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Hard X-ray photon index as an indicator of bolometric correction in active galactic nuclei

We propose the rest-frame 2-10 keV photon index, \ga, acting as an indicator of the bolometric correction, \lb/$L_{\rm 2-10keV}$ (where \lb~ is the bolometric luminosity and $L_{\rm 2-10keV}$ is the rest-frame 2-10 keV luminosity), in radio-quiet active galactic nuclei (AGNs). Correlations between \ga~ and both bolometric correction and Eddington ratio are presented, based on simultaneous X-ray, UV, and optical observations of reverberation -mapped AGNs. These correlations can be compared with those for high-redshift AGNs to check for any evolutionary effect. Assuming no evolutionary effect in AGNs' spectral properties, together with the independent estimates of $L_{\rm 2-10keV}$, the bolometric correction, Eddington ratio, and black hole (BH) mass can all be estimated from these correlations for high-redshift AGNs, with the mean uncertainty of a factor of 2-3. If there are independent estimates of BH masses, \ga~ for high-redshift AGNs can be used to determine their true \lb~ and $L_{\rm 2-10keV}$, and in conjunction with the redshift, can be potentially used to place constraints on cosmology by comparison with the rest-frame 2-10 keV flux. We find that the true $L_{\rm 2-10keV}$ estimated from \ga~ for the brightest Type I AGNs with $z<1$ in the Lockman Hole is generally in agreement with the observed $L_{\rm 2-10keV}$. However, there are still many uncertainties, such as the accurate determination of the intrinsic \ga~ for distant AGNs and the large uncertainty in the luminosities obtained, which call for significant further study before ``AGN cosmology'' can be considered a viable technique.