Summary of models. Column 1 gives the model name. Columns 2, 3, 4, and 5 define the initial condition model underlying the input sevn catalogue, in terms of the initial distributions of primary stellar mass, mass ratio, orbital period, and eccentricity, respectively (see Section 2.1). Here, the primary star IMF is specified with the power-law slope at the high-mass end (given the fixed ZAMS mass range |$m_{\star ,1}\in [5,550]\ \rm M_\odot$|⁠), which is identical to that of the Pop III IMF adopted in a-sloth to model stellar feedback. Columns 6 and 7 define the NSC (and DF) model with the NSC occupation fraction as a function of galaxy (stellar) mass and the minimum galaxy mass for DF and NSC formation at |$z\gt 5$|⁠. See Section 3.1 for a detailed description of how these parameters are used to determine whether Pop III BBHs can fall into the NSC by DF in a galaxy. Column 8 shows the BBH formation efficiency |$\epsilon _{\rm BBH}$|⁠, i.e. the number of BBHs formed per unit stellar mass. Similarly, Column 9 shows the BBH merger efficiency |$\epsilon _{\rm GW}^{\rm all\ (field/NSC)}$|⁠, i.e. the number of BBH mergers at |$z\gt 0$| per unit stellar mass, for all (field/NSC) mergers. Column 10 shows the fraction of mergers in NSCs |$f_{\rm NSC}=\epsilon _{\rm GW}^{\rm NSC}/\epsilon _{\rm GW}^{\rm all}$|⁠. Column 11 shows the fraction |$f_{\rm GW}^{\rm field/NSC}$| of BBHs that merge at |$z\gt 0$| in galaxy fields/NSCs. The last column shows the fraction |$f_{\rm infall}$| of Pop III BBHs that fall into NSCs.