Redshift-evolutionary X-Ray and UV luminosity relation of quasars from Gaussian copula

Copula is proposed to describe the intercorrelation between statistical variables (Nelsen, 2007). It can join or “couple” multivariate distribution functions with one-dimensional marginal distribution functions. Supposing that there are three variables $x_{1}$, $x_{2}$ and $x_{3}$ with the marginal cumulative distribution functions (CDFs) being $F_{1}(x_{1})$, $F_{2}(x_{2})$ and $F_{3}(x_{3})$, respectively, the joint distribution function of these three variables can be described by using copula function $C$:

The key point of Eq. (1) is that by using copula one can model the dependence structure and the marginal distribution separately. In the following, we set $u_{i}=F_{i}(x_{i})$ ($i=1,2,3$) for simplifying the expressions. The joint probability density function (PDF) $h(x_{1},x_{2},x_{3})$ of $H(x_{1},x_{2},x_{3})$ can be obtained through

where $f_{i}(x_{i})$ ($i=1,2,3$) are the marginal PDFs of $F_{i}(x_{i})$, and $c(u_{1},u_{2},u_{3})$ is the density function of $C(F_{1}(x_{1}),F_{2}(x_{2}),F_{3}(x_{3}))$.

Two main copula families are elliptic copulas and Archimedean copulas. The Gaussian copula belongs to a type of elliptic copulas and has a symmetric tail correlation. Since the Gaussian copula is the simplest and it can be analyzed analytically in many cases, we choose the Gaussian copula in our discussion and then we have

where $\bm{\theta}$ denotes the parameters of the copula function, and $\Psi_{3}$ and $\Psi_{1}$ are the standard three-dimensional and one-dimensional Gaussian CDFs, respectively, which are defined as

and

Here

$\Sigma$ is the covariance matrix, which relates to the correlation coefficients $\bm{\theta}=\{\rho_{12},\rho_{13},\rho_{23}\}$ ($\Sigma_{ij}=\Sigma_{ji}=\rho_{ij},\Sigma_{ii}=1$), and $\mathrm{erfc}$ and $\mathrm{erfc}^{-1}$ are complementary error function and its inverse, respectively.

The density function of the Gaussian copula can be calculated through

where $\bm{\Psi^{-1}}\equiv\{\Psi_{1}^{-1}(u_{1}),\Psi_{1}^{-1}(u_{2}),\Psi_{1}^{-1}(u_{3})\}$. Then, the conditional PDF of $x_{1}$ denotes the probability of variable $x_{1}$ when $x_{2}$ and $x_{3}$ are fixed, which can be expressed as:

When the variable $x_{i}$ ($i=1,2,3$) obeys the Gaussian distribution with the mean value being $\mu_{i}$ and the standard deviation $\sigma_{i}$, its CDF $u_{i}$ can be expressed by

Thus, from the Eq. (6), one can obtain

Combining Eqs. (7), (8), and (10), we can obtain

where $\sigma_{x_{1}\mid x_{2},x_{3}}$ is the standard deviation of $f_{x_{1}}$, and

with $\hat{x}_{i}\equiv(x_{i}-\mu_{i})/\sigma_{i}$. Since $x_{1}$ follows the Gaussian distribution, the maximum probability of $x_{1}$ can be obtained from $S(x_{1},x_{2},x_{3};\bm{\theta})=0$. Thus, we can achieve the relation between variables $x_{1}$, $x_{2}$ and $x_{3}$ by solving $S(x_{1},x_{2},x_{3};\bm{\theta})=0$, and find that the relation can be simplified to be

where

Risaliti & Lusso (2015) have found that there is a nonlinear relation between the X-ray luminosity $L_{X}$ and UV luminosity $L_{UV}$ in observational data of quasars. The $L_{X}-L_{UV}$ relation has the form

where $\beta$ and $\gamma$ are two free parameters to be determined from the data, and $\log\equiv\log_{10}$. Expressing the luminosity in terms of the flux, one can obtain

where $F_{X}$ and $F_{UV}$ are the X-ray and UV fluxes, respectively, and $\beta^{\prime}(z)=2(\gamma-1)\log(d_{L})+\beta+(\gamma-1)\log(4\pi)$. Here $d_{L}$ is the luminosity distance. We can fit parameters $\gamma$ and $\beta$ by using the quasar data, if a concrete cosmological model is chosen.

Now, we will use the copula function to construct the correlation between $L_{X}$, $L_{UV}$ and the redshift. First, we assume that both $\log(L_{X})$ and $\log(L_{UV})$ follow Gaussian distributions with the distributional functions being

Here $x=\log(L_{UV})$, $y=\log(L_{X})$, $\bar{a}_{x}$ and $\bar{a}_{y}$ represent the mean values, and $\sigma_{x}$ and $\sigma_{y}$ are the standard deviations. We have checked that these assumptions are very reasonable by performing a statistical distribution test (Cramér-von Mises test (Cramér, 1928; Von Mises, 1928)) on $x$ and $y$ based on the $\Lambda$CDM model. Then, the redshift distribution of quasars is considered. In the upper panel of Fig. 1 we show the probability density distribution of 2421 quasar data points (Lusso et al., 2020) in the redshift $z$ space, which satisfies the gamma distribution apparently. The probability density distribution is plotted by requiring that the total area enclosed by the curve and the $z$ axis is normalized to one. Thus, if the value of the vertical axis is $P$ at redshift $z$, the probability of the data in the redshift region $(z-\delta z,z+\delta z)$ is $2P\delta z$, where $\delta z$ is a very small variation and thus the value of the vertical axis in the redshift region $(z-\delta z,z+\delta z)$ can be regarded as constant. Since the log-transformation is a common way to transform the non-Gaussian distribution into the Gaussian one, we consider the $z_{*}$ space, where $z_{*}\equiv\ln(\bar{a}+z)$ with $\bar{a}$ being a constant and $\ln\equiv\log_{e}$, and find that the distribution of quasars can be described approximately by using a Gaussian distribution $f(z_{*})$, which is shown in the down panel of Fig. 1. Thus, $f(z_{*})$ has the form

We obtain that $\bar{a}=5$ is a very good set after choosing some different values of $\bar{a}$ for comparison since it can lead to very consistent fitting results in the following analysis.

According to Eq. (13), we obtain the three-dimensional relation, which has the form

Apparently $\alpha$ is a new free parameter. Eq. (19) is different from the standard $L_{X}-L_{UV}$ relation (Eq. (15)) with the addition of a redshift-dependent term. When $\alpha=0$ the standard relation is recovered. When $\bar{a}=1$, our relation reduces to the one given in (Dainotti et al., 2022) obtained by assuming that the luminosities of quasars are corrected by a redshift dependent function $(1+z)^{\alpha}$. Converting the luminosity to the flux, one has

Eq. (20) is the main result of this Paper. In the following, we will discuss the viability of the three-dimensional $L_{X}-L_{UV}$ relation from copula.

Since $\beta^{\prime}$ is dependent on the luminosity distance $d_{L}$, we choose the fiducial cosmological model to give $d_{L}$. We consider two models: the specially flat $\Lambda$CDM with ($H_{0}=70~{}\mathrm{km~{}s^{-1}~{}Mpc^{-1}}$, $\Omega_{\mathrm{m0}}=0.3$) and with ($H_{0}=70~{}\mathrm{km~{}s^{-1}~{}Mpc^{-1}}$, $\Omega_{\mathrm{m0}}=0.4$), respectively. Since the furthest type Ia supernovae reaches to redshift $z\sim 2$ (Scolnic et al., 2018), we choose the quasar data within the low-redshift region $z<2$ to determine the values of the coefficients in the three-dimensional and standard $L_{X}-L_{UV}$ relations. Extrapolating these results to the high-redshift ($z>2$) data, we can obtain the Hubble diagram of high-redshift quasar data at $z\sim 7.5$. Finally, the high-redshift quasar data will be used to constrain the $\Lambda$CDM model. If the value of $\Omega_{\mathrm{m0}}$ assumed in the fiducial model is compatible with the one from the high-redshift data, the low-redshift and high-redshift data give consistent results and thus we regard this relation as viable and then the quasars can be taken as standard candles.

The data used in our analysis contain 2421 X-ray and UV flux measurements of quasars (Lusso et al., 2020), which cover the redshift range of $z\in[0.009,7.541]$. We use 1917 low-redshift data points for calibration. The values of the coefficients in the $L_{X}-L_{UV}$ relation can be obtained by minimizing $-\ln(\mathcal{L})$, where $\mathcal{L}$ is the D’Agostinis likelihood function (D’Agostini, 2005)

Here $\sigma_{i}$ represent the measurement errors in $\log(F_{X})$, $\delta$ is an intrinsic dispersion, function $\Phi$ is defined in Eq. (20), and $d_{L}$ is the luminosity distance predicted by the $\Lambda$CDM model, which is calculated as

In order to compare the three-dimensional $L_{X}-L_{UV}$ relation and the standard one, we use the Akaike information criterion (AIC) (Akaike, 1974, 1981) and the Bayes information criterion (BIC) (Schwarz, 1978), which are, respectively, defined as

where $p$ is the number of free parameters and $N$ is the number of data points. We need to calculate $\Delta$AIC(BIC) ($\Delta$AIC (BIC)$=$ AIC (BIC) $-$AIC_min (BIC_min)) of the two relations when comparing them. We have strong evidence against the relation that has a large AIC(BIC) if $\Delta\mathrm{AIC(BIC)}>10$ (Hu & Jeffreys, 1998).

The likelihood analysis is performed by using the Markov Chain Monte Carlo (MCMC) method as implemented in the $emcee$ package in $python$ 3.8 (Foreman-Mackey et al., 2013). Table 1 and Figure 2 show the results. From them, we can conclude that

• •

  The value of $\alpha$ deviates from zero more than $3\sigma$, which indicates that the observational data support apparently the redshift-dependent luminosity relation.

• •

  Since both $\Delta$AIC and $\Delta$BIC are larger than $20$, the three-dimensional relation from the copula function is favored strongly by the information criterions.

• •

  The intrinsic dispersion has a negligible difference for all cases. And the values of $\beta$ are almost independent of the cosmological models.

Extrapolating the values of the coefficients of the luminosity relations and the intrinsic dispersion from the low-redshift data to the high-redshift samples, we can construct the Hubble diagram of quasars. The distance modulus and its errors are obtained from the formula given in the Appendix. In Fig. 3, the Hubble diagrams of quasars obtained from two different relations (standard and three dimension) and two different fiducial models are shown. Apparently, the distance modulus from the three-dimensional relation is closer to the theoretical curve than that from the standard relation.

We use the distance modulus of high-redshift quasars to constrain the $\Lambda$CDM model by minimizing $\chi^{2}$

Here $\mu(z)=25+5\log\left[\frac{d_{L}(z)}{\mathrm{Mpc}}\right]$ is the distance modulus. The results are shown in Fig. 4 and summarized in Tab. 2. It is easy to see that the data from the three-dimensional relation can give a constraint on $\Omega_{\mathrm{m0}}$ consistent with that assumed in the fiducial model at the $1\sigma$ confidence level (CL), while the data from the standard relation cannot, and the deviations are larger than $3\sigma$. The AIC and BIC also favor strongly the three-dimensional relation. Therefore, we can conclude that the three-dimensional luminosity relation from copula is obviously superior to the standard relation. Using this three-dimensional relation, the quasars can be treated as the standard candle to probe the cosmic evolution.

To compare two relations with data more clearly, we plot the quasar data points in the $\log(L_{X})$-$\log(L_{UV})$ and $\log(L_{X})^{\prime}$-$\log(L_{UV})$ planes in Fig. 5, where $\log(L_{X})^{\prime}\equiv\log(L_{X})-\alpha\ln(z+5)$. The solid lines are luminosity relations calibrated by using the $\Lambda$CDM model with $\Omega_{m0}=0.3$ from the low-redshift data. It is easy to see that the three-dimensional and redshift-evolutionary relation is more consistent with the high-redshift data than the standard one.

In the previous subsection, the fiducial cosmological model is used to study the reliability of the two different $L_{X}-L_{UV}$ relations. Here, we discard this condition. The coefficients of the relation and the cosmological parameters of $\Lambda$CDM model will be fitted simultaneously by minimizing the value of the D’Agostinis likelihood function (Eq. (21)).

Figure 6 shows one-dimensional probability density plots and contour plots of $\Omega_{\mathrm{m0}}$ and the coefficients of the two luminosity relations. The marginalized mean values with $1\sigma$ CL are summarized in Tab. 3. We find that the data favor apparently the redshift-evolutionary relation since $\alpha$ deviates from zero more than $3\sigma$. When the three-dimensional relation is used the quasar data can give an effective constraint on $\Omega_{\mathrm{m0}}$ ($\Omega_{\mathrm{m0}}=0.510^{+0.163}_{-0.254})$, which is consistent with what were given by the current popular data including type Ia supernova and the cosmic microwave background radiation and so on. However the quasar data only gives a lower bound limit on $\Omega_{\mathrm{m0}}$ for the standard relation. Furthermore, the AIC and BIC information criterions support strongly the three-dimensional relation since both $\Delta$AIC and $\Delta$BIC are larger than 50.