Revisiting the energy distribution and formation rate of CHIME fast radio bursts

Based on the first CHIME/FRB catalogue, three volume-limited samples of fast radio bursts (FRBs) are built, with samples 1, 2, and 3 corresponding to a fluence cut of 5, 3, and 1, respectively. The Lynden-Bell's c$^-$ method was applied to study their energy function and event rate evolution with redshift ($z$). Using the non-parametric Kendall's $\tau$ statistics, it is found that the FRB energy ($E$) strongly evolves with redshift as $E(z)\varpropto(1+z)^{1.24}$ for sample 1, $E(z)\varpropto(1+z)^{0.98}$ for sample 2, and $E(z)\varpropto(1+z)^{1.99}$ for sample 3. After removing the redshift dependence, the local energy distributions of the three samples can be well described by a broken power-law form with a broken energy of $\sim10^{40} \rm erg$. Meanwhile, the redshift distributions of samples 1 and 2 are identical but different from that of sample 3. Interestingly, we find that the event rates of samples 1 and 2 are independent of redshift, and sample 3 decreases as a single power-law form with an index of -2.41. The local event rates of the three samples of CHIME FRBs are found to be consistently close to $\sim 10^4\rm{\,Gpc^{-3}yr^{-1}}$, which is comparable with some previous estimates. In addition, we notice that the event rate of sample 3 FRBs with lower energies not only exceeds the star formation rate at the lower redshifts but also always declines with the increase in redshift. We suggest that the excess of FRB rates compared with the star formation rate at low redshift mainly results from the low-energy FRBs that could originate in the older stellar populations.