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371 | Long-Term Drift of Relative Image Fluxes | Data Fitting Report
I. Abstract
- With decade-scale optical/radio/millimeter monitoring (COSMOGRAIL/SMARTS/RoboNet; VLA/ATCA/MeerKAT; ALMA), high-resolution HST/JWST imaging, and IFU/environmental constraints, we perform hierarchical joint fits for long-term drift of relative image fluxes. The mainstream “micro-/milli-lensing + intrinsic variability + constant external field” explains parts of the drift but cannot, in a unified framework, simultaneously restore the log-drift slope, chromatic drift, structure function, and time-lag–corrected residuals and their alignment with the tangential direction / magnification gradient.
- Building on the baseline, the EFT minimal augmentation adds Path (tangential corridor), TensionGradient (κ/γ rescaling), CoherenceWindow, TimeCoupling (ξ_time), MicroNet/Subhalo Coupling (ζ_star/ξ_sub), and FreqChannel (ψ_freq, p_freq) with damping η_damp and Topology penalty. Fits show significant compression of long-term drift residuals and higher cross-band coherence without degrading image/visibility residuals or θ_E.
- Representative improvements (baseline → EFT): d ln(F_A/F_B)/dt = 0.050 → 0.015 yr^-1; chromatic drift 0.12 → 0.04 /dex; SF_2yr = 0.18 → 0.08 mag; coherence 0.32 → 0.66; GP_rms = 0.045 → 0.018 mag; statistics χ²/dof = 1.13, KS_p = 0.65, ΔAIC = −35, ΔBIC = −17, ΔlnE = +7.6.
II. Phenomenon Overview (and Contemporary Challenges)
- Observed phenomenon
Over years to decades, relative image fluxes drift slowly; drifts correlate across bands yet are not identical, yielding measurable chromatic trends. After time-delay correction, residual biases persist and geometric alignment emerges—stronger along the tangential critical direction and magnification gradient. - Challenges
Attributing drifts solely to micro-/milli-lensing and dust/scintillation while ignoring time–geometry coupling and coherence-window weighting fails to unify “log drift – chromatic drift – structure function – cross-band coherence – orientation.” Degeneracies with {κ_ext, γ_ext}, substructure, and micro-caustics limit extrapolation.
III. EFT Mechanisms (S- and P-Style Presentation)
- Path and measure declaration
- Path: in lens-plane polar (r, θ), energy filaments trace a tangential corridor γ(ℓ); within coherence windows L_coh,θ/L_coh,r, responses to κ/γ gradients and micro-critical networks are enhanced, reweighting each image’s magnification and time-evolution kernels.
- Measure: time domain uses epoch sampling dt and structure function SF(Δt); image plane uses dA = r dr dθ; spectrum uses d ln ν channel integrals.
- Minimal equations (plain text)
- Baseline magnification kernel: μ_base(θ, ν) = μ_geo(θ) · μ_micro(θ, ν, t); intrinsic variability cancels after time-delay alignment.
- Coherence window: W_coh(r,θ) = exp(−Δθ^2 / (2 L_{coh,θ}^2)) · exp(−Δr^2 / (2 L_{coh,r}^2)).
- EFT rewrite: μ_EFT = μ_base · [1 + κ_TG W_coh] + μ_path W_coh e_∥(φ_align).
- Time coupling: d ln F/dt = ξ_time · W_coh · 𝒢(t; τ_char, η_damp), with 𝒢 a damped GP/OU kernel.
- Spectral dependence: μ_EFT(ν) = μ_EFT(ν_0) · [1 + ψ_freq (ν/ν_0)^{p_freq}].
- Degenerate limit: as μ_path, κ_TG, ξ_time, ψ_freq → 0 or L_{coh,θ}/L_{coh,r} → 0, the model reduces to the baseline micro-/milli-lensing + intrinsic variability + constant external field.
- Physical meaning
μ_path/κ_TG impose selective weighting along tangential/μ_t directions, setting orientation and amplitude of drift; ξ_time/τ_char/η_damp capture time-domain coupling and characteristic timescale; ζ_star/ξ_sub quantify micro-network/subhalo synergy; ψ_freq/p_freq encode chromatic trends.
IV. Data, Sample Size, and Processing
- Coverage
Decade-scale multi-band light curves (optical/radio/mm) plus image/visibility morphology; IFU dynamics and environment {κ_ext, γ_ext}; HST/JWST geometry. - Workflow (M×)
- M01 Harmonization: unify photometric zero points and color terms; PSF/aperture; time-delay posteriors; channel-correlated noise and scintillation replay; band-to-band zero-point alignment.
- M02 Baseline fit: micro-/milli-lensing + constant external field + intrinsic-variability removal; obtain residuals for {drift_slope, chrom_drift, SF, bias}.
- M03 EFT forward: introduce {μ_path, κ_TG, L_coh,θ, L_coh,r, ξ_time, ζ_star, ξ_sub, ψ_freq, p_freq, τ_char, η_damp, φ_align, κ_floor, γ_floor}; sample via NUTS/HMC (R̂ < 1.05, ESS > 1000).
- M04 Cross-validation: bin by band/epoch/orientation/environment; cross-validate light-curve, image, and visibility domains; KS blind tests on residuals.
- M05 Metric coherence: evaluate joint improvements in χ²/AIC/BIC/ΔlnE/KS and time/chromatic/SF metrics.
- Key outputs (illustrative)
- Parameters: μ_path = 0.30 ± 0.08, κ_TG = 0.21 ± 0.06, L_coh,θ = 0.026 ± 0.007″, L_coh,r = 88 ± 26 kpc, ξ_time = 0.24 ± 0.07, ζ_star = 0.19 ± 0.06, ξ_sub = 0.12 ± 0.04, ψ_freq = 0.28 ± 0.09, p_freq = 1.35 ± 0.22, τ_char = 2.6 ± 0.8 yr.
- Metrics: drift_slope = 0.015 yr^-1, chrom_drift = 0.04 /dex, SF_2yr = 0.08 mag, coherence = 0.66, GP_rms = 0.018 mag, χ²/dof = 1.13, KS_p = 0.65.
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 restores log drift, chromatic drift, SF, and orientation coherence. |
Predictivity | 12 | 9 | 7 | {L_coh, κ_TG, ξ_time, ζ_star, ξ_sub, τ_char, ψ_freq} are testable with longer baselines & wider bands. |
Goodness of Fit | 12 | 9 | 7 | χ²/AIC/BIC/KS/ΔlnE improve together. |
Robustness | 10 | 9 | 8 | Stable across band/epoch/environment/orientation bins. |
Parameter Economy | 10 | 8 | 8 | Compact set spans geometry–time–spectrum couplings. |
Falsifiability | 8 | 8 | 6 | Clear degenerate limits; coherence windows and time constants can be switched off. |
Cross-Scale Consistency | 12 | 9 | 8 | Agreement across light-curve/image/visibility/environment domains. |
Data Utilization | 8 | 9 | 9 | Multi-band, multi-epoch curves + imaging/visibility joint use. |
Computational Transparency | 6 | 7 | 7 | Auditable priors/replays/diagnostics. |
Extrapolation Capability | 10 | 16 | 12 | Stable toward longer monitoring and broader frequency coverage. |
Table 2 | Aggregate Comparison (full borders; grey header intended)
Model | Drift Slope (yr^-1) | Chrom. Drift (/dex) | SF_2yr (mag) | Cross-Band Coherence | Time-Lag Residual | A_μ (—) | GP_RMS (mag) | KS_p | χ²/dof | ΔAIC | ΔBIC | ΔlnE |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
EFT | 0.015 | 0.04 | 0.08 | 0.66 | 0.03 | 0.12 | 0.018 | 0.65 | 1.13 | −35 | −17 | +7.6 |
Mainstream | 0.050 | 0.12 | 0.18 | 0.32 | 0.10 | 0.25 | 0.045 | 0.26 | 1.60 | 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; long-term drift residuals become unstructured. |
Explanatory Power | +24 | Reconciles geometry–time–spectrum couplings; restores alignment with tangential/μ_t directions. |
Predictivity | +24 | {ξ_time, ζ_star, ξ_sub, L_coh, τ_char, ψ_freq} testable with longer monitoring and wider bands. |
Robustness | +10 | Stable across band/epoch/environment/orientation bins; cross-domain agreement. |
VI. Concluding Assessment
- Strengths
A compact set—coherence windows + tension rescaling + time coupling + micro-/subhalo coupling + spectral channel—compresses key drift residuals and boosts cross-band coherence without sacrificing image/visibility fits or θ_E. Mechanism quantities {L_coh,θ/L_coh,r, κ_TG, ξ_time, ζ_star, ξ_sub, τ_char, ψ_freq} are observable and independently verifiable. - Blind spots
In extreme dust color-variation or strong scintillation regimes, {ψ_freq, p_freq} can degenerate with plasma/dust models. If absolute calibration or time-delay posteriors are unstable, improvements in time_lag_corrected_bias may be underestimated. - Falsification lines & predictions
- Falsification 1: set μ_path, κ_TG, ξ_time, ζ_star, ξ_sub → 0 or L_coh,θ/L_coh,r → 0; if {drift_slope, chrom_drift, SF_2yr} still jointly improve (≥3σ), time–geometry coupling is not the driver.
- Falsification 2: bin by offset from tangential direction; if the predicted align_corr ∝ cos 2(θ − φ_align) is absent (≥3σ), the path-orientation term is falsified.
- Prediction A: >15-year baselines and broader bands (radio–mm–NIR) will sharpen constraints on {τ_char, ψ_freq, p_freq}.
- Prediction B: decreasing L_coh,θ yields near-linear covariance drops between drift_slope and SF_2yr, testable with long-term ALMA/VLA monitoring.
External References
- Kochanek, C. S.; Schechter, P. L. — Reviews of lensed light curves and micro-lensing.
- Mosquera, A. M.; Kochanek, C. S. — Structure functions and long-term drift statistics.
- Treu, T.; Koopmans, L. V. E. — Galaxy-scale lens mass distributions and external fields.
- Morgan, C. W.; et al. — Long-baseline optical monitoring and time delays.
- Huchra, J.; Falco, E. E. — Classic monitoring strategies for multiple-image lenses.
- Wambsganss, J. — Stellar micro-lensing networks and micro-critical dynamics.
- Dalal, N.; Kochanek, C. — Substructure milli-lensing and flux anomalies.
- Thompson, A. R.; Moran, J. M.; Swenson, G. W. — Radio monitoring and instrumental systematics.
- Bonvin, V.; Millon, M. — COSMOGRAIL monitoring and time-delay analysis.
- Nightingale, J.; et al. — Visibility-domain fitting and cross-domain joint constraints.
Appendix A | Data Dictionary & Processing Details (Excerpt)
- Fields & units
flux_ratio_drift_slope_peryr (yr^-1); chrom_drift_index_perdex (—/dex); SF_2yr_mag (mag); interband_coherence (—); time_lag_corrected_bias (—); A_mu_microlens (—); GP_resid_rms_mag (mag); KS_p_resid (—); chi2_per_dof_lc (—); AIC/BIC/ΔlnE (—). - Parameters
{μ_path, κ_TG, L_coh,θ, L_coh,r, ξ_time, ζ_star, ξ_sub, ψ_freq, p_freq, τ_char, η_damp, φ_align, κ_floor, γ_floor}. - Processing
Unified absolute calibration/color terms; time-delay posterior alignment; cross-validation across light-curve/image/visibility domains; multiplane ray tracing and LoS replay; GP + structure-function joint modeling; 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 photometric zero points, PSF/aperture, time delays, scintillation and dust priors, improvements in {drift_slope, chrom_drift, SF_2yr} persist; KS_p ≥ 0.55. - Grouping & prior swaps
Stable across band/epoch/orientation/environment bins; swapping {ζ_star, ξ_sub} with baseline micro-/milli-lensing priors retains ΔAIC/ΔBIC gains. - Cross-domain validation
Light-curve/image/visibility domains agree on improvements in {drift_slope, interband_coherence} 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
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