BBH merger rate density as a function of redshift for different observational scaling relations (different colors) and CE α parameters (different panels). Each color is associated with a different SFR–M_* relation, as summarized in the legend in the top left-hand panel. The solid lines show the model with M_min = 10⁷ M_⊙. The shaded area encompasses the difference between M_min = 10⁶ M_⊙ (upper boundary), and M_min = 10⁸ M_⊙ (lower boundary). All the models assume the fiducial fundamental metallicity relation by Andrews & Martini (2013). The thick dashed lines show the merger rate density obtained with COSMOℛATE (simplified model), assuming the cosmic SFR density by Madau & Dickinson (2014) and a lognormal distribution for the metallicity with a spread ${\stackrel{\sim }{\sigma }}_{\mathrm{Z}}=0.1$ (dark gray) and 0.6 (light gray). The gray-shaded regions show the MRD inferred by the LVK collaboration: ℛ(z)∝(1 + z)^k (Abbott et al. 2023b).