34-EFT.WP.Astro.Acceleration v1.0 | Chapter 11 FRB: Dispersion, Polarization & Delays

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Chapter 11 FRB: Dispersion, Polarization & Delays

I. Abstract & Scope
This chapter treats fast radio bursts (FRBs) in dynamic spectrum, polarization, and time-of-arrival (ToA) metrics, establishing a unified workflow M64-* for dispersion measure DM, rotation measure RM, polarization fraction Pi, temporal broadening tau_scat(ν), and path corrections. Both ToA forms are recorded in parallel with explicit path gamma(ell) and measure d ell. All symbols use English notation with backticks; SI units; composite expressions are parenthesized.

II. Dependencies & References

• Unified symbols & units: Chapter 2 Tab. 2-1 and P12-*.
• Kinematics & channels (for source-region constraints/comparators): Chapter 3 S20-, Chapter 4 S30-, Chapter 5 S40-, Chapter 6 S45-.
• Spectrum formation & transport (for joint fits): Chapter 7 S50-, Chapter 8 S52-.
• Timebase, path & redshift corrections: Metrology.TimeBase v1.0, Metrology.PathCorrection v1.0, Propagation.PathRedshift v1.0, Packets.Light v1.0.

III. Normative Anchors (added in this chapter, M64-*)

• M64-0 (Timebase & Dedispersion Alignment): t_src = ( t_obs - t0 - T_arr(ν) ) / ( 1 + z ), with ToA recorded in both forms and delta_form flagged:
  T_arr = ( 1 / c_ref ) * ( ∫_{gamma(ell)} n_eff(ν, r) d ell ) and T_arr = ( ∫_{gamma(ell)} ( n_eff(ν, r) / c_ref ) d ell ).
• M64-1 (Dynamic-Spectrum Packaging): dataset cards for I(ν_i, t_j) and Stokes {Q,U,V}, with standardized bandwidth, time resolution, channelization, and calibration conventions.
• M64-2 (Coarse–Fine Grid for DM/RM): two-stage grid search over {DM, RM} minimizing a joint objective L_disp + L_PA + L_band; output fine-grid posteriors and evidence.
• M64-3 (Polarization & Depolarization Modeling): per window fit Pi(ν,t), PA(ν,t) with depolarization kernel D_dep(ν; τ_scat, Δν_chan, Δt).
• M64-4 (Broadening & Multi-Screen Propagation): separable form tau_scat(ν) = tau_scat0 * ( ν / ν0 )^{-alpha_scat}; multi-screen path with {w_k, tau_scat,k, DM_k, RM_k} combined coherently.
• M64-5 (Path Decomposition): DM = DM_MW + DM_IGM + DM_host/(1+z) + DM_loc; RM = RM_MW + RM_IGM + RM_host/(1+z)^2 + RM_loc, with interval priors.
• M64-6 (Joint Likelihood): L = L_dynspec + L_PA + L_Pi + L_scat + L_ToA; return {posterior, evidence} and delta_form.
• M64-7 (Deliverables): {DM, RM, tau_scat(ν), Pi(ν,t), PA(ν,t), residuals(ν,t), delta_form} plus reproducible configuration.

IV. Body Structure

I. Background & Problem Statement
FRB impulsiveness and strong dispersion require dedispersion and path corrections on a unified timebase. Polarization’s λ^2 rotation and multipath broadening jointly shape the dynamic spectrum. This chapter’s “dynamic spectrum + polarization + dual-form ToA” modeling separates medium-integrated quantities from local propagation kernels and harmonizes with upstream channels and downstream transport.

II. Key Equations & Derivations (S-series)

• S64-1 (Two ToA Forms with Frequency Dependence):
  T_arr(ν) = ( 1 / c_ref ) * ( ∫_{gamma(ell)} n_eff(ν, r) d ell ) = ( ∫_{gamma(ell)} ( n_eff(ν, r) / c_ref ) d ell ), with delta_form recorded.
• S64-2 (General Dispersion Delay): for ΔT_disp(ν1, ν2) = T_arr(ν1) − T_arr(ν2) and weak-dispersion approximation,
  ΔT_disp(ν) ≈ K_DM * DM_eff * ν^{-2} + O(ν^{-4}), where DM_eff is a path-weighted effective dispersion and K_DM is a units-consistent constant.
• S64-3 (DM Definition & Units): DM = ∫_{gamma(ell)} n_e(r) d ell, SI primary unit m^-2 (astrophysical convenience unit pc·cm^-3 listed alongside).
• S64-4 (Faraday Rotation & RM): PA(λ) = PA_0 + RM * λ^2, with RM = K_RM * ∫_{gamma(ell)} n_e(r) * B_parallel(r) d ell and PA in radians.
• S64-5 (Multipath Broadening): tau_scat(ν) = tau_scat0 * ( ν / ν0 )^{-alpha_scat}, alpha_scat > 0; multi-screen combination by weighted sums.
• S64-6 (Observation Kernel Convolution):
  I_obs(ν,t) = ( S_int(ν,t) ⊗ K_scat(ν,t) ) ⊗ G_DM(ν,t) ⊗ R_inst(ν,t), with depolarization factor D_dep ∈ (0,1] acting on {Q,U,V}.
• S64-7 (Path Decomposition & Redshift Factors): DM_host and RM_host scale by (1+z)^{-1} and (1+z)^{-2} respectively; sum components {MW, IGM, host, local}.

III. Methods & Flows (M-series)

• M64-1 (Time Alignment & Dedispersion): align t_src via M64-0; maximize pulse sharpness over coarse {DM_i} and refine to the posterior peak.
• M64-2 (RM Estimation): after DM refinement, estimate RM via synthesis or direct frequency-domain fitting of {Q,U}, yielding {RM, PA_0} and depolarization-kernel parameters.
• M64-3 (Broadening & Multi-Screen): fit tau_scat(ν) amplitude and slope alpha_scat; if residuals show chromatic/asymmetric tails, enable multi-screen {w_k, tau_scat,k}.
• M64-4 (Joint Dynamic-Spectrum Fit): minimize L_dynspec + L_Pi + L_PA + L_scat + L_ToA; output {DM, RM, tau_scat, Pi, PA} with residual maps.
• M64-5 (Uncertainty Propagation): sample {DM, RM, tau_scat, D_dep, delta_form} and propagate to T_arr, dedispersed pulse shape, and ToA residuals.
• M64-6 (Path Decomposition with Priors): decompose into {DM_k, RM_k} (MW/IGM/host/local) under physical priors; report component CIs and evidence.
• M64-7 (Reporting & Delivery): deliver {DM, RM, tau_scat(ν), Pi(ν,t), PA(ν,t), delta_form, masks} with a reproducibility pack.

IV. Cross-References within/beyond this Volume

• Timebase & path corrections: Metrology.TimeBase v1.0, Metrology.PathCorrection v1.0, Propagation.PathRedshift v1.0, Packets.Light v1.0.
• Spectrum & transport: Chapter 7 S50-; Chapter 8 S52- (when jointed with HE/multi-messenger).
• Channels & comparators (if inferring source-region physics): Chapters 3–6 (A_acc and dominance factors).

V. Validation, Criteria & Counterexamples

1. Positive criteria:
   • After dedispersion, pulse kurtosis maximized and ΔT_disp(ν) follows ν^{-2} linearly with the inferred DM.
   • PA(λ) linear in λ^2 and consistent with RM; sub-band residuals show no systematic offsets.
   • Pi(ν,t) frequency/time dependences explained by D_dep(ν; τ_scat, Δν, Δt); tau_scat(ν) matches tail morphology.
2. Negative criteria:
   • Evidence does not drop when omitting RM or tau_scat.
   • A single DM cannot explain multi-subband ToA residuals simultaneously.
   • Dual ToA forms yield nearly identical residuals and no evidence gap (delta_form uninformative), indicating path-correction nonessentiality.
3. Contrasts:
   • Compare {single-screen, two-screen, multi-screen} broadening; {constant RM, time-varying RM}; {global DM, segmented DM}.
   • Fit both ToA forms in parallel and compare evidence gaps via delta_form.

VI. Summary & Handoff
This chapter completes M64-* for FRB dispersion, polarization, and ToA modeling, delivering {DM, RM, tau_scat, Pi, PA} with uncertainties and unified dual-form ToA recording and path decomposition. Chapter 12 moves to simulations and benchmarks (SimStack) for reproducible experiments and regression tests.

V. Figures & Tables (this chapter)

• Tab. 11-1 Local Symbol Table (this chapter)

• Tab. 11-2 Model Parameters & Default Priors (examples)

• Tab. 11-3 Tasks & Deliverables