Model Comparison
This page ranks the three models with the Akaike and Bayesian information criteria on the combined dataset (N = 1,603 points: 1,590 SNe + 13 BAO). CPL is preferred by both criteria; Ξosc has the lowest raw χ² but pays for its two extra parameters.
Key results
| Model | χ² | k | AIC | BIC | ΔAIC | Acceptance |
|---|---|---|---|---|---|---|
| CPL | 1531.824 | 4 | 1539.824 | 1561.343 | 0.000 | 0.594 |
| Ξosc | 1529.763 | 6 | 1541.763 | 1574.041 | 1.939 | 0.180 |
| ΛCDM | 1554.165 | 2 | 1558.165 | 1568.924 | 18.341 | 0.715 |
Rows are sorted by AIC; the first row has the lowest AIC. ΔAIC is relative to it.
Definitions
k is the number of free parameters (2 for ΛCDM, 4 for CPL, 6 for Ξosc).
How χ² is obtained
run_mcmc does not minimise χ². It takes the sample with the highest log-probability in the retained chain and sets chi2_min = −2 ln p. With flat priors this is χ² at the best sampled point.
The value can only equal or exceed the true minimum: a longer chain could find a slightly better point. For Ξosc, whose ε and T posteriors extend to their prior bounds, the best sampled point is the least certain of the three. The AIC and BIC inherit that uncertainty.
Reading the table
| Comparison | Result | Meaning |
|---|---|---|
| ΛCDM vs the dynamical models | ΔAIC = 18.3 | ΛCDM is clearly worse by AIC |
| CPL vs Ξosc (AIC) | ΔAIC = 1.94 | close; CPL slightly ahead |
| ΛCDM vs Ξosc (BIC) | BIC 1568.924 vs 1574.041 | the BIC puts ΛCDM ahead of Ξosc |
| Raw χ² | Ξosc 1529.763 vs CPL 1531.824 | Ξosc fits best, by a small margin |
Interpretation. The data suggest departures from w = −1 (both dynamical models beat ΛCDM by AIC), but they do not choose between a smooth CPL evolution and an oscillation. The oscillating model is the most flexible and is not rewarded for it by the BIC.
Limits of AIC and BIC here
- Prior volume is ignored. T, φ₀ and (near its bound) ε are weakly constrained; an information criterion that counts them as k = 6 does not capture how much of the prior volume they leave unconstrained. A Bayesian evidence (e.g. nested sampling) would — it is listed as future work in the project README.
- N counts data points, not independent constraints. Supernova errors are correlated through the covariance matrix.
- Models are not nested (Three Model Families), so differences in χ² do not follow a χ² distribution.
- Fixed r_d and the fixed reference cosmology for the Ξosc time axis apply to every row (Known Limitations).