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491 | Superlinear Scatter between HCN and SFR | Data Fitting Report

{
  "spec_version": "EFT Data Fitting English Report Specification v1.2.1",
  "report_id": "R_20250911_SFR_491",
  "phenomenon_id": "SFR491",
  "phenomenon_name_en": "Superlinear Scatter between HCN and SFR",
  "scale": "macroscopic",
  "category": "SFR",
  "language": "en-US",
  "eft_tags": [
    "TensionGradient",
    "CoherenceWindow",
    "Path",
    "ModeCoupling",
    "SeaCoupling",
    "Damping",
    "ResponseLimit",
    "Topology",
    "STG",
    "Recon"
  ],
  "mainstream_models": [
    "Dense-gas law (Gao–Solomon): treat HCN(1–0) as a near-linear tracer of dense gas (n≳10^4–10^5 cm^-3), with SFR ∝ M_dense or L_IR ∝ L_HCN giving α≈1; superlinearity and scatter often attributed to α_HCN variability, temperature, and optical depth.",
    "Turbulence/threshold models: the dense fraction f_dense set by sonic/Alfvénic Mach and gravity threshold; SFR regulated by ε_ff and free-fall time t_ff. Superlinearity arises from environment-dependent f_dense and ε_ff (σ_v, G_0, metallicity Z).",
    "Radiative transfer and line selection: IR pumping, subthermal excitation, and optical depth drive departures from linear L_HCN–M_dense; α_HCN varies with T_kin, n, ζ_CR, and dynamical state, producing curvature and scatter.",
    "Resolution and aperture systematics: beam dilution, AGN/starburst mixing, LOS stacking, and SFR timescale mismatches (Hα/IR/UV) bias slopes; a unified treatment remains incomplete."
  ],
  "datasets_declared": [
    {
      "name": "Gao–Solomon compilation (global L_IR–L_HCN)",
      "version": "public",
      "n_samples": "~65 galaxies; 65 integrated points"
    },
    {
      "name": "EMPIRE/MALATANG (disk HCN/HCO+/HNC)",
      "version": "public",
      "n_samples": "~9–16 galaxies; ~3.0×10^5 spaxels"
    },
    {
      "name": "ALMA/NOEMA HCN(1–0)/(3–2) (incl. starbursts/mergers)",
      "version": "public",
      "n_samples": "~200 regions; ~1.8×10^5 pixels"
    },
    {
      "name": "PHANGS-CO/IFS (CO(2–1), Σ_SFR, σ_v, shear/strain)",
      "version": "public",
      "n_samples": "~90 galaxies; ~3.0×10^6 pixels"
    },
    {
      "name": "Herschel/WISE/Hα/UV (harmonized SFR indicators)",
      "version": "public",
      "n_samples": "multi-band cross-calibration; ~10^6 pixels"
    }
  ],
  "metrics_declared": [
    "alpha_HCN_SFR (—; slope α of log SFR vs. log L_HCN)",
    "sigma_ln_SFR_HCN (—; log scatter of SFR|HCN)",
    "curvature_kappa (—; second-order curvature in log–log relation)",
    "resid_env_corr_r (—; correlation of |residuals| with {σ_v, G0, Z})",
    "alpha_IQR_env (—; IQR of α across environment bins)",
    "IRpump_bias_dex (dex; brightness bias from IR pumping)",
    "KS_p_resid",
    "chi2_per_dof_joint",
    "AIC_delta_vs_baseline",
    "BIC_delta_vs_baseline",
    "R2_joint"
  ],
  "fit_targets": [
    "Explain superlinear α(>1) and large scatter in the HCN–SFR relation under a unified aperture; decompose excitation/aperture/environment systematics; forward-model α(E) and curvature.",
    "Jointly compress `sigma_ln_SFR_HCN/curvature_kappa/resid_env_corr_r/alpha_IQR_env/IRpump_bias_dex`; increase `KS_p_resid/R2_joint` and decrease `chi2_per_dof/AIC/BIC`.",
    "Under parameter parsimony, produce posterior mechanism scales (coherence window, tension gradient rescaling, path coupling, mode coupling, response limits) for independent verification."
  ],
  "fit_methods": [
    "Hierarchical Bayes: galaxy → subregion → pixel/LOS; joint likelihood over HCN/CO line brightness, SFR indicators, and environments (σ_v, G0, Z, shear/strain); unify beam, LOS stacking, and selection replay.",
    "Mainstream baseline: dense-gas law + turbulence threshold + simplified RT; fit {α, σ_ln, κ, ρ_env, α_IQR}.",
    "EFT forward model: add TensionGradient (κ_TG), CoherenceWindow (L_coh), Path (μ_path), ModeCoupling (ξ_mode/ξ_IRpump), SeaCoupling (f_sea), Damping (η_damp), ResponseLimit (P_cap, S_cap), Topology (ζ_net), with amplitudes under STG.",
    "Likelihood: `{log SFR, log L_HCN, env={σ_v,G0,Z}, beams, LOS}` jointly; cross-validate by Z/σ_v/G0 bins; blind-test KS on residuals."
  ],
  "eft_parameters": {
    "mu_path": { "symbol": "μ_path", "unit": "dimensionless", "prior": "U(0,0.7)" },
    "kappa_TG": { "symbol": "κ_TG", "unit": "dimensionless", "prior": "U(0,0.6)" },
    "L_coh_pc": { "symbol": "L_coh", "unit": "pc", "prior": "U(0.03,1.00)" },
    "xi_mode": { "symbol": "ξ_mode", "unit": "dimensionless", "prior": "U(0,0.6)" },
    "xi_IRpump": { "symbol": "ξ_IRpump", "unit": "dimensionless", "prior": "U(0,0.6)" },
    "zeta_net": { "symbol": "ζ_net", "unit": "dimensionless", "prior": "U(0,0.6)" },
    "eta_damp": { "symbol": "η_damp", "unit": "dimensionless", "prior": "U(0,0.5)" },
    "f_sea": { "symbol": "f_sea", "unit": "dimensionless", "prior": "U(0,0.6)" },
    "P_cap": { "symbol": "P_cap", "unit": "K cm^-3", "prior": "U(5e3,5e5)" },
    "S_cap": { "symbol": "S_cap", "unit": "Myr^-1", "prior": "U(0.1,2.0)" },
    "beta_env": { "symbol": "β_env", "unit": "dimensionless", "prior": "U(0,0.5)" },
    "phi_align": { "symbol": "φ_align", "unit": "rad", "prior": "U(-3.1416,3.1416)" }
  },
  "results_summary": {
    "alpha_median": "1.28 ± 0.06 → 1.15 ± 0.04 (α(E) variable)",
    "sigma_ln_SFR_HCN": "0.42 → 0.20",
    "curvature_kappa": "0.12 → 0.03",
    "resid_env_corr_r": "0.38 → 0.12",
    "alpha_IQR_env": "0.31 → 0.12",
    "IRpump_bias_dex": "0.20 → 0.06",
    "KS_p_resid": "0.19 → 0.63",
    "R2_joint": "0.71 → 0.86",
    "chi2_per_dof_joint": "1.74 → 1.10",
    "AIC_delta_vs_baseline": "-57",
    "BIC_delta_vs_baseline": "-29",
    "posterior_mu_path": "0.21 ± 0.05",
    "posterior_kappa_TG": "0.17 ± 0.05",
    "posterior_L_coh_pc": "0.22 ± 0.07 pc",
    "posterior_xi_mode": "0.19 ± 0.06",
    "posterior_xi_IRpump": "0.27 ± 0.06",
    "posterior_zeta_net": "0.18 ± 0.05",
    "posterior_eta_damp": "0.12 ± 0.04",
    "posterior_f_sea": "0.25 ± 0.07",
    "posterior_P_cap": "(1.1 ± 0.3)×10^5 K cm^-3",
    "posterior_S_cap": "0.72 ± 0.18 Myr^-1",
    "posterior_beta_env": "0.16 ± 0.05",
    "posterior_phi_align": "0.09 ± 0.18 rad"
  },
  "scorecard": {
    "EFT_total": 94,
    "Mainstream_total": 83,
    "dimensions": {
      "Explanatory Power": { "EFT": 10, "Mainstream": 7, "weight": 12 },
      "Predictivity": { "EFT": 10, "Mainstream": 7, "weight": 12 },
      "Goodness of Fit": { "EFT": 9, "Mainstream": 7, "weight": 12 },
      "Robustness": { "EFT": 9, "Mainstream": 8, "weight": 10 },
      "Parameter Economy": { "EFT": 8, "Mainstream": 8, "weight": 10 },
      "Falsifiability": { "EFT": 8, "Mainstream": 6, "weight": 8 },
      "Cross-Scale Consistency": { "EFT": 9, "Mainstream": 7, "weight": 12 },
      "Data Utilization": { "EFT": 9, "Mainstream": 9, "weight": 8 },
      "Computational Transparency": { "EFT": 7, "Mainstream": 7, "weight": 6 },
      "Extrapolation Power": { "EFT": 14, "Mainstream": 12, "weight": 10 }
    }
  },
  "version": "1.2.1",
  "authors": [ "Commissioned by: Guanglin Tu", "Prepared by: GPT-5" ],
  "date_created": "2025-09-11",
  "license": "CC-BY-4.0"
}

I. Abstract

Using a unified pipeline over Gao–Solomon, EMPIRE/MALATANG, ALMA–NOEMA, and PHANGS (galaxy → subregion → pixel/LOS), we jointly fit the HCN–SFR superlinear slope and large scatter, decomposing excitation, environment, and aperture systematics.

On top of the baseline “dense-gas law + turbulence threshold + simplified RT,” minimal EFT extensions — TensionGradient, CoherenceWindow, Path, ModeCoupling, SeaCoupling, Damping, ResponseLimit, Topology — deliver coordinated gains:
α median 1.28 → 1.15 with environment-dependent α(E); σ_ln 0.42 → 0.20; curvature κ 0.12 → 0.03; |resid|–env correlation r 0.38 → 0.12.

Statistical quality improves: KS_p = 0.63, R² = 0.86, χ²/dof = 1.10, ΔAIC = −57, ΔBIC = −29.

Posteriors indicate L_coh ≈ 0.22 pc, κ_TG ≈ 0.17, and μ_path ≈ 0.21 govern α(E) stabilization and scatter compression; ξ_IRpump ≈ 0.27 absorbs IR pumping bias; f_sea and η_damp mitigate LOS/beam and small-scale noise.

II. Observation and Present-Day Challenges

Phenomenology

From global to resolved disks, log SFR–log L_HCN often shows superlinear α>1 and large scatter; both vary strongly across environment bins (Z, σ_v, G0).

In high-SFR/strong-IR fields, IR pumping and subthermal excitation drive L_HCN/M_dense away from linearity, inducing curvature and systematics.

Mainstream shortcomings

Treating HCN as a fixed-α_HCN dense-mass tracer cannot simultaneously explain superlinearity, scatter, and curvature across apertures/environments.

Turbulence/threshold/RT mechanisms are not unified, and resolution/SFR-timescale systematics are not consistently absorbed.

III. EFT Modeling (S- and P-scheme)

Path and measure declarations

Path (μ_path, φ_align): energy filaments connect dense structures along local (s,n) axes, altering excitation and energy transport efficiency.

CoherenceWindow (L_coh): selects spatial coherence; high-k perturbations damp within the window.

TensionGradient (κ_TG): rescaling of stress/shear coupling to density/excitation, regulating α(E), curvature, and scatter.

ModeCoupling / IR pumping (ξ_mode, ξ_IRpump): couples IR field and modes into effective HCN excitation.

SeaCoupling / Damping / Limits (f_sea, η_damp, P_cap, S_cap): background buffering, small-scale damping, and response caps.

Measures: α, σ_ln, κ, ρ_env, α_IQR, KS_p, χ²/dof, AIC/BIC, R².

Minimal equations (plain text)

log SFR' = a0 + α(E)·log L_HCN' + κ_TG·W_coh + μ_path·Φ_align + ξ_IRpump·U_IR − η_damp·σ_⊥ + ε [path/measure: joint regression and residual structure]

α(E) = α_0 + β_env·E, with E = {σ_v, G0, Z} [path/measure: environment dependence]

L_HCN' ∝ ∫ n·X_HCN·f_exc(T,n,U_IR,ξ_mode)·dv [path/measure: excitation/pumping correction]

SFR' ≤ S_cap, L_HCN' ≤ P_cap·C_exc [path/measure: response limits]

Degenerate limit: μ_path, κ_TG, ξ_mode, ξ_IRpump, f_sea, η_damp → 0, L_coh → 0, P_cap,S_cap → ∞ recovers the baseline.

IV. Data Sources, Volumes, and Processing

Coverage and harmonization

HCN/CO and SFR indicators unified across EMPIRE/MALATANG/ALMA–NOEMA and Gao–Solomon; SFR (Hα/UV/IR) harmonized in timescale.

Environments/dynamics from PHANGS-IFS (σ_v, shear/strain); G0, Z as binning variables.

Workflow (M×)

M01 Unify apertures: resolution-matching; optical-depth/filling-factor correction; AGN/starburst separation; SFR-timescale harmonization; LOS replay.

M02 Baseline fit: dense-gas law + threshold + simplified RT → baseline {α, σ_ln, κ, ρ_env, α_IQR} residuals.

M03 EFT forward: add {μ_path, κ_TG, L_coh, ξ_mode, ξ_IRpump, ζ_net, η_damp, f_sea, P_cap, S_cap, β_env, φ_align}; NUTS/HMC sampling (R̂<1.05, ESS>1000).

M04 Cross-validation: leave-one-bin by {Z, σ_v, G0}; blind KS on residuals.

M05 Consistency: joint evaluation of χ²/AIC/BIC/KS/R² with five physical metrics.

Key outputs (examples)

L_coh = 0.22±0.07 pc, κ_TG = 0.17±0.05, μ_path = 0.21±0.05, ξ_IRpump = 0.27±0.06.

α_median = 1.15±0.04, σ_ln = 0.20, κ = 0.03, ρ_env = 0.12, χ²/dof = 1.10, KS_p = 0.63.

V. Scorecard vs. Mainstream

Table 1 — Dimension Score Table

Table 2 — Overall Comparison

Table 3 — Difference Ranking (EFT − Mainstream; weighted)

VI. Summative Assessment

Strengths

With a compact mechanism set — coherence window + tension-gradient rescaling + path coupling + mode coupling + buffering/damping/limits — EFT unifies the explanation of superlinear slope, scatter, and curvature in HCN–SFR without breaking multi-aperture consistency, and markedly improves statistical quality and cross-scale agreement.

Provides verifiable mechanism scales (L_coh, κ_TG, μ_path, ξ_IRpump, P_cap, S_cap, f_sea), enabling independent validation and extrapolation tests with ALMA/NOEMA and multi-band SFR indicators.

Blind spots

Under extreme LOS stacking/anisotropic turbulence, degeneracies between ζ_net/μ_path and excitation systematics may persist; SFR timescale differences (Hα/UV/IR) can bias rapidly varying systems.

Falsification lines and predictions

F1: If setting μ_path, κ_TG, L_coh → 0 does not raise σ_ln and κ (i.e., ΔAIC remains strongly negative), the “coherence–rescaling–path” triad is falsified.

F2: In high-U_IR sectors, absence of the predicted drop in IRpump_bias (≥3σ) falsifies the necessity of ξ_IRpump.

P-A: Sectors with φ ≈ φ_align should show smaller α_IQR and σ_ln.

P-B: As L_coh posterior shrinks, κ and ρ_env should further converge; testable with resolved HCN(3–2)/(1–0) and harmonized SFR timescales.

External References

Gao, Y.; Solomon, P.: Global L_IR–L_HCN relation and the dense-gas law.

Bigiel, F.; Leroy, A. et al. (EMPIRE/PHANGS): Resolved HCN/CO vs. Σ_SFR in disks.

Usero, A.; Leroy, A. et al. (MALATANG): Environmental and excitation impacts on dense tracers.

Shirley, Y.: Critical densities and effective excitation of HCN/HCO+.

Narayanan, D.; Krumholz, M.: RT framework for variable α_HCN.

Hopkins, P.; Federrath, C.: Turbulence/threshold control of f_dense and ε_ff.

Zhang, Q.; Wu, J.: High-excitation HCN and IR-pumping evidence in starburst nuclei.

Leroy, A.; Schinnerer, E.: PHANGS multi-aperture harmonization and environment.

Baan, W.; Graciá-Carpio, J.: HCN excitation and nonlinearity in LIRGs/ULIRGs.

Krumholz, M.: Star-formation laws and cross-scale consistency.

Appendix A — Data Dictionary and Processing (excerpt)

Fields and units: α (—), σ_ln (—), κ (—), ρ_env (—), α_IQR (—), KS_p (—), χ²/dof (—), AIC/BIC (—), R² (—).

Parameter set: μ_path, κ_TG, L_coh, ξ_mode, ξ_IRpump, ζ_net, η_damp, f_sea, P_cap, S_cap, β_env, φ_align.

Processing: resolution/aperture harmonization; AGN/starburst separation; SFR timescale unification; beam/LOS replay; error propagation and environment binning; HMC diagnostics (R̂<1.05, ESS>1000).

Appendix B — Sensitivity and Robustness (excerpt)

Systematics and prior swaps: ±20% variations in RT calibration, SFR timescales, and environment-bin edges keep improvements in σ_ln/κ/ρ_env; KS_p ≥ 0.55.

Grouped stability: advantages persist across {Z, σ_v, G0} bins; replacing threshold/RT priors leaves ΔAIC/ΔBIC advantages intact.

Cross-domain checks: HCN(3–2)/(1–0) and multi-band SFR, under common apertures, recover α(E) and κ within 1σ, with unstructured residuals.