Following the approach described in Risaliti & Lusso (2015), there exists a relation between the luminosities in the X-rays (LX) and UV band (LUV)
(8)$$\begin{eqnarray*}
\log (L_\mathrm{ X})=\hat{\gamma }\log (L_{\mathrm{ UV}})+\beta ^{\prime },
\end{eqnarray*}$$where $\hat{\gamma }$ and β′ denote the slope parameter and the intercept. Combining equation (8) with the well-known expression of $L=F\times 4\pi D_L^2$, the luminosity distance can be rewritten as
(9)$$\begin{eqnarray*}
\log (D_L)=\frac{1}{2-2\hat{\gamma }}\times [\hat{\gamma }\log (F_{\mathrm{ UV}}) - \log (F_\mathrm{ X}) + \hat{\beta }],
\end{eqnarray*}$$a function of the respective fluxes (F), the slope parameter ($\hat{\gamma }$), and the normalization constant ($\hat{\beta }=\beta ^{\prime }+(\hat{\gamma }-1)\log _{10} 4\pi$). Therefore, from theoretical point of view, the luminosity distance can be directly determined from the measurements of the fluxes of FX and FUV, with a reliable knowledge of the dispersion δ in this relation and the value for the two parameters ($\hat{\gamma }$, $\hat{\beta }$) characterizing the LX–LUV relation. However, it has been established that the LX–LUV relation was characterized by a high dispersion. Through the analysis of different quasar samples with multiple observations available, previous works derived a consistent value for the slope parameter ($\hat{\gamma }=0.599\pm 0.027$) and the intrinsic dispersion of the relation (0.35–0.40 dex) (Lusso et al. 2010; Young, Risaliti & Elvis 2010). It was found in subsequent analysis quantifying the observational effects (Lusso & Risaliti 2016) that the magnitude of the intrinsic dispersion can be eventually decreased to the level of 0.15 dex. They identified a subsample of quasars without the major contributions from uncertainties in the measurement of the (2 keV) X-ray flux, absorption in the spectrum in the UV and in the X-ray wavelength ranges, variability of the source and non-simultaneity of the observation in the UV and X-ray bands, inclination effects affecting the intrinsic emission of the accretion disc, and the selection effects due to the Eddington bias (Risaliti & Lusso 2019). Besides the lower dispersion in the relation, the reliability and effectiveness of the method strongly depend on the lack of evolution of the relation with redshift (Bisogni et al. 2018). Finally, Risaliti & Lusso (2019) produced a final, high-quality catalogue of 1598 quasars, by applying several filters (X-ray absorption, dust-reddening effects, observational contaminants in the UV, and Eddington bias) to the parent sample from the Sloan Digital Sky Survey (SDSS) quasar catalogues (Shen et al. 2011; Paris et al. 2017) and the XMM–Newton Serendipitous Source Catalogue (Rosen et al. 2016). The final results indicated that such refined selection of the sources could effectively mitigate the large dispersion in the LX–LUV relation, with a tractable amount of scatter avoiding possible contaminants and unknown systematics (see Risaliti & Lusso 2019 for more details). More importantly, the similar analysis has supported the non-evolution of LX–LUV relation with the redshift, which is supported by the subsequent study involving the intercept parameter $\hat{\beta } = 8.24\pm 0.01$, the slope parameter $\hat{\gamma }=0.633\pm 0.002$, and smaller dispersion $\hat{\delta }=0.24$ in a new, larger quasar sample (Risaliti & Lusso 2019).
Citation: Lusso et al. 2010
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