Exploring primordial curvature perturbation on small scales with the lensing effect of fast radio bursts

Cosmological observations, e.g., cosmic microwave background, have precisely measured the spectrum of primordial curvature perturbation on larger scales, but smaller scales are still poorly constrained. Since primordial black holes (PBHs) could form in the very early Universe through the gravitational collapse of primordial density perturbations, constrains on the PBH could encodes much information on primordial fluctuations. In this work, we first derive a simple formula for lensing effect to apply PBH constraints with the monochromatic mass distribution to an extended mass distribution. Then, we investigate the latest fast radio burst observations with this relationship to constrain two kinds of primordial curvature perturbation models on the small scales. It suggests that, from the null search result of lensed fast radio burst in currently available observations, the amplitude of primordial curvature perturbation should be less than $8\times 10^{-2}$ at the scale region of $10^5-10^6~\rm Mpc^{-1}$. This corresponds to an interesting mass range relating to binary black holes detected by LIGO-Virgo-KAGRA and future Einstein Telescope or Cosmic Explorer.