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Supermassive stars (SMSs; M>10^5 Msun) and their remnant black holes arepromising progenitors for supermassive black holes (SMBHs) observed in theearly universe at z>7. It has been postulated that SMSs forms through veryrapid mass accretion onto a protostar at a high rate exceeding 0.01 Msun/yr.According to recent studies, such rapidly accreting protostars evolve into"supergiant protostars", i.e. protostars consisting of a bloated envelope and acontracting core, similar to giant star. However, like massive stars as well asgiant stars, both of which are known to be pulsationally unstable, supergiantprotostars may also be also unstable to launch strong pulsation-drivenoutflows. If this is the case, the stellar growth via accretion will behindered by the mass loss. We here study the pulsational stability of thesupergiant protostars in the mass range M<10^3 Msun through the method of thelinear perturbation analysis. We find that the supergiant protostars with M>600Msun and very high accretion rate Mdot>1.0 Msun/yr are unstable due to thekappa mechanism. The pulsation is excited in the He^+ ionization layer in theenvelope. Even under a conservative assumption that all the pulsation energy isconverted into the kinetic energy of the outflows, the mass-loss rate is ~10^-3Msun/yr, which is lower than the accretion rate by more than two orders ofmagnitude. We thus conclude that the supergiant protostars should grow stablyvia rapid accretion at least in the mass range we studied. As long as the rapidaccretion is maintained in the later stage, protostars will become SMSs, whicheventually produce seeds for the high-z SMBHs.

Pulsational instability of supergiant protostars: Do they grow supermassive by accretion?