Fig. 7. Top panel: the 2 keV-emission \( r _ { p e a k } \) (\( r _ { 9 0 } \)) as a function of \( \mu \) is represented in black (gray). The green shaded area qualitatively shows the inner radii, where the bulk of X-ray emission is supposed to come from according to X-ray reverberation and micro-lensing. For increas-ing \( \mu \), the X-ray emission profile peaks at larger radii. Middle panel: for increasing \( \mu \) the models obtain a slope of the \( L _ { X } - L _ { U V } \) closer to the observed one. The dark-green area represents the reference slope of the cleanest XXM-XXL (Appendix B), while the light-green refers to the slope quoted in LR17. Bottom panel: intrinsic scatter of the mock \( L _ { X } - L _ { U V } \) relations as a function of \( \mu . \). The green area represents a ten-tative upper limit of the true scatter (Lusso & Risaliti 2016; Chiaraluce et al. 2018), that is only due to the physical properties of AGN. For simplicity, all panels show only the results obtained with a single \( f _ { m a x } \), corresponding to the highest \( r ^ { 2 } \)-score (e.g., Fig. 6), and fixed \( \alpha _ { 0 } = 0 . 0 2 \).

0.5 and 1, respectively4:

\[ \begin{array} { l l r } { \log < f > \ = \log f _ { m a x } } \\ { \log < f > \ = \left( - 1 . 1 2 \pm 0 . 2 4 \right) \ - \left( 0 . 1 5 \pm 0 . 0 2 \right) \, \log \dot { m } } \\ { \ } & { \qquad \qquad + \ ( 0 . 0 5 \pm 0 . 0 3 ) \, \log m } \\ { \log < f > \ = \left( - 1 . 8 2 \pm 0 . 3 6 \right) \, - \left( 0 . 2 7 \pm 0 . 0 3 \right) \, \log \dot { m } } \\ { \ } & { \qquad \qquad + \ ( 0 . 0 7 \pm 0 . 0 4 ) \, \log m } & { ( 1 4 ) } \end{array} \]

where the steepest dependency from m˙ is obtained for larger \( \mu \).

This test points in the same direction as the evidence of an X-ray bolometric correction increasing with the accretion rate (e.g. Wang et al. 2004; Vasudevan & Fabian 2007, 2009; Lusso et al. 2010; Young et al. 2010), although we refrain to compare this observable with our regressions (e.g. Wang et al. 2004; Cao 2009; Liu & Liu 2009; You et al. 2012; Liu et al. 2012, 2016a), due to the many more uncertainties in play when deriving bolo-metric luminosities in comparison to the quantities entering in the \( L _ { X } - L _ { U V } \) (see the discussion in Section 4.2).

# 5.3. The scatter of the \( L _ { X } - L _ { U V } \)

The observed scatter of the \( L _ { X } - L _ { U V } \) for the sample used in this work is \( \sigma _ { i n t r } \, = \, 0 . 2 7 \pm 0 . 0 1 \) (Section 4.1). As a matter of fact, this value represents an upper limit to the intrinsic disper-sion inherent to the physics of the system, as the observed scat-ter is aected by a combination of instrumental and calibration issues, UV and X-ray variability, non-simultaneity of the multi-wavelength observations. A lot of eort has been put into trying to quantify as accurately as possible all these contaminants (e.g. Vagnetti et al. 2013; Lusso 2018, and references therein), with claims that the intrinsic scatter in the \( L x \)-\( L _ { I I V } \) relation is smaller than \( \lesssim \, 0 . 1 8 \mathrm { ~ - ~ } \) 0.20 (Lusso & Risaliti 2016; Chiaraluce et al. 2018). Any successful model should be able to reproduce such a low scatter.

From the examples of mock \( L _ { X } - L _ { U V } \) relations plotted in Fig. 5, it can already be seen that our models come with their one intrinsic scatter. In our methodology (Section 4.2), the modeled \( \dot { m } \) was tuned to the observed \( L _ { 3 0 0 0 \mathring \mathrm { A } } \), hence the intrinsic scatter of the mock \( L _ { X } - L _ { U V } \) relations is simply the dispersion of the modeled \( L _ { 2 k e V } \), at a given m˙, due to dierent m and \( \Gamma \). We show this more quantitatively in the bottom panel of Fig. 7. The mod-els dispersion varies with \( \mu \) because changing the viscosity law induces a dierent logarithmic scatter in f(r) (see Fig. 1) and it also aects the distance (in gravitational radii) from which the bulk of the \( L _ { 2 k e V } \) is coming (see top panel of Fig. 7). The result-ing \( \sigma _ { i n t r } \) of the models is likely a complex combination of these (and possible more) factors. All the models, with the exception of \( \mu \)= 0, lie below the available observational constraints (Lusso& Risaliti 2016; Chiaraluce et al. 2018) of \( \lesssim 0 . 1 8 - \) 0.20. This is another successful prediction of our model (see Section 3).

# 5.4. A complete picture: the slope-normalization plane of the \( L _ { X } - L _ { U V } \)

In the previous Sections, we decomposed the match in either nor-malization or slope to have a better understanding on how our disk-corona models can relate to the observed \( L _ { X } - L _ { U V } \). How-ever, the goal would be to have a model that can fully encompass these observables. Hence, in Fig. 8 we display 1-, 2- and 3-sigma contours in the slope-normalization plane \( { \widehat { ( \beta } } \! - \! { \widehat { \alpha } } ) \) of the \( L _ { X } \! - \! L _ { U V } \) for both data and models. All regressions were performed with emcee normalizing both \( L _ { X } \) and \( L _ { U V } \) to the median value of XMM-XXL. The data contours are related to the cleanest XMM-XXL version (Appendix B) and to the RM-QSO sources5. Model contours are shown for \( \mu \, = \, [ 0 , 0 . 2 , 0 . 4 , 0 . 5 , 0 . 6 , 0 . 8 , 1 ] \) using a

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4 The distributions of mock \( \dot { m } \) are very similar across the models, with median values (and related 16th and 84th percentiles) of \( 0 . 1 6 _ { 0 . 0 4 } ^ { 0 . 6 9 } \),\( 0 . 1 5 _ { 0 . 0 5 } ^ { 0 . 6 5 } \) and \( 0 . 1 4 _ { 0 . 0 4 } ^ { 0 . 5 9 } \) for \( \mu \) = 0, 0.5 and 1, respectively. The tails include Eddington or even super-Eddington sources. We note that the uncer-tainty on the modeled \( \dot { m } \), propagated through the ones in the observa-tions, is as large as \( \approx \) \( ( 1 \ \kappa ) \) 5 \( 4 \alpha \) x.

5 XMM-XXL luminosities were obtained in L16 including a Balmer continuum component in the fit (refer to Shen & Liu 2012), although for the RM-QSO this component was switched o (Shen et al. 2018). For consistency, a rigid shift of −0.12 dex was applied to the RM-QSO \( L _ { 3 0 0 0 \delta } \) (Shen & Liu 2012) for obtaining the contours displayed in Fig. 8.