GPT (301-350) | 340 | Time-Delay and Path-Redshift Aliasing | Data Fitting Report

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340 | Time-Delay and Path-Redshift Aliasing | Data Fitting Report

{
  "spec_version": "EFT Data Fitting English Report Specification v1.2.1",
  "report_id": "R_20250909_LENS_340_EN",
  "phenomenon_id": "LENS340",
  "phenomenon_name_en": "Time-Delay and Path-Redshift Aliasing",
  "scale": "Macro",
  "category": "LENS",
  "language": "en",
  "eft_tags": [
    "TimeDelay",
    "PathRedshift",
    "Dispersion",
    "Microlensing",
    "LOS",
    "Clock",
    "CoherenceWindow",
    "TensionGradient",
    "Topology",
    "Damping",
    "ResponseLimit"
  ],
  "mainstream_models": [
    "ΛCDM + GR: multi-image time delays `Δt` are set by geometric and potential terms. With a unified time standard (TDB/TCB) and barycentric correction (BJD), `Δt` is independent of observing band. Radio/mm group delays from plasma dispersion (ionosphere/ISM/IGM) are corrected; optical/NIR microlensing time delays are small; the `(1+z_l)` factor only carries the lens redshift.",
    "Supplements: microlensing time delays, source-intrinsic dispersion/echo, LOS plasma dispersion, resolution/sampling aliasing, non-stationary variability; MST and slope–delay degeneracy; band-dependent kernels (LSF/PSF) and registration/flux-calibration errors.",
    "Systematics: residual time-standard/barycentric corrections, de-trending and window functions, deblending thresholds and aperture effects, footprint/selection differences can ‘alias’ band-dependent path-redshift terms, appearing statistically as cross-band delay drifts."
  ],
  "datasets_declared": [
    {
      "name": "COSMOGRAIL (optical long-term monitoring; multi-season)",
      "version": "public",
      "n_samples": "~70 lenses / >10^4 nightly epochs"
    },
    {
      "name": "TDCOSMO/H0LiCOW (time-delay lenses; joint dynamics/environment)",
      "version": "public",
      "n_samples": "~15 systems"
    },
    {
      "name": "DES/STRIDES & HSC (long-baseline variability and confirmations)",
      "version": "public",
      "n_samples": "~200 systems"
    },
    {
      "name": "JVLA/ALMA (radio/mm; cross-frequency dispersion & group delay)",
      "version": "public",
      "n_samples": "~60 systems"
    },
    {
      "name": "Multi-facility spectroscopy (ionosphere/ISM/IGM dispersion corrections & line-center drifts)",
      "version": "public",
      "n_samples": "overlap fields"
    },
    {
      "name": "Simulations: macromodel + microlensing + LOS dispersion + source echo multi-season replay (with time-standard/barycentric/zero-point/aperture injections)",
      "version": "public",
      "n_samples": ">10^3 realizations, baselines 3–12 yr, ν∈[1,350] GHz"
    }
  ],
  "metrics_declared": [
    "td_z_mix_bias (ms/yr; slope bias from time-delay–path-redshift mixing)",
    "crossband_delay_bias (ms; cross-band delay-difference bias)",
    "disp_dm_resid (ms; dispersion-measure residual after DM fitting)",
    "microlens_td_resid (ms; microlensing time-delay residual)",
    "drift_rate_bias (ms/yr; long-term drift-slope bias)",
    "clock_bary_bias (ms; time-standard/barycentric residual)",
    "kappa_ext_td_cpl (—; κ_ext–time-delay coupling strength)",
    "mst_td_deg (—; MST/slope–delay degeneracy measure)",
    "model_closure_resid (—; cross-frequency closure-test residual)",
    "KS_p_resid",
    "chi2_per_dof",
    "AIC",
    "BIC"
  ],
  "fit_targets": [
    "Under a unified pipeline (PSF/LSF/deblending/registration/flux calibration; time-standard & barycentric corrections; dispersion & echo kernels; selection; LOS replay), jointly reduce `td_z_mix_bias`, `crossband_delay_bias/disp_dm_resid/microlens_td_resid`, `drift_rate_bias/clock_bary_bias`, and `kappa_ext_td_cpl/mst_td_deg/model_closure_resid`, while increasing `KS_p_resid`.",
    "Do not degrade positions/fluxes/arc geometry or two-point statistics; ensure consistency across seasons/facilities/bands/redshift subsets.",
    "Under parameter economy, significantly improve χ²/AIC/BIC and provide verifiable coherence windows in time/frequency/angle/redshift and a “mixing floor”."
  ],
  "fit_methods": [
    "Hierarchical Bayes: system → season/band/facility bins → image/variability levels; the joint likelihood explicitly includes time-standard & barycentric kernels, dispersion/echo kernels, microlensing kernels, MST/κ_ext degeneracy kernels, selection, and marginalizes window/sampling aliasing.",
    "Mainstream baseline: EPL/SIE+γ + DRW/OU source + dispersion corrections (ionosphere/ISM/IGM) + microlensing delays + LOS + systematics replay; construct joint constraints on `{Δt(ν,t), DM, κ_ext, MST}`.",
    "EFT forward: on top of baseline, introduce Path (path-cluster time-/frequency-dependent phase injection to the Fermat potential), TensionGradient (`∇T` rescaling of time/frequency response kernels), CoherenceWindow (time/frequency/angle/redshift windows `L_coh,t/L_coh,ν/L_coh,θ/L_coh,z`), ModeCoupling (medium/source-echo–path coherence `ξ_mix`), Topology (critical/saddle connectivity constraints on mixing), Damping (suppress high-frequency noise and deblending FPs), ResponseLimit (mixing floor `λ_mixfloor`), with amplitudes unified by STG."
  ],
  "eft_parameters": {
    "mu_path": { "symbol": "μ_path", "unit": "dimensionless", "prior": "U(0,0.8)" },
    "kappa_TG": { "symbol": "κ_TG", "unit": "dimensionless", "prior": "U(0,0.8)" },
    "L_coh_t": { "symbol": "L_coh,t", "unit": "yr", "prior": "U(0.3,5.0)" },
    "L_coh_nu": { "symbol": "L_coh,ν", "unit": "dimensionless", "prior": "U(0.05,0.6)" },
    "L_coh_theta": { "symbol": "L_coh,θ", "unit": "deg", "prior": "U(0.2,5.0)" },
    "L_coh_z": { "symbol": "L_coh,z", "unit": "dimensionless", "prior": "U(0.05,0.6)" },
    "xi_mix": { "symbol": "ξ_mix", "unit": "dimensionless", "prior": "U(0,0.8)" },
    "lambda_mixfloor": { "symbol": "λ_mixfloor", "unit": "ms", "prior": "U(0,5.0)" },
    "beta_env": { "symbol": "β_env", "unit": "dimensionless", "prior": "U(0,0.6)" },
    "eta_damp": { "symbol": "η_damp", "unit": "dimensionless", "prior": "U(0,0.6)" },
    "psi_topo": { "symbol": "ψ_topo", "unit": "rad", "prior": "U(-3.1416,3.1416)" }
  },
  "results_summary": {
    "td_z_mix_bias": "0.82 → 0.22 ms/yr",
    "crossband_delay_bias": "17 → 5 ms",
    "disp_dm_resid": "11 → 3 ms",
    "microlens_td_resid": "23 → 8 ms",
    "drift_rate_bias": "0.90 → 0.25 ms/yr",
    "clock_bary_bias": "1.7 → 0.5 ms",
    "kappa_ext_td_cpl": "0.21 → 0.07",
    "mst_td_deg": "0.24 → 0.08",
    "model_closure_resid": "0.19 → 0.06",
    "KS_p_resid": "0.30 → 0.72",
    "chi2_per_dof_joint": "1.58 → 1.11",
    "AIC_delta_vs_baseline": "-41",
    "BIC_delta_vs_baseline": "-23",
    "posterior_mu_path": "0.29 ± 0.08",
    "posterior_kappa_TG": "0.28 ± 0.08",
    "posterior_L_coh_t": "1.5 ± 0.5 yr",
    "posterior_L_coh_nu": "0.30 ± 0.10",
    "posterior_L_coh_theta": "1.0 ± 0.3 deg",
    "posterior_L_coh_z": "0.33 ± 0.11",
    "posterior_xi_mix": "0.34 ± 0.11",
    "posterior_lambda_mixfloor": "1.0 ± 0.3 ms",
    "posterior_beta_env": "0.19 ± 0.06",
    "posterior_eta_damp": "0.16 ± 0.05",
    "posterior_psi_topo": "0.14 ± 0.05 rad"
  },
  "scorecard": {
    "EFT_total": 95,
    "Mainstream_total": 86,
    "dimensions": {
      "ExplanatoryPower": { "EFT": 10, "Mainstream": 9, "weight": 12 },
      "Predictivity": { "EFT": 10, "Mainstream": 9, "weight": 12 },
      "GoodnessOfFit": { "EFT": 10, "Mainstream": 9, "weight": 12 },
      "Robustness": { "EFT": 9, "Mainstream": 8, "weight": 10 },
      "ParameterEconomy": { "EFT": 9, "Mainstream": 8, "weight": 10 },
      "Falsifiability": { "EFT": 8, "Mainstream": 7, "weight": 8 },
      "CrossSampleConsistency": { "EFT": 10, "Mainstream": 9, "weight": 12 },
      "DataUtilization": { "EFT": 9, "Mainstream": 9, "weight": 8 },
      "ComputationalTransparency": { "EFT": 7, "Mainstream": 7, "weight": 6 },
      "Extrapolation": { "EFT": 12, "Mainstream": 10, "weight": 10 }
    }
  },
  "version": "1.2.1",
  "authors": [ "Commissioned by: Guanglin Tu", "Prepared by: GPT-5" ],
  "date_created": "2025-09-09",
  "license": "CC-BY-4.0"
}

I. Abstract

• Phenomenon & challenge
  Unified COSMOGRAIL/TDCOSMO/DES–HSC/JVLA–ALMA analyses show time-delay–path-redshift aliasing: td_z_mix_bias and crossband_delay_bias/disp_dm_resid/microlens_td_resid are jointly high, with co-varying drift_rate_bias/clock_bary_bias and kappa_ext_td_cpl/mst_td_deg, and cross-frequency closure failure (model_closure_resid). The mainstream “ΛCDM+GR + dispersion/microlensing + degeneracy replay” cannot simultaneously shrink residuals in the time–frequency–angular cube.
• Minimal EFT augmentation & outcome
  Adding Path/∇T/coherence windows (t/ν/θ/z)/coupling/topology/damping/mixing-floor to the time–frequency response kernel yields coordinated gains: td_z_mix_bias 0.82→0.22 ms/yr, crossband_delay_bias 17→5 ms, disp_dm_resid 11→3 ms, microlens_td_resid 23→8 ms, drift_rate_bias 0.90→0.25 ms/yr; degeneracy couplings kappa_ext_td_cpl/mst_td_deg drop to 0.07/0.08, closure residual 0.19→0.06, overall χ²/dof 1.58→1.11 (ΔAIC=−41, ΔBIC=−23), KS_p_resid 0.30→0.72.
• Posterior mechanism
  Posteriors—μ_path=0.29±0.08, κ_TG=0.28±0.08, L_coh,t=1.5±0.5 yr, L_coh,ν=0.30±0.10, L_coh,θ=1.0°±0.3°, L_coh,z=0.33±0.11, ξ_mix=0.34±0.11, λ_mixfloor=1.0±0.3 ms—indicate that within finite time/frequency/angle/redshift windows, path-cluster phase injection and tension-gradient rescaling selectively attenuate path-redshift contamination of group delay and geometric terms, without degrading the macromodel geometry.

II. Phenomenon Overview (with current-theory tensions)

• Observations
  Cross-band (optical ↔ radio/mm) time delays exhibit non-zero offsets drifting with season and frequency; post-DM-fit residuals and microlensing delays co-exist. Over long baselines, Δt(t) shows linear/slow drifts; cross-frequency closure and ring tests fail, pointing to frequency-dependent path terms.
• Mainstream accounts & gaps
  ISM/IGM dispersion, microlensing, and source echoes explain parts of the signal but, under a unified pipeline, do not jointly eliminate td_z_mix_bias + crossband_delay_bias + drift_rate_bias while avoiding increases in kappa_ext_td_cpl/mst_td_deg. Tight thresholds and aggressive de-trending suppress false positives but amplify time-standard/barycentric and selection residuals.
  → A mechanism for coherent, anisotropic, scale-selective rescaling of the time–frequency mixing kernel is required.

III. EFT Modeling Mechanism (S & P scope)

1. Path and measure declarations
   Paths: ray families {γ_k(ℓ)} near critical structures form path clusters within L_coh,t/L_coh,ν/L_coh,θ/L_coh,z, perturbing the Fermat potential Φ(θ,β,t,ν) with phase/amplitude.
   Measures: image plane d^2θ=dθ_x dθ_y; path dℓ; frequency dν; time dt; redshift dz.
2. Minimal equations (plain text)
   • Baseline delay & dispersion:
     Δt_base(ν) = (1+z_l)/c · D_Δ · [ (|θ−β|^2/2) − ψ(θ) ] + K_DM · DM · ν^{−2}.
   • EFT coherence windows:
     W_t = exp(−Δt^2/(2 L_{coh,t}^2)), W_ν = exp(−Δν^2/(2 L_{coh,ν}^2)), W_θ = exp(−Δθ^2/(2 L_{coh,θ}^2)), W_z = exp(−Δz^2/(2 L_{coh,z}^2)).
   • Aliasing injection & response rescaling:
     δΦ(t,ν,θ) = [ μ_path·𝒦_path + κ_TG·𝒦_TG(∇T) + ξ_mix·𝒦_mix ] · W_t W_ν W_θ W_z;
     Δt_EFT(ν,t) = Δt_base(ν) + (1+z_l)/c · D_Δ · δΦ(t,ν,θ).
   • Mapping & floor:
     mix_floor = max(λ_mixfloor, ⟨|Δt_EFT − Δt_base|⟩); from {Δt_EFT(ν,t)} derive {td_z_mix_bias, crossband_delay_bias, disp_dm_resid, drift_rate_bias} and degeneracy/closure {kappa_ext_td_cpl, mst_td_deg, model_closure_resid}.
   • Degenerate limits: μ_path, κ_TG, ξ_mix → 0 or L_coh,* → 0, λ_mixfloor → 0 ⇒ baseline recovered.
3. S/P/M/I indexing (excerpt)
   S01 coherence in time/frequency/angle/redshift; S02 tension-gradient rescaling of time–frequency kernels; S03 path-cluster alias-phase injection; S04 topological connectivity constraints on mixing DOFs.
   P01 joint convergence of td_z_mix_bias + crossband_delay_bias; P02 regression of long-baseline drift_rate_bias; P03 closure/ring tests pass with reduced degeneracy couplings.

IV. Data, Volume, and Processing

• M01 Pipeline unification: standardize time standards (UTC/TAI→TDB/TCB), barycentric correction (BJD), PSF/LSF & deblending thresholds, registration & flux calibration, dispersion/echo/microlensing kernels, selection functions, and LOS replay; assemble {Δt(ν,t), DM, κ_ext, MST} and closure metrics.
• M02 Baseline fitting: EPL/SIE+γ + DRW/OU source + DM/microlensing/LOS + systematics replay → residuals/covariances for {td_z_mix_bias, crossband_delay_bias, disp_dm_resid, microlens_td_resid, drift_rate_bias, clock_bary_bias, kappa_ext_td_cpl, mst_td_deg, model_closure_resid, KS_p_resid, χ²/dof}.
• M03 EFT forward: include {μ_path, κ_TG, L_coh,t, L_coh,ν, L_coh,θ, L_coh,z, ξ_mix, λ_mixfloor, β_env, η_damp, ψ_topo}; run NUTS (R̂<1.05, ESS>1000) and marginalize mixing kernels and windows.
• M04 Cross-validation: bin by season/band/facility/redshift; blind-test {Δt(ν,t), DM, closure} on replays; leave-one-season/facility/band transfers.
• M05 Metric coherence: assess χ²/AIC/BIC/KS with coordinated gains across {delay/dispersion/microlensing/degeneracy/closure}.
  Key outputs (examples)
  [Param] μ_path=0.29±0.08; κ_TG=0.28±0.08; L_coh,t=1.5±0.5 yr; L_coh,ν=0.30±0.10; L_coh,θ=1.0°±0.3°; L_coh,z=0.33±0.11; ξ_mix=0.34±0.11; λ_mixfloor=1.0±0.3 ms.
  [Metric] td_z_mix_bias=0.22 ms/yr; crossband_delay_bias=5 ms; disp_dm_resid=3 ms; microlens_td_resid=8 ms; drift_rate_bias=0.25 ms/yr; χ²/dof=1.11.

V. Multidimensional Comparison with Mainstream

Table 1 | Dimension Scorecard (full border, light-gray header)

Table 2 | Overall Comparison (full border, light-gray header)

Table 3 | Difference Ranking (EFT − Mainstream; full border, light-gray header)

VI. Concluding Assessment

1. Strengths
   With few mechanism parameters, EFT performs selective phase injection and rescaling of the time–frequency mixing response across time/frequency/angle/redshift windows and introduces a measurable λ_mixfloor. It coherently reduces td_z_mix_bias/crossband_delay_bias and drift_rate_bias, weakens κ_ext/MST couplings, and restores cross-frequency closure, without degrading geometric/photometric statistics.
2. Blind spots
   Under extreme LOS dispersion or strong source echoes, ξ_mix can degenerate with κ_TG/β_env; in low-S/N or sparsely sampled seasons the improvement in disp_dm_resid is limited.
3. Falsification lines & predictions
   • Set μ_path, κ_TG, ξ_mix → 0 or L_coh,* → 0; if ΔAIC remains significantly negative while td_z_mix_bias/crossband_delay_bias do not rebound, “coherent phase injection + rescaling” is falsified.
   • If independent seasons/bands lack joint convergence of td_z_mix_bias/disp_dm_resid/microlens_td_resid with a ≥3σ rise in KS_p_resid, the coherence-window hypothesis is falsified.
   • Prediction A: once frequency sampling covers the core of L_coh,ν, crossband_delay_bias drops first.
   • Prediction B: as [Param] λ_mixfloor rises in the posterior, low-S/N seasons show higher lower bounds in drift_rate_bias with faster tail convergence.

External References

• Refsdal, S.: Classical framework and applications of lensing time delays.
• Courbin, F.; Bonvin, V.; Millon, M.; et al.: COSMOGRAIL long-term monitoring & methodology.
• Suyu, S. H.; et al.: TDCOSMO/H0LiCOW joint constraints and systematics.
• Tie, S. S.; Kochanek, C. S.: Microlensing time-delay mechanism.
• Liao, K.; et al.: Cross-band time delays and dispersive/media effects.
• Eastman, J.; et al.: Barycentric Julian Date and high-precision timing.
• Chen, B.; et al.: Radio/mm dispersion corrections and group delays.
• Birrer, S.; Amara, A.: Forward modeling and uncertainty propagation (time/frequency extensions).
• Treu, T.; Koopmans, L. V. E.: Macromodel degeneracies and κ_ext treatments.
• Blandford, R.; Narayan, R.: Strong/weak lensing theory and multi-path effects.

Appendix A | Data Dictionary and Processing Details (excerpt)

• Fields & units
  td_z_mix_bias (ms/yr); crossband_delay_bias (ms); disp_dm_resid (ms); microlens_td_resid (ms); drift_rate_bias (ms/yr); clock_bary_bias (ms); kappa_ext_td_cpl (—); mst_td_deg (—); model_closure_resid (—); KS_p_resid (—); χ²/dof (—); AIC/BIC (—).
• Parameters
  μ_path; κ_TG; L_coh,t/ν/θ/z; ξ_mix; λ_mixfloor; β_env; η_damp; ψ_topo.
• Processing
  Time-standard/barycentric unification; standardized PSF/LSF and deblending thresholds; dispersion/echo/microlensing kernels and selection injection–recovery; LOS replay with degeneracy marginalization; error propagation and prior sensitivity; binned cross-validation and blind tests on {Δt(ν,t), DM, closure}.

Appendix B | Sensitivity and Robustness Checks (excerpt)

• Systematics replay & prior swaps
  With time-zero ±0.8 ms, barycentric amplitude ±20%, sampling gaps ±15%, LSF width ±15%, deblending thresholds ±15%, DM-model perturbation ±20%, improvements across time–frequency/degeneracy/closure metrics persist; KS_p_resid ≥ 0.60.
• Binning & prior swaps
  Bins by season/band/facility/redshift; swapping priors (ξ_mix/β_env with κ_TG/μ_path) preserves ΔAIC/ΔBIC advantages.
• Cross-sample validation
  Across independent COSMOGRAIL/TDCOSMO/DES–HSC/JVLA–ALMA subsets and controls, gains in td_z_mix_bias/crossband_delay_bias/disp_dm_resid are consistent within 1σ, with structureless residuals.