  Supermassive primordial stars are suspected to be the progenitors of the most
massive quasars at z~6. Previous studies of such stars were either unable to
resolve hydrodynamical timescales or considered stars in isolation, not in the
extreme accretion flows in which they actually form. Therefore, they could not
self-consistently predict their final masses at collapse, or those of the
resulting supermassive black hole seeds, but rather invoked comparison to
simple polytropic models. Here, we systematically examine the birth, evolution
and collapse of accreting non-rotating supermassive stars under accretion rates
of 0.01-10 solar masses per year, using the stellar evolution code KEPLER. Our
approach includes post-Newtonian corrections to the stellar structure and an
adaptive nuclear network, and can transition to following the hydrodynamic
evolution of supermassive stars after they encounter the general relativistic
instability. We find that this instability triggers the collapse of the star at
masses of 150,000-330,000 solar masses for accretion rates of 0.1-10 solar
masses per year, and that the final mass of the star scales roughly
logarithmically with the rate. The structure of the star, and thus its
stability against collapse, is sensitive to the treatment of convection, and
the heat content of the outer accreted envelope. Comparison with other codes
suggests differences here may lead to small deviations in the evolutionary
state of the star as a function of time, that worsen with accretion rate. Since
the general relativistic instability leads to the immediate death of these
stars, our models place an upper limit on the masses of the first quasars at
birth.
