Variabilities of Gamma-ray Bursts from the Dynamics of Fallback Material after Tidal Disruption

The gamma-ray burst (GRB) GRB 211211A and GRB 060614, believed to originate from the merger of compact objects, exhibit similarities to the jetted tidal disruption event (TDE) Sw J1644+57, by showing violent variabilities in the light-curve during the decay phase. Previous studies suggest that such fluctuations in TDE may arise from the fallback of tidal disrupted debris. In this paper, we introduce the fluctuations of the mass distribution ${\rm d}M/{\rm d}E$ for the debris ejected during the tidal disruption (with energy $E$) and study their impact on jet power. Turbulence induced by tidal force and the self-gravity of the debris may imprint variabilities in ${\rm d}M/{\rm d}E$ during fallback. We model these fluctuations with a power density spectrum $\propto f_{\rm E}^{\beta}$, where $f_{\rm E} = 1/E$ and $\beta$ is the power-law index. We find that the resulting light curve can preserve the fluctuation characteristics from ${\rm d}M/{\rm d}E$. In addition, the observed fluctuations in the light-curves can be reproduced for a given suitable $\beta$. Based on the observations, we find that the value of $\beta$ should be around $-1$.