Linear adiabatic analysis for general relativistic instability in primordial accreting supermassive stars

Accreting supermassive stars of $\gtrsim 10^{5}\,M_\odot$ will eventually collapse directly to a black hole via the general relativistic (GR) instability. Such direct collapses of supermassive stars are thought to be a possible formation channel for supermassive black holes at $z > 6$. In this work, we investigate the final mass of accreting Population III stars with constant accretion rates between $0.01$ and $1000\,M_\odot$yr$^{-1}$. We determine the final mass by solving the differential equation for the general relativistic linear adiabatic radial pulsations. We find that models with accretion rates $\gtrsim 0.05\,M_\odot$yr$^{-1}$ experience the GR instability at masses depending on the accretion rates. The critical masses are larger for higher accretion rates, ranging from $8\times10^{4}\,M_\odot$ for $0.05\,M_\odot$yr$^{-1}$ to $\sim10^6\,M_\odot$ for $1000\,M_\odot$yr$^{-1}$. The $0.05\,M_\odot$yr$^{-1}$ model reaches the GR instability at the end of the core hydrogen burning. The higher mass models with the higher accretion rates reach the GR instability during the hydrogen burning stage.