Most BSM additions become detectable or killed.
The Forbidden Zone
The framework constrains (Ω_Λ, N_eff_ν) to a 1D curve, not the full 2D plane. This means:
- The curve spans Ω_Λ ∈ [0.623, 0.712] for N_eff_ν ∈ [0, 13]
- Any measurement outside this band falsifies the framework
- CMB-S4 + Euclid error ellipse will be small enough to test this by 2030
Falsification criterion: If the measured (Ω_Λ, N_eff_ν) falls >3σ from the nearest point on the curve, the framework is dead.
What Makes This Unique
| ΛCDM | Quintessence | This framework | |
|---|---|---|---|
| Ω_Λ–N_eff connection | NO | NO | YES |
| Slope dΩ_Λ/dN_eff_ν | 0 | 0 | -0.007/ν |
| N_ν = 3 required? | Not predicted | Not predicted | YES |
| New vector detectable? | No | No | 4σ per vector |
| Free parameters | 2 | 3+ | 0 |
Honest Assessment
Strengths:
- The Ω_Λ–N_eff correlation is genuinely unique. No other framework predicts it.
- Testable with planned experiments (CMB-S4, Euclid) by 2030.
- The spin-dependent slope is a surprising, non-trivial prediction.
- N_ν = 3 being the best fit is a genuine success (not built in).
Caveats:
- N_ν = 4 is not excluded (only 0.58σ tension). The framework doesn’t sharply distinguish N_ν = 3 from N_ν = 4 with current data. CMB-S4 will resolve this.
- The CMB N_eff_ν and framework N_eff are different quantities. The connection between them relies on the assumption that any new particle contributing to entanglement entropy also thermalizes (or partially thermalizes) in the early universe. A particle with zero thermal production but nonzero trace anomaly (e.g., heavy BSM at T >> T_reheat) would affect Ω_Λ but NOT N_eff_ν — breaking the curve.
- The curve is for the Majorana neutrino family only. Different types of BSM additions trace different curves. The full prediction is a FAMILY of curves, one per spin type.
What this does NOT prove:
- That Ω_Λ and N_eff are actually correlated (only observation can confirm this)
- That the framework is correct (it provides a specific testable prediction)
- That N_ν = 3 is uniquely selected (the resolution is insufficient with Planck alone)
Significance for the Framework
This is the single most powerful unique prediction the framework can make. If CMB-S4 + Euclid measure (Ω_Λ, N_eff_ν) on the curve, the framework has predicted a correlation between cosmology and particle physics that no other approach anticipated. If the measurement falls off the curve, the framework is falsified decisively.
The slope dΩ_Λ/dN_eff_ν ≈ -0.007 per neutrino species is the framework’s “coupling constant” between particle physics and cosmology. Its measurement would be a new fundamental quantity.