Merger rate density of Population III binary black holes below, above, and in the pair-instability mass gap

We present the merger rate density of Population (Pop.) III binary black holes (BHs) by means of a widely-used binary population synthesis code BSE with extensions to very massive and extreme metal-poor stars. We consider not only low-mass BHs (lBHs: $5-50M_{\odot }$) but also high-mass BHs (hBHs: $130-200 M_\odot$), where lBHs and hBHs are below and above the pair-instability mass gap ($50-130M_{\odot }$), respectively. Pop. III BH-BHs can be categorized into three subpopulations: BH-BHs without hBHs (hBH0s: $m_{\rm tot} \lesssim 100 M_\odot$),withonehBH(hBH1s:$m_{\rm tot} \sim 130-260 M_\odot$), and with two hBHs (hBH2s: $m_{\mathrm{t}\mathrm{o}\mathrm{t}}\sim 270-400M_{\odot }$), where $m_{\mathrm{t}\mathrm{o}\mathrm{t}}$ is the total mass of a BH-BH. Their merger rate densities at the current universe are $\sim 0.1$ yr${}^{-1}$ Gpc${}^{-3}$ for hBH0s, and $\sim 0.01$ yr${}^{-1}$ Gpc${}^{-3}$ for the sum of hBH1s and hBH2s, provided that the mass density of Pop. III stars is $\sim {10}^{13}{M}_{\odot }$ Gpc${}^{-3}$. These rates are modestly insensitive to initial conditions and single star models. The hBH1 and hBH2 mergers can dominate BH-BHs with hBHs discovered in near future. They have low effective spins $\lesssim 0.2$ in the current universe. The number ratio of the hBH2s to the hBH1s is high, $\gtrsim 0.1$. We also find BHs in the mass gap (up to $\sim 85M_{\odot }$) merge. These merger rates can be reduced to nearly zero if Pop. III binaries are always wide ($\gtrsim 100R_{\odot }$), and if Pop. III stars always enter into chemically homogeneous evolution. The presence of close Pop. III binaries ($\sim 10R_{\odot }$) are crucial for avoiding the worst scenario.