Impact of propagation effects on the spectro-temporal properties of Fast Radio Bursts

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 is an inverse relation between frequency-drift rate and temporal width of sub-bursts. We extend our analysis to include two frequency-dependent propagation effects: (i) multipath scattering, characterized by a pulse-broadening timescale $τ_\mathrm{sc} \propto ν^{-4}$, and (ii) residual dispersion, parameterized by $Δ\mathrm{DM}\propto ν^{-2}$. Our analysis shows that scattering preserves the inverse relation between sub-burst slope and duration, but increases the scaling coefficient when $τ_\mathrm{sc}$ exceeds the intrinsic width ($t_\mathrm{w}$) of sub-bursts. Residual DM errors act asymmetrically: under-dedispersion flattens the sub-burst slope, whereas over-dedispersion causes a non-linear increase and eventually a change of sign. When both effects are present, scattering counterbalances the steepening induced by over-dedispersion and augments the flattening produced by under-dedispersion, yielding characteristically distorted curves. We repeat measurements for ultra-short duration bursts (ultra-FRBs) with $t_\mathrm{w} = 50\ μ\mathrm{s}$ at 1 GHz and found them to be far more sensitive to propagation errors. Deviations become measurable for $\left | Δ\mathrm{DM} \right |\sim0.05$ pc cm$^{-3}$ and for $τ_\mathrm{sc} \sim0.1$ 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 the observed spectro-temporal properties of FRBs, thereby recovering true correlations among their intrinsic parameters.