V2.357 - The N_eff-Lambda Joint Constraint Curve

V2.357: The N_eff-Lambda Joint Constraint Curve

Question

In LCDM, N_eff and Omega_Lambda are independent free parameters — any combination is allowed. In this framework, they are functionally related through the field content. Does this create a testable prediction unique to this framework?

The Key Insight

The framework predicts:

R = |delta_total| / (6 * alpha_s * N_eff_comp)

Any extra light species changes BOTH:

Cosmological N_eff (measured by CMB — extra radiation)
Omega_Lambda = R (predicted by the framework — field content changes delta and N_eff_comp)

This creates a one-dimensional curve in (N_eff, Omega_Lambda) space. LCDM has no such curve — it’s a free 2D plane. Measuring both N_eff and Omega_Lambda jointly tests this framework in a way LCDM cannot be tested.

Results

The SM Point

N_eff(cosmo) = 3.044
Omega_Lambda = R = 0.6877
Observed: N_eff = 2.99 +/- 0.17, Omega_Lambda = 0.685 +/- 0.007
Pull(N_eff) = +0.32 sigma, Pull(Omega) = +0.41 sigma

Response Function: dR/dN_eff

The slope tells us how Omega_Lambda responds to changes in N_eff for each species type:

Species	dR/dN_eff	Direction
Weyl fermion	-0.00736	N_eff up, Omega_Lambda down
Real scalar (T=T_nu)	-0.00424	N_eff up, Omega_Lambda down
Real scalar (early decoupled)	-0.17604	N_eff up, Omega_Lambda steeply down
Vector boson (T=T_nu)	+0.01199	N_eff up, Omega_Lambda up

Critical finding: Vectors move R in the opposite direction from fermions and scalars. The sign of dR/dN_eff identifies the spin of the extra species.

Why Vectors Are Opposite

For fermions/scalars: |delta_per| is small relative to n_comp, so adding them dilutes the overall |delta|/N_eff ratio — R decreases.

For vectors: |delta_vector| = 31/45 = 0.689 is large. Adding a vector increases |delta_total| faster than it increases N_eff — R increases.

This means the direction of the (N_eff, Omega_Lambda) shift identifies whether extra radiation is fermionic, scalar, or vector.

Joint Exclusion with Current Data

Species	Best-fit n_extra	Min chi2	Significance
Weyl fermion	-0.03	0.19	0.43 sigma
Real scalar (T=T_nu)	-0.05	0.19	0.43 sigma
Vector boson (T=T_nu)	-0.02	0.19	0.44 sigma

Current data is consistent with SM (n_extra = 0) at < 0.5 sigma for all species types.

CMB-S4 x Euclid Forecast

With CMB-S4 (sigma_N_eff = 0.03) and Euclid (sigma_Omega_Lambda = 0.002):

Species	n_extra excluded at 3 sigma	Delta_N_eff at 3 sigma
Weyl fermion	0.089	+0.089
Real scalar (T=T_nu)	0.079	+0.090
Vector boson (T=T_nu)	0.039	+0.089

Vectors are 2x more constrained than fermions because their larger |delta| makes them easier to detect via the Omega_Lambda shift.

The Cross-Check Prediction

If CMB-S4 measures N_eff = 3.044 +/- 0.03:

Framework predicts Omega_Lambda = 0.6877 +/- 0.0002 (from N_eff uncertainty alone)
Combined with theory error: Omega_Lambda = 0.6877 +/- 0.0015

If Euclid measures Omega_Lambda = 0.685 +/- 0.002:

Framework predicts N_eff = 3.044 +/- 0.110

Measure one, predict the other. No other framework makes this prediction.

Falsification Scenarios

N_eff = 3.044, Omega_Lambda = 0.670: If N_eff is SM but Omega_Lambda is low, the framework is excluded at 8.9 sigma (Euclid). LCDM can accommodate this trivially.
N_eff = 3.2, Omega_Lambda = 0.685: If extra radiation is found but Omega_Lambda doesn’t shift, the framework predicts R = 0.6866 for Weyl fermions — a 1.0 sigma gap on Euclid. Marginal but testable.
N_eff = 3.5, Omega_Lambda = 0.685: Large extra radiation with no Omega_Lambda shift would be devastating — framework requires R = 0.6844.
N_eff up, Omega_Lambda up: If both increase, the extra species must be a vector (dark photon). If both increase but the ratio doesn’t match dR/dN_eff = +0.012 for vectors, the framework is falsified.

What Makes This Unique

Feature	This framework	LCDM
N_eff and Omega_Lambda	Correlated (1D curve)	Independent (2D plane)
Extra species detection	Spin identified by direction of shift	Only N_eff changes; Omega_Lambda unaffected
Testable with	CMB-S4 x Euclid cross-correlation	N_eff measurement alone
Number of free parameters	0	2 (N_eff and Omega_Lambda independent)

Honest Limitations

Small slopes: dR/dN_eff ~ 0.007-0.012, so the Omega_Lambda shift from a Delta_N_eff = 0.1 change is only 0.0007-0.0012 — comparable to Euclid’s sigma. The joint test adds modest power over N_eff alone (gain factor ~1.0x for near-future experiments).
Degeneracies: Multiple species combinations can produce the same (N_eff, Omega_Lambda) point. The curve is species-dependent, not unique.
Theory error: The framework’s R = 0.6877 already has a +0.4 sigma pull from observed Omega_Lambda = 0.685. Any systematic in R would shift the entire curve.
Decoupling temperature matters: A scalar decoupled above the EW scale contributes almost nothing to cosmological N_eff but still shifts R. The curve depends on thermal history assumptions.
Not yet a strong discriminator: The gain from joint testing over N_eff alone is ~1.0x with CMB-S4 + Euclid. The real power is qualitative — the existence of the correlation, not its quantitative strength.

The Bottom Line

This framework makes a prediction no other framework makes: N_eff and Omega_Lambda are correlated. The correlation is weak (dR/dN_eff ~ 0.01) but in principle testable. More importantly, the sign of the correlation identifies the spin of extra species — fermions/scalars push Omega_Lambda down while vectors push it up. This is a qualitative prediction that distinguishes this framework from LCDM at a structural level, even if the quantitative signal is small.

Files

src/neff_lambda.py: Core module with ExtraSpecies class, curve computation, joint exclusion
tests/test_neff_lambda.py: 11 tests, all passing
run_experiment.py: Full experiment driver with 8 analysis sections
results.json: Machine-readable results