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quasars rank among the most luminous objects in the universe and are believed to be powered by smbhs ( e.g. , @xcite ) . they constrain the formation and evolution of galaxies and smbhs throughout cosmic time . the similarity between the cosmic star formation history ( e.g. , @xcite ) and the evolution of quasar abundances ( e.g. , @xcite ) suggests an intriguing link between galaxy formation and black hole growth . this is strengthened by tight correlations measured locally between the masses of the black holes and the global properties of the spheroid components of their hosts , such as their luminosities and masses ( @xcite ) , light concentration ( @xcite ) , and velocity dispersions ( @xcite ) . distant , highly luminous quasars are important cosmological probes for studying the first generation of galaxies , the star formation history and metal enrichment in the early universe , the growth of the first supermassive black holes ( smbhs ) , the role of feedback from quasars and black holes in galaxy evolution , and the epoch of reionization . the sloan digital sky survey ( sdss , @xcite ) has contributed significantly to the discovery of high redshift quasars . currently , there are over 1000 quasars known at @xmath11 and 12 at @xmath12 ( @xcite ) . as reviewed by @xcite , quasars at @xmath13 are characterized by : ( 1 ) a low space density ( @xmath14 comoving ) ; ( 2 ) high luminosities ( absolute luminosity at rest - frame @xmath15 ) , believed to be powered by smbhs of @xmath1 ; ( 3 ) gunn - peterson absorption troughs @xcite in their spectra , which place these quasars at the end of the epoch of reionization ( e.g. , @xcite ) ; and ( 4 ) a lack of evolution in the spectral energy distribution compared to lower - shift counterparts ( e.g. , @xcite ) , which demonstrates the existence of `` mature '' quasars at early times and comparable metal enrichment in quasars at all cosmic epochs . the most distant quasar known , sdss j1148 + 5251 ( hereafter j1148 + 5251 ) , was discovered by sdss at z = 6.42 @xcite . it is extremely bright optically with @xmath16 = -27.8 , and deep imaging surveys in both optical and radio ( @xcite ) show no sign of gravitational lensing or other companions at the same redshift in the vicinity . over the past few years , this quasar has been extensively studied at many wavelengths . near - infrared observations by @xcite and @xcite imply a bolometric luminosity of @xmath17 powered by accretion onto a smbh of mass 15 @xmath18 . radio observations by @xcite and @xcite suggest that the host is a hyper - luminous far - infrared ( fir ) galaxy , with @xmath19 , and these authors estimate a star formation rate of @xmath20 by assuming most of the fir luminosity comes from young stars . emission from carbon monoxide ( co ) has been detected ( @xcite ) corresponding to a mass of @xmath21 . dust has been seen by several groups ( e.g. , @xcite ) with an estimated mass of @xmath22 . in particular , _ spitzer _ observations by @xcite and @xcite indicate that the dust is heated by an active galactic nucleus ( agn ) . furthermore , the detection of iron by @xcite , the carbon fine structure line [ cii ] by @xcite and excess oi absorption by @xcite indicate near - solar metallicity in the quasar host . these observations raise many fundamental questions for models of quasar and galaxy formation : where do such high - redshift , luminous quasars originate ? how do they form ? what are the mechanisms and physical conditions for smbh growth ? and , do these quasar hosts obey the same smbh host correlations as observed in the local universe ?    interpretations of various observations of j1148 + 5251  have painted a rather conflicting picture for the formation site of the quasar halo and the smbh  host relationship . the low abundance of these quasars leads to the view that they are hosted by massive halos ( @xmath23 ) in the rarest density peaks of the dark matter distribution @xcite . however , it has been argued , based on the lack of companion galaxies in the field , that this quasar may reside in a much lower mass halo in a less overdense region ( @xcite ) . moreover , @xcite suggest , based on the dynamical mass estimate from co measurements , that j1148 + 5251contains a small stellar spheroid , and that the smbh may have largely formed before the host galaxy . however , the detection of metal lines ( @xcite ) , along with dust ( @xcite ) , indicates that the interstellar medium ( ism ) of j1148 + 5251  was significantly enriched by heavy elements produced through massive star formation at rates of @xmath24 occurring as early as @xmath25 , and that large stellar bulges form before accreting smbhs undergo luminous quasar activity .    in an expanding universe that is dominated by cold dark matter and is accelerated by dark energy , the @xmath2-cold dark matter ( @xmath2cdm ) cosmology , the leading theoretical model for structure formation , assumes that structure grows from weak density fluctuations amplified by gravity , with small objects collapsing first and subsequently merging to form progressively more massive ones , a process known as `` hierarchical assembly '' ( e.g. , see @xcite for a review ) . the formation of galaxies and quasars is therefore determined by the abundance of dark matter halos ; i.e. , the number density of halos as a function of mass and redshift . the most widely used analytic model for the halo mass function was first developed by @xcite ( hereafter ps ) , which is based on gaussian density perturbations , linear gravitational growth , and spherical collapse of dark matter . using the ps mass functions , @xcite studied the abundance of rare objects , such as luminous quasars at high redshifts . these authors predicted a sharp `` cut - off '' of the quasar population at @xmath26 . however , while the initial , linear growth of density perturbations can be calculated analytically , the gravitational collapse and subsequent hierarchical build - up of structure is a highly nonlinear process that can be followed only through numerical simulations . it has been shown by previous numerical studies ( e.g. , @xcite ) , and more recently by the state - of - the - art _ millennium simulation _ by @xcite , that the ps formula underestimates the abundance of high - mass halos by up to an order of magnitude . therefore , whether or not rare quasars such as j1148 + 5251  can form in the @xmath2cdm cosmology remains an open question and an important test of the theory .    to date , only a limited number of analytical or semi - analytic models have addressed the early formation of a @xmath27 smbh at @xmath0 ( @xcite ) . these approaches use merger trees of dark matter halos generated using the ps theory , and assume a black hole accretion rate at or above the eddington limit . however , as discussed above , the ps  based approach may be inaccurate . moreover , it is not clear whether sufficient physical conditions for such large accretion rates exist in quasar systems as the hydrodynamics of the large - scale gas flow and feedback from black holes have not been incorporated in earlier modeling . it is believed that the growth of smbhs is closely linked to galaxy formation ( e.g. , @xcite ) , and that the growth is self - regulated by feedback ( e.g. , @xcite ) . remarkably , self - regulated models with smbh feedback in the form of thermal energy coupled to the ambient gas have been demonstrated to successfully reproduce many observations of galaxies , including the @xmath28 relation ( @xcite ) , galaxy colors ( @xcite ) , x - ray gas emission @xcite , elliptical kinematics @xcite and the fundamental plane @xcite , quasar properties ( @xcite ) , luminosity functions ( @xcite ) , and populations ( @xcite ) , and the luminosity function of low - level agn @xcite . furthermore , these simulations of binary mergers identify a plausible , merger - driven formation mechanism for massive black holes and luminous quasars ( e.g. , @xcite ) .    here , we present a model that accounts for the smbh growth , quasar activity and host galaxy properties of the most distant quasar known . in our scenario , the quasar and its host galaxy form in a massive halo that originates from a rare density peak in the standard @xmath2cdm paradigm , and they grow hierarchically through multiple gas - rich mergers , supporting an average star formation rate of @xmath24 that peaks at @xmath29 . once the progenitors undergo sufficient dynamical friction to coalesce , the multiple smbhs from the progenitor galaxies merge and exponentially increase their mass and feedback energy via accretion . at @xmath30 when the smbh mass exceeds @xmath31 , black hole accretion drives a sufficiently energetic wind to clear obscuring material from the central regions of the system and powers an optically luminous quasar similar to j1148 + 5251 . we have devised a set of novel multi - scale simulations , which include cosmological n - body calculations on large scales and hydrodynamic simulations of galaxy mergers on galactic scales , coupled with the self - regulated growth of smbhs , enabling us to follow galaxy assembly and quasar formation at @xmath0 . this paper is organized as follows . in   2 , we describe our computational methods and models , which includes a set of large scale cosmological n - body simulations , and hydrodynamical galaxy mergers along the merging history of the quasar halo . in   3 , we present the formation and evolution of the quasar and its host galaxy , including the assembly of the galaxy progenitors , star formation , and smbh growth , as well as the smbh  host correlations , and properties of the quasar such as luminosities and lifetimes . we discuss feedback from starburst - driven winds , quasar abundances for cosmological models with different parameters , the implication of black hole mergers , and galaxies in the epoch of reionization in   4 , and summarize in   5 . rare , high - redshift quasars originate in highly overdense regions in the initial dark matter density distribution and grow through hierarchical mergers , as predicted by the @xmath2cdm theory . simulations of high - redshift quasar formation must model a large cosmological volume to accommodate the low abundance of this population , have a large dynamic range to follow the hierarchical build - up of the quasar hosts , and include the hydrodynamics of the gas flows in galaxy mergers . the cutting - edge _ millennium simulation _ by @xcite follows structure formation in a box with side length of @xmath32 using @xmath33 dark matter particles . it reproduces the large - scale galaxy distribution as observed @xcite , and identifies an early quasar halo candidate at @xmath34 which ends up in the richest cluster at the present day . however , even such a large dynamic range still falls short of being able to follow the formation and evolution of the rarest quasars observed at the highest redshifts . moreover , in order to address the properties of quasars and host galaxies , gasdynamics and physical processes related to star formation and black hole growth must be included . to satisfy these requirements , we have performed a set of novel multi - scale simulations that enable us to resolve individual mergers on galactic scales and retain the context of large - scale structure formation , as well as the evolution of black holes and stars . first , we perform a coarse dark matter simulation in a volume of @xmath35 designed to accommodate the low number density of @xmath36 quasars . the largest halo at @xmath37 , within which the descendants of early , luminous quasars are assumed to reside @xcite , is then selected for resimulation with higher resolution using a multi - grid zoom - in technique developed by @xcite . the merging history of the largest halo at @xmath0 , which has reached a mass of @xmath38 through 7 major ( mass ratio @xmath39 5:1 ) mergers between redshifts 14.4 and 6.5 , is extracted . these major mergers are again re - simulated hydrodynamically using galaxy models scaled appropriately for redshift @xcite and adjusted to account for mass accretion through minor mergers . the simulations include prescriptions for star formation @xcite , and for smbh growth and feedback ( @xcite ) , as described below . our multi - scale simulations were performed using the parallel , n - body / smoothed particle hydrodynamics ( sph ) code gadget2 developed by @xcite that is well tested in a wide range of applications from large scale structure formation to star formation . for the computation of gravitational forces , the code uses the `` treepm '' method @xcite that combines a `` tree '' algorithm @xcite for short - range forces and a fourier transform particle - mesh method @xcite for long - range forces . the pm - method works efficiently in large scale cosmological simulations , while the tree - method provides accurate forces for the large dynamic range of galaxy merger simulations . gadget2 implements the entropy - conserving formulation of sph @xcite with adaptive particle smoothing , as in @xcite . radiative cooling and heating processes are calculated assuming collisional ionization equilibrium @xcite . star formation is modeled in a multi - phase ism , with a rate that follows the schmidt - kennicutt law ( @xcite ) . feedback from supernovae is captured through a multi - phase model of the ism by an effective equation of state for star - forming gas @xcite . a prescription for supermassive black hole growth and feedback is also included , where black holes are represented by collisionless `` sink '' particles that interact gravitationally with other components and accrete gas from their surroundings . the accretion rate is estimated from the local gas density and sound speed using a spherical bondi - hoyle @xcite model that is limited by the eddington rate . feedback from black hole accretion is modeled as thermal energy injected into surrounding gas , as described in @xcite and @xcite . the simulations presented in this paper adopt the @xmath2cdm model with cosmological parameters chosen according to the first year wilkinson microwave anisotropy probe data ( wmap1 , @xcite ) , ( @xmath40 , @xmath41 , @xmath42 , @xmath43 , @xmath44 , @xmath45)= ( 0.3 , 0.04 , 0.7 , 0.7 , 1 , 0.9 ) . here , @xmath40 is the total matter density in units of the critical density for closure , @xmath46 . similarly , @xmath41 and @xmath47 denote the densities of baryons and dark energy at the present day . the hubble constant is parameterized as @xmath48 , while @xmath45 is the root - mean - squared linear mass fluctuation within a sphere of radius @xmath49 extrapolated to @xmath37 . we have also done the same cosmological simulations with wmap third year results ( wmap3 , @xcite ) , ( @xmath40 , @xmath41 , @xmath42 , @xmath43 , @xmath44 , @xmath45)= ( 0.236 , 0.042 , 0.759 , 0.732 , 0.95 , 0.74 ) for comparison . the quasars at @xmath0 have an extremely low comoving space density , @xmath50 , and are believed to reside in massive dark matter halos with @xmath51 @xcite . cosmological simulations of quasar formation must therefore model a volume of @xmath52 to account for the rarity of such objects . however , in order to resolve a @xmath53 halo at @xmath0 in a cosmological simulation with uniform resolution , a dark matter particle mass at least as small as @xmath54 and particle numbers of @xmath55 are required . tracking the merger history of such halos requires @xmath56 orders of magnitude higher resolution and would be computationally prohibitive with standard techniques . we achieve the mass resolution requirements for the merger history of a @xmath53 halo at @xmath57 by means of a two - step re - simulation . first , coarse dark matter cosmological simulations are performed to identify a candidate halo for the quasar host . a cubic volume @xmath58 on a side is simulated with @xmath59 particles , achieving mass and force resolutions of @xmath60 and @xmath61 ( comoving ) , respectively . to generate the initial conditions , we use the boltzmann code cmbfast by @xcite to compute a linear theory power spectrum for our chosen cosmology . a random realization of the power spectrum is constructed in fourier space , sampling modes in a sphere up to the nyquist frequency of the mesh . the particle distribution is evolved forward in time to @xmath37 from its initial displacement at @xmath62 determined using the zeldovich approximation .    at the end of the simulation , halos are identified using a `` friends - of - friends '' ( fof ) group - finding algorithm @xcite with a fixed comoving linking length equal to 0.2 times the mean dark matter interparticle separation and a minimum of 32 particles per group @xcite . the mean overdensity of the groups corresponds approximately to the expected density of virialized halos @xcite . from the more than 126000 groups identified in the @xmath63 volume at @xmath37 the largest halo with @xmath64 is selected as a candidate halo for modeling the formation of a quasar at @xmath65 .    a multi - grid technique developed by @xcite and @xcite is used to `` zoom in '' with high resolution on the selected halo region which has an effective side length of @xmath66 . large - scale tidal forces are captured by binning exterior particles into cells according to their distance from the high - resolution region . to ensure proper treatment of small - scale structure , the initial displacements of the high resolution particles are calculated assuming a higher initial redshift of @xmath67 and normalized to @xmath45 at @xmath37 . the re - simulation uses @xmath68 particles , with @xmath69 particles inside the high - resolution region . with this technique , the mass resolution increases by almost four orders of magnitude to @xmath70 while the spatial resolution reaches @xmath71 . figure  [ figmap ] shows snapshots of both the coarse and high - resolution zoom - in runs that locate the quasar halo candidate . in the coarse run , the `` cosmic web '' is clearly seen , although the distribution appears nearly homogeneous on such large scales . in the zoom - in run , filamentary structures are prominent . dark matter collapses along the filaments , and the largest halo forms in the deepest potential wells at the intersections of the filaments . the high resolution of the zoom - in run enables the identification of more halos with lower masses both at @xmath37 and at high redshifts as early as @xmath72 , which is sufficient to identify the halo progenitors of the candidate quasar at @xmath0 . it appears that the halo progenitor of the largest one at the present day is also the most massive halo at @xmath0 , when it reaches a mass of @xmath73 , making it a plausible candidate for hosting a rare @xmath74 quasar . the impact that variations in the cosmological parameters can have on large - scale structure formation can be understood from the theoretical mass function of halos , as derived by @xcite and later developed by @xcite . the comoving number density @xmath75 of halos of mass between @xmath76 and @xmath77 can be described as , @xmath78 \ , , \label{eq_mf}\ ] ] where @xmath79 is the local mean mass density , @xmath80 is the critical density of collapse at redshift @xmath81 linearly extrapolated to the present day , and @xmath82 is the mass variance , which is a function of the power spectrum @xmath83 with wavenumber k and the window function @xmath84 , @xmath85 . the abundance of halos depends on the two functions @xmath82 and @xmath86 , each of which involves the cosmological parameters , in particular @xmath45 , @xmath87 and @xmath88 . these parameters determine the formation epoch of a halo and its mass .    the recently - released third year wmap3 results @xcite have lower values of @xmath45 , @xmath89 and @xmath87 , compared to wmap1 @xcite . the smaller @xmath45 from wmap3 would lower the amplitude of the power spectrum , which in turn reduces @xmath82 . furthermore , a smaller @xmath87 would reduce @xmath80 and hence delay halo formation . so , compared to wmap1 , at a given redshift the wmap3 parameters would yield a lower abundance of halos with mass @xmath90 , where @xmath91 is the halo mass corresponding to the characteristic luminosity in the schechter luminosity function for galaxies @xcite ; while for @xmath92 , it predicts a slightly larger halo abundance .    to test the sensitivity of our model to the new wmap results , we have performed the same set of cosmological simulations with parameters from the wmap3 measurements @xcite . we find that indeed the changes implied by the new parameters primarily affect the formation time and the mass of the candidate quasar halo . for the same random phases in the initial conditions , the location of the most massive halo at @xmath37 remains the same in both the wmap1 and wmap3 runs , except that its mass is reduced by a factor of @xmath93 for the wmap3 parameters . similarly , the mass of the largest halo at @xmath94 is altered by roughly the same factor . other notable changes include : ( 1 ) the formation epoch of the first halo is shifted from @xmath95 in the wmap1 run to @xmath96 in the wmap3 run , and ( 2 ) the merging history of the largest halo at @xmath0 moves to lower redshifts in the wmap3 run , but the number of major mergers remains the same . figure  [ figmf ] shows the halo mass functions from different cosmological simulations . the ps mass function @xcite , as well as the one corrected to match numerical simulations by @xcite ( st ) , are also shown for comparison . one important feature in this figure is that the coarse runs agree well with the st mass function , but show a larger comoving density at the high mass end than that predicted by the ps theory . our results show that the ps formula underestimates the abundance of high - mass halos by nearly an order of magnitude at @xmath97 , and the discrepancy between the ps calculation and numerical simulations becomes larger at higher redshifts , confirming previous findings ( e.g. , @xcite ) . this may explain why previous models using the ps formula to study the abundance of luminous quasars , which presumably form in massive halos , under - predicted the number of bright quasars at @xmath98 ( e.g. , @xcite ) . furthermore , these results also suggest that the commonly used analytical merger tree generated using the ps formula may not be suitable to study quasar formation at high redshifts . there are two clear `` shifts '' of the mass function caused by resolution and cosmological parameters . those from runs with higher resolution extend to higher redshifts , and at the same redshift , the wmap1 runs produce more massive halos than the wmap3 ones . as shown in figure  [ figmf ] , the coarse runs produce mass functions only up to @xmath99 owing to limited mass resolution , while the zoom - in runs can produce quite reasonable mass functions as early as @xmath100 . because the zoom - in runs were deliberately centered on the highest density peak of the @xmath35 box , they each contain a very massive halo ( @xmath101 at @xmath37 ) by construction . this explains why the highest mass bin ( which contains only one halo in this case ) is @xmath56 orders of magnitude larger than the theoretical curves ( st , ps ) , which apply only to a _ random _ region that has a much lower density fluctuation .    to summarize , at a given redshift , runs with the wmap3 parameters yield slightly less massive halos than ones performed with the wmap1 values . or , to put it differently , objects in the wmap3 cosmology will have masses similar to those for wmap1 , but at slightly later times ( i.e. lower redshifts ) . in what follows , we are primarily concerned with investigating the plausibility of forming @xmath57 quasars through the self - regulated growth of smbhs in hierarchical mergers , rather than precisely reproducing the properties of an individual quasar at a given redshift , such as j1148 + 5251 . most of our results are therefore based on runs with the wmap1 parameters , to ease comparison with previous numerical work . if it were firmly established that e.g. @xmath45 is in reality smaller than its wmap1 value , then a more exact match to a particular quasar could presumably be obtained by considering larger simulation volumes and identifying a suitable candidate host that is slightly rarer than the one we have chosen to focus on here . traced by mergers at different redshifts with mass ratio @xmath102 , which is defined as the mass ratio between the halo and progenitor at a given time . each of the progenitors joins in this big merging event at a given redshift , interacts with the system , and subsequently merges with others at later times . the quasar host at @xmath0 is built up by seven successive major mergers of progenitors g1 , g2 , ... g8 from @xmath96 to @xmath103 , as illustrated by the red lines in this plot . the first interaction between g1 and g2 takes place at @xmath96 , then g3 and g4 join in the system at @xmath104 , followed by g5 , g6 , g7 and g8 at later times ( see text for more details ) . the timeline of these events , the mass and other properties of these progenitors are listed in table  1.,width=336 ]    to follow the hierarchical mass assembly of the host galaxy over cosmic time , the merger tree of the halo is extracted from the cosmological simulation . this tree provides key information for computing the physical properties of the progenitor galaxy population . while the merger history of the halo includes a spectrum of progenitor masses , the most massive progenitors contribute the majority of the halo mass over the redshift range considered . we trace the merger history of the most massive progenitor at each redshift by using particle tags to identify progenitor systems at earlier redshifts in the simulation . groups that contribute at least 10% of the halo mass at a given time step are considered as the progenitors of the halo and are recorded . the procedure is repeated until the last progenitor is reached , producing the merging history . figure  [ figtree ] illustrates the merging history of the largest halo at @xmath37 in our cosmological simulation , which has a mass of @xmath105 . it is also the largest one at @xmath106 with a mass of @xmath107 . this schematic plot outlines the redshift of merger event , and the mass ratio of the halo to its galaxy progenitors at a given time . it shows that this halo grows rapidly through hierarchical mergers early on , with seven major mergers ( mass ratio of the merging pairs @xmath108 ) from @xmath96 to @xmath109 that build up a substantial fraction of the halo mass at @xmath0 .    in modeling the development of a @xmath57 quasar , we are primarily interested in `` major '' mergers , where the mass ratio of the merging galaxies is not too far from unity , for several reasons . first , it is believed that major mergers play the most important role in the formation and evolution of massive galaxies ( e.g. , @xcite ) . second , and of greater concern to us in this paper , in our picture for quasar fueling , gas in a rotationally supported disk loses angular momentum through gravitational torques excited by tidal forces in a merger , driving the growth of supermassive black holes . this process operates most effectively in a major merger because the tidal deformation of each galaxy is significant in such an event @xcite . collisions involving galaxies with a mass ratio as large as @xmath110 can induce gas inflows in disks @xcite , but only for limited orbital configurations . for these reasons , we focus on mergers from the merger tree having a mass ratio @xmath108 , as outlined by the red color in figure  [ figtree ] .    in the resimulation of the merger tree as described in   [ sec_mt ] , we take into account mass accretion of the halo by adding mass proportionally to each of the eight progenitor galaxies in the major mergers . this approach preserves the progenitor mass ratios and approximately preserves the dynamics of the major mergers @xcite . lccccccc [ tab1 ] g1 & 14.4 & 6.3 & 234.1 & 1.0 & 0.15 &  & + g2 & 14.4 & 5.3 & 220.3 & 1.0 & 0.15 & 0.2 & 7.1 + g3 & 13.0 & 15.0 & 297.8 & 1.0 & 0.51 & 0.2 & 8.5 + g4 & 13.0 & 17.7 & 314.6 & 1.0 & 0.51 & 0.3 & 10.7 + g5 & 10.5 & 49.1 & 401.0 & 1.0 & 6.56 & 0.4 & 11.3 + g6 & 9.4 & 79.6 & 448.6 & 0.9 & 23.3 & 0.5 & 18.2 + g7 & 8.5 & 160.0 & 540.4 & 0.9 & 89.2 & 0.7 & 25.2 + g8 & 8.5 & 207.7 & 589.5 & 0.9 & 89.2 & 1.0 & 34.5 +    in order to model the formation and evolution , and properties of the quasar candidate , the merger tree is then re - simulated hydrodynamically with galaxy models that consist of an extended dark matter halo , a rotationally supported , exponential disk of gas and stars and a central supermassive black hole . we follow the evolution of the system built up by seven major mergers hierarchically from @xmath96 to @xmath0 , as shown in figure  [ figtree ] . technically , this is a series of successive merger simulations . the first simulation includes g1 and g2 interacting at @xmath111 . it stops at @xmath112 and a new galaxy g3 is added into the system . during this process , all the dynamical properties of the pre - existing system ( e.g. g1 and g2 in this case ) are preserved , while g3 is added based on its properties and orbital parameters derived from cosmological simulations . then a second merger simulation with g1 , g2 and g3 starts . such a procedure is repeated until all the progenitors enter the system . in the end , the simulation includes all eight galaxies . eventually all these galaxies and black holes merge together . the duration of each merger simulation is determined by the merger tree . the redshift at which each progenitor galaxy enters the merger tree , the properties of each progenitor galaxy , and the numerical parameters of the merger simulations are listed in table  1 . below we describe the specification of these parameters . the structure of the galaxy models is motivated from leading theories of dissipational disk galaxy formation in cdm cosmologies that , as shown by @xcite , are successful in reproducing the observed properties of both present - day disk galaxies and damped ly@xmath113 absorbers in quasar spectra at high redshift . the initial galaxy models are constructed in dynamical equilibrium using a well - tested method ( @xcite ) . a halo is identified with a virial mass @xmath114 and a virial radius @xmath115 within which the overdensity @xmath116 , where @xmath79 and @xmath117 are the mean and critical density , respectively . the density profile of the dark matter halo follows a hernquist profile @xcite , scaled to match that found in cosmological simulations @xcite , as described in @xcite :    @xmath118    where @xmath119 is a parameter that relates the @xcite profile parameters to the appropriate nfw halo scale - length @xmath120 and concentration @xmath121 ( @xmath122 ) , @xmath123}\,.\ ] ]    the exponential disk of stars and gas are then constructed as in @xcite and @xcite . the properties of the galaxy , including the virial mass @xmath114 , virial radius @xmath115 and halo concentration @xmath121 are scaled appropriately with redshift , as described in @xcite . in particular , for a progenitor with virial velocity @xmath124 at redshift @xmath81 , @xmath114 and @xmath115 are calculated following @xcite , while @xmath121 is adopted from @xcite , as briefly outlined below :    @xmath125^{1/2}\ , , \\    { c_{\rm{vir } } } & = & 9 \left[\frac{{m_{\rm{vir}}}}{m_0}\right]^{-0.13 }    \left(1+z\right)^{-1 } \,.\end{aligned}\ ] ]    where @xmath126 is the gravitational constant , and @xmath127 is the linear collapse mass at the present epoch . we assume a baryon fraction of @xmath128 for these high - redshift galaxies based on the wmap1 result @xcite . the gas fraction of each progenitor is extrapolated from the results of semi - analytic models of galaxy formation @xcite , with 100% gas disks at @xmath129 and 90% at @xmath130 . the multiphase ism is envisioned to consist of cold clouds embedded in a hot , tenuous gas in pressure equilibrium . stars form out of the cold clouds by gravitational instability @xcite with a rate that is proportional to the surface density of the gas ( @xcite ; @xcite ) .    in the adopted ism model for the gas , the equation of state ( eos ) is controlled by a parameter @xmath131 that linearly interpolates between isothermal gas ( @xmath132 ) and a strongly pressurized multiphase ism ( @xmath133 ) . this eos describes the dynamics of star - forming gas and accounts for the consequences of stellar feedback on galactic scales , and enables us to construct equilibrium disk models even with large gas fractions @xcite . supernova feedback is modeled through thermal energy input into surrounding gas and can help evaporate the cold clouds to replenish the hot phase . for the simulation presented here a value of @xmath134 is used , but test simulations using @xmath135 produce average star formation and black hole accretion rates that converge to within 15% . the supermassive black holes are represented by collisionless `` sink '' particles . they interact with other particles gravitationally , and accrete the gas . accretion of gas onto the black holes is modeled using a bondi - hoyle - lyttleton parameterization ( @xcite ) , in which the black holes accrete spherically from a stationary , uniform distribution of gas , as described in @xcite and @xcite :    @xmath136    where @xmath137 is the black hole mass , @xmath138 and @xmath139 are the density and sound speed of the gas , respectively , @xmath113 is a dimensionless parameter of order unity , and @xmath140 is the velocity of the black hole relative to the gas . we assume the accretion has an upper limit by the eddington rate , @xmath141 where @xmath142 is the proton mass , @xmath143 is the thomson cross - section , and @xmath144 is the radiative efficiency . the latter determines the conversion efficiency of mass accretion into energy released as radiated luminosity . we adopt a fixed value of @xmath145 , which is the mean value for radiatively efficient @xcite accretion onto a schwarzschild black hole . in the simulations , the accretion rate is then the minimum of these two rates , @xmath146 . the feedback from the black holes is associated with the mass accretion . we assume that a small fraction ( @xmath147 ) of the radiated energy couples to the surrounding gas isotropically as feedback in form of thermal energy . this fraction is a free parameter , determined by matching the observed @xmath28 relation @xcite . for more discussions on this prescription , see @xcite . this feedback scheme self - regulates the growth of the black hole , and has been demonstrated to successfully reproduce many observed properties of elliptical galaxies , as mentioned earlier .      to grow a black hole up to @xmath148 in less than 800 million years , a wide range in seed masses , from @xmath149 to @xmath150 , have been suggested ( e.g. , @xcite ) . the formation of the black hole seeds remains an open question , and several scenarios have been proposed . in particular , @xcite show that rapid collapse of massive popiii stars due to pair instability could produce black hole of @xmath151 ; @xcite suggest that hot and dense gas clump may collapse monolithicly to form a massive black hole of @xmath152 in metal - free galaxies with a virial temperature of @xmath153 k ; while @xcite propose that @xmath154 black holes could form by direct collapse of self - gravitating gas due to global instabilities in protogalactic halos , they then grow to @xmath155 with super - eddington accretion . we adopt the picture where black hole seeds are the remnants of the first stars ( @xcite ) . the remnant black hole mass is currently uncertain and widely debated . recent theory of popiii star formation predicts a mass range of @xmath156 , but there are two regimes where a smbh could form , either @xmath157 or @xmath158 ( @xcite , see also @xcite for a recent discussion ) . we have tested the seed mass in the range of 100300  @xmath159 and find that the exponential growth of the black holes during the merger makes our results insensitive to the choice in that range . we therefore assume that the black hole seed starts with an initial mass of @xmath160 after the collapse of the first star at @xmath62 . these seed black holes then grow in the centers of @xmath152 halo which contains a large amount of high density primordial gas , as current theories predict that only one star forms per such mini - halo . the dense gas in the central region provides abundant fuel for bh accretion . to account for their evolution before the major mergers take place , the black holes are assumed to grow at the eddington rate until their host galaxies enter the simulated merger tree . such an approximation is supported by the fact that the eddington ratio in the simulations depends on the galaxy interaction and strength of the feedback from the black holes . in our simulations , most black holes grow at nearly the eddington rate in the early stages of a galaxy interaction when the feedback is weak . however , when the interaction and the feedback become stronger , the eddington ratios fluctuate by orders of magnitude . so a constant accretion rate at the eddington limit is no longer appropriate , as we show below . under this assumption , the first progenitor galaxies ( g1 and g2 ) of the quasar host have black hole seeds of order @xmath161 by the time it enters the merger tree at @xmath162 . however , we should emphasize that this assumption serves only as an upper limit of the early growth of the black holes . our results in the next sections imply that even if all the black hole seeds had a uniform mass of @xmath163 when they enter the merger tree , it is still possible to build a massive one to @xmath148 at @xmath3 through gas - rich mergers .    in our model , mergers are invoked in the formation of the most massive black holes of @xmath164 because that requires large supplies of gas . early on , however , this may not be necessary to grow the black hole seeds from @xmath165 to the @xmath163 we start from , because the accretion rate is small so other gas fueling could be sufficient . as demonstrated in @xcite , faint agns could be fueled by stochastic accretion of cold gas that does not involve mergers . a similar process could go on in the black hole seeds left by the popiii stars at very high redshifts . we should point out that in our simulations , it is necessary for galaxy progenitors in the merger tree to have reasonable massive black hole seeds ( @xmath163 ) initially in order to build a @xmath148 black hole at @xmath3 . however , our results are insensitive to specific formation recipes of the seeds . the formation of seed black holes at high redshifts is a challenging problem , and some of the proposed scenarios mentioned above may indeed be necessary to make our seeds . however , currently there is no observation available to test these models .    in the picture we adopt in which the seeds come from the first stars , the early accretion may be complicated by the feedback from the stars . we note that recent studies by @xcite and @xcite show that hii regions form around the first stars , and that the halo gas would be photo - ionized , photo - heated , and evacuated by the radiation feedback from the stars . @xcite suggest that such feedback would deplete the gas in the central region , and would delay the black hole accretion by up to @xmath166 yrs . however , this destruction effect depends sensitively on the lifetime of these massive stars , and more importantly on the environment which determines both the gas density profile and gas replenish from inflow of the expelled gas or neighboring halos . in the simulations presented in @xcite , the box size is only @xmath167 , too small to contain the large scale gravitational potential and the large wavelength density modes that drive gas infall , so the initial gas density is low and the destruction timescale is long in this case . however , the quasar halo in our simulation resides in the highest density peak in a volume of @xmath168 , where the halo potential and gas density , as well we the accretion rate are much higher @xcite . for a @xmath160 black hole , the accretion rate at eddington limit is only @xmath169 , which corresponds to the bondi accretion of molecular gas with a typical temperature of @xmath170 k at density @xmath171 , as implied from equation ( [ eq_bondi ] ) . such a density requirement is satisfied with the initial conditions of our model . therefore , the gas re - incorporation timescale in our case may be substantially shorter than that estimated in @xcite . we will investigate in a future project the growth and evolution of the early black holes after the death of the first stars in such a cosmological environment , using hydro - radiation simulations that include both radiative transfer and black hole accretion with ultra - high resolutions . the merger tree contains eight galaxies engaging in seven major mergers at different times . for each merger event , the initial orbits of the incoming progenitors are set to be parabolic , consistent with the majority of the major mergers in our cosmological simulation and with previous findings @xcite . the orientation of each merging galaxy is selected randomly . the initial separation between each merging pair is set to @xmath172 , where @xmath173 is the virial radius of the incoming system , while the pericentric distance is chosen as @xmath174 , where @xmath175 is the radial disk scale length of the incoming system . we have tested different choices of @xmath176 and orientations , and found that the impact of these parameters is minor because the orbital properties of the progenitors change rapidly through interactions with the multiple galaxies in the system .    throughout the merger simulation , the mass and force resolutions are fixed for each particle type , and the total initial particle number of @xmath177 results in particle masses of @xmath178 for the halo and @xmath179 for both the gas and stars . the gravitational softening lengths are @xmath180 for halo particles and @xmath181 for both gas and stars . in the simulations , it is impossible to resolve individual stars , and the accretion radii of some small black holes are under - resolved . however , with the sub - resolution implementation in our models , we can calculate time - averaged rates of star formation and black hole accretion from the large - scale properties of the gas , which are well resolved in our simulations . resolution studies of a single merger @xcite with particle numbers from @xmath182 to @xmath183 show that resolution affects some fine structures of the gas and the instantaneous growth rates of star and black holes , but the time - averaged properties of the system converge to within 20% .       as a function of distance @xmath184 ( @xmath115 is the virial radius ) to the halo center for various models in our merger simulations . this plot includes the isolated halo progenitors g1  g8 in table  1 , as well as the first merger remnant at @xmath185 and the last one at @xmath186 , as labeled in the legend . the shaded region indicates the range of the escape velocities of the mergers in our simulations , with the values in the central regions being @xmath187 . , width=336 ]    in a galaxy merger with black holes , the black holes may merge into one , or may be ejected by gravitational recoil in the final stage . their fate depends on the halo escape velocity @xmath188 . if the recoil velocity is larger than @xmath188 , then the black hole will be kicked out of the halo . we follow @xcite to calculate this important parameter @xmath189 . it is defined by @xmath190 where @xmath191 is the gravitational potential at a given radius @xmath192 . because the halo is spherical , the potentials of different spherical shells add linearly , so @xmath191 is contributed by two parts , i.e. , shells within @xmath192 ( @xmath193 ) and outside ( @xmath194 ) : @xmath195\,.\ ] ] where @xmath196 is again the @xcite density profile of dark matter halo as in equation  [ eq_hernquist ] . figure  [ figvesc ] shows the escape velocities of the halo progenitors g1  g8 in table  1 , as well as two merger remnants at @xmath197 and @xmath186 , respectively . the escape velocity depends on the halo mass , redshift , and distance from the halo center . the @xmath188 remains constant in the central region , begins to decline around @xmath198 . at the center , @xmath199 , while at the virial radius @xmath115 , the escape velocity is comparable to the virial velocity ( by a factor of @xmath200 ) . the isolated halo progenitors g1  g8 have a @xmath188 range of @xmath201  @xmath202 . the first merger halo at @xmath185 , which has a mass of @xmath203 as the merger of g1 and g2 , has a central escape velocity of @xmath204 , while the final merger halo at @xmath186 , which has a mass of @xmath205 , has @xmath206 . the shaded region indicates the range of the halo escape velocities of the mergers in our simulations . in particular , the escape speed in the halo central region has a range of @xmath207 . this range is important for analysis of black hole ejection from gravitational recoil in the black hole binaries in   [ subsec_bhgrowth ] and   [ subsec_bhbinary ] . the vigorous merging history of the quasar host is illustrated through selected snapshots of the gas and stellar distributions in figure  [ figpic_gas ] and figure  [ figpic_star ] , respectively . the progenitors at high redshifts are very compact and gas rich . as the host galaxy of the quasar builds up hierarchically , strong gravitational interactions between the merging galaxies lead to tidal tails , strong shocks and efficient gas inflow that triggers large - scale starbursts , a phenomenon that has been demonstrated by many numerical simulations ( e.g. , @xcite ) , as reviewed by @xcite . the highly concentrated gas fuels rapid accretion onto the smbhs ( @xcite ) . between @xmath208  149 , the merging systems are physically small and the interactions occur on the scale of tens of kiloparsecs . by @xmath208  97 , when the last major mergers take place , the scale and strength of interactions have increased dramatically . galaxies are largely disrupted in close encounters , tidal tails of gas and stars extend over hundreds of kiloparsecs , and intense bursts of star formation are triggered . the black holes continue to grow rapidly during this period but are heavily obscured by a significant amount of circumnuclear gas . during galaxy mergers , the black holes follow their hosts to the center of the system and can interact closely with each other . it has been shown that black hole binaries decay rapidly in a gaseous environment and can merge within @xmath209 yrs ( @xcite ) . because the galaxies in our simulations are very gas rich and the gas is highly concentrated during the mergers , we therefore assume that the black holes merge efficiently owing to strong dynamical friction with the gas @xcite . we will return to more discussions of this process in   [ subsec_bhgrowth ] and   [ subsec_bhbinary ] . at redshift @xmath210 the progenitor galaxies coalesce , inducing high central gas densities that bring the smbh accretion and feedback to a climax . the smbh feedback then drives a powerful galactic wind that clears the obscuring material from the center of the system . the largest smbh becomes visible as an optically - bright quasar @xcite during this phase , after which quasar feedback quenches star formation and self - regulates smbh accretion . consequently , both star formation and quasar activity die down , leaving a remnant which reddens rapidly , as illustrated schematically in figure  [ figpic_star ] . the evolution of the star formation rates ( sfrs ) of each individual galaxy , and total sfr of the whole system are shown in figure  [ figsfr ] ( _ top panel _ ) . the system forms stars rapidly as these compact and gas rich progenitors undergo strong interactions . the total sfr ranges from @xmath211 to @xmath212 between redshifts @xmath213  98 when the galaxies begin their final major mergers , while the sfrs of individual galaxies fall below a few@xmath214 , within the starburst intensity limit of @xmath215 proposed by @xcite and @xcite . at @xmath216 the star formation rate decreases gradually owing to a depletion of the gas supply and progressively stronger feedback from the smbhs . at the time of final coalescence ( @xmath210 ) the star formation rate is @xmath217 , an order of magnitude lower than for estimates of j1148 + 5251(@xcite ) . we note , however , that the estimates by these authors are based on the assumption that the fir luminosity is dominated by young stars , and they can not rule out the possibility that agn may contribute significantly to the luminosity .    in a forthcoming paper @xcite , we have calculated the infrared properties of the quasar system using a 3-d monte carlo radiative transfer code that incorporates adaptive grids and treats dust emission self - consistently . we find that the far - infrared luminosity of our quasar is not dominated by young stars but instead has a substantial quasar contribution of over @xmath218 . this finding is supported by observations of j1148 + 5251  in near - ir ( e.g. , @xcite , and more recently @xcite ) , which show a remarkably flat spectral energy distribution and suggest an agn origin for the flux excess . furthermore , adopting a total gas mass of @xmath219 ( @xcite ) in j1148 + 5251 , a simple application of the schmidt - kennicutt star formation law ( @xcite ) gives a star formation rate of @xmath220 , close to what we find here .    within only about 600 myrs from @xmath162 to @xmath65 , the system accumulates a stellar mass of @xmath221 as shown in figure  [ figsfr ] ( _ bottom panel _ ) . the specific star formation rate ( ssfr ) , or `` b - parameter '' ( e.g. , @xcite ) , is defined as @xmath222 . it is a measure of the fraction of the total stellar mass currently forming at a specific time . the ssfr is typically larger in high - redshift galaxies than in ones at low redshifts owing to vigorous star formation . during the past several years , there has been rapid progress in observing galaxies at @xmath12 using the _ hubble space telescope _ ( hst ) , and the _ spitzer space telescope _ ( spitzer ) coupled with ground - based observatories ( e.g. , ) , and hundreds of these distant objects have been detected . these frontier observations suggest that the universe experienced rapid star formation during the redshift interval @xmath223 , and the development of large stellar systems in the mass range of @xmath224 @xmath225 . in particular , several groups ( @xcite ) find ssfrs in the range of @xmath226@xmath227 in their observations , consistent with our simulations .      ) of each galaxy ( black curve ) . the blue hatched region indicates the range of 25%75% of the gas metallicity . , width=336 ]    , from both gas ( _ top panel _ ) and stars ( _ bottom panel _ ) , respectively . the images are projected metallicity adaptively smoothed over 32 particles ( analog to the sph kernel in a 2-dimensional plane ) . the black dot indicates the center of the quasar.,width=336 ]    rapid star formation in the quasar progenitors produces an abundant mass of heavy elements to enrich the ism . observations of j1148 + 5251  show solar metallicity in the system ( @xcite ) . figure  [ figmett ] shows the metallicity in our simulated quasar system at different times . note that the dips and jumps in the curves owe to incoming new galaxies which bring in metal - poor pristine gas and newly - formed stars . the quasar host reaches solar metallicity as early as @xmath228 , and maintains similar levels to later times . the spatial distribution of metallicity from both gas and stars at the peak quasar phase at @xmath229 is shown in figure  [ figmetr ] . the metals are widely spread owing to outflow from the quasar feedback and gas infall toward the merger center . the metallicity in the central region of the merger remnant is slightly above the solar value . in some outer regions , because the gas and stars are still falling back to the system center , the infalling material triggers small - scale bursts of star formation . so the metallicity in these blobs appears to be super - solar , as shown in figure  [ figmetr ] . calculations of carbon monoxide emission using non - local thermodynamic equilibrium radiative transfer codes @xcite by @xcite show co luminosities , excitation patterns , and morphologies within the central @xmath230 of the quasar host center that are consistent with observations of j1148 + 5251  ( @xcite ) . these results were derived using galactic co abundances , and thus support our conclusions that significant metal enrichment takes place early in the quasar host , as a result of strong star formation in the progenitors . ( _ middle panel _ ) , and black hole mass ( _ bottom panel _ ) . note that the black curves represent totals , while colored curves show individual black holes , as indicated in the legend.,width=336 ]    in the simulations , the quasar host at @xmath0 is built up by eight progenitors , each containing a black hole in the center . figure  [ figar ] shows the evolution of the black hole accretion rate , the eddington ratio , and the integrated masses of the whole system and individual black holes . the total black hole accretion rate grows steadily during the hierarchical assembly of the host galaxy and peaks at @xmath231 around @xmath210 during the final coalescence . the eddington ratio , @xmath232 , of each individual black hole varies with time , depending on the galaxy interaction and feedback from the black holes . the black holes maintain accretion at the eddington limit for only a fraction ( @xmath233 ) of the time . at the peak of quasar activity , the eddington ratio of the most massive black hole is near unity , while that of the other black hole is only 0.1 . however , collectively , the whole system appears to accrete at @xmath234 at @xmath235 , as implied in figure  [ figar ] . studies of black hole accretion ( e.g. , @xcite ) show that the eddington ratio has a wide range of 0.011.0 , and it varies with both luminosity and redshift . luminous systems tend to have higher @xmath232 than less - luminous counterparts , and at @xmath11 , most quasars shine at nearly eddington luminosity . our results suggest that _ individual _ black holes do not always necessarily accrete at the eddington rate . however , since high - redshift , luminous quasars may form through mergers of several galaxy progenitors containing black holes as in our case , therefore the growth of the quasar represents a _ collective _ contribution from each individual black hole . the total black hole mass increases from @xmath236 at @xmath237 to about @xmath238 at @xmath229 , close to that estimated for j1148 + 5251  by @xcite and @xcite .    in the simulations , we do not have sufficient resolution nor the relativistic physics to consider the ejection of black holes by gravitational recoil . the black holes are assumed to merge efficiently once their separation is below the spatial resolution . in the final stage of black hole mergers , the emission of gravitational wave carries linear momentum , which could cause the black holes to recoil ( e.g. , @xcite ) . if the recoil velocity is larger than the halo escape velocity , then the black holes will be kicked out from their halo ( e.g. , ) . previous studies by @xcite suggest that constant or super - eddington accretion is required to produce @xmath148 black holes at @xmath57 if ejection of black holes is included . in particular , @xcite suggests a black hole will be ejected if the kick velocity @xmath239 for the coalescing smbh binary is larger than twice the halo velocity dispersion @xmath240 , @xmath241 , as the dynamical friction timescale for the kicked black hole to return to the halo center is longer than the hubble time @xcite . by applying this ejection criterion to a press - schechter merger tree of a @xmath242 halo within which j1148 + 5251 is assumed to reside , @xcite finds that the smbh of the quasar gains most of its mass rapidly from seed holes during @xmath243 due to black hole ejection , and the smbh likely accretes with super - eddington rate in order to build a mass as that of j1148 + 5251 .    however , the halo escape velocities or velocity dispersions in our model are much larger than the currently best estimates of the kick velocity . the quasar halo in our simulations has active merging history from redshifts @xmath244 to @xmath186 , and the halo progenitors have masses much higher than those considered in the previous studies @xcite . the quasar halo builds its mass from @xmath245 at @xmath246 ( the sum of progenitors g1 and g2 in table  1 ) to @xmath205 at @xmath65 . as shown in figure  [ figvesc ] ( the shaded region ) , the central escape velocity of the mergers in our simulations is in the range @xmath247 . currently , the maximum kick velocity for unequal - mass , non - rotating bh binary is in the range of @xmath248 @xmath249 from both analytic post - newtonian approximation ( e.g. , @xcite ) and the ground - breaking full relativistic numerical simulations ( e.g. , @xcite ) . for equal - mass , spinning bh binary , @xcite estimate a range of @xmath170  @xmath250 using bh perturbation theory , and @xcite derive a formula from relativistic simulations , @xmath251 , where @xmath252 is the bh spin . this gives a maximum kick of @xmath253 for maximal spin . although it is also reported that the recoil velocity can be as large as thousands @xmath254 @xcite for bh binary in the orbital plane with opposite - directed spin . however , as pointed out by @xcite , such a configuration is rather uncommon , especially in gas - rich galaxy mergers , because torques from accreting gas suffice to align the orbit and spins of both black holes with the large - scale gas flow . the resulting maximum kick velocity from such a configuration is @xmath255 . overall , the kick velocity from the latest calculations of black hole binary is in the range of @xmath170 @xmath256 , falling safely below the escape velocities of the quasar halos in our simulations , so black hole ejection may be insignificant in our case . moreover , we find that our model can produce a @xmath148 smbh even if ejection is allowed . from figure  [ figar ] ( bottom panel ) , the @xmath148 smbh is dominated by bh5 , most of its mass comes from gas accretion . even if the less massive black holes , for example bh7 or bh8 was ejected , the most massive one bh5 is still able to reach @xmath1 in the end . furthermore , even if all the seeds started with @xmath257 in the merger tree , the result would be about the same . we therefore conclude that the results from our modeling are robust . suppermassive black holes of @xmath1 can grow rapidly through gas accretion and mergers hierarchically in the early universe , constant or super - eddington accretion is not necessary , unless the recoil velocity of the coalescing black hole binary is extremely high such that most of the black hole seeds in our simulations are ejected ( e.g. , @xmath258 ) .       at the present day , as parameterized by @xcite.,width=336 ]    tight correlations between supermassive black holes and hosts have been observed in local galaxies ( e.g. , @xcite ) , but the inference of these relationships at higher redshifts remains an open question . because the eight galaxies in our system interact vigorously with each other , the stellar components are widely spread and mixed , it is impossible to separate individual galaxy - smbh pairs , so we only consider the correlations in _ total _ quantity of the whole system . a comparison of the total stellar mass and total black hole mass is shown in figure  [ figmas ] . at early time , both the stars and black holes grow rapidly through galaxy mergers . shortly after the peak quasar phase , strong feedback suppresses both the accretion and star formation , the masses of the black holes and stars become saturated gradually , and in the end satisfy @xmath259 , similar to the correlation measured in nearby galaxies ( @xcite ) . our results are consistent with findings by @xcite and @xcite , and demonstrate that the observed @xmath10 scaling relation is a result of the coeval growth of the smbh and its host galaxy , and that it holds across different cosmic times . we note , however , that the velocity dispersion of the stars in the remnant center is about @xmath260 ( after the system relaxes ) owing to the deep potential well of the merger system , so the @xmath28 relation falls below the correlation observed locally ( @xcite ) . single mergers of progenitor galaxies constructed in a redshift range of z=06 by @xcite appear to follow the observed @xmath28 correlation with a weak redshift dependence of the normalization , which results from an increasing velocity dispersion of the progenitors at higher redshift . the multiple mergers we derive from cosmological simulations take place at much higher redshifts and hence the progenitors have larger velocity dispersions , implying a larger deviation from the local @xmath28 relation than in the work of @xcite . however , because we do not follow subsequent mergers and accretion into the host halo below @xmath57 , the implications of this result for the evolution of the @xmath28 relation are unclear . observations of active galaxies have yielded ambiguous results about the smbh  spheroid relationship . for example , @xcite report a lower zero - point of the @xmath28 of local active galaxies than that of the inactive sample @xcite ; at @xmath261 , @xcite found the same @xmath28 relation in the redshift range @xmath26213 , while others ( e.g. , @xcite ) show correlations with various offsets . in particular , @xcite estimate a dynamical mass of @xmath263 using the co linewidth measured in j1148 + 5251 , and suggest that the bulge is under - massive by at least one order of magnitude compared to the local @xmath10 relation . however , the co calculation by @xcite finds that the co linewidth of the quasar in our simulation is larger than the mean 280 km s@xmath264 measured by @xcite and @xcite by almost an order of magnitude , and that the derived dynamical mass is @xmath265 , putting the simulated quasar on the @xmath10 correlation . @xcite further suggest that the observed emission line may be sitting on top of a much broader line , which may be tested by future observations with large bandwidths . the different relations reported from the observations may reflect a divergence of the methods used to estimate the black hole mass and stellar properties , or may represent different evolutionary stages of the systems @xcite . more observations and better measurements of black hole mass and properties of host bulges will be crucial to study the smbh  host relations in high - redshift quasar systems @xcite and to test our hypothesis . .,width=336 ]    both the bolometric and attenuated luminosities of the quasar and the host galaxy in the simulations can be readily calculated following the methodology of @xcite . the bolometric luminosity @xmath266 of stars is calculated using the stellar population synthesis model of @xcite , while that of a black hole is calculated as @xmath267 , where @xmath268 is the radiative efficiency , @xmath269 is the black hole accretion rate , and @xmath270 is the speed of light . in this calculation , the black holes are assumed to be non - rotating . if the black holes are spinning , their radiative efficiencies and luminosities would be higher due to the shrink of the innermost stable circular orbit , by up to a factor of 4 for maximal rotation . the b - band luminosity of each source is corrected for attenuation by absorption from the ism along the line - of - sight . we first calculate the line - of - sight column - density of the gas from each source to a distant observer . for each black hole we generate 1000 radial sight lines originating at the black hole particle location and uniformly spaced in the @xmath271 solid angle @xmath272 , while for the stars , an accurate estimate of the luminosity is possible with only one sight line per source owing to the extended distribution . along each ray , the gas column density is calculated using a radial spacing of @xmath273 , where @xmath274 and @xmath275 is the local sph smoothing length . the distribution of line - of - sight properties converges for @xmath276 rays and at a distance of @xmath277 . in the calculation , only the diffuse - phase density is considered because of its large volume filling factor @xmath278 , allowing for a determination of the lower limit on the column density along a particular line of sight .    adopting the mean observed intrinsic quasar continuum spectral energy distribution @xcite gives a b - band luminosity which is well approximated by the following equation given by @xcite , @xmath279 , where @xmath280 , and @xmath281 . we then use the milky way gas - to - dust ratio scaled by metallicity , @xmath282 to determine the extinction along a given line of sight for this band . in the above calculation , we do not include a full treatment of radiative transfer , and therefore do not model scattering or re - processing of radiation by dust in the infrared . however , for the b - band luminosity , results using a 3-d monte carlo radiative transfer code are close to those calculated using the methods we present here @xcite .    figure  [ figlum ] shows both the bolometric and attenuated b - band luminosities of the quasar , compared with observations of j1148 + 5251 . the system is intrinsically bright with a total luminosity @xmath283 , and the host appears as an ultraluminous infrared galaxy ( ulirg ) with @xmath284 for most of the time . at high redshifts , @xmath285 , starlight dominates the total luminosity . however , black holes take over at a later time . the quasar light - curve increases dramatically , peaking at @xmath30 , when it is powered by the most massive black hole accreting at near the eddington rate . the estimated @xmath266 of j1148 + 5251  differs from that of the simulated quasar by less than the uncertainty in the luminosity estimate . the rest - frame b - band absolute magnitude reaches @xmath286 , almost one magnitude fainter than that of j1148 + 5251  derived from 1450@xmath287  data @xcite . however , we should emphasize that in this paper , our main goal is to investigate the plausibility of forming luminous @xmath13 quasars through hierarchical mergers , rather than precisely reproducing the properties of an individual quasar such as j1148 + 5251 , so the disagreement shown in figure  [ figlum ] should not be taken too literally . moreover , the _ exact _ luminosity can change by a factor of several from relatively trivial or random details in the simulations . if the black hole spin is taken into account , then the simulated luminosities would increase by a factor of up to 4 , which would match the observation of j1148 + 5251  better . feedback - driven outflows create un - obscured lines - of - sight , allowing the growing central smbh to be visible as an optically - bright quasar between redshifts @xmath288 and @xmath74 . at the peak of the quasar activity , more than 50% of the 1000 sight lines have @xmath289 . the absorbed light is re - emitted at infrared wavelengths by dust . we find that the luminosity in the far infrared @xcite is close to @xmath290 estimated for j1148 + 5251  by @xcite . moreover , we find that up to 80% of the fir light comes from the black hole , while stars contribute only @xmath291 . this may explain why the star formation rate at @xmath292 during the peak quasar phase is an order of magnitude lower than the estimate from fir observations @xcite , which will be contaminated by the agn . another prominent feature of figure  [ figlum ] is a clear phase transition from starburst to quasar . it has long been suggested that ulirgs are powered by starbursts in galaxy mergers ( for reviews , see @xcite ) , and that bright quasars are the descendants of ulirgs ( @xcite ) . this conjecture has been supported by observations of quasar hosts ( e.g. , @xcite ) , and theoretical modeling @xcite . in @xcite , we calculate the spectral energy distributions ( seds ) of the quasar system and its galaxy progenitors . we find that the seds of the system at @xmath285 are characterized by those of starburst galaxies @xcite , while at the peak quasar phase , the seds resemble those observed in @xmath0 quasars @xcite . we also find that the system evolves from cold to warm ulirg as it transforms from starburst to quasar phase . our results provide further theoretical evidence for the ulirg quasar connection in quasar systems in the early universe . the quasar lifetimes depend on the observed luminosity threshold , as proposed by @xcite . in our simulation , at the peak luminosity of @xmath293 , the quasar lifetime is roughly @xmath294 yrs , as shown in figure  [ figtq ] . again , if black hole spin is included in the calculation , the luminosity of the quasar would increase by a factor of several , and the quasar lifetime would be longer . however , when increasing the radiative efficiency , the salpeter time ( @xmath295-folding time for eddington - limited black hole growth , @xcite ) is increased by an identical factor , meaning it would also require a longer time to reach the same mass . if high - redshift quasars are rapidly rotating , then , our calculations demand that either the seeds be _ much _ more massive at @xmath12 , or that they accrete in a super - eddington manner . in other words , if the observed sloan quasars at @xmath0 shine with eddington luminosity but are rotating rapidly , then our model suggests that their masses would be considerably smaller than estimated . we note that recent _ spitzer _ observations by @xcite show that 2 out of 13 quasars at @xmath57 have a remarkably low nir - to - optical flux ratio compared to other quasars at different redshifts , and these authors suggest that the two quasars may have different dust properties . according to our model , however , these two outliers may be young quasars that have just experienced their first major starburst but have not yet reached peak quasar activity , so the light from star formation may be dominant , or comparable to that from the accreting smbh still buried in dense gas . this may explain the low nir flux , as well as the b - band luminosity and the narrow @xmath296 emission line , which are primarily produced by the starburst . we will address this question further in a future paper with detailed modeling and ir calculations @xcite . our multi - scale simulations that include large - scale cosmological n - body calculations and hydrodynamic simulations of galaxy mergers , as well as a self - regulated model for black hole growth , have successfully produced a luminous quasar at @xmath3 with a black hole mass of @xmath297 and a number of properties similar to those of j1148 + 5251 , the most distant quasar detected at @xmath298 @xcite . our approach differs from previous semi - analytic studies by @xcite and @xcite in the following ways :    1 . we use a realistic merger tree derived directly from multi - scale , high - resolution cosmological simulations . the previous studies used merger trees of dark matter halos generated with the extended press - schechter theory @xcite , which may underestimate the abundance of high - mass halos by up to one order of magnitude , as shown in   [ subsec_mf ] . also , the merger trees in those studies started from much higher redshifts than what we consider here . in our model , the quasar halo is the largest one in a volume of @xmath299 . it has a mass of @xmath300 at @xmath301 built up through seven major mergers from @xmath244 to @xmath302 . we follow the evolution of the system and treat the gas dynamics , star formation , and black hole growth properly . this approach is critical to investigation of the properties of both black holes and host galaxies , and their evolution ( e.g. , @xcite ) , but it was not included in those previous studies on formation of @xmath0 quasars . we employ a self - regulated model for the growth of supermassive black holes , in which the accretion is regulated by the black hole feedback , and the rate is under the eddington limit . in the previous studies , the black hole growth was unregulated , but instead a constant- or super - eddington accretion rate was used . 4 .   in our simulations , we do not consider black hole ejection caused by gravitational recoil owing to insufficient resolution and lack of relativistic physics . however , the halo escape velocities in our simulations are in the range of @xmath303 @xmath304 , much larger than the kick velocity @xmath170  @xmath253 ( e.g. , @xcite , see   [ subsec_vesc ] and   [ subsec_bhgrowth ] for more details ) . therefore , black hole ejection may be negligible in our case . previous studies had much smaller halo progenitors at higher redshifts than ours , so the black hole seeds would be more likely subject to ejection from their halos . this leads to the conclusion in these studies that constant or super - eddington accretion is needed owing to significant black hole ejection . our results are robust within the best estimates currently available for the recoil velocity of the black hole binary . the black hole seeds in the galaxy progenitors in our simulations are massive ( e.g. , @xmath163 at @xmath305 ) . the sub - resolution recipe in our model does not allow us to resolve the actual formation and accretion of such black holes below this mass scale . the formation of these seeds is an unsolved problem , but our results do not depend on the specific prescription of the formation process . we adopt a picture in which the seed holes come from the remnants of the first stars ( which have a mass @xmath160 at @xmath306 ) and grow under eddington limit until they enter the merger tree we simulated . if the growth is delayed by radiation feedback from the popiii stars ( e.g. , @xcite ) , then super - eddington accretion , or other proposed scenarios ( e.g. , @xcite ) may be necessary to form massive seeds of @xmath307 in the protogalaxies . overall , we conclude that the results from our simulations , which are more realistic and more detailed than the models previously done , are robust . suppermassive black holes of @xmath1 can form rapidly through gas - rich hierarchical mergers under eddington limit , even within a short period of time . we find that constant or super - eddington accretion is not necessary unless the above assumptions in our modeling break , i.e. , there are no massive black hole seeds of @xmath308 available at @xmath309 , or the recoil velocity of the coalescing black hole binary is extremely high ( e.g. , @xmath310 ) . under these extreme circumstances , some `` exotic '' processes such as super - eddington accretion may be necessary to grow a @xmath1 smbh within a few hundred million years . however , we should note , as pointed out by @xcite , that most gas - rich galaxy mergers have a configuration such that the orbit and spins of both black holes are aligned with the large - scale gas flow owing to torques from accreting gas . such a configuration has a maximum kick velocity @xmath255 , which is well below the escape velocity of a @xmath311 dwarf galaxy , as well as those of the halos in our modeling .      during the galaxy mergers , the black holes follow their host halos to the system center and can form binaries ( or multiple systems ) . the coalescence of a black hole binary includes three distinct phases : inspiral , merger , and ringdown ( e.g. , @xcite ) . whether black hole binaries can coalesce on short timescales is a matter of debate . in a stellar environment , it has been argued that a binary hardens very slowly owing to an eventual depletion of stars that cause the binary to lose angular momentum ( e.g. , @xcite ) . in a gaseous environment , however , numerical simulations by @xcite and @xcite show that the binaries decay rapidly owing to strong dynamical friction with the gas , and they likely merge within @xmath312 years . because our galaxies are very gas rich and have large central concentrations of gas during the mergers , we assume the black hole particles coalesce once their separation decreases below our spatial resolution ( @xmath313 ) and their relative speed falls below the local gas sound speed @xcite .    in the simulations , we do not have sufficient resolution nor the relativistic physics to consider the ejection of black holes by gravitational recoil during the merger phase . however , as discussed in [ subsec_vesc ] and   [ subsec_bhgrowth ] , the halo escape velocities in our simulations are much larger than the maximum kick velocity for black hole binary estimated from the latest relativistic calculations . so black hole ejection is likely unimportant in our modeling . to accurately address gravitational recoil in the galaxy merger simulations , we need to include general relativity , resolve the dynamics of black hole binaries with extremely high resolution , and calculate the halo potential in a cosmological context ( in which halo potential distribution may be different from that of a single object ) . however , such a comprehensive treatment is impossible at the moment . we therefore assume the black holes merge quickly once they reach the stage of gravitational radiation . these coalescing supermassive black holes will be strong sources of gravitational radiation detectable by the laser interferometer space antenna ( _ lisa _ , @xcite ) , as suggested by many authors ( e.g. , @xcite ) . by tracing the merging history of the smbhs , _ lisa _ could shed light on the distribution , structures and evolution of the associated dark matter halos . because luminous , high - redshift quasars are likely sites of vigorous hierarchical mergers , they may be the best targets for _ lisa _ to explore the early universe .      ) , and the specifications of the wind model is : wind efficiency @xmath314 , wind energy coefficient @xmath315 , wind free travel length @xmath316 , and a wind velocity @xmath317.,width=336 ]    vigorous star formation would induce a galactic wind and mass outflow , a phenomenon that has been observed to prevail in both local star - forming galaxies as indicated by blueshifted optical absorption lines ( e.g. , @xcite ) , and lyman break galaxies at @xmath318 as indicated by blueshifted interstellar absorption lines and redshifted ly@xmath113 emission lines ( e.g. , @xcite ) , as well as ly@xmath113 emitters at @xmath319 @xcite . these galactic winds are generally thought to play a significant role in galaxy evolution ( e.g. , see @xcite and @xcite for recent reviews ) . the strong starburst preceding the major quasar phase in our simulations may drive strong galactic winds and affect the black hole growth . to investigate the impact of the feedback from a starburst - driven wind on the growth of the black hole , we have done the same merger simulation with lower resolution ( @xmath320 ) and with a canonical wind model from @xcite : the wind efficiency @xmath314 , which measures the coefficient of the star formation that determines the mass outflow ; the energy fraction from supernovae injected into the wind @xmath315 , wind free travel length @xmath316 , and a wind velocity @xmath321 . as demonstrated by @xcite , this wind model is able to reproduce the starbursts as observed in lyman break galaxies , and therefore is suitable to our study . we find that the impact of the starburst - driven wind on the quasar evolution is minor , as shown in figure  [ figwind ] . both the histories of star formation and black hole growth remain roughly the same as in the simulation without a starburst wind , only the amplitude is lowered by a factor of @xmath200 . similarly , the final masses of the black hole and the stars are reduced by roughly the same factor , but the quasar host is still on the @xmath10 correlation . the peak quasar phase is delayed to @xmath0 . overall , the starburst wind affects the gas dynamics locally , but owing to the deep potential of the system , its impact on the process of quasar formation is minor . our results support the finding by @xcite that feedback from starburst - driven winds alone is ineffective at regulating the growth of the central black hole , so feedback from the black hole plays the dominant role in the formation and evolution of quasars .       from the zoom - in simulations with parameters from both wmap1 ( solid line ) and wmap3 ( dashed line ) . the volume of the high - resolution zoom - in region is @xmath322.,width=336 ]    because we have so far simulated only one quasar in a volume of @xmath323 , we are not yet able to constrain the expected abundance of quasars @xmath57 . as mentioned in   2.3 , at a given redshift , cosmological simulations with parameters from wmap1 produce more massive halos than runs with wmap3 owing to a larger value of @xmath45 . figure  [ fignum ] shows the number of halos at @xmath57 from the zoom - in runs with parameters from both wmap1 and wmap3 . there are about three dozen halos with mass @xmath324 in the wmap1 run , while in the wmap3 run there are only a handful of such halos . however , since in our picture the quasar activity depends not only on the halo mass , but also on the merging history , an accurate estimate of the quasar abundance and luminosity function would require hydrodynamical simulations of all the quasar candidates in a large box , which are currently unavailable . nevertheless , all conditions being equal , the change from the wmap3 parameters would produce fewer luminous quasars at @xmath0 . this suggests that in a wmap3 cosmology , the quasar observed with the largest redshift , j1148 + 5251 , might have formed in a slightly higher overdensity peak than that we have presented here . in that event , if the wmap3 determination of @xmath45 were correct , we would need to identify a rarer density fluctuation to match j1148 + 5251  at its observed redshift . however , this does not change our conclusion that the most distant and luminous quasars can form from hierarchical galaxy mergers in the @xmath2cdm cosmology . imaging surveys of j1148 + 5251  show that there is no other luminous quasar from the same epoch in the field ( @xcite ) . in our simulations , around the peak of quasar activity at @xmath292 , there are no other halos of mass @xmath325 within a few mpc of this quasar . however , the numerous major mergers this halo experienced prior to the peak quasar activity demonstrates that this region was once highly clustered with massive halos , but they merged to become a bigger one by @xmath326 .    as seen from figure  [ figtree ] , this quasar halo will undergo a handful of major mergers at a later time from @xmath327  41 , and eventually end up as a cd - like galaxy at the center of a rich cluster . since we do not follow hydrodynamically the evolution of the quasar at @xmath328 , the physical conditions of these mergers remain undetermined . it is not clear whether this halo would experience more episodes of starburst or quasar activity later on during these mergers . therefore , the final black hole mass and other properties of this quasar at the present day are deferred to future simulations that follow its evolution to @xmath37 . the epoch of reionization ( eor ) is an important landmark event in cosmic history that constrains the formation of the first luminous objects @xcite . the recent results of wmap3 indicate that the universe was 50% reionized at @xmath329 @xcite , while studies of gunn - peterson absorption @xcite suggest that reionization began as early as @xmath305 and ended at @xmath0 @xcite . at present , it is believed that the reionization sources are star - forming galaxies since there are insufficient quasars at @xmath330 as indicated by the steep quasar luminosity function @xcite . the galaxy progenitors of the quasar in our simulations underwent extreme and prolonged starbursts before @xmath3 . less extreme galaxies in this epoch may also have vigorous star formation histories . detecting these galaxies and determining their contribution to reionization will be crucial to understanding the eor ( @xcite ) . as reviewed by @xcite , recent observations using both broad - band colors ( e.g. , @xcite ) and narrow - band ly@xmath113 emission ( e.g. , @xcite ) have detected @xmath331 galaxies at @xmath0 and a handful at @xmath332 @xcite . the low luminosity density of galaxies currently detected at @xmath333 seems insufficient to reionize the universe . however , ongoing surveys with the hst and spitzer telescopes , and future missions such as the _ dark ages z ly@xmath113 explorer _ ( dazle , @xcite ) and the _ james webb space telescope _ ( jwst , @xcite ) will search deeper and further for high - redshift objects , and may eventually unveil ionizing sources in the eor . we have presented a model that accounts for the smbh growth , quasar activity and host galaxy properties of the most distant quasar observed at @xmath334 , by following the hierarchical assembly of the quasar halo in the standard @xmath2cdm cosmology . we employ a set of multi - scale simulations that include large - scale cosmological n - body calculations and hydrodynamic simulations of galaxy mergers , and a recipe for black hole growth self - regulated by feedback . we first perform a coarse n - body simulation in a volume of @xmath35 to identify the largest halo at @xmath37 , which is assumed to be the descendant of the earliest luminous quasar . we then `` zoom in '' on the halo and resimulate the region with higher resolution sufficient to extract its merging history starting from very high redshift . the largest halo at @xmath0 reaches a mass of @xmath335 through 7 major mergers between @xmath208 14.46.5 . these major mergers are again re - simulated hydrodynamically using galaxy models and recipes for star formation , smbh growth , and feedback . we find that the quasar host galaxy builds up rapidly through gas - rich major mergers , with star formation rates up to @xmath336 , reaching a stellar mass of @xmath265 at @xmath9 . the black holes grow through gas accretion under the eddington limit in a self - regulated manner owing to their own feedback . as the galaxies merge , the black holes coalescence to form a dominant black hole , reaching a peak accretion rate of @xmath337 and a mass of @xmath338 at @xmath339 . feedback from black hole accretion clears away the obscuring gas from the central regions , making the quasar optically visible from @xmath208  7.56 . at the peak of the quasar phase , the star formation rate , metallicity , black hole mass , as well as quasar luminosities of the simulated system are consistent with observations of j1148 + 5251 . our results demonstrate that rare and luminous quasars at high redshifts can form in the standard @xmath2cdm cosmology through hierarchical , gas - rich mergers , within the available cosmic time up to the early epoch of @xmath340 , without requiring exotic processes . our model should also provide a viable formation mechanism for other distant , luminous quasars . moreover , we predict that quasar hosts at high redshifts follow similar @xmath10 correlation observed locally as a result of the coeval evolution of the smbhs and host galaxies . better measurements of black hole masses and host properties with future observations will therefore be crucial to test our prediction . furthermore , we predict that the progenitors of the distant quasars undergo strong and prolonged starbursts with rates @xmath341 at higher redshifts @xmath285 , that would contribute to the reionization of the universe . detecting these early galaxies and unveiling the epoch of reionization will be an important goal of current and future missions in observational cosmology . , m. , taniguchi , y. , murayama , t. , nagao , t. , veilleux , s. , shioya , y. , fujita , s.  s. , kakazu , y. , komiyama , y. , okamura , s. , sanders , d.  b. , oyabu , s. , kawara , k. , ohyama , y. , iye , m. , kashikawa , n. , yoshida , m. , sasaki , t. , kosugi , g. , aoki , k. , takata , t. , saito , y. , kawabata , k.  s. , sekiguchi , k. , okita , k. , shimizu , y. , inata , m. , ebizuka , n. , ozawa , t. , yadoumaru , y. , taguchi , h. , ando , h. , nishimura , t. , hayashi , m. , ogasawara , r. , & ichikawa , s .- i . 2002 , , 576 , l25                            , r.  h. , fan , x. , white , r.  l. , strauss , m.  a. , narayanan , v.  k. , lupton , r.  h. , gunn , j.  e. , annis , j. , bahcall , n.  a. , brinkmann , j. , connolly , a.  j. , csabai , i. , czarapata , p.  c. , doi , m. , heckman , t.  m. , hennessy , g.  s. , ivezi ,  . , knapp , g.  r. , lamb , d.  q. , mckay , t.  a. , munn , j.  a. , nash , t. , nichol , r. , pier , j.  r. , richards , g.  t. , schneider , d.  p. , stoughton , c. , szalay , a.  s. , thakar , a.  r. , & york , d.  g. 2001 , , 122 , 2850                                                                          , m. , stern , d. , giavalisco , m. , ferguson , h.  c. , tsvetanov , z. , chornock , r. , cristiani , s. , dawson , s. , dey , a. , filippenko , a.  v. , moustakas , l.  a. , nonino , m. , papovich , c. , ravindranath , s. , riess , a. , rosati , p. , spinrad , h. , & vanzella , e. 2004 , , 600 , l99                            , x. , hennawi , j.  f. , richards , g.  t. , strauss , m.  a. , schneider , d.  p. , donley , j.  l. , young , j.  e. , annis , j. , lin , h. , lampeitl , h. , lupton , r.  h. , gunn , j.  e. , knapp , g.  r. , brandt , w.  n. , anderson , s. , bahcall , n.  a. , brinkmann , j. , brunner , r.  j. , fukugita , m. , szalay , a.  s. , szokoly , g.  p. , & york , d.  g. 2004 , , 128 , 515    , x. , narayanan , v.  k. , lupton , r.  h. , strauss , m.  a. , knapp , g.  r. , becker , r.  h. , white , r.  l. , pentericci , l. , leggett , s.  k. , haiman , z. , gunn , j.  e. , ivezi ,  . , schneider , d.  p. , anderson , s.  f. , brinkmann , j. , bahcall , n.  a. , connolly , a.  j. , csabai , i. , doi , m. , fukugita , m. , geballe , t. , grebel , e.  k. , harbeck , d. , hennessy , g. , lamb , d.  q. , miknaitis , g. , munn , j.  a. , nichol , r. , okamura , s. , pier , j.  r. , prada , f. , richards , g.  t. , szalay , a. , & york , d.  g. 2001 , , 122 , 2833    , x. , strauss , m.  a. , schneider , d.  p. , becker , r.  h. , white , r.  l. , haiman , z. , gregg , m. , pentericci , l. , grebel , e.  k. , narayanan , v.  k. , loh , y .- s . , richards , g.  t. , gunn , j.  e. , lupton , r.  h. , knapp , g.  r. , ivezi ,  . , brandt , w.  n. , collinge , m. , hao , l. , harbeck , d. , prada , f. , schaye , j. , strateva , i. , zakamska , n. , anderson , s. , brinkmann , j. , bahcall , n.  a. , lamb , d.  q. , okamura , s. , szalay , a. , & york , d.  g. 2003 , , 125 , 1649                    , k. , bender , r. , bower , g. , dressler , a. , faber , s.  m. , filippenko , a.  v. , green , r. , grillmair , c. , ho , l.  c. , kormendy , j. , lauer , t.  r. , magorrian , j. , pinkney , j. , richstone , d. , & tremaine , s. 2000 , , 539 , l13    , m. , dickinson , m. , ferguson , h.  c. , ravindranath , s. , kretchmer , c. , moustakas , l.  a. , madau , p. , fall , s.  m. , gardner , j.  p. , livio , m. , papovich , c. , renzini , a. , spinrad , h. , stern , d. , & riess , a. 2004 , , 600 , l103                                                                          , a. , parry , i. , bland - 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the discovery of luminous quasars at redshift @xmath0 indicates the presence of supermassive black holes ( smbhs ) of mass @xmath1 when the universe was less than one billion years old . 
 this finding presents several challenges for theoretical models , because whether such massive objects can form so early in the @xmath2-cold dark matter ( @xmath2cdm ) cosmology , the leading theory for cosmic structure formation , is an open question . 
 furthermore , whether the formation process requires exotic physics such as super - eddington accretion remains undecided . here , we present the first multi - scale simulations that , together with a self - regulated model for the smbh growth , produce a luminous quasar at @xmath3 in the @xmath2cdm paradigm . 
 we follow the hierarchical assembly history of the most massive halo in a @xmath4 volume , and find that this halo of @xmath5 forming at @xmath3 after several major mergers is able to reproduce a number of observed properties of sdss j1148 + 5251 , the most distant quasar detected at @xmath6 @xcite . moreover , the smbhs grow through gas accretion below the eddington limit in a self - regulated manner owing to feedback . 
 we find that the progenitors experience vigorous star formation ( up to @xmath7 ) preceding the major quasar phase such that the stellar mass of the quasar host reaches @xmath8 at @xmath9 , consistent with observations of significant metal enrichment in sdss j1148 + 5251 . 
 the merger remnant thus obeys similar @xmath10 scaling relation observed locally as a consequence of coeval growth and evolution of the smbh and its host galaxy . 
 our results provide a viable formation mechanism for @xmath0 quasars in the standard @xmath2cdm cosmology , and demonstrate a common , merger - driven origin for the rarest quasars and the fundamental @xmath10 correlation in a hierarchical universe .