What does FRB light-curve variability tell us about the emission mechanism?

A few fast radio bursts' (FRBs) light curves have exhibited large intrinsic modulations of their flux on extremely short ($t_{\rm r}\sim 10\, \mu$s) time-scales, compared to pulse durations (t_FRB ~ 1 ms). Light-curve variability time-scales, the small ratio of rise time of the flux to pulse duration, and the spectro-temporal correlations in the data constrain the compactness of the source and the mechanism responsible for the powerful radio emission. The constraints are strongest when radiation is produced far (≳10¹⁰ cm) from the compact object. We describe different physical set-ups that can account for the observed t_r/t_FRB ≪ 1 despite having large emission radii. The result is either a significant reduction in the radio production efficiency or distinct light-curve features that could be searched for in observed data. For the same class of models, we also show that due to high-latitude emission, if a flux f₁(ν₁) is observed at t₁ then at a lower frequency ν₂ < ν₁ the flux should be at least (ν₂/ν₁)²f₁ at a slightly later time (t₂ = t₁ν₁/ν₂) independent of the duration and spectrum of the emission in the comoving frame. These features can be tested, once light-curve modulations due to scintillation are accounted for. We provide the time-scales and coherence bandwidths of the latter for a range of possibilities regarding the physical screens and the scintillation regime. Finally, if future highly resolved FRB light curves are shown to have intrinsic variability extending down to ${\sim}\mu$s time-scales, this will provide strong evidence in favour of magnetospheric models.