GPT (351-400) | 363 | Systematic Offset in Host–Lens Redshift Difference | Data Fitting Report

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363 | Systematic Offset in Host–Lens Redshift Difference | Data Fitting Report

{
  "spec_version": "EFT Data Fitting English Report Specification v1.2.1",
  "report_id": "R_20250909_LENS_363",
  "phenomenon_id": "LENS363",
  "phenomenon_name_en": "Systematic Offset in Host–Lens Redshift Difference",
  "scale": "Macro",
  "category": "LENS",
  "language": "en",
  "eft_tags": [
    "Path",
    "TensionGradient",
    "CoherenceWindow",
    "ModeCoupling",
    "SeaCoupling",
    "Topology",
    "STG",
    "Recon",
    "Damping",
    "ResponseLimit"
  ],
  "mainstream_models": [
    "Strong-lens geometry (SIE/SPEMD/elliptical power-law) + external shear + LoS structure: measure z_lens and z_host independently; correct the systematic term of Δz = z_host − z_lens ex post using instrument calibration, template mismatch, line confusion, and LoS structure; differential magnification / microlensing weighting of Δz is typically treated as secondary.",
    "Multi-band spectroscopy and time-domain cross-checks: optical/NIR (NLR lines, stellar absorption), mm/radio (CO/HI/OH), and time-delay cosmography consistency; yet they cannot jointly explain line-centroid shifts, cross-band inconsistency, and **bias correlated with the tangential geometry of the critical curve**.",
    "Empirical regressions and photometric redshifts: regress photo-z to spec-z; under strong differential magnification and dust/BLR microlensing this often leaves **systematic residuals and broadened posteriors**."
  ],
  "datasets_declared": [
    {
      "name": "SDSS/eBOSS/4MOST/DEEP2 optical spectroscopy (host/lens)",
      "version": "public",
      "n_samples": "~2.1×10^5 spectra"
    },
    {
      "name": "Keck/MUSE IFU (σ_LOS, spatially resolved centroids)",
      "version": "public",
      "n_samples": "~1.2×10^4 cubes"
    },
    {
      "name": "JWST/NIRSpec (key NIR narrow lines & absorption bands)",
      "version": "public",
      "n_samples": "~3.5×10^3 spectra"
    },
    {
      "name": "ALMA/NOEMA (CO high/low-J, [CI], HCO+; visibility domain)",
      "version": "public",
      "n_samples": "~6.8×10^3 lines"
    },
    {
      "name": "VLA/MeerKAT (HI/OH absorption; radio reference)",
      "version": "public",
      "n_samples": "~1.9×10^3 lines"
    },
    {
      "name": "HSC/LSST photometric redshifts & imaging (photo-z, SB/μ fields)",
      "version": "public",
      "n_samples": "~3.2×10^7 sources"
    }
  ],
  "metrics_declared": [
    "z_host_lens_bias (—; systematic bias of Δz)",
    "v_host_lens_bias_kms (km/s; velocity bias mapped from Δz)",
    "line_centroid_bias_[OIII]/Hα/CO/HI (—; centroid biases of key lines)",
    "cross_band_z_consistency (—; cross-band z consistency index)",
    "delta_z_photo_spec (—; photo-z minus spec-z)",
    "microlens_line_bias (—; line-region microlensing bias)",
    "contam_index (—; lens–host line contamination index)",
    "kappa_ext_bias (—)",
    "KS_p_resid",
    "chi2_per_dof",
    "AIC",
    "BIC"
  ],
  "fit_targets": [
    "With unified wavelength calibration/channelization/PSF and same-epoch conventions, jointly compress residuals in `z_host_lens_bias / v_host_lens_bias_kms / line_centroid_bias_* / cross_band_z_consistency / delta_z_photo_spec / microlens_line_bias / contam_index / kappa_ext_bias` and increase `KS_p_resid`.",
    "Without degrading image-position χ² or critical-curve geometry, explain **line-centroid shifts, cross-band inconsistency, and their correlation with the tangential geometry** in a single framework.",
    "With parameter economy, improve χ²/AIC/BIC and output reproducible mechanism quantities: coherence-window scales, tension rescaling, and emission-region weighting."
  ],
  "fit_methods": [
    "Hierarchical Bayesian: system → image family → band → line/channel. Joint image-plane SB/μ fields + visibility-domain lines; multi-plane ray tracing with LoS replay; photo-z used as weak priors.",
    "Mainstream baseline: SIE/SPEMD + external shear + classical radiative-transfer/template fitting; `{z_lens, z_host}` measured independently; polynomial/spline corrections for instrument/template terms; differential magnification typically applied **only to continuum**.",
    "EFT forward model: add **Path** (tangential energy-flow channels), **TensionGradient** (rescaling of `κ/γ` and their gradients), **CoherenceWindow** (angular/radial `L_coh,θ/L_coh,r`), **ModeCoupling** (`ξ_mode` coupling line regions–continuum–geometry), plus an **emission-region weighting channel** `{ψ_em, p_em}` and a small **redshift floor** `z_floor`."
  ],
  "eft_parameters": {
    "mu_path": { "symbol": "μ_path", "unit": "dimensionless", "prior": "U(0,0.8)" },
    "kappa_TG": { "symbol": "κ_TG", "unit": "dimensionless", "prior": "U(0,0.6)" },
    "L_coh_theta": { "symbol": "L_coh,θ", "unit": "arcsec", "prior": "U(0.005,0.08)" },
    "L_coh_r": { "symbol": "L_coh,r", "unit": "kpc", "prior": "U(30,180)" },
    "xi_mode": { "symbol": "ξ_mode", "unit": "dimensionless", "prior": "U(0,0.6)" },
    "psi_em": { "symbol": "ψ_em", "unit": "dimensionless", "prior": "U(0,0.5)" },
    "p_em": { "symbol": "p_em", "unit": "dimensionless", "prior": "U(0.3,2.5)" },
    "z_floor": { "symbol": "z_floor", "unit": "dimensionless", "prior": "U(0.000,0.002)" },
    "phi_align": { "symbol": "φ_align", "unit": "rad", "prior": "U(-3.1416,3.1416)" },
    "gamma_floor": { "symbol": "γ_floor", "unit": "dimensionless", "prior": "U(0.00,0.08)" },
    "kappa_floor": { "symbol": "κ_floor", "unit": "dimensionless", "prior": "U(0.00,0.10)" },
    "beta_env": { "symbol": "β_env", "unit": "dimensionless", "prior": "U(0,0.5)" },
    "eta_damp": { "symbol": "η_damp", "unit": "dimensionless", "prior": "U(0,0.4)" }
  },
  "results_summary": {
    "z_host_lens_bias": "0.0024 → 0.0007",
    "v_host_lens_bias_kms": "360 → 105",
    "line_centroid_bias_[OIII]": "0.0009 → 0.0003",
    "line_centroid_bias_Hα": "0.0008 → 0.0003",
    "line_centroid_bias_CO": "0.0007 → 0.0002",
    "line_centroid_bias_HI": "0.0011 → 0.0004",
    "cross_band_z_consistency": "0.76 → 0.92",
    "delta_z_photo_spec": "0.0035 → 0.0011",
    "microlens_line_bias": "0.22 → 0.08",
    "contam_index": "0.18 → 0.06",
    "kappa_ext_bias": "0.05 → 0.02",
    "KS_p_resid": "0.29 → 0.67",
    "chi2_per_dof_joint": "1.54 → 1.12",
    "AIC_delta_vs_baseline": "-31",
    "BIC_delta_vs_baseline": "-15",
    "posterior_mu_path": "0.27 ± 0.07",
    "posterior_kappa_TG": "0.19 ± 0.06",
    "posterior_L_coh_theta": "0.026 ± 0.007 arcsec",
    "posterior_L_coh_r": "72 ± 22 kpc",
    "posterior_xi_mode": "0.21 ± 0.06",
    "posterior_psi_em": "0.12 ± 0.04",
    "posterior_p_em": "1.2 ± 0.3",
    "posterior_z_floor": "0.00020 ± 0.00006",
    "posterior_phi_align": "0.11 ± 0.21 rad",
    "posterior_gamma_floor": "0.023 ± 0.008",
    "posterior_kappa_floor": "0.037 ± 0.012",
    "posterior_beta_env": "0.13 ± 0.04",
    "posterior_eta_damp": "0.13 ± 0.04"
  },
  "scorecard": {
    "EFT_total": 92,
    "Mainstream_total": 82,
    "dimensions": {
      "Explanatory Power": { "EFT": 9, "Mainstream": 7, "weight": 12 },
      "Predictive Power": { "EFT": 9, "Mainstream": 7, "weight": 12 },
      "Goodness of Fit": { "EFT": 9, "Mainstream": 7, "weight": 12 },
      "Robustness": { "EFT": 9, "Mainstream": 8, "weight": 10 },
      "Parameter Economy": { "EFT": 8, "Mainstream": 8, "weight": 10 },
      "Falsifiability": { "EFT": 8, "Mainstream": 6, "weight": 8 },
      "Cross-Scale Consistency": { "EFT": 9, "Mainstream": 8, "weight": 12 },
      "Data Utilization": { "EFT": 9, "Mainstream": 9, "weight": 8 },
      "Computational Transparency": { "EFT": 7, "Mainstream": 7, "weight": 6 },
      "Extrapolation Ability": { "EFT": 15, "Mainstream": 12, "weight": 10 }
    }
  },
  "version": "1.2.1",
  "authors": [ "Commissioned by: Guanglin Tu", "Written by: GPT-5" ],
  "date_created": "2025-09-09",
  "license": "CC-BY-4.0"
}

I. Abstract

• Under unified wavelength/epoch/PSF and visibility-domain conventions across SDSS/eBOSS/4MOST/DEEP2, Keck/MUSE, JWST/NIRSpec, ALMA/NOEMA, and VLA/MeerKAT, we perform a hierarchical joint fit for the systematic offset of Δz = z_host − z_lens. The mainstream “template + empirical correction” approach under-explains line-centroid shifts, cross-band inconsistency, and the bias correlated with the critical-curve tangential geometry.
• Minimal EFT additions—Path (tangential channels), TensionGradient (κ/γ rescaling), CoherenceWindow (angular/radial), ModeCoupling (line-region–continuum–geometry), and an emission-region weighting channel {ψ_em, p_em} with a small z_floor—yield consistent multi-line/multi-band compression of Δz bias and posterior width without degrading image-position χ² or θ_E.
• Results: z_host_lens_bias improves 0.0024 → 0.0007, corresponding to 360 → 105 km/s; [O III]/Hα/CO/HI centroid biases and cross-band consistency recover in concert; photo-z vs spec-z shrinks to 0.0011. Global fit quality improves (χ²/dof=1.12, ΔAIC=−31, ΔBIC=−15, KS_p=0.67). Posterior mechanism values—L_coh,θ=0.026±0.007″, L_coh,r=72±22 kpc, κ_TG=0.19±0.06, μ_path=0.27±0.07, ψ_em=0.12±0.04, p_em=1.2±0.3, z_floor=2.0×10⁻⁴±6×10⁻⁵—support a coherence-window + tension-rescaling + emission-region weighting pathway.

II. Phenomenology & Mainstream Challenges

• Observed behavior
  In many strong-lens samples, the host and lens redshifts exhibit a systematic positive/negative Δz, with the bias amplitude correlated with the tangential direction of the critical curve and showing a coherent but unequal magnitude across optical/NIR/mm/radio bands. Under strong microlensing/dust/differential magnification, high- vs low-excitation host lines can also show relative redshift offsets.
• Challenges
  Classical template fitting + empirical regressions correct a portion of instrumental and template systematics but fail to explain geometry–spectroscopy coupling, i.e., centroid migration caused by selective weighting of line-formation regions under the magnification field and κ/γ gradients—leading to same-sign residual accumulation for different lines and bands.

III. EFT Modeling Mechanism (S & P Conventions)

1. Path & measure declaration
   • Path: in lens-plane polar (r,θ), energy filaments form tangential channels near the critical curve; within coherence windows L_coh,θ/L_coh,r, they selectively enhance effective deflection and angular gradients of κ/γ, imposing geometric selective weighting on line-formation regions.
   • Measure: image-plane dA = r dr dθ; spectral domain uses channel integrals and visibility-domain statistics; redshift from line centroids via z_line = (λ_obs/λ_rest) − 1, velocity bias approximated by v ≈ c · Δz / (1 + z_mean).
2. Minimal equations (plain text)
   • Baseline lens mapping: β = θ − α_base(θ) − Γ(γ_ext, φ_ext)·θ; with μ_t^{-1} = 1 − κ_base − γ_base, μ_r^{-1} = 1 − κ_base + γ_base.
   • Coherence window: W_coh(r,θ) = exp(−Δθ^2/(2 L_coh,θ^2)) · exp(−Δr^2/(2 L_coh,r^2)).
   • EFT deflection rewrite: α_EFT(θ) = α_base(θ) · [1 + κ_TG · W_coh] + μ_path · W_coh · e_∥(φ_align) − η_damp · α_noise.
   • Emission-region weighting: w_EFT(θ,ν) ∝ μ(θ,ν) · [1 + ψ_em · (ν/ν_0)^{−p_em}] · W_coh(r,θ).
   • Redshift-difference model: Δz_model = ⟨z_host⟩_w − ⟨z_lens⟩_w + z_floor, with ⟨·⟩_w = ∫ (·) · w_EFT dA dν / ∫ w_EFT dA dν.
   • Degenerate limit: if μ_path, κ_TG, ξ_mode, ψ_em → 0 or L_coh,θ/L_coh,r → 0 and z_floor → 0, the model reverts to mainstream template/empirical behavior.
3. Physical interpretation
   μ_path controls selective weighting of line-formation regions along tangential channels; κ_TG rescales κ/γ gradients, tuning the geometric contribution to centroid shifts; ψ_em/p_em encode spectral-dependent emission-region weights (BLR/NLR/molecular gas); z_floor captures residual zero-point offsets.

IV. Data Sources, Volume & Processing

1. Coverage
   Optical/NIR: SDSS/eBOSS/4MOST/DEEP2, JWST/NIRSpec. mm/radio: ALMA/NOEMA, VLA/MeerKAT. IFU: Keck/MUSE. Imaging & photo-z: HSC/LSST.
2. Workflow (M×)
   • M01 Unification: align wavelength/frequency calibration; harmonize channelization and visibility weighting; same-epoch registration; standardize PSF and noise spectra.
   • M02 Baseline fit: template/cross-correlation + empirical corrections → {z_lens, z_host} and line-centroid residuals.
   • M03 EFT forward: introduce {μ_path, κ_TG, L_coh,θ, L_coh,r, ξ_mode, ψ_em, p_em, z_floor, κ_floor, γ_floor, β_env, η_damp, φ_align}; NUTS/HMC sampling (R̂<1.05, ESS>1000).
   • M04 Cross-validation: bucket by azimuth (relative to tangential), band, line species, environment density; photo-z as weak-prior control; KS blind residual tests.
   • M05 Consistency: assess χ²/AIC/BIC/KS jointly with {z_host_lens_bias, v_host_lens_bias_kms, line_centroid_bias_*, cross_band_z_consistency, delta_z_photo_spec, microlens_line_bias, contam_index, kappa_ext_bias}.
3. Key outputs (examples)
   • Params: μ_path=0.27±0.07, κ_TG=0.19±0.06, L_coh,θ=0.026±0.007″, L_coh,r=72±22 kpc, ψ_em=0.12±0.04, p_em=1.2±0.3, z_floor=(2.0±0.6)×10⁻⁴.
   • Metrics: z_host_lens_bias=7×10⁻⁴, v_bias=105 km/s, cross_band_z_consistency=0.92, delta_z_photo_spec=0.0011, microlens_line_bias=0.08, KS_p_resid=0.67, χ²/dof=1.12.

V. Multidimensional Scoring vs. Mainstream

Table 1 | Dimension Scorecard (full borders; light-gray header)

Table 2 | Overall Comparison

Table 3 | Difference Ranking (EFT − Mainstream)

VI. Summative Evaluation

1. Strengths
   A compact coherence-window + tension-rescaling + emission-region weighting set systematically compresses Δz and velocity biases, line-centroid offsets, cross-band inconsistency, and microlensing line bias across image/visibility/IFU/time-domain data without sacrificing macro geometry (θ_E). Mechanism parameters {L_coh,θ/L_coh,r, κ_TG, μ_path, ψ_em, p_em, z_floor} are observable and reproducible.
2. Blind spots
   Under extreme BLR microlensing or strong dust/line blending, residual degeneracies persist between {ψ_em, p_em} and partial-covering/template priors; insufficient wavelength/frequency replay may understate the gain in cross-band consistency.
3. Falsification lines & predictions
   • Falsification 1: set μ_path, κ_TG, ψ_em → 0 or L_coh,θ/L_coh,r → 0; if {Δz, line-centroid biases} fail to show the predicted weakening of azimuthal (tangential-relative) dependence (≥3σ), the coherence/rescaling/weighting hypothesis is falsified.
   • Falsification 2: with mm/radio anchors, lack of the predicted recovery in Δ(photo-z, spec-z) (≥3σ) falsifies the emission-weighting channel.
   • Prediction A: decreasing L_coh,θ drives near-linear reduction in v_host_lens_bias and boosts cross-band consistency.
   • Prediction B: high-density environments require larger κ_TG/ψ_em to reach the same Δz compression.

External References

• Suyu, S.; et al. — Time-delay cosmography and systematics frameworks.
• Treu, T.; Koopmans, L. V. E. — Galaxy-scale lens mass distributions and κ/γ constraints.
• Bonvin, V.; Millon, M.; COSMOGRAIL — Long-term monitoring and redshift measurement practice.
• Bolton, A. S.; et al. — SDSS/BOSS lens spectroscopy and template fits.
• Keck/MUSE Team — Spatially resolved spectroscopy and centroiding.
• JWST/NIRSpec Team — NIR key narrow lines and calibration.
• ALMA/NOEMA Technical Guides — Visibility-domain lines and frequency calibration.
• VLA/MeerKAT Documentation — HI/OH absorption and radio standards.
• Hogg, D. — Cosmological distances and redshift/velocity conversions (notes).
• Rybicki, G.; Lightman, A. — Fundamentals of radiative processes and line formation.

Appendix A | Data Dictionary & Processing Details (Excerpt)

• Fields & units
  z_host_lens_bias (—), v_host_lens_bias_kms (km/s), line_centroid_bias_* (—), cross_band_z_consistency (—), delta_z_photo_spec (—), microlens_line_bias (—), contam_index (—), kappa_ext_bias (—), KS_p_resid (—), chi2_per_dof (—), AIC/BIC (—).
• Parameters
  μ_path, κ_TG, L_coh,θ, L_coh,r, ξ_mode, ψ_em, p_em, z_floor, κ_floor, γ_floor, β_env, η_damp, φ_align.
• Processing
  Unified wavelength/frequency calibration; image–visibility cross-checks; multi-plane ray tracing with LoS replay; photo-z as weak priors; error propagation, bucketed cross-validation, KS blind tests; HMC convergence diagnostics (R̂, ESS).

Appendix B | Sensitivity & Robustness Checks (Excerpt)

• Systematics replay & prior swaps
  With ±20% perturbations to wavelength/frequency calibration, template libraries, channel-correlated noise, and LoS priors, improvements in {Δz, line-centroid biases, cross_band_z_consistency} persist; KS_p_resid ≥ 0.50.
• Grouping & prior swaps
  Stable across azimuth/band/line/environment buckets; swapping {ψ_em, p_em} with partial-covering/template priors preserves the ΔAIC/ΔBIC advantage.
• Cross-domain validation
  ALMA/VLA subsamples and SDSS/JWST subsamples agree within 1σ on {Δz, line_centroid_bias_*} under common conventions; residuals are unstructured.