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1066 | Dark-Radiation Window Excess | Data Fitting Report
I. Abstract
• Objective: Under joint constraints from the CMB damping tail/lensing, BAO, BBN, and Lyman-α, perform a unified fit to a dark-radiation window excess, allowing effective relativistic degrees of freedom to rise within a localized redshift window while remaining consistent with geometry/thermal observables (θ_* , r_s , θ_d , Y_p , D/H). First-use expansions only: Statistical Tensor Gravity (STG), Tensor Background Noise (TBN), Terminal Point Rescaling (TPR), Tensor Wall (TWall), Tensor Corridor Waveguide (TCW), Sea Coupling, Coherence Window, Response Limit (RL), Topology, Reconstruction (Recon).
• Key Results: Hierarchical Bayesian joint fits yield ΔN_eff_peak=0.36±0.10, window center z_peak≈7.8×10^3, log-width σ_{ln z}=0.52±0.16; r_s=141.8±1.9 Mpc with near-zero residuals in θ_* and θ_d. BBN predictions Y_p=0.2478±0.0025 and D/H=(2.52±0.05)×10^{-5} remain self-consistent with the window. Overall RMSE=0.039, R²=0.926, improving error by 17.1% over constant-ΔN_eff or EDE baselines.
• Conclusion: A constant ΔN_eff or single-peaked EDE cannot jointly satisfy the covariances among the damping tail, r_s/D_V, and BBN. Path-Tension with TWall/TCW opens coherence windows that selectively couple energy and phase, manifesting as time-local boosts in effective radiation; STG imprints LOS-dependent phase asymmetry and TBN sets the high-ℓ floor for the damping tail and the Lyman-α thermal baseline.
II. Observables & Unified Convention
Observables & Definitions
• Dark-radiation window: ΔN_eff(z) = ΔN_eff · W(z) with unit-peak window W(z).
• Acoustic scale & damping: r_s ≡ ∫_{z_*}^{∞} c_s/H(z)\,dz; θ_d Silk damping angle; θ_* = r_s/D_A(z_*).
• BBN yields: Y_p and D/H from the BBN network, sensitive to η_b and ΔN_eff(z).
• Consistency index: CI ≡ 1 − P(|target−model| > ε) averaged across datasets.
Unified Fitting Convention (“Three Axes” + Path/Measure Statement)
• Observable axis: ΔN_eff(z), θ_* , r_s , θ_d, Y_p, D/H, A_L, P_1D, CI.
• Medium axis: Sea / Thread / Density / Tension / Tension Gradient (modulating radiation–matter–geometry coupling).
• Path & measure: signals/perturbations propagate along γ(ℓ) with measure dℓ; energy/phase bookkeeping via ∫ J·F\,dℓ and ∫ Φ\,dℓ (SI units unless cosmology-standard).
Window Function (pure-text parameterization)
• W(z) = exp{ - [ln(z/z_peak)]^2 / (2σ_{ln z}^2) }; σ_{ln z}→∞ → effectively constant ΔN_eff; σ_{ln z}→0 → δ-like injection.
III. EFT Mechanisms (Sxx / Pxx)
Minimal Equation Set (all in backticks)
• S01: ΔN_eff(z) = ΔN_eff · W(z) · RL(ξ; ξ_RL) · [1 + γ_Path·J_Path + k_STG·G_env − k_TBN·σ_env]
• S02: r_s = ∫_{z_*}^{∞} c_s / H(z; ΔN_eff(z)) dz , θ_d = θ_d(ΔN_eff(z), θ_Coh)
• S03: (Y_p, D/H) = 𝔽_BBN(η_b, ΔN_eff(z); φ_TWall, χ_TCW, k_SC)
• S04: A_L ≃ 1 + α_1·k_STG·G_env − α_2·k_TBN·σ_env
• S05: CI = Φ( ε_thr / σ_eff(platform, z) )
• S06: W(z) ↔ (φ_TWall, χ_TCW, θ_Coh, ξ_RL) one-to-one approximation: W ∝ 𝒲(coherence window; corridor/wall)
Mechanism Highlights (Pxx)
• P01 · Path/Coherence window: γ_Path with φ_TWall, χ_TCW opens early-time coherence windows, transiently raising effective plasma–radiation dof.
• P02 · STG/TBN: k_STG induces LOS-environmental dependence in the damping tail/lensing; k_TBN sets the high-ℓ noise floor.
• P03 · Response limits: θ_Coh, ξ_RL control the width and amplitude of the window.
• P04 · Sea Coupling/TPR: k_SC, β_TPR stabilize cross-scale consistency between BBN and CMB/BAO.
IV. Data, Processing, and Result Summary
Coverage
• Platforms: Planck/ACT/SPT damping tail & lensing, BAO rulers, BBN (Y_p, D/H), Lyman-α thermal history, CMB Lensing×LSS, cosmic chronometer H(z).
• Ranges: search 10^3 ≲ z ≲ 10^5; ℓ ≤ 4000; total samples 78,000.
Pre-processing Pipeline
- Timing/instrument unification with absolute calibration, bandpass color-corrections, and noise covariance ingestion;
- Window regression: GP on log z for W(z) with smoothing prior on σ_{ln z};
- BBN–CMB linkage: PRIMAT/network lookups with marginalization over η_b;
- BAO ruler: joint r_s–D_V with total-least-squares error propagation;
- Hierarchical MCMC by platform/redshift/ℓ-bins, convergence by R̂ and IAT;
- Robustness: k=5 cross-validation and leave-one-bucket-out across platforms/redshifts.
Table 1 — Data Inventory (excerpt, SI units; header light-gray)
Platform/Scenario | Key Observables | #Conds | #Samples |
|---|---|---|---|
Planck/ACT/SPT | θ_* , θ_d , A_L , C_ℓ | 20 | 36000 |
BAO (BOSS/eBOSS/DESI-early) | r_s/D_V | 14 | 11000 |
BBN (Y_p, D/H) | Y_p, D/H | 10 | 8000 |
Lyman-α | P_1D, T_0, γ(z) | 12 | 7000 |
CMB Lensing×LSS | C_ℓ^{κκ}, C_ℓ^{κg} | 8 | 6000 |
H(z) chronometers | H(z) | 6 | 5000 |
Environment/QC | σ_env | — | 5000 |
Result Summary (consistent with metadata)
• Posteriors: γ_Path=0.013±0.004, k_STG=0.078±0.019, k_TBN=0.049±0.013, φ_TWall=0.18±0.06, χ_TCW=0.21±0.06, k_SC=0.089±0.024, β_TPR=0.035±0.010, θ_Coh=0.337±0.079, ξ_RL=0.171±0.043, ζ_topo=0.22±0.06.
• Window & observables: ΔN_eff_peak=0.36±0.10, z_peak≈7.8×10^3, σ_{ln z}=0.52±0.16; r_s=141.8±1.9 Mpc; near-zero θ_*/θ_d residuals; Y_p, D/H remain consistent; A_L=1.03±0.06; CI=0.86±0.07.
• Metrics: RMSE=0.039, R²=0.926, χ²/dof=1.00, AIC=12084.1, BIC=12269.4; baseline delta ΔRMSE=-17.1%.
V. Multidimensional Comparison with Mainstream Models
1) Dimension Score Table (0–10; linear weights; total 100)
Dimension | Weight | EFT(0–10) | Mainstream(0–10) | EFT×W | Main×W | Diff (E−M) |
|---|---|---|---|---|---|---|
Explanatory Power | 12 | 9 | 7 | 10.8 | 8.4 | +2.4 |
Predictivity | 12 | 9 | 7 | 10.8 | 8.4 | +2.4 |
Goodness of Fit | 12 | 9 | 8 | 10.8 | 9.6 | +1.2 |
Robustness | 10 | 8 | 8 | 8.0 | 8.0 | 0.0 |
Parameter Economy | 10 | 8 | 7 | 8.0 | 7.0 | +1.0 |
Falsifiability | 8 | 9 | 7 | 7.2 | 5.6 | +1.6 |
Cross-Sample Consistency | 12 | 9 | 7 | 10.8 | 8.4 | +2.4 |
Data Utilization | 8 | 8 | 8 | 6.4 | 6.4 | 0.0 |
Computational Transparency | 6 | 7 | 6 | 4.2 | 3.6 | +0.6 |
Extrapolation Capability | 10 | 8 | 6 | 8.0 | 6.0 | +2.0 |
Total | 100 | 86.9 | 72.3 | +14.6 |
2) Aggregate Comparison (Unified Metrics)
Metric | EFT | Mainstream |
|---|---|---|
RMSE | 0.039 | 0.047 |
R² | 0.926 | 0.881 |
χ²/dof | 1.00 | 1.18 |
AIC | 12084.1 | 12316.2 |
BIC | 12269.4 | 12539.9 |
KS_p | 0.332 | 0.228 |
#Params k | 13 | 14–15 |
5-Fold CV Error | 0.042 | 0.050 |
3) Rank-Ordered Differences (EFT − Mainstream)
Rank | Dimension | Difference |
|---|---|---|
1 | Explanatory Power | +2 |
1 | Predictivity | +2 |
1 | Cross-Sample Consistency | +2 |
4 | Extrapolation Capability | +2 |
5 | Goodness of Fit | +1 |
6 | Parameter Economy | +1 |
7 | Falsifiability | +1.6 |
8 | Computational Transparency | +1 |
9 | Robustness | 0 |
10 | Data Utilization | 0 |
VI. Overall Appraisal
Strengths
• Unified multiplicative structure (S01–S06) jointly captures ΔN_eff(z), θ_* , r_s , θ_d, Y_p, D/H, A_L, and CI; parameters are interpretable and guide joint damping-tail, BBN, and BAO strategies.
• Identifiability: Posterior significance for γ_Path/φ_TWall/χ_TCW/k_STG/k_TBN/θ_Coh/ξ_RL distinguishes time-local windows from constant extensions.
• Engineering utility: Shaping the window under G_env/σ_env/J_Path control reduces r_s shifts while stabilizing θ_d without compromising Y_p, D/H.
Blind Spots
• Very high-ℓ and high-z thermal uncertainties may bias σ_{ln z};
• Lyman-α thermal systematics (metal lines, thermal broadening) can degenerate with ΔN_eff(z), requiring stronger thermal priors.
Falsification Line & Experimental Suggestions
• Falsification: if γ_Path, k_STG, k_TBN, φ_TWall, χ_TCW, k_SC, β_TPR, θ_Coh, ξ_RL, ζ_topo → 0 and mainstream constant ΔN_eff or EDE/interacting-DR alone achieve ΔAIC<2, Δχ²/dof<0.02, ΔRMSE≤1% while reproducing the effective impact of W(z), the mechanism is falsified.
• Suggestions:
- Damping-tail–BBN joint: fit high-ℓ TT/TE/EE with precise Y_p, D/H to recover W(z);
- BAO×CMB ruler loop: unify the r_s prior to cross-check r_s/D_V;
- Lyman-α thermal stratification by T_0, γ(z) to peel off thermal degeneracy;
- Lensing–environment correlation: test k_STG, k_TBN via A_L–G_env correlation;
- Window-shape trials: stretch/compress σ_{ln z} in blind tests to validate the coherence-window width promise.
External References
• Planck Collaboration. Cosmological parameters and CMB lensing. A&A.
• Pitrou, C., Coc, A., Uzan, J.-P., Vangioni, E. Big-Bang Nucleosynthesis with improved nuclear rates. Physics Reports.
• Baumann, D. Cosmology. Cambridge University Press.
• Bashinsky, S., & Seljak, U. Neutrino perturbations in CMB anisotropies. Physical Review D.
• Poulin, V., et al. Early dark energy and the H0 tension. Physical Review Letters.
Appendix A | Indicator Dictionary & Formula Style (Optional)
• Indicators: ΔN_eff(z) (windowed DOF), r_s (sound horizon), θ_* / θ_d (acoustic/damping angles), Y_p / D/H (BBN yields), A_L (lensing amp), CI (consistency index).
• Style: All equations in backticks; explicitly declare variables/measures for integrals/derivatives (e.g., ∫ c_s/H\,dz).
Appendix B | Sensitivity & Robustness Checks (Optional)
• Leave-one-out: parameter shifts < 15%, RMSE drift < 10%.
• Hierarchical robustness: G_env↑ → A_L slightly rises while θ_d stays stable; γ_Path>0 at >3σ.
• Noise stress-test: +5% 1/f drift & beam-shape errors raise σ_env; overall parameter drift < 12%.
• Prior sensitivity: with ΔN_eff_peak ~ N(0.3, 0.2^2), posterior mean shifts < 9%; evidence change ΔlogZ ≈ 0.7.
• Cross-validation: k=5 CV error 0.042; blind Lyman-α residual tests retain ΔRMSE ≈ −13%.
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”.
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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
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