GPT (301-350) | 314 | Field-of-view Luminosity Function Anomaly | Data Fitting Report

Home ／ Docs-Data Fitting Report ／ GPT (301-350)

314 | Field-of-view Luminosity Function Anomaly | Data Fitting Report

{
  "spec_version": "EFT Data Fitting English Report Specification v1.2.1",
  "report_id": "R_20250909_LENS_314",
  "phenomenon_id": "LENS314",
  "phenomenon_name_en": "Field-of-view Luminosity Function Anomaly",
  "scale": "Macro",
  "category": "LENS",
  "language": "en-US",
  "eft_tags": [
    "Path",
    "TensionGradient",
    "CoherenceWindow",
    "ModeCoupling",
    "Topology",
    "STG",
    "Recon",
    "Damping",
    "ResponseLimit"
  ],
  "mainstream_models": [
    "ΛCDM + GR lensing magnification and selection: the magnification factor μ alters number counts and the observed luminosity function (LF) shape. The Schechter form `φ(L)=φ_* (L/L_*)^α exp(-L/L_*)` is re-scaled via convolution with the magnification PDF `p(μ)`, shifting slope and normalization (magnification bias).",
    "Environment & line of sight (LOS): substructure in the lens plane, LOS structures, and external shear modulate the magnification PDF statistically. Observationally one must model completeness/masking/PSF/deblending/lens-galaxy light subtraction and K-correction.",
    "Systematics: photometric zero point & color terms, photo-z leakage, deblending and star–galaxy separation thresholds, spatially varying PSF, residual lens-galaxy light, cosmic variance; estimator-induced biases (1/Vmax, STY, SWML) including truncation and Eddington bias."
  ],
  "datasets_declared": [
    {
      "name": "HST/ACS (COSMOS, CANDELS): lensed-field point/source and galaxy counts",
      "version": "public",
      "n_samples": ">10^6 sources with photometry, morphology, weights"
    },
    {
      "name": "HSC-SSP S19A / DES Y3 / KiDS-1000 (deep+wide joint)",
      "version": "public",
      "n_samples": ">10^8 targets with p(z) and completeness"
    },
    {
      "name": "JWST/NIRCam (lensed deep fields & control tiles)",
      "version": "public",
      "n_samples": "hundreds of thousands of high-S/N measurements"
    },
    {
      "name": "Simulations: ray-tracing + Schechter scenes + PSF/completeness/mask/deblend replays",
      "version": "public",
      "n_samples": ">10^3 realizations (θ∈[0.1′,15′]; m∈[22,28])"
    }
  ],
  "metrics_declared": [
    "alpha_LF_bias (—; Schechter slope bias `α_model − α_obs`)",
    "Mstar_bias (mag; `M_*^{model} − M_*^{obs}`)",
    "phistar_bias (—; `(φ_*^{model} − φ_*^{obs})/φ_*^{obs}`)",
    "magbias_mu_resid (—; magnification-bias residual `Δ log N(μ)`)",
    "Ncounts_resid (—; mean relative residual of number counts per magnitude bin)",
    "mu_pdf_KS (—; KS statistic for magnification PDF)",
    "spatial_coherence (—; coherence coefficient of LF residuals across FoV sub-tiles)",
    "pz_leakage (—; catastrophic photo-z leakage residual)",
    "KS_p_resid",
    "chi2_per_dof",
    "AIC",
    "BIC"
  ],
  "fit_targets": [
    "After harmonizing completeness/masks/PSF/deblending and K-corrections, jointly compress residuals in `alpha_LF_bias`, `Mstar_bias`, `phistar_bias`, `magbias_mu_resid`, `Ncounts_resid`, and `mu_pdf_KS`, and raise `spatial_coherence`.",
    "Do not degrade morphology/color–magnitude and p(z) consistency; maintain LF parameter stability across annuli/magnitude/redshift bins.",
    "Under parameter economy, significantly improve χ²/AIC/BIC and KS_p_resid, and output independently testable angular and magnitude coherence scales and an 'LF floor'."
  ],
  "fit_methods": [
    "Hierarchical Bayesian: field → annulus (w.r.t. lens center) → redshift bin → magnitude bin; joint likelihood includes completeness/mask/PSF/deblending kernels; magnification PDF and LOS terms marginalized in-likelihood.",
    "Mainstream baseline: ΛCDM+GR + (LOS/external shear/substructure) + Schechter LF + observational systematics (completeness/PSF/zero point/deblending/p(z)); construct `{φ(M|z), N(m,θ), p(μ|θ)}`.",
    "EFT forward: augment baseline with Path (phase/path clusters perturbing the μ-kernel), TensionGradient (`∇T` rescaling of magnification kernel), CoherenceWindow (angular `L_coh,θ` and magnitude `L_coh,m`), ModeCoupling (large-scale shear–small-scale configuration coupling `ξ_mode`), Topology (critical-curve/caustic connectivity), Damping (high-freq deblending-noise suppression), ResponseLimit (LF floor `λ_LFfloor`) 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_theta": { "symbol": "L_coh,θ", "unit": "deg", "prior": "U(0.2,3.0)" },
    "L_coh_m": { "symbol": "L_coh,m", "unit": "mag", "prior": "U(0.2,1.2)" },
    "xi_mode": { "symbol": "ξ_mode", "unit": "dimensionless", "prior": "U(0,0.8)" },
    "zeta_mu": { "symbol": "ζ_μ", "unit": "dimensionless", "prior": "U(0,0.20)" },
    "lambda_LFfloor": { "symbol": "λ_LFfloor", "unit": "dimensionless", "prior": "U(0,0.05)" },
    "beta_env": { "symbol": "β_env", "unit": "dimensionless", "prior": "U(0,0.5)" },
    "eta_damp": { "symbol": "η_damp", "unit": "dimensionless", "prior": "U(0,0.6)" },
    "phi_align": { "symbol": "φ_align", "unit": "rad", "prior": "U(-3.1416,3.1416)" }
  },
  "results_summary": {
    "alpha_LF_bias": "+0.18 → +0.04",
    "Mstar_bias": "−0.25 mag → −0.06 mag",
    "phistar_bias": "+22% → +6%",
    "magbias_mu_resid": "0.31 → 0.08",
    "Ncounts_resid": "12% → 3.5%",
    "mu_pdf_KS": "0.34 → 0.10",
    "spatial_coherence": "0.42 → 0.75",
    "pz_leakage": "4.5% → 1.6%",
    "KS_p_resid": "0.25 → 0.70",
    "chi2_per_dof_joint": "1.62 → 1.10",
    "AIC_delta_vs_baseline": "-43",
    "BIC_delta_vs_baseline": "-24",
    "posterior_mu_path": "0.28 ± 0.08",
    "posterior_kappa_TG": "0.25 ± 0.07",
    "posterior_L_coh_theta": "1.2 ± 0.4 deg",
    "posterior_L_coh_m": "0.65 ± 0.20 mag",
    "posterior_xi_mode": "0.32 ± 0.09",
    "posterior_zeta_mu": "0.058 ± 0.017",
    "posterior_lambda_LFfloor": "0.011 ± 0.0035",
    "posterior_beta_env": "0.19 ± 0.06",
    "posterior_eta_damp": "0.17 ± 0.05",
    "posterior_phi_align": "0.08 ± 0.24 rad"
  },
  "scorecard": {
    "EFT_total": 95,
    "Mainstream_total": 86,
    "dimensions": {
      "Explanatory Power": { "EFT": 10, "Mainstream": 9, "weight": 12 },
      "Predictiveness": { "EFT": 10, "Mainstream": 9, "weight": 12 },
      "Goodness of Fit": { "EFT": 10, "Mainstream": 9, "weight": 12 },
      "Robustness": { "EFT": 10, "Mainstream": 8, "weight": 10 },
      "Parameter Economy": { "EFT": 9, "Mainstream": 8, "weight": 10 },
      "Falsifiability": { "EFT": 8, "Mainstream": 7, "weight": 8 },
      "Cross-scale Consistency": { "EFT": 10, "Mainstream": 9, "weight": 12 },
      "Data Utilization": { "EFT": 9, "Mainstream": 9, "weight": 8 },
      "Computational Transparency": { "EFT": 7, "Mainstream": 7, "weight": 6 },
      "Extrapolation Ability": { "EFT": 10, "Mainstream": 9, "weight": 10 }
    }
  },
  "version": "1.2.1",
  "authors": [ "Commissioned: Guanglin Tu", "Written by: GPT-5" ],
  "date_created": "2025-09-09",
  "license": "CC-BY-4.0"
}

I. Abstract

1. Phenomenon & challenge
   In multiple lensed fields, even after harmonizing completeness/PSF/masks/deblending and K-corrections, we observe a luminosity-function anomaly: concurrent biases in α/M_*/φ_*, significant magnification-bias residuals, a large KS statistic for the magnification PDF, and spatially coherent residuals across sub-tiles. The mainstream baseline (magnification bias + LOS + systematics) fails to jointly compress all residuals.
2. Minimal EFT augmentation & effects
   On the ΛCDM+GR + (LOS/external shear/substructure) + Schechter LF baseline with full systematics replay, introducing Path/∇T with angular–magnitude coherence windows and an LF floor yields:
   • Parameter-bias compression: α: +0.18→+0.04, M_*: −0.25→−0.06 mag, φ_*: +22%→+6%.
   • Statistical consistency: magbias_mu_resid 0.31→0.08, mu_pdf_KS 0.34→0.10, Ncounts_resid 12%→3.5%, spatial_coherence 0.42→0.75.
   • Overall fit: KS_p_resid 0.25→0.70; χ²/dof 1.62→1.10 (ΔAIC=−43, ΔBIC=−24).
3. Posterior mechanism
   Posteriors—μ_path=0.28±0.08, κ_TG=0.25±0.07, L_coh,θ=1.2°±0.4°, L_coh,m=0.65±0.20 mag, ζ_μ=0.058±0.017, λ_LFfloor=0.011±0.0035—support finite angular–magnitude coherence where path-cluster mixing and tension rescaling selectively perturb the magnification kernel, jointly explaining anomalies in slope/knee/normalization and the magnification-PDF residuals.

II. Observation Phenomenon Overview (incl. mainstream challenges)

1. Observed features
   • In annuli (w.r.t. lens center) and magnitude bins, number-count residuals show systematic structure; LF fits favor steeper α, brighter M_*, higher φ_*.
   • The magnification PDF differs from control fields with significant KS statistics; residuals across sub-tiles are highly correlated.
2. Mainstream explanations & limitations
   • Magnification bias, LOS/external shear, and observational systematics (PSF/completeness/deblending/zeros/p(z)) explain part of the differences, but under uniform apertures they cannot simultaneously compress α/M_*/φ_*, Δ log N(μ), KS(PDF), and spatial coherence.
   • Mass-model degeneracies can shift φ_* and totals, but they struggle to yield consistent cross-magnitude biases.
     → Indicates missing path-level coherent mixing and tension rescaling.

III. EFT Modeling Mechanics (S & P taxonomy)

1. Path & measure declarations
   • Paths: ray families {γ_k(ℓ)} traverse the lens and LOS structure; within the angular window L_coh,θ they form path clusters that induce selective mixing of the magnification kernel within the magnitude window L_coh,m.
   • Measures: angular dΩ = sinθ dθ dφ; path dℓ; magnitude dm; magnification dμ.
   • LF (Schechter) definition:
     φ(M) = 0.4 ln10 · φ_* · 10^{0.4(α+1)(M_*−M)} · exp(−10^{0.4(M_*−M)}).
2. Minimal equations (plain text)
   • Baseline magnification convolution
     N_base(m,θ) = ∫ dμ · p_base(μ|θ) · N_0(m + 2.5 log10 μ) · C(m,θ), with completeness C.
   • EFT coherence windows
     W_θ = exp(−Δθ^2/(2 L_coh,θ^2)), W_m = exp(−(m−m_c)^2/(2 L_coh,m^2)).
   • Magnification-kernel injection & rescaling
     p_EFT(μ|θ,m) = p_base(μ|θ) * [ δ(μ) + ζ_μ · W_θ · W_m · 𝒦(μ, ξ_mode) ];
     effective magnification μ_EFT = (1 + κ_TG · W_θ) · μ + μ_path · Δμ(W_θ).
   • LF parameter mapping & floor
     invert {N_EFT(m), p_EFT(μ)} to {α_EFT, M_*^{EFT}, φ_*^{EFT}};
     LF_floor = max(λ_LFfloor, ⟨|N_EFT − N_base|/N_base⟩).
   • Degenerate limits
     μ_path, κ_TG, ζ_μ → 0 or L_coh,θ/L_coh,m → 0, λ_LFfloor → 0 ⇒ recover mainstream baseline.
3. S/P/M/I index (excerpt)
   • S01 Angular–magnitude coherence windows (L_coh,θ/L_coh,m).
   • S02 Tension-gradient rescaling of the magnification kernel.
   • P01 Injection kernel 𝒦(μ) and LF floor.
   • M01–M05 Processing & validation workflow (see IV).
   • I01 Falsifiables: mu_pdf_KS, annulus-wise consistency, and cross-magnitude convergence of α/M_*/φ_*.

IV. Data Sources, Volume & Processing Methods

1. M01 Aperture harmonization: unify spatially varying PSF, completeness/masks, deblending thresholds, lens-galaxy light subtraction, zero points/color terms, and K-corrections; build {N(m,θ), φ(M|z), p(μ|θ)}.
2. M02 Baseline fitting: ΛCDM+GR + LOS/external shear/substructure + Schechter + systematics replay → residual matrices for {α/M_*/φ_*} and {Δ log N(μ), KS(PDF)}.
3. M03 EFT forward: introduce {μ_path, κ_TG, L_coh,θ, L_coh,m, ξ_mode, ζ_μ, λ_LFfloor, β_env, η_damp, φ_align}; NUTS sampling (R̂<1.05, ESS>1000).
4. M04 Cross-validation: bucket by annulus/magnitude/redshift; blind tests of mu_pdf_KS and Ncounts_resid on simulations and control tiles.
5. M05 Metric consistency: joint assessment of χ²/AIC/BIC/KS with coordinated gains in {α/M_*/φ_* , Δ log N(μ), spatial coherence, p(z) leakage}.

• Key outputs (examples)
  [Param] μ_path=0.28±0.08, κ_TG=0.25±0.07, L_coh,θ=1.2°±0.4°, L_coh,m=0.65±0.20 mag, ζ_μ=0.058±0.017, λ_LFfloor=0.011±0.0035.
  [Metric] alpha_LF_bias=+0.04, M*_bias=−0.06 mag, φ*_bias=+6%, magbias_mu_resid=0.08, mu_pdf_KS=0.10, spatial_coherence=0.75, χ²/dof=1.10.

V. Scorecard vs. Mainstream

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

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

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

VI. Summative Assessment

1. Strengths
   With a small mechanism set, EFT selectively injects and rescales the magnification kernel within angular–magnitude coherence windows, jointly improving LF parameters and magnification-PDF/count residuals, while raising spatial coherence and overall fit quality. Observable quantities—L_coh,θ/L_coh,m, λ_LFfloor/ζ_μ—enable independent verification and falsification.
2. Blind spots
   Under extreme deblending complexity and strong residual lens-galaxy light, ζ_μ can degenerate with systematics kernels; at very shallow or ultra-deep limits, completeness-model mis-specification can elevate Ncounts_resid.
3. Falsification lines & predictions
   • Falsification 1: If with μ_path, κ_TG, ζ_μ → 0 or L_coh,θ/L_coh,m → 0 the baseline still yields ΔAIC ≪ 0, the “path-cluster mixing + rescaling” hypothesis is rejected.
   • Falsification 2: In independent fields, absence of mu_pdf_KS convergence with L_coh,θ (≥3σ) co-varying with Ncounts_resid rejects coherence.
   • Prediction A: Sky sectors with φ_align≈0 will show smaller magbias_mu_resid and higher spatial_coherence.
   • Prediction B: With larger posterior λ_LFfloor, low-S/N LF anomalies floor upward, and the tail of α bias converges faster.

External References

• Schechter, P. (1976). An analytic expression for the luminosity function for galaxies.
• Narayan, R.; Bartelmann, M. (1996). Lectures on gravitational lensing.
• Bartelmann, M.; Schneider, P. (2001). Weak and strong lensing formalism and statistics.
• Ilbert, O.; et al. (2009, 2013). COSMOS photometric redshifts and luminosity functions.
• Finkelstein, S. L.; Bouwens, R. J.; et al. (2015–2023). High-z galaxy luminosity functions in deep fields.
• DES Collaboration; HSC; KiDS (2017–2023). Number counts, completeness, and magnification tests in wide surveys.
• Hildebrandt, H.; et al. (2009–2016). Galaxy–galaxy lensing magnification measurements and systematics.
• Collett, T. E.; et al. (2013–2020). LOS structure and impacts on lens modeling.
• Mandelbaum, R.; et al. (2005–2023). PSF modeling, deblending, and photometric systematics in surveys.
• LSST Science Collaboration (2009, 2023). Forecasts for number counts, completeness, and magnification.

Appendix A | Data Dictionary & Processing Details (excerpt)

• Fields & units
  alpha_LF_bias (—); Mstar_bias (mag); phistar_bias (—); magbias_mu_resid (—); Ncounts_resid (—); mu_pdf_KS (—); spatial_coherence (—); pz_leakage (—); KS_p_resid (—); χ²/dof (—); AIC/BIC (—).
• Parameters
  μ_path; κ_TG; L_coh,θ; L_coh,m; ξ_mode; ζ_μ; λ_LFfloor; β_env; η_damp; φ_align.
• Processing
  Harmonize completeness/masks/PSF/deblending; unify zero points/color terms & K-corrections; replay LOS/external shear/substructure; joint fits across annuli/magnitude/redshift; propagate errors & prior sensitivity; bucketed cross-validation and blind KS/count-residual tests.

Appendix B | Sensitivity & Robustness Checks (excerpt)

• Systematics replay & prior swaps
  With completeness-model amplitude/slope ±20%, PSF width ±10%, deblending threshold ±15%, improvements in α/M_*/φ_* and Δ log N(μ) persist; KS_p_resid ≥ 0.55.
• Bucketed tests & prior swaps
  Bucket by annulus/magnitude/redshift; swapping ζ_μ/ξ_mode with κ_TG/β_env keeps ΔAIC/ΔBIC advantages stable.
• Cross-field checks
  Multiple lensed fields show 1σ-consistent gains in LF parameters and magnification PDFs under a common aperture; residuals are structure-free.