V2.599: Bayesian Model Selection — Framework vs ΛCDM vs w₀w_aCDM

Motivation

V2.596 showed the framework passes DESI BAO alone at 0.6σ. But BAO alone is not the full picture. The real test is whether the framework survives when ALL major cosmological datasets are combined: DESI BAO + Planck CMB + Pantheon+ SNe. We apply Bayesian model selection (BIC/AIC) to ask whether Occam’s razor favors the framework’s zero-parameter prediction.

Method

Three models compared:

1. Framework: Ω_m = 0.3123 fixed, w = −1. Only free parameter: H₀r_d (distance scale). Zero cosmological free parameters.
2. ΛCDM: Ω_m free, w = −1. Two free parameters: Ω_m, H₀r_d.
3. w₀w_aCDM: Ω_m, w₀, w_a all free. Four free parameters.

Data:

• DESI DR1 BAO: 12 measurements at z = 0.30–2.33
• Planck 2018 CMB compressed likelihood: shift parameter R = 1.7502 ± 0.0046, acoustic scale l_A = 301.47 ± 0.09
• Pantheon+ SNe Ia: Ω_m = 0.334 ± 0.018

Total: 15 effective data points.

Results

Model Fits

Model	Ω_m	w₀	w_a	H₀	χ²	k_free	χ²/dof
Framework	0.312 (fixed)	−1.00	0.00	66.8	33.0	1	2.36
ΛCDM	0.296	−1.00	0.00	68.3	24.2	2	1.86
w₀w_aCDM	0.337	−0.45	−1.59	64.0	15.3	4	1.39

χ² Breakdown by Dataset

Model	χ²_BAO	χ²_CMB	χ²_SNe	χ²_total
Framework	29.6	1.9	1.5	33.0
ΛCDM	19.2	0.5	4.6	24.2
w₀w_aCDM	15.2	0.1	0.0	15.3

Bayesian Model Selection

Model	BIC	ΔBIC	BIC weight	Jeffreys interpretation
w₀w_aCDM	26.1	0.0	84.7%	—
ΛCDM	29.6	3.5	14.6%	positive evidence against
Framework	35.7	9.6	0.7%	strong evidence against

Honest Assessment: The Framework is in Tension

The combined data disfavor the framework at 3.0σ (ΔBIC = 6.1 vs ΛCDM, 9.6 vs w₀w_aCDM).

This is a serious result that demands honest analysis.

Where Does the Tension Come From?

1. BAO alone (V2.596): Framework passes at 0.6σ. χ²_BAO = 14.5 at optimal H₀r_d.
2. Combined data: χ²_BAO jumps to 29.6. The CMB compressed likelihood pulls H₀r_d to a value that worsens the BAO fit.
3. The root cause: With Ω_m fixed at 0.312, the framework cannot simultaneously satisfy the BAO-preferred H₀r_d and the CMB-preferred H₀r_d. ΛCDM resolves this by adjusting Ω_m; the framework cannot.

This is a BAO-CMB cross-constraint tension specific to the framework’s fixed Ω_m.

Caveats (which do NOT eliminate the tension)

1. CMB compressed likelihood: The Planck compressed parameters (R, l_A) are derived from the full Planck analysis with many nuisance parameters. Using them as a simple Gaussian prior may introduce systematic bias. The tension might soften with the full Planck likelihood.

2. Pantheon+ Ω_m preference: SNe prefer Ω_m = 0.334 ± 0.018, higher than both the framework (0.312) and ΛCDM best-fit (0.296). This is a known SNe-CMB tension that affects all models.

3. No covariance: We treat the 12 DESI data points as independent. The actual DESI analysis includes correlations between D_M and D_H at the same redshift, which would typically reduce χ² by 10–30%.

4. w₀w_aCDM instability: The best-fit w₀ = −0.45, w_a = −1.59 implies phantom crossing (w < −1 in the past), which most physical models cannot accommodate. The improved χ² may reflect overfitting rather than real physics.

What Would Resolve the Tension

• Including DESI covariance matrix: Would reduce all χ² values, likely reducing the framework’s tension by ~1σ.
• DESI DR2/DR3: If the LRG1 D_H outlier (the dominant contributor) is revised, the tension drops.
• Graviton mode uncertainty: If n_grav = 9 instead of 10, Ω_m shifts to ~0.305, closer to DESI’s preference.
• Full Planck likelihood: The compressed likelihood may overstate the CMB constraint.

What Would NOT Resolve the Tension

• The tension is real in the sense that the framework’s Ω_m = 0.312 is higher than the combined data prefer.
• No adjustment within the framework (without changing field content) can shift Ω_m below 0.312.
• If future data converge on Ω_m ≈ 0.295, the framework faces a 3σ problem that will only grow.

Comparison with V2.596 (BAO-only)

Test	Framework tension	Verdict
DESI BAO alone (V2.596)	0.6σ	PASSES
DESI BAO + CMB + SNe (this work)	3.0σ	TENSION

The framework passes BAO alone but shows tension when CMB constraints are added. This is because the CMB constrains H₀ in a way that’s incompatible with the framework’s BAO distances at fixed Ω_m = 0.312.

Interpretation

This is a yellow flag, not a red flag. Three points:

1. 3σ is not 5σ. The framework is not falsified. But the tension is significant enough that it cannot be dismissed.

2. The w₀w_aCDM preference is the real story. The data prefer evolving dark energy at 2.5σ over ΛCDM. This is a known DESI result. The framework’s prediction w = −1 shares ΛCDM’s tension with this hint. If evolving DE is confirmed by DESI DR2+, BOTH the framework AND ΛCDM are in trouble.

3. The framework’s Ω_m is its Achilles heel. The fixed value 0.312 is slightly high compared to the combined-data preference of ~0.296. The 5% difference is currently at 3σ but will grow as data improve. This is the most important number to watch.

What This Means for the Science

The Bayesian analysis reveals an uncomfortable truth: the framework’s zero-parameter prediction, while elegant, is under genuine pressure from combined cosmological data. The tension is mild (3σ, well below falsification) and has known systematic uncertainties (no covariance, compressed CMB). But it is real and cannot be wished away.

The framework’s survival depends on:

1. The LRG1 D_H outlier being a fluctuation (reduces BAO tension)
2. The evolving dark energy hint NOT being confirmed (keeps w = −1 alive)
3. Full covariance analysis softening the combined constraint

This is what honest science looks like: reporting tension as clearly as we report success.