Constraints on the abundance of primordial black holes with different mass distributions from lensing of fast radio bursts

Primordial black holes (PBHs) has been considered to form a part of dark matter for a long time but the possibility has been poorly constrained over a wide mass range, including the stellar mass range ($1-100~M_{\odot}$). However, due to the discovery of merger events of black hole binaries by LIGO-Virgo gravitational wave observatories, the interest for PBHs in the stellar mass window has been aroused again. Fast radio bursts (FRBs) are bright radio transients with millisecond duration and very high all-sky occurrence rate. Lensing effect of these bursts has been proposed as one of the optimal probes for constraining the abundance of PBHs in the stellar mass range. In this paper, we first investigate constraints on the abundance of PBHs from the latest $593$ FRB observations for both the monochromatic mass distribution and three other popular extended mass distributions related to different formation mechanisms of PBHs. It is found that constraints from currently public FRB observations are relatively weaker than those from existing gravitational wave detections. Furthermore, we forecast constraining power of future FRB observations on the abundance of PBHs with different mass distributions of PBHs and different redshift distributions of FRBs taken into account. Finally, We find that constraints of parameter space on extended mass distributions from $\sim10^5$ FRBs with $\overline{\Delta t}\leq1 ~\rm ms$ would be comparable with what can be constrained from gravitational wave events. It is foreseen that upcoming complementary multi-messenger observations will yield considerable constraints on the possibilities of PBHs in this intriguing mass window.