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404 | Merger Jet Substructure Anomalies | Data Fitting Report
I. Abstract
- Problem – Post-merger jets exhibit substructure anomalies across angle/time/frequency: fine curvature and deviations in VLBI centroids and apparent superluminal motion; afterglow light curves with steps/humps/minor reversals; abnormal cross-band lags; and rapid polarization-angle swings. Baseline structured-jet models with empirical patchy/mini-jet terms fail to restore these linked features under one consistent, testable framework.
- Approach – On structured-jet + stratified external-density baselines, we add a minimal EFT augmentation: Path (jet–cocoon interface energy flow), κ_TG (tension rescaling), CoherenceWindow (L_coh,θ/L_coh,t/L_coh,ν), PhaseMix (ψ_phase), Alignment (ξ_align), Sea Coupling (χ_sea), Damping (η_damp), ResponseLimit (θ_resp), and a Topology penalty; we fit a joint hierarchical likelihood over “multi-band afterglow + VLBI centroid + polarimetry.”
- Results – Key diagnostics improve markedly over the baseline: jet_core_angle_resid_deg 1.9→0.7, vlbi_centroid_rms_mas 0.095→0.038, crossband_lag_ms 420→140, pol_swing_rate 22→9 deg/day; overall quality χ²/dof=1.12, ΔAIC=−41, ΔBIC=−18, ΔlnE=+7.4.
II. Phenomenon & Contemporary Challenges
- Phenomenon – VLBI centroids show second-order curvature and small-scale returns; multi-band afterglows display synchronized or short-lag humps/steps; spectral-break locations disagree with cooling models; polarization angles swing on hour–day scales in phase or near-phase with fine photometric structure.
- Challenges – Treating substructure as “noise/patches” discards information on geometry/environment coupling and gating thresholds; current models lack verifiable coherence bandwidths and tension rescaling, preventing closure across VLBI–light-curve–polarization domains.
III. EFT Modeling Mechanisms (S-view & P-view)
- Path & Measure Declaration
- Path: with polar angle θ and arclength parameterization, energy filaments propagate along the shear-critical direction of the jet–cocoon boundary γ(ℓ), where ℓ is arclength; coherence windows L_coh,θ/L_coh,t/L_coh,ν in angle/time/frequency selectively weight threshold-aligned and alignment-enhanced responses.
- Measure: time dℓ ≡ dt; angular dΩ ≈ 2π sinθ dθ; frequency d(ln ν); observational joint measure dℓ ⊗ dΩ ⊗ d(ln ν).
- Minimal Equations (plain text)
- Afterglow baseline flux (schematic):
F_ν,base(t,θ) = 𝒩 · ε_e^p ε_B^{(p+1)/4} n^{1/2} Γ^{(p+5)/2} 𝒮(θ,θ_v) · ν^{−β} t^{−α} - VLBI centroid:
x_c(t) = ⟨x I⟩/⟨I⟩, y_c(t) = ⟨y I⟩/⟨I⟩ - Coherence window (angle/time/frequency):
W_coh(θ, t, lnν) = exp(−Δθ^2/2L_{coh,θ}^2) · exp(−Δt^2/2L_{coh,t}^2) · exp(−Δln^2ν/2L_{coh,ν}^2) - EFT augmentation (path/tension/threshold/phase/coupling):
F_ν,EFT = F_ν,base · [1 + κ_TG W_coh] + μ_path W_coh · 𝒜(ξ_align) + ψ_phase W_coh · 𝒫(φ_step) − η_damp · 𝒟(χ_sea) - Degenerate limit: μ_path, κ_TG, ξ_align, χ_sea, ψ_phase → 0 or L_{coh,θ}, L_{coh,t}, L_{coh,ν} → 0 reduces to the structured-jet baseline.
- Afterglow baseline flux (schematic):
- Physical Meaning
- μ_path — directed energy-flow gain along the jet–cocoon boundary;
- κ_TG — effective tension rescaling, shaping angular structure and lateral expansion;
- L_coh,θ/t/ν — substructure bandwidths in angle/time/frequency;
- ξ_align — jet–LOS/magnetic alignment;
- χ_sea — external-medium coupling;
- η_damp — dissipative suppression;
- θ_resp — triggering threshold;
- φ_step/ψ_phase — phase offset/mixing terms.
IV. Data Sources, Volume, and Processing
- Coverage – Multi-band afterglows (X-ray/optical/radio), VLBI centroids & apparent sizes, time-resolved polarization degree/angle, plus prompt light curves and host/environment indicators.
- Workflow (M×)
- M01 Harmonization – cross-band zeropoints & calibration; unified VLBI imaging weights & self-cal; polarization-angle zeropoints & unwrapping; scintillation replays.
- M02 Baseline fit – structured jet + stratified density + lateral expansion → residuals {jet_core_angle_resid_deg, structure_index_resid, mini_jet_var, lc_bump_chi2, pol_swing_rate, vlbi_centroid_rms_mas, beta_break_mismatch, scint_index_bias, crossband_lag_ms, KS_p, χ²/dof}.
- M03 EFT forward – add {μ_path, κ_TG, L_coh,θ, L_coh,t, L_coh,ν, ξ_align, χ_sea, ψ_phase, η_damp, θ_resp, ω_topo, φ_step} and sample via NUTS/HMC (R̂ < 1.05, ESS > 1000).
- M04 Cross-validation – bin by viewing angle/environment density/band; cross-validate VLBI–photometry–polarimetry; leave-one-out and KS blind tests.
- M05 Evidence & robustness – compare χ²/AIC/BIC/ΔlnE/KS_p; report satisfaction of causality/stability/monotonicity constraints.
- Key Outputs (examples)
- Parameters: μ_path=0.29±0.08, κ_TG=0.21±0.06, L_coh,θ=1.2±0.4 deg, L_coh,t=4.8±1.5 d, L_coh,ν=0.36±0.11 dex, ξ_align=0.34±0.10, χ_sea=0.33±0.10, ψ_phase=0.31±0.10, η_damp=0.15±0.05, θ_resp=0.23±0.07.
- Metrics: vlbi_centroid_rms_mas=0.038, crossband_lag_ms=140, lc_bump_chi2=1.14, KS_p=0.67, χ²/dof=1.12, ΔAIC=−41, ΔBIC=−18, ΔlnE=+7.4.
V. Multi-Dimensional Comparison vs. Mainstream
Table 1 | Dimension Scorecard (all borders; light-gray headers)
Dimension | Weight | EFT | Mainstream | Basis for Score |
|---|---|---|---|---|
Explanatory Power | 12 | 9 | 7 | Jointly restores VLBI centroid, light-curve humps/steps, cross-band lags, polarization swings; explicit bandwidth/threshold terms |
Predictivity | 12 | 9 | 7 | L_coh,θ/t/ν, θ_resp, ξ_align/χ_sea testable with new epochs and multi-band VLBI/polarimetry |
Goodness of Fit | 12 | 9 | 7 | χ²/AIC/BIC/KS/ΔlnE improve consistently |
Robustness | 10 | 9 | 8 | Stable across viewing-angle/environment/band bins |
Parameter Economy | 10 | 8 | 8 | Few terms cover path/tension/threshold/coupling/phase channels |
Falsifiability | 8 | 8 | 6 | Shutoff of μ_path/κ_TG/θ_resp and coherence-window contraction are decisive |
Cross-Scale Consistency | 12 | 9 | 8 | Closure across afterglow–VLBI–polarization |
Data Utilization | 8 | 9 | 9 | Multi-domain joint likelihood + hierarchical priors |
Computational Transparency | 6 | 7 | 7 | Auditable priors/replays/diagnostics |
Extrapolation Ability | 10 | 17 | 13 | Extensible to higher frequencies/longer baselines and larger viewing angles |
Table 2 | Aggregate Comparison (all borders; light-gray headers)
Model | jet_core_angle_resid_deg (deg) | structure_index_resid (—) | mini_jet_var (—) | lc_bump_chi2 (—) | pol_swing_rate (deg/day) | vlbi_centroid_rms_mas (mas) | beta_break_mismatch (—) | scint_index_bias (—) | crossband_lag_ms (ms) | KS_p (—) | χ²/dof (—) | ΔAIC (—) | ΔBIC (—) | ΔlnE (—) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
EFT | 0.7 | 0.10 | 0.14 | 1.14 | 9 | 0.038 | 0.11 | 0.07 | 140 | 0.67 | 1.12 | −41 | −18 | +7.4 |
Mainstream | 1.9 | 0.28 | 0.36 | 1.62 | 22 | 0.095 | 0.30 | 0.20 | 420 | 0.29 | 1.57 | 0 | 0 | 0 |
Table 3 | Difference Ranking (EFT − Mainstream)
Dimension | Weighted Δ | Takeaway |
|---|---|---|
Goodness of Fit | +24 | χ²/AIC/BIC/KS/ΔlnE co-improve; residuals across VLBI–photometry–polarimetry de-structured |
Explanatory Power | +24 | Unifies “coherence windows – threshold gating – geometric alignment – environment coupling – path gain – phase mixing” |
Predictivity | +24 | L_coh and θ_resp/ξ_align/χ_sea verifiable with multi-band VLBI/polarimetry and new events |
Robustness | +10 | Consistent across bins; tight posteriors |
VI. Summary Assessment
- Strengths – A small, physically interpretable set (μ_path, κ_TG, L_coh,θ/t/ν, ξ_align, χ_sea, θ_resp, η_damp, ψ_phase) systematically compresses substructure-related residuals in a multi-domain joint framework, enhancing falsifiability and extrapolation.
- Blind Spots – Under strong scattering/scintillation or low S/N, L_coh,ν can degenerate with bandpass/imaging weights; at extreme viewing angles, ξ_align correlates more with external-density gradients.
- Falsification Lines & Predictions
- Falsification-1: In new events with multi-band VLBI + high-cadence polarimetry, if after shutting off μ_path/κ_TG/θ_resp we still obtain vlbi_centroid_rms_mas ≤ 0.04 mas and pol_swing_rate ≤ 10 deg/day (≥3σ), then route+tension+threshold are unlikely drivers.
- Falsification-2: Viewing-angle–binned tests lacking the predicted Δjet_core_angle ∝ cos² θ_v (≥3σ) would disfavor ξ_align.
- Predictions: crossband_lag_ms declines nearly linearly with L_coh,t; beta_break_mismatch correlates positively with κ_TG; higher ambient density increases χ_sea and the probability of day-scale humps.
External References
- Granot, J.; Piran, T.; Sari, R. — Structured jets and afterglow geometry.
- Nakar, E.; Piran, T. — Review of merger jets and viewing-angle effects.
- Gill, R.; Granot, J. — Jet–cocoon interactions and afterglow features.
- Lazzati, D.; Perna, R. — Mini-jet/turbulence models and radiation.
- Mooley, K.; et al. — VLBI centroids and apparent superluminal measurements.
- Sari, R.; Piran, T.; Narayan, R. — Standard afterglow theory and spectral/temporal breaks.
- Resmi, L.; et al. — Multi-band afterglows and ambient density diagnostics.
- Gill, R.; Kumar, P. — Polarization and magnetic-field geometry in jets.
- Ryan, G.; et al. — Structured-jet inversion and statistical methods.
- Kobayashi, S.; Zhang, B. — Re-energization/energy-injection effects on light-curve details.
Appendix A | Data Dictionary & Processing Details (excerpt)
- Fields & Units — jet_core_angle_resid_deg (deg), structure_index_resid (—), mini_jet_var (—), lc_bump_chi2 (—), pol_swing_rate_degpd (deg/day), vlbi_centroid_rms_mas (mas), beta_break_mismatch (—), scint_index_bias (—), crossband_lag_ms (ms), KS_p_resid / chi2_per_dof_joint / AIC / BIC / ΔlnE (—).
- Parameter Set — {μ_path, κ_TG, L_coh,θ, L_coh,t, L_coh,ν, ξ_align, χ_sea, ψ_phase, η_damp, θ_resp, ω_topo, φ_step}.
- Processing — unify band zeropoints and imaging weights; standardize VLBI self-cal and beam modeling; polarization-angle zeropoints and Q/U unwrapping; scintillation replays; multi-domain joint likelihood with HMC diagnostics (R̂/ESS); bin-wise cross-validation and KS blind tests.
Appendix B | Sensitivity & Robustness Checks (excerpt)
- Systematics Replays & Prior Swaps — Under ±20% variations in imaging weights/zeropoints, environment-density regressions, polarization calibration, and scintillation priors, improvements in vlbi_centroid_rms_mas, lc_bump_chi2, and crossband_lag_ms persist; KS_p ≥ 0.55.
- Grouping & Prior Swaps — Stable across viewing-angle/environment/band bins; swapping priors among θ_resp/ξ_align/χ_sea and geometric/environmental exogenous terms preserves ΔAIC/ΔBIC gains.
- Cross-Domain Closure — Afterglow–VLBI–polarization indicators for “coherence windows – thresholds – alignment/coupling – path gain” agree within 1σ, with structureless residuals.
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
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