Supermassive Star Formation via Super Competitive Accretion in Slightly
  Metal-enriched Clouds
  Direct collapse black hole (DCBH) formation with mass $\gtrsim
10^{5}~M_{\odot}$ is a promising scenario for the origin of high-redshift
supermassive black holes. It has usually been supposed that the DCBH can only
form in the primordial gas since the metal enrichment enhances the cooling
ability and causes the fragmentation into smaller pieces. What actually happens
in such an environment, however, has not been explored in detail. Here, we
study the impact of the metal enrichment on the clouds, conducting
hydrodynamical simulations to follow the cloud evolution in cases with
different degree of the metal enrichment $Z/Z_{\odot}=10^{-6}-10^{-3}$. Below
$Z/Z_{\odot}=10^{-6}$, metallicity has no effect and supermassive stars form
along with a small number of low-mass stars. With more metallicity $Z/Z_{\odot}
\gtrsim 5 \times 10^{-6}$, although the dust cooling indeed promotes
fragmentation of the cloud core and produces about a few thousand low-mass
stars, the accreting flow preferentially feeds the gas to the central massive
stars, which grows supermassive as in the primordial case. We term this
formation mode as the {\it super competitive accretion}, where only the central
few stars grow supermassive while a large number of other stars are competing
for the gas reservoir. Once the metallicity exceeds $10^{-3}~Z_{\odot}$ and
metal-line cooling becomes operative, the central star cannot grow supermassive
due to lowered accretion rate. Supermassive star formation by the super
competitive accretion opens up a new window for seed BHs, which relaxes the
condition on metallicity and enhances the seed BH abundance.
