Revisiting Chaplygin gas cosmologies with the recent observations of high-redshfit quasars

The GCG model, which is extended from the CG model, has been generally studied to explain the accelerating universe Zhangjingfei32 ; Bento2003 ; zhu2004 ; Lixiaolei22 ; lixiaolei50 ; lixiaolei ; Lian2021 . In this model, the dark energy and dark matter could be unified with an exotic equation, which is introduced as

where $p_{gcg}$ and $\rho_{\mathrm{gcg}}=\rho_{de}+\rho_{dm}$ present the pressure and density of Chaplygin gas, respectively. $A$ is a positive constant and $0\leq\alpha\leq 1$. When $\alpha=1$, the GCG model reduces to the CG model, and when $\alpha=0$, the GCG model reduces to the $\Lambda$CDM model. The energy density of the GCG model is expressed as

where $a$ is a scale factor, which is related to the observable redshift as $a=\frac{1}{1+z}$, $A_{\mathrm{s}}\equiv A/\rho_{\mathrm{gcg}0}^{1+\alpha}$ is a dimensionless parameter, and $\rho_{\mathrm{gcg}0}$ is the present energy value of the GCG density. $A_{s}$ can be written by the effective total matter density $\Omega_{m}$ and $\alpha$ as

Therefore, we can derive the normalized Hubble parameter $E(z)$ for this model as

where $E(z)=H^{2}(z)/H^{2}_{0}$ and the parameter set is $\mathbf{p}\equiv\left(\Omega_{m},A_{\mathrm{s}},\alpha,H_{0}\right)$.

The MCG model is also a unified dark matter and dark energy model, which is a modification of the GCG model. It has been widely discussed in many perspectives lixinxu16 ; lixinxu14 ; lixinxu15 ; xulixin2012modified ; li2019MCG ; debnath2021MCG . This class of equation of state is expressed as,

where $\rho_{\mathrm{gcg}}=\rho_{DE}+\rho_{DM}$, $A$ is a positive constant, $B$ is a free parameter, and $0\leq\alpha\leq 1$. When $B=0$, this model corresponds to the GCG model, whereas when $A=0$, it reduces to the standard equation of state of a perfect fluid. Especially, it turns to $\Lambda$CDM model with $B=0$ and $\alpha=0$ and it reduces to CG model with $B=0$ and $\alpha=1$. Considering energy conservation, we can obtain the energy density as

where $A_{s}=A/(1+B)\rho_{\mathrm{mcg}0}^{1+\alpha}$, $B\neq-1$ and $\rho_{\mathrm{mcg}0}$ is the present energy value of the MCG density. Therefore, we can rewrite the normalized Hubble parameter $E(z)=H(z)/H_{0}$ for the MCG model as

For MCG, the parameter set is $\mathbf{p}\equiv\left(\Omega_{m},A_{\mathrm{s}},B,\alpha,H_{0}\right)$.

The NGCG model has been studied in previous work, such as zhangjingfei34 ; liaokai2013 ; Zhangjingfei2019 ; salahedin2020NGCG ; Almamon2021 . In the NGCG model, it assumes that the exotic background fluid interpolates between a dust-dominated epoch $\rho\sim a^{-3}$ and a cosmological constant-dominated epoch $\rho\sim a^{-3\left(1+\omega\right)}$, which is portrayed as a unification of X-type dark energy and dark matter. Specifically, when $\omega=-1$, the NGCG model reduces to the GCG model, while $\omega=-1$ and $\alpha=0$, it reduces to the XCDM model. The equation of state of NGCG is given by,

where $\tilde{A}(a)=-wAa^{-3(1+w)(1+\alpha)}$ is a function of the scale factor, and $\alpha$ is a free parameter spanning 0 to 1. The energy density of the NGCG fluid is

where $A_{s}=\frac{1-\Omega_{m}}{1-\Omega_{\mathrm{b}}}$. Finally, we can get the form of $E(z)=H(z)/H_{0}$ of the NGCG model,

Hence, for the NGCG model, the parameter set that we adopt is $\mathbf{p}\equiv\left(\Omega_{m},\omega,\alpha,H_{0}\right)$.

To tackle the late accelerated expansion of the universe, a hybrid model that consists of a fusion of viscous effects and the features of Chaplygin gas, the VGCG model was studied in VGCG2006 ; LiweiVGCG ; liwei2015 ; almada2021VGCG . This model is able to avoid causality problems that arise when only dissipative fluid is considered and alleviate the blowup in the DM power spectrum for GCG models almada2021VGCG . The equation of state of the VGCG model is given by

One can obtain the standard $\Lambda$CDM model when $\alpha=0$ and $\zeta=0$, and this model reduces to the GCG model with $\zeta=0$. Then, we can deduce its energy density as,

where $B_{s}=A/\rho_{\mathrm{vgcg}0}^{1+\alpha}$, $0\leq B_{s}\leq 1$ and $\zeta<\frac{1}{\sqrt{3}}$. The dimensionless Hubble parameter $E(z)=H(z)/H_{0}$ is expressed as

It is straightforward that the parameter set of the VGCG model is $\mathbf{p}\equiv(\Omega_{m},B_{s},\alpha,\zeta,H_{0})$.

The latest compilation of quasar (QSO[XUV]) from X-ray and UV flux measurements is recognized via the X-ray luminosity and UV luminosity ($L_{X}-L_{UV}$) relation 2019NatAs…3..272R and used to constrain cosmological model parameters Risaliti2015 ; Lian2021 . The $L_{X}-L_{UV}$ relation is given by,

where the slopes $\gamma$ and $\beta$ are free parameters that can be measured from the dataset. When we express luminosities in terms of fluxes, $F=L/4\pi D_{L}(z)^{2}$, Eq. (15) becomes

where $F_{X}$ and $F_{UV}$ are the quasar X-ray and UV fluxes, respectively, and $D_{L}$ is the luminosity distance, which is determined via

where $E(z)$ depends on different cosmological models.

To obtain the likelihood function, we use Eq. (16) and Eq. (17) in a specific model as

where $\ln=\log_{e}$, $s_{i}^{2}=\sigma_{i}^{2}+\delta^{2}$, and where $\sigma_{i}$ and $\delta$ are the data error on the observed flux and the global intrinsic dispersion, respectively. In addition, according to 2019NatAs…3..272R ; khadka2021 , we employ the QSO from X-ray and UV fluxes in the analysis with the chi-square statistic

To use the Pantheon sample, first, we should determine the corresponding observable value and its theoretical value. The observable value given in the Pantheon sample is a corrected magnitude; see Table A17 of pantheon for more details, expressed by

where $\mu$ is the distance modulus, $m_{B}$ is the apparent B-band magnitude, and $M$ is the absolute B-band magnitude of fiducial SNe Ia. There is a correction term $K=\alpha x_{1}-\beta c+\Delta_{M}+\Delta_{B}$ that includes the corrections related to four different sources (for more details, see pantheon ). The theoretical value is given by,

where the constant term $Y_{0}=M+5log(\frac{cH_{0}^{-1}}{Mpc})+25$, which should be marginalized by the methodology presented in Giostri2019 . The chi-square for the Pantheon sample can be given by

where $\Delta\overrightarrow{Y}_{i}=[Y^{obs}_{i}-Y^{th}(z_{i};Y_{0},\textbf{p})]$ and the covariance matrix C of the sample includes the contributions from both the statistical and systematic errors pantheon .

Cao20017qsoas extracted 120 compact radio quasars (QSO[AS]) based on a 2.29 GHz VLBI all-sky survey of 613 milliarcsecond ultracompact radio sources, covering a redshift range from 0.46 to 2.76. The observable value angular sizes $\theta_{obs}(z)$ is related to the intrinsic length $\ell_{m}$ and the angular diameter distance $D_{A}(z)$ Cao:2015APJ ; AS3 . The corresponding theoretical angular size is defined by

where $\ell_{m}$ is the intrinsic metric linear size, which is calibrated to $11.03\pm 0.25$ pc by an independent method introduced in caoshuo2017AA , and $D_{A}(z)$ is the angular diameter distance

where $D_{L}(z)$ is defined by Eq.(17). Therefore, we calculate the chi-square function by

where $\theta(z_{i};\hat{p})$ is the theoretical value of the angular size and the total uncertainty can be expressed as $\sigma^{2}_{i}=\sigma^{2}_{stat,i}+\sigma^{2}_{sys,i}$.

The BAO data is also a powerful cosmological probe eisenstein1998 ; eisenstein20005 , which is extracted from galaxy redshift surveys. Here, we use 11 BAO measurements summarized in Table 1. The observable quantities used in the measurements are expressed in terms of the transverse co-moving distance $D_{M}(z)$, the volume-average angular diameter distance $D_{V}(z)$, the Hubble rate $H(z)\equiv H_{0}E(z)$, the Hubble distance $D_{H}\equiv c/H(z)$, the sound horizon at the drag epoch $r_{s}$, and its fiducial value $r_{\rm{s,fid}}$. In a flat universe, the transverse co-moving distance $D_{M}(z)$ equals the line-of-sight co-moving distance $D_{C}(z)$, which is expressed as

where $c$ is the velocity of light. The volume-average angular diameter distance is

Following ryan2019 , we use the fitting formula of eisenstein1998 to compute $r_{s}$ and calculate $r_{s,fid}$ by using the fiducial cosmological model.

Most of data we used are correlated; however, those from carter2018 ; DES2018 ; ata2018 ) are uncorrelated. For the uncorrelated data points, the chi-square statistic is expressed as

where $A_{th}(p,z_{i})$ denotes the model predictions at the effective redshift, $A_{obs}(z_{i})$ is the observational value and $\sigma_{i}$ is the error bar of the measurements. For the correlated data points from Alam2017 ; dsa2019 , it requires

where $C^{-1}$ is the inverse of the covariance matrix. The corresponding covariance matrix of Alam2017 is available from the SDSS website, and that of dsa2019 is presented in Caoshulei2020 .

In the cosmological analysis, the probability distributions of model parameters are obtained with an affine invariant Markov chain Monte Carlo (MCMC) ensemble sampler (emcee) emcee , where the statistic can be determined with

where $p$ is the set of model parameters from different cosmological models.

We present the 1D probability distributions and 2D contours with 1$\sigma$ and 2$\sigma$ confidence levels (CLs) for the GCG model in Fig. 2 and list the best-fit parameters at the 1$\sigma$ confidence level in Table 2. The standard candle data gives $\Omega_{m}=0.53^{+0.53}_{-0.29}$, $A_{s}=0.78\pm 0.06$,$\alpha=0.46^{+0.57}_{-0.42}$ and $H_{0}=68.27^{+6.98}_{-4.76}$ km/s/Mpc, while the standard ruler data obtains $\Omega_{m}=0.33\pm 0.02$, $A_{s}=0.60\pm 0.10$, $\alpha=-0.33^{+0.27}_{-0.24}$ and $H_{0}=65.81^{+2.26}_{-2.28}$ km/s/Mpc. First, it is clear that the value of $\Omega_{m}$ obtained from standard candles shows a deviation from the Planck collaboration ($\Omega_{m}=0.3103\pm 0.0057$) planck2018result . This is because a larger value of the matter density parameter is favored by the recent QSO[XUV] compilation in most cosmological models at higher redshifts ($2.5<z<5$), which has been discussed in previous work Risaliti2015 ; khadka2020_1 . In addition, $\alpha$ is an important parameter $0\leq\alpha\leq 1$, where $\alpha=0$ denotes the $\Lambda$CDM model and $\alpha=1$ denotes the CG model. Although previous studies showed that the CG model is ruled out by observations, we find that the CG model is accepted by QSO[XUV]+SNe Ia at a 68% CL. In addition, the standard ruler data favor the $\Lambda$CDM model at a 95% CL, as well as the combination sample. For the Hubble constant, our constraint results are in good agreement with the Planck collaboration ($H_{0}=67.66\pm 0.42$ km/s/Mpc) planck2018result , although the values obtained from standard rulers are lower than that from other probes. In addition, the standard ruler data could bring down the error bars of $\Omega_{m}$, $\alpha$ and $H_{0}$ compared with the standard candles. This indicates that QSO[AS]+BAO could give a more restrictive constraint on cosmological parameters. Moreover, it is necessary to refer to the previous results, such as $A_{s}=0.70^{+0.16}_{-0.17}$ and $\alpha=-0.09^{+0.54}_{-0.33}$ constrained from the X-ray gas mass fraction, Type Ia supernovae and Type IIb radio galaxies in zhu2004 and $\alpha=-0.14^{+0.30}_{-0.19}$ obtained by SNe Ia+H(z)+CMB in wupuxun2007 , which are consistent with our results from combination data and favor the standard $\Lambda$CDM model. It is worth mentioning that Lian2021 gave $\Omega_{m}=0.416^{+0.088}_{-0.068}$, $\alpha=2.360^{+1.803}_{-1.793}$ and $H_{0}=69.254^{+4.427}_{-4.970}$ km/s/Mpc from the QSO[XUV]+QSO[AS], which is in good agreement with our results from QSO[XUV]+SNe Ia and includes the CG model at a 68% CL. This suggests that the latest QSO compilation from X-ray and UV flux measurements slightly favors the CG model and prefers a larger value of $\Omega_{m}$.

In the case of the MCG model, the results are presented in Fig. 3 and Table 2. The standard candle data generates $\Omega_{m}=0.47^{+0.36}_{-0.27}$, $A_{s}=0.81^{+0.06}_{-0.09}$, $B=0.12^{+0.26}_{-0.21}$, $\alpha=0.20^{+0.58}_{-0.39}$ and $H_{0}=68.28^{+7.23}_{-3.48}$ km/s/Mpc, while the standard ruler data provides $\Omega_{m}=0.33\pm 0.02$, $A_{s}=0.61^{+0.10}_{-0.14}$, $B=-0.12^{+0.18}_{-0.09}$, $\alpha=0.05^{+0.87}_{-0.52}$ and $H_{0}=66.27^{+2.01}_{-2.30}$ km/s/Mpc. The value of $\Omega_{m}$ obtained from QSO[XUV]+SNe Ia is still higher than that from other probes, which is the same as the case of the GCG model and still consistent with that from planck2018result at a 68.3% CL. In the framework of the MCG model, considering the fact that the parameter $B$ reflects the deviation from the GCG model (the MCG model reduces to the GCG model when $B=0$), the GCG model is accepted by current observations at a 95% CL in all cases. However, the MCG model shows a tiny deviation from the GCG model by the combination sample at a 68% CL. For the key parameter $\alpha$ that quantifies the deviation from the CG model and $\Lambda$CDM model, it is clear that the $\Lambda$CDM model, $B=0$ and $\alpha=0$, is accepted by standard candles and standard rulers at 68% CLs, while the CG model, $B=0$ and $\alpha=1$, is favored by the combination sample at a 95% CL. However, in the case of the combination sample, both the $\Lambda$CDM and CG models are favored within a 68% CL. In other words, this suggests that the $\Lambda$CDM model is more favored by standard candles and standard rulers, respectively, but the CG model is slightly preferred by the combination. Focusing on the Hubble constant, the constraint results agree well with Planck collaboration planck2018result . Moreover, our results from combination samples are consistent with the results obtained from SNe Ia+BAO+CMB xulixin2012modified with $\alpha=0.000727^{+0.00142}_{-0.00140}$, $B_{s}=0.782^{+0.0163}_{-0.0162}$ and $B=0.000777^{+0.000201}_{-0.000302}$ and from H(z)+BAO+CMB+SNe Ia Thakur2019 with $\Omega_{m}=0.284^{+0.013}_{-0.014}$, $\alpha=0.046^{+0.107}_{-0.102}$ and $B=0.0026\pm 0.005$ at the 1$\sigma$ confidence level. This proves that most cosmological probes favor the $\Lambda$CDM model; however, the inclusion of the QSO sample from X-ray and UV flux measurements Risaliti2015 at higher redshifts changes to slightly favor the CG model.

In Fig. 4 and Table 2, we show the constraint results of the NGCG model. Compared with the standard candle dataset with $\Omega_{m}=0.30^{+0.16}_{-0.14}$, $\omega=-1.10^{+0.24}_{-0.39}$, $\alpha=0.23^{+0.89}_{-0.53}$ and $H_{0}=68.50^{+7.19}_{-5.21}$ km/s/Mpc, the standard ruler data obtains $\Omega_{m}=0.34\pm 0.02$, $\omega=-0.79^{+0.13}_{-0.14}$, $\alpha=-0.12^{+0.12}_{-0.17}$ and $H_{0}=65.16^{+2.38}_{-2.18}$ km/s/Mpc. The most notable thing is that $\Omega_{m}=0.30^{+0.16}_{-0.14}$ from standard candles is consistent with Planck collaboration ($\Omega=0.3103\pm 0.0057$) planck2018result , however this is contrary in the scenarios of the GCG, MCG and VGCG models. khadka2020_1 constrained $\Omega_{m}\sim 0.3$ in the XCDM model from only compiled X-ray and UV flux measurements of 1598 quasars, while $\Omega_{m}\sim 0.5-0.6$ in the $\Lambda$CDM and $\phi$CDM models. There are similarities between the NGCG and XCDM models because the parameter $\omega$ in the NGCG model is proposed by a similar idea to that in the XCDM model. Hence, we obtain a normal value of $\Omega_{m}$ in the framework of NGCG, which indicates that X-ray and UV flux measurements of 1598 quasar compilations could help to determine the dark energy and dark matter. It should be noted that $\omega$ is a free constant and NGCG2006 proposed the probability that dark energy behaves in a quintessence-like form with $\omega>-1$ and phantom-like form with $\omega<$ $-1$. The 1$\sigma$ range $\omega\in(-1.05,-0.95)$ from the combination sample implies that there is an equal chance that dark energy behaves as a quintessence-like form or phantom-like form. In all cases, it suggests that the GCG model (i.e., $\omega=-1$) and XCDM model (i.e., $\omega=-1$ and $\alpha=0$) are still supported by the observational data at a 95% CL. In addition, it is remarkable that the CG model, $\omega=-1$ and $\alpha=1$, is accepted by the standard candle data at a 68% CL. The Hubble constant obtained in our analysis is more consistent with the results of planck2018result at a 68% CL. Furthermore, we make a comparison with the previous findings in the literature. For instance, liaokai2013 derived $\Omega_{de}=0.7297^{+0.0229}_{-0.0276}$, $\omega=-1.0510^{+0.1563}_{-0.1685}$ and $\eta=1+\alpha=1.0117^{+0.0469}_{-0.0502}$ with SNe Ia+BAO+WMAP+H(z) data; Zhangjingfei2019 obtained $\Omega_{de}=0.6879\pm 0.0078$, $\omega=-1.02\pm 0.045$, $\alpha=-0.0029\pm 0.0097$ and $H_{0}=67.78\pm 0.87$ km/s/Mpc with a joint sample of SNe Ia+BAO+CMB; and salahedin2020NGCG stated $\Omega_{m}=0.2508^{+0.0081}_{-0.0097}$, $\omega=-1.041\pm 0.045$, $A_{s}=0.7371^{+0.0097}_{-0.0086}$, $\eta=1+\alpha=0.9443\pm 0.0097$ and $H_{0}=70.15\pm 0.84$ km/s/Mpc with SNe Ia+BAO+CMB+BBN+H(z) data. This indicates that the value of $\Omega_{m}$ from current observations, i.e., SNe Ia, BAO, CMB and H(z), is generally smaller than $\Omega_{m}=0.3103\pm 0.057$ from planck2018result ; however, the inclusion of QSO[XUV] and QSO[AS] changes the value of $\Omega_{m}$ to 0.3-0.34 in our work. It indicates that the inclusion of quasar data could help us to study dark matter and dark energy.

The best-fit values for the VGCG model from different observations are shown in Fig. 5 and Table 2. The standard candle data obtains $\Omega_{m}=0.47^{+0.36}_{-0.31}$, $B_{s}=0.82^{+0.13}_{-0.19}$, $\alpha=0.41^{+0.74}_{-0.49}$, $\zeta=-0.0017^{+0.10}_{-0.08}$ and $H_{0}=68.58^{+6.82}_{-5.16}$ km/s/Mpc, while the standard ruler data shows $\Omega_{m}=0.33\pm 0.02$, $B_{s}=0.55^{+0.17}_{-0.16}$, $\alpha=-0.08^{+0.99}_{-0.57}$, $\zeta=0.07^{+0.06}_{-0.12}$ and $H_{0}=66.32^{+2.16}_{-2.42}$ km/s/Mpc. It indicates that the value of $\Omega_{m}$ is still larger than that of planck2018result , since the QSO[XUV] data favors higher $\Omega_{m}$ in most dark energy models Risaliti2015 ; khadka2020_1 . $\zeta$ is the viscosity term that affects the CMB power spectrum about the matter density on the height of the acoustic peaks. From the results shown in Table 2, it implies that $\zeta$ is very small, which could alleviate the oscillations causing the blowup in the DM power spectrum in the GCG models. Moreover, the GCG model (i.e., $\zeta=0$) is still favored by the available observations. On the other hand, $\alpha$ is an important parameter that reflects the deviation from the CG model and $\Lambda$CDM model. In all cases, the CG model cannot be ruled out by current observations at a 68% CL, while $\Lambda$CDM is still accepted by the observations at a 68% CL. In other words, it indicates that QSO[XUV], SNe Ia, QSO[AS] and BAO data could not give accurate constraints on $\alpha$. In addition, our results on the Hubble constant approve the value of Planck collaboration ($H_{0}=67.66\pm 0.42$ km/s/Mpc) planck2018result at a 68% CL. It is reasonable to compare to previous studies, such as liwei2013VGCG declared $\zeta=0.000708^{+0.00151}_{-0.00155}$ from SNe Ia+BAO+WMAP and LiweiVGCG announced $\zeta=0.0000138^{+0.00000614}_{-0.0000105}$ from SNLS3 +BAO+HST. In a recent work, almada2021VGCG used a joint sample of SLS+SNe Ia +BAO+OHD+HIIG and obtained $B_{s}=0.50^{+0.05}_{-0.06}$, $\alpha=0.99^{+0.61}_{-0.58}$, $\zeta=0.13^{+0.02}_{-0.03}$ and $h=0.69\pm 0.01$. They concluded that the GCG model, $\zeta=0$, is disfavored by SLS+SNe Ia+BAO+OHD+HIIG at a 68% CL, which is different from our results with $\zeta=0.07^{+0.04}_{-0.08}$. Moreover, we find that the inclusion of cosmic microwave background data could give a more precise constraint on $\zeta$.

From our constraint results on the matter density parameter $\Omega_{m}$ in different CG models, it is clear that the standard candle data combining QSO[XUV] with SNe Ia prefers larger values of $\Omega_{m}$ ranging from $0.47-0.53$ except the NGCG model. In khadka2020_1 , it states that the QSO[XUV] data at $z\sim 2-5$ prefers larger values of $\Omega_{m}\sim 0.5-0.6$. Other studies have concentrated on exploring the tension between high redshift quasar measurements and other observations, such as BAO measurements in Risaliti2015 ; razaei2020 ; yangtao2020cosmography ; lixiaolei2021hubble . It implies that there is an unknown systematic error in the high redshift observations or a stimulus of the new physics and astronomy. Therefore, more accurate cosmological probes are required to solve the problem of the $\Omega_{m}$ inconsistency from high and low redshift observations. On the other hand, it is also rewarding to comment on the possible alleviation of the $H_{0}$ tension by the VGCG and MCG model. Based on our results presented in Table 2, the constraint on the Hubble constant lies in the range of $H_{0}=68.28^{+7.23}_{-3.48}$ km/s/Mpc to $H_{0}=68.58^{+6.82}_{-5.16}$ km/s/Mpc for the standard candles, as well as $H_{0}=68.09^{+1.11}_{-1.06}$ km/s/Mpc to $H_{0}=68.21^{+1.19}_{-1.04}$ km/s/Mpc for the combined sample. It is noteworthy that these two Chaplygin gas models suggest a central value of the Hubble constant between the Planck experiment planck2018result , $H_{0}=67.4\pm 0.5$ km/s/Mpc and the SH0ES experiment riessH0 , $H_{0}=74.03\pm 1.42$ km/s/Mpc.

This new class of information-theoretic divergence measures based on Jensen’s inequality and the Shannon entropy, called “Jensen-Shannon Divergence”, could assign the similarity between two probability distributions Lin1991JSD ; JSD2012 ; Lian2021 . It should be mentioned that JSD is used to assess two different cosmological models by the common parameters; here, we choose the matter density $\Omega_{m}$ and the Hubble constant $H_{0}$ to distinguish the four CG models as well as the $\Lambda$CDM model. In general, the JSD is symmetric and ranges from 0 to 1, which can be written as

where $s=1/2(p+q)$. $p(x)$ and $q(x)$ are two probability distributions of two different models and $D_{KL}$ denotes the Kulback-Leibler divergence (KLD), which can be expressed as

It is clear that a smaller value of JSD indicates that the two models are similar. Fig. 6 and Fig. 7 display the posterior distributions of $\Omega_{m}$ and $H_{0}$. Table 3 presents the JSD values between the $\Lambda$CDM model and four nonstandard models by using different observations with respect to $\Omega_{m}$ and $H_{0}$. For standard candle data, the posterior distributions of $\Omega_{m}$ and $H_{0}$ in the NGCG model agree more with the $\Lambda$CDM model in terms of the JSD values, while the MCG model shows a larger distance from the $\Lambda$CDM model. In the scenario of standard ruler data, the value of JSD concerning $\Omega_{m}$ shows that the GCG model agrees more with the $\Lambda$CDM model; however, concerning $H_{0}$, all four nonstandard models are distant from the $\Lambda$CDM model, where the VGCG model is closest to the $\Lambda$CDM model. In the case of the combination sample, for $\Omega_{m}$, the GCG model and NGCG model are more closer to the $\Lambda$CDM model due to the smaller values of JSD, while for $H_{0}$, the MCG model and VGCG model are closest to the $\Lambda$CDM model.

In the framework of a specific cosmological model, the Hubble parameter $H(z)$ and the deceleration parameter $q(z)$ can be expressed,

where $a$ is the scale factor $a=1/1+z$. As $H(z)$ and $q(z)$ cannot effectively distinguish different cosmological models, it requires a higher order of time derivatives of $a$. To investigate more dark energy models, except for the cosmological constant model, the author of statefinder focused on a new geometrical diagnostic pair $(r,s)$ constructed from the $a(t)$ and its third time derivatives beyond, where $r(z)$ is a natural next step beyond $H(z)$ and $q(z)$, and $s(z)$ is a linear combination of $r(z)$ and $q(z)$. This approach has been widely adopted in comparing different cosmological models lixiaolei ; xutengpeng2018 ; dubey2021statefinder ; panyu2021statefinder .

The statefinder pair $(r,s)$ is also related to the equation of state of dark energy and its first time derivative, which can be expressed as

For a given model, the statefinder diagnostic can be obtained by

and

and

Based on the best-fit model parameters derived from the combined QSO[XUV]+SNe Ia+QSO[AS]+BAO data, we calculate the statefinder pairs $(r,s)$ for the $\Lambda$CDM model and four CG models and present the results in Fig. 8. Specifically, the parameter $r$ is more effective in distinguishing different cosmological models. It is noteworthy that although the corresponding values for the MCG model and VGCG model significantly deviate from the $\Lambda$CDM model at the present epoch, both of them eventually converge to the standard cosmological model. On the other hand, it is obvious that in the framework of the GCG model and NGCG model, the statefinder pairs $(r,s)$ exhibit similar behaviors at present and evolve along different trajectories; however, only the GCG model ultimately converges on the point of $(r,s)=(1,0)$.

The evolutionary trajectories in the $r-q$ plane are displayed in Fig. 9. Although the curves of each cosmological model originate from different points, they finally converge to the same point $(r,q)=(1,-1)$ except for the NGCG model. We clearly see that the GCG and NGCG models evolve along similar trajectories with the $\Lambda$CDM model. In addition, we find that the GCG model and NGCG model presume values in the range $r>1$ and $q>0$ at early times and therefore represent as Chaplygin gas-type dark energy models. Moreover, the MCG model and VGCG model start from the regions $r<1$ and $q>0$ belonging to Quintessence dark energy models, while the MCG model quickly reverts back into the Chaplygin gas-type dark energy model at later times. There are notable flips from positive to negative in the value of $q$, which explains the recent phase transition of these models and proves the accelerating universe exactly.

From Sect. 5.1 and Sect. 5.2, we cannot clearly determine these four CG models with the $\Lambda$CDM model. When comparing and distinguishing different competing models, certain information criteria, such as the Akaike information criterion AIC , the Bayes information criterion BIC , and the deviance information criterion DIC , would be crucial.

The AIC is based on information theory, the BIC is based on Bayesian inference, and the DIC combines heritage from both Bayesian methods and information theory DIC . Compared with DIC, the AIC and BIC are too simple to select which model performs better by only requiring the maximum likelihood and the number of parameters within a given model rather than the likelihood throughout the parameter space 2011PhRvD..84b3005C ; 2012ApJ…755…31C ; therefore, we apply DIC to model selection in this paper. Moreover, $\Delta$DIC is an important value which denotes the difference in values of DIC between cosmological models. In our analysis, we calculate the values of DIC and $\Delta$DIC with respect to four Chaplygin gas models and $\Lambda$CDM model for same observations. In particular, negative values of $\Delta$DIC indicates that the model fits the observations better than $\Lambda$CDM model.

The DIC was introduced by DIC and defined as

where $D(\theta)=-2\ln\mathcal{L}(\theta)+C$, $p_{D}=\overline{D(\theta)}-D(\bar{\theta})$, $C$ is a ‘standardizing’ constant depending only on the data that will vanish from any derived quantity and $D$ is the deviance of the likelihood. The definition of DIC (i.e., Eq. (38)) is motivated by the form of the AIC, replacing the maximum likelihood $\mathcal{L}_{max}$ with the mean parameter likelihood $\mathcal{L}(\bar{\theta})$ and replacing the number of parameters $k$ with the effective number of parameters $p_{D}$, which represents the number of parameters that can be usefully constrained by a particular dataset. By using the effective number of parameters, the DIC also overcomes the problem of the BIC that they do not discount parameters that are unconstrained by the data DIC . In the DIC analysis, the favorite model is the one with the minimum DIC value.

We introduce the DIC to evaluate which model is more consistent with the observational data. As for standard candle data, it suggests that the DIC criterion advocates on the MCG model. From standard rulers and the combination sample, the VGCG model seems to be preferred by the smallest values of DIC. In addition, the GCG and NGCG model are seriously punished by the DIC. In particular, we use the model selection DIC criterion to specify which model is preferred by the currently available observations, rather than selecting the single best-fit cosmological model. As shown in the recent observational constraints on $f(T)$ gravity PhysRevD.100.083517 , the exponential $f(T)$ model presents a small deviation from $\Lambda$CDM paradigm, based on the SNe Ia Pantheon sample, Hubble constant measurements from cosmic chronometers, the CMB shift parameter and redshift space distortion measurements. Our findings demonstrate that the MCG model and VGCG model behave better than the concordance $\Lambda$CDM model. We remark here that the $\Lambda$CDM cosmological model, built on the assumptions of a cosmological constant and cold dark matter, shows a $\sim 4\sigma$ tension with the high-redshift Hubble diagram of SNe Ia, QSO and gamma-ray bursts (GRB) 2019NatAs…3..272R . Such irreconcilable tension between high-redshift QSOs and flat $\Lambda$CDM, which has been recently traced and extensively discussed 2020PhRvD.102l3532Y ; Lian2021 in the framework of log polynomial expansion and modified gravity theories, highlights the seriousness of the conflict with dark energy within the flat $\Lambda$CDM model. However, it is still interesting to see if future high-redshift datasets show similar tension with flat $\Lambda$CDM cosmology, given the limited sample size and current quality of the available observational data.