34-EFT.WP.Astro.Acceleration v1.0 | Chapter 10 GRB: Lightcurves, Spectra & Timing

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Chapter 10 GRB: Lightcurves, Spectra & Timing

I. Abstract & Scope
This chapter treats gamma-ray bursts (GRBs) in both prompt and afterglow phases, providing a unified multi-band lightcurve–spectrum–timing workflow M62-*. Time-variable drivers from reconnection and shear, {A_rec(E,t), A_shear(E,t)}, are mapped to N(E,t) and Phi(E,t). Spectral-peak evolution, hardness–intensity trends, inter-band lags, and polarization serve as channel criteria. Any time-of-arrival (ToA) quantity records both forms with explicit path gamma(ell) and measure d ell.

II. Dependencies & References

• Unified symbols & units: Chapter 2 Tab. 2-1 and P12-*.
• Kinematics & channels: Chapter 3 S20-; reconnection & shear: Chapter 4 S30-, Chapter 5 S40-; comparators: Chapter 6 S45-.
• Spectrum formation & transport: Chapter 7 S50-, Chapter 8 S52-.
• Timebase and redshift/path corrections: Metrology.TimeBase v1.0, Propagation.PathRedshift v1.0, Packets.Light v1.0.

III. Normative Anchors (added in this chapter, M62-*)

• M62-0 (Timebase Alignment): t_src = ( t_obs - t0 - T_arr ) / ( 1 + z ), with ToA recorded in both forms:
  T_arr = ( 1 / c_ref ) * ( ∫_{gamma(ell)} n_eff d ell ) and T_arr = ( ∫_{gamma(ell)} ( n_eff / c_ref ) d ell ), along with delta_form.
• M62-1 (Lightcurve Segmentation): segment into {prompt pulses}{plateau}{normal decay}{jet break}, define windows {Δt_k}, and assume locally constant parameters within each window.
• M62-2 (Multi-Band Data Packaging): standardized cross-instrument dataset cards for Phi(E_i, t_j), covariance, systematics, units, and energy windows.
• M62-3 (Channel-Driver Packaging): A_rec(E,t) = A_rec^0(t) * f_rec(E;θ), A_shear(E,t) = A_sh^0(t) * f_sh(E;θ); A_acc(E,t) = A_rec + A_shear.
• M62-4 (Joint Spectro-Temporal Solution): within each Δt_k, compute N(E,t) and Phi(E,t) using Chapter 7 S50-* semi-closed solutions, harmonized with Chapter 8 {D(E), A_loss(E), tau_esc(E)}.
• M62-5 (Lag & Hardness Diagnostics): compute inter-band lag tau_lag(E_1,E_2) and hardness H(t)=F_{high}(t)/F_{low}(t); output {tau_lag(t), dH/dt}.
• M62-6 (Polarization Criteria): fit Pi(E,t) and polarization-angle (PA) evolution to discriminate reconnection (rapid PA jumps) vs shear (smooth drift).
• M62-7 (Joint Likelihood): L = L_lc + L_spec + L_lag + L_polar + L_ToA; return {posterior, evidence} and weights {w_rec(t,E), w_shear(t,E)}.
• M62-8 (Dominant-Band Masks): extract eta_dom(E,t) masks per Chapter 6 and feed back to spectrum/transport steps.
• M62-9 (Deliverables): {E_pk(t), alpha_spec(t), tau_lag(E), Pi(t), masks, delta_form} plus reproducible configs.

IV. Body Structure

I. Background & Problem Statement
Prompt fast pulses and segmented afterglow decays require mapping time-variable {A_rec, A_shear} into N(E,t) on a unified source-frame timeline. A “segmented windows + semi-closed spectral solution” closure ensures numerical stability and enables joint multi-band/multi-instrument fitting. Lags, hardness–intensity behavior, and polarization provide positive/negative channel criteria.

II. Key Equations & Derivations (S-series)

• S62-1 (Quasi-Steady Spectral Evolution):
  ∂/∂t N(E,t) + ∂/∂E ( b(E,t) * N(E,t) ) + N/tau_esc(E,t) + N/tau_loss(E,t) = Q(E,t),
  with b(E,t) = E * ( A_acc(E,t) - A_loss(E,t) ).
• S62-2 (Flux Mapping): Phi(E,t) = C_geom(t) * C_prop(E,t) * N(E,t).
• S62-3 (Spectral Peak & Break): E_pk(t) ≈ E_br(t), defined by tau_acc(E_pk,t) = min{ tau_loss(E,t), tau_esc(E,t) }.
• S62-4 (Inter-Band Lag): tau_lag(E_1,E_2) = argmax_τ xcorr( F_{E_1}(t), F_{E_2}(t+τ) ).
• S62-5 (Hardness): H(t) = F_{high}(t) / F_{low}(t); sign(dH/dt) correlates with channel dominance.
• S62-6 (Two ToA Forms): record both T_arr forms with explicit gamma(ell) and d ell, and archive delta_form in any time-related metric.

III. Methods & Flows (M-series)

• M62-1 (Time Alignment & Segmentation): align to t_src via t0, redshift z, and T_arr; adaptively segment {Δt_k} by structure (rise/peak/decay).
• M62-2 (Joint Spectral–Temporal Fit): in each Δt_k, drive S62-1 with the separable A_acc(E,t) parameterization, compute {N, Phi}, and fit multi-band data.
• M62-3 (Lag & Polarization Tasks): compute {tau_lag, H(t), Pi(t)} and correlate with {w_rec, w_shear} over time.
• M62-4 (ToA & Path Corrections): apply both ToA forms to Phi(E,t) and timing metrics; record delta_form and include in L_ToA.
• M62-5 (Evidence & Disentangling): combine with Chapter 6 dominance to report evidence ratios for {reconnection only, shear only, mixed} and emit energy–time masks.

IV. Cross-References within/beyond this Volume

• Channel drivers & kinematics: Chapter 3 S20-, Chapter 4 S30-, Chapter 5 S40-*.
• Comparators & dominance: Chapter 6 S45-*.
• Spectrum & transport: Chapter 7 S50-, Chapter 8 S52-.
• Timebase & path/redshift corrections: Metrology.TimeBase v1.0, Propagation.PathRedshift v1.0.

V. Validation, Criteria & Counterexamples

1. Positive criteria:
   • Prompt: rapid co-variation of E_pk(t) and alpha_spec(t) consistent with A_rec dominance.
   • Afterglow: smooth monotonic H(t) evolution and slow Pi(t) drift consistent with A_shear dominance.
   • tau_lag(E_1,E_2) sign and amplitude track energy bands and {w_rec,w_shear}.
2. Negative criteria:
   • Setting A_rec → 0 or A_shear → 0 does not reduce joint evidence.
   • Evolution of E_pk(t) and alpha_spec(t) inconsistent with the ordering of {tau_acc, tau_loss, tau_esc} intersections.
   • Two ToA forms yield statistically equivalent timing with no evidence gap, indicating path corrections are nonessential.
3. Contrasts:
   • With {D, A_loss, tau_esc} fixed, compare {reconnection only, shear only, mixed} on {E_pk, alpha_spec, tau_lag, Pi}.
   • Cross-validate and refit per Δt_k to assess robustness.

VI. Summary & Handoff
This chapter establishes M62-* for unified GRB lightcurve–spectrum–timing modeling and channel criteria, mapping time-variable {A_rec, A_shear} to {N(E,t), Phi(E,t)} with multi-signal diagnostics. Chapter 11 addresses FRBs with dispersion, polarization, and delays for path corrections and joint constraints.

V. Figures & Tables (this chapter)

• Fig. 10-1 Multi-band segmentation & timebase schematic (t_obs → t_src; dual-form T_arr alignment).
• Fig. 10-2 Co-evolution of E_pk(t), alpha_spec(t), and {w_rec,w_shear}.
• Tab. 10-1 Local Symbol Table

• Tab. 10-2 GRB dataset-card fields (Instrument / Band / Unit / Calibration / Systematics / Covariance / See).
• Tab. 10-3 Model parameters & default priors (A_rec^0(t_k), A_sh^0(t_k), shape params θ, E0, cutoff E_max(t_k), etc.).
• Tab. 10-4 Diagnostic & criteria checklist (tau_lag, dH/dt, Pi, evidence ratios, dominance masks).