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Previous studies suggest that the growth of supermassive black holes (SMBHs) may be fundamentally related to host-galaxy stellar mass (M ). To investigate this SMBH growth-M relation in detail, we calculate long-term SMBH accretion rate as a function of M and redshift BHAR(M , z) over ranges of log (M /M ) = 9.5-12 and z = 0.4-4. Our BHAR(M , 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 , BHAR is higher at high redshift. This redshift dependence is stronger in more massive systems for log (M /M ) 11.5, 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 BHAR and average star formation rate (SFR) rises towards high M at a given redshift. This BHAR/SFR dependence on M does not support the scenario that SMBH and galaxy growth are in lockstep. We calculate SMBH mass history M BH (z) based on our BHAR(M , z) and the M (z) from the literature, and find that the M BH -M relation has weak redshift evolution since z 2. The M BH /M ratio is higher towards massive galaxies: it rises from 1/5000 at log M 10.5 to 1/500 at log M 11.2. Our predicted M BH /M ratio at high M is similar to that observed in local giant ellipticals, suggesting that SMBH growth from mergers is unlikely to dominate over growth from accretion.
Yang et al. (2017) studied this question.