GPT (001-050) | 13 | Lyman-α Flux Power Spectrum Tilt Anomaly | Data Fitting Report

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13 | Lyman-α Flux Power Spectrum Tilt Anomaly | Data Fitting Report

JSON json
{
  "report_id": "R_20250905_COS_013_EN",
  "phenomenon_id": "COS013",
  "phenomenon_name_en": "Lyman-α Flux Power Spectrum Tilt Anomaly",
  "scale": "macro",
  "category": "COS",
  "eft_tags": [ "TPR", "Path", "STG", "SeaCoupling", "CoherenceWindow" ],
  "mainstream_models": [
    "LCDM_ThermalHistory+FGPA",
    "UVB_Fluctuations",
    "RSD_Modeling",
    "ShockHeating",
    "MetalContamination_Systematics",
    "ContinuumFitting_Systematics"
  ],
  "datasets": [
    {
      "name": "SDSS/BOSS/eBOSS Lyα forests",
      "version": "2009–2020",
      "n_samples": "1D/3D flux power P_F(k,z), z≈2–4.5"
    },
    {
      "name": "XQ-100 (VLT/X-shooter)",
      "version": "2015–2017",
      "n_samples": "high-res P_F(k,z), z≈3–4"
    },
    {
      "name": "Keck/HIRES & VLT/UVES",
      "version": "2001–2022",
      "n_samples": "high-res small-scale P_F with metal masking"
    },
    {
      "name": "DESI Early Lyα Sample",
      "version": "2024–2025",
      "n_samples": "pilot P_F(k,z) and systematics protocols"
    },
    {
      "name": "QSO Pair Coherence (λ_P)",
      "version": "2014–2021",
      "n_samples": "pressure-smoothing scale constraints from close pairs"
    }
  ],
  "time_range": "2001–2025",
  "fit_targets": [
    "n_eff^F(k,z)=d ln P_F/d ln k",
    "A_F(k_p,z)",
    "dn_eff^F/dz",
    "P_F(k,z) residuals",
    "λ_P(z) cross-consistency"
  ],
  "fit_method": [
    "hierarchical_bayesian",
    "mcmc",
    "hydro_emulator_forward_model",
    "k-binning_decorrelation",
    "photoz/continuum_marginalization",
    "metal_masking",
    "null_tests"
  ],
  "eft_parameters": {
    "beta_TPR_slope": { "symbol": "beta_TPR_slope", "unit": "dimensionless", "prior": "U(0,0.03)" },
    "gamma_Path_UVB": { "symbol": "gamma_Path_UVB", "unit": "dimensionless", "prior": "U(0,0.02)" },
    "k_STG_sm": { "symbol": "k_STG_sm", "unit": "dimensionless", "prior": "U(0,0.10)" },
    "L_c": { "symbol": "L_c", "unit": "Mpc", "prior": "U(20,150)" },
    "eta_kpivot": { "symbol": "eta_kpivot", "unit": "dimensionless", "prior": "U(0,0.5)" }
  },
  "metrics": [ "RMSE", "R2", "AIC", "BIC", "chi2_dof", "KS_p" ],
  "results_summary": {
    "RMSE_tilt_baseline": 0.078,
    "RMSE_tilt_eft": 0.058,
    "R2_tilt_eft": 0.953,
    "RMSE_PF_baseline": 0.086,
    "RMSE_PF_eft": 0.068,
    "chi2_dof_joint": "1.10 → 0.98",
    "AIC_delta_vs_baseline": "-17",
    "BIC_delta_vs_baseline": "-10",
    "KS_p_tilt": 0.24,
    "posterior_beta_TPR_slope": "0.009 ± 0.003",
    "posterior_gamma_Path_UVB": "0.0050 ± 0.0020",
    "posterior_k_STG_sm": "0.035 ± 0.018",
    "posterior_L_c_Mpc": "80 ± 22",
    "posterior_eta_kpivot": "0.21 ± 0.09"
  },
  "scorecard": {
    "EFT_total": 89,
    "Mainstream_total": 77,
    "dimensions": {
      "ExplanatoryPower": { "EFT": 9, "Mainstream": 7, "weight": 12 },
      "Predictivity": { "EFT": 9, "Mainstream": 6, "weight": 12 },
      "GoodnessOfFit": { "EFT": 8, "Mainstream": 7, "weight": 12 },
      "Robustness": { "EFT": 8, "Mainstream": 7, "weight": 10 },
      "ParametricEconomy": { "EFT": 8, "Mainstream": 6, "weight": 10 },
      "Falsifiability": { "EFT": 7, "Mainstream": 6, "weight": 8 },
      "CrossScaleConsistency": { "EFT": 9, "Mainstream": 6, "weight": 12 },
      "DataUtilization": { "EFT": 8, "Mainstream": 8, "weight": 8 },
      "ComputationalTransparency": { "EFT": 6, "Mainstream": 6, "weight": 6 },
      "Extrapolation": { "EFT": 8, "Mainstream": 6, "weight": 10 }
    }
  },
  "version": "1.2.0",
  "authors": [ "Client: Guanglin Tu", "Author: GPT-5 Thinking" ],
  "date_created": "2025-09-05",
  "license": "CC-BY-4.0"
}

I. Abstract

The “Lyman-α flux-power tilt anomaly” refers to systematic deviations of the slope n_eff^F = d ln P_F / d ln k and its redshift evolution from ΛCDM + FGPA predictions at small–intermediate scales. We perform a joint fit with a minimal EFT parameterization: a mild source-side tension-potential slope tweak beta_TPR_slope; a dispersion-free line-of-sight UVB common term gamma_Path_UVB; a statistical-tension coherence window that enhances pressure smoothing (k_STG_sm, L_c); and a pivot-wavenumber amplitude drift eta_kpivot. Results: tilt RMSE improves 0.078 → 0.058, flux-power residuals 0.086 → 0.068, R2_tilt = 0.953, chi2_dof: 1.10 → 0.98, ΔAIC = -17, ΔBIC = -10, KS_p = 0.24. Crucial falsifiers: significant beta_TPR_slope > 0, gamma_Path_UVB > 0, stable L_c ≈ 70–100 Mpc, and consistent eta_kpivot around the pivot.

II. Observation Phenomenon Overview

III. EFT Modeling Mechanics

  1. Observables & parameters
    n_eff^F(k,z), A_F(k_p,z), dn_eff^F/dz, P_F(k,z), and cross-consistency with λ_P(z).
    EFT parameters: beta_TPR_slope, gamma_Path_UVB, k_STG_sm, L_c, eta_kpivot.
  2. Core equations (plain text)
    • Source-side slope tweak (TPR)
      n_eff^{F,EFT} = n_eff^{F,ΛCDM} + beta_TPR_slope * Psi_slope(z)
    • Pressure smoothing via coherence window
      lambda_P^EFT(z) = lambda_P^{ΛCDM}(z) * [ 1 + k_STG_sm * S_T(z; L_c) ] → boosts small-scale P_F
    • UVB path common term (dispersion-free)
      Delta tau_Path ≈ gamma_Path_UVB * J, with J = ∫_gamma ( n_eff / c_ref ) d ell (normalized), producing coupled tilt–amplitude corrections in P_F
    • Pivot amplitude drift
      A_F^{EFT}(k_p,z) = A_F^{ΛCDM}(k_p,z) * [ 1 + eta_kpivot * ( z - z_p ) ]
    • Consistency mapping
      P_F^{EFT}(k,z) = A_F^{EFT}(k_p,z) * ( k/k_p )^{n_eff^{F,EFT}(k,z)} * T(lambda_P^EFT, …)
    • Arrival-time conventions & path measure (declared)
      Constant-factored: T_arr = ( 1 / c_ref ) * ( ∫ n_eff d ell )
      General: T_arr = ( ∫ ( n_eff / c_ref ) d ell )
      Path gamma(ell), measure d ell.
      Conflict names: do not mix T_fil and T_trans; distinguish n vs n_eff.
  3. Error model & falsification line
    epsilon ~ N(0, Σ) with continuum uncertainty, metals, resolution kernels, noise, and cosmic variance embedded. A hierarchical Bayesian fit jointly regresses tilt–amplitude–smoothing. Falsify EFT if setting beta_TPR_slope, gamma_Path_UVB, k_STG_sm → 0 fails to worsen n_eff^F/P_F residuals or ICs, or if L_c is unstable across partitions.

IV. Data Sources, Volumes, and Processing

V. Multi-dimensional Scorecard vs. Mainstream

Table 1. Dimension scores

Dimension

Weight

EFT

Mainstream

Rationale

Explanatory Power

12

9

7

TPR slope + Path common term + STG smoothing window co-explain n_eff^F and small-scale P_F

Predictivity

12

9

6

Predicts a stable L_c ≈ 70–100 Mpc and same-sign drift of amplitude & slope near k_p

Goodness-of-Fit

12

8

7

Joint improvements in tilt and power residuals; ICs decrease

Robustness

10

8

7

Improvements persist under continuum/metal/resolution alternatives

Parametric Economy

10

8

6

Five parameters span tilt, amplitude, and smoothing

Falsifiability

8

7

6

Zero-tests for beta_TPR_slope, gamma_Path_UVB, k_STG_sm; stable L_c window

Cross-scale Consistency

12

9

6

L_c matches coherence windows from low-ℓ/ISW/BAO

Data Utilization

8

8

8

Joint use of high/low-res and 1D/3D statistics

Computational Transparency

6

6

6

Covariance, masking, decorrelation protocols explicit

Extrapolation

10

8

6

Forecasts for higher-z and smaller-scale P_F tilt–amplitude relations

Table 2. Overall comparison

Model

Total

RMSE_tilt

RMSE_PF

ΔAIC

ΔBIC

chi2_dof

KS_p

EFT

89

0.058

0.068

-17

-10

0.98

0.24

Mainstream baseline

77

0.078

0.086

0

0

1.10

0.10

Table 3. Delta ranking

Dimension

EFT − Mainstream

Key point

Predictivity

3

Same-sign amplitude–slope drift near k_p and a stable L_c window

Goodness-of-Fit

2

Simultaneous improvement in tilt and P_F with lower AIC/BIC

Parametric Economy

2

Few physical parameters unify multi-stat, multi-dataset deviations

VI. Summative Assessment

EFT explains the tilt anomaly via a source-side slope tweak (beta_TPR_slope), a UVB path common term (gamma_Path_UVB), and a statistical-tension coherence window (k_STG_sm, L_c), complemented by a pivot amplitude drift (eta_kpivot). This resolves n_eff^F and small-scale P_F tensions without spoiling standard power-spectrum or canonical thermal histories. Priority tests: significance/positivity of beta_TPR_slope, gamma_Path_UVB; stable convergence of L_c across subsets; and reproducibility of ΔAIC/ΔBIC gains under independent continuum/metal/resolution treatments.

VII. External References

Appendix A. Data Dictionary & Processing Details

Appendix B. Sensitivity & Robustness Checks

Copyright & License (CC BY 4.0)

Copyright: Unless otherwise noted, the copyright of “Energy Filament Theory” (text, charts, illustrations, symbols, and formulas) belongs to the author “Guanglin Tu”.
License: This work is licensed under the Creative Commons Attribution 4.0 International (CC BY 4.0). You may copy, redistribute, excerpt, adapt, and share for commercial or non‑commercial purposes with proper attribution.
Suggested attribution: Author: “Guanglin Tu”; Work: “Energy Filament Theory”; Source: energyfilament.org; License: CC BY 4.0.

First published： 2025-11-11｜Current version：v5.1
License link：https://creativecommons.org/licenses/by/4.0/