Formation of primordial supermassive stars by burst accretion

A promising formation channel of SMBHs at redshift 6 is the so-called DC model, which posits that a massive seed BH forms through gravitational collapse of a $\sim 10^5~M_\odot$ SMS. We study the evolution of such a SMS growing by rapid mass accretion. In particular, we examine the impact of time-dependent mass accretion of repeating burst and quiescent phases that are expected to occur with a self-gravitating circumstellar disk. We show that the stellar evolution with such episodic accretion differs qualitatively from that expected with a constant accretion rate, even if the mean accretion rate is the same. Unlike the case of constant mass accretion, whereby the star expands roughly following $R_* \simeq 2.6 \times 10^3 R_\odot (M_*/100~M_\odot)^{1/2}$, the protostar can substantially contract during the quiescent phases between accretion bursts. The stellar effective temperature and ionizing photon emissivity increase accordingly as the star contracts, which can cause strong ionizing feedback and halt the mass accretion onto the star. With a fixed duration of the quiescent phase $\Delta t_{\rm q}$, such contraction occurs in early evolutionary phases, i.e. for $M_* \lesssim 10^3~M_\odot$ with $\Delta t_{\rm q} \simeq 10^3$ yr. For later epochs and larger masses but the same $\Delta t_{\rm q}$, contraction is negligible even during quiescent phases. With larger quiescent times $\Delta t_{\rm q}$, however, the star continues to contract during quiescent phases even for the higher stellar masses. We show that such behavior is well understood by comparing the interval time and the thermal relaxation time for a bloated surface layer. We conclude that the UV radiative feedback becomes effective if the quiescent phase associated by the burst accretion is longer than $\sim 10^3$ yr, which is possible in an accretion disk forming in the direct collapse model.