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

{We present an investigation into how well the properties of the accretion flow on to a supermassive black hole may be coupled to those of the overlying hot corona. To do so, we specifically measure the characteristic spectral index, {$\Gamma$}, of a power-law energy distribution, over an energy range of 2-10 keV, for X-ray selected, broad-lined radio-quiet active galactic nuclei (AGN) up to z {\tilde} 2 in Cosmic Evolution Survey (COSMOS) and Extended Chandra Deep Field South (E-CDF-S). We test the previously reported dependence between {$\Gamma$} and black hole mass, full width at half-maximum (FWHM) and Eddington ratio using a sample of AGN covering a broad range in these parameters based on both the Mg II and H{$\alpha$} emission lines with the later afforded by recent near-infrared spectroscopic observations using Subaru/Fibre Multi Object Spectrograph. We calculate the Eddington ratios, {$\lambda$}$_{Edd}$, for sources where a bolometric luminosity (L$_{Bol}$) has been presented in the literature, based on spectral energy distribution fitting, or, for sources where these data do not exist, we calculate L$_{Bol}$ using a bolometric correction to the X-ray luminosity, derived from a relationship between the bolometric correction and L$_{X}$/L$_{3000}$. From a sample of 69 X-ray bright sources ({\gt}250 counts), where {$\Gamma$} can be measured with greatest precision, with an estimate of L$_{Bol}$, we find a statistically significant correlation between {$\Gamma$} and {$\lambda$}$_{Edd}$, which is highly significant with a chance probability of 6.59{\times} 10$^{-8}$. A statistically significant correlation between {$\Gamma$} and the FWHM of the optical lines is confirmed, but at lower significance than with {$\lambda$}$_{Edd}$ indicating that {$\lambda$}$_{Edd}$ is the key parameter driving conditions in the corona. Linear regression analysis reveals that {$\Gamma$} = (0.32 {\plusmn} 0.05) log$_{10}${$\lambda$}$_{Edd}$ + (2.27 {\plusmn} 0.06) and {$\Gamma$} = (-0.69 {\plusmn} 0.11) log$_{10}$(FWHM/km s$^{-1}$) + (4.44 {\plusmn} 0.42). Our results on {$\Gamma$}-{$\lambda$}$_{Edd}$ are in very good agreement with previous results. While the {$\Gamma$}-{$\lambda$}$_{Edd}$ relationship means that X-ray spectroscopy may be used to estimate black hole accretion rate, considerable dispersion in the correlation does not make this viable for single sources, however could be valuable for large X-ray spectral samples, such as those to be produced by eROSITA. }