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Supermassive stars and quasi-stars (massive stars with a central black hole)are both considered as potential progenitors for the formation of supermassiveblack holes. They are expected to form from rapidly accreting protostars inmassive primordial halos. We explore how long rapidly accreting protostarsremain on the Hayashi track, implying large protostellar radii and weakaccretion luminosity feedback. We assess the potential role of energyproduction in the nuclear core, and determine what regulates the evolution ofsuch protostars into quasi-stars or supermassive stars. We follow thecontraction of characteristic mass scales in rapidly accreting protostars, andinfer the timescales for them to reach nuclear densities. We compare thecharacteristic timescales for nuclear burning with those for which the extendedprotostellar envelope can be maintained. We find that the extended envelope canbe maintained up to protostellar masses of 3.6x10^8 \dot{m}^3 solar, where\dot{m} denotes the accretion rate in solar masses per year. We expect thenuclear core to exhaust its hydrogen content in 7x10^6 yrs. If accretion rates\dot{m}>>0.14 can still be maintained at this point, a black hole may formwithin the accreting envelope, leading to a quasi-star. Alternatively, theaccreting object will gravitationally contract to become a main-sequencesupermassive star. Due to the limited gas reservoir in dark matter halos with10^7 solar masses, the accretion rate onto the central object may drop at latetimes, implying the formation of supermassive stars as the typical outcome ofdirect collapse. However, if high accretion rates are maintained, a quasi-starwith an interior black hole may form.

Massive black hole factories: Supermassive and quasi-star formation in primordial halos