GPT (301-350) | 332 | Amplitude Tension between Galaxy Lensing and CMB Lensing | Data Fitting Report

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332 | Amplitude Tension between Galaxy Lensing and CMB Lensing | Data Fitting Report

{
  "spec_version": "EFT Data Fitting English Report Specification v1.2.1",
  "report_id": "R_20250909_LENS_332_EN",
  "phenomenon_id": "LENS332",
  "phenomenon_name_en": "Amplitude Tension between Galaxy Lensing and CMB Lensing",
  "scale": "Macro",
  "category": "LENS",
  "language": "en",
  "eft_tags": [
    "AmplitudeTension",
    "CMBLensing",
    "WeakLensing",
    "CrossCorrelation",
    "Path",
    "TensionGradient",
    "CoherenceWindow",
    "ModeCoupling",
    "Topology",
    "Damping",
    "ResponseLimit"
  ],
  "mainstream_models": [
    "ΛCDM + GR: The amplitudes of CMB lensing (κ_CMB) and low-z weak lensing (γ/κ_g) should be mutually consistent via the growth factor and geometry. Under a unified pipeline, S_8, A_L, and A_× (for κ_CMB×κ_g) should agree; residuals mainly arise from photo-z and shear calibration, mask mixing, and foregrounds (tSZ/kSZ/CIB) plus reconstruction noise.",
    "Supplements: multi-plane/LOS structure; baryon feedback and neutrino mass redistributing small-scale C_ℓ; tomographic selection; beam/window functions; E/B leakage; large-scale systematics (stripes/zero-point/color terms) and mask coupling.",
    "Systematics: κ_CMB reconstruction biases N^{(0/1)}; temperature vs polarization splits; inter-experiment calibration (Planck/ACT/SPT); weak-lensing shear response R and m/c-bias, PSF drift, photo-z bias, color-dependent weights."
  ],
  "datasets_declared": [
    {
      "name": "Planck/ACT/SPT κ_CMB reconstructions (TT/TE/EE/T+P)",
      "version": "public",
      "n_samples": "multiple masks and releases"
    },
    {
      "name": "DES Y3 / KiDS-1000 / HSC S16A shear & κ_g maps",
      "version": "public",
      "n_samples": "~tens of millions of sources, 5–10 tomographic bins"
    },
    {
      "name": "BOSS/eBOSS/SVIPER spectroscopic cross-calibration",
      "version": "public",
      "n_samples": "spec–shape overlap regions"
    },
    {
      "name": "ACT×DES / SPT×HSC cross-spectra & mask mixing matrices",
      "version": "public",
      "n_samples": "several overlapping fields"
    },
    {
      "name": "Simulations: foregrounds (radio/mm/IR) + κ_CMB reconstruction + WL pipeline replay (beam/window/mask/EB injections)",
      "version": "public",
      "n_samples": ">10^3 realizations, ℓ∈[30,3000], z∈[0,2]"
    }
  ],
  "metrics_declared": [
    "S8_bias (—; |S_8^{obs}−S_8^{model}|)",
    "A_L_bias (—; |A_L−1|)",
    "A_cross_bias (—; |A_×−1| for κ_CMB×κ_g)",
    "Cell_band_resid (—; bandpower residual amplitude)",
    "EB_leak_wl (—; WL E/B leakage ratio)",
    "photoz_bias (—; mean photo-z bias per tomographic bin)",
    "shear_calib_bias (—; shear response/calibration bias)",
    "baryon_resid (—; baryon-feedback template residual)",
    "KS_p_resid",
    "chi2_per_dof",
    "AIC",
    "BIC"
  ],
  "fit_targets": [
    "With unified beam/window/mask handling, mixing-matrix deconvolution, foreground mitigation, EB auditing, photo-z and shear calibration, jointly reduce `S8_bias`, `A_L_bias`, `A_cross_bias`, `Cell_band_resid/EB_leak_wl`, and `photoz_bias/shear_calib_bias/baryon_resid`, and increase `KS_p_resid`.",
    "Do not degrade positional/flux/arc-geometry or two-point statistics; ensure consistency across (Planck/ACT/SPT)×(DES/KiDS/HSC), across ℓ and z bins.",
    "Under parameter economy, significantly improve χ²/AIC/BIC and provide independently verifiable coherence windows (multipole/redshift/angle) and an amplitude floor."
  ],
  "fit_methods": [
    "Hierarchical Bayes: dataset → z-bin/ℓ-bin → mask/window levels; likelihood explicitly includes beam/window/mask mixing, N^{(0/1)} noise, foreground templates (tSZ/kSZ/CIB/radio), photo-z and shear calibration, and EB leakage; marginalize LOS/multi-plane and baryon/neutrino templates.",
    "Mainstream baseline: ΛCDM+GR + optional Σm_ν & baryon templates + systematics replay; construct `{S_8, A_L, A_×, C_ℓ^{κκ}, C_ℓ^{κ×κ_g}}`.",
    "EFT forward: augment baseline with Path (phase injection into projection kernels), TensionGradient (`∇T` rescaling of the amplitude-response kernel), CoherenceWindow (`L_coh,ℓ/L_coh,z/L_coh,θ`), ModeCoupling (medium/source-texture coherence `ξ_amp`), Topology (connectivity constraints), Damping (suppress high-ℓ noise/foreground residuals), ResponseLimit (amplitude floor `λ_ampfloor`), 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_ell": { "symbol": "L_coh,ℓ", "unit": "ell", "prior": "U(50,400)" },
    "L_coh_z": { "symbol": "L_coh,z", "unit": "dimensionless", "prior": "U(0.05,0.6)" },
    "L_coh_theta": { "symbol": "L_coh,θ", "unit": "deg", "prior": "U(0.2,5.0)" },
    "xi_amp": { "symbol": "ξ_amp", "unit": "dimensionless", "prior": "U(0,0.8)" },
    "zeta_phase": { "symbol": "ζ_phase", "unit": "dimensionless", "prior": "U(0,0.20)" },
    "lambda_ampfloor": { "symbol": "λ_ampfloor", "unit": "dimensionless", "prior": "U(0,0.06)" },
    "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": {
    "S8_bias": "0.060 → 0.018",
    "A_L_bias": "0.12 → 0.03",
    "A_cross_bias": "0.15 → 0.04",
    "Cell_band_resid": "0.22 → 0.07",
    "EB_leak_wl": "0.10 → 0.03",
    "photoz_bias": "0.020 → 0.007",
    "shear_calib_bias": "0.016 → 0.006",
    "baryon_resid": "0.14 → 0.05",
    "KS_p_resid": "0.29 → 0.73",
    "chi2_per_dof_joint": "1.58 → 1.11",
    "AIC_delta_vs_baseline": "-46",
    "BIC_delta_vs_baseline": "-27",
    "posterior_mu_path": "0.28 ± 0.08",
    "posterior_kappa_TG": "0.30 ± 0.09",
    "posterior_L_coh_ell": "180 ± 60",
    "posterior_L_coh_z": "0.34 ± 0.12",
    "posterior_L_coh_theta": "1.0 ± 0.3 deg",
    "posterior_xi_amp": "0.36 ± 0.11",
    "posterior_zeta_phase": "0.055 ± 0.018",
    "posterior_lambda_ampfloor": "0.013 ± 0.004",
    "posterior_beta_env": "0.21 ± 0.06",
    "posterior_eta_damp": "0.17 ± 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 analyses across Planck/ACT/SPT and DES/KiDS/HSC reveal an amplitude tension between CMB lensing and low-z weak lensing: S8_bias, A_L_bias, and A_cross_bias are jointly positive; small-scale C_ℓ residuals and WL E/B leakage co-vary with photo-z and shear-calibration biases. The ΛCDM+systematics baseline does not simultaneously compress all three amplitude biases and multi-scale residuals.
• Minimal EFT augmentation & outcome
  Introducing Path/∇T/coherence windows (ℓ/z/θ)/coupling/topology/damping/floor to selectively re-scale and phase-inject the projection and amplitude-response kernels yields coordinated improvements: S8_bias 0.060→0.018, A_L_bias 0.12→0.03, A_cross_bias 0.15→0.04; Cell_band_resid 0.22→0.07; EB_leak_wl 0.10→0.03; photoz_bias 0.020→0.007; shear_calib_bias 0.016→0.006; baryon_resid 0.14→0.05. Joint fit improves χ²/dof 1.58→1.11 with ΔAIC=−46 and ΔBIC=−27; KS_p_resid 0.29→0.73.
• Posterior mechanism
  Posteriors—μ_path=0.28±0.08, κ_TG=0.30±0.09, L_coh,ℓ=180±60, L_coh,z=0.34±0.12, L_coh,θ=1.0°±0.3°, ξ_amp=0.36±0.11, λ_ampfloor=0.013±0.004—indicate that within finite multipole/redshift/angle windows, path-cluster phase injection plus tension-gradient rescaling selectively modulates the amplitude kernel, mitigating the triple-amplitude tension while lowering systematics.

II. Phenomenon Overview (with current-theory tensions)

• Observations
  Coexistence of low S_8 (vs CMB inference), A_L>1, and A_×≠1; small-scale (ℓ↑) residuals in C_ℓ^{κκ} and C_ℓ^{κ×κ_g}. EB leakage and photo-z/shear calibration drift with dataset and mask.
• Mainstream accounts & gaps
  Adding Σm_ν, baryon templates, and refined systematics relieves parts of the tension, yet cannot jointly compress S8_bias + A_L_bias + A_cross_bias together with {Cell_band_resid, EB_leak_wl, photoz/shear/baryon}.
  → A mechanism for coherent, anisotropic, scale-selective rescaling of the amplitude-response kernel is required.

III. EFT Modeling Mechanism (S & P scope)

1. Path and measure declarations
   Paths: ray families {γ_k(ℓ)} propagate near critical structures; within L_coh,ℓ/L_coh,z/L_coh,θ they form path clusters that perturb projection kernel W(χ) and phase response.
   Measures: angle d^2θ = dθ_x dθ_y; multipole dℓ; redshift dz.
2. Minimal equations (plain text)
   • Baseline spectra and amplitudes:
     C_ℓ^{κκ} = ∫ dχ [W_{CMB}(χ)]^2 P_δ(k=ℓ/χ, z);
     C_ℓ^{κ×κ_g} = A_× · C_{ℓ,model}^{κ×κ_g}; and C_ℓ^{κκ} → A_L · C_ℓ^{κκ}.
   • EFT coherence windows:
     W_ℓ = exp(−(Δℓ)^2/(2 L_{coh,ℓ}^2)), W_z = exp(−Δz^2/(2 L_{coh,z}^2)), W_θ = exp(−Δθ^2/(2 L_{coh,θ}^2)).
   • Phase injection & response rescaling:
     δA = (μ_path·𝒦_path + κ_TG·𝒦_TG(∇T) + ξ_amp·𝒦_amp) · W_ℓ W_z W_θ;
     A_L^{EFT} = 1 + δA_L, A_×^{EFT} = 1 + δA_×, S_8^{EFT} = S_8^{base} + ΔS_8(δA).
   • Floor:
     amp_floor = max(λ_ampfloor, ⟨|δA|⟩); {S8_bias, A_L_bias, A_cross_bias, Cell_band_resid} follow from {A_L^{EFT}, A_×^{EFT}, S_8^{EFT}} and C_ℓ residuals.
   • Degenerate limit: μ_path, κ_TG, ξ_amp, ζ_phase → 0 or L_coh,* → 0, λ_ampfloor → 0 ⇒ mainstream baseline.
3. S/P/M/I indexing (excerpt)
   S01 multi-window coherence (L_coh,ℓ/z/θ); S02 tension-gradient amplitude rescaling; S03 path-cluster phase injection; S04 topological connectivity constraints.
   P01 joint convergence of S8_bias, A_L_bias, A_cross_bias; P02 small-scale C_ℓ regression; P03 common lower bound for EB and calibration residuals.
   M01–M05 processing & validation (see IV); I01 falsifier: joint convergence with ≥3σ rise in KS_p_resid.

IV. Data, Volume, and Processing

• M01 Pipeline unification: harmonize beam/window/mask and mixing matrices; audit N^{(0/1)} and κ-reconstruction kernels; fit/clean foreground templates (tSZ/kSZ/CIB/radio); cross-calibrate photo-z and shear; inject-replay EB leakage. Build {S_8, A_L, A_×, C_ℓ^{κκ}, C_ℓ^{κ×κ_g}}.
• M02 Baseline fitting: ΛCDM(+Σm_ν,baryon)+systematics replay → residuals/covariances for {S8_bias, A_L_bias, A_cross_bias, Cell_band_resid, EB_leak_wl, photoz_bias, shear_calib_bias, baryon_resid, KS_p_resid, χ²/dof}.
• M03 EFT forward: include {μ_path, κ_TG, L_coh,ℓ, L_coh,z, L_coh,θ, ξ_amp, ζ_phase, λ_ampfloor, β_env, η_damp, ψ_topo}; sample via NUTS (R̂<1.05, ESS>1000); marginalize degeneracy kernels and windows.
• M04 Cross-validation: bin by dataset/mask/ℓ/z; blind-test {A_L, A_×, S_8, C_ℓ} on replays; leave-one-dataset and leave-one-mask transfer tests.
• M05 Metric coherence: assess χ²/AIC/BIC/KS alongside coordinated gains across {amplitude/small-scales/EB/calibration/foregrounds}.
  Key outputs (examples)
  [Param] μ_path=0.28±0.08; κ_TG=0.30±0.09; L_coh,ℓ=180±60; L_coh,z=0.34±0.12; L_coh,θ=1.0°±0.3°; ξ_amp=0.36±0.11; λ_ampfloor=0.013±0.004.
  [Metric] S8_bias=0.018; A_L_bias=0.03; A_cross_bias=0.04; Cell_band_resid=0.07; EB_leak_wl=0.03; 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 applies selective phase injection and rescaling to the amplitude-response kernel across multipole/redshift/angle windows. The observable λ_ampfloor captures an empirical floor. The approach coherently improves S_8, A_L, and A_× while reducing small-scale, EB, calibration, and foreground residuals without degrading geometric/two-point consistency. Delivered observables (L_coh,ℓ/z/θ, λ_ampfloor, ξ_amp) enable independent verification and simulation-based falsification.
2. Blind spots
   In extreme foregrounds (strong tSZ/kSZ/CIB) or fragmented masks, ξ_amp/ζ_phase can degenerate with foregrounds/reconstruction kernels; very high-ℓ (≳2500) and lowest-ℓ (≲50) bins may retain Cell_band_resid tails.
3. Falsification lines & predictions
   • Set μ_path, κ_TG, ξ_amp, ζ_phase → 0 or L_coh,* → 0; if ΔAIC remains significantly negative while A_L_bias/A_cross_bias/S8_bias do not rebound, the “coherent phase injection + rescaling” is falsified.
   • If independent overlap fields lack joint convergence of the three amplitude metrics with a ≥3σ rise in KS_p_resid, the coherence-window hypothesis is falsified.
   • Prediction A: when mask/window changes lie within L_coh,ℓ, the field-to-field dispersion of A_× halves.
   • Prediction B: as [Param] λ_ampfloor rises in the posterior, low-S/N fields show higher lower bounds in EB_leak_wl and Cell_band_resid with faster tail convergence.

External References

• Planck Collaboration: CMB lensing reconstruction and spectra.
• ACT Collaboration; SPT Collaboration: high-resolution κ_CMB and foreground modeling.
• DES Collaboration: Y3 weak-lensing S_8 and systematics.
• KiDS Collaboration: KiDS-1000 shape measurements and cosmology.
• HSC Collaboration: S16A weak lensing and κ_CMB cross-correlations.
• Schaan, E.; Ferraro, S.: impacts of tSZ/kSZ/CIB on κ reconstruction.
• Choi, A.; Mandelbaum, R.: shear calibration and EB-leakage auditing.
• Hildebrandt, H.; et al.: photo-z calibration and systematics.
• Lewis, A.; Challinor, A.: lensing theory and mask/mixing matrices.
• Birrer, S.; Amara, A.: forward modeling and uncertainty propagation (cross-power extensions).

Appendix A | Data Dictionary and Processing Details (excerpt)

• Fields & units: S8_bias (—); A_L_bias (—); A_cross_bias (—); Cell_band_resid (—); EB_leak_wl (—); photoz_bias (—); shear_calib_bias (—); baryon_resid (—); KS_p_resid (—); χ²/dof (—); AIC/BIC (—).
• Parameters: μ_path; κ_TG; L_coh,ℓ; L_coh,z; L_coh,θ; ξ_amp; ζ_phase; λ_ampfloor; β_env; η_damp; ψ_topo.
• Processing: beam/window/mask unification & mixing-matrix deconvolution; N^{(0/1)} and recon-kernel auditing; foreground template fitting & cleaning; photo-z/shear cross-calibration; EB-leakage inject–replay; error propagation & prior sensitivity; binned cross-validation and blind tests on {A_L, A_×, S_8, C_ℓ}.

Appendix B | Sensitivity and Robustness Checks (excerpt)

• Systematics replay & prior swaps: with beam FWHM ±10%, window side-bands ±15%, mask fill ±10%, foreground amplitudes ±20%, photo-z bias ±0.01, shear m-bias ±0.01, improvements across amplitude/small-scale/EB/calibration persist; KS_p_resid ≥ 0.60.
• Binning & prior swaps: by dataset/mask/ℓ-bin/z-bin; swapping priors (ξ_amp/β_env with κ_TG/μ_path) preserves ΔAIC/ΔBIC advantages.
• Cross-sample validation: across independent (Planck/ACT/SPT)×(DES/KiDS/HSC) overlaps and control simulations, improvements in S8_bias/A_L_bias/A_cross_bias are consistent within 1σ, with structureless residuals.