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The spin of a supermassive black hole (SMBH) is directly related to theradiative efficiency of accretion on to the hole, and therefore impacts theamount of fuel required for the black hole to reach a certain mass. Thus, aknowledge of the SMBH spin distribution and evolution is necessary to develop acomprehensive theory of the growth of SMBHs and their impact on galaxyformation. Currently, the only direct measurement of SMBH spin is throughfitting the broad Fe K line in AGNs. The evolution of spins could be determinedby fitting the broad line in the integrated spectra of AGNs over differentredshift intervals. The accuracy of these measurements will depend on theobserved integrated line strength. Here, we present theoretical predictions ofthe integrated relativistic Fe K line strength as a function of redshift andAGN luminosity. The equivalent widths of the integrated lines are much lessthan 300 eV. Searches for the integrated line will be easiest for unobscuredAGNs with 2-10 keV luminosities between 44 < log L_{X} <= 45. The totalintegrated line makes up less than 4% of the X-ray background, but its shape issensitive to the average SMBH spin. By following these recommendations, futureInternational X-ray Observatory surveys of broad Fe K lines should be able todetermine the spin evolution of SMBHs.

The Integrated Relativistic Iron Line from Active Galactic Nuclei: Chasing the Spin Evolution of Supermassive Black Holes