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

We present a mathematical analysis of the spectro-temporal properties of Fast Radio Bursts (FRBs), focusing on the distortions introduced by propagation effects such as scattering and inaccurate de-dispersion. By examining the impact of different scattering timescales and residual dispersion measures (DMs), both independently and in combination, we identify systematic trends in the sub-burst slope law as defined within the framework of the Triggered Relativistic Dynamical Model (TRDM). These effects primarily alter the measurements of the sub-burst slope and duration, thereby also modifying their correlations with other properties, such as central frequency and bandwidth. Our results show that scatter-induced temporal broadening affects duration more than slope, with weak to moderate scattering subtly modifying the sub-burst slope law and strong scattering causing significant deviations. Residual dispersion preferentially modifies the slope, further changing the trends predicted by the sub-burst slope law. Ultra-short bursts (or ultra-FRBs) emerge as particularly susceptible to these effects even at relatively high frequencies, underscoring the need for precise treatment of scattering and accurate dedispersion before performing analyses. Our findings emphasize the necessity for higher frequency observations (especially for ultra-FRBs) to improve the DM estimates as well as the measurements of spectro-temporal properties.