On the AGN radio luminosity distribution and the black hole fundamental plane

We have studied the dependence of the AGN nuclear radio (1.4 GHz) luminosity on both the AGN 2-10 keV X-ray and the host-galaxy K-band luminosity. A complete sample of 1268 X-ray selected AGN (both type 1 and type 2) has been used, which is the largest catalogue of AGN belonging to statistically well defined samples where radio, X and K band information exists. At variance with previous studies, radio upper limits have been statistically taken into account using a Bayesian Maximum Likelihood fitting method. It resulted that a good fit is obtained assuming a plane in the 3D L$_R$-L$_X$-L$_K$ space, namely logL$_R$= $\xi_X$logL$_X$+ $\xi_K$logL$_K$ + $\xi_0$, having a $\sim$1 dex wide (1$\sigma$) spread in radio luminosity. As already shown by \citet{lafranca10}, no evidence of bimodality in the radio luminosity distribution was found and therefore any definition of radio loudness in AGN is arbitrary. Using scaling relations between the BH mass and the host galaxy K-band luminosity, we have also derived a new estimate of the BH fundamental plane (in the L$_{5 GHz}$-L$_X$-M$_{BH}$ space). Our analysis shows that previous measures of the BH fundamental plane are biased by $\sim$0.8 dex in favor of the most luminous radio sources. Therefore, many AGN studies, where the BH fundamental plane is used to investigate how AGN regulate their radiative and mechanical luminosity as a function of the accretion rate, or many AGN/galaxy co-evolution models, where radio-feedback is computed using the AGN fundamental plane, should revise their conclusions.