Lensed fast radio bursts as a probe of a time-varying gravitational potential induced by wave dark matter

Ultralight bosonic wave dark matter (DM) is preponderantly contesting the conventional cold DM paradigm in predicting diverse and rich phenomena on small scales. For a DM halo made of ultralight bosons, the wave interference naturally induces slow de Broglie timescale fluctuations of the gravitational potential. In this Letter, we first derive an estimation for the effect of a time-varying gravitational potential on photon propagation. Our numerical simulations suggest that the time-varying potential of a ${10}^{11}{M}_{\ensuremath{\bigodot}}$ halo composed of ${10}^{\ensuremath{-}22}\text{ }\text{ }\mathrm{eV}$ bosons would stretch or compress a time series signal by a factor of ${10}^{\ensuremath{-}10}$. Here, we propose that, due to the precise measurements of their arrival times, lensed repeating fast radio bursts (FRBs) have the potential to effectively validate temporal variations in gravitational potential by monitoring their images over a period of approximately $\mathcal{O}(1)\text{ }\text{ }\mathrm{years}$. With rapidly growing FRB observations, this method would serve as a promising method to directly probe the wave nature of galactic DM halos.