Growth of Intermediate Massive Black Holes in the Hierarchical Formation of Small Spiral Galaxies in the High-z Universe

Combining a theoretical model of mass accretion onto a galactic center with a high-resolution $N$-body/SPH simulation, we investigate the formation of an intermediate massive black hole (IMBH) during the hierarchical formation of a small spiral galaxy (with a total mass of $10^{10}M_{\odot}$) in the high-$z$ universe. We found that the rate of average mass accretion to the nucleus due to the radiation drag exerted by newly formed stars in the forming galaxy is $\approx 10^{-5}M_{\odot}$yr$^{-1}$. As a result of this accretion, an IMBH with $\approx 10^{4}M_{\odot}$ can be formed in the center of the spiral galaxy at $z\sim 4$. We found that a central BH coevolves with the dark matter halo from $z\sim 15$ to $z\sim 2$. The mass ratio of the BH to the dark matter halo is nearly constant $\approx (1-3) \times 10^{-6}$ from $z\sim 10$ to $z\sim 2$. This is because that change in the dark matter potential enhances star formation in the central part of the galaxy, and as a result the BH evolves due to mass accretion via the radiation drag. Therefore, our model naturally predicts a correlation between massive BHs and dark matter halos. Moreover, it is found that the final BH-to-bulge mass ratio ($\approx 5\times 10^{-5}$) in a small spiral galaxy at high-$z$ is much smaller than that in the large galaxies ($\approx 10^{-3}$). Our results also suggest that the scatter in the observed scaling relations between the bulge mass and black hole mass are caused by a time lag between BH growth and growth of bulge. We also predict that the X-ray luminosity of AGN is positively correlated with the CO luminosity in the central region. By comparing our results with the properties of Lyman break galaxies (LBGs), it is predicted that some LBGs have massive BHs of $\approx 10^{6}-10^{7}M_{\odot}$.