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tokens: property of the galaxy allows us to isolate its effects and explore its consequences. Here, we will discuss two criteria. The first is the metallicity threshold, already mentioned, that provides an upper mass limit on galaxies that will be able to host a GRB (see Equation ( 1 )). The second criterion we explore is that a given galaxy will be more likely to host a GRB if it has a high specific SFR. We implement this by assuming that to host a GRB, a galaxy will need to have a specific rate above a given threshold value that we further assume to be independent of redshift. Again, this provides an upper stellar-mass limit for galaxies hosting GRBs. Figure 2 shows examples of the critical masses as a function of redshift for three different values of S * thresholds, as well as three different metallicity thresholds. The critical mass increases more rapidly with increasing redshift for the S * thresholds than for the metallicity thresholds. In addition, at a fixed redshift, the critical mass increases with an increasing metallicity threshold, but decreases with an increasing specific rate threshold. Galaxies with masses above the critical mass values have, in this model, a very low probability of producing a GRB. Figure 2. Critical galaxy stellar mass (in solar masses) as a function of redshift, z , for three different values of S * (in Gyr −1 , solid curves) and three different metallicity thresholds (dashed curves). The critical mass increases faster with increasing redshift for the S * thresholds. For galaxies more massive than the critical mass, GRB production is suppressed. For comparison we also show the redshift evolution of M 0 and M 1 (dashed–dotted). It is clear that for both threshold types ( S * and metallicity), in fact, very few if any GRBs would be produced by galaxies in the mass range out to redshifts z ≲ 2. Indeed, galaxies of mass ∼10 11 M would not be hosting a GRB out to a redshift of z ∼ 3–4 in either scenario, as observed by Perley et al. ( 2016b ). Only in the case of relatively low values of S * and rather high values of the metallicity threshold does the critical mass reach the characteristic mass, M 1 , of Equation ( 2 ), at the highest redshifts shown. Massive galaxies, around M 1 or higher, are only likely to host a GRB at high redshift and thus contribute to the GRB-probed star formation if of high metallicity or of low specific SFR. 3. The SFR Fraction We now proceed as in Robertson Ellis ( 2012 ) and define the fraction of star formation occurring in galaxies with M M cr as:Here, P ( x , y ) = Γ( x , y )/Γ( x ), with Γ( x ) the gamma function,is the lower incomplete gamma function, and the second equality in Equation ( 5 ) holds for the functional forms adopted for and SFR( M , z ). Recall that ψ ( z ) is also the fraction of star formation producing a GRB (Robertson Ellis 2012 ). In Equation ( 5 ), t cr = M cr (1/ M 0 + 1/ M 1 ), and the redshift dependence of ψ ( z ) is only through the mass scales if the