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We investigate the formation of merging binary black holes (BHs) throughisolated binary evolution, performing binary population synthesis calculationscovering an unprecedentedly wide metallicity range of Population (Pop) I, II,III, and extremely metal-poor (EMP) binary stars. We find that the predictedmerger rate density and primary BH mass ($m_1$) distribution are consistentwith the gravitational wave (GW) observations. Notably, Pop III and EMP ($<10^{-2}$ $Z_\odot$) binary stars yield most of the pair instability (PI) massgap events with $m_1 = 65$--$130$ $M_\odot$. Pop III binary stars contributemore to the PI mass gap events with increasing redshift, and all the PI massgap events have the Pop III origin at redshifts $\gtrsim 8$. Our result can beassessed by future GW observations in the following two points. First, thereare no binary BHs with $m_1=100$--$130$ $M_\odot$ in our result, and thus the$m_1$ distribution should suddenly drop in the range of $m_1=100$--$130$$M_\odot$. Second, the PI mass gap event rate should increase toward higherredshift up to $\sim 11$, since those events mainly originate from the Pop IIIbinary stars. We find that the following three assumptions are needed toreproduce the current GW observations: a top-heavy stellar initial massfunction and the presence of close binary stars for Pop III and EMP binarystars, and inefficient convective overshoot in the main-sequence phase ofstellar evolution. Without any of the above, the number of PI mass gap eventsbecomes too low to reproduce current GW observations.

Merger rate density of binary black holes through isolated Population I, II, III and extremely metal-poor binary star evolution