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tokens: limit. The simple method above ignores that limits will be more stringent when considering larger volumes over which the detection probability of individual repeating FRBs is smaller. Furthermore, it cannot account for changing population densities with redshift, i.e. Φ → Φ( z ). The probability p ≥2 ( F th , z ) of detecting two or more bursts from a given FRB depends on both the distance z and the beam sensitivity B , which governs the detection threshold F th = F 0 / B as discussed in Section 2.3 . The expected number of observed repeating FRBs, Λ rep , in a given survey is then given by where ε( z ) weights the sensitive solid angle over beam sensitivity according to the detection probability. Unlike equation ( 14 ), the integral in equation ( 21 ) extends over all z , allowing for many repeating FRBs at large distances to contribute to the expected number of events. The factor p ≥2 ( B, z ) in the integrand gives different weights per unit volume – generally, this will be close to unity for z ≲ z lim , and fall more rapidly than increasing volume beyond this. To calculate p ≥2 ( B, z ), it is most useful to first calculate the expected number of bursts, λ( B, z ), so that p ≥2 ( B, z ) ≡ p ≥2 (λ( B, z )). λ( B, z ) can be found for the burst energy distribution of equation ( 1 ) by inverting equation ( 9 ), with z lim → z and λ lim → λ, i.e. In the case of a Poissonian distribution, p ≥2 (λ) is given simply by although the result will be different for different distributions (see e.g. Section 4 ). Evaluating p ≥2 allows equation ( 21 ) to be evaluated for any hypothesized Φ. It is useful to write Φ as the product of a z -dependent factor φ( z ) and its value in the current epoch, Φ 0 , i.e.: This allows limits on Φ 0 to be set for any given observed number of repeating FRBs by choosing appropriate Λ rep = Λ lim . The limit Φ lim on Φ 0 can be calculated as The implicit assumption here is that Φ lim will be an upper limit on the FRB density. However, this need not be the case, and both upper and lower limits could be calculated by choosing appropriate Λ lim should one or more repeating FRBs be detected, as is the case for the recent results from the Canadian Hydrogen Intensity Mapping Experiment (CHIME; CHIME/FRB Collaboration et al. 2019b ). 6.1 Volumetric limits from the ASKAP/CRAFT lat50 survey In order to limit Φ 0 , the z -dependence of the FRB population density φ( z ) is chosen to be an integer power n of the star formation rate (SFR). As per Macquart Ekers ( 2018b ), values of n = 0 (no dependence), n = 1 (stellar origin), and n = 2 (approximately tracing AGN activity) are considered. Evidently, this does not encompass models with significant delay-time, e.g. long-duration mergers, as tested by Cao, Yu Zhou ( 2018 ). The normalized SFR is taken from Madau Dickinson ( 2014 ), giving For the standard parameter set, n = 0, 1, 2, and using Φ 0 = 1 Mpc −3 , the integrand in equation ( 22 ) is shown in Fig. 11 . This is proportional to