Cosmological Evolution of Fast Radio Bursts and The Star Formation Rate

We investigate the cosmological evolution of the luminosity and redshift of FRBs. As is the case for all extragalactic sources, we are dealing with data that are truncated by observational selection effects, the most important being the flux limit, which introduces the so-called Eddington-Malmquist bias. In addition, for FRBs, there is a significant uncertainty in the redshifts obtained from the observed dispersion measures (DMs). To correct for the truncation we use the non-parametric, non-binning Efron-Petrosian and Lynden-Bell methods, which give unbiased distributions of luminosities and redshifts and their cosmological evolution. To quantify the redshift uncertainty, we use a data set which accounts for uncertainties of contribution to the DM of the host galaxy, and DM uncertainties due to fluctuation in the intergalactic medium. This data, in addition to a mean redshift, gives the one-sigma errors. We construct three samples with lower, mean, and upper redshifts and apply the above methods to each. For the three samples, we find similar (1) $\sim 3\sigma$ evidence for luminosity evolution, (2) a luminosity function that can be fit by a simple broken power law, and (3) a comoving density formation rate that decreases rapidly with redshift unlike the cosmic star formation rate (SFR). This rate is similar to that of short gamma-ray bursts, which are believed to result from compact star mergers with a formation rate delayed relative to the SFR. This may further support the hypothesis that magnetars are the progenitors of FRBs.