V2.430: Joint Ω_Λ–N_eff Constraint — The Two-Observable Test

The Idea

This is the framework’s most powerful unique test. Euclid measures Ω_Λ. CMB-S4 measures N_eff. In ΛCDM, these are independent — Ω_Λ is a free parameter that doesn’t care about N_eff. In this framework, both observables depend on the SAME field content:

$\Omega_\Lambda = R(\text{fields}), \qquad N_\text{eff} = f(\text{fields})$

If extra light species exist, they shift BOTH observables simultaneously. The TYPE of species (scalar, fermion, vector) determines a SPECIFIC trajectory in the (N_eff, Ω_Λ) plane with a different slope for each spin. No other framework predicts any relationship between Ω_Λ and N_eff.

Key Results

Per-Spin Slopes

Species type	dΩ_Λ/dN_eff	Direction	Physical mechanism
Real scalar	−0.0083	Ω_Λ drops	Small
Weyl fermion	−0.0074	Ω_Λ drops	Similar to scalar but steeper per ΔN_eff
Dirac fermion	−0.0089	Ω_Λ drops	Steepest negative slope
Gauge vector	+0.0240	Ω_Λ rises	Large

The vector slope is positive — adding vector bosons INCREASES Ω_Λ while adding radiation. This is the opposite of scalars and fermions, and provides a clear spin diagnostic.

If CMB-S4 Detects ΔN_eff = 0.1

If the species is…	Ω_Λ predicted	ΔΩ_Λ	Euclid detection?
Scalar	0.6869	−0.0008	No (0.8σ shift)
Weyl fermion	0.6870	−0.0007	No (0.7σ shift)
Gauge vector	0.6901	+0.0024	Yes (2.4σ shift)

Even a marginal ΔN_eff detection from CMB-S4 produces a detectable Ω_Λ shift for vectors. The framework turns a particle physics measurement into a cosmological prediction.

The Falsification Protocol

Case A: CMB-S4 confirms N_eff = 3.044 ± 0.03 (SM)

• Framework predicts Ω_Λ = 0.6877 ± 0.0003 (from SM+grav)
• Euclid must measure Ω_Λ ∈ [0.682, 0.691]
• Outside this band → FALSIFIED

Case B: CMB-S4 detects ΔN_eff ≈ 0.1

• Three trajectories possible (scalar, fermion, vector)
• Euclid identifies the spin via the Ω_Λ shift direction (+/−) and magnitude
• If Ω_Λ doesn’t match ANY spin trajectory → FALSIFIED

Case C: ΔN_eff ≈ 1 (sterile neutrino)

• Majorana: Ω_Λ = 0.6805, Dirac: Ω_Λ = 0.6790, Scalar: Ω_Λ = 0.6795
• The framework requires a SPECIFIC correlated shift in both observables

The Temperature Insight

A critical subtlety: N_eff(CMB) depends on the species temperature (∝ T⁴), but Ω_Λ does NOT (δ and α are UV quantities, mass/temperature-independent).

This means:

• Hot species (T = T_ν): shift both N_eff and Ω_Λ → diagonal trajectory
• Cold species (T ≪ T_ν): shift Ω_Λ only, N_eff unchanged → vertical trajectory
• ΛCDM: Ω_Λ never changes → horizontal line

A cold dark scalar would shift Ω_Λ by −0.005 with NO N_eff change. This is a pure Ω_Λ signal detectable at 5σ by Euclid, invisible to CMB-S4. ΛCDM predicts no such shift.

CMB-S4 + Euclid Joint Discovery Power

For ΔN_eff = 0.5 (5× CMB-S4 threshold):

Species	Ω_Λ shift	Detectable by Euclid?
Scalar	−4.1σ	Yes
Weyl	−3.7σ	Yes
Vector	+11.9σ	Yes

All three spin types produce detectable Ω_Λ shifts for ΔN_eff ≥ 0.5.

Why This Is the Most Powerful Test

Framework	Prediction in (N_eff, Ω_Λ) plane
ΛCDM	Horizontal line (Ω_Λ = free parameter)
Quintessence	No predicted trajectory
String landscape	No predicted trajectory
This framework	Specific slope per spin type

This is the ONLY framework in physics that predicts dΩ_Λ/dN_eff ≠ 0.

Honest Limitations

1. Scalars and fermions have similar slopes (−0.008 vs −0.007). Distinguishing between them requires σ(Ω_Λ) < 0.001, which is next-generation precision.

2. Temperature degeneracy: A cold species shifts Ω_Λ without changing N_eff. This looks like a different Ω_Λ, not a new species. Only if N_eff ALSO shifts can we identify the trajectory.

3. The baseline uncertainty: SM+grav(n=10) gives R = 0.6877, already +0.4σ from Ω_Λ_obs. With Euclid precision (σ=0.001), this becomes +3.0σ — either the GF core (R=0.6851, +0.4σ) is the true prediction, or the graviton counting needs refinement.

4. Fractional species: The trajectories pass through non-integer species counts. The physical prediction is at integer values — a smooth curve between integers is extrapolation.

Files

• src/joint_constraint.py: Core computation with slopes, trajectories, exclusion contours
• tests/test_joint_constraint.py: 10 tests, all passing
• run_experiment.py: Full 8-part analysis