The origin of the dust in high-redshift quasars: the case of SDSS J1148+5251

We present a semi-analytical model for the formation and evolution of a high-redshift quasar (QSO). We reconstruct a set of hierarchical merger histories of a 10(13)-M-circle dot dark matter halo and model the evolution of the corresponding galaxy and of its central supermassive black hole. The code GAMETE/QSODUST consistently follows (i) the black hole assembly via both coalescence with other black holes and gas accretion; (ii) the build-up and star formation history of the quasar host galaxy, driven by binary mergers and mass accretion; (iii) the evolution of gas, stars and metals in the interstellar medium (ISM), accounting for mass exchanges with the external medium (infall and outflow processes); (iv) the dust formation in supernova (SN) ejecta and in the stellar atmosphere of asymptotic giant branch (AGB) stars, dust destruction by interstellar shocks and grain growth in molecular clouds; and (v) the active galactic nucleus feedback which powers a galactic-scale wind, self-regulating the black hole growth and eventually halting star formation. We use this model to study the case of SDSS J1148+5251 observed at redshift 6.4. We explore different star formation histories for the QSO host galaxy and find that Population III stars give a negligible contribution to the final metal and dust masses due to rapid enrichment of the ISM to metallicities >Z(cr) = 10(-6)-10(-4) Z(circle dot) in progenitor galaxies at redshifts > 10. If Population II/I stars form with a standard initial mass function (IMF) and with a characteristic stellar mass of m(ch) = 0.35 M-circle dot, a final stellar mass of (1-5) x 10(11) M-circle dot is required to reproduce the observed dust mass and gas metallicity of SDSS J1148+5251. This is a factor of 3-10 higher than the stellar mass inferred from observations and would shift the QSO closer or on to the stellar bulge-black hole relation observed in the local Universe; alternatively, the observed chemical properties can be reconciled with the inferred stellar mass, assuming that Population II/I stars form according to a top-heavy IMF with m(ch) = 5M(circle dot). We find that SNe dominate the early dust enrichment and that, depending on the shape of the star formation history and on the stellar IMF, AGB stars contribute at z < 8-10. Yet, a dust mass of (2-6) x 10(8) M-circle dot estimated for SDSS J1148+ 5251 cannot be reproduced considering only stellar sources, and the final dust mass is dominated by grain growth in molecular clouds. This conclusion is independent of the stellar IMF and star formation history.