Fooled by mass? Recovering the $Γ-λ_{Edd}$ relation of quasars from their X-ray variability
We measure the relation between the X-ray photon index $Γ$ and the Eddington ratio $λ_{Edd}$ in unobscured type-1 quasars, using as parent sample the cross-match of the SDSS DR16Q catalogue with the 4XMM-DR14 serendipitous catalogue. After cleaning the sample of potential observational biases, we obtain 7835 objects, one order of magnitude larger than most previous analyses in the literature. A single regression of $Γ$ against $\logλ_{Edd}$, with single-epoch virial black-hole masses, returns slopes that depend on the mass indicator (H$β$, MgII, CIV) and on the bolometric correction, and that are generally flat. We interpret this as regression dilution: the systematic uncertainty on virial masses is much larger than the intrinsic spread of $λ_{Edd}$ in SDSS quasars. We therefore introduce a different approach, which removes the black-hole mass from the problem. For quasars observed by XMM-Newton at multiple epochs, the mass of each object is fixed, and changes in its X-ray luminosity trace changes in its accretion state. We fit the $Γ$-$\log(L_{2keV})$ relation of each object (for a sample of N=893) and combine the per-object posteriors. This recovers the mean response of the population $a_X = 0.35\pm0.03$, which steepens to $\approx0.43$ when only epochs with precisely measured photon indices are used. Translated to the Eddington ratio, this corresponds to $a_λ\approx0.20$-0.25, depending on the choice of bolometric correction and on the LX-LUV slope. The $Γ$--$λ_{Edd}$ relation therefore exists, but it can be uncovered only by exploiting the variability of individual sources, which removes the black-hole mass from the problem. Moreover, its shallowness implies that the differences in Eddington ratio, as currently measured, cannot by themselves account for the variety of photon indices observed among unobscured quasars.