BeeTheory · Statistical Analysis · 2025
Reading the Fit
χ²/dof = 0.475
A number between 0 and ∞ that tells you whether your model fits the data well, too well, or not at all. Here is what it means in the BeeTheory galactic simulation — and what we would need to claim a genuinely solid result.
What Is χ²/dof?
When a model predicts values Vmodel(Ri) and the data provides observed values Vobs(Ri) with measurement uncertainties σi, the reduced chi-squared is:
\(\frac{\chi^2}{\mathrm{dof}}=\frac{1}{N-p}\sum_{i=1}^{N}\left(\frac{V_{\mathrm{model}}(R_i)-V_{\mathrm{obs}}(R_i)}{\sigma_i}\right)^2\)where N is the number of data points and p is the number of free parameters.
Here: N = 16 Gaia 2024 points, p = 2 fitted parameters, K and α, so dof = 14.
Each term in the sum is a pull: the residual expressed in units of the measurement uncertainty.
A pull of 0.5 means the model is off by half a sigma — excellent. A pull of 2.0 means a two-sigma discrepancy — worth investigating.
\(\frac{\chi^2}{\mathrm{dof}}\approx1\quad\Rightarrow\quad\text{model fits the data at the level of their uncertainties}\) \(\frac{\chi^2}{\mathrm{dof}}\ll1\quad\Rightarrow\quad\text{model fits too well: uncertainties may be overestimated, or overfitting}\) \(\frac{\chi^2}{\mathrm{dof}}\gg1\quad\Rightarrow\quad\text{model is wrong, or uncertainties are underestimated}\)Our Number: 0.475
0.475
χ² / dof · N = 16 · p = 2 · dof = 14
Statistically excellent
Model is not wrong
Uncertainties may be slightly large
χ²/dof scale
over-fit
excellent
good
acceptable
poor
A value of 0.475 sits in the excellent zone. The model is not wrong — the average residual is only 0.69σ across all 16 data points.
Physically, this means BeeTheory predicts the circular velocity to within less than one measurement uncertainty at virtually every observed radius.
But “excellent” is not the same as “proven”
χ²/dof = 0.475 could also mean the measurement uncertainties σi are slightly overestimated. If the true errors were 30% smaller, the same model would give χ²/dof ≈ 1.0.
We cannot distinguish between “model is very good” and “errors are slightly generous” from χ² alone. This is a standard statistical ambiguity.
The Residuals — Point by Point
The raw number 0.475 hides information. Looking at individual pulls reveals the structure of the fit: which points the model nails, and where it struggles.
Pull plot: \((V_{\mathrm{model}}-V_{\mathrm{obs}})/\sigma_i\) for each Gaia 2024 data point
| R (kpc) | Vobs | σ | Vmodel | Residual | Pull |
|---|---|---|---|---|---|
| Loading residuals… | |||||
15 / 16 points within 1σ
Every data point except R = 27.3 kpc has |pull| < 1.0. In a well-calibrated model with Gaussian errors, we expect about 68% of points within 1σ — here we have 94%.
This suggests the model is excellent or the error bars on inner points are slightly too large.
The outlier at R = 27.3 kpc — pull = +1.53σ
The model predicts Vc = 195.3 km/s, while Gaia measures 173 ± 17 km/s.
The discrepancy is 22.3 km/s, or 1.53σ. This is not statistically alarming, but it is physically significant: the model declines too slowly at the largest radius.
The Best-Fit Parameter K = 0.01349
The coupling constant K = 0.01349 kpc⁻¹ is the amplitude of the BeeTheory dark mass field generated per unit of visible mass.
It was determined by minimising χ² over the 16 Gaia data points with α = 0.0744 kpc⁻¹ fixed from the combined Newby + Gaia fit.
\(K=0.01349\,\mathrm{kpc}^{-1}\) \(\lambda=K\ell^2=\frac{K}{\alpha^2}=\frac{0.01349}{(0.0744)^2}=2.44\) \(\rho_{\mathrm{dark}}(R_\odot=8\,\mathrm{kpc})=0.37\,\mathrm{GeV/cm^3}\)The observed local dark matter density is:
\(\rho_{\mathrm{obs}}(R_\odot)=0.39\pm0.03\,\mathrm{GeV/cm^3}\)The dimensionless coupling λ = 2.44 sits in the range λ ∈ [2, 7] observed across BeeTheory fits, including single-component, multi-component, and atomic H₂ comparisons.
This universality — the same order of magnitude from femtometre scales to kiloparsec scales — is the strongest internal consistency check of the BeeTheory framework.
Why K is smaller here than in single-disk fits
Previous fits using only the thin disk as source gave Ksingle = 0.038 kpc⁻¹. With all six galactic components contributing to the dark field, the total baryonic source is 2.8× larger.
Therefore K must be proportionally smaller to produce the same dark mass.
\(K_{\mathrm{multi}}\approx\frac{K_{\mathrm{single}}}{2.8}=\frac{0.038}{2.8}=0.0136\,\mathrm{kpc}^{-1}\)This is exactly what the fit gives. It is a consistency check, not a coincidence.
Why 27.3 kpc Is Hard for Every Model
The outermost Gaia 2024 data point, Vc(27.3 kpc) = 173 ± 17 km/s, is not just difficult for BeeTheory. It is the hardest point for every dark matter model applied to the Milky Way rotation curve.
| Model | Vc(27.3 kpc) predicted | Residual | Pull | χ²/dof |
|---|---|---|---|---|
| BeeTheory | 195.3 km/s | +22.3 | +1.53σ | 0.475 |
| NFW profile | ~198 km/s | +25 | +1.5σ | 0.44 |
| Einasto α = 0.91 | ~191 km/s | +18 | +1.1σ | 0.38 |
| Isothermal halo | ~218 km/s | +45 | +2.6σ | 2.6 |
No standard two-parameter model reproduces Vc(27.3 kpc) = 173 km/s within 1σ.
There are three plausible explanations.
- The measurement itself: R = 27.3 kpc is the most distant point, with the fewest kinematic tracers and the largest systematic uncertainties. Gaia DR4 may shift the value.
- A gas disk contribution: The HI disk extends to around 25 kpc and contributes to the baryonic mass at large radii. Including it as a separate component could steepen the decline.
- A smaller coherence length: α = 0.12 kpc⁻¹, or ℓ = 8.3 kpc, fits the outermost point better but worsens the inner plateau.
What a Genuinely Solid Result Would Look Like
The current fit is statistically good. But good in a model with 2 free parameters and 16 data points is not the same as proven.
0.475
Current χ²/dof, 16 points, 2 parameters
< 0.8
Target: good fit on extended dataset
~1.0
Ideal: calibrated across galaxy samples
Requirements for a Solid Validation
- ✓ Good χ²/dof on the training dataset: Achieved. χ²/dof = 0.475 on Gaia 2024.
- ✓ Correct local dark matter density: Achieved. 0.37 GeV/cm³ predicted vs 0.39 ± 0.03 observed.
- ✓ Consistent dimensionless coupling: Achieved. λ = 2.44 compared with λ ≈ 3.5 from H₂ calibration.
- ! Reproduce the outermost Gaia point within 1σ: Not fully achieved. The residual at R = 27.3 kpc is +1.53σ.
- ✗ Blind prediction on other galaxies: Not yet done. The SPARC catalogue provides the natural test.
- ✗ Derivation of ℓ from first principles: Not yet done. ℓ is currently fitted, not derived.
- ✗ Cluster-scale validation: Not yet done. Bullet Cluster lensing is a key test.
- → Extended rotation curve beyond 30 kpc: Gaia DR4 should provide a critical near-term test.
The honest scientific status
BeeTheory in its current form is a physically motivated phenomenological framework that fits the Milky Way rotation curve as well as the best empirical dark matter profiles, with the advantage of having a physical mechanism.
It is not yet a theory in the strict sense, because K and ℓ are fitted rather than derived.
The path to a full theory is clear: derive ℓ = f(Lsource) from the wave-mass postulate, test universally on SPARC, and make a blind prediction for Gaia DR4.
Data: Ou, X. et al., MNRAS 528, 2024. Model: BeeTheory v2, Dutertre 2023, multi-component fit. SPARC reference: Lelli, McGaugh, Schombert, AJ 152, 2016. Bullet Cluster: Clowe et al., ApJL 648, 2006.
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