Effects of spin on constraining the seeds and growth of $\gtrsim {10}^9{M}_{\odot }$ supermassive black holes in $z>6.5$ Quasars

The existence of $\gtrsim {10}^9{M}_{\odot }$ supermassive black holes (SMBHs) at redshift $z>6$ raises the problem of how such SMBHs can grow up within the cosmic time ($<1$,Gyr) from small seed BHs. In this letter, we use the observations of $14$ Quasars at $z>6.5$ with mass estimates to constrain their seeds and early growth, by self-consistently considering the spin evolution and the possibility of super-Eddington accretion. We find that spin plays an important role in the growth of early SMBHs, and the constraints on seed mass and super-Eddington accretion fraction strongly depend on the assumed accretion history. If the accretion is coherent with single (or a small number of) episode(s), leading to high spins for the majority of accretion time, then the SMBH growth is relatively slow; and if the accretion is chaotic with many episodes and in each episode the total accreted mass is much less than the SMBH mass, leading to moderate/low spins, then the growth is relatively fast. The constraints on the seed mass and super-Eddington accretion fraction are degenerate. A significant fraction ($\gtrsim 0.1\mathrm{\%}-1\mathrm{\%}$ in linear scale but $\sim 3-4$ dex in logarithmic scale for ${10}^3-{10}^4{M}_{\odot }$ seeds) of super-Eddington accretion is required if the seed mass is not $\gg {10}^5{M}_{\odot }$, and the requirements of high seed-mass and/or super-Eddington accretion fraction are moderately relaxed if the accretion is chaotic.