The formation rate and luminosity function of fast radio bursts

Fast radio bursts (FRBs) are millisecond-duration flashes with unknown origins. Its formation rate is crucial for unveiling physical origins. However, the luminosity and formation rate are degenerated when directly fitting the redshift distribution of FRBs. In contrast to previous forward-fitting methods, we use the Lynden-Bell's $c^{-}$ method to derive luminosity function and formation rate of FRBs without any assumptions. Using the non-repeating FRBs from the first CHIME/FRB catalog, we find a relatively strong luminosity evolution, and luminosity function can be fitted by a broken power-law model with a break at $1.33\times10^{41}\ \mathrm{erg}\ \mathrm{s}^{-1}$. The formation rate declines rapidly as $(1+z)^{-4.9\pm0.3}$ with a local rate $1.13\times10^4\ \mathrm{Gpc}^{-3}\ \mathrm{yr}^{-1}$. This monotonic decrease is similar to the rate of short gamma-ray bursts. After comparing it with star formation rate and stellar mass density, we conclude that the old populations including neutron stars and black holes, are closely related to the origins of FRBs. Monte Carlo simulations are used to test our results. The distributions of mock sample are consistent with the observational data.