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

Supermassive stars (SMSs; M-* greater than or similar to 10(5) M-circle dot) and their remnant black holes are promising progenitors for supermassive black holes (SMBHs) observed in the early universe at z greater than or similar to 7. It has been postulated that SMSs form through very rapid mass accretion on to a protostar at a high rate exceeding 0.01 M-circle dot yr(-1). According to recent studies, such rapidly accreting protostars evolve into 'supergiant protostars', i.e. protostars consisting of a bloated envelope and a contracting core, similar to giant star. However, like massive stars as well as giant stars, both of which are known to be pulsationally unstable, supergiant protostars may also be unstable to launch strong pulsation-driven outflows. If this is the case, the stellar growth via accretion will be hindered by the mass-loss. We here study the pulsational stability of the supergiant protostars in the mass range M-* greater than or similar to 10(3) M-circle dot through the method of the linear perturbation analysis. We find that the supergiant protostars with M-* greater than or similar to 600 M-circle dot and very high accretion rate (M) over dot(acc) greater than or similar to 1.0 M-circle dot yr(-1) are unstable due to the kappa mechanism. The pulsation is excited in the He+ ionization layer in the envelope. Even under a conservative assumption that all the pulsation energy is converted into the kinetic energy of the outflows, the mass-loss rate is similar to 10(-) M-3(circle dot) yr(-1), which is lower than the accretion rate by more than two orders of magnitude. We thus conclude that the supergiant protostars should grow stably via rapid accretion at least in the mass range we studied. As long as the rapid accretion is maintained in the later stage, protostars will become SMSs, which eventually produce seeds for the high-z SMBHs.