Constraints on the radiatively inefficient accretion history of active galactic nuclei from the hard cosmological X-ray background

The transition of a standard thin disk into a radiatively inefficient accretion flow (RIAF) is expected to occur when its accretion rate is lower than a critical value m(crit) (m = M/M-Edd). RIAFs are very hot, and they radiate mostly in the hard X-ray band (>= 100 keV). Assuming that the accretion disk in every bright active galactic nucleus (AGN) will finally undergo an RIAF phase when m < m(crit), we calculate the contribution of the RIAFs in AGNs to the cosmological X-ray background at 10 - 1000 keV. We find that the timescale t(RIAF) of an RIAF accreting at similar to m(crit) should be shorter than similar to 10(-2) t(b) if m(crit) = 0.01, where t(b) is the lifetime of bright AGNs; that is, m declines from crit to a rate significantly lower than crit within tRIAF. The derived timescale t(RIAF) is affected by the parameters adopted in the model calculations, which are also discussed.