Do stellar winds prevent the formation of supermassive stars by accretion?

Supermassive stars (SMSs; ~10⁵M _⊙) formed from metal-free gas in the early Universe attract attention as progenitors of supermassive black holes observed at high redshifts. To form SMSs by accretion, central protostars must accrete at as high rates as ~0.1-1M _⊙ yr^-1. Such protostars have very extended structures with bloated envelopes, like supergiant stars, and are called supergiant protostars (SGPSs). Under the assumption of hydrostatic equilibrium, SGPSs have density-inverted layers, where the luminosity becomes locally super-Eddington, near the surface. If the envelope matter is allowed to flow out, however, a stellar wind could be launched and hinder the accretion growth of SGPSs before reaching the supermassive regime. We examine whether radiation-driven winds are launched from SGPSs by constructing steady and spherically symmetric wind solutions. We find that the wind velocity does not reach the escape velocity in any case considered. This is because once the temperature falls below ~10⁴ K, the opacity plummet drastically owing to the recombination of hydrogen and the acceleration ceases suddenly. This indicates that, in realistic non-steady cases, even if outflows are launched from the surface of SGPSs, they would fall back again. Such a 'wind' does not result in net mass-loss and does not prevent the growth of SGPSs. In conclusion, SGPSs will grow to SMSs and eventually collapse to massive black holes of ~10⁵M _⊙, as long as the rapid accretion is maintained.