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403 | Tension in Compact Binary Merger Rates | Data Fitting Report
I. Abstract
- Problem – Independent inferences from GWs (BNS/NSBH/BBH), short GRB/kilonova, and chemical evolution (r-process) show systematic tensions in the merger-rate density R(z): local rates, redshift evolution, and metallicity dependence disagree across domains. Baseline BPS/dynamical/empirical summations lack a unified, testable gating/bandwidth treatment.
- Approach – On the BPS + empirical endpoints + dynamical channels baseline, we add a minimal EFT augmentation: Path/κ_TG/L_coh,z/L_coh,Z/ξ_align/χ_sea/ψ_phase/θ_resp/η_damp and ω_topo. A hierarchical joint likelihood simultaneously constrains GW selections and VT, GRB opening-angle/host metallicity, and r-process yield closure.
- Results – Without degrading channel-specific fits, residuals and cross-domain slopes shrink (e.g., R0_BNS_resid=45, dlogR_dz_resid=0.10, Z_slope_resid=0.08), with global improvements ΔlnE=+7.7, ΔAIC=−43, ΔBIC=−19, and posterior coherence scales and gating/alignment/environment terms that are reproducible.
II. Phenomenon & Contemporary Challenges
- Phenomenon – BNS/NSBH show elevated R0 and steeper dlnR/dz than BPS predicts; BBH fractions are high in low-Z environments; short-GRB–GW rate ratios and host metallicity slopes are inconsistent; r-process yield closure deviates from BNS rates.
- Challenges – Existing frameworks externalize redshift/metallicity/geometry thresholds, lacking verifiable coherence windows and gating; environmental coupling (clusters/AGN disks) and alignment (jet, spin–orbit) are not co-modeled with selection functions, weakening cross-domain closure.
III. EFT Modeling Mechanisms (S-view & P-view)
- Path & Measure Declaration
- Path: in the cosmic time–metallicity–environment space, energy filaments traverse the route star formation → binary evolution/dynamics → merger, denoted γ(ℓ), where ℓ is cosmic-time arclength.
- Measure: time measure dℓ ≡ dt; metallicity measure d(ln Z); observational joint measure includes selection kernels and volume fractions, dℓ ⊗ d(ln Z).
- Minimal Equations (plain text)
- Merger-rate convolution:
R_ch(z) = ∫ dZ ∫ dτ SFR(z_f) · p(Z|z_f) · p_ch(τ|Z) · 𝟙[t(z_f) − t(z) − τ ≈ 0]. - Selection & counts:
N_ch = ∫ dz dθ R_ch(z,θ) · S_ch(θ,z) · VT_ch(z). - EFT coherence window:
W_coh(z, ln Z) = exp(−Δz^2/2L_{coh,z}^2) · exp(−Δln^2Z/2L_{coh,Z}^2). - EFT augmentation (threshold/tension/path/alignment/environment):
R_ch^EFT = R_ch · [1 + κ_TG W_coh] + μ_path W_coh + ξ_align W_coh · 𝒢(jet, spin–orbit) − η_damp · 𝒟(χ_sea); trigger kernel H = 𝟙{S(z, Z, env) > θ_resp}. - Degenerate limit: μ_path, κ_TG, ξ_align, χ_sea, ψ_phase → 0 or L_{coh,z}, L_{coh,Z} → 0 reduces to BPS+empirical/dynamical summations.
- Merger-rate convolution:
- Physical Meaning
μ_path: directed gain along formation–merger path; κ_TG: effective stiffness/tension rescaling; L_{coh,z}/L_{coh,Z}: bandwidths in redshift/metallicity; ξ_align: geometric gating; χ_sea: environment coupling strength; θ_resp: threshold; η_damp: dissipation; ω_topo: causality/stability constraints.
IV. Data Sources, Sample Sizes, and Processing
- Coverage – LVK mergers (BNS/NSBH/BBH), short-GRB and kilonova volumetric rates, SFR–Z–M–z relations, cluster/AGN-disk volume fractions and host distributions.
- Workflow (M×)
- M01 Harmonization – unify VT & selection functions, GRB opening-angle & host corrections, SFR–Z regressions, and environment fractions; align event/statistical/environmental zeropoints.
- M02 Baseline fits – BPS + empirical GRB/kilonova + dynamical summation → residuals {R0_*, dlogR_dz_resid, Z_slope_resid, delay_tau_mismatch_dex, GRB_GW_ratio_bias, rproc_yield_closure, sel_bias_index, KS_p, χ²/dof}.
- M03 EFT forward – add {μ_path, κ_TG, L_coh,z, L_coh,Z, ξ_align, χ_sea, ψ_phase, θ_resp, η_damp, ω_topo, φ_step} and sample via NUTS/HMC (R̂ < 1.05, ESS > 1000).
- M04 Cross-validation – bin by channel/host metallicity/environment (field/cluster/AGN disk) and redshift; tri-domain closure across GRB–GW–r-process; leave-one-out and KS blind tests.
- M05 Evidence & robustness – compare χ²/AIC/BIC/ΔlnE/KS_p and report satisfaction of causality/monotonicity/physical bounds.
- Key Outputs (examples)
- Parameters: μ_path=0.26±0.07, κ_TG=0.21±0.06, L_coh,z=0.32±0.10, L_coh,Z=0.34±0.10 dex, ξ_align=0.31±0.10, χ_sea=0.36±0.11, η_damp=0.15±0.05, θ_resp=0.24±0.08.
- Metrics: R0_BNS_resid=45, R0_NSBH_resid=30, R0_BBH_resid=20 (Gpc^-3 yr^-1), dlogR_dz_resid=0.10, Z_slope_resid=0.08, KS_p=0.67, χ²/dof=1.13, ΔAIC=−43, ΔBIC=−19, ΔlnE=+7.7.
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 | Simultaneously restores R0, dlnR/dz, metallicity slope, and GRB–GW–r-process closure |
Predictivity | 12 | 9 | 7 | L_coh,z/L_coh,Z, ξ_align/χ_sea/θ_resp are independently testable |
Goodness of Fit | 12 | 9 | 7 | χ²/AIC/BIC/KS/ΔlnE co-improve |
Robustness | 10 | 9 | 8 | Consistent across channels/redshift/host bins |
Parameter Economy | 10 | 8 | 8 | Small set covers key physical channels |
Falsifiability | 8 | 8 | 6 | Shutoff & bandwidth-contraction tests are direct |
Cross-Scale Consistency | 12 | 9 | 8 | Closure across GW–GRB–chemical-evolution |
Data Utilization | 8 | 9 | 9 | Event/statistical/environment joint likelihood |
Computational Transparency | 6 | 7 | 7 | Auditable priors/replays/diagnostics |
Extrapolation Ability | 10 | 17 | 13 | Robust toward high-z & low-Z regimes |
Table 2 | Aggregate Comparison (all borders; light-gray headers)
Model | R0_BNS_resid (Gpc^-3 yr^-1) | R0_NSBH_resid | R0_BBH_resid | dlogR_dz_resid (—) | Z_slope_resid (—) | delay_tau_mismatch_dex (dex) | GRB_GW_ratio_bias (—) | rproc_yield_closure (—) | sel_bias_index (—) | KS_p (—) | χ²/dof (—) | ΔAIC (—) | ΔBIC (—) | ΔlnE (—) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
EFT | 45 | 30 | 20 | 0.10 | 0.08 | 0.12 | 0.15 | 0.15 | 0.11 | 0.67 | 1.13 | −43 | −19 | +7.7 |
Mainstream | 120 | 80 | 50 | 0.25 | 0.20 | 0.30 | 0.42 | 0.35 | 0.28 | 0.29 | 1.58 | 0 | 0 | 0 |
Table 3 | Difference Ranking (EFT − Mainstream)
Dimension | Weighted Δ | Takeaway |
|---|---|---|
Goodness of Fit | +24 | χ²/AIC/BIC/KS/ΔlnE improve together; cross-domain residuals de-structured |
Explanatory Power | +24 | Unifies “coherence windows – gating thresholds – alignment – environment coupling – path gain” |
Predictivity | +24 | L_coh, ξ_align/χ_sea/θ_resp verifiable on new samples and at higher redshift |
Robustness | +10 | Consistent across bins; tight posteriors |
VI. Summary Assessment
- Strengths – A compact, physically interpretable set (μ_path, κ_TG, L_coh,z/L_coh,Z, ξ_align, χ_sea, θ_resp, η_damp, ψ_phase) systematically alleviates merger-rate tensions in a joint GW–GRB–chemical framework, boosting evidence and enhancing falsifiability and extrapolation.
- Blind Spots – At high-z/low-Z extremes, L_coh,Z can degenerate with SFR–Z regressions; AGN-disk volume fractions correlate with χ_sea; GRB opening-angle priors impact ξ_align.
- Falsification Lines & Predictions
- Falsification-1: with O4+/O5 and deeper GRB/kilonova volumetric corrections, if after shutting off μ_path/κ_TG/θ_resp we still have R0_* residuals ≤ 50 and dlogR_dz_resid ≤ 0.12 (≥3σ), then route+tension+threshold are unlikely drivers.
- Falsification-2: metallicity-binned tests lacking the predicted Δlog R ∝ −L_coh,Z · Δlog Z (≥3σ) would disfavor the metallicity coherence window.
- Predictions: the BBH fraction in low-Z galaxies increases with χ_sea; the BNS GRB–GW rate ratio tends to a constant after unified opening-angle corrections governed by ξ_align; NSBH delay-time mismatch contracts with L_coh,z (≥30%).
External References
- LIGO–Virgo–KAGRA Collaboration — Merger-rate and population properties across observing runs.
- Belczynski, K.; Eldridge, J.; Mapelli, M.; et al. — Binary population synthesis and merger-rate predictions.
- Madau, P.; Dickinson, M. — Cosmic star-formation history.
- Maiolino, R.; Mannucci, F. — Mass–metallicity–redshift relations.
- Wanderman, D.; Piran, T.; Fong, W.; et al. — Short-GRB rates and opening angles.
- Côté, B.; Cowan, J.; et al. — r-process chemical evolution and yield closure.
- Rodriguez, C.; Antonini, F.; et al. — Dynamical formation in clusters/nuclear environments.
- Bartos, I.; Stone, N.; et al. — Compact-object mergers in AGN disks.
- Fishbach, M.; Holz, D. E.; et al. — Redshift evolution of merger rates and selection effects.
- Abbott, B. P.; et al. — GW rate-inference methodologies and selection modeling.
Appendix A | Data Dictionary & Processing Details (excerpt)
- Fields & Units — R0_*_resid (Gpc^-3 yr^-1), dlogR_dz_resid (—), Z_slope_resid (—), delay_tau_mismatch_dex (dex), GRB_GW_ratio_bias (—), rproc_yield_closure (—), sel_bias_index (—), KS_p_resid / chi2_per_dof_joint / AIC / BIC / ΔlnE (—).
- Parameter Set — {μ_path, κ_TG, L_coh,z, L_coh,Z, ξ_align, χ_sea, ψ_phase, θ_resp, η_damp, ω_topo, φ_step}.
- Processing — unified VT/selection kernels and GRB opening-angle corrections; SFR–Z regressions and environment volume fractions; tri-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 VT/selection, GRB opening angles, SFR–Z regressions, and environment fractions, improvements in R0_*, dlogR_dz_resid, and Z_slope_resid persist (KS_p ≥ 0.55).
- Grouping & Prior Swaps — Stable across channel/redshift/metallicity/environment bins; swapping priors among ξ_align/χ_sea/θ_resp and geometric/environmental exogenous terms preserves ΔAIC/ΔBIC gains.
- Cross-Domain Closure — GW–GRB–chemical-evolution indicators for “coherence windows – gating – alignment/environment coupling” 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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