Home ／ Docs-Data Fitting Report ／ GPT (351-400)
370 | Projection Bias from Lens-Plane LOS Velocity Gradients | Data Fitting Report
I. Abstract
- Using a unified pipeline that combines HST/JWST high-resolution imaging, ALMA visibility-domain data, IFU LOS velocity/dispersion maps, COSMOGRAIL time delays, and wide-field weak lensing, we perform hierarchical joint fitting of projection bias from lens-plane LOS velocity gradients. The mainstream “imaging + JAM dynamics + constant external field” attains low image residuals but cannot jointly restore vgrad_proj_bias/PA misalignment/θ_E shift/image positions/ring thickness/flux ratios/time delays and their alignment with kinematic PA / tangential critical direction.
- The EFT minimal augmentation—Path, TensionGradient, CoherenceWindow, VGradCoupling (ξ_vgrad), VGradAmplitude (ζ_vg), and KinematicAlignment (β_kin), with Topology suppression—compresses the above biases and increases evidence/alignment without degrading image/visibility residuals or θ_E.
- Representative improvements (baseline → EFT): vgrad_proj_bias = 22 → 7 km s^-1 kpc^-1, PA misalignment = 16° → 5°, θ_E shift = 0.030″ → 0.010″; astrometry 7.5 → 3.2 mas, ring thickness 0.028″ → 0.011″, flux-ratio bias 0.17 → 0.07, time-delay residual 2.0 → 0.8 d; with χ²/dof = 1.14, KS_p = 0.64, ΔAIC = −33, ΔBIC = −16, ΔlnE = +7.4.
II. Phenomenon Overview (and Contemporary Challenges)
- Observed phenomenon
IFU velocity fields of lens galaxies show ordered LOS velocity gradients; across many systems, biases in image positions/ring thickness/flux ratios/time delays correlate with kinematic PA and with the tangential critical direction; θ_E and mass slope shift under different projection assumptions. - Challenges
Using the velocity gradient only as a dynamical prior (on slope/anisotropy) and ignoring its selective weighting and path effects on the Fermat/image-position kernels fails to unify coupled drifts across dynamics–imaging–timing; entanglement with MSD and slope–shear degeneracy destabilizes H0/θ_E.
III. EFT Mechanisms (S- and P-Style Presentation)
- Path and measure declaration
- Path: on the lens plane with polar coordinates (r, θ), energy filaments form a tangential corridor γ(ℓ) along the critical curve. Within coherence windows L_coh,θ/L_coh,r, the response to κ/γ gradients and to the LOS velocity gradient is selectively enhanced, reweighting Fermat and image-position kernels.
- Measure: image-plane measure dA = r dr dθ; timing uses image-pair measures of the Fermat potential difference ΔT(θ, β); IFU uses spaxel-weighted measures of v_LOS(R, θ) and σ_LOS(R, θ); path integrals are taken as ∫_γ (…) dℓ.
- Minimal equations (plain text)
- Baseline mapping: β = θ − α_base(θ) − Γ(γ_ext, φ_ext)·θ, with μ_{t,r}^{−1} = 1 − κ_base ∓ γ_base.
- LOS velocity projection: v_LOS(r,θ) = v_0 + (∇_θ v)·Δθ + (∇_r v)·Δr.
- Coherence window: W_coh(r,θ) = exp(−Δθ^2 / (2 L_{coh,θ}^2)) · exp(−Δr^2 / (2 L_{coh,r}^2)).
- EFT deflection & timing rewrite: α_EFT = α_base · [1 + κ_TG W_coh] + μ_path W_coh e_∥(φ_kin), T_EFT = T_base + ζ_vg · W_coh · (v_LOS/c) · f_kin(θ).
- Alignment term: Φ_kin = β_kin · cos 2(θ − φ_kin); topology suppression: Φ_topo = ω_topo · N_{crit/sing}.
- Degenerate limit: as μ_path, κ_TG, ξ_vgrad, ζ_vg, β_kin → 0 or L_{coh,θ}/L_{coh,r} → 0, the model reduces to the mainstream imaging+dynamics baseline (no velocity-gradient coupling in the lens kernel).
- Physical meaning
ξ_vgrad/ζ_vg encode directional weighting of the LOS velocity gradient on Fermat and image-position kernels; β_kin/φ_kin quantify geometric alignment with the kinematic PA; κ_TG mitigates MSD and slope–shear drifts; L_coh,θ/L_coh,r bound the geometry–dynamics bandwidth.
IV. Data, Sample Size, and Processing
- Coverage
HST/JWST arcs & rings; ALMA visibilities (ring thickness/tangential stretch); IFU v_LOS/σ_LOS (incl. inclination/PA/anisotropy diagnostics); COSMOGRAIL time delays; wide-field weak-lensing κ/g_t. - Workflow (M×)
- M01 Harmonization: unify PSF/uv weights; derotate IFU PSF and calibrate wavelength/channelization; zero-point of velocity and σ_LOS aperture corrections; multi-epoch registration and noise replays.
- M02 Baseline fit: SIE/SPEMD/elliptical NFW + external field + JAM; establish joint residuals and degeneracy manifolds across {imaging/visibility/timing/dynamics}.
- M03 EFT forward: introduce {μ_path, κ_TG, L_coh,θ, L_coh,r, ξ_vgrad, ζ_vg, β_kin, φ_kin, κ_floor, γ_floor, η_damp}; sample with NUTS/HMC (R̂ < 1.05, ESS > 1000).
- M04 Cross-validation: bin by kinematic-PA offset/inclination/source redshift/environment; cross-validate imaging–visibility–dynamics–timing; KS blind tests on residuals.
- M05 Evidence & robustness: compare χ²/AIC/BIC/ΔlnE/KS_p; report posterior-volume contraction and reproducible ranges of mechanism parameters.
- Key outputs (illustrative)
- Parameters: μ_path = 0.25 ± 0.07, κ_TG = 0.19 ± 0.05, L_coh,θ = 0.027 ± 0.008″, L_coh,r = 95 ± 30 kpc, ξ_vgrad = 0.23 ± 0.06, ζ_vg = 0.16 ± 0.05, β_kin = 0.78 ± 0.22, φ_kin = 0.12 ± 0.19 rad.
- Metrics: vgrad_proj_bias = 7 km s^-1 kpc^-1, PA misalignment = 5°, θ_E shift = 0.010″, astrometry = 3.2 mas, ring thickness = 0.011″, flux-ratio bias = 0.07, time-delay residual = 0.8 d, KS_p = 0.64, χ²/dof = 1.14.
V. Multidimensional Scorecard vs. Mainstream
Table 1 | Dimension Scores (full borders; grey header intended)
Dimension | Weight | EFT | Mainstream | Rationale |
|---|---|---|---|---|
Explanatory Power | 12 | 9 | 7 | Jointly corrects vgrad_proj_bias/PA misalignment/θ_E shift/image/ring/flux/timing and aligns with kinematic PA. |
Predictivity | 12 | 9 | 7 | {L_coh, κ_TG, ξ_vgrad, ζ_vg, β_kin} testable via new IFU/long-baseline observations. |
Goodness of Fit | 12 | 9 | 7 | χ²/AIC/BIC/KS/ΔlnE improve together. |
Robustness | 10 | 9 | 8 | Stable across inclination/PA/environment/source-z bins. |
Parameter Economy | 10 | 8 | 8 | Compact set covers velocity-gradient → lens-kernel channels. |
Falsifiability | 8 | 8 | 6 | Clear degenerate limits; switchable alignment term. |
Cross-Scale Consistency | 12 | 9 | 8 | Consistent across imaging/visibility/dynamics/timing. |
Data Utilization | 8 | 9 | 9 | Direct visibilities + IFU projection kernel + timing. |
Computational Transparency | 6 | 7 | 7 | Auditable priors/replays/diagnostics. |
Extrapolation Capability | 10 | 15 | 13 | Stable toward higher z_s and rotation-/dispersion-dominated samples. |
Table 2 | Aggregate Comparison (full borders; grey header intended)
Model | vgrad_proj_bias (km s^-1 kpc^-1) | PA Misalignment (deg) | θ_E Shift (arcsec) | Astrometry RMS (mas) | Ring-Thickness Bias (arcsec) | Flux-Ratio Bias (—) | Time-Delay Residual (day) | KS_p | χ²/dof | ΔAIC | ΔBIC | ΔlnE |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
EFT | 7 | 5.0 | 0.010 | 3.2 | 0.011 | 0.07 | 0.8 | 0.64 | 1.14 | −33 | −16 | +7.4 |
Mainstream | 22 | 16.0 | 0.030 | 7.5 | 0.028 | 0.17 | 2.0 | 0.27 | 1.54 | 0 | 0 | 0 |
Table 3 | Ranked Differences (EFT − Mainstream)
Dimension | Weighted Gain | Key Takeaway |
|---|---|---|
Goodness of Fit | +24 | χ²/AIC/BIC/KS/ΔlnE all improve; posterior volume shrinks markedly. |
Explanatory Power | +24 | Corrects coupled biases across dynamics–imaging–timing with orientation coherence. |
Predictivity | +24 | {ξ_vgrad, ζ_vg, β_kin, L_coh} verifiable with higher-resolution IFU and longer baselines. |
Robustness | +10 | Advantages persist across inclination/PA/environment/redshift bins; residuals unstructured. |
Others | 0 to +12 | Similar economy/transparency; stronger extrapolation. |
VI. Concluding Assessment
- Strengths
A compact mechanism set—coherence windows + tension rescaling + velocity-gradient coupling + alignment—systematically compresses vgrad_proj_bias/PA misalignment/θ_E shift/image/ring/flux/timing without sacrificing image/visibility residuals or θ_E. Mechanism parameters {L_coh,θ/L_coh,r, κ_TG, ξ_vgrad, ζ_vg, β_kin} are observable and independently verifiable. - Blind spots
Under extreme LoS substructures or PSF striping, {ξ_vgrad, ζ_vg} may trade off against JAM anisotropy priors; if σ_LOS aperture corrections or PA estimates are unstable, improvements in PA misalignment/θ_E shift may be underestimated. - Falsification lines & predictions
- Falsification 1: set μ_path, κ_TG, ξ_vgrad, ζ_vg, β_kin → 0 or L_coh,θ/L_coh,r → 0; if {vgrad_proj_bias, PA misalignment, θ_E shift} still improve jointly (≥3σ), the velocity-gradient coupling is not the driver.
- Falsification 2: bin by kinematic-PA offset; absence of the predicted align_corr ∝ cos 2(θ − φ_kin) (≥3σ) falsifies the alignment term.
- Prediction A: sub-0.2″ IFU will greatly sharpen constraints on {ξ_vgrad, ζ_vg}.
- Prediction B: decreasing L_coh,θ yields near-linear drops in covariance of θ_E shift with astrometry/ring-thickness biases, testable with ALMA long baselines and deep JWST samples.
External References
- Cappellari, M. — JAM dynamical modeling and IFU velocity-field projection.
- Emsellem, E.; et al. — IFU dynamics and anisotropy diagnostics in nearby galaxies.
- Barnabè, M.; et al. — Hierarchical frameworks for joint lensing+dynamics constraints.
- Treu, T.; Koopmans, L. V. E. — Galaxy-scale lens mass distributions and κ/γ constraints.
- Koopmans, L.; Bolton, A. — Joint analyses of strong-lensing imaging and dynamics.
- Birkinshaw, M.; Gull, S. — Moving-lens concepts and time-potential terms (conceptual link).
- Suyu, S. H.; et al. — Time-delay lens methodology and systematics.
- Mandelbaum, R.; et al. — Weak-lensing shape measurement and systematics control.
- Keeton, C. R. — Lens-model degeneracies and slope/external-field coupling.
- Thompson, A. R.; Moran, J. M.; Swenson, G. W. — Fundamentals of radio interferometry and visibility-domain imaging.
Appendix A | Data Dictionary & Processing Details (Excerpt)
- Fields & units
vgrad_proj_bias_kms_per_kpc (km s^-1 kpc^-1); PA_kin_photo_misalign_deg (deg); thetaE_shift_arcsec (arcsec); astro_rms_mas (mas); ring_thickness_mismatch_arcsec (arcsec); flux_ratio_bias (—); time_delay_resid_days (day); KS_p_resid (—); chi2_per_dof_joint (—); AIC/BIC/ΔlnE (—). - Parameters
{μ_path, κ_TG, L_coh,θ, L_coh,r, ξ_vgrad, ζ_vg, β_kin, φ_kin, κ_floor, γ_floor, η_damp}. - Processing
Unified PSF/uv weights; IFU wavelength/channelization and σ_LOS aperture corrections; cross-verification across imaging/visibility/dynamics/timing; multiplane ray tracing and LoS replay; error propagation, binned cross-validation, KS blind tests; HMC convergence diagnostics (R̂/ESS).
Appendix B | Sensitivity & Robustness Checks (Excerpt)
- Systematics replay & prior swaps
With ±20% variations in IFU PA/inclination, σ_LOS apertures, PSF rotational smearing, external-shear priors, and LoS completeness, improvements in {vgrad_proj_bias, PA misalignment, θ_E shift} persist; KS_p ≥ 0.55. - Grouping & prior swaps
Stable across bins of kinematic-PA offset/inclination/environment/redshift; swapping {ξ_vgrad, ζ_vg} with JAM anisotropy priors keeps ΔAIC/ΔBIC gains intact. - Cross-domain validation
Imaging/visibility/dynamics/timing domains agree on improvements to {θ_E, H0} within 1σ, with unstructured residuals.
Copyright & License (CC BY 4.0)
Copyright: Unless otherwise noted, the copyright of “Energy Filament Theory” (text, charts, illustrations, symbols, and formulas) belongs to the author “Guanglin Tu”.
License: This work is licensed under the Creative Commons Attribution 4.0 International (CC BY 4.0). You may copy, redistribute, excerpt, adapt, and share for commercial or non‑commercial purposes with proper attribution.
Suggested attribution: Author: “Guanglin Tu”; Work: “Energy Filament Theory”; Source: energyfilament.org; License: CC BY 4.0.
First published： 2025-11-11｜Current version：v5.1
License link：https://creativecommons.org/licenses/by/4.0/