The formation of globular clusters with top-heavy initial mass functions

We study the formation of globular clusters in massive compact clouds with the low-metallicity of $Z=10^{-3}~Z_{\odot}$ by performing three-dimensional radiative-hydrodynamics simulations. Considering the uncertainty of the initial mass function (IMF) of stars formed in low-metallicity and high-density clouds, we investigate the impacts of the IMF on the cloud condition for the GC formation with the range of the power-law index of IMF as $\gamma = 1-2.35$. We find that the threshold surface density ($\Sigma_{\rm thr}$) for the GC formation increases from $800~M_{\odot} \; {\rm pc^{-2}}$ at $\gamma = 2.35$ to $1600~M_{\odot}\; {\rm pc^{-2}}$ at $\gamma = 1.5$ in the cases of clouds with $M_{\rm cl} = 10^6~M_{\odot}$ because the emissivity of ionizing photons per stellar mass increases as $\gamma$ decreases. For $\gamma < 1.5$, $\Sigma_{\rm thr}$ saturates with $\sim 2000~M_{\odot}\; {\rm pc^{-2}}$ that is quite rare and observed only in local starburst galaxies due to e.g., merger processes. Thus, we suggest that formation sites of low-metallicity GCs could be limited only in the very high-surface density regions. We also find that $\Sigma_{\rm thr}$ can be modelled by a power-law function with the cloud mass ($M_{\rm cl}$) and the emissivity of ionizing photons ($s_*$) as $\propto M_{\rm cl}^{-1/5} s_{*}^{2/5}$. Based on the relation between the power-law slope of IMF and $\Sigma_{\rm thr}$, future observations with e.g., the James Webb Space Telescope can allow us to constrain the IMF of GCs.