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Table 1.

Size of the bias for different redshift distributions and luminosity functions.

Source class Source density #Sources lumi. variations Flux of source det. with one event
Eff. density
(at z = 0) (within z <  4) width of gaussian 5% perc. median 95% perc.
(Mpc−3) (dex) (Mpc−3)
FSRQs 6 × 10−10 530 0 (standard candle) 4 × 10−3 0.04 0.5 6 × 10−10
Ajello et al. (2014) 1 (lognorm. σ = 1) 6 × 10−3 0.11 1.1 10−10
BL Lac objects 2 × 10−7 1.2 × 104 0 1.9 × 10−4 6 × 10−3 0.2 8 × 10−9
Ajello et al. (2014) 1 3 × 10−4 0.03 0.7 9 × 10−10
Galaxy clusters 3 × 10−5 1.9 × 106 0 1.1 × 10−6 3 × 10−5 6 × 10−3 2 × 10−6
Zandanel et al. (2015) 1 3 × 10−6 5 × 10−4 0.2 1.4 × 10−7
Starburst galaxies 3 × 10−5 1.8 × 107 0 1.3 × 10−7 1.7 × 10−6 3 × 10−4 4 × 10−5
Gruppioni et al. (2013) 1 2 × 10−7 3 × 10−5 1.4 × 10−2 2 × 10−6

Notes. For each source class the reference to the used redshift distribution, the source density at z = 0, and the resulting number of sources is given. In addition, two different luminosity functions were tested. The first line represents the standard candle scenario, while in the second line the calculation was repeated assuming rather large luminosity variations described by a lognormal distribution with a width of one order of magnitude. The 90% confidence region for one detected event is characterized by quoting the median and the 5% and 95% percentiles of the probability distribution. To first order the effect of the redshift distributions and luminosity variations can be absorbed into an effective source density. For these densities the bias is equally large (same median) for a population of standard candle sources without evolution (see Fig. 2). The effective density for BL Lac objects without luminosity fluctuations, 8 × 10−9 Mpc−3, is used for the calculation in Sect. 2. As before fluxes are given as the expected number of detected events and the emission of the total population is normalized to ten detected events.

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