Early Growth of Massive Black Holes in Quasars

Episodic activity of quasars is driving growth of supermassive black holes (SMBHs) via accretion of baryon gas. In this Letter, we develop a simple method to analyse the duty cycle of quasars up to redshift $z\sim 6$ universe from luminosity functions (LFs). We find that the duty cycle below redshift $z\sim 2$followsthecosmichistoryofstarformationrate(SFR)density\mathrm{.}Beyond$z\sim 2$, the evolutionary trends of the duty cycle are just opposite to that of the cosmic SFR density history, implying the role of feedback from black hole activity. With the duty cycle, we get the net lifetime of quasars ($z\le 5$)about$\sim 10^9$yrs. Based on the local SMBHs, the mean mass of SMBHs is obtained at any redshifts and their seeds are of ${10}^5\text{sunm}$ at the reionization epoch ($z_{\mathrm{r}\mathrm{e}}$) of the universe through the conservation of the black hole number density in comoving frame. We find that primordial black holes ($\sim {10}^3\text{sunm}$) are able to grow up to the seeds via a moderate super-Eddington accretion of $\sim 30$ times of the critical rate from $z=24$ to $z_{\mathrm{r}\mathrm{e}}$. Highly super-Eddington accretion onto the primordials is not necessary.