2026ApJ...998...18K

We present a mathematical analysis of propagation-induced distortions in the spectro-temporal properties of fast radio bursts (FRBs). Within the triggered relativistic dynamical model, we derive a centroid-based formulation of the sub-burst slope law, which describes an inverse relation between the sub-burst frequency drift rate and the intrinsic temporal width t_w. We extend this framework to include two frequency-dependent propagation effects: (i) multipath scattering, characterized by a pulse-broadening timescale τ_sc ∝ ν^–4, and (ii) residual dispersion, parameterized by ΔDM ∝ ν^–2. Our analysis shows that scattered bursts exhibit the same qualitative inverse slope-duration relation but differ quantitatively through a larger scaling coefficient when τ_sc > t_w. Residual DM errors act asymmetrically: under-dedispersion flattens the sub-burst slope, whereas over-dedispersion produces a nonlinear increase and eventual sign change. When both effects are present, scattering counterbalances the steepening induced by over-dedispersion and augments the flattening from under-dedispersion, yielding characteristically distorted curves. We repeat analysis for ultrashort duration bursts (ultra-FRBs) with t_w = 50 μs at 1 GHz and find them to be far more sensitive to propagation errors. Deviations become measurable for $\left|{\rm{\Delta }}{\rm{DM}}\right|\sim 0.05$ pc cm^–3 and for τ_sc ∼ 0.1 ms at 1 GHz, levels that have negligible impact on the standard-width sub-bursts. Our analysis provides practical diagnostics to disentangle propagation effects from observed spectro-temporal properties, thereby recovering intrinsic correlations among FRB parameters.