Linking black-hole growth with host galaxies: The accretion-stellar mass relation and its cosmic evolution

Previous studies suggest that the growth of supermassive black holes (SMBHs) may be fundamentally related to host-galaxy stellar mass ($M_\star$). To investigate this SMBH growth-$M_\star$ relation in detail, we calculate long-term SMBH accretion rate as a function of $M_\star$ and redshift [$\overline{\rm BHAR}(M_\star, z)$] over ranges of $\log(M_\star/M_\odot)=\text{9.5--12}$ and $z=\text{0.4--4}$. Our $\overline{\rm BHAR}(M_\star, z)$ is constrained by high-quality survey data (GOODS-South, GOODS-North, and COSMOS), and by the stellar mass function and the X-ray luminosity function. At a given $M_\star$, $\overline{\rm BHAR}$ is higher at high redshift. This redshift dependence is stronger in more massive systems (for $\log(M_\star/M_\odot)\approx 11.5$, $\overline{\rm BHAR}$ is three decades higher at $z=4$ than at $z=0.5$), possibly due to AGN feedback. Our results indicate that the ratio between $\overline{\rm BHAR}$ and average star formation rate ($\overline{\rm SFR}$) rises toward high $M_\star$ at a given redshift. This $\overline{\rm BHAR}/\overline{\rm SFR}$ dependence on $M_\star$ does not support the scenario that SMBH and galaxy growth are in lockstep. We calculate SMBH mass history [$M_{\rm BH}(z)$] based on our $\overline{\rm BHAR}(M_\star, z)$ and the $M_\star(z)$ from the literature, and find that the $M_{\rm BH}$-$M_\star$ relation has weak redshift evolution since $z\approx 2$. The $M_{\rm BH}/M_\star$ ratio is higher toward massive galaxies: it rises from $\approx 1/5000$ at $\log M_\star\lesssim 10.5$ to $\approx 1/500$ at $\log M_\star \gtrsim 11.2$. Our predicted $M_{\rm BH}/M_\star$ ratio at high $M_\star$ is similar to that observed in local giant ellipticals, suggesting that SMBH growth from mergers is unlikely to dominate over growth from accretion.