We perform a binary population synthesis calculation incorporating very massive population (Pop.) III stars up to 1500 $M_odot$, and investigate the nature of binary black hole (BBH) mergers. Above the pair-instability mass gap, we find that the typical primary black hole (BH) mass is 135-340 $M_odot$. The maximum primary BH mass is as massive as 686 $M_odot$. The BBHs with both of their components above the mass gap have low effective inspiral spin $sim$ 0. So far, no conclusive BBH merger beyond the mass gap has been detected, and the upper limit on the merger rate density is obtained. If the initial mass function (IMF) of Pop. III stars is simply expressed as $xi_m(m) propto m^{-alpha}$ (single power law), we find that $alpha gtrsim 2.8$ is needed in order for the merger rate density not to exceed the upper limit. In the future, the gravitational wave detectors such as Einstein telescope and Pre-DECIGO will observe BBH mergers at high redshift. We suggest that we may be able to impose a stringent limit on the Pop. III IMF by comparing the merger rate density obtained from future observations with that derived theoretically.
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