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.

A&A proofs: manuscript no. paper\_disc\_corona\_LXvsLUV

Fig. 8. 1-, 2- and 3-sigma contours of the emcee regressions in the slope-normalization \( ( { \widehat { \beta } } - { \widehat { \alpha } } ) \) plane of the \( L _ { X } - L _ { U V } \) for both data and models, normalizing all \( L _ { X } \) and \( L _ { I I V } \) to the corresponding median val-ues of XMM-XXL. Dark green contours are related to the cleanest XMM-XXL sample (Appendix B) and the light green ones to the RM-QSO sources. The contour of the models are color coded for \( \mu \) \( = \)\( [ 0 , 0 . 2 , 0 . 4 , 0 . 5 , 0 . 6 , 0 . 8 , 1 ] \), as shown in the legend. For simplicity, we report for each \( \mu \) only results obtained with a single \( f _ { m a x } \), correspond-ing to the highest \( r ^ { 2 } \)-score, and fixed \( \alpha _ { 0 } = 0 . 0 2 \). Models that reproduce the observed slope \( _ { - } \) are also the ones that show weaker coronae (lower normalization \( \widehat { \alpha } \)).

single \( f _ { m a x } \), corresponding to the highest \( r ^ { 2 } \)-score (e.g., Fig. 6) for each \( \mu \), and a fixed \( \alpha _ { 0 } = 0 . 0 2 \), for simplicity.

Fig. 8 shows that models reproducing the observed slope, namely the ones with higher \( \mu \) (as in middle panel of Fig. 7), are also the ones that show weaker coronae (lower normalization \( \bar { \alpha } \) b) and overly extended \( L _ { 2 k e V } \)-emission (i.e. higher \( r _ { p e a k } \) and \( r _ { 9 0 } \), top panel of Fig. 7).

# 5.5. The 3D plane: \( L _ { X } \) vs \( L r r \) vs m

As shown by LR17, the \( L _ { X } - L _ { U V } \) relation for AGN is rather a three-dimensional problem, with the mass (or its proxy given by the full-width half-maximum of broad emission lines) playing a significant role as well. The observed \( L _ { X } - L _ { U V } - m \) plane from XMM-XXL can be fit by:

\[ \begin{array} { c } { { \log L _ { 2 k e V } - 2 5 = ( - 0 . 9 1 \pm 0 . 1 3 ) + ( 0 . 3 9 \pm 0 . 0 3 ) ( \log L _ { 3 0 0 0 \hat { \mathbf { A } } } - 2 5 ) } } \\ { { + \left( 0 . 2 3 \pm 0 . 0 4 \right) ( \log m - 7 ) } } \\ { { ( 1 5 ) } } \end{array} \]

and the mock \( L _ { X } - L _ { U V } - m \) from models with \( \mu \)= 0, 0.5 and 1, respectively:

\[ \begin{array} { c } { { \log L _ { 2 k e V } - 2 5 = ( - 3 . 4 9 \pm 0 . 1 5 ) + ( 1 . 0 8 \pm 0 . 0 3 ) ( \log L _ { 3 0 0 0 \AA } - 2 5 ) } } \\ { { - \, ( 0 . 2 7 \pm 0 . 0 3 ) ( \log m - 7 ) } } \\ { { { } } } \\ { { \log L _ { 2 k e V } - 2 5 = ( - 2 . 4 1 \pm 0 . 1 5 ) + ( 0 . 7 3 \pm 0 . 0 1 ) ( \log L _ { 3 0 0 0 \AA } - 2 5 ) } } \\ { { + \, ( 0 . 0 1 3 \pm 0 . 0 0 4 ) ( \log m - 7 ) } } \\ { { { } } } \\ { { \log L _ { 2 k e V } - 2 5 = ( - 2 . 2 8 \pm 0 . 0 8 ) + ( 0 . 5 7 \pm 0 . 0 2 ) ( \log L _ { 3 0 0 0 \AA } - 2 5 ) } } \\ { { { } } } \\ { { + \, ( 0 . 1 4 \pm 0 . 0 2 ) ( \log m - 7 ) } } \end{array} \]

Fig. 9. Same as Fig. 8, with the addition of empty contours for \( \mu \)=0.4, 0.5 and 0.6 (color coded in the legend) obtained with maximally spinning black holes (i.e. with \( \epsilon _ { 0 } = 0 . 3 \) and \( r _ { 0 } \, = \, 1 . 2 4 r _ { g } \). The dashed lines connect them to the non-spinning analogous realizations. Dark-red density spots represent the location of the center of dierent contours of the standard \( \mu \) = 0.5 case, in which the only dierence is the adoption of \( \eta \) (downward scattering component) varying among 0.4, 0.5 and 0.6, going from higher to lower \( \widehat { \alpha } \), respectively.

The comparison in the 3D plane states that the exact dependency is not obtained by any of the models, with \( \mu \) = 1 being the clos-est in qualitatively retrieving the coecients for \( L _ { 3 0 0 0 \mathring { \mathrm { A } } } \) and m. We note that the mass is taken from the observations, thus this mismatch states that the luminosities in the model do not depend on the mass in the correct way.

# 5.6. The impact of the accretion efficiency

Throughout this work we adopted an eciency \( \epsilon _ { 0 } = 0 . 0 5 7 \), typ-ical of non-rotating black holes (e.g. Shapiro 2005), for sim-plicity. Nonetheless, a high spin seems to be preferred to model the blurred relativistic iron line, detected both in the local Uni-verse (Nandra et al. 2007; Reynolds 2013) and up to \( z \sim \) 4 (e.g. Baronchelli et al. 2018). Moreover, flux-limited samples are known to be biased in preferentially detecting high-spinning black holes (Brenneman et al. 2011; Vasudevan et al. 2016), sim-ply because they are brighter than their non-rotating analogous (see Reynolds 2019).

Then, we tested the model using maximally-spinning black holes, with radiative eciency 0.3 and ISCO down to \( r _ { 0 } = \)\( 1 . 2 4 r _ { g } \)(Thorne 1974). This has a major impact on the normal-ization axis of the \( L _ { X } - L _ { U V } . \). Everything else in the source being equal, in a spinning black hole matter can be accreted down to smaller distances with respect to their non-rotating companions, thus the accretion power in the system is much higher. As a mat-ter of fact, changing the radiative eciency has an impact on the numerical equation that regulates f(r): for the same m and \( \dot { m } \) and \( r > \) 3 the values of f is higher, and the transition radius between \( P _ { r a d } - \) and \( P _ { g a s } \)-dominated regions moves at lower radii. This self-consistently aects the disk equations via the \( ( 1 - \tilde { f } ) \) factor (see Appendix A), hence the surface temperature is de-creased at higher radii, where most of the disk emission at 3000Åcomes from. Then, the modeled \( \dot { m } \) value needed to match the observed \( L _ { 3 0 0 0 \mathring \mathrm { A } } \) is higher (see Section 4.2) and, consequently, \( L _ { 2 k e V } \propto f Q _ { + } \) is higher.