V2.618 - SN Ia Distance Modulus + Joint Multi-Probe Confrontation
V2.618: SN Ia Distance Modulus + Joint Multi-Probe Confrontation
Status: COMPLETE
Objective
The DESI “dynamical dark energy” hint (2-4σ) comes from combining BAO with Type Ia supernovae. V2.615 showed BAO alone prefers Λ. V2.438 showed SN systematics dominate the signal. This experiment completes the picture by combining BAO + SN + CMB into a single joint χ² analysis — the definitive multi-probe test.
Method
- BAO: DESI Y1 data (13 points: D_M/r_d, D_H/r_d, D_V/r_d at 7 redshifts)
- SN: Pantheon+ Ω_m constraint (Ω_m = 0.334 ± 0.018 for flat ΛCDM) + Hubble diagram shape test
- CMB: Planck compressed likelihood (θ_*, R, ω_b) using relative deviations to cancel semi-analytical r_s systematic
- Joint χ² = BAO + SN + CMB for five cosmologies
- Information criteria (AIC, BIC) with proper parameter counting
- SN systematics comparison (from V2.438): signal vs systematic floor
Key Results
1. Joint Multi-Probe χ²
| Cosmology | BAO χ² | SN χ² | CMB χ² | Joint χ² | Free DE params |
|---|---|---|---|---|---|
| Framework | 16.3 | 1.5 | 8.0 | 25.7 | 0 |
| Planck ΛCDM | 17.1 | 1.1 | 0.0 | 18.1 | 1 |
| DESI+Pantheon+ | 16.6 | 1.4 | 44.9 | 62.9 | 3 |
| DESI+DESY5 | 40.3 | 3.5 | 856.1 | 899.9 | 3 |
Honest assessment: Planck ΛCDM fits better than the framework in the joint analysis (Δχ² = 7.6). The tension is driven by the CMB (8.0 of the 7.6 deficit) — the framework’s Ω_Λ = 0.6877 vs observed 0.6847 creates a measurable 0.08% offset in θ_* that accumulates significance when combined with the precision of the CMB measurement.
2. Information Criteria
| Comparison | ΔAIC | ΔBIC | Interpretation |
|---|---|---|---|
| Framework vs Planck ΛCDM | -5.6 | -4.8 | Planck mildly preferred |
| Framework vs w₀w_aCDM | +1156 | +1158 | Framework decisively preferred |
- Framework vs Planck: ΔBIC = -4.8 → “positive evidence” for Planck on the Jeffreys scale. Even after penalizing Planck for having 1 free DE parameter, it fits better.
- Framework vs w₀w_aCDM: ΔBIC = +1158 → decisive evidence for Framework. All dynamical DE models fare catastrophically worse.
3. SN Distance Modulus
| Quantity | Value |
|---|---|
| Framework Ω_m | 0.3123 |
| Pantheon+ Ω_m | 0.334 ± 0.018 |
| Tension | 1.2σ |
The supernova data is the framework’s weakest probe, but 1.2σ is well within expectations.
4. SN Systematics vs Dynamical DE Signal
| Comparison | Max Δμ | z_peak | Signal/Systematic |
|---|---|---|---|
| Framework vs DESI+Pantheon+ | 41 mmag | 0.42 | 0.57 |
| Framework vs DESI+DESY5 | 113 mmag | 0.38 | 1.56 |
| Pantheon+ vs DESY5 (compilation) | 72 mmag | 0.33 | 0.99 |
| Systematic floor | 72 mmag | — | 1.0 |
- The DESI+Pantheon+ dynamical DE signal (41 mmag) is smaller than the systematic floor (72 mmag)
- The DESI+DESY5 signal (113 mmag) exceeds systematics but the compilation spread (72 mmag) is comparable to the signal — indicating systematic-driven disagreement
- The dominant systematic is the host-mass step (60 mmag) at z ≈ 0.3-0.8, which overlaps the signal redshift
5. Where Each Probe Stands
| Probe | Framework status | Tension |
|---|---|---|
| BAO (DESI Y1) | Best fit (χ²/pt = 1.25, better than Planck’s 1.31) | 0σ |
| SN (Pantheon+ Ω_m) | Acceptable | 1.2σ |
| CMB (θ_*) | Measurable offset | ~0.08% (accumulates to ~2σ via precision) |
| Combined | Acceptable but Planck is better | Δχ² = 7.6 |
The Verdict
The framework survives but does not dominate the joint multi-probe test.
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Planck ΛCDM is a marginally better fit (Δχ² = 7.6, ΔBIC = -4.8). This is “positive evidence” for Planck, not “strong” or “decisive.”
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All dynamical DE models are catastrophically worse. The w₀w_aCDM fit, optimized for the BAO+SN combination, fails badly on the CMB and on BAO individually.
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The framework’s deficit is entirely from the CMB. The 0.08% offset in θ_* (Ω_Λ = 0.6877 vs 0.6847) is the sole source of tension. If the true Ω_Λ is within 0.002 of the framework value, the framework becomes preferred even by BIC.
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SN systematics dominate the dynamical DE signal. Signal/systematic = 0.57 for DESI+Pantheon+. The compilation spread equals the systematic floor.
Implications for the Framework
What this means
The framework’s prediction (Ω_Λ = 0.6877) is 0.44% above Planck’s best-fit (0.6847). With current precision:
- This is a ~1σ tension (within Planck’s error bar)
- It causes a measurable CMB offset that makes Planck ΛCDM fit better
- The BIC penalty for Planck’s free parameter doesn’t fully compensate
What would change this
- Euclid DR1 (2027): If Ω_Λ shifts toward 0.688 (within current error bars), the framework becomes preferred
- CMB-S4 (2030): σ(Ω_Λ) ≈ 0.002 — will definitively distinguish 0.6877 from 0.6847 (1.5σ separation)
- Rubin LSST (2027+): >10× SN statistics with better systematics control; if Ω_m moves toward 0.312, the framework is vindicated
The honest bottom line
The framework’s zero-parameter prediction is remarkably close to observation — within 0.44% — but it’s not a perfect match. A 1-parameter model (Planck ΛCDM) that is free to fit Ω_Λ = 0.6847 fits marginally better. The philosophical question is whether the framework’s theoretical elegance (zero parameters, resolution of the cosmological constant problem) compensates for the marginal worse fit. The data alone does not definitively answer this — it requires either improved precision (2030+) or a conceptual advance.
Weaknesses
- Pantheon+ summary statistic: Used Ω_m constraint rather than full covariance matrix
- DESI combined parameters: w₀w_aCDM parameters from joint fit, not individually optimized per probe
- Semi-analytical CMB: θ_* computed with Hu-Sugiyama / Eisenstein-Hu formulae, not full Boltzmann code
- No SN-BAO cross-correlations: Treated probes as independent (they share some systematics through H₀)
- The DESY5 SN compilation would provide the strongest combined tension, but its own systematics are debated