Formation of supermassive black hole seeds in nuclear star clusters via
  gas accretion and runaway collisions
  More than two hundred supermassive black holes (SMBHs) of masses $\gtrsim
10^9\,\mathrm{M_{\odot}}$ have been discovered at $z \gtrsim 6$. One promising
pathway for the formation of SMBHs is through the collapse of supermassive
stars (SMSs) with masses $\sim 10^{3-5}\,\mathrm{M_{\odot}}$ into seed black
holes which could grow upto few times $10^9\,\mathrm{M_{\odot}}$ SMBHs observed
at $z\sim 7$. In this paper, we explore how SMSs with masses $\sim
10^{3-5}\,\mathrm{M_{\odot}}$ could be formed via gas accretion and runaway
stellar collisions in high-redshift, metal-poor nuclear star clusters (NSCs)
using idealised N-body simulations. We explore physically motivated accretion
scenarios, e.g. Bondi-Hoyle-Lyttleton accretion and Eddington accretion, as
well as simplified scenarios such as constant accretions. While gas is present,
the accretion timescale remains considerably shorter than the timescale for
collisions with the most massive object (MMO). However, overall the timescale
for collisions between any two stars in the cluster can become comparable or
shorter than the accretion timescale, hence collisions still play a crucial
role in determining the final mass of the SMSs. We find that the problem is
highly sensitive to the initial conditions and our assumed recipe for the
accretion, due to the highly chaotic nature of the problem. The key variables
that determine the mass growth mechanism are the mass of the MMO and the gas
reservoir that is available for the accretion. Depending on different
conditions, SMSs of masses $\sim10^{3-5} \,\mathrm{M_{\odot}}$ can form for all
three accretion scenarios considered in this work.
