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The cosmic evolution of massive black holes and galaxy spheroids: global constraints at redshift z<~1.2

We study the observational constraints on the cosmic evolution of the relationships between the massive black hole (MBH) mass (M_bh) and the stellar mass (M^*_sph; or velocity dispersion σ) of the host galaxy/spheroid. Assuming that the M_bh-M^*_sph (or M_bh-σ) relation evolves with redshift as \propto (1+z)^Γ, the MBH mass density can be obtained from either the observationally determined galaxy stellar mass functions or velocity dispersion distribution functions over redshift z~0-1.2 for any given Γ. The MBH mass density at different redshifts can also be inferred from the luminosity function of QSOs/AGNs provided known radiative efficiency ε. By matching the MBH density inferred from galaxies to that obtained from QSOs/AGNs, we find that Γ=0.64^{+0.27}_{-0.29} for the M_bh-M^*_sph relation and Γ=-0.21^{+0.28}_{-0.33} for the M_bh-σrelation, and ε=0.11^{+0.04}_{-0.03}. Our results suggest that the MBH mass growth precedes the bulge mass growth but the galaxy velocity dispersion does not increase with the mass growth of the bulge after the quench of nuclear activity, which is roughly consistent with the two-phase galaxy formation scenario proposed by Oser et al. (2012) in which a galaxy roughly double its masses after z=1 due to accretion and minor mergers while its velocity dispersion drops slightly.