Wavelength Dependences of the Optical/UV and X-ray Luminosity Correlations of Quasars

The study of the optical/UV and X-ray luminosity correlation requires a large, unobscured quasar sample with high-quality optical/UV and X-ray data. The parent sample of this work is taken from Lusso et al. (2020), which contains 2421 quasars with optical spectroscopy from the Sloan Digital Sky Survey Data Release 14 (SDSS DR14) and X-ray observations from either XMM-Newton or Chandra. This parent sample is one of the cleanest quasar samples, in the sense that all quasars with strong extinction (i.e. dust reddening and X-ray absorption), host galaxy contamination and Eddington bias have been excluded (see Lusso et al. 2020 for more details). Since the aim of this work is to explore the wavelength dependence of the inter-band correlation, it is necessary to use the SDSS spectra with the best absolute flux calibration. In fact, there is a known issue with the flux calibration of quasar spectroscopy in DR12 and prior data releases (see e.g. Milaković et al., 2021, for details). Briefly, BOSS quasar targets are observed with an offset on the focal plane to increase the S/N around the Ly$\alpha$, but since the standard stars are not observed in the same fashion, the derived flux correction is not suited for quasars. This results in an overall bluer quasar spectra. A correction has been developed and applied to the individual exposures and the final coadded spectra in DR14 and later data releases, which improved the flux calibration, although there are still significant residuals¹¹1https://www.sdss3.org/dr9/spectro/caveats.php (e.g. Wu & Shen, 2022, see their Section 2). We thus crossmatch the parent sample with the SDSS DR7 catalogue, finding 1278 unique sources. All of these sources have the sciencePrimary flag equal to 1, indicating that the spectra are good for scientific analysis²²2https://skyserver.sdss.org/dr7/en/help/docs/glossary.asp. There are 128 sources having multiple SDSS spectra, for which we choose the spectrum with the highest signal-to-noise (S/N) for our subsequent analysis. We then visually inspect all the SDSS spectra and further exclude 105 sources whose spectra contain data gaps, or if the averaged S/N is $<$ 10 (i.e. sn_median_all>10, which is the S/N per observation over the entire wavelength range and for all observations), or the continuum shows a suspicious down-turn towards the blue end of the spectrum which is most likely due to reddening. The final clean sample contains 1169 quasars.

The SDSS DR7 spectra were downloaded from the SDSS Data Archive Server³³3http://das.sdss.org/www/html/das2.html. For the purpose of investigating the detailed wavelength dependence of various correlation parameters, we must rebin all de-redshifted SDSS spectra on common wavelengths. A relatively low spectral resolution of $\lambda/\Delta\lambda=200$ is adopted to increase the S/N in each wavelength bin. Using a larger $\lambda/\Delta\lambda$ will mainly result in more noisy spectra, but will not change the results of this work. The wavelength coverage of SDSS is 3800 – 9200 Å, but the instrument throughput decreases towards the edges⁴⁴4http://classic.sdss.org/dr7/instruments/spectrographs/index.html where the S/N also decreases, and so we restrict the spectral range to 4000 – 8600 Å for the following analysis. All spectra are de-reddened for Galactic extinction along their line of sight, using the dust map of Schlegel, Finkbeiner & Davis (1998) and the reddening curve of Fitzpatrick & Massa (2007), and then de-redshifted to their respective individual rest-frames.

The catalogue table of Lusso et al. (2020)⁵⁵5https://cdsarc.cds.unistra.fr/viz-bin/cat/J/A+A/642/A150 contains the X-ray flux at 2 keV (rest-frame) and the UV flux at 2500 Å from photometry. However, the photometric flux may include contamination from both emission and absorption lines (e.g. strong UV Fe ii lines) in the filter’s bandpass, so we considered the spectral flux to study the correlation in our analysis. We then check whether there is statistical agreement between the photometric and the spectral flux values. We first convert all the fluxes to luminosities, and then cross-correlate the photometric luminosity with the spectral luminosity at 2500 Å. Note that only 902 sources in our sample are used for this analysis, because these all lie in the redshift range of 0.60 – 2.44 (corresponding to 4000–8600 Å), and so the SDSS spectra can the rest-frame 2500 Å.

As shown in Figure 1 left panel, the two estimates of luminosities at 2500 Å are very consistent with each other. We apply two regression algorithms to check the correlation. The first is the orthogonal distance regression (ODR) algorithm, which treats the two variables symmetrically. The second is the LINMIX algorithm⁶⁶6https://linmix.readthedocs.io/en/latest/src/linmix.html (Kelly 2007), which uses Monte Carlo Markov Chains (MCMC) and returns a reliable estimate of the intrinsic scatter. Both algorithms take into account variable uncertainties (see Lusso & Risaliti 2016 for more details). The ODR algorithm obtains a slope of 1.0046 $\pm$ 0.0044 and an intercept of -0.119 $\pm$ 0.136 (blue dash line), and the LINMIX algorithm finds a slope of 0.998 $\pm$ 0.005 and an intercept of 0.094 $\pm$ 0.148 (green dash line). Both are consistent with the $X=Y$ relation (red solid line). The intrinsic scatter is found to be 6.63 $\pm$ 0.17 per cent. These results confirm that the photometric and spectral fluxes are all acceptably well calibrated, and the contamination by nearby emission (or absorption) lines in the photometric flux is not significant. Therefore our SDSS spectra can be used to study the wavelength dependence of the correlations, and the results can then be compared with previous studies based on the photometric flux at 2500 Å.

The redshift distributions of the luminosities at 2500 Å and 2 keV are shown in the right panel of Figure 1. These are similar to the parent sample in Lusso et al. (2020). Since we applied a S/N cut, some high-redshift sources in the parent sample were excluded due to their low spectral quality, so the redshift range of the current sample is 0.13 – 4.51 with a mean redshift of 1.37 (see the distributions of various parameters in Table 1 and Figure 2a). The mean spectral S/N is 66.8 for the entire sample, confirming that the sample has good SDSS spectral quality. The luminosity at 2 keV covers the range of 42.7 – 46.2 with a mean value of 44.3 (erg s^-1 in logarithm), and the luminosity at 2500 Å has a range of 44.5 – 47.5 with a mean value of 45.8.

Then the SDSS DR14 catalogue of Rakshit, Stalin & Kotilainen (2020) is used to extract the black hole masses for the entire sample. These are the fiducial single-epoch virial masses measured from various broad emission lines in the optical/UV band. The mass range covered by our sample is 7.04 – 11.09 with a mean value of 8.92 in logarithm. Then, given the optical/UV luminosity and the black hole mass, the mass accretion rate through the outer disc ($\dot{m}=\dot{M}/\dot{M}_{\rm Edd}$) can be measured for every source (e.g. Davis & Laor 2011; Jin et al. 2017, 2022). This is done by fitting the SDSS spectrum with the AGN broadband spectral energy distribution (SED) model optxagnf (Done et al. 2012), with the black hole mass fixed at the viral mass, spin fixed at zero and $R_{cor}$ fixed at the innermost stable circular orbit i.e. assuming the accretion flow is a standard disc⁷⁷7A detailed SED study of the sample will be published in another paper.. Note that $\dot{m}$ is not necessarily equal to the Eddington ratio ($L_{\rm bol}/L_{\rm Edd}$), because the bolometric luminosity ($L_{\rm bol}$) also depends on the radiative efficiency of the accretion flow, which in turn depends on the physical properties of the flow and the black hole spin (e.g. Davis & Laor 2011; Done et al. 2013; Kubota & Done 2019). The viewing angle may also affect the observed $L_{\rm bol}$ (e.g. Jin et al. 2017, 2022). Hence we prefer to use $\dot{m}$ as a more robust indicator of the mass accretion state of an AGN. We find that $\dot{m}$ of the sample covers a range of -2.70 – 1.09 with a mean value of -1.14 in logarithm (i.e. the mean $\dot{m}$ is 7.2 per cent).

Besides the overall statistical properties presented above, we also check the sample distribution and quality at various wavelengths. Firstly, we examine the sample size at different wavelengths in the rest-frame, as shown in Figure 3a. The maximal number (926 quasars) is found at the wavelength of 2672 Å, which then decreases towards both sides. There are 903 quasars at 2500 Å. We choose a conservative threshold of $n=50$ to ensure that the correction analysis at each wavelength is statistically robust. This threshold corresponds to a wavelength cut at 6360 Å, and so we restrict the following analysis to the rest-frame wavelength range of 1000 – 6360 Å.

Then, we check the redshift distribution and the mean S/N of different sources in each wavelength bin, as shown Figures 3b and  3c. The sample’s redshift range increases naturally towards short wavelengths. At $\lambda\sim$ 1000 Å, the redshifts are mostly above 3. The redshift range at 2500 Å is 0.61 – 2.42. For the S/N, the mean value reaches its maximum of 67 at the Mg ii line, and decreases towards both sides. The mean S/N at 2500 Å is 51.5. The wavelength range with the mean S/N $\geq$ 10 is 1280 – 5550 Å, where the results of correlation analysis should be more reliable.

Finally, we examine the composite optical/UV spectrum for the entire sample. As described in Vanden Berk et al. (2001), the composite spectrum is sensitive to the method of assembly. Since the optical/UV underlying continua of quasars generally have a power law shape, we choose the geometric mean algorithm, so that the continuum’s shape of the composite spectrum reflects the mean slope of the input sample. To further suppress the bias due to the sample variation at different wavelengths, we divide the sample into three subsamples for three redshift ranges (0.13 – 0.8, 0.8 – 1.8, 1.8 – 4.51), and produce a composite spectrum in each interval. With these three composite spectra, we use the geometric mean to connect different spectra in the two wavebands of 2300 – 3000 Å and 1470 – 3000 Å, and finally derive a total composite spectrum. The absolute flux of the composite spectrum is trivial, so the choice of the above wavelength ranges are arbitrary. The resulting composite spectrum⁸⁸8Our new quasar composite spectrum can be downloaded from https://www.dropbox.com/s/mk3zgwnxe9w66tr/quasar_compspec_Jin2023.txt?dl=0 is shown in Figure 3d, and is compared to the quasar composite spectrum of Vanden Berk et al. (2001) which contains 2204 quasars covering the redshift range of 0.044 – 4.789. We find that these two composite spectra are very similar. The main difference is that the underlying continuum of our new composite spectrum is slightly steeper, which is likely because the parent sample of this study has been filtered more strongly to exclude sources with significant extinction and host galaxy contamination (Lusso et al. 2020).

While the 2 keV luminosity may physically represent the intensity of the corona emission, including both the hot corona and probably some fraction of the warm corona, the 2500 Å luminosity could be dominated by emission from the disc itself (although an extended warm corona may also contribute significantly to the UV emission for some AGN, see Gardner & Done 2017). Thus, by taking the luminosity at 2500 Å, instead of 2 keV as the correlation variable, we can examine the self-correlation of the disc emission in the optical/UV, as well as any correlations for various emission lines with the disc continuum.

The photometric luminosity at 2500 Å ($L_{\rm 2500\textup{\AA},phot}$) is used because it is available for the entire sample (as listed in the catalogue published by Lusso et al., 2020). This new version of correlation spectrum is named the optical/UV self-correlation spectrum (OUSCS). Likewise, we first create the first type of OUSCS, namely OUSCS-1, as shown in Figure 4b (black solid line). The self-correlation with $L_{\rm 2500\textup{\AA},phot}$ is approaching unity across the entire waveband, and is much stronger than the correlations with $L_{\rm 2keV}$, and the self-correlation is also stronger in the UV band than optical band.

There are many absorption features in OUSCS-1, indicating that the correlation of the emission lines with the UV luminosity is weaker than with their underlying continua. To examine the effect of sample bias, we also calculate the OUSCS-1 from the disc model, as shown by the red solid line in Figure 4b. The comparison between the observed and model OUSCS-1 suggests that the overall shape is driven by the disc emission, whilst the small-scale residual fluctuations in the observed OUSCS-1 have a different origin than the disc continuum (e.g. emission lines). The observed OUSCS-1 remains flat below 2000 Å, which is mostly because the input disc model does not provide a good match of the observed spectra in the UV band. This finding is consistent with the ones published by Mitchell et al. (2023). By analysing a sample of $\simeq$700 SDSS AGN, they found that the data do not match the predictions made by any current accretion flow model. Specifically, they observed that either the disc is completely covered by a warm Comptonisation layer, whose properties change with accretion rate, or the accretion flow structure is different to that of the standard disc models (see their Figure 10 and discussion in their Section 5).

The gradual decrease of the observed OUSCS-1 towards longer wavelengths also seems to follow the decrease of the luminosity range (blue dash line), so we create OUSCS within the aforementioned seven different wavebands, each of which is based on the same subsample (see Figure 6 Panel-b1). These individual short OUSCS are then joined together, and renormalized to OUSCS-1 within 4900 – 5100 Å, to create the second type of OUSCS, namely OUSCS-2.

Figure 6 Panel-b2 compares the two types of OUSCS. Likewise, OUSCS-2 is flatter than OUSCS-1 after suppressing the bias of luminosity range. The correlations remain good across the entire optical/UV band, and the UV continuum show slightly better correlations than optical. The statistical error of OUSCS is $\sim$ 0.001, so the difference between optical and UV in OUSCS-2 is also statistically significant. The peak correlation lies within 2500 – 3500 Å. In addition, OUSCS-2 also shows various absorption-like features. Similar to OUXCS-2, the broad component of H$\beta$ shows stronger correlation, while the correlation of the narrow component is weaker. However, C iv, C iii], Mg ii and [O iii]$\lambda\lambda$4959/5007 all appear absorption-like, suggesting that their correlations with $L_{\rm 2500\textup{\AA},phot}$ are weaker than with their underlying continua. The absorption-like features at C iv and [O iii] are particularly strong. This is probably due to their higher ionization energies ($>$ 50 eV), which also leads to their higher correlations with the 2 keV luminosity in OUXCS. We discuss these in more detail in Section 5.5.

Based on the results presented in the previous sections we can address the question proposed at the beginning of this paper: whether the conventional choice of 2500 Å is a good choice to represent the optical/UV luminosity and its correlation with the X-rays, for the large quasar sample. As shown by the OUXCS-2 in Figure 6 Panel-a2 , there is a strong correlation with the X-ray luminosity at every wavelength within 1280 – 5550Å if the spectral quality is high. The correlation coefficients found at other wavelengths are consistent with those at 2500 Å. The weak wavelength-dependence of the regression parameters shown in Figure 7 also suggest that the slope and intercept found at 2500 Å are similar to those found at the other wavelengths. The regression parameters only change significantly at the wavelengths covering strong emission lines. But after these lines are subtracted, their underlying continua still show similar correlations as seen at other wavelengths. Therefore, we conclude that results of optical/UV and X-ray inter-band correlations found by previous studies using 2500 Å  are indeed robust and representative for the entire optical/UV continuum, provided that the quasar sample is composed of unobscured objects with high-quality data. Our results also highlight the efficiency of the filtering steps we applied for selecting unobscured quasar samples, as described in Lusso et al. (2020).

The luminosity at the rest frame 2500 Å ($L_{\rm 2500\textup{\AA}}$) can be measured in different ways: from the photometric spectral energy distribution (i.e. $L_{\rm 2500\textup{\AA},phot}$, thus including the contribution of both the continuum and emission/absorption lines), or directly from the spectrum (i.e. $L_{\rm 2500\textup{\AA},spec}$). Note that there are only 902 sources whose SDSS spectra cover 2500 Å in the rest-frame, so that only this subsample has $L_{\rm 2500\textup{\AA},spec}$ measurement.

For our sample, the systematic difference between $L_{\rm 2500\textup{\AA},phot}$ and $L_{\rm 2500\textup{\AA},spec}$ is only 6.6 per cent (see Figure 1), so their regression results are expected to be similar. Indeed, Figures 9a and 9b show the regression results for the two measurements of $L_{\rm 2500\textup{\AA}}$. No significant difference is found between the regression parameters, including a similar level of intrinsic dispersion.

We note that the flux at 2500 Å contains both the continuum and the blend of Fe ii UV emission lines (Vestergaard & Wilkes 2001), so it is also useful to examine whether the UV Fe ii emission changes the regression results at 2500 Å significantly. For this purpose, we use a series of line-free wavelengths to determine the UV continuum, as shown in Figure 10. We then visually inspect each spectrum to ensure the quality. Then $L_{\rm 2500\textup{\AA}}$ is measured from the continuum (i.e. $L_{\rm 2500\textup{\AA},cont}$). In most cases we use the flux at 2230 Å as one continual point, so it results in a reduction in the sample size. Some sources are also excluded due to the poor quality on the blue end of the spectrum. Thus the final subsample with $L_{\rm 2500\textup{\AA},cont}$ measurement has only 778 sources. We find that $L_{\rm 2500\textup{\AA},spec}$ is statistically larger than $L_{\rm 2500\textup{\AA},cont}$ by only 6.4 $\pm$ 5.4 per cent. Figures 9c shows the regression result for $L_{\rm 2500\textup{\AA},cont}$, and the results are still consistent with the other two measurements of $L_{\rm 2500\textup{\AA}}$, including a similar intrinsic dispersion of 0.231 $\pm$ 0.006 dex.

Therefore we conclude that the regression results for the $L_{\rm 2500\textup{\AA}}$ vs. $L_{\rm 2keV}$ relation are not affected by different measurements of $L_{\rm 2500\textup{\AA}}$ in this study.

Lusso & Risaliti (2017) adopted a simplified disc-corona model and assumed that the radius of the broad line region (BLR) is proportional to the square-root of the disc luminosity, and then proposed the theoretical relation: $L_{\rm 2keV}\propto L_{\rm optuv}^{4/7}\nu_{\rm fwhm}^{4/7}$, where $\nu_{\rm fwhm}$ is the virial velocity of the BLR. This relation was confirmed by their statistical analysis of a sample of 545 optically selected quasars. In this work, the Mg ii line was used to measure $\nu_{\rm fwhm}$. Since line-profile fitting has been conducted for all the sources (see Section 4), we can perform 2-dimensional (2D) regression in the $L_{\rm 2keV}$ – $L_{\rm 2500\textup{\AA}}$ – $\nu_{\rm fwhm}$ plane to test the above theoretical relation. This 2D regression is defined as:

where $\gamma_{1}$, $\gamma_{2}$ and $\beta_{1}$ are the free parameters to be fitted. Thus we have $\gamma_{1}$ and $\gamma_{2}$ equal to 4/7 for the theoretical relation. We also define $\delta_{1}$ as the intrinsic dispersion of this plane, to distinguish it from the intrinsic dispersion ($\delta$) of the $L_{\rm 2keV}$ – $L_{\rm 2500\textup{\AA}}$ relation. We conducted the 2D regression using the $\nu_{\rm fwhm}$ of C iv, C iii], Mg ii and H$\beta$, respectively. The coefficients are listed in Table 4. The planes seen edge-on are shown in Figure 11.

The subsample of Mg ii has 836 quasars, each has measurements of $L_{\rm 2keV}$, $L_{\rm 2500\textup{\AA},phot}$ and Mg ii line width. We find $\gamma_{2}=0.623\pm 0.017$ and $\gamma_{2}=0.501\pm 0.046$, which are statistically consistent with both the theoretical values and the results reported by Lusso & Risaliti (2017) for the same line but a different, larger sample. We also find that, for the same subsample, the dispersion of the $L_{\rm 2keV}$ – $L_{\rm 2500\textup{\AA}}$ relation decreases by $0.012\pm 0.003$ dex when the $\nu_{\rm fwhm}$ of Mg ii is included in the regression (see Table 4).

For the H$\beta$ line, the subsample of 209 quasars displays a relatively small dynamical range in luminosity (most of the data lies in the range 27.5 – 28.5, $\sim$1 dex, see Figure 14), and the intrinsic dispersion of the relation appears to dominate the distribution (see Figure 11d). We find a similar index of $\gamma_{1}=0.627\pm 0.045$ for $L_{\rm 2500\textup{\AA},phot}$. But $\gamma_{2}$ is found to be $0.165\pm 0.162$. However the large uncertainty makes it difficult to draw any firm conclusion about the index of $\nu_{\rm fwhm}$.

The results for C iv and C iii], however, are significantly different. We find $\gamma_{2}$ is $-0.352\pm 0.067$ for C iv and $-0.302\pm 0.049$ for C iii], both of which are negative, and opposite to the indices found for Mg ii, H$\beta$ and the theoretical value. The negative $\gamma_{2}$ indicates that larger line-widths of C iv and C iii] correspond to weaker X-ray luminosity relative to UV. Considering the fact that the line-widths of C iv and C iii] do not only include components of virial velocity, but also a significant outflow velocity of the BLR (Gaskell 1982; Wilkes 1984; Leighly 2004), the negative $\gamma_{2}$ is actually consistent with the anti-correlation found between the optical-to-X-ray spectral index ($\alpha_{\rm ox}$: Tananbaum et al. 1979; Lusso et al. 2010) and the blueshift velocity of C iv (Timlin et al. 2020; Lusso et al. 2021). It is well known that AGN properties exhibit different sets of correlations. The primary set is the so-called ‘Eigenvector-1’ (Boroson & Green 1992), which is found to be driven by the Eddington ratio (e.g. Marziani et al. 2003; Grupe 2004; Jin, Ward & Done 2012c). The blueshift of C iv also belongs to Eigenvector-1, and a higher Eddington ratio corresponds to a larger blueshift velocity of this line (Sulentic et al. 2007). In addition, it is found that a higher Eddington ratio corresponds to a lower ratio of X-ray emission relative to UV (e.g. Vasudevan & Fabian 2007; Lusso et al. 2010; Grupe et al. 2010; Jin, Ward & Done 2012c). Therefore, these previously known correlations with the Eddington ratio can naturally result in a negative $\gamma_{2}$ of C iv and C iii].

The OUXCS and OUSCS produced by the direct-correlation method shows that, compared to the underlying continuum, the primary emission lines have different correlations with the 2 keV and 2500 Å luminosities. This is consistent with the results found by fitting the lines, separating the line flux from the continuum, and then comparing their individual correlations. However, these two methods have fundamentally different characteristics.

The line-fitting method can separate the emission lines from the continuum, and test their correlations separately with some other parameters (such as 2 keV luminosity and 2500Å luminosity), but this method may also introduce the following issues.

Firstly, the uncertainty caused by the line profile fitting. There are various complexities in the line profile fitting. For example, the broad and narrow components are blended; the broad H$\beta$, Fe ii and [O iii] are also blended; narrow absorption lines can be present with different velocity-shifts in C iv and C iii], although these absorption features are not visible in the mean spectrum in Figure 3d. These can all lead to a measurement uncertainty of the line flux. However, the results for individual lines in Section 4 are not affected by the line decomposition because the total line flux is used.

Secondly, the relative error of the line flux. The LINMIX algorithm is sensitive to the relative errors of the data points. Once the lines are separated from the continuum, the relative errors of the line fluxes increase, especially when the lines are weak. This can lead to different regression parameters.

Thirdly, different correlations at different velocity shifts of one line. For example, C iv has a significant blue wing, but OUXCS does not show a similarly strong correlation in the blue side of the line. This means that the correlation of the entire line flux will be the mean value of the correlations for components present at all velocity shifts.

In comparison, the direct-correlation method examines the correlation with a specific parameter (such as 2 keV luminosity or 2500Å luminosity) at each wavelength. At the wavelength of an emission line, the correlation given is the overall result of the emission line, the underlying continuum and nearby blended lines (such as the Fe ii lines). If the correlation of an emission line shows a similar trend to that of the continuum, but with a statistically stronger correlation, it appears as an ‘emission line’ in the correlation spectrum as well, just like C iv, C iii], Mg ii, H$\beta$ in OUXCS. Conversely, if the correlation of the line is different from or weaker than that of the continuum, it appears as an ‘absorption line’ in the correlation spectrum, just like [O iii], C iv, Mg ii and the narrow H$\beta$ in OUSCS. Therefore, the direct-correlation method can be used to study the difference in correlation between emission lines and the continuum without detailed spectral fitting.

The advantage of this method can be reflected by the correlation parameters seen at the optical Fe ii complex in the OUXCS (Figures 6 Panel-a2 and 7). Usually, it is very difficult to explore the correlation of Fe ii lines with other parameters unless they are very strong (e.g. in some super-Eddington NLS1s, Du & Wang 2019) and not blended with nearby lines (e.g. [O iii] $\lambda\lambda$4959/5007). With the direct-correlation method, we can reveal the correlation at the Fe ii complex without line fitting or de-blending. However, the disadvantage is that if different lines in a certain waveband are mixed together, the result of direct correlation will contain the correlations of different lines, making it difficult to distinguish which line has a greater contribution.

We can also apply the direct-correlation method to each emission line. Based on the line fitting described in Section 4, we first remove the best-fit local continuum and Fe ii lines, and then directly correlate the spectral lines with the 2 keV luminosity and 2500Å luminosity, respectively. Figure 12 shows the direct correlation results for different emission lines (plotted in black in each panel). Firstly, by comparing with the composite line profile (plotted in orange), we find that the correlation between the narrow line component and 2 keV is weaker than that of the broad line component. Secondly, the direct correlation results of the line profiles are very different from the line profiles observed in OUXCS and OUSCS (plotted in light blue), which indicates that the shapes of OUXCS and OUSCS are the overall results of different spectral components. For example, when we look at Panels a1, a2, a3 and a4 separately, we find that the blue wing of C iv and C iii], as well as the narrow component of Mg ii and H$\beta$, are relatively weaker in OUXCS than those observed in the direct correlation line profiles, indicating that the correlations of these line components are more different from the continuum.

Therefore, we can see that the direct correlation method is simple and straightforward without introducing the complexities of line fitting, and it also provides new clues for understanding the profile and origin of different lines which we discuss in more detail in the next section.

The direct-correlation line profiles in Figure 12 show the correlation of emission line at different velocities. In comparison, the lines observed in OUXCS and OUSCS show if and how significant are their correlations (with 2 keV luminosity or 2500 Å luminosity) different from those of the continuum. We summarize these correlation properties and, whenever possible, explain them in terms of the AGN unified model (Antonucci 1993) and the shape of the spectral energy distribution (SED) of the ionizing photons , thereby inferring the origin of different lines and line components.

(1) By comparing the direct-correlation line profiles of 2 keV with 2500 Å in Figure 12, we find that the correlations of C iv, C iii], Mg ii, H$\beta$ lines with the 2500 Å luminosity are systematically stronger than those with 2 keV luminosity.

These results can be understood from two aspects. Firstly, the ionization potentials of C iv, C iii], [O iii] and Mg ii are 47.9 eV, 24.4 eV, 35.1 eV and 7.6 eV, respectively. These energies correspond to wavelengths of 259.5Å, 509.5Å, 354.2Å and 1635.7Å. Therefore, these lines should be correlated with the far-UV and X-ray emission. The fact that they are all well correlated with the 2500 Å luminosity suggests that the good optical/UV self-correlation seen in OUSCS-2 (Figure 6 Panel-b2) should extend into the far-UV regime of at least a few hundreds of angstrom, which is likely also dominated by the accretion disc emission. Secondly, X-rays are easily absorbed and only account for a low proportion of the broadband spectral energy distribution (SED). For example, based on the classic flat equatorial distribution of BLR, X-rays may be significantly absorbed by the disc wind or the disc itself before reaching the BLR. Therefore, it is reasonable to expect that these BLR lines should correlate better with the UV continuum than X-ray.

(2) By comparing the direct-correlation line profile (black) and composite line profile (orange) in each panel of Figure 12, we find that the correlation between the narrow component of the broad emission lines and 2 keV or 2500 Å is generally worse than that of the broad component. However, the direct-correlation line profile of Mg ii for 2500 Å are almost the same as the composite line profile, indicating that the narrow component of Mg ii also have a similarly good correlation with 2500 Å luminosity.

These results can be understood as the broad component being mainly produced by the ionization of disc photons, while the narrow component may also include contaminations from the host galaxy star light or being affected by other factors (such as covering factor), so the correlation of the narrow component with 2 keV and 2500 Å is worse. This is also consistent with the results of Jin, Ward & Done (2012b).

(3) By comparing the direction-correlation line profiles (black) in Panels a1 to a4 of Figure 12 with OUXCS (light blue), we find that the correlation on the blue side of C iv is somewhat different from the continuum, and the central narrow components of Mg ii and H$\beta$ show similar behaviours. We discuss these results together with the next point below.

(4) By comparing the direction-correlation line profiles (black) and OUXCS (light blue) in Panels b1 to b4 of Figure 12, we find that although strong correlations with 2500 Å are observed across the entire wavelength range, the lines and their continua actually behave differently, but only in the cases of high-ionization lines (i.e. C iv, C iii]) and narrow component of low-ionization lines (H$\beta$, Mg ii). This is also confirmed by the linear regression parameters given in Table 2.

The OUSCS has shown that the optical/UV continuum is highly self-correlated. Then these strong correlations should be transmitted through the ionizing source to different emission lines. However, point-3 and 4 above indicate that the forms of correlation between different lines (or different line components) and their ionizing sources are not the same. This implies that there is more complexity in the connection between these emission line regions and their respective ionizing sources, such as the difference of covering factor. For example, high-ionization lines such as C iv are considered to be located closer to the ionizing source (e.g. Grier et al. 2019), and so are more sensitive to the change of inner disc structure and wind (e.g. Jin et al. 2022).

(5) Comparing to the results above, [O iii] is a clear exception. OUXCS shows that its correlation with 2 keV is significantly better than that of the continuum and broad lines. This is consistent with previous studies of [O iii] vs. X-ray correlation for various AGN samples (e.g. Heckman et al. 2005; Panessa et al. 2006; Lamastra et al. 2009; Jin et al. 2012a; Jin, Ward & Done 2012b; Ueda et al. 2015). Meanwhile, OUSCS shows that the correlation between [O iii] and 2500 Å is significantly worse. However, in Figure 12 and Table 2, the correlation between [O iii] and 2 keV is not significantly better. This is likely due to the additional uncertainties introduced by removing the best-fit Fe ii lines and the underlying continuum. Thus it also demonstrates the advantage of OUXCS, which is that it does not require spectral fitting.

Compared with the narrow component of broad lines, the distribution of [O iii]-emitting gas may be more spherical, i.e. having a significantly larger covering factor to the high-energy emission from the corona. As a result [O iii] lines indeed show stronger correlations with the X-rays than C iv, but weaker correlations with the optical/UV continua.

(6) The optical Fe ii complex on both sides of H$\beta$ seems to show different correlation with e.g. the broad H$\beta$. It is known that the intensity of optical Fe ii is strongly correlated with the Eddington ratio (i.e. Eigenvector-1, Boroson & Green 1992), so it does not correlate with the optical luminosity alone, but also with the black hole mass. Furthermore, the origin of optical Fe ii is still not clear (Wilkes, Elvis & McHardy 1987; Hu et al. 2015; Lawrence et al. 1997; Du & Wang 2019). It may not simply be the photoionization, but may also involve collisional excitation of gas turbulence (e.g. Baldwin et al. 2004). Thus it is not surprising if the Fe ii lines show a weaker/different correlation with the continuum luminosity.

The above explanations about different line correlations are very preliminary, but still qualitatively consistent with the classic picture of AGN unified model and BLR. However, each line requires more detailed analysis in order to fully understand their different correlations and origins, but this is beyond the scope of this work.