The slope of the source-count distribution for fast radio bursts

The slope of the source-count distribution of fast radio burst (FRB) fluences, alpha, has been estimated using a variety of methods. Hampering all attempts have been the low number of detected FRBs, and the difficulty of defining a completeness threshold for FRB surveys. In this work, we extend maximum likelihood methods for estimating a, using detected and threshold signal-to-noise ratios applied to all FRBs in a sample without regard to a completeness threshold. Using this method with FRBs detected by the Parkes radio telescope, we find alpha = -1.18 +/- 0.24 (68 per cent confidence interval, CI), i.e. consistent with a non-evolving Euclidean distribution (alpha = -1.5). Applying these methods to the Australian Square Kilometre Array Pathfinder (ASKAP) Commensal Real-time ASKAP Fast Transients (CRAFT) FRB survey finds alpha = -2.2 +/- 0.47 (68 per cent CI). A full maximum likelihood estimate finds an inconsistency with the Parkes rate with a p-value of 0.86 per cent (2.6 sigma). If not due to statistical fluctuations or biases in Parkes data, this is the first evidence for deviations from a pure power law in the integral source-count distribution of FRBs. It is consistent with a steepening of the integral source-count distribution in the fluence range 5-40 Jy ms, for instance due to a cosmological population of FRB progenitors evolving more rapidly than the star formation rate, and peaking in the redshift range 1-3.