Does the obscured AGN fraction really depend on luminosity?

We use a sample of 151 local non-blazar AGN selected from the INTEGRAL all-sky hard X-ray survey to investigate if the observed declining trend of the fraction of obscured (i.e. showing X-ray absorption) AGN with increasing luminosity is mostly an intrinsic or selection effect. Using a torus-obscuration model, we demonstrate that in addition to negative bias, due to absorption in the torus, in finding obscured AGN in hard X-ray flux limited surveys, there is also positive bias in finding unobscured AGN, due to Compton reflection in the torus. These biases can be even stronger taking into account plausible intrinsic collimation of hard X-ray emission along the axis of the obscuring torus. Given the AGN luminosity function, which steepens at high luminosities, these observational biases lead to a decreasing observed fraction of obscured AGN with increasing luminosity even if this fraction has no intrinsic luminosity dependence. We find that if the central hard X-ray source in AGN is isotropic, the intrinsic (i.e. corrected for biases) obscured AGN fraction still shows a declining trend with luminosity, although the intrinsic obscured fraction is significantly larger than the observed one: the actual fraction is larger than $\sim 85$% at $L\lesssim 10^{42.5}$ erg/s (17--60 keV), and decreases to $\lesssim 60$% at $L\gtrsim 10^{44}$ erg/s. In terms of the half-opening angle, $θ$, of an obscuring torus, this implies that $θ\lesssim 30$ deg in lower-luminosity AGN, and $θ\gtrsim 45$ deg in higher-luminosity ones. If, however, the emission from the central SMBH is collimated as $dL/dΩ\propto\cosα$, the intrinsic dependence of the obscured AGN fraction is consistent with a luminosity-independent torus half-opening angle $θ\sim 30$ deg.