A Null Test for Correlations Between Residual Rotation-Curve Parameters and Large-Scale Velocity-Field Kinematics
A Null Test for Correlations Between Residual Rotation-Curve Parameters and Large-Scale Velocity-Field Kinematics
1. Introduction
1.1. Motivation
The rotation curves of disk galaxies exhibit systematic structure well-described by baryonic mass distributions over a wide range of scales (Lelli et al. 2016; McGaugh et al. 2016). However, residual deviations from baryonic scaling relations persist. The parameter Δγ_resid — defined below as the residual from a baryonic regression — isolates these deviations. Whether Δγ_resid correlates with large-scale environmental properties remains an open empirical question.
This paper tests for such correlations using the SPARC galaxy catalog and the 2M++ reconstructed velocity field (Carrick et al. 2015). We focus on kinematic variables: directional alignment with the bulk flow, shear tensor eigenvalues, and vorticity magnitude. All tests explicitly control for distance-dependent structure.
1.2. Scope
We test low-order kinematic observables:
- cos θ: alignment with data-driven bulk flow axis
- λ₁, λ₂, λ₃: eigenvalues of the traceless shear tensor (sorted descending)
- |ω|: magnitude of the vorticity vector
Higher-order moments and alternative tracers are not tested. The 4 Mpc/h Gaussian smoothing kernel applied to the 2M++ reconstruction inherently limits this test to macro-scale environmental properties.
1.3. Epistemological Stance
This analysis is explicitly exploratory and falsification-oriented. We do not claim discovery; rather, we test whether detectable coupling exists under well-defined observables and controls. Multiple tests are performed without correction; results are interpreted as exploratory.
2. Data and Sample
2.1. SPARC Galaxy Catalog
The SPARC database (Lelli et al. 2016) contains 175 late-type galaxies with Spitzer 3.6 μm photometry and extended H I rotation curves. For each galaxy, we extract:
- V_flat: terminal rotation curve velocity (km/s)
- L_[3.6]: 3.6 μm luminosity (stellar mass proxy)
- log₁₀(SFR_NUV): logarithm of near-ultraviolet star formation rate (from GALEX)
- Distance: published distance estimate (Mpc)
- Coordinates: Right Ascension, Declination
2.2. 2M++ Reconstructed Velocity Field
The 2M++ redshift survey (Carrick et al. 2015) provides a three-dimensional Cartesian grid (257³, box size 800 Mpc/h) of:
- δ: luminosity-weighted density contrast
- v_x, v_y, v_z: peculiar velocity components (km/s) in Galactic Cartesian coordinates
The velocity field is smoothed with a Gaussian kernel of scale 4 Mpc/h. Grid spacing is 1.5625 Mpc/h. The reconstruction accounts for redshift-space distortions via iterative reconstruction.
2.3. Definition of Δγ_resid
We define a dimensionless parameter γ that quantifies deviations from a reference baryonic scaling relation. To isolate potential environmental contributions, we residualize γ against three internal galaxy properties in logarithmic space:
The residualization uses a Huber-robust linear regression (ε = 1.35) to reduce the influence of outliers. The fit yields R² = 0.0433 (N = 80), indicating that baryonic predictors explain approximately 4% of the variance in γ — leaving 96% of the variance unaccounted for.
2.4. Final Matched Sample
After merging SPARC with 2M++ interpolated velocities and requiring non-missing values for all variables, the final sample contains 80 galaxies. Sample size is modest; weak couplings may remain undetectable.
3. Methodology
3.1. Coordinate Transformation and Grid Interpolation
Galaxy coordinates (RA, Dec, Distance) are transformed to Supergalactic Cartesian coordinates (SGX, SGY, SGZ) using astropy, then to Galactic Cartesian coordinates (X, Y, Z) via a rotation matrix. Galactic coordinates are converted to 2M++ grid indices:
where 1.5625 Mpc/h is the grid spacing and h = 0.7 is assumed. Values are clipped to the valid index range [0, 256]. Velocities are interpolated at each galaxy position using trilinear interpolation.
3.2. Interpolation Self-Consistency Audit
To verify correct grid indexing, we test interpolation at three known grid points: (0,0,0), (128,128,128), and (256,256,256). The interpolator returns the exact stored values (difference < 10⁻⁵) at all test points, confirming correct axis ordering and boundary handling.
3.3. Kinematic Variable Definitions
3.3.1. Directional Alignment (cos θ)
The data-driven bulk flow vector is defined as the mean of interpolated velocities over all galaxies:
For each galaxy, the alignment cosine is:
Values range from −1 (anti-aligned) to +1 (aligned).
3.3.2. Shear Tensor Eigenvalues (λ₁, λ₂, λ₃)
The velocity gradient tensor ∂v_i/∂x_j is computed on the 2M++ grid using finite differences. The shear tensor Σ_ij is the traceless symmetric part:
Eigenvalues are computed at each galaxy position and sorted descending: λ₁ ≥ λ₂ ≥ λ₃. Here λ₁ represents the maximum stretching rate, λ₃ the maximum compression rate.
3.3.3. Vorticity Magnitude (|ω|)
The vorticity vector is defined as:
The magnitude |ω| is computed at each galaxy position.
3.4. Partial Spearman Correlation
To isolate associations independent of distance, we compute partial Spearman correlations controlling for distance D. Using the algebraic formula:
where x = Δγ_resid, y = kinematic variable (cos θ, λ_i, |ω|), and z = distance. This removes monotonic distance-dependent trends from both variables before correlation.
3.5. Permutation Tests
For each raw Spearman correlation, we generate a null distribution by shuffling Δγ_resid values (N_perm = 1000) and recomputing ρ. The permutation p-value is:
3.6. Bootstrap Resampling
To assess stability under observational uncertainties, we perform bootstrap resampling (N_bootstrap = 1000) of the galaxy sample, recomputing the partial Spearman correlation for each resampled dataset. The 95% confidence interval for ρ_partial is reported. Null results are considered robust if the confidence interval spans zero and the median bootstrap ρ remains near zero.
3.7. Exploratory Framing
All p-values are reported as exploratory. No correction for multiple tests (five kinematic variables) is applied. Results with p < 0.05 are interpreted as "non-zero association detected (exploratory)" rather than "statistically significant discovery."
4. Results
4.1. Interpolation Audit
Interpolation at grid indices (0,0,0), (128,128,128), and (256,256,256) returned the exact stored values (difference < 10⁻⁵). Grid indexing, axis ordering, and boundary handling are correct.
4.2. Distance–Kinematic Confound Matrix
| Variable | Spearman ρ | p-value |
|---|---|---|
| λ₃ (compression) | 0.823 | <0.001 |
| λ₁ (stretching) | -0.755 | <0.001 |
| |ω| (vorticity) | -0.476 | <0.001 |
| cos θ | -0.271 | 0.015 |
4.3. Raw Spearman Correlations
| Variable | ρ_raw | p_raw |
|---|---|---|
| cos θ | -0.118 | 0.299 |
| λ₁ | -0.207 | 0.066 |
| λ₂ | 0.111 | 0.329 |
| λ₃ | 0.219 | 0.051 |
| |ω| | -0.201 | 0.074 |
4.4. Partial Spearman Correlations
| Variable | ρ_partial | p_partial | Bootstrap 95% CI |
|---|---|---|---|
| cos θ | -0.064 | 0.574 | [-0.27, 0.15] |
| λ₁ | -0.074 | 0.520 | [-0.28, 0.14] |
| λ₂ | 0.045 | 0.693 | [-0.17, 0.26] |
| λ₃ | 0.080 | 0.482 | [-0.13, 0.29] |
| |ω| | -0.116 | 0.307 | [-0.32, 0.10] |
4.5. Permutation and Bootstrap Validation
The raw λ₃ permutation p-value (0.048) indicates a borderline effect, but the partial correlation (ρ = 0.080, p = 0.482) and bootstrap confidence interval spanning zero demonstrate that this raw trend is fully explained by distance.
4.6. Summary Table
| Variable | ρ_raw | p_raw | ρ_partial | p_partial | p_perm |
|---|---|---|---|---|---|
| cos θ | -0.118 | 0.299 | -0.064 | 0.574 | 0.307 |
| λ₁ | -0.207 | 0.066 | -0.074 | 0.520 | 0.070 |
| λ₂ | 0.111 | 0.329 | 0.045 | 0.693 | 0.320 |
| λ₃ | 0.219 | 0.051 | 0.080 | 0.482 | 0.048 |
| |ω| | -0.201 | 0.074 | -0.116 | 0.307 | 0.082 |
5. Discussion
5.1. No Significant Coupling Detected
After controlling for distance, none of the tested kinematic variables show a statistically significant association with Δγ_resid. Effect sizes are weak (|ρ| < 0.12), well within sampling noise for N = 80. Permutation tests and bootstrap confidence intervals confirm the robustness of these nulls. Given that 96% of the variance in γ remains unexplained by baryonic predictors, the absence of detectable environmental coupling is particularly notable.
5.2. Distance–Shear Confound as Methodological Finding
The strong correlations between distance and shear eigenvalues (|ρ| up to 0.823) constitute a key methodological result. They demonstrate that naive environmental correlation analyses using the 2M++ reconstruction are highly susceptible to distance-driven false positives unless explicit distance control is applied. The partial correlation approach successfully stripped this contamination, revealing the underlying null.
5.3. Comparison with Previous Environmental Tests
| Test | Result | Reference |
|---|---|---|
| Δγ vs δ (CF4) | Null (ρ = 0.08, p = 0.34) | Baseline v1.0 |
| Δγ vs δ (2M++) | Null (ρ = 0.08, p = 0.34) | Baseline v1.2 |
| Δγ vs |V_pec| | Null (ρ = 0.10, p = 0.22) | Watchdog-v1.0 |
| Δγ vs kinematic vars | Null (partial |ρ| < 0.12, p > 0.30) | This work |
5.4. Limitations
Key limitations of this analysis include:
- Sample size (N=80): Weak couplings may be undetectable.
- 2M++ reconstruction: Smoothed (4 Mpc/h), incomplete at depth.
- Smoothing kernel: Limits test to macro-scale properties; local sub-megaparsec kinematics unprobed.
- Distance uncertainties: Not fully propagated (bootstrap partially addresses this).
- Low-order kinematics only: Higher-order couplings untested.
- SPARC selection: Not volume-complete; selection effects may remain.
- Multiple testing: No correction applied; p-values are exploratory.
5.5. What This Analysis Does Not Claim
- Not claimed: Δγ_resid has no environmental coupling anywhere.
- Not claimed: Any ontology is falsified.
- Not claimed: Higher-order couplings (e.g., spin–vorticity alignment) are absent.
- Not claimed: Different tracers (e.g., SDSS, larger volume) would yield the same null.
- Claimed only: Within the SPARC × 2M++ overlap sample (N=80), after controlling for distance, no statistically significant association is detected between Δγ_resid and the tested low-order kinematic variables.
6. Conclusion
6.1. Strongest Defensible Statement
6.2. Implications for Future Work
- Higher-order kinematic couplings (e.g., spin–vorticity alignment): Untested; requires galaxy orientation data.
- Larger volume catalogs (e.g., SDSS): May reveal weak couplings at larger scales.
- Direct velocity measurements (vs. reconstructed): Would reduce smoothing uncertainties.
Without new theoretical motivation or empirical anomaly, further tests using the same low-order observables and reconstruction framework are not justified.
7. Data and Code Availability
All data and code are archived in a locked Google Drive directory:
- Locked baseline: baseline_v1.2/sparc_residualized_v1.2.csv
- Watchdog script: WATCHDOG-KINEMATICS v1.1 (available upon request)
- Output data: sparc_kinematics_watchdog_v1.1.csv
The 2M++ data are publicly available from Carrick et al. (2015). SPARC data are publicly available from Lelli et al. (2016).
8. Acknowledgments
This analysis was conducted using an adversarial multi-AI auditing framework, with contributions from ChatGPT, DeepSeek, Gemini, Copilot, and independent reviewer AIs. Their role was limited to methodological critique, code debugging, and statistical validation; no AI contributed to ontological interpretation beyond empirical constraint. The FRCMFD collaboration thanks the developers of SPARC and 2M++ for making their data publicly available.
9. References
Carrick, J., et al. 2015, MNRAS, 450, 317.
Lelli, F., McGaugh, S. S., & Schombert, J. M. 2016, AJ, 152, 157.
McGaugh, S. S., Lelli, F., & Schombert, J. M. 2016, Phys. Rev. Lett., 117, 201101.
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