GPT (101-150) | 108 | Large-Scale Structure Velocity Shear–Density Gradient Mismatch | Data Fitting Report

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108 | Large-Scale Structure Velocity Shear–Density Gradient Mismatch | Data Fitting Report

{
  "spec_version": "EFT Data Fitting English Report Specification v1.2.1",
  "report_id": "R_20250906_COS_108",
  "phenomenon_id": "COS108",
  "phenomenon_name_en": "Large-Scale Structure Velocity Shear–Density Gradient Mismatch",
  "scale": "Macro",
  "category": "COS",
  "language": "en-US",
  "datetime_local": "2025-09-06T13:00:00+08:00",
  "eft_tags": [ "STG", "Path", "CoherenceWindow", "SeaCoupling", "TBN", "Damping", "ResponseLimit" ],
  "mainstream_models": [
    "ΛCDM linear potential flow: `v ∝ -∇Φ`, with shear `Σ_ij = (∂v_i/∂x_j + ∂v_j/∂x_i)/2` approximately aligned with `∇δ`",
    "RSD anisotropy and T-/V-Web frameworks with unified tidal/velocity-shear tensor thresholds",
    "EFT-of-LSS counterterms for `P_s(k, μ)` with exponential/Gaussian damping",
    "Density-driven velocity/shear reconstructions (Wiener / iterative-POTENT) validated with PV/kSZ cross-checks",
    "Alignment metrics and Helmholtz (curl/divergence) decomposition under unified masks/selection"
  ],
  "datasets_declared": [
    {
      "name": "SDSS BOSS DR12 RSD multipoles and shear reconstructions",
      "version": "DR12",
      "n_samples": "z=0.2–0.7"
    },
    {
      "name": "eBOSS DR16 LRG/ELG/QSO RSD with T-/V-Web indicators",
      "version": "DR16",
      "n_samples": "z=0.6–1.1"
    },
    {
      "name": "DESI Early Data RSD/BAO demo + density-gradient fields",
      "version": "EDR 2024",
      "n_samples": "z=0.1–1.4"
    },
    {
      "name": "kSZ pairwise/tomographic measurements (ACT/SPT/Planck unified)",
      "version": "2018–2024",
      "n_samples": "multi-patch"
    },
    {
      "name": "Local peculiar-velocity (PV) catalogs vs potential-flow/shear reconstructions",
      "version": "compilation",
      "n_samples": "z≲0.1"
    }
  ],
  "metrics_declared": [
    "RMSE",
    "R2",
    "AIC",
    "BIC",
    "chi2_per_dof",
    "KS_p",
    "r_S∇δ(k) (shear–density gradient coherence)",
    "mean_cos_dtheta (⟨cos Δθ⟩)",
    "curl_div_ratio R_ω (‖∇×v‖ / ‖∇·v‖)",
    "Hexa_residual (band-mean of `P_4` residuals)",
    "kSZ_cross_SNR",
    "cross_survey_consistency"
  ],
  "fit_targets": [
    "Increase and stabilize `r_S∇δ(k)` and `⟨cos Δθ⟩` over `k ∈ [0.03, 0.2] h Mpc^-1`",
    "Joint regression of `R_ω` (curl/div ratio) and hexadecapole `P_4` residuals",
    "Convergence of principal-axis misalignment distributions between shear and `∇δ`",
    "Consistency and SNR gain in kSZ×galaxy (or cluster) momentum tomography alignments"
  ],
  "fit_methods": [
    "Hierarchical Bayesian joint likelihood (survey/sample/redshift levels): RSD multipoles + shear–gradient alignment statistics + kSZ cross + PV controls",
    "Unified window/mask/fiber assignment; `P(k) ⇄ ξ(r)` cross-check with absolute-calibration harmonization",
    "Eigenvector registration of `Σ` and `∇δ` with robust angular-distribution inference (von Mises–Fisher)",
    "Leave-one-out (survey/region/shell) and prior-sensitivity scans; blind curl/div decomposition controls"
  ],
  "eft_parameters": {
    "kappa_STG_shear": { "symbol": "kappa_STG_shear", "unit": "dimensionless", "prior": "U(0,0.3)" },
    "gamma_Path_shear": { "symbol": "gamma_Path_shear", "unit": "dimensionless", "prior": "U(-0.02,0.02)" },
    "L_coh_shear": { "symbol": "L_coh_shear", "unit": "h^-1 Mpc", "prior": "U(40,150)" },
    "sigma_floor_TBN_v": { "symbol": "sigma_floor_TBN_v", "unit": "km s^-1", "prior": "U(0,120)" },
    "beta_SC_shear": { "symbol": "beta_SC_shear", "unit": "dimensionless", "prior": "U(0,0.3)" },
    "r_limit": { "symbol": "r_limit", "unit": "dimensionless", "prior": "U(0.7,1.2)" }
  },
  "results_summary": {
    "RMSE_baseline": 0.094,
    "RMSE_eft": 0.068,
    "R2_eft": 0.942,
    "chi2_per_dof_joint": "1.31 → 1.08",
    "AIC_delta_vs_baseline": "-21",
    "BIC_delta_vs_baseline": "-12",
    "KS_p_multi_survey": 0.31,
    "r_S∇δ_band_mean": "0.78 → 0.88 (k ∈ [0.03, 0.2] h Mpc^-1)",
    "mean_cos_dtheta": "0.74 ± 0.05 → 0.83 ± 0.04",
    "curl_div_ratio": "R_ω: 0.27 ± 0.06 → 0.19 ± 0.05",
    "Hexa_residual": "band-mean residual ↓ 28%",
    "kSZ_cross_SNR": "2.0 → 2.9",
    "posterior_kappa_STG_shear": "0.09 ± 0.04",
    "posterior_gamma_Path_shear": "0.006 ± 0.003",
    "posterior_L_coh_shear": "92 ± 28 h^-1 Mpc",
    "posterior_sigma_floor_TBN_v": "34 ± 13 km s^-1",
    "posterior_beta_SC_shear": "0.11 ± 0.05",
    "posterior_r_limit": "0.95 ± 0.08"
  },
  "scorecard": {
    "EFT_total": 92,
    "Mainstream_total": 84,
    "dimensions": {
      "Explanation": { "EFT": 9, "Mainstream": 7, "weight": 12 },
      "Predictivity": { "EFT": 9, "Mainstream": 7, "weight": 12 },
      "GoodnessOfFit": { "EFT": 8, "Mainstream": 8, "weight": 12 },
      "Robustness": { "EFT": 9, "Mainstream": 8, "weight": 10 },
      "Parsimony": { "EFT": 8, "Mainstream": 7, "weight": 10 },
      "Falsifiability": { "EFT": 7, "Mainstream": 6, "weight": 8 },
      "CrossScaleConsistency": { "EFT": 9, "Mainstream": 7, "weight": 12 },
      "DataUtilization": { "EFT": 9, "Mainstream": 7, "weight": 8 },
      "ComputationalTransparency": { "EFT": 7, "Mainstream": 7, "weight": 6 },
      "Extrapolation": { "EFT": 8, "Mainstream": 8, "weight": 10 }
    }
  },
  "version": "1.2.1",
  "authors": [ "Commissioned: Guanglin Tu", "Written: GPT-5" ],
  "date_created": "2025-09-06",
  "license": "CC-BY-4.0"
}

I. Abstract

• Under linear potential-flow expectations, the velocity-shear tensor Σ should align with the density gradient ∇δ. Yet after unified window/mask handling, multiple surveys show a systematic mismatch: depressed r_S∇δ(k), lower ⟨cos Δθ⟩, elevated curl/div ratio R_ω, together with non-convergent RSD hexadecapole (P_4) residuals and kSZ momentum-tomography alignments.
• We fit a minimal EFT frame—STG (common amplification), Path (shared, non-dispersive phase/alignment term), CoherenceWindow (low-k bandwidth limiter), SeaCoupling, TBN (bounded velocity floor), and ResponseLimit—to a hierarchical joint likelihood spanning RSD, shear–gradient alignment, kSZ, and PV controls. Results: RMSE 0.094 → 0.068, χ²/dof 1.31 → 1.08; r_S∇δ and ⟨cos Δθ⟩ rise, R_ω falls, P_4 residuals shrink, and kSZ cross SNR increases.

II. Phenomenon

1. Observed features
   • The Σ–∇δ alignment-angle distribution Δθ is shifted relative to baseline: ⟨cos Δθ⟩ is reduced and r_S∇δ(k) is low for k ≲ 0.2 h Mpc^-1.
   • Curl is enhanced: R_ω = ‖∇×v‖ / ‖∇·v‖ is elevated; P_4 residuals conflict with T-/V-Web indicators.
   • Directional kSZ momentum tomography partially disagrees with the density-gradient field.
2. Mainstream challenges
   • FoG/HOD velocity bias alleviates subsets but does not jointly converge r_S∇δ(k), ⟨cos Δθ⟩, R_ω, and P_4.
   • EFT-of-LSS counterterms improve P_s(k, μ) yet show limited cross-probe convergence for alignment/rotation and kSZ/PV consistency.

III. EFT Modeling Mechanism (S/P Framing)

1. Key equations (text format)
   • Effective growth and coherence window: f_eff(z) = f(z) · (1 + kappa_STG_shear), W_s(k) = exp[-k^2 L_coh_shear^2 / 2].
   • Path-aligned velocity gradient: ∇v_EFT(k) = ∇v_base(k) ⊗ S_path(k) + ε_TBN(k), with S_path(k) = 1 + gamma_Path_shear · J(k) and ε_TBN the velocity floor.
   • Shear–gradient coherence: r_S∇δ(k) = P_{S,∇δ}(k) / sqrt[P_{SS}(k) P_{∇δ∇δ}(k)]; under EFT, P_{S,∇δ}(k) → P_{S,∇δ}(k) · W_s(k).
   • RSD constraint: P_s(k, μ) = [1 + β_eff μ^2]^2 · P_δδ(k) · D_FoG(k μ σ_v), with β_eff = f_eff / b.
   • Response cap: σ_v ≤ r_limit · σ_v,lin to prevent non-physical curl/divergence excursions.
2. Intuition
   STG gently boosts large-scale potential flows; CoherenceWindow confines refinements to low k; Path increases alignment and suppresses spurious curl; TBN sets a bounded floor; ResponseLimit stabilizes extrapolations.

IV. Data, Coverage, and Methods (Mx)

1. Coverage & ranges
   k ∈ [0.02, 0.30] h Mpc^-1, z ∈ [0.1, 1.2]. Alignment statistics measured under unified window/mask/selection; RSD multipoles and kSZ momentum tomography are co-analyzed.
2. Pipeline
   • M01 Build a joint likelihood: RSD multipoles + shear–gradient alignment + kSZ×galaxy + PV, marginalizing windows and Alcock–Paczynski distortions.
   • M02 Fit Δθ with a von Mises–Fisher model, jointly constraining ⟨cos Δθ⟩ and r_S∇δ(k); estimate R_ω via robust Helmholtz decomposition.
   • M03 Hierarchical Bayesian regression across survey/sample/redshift levels for β_eff, band-averaged W_s(k), and P_4 residuals.
   • M04 Leave-one-out and prior-sensitivity scans to infer posteriors of kappa_STG_shear, gamma_Path_shear, L_coh_shear, sigma_floor_TBN_v, beta_SC_shear, r_limit.
3. Key output flags
   [param: kappa_STG_shear = 0.09 ± 0.04], [param: L_coh_shear = 92 ± 28 h^-1 Mpc], [metric: r_S∇δ (band-mean) = 0.88], [metric: R_ω = 0.19 ± 0.05], [metric: chi2_per_dof = 1.08].

V. Path and Measure Declaration (Arrival Time)

• Arrival-time aperture: T_arr = ∫ (n_eff / c_ref) · dℓ. The measure dℓ arises from the unified window operator; S_path contributes non-dispersively to ∇v_EFT and P_s.
• Units: 1 Mpc = 3.0856776e22 m; report k in h Mpc^-1 and velocities in km s^-1.

VI. Results and Comparison with Mainstream Models

Table 1. Dimension Scorecard

Table 2. Overall Comparison

Table 3. Delta Ranking

VII. Conclusion and Falsification Plan

• Conclusion
  The minimal STG + Path + CoherenceWindow + SeaCoupling + TBN + ResponseLimit EFT frame provides small, testable low-k alignment/phase refinements with a bounded velocity floor, jointly explaining the core signatures of “velocity shear–density gradient mismatch”: depressed r_S∇δ, shifted alignment-angle distribution, enhanced curl, and RSD hexadecapole residuals. As parameters → 0, the model reverts to the mainstream baseline.
• Falsification
  In larger-volume, stricter-window datasets, if forcing kappa_STG_shear = 0, gamma_Path_shear = 0, sigma_floor_TBN_v = 0, L_coh_shear → 0 still reproduces the observed r_S∇δ, ⟨cos Δθ⟩, R_ω, and P_4 behavior, the EFT mechanism is falsified. Conversely, stable recovery of kappa_STG_shear ≈ 0.06–0.12, L_coh_shear ≈ 70–120 h^-1 Mpc, and sigma_floor_TBN_v ≈ 20–60 km s^-1 across independent samples would support the mechanism.

External References

• RSD multipole modeling and EFT-of-LSS harmonization for P_s(k, μ).
• Velocity-field curl/div decomposition and shear-tensor estimation methodologies.
• Density-gradient and potential-flow/shear reconstructions (Wiener / iterative-POTENT).
• Applications of pairwise and tomographic kSZ to momentum-field reconstruction.
• T-/V-Web environment classification and comparative statistics of tidal/velocity-shear tensors.

Appendix A. Data Dictionary and Processing Details

• Fields & units
  r_S∇δ(k) (dimensionless), ⟨cos Δθ⟩ (dimensionless), R_ω (dimensionless), P_ℓ(k) (dimensionless), kSZ_cross_SNR (dimensionless), χ²/dof (dimensionless).
• Parameters
  kappa_STG_shear, gamma_Path_shear, L_coh_shear, sigma_floor_TBN_v, beta_SC_shear, r_limit.
• Processing
  Unified windows/masks/selection and calibration; joint RSD + alignment + kSZ + PV likelihood; von Mises–Fisher alignment fits; Helmholtz decomposition for R_ω; hierarchical Bayes and leave-one-out.
• Key output flags
  [param: L_coh_shear = 92 ± 28 h^-1 Mpc], [param: kappa_STG_shear = 0.09 ± 0.04], [metric: r_S∇δ (band-mean) = 0.88], [metric: chi2_per_dof = 1.08].

Appendix B. Sensitivity and Robustness Checks

• Prior sensitivity
  Switching between uniform/normal priors yields < 0.3σ posterior drifts for kappa_STG_shear, L_coh_shear, and sigma_floor_TBN_v.
• Blind / leave-one-out tests
  Dropping one survey/region/shell preserves conclusions; intervals for r_S∇δ, ⟨cos Δθ⟩, R_ω, and P_4 residuals remain overlapping.
• Alternative statistics
  Re-binning, profile-likelihood variants, and alternative threshold/HOD priors retain directions and significances; regression magnitudes remain consistent.