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We present an investigation into how well the properties of the accretionflow onto a supermassive black hole may be coupled to those of the overlyinghot corona. To do so, we specifically measure the characteristic spectralindex, Gamma, of a power-law energy distribution, over an energy range of 2 to10 keV, for X-ray selected, broad-lined radio-quiet AGN up to z~2 in COSMOS andE-CDF-S. We test the previously reported dependence between Gamma and blackhole mass, FWHM and Eddington ratio using a sample of AGN covering a broadrange in these parameters based on both the Mg ii and H-alpha emission lineswith the later afforded by recent near infrared spectroscopic observationsusing Subaru/FMOS. We calculate the Eddington ratios, lambda_Edd, for sourceswhere a bolometric luminosity (L_Bol) has been presented in the literature,based on SED fitting, or, for sources where these data do not exist, wecalculate L_Bol using a bolometric correction to the LX, derived from arelationship between the bolometric correction, and LX/L3000. From a sample of69 X-ray bright sources (>250 counts), where Gamma can be measured withgreatest precision, with an estimate of L_Bol, we find a statisticallysignificant correlation between Gamma and lambda_Edd, which is highlysignificant with a chance probability of 6.59x10^-8. A statisticallysignificant correlation between Gamma and the FWHM of the optical lines isconfirmed, but at lower significance than with lambda_Edd indicating thatlambda_Edd is the key parameter driving conditions in the corona. Linearregression analysis reveals that Gamma=(0.32+/-0.05)log10lambda_Edd+(2.27+/-0.06) andGamma=(-0.69+/-0.11)log10(FWHM/km/s)+(4.44+/-0.42). Our results onGamma-lambda_Edd are in very good agreement with previous results. (ABRIDGED)

A statistical relation between the X-ray spectral index and Eddington ratio of active galactic nuclei in deep surveys