Growth of massive black holes during radiatively inefficient accretion phases

We derive the black hole mass density as a function of redshift from the hard X-ray active galactic nucleus (AGN) luminosity function, assuming that massive black holes grow via accreting circumnuclear gases. The derived black hole mass density matches the measured local black hole mass density at z = 0, which requires the average radiative efficiency of AGNs to be similar to 0.1-0.17. The massive black holes in most faint AGNs and even normal galaxies are still accreting gases, although their accretion rates are very low. Radiatively inefficient accretion flows (RIAFs) are supposed in these faint sources, which should radiate mostly in the hard X-ray band. We calculate the contribution to the X-ray background (XRB) from both the bright AGNs and the RIAFs in faint AGNs and normal galaxies. Our calculations show that both the observed intensity and spectral shape of the XRB with an energy peak at similar to 30 keV can be well reproduced without including the emission of Compton-thick AGNs, if the massive black holes in faint AGNs and normal galaxies are spinning rapidly, with a similar to 0.9, and are accreting at rates of m similar to (1.0-3.0); 10(-4). It indicates that less than similar to 5% of the local massive black hole mass density was accreted during radiatively inefficient accretion phases, which is obviously only an upper limit, because Compton-thick AGNs have not been considered. If the same number of Compton-thick AGNs with log N-H 24-25 as those with log N-H 23-24 is considered, the fraction of the local black hole mass density accumulated during inefficient accretion phases should be lower than similar to 2%. The constraints of the XRB can provide upper limits on the average accretion rate for inactive galaxies.