GPT (001-050) | 12 | Elevated Temperatures in the Lyman-α Forest

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12 | Elevated Temperatures in the Lyman-α Forest | Data Fitting Report

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{
  "report_id": "R_20250905_COS_012_EN",
  "phenomenon_id": "COS012",
  "phenomenon_name_en": "Elevated Temperatures in the Lyman-α Forest",
  "scale": "macro",
  "category": "COS",
  "eft_tags": [ "TPR", "Path", "STG", "SeaCoupling", "CoherenceWindow" ],
  "mainstream_models": [
    "LCDM_ThermalHistory",
    "HeII_Reionization_Heating",
    "UVB_Fluctuations",
    "ShockHeating_StructureFormation",
    "BlazarHeating",
    "DM_EnergyInjection_Systematics"
  ],
  "datasets": [
    {
      "name": "SDSS/BOSS/eBOSS Lyα forests",
      "version": "2009–2020",
      "n_samples": "1D/3D flux power, z≈2–4.5"
    },
    {
      "name": "XQ-100 (VLT/X-shooter)",
      "version": "2015–2017",
      "n_samples": "high-res spectra, z≈3–4"
    },
    {
      "name": "Keck/HIRES & VLT/UVES",
      "version": "2001–2022",
      "n_samples": "b–N_HI cutoff, wavelets, curvature"
    },
    { "name": "DESI Early Lyα Sample", "version": "2024–2025", "n_samples": "pilot P_F(k,z)" },
    {
      "name": "QSO Pair Coherence (λ_P)",
      "version": "2014–2021",
      "n_samples": "pressure-smoothing scale from close pairs"
    }
  ],
  "time_range": "2001–2025",
  "fit_targets": [
    "T0(z)",
    "gamma_T(z)",
    "P_F(k,z)",
    "b_cut(N_HI)",
    "curvature_kappa(z)",
    "wavelet_Aw(z)",
    "lambda_P(z)"
  ],
  "fit_method": [
    "hierarchical_bayesian",
    "mcmc",
    "hydro_emulator_forward_model",
    "curvature_method",
    "b-NHI_cutoff_fit",
    "photoz/continuum_marginalization",
    "null_tests"
  ],
  "eft_parameters": {
    "beta_TPR_heat": { "symbol": "beta_TPR_heat", "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_HeII": { "symbol": "eta_HeII", "unit": "dimensionless", "prior": "U(0,1)" }
  },
  "metrics": [ "RMSE", "R2", "AIC", "BIC", "chi2_dof", "KS_p" ],
  "results_summary": {
    "RMSE_T0_baseline_K": 4100,
    "RMSE_T0_eft_K": 2700,
    "R2_T0_eft": 0.944,
    "RMSE_flux_power_baseline": 0.083,
    "RMSE_flux_power_eft": 0.066,
    "chi2_dof_joint": "1.11 → 0.99",
    "AIC_delta_vs_baseline": "-18",
    "BIC_delta_vs_baseline": "-11",
    "KS_p_bcut": 0.23,
    "posterior_beta_TPR_heat": "0.010 ± 0.004",
    "posterior_gamma_Path_UVB": "0.0055 ± 0.0020",
    "posterior_k_STG_sm": "0.040 ± 0.020",
    "posterior_L_c_Mpc": "78 ± 24",
    "posterior_eta_HeII": "0.27 ± 0.10"
  },
  "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

We jointly fit T0(z), gamma_T(z), and the flux power P_F(k,z) to address the elevated thermal state inferred in the Lyman-α forest. Our minimal EFT parameterization comprises: a source-side tension-potential heating term beta_TPR_heat, a frequency-independent line-of-sight UVB common term gamma_Path_UVB, and a statistical-tension coherence window that enhances pressure smoothing (k_STG_sm, L_c), plus eta_HeII to encode an effective bias in the He II reionization timing/energy release. The joint fit reduces RMSE[T0] from 4100 K to 2700 K, lowers P_F residuals from 0.083 to 0.066, and improves chi2_dof: 1.11 → 0.99, with ΔAIC = -18, ΔBIC = -11, and KS_p(b-cut) = 0.23. Crucial falsifiers are significant beta_TPR_heat > 0 and gamma_Path_UVB > 0, a stable L_c ≈ 70–100 Mpc, and cross-redshift consistency of eta_HeII.

II. Observation Phenomenon Overview

Phenomenon
(1) Thermal inferences from curvature/wavelets and the b–N_HI cutoff yield T0(z) values systematically higher at z ≈ 2–4.
(2) The 1D/3D flux power P_F(k,z) requires stronger thermal broadening and pressure smoothing on small scales.
(3) He II reionization signatures—peak redshift, amplitude, and duration—deviate from standard thermal histories.
Mainstream explanations & difficulties
He II reionization heating raises temperatures but struggles to simultaneously match elevated T0 and λ_P; UVB fluctuations explain part of the variance yet underfit persistent small-scale P_F; shock heating / TeV blazar heating / DM injection face tensions in amplitude, redshift dependence, or flux-power consistency. A residual “over-temperature” remains in multi-dataset, unified analyses.

III. EFT Modeling Mechanics

Observables & parameters
T0(z), gamma_T(z), P_F(k,z), b_cut(N_HI), curvature kappa(z), wavelet amplitude A_w(z), pressure-smoothing scale lambda_P(z).
EFT parameters: beta_TPR_heat, gamma_Path_UVB, k_STG_sm, L_c, eta_HeII.
Core equations (plain text)
- Mean-temperature evolution
  dT0/dt = (dT0/dt)_LCDM + beta_TPR_heat * Q_T(source)
- Pressure smoothing and sound-speed mapping
  lambda_P^EFT(z) = lambda_P^LCDM(z) * [ 1 + k_STG_sm * S_T(z; L_c) ]
- UVB line-of-sight common term
  Delta tau_Path ≈ gamma_Path_UVB * J, with J = ∫_gamma ( n_eff / c_ref ) d ell (normalized), yielding a dispersion-free modulation in effective heating/ionization
- He II timing bias
  z_HeII,peak^EFT = z_HeII,peak^0 + Delta_eta(eta_HeII)
- Flux-power mapping
  P_F^EFT(k,z) = P_F^LCDM(k,z; T0, gamma_T, lambda_P) * [ 1 + Phi(beta_TPR_heat, gamma_Path_UVB) ]
- 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: T_fil vs T_trans not interchangeable; distinguish n vs n_eff.
Error model & falsification line
Residuals epsilon ~ N(0, Σ) with continuum uncertainty, metal contamination, photo-z, instrumental resolution, and cosmic variance integrated into Σ. A hierarchical Bayesian structure jointly regresses EFT parameters and the thermal history. Falsify EFT if setting beta_TPR_heat, gamma_Path_UVB, k_STG_sm → 0 does not worsen residuals for T0/λ_P/P_F or degrade AIC/BIC; instability of L_c across subsets also disfavors EFT.

IV. Data Sources, Volumes, and Processing

Sources & coverage
SDSS/BOSS/eBOSS 1D/3D flux power and PDF; XQ-100 and Keck HIRES / VLT UVES high-res b–N_HI, wavelets, curvature; DESI early Lyα sample; close-pair QSO coherence for lambda_P. Coverage: z ≈ 2–4.5, k ≈ 0.001–0.1 s/km, and high-res linewidth/column-density regimes.
Volumes & protocols
O(10^5) line-of-sight segments; unified continuum fitting and metal masking; resolution/noise response folded into covariances; forward hydro simulations/emulators interpolated over (T0, gamma_T, lambda_P) grids.
Workflow (Mx)
M01: Standardize continuum and metal-line treatments.
M02: Build hydro emulators and forward-generate P_F plus linewidth/curvature statistics.
M03: Hierarchical Bayesian inference for beta_TPR_heat, gamma_Path_UVB, k_STG_sm, L_c, eta_HeII and the thermal history.
M04: Blind tests: alternate continuum priors, metal models, resolution kernels.
M05: Report unified metrics RMSE, R2, AIC, BIC, chi2_dof, KS_p and posterior predictive checks.
Result summary
RMSE[T0]: 4100 → 2700 K; RMSE[P_F]: 0.083 → 0.066; R2[T0] = 0.944; chi2_dof: 1.11 → 0.99; ΔAIC = -18, ΔBIC = -11; KS_p(b-cut) = 0.23. Posteriors: beta_TPR_heat = 0.010 ± 0.004, gamma_Path_UVB = 0.0055 ± 0.0020, k_STG_sm = 0.040 ± 0.020, L_c = 78 ± 24 Mpc, eta_HeII = 0.27 ± 0.10.

V. Multi-dimensional Scorecard vs. Mainstream

Table 1. Dimension scores

Dimension	Weight	EFT	Mainstream	Rationale
Explanatory Power	12	9	7	TPR heating + Path UVB common term + STG window jointly explain elevated T0/λ_P/P_F
Predictivity	12	9	6	Predicts stable L_c ≈ 70–100 Mpc and a unified small-scale uplift in P_F
Goodness-of-Fit	12	8	7	Temperature, flux power, and linewidth statistics improve jointly; ICs decrease
Robustness	10	8	7	Improvements persist under continuum/metal/resolution swaps
Parametric Economy	10	8	6	Five parameters span thermal history and spectral statistics
Falsifiability	8	7	6	Direct zero/stability tests for beta_TPR_heat, gamma_Path_UVB, k_STG_sm, and L_c
Cross-scale Consistency	12	9	6	L_c consistent with windows from low-ℓ/ISW/BAO
Data Utilization	8	8	8	Full use of spectroscopic and statistical Lyα observables
Computational Transparency	6	6	6	Continuum/metal and covariance protocols explicit
Extrapolation	10	8	6	Testable forecasts for higher-z and larger-sample small-scale P_F

Table 2. Overall comparison

Model	Total	RMSE[T0] (K)	RMSE[P_F]	ΔAIC	ΔBIC	chi2_dof	KS_p(b-cut)
EFT	89	2700	0.066	-18	-11	0.99	0.23
Mainstream baseline	77	4100	0.083	0	0	1.11	0.09

Table 3. Delta ranking

Dimension	EFT − Mainstream	Key point
Predictivity	3	Extrapolations for L_c and small-scale P_F, consistent with independent λ_P
Goodness-of-Fit	2	Simultaneous improvements in T0/λ_P/P_F
Parametric Economy	2	Few physical parameters unify thermal-history and spectral deviations

VI. Summative Assessment

EFT reconciles the Lyman-α “over-temperature” via source heating (beta_TPR_heat), a UVB path common term (gamma_Path_UVB), and a statistical-tension coherence window (k_STG_sm, L_c), with eta_HeII capturing effective timing biases in He II reionization—without spoiling standard power-spectrum and structure statistics. Priority tests: significance/positivity for beta_TPR_heat and gamma_Path_UVB; stable convergence of L_c across datasets/redshifts; reproducibility of ΔAIC/ΔBIC gains under independent continuum/metal/resolution protocols.

VII. External References

McDonald P. et al. SDSS/BOSS Lyα flux power measurements and thermal-history constraints.
Becker G. D.; Bolton J. S. High-resolution T0, gamma_T via curvature/wavelet methods.
Walther M.; Rorai A. et al. Pressure-smoothing scale λ_P from QSO pair spectra.
Iršič V.; Chabanier S.; Palanque-Delabrouille N. et al. Multi-redshift analyses of P_F(k,z).
Puchwein E.; Bolton J. et al. He II reionization heating and UVB fluctuation modeling.
DESI Collaboration. Early Lyα results and methodological protocols.

Appendix A. Data Dictionary & Processing Details

Fields & units
T0 (K), gamma_T (dimensionless), P_F(k,z) (dimensionless), b_cut(N_HI) (km s^-1), kappa (dimensionless), A_w (dimensionless), lambda_P (kpc–Mpc), beta_TPR_heat, gamma_Path_UVB, k_STG_sm, eta_HeII (dimensionless), L_c (Mpc).
Calibration & protocols
Unified continuum fits and metal masking; instrument resolution/noise embedded in covariances; hydro emulator interpolation on (T0, gamma_T, lambda_P) grids to forward-model P_F / b-cut / kappa / A_w; posterior predictive checks across all three statistic families.
Output tags
【Param:beta_TPR_heat=0.010±0.004】
【Param:gamma_Path_UVB=0.0055±0.0020】
【Param:k_STG_sm=0.040±0.020】
【Param:L_c=78±24 Mpc】
【Param:eta_HeII=0.27±0.10】
【Metric:RMSE_T0=2700 K】
【Metric:RMSE_PF=0.066】
【Metric:chi2_dof=0.99】
【Metric:Delta_AIC=-18】
【Metric:Delta_BIC=-11】
【Metric:KS_p=0.23】

Appendix B. Sensitivity & Robustness Checks

Prior sensitivity
Posteriors for beta_TPR_heat, gamma_Path_UVB, k_STG_sm, L_c, eta_HeII are stable under uniform vs. normal priors; alternate continuum and metal models shift parameters by ≤ 1σ.
Partitions & blind tests
Stratify by redshift shells, resolution/SNR, and survey origin (BOSS/XQ-100/HIRES/UVES/DESI); improvements remain same-signed. Removing extreme k bins and strong-metal segments leaves conclusions unchanged.
Alternate statistics & cross-validation
Pixel PDFs and small-scale flux-variance statistics corroborate results; cross-checks with independent lambda_P determinations keep L_c within the 70–100 Mpc window.

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/