Based on the physical model of a supermassive black hole ( SMBH ) growth via gas accretion in a circumnuclear disk ( CND ) proposed by Kawakatu & Wada ( 2008 ) , we describe the formation of high- z ( z > 6 ) quasars ( QSOs ) whose BH masses are M _ { BH } > 10 ^ { 9 } M _ { \odot } .
We derive the necessary conditions to form QSOs at z > 6 by only gas accretion : ( i ) A large mass supply with M _ { sup } > 10 ^ { 10 } M _ { \odot } from host galaxies to CNDs , because the final BH mass is only 1 - 10 \% of the total supplied mass from QSO hosts .
( ii ) High star formation efficiency for a rapid BH growth which is comparable to high- z starburst galaxies such as submillimeter galaxies ( SMGs ) .
We also find that if the BH growth is limited by the Eddington accretion , the final BH mass is greatly suppressed when the period of mass-supply from hosts , t _ { sup } is shorter than the Eddington timescale .
Thus , the super-Eddington growth is required for the QSO formation as far as t _ { sup } , which is determined by the efficiency of angular momentum transfer , is shorter than \sim 10 ^ { 8 } { yr } .
The evolution of the QSO luminosity depends on the redshift z _ { i } at which accretion onto a seed BH is initiated .
In other words , the brighter QSOs at z > 6 favor the late growth of SMBHs ( i.e. , z _ { i } \approx 10 ) rather than early growth ( i.e. , z _ { i } \approx 30 ) .
For z _ { i } \approx 10 , t _ { sup } \simeq 10 ^ { 8 } { yr } is shorter than that of the star formation in the CND .
Thus , the gas in the CND can accrete onto a BH more efficiently , compared with the case for z _ { i } \approx 30 ( or t _ { sup } \approx 10 ^ { 9 } { yr } ) .
Moreover , we predict the observable properties and the evolution of QSOs at z > 6 .
In a QSO phase , there should exist a stellar rich massive CND , whose gas mass is about 10 \% of the dynamical mass inside \sim 0.1 - 1 { kpc } .
On the other hand , in a phase where the BH grows ( i.e. , a proto-QSO phase ) , the proto-QSO has a gas rich massive CNDs whose gas mass is comparable to the dynamical mass .
Compared with the observed properties of the distant QSO SDSS J1148+5251 observed at z = 6.42 , we predict that SDSS J1148+5251 corresponds to the scenario of the late growth of SMBH with z _ { i } \sim 10 , which is accompanied by a massive CNDs with M _ { g } \approx 5 \times 10 ^ { 10 } M _ { \odot } and the luminous nuclear starburst L _ { SB } at infrared band with L _ { SB } \approx 10 ^ { 47 } { erg } { s } ^ { -1 } .
Moreover , we predict that the progenitor of SDSS J1148+5251 can be the super-Eddington object .
These predictions can be verified by ALMA , SPICA and JWST .