Predicted maximum GRB rates observed by EXIST. We adopted Salpeter IMF, v_wind = 100      km   s^-1, f_GRB = 0.01, f_∗ = 0.01; for Pop III.2 GRBs, dashed black line; and Gaussian IMF, v_wind = 50      km   s^-1, f_∗ = 0.1, f_GRB = 0.1; for Pop III.1 GRBs, dotted blue line. Dotted red lines represent the same with LF’s bright end power law index 1.7 and 0.8.