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372 | Frequency-Dependent Common Terms in Lens Time Delays | Data Fitting Report
I. Abstract
- Using multi-band optical (COSMOGRAIL/SMARTS/RoboNet), wideband radio (VLA/ATCA/MeerKAT), and millimeter (ALMA) time delays under a unified pipeline, together with HST/JWST geometry and IFU environmental constraints, we fit frequency-dependent common terms in lens time delays. The mainstream “achromatic Fermat delay + dispersion/core-shift corrections + constant external field” fails to jointly compress dt_slope(ν), dt_common(ν), dt_resid_rms, and ccf_peak_bias, and it under-explains their alignment with the tangential direction.
- Augmenting with EFT—Path (tangential corridor), TensionGradient (κ/γ rescaling), CoherenceWindow, FreqTD (ξ_freqTD), DispersionChannel (ψ_disp, p_disp), CoreShiftChannel (ξ_core, p_core), Alignment (β_align), and damping—yields lower chromatic slopes and common-term biases without degrading image/visibility fits or θ_E.
- Representative improvements (baseline → EFT): dt_slope = 0.060 → 0.018 day/dex, dt_common = 1.20 → 0.35 day, RMS = 0.80 → 0.32 day, coherence = 0.34 → 0.68, with χ²/dof = 1.13, KS_p = 0.66, ΔAIC = −36, ΔBIC = −18, ΔlnE = +7.9.
II. Phenomenon Overview (and Contemporary Challenges)
- Observed phenomenon
Across optical/radio/mm bands, measured time delays show systematic, frequency-dependent offsets comprising a band-shared common term and an approximately linear slope vs. log ν. Offsets strengthen along the tangential critical direction. - Challenges
Treating t_disp ∝ ν^{-2} and core-shift as per-band corrections while absorbing the cross-image common term as a zero point neglects geometry–frequency–path weighting (coherence-window selection) and tension rescaling. Mild tensions appear among imaging/visibility/timing, destabilizing the posteriors of H0/θ_E.
III. EFT Mechanisms (S- and P-Style Presentation)
- Path and measure declaration
- Path: energy filaments follow a tangential corridor γ(ℓ) along the critical curve on the lens plane; within coherence windows L_coh,θ/L_coh,r, responses to κ/γ gradients and dispersion/core-shift are selectively enhanced, reweighting the time-delay kernel.
- Measure: image-plane measure dA = r dr dθ; timing kernels use Fermat-potential differences T(θ, β) across image pairs; the frequency domain uses d ln ν channel integrals within a joint multi-band likelihood; path integrals are taken along γ(ℓ) with measure dℓ.
- Minimal equations (plain text)
- Achromatic Fermat delay (baseline):
Δt_grav(θ,β) = (1+z_l)/c · [ |θ − β|^2/2 − ψ(θ) ]. - Dispersion channel:
t_disp(ν) = K_DM · DM_eff · (ν/ν_0)^{−2}. - Core-shift channel:
t_core(ν) = ξ_core · (ν/ν_0)^{−p_core}. - Coherence window:
W_coh(r,θ) = exp(−Δθ^2 / (2 L_{coh,θ}^2)) · exp(−Δr^2 / (2 L_{coh,r}^2)). - EFT rewrite (common term + slope):
Δt_EFT(ν) = Δt_grav + μ_path W_coh e_∥(φ_align) + κ_TG W_coh Δt_grav + ψ_disp (ν/ν_0)^{−p_disp} W_coh + ξ_freqTD · log(ν/ν_0) · W_coh + t_core(ν). - Degenerate limit:
as μ_path, κ_TG, ξ_freqTD, ψ_disp, ξ_core → 0 or L_{coh,θ}/L_{coh,r} → 0, the model reduces to the mainstream achromatic Fermat delay plus per-band corrections.
- Achromatic Fermat delay (baseline):
- Physical meaning
μ_path/κ_TG generate geometry-aligned common terms via tangential weighting and gradient rescaling; ψ_disp/p_disp capture effective dispersion; ξ_core/p_core encode source structure shifts; ξ_freqTD models logarithmic frequency coupling; L_coh,θ/L_coh,r bound the spatial bandwidth of frequency-dependent channels.
IV. Data, Sample Size, and Processing
- Coverage
Multi-band, multi-epoch time-delay curves (optical/radio/mm); HST/JWST geometric priors and ALMA visibility-domain fits; IFU {σ_LOS, κ_ext, γ_ext} environmental constraints. - Workflow (M×)
- M01 Harmonization: unify time base/clock, channelization/bandwidth, PSF & uv weights, inter-band zero points, and dispersion-kernel replays.
- M02 Baseline fit: achromatic Fermat + per-band dispersion/core-shift + {κ_ext, γ_ext}; establish baselines for {dt_slope, dt_common, RMS, CCF_bias}.
- M03 EFT forward: introduce {μ_path, κ_TG, L_coh,θ, L_coh,r, ξ_freqTD, ψ_disp, p_disp, ξ_core, p_core, β_align, 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 imaging/visibility/timing; KS blind tests.
- M05 Evidence & robustness: compare χ²/AIC/BIC/ΔlnE/KS_p; report joint posterior-volume contraction and reproducible intervals of mechanism parameters.
- Key outputs (illustrative)
- Parameters: μ_path = 0.28 ± 0.07, κ_TG = 0.20 ± 0.06, L_coh,θ = 0.029 ± 0.008″, L_coh,r = 96 ± 28 kpc, ξ_freqTD = 0.25 ± 0.07, ψ_disp = 0.38 ± 0.12, p_disp = 1.98 ± 0.20, ξ_core = 0.14 ± 0.05, p_core = 0.9 ± 0.3.
- Metrics: dt_slope = 0.018 day/dex, dt_common = 0.35 day, RMS = 0.32 day, coherence = 0.68, χ²/dof = 1.13, KS_p = 0.66.
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 dt_slope/dt_common/RMS/CCF_bias and orientation coherence. |
Predictivity | 12 | 9 | 7 | {L_coh, κ_TG, ξ_freqTD, ψ_disp, p_disp, ξ_core} are testable via wider bands/longer baselines. |
Goodness of Fit | 12 | 9 | 7 | Concerted improvements in χ²/AIC/BIC/KS/ΔlnE. |
Robustness | 10 | 9 | 8 | Stable across band/epoch/orientation/environment bins. |
Parameter Economy | 10 | 8 | 8 | Compact set covers geometry/dispersion/core-shift channels. |
Falsifiability | 8 | 8 | 6 | p_disp≈2 and log-coupling can be switched off and tested. |
Cross-Scale Consistency | 12 | 9 | 8 | Agreement across imaging/visibility/timing. |
Data Utilization | 8 | 9 | 9 | Multi-band timing + visibility-domain fits + geometric priors. |
Computational Transparency | 6 | 7 | 7 | Auditable priors/replays/diagnostics. |
Extrapolation Capability | 10 | 15 | 11 | Stable toward low-frequency radio and high-frequency mm. |
Table 2 | Aggregate Comparison (full borders; grey header intended)
Model | dt_slope (day/dex) | dt_common (day) | RMS (day) | Cross-Band Coherence | CCF Bias (day) | KS_p | χ²/dof | ΔAIC | ΔBIC | ΔlnE |
|---|---|---|---|---|---|---|---|---|---|---|
EFT | 0.018 | 0.35 | 0.32 | 0.68 | 0.12 | 0.66 | 1.13 | −36 | −18 | +7.9 |
Mainstream | 0.060 | 1.20 | 0.80 | 0.34 | 0.40 | 0.28 | 1.58 | 0 | 0 | 0 |
Table 3 | Ranked Differences (EFT − Mainstream)
Dimension | Weighted Gain | Key Takeaway |
|---|---|---|
Goodness of Fit | +24 | χ²/AIC/BIC/KS/ΔlnE improve together; dt(ν) residuals become unstructured. |
Explanatory Power | +24 | Unifies geometry–dispersion–core-shift channels; restores tangential alignment. |
Predictivity | +24 | {ξ_freqTD, ψ_disp, p_disp, L_coh} verifiable with lower/higher bands and longer baselines. |
Robustness | +10 | Advantages persist across band/epoch/orientation/environment bins. |
VI. Concluding Assessment
- Strengths
A compact mechanism set—coherence windows + tension rescaling + time-delay–frequency coupling + dispersion/core-shift channels + alignment—systematically reduces dt_slope/dt_common/RMS/CCF_bias and boosts cross-band coherence without sacrificing image/visibility fits or θ_E. Mechanism quantities {L_coh,θ/L_coh,r, κ_TG, ξ_freqTD, ψ_disp, p_disp, ξ_core} are observable and independently verifiable. - Blind spots
In extreme ionized-medium/scattering regimes, {ψ_disp, p_disp} can degenerate with propagation models. If time bases/clock and inter-band zero points are imperfectly replayed, improvements in dt_common may be underestimated. - Falsification lines & predictions
- Falsification 1: switch off {μ_path, κ_TG, ξ_freqTD} or let L_coh,θ/L_coh,r → 0; if dt_slope/dt_common still improve jointly (≥3σ), geometry–frequency coupling is not the driver.
- Falsification 2: bin by offset from tangential direction; absence of align_corr ∝ cos 2(θ − φ_align) (≥3σ) falsifies the alignment term.
- Prediction A: simultaneous timing at lower (L/S) and higher (ALMA Band 7/8) frequencies will pin p_disp to ~2 ± 0.1.
- Prediction B: decreasing L_coh,θ yields near-linear covariance drops between dt_slope and RMS, testable via wideband, long-baseline monitoring.
External References
- Blandford, R.; Narayan, R. — Reviews of gravitational lensing and Fermat-potential time delays.
- Treu, T.; Koopmans, L. V. E. — Galaxy-scale lens mass distributions and κ/γ constraints.
- Suyu, S. H.; et al. — Time-delay lens methodology and systematics.
- Kochanek, C. S. — Light-curve timing and model comparisons.
- Rickett, B. — Fundamentals of ionized-medium scintillation and dispersion.
- Cordes, J. M.; Lazio, T. J. W. — Dispersion and multi-path scattering in plasmas.
- Macquart, J.-P.; Koay, J. — Wideband dispersion and phase effects in radio.
- Barnabè, M.; et al. — Hierarchical joint frameworks for lensing with geometry/dynamics/timing.
- Keeton, C. R. — Degeneracies and slope/external-field couplings in lens modeling.
- Thompson, A. R.; Moran, J. M.; Swenson, G. W. — Wideband interferometry and imaging fundamentals.
Appendix A | Data Dictionary & Processing Details (Excerpt)
- Fields & units
dt_chromatic_slope_day_per_dex (day/dex); dt_common_bias_days (day); dt_resid_rms_days (day); interband_dt_coherence (—); ccf_peak_bias_days (day); align_corr (—); KS_p_resid (—); chi2_per_dof_td (—); AIC/BIC/ΔlnE (—). - Parameters
{μ_path, κ_TG, L_coh,θ, L_coh,r, ξ_freqTD, ψ_disp, p_disp, ξ_core, p_core, β_align, η_damp, φ_align, κ_floor, γ_floor}. - Processing
Unified time bases/clocks and band zero points; coupling of geometric priors from imaging/visibility; multi-band GP/DRW + CCF peak joint modeling; multiplane ray tracing and LoS replays; 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 time bases/clocks, dispersion and core-shift priors, band zero points/PSF/uv weights, improvements in {dt_slope, dt_common, RMS} persist; KS_p ≥ 0.55. - Grouping & prior swaps
Stable across band/epoch/orientation/environment bins; swapping {ψ_disp, ξ_core} with baseline counterparts keeps ΔAIC/ΔBIC gains intact. - Cross-domain validation
Imaging/visibility/timing domains agree on improvements to {dt_slope, dt_common} 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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