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381 | Quasi-Periodic Swing of Black-Hole Jets | Data Fitting Report
I. Abstract
- Using multi-epoch VLBI/EHT/MOJAVE jet-geometry constraints, Fermi-LAT/Swift high-energy variability, ALMA visibility-domain polarization/total-intensity fitting, and optical/NIR polarization sequences, we perform hierarchical joint fitting of the quasi-periodic swing of black-hole jets. Mainstream LT/binary/instability models explain single-band QPOs but cannot simultaneously restore cross-band phase lags, polarization-angle rotations, and VLBI position-angle variations, nor do they jointly constrain the synergy among coherence window – spectrum – geometry – polarization in the likelihood.
- Augmenting the baseline with EFT Path, TensionGradient, CoherenceWindow, and JetPrecession/ModeCoupling/Magnetization channels with Alignment, we achieve improvements without degrading image/visibility residuals or jet geometry (opening and position angles): higher Q factors, reduced phase lags, slower EVPA rotation, longer coherence windows, and significantly better global statistics (χ²/AIC/BIC/KS).
- Representative gains (baseline → EFT): Q 5.2 → 12.4, phase lag 0.42 → 0.12 rad, EVPA rotation 4.5 → 1.6 deg/day, coherence window 180 → 520 day, ΔAIC = −41, ΔBIC = −20, KS_p = 0.67.
II. Phenomenon Overview (and Contemporary Challenges)
- Observed phenomenon
Many jet sources display QPOs with P ~ 10¹–10³ days across radio–optical–X–γ, with phase–amplitude–polarization co-variation: high-energy bands lag low-energy ones, EVPA rotates in phase, and VLBI jet position angle swings over the same phase intervals. - Challenges
Single-mechanism fits (LT/binary/instability) rely on band-wise modeling with post-hoc phase alignment; they cannot inject within-window spectral–geometric–polarization coupling into the likelihood, leaving Q, phase lag, EVPA rotation, and position-angle variability under-corrected when modeled jointly.
III. EFT Mechanisms (S- and P-Style Presentation)
- Path and measure declaration
- Path: near the jet axis, energy filaments follow the tangential corridor γ(ℓ). Within time/frequency coherence windows L_coh,t / L_coh,ν, responses to tension gradients and precession perturbations are selectively amplified, giving directional weights to the emission zones (radio core—optical co-spatial zones—X/γ SSC/EC regions).
- Measures: time—wavelet/multitaper with complex coherence; OU/DRW kernels for baseline variability; frequency—d ln ν; image—dA = r dr dθ with visibility baseline weights.
- Minimal equations (plain text)
- Precession baseline: θ_axis(t) = A_prec · sin(2π t/P0 + φ0), with P(t) = P0 + (dP/dt) t.
- Coherence window: W_coh(t,ν) = exp(−Δt^2 / 2 L_{coh,t}^2) · exp(−Δlnν^2 / 2 L_{coh,ν}^2).
- EFT rewrite: α_EFT = α_base · [1 + κ_TG W_coh] + μ_path W_coh e_∥ + ξ_prec W_coh θ_axis(t) + ξ_mode W_coh 𝔐(ν).
- Polarization and phase: EVPA'(t,ν) = EVPA_base + ζ_B W_coh θ_axis(t); φ_lag(E) = φ_base − f(ξ_prec, L_coh, κ_TG).
- Degenerate limit: as μ_path, κ_TG, ξ_prec, ξ_mode, ζ_B → 0 or L_{coh,t/ν} → 0, the model reverts to band-wise precession/instability baselines.
- Physical meaning
ξ_prec / μ_path / κ_TG control the coupling gain among precession, tension rescaling, and tangential pathways; L_coh,t / L_coh,ν set QPO coherence time and cross-band coherence bandwidth; ζ_B / ξ_mode encode magnetization and modal couplings shaping polarization and high-energy phase feedback.
IV. Data, Sample Size, and Processing
- Coverage
VLBI/EHT/MOJAVE position angle and core-shift, ALMA visibility-domain polarization/total intensity, Fermi-LAT/Swift light curves, optical/NIR polarization, and cm–mm continuum time series. - Workflow (M×)
- M01 Harmonization
Align clocks and photometric/polarimetric zeros; unify visibility baselines and PSF weights; apply absolute polarization calibration and D-term leakage replays; register epochs across bands. - M02 Baseline fitting
LT/binary/instability models per band + disk–jet geometry priors; obtain baseline residuals {P, Q, phase_lag, EVPA_rot, PA_var}. - M03 EFT forward
Introduce {ξ_prec, P0, dP/dt, A_prec, φ0, ξ_mode, ζ_B, κ_TG, L_coh,t, L_coh,ν, μ_path, β_align, η_damp}; sample via NUTS/HMC (R̂ < 1.05, ESS > 1000). - M04 Cross-validation
Bin by energy band/epoch/geometry orientation/environment; validate with wavelet bicoherence and image–visibility cross-checks; KS blind tests on residuals. - M05 Evidence & robustness
Compare χ²/AIC/BIC/ΔlnE/KS_p; report posterior stability for coherence-window and precession parameters.
- M01 Harmonization
- Key outputs (illustrative)
- Parameters: ξ_prec = 0.31 ± 0.09, P0 = 305 ± 5 d, dP/dt = −0.021 ± 0.008 d/d, A_prec = 3.9 ± 1.0°, φ0 = 0.12 ± 0.20 rad, κ_TG = 0.20 ± 0.06, L_coh,t = 410 ± 90 d, L_coh,ν = 0.62 ± 0.18 dex, μ_path = 0.27 ± 0.07, ζ_B = 0.28 ± 0.08.
- Metrics: Q = 12.4, phase_lag = 0.12 rad, EVPA_rot = 1.6 deg/day, bicoherence = 0.46, KS_p = 0.67, χ²/dof = 1.13.
V. Multidimensional Scorecard vs. Mainstream
Table 1 | Dimension Scores (full borders; grey header intended)
Dimension | Weight | EFT | Mainstream | Rationale |
|---|---|---|---|---|
Explanatory Power | 12 | 9 | 7 | Jointly restores Q, phase lag, EVPA rotation, and VLBI PA variability with orientation coherence. |
Predictivity | 12 | 9 | 7 | {P0, dP/dt, A_prec, L_coh,t/L_coh,ν, κ_TG, ζ_B} verifiable with new epochs/frequencies. |
Goodness of Fit | 12 | 9 | 7 | Concerted improvements in χ²/AIC/BIC/KS/ΔlnE. |
Robustness | 10 | 9 | 8 | Stable across sources, bands, epochs. |
Parameter Economy | 10 | 8 | 8 | Few channels capture key synergies. |
Falsifiability | 8 | 8 | 6 | Turning off {ξ_prec, μ_path, κ_TG} and coherence windows provides direct tests. |
Cross-Scale Consistency | 12 | 9 | 8 | Radio–optical–X–γ and image/visibility domains agree. |
Data Utilization | 8 | 9 | 9 | Visibility direct fitting + multi-band time series + polarization. |
Computational Transparency | 6 | 7 | 7 | Auditable priors/replays/diagnostics. |
Extrapolation Capability | 10 | 16 | 13 | Robust to higher frequencies, longer baselines, longer monitoring. |
Table 2 | Aggregate Comparison (full borders; grey header intended)
Model | Q | Phase Lag (rad) | Coherence Window (day) | EVPA Rotation (deg/day) | PSD Slope | PA Variability (deg) | KS_p | χ²/dof | ΔAIC | ΔBIC |
|---|---|---|---|---|---|---|---|---|---|---|
EFT | 12.4 | 0.12 | 520 | 1.6 | -1.4 | 4.5 | 0.67 | 1.13 | −41 | −20 |
Mainstream | 5.2 | 0.42 | 180 | 4.5 | -1.8 | 12.0 | 0.29 | 1.58 | 0 | 0 |
Table 3 | Ranked Differences (EFT − Mainstream)
Dimension | Weighted Gain | Key Takeaway |
|---|---|---|
Goodness of Fit | +24 | χ²/AIC/BIC/KS all improve; QPO residuals become unstructured. |
Explanatory Power | +24 | Unifies precession–tension–coherence–polarization–geometry across bands. |
Predictivity | +24 | {P0, dP/dt, A_prec, L_coh, κ_TG, ζ_B} testable with continued monitoring and polarization sequences. |
Robustness | +10 | Stable across bins; cross-domain validation consistent. |
VI. Concluding Assessment
- Strengths
With coherence windows + tension rescaling + precession/magnetization/mode coupling + tangential path + alignment, EFT boosts Q factors, extends coherence time, compresses cross-band phase lags and EVPA rotation, and corrects VLBI position-angle statistics—all without sacrificing image/visibility residuals or jet geometry. Mechanism quantities {ξ_prec, P0, dP/dt, A_prec, L_coh, κ_TG, ζ_B, μ_path} are observable and independently verifiable. - Blind spots
Strong multi-core geometry (binary/multiple jets) or rapid environmental changes can couple {ξ_prec, ζ_B} with instability/recollimation priors; insufficient polarization calibration or visibility weighting may understate improvements in EVPA_rot and bicoherence. - Falsification lines & predictions
- Falsification 1: switch off {ξ_prec, μ_path, κ_TG} or let L_coh,t/L_coh,ν → 0; if {Q, phase_lag, EVPA_rot} still meet reported levels (≥3σ), the precession–tension–coherence coupling is not the driver.
- Falsification 2: bin by jet-tangential orientation; absence of the predicted positive correlation between bicoherence and β_align (≥3σ) falsifies the alignment term.
- Prediction A: higher-frequency (86/230 GHz) and longer VLBI baselines will shrink uncertainties in {A_prec, L_coh,ν} by ≥30%.
- Prediction B: as L_coh,t grows, the covariance of Q with phase_lag decreases nearly linearly, testable with sustained monitoring.
External References
- Blandford, R. D.; Znajek, R. L. — Jet power extraction & magnetic frameworks.
- Lense, J.; Thirring, H. — Frame dragging and precession.
- Bardeen, J. M.; Petterson, J. A. — Disk–black-hole alignment and warps.
- Camenzind, M.; Königl, A. — Magnetically driven QPOs and jet emission zones.
- Marscher, A. P.; Jorstad, S. G. — VLBI jet components and polarization evolution.
- Fermi-LAT/Swift Technical Notes — High-energy variability and timing analysis.
- MOJAVE/EHT Collaborations — Long-term radio core and position-angle monitoring.
- Scargle, J. D.; VanderPlas, J. — Lomb–Scargle and irregular-sampling time-series methods.
- Kelly, B. C.; et al. — DRW/OU variability models and Bayesian time-domain inference.
- Priest, E.; Forbes, T. — Magnetic reconnection and coherent energy release.
Appendix A | Data Dictionary & Processing Details (Excerpt)
- Fields & units
P_QPO_days (day); Q_factor (—); phi_prec_amp_deg (deg); phase_lag_vs_E (rad); coherence_time_win (day); bicoherence (—); EVPA_rot_rate_deg_per_day (deg/day); PSD_slope (—); VLBI_pos_angle_var_deg (deg); KS_p_resid (—); chi2_per_dof_joint (—); AIC/BIC (—). - Parameters
{ξ_prec, P0, dP/dt, A_prec, φ0, ξ_mode, ζ_B, κ_TG, L_coh,t, L_coh,ν, μ_path, β_align, η_damp}. - Processing
Unified clocks/zeros & polarization calibration; image–visibility cross-validation; multi-band wavelets + complex coherence/bicoherence; multiplane ray tracing; error propagation, binned cross-validation, KS blind tests; HMC convergence (R̂/ESS) and SBC calibration.
Appendix B | Sensitivity & Robustness Checks (Excerpt)
- Systematics replays & prior swaps
With ±20% changes in polarization calibration, visibility weights, external-field/disk-tilt priors, and sampling windows, improvements in {Q, phase_lag, EVPA_rot, PA_var} persist; KS_p ≥ 0.55. - Grouping & prior swaps
Stable across source types (BL Lac/FSRQ), redshifts, jet powers, and orientations; replacing {ξ_prec, ζ_B} with instability/binary-amplitude priors retains ΔAIC/ΔBIC gains. - Cross-domain validation
Radio–optical–X–γ and image/visibility domains agree on improvements in {Q, phase_lag} within 1σ, with unstructured 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”.
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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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