Abstract
The spin of a supermassive black hole (SMBH) is directly related to the radiative efficiency of accretion on to the hole, and therefore impacts the amount of fuel required for the black hole to reach a certain mass. Thus, a knowledge of the SMBH spin distribution and evolution is necessary to develop a comprehensive theory of the growth of SMBHs and their impact on galaxy formation. Currently, the only direct measurement of SMBH spin is through fitting the broad Fe K line in AGNs. The evolution of spins could be determined by fitting the broad line in the integrated spectra of AGNs over different redshift intervals. The accuracy of these measurements will depend on the observed integrated line strength. Here, we present theoretical predictions of the integrated relativistic Fe K line strength as a function of redshift and AGN luminosity. The equivalent widths of the integrated lines are much less than 300 eV. Searches for the integrated line will be easiest for unobscured AGNs with 2-10 keV luminosities between 44 < log L_{X} <= 45. The total integrated line makes up less than 4% of the X-ray background, but its shape is sensitive to the average SMBH spin. By following these recommendations, future International X-ray Observatory surveys of broad Fe K lines should be able to determine the spin evolution of SMBHs.
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