Quasars as standard candles I: The physical relation between disc and coronal emission

A tight non-linear relation exists between the X-ray and UV emission in quasars (i.e. $L_{\rm X}\propto L_{\rm UV}^{\gamma}$), with a dispersion of $\sim$0.2~dex over $\sim$3~orders of magnitude in luminosity. Here we propose a modified version of this relation which involves the broad emission lines' full-width at half maximum, $L_{\rm X}\propto L_{\rm UV}^{\hat\gamma}\upsilon_{\rm fwhm}^{\hat\beta}$. We interpret this new relation through a simple, {\it ad-hoc} model of accretion disc corona, derived from previous works where it is assumed that reconnection and magnetic loops above the accretion disc can account for the production of the primary X-ray radiation. We find that the monochromatic optical-UV (2500 \AA) and X-ray (2 keV) luminosities depend on the black hole mass and accretion rate as $L_{\rm UV}\propto M_{\rm BH}^{4/3} (\dot{M}/\dot{M}_{\rm Edd})^{2/3}$ and $L_{\rm X}\propto M_{\rm BH}^{19/21} (\dot{M}/\dot{M}_{\rm Edd})^{5/21}$, respectively. Assuming a broad line region size function of the disc luminosity $R_{\rm blr}\propto L_{\rm disc}^{0.5}$ we finally have that $L_{\rm X}\propto L_{\rm UV}^{4/7} \upsilon_{\rm fwhm}^{4/7}$. Such relation is remarkably consistent with the slopes and the normalization obtained from a fit of a sample of 545 optically selected quasars from SDSS DR7 cross matched with the latest XMM-Newton catalogue 3XMM-DR6. The homogeneous sample used here has a dispersion of 0.21 dex, which is much lower than previous works in the literature and suggests a tight physical relation between the accretion disc and the X-ray emitting corona. The proposed new relation does not evolve with time, and thus it can be employed as a cosmological indicator to robustly estimate cosmological parameters.