Constraints on Cosmological Models from Quasars Calibrated with Type Ia Supernova by a Gaussian Process

In this paper, we use quasars calibrated from type Ia supernova (SN Ia) to constrain cosmological models. We consider three different X-ray luminosity ($L_{X}$) - ultraviolet luminosity ($L_{UV}$) relations of quasars, i.e., the standard $L_{X}$-$L_{UV}$ relation and two redshift-evolutionary relations (Type I and Type II) respectively constructed from copula and considering a redshift correction to the luminosity of quasars. Only in the case of the Type I relation, quasars can always provide effective constraints on the $\Lambda$CDM model. Furthermore, we show that, when the observational Hubble data (OHD) are added, the constraints on the absolute magnitude $M$ of SN Ia and the Hubble constant $H_0$ can be obtained. In the $\Lambda$CDM model, the OHD measurements plus quasars with the Type I relation yields $M$ =$-19.321^{+0.085}_{-0.076}$, which is in good agreement with the measurement from SH0ES ($M=-19.253\pm{0.027}$), and $H_0$ = $70.80\pm3.6~\mathrm{km~s^{-1}Mpc^{-1}}$, falling between the measurements from SH0ES and the Planck cosmic microwave background radiation data.


INTRODUCTION
The ΛCDM (cosmological constant Λ plus cold dark matter) model is the simplest cosmological model to explain the accelerating cosmic expansion.Although consistent with many observations (Perlmutter et al. 1999;Riess et al. 1998;Alam et al. 2021;Brout et al. 2022), it still suffers the Hubble tension (Riess 2020;Di Valentino et al. 2021;Perivolaropoulos et al. 2022;Dainotti et al. 2021Dainotti et al. , 2022a;;Liu, Yu & Wu 2023), which refers to the more than 5 disagreement between the constraints on the Hubble constant  0 from the nearby type Ia supernova (SN Ia) and the cosmic microwave background (CMB) radiation data, respectively.Using the latest SN Ia measurements, Riess et al. (2022) obtain  0 = 73.04 ± 1.04 km s −1 Mpc −1 in a modelindependent manner, while the CMB data from the Planck 2018 survey yield  0 = 67.36 ± 0.54 km s −1 Mpc −1 (Planck Collaboration 2020) in the framework of the ΛCDM model.Many other observational data have been utilized to search the possible origins of the  0 tension, but a satisfactory explanation remains elusive.Since the redshift range of commonly used observations, including SN Ia (Scolnic et al. 2022), observational Hubble parameter data (OHD) (Moresco et al. 2020), baryon acoustic oscillations (Eisenstein et al. 2005), and strong gravitational lenses (Suyu et al. 2010(Suyu et al. , 2013)), reaches only about  ∼ 2, while the CMB data is near  ∼ 1100, cosmological ★ Co-first author † hwyu@hunnu.edu.cn‡ liangn@bnu.edu.cn§ pxwu@hunnu.edu.cndata in the mid-redshift region (2 < ∼  < ∼ 1100) might offer important insights into the origins of the Hubble tension.
Quasars (quasi-stellar objects) are the extremely luminous and persistent sources of light found in the universe, which can be detected at  > 7 (Mortlock et al. 2011;Bañados et al. 2017;Wang et al. 2021).To use quasars as the standard candles in cosmology, the luminosity relation of quasars needs to be constructed.In this regard, an empirical non-linear relation between the ultraviolet (UV) luminosity and the X-ray luminosity (  -  ) has been proposed in (Tananbaum et al. 1979;Zamorani et al. 1981;Avni & Tananbaum 1986;Risaliti & Lusso 2015), and it has been widely applied to investigate the high-redshift universe using quasars (Risaliti & Lusso 2015, 2019;Lusso & Risaliti 2016, 2017;Lusso et al. 2019Lusso et al. , 2020;;Velten & Gomes 2020;Khadka & Ratra 2020a,b, 2021;Khadka et al. 2023;Wei & Melia 2020;Li et al. 2021Li et al. , 2024;;Hu & Wang 2022a; Bargiacchi et al. 2022;Dainotti et al. 2023a;Yang, Banerjee & Colgain 2020).Recently, Khadka & Ratra (2020a) showed some evidences of redshift evolution of the X-ray and UV relation.By considering a power-law redshift correction ((1 + )  ) to the quasar luminosity, Dainotti et al. (2022b) obtained a three-dimensional and redshift-evolutionary version of the   -  relation.In (Dainotti et al. 2022b), the coefficient of the redshift correction term is determined by using the Efron-Petrosian (EP) method (Efron & Petrosian 1992).This coefficient will be treated as a free parameter in the following analysis.Wang et al. (2022) introduced an improved threedimensional   -  relation with a redshift-dependent term given by a powerful statistical tool called copula 1 .The increased reliability of the   -  relations as a cosmological tool has been established in (Wang et al. 2022(Wang et al. , 2024;;Dainotti et al. 2022b;Lenart et al. 2023) Similar to the Gamma-ray burst (GRB) cosmology, the quasar cosmology also suffers the so-called circularity problem since a fiducial cosmological model is usually assumed to calibrate the empirical relation when quasars are used to constrain the cosmological models.Inspired by the idea of distance ladder with the Cepheids and SN Ia, the low-redshift calibration has been proposed to calibrate the GRB relations from SN Ia to build the Hubble diagram of GRB (Liang et al. 2008(Liang et al. , 2010(Liang et al. , 2011;;Kodama et al. 2008;Capozziello & Izzo 2008;Wei & Zhang 2009;Wei 2010;Liu & Wei 2015;Wang et al. 2016;Demianski et al. 2017;Liu et al. 2022b;Liang et al. 2022;Li et al. 2024;Mu et al. 2023;Xie et al. 2023).In addition, the simultaneous fitting or global fitting method has also been established to avoid the circularity problem in the GRB cosmology (Ghirlanda et al. 2004;Li et al. 2008;Dainotti et al. 2023b;Cao & Ratra 2024).Here, we want to apply the low-redshift calibration method in the quasar cosmology and calibrate the luminosity relations of quasars by using SN Ia. 2 To achieve the luminosity distance at the redshift of quasars from SN Ia, we need to employ a cosmological model-independent method to reconstruct the Hubble diagram of SN Ia.In this regard, let us note that the Gaussian process (GP) is a fully Bayesian statistical method used for reconstructing the Hubble diagram from existing data and predicting unknown data (Seikel et al. 2012a), which can effectively reduce errors in the reconstructed results.In recent years, the GP method has found extensive applications in the fields of cosmology and astrophysics (Seikel et al. 2012b;Busti et al. 2014;Yu & Wang 2016;Lin et al. 2018;Wei 2018;Yu et al. 2018;Pan et al. 2020;Hu & Wang 2022b).
Thus, in this paper, the GP method will be used to reconstruct the Hubble diagram of the 1590 Pantheon+ SN Ia data points, which are obtained by removing the data in the redshift region of  < 0.01 from 1701 SN light curves of 1550 spectroscopically confirmed SN Ia with the maximum redshift being about  ∼ 2.3 (Scolnic et al. 2022), and then the luminosity distances of low-redshift quasars, which are the subset of the dataset comprising 2421 X-ray and UV flux measurements of quasars (Lusso et al. 2020), can be derived from the SN Ia Hubble diagram.From an initial dataset of 21,785 data points, a total of 2,421 quasars have been selected.These quasars originate from seven different samples, observed using instruments such as Chandra and XMM-Newton.Quasars displaying UV reddening, significant host-galaxy contamination in the near-infrared, or poor photometry data were excluded.Additionally, adjustments for Eddington bias were considered.Following these corrections, Lusso et al. (2020) compiled the final, refined sample of 2,421 quasars, covering a redshift range from 0.009 to 7.52.It is important to note that no -correction was applied to the quasars (Bloom et al. 2001;Lusso et al. 2020).These low-redshift quasars can be used to determine the coefficients in different   -  relations.Extrapolating the results to the high-redshfit quasars, we can obtain the luminosity distances of all quasars model-independently, and use these distances to constrain 1 Copula is a powerful tool developed in modern statistics to describe the correlation between multivariate random variables (Nelsen 2007) and it has been used to construct the redshift-evolutionary relation of Gamma-ray bursts (Liu et al. 2022a,b). 2As demonstrated in (Wang et al. 2022;Lenart et al. 2023), it has been established that certain cosmological parameters, such as Ω m0 , are linearly independent of the correction coefficients.Hence, there is no impediment to constraining these parameters without calibration.However, calibration can significantly improve the precision of cosmological parameter estimation.the cosmological models.Since the quasar sample only cannot give constraints on the absolute magnitude () of SN Ia and the Hubble constant ( 0 ) simultaneously, we also add 32 OHD measurements obtained by the cosmic chronometers method (Moresco et al. 2020), which relate to the evolution of differential ages of passive galaxies at different redshifts (Jimenez & Loeb 2002), to obtain constraints on  and  0 .
The rest of this paper is organized as follow.In Section 2, we respectively calibrate, by using the low-redshift calibration method, three different   -  relations from SN Ia: the standard relation and two redshift-evolutionary relations (Type I and Type II) constructed from copula and considering a redshift correction to the luminosity of quasars.The constraints on different cosmological models from quasars and OHD are given in Section 3. Section 4 shows the discussion on results.Conclusions are summarized in Section 5.

𝐿
and   =   4  2  are the observed flux of Xray and UV, respectively, and   is the luminosity distance, which contains the information of cosmological models.
Recently, the standard   -  relation has been generalized to contain possible redshift-evolutionary effects: Here  is a coefficient, and  ≠ 0 represents that the relation evolves with redshift.The constant ᾱ equals to 1 or 5.When ᾱ = 5, the relation given in Eq. ( 3), which is named the Type I relation in this paper, corresponds to that constructed by using the copula function (Wang et al. 2022).If ᾱ = 1, the corresponding relation is called the Type II relation, and is obtained by assuming that the luminosity of quasars is corrected via a redshift-dependent function (1 + )  (Dainotti et al. 2015(Dainotti et al. , 2022b)).
To apply quasars in cosmology, we need to obtain their luminosity distances from the observables of quasars by using the relations shown in Eqs.(2, 3).Apparently, the values of the coefficients (, , ) need to be determined first.Here, we adopt the idea of distance ladder and use the SN Ia data to calibrate the   -  relations.Since the data points of SN Ia concentrate in the low-redshift ( ≤ 1.4) region, the SN Ia data from the latest Pantheon+ sample at  ≤ 1.4 (Scolnic et al. 2022) are utilized to determine ,  and .To avoid the effect of cosmological models on the calibration, we use the GP method (Seikel et al. 2012a) to construct the smooth curve of the apparent magnitude ()-redshift relation of SN Ia.GP is a generalization of Gaussian distribution, which refers to the distribution of random variables, and describes the distribution over functions.With the GP method, the smooth functions from a set of discrete data points can be constructed (Seikel et al. 2012a).Here we use an infinitely differentiable squared exponential covariance function to reconstruct the smooth functions of the apparent magnitude of SN Ia.The covariance function is given by: where the hyperparameters   and  are optimized by maximizing the marginal likelihood 3 .When the Pantheon+ sample (Scolnic et al. 2022) is used, we exclude those data whose redshifts are less than 0.01 since the nearby SN Ia sample may be impacted by the peculiar velocities (Brout et al. 2022).The reconstructed - relation with an uncertainty of 1 by using the GP is shown in Fig. 1.This figure indicates that the reconstructed result is well consistent with the data distribution.
The distance module  of SN Ia is related to the luminosity distance (  ) and the absolute magnitude () through Thus, the luminosity distance of quasars that fall in the redshift under  = 1.4 may be obtained from the reconstructed () after the value of  is determined.However, the value of  cannot be directly obtained using only the SN Ia sample and thus  is treated as a free parameter.Substituting Eq. ( 5) into the   -  relations, one can obtain which shows that there is a strong degeneracy between the absolute magnitude  and the coefficient .Thus, we cannot constrain the values of  and  simultaneously by using the observational data.
To address this issue without assuming any prior values of , we 3 In this work, we use the public PYTHON package GaPP to reconstruct the apparent magnitude () -redshift relation of SN Ia sample in low-redshift ( < 1.4) region.This code is available at https://github.com/astrobengaly/GaPP.
introduce a new coefficient:  ′ = −2( − 1)  5 + 5 +  + ( − 1) log(4).Then, the relation Eq. ( 6) is reduced to log (  ) = 2( − 1) Using the apparent magnitude of SN Ia reconstructed at a redshift of quasars and the observed log (  ) obs at this redshift, we can obtain log(  ) re from Eq. ( 7).Comparing this log(  ) re with the observed X-ray flux of quasars can give the values of ,  ′ and  by maximizing the D'Agostinis likelihood function (D'Agostini 2005): , where  is the intrinsic dispersion, θ ≡ {,  ′ , }, and with  log(  ) obs, ,  log(  ) obs, , and   being the uncertainties of log (  ) obs, , log (  ) obs, , and the reconstructed , respectively.In Eq. ( 8), number  = 1326 is the number of quasars in the low-redshift ( ≤ 1.4) region.To find the maximum likelihood, we utilize the Python package  (Foreman-Mackey et al. 2013), which bases on the Metropolis-Hastings algorithm and implements the Markov Chain Monte Carlo (MCMC) numerical fitting method.The calibrated results for the coefficients in various   -  relations are presented in Tab. 1 and Fig. 2. Obviously, only the value of  is almost independent of the relations.The values of  are almost the same in the Type I and II relations, but they deviate from that in the standard   -  relation.The Type I and standard relations give almost the same value of  ′ , which is smaller that the one in the Type II relation.The Type I and II relations give different values of , but both of them deviate from zero at more than 3, which indicates that the observations seem to support the redshift-evolutionary relations.
To investigate the effect of different redshift borders, we also consider quasars in two different redshift regions ( ≤ 1 and  ≤ 1.8).The results are shown in Tab. 2. Comparing Tab. 1 with Tab. 2, we find that the relation coefficients and the intrinsic dispersions across these redshift regions vary slightly but remain consistent within a 1 CL.Thus, the choice of redshift borders exerts a negligible effect on the calibration of the luminosity relations.In the following analyses, we will use the results from quasars with  ≤ 1.4.
To further check whether there exist selection biases and redshift evolution in quasar data (Singal et al. 2022(Singal et al. , 2013(Singal et al. , 2019)), we follow the process given in (Dainotti et al. 2015(Dainotti et al. , 2022b) ) and use the EP method (Efron & Petrosian 1992) to calculate the value of  parameter which is introduced to correct the luminosity via  corrected =  observed /( ᾱ + )  in Type I and II relations, and the corresponding value of .If the obtained value of  is consistent with that from the low-redshift calibration, then the results presented in Table 1 are not influenced by selection biases or redshift evolution in quasars.When the correction to the luminosity is considered, we need to replace   and   with   /( ᾱ+)   and   /( ᾱ+)   in Eq. (1).Then, we obtain the following simple relation between  and   and   : Clearly, the value of  can be inferred from the values of   and   when  is known.The values of   and   can be obtained by using the EP method, which is given in a publicly available Mathematica code: Selection biases and redshift evolu- tion in relation to cosmology 4 .Here we set a fiducial model: the flat ΛCDM model with  0 = 73.04km s −1 Mpc −1 and Ω m0 = 0.33 for calculating the luminosity of quasars, and choose the flux limit  lim = 6 × 10 −33 erg s −1 cm −2 Hz −1 for the X-rays and  lim = 4.5 × 10 −29 erg s −1 cm −2 Hz −1 for the UV.We finally obtain   = 7.10 ± 0.17 and   = 9.76 ± 0.22 for the Type I relation, and   = 2.64 ± 0.06 and   = 3.62 ± 0.08 for the Type II relation.Using the fiducial model to estimate the luminosity distance, the value of parameter  can then be determined from Eq. ( 8) with the whole quasar sample, and the results are  = 0.579 ± 0.011 and  = 0.577 ± 0.011 for the Type I and II relations, respectively.Finally, we use Eq. ( 9) to calculate the mean value of  with 1 uncertainty, and obtain  = 0.629 ± 0.103 and  = 0.239 ± 0.0375 for the Type I and II relations, respectively.These results deviate from zero at more than 5 and are compatible with those obtained from the low-redshift calibration and shown in Tab. 1 within 2 CL.This compatibility suggests that the nonzero value of  from the low-redshift calibration is unlikely to result from selection biases or redshift evolution in quasars.It is worth noting that the bias due to extinction on quasar luminosity distances has been studied recently in (Zajaček et al. 2024).

CONSTRAINTS ON COSMOLOGICAL MODELS
Extrapolating the values of ,  ′ ,  and  from the low-redshift data to the high-redshift regions, we can obtain the luminosity distance of all quasar data.These data can now be employed to constrain the cosmological models by finding the maximum likelihood of the following function: Here  denotes the number of data, and σ is the derived uncertainty of log (  ) by using the error propagation formula where    is the covariance matrix, and   ,   ′ , and   are the uncertainties of coefficients ,  ′ , and  estimated from the GP, respectively.In Eq. ( 10), p represents the cosmological model parameters and  is the apparent magnitude derived from the cosmological model, which relates to the dimensionless luminosity distance   through with The dimensionless luminosity distance is defined as in the spatially flat universe.Here,  (; p) is the dimensionless Hubble parameter.For the CDM model,  (; p) has the form where Ω m0 and  are the current matter density and the equation of state (EOS) of dark energy, respectively.This model will reduce to the ΛCDM when  = −1.Therefore, we have p ≡ {Ω m0 } for the ΛCDM model and p ≡ {, Ω m0 } for the CDM model.Here the prior of  is set as a uniform distribution in 1/(Ω m0 − 1) ≤  ≤ −1/3, which comes from the accelerated cosmic expansion (Riess et al. 1998;Perlmutter et al. 1999) and the null energy condition (Visser & Barcelo 2000;Lenart et al. 2023).Since  and  0 can not be constrained simultaneously when only quasars are used, M is treated as a nuisance parameter, and is marginalized in our analysis.

Constraints on cosmological models from Quasars
In Sec.II, we have used the SN Ia data in the redshift region of  ≤ 1.4 to calibrate the quasars in this redshift region and extrapolate the results to the high-redshift ( > 1.4) quasars.Thus, the highredshift quasars, which contain 1095 observations, are utilized firstly to constrain the ΛCDM and CDM models.Furthermore, we also consider the full-redshift quasar data (2421 data points) to limit these two models.The results are shown in Figs. 3 and 4, and summarized in Tab. 3. It is easy to see that for the ΛCDM model when the standard   -  relation is used, quasars only give a lower bound on Ω m0 .The Since the cosmological parameters Ω m0 and  cannot be well constrained simultaneously, we further consider a fitting for  with Ω m0 fixed to be 0.30.The results are shown in Fig. 5. Apparently, for all three relations the high-redshift quasars only give the lower limit of .When the full-redshift quasars are used, we obtain that  = −1.08 +0.17  −0.34 for Type I relation and  = −1.02+0.63 −0.41 for Type II relation.If the standard relation is considered, a large  (−0.381 <  < −1/3) is obtained.

Constraints on cosmological models from Quasars + OHD
Since quasars cannot provide the constraints on  and  0 due to the degeneracy between them, we further add the OHD from the cosmic age difference method (Jimenez & Loeb 2002) into our analysis.The updated 32 OHD measurements cover a redshift range of 0.07 <  < 1.965 (Moresco et al. 2020), which contains 17 uncorrelated and 15 correlated measurements.The 17 uncorrelated data are given in (Zhang et al. 2014;Simon, Verde & Jimenez 2005;Ratsimbazafy et al. 2017) and the 15 correlated measurements are sourced from (Moresco et al. 2012;Moresco 2015;Moresco et al. 2016) with the covariance matrix being given by Moresco et al. (2020).To constrain the cosmological models, the minimization of the  2 method is used: Here   represents the observed uncertainty of 17 uncorrelated measurements, Δ Ĥ =  th (; p) −  obs () represents the difference vector between the observed data and the theoretical values for the 15 correlated measurements, and C −1  is the inverse of the covariance matrix. obs represents the observed value of the Hubble parameter, while  th (; p) ≡  0  (; p) is the corresponding theoretical one calculated from the cosmological model.
The constraints on the cosmological models from quasars+OHD can be obtained from the total likelihood: The results are shown in Figs. 6 and 7, and summarized in Tab. 4. For the ΛCDM model, we find that in the case of the standard   -  relation the combination of the full-redshift quasars and the OHD measurements favors a smaller  0 , a larger Ω m0 , and a larger  than the high-redshift quasars plus OHD.The differences between the values of the cosmological model parameters and the absolute magnitude respectively from the high-and full-redshift quasars found in the standard relation become negligible when the Type II or Type I relation is used instead.In the Type II relation, the value of  0 is consistent with the result from the Planck CMB observations (Planck Collaboration 2020), while the value of  is smaller than the SH0ES measurement (Riess et al. 2022).When the Type I relation is considered, we find that the  0 value from full-redshift quasars locates between the Planck result and the SH0ES measurement (Riess et al. 2022;Planck Collaboration 2020), and the  value is compatible with the SH0ES measurement (Riess et al. 2022).
For the CDM model, we find that there is a large difference between the constraints on  0 obtained from the high-and fullredshift quasars in the standard   -  relation, and this difference reduces to be very small when the Type I or II relation is considered.For the standard relation, the full-redshift quasars favor a smaller  0 than the high-redshift data, but the results obtained in the Type I and II relations are opposite.For Type I and II relations, the constraints on Ω m0 obtained from the high-and full-redshift quasars are very close, while the values obtained from Type I relation are always smaller than those obtained from Type II one.The standard relation gives two significantly different Ω m0 from the full-and high-redshift quasars, respectively, which are larger than those obtained in Type I and II relations.In the three relations, quasars can always give  (Dainotti et al. 2023c).The values of  achieved in (Dainotti et al. 2023c;Lenart et al. 2023;Wang et al. 2022) and this paper are significant larger than  = 0.007 ± 0.004 from the golden sample of 983 quasars (Dainotti et al. 2023a).Interestingly, the constraint on Ω m0 from quasars with Type I relation is well consistent with Ω m0 = 0.240 ± 0.064 from 1253 gold quasar sample (Dainotti et al. 2024) and Ω m0 = 0.295 +0.013 −0.012 from SN Ia + quasars (Bargiacchi et al. 2022).Recently, Bargiacchi, Dainotti & Capozziello (2023) found that the high-redshift evolution of our universe from GRBs and quasars in the case of Type II relation with the effect of redshift evolution being fixed by using the EP method has a strong tension with the prediction of the ΛCDM model.Whether this tension still exists for quasars with Type I relation is an interesting topic and needs to be investigated in the future.

CONCLUSIONS
In this paper, we compare the constraints on the cosmological models from quasars with three different   -  relations: the standard relation proposed by Risaliti & Lusso (2015), and the Type I and Type II redshift-evolutionary relations constructed by Wang et al. (2022) and Dainotti et al. (2022b), respectively.We employ the GP method to calibrate these relations from the latest Pantheon+ SN Ia data within the low-redshift region ( < 1.4).These results align with that reported by Wang et al. (2022), and suggest a potential trend of redshift evolution in the   -  relation.
Extrapolating the calibrated relations from the low-redshift quasars to the quasars that lie in high-redshift region, we obtain the luminosity distances of quasars and then constraints from them on the spatially flat ΛCDM and CDM models.We find that the standard relation provides only a lower bound on Ω m0 in both the ΛCDM and CDM When the additional OHD sample is considered, the observations can give tight constraints on the ΛCDM and CDM models in all three relations, while the ΛCDM model constrained from quasars with the Type I relation still yields the tightest results.For this preferred model, we find that the constraints on Ω m0 are 0.332 +0.050 −0.073 for the high-redshift quasars and 0.289 +0.038  −0.051 for the full-redshift quasars, which are consistent with the CMB results (Planck Collab-oration 2020).Moreover, since the addition of the OHD sample can break the degeneracy between parameters  and  0 , we obtain the following constraints on them.In the ΛCDM model, the OHD measurements plus the full-redshift quasars with the Type I relation give the absolute magnitude  to be −19.321+0.085 −0.076 , which aligns well with that obtained from SH0ES ( = −19.253± 0.027) (Riess et al. 2022), and the Hubble constant  0 to be 70.80 ± 3.6 km s −1 Mpc −1 , which lies between the measurements of SH0ES (Riess et al. 2022) and CMB (Planck Collaboration 2020).

Figure 1 .
Figure 1.The blue curves depict the reconstructed function with 1 uncertainty from the SN Ia data (red dots) through Gaussian process.The red dashed line represents  = 1.4.

Figure 2 .
Figure 2. One-dimensional probability density distributions and twodimensional contours of ,  ′ , , and  from the low-redshift quasars.The blue, orange, and green contours represent the Type I, Type II, and standard relations, respectively.
models.For the ΛCDM model, Type II relation gives effective constraints on the cosmological model parameters only when the fullredshift quasars are used.In contrast, Type I relation always can offer tight constraints on the cosmological model parameters, since the mean values of Ω m0 with 1 CL are Ω m0 = 0.49 +0.21 −0.33 for the highredshift quasars and Ω m0 = 0.253 +0.046 −0.067 for the full-redshift data.For the CDM model, Type I relation yields the Ω m0 = 0.249 ± 0.082 and  = −0.98 +0.25 −0.34 when the full-redshift data is used, which are consistent with those obtained from the CMB data (Planck Collaboration 2020).Additionally, Type I relation always tends to give smaller values of −2 ln L than those of the standard and Type II relations in both the ΛCDM and CDM models.

Table 1 .
Marginalized one-dimensional constraints on parameters with 1 CL from the low-redshift ( ≤ 1.4) quasars

Table 2 .
Marginalized one-dimensional constraints on parameters with 1 CL from the quasars in two different redshift regions full-redshift quasars with Type II relation give an effective constraint on Ω m0 , while the high-redshift ones do not.When Type I relation is used, both the full-and high-redshift quasars can constrain Ω m0 effectively.For the CDM model, the effective constraints on Ω m0 and  can be achieved only from the full-redshift quasars with Type I or II relation.In both the ΛCDM and CDM models, quasars with Type I relation always tend to give smaller Ω m0 and −2 ln L than those with the standard and Type II relations.

Table 3 .
Constraints on the ΛCDM and CDM models from Quasars.

Table 4 .
Constraints on the ΛCDM and CDM models from quasars and OHD.The marginalized mean values, the standard deviations, and the 68% CL.