Abstract

Context. Supermassive stars and quasi-stars (massive stars with a central black hole) are both considered as potential progenitors for the formation of supermassive black holes. They are expected to form from rapidly accreting protostars in massive primordial halos. Aims: We explore how long rapidly accreting protostars remain on the Hayashi track, implying large protostellar radii and weak accretion luminosity feedback. We assess the potential role of energy production in the nuclear core, and determine what regulates the evolution of such protostars into quasi-stars or supermassive stars. Methods: We followed the contraction of characteristic mass shells in rapidly accreting protostars, and inferred the timescales for them to reach nuclear densities. We compared the characteristic timescales for nuclear burning with those for which the extended protostellar envelope can be maintained. Results: We find that the extended envelope can be maintained up to protostellar masses of 3.6 {\times} 10$^{8}$ {\.m}$^{3}$ M$_{⊙}$, where {\.m} denotes the accretion rate in solar masses per year. We expect the nuclear core to exhaust its hydrogen content in 7 {\times} 10$^{6}$ yr. If accretion rates {\.m} {\Gt} 0.14 can still be maintained at this point, a black hole may form within the accreting envelope, leading to a quasi-star. Alternatively, the accreting object will gravitationally contract to become a main-sequence supermassive star. Conclusions: Due to the limited gas reservoir in typical 10$^{7}$ M$_{⊙}$ dark matter halos, the accretion rate onto the central object may drop at late times, implying the formation of supermassive stars as the typical outcome of direct collapse. However, if high accretion rates are maintained, a quasi-star with an interior black hole may form.