  Supermassive stars (SMSs; M>10^5 Msun) and their remnant black holes are
promising progenitors for supermassive black holes (SMBHs) observed in the
early universe at z>7. It has been postulated that SMSs forms through very
rapid 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 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 also 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<10^3 Msun through the method of the
linear perturbation analysis. We find that the supergiant protostars with M>600
Msun and very high accretion rate Mdot>1.0 Msun/yr 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 ~10^-3
Msun/yr, 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.
