Observations favor the redshift-evolutionary $L_{X}$-$L_{UV}$ relation of quasars from copula

Quasars are extremely luminous and persistent energy sources powered by supermassive black holes. The luminosities of quasars are so great that the maximum redshift of quasars can reach $z>7$ (Mortlock et al., 2011; Bañados et al., 2018; Lyke et al., 2020; Wang et al., 2021). If quasars can be regarded as standard candles, they will cover the redshift desert of cosmological data, and may play an important role in understanding the property of dark energy and the possible origin of the Hubble constant ($H_{0}$) tension. To use data from quasars for cosmological purposes, one needs to construct a luminosity relation to determine the distance of quasars. Several empirical relations have been proposed (Baldwin, 1977; Paragi et al., 1999; Chen & Ratra, 2003; Watson et al., 2011; La Franca et al., 2014; Wang et al., 2014; Cao et al., 2021). Among them, the nonlinear relation between the X-ray luminosity ($L_{X}$) and the ultraviolet (UV) luminosity ($L_{UV}$) (Risaliti & Lusso, 2015, 2019; Lusso & Risaliti, 2016; Lusso et al., 2020) is a very popular one. The $L_{X}$-$L_{UV}$ relation has been utilized to construct the Hubble diagram of the quasars up to $z\sim 7.5$, and has been applied in the quasar cosmology widely (Khadka & Ratra, 2020a, b; Wei & Melia, 2020; Khadka & Ratra, 2021; Li et al., 2021; Lian et al., 2021; Bargiacchi et al., 2022).

With the $L_{X}$-$L_{UV}$ relation and the logarithmic polynomial expansion of the cosmic distance, it has been found that the distance modulus/redshift relation of the quasars at $z>1.4$ has a more than 4$\sigma$ deviation from the prediction of the cosmological constant plus the cold dark matter ($\Lambda$CDM) model with $\Omega_{m0}=0.3$ (Risaliti & Lusso, 2019; Lusso et al., 2019) ¹¹1It should be pointed out that controversies have arisen regarding this deviation (Yang et al., 2020; Banerjee et al., 2021; Bargiacchi et al., 2021; Khadka & Ratra, 2022; Sacchi et al., 2022; Khadka et al., 2022; Dainotti et al., 2022; Lenart et al., 2023; Wang et al., 2022; Mehrabi & Basilakos, 2020; Velten & Gomes, 2020; Petrosian et al., 2022). Some researches indicated that this deviation may originate from the divergence of the logarithmic polynomial expansion at the high-redshift regions (Yang et al., 2020; Banerjee et al., 2021)., where $\Omega_{m0}$ is the present dimensionless matter density parameter. This deviation may indicate that the standard $L_{X}$-$L_{UV}$ relation of quasars may not be accurate. Recently, Khadka & Ratra (2022) found that part of the quasar data shows evidence of redshift evolution of the $L_{X}$-$L_{UV}$ relation, suggesting a possible redshift-evolutionary $L_{X}$-$L_{UV}$ relation. Dainotti et al. (2022) first obtained a three-dimensional and redshift-evolutionary $L_{X}$-$L_{UV}$ relation by considering a redshift dependent correction to the luminosities of quasars. We also constructed a three-dimensional and redshift-evolutionary $L_{X}$-$L_{UV}$ relation by using the powerful statistical tool called copula (Wang et al., 2022). For the standard and two redshift-evolutionary $L_{X}$-$L_{UV}$ relations, it remains interesting to determine which one is favored by observations.

To compare three different $L_{X}$-$L_{UV}$ relations with observations, the theoretical value of the luminosity distance $d_{L}$ must be given. One way to achieve this goal is to use a cosmological model to derive the luminosity distance. The obtained results will then be model-dependent. The other one is to resort to the so-called cosmography, which is cosmological-model-independent. However, the usual cosmographic method, which is based on the Taylor expansion or the Padé approximation of the Hubble expansion rate, suffers the divergence problem in the high-redshift regions. Recently, Huang (2020) proposed the PAge approximation and found that it can describe the global expansion history of our universe with high accuracy. The PAge approximation can be considered as a cosmological-model-independent method for the high-redshift cosmography (Huang, 2020; Huang et al., 2021; Cai et al., 2022a) and has been applied in cosmology widely. For example, the PAge approximation has been used to explore the supernova magnitude evolution (Huang, 2020), test the high redshift gamma-ray burst luminosity correlations (Huang et al., 2021), reaffirm the cosmic expansion history (Luo et al., 2020), and investigate the $S_{8}$ tension (Huang et al., 2022) and the $H_{0}$ tension (Cai et al., 2022b, c). More recently, by dividing the quasar data into the low-redshift and high-redshift parts and using them to constrain the coefficients of the $L_{X}$-$L_{UV}$ relation and the PAge parameters, Li et al. (2022) found that there are apparent inconsistencies between the results from the low-redshift data and the high-redshift ones respectively, and concluded that the deviation between the distance modulus/redshift relation of the high-redshift quasars and the prediction from the $\Lambda$CDM model found in Ref. (Risaliti & Lusso, 2019) probably originates from the redshift-evolutionary effects of the $L_{X}$-$L_{UV}$ relation.

In this paper, we check whether the $L_{X}$-$L_{UV}$ relation is redshift-evolutionary and compare three different $L_{X}$-$L_{UV}$ relations (Dainotti et al., 2022; Wang et al., 2022; Risaliti & Lusso, 2015) by using the PAge approximation with the observational data comprising of the 2421 quasar sample (Lusso et al., 2020), the Hubble parameter ($H(z)$) measurements (Moresco et al., 2020) and the Pantheon+ type Ia supernova (SNe Ia) data (Brout et al., 2022). We find that the three-dimensional and redshift-evolutionary $L_{X}$-$L_{UV}$ relation from copula (Wang et al., 2022) is favored by the observational data.

The rest of the paper is organized as follows. In section 2, we introduce the PAge approximation, the three different $L_{X}$-$L_{UV}$ relations and the data sets. The redshift-evolutionary effects of $L_{X}$-$L_{UV}$ relations are examined in section 3. A comparison of the three different relations is given in section 4 and the selection effects are discussed in section 5. We summarize our conclusions in section 6.

After dividing all the quasar data into the high-redshift part and the low-redshift one, Li et al. (2022) used them to constrain the standard $L_{X}$-$L_{UV}$ relation with the PAge approximation and found that there exists tension between the results from the high-redshift and low-redshift data respectively. They concluded that this tension may originate from the redshift-evolutionary effects of the $L_{X}$-$L_{UV}$ relation. In this section, we discuss the constraints on the two redshift-evolutionary $L_{X}$-$L_{UV}$ relations with the PAge approximation using data from the low-redshift ($z<1.4$) and high-redshift ($z>1.4$) quasars respectively. For comparison, the standard $L_{X}$-$L_{UV}$ relation is also considered in our analysis. Since the Hubble constant $H_{0}$ is degenerate with parameter $\beta$ in the relations and the quasars have large dispersions, we also include the $H(z)$ measurements and the SNe Ia data.

The best-fitting values with uncertainties are obtained by using the Markov Chain Monte Carlo (MCMC) method, which is available via the $emcee$ package in Python 3.7 (Foreman-Mackey et al., 2013). Table 1 and Figure 1 show the marginalized results and the probability contour plots, respectively. From them, we obtain the following main results:

• •

  For the standard relation, the constraints from the low-redshift and high-redshift data on all the parameters are inconsistent. This agrees with the conclusion obtained in (Li et al., 2022).

• •

  For the redshift-evolutionary relations, the cosmographic parameters ($t_{0}$, $H_{0}$ and $q_{0}$) and the absolute magnitude ($M_{B}$) are consistent in two different redshift ranges. However, the values of the relation coefficients ($\beta$, $\gamma$ and $\alpha$) and the intrinsic dispersion ($\delta$) from the low-redshift and high-redshift data respectively are discrepant. This is similar to what obtained in the standard relation case.

• •

  The redshift-evolutionary coefficient $\alpha$ deviates from zero at more than $2\sigma$, which implies that the observational data support a redshift-dependent relation. Furthermore, the redshift-evolutionary character is more apparent in the Type I relation than that in the Type II case.

Therefore, the redshift-evolutionary $L_{X}$-$L_{UV}$ relations are favored as opposed to the standard one since consistent PAge parameters can be obtained in the redshift-evolutionary relations and the value of $\alpha$ deviates from zero at more than $2\sigma$.

To further compare the three different $L_{X}$-$L_{UV}$ relations, we will use all $H(z)$, SNe Ia and quasar data to constrain the PAge parameters ($t_{0}$, $H_{0}$ and $q_{0}$), the absolute magnitude ($M_{B}$), the coefficients of the relations ($\beta$, $\gamma$ and $\alpha$) and the intrinsic dispersion ($\delta$) simultaneously. We will utilize the Akaike information criterion (AIC) (Akaike, 1974, 1981) and the Bayes information criterion (BIC) (Schwarz, 1978) to find the relation favored by the observations. The AIC and BIC are defined, respectively, as

where $\mathcal{L}_{\rm max}$ is the maximum value of the likelihood function, $p$ represents the number of free parameters and $N$ is the number of data. We calculate $\Delta$AIC(BIC) of the three relations, which denotes the difference in the AIC(BIC) of a given relation relative to the reference relation (the Type I relation ), i.e., $\rm\Delta AIC(BIC)=AIC(BIC)-AIC_{ref}(BIC_{ref})$, for the relation comparison. If $0<\Delta\mathrm{AIC(BIC)}\leq 2$, it is difficult to single out a better relation. If $2<\Delta\mathrm{AIC(BIC)}\leq 6$ we have mild evidence against the given relation, and we have strong evidence against the given relation if $\Delta\mathrm{AIC(BIC)}>6$. The parameter values and $\Delta$AIC(BIC) obtained from MCMC are summarized in Table 2, and the posterior distribution contours are shown in Figure 2. In the last column of Table 2, the constraints on the PAge parameters and the absolute magnitude obtained from data without the quasars are also displayed in order to assess the influence of the quasars.

One can see that the constraints on the PAge parameters ($H_{0}$, $t_{0}$ and $q_{0}$) and the absolute magnitude $M_{B}$ from $H(z)$+SNe Ia+quasars with the Type I relation are well consistent with those from data without the quasars. The values of the PAge parameters and the absolute magnitude are compatible with each other for the Type I and Type II relations, while they apparently deviate for the standard relation. The values of $t_{0}$ for the Type I and Type II relations are more comparable with the result from the Planck 2018 CMB observations ($t_{0}=13.801\pm 0.024$ Gyr) (Planck Collaboration et al., 2020) than those for the standard relation. The deceleration parameter $q_{0}$ for the Type I and Type II relations is slightly larger than that in the $\Lambda$CDM model ($q_{0}=3\Omega_{m0}/2-1=-0.528\pm 0.011$) from the Planck 2018 CMB observations (Planck Collaboration et al., 2020), while its value for the standard relation is smaller than $q_{0}=-0.528\pm 0.011$. Furthermore, we find that the value of the Hubble constant $H_{0}$ for the Type I and Type II relations is lager than that for the standard relation, and locates in the region between the Planck 2018 CMB observations (Planck Collaboration et al., 2020) and the local distance ladders from SH0ES (Riess et al., 2022), which can be seen in Figure 3, where the values of $H_{0}$ from the quasars+$H(z)$+SNe Ia, the Planck 2018, and the SH0ES are shown.

The intrinsic dispersion $\delta$ and the relation coefficient $\gamma$ have almost the same values in the Type I and Type II relations, which are larger than those obtained in the standard relation. The value of $\gamma$ in the two three-dimensional relations is consistent with $\gamma=0.586\pm 0.061$ obtained through the narrow redshift bin method (Lusso et al., 2020). For the relation coefficient $\beta$, the Type II relation has the maximum value and the standard relation has the minimum one. The deviations of $\beta$ among three different relations are of more than $2\sigma$. The observations favor strongly the redshift-evolutionary relations since $\alpha=0$ is ruled out at more than $3\sigma$. The value of $\alpha$ in the Type I relation is apparently larger than that in the Type II relation, which indicates that the redshift-evolutionary character in the Type I relation is more apparent than that in the Type II relation. The $\alpha$ value in the Type I relation is comparable with $\alpha=0.580^{+0.084}_{-0.099}$ obtained in the $\Lambda$CDM model (Wang et al., 2022). From Table 2, we find that the Type I three-dimensional relation has the minimum AIC(BIC). The $\Delta$AIC(BIC) of the standard relation is much larger than 6, which indicates that we have strong evidence against it. The $\Delta$AIC(BIC) of the Type II three-dimensional relation is 3.865. Thus, the Type I three-dimensional relation has a slight advantage.

To check whether our result that the redshift-evolutionary relation is favored by the observations is caused by the selection effects involved in the measurement process, we follow the method given in (Li, 2007; Singh et al., 2022) and use firstly Monte Carlo simulations to simulate a realistic population of quasars as a function of redshift. Since the 2421 quasar data points satisfy the Gamma distribution in the $z$ space (Lusso et al., 2020), we use this Gamma distribution to generate the redshift distribution of the simulated data. At redshift $z$, the value of $\log(F_{UV})$ is assumed to satisfy the Gaussian distribution shown in Eq. (8) with the mean value and the standard deviation being $-27.503$ and $0.488$, respectively, which are obtained from the real data. We further assume that the error ($\sigma_{{UV}}$) of $\log(F_{UV})$ satisfies the lognormal distribution and find, by using the real data, the mean value and the standard deviation being $-4.342$ and $0.566$ respectively. Since the real data show a correlation between $z$ and $\log(F_{UV})$ with the correlation coefficient being $\rho=-0.255$, we also consider this correlation when generating the ($z,\log(F_{UV})$) pair. Then, we remove the simulated data below the UV flux limit, i.e., the minimum observable flux $\log(F_{UV})_{min}=-28.759$. The flux limit reflects the detection capability of the instrument, and quasars are unobservable by current detectors if their $\log(F_{UV})$ are less than $\log({F_{UV}})_{min}$. For any pair of ($z,\log(F_{UV})$), we generate $\log(F_{X})$ by assuming its distribution function to be $f(\log(F_{X}))=\frac{1}{\sqrt{2\pi}\delta}\exp\left\{-\frac{[\log(F_{X})-\Phi(\log(F_{UV}),d_{L}({\bf p}))]^{2}}{2\delta^{2}}\right\}$ and the luminosity relation to be the standard $L_{X}$-$L_{UV}$ relation, where the intrinsic dispersion $\delta$, the relation coefficients and the PAge parameters ${\bf p}$ are taken as fixed values shown in the third column of Table 2.

With the above approach, we generate 2500 quasars. Using these simulated data, we can fit the intrinsic dispersion and the Type I relation coefficients with the fixed PAge parameters. This process is repeated twenty times, resulting in the average values of the intrinsic dispersion and the relation coefficients to be: $\bar{\delta}=0.2254\pm 0.0032$, $\bar{\beta}=6.46\pm 0.21$, $\bar{\gamma}=0.657\pm 0.009$, and $\bar{\alpha}=0.050\pm 0.054$. Apparently, they have slightly smaller errors than those shown in the first column of Table 2. This is because the errors of the PAge parameters are ignored when fitting the intrinsic dispersion and the relation coefficients from the simulated data. In Figure 4, the twenty values of the relation coefficient $\alpha$ from the simulated data are displayed to compare with $\alpha=0$ used in the simulation. It is easy to see that $\alpha=0$ is included in the 1$\sigma$ CL of the average value of twenty $\alpha$. Comparing $\bar{\alpha}=0.050\pm 0.054$ with $\alpha=0.672^{+0.063}_{-0.073}$ shown in Table 2, we find that the selection effects are ruled out as the possible cause of redshift-evolution of the $L_{X}$-$L_{UV}$ relation.

Recently, a three-dimensional and redshift-evolutionary $L_{X}$-$L_{UV}$ relation of quasars was constructed from the Gaussian copula (Wang et al., 2022). In this paper, we compare, by using the observational data from quasars, the Hubble parameter measurements and the Pantheon+ SNe Ia sample, this redshift-evolutionary $L_{X}$-$L_{UV}$ relation with another redshift-evolutionary $L_{X}$-$L_{UV}$ relation constructed by considering a redshift dependent correction to the luminosities of quasars and the standard $L_{X}$-$L_{UV}$ relation. We do not assume any cosmological model but use the PAge approximation to describe the cosmic background evolution. The quasar data are divided into the low-redshift part and the high-redshift one. The low-redshift and high-redshift data are then utilized respectively to constrain the relation coefficients, the SNe Ia absolute magnitude, the intrinsic dispersion of quasars and the PAge parameters. For the two redshift-evolutionary relations, we find that the low-redshift and high-redshift data can give consistent results on the PAge parameters and the absolute magnitude of the SNe Ia. While, the values of the relation coefficients respectively from the low-redshift and high-redshift data are discrepant. If the standard relation is considered, all the results from the low-redshift and high-redshift data are incomparable. Furthermore, we find that the observations favor the redshift-evolutionary relations at more than $2\sigma$. Thus, the redshift-evolutionary relations of quasars are favored by the observations as opposed to the standard one.

To further compare three different relations, we use the quasar, $H(z)$ and SNe Ia data to simultaneously constrain the PAge parameters, the intrinsic dispersion, the relation coefficients and the absolute magnitude of the SNe Ia. We find that the constraints obtained with the two redshift-evolutionary relations on the PAge parameters, the absolute magnitude, the intrinsic dispersion and the relation coefficient $\gamma$ are well consistent with each other, but they deviate significantly from those obtained in the standard relation. For the relation coefficient $\beta$, different relations lead to different values and the deviations between them are more than $2\sigma$. The observations favor strongly the redshift-evolutionary relations since $\alpha=0$ is ruled out more than $3\sigma$. According to the AIC and BIC, we find that there is strong evidence against the standard relation and mild evidence against the redshift-evolutionary relation obtained by considering a correction to luminosities of quasars. We also confirm that the redshift-evolution of the $L_{X}$-$L_{UV}$ relation can not be caused by the selection effects. Therefore, we can conclude that among the three different $L_{X}$-$L_{UV}$ relations the redshift-evolutionary relation from copula is the most favored one.