Implications from ASKAP Fast Radio Burst Statistics

Although there has recently been tremendous progress in studies of fast radio bursts (FRBs), the nature of their progenitors remains a mystery. We study the fluence and dispersion measure (DM) distributions of the ASKAP sample to better understand their energetics and statistics. We first consider a simplified model of a power-law volumetric rate per unit isotropic energy dN/dE ~ E^{-gamma} with a maximum energy E_max in a uniform Euclidean Universe. This provides analytic insights for what can be learnt from these distributions. We find that the observed cumulative DM distribution scales as N(>DM) ~ DM^{5-2*gamma} (for gamma > 1) until a maximum value DM_max above which bursts near E_max fall below the fluence threshold of a given telescope. Comparing this model with the observed fluence and DM distributions, we find a reasonable fit for gamma ~ 1.7 and E_max ~ 10^{33} erg/Hz. We then carry out a full Bayesian analysis based on a Schechter rate function with cosmological factor. We find roughly consistent results with our analytical approach, although with large errors on the inferred parameters due to the small sample size. The power-law index and the maximum energy are constrained to be gamma = 1.6 +/- 0.3 and log(E_max) [erg/Hz] = 34.1 +1.1 -0.7 (68% confidence), respectively. From the survey exposure time, we further infer a cumulative local volumetric rate of log N(E > 10^{32} erg/Hz) [Gpc^{-3} yr^{-1}] = 2.6 +/- 0.4 (68% confidence). The methods presented here will be useful for the much larger FRB samples expected in the near future to study their distributions, energetics, and rates.Comment: ApJ accepted. Expanded beyond the scope of the earlier version into 8 pages, 7 figures. Following referees' comments, we included a full Bayesian analysis based on a Schechter rate function with cosmological factor. The PDF of the inferred model parameters are presented by MCMC sampling in Figure 4 (the most important result). We also discussed the completeness of ASKAP sample in Section