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 { BHAR } ( M _ { \star } ,z ) ] over ranges of \log ( M _ { \star } / M _ { \odot } ) = \text { 9.5 - -12 } and z = \text { 0.4 - -4 } .
Our \overline { 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 { 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 { 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 { BHAR } and average star formation rate ( \overline { SFR } ) rises toward high M _ { \star } at a given redshift .
This \overline { BHAR } / \overline { SFR } dependence on M _ { \star } does not support the scenario that SMBH and galaxy growth are in lockstep .
We calculate SMBH mass history [ M _ { BH } ( z ) ] based on our \overline { BHAR } ( M _ { \star } ,z ) and the M _ { \star } ( z ) from the literature , and find that the M _ { BH } - M _ { \star } relation has weak redshift evolution since z \approx 2 .
The M _ { 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 _ { 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 .