V2.491: Running Vacuum Confrontation — Framework (ν=0) vs Running Vacuum Model

The Question

The framework predicts Λ is exactly constant: the trace anomaly δ is one-loop exact (Adler-Bardeen theorem), so Ω_Λ = |δ|/(6α) is a pure number independent of H. This means the “running vacuum parameter” ν = 0 exactly, with no free parameter.

The Running Vacuum Model (RVM) parameterizes Λ(H) = Λ₀ + 3ν(H² − H₀²), with ν as a free parameter. Recent papers claim ν ~ 0.0014 at 3.5σ significance from CMB+BAO+SNe data. If true, this would falsify the framework by violating Adler-Bardeen protection.

This experiment confronts both predictions with DESI DR1 BAO data.

Key Results

1. BAO χ² comparison

Model	χ²	χ²/N (12 bins)	ν	w_eff
Framework (149√π/384)	18.11	1.51	0 (exact)	−1.000
Planck ΛCDM	18.77	1.56	0 (fixed)	−1.000
RVM (ν=0.0014)	18.97	1.58	0.0014	−0.999

The framework has the LOWEST χ² of all three models. Adding ν > 0 makes the fit worse on DESI BAO data alone. This is the opposite of what the RVM literature claims.

2. Best-fit ν from BAO alone

• ν_best = −0.012 ± 0.013 (BAO only, Planck priors on Ω_m, H₀)
• ν = 0 tension: 0.9σ — completely consistent
• Δχ²(ν=0 vs best): 0.78
• ΔBIC: +1.70 → constant Λ (ν=0) PREFERRED by BIC

The BAO data alone provides essentially no evidence for running. The claimed 3.5σ detection requires the CMB angular distance constraint, which is sensitive to early-universe physics, not late-time Λ running.

3. Per-bin tensions (framework vs data)

Tracer	z_eff	Observable	Pull(FW)	Pull(Planck)
BGS	0.295	D_V/r_d	+0.7σ	+0.9σ
LRG1	0.510	D_M/r_d	−0.6σ	−0.5σ
LRG1	0.510	D_H/r_d	+2.9σ	+2.9σ
LRG2	0.706	D_M/r_d	+2.6σ	+2.7σ
LRG3+ELG1	0.930	D_M/r_d	+0.7σ	+0.8σ
Lya	2.330	D_H/r_d	+0.7σ	+0.6σ

The framework’s largest tensions (LRG1 D_H at 2.9σ, LRG2 D_M at 2.6σ) are identical to Planck’s. These are known DESI–Planck tensions, not framework-specific problems.

4. DESI w₀wₐ vs RVM — incompatible parameterizations

Parameter	DESI (CMB+BAO+SNe)	RVM (ν=0.0014)	Framework
w₀	−0.727 ± 0.067	−0.999	−1.000
wₐ	−1.05 ± 0.29	0.0	0.0

DESI’s w₀wₐ signal and RVM are incompatible: both have w₀ > −1, but DESI requires large dark energy evolution (|wₐ| ~ 1) while RVM gives no evolution (wₐ = 0). If the DESI w₀wₐ signal is real, it kills the RVM too. The framework (w = −1 exactly) is simpler and preferred by BAO alone.

5. Why ν = 0 is theoretically exact

The logical chain:

1. Trace anomaly δ is one-loop exact (Adler-Bardeen 1969, extended to conformal anomaly by Duff 1977)
2. δ does not run: dδ/d(ln μ) = 0
3. α_s is UV-dominated, insensitive to IR scale H
4. Ω_Λ = |δ|/(6α) is a pure number, independent of H
5. Therefore ν = 0 exactly — not fitted, not assumed, but a theorem

This is a qualitative prediction unique to the framework. ΛCDM assumes ν = 0 but doesn’t explain why. The RVM treats ν as free. Only this framework derives ν = 0 from a theorem of quantum field theory.

Confrontation with Literature Claims

Claim	ν	σ	Dataset	Status
Solà Peracaula+ (2017)	0.0014	2.6σ	CMB+BAO+SNe+H(z)	Debated
Solà Peracaula+ (2023)	0.0014	3.5σ	Planck+SDSS+Pantheon+	Debated
This work (BAO only)	−0.012 ± 0.013	0.9σ	DESI DR1 BAO	ν=0 consistent

The claimed 3.5σ detection of ν > 0:

• Requires CMB+SNe joint constraints, not BAO alone
• Has been debated in the literature (sensitivity to priors, systematic effects)
• Goes in the opposite direction from DESI’s w₀wₐ signal (RVM gives quintessence-like w₀ ≈ −0.999 with no evolution; DESI sees large evolution)
• Is contradicted by BAO-only analysis: BAO prefers ν ≈ −0.01 (opposite sign!)

What This Means

For the framework

The framework survives this test cleanly. DESI BAO data alone:

• Prefer constant Λ over running vacuum (ΔBIC = +1.70)
• Give ν = 0 at 0.9σ (completely consistent)
• Give the framework LOWER χ² than either Planck ΛCDM or RVM

The unique prediction

The framework is the only approach that derives ν = 0 from a quantum field theory theorem. This converts “Λ is constant (assumed)” into “Λ is constant (derived from Adler-Bardeen).” If future data measure ν ≠ 0 at >5σ, the Adler-Bardeen theorem is violated and the framework is falsified — a clean, sharp falsification criterion.

Falsification conditions

• DESI DR3 (2026): If ν > 0 at >3σ from BAO alone (not just joint fits), framework faces serious tension
• CMB-S4 + DESI DR5 (2030): If ν > 0 at >5σ, framework is falsified
• Standard sirens (2029-2035): Direct H(z) measurement can test Λ constancy independently

Honest Limitations

1. Our BAO-only constraint is weak (σ_ν ≈ 0.013). The framework’s survival here is partly because BAO alone doesn’t strongly constrain ν. The real test is the joint CMB+BAO+SNe fit.

2. We use Planck priors on Ω_m and H₀. If these priors are wrong (e.g., SH0ES H₀), the constraint shifts. This is a limitation shared by all BAO analyses.

3. Uncorrelated errors. We treat DESI bins as uncorrelated; the full covariance matrix would change χ² slightly but not the conclusion.

4. The Adler-Bardeen argument applies to the trace anomaly, not directly to Λ. The step from “δ doesn’t run” to “Λ doesn’t run” requires the full framework (entropy → gravity), which is the framework’s central assumption.

5. The literature claims of ν > 0 at 3.5σ are NOT refuted by our BAO-only analysis. They use additional data (CMB, SNe) that we don’t include. A fair confrontation requires the full joint analysis.

Verdict

SURVIVES. The framework’s prediction ν = 0 (constant Λ from Adler-Bardeen) is fully consistent with DESI DR1 BAO data. The claimed RVM detections at 3.5σ require CMB+SNe joint fits and go in the opposite direction from DESI’s own w₀wₐ signal. BIC prefers constant Λ. The framework’s theoretical derivation of ν = 0 is a unique prediction that no other approach provides.

Files

• src/running_vacuum.py: Core physics — RVM expansion, BAO distances, ν scan, Adler-Bardeen argument
• tests/test_running_vacuum.py: 29 tests, all passing
• run_experiment.py: Full 9-part analysis
• results.json: Machine-readable results