tokens: β > 1 where L max remains irrelevant. Thus, equation ( 4 ) reduces to EE detection probability: Perley et al. ( 2009 ) caution that many non-EE short bursts are faint, and the presence of an extended emission might have gone undetected resulting in their classification as non-EE bursts. Norris et al. ( 2010 ), who did a detailed study on the sensitivity of BAT to extended emission conclude that BAT should be able to detect extended emission to intensities ∼10 −2 counts cm −2 s −1 , however it is only seen up to around ∼0.005 counts cm −2 s −1 indicating a true absence of EE in some bursts. Going by the latter, one does not necessarily have to incorporate an additional EE detection probability. However, we take a conservative stand and try to account for a possible selection bias against the detection of the faint extended emission component. The redshift-dependent EE detection probability should involve the luminosity function of extended emission, but with the limited sample it is difficult to arrive at the same. Therefore, we considered a single constant luminosity for extended emission. This approximation is simplistic, given the two orders of magnitude spread in EE luminosity (see Fig. 3 ). For the threshold of BAT, we used ε BAT = 0.005 counts cm −2 s −1 , motivated by the typical average fluxes EE is seen in BAT as mentioned by Norris et al. ( 2010 ). Thus, equation ( 4 ), in this particular case, reduces to as, P () becomes a step function, H ( L EE − 4π d L () 2 ε BAT ) which is zero if L EE 4π d L () 2 ε BAT and unity otherwise. We ran our simulations for two values of L EE around the average luminosity of the six bursts (1.5 × 10 56 counts s −1 ), L EE = 10 56 , and 2 × 10 56 counts s −1 resulting in a max of 1.8 and 2.53, respectively. The results we obtain from these are nearly identical. Figure 3. Average luminosity of the EE component, from the data presented in Norris et al. ( 2010 ). Pen symbol is GRB 090510A. The error bars on other bursts are smaller than the symbol. One of our weak-EE bursts, 080123, is not a part of the data of Norris et al. ( 2010 ). We included GRB160410 following the GCN report by Sakamoto et al. ( 2016 ). We used a det = 0.16 sq cm to convert numbers from Sakamoto et al. ( 2016 ) and calculated the average luminosity. It can be seen that L EE varies by around two orders of magnitude. For our analysis we fixed values near the mean of these bursts (dotted and dash–dotted lines). The yellow solid line is the threshold, . We also considered a P () = 1 throughout, i.e. BAT does not miss any EE component due to sensitivity issues. Throughout this paper, we have used a standard cosmological model with H 0 = 71 km s −1 Mpc −1 , Ω m = 0.3, and Ω λ = 0.7. 4 ANALYSIS AND RESULTS In this section, we describe our analysis of the redshift distribution for both EE and non-EE bursts, and compare the parameters focusing mostly on the delay time distribution function. We begin by giving a detailed account of how we set up
ner_tags: B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O B-Telescope O O O O O B-Telescope O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Telescope O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O B-Telescope O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O B-CelestialObject I-CelestialObject O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O B-CelestialObject O O B-Dataset I-Dataset O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-TextGarbage I-TextGarbage O O O O B-Formula I-Formula I-Formula I-Formula O O B-Telescope O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O