The discovery of luminous quasars at redshift z ∼ 6 indicates the presence of supermassive black holes (SMBHs) of mass ∼ 10 9 M ⊙ 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 Λcold dark matter (ΛCDM) 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 z ∼ 6.5 in the ΛCDM paradigm. We follow the hierarchical assembly history of the most massive halo in a ∼ 3 Gpc 3 volume, and find that this halo of ∼ 8 × 10 12 M ⊙ forming at z ∼ 6.5 after several major mergers is able to reproduce a number of observed properties of SDSS J1148+5251, the most distant quasar detected at z = 6.42 . 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 10 4 M ⊙ yr −1 ) preceding the major quasar phase such that the stellar mass of the quasar host reaches 10 12 M ⊙ at z ∼ 6.5, consistent with observations of significant metal enrichment in SDSS J1148+5251. The merger remnant thus obeys similar M BH −M bulge 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 z ∼ 6 quasars in the standard ΛCDM cosmology, and demonstrate a common, merger-driven origin for the rarest quasars and the fundamental M BH −M bulge correlation in a hierarchical Universe.

Abstract. The thermodynamic state of star-forming gas determines its fragmentation behavior and thus plays a crucial role in determining the stellar initial mass function (IMF). We address the issue by studying the effects of a piecewise polytropic equation of state (EOS) on the formation of stellar clusters in turbulent, self-gravitating molecular clouds using three-dimensional, smoothed particle hydrodynamics simulations. In these simulations stars form via a process we call gravoturbulent fragmentation, i.e., gravitational fragmentation of turbulent gas. To approximate the results of published predictions of the thermal behavior of collapsing clouds, we increase the polytropic exponent γ from 0.7 to 1.1 at a critical density n c , which we estimated to be 2.5 × 10 5 cm −3 . The change of thermodynamic state at n c selects a characteristic mass scale for fragmentation M ch , which we relate to the peak of the observed IMF. A simple scaling argument based on the Jeans mass M J at the critical density n c leads to M ch ∝ n −0.95 c . We perform simulations with 4.3 × 10 4 cm −3 < n c < 4.3 × 10 7 cm −3 to test this scaling argument. Our simulations qualitatively support this hypothesis, but we find a weaker density dependence of M ch ∝ n −0.5±0.1 c . We also investigate the influence of additional environmental parameters on the IMF. We consider variations in the turbulent driving scheme, and consistently find M J is decreasing with increasing n c . Our investigation generally supports the idea that the distribution of stellar masses depends mainly on the thermodynamic state of the star-forming gas. The thermodynamic state of interstellar gas is a result of the balance between heating and cooling processes, which in turn are determined by fundamental atomic and molecular physics and by chemical abundances. Given the abundances, the derivation of a characteristic stellar mass can thus be based on universal quantities and constants.

Using a model for the self-regulated growth of supermassive black holes in mergers involving gas-rich galaxies, we study the relationship between quasars and the population of merging galaxies and thereby predict the merger-driven star formation rate density of the Universe. In our picture, mergers drive gas inflows, fueling nuclear starbursts and "buried" quasars until feedback disperses the gas, allowing the quasar to be briefly visible as a bright optical source. As black hole accretion declines, the quasar dies and the stellar remnant relaxes passively with properties and correlations typical of red, elliptical galaxies. By simulating the evolution of such events, we demonstrate that the observed statistics of merger rates/fractions, luminosity functions, mass functions, star formation rate distributions, quasar luminosity functions, quasar host galaxy luminosity functions, and elliptical/red galaxy luminosity and mass functions are self-consistent. We use our simulations to de-convolve both the quasar and merging galaxy luminosity functions to determine the birthrate of black holes of a given final mass and merger rates as a function of the total stellar mass and the mass of new stars formed during a merger. From this, we predict the merging galaxy luminosity function in various observed wavebands (e.g. UV, optical, and near-IR), color-magnitude relations, mass functions, absolute and specific star formation rate distributions and star formation rate density, and quasar host galaxy luminosity function, as a function of redshift from z = 0 − 6. We invert this relationship to predict e.g. quasar luminosity functions from observed merger luminosity functions or star formation rate distributions. Our results show good agreement with observations, but idealized models of quasar lightcurves give inaccurate estimates and are ruled out by comparison of merging galaxy and quasar observations at > 99.9% confidence, provided that quasars are triggered in mergers. Using only observations of quasars, we estimate the contribution of mergers to the star formation rate density of the Universe to high redshifts, z ∼ 4, and constrain the evolution in the characteristic initial gas fractions of quasar and spheroid-producing mergers.