[O III]$\lambda 5007$ and X-ray Properties of a Complete Sample of Hard X-ray Selected AGNs in the Local Universe

We study the correlation between the [O III]$\lambda 5007$ and X-ray luminosities of local Active Galactic Nuclei (AGNs), using a complete, hard X-ray ($>10$ keV) selected sample in the Swift/BAT 9-month catalog. From our optical spectroscopic observations at the South African Astronomical Observatory and the literature, a catalog of [O III]$\lambda 5007$ line flux for all 103 AGNs at Galactic latitudes of $|b|>15^\circ$ is complied. Significant correlations with intrinsic X-ray luminosity ($L_{\rm X}$) are found both for observed ($L_{\rm [O~III]}$) and extinction-corrected ($L_{\rm [O~III]}^{\rm cor}$) luminosities, separately for X-ray unabsorbed and absorbed AGNs. We obtain the regression form of $L_{\rm [O~III]}$ $\propto L_{\rm 2-10\; keV}^{1.18\pm0.07}$ and $L_{\rm [O~III]}^{\rm cor}$ $\propto L_{\rm 2-10\; keV}^{1.16\pm0.09}$ from the whole sample. The absorbed AGNs with low ($<$0.5\%) scattering fractions in soft X-rays show on average smaller $L_{\rm [O~III]}/L_{\rm X}$ and $L_{\rm [O~III]}^{\rm cor}/L_{\rm X}$ ratios than the other absorbed AGNs, while those in edge-on host galaxies do not. These results suggest that a significant fraction of this population are buried in tori with small opening angles. By using these $L_{\rm [O~III]}$ vs. $L_{\rm X}$ correlations, the X-ray luminosity function of local AGNs (including Compton thick AGNs) in a standard population synthesis model gives much better agreement with the [O III]$\lambda 5007$ luminosity function derived from the Sloan Digital Sky Survey than previously reported. This confirms that hard X-ray observations are a very powerful tool to find AGNs with high completeness.