Constraints on Cosmological Models from Quasars Calibrated with Type Ia Supernova by a Gaussian Process
Haixiang Zhang Yang Liu Hongwei Yu Xiaodong Nong Nan Liang Puxun Wu


firstpage–lastpage
Strong coupling non-Markovian quantum thermodynamics using exact dynamics of a central spin surrounded by a spin bath and the role of a canonical Hamiltonian
    Subhashish Banerjee
    April 29, 2024
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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 ΛCDM 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 ΛCDM 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±0.027),
and   H_0 = 70.80±3.6 km s^-1Mpc^-1, falling  between the measurements  from SH0ES and the Planck cosmic microwave background radiation data. 






Quasars: general - (cosmology:) dark energy - cosmology: observations








§ 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 <cit.>, it still suffers the  Hubble tension <cit.>, which refers to the more than 5σ disagreement between the constraints on the Hubble constant H_0 from the nearby type Ia supernova (SN Ia) and the  cosmic microwave background (CMB) radiation data, respectively. Using the latest SN Ia measurements, <cit.> obtain H_0=73.04±1.04 km s^-1Mpc^-1 in  a model-independent manner, while  the CMB data from the Planck 2018 survey yield H_0=67.36±0.54 km s^-1Mpc^-1 <cit.> in the framework of the ΛCDM model. Many other observational data have been utilized to search  the possible origins of the H_0 tension, but a satisfactory explanation remains elusive.  Since the redshift range of commonly used observations, including SN Ia <cit.>,  observational Hubble parameter data (OHD) <cit.>, baryon acoustic oscillations <cit.>, and strong gravitational lenses <cit.>, reaches only about z∼2, while the CMB data is near z∼1100, cosmological data in the mid-redshift region (2 z 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  z>7 <cit.>. 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 (L_X-L_UV) has been proposed in <cit.>, and it has been widely applied to investigate the high-redshift universe using quasars <cit.>.   Recently, <cit.> showed some evidences of redshift evolution of the X-ray and UV relation. By considering a power-law redshift correction ((1+z)^k) to the quasar luminosity, <cit.> obtained a three-dimensional and redshift-evolutionary version of the L_X-L_UV relation.  In <cit.>, the coefficient of the redshift correction term is determined by using the Efron-Petrosian (EP) method <cit.>. This coefficient will be treated as a free parameter  in the following analysis. <cit.> introduced an improved three-dimensional L_X-L_UV relation with a redshift-dependent term given by a powerful statistical tool called copula[Copula is a powerful tool developed in modern statistics to describe the correlation between multivariate random variables <cit.> and it has been used to construct the  redshift-evolutionary relation of Gamma-ray bursts <cit.>.]. The increased reliability of the L_X-L_UV relations as a cosmological tool has been established  in <cit.>
 
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 <cit.>.   In addition, the simultaneous fitting or global fitting method has also been established to avoid the circularity problem in the GRB cosmology <cit.>. 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.[ 
As demonstrated in <cit.>, 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.] 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 <cit.>, 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 <cit.>.

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 z<0.01 from  1701 SN light curves of 1550 spectroscopically confirmed SN Ia with the maximum redshift being about z ∼ 2.3  <cit.>, 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 <cit.>,  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, <cit.> 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  K-correction was applied to the quasars <cit.>. These low-redshift quasars can be used to determine the coefficients in different L_X-L_UV 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 the cosmological models. 
Since the quasar sample only cannot give constraints on the absolute magnitude (M) of SN Ia and the Hubble constant (H_0) simultaneously, we also add  32 OHD measurements  obtained by the cosmic chronometers  method <cit.>, which relate to the evolution of differential ages of passive galaxies at different redshifts <cit.>,
to obtain  constraints on M and  H_0.

The rest of this paper is organized as follow. In Section <ref>, we respectively calibrate, by using the low-redshift calibration method, three different L_X-L_UV 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 <ref>. Section <ref> shows the discussion on results.
Conclusions are summarized in Section <ref>.



§ CALIBRATING THE L_X-L_UV RELATIONS FROM SN IA

The standard L_X-L_UV relation, which is a non-linear relation between the X-ray luminosity (L_X) and the  UV luminosity (L_UV)  of quasars <cit.>, takes the form

    log(L_X)=β+γlog(L_U V).

Here  β and γ are two coefficients , and “log” denotes the logarithm to  the base 10. Expressing the luminosity in terms of the flux, one can obtain

    log(F_X)   =   2(γ-1) log(d_L)+β+(γ-1) log (4 π) 
          +γlog(F_U V).

 Here  F_X=L_X/4π d_L^2 and F_UV=L_UV/4π d_L^2 are the observed flux of X-ray and UV, respectively, and d_L is the luminosity distance, which contains the information of cosmological models.

Recently, the standard L_X-L_UV relation has been generalized  to contain possible redshift-evolutionary effects:

    log(F_X)   =   2(γ-1) log(d_L)+β+(γ-1) log (4 π) 
           +γlog(F_U V)+αln(α̅+z).

Here α is a coefficient, and α≠ 0 represents that the relation evolves with redshift. The constant α̅ equals to 1 or 5. When  α̅=5, the relation given in Eq. (<ref>), which is named  the Type I relation in this paper, corresponds to that constructed by using the copula function <cit.>. 
If α̅=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+z)^k <cit.>.





To apply quasars in cosmology, we need to obtain their luminosity distances from the observables of quasars by using  the relations shown in Eqs. (<ref>, <ref>). Apparently, the values of the coefficients (α, β, γ)  need to be determined first. Here, we adopt  the idea of distance ladder and  use the SN Ia data to calibrate the L_X-L_UV relations. Since the data points of SN Ia concentrate in the low-redshift (z≤1.4) region, the  SN Ia data from the latest Pantheon+ sample at z≤1.4 <cit.> are utilized to determine α, β and γ. 
To avoid the effect of cosmological models on the calibration, we use  the GP method <cit.> to construct the smooth curve of the apparent magnitude (m)-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 <cit.>. 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:

    k(z, z̃)=σ_f^2exp[-(z-z̃)^2/2 l^2],

where the hyperparameters σ_f and l are optimized by maximizing the marginal likelihood [In this work, we use the public PYTHON package GaPP to reconstruct the apparent magnitude (m) -redshift relation of SN Ia sample in low-redshift (z<1.4) region. This code is available at https://github.com/astrobengaly/GaPPhttps://github.com/astrobengaly/GaPP.].
When the Pantheon+ sample <cit.>  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 <cit.>.
The reconstructed m-z relation with an uncertainty of 1σ by using the GP is shown in Fig. <ref>. This figure indicates that the reconstructed result is well consistent  with the data distribution.


The distance module μ of SN Ia  is related to the  luminosity distance (d_L) and the absolute magnitude (M) through

    μ=m-M=5 log(d_L/Mpc)+25.

Thus, the luminosity distance of quasars that fall in the redshift under z=1.4 may be obtained from the reconstructed m(z) after  the value of M is determined.
However, the value of M cannot be directly obtained using only the SN Ia sample and thus M is treated as a free parameter. Substituting Eq. (<ref>) into the   L_X-L_UV relations, one can obtain

    log(F_X)   =2(γ-1) m-M-25/5+β
    	+(γ-1) log (4 π)
       +γlog(F_U V)+ αln(α̅+z),

which shows  that there is a strong degeneracy between the absolute magnitude M and the coefficient β.
Thus, we cannot constrain the values of M and β simultaneously by using the observational data.
To address this issue without assuming any prior values of M, we introduce a new coefficient: β'=-2(γ-1)(M/5+5)+β+(γ-1)log(4π). Then, the relation  Eq. (<ref>) is reduced to

    log(F_X)  =2(γ-1) m/5+β'+γlog(F_U V)+αln(α̅+z).

 Using the apparent magnitude of SN Ia reconstructed at a redshift of quasars and  the observed log(F_U V)_obs at this redshift, we can obtain  log (F_X)_re from Eq. (<ref>).
Comparing this log (F_X)_re with the observed X-ray flux of quasars can give the values of α, β' and γ  by maximizing the D'Agostinis likelihood function <cit.>:


    ℒ(δ,θ ) =  ∏_i=1^N1/√(2π(σ^2_i+δ^2))×
          exp{-[log(F_X)_obs,i-log(F_X)_re(log(F_UV)_obs,i; θ)]^2/2 (σ^2_i+δ^2)},


where δ is the intrinsic dispersion, θ≡{α, β', γ}, and σ_i^2=σ_log(F_X)_obs,i^2+γ^2σ_log(F_U V)_obs,i^2+(2 γ-2/5)^2σ_m^2 with σ_log(F_X)_obs,i, σ_log(F_U V)_obs,i, and σ_m being the uncertainties of log(F_X)_obs,i, log(F_UV)_obs,i, and the reconstructed m, respectively.
In Eq. (<ref>), number N=1326 is the number of quasars in the low-redshift (z≤ 1.4) region.
To find the maximum  likelihood, we utilize the Python package emcee <cit.>, 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 L_X-L_UV relations are presented in Tab. <ref> and Fig. <ref>.
Obviously, only the value of δ is almost independent of the  relations.  The values of γ are almost the same in  the Type I and II relations, but they deviate from that in the standard L_X-L_UV relation. The Type I and standard relations give almost the same value of β', which is smaller that the one in the Type II relation. 
The Type I and II relations give different values of  α, but both of them deviate from zero at more than 3σ, 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 (z≤1 and z≤ 1.8). The results are shown in Tab. <ref>.  Comparing Tab. <ref> with Tab. <ref>, we find that the relation coefficients and the intrinsic dispersions across these redshift regions vary slightly but remain consistent within a 1σ 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 z≤1.4. 



To further check whether there exist selection biases and redshift evolution in  quasar data <cit.>, we follow the process given in <cit.> and use the EP method <cit.> to calculate the value of k parameter which is introduced to correct the luminosity via L_corrected=L_observed/(α̅+z)^k in Type I and II relations, and the corresponding value of α. If the obtained value of α is consistent with that from the low-redshift calibration,  then the results presented in Table <ref> are not influenced by selection biases or redshift evolution in quasars. When the correction to the luminosity is considered, we need to 
replace  L_X and L_UV with L_X/(α̅+z)^k_X and L_UV/(α̅+z)^k_UV in Eq. (<ref>).  Then, we obtain the following simple relation between α and k_X and k_UV:

    α=k_X-γ k_UV/ln10.

Clearly, the value of α can be inferred from the values of k_X and k_UV when γ is known.
The values of k_X and k_UV can be  obtained by using the EP method, which is given in a publicly available Mathematica code: Selection biases and redshift evolution in relation to cosmology[https://notebookarchive.org/2023-05-8b2lbrhhttps://notebookarchive.org/2023-05-8b2lbrh].
Here we set a fiducial model: the  flat ΛCDM model with H_0=73.04 km s^-1Mpc^-1 and  Ω_m0=0.33 for calculating the luminosity of quasars, and choose the flux limit F_lim=6×10^-33 erg s^-1cm^-2Hz^-1 for the X-rays and F_lim=4.5×10^-29 erg s^-1cm^-2Hz^-1 for the UV.
We finally obtain  k_X=7.10±0.17 and k_UV=9.76±0.22 for the Type I relation, and k_X=2.64±0.06 and k_UV=3.62±0.08 for the Type II relation.
Using the fiducial model  to estimate the luminosity distance, the value of parameter γ can then be determined from  Eq. (<ref>) with the whole quasar sample, and the results are  γ=0.579± 0.011 and γ=0.577± 0.011 for the Type I and II relations, respectively.
Finally, we use Eq. (<ref>) to calculate  the mean value of α with 1σ uncertainty,  and obtain α=0.629±0.103 and α=0.239±0.037[This result is different from 0.342± 0.041 obtained in <cit.> since the luminosity is correced as L'=L/g(z) with g(z)=z^kz^k_crit/( z^k+z^k_crit) rather than g(z)=(1+z)^k in <cit.>.] for the Type I and II relations, respectively.
These results deviate from zero at  more than 5σ and are  compatible with those  obtained from the low-redshift calibration and shown in Tab. <ref>  within 2σ CL. This compatibility suggests that the nonzero value of  α
 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 <cit.>.









§ CONSTRAINTS ON COSMOLOGICAL MODELS

Extrapolating the values of α, β', γ and δ 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:


    ℒ̃ (p)    =   ∏_i=1^N1/√(2 π(σ̃_i^2+δ^2))×
          exp{-[log(F_X)_obs,i-log(F_X)_th( 
    d_L(z_i, p))]^2/2(σ̃_i^2+δ^2)}.


Here N denotes the number of data, and σ̃_i is the derived uncertainty of log(F_X) by using the error propagation formula


    σ̃_i^2   =   σ_log(F_X)_obs,i^2+(∂log(F_X)_t h/∂γ)_i^2σ_γ^2
       +   (∂log(F_X)_t h/∂β')_i^2σ_β'^2
    	+(∂log(F_X)_t h/∂α)_i^2σ_α^2
       +   (∂log(F_X)_t h/∂log(F_U V)_obs)_i^2σ_log(F_U V)_obs^2
       +   2 ∑_j=1^3 ∑_k=j+1^3(∂log(F_X)_t h/∂θ_j∂log(F_X)_t h/∂θ_k)_i C_j k,


where C_j k is the covariance matrix, and σ_α,
σ_β', and σ_γ are the uncertainties of coefficients α, β', and γ estimated from the GP, respectively.
In Eq. (<ref>), p represents the cosmological model parameters and m is the apparent magnitude derived  from the cosmological model, which relates to  the dimensionless luminosity distance D_L through

    m(z, p)=5 log D_L(z;p) + ℳ,

with

    ℳ=25 + M - 5logc/H_0.

The dimensionless luminosity distance  is defined as

    D_L(z;p)=H_0/c d_L(z;p)=  (1+z) ∫_0^zdz̃/E(z̃;p)

in the spatially flat universe. Here, E(z;p) is the dimensionless Hubble parameter.
For the wCDM model, E(z;p) has the form

    E(z;p)^2=Ω_m0(1+z)^3+(1-Ω_m0)(1+z)^3(1+w),

where Ω_m0 and w are the current matter density and the equation of state (EOS) of dark energy, respectively.
This model will reduce to the  ΛCDM when w=-1. Therefore, we have p≡{Ω_m0} for the ΛCDM model and p≡{w,Ω_m0} for the wCDM model. 
Here the prior of w is set as a uniform distribution in 1/(Ω_m0-1)≤ w≤ -1/3, which comes from the accelerated cosmic expansion <cit.> and the null energy  condition <cit.>.
Since  M and H_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 z≤ 1.4 to calibrate the quasars in this redshift region and extrapolate the results to the high-redshift (z>1.4) quasars. Thus, the high-redshift quasars, which contain 1095 observations,  are utilized firstly to constrain the ΛCDM and wCDM models. Furthermore, we also consider the full-redshift quasar data (2421 data points) to limit these two models. The results are shown in Figs. <ref> and <ref>, and summarized in  Tab. <ref>.


It is easy to see that  for the ΛCDM model when the standard L_X-L_UV relation is used, quasars only  give a lower bound on Ω_m0. The 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 wCDM model,   the effective constraints on Ω_m0 and w can be achieved only from the full-redshift quasars with  Type I or II relation.  In both the ΛCDM and wCDM models,  quasars with  Type I relation always tend to give smaller Ω_m0 and -2lnℒ  than those with the  standard and Type II relations.



Since the cosmological  parameters Ω_m0 and w cannot be well constrained simultaneously, we further consider a fitting for  w with  Ω_m0 fixed to be 0.30.
The results are shown in Fig. <ref>.
Apparently, for all three relations the high-redshift quasars only give the lower limit of w.  When the full-redshift quasars are used, we obtain that w=-1.08^+0.17_-0.34 for Type I relation and w=-1.02^+0.63_-0.41 for Type II relation.  If the standard relation is considered,  a large w (-0.381<w<-1/3) is obtained.






 §.§ Constraints on cosmological models from   Quasars + OHD


Since  quasars cannot provide the constraints on M and H_0 due to the degeneracy between them, we further add the OHD from the cosmic age difference method <cit.>  into our analysis.
The updated 32 OHD measurements cover a redshift range of 0.07<z<1.965 <cit.>, which contains 17 uncorrelated and 15 correlated measurements. The 17 uncorrelated data are given in <cit.> and the 15 correlated measurements are sourced from <cit.> with the covariance matrix being  given by <cit.>.
To constrain the cosmological models, the minimization of the χ^2 method is used:

    χ_OHD^2(p)=∑_i=1^17[H_th(z_i;p)-H_obs(z_i)]^2/σ_H, i^2+ΔĤ^T𝐂_H^-1ΔĤ.

Here σ_H represents the observed uncertainty of 17 uncorrelated measurements, ΔĤ=H_th(z;p)-H_obs(z) represents the difference vector between the observed data and the theoretical values for the 15 correlated measurements, and 𝐂_H^-1 is the inverse of the covariance matrix. H_obs represents the observed value of the Hubble parameter, while H_th(z;p)≡ H_0E(z;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:

    lnℒ_total (p)=  lnℒ̃(p)-1/2χ_OHD^2(p).

The results are shown in Figs. <ref> and <ref>, and summarized in Tab. <ref>.

For the ΛCDM model,  we find that in the case of the standard L_X-L_UV relation the combination of the full-redshift quasars and the OHD measurements favors a smaller H_0, a larger Ω_m0, and a larger M 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 H_0 is consistent with the result from the Planck CMB observations <cit.>, while the value of M is smaller than the SH0ES measurement <cit.>. 
When the Type I relation is considered, we find that the H_0 value from full-redshift quasars locates between the Planck result and the SH0ES measurement <cit.>, and the M value is compatible with the SH0ES measurement <cit.>.

For the wCDM model, we find that there is a large difference between the constraints on H_0 obtained from the high- and full-redshift quasars in the standard  L_X-L_UV 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 H_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  constraints on w consistent with -1 at 2σ CL.
The constraints on M obtained from both the high- and full-redshift quasars in these three relations are nearly identical to that obtained in the ΛCDM model.  Furthermore, quasars with the Type I relation have the minimum  value of -2lnℒ.





















§ DISCUSSION



Except for the low-redshift calibration, the global fitting method is usually used to avoid the so-called circularity problem. Using the total 2421 quasar datapoints to constrain the spatially flat ΛCDM model and the coefficients in the L_X-L_UV relations simultaneously, it has been found that Ω_m0>0.924, δ=0.228±0.004, β=7.021±0.249 and γ=0.639±0.008 for the standard relation, and Ω_m0=0.510^+0.163_-0.254, δ=0.225±0.004, β=7.825±0.316, γ=0.579±0.011 and α=0.580^+0.084_-0.099 for Type I relation <cit.>. Apparently, the values of relation coefficients and intrinsic dispersion from the global fitting are well consistent with  those shown Tab. <ref>, and the values of Ω_m0 are compatible with those (see Tab. <ref>) from the high-redshift quasars. In the flat ΛCDM model, quasars in the redshift region 0.0041 ≤ z ≤ 1.686 give that the best fitting values are  Ω_m0=0.995, δ=0.329 and γ=0.559 <cit.>. These values of δ and γ are compatible with those shown in Tab. <ref>. 
Assuming a cosmological model with parameter values based on the computations with SNe Ia, <cit.> obtained that δ=0.231±0.004,  β=6.817±0.265, γ=0.648±0.009 for the standard relation, and  δ=0.231±0.004,  β=8.278±0.362, γ=0.591±0.013 for the Type II relation with α fixed from the EP method.  These results are also consistent with ours obtained from the low redshift calibration. Comparing Fig. <ref> with Fig. 2a and 2b in <cit.> and Figs. 2 and 3 in <cit.>, we find that the values of intrinsic dispersion and  coefficients of the standard and Type II relations  obtained in this paper are compatible with what were determined from the combination of Pantheon SN Ia, GRBs, quasars and BAO in the ΛCDM model. While, the values of H_0 in the standard and Type II relations are slightly smaller than what were obtained in <cit.>. The values of δ achieved in <cit.> and this paper are significant larger than δ=0.007±0.004 from the golden sample of 983 quasars <cit.>. 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 <cit.> and Ω_m0=0.295^+0.013_-0.012 from SN Ia + quasars <cit.>. Recently,  <cit.> 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 L_X-L_UV relations: the standard relation proposed by <cit.>, and the Type I and Type II redshift-evolutionary relations constructed by <cit.> and <cit.>, respectively.  We employ the GP method to calibrate these relations from  the latest Pantheon+ SN Ia data within the low-redshift region (z<1.4).  These results align with that reported by <cit.>, and suggest a potential trend of redshift evolution in the L_X-L_UV 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 wCDM models.
We find that the standard relation provides only a lower bound  on Ω_m0 in both the ΛCDM and  wCDM models. 
For the ΛCDM model, Type II relation gives effective constraints on the cosmological model parameters only when the full-redshift 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 1σ CL are Ω_m0=0.49^+0.21_-0.33 for the high-redshift quasars and Ω_m0=0.253^+0.046_-0.067 for the full-redshift data.
For the wCDM model,  Type I relation yields the Ω_m0=0.249±0.082 and w=-0.98^+0.25_-0.34 when the full-redshift data is used, which are consistent with those obtained from the CMB data <cit.>.
Additionally, Type I relation always tends to give smaller values of -2lnℒ than those of  the standard and Type II relations in both the ΛCDM and wCDM models.

When the additional OHD sample is considered,  the observations can  give tight constraints on the ΛCDM and wCDM 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 <cit.>.
Moreover, since the addition of the OHD sample can break the degeneracy between parameters M and H_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 M to be -19.321^+0.085_-0.076, which aligns well with that obtained from SH0ES (M=-19.253±0.027) <cit.>, and  the Hubble constant H_0  to be 70.80±3.6 km s^-1Mpc^-1, which lies between the measurements of SH0ES <cit.>  and CMB <cit.>.




§ ACKNOWLEDGMENTS

This project was supported by  the NSFC under Grants Nos. 12275080 and 12075084,  the innovative research group of Hunan Province under Grant No. 2024JJ1006, and  the Guizhou Provincail Science and Technology Foundation (QKHJC-ZK[2021] Key 020).



§ DATA AVAILABILITY

Data are available at the following references:
the 2421 X-ray and UV flux measurements of quasars
from <cit.>,
the Pantheon+ SNe Ia sample from <cit.>,
and the latest OHD obtained with the cosmic chronometer method from <cit.>.

.












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