The luminosity-volume test for cosmological fast radio bursts

We perform the luminosity--volume test, also known as $\langle V/V_{MAX}\rangle$, to Fast Radio Bursts (FRBs). We compare the 23 FRBs, recently discovered by ASKAP, with 20 of the FRBs found by Parkes. These samples have different flux limits and correspond to different explored volumes. We assume that their dispersion measure indicates their redshift and apply the appropriate cosmological corrections to the spectrum and rate in order to compute the $\langle V/V_{MAX}\rangle$ for the ASKAP and Parkes samples. For a radio spectrum of FRBs $F_\nu \propto \nu^{-1.6}$, we find $\langle V/V_{MAX}\rangle=0.66\pm 0.05$ for the ASKAP sample, that includes FRBs up to $z=0.7$, and 0.52$\pm 0.04$ for Parkes, that extends up to $z=2.1$. The ASKAP value suggests that the population of FRB progenitors evolves faster than the star formation rate, while the Parkes value is consistent with it. Even a delayed (as a power law or gaussian) star formation rate cannot reproduce the $\langle V/V_{MAX}\rangle$ of both samples. If FRBs do not evolve in luminosity, the $\langle V/V_{MAX}\rangle$ values of ASKAP and Parkes sample are consistent with a population of progenitors whose density strongly evolves with redshift as $\sim z^{2.8}$ up to $z \sim 0.7$. We discuss possible scenarios accounting for our results.