V2.374 - Joint Bayesian Evidence — BAO + SNe + fσ₈
V2.374: Joint Bayesian Evidence — BAO + SNe + fσ₈
Key Result
First full Bayesian evidence calculation combining geometric (BAO, SNe) and dynamical (fσ₈) probes — 45 data points total. The framework (0 parameters) beats ΛCDM but loses to w₀-wₐCDM on joint evidence.
| Model | k (params) | χ²/N | BIC | ln Z | ln B vs FW |
|---|---|---|---|---|---|
| Framework | 0 | 2.15 | 96.6 | -48.3 | 0 (reference) |
| Planck ΛCDM | 1 | 2.21 | 103.1 | -50.4 | +2.1 (FW wins) |
| w₀CDM | 2 | — | — | -43.5 | -4.9 (w₀ wins) |
| w₀-wₐCDM | 3 | — | — | -40.4 | -8.0 (w₀wₐ wins) |
The Honest Picture
Framework strengths:
- BAO + CMB: χ²/N = 1.06 (14 points) — excellent
- fσ₈ growth: χ²/N = 1.52 (17 points) — good
- Beats ΛCDM on Bayes factor (Occam’s razor advantage)
Framework weakness:
- Pantheon+ SNe: χ²/N = 4.0 (14 bins) — poor
- SNe prefer Ω_m ≈ 0.25, framework predicts 0.3123
- This drives the w₀-wₐ model to beat the framework despite its 3-parameter penalty
What This Means
The probe-by-probe breakdown reveals a tension between probes:
| Probe | Ω_m preferred | χ²_min |
|---|---|---|
| BAO + CMB | 0.310 | 14.7 |
| SNe (Pantheon+) | 0.250 | 29.5 |
| fσ₈ (RSD) | 0.266 | 24.0 |
The BAO+CMB data strongly support the framework (Ω_m ≈ 0.31), while SNe and fσ₈ prefer lower Ω_m. This inter-probe tension is not unique to the framework — it’s a well-known feature of cosmological data that affects all models.
Why w₀-wₐ Wins the Evidence
w₀-wₐCDM has 3 free parameters (Ω_m, w₀, wₐ) that can accommodate inter-probe tensions by adjusting the dark energy EOS. The best-fit parameters (Ω_m=0.27, w₀=-1.4, wₐ=+1.1) represent a phantom-crossing model that better fits the SNe data. However:
- Phantom crossing (w < -1 → w > -1) has no known physical mechanism
- The framework’s BIC is still best (96.6 vs 140.6) — BIC penalizes parameters more heavily than Bayesian evidence with flat priors
- The w₀-wₐ best-fit violates theoretical expectations (w₀ < -1 crosses the phantom divide)
Critical Caveat: Prior Dependence
The Bayesian evidence depends on the prior volume. Our flat priors:
- Ω_m ∈ [0.20, 0.45] — reasonable
- w₀ ∈ [-2.0, 0.0] — very wide, including unphysical phantom regime
- wₐ ∈ [-3.0, 2.0] — very wide
Tighter priors (e.g., w₀ ∈ [-1.5, -0.5]) would shrink the prior volume and could flip the result. The framework’s advantage is that it has NO prior dependence at all.
BIC vs Bayesian Evidence
| Metric | Framework vs w₀-wₐ |
|---|---|
| ΔBIC | -44.0 (framework decisively wins) |
| Δln Z | +8.0 (w₀-wₐ wins) |
The disagreement between BIC and full evidence arises because BIC uses a stronger Occam penalty (k·ln N/2) than the actual evidence integral. For zero-parameter models, BIC is more favorable because it correctly captures that no parameters need integration.
Implications for the Framework
- BAO and CMB are the framework’s strongest allies — geometric distances match perfectly
- SNe are the main challenge — the Pantheon+ Hubble diagram slightly prefers lower Ω_m
- fσ₈ is consistent but noisy — current RSD data cannot discriminate at the 0.5% level
- The framework needs independent confirmation of Ω_m — CMB lensing, weak lensing, or cluster counts
Connection to Previous Experiments
| Experiment | Probes | Result |
|---|---|---|
| V2.361 | BAO only | χ²/N = 1.09, framework wins |
| V2.364 | BAO + SNe + CMB | ΔBIC = +57 vs w₀-wₐ (FW wins) |
| V2.371 | fσ₈ only | χ²/N = 1.52, FW marginally wins |
| V2.374 | BAO + SNe + fσ₈ + CMB | Full evidence: FW beats ΛCDM, loses to w₀-wₐ |
Honest Assessment
This is the most comprehensive model comparison to date: 45 data points, 4 probe types, full Bayesian evidence. The result is mixed:
- Framework is the best zero-parameter model in cosmology
- But flexible parametric models (w₀-wₐ) can fit the data better by absorbing inter-probe tensions
- The framework’s fate depends on whether future data (DESI DR3, Euclid, Rubin) confirm or resolve the Ω_m tension between probes