∎

e1e-mail<EMAIL_ADDRESS>e2e-mail<EMAIL_ADDRESS>
11institutetext: Department of Astronomy, Beijing Normal University, Beijing
100875, China 22institutetext: School of Physics and Optoelectronic, Yangtze
University, Jingzhou 434023, China 33institutetext: School of Physics and
Technology, Wuhan University, Wuhan 430072, China

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

Jie Zhengaddr1  Shuo Caoe1,addr1  Yujie Lianaddr1  Tonghua Liuaddr2  Yuting
Liuaddr1  Zong-Hong Zhue2,addr1,addr3

(Received: date / Accepted: date)

###### Abstract

In this paper, we use the latest observations of quasars covering the redshift
range of $0.04<z<5.1$ to investigate a series of Chaplygin gas models as
candidates for unified dark matter and dark energy. Based on different
combinations of available standard candle and standard ruler data, we put
constraints on the generalized Chaplygin gas (GCG), modified Chaplygin gas
(MCG), new generalized Chaplygin gas (NGCG) and viscous generalized Chaplygin
gas (VGCG) models. Moreover, we apply Jensen-Shannon divergence (JSD),
statefinder diagnostics, and the deviance information criterion (DIC) to
distinguish these CG models, based on the statistical results derived from
Markov chain Monte Carlo method. The results show that (1) The standard ruler
data could provide more stringent constraints on the cosmological parameters
of different CG models considered in this analysis. Interestingly, the matter
density parameter $\Omega_{m}$ and Hubble constant $H_{0}$ derived from the
available data are well consistent with those from the Planck 2018 results;
(2) Based on the statistical criteria JSD, our findings demonstrate the well
consistency between Chaplygin gas and the concordance $\Lambda$CDM model.
However, in the framework of statefinder diagnostics, the GCG and NGCG models
cannot be distinguished from $\Lambda$CDM, while MCG and VGCG models show
significant deviation from $\Lambda$CDM in the present epoch; (3) According to
the the statistical criteria DIC, we show that the MCG and VGCG models have
substantial observational support from high-redshfit quasars, whereas the GCG
and NGCG models miss out on the less observational support category but can
not be ruled out.

††journal: Eur. Phys. J. C

##  1 Introduction

The analysis of various observational data, including Type Ia supernovae (SNe
Ia) SNe1 ; SNe2 , baryon acoustic oscillation (BAO) eisenstein20005 , and
cosmic microwave background (CMB) CMB  suggest that the present universe is
undergoing an accelerated phase of expansion acc_universe . Different
suggestions have been put forward to understand this phenomenon, with the
inclusion of exotic dark energy (DE) with negative pressure on the right-hand
side of the Einstein equation. The earliest and simplest model for DE is the
cosmological standard $\Lambda$CDM model, which is in good agreement with
recent observations but embarrassed by the well known coincidence problem and
the fine-tuning problem weinberg1 ; weinberg2 . Meanwhile, the existence of
dark matter (DM), which constitutes the major component of the matter density
in our Universe, is the other primary indicator for the limitation of our
knowledge of physics laws Cao2021 ; 2022A&A…659L…5C . In recent times,
scholars proposed that a fluid called Chaplygin gas could provide a possible
solution to unify two uncharted territories, mimicing the effects of DM in the
early times and DE in the late times kamenshchik2001 . Specially, the
Chaplygin gas obeys the exotic equations of state:

$p=-\frac{A}{\rho},$ (1)

where $p$ and $\rho$ denote the pressure and energy density, respectively. $A$
is a positive constant. Unlike quintessence, which describes the transition
from the quasi-exponential expansion of the early universe to a power law
expansion to explain the present acceleration of the universe but fails to
avoid fine-tuning in explaining the cosmic coincidence problem, the Chaplygin
gas (CG) model provided an alternative way to account for the accelerating
universe by describing a transition from an epoch filled with dust-like matter
to an accelerating universe. Additionally, they predicted that the
cosmological constant was variable. In particular, the Chaplygin gas behaves
as a pressureless fluid at higher redshifts and as a cosmological constant at
lower redshifts, which tends to promote expansion. In addition, the equation
of state of CG shows a well-defined connection with string and brane theories
kamenshchik2001 ; Bento2003 . However, several fatal drawbacks appeared in CG
models. There is unexpected blowup in the DM power spectrum AH32 ; AH33  in
the framework of the CG model, and the CG model is in disagreement with the
observations, such as Type Ia supernovae GCG_SNe1 ; GCG_SNe2 ; GCG_SNe3 ,
X-ray gas mass fraction of clusters zhu2004 , Hubble parameter-redshift
dataGCG_hz  and gamma-ray bursts GCG_Gamma . Therefore, generalized Chaplygin
gas (GCG) model was proposed Amendola2003 ; Bento2003 , which is capable of
explaining the background dynamics of the early and late universe and is in
good agreement with recent observations. The effective equation of state of
GCG, given by $p=\alpha\rho$, proves the evolution of a universe evolving from
a phase dominated by non-relativistic matter to a phase dominated by a
cosmological constant through an intermediate period. There are some
undesirable features of the GCG power spectrum caused by adiabatic pressure
perturbation, which is produced from a nonzero $\alpha$ Amendola2003 ;
Thakur2019 . As a result, MCG2002 ; MCG2004  proposed the “modified” Chaplygin
gas (MCG) model, which considered an interpolation between standard fluids at
high energy densities and Chaplygin gas fluids at lower energy densities.
Another generalization is dubbed new generalized Chaplygin gas model (NGCG),
which was proposed by NGCG2006 . Since the equation of state of dark energy
still cannot be determined exactly, they argued that the GCG model could be
accommodated to any possible X-type dark energy with constant $\omega$, dual
to an interacting XCDM parametrization scenario. In the framework of the NGCG
model, it is not only described by Chaplygin gas fluid but also exhibits dust-
like matter in the early universe and X-type dark energy in the late universe.
Up to now, the nature of dark energy and dark matter is still unknown. It is
reasonable to consider other forms of dark energy models or further generalize
the GCG model. For instance, VGCG2006  considered a phenomenological model
that consists of viscous effects and the features of GCG, dubbed viscous
generalized Chaplygin gas (VGCG), which is able to eliminate the problems
raised by only dissipative fluids and explain the dynamics of the universe.

With so many GG cosmologies proposed in the literature, it is rewarding to
determine which model is strongly supported by the currently available
astrophysical probes. There are two general types of distance indicators at
present: standard candles (SNe Ia and quasars), which are related to the
luminosity distance $D_{L}(z)$ and standard rulers (BAO and CMB) that usually
provide information on the large scale of the Universe. In this work, we adopt
two different catalogs of data, standard candles and standard rulers, to
determine how different samples affect the estimation of cosmological
parameters. Here, we turn to a new standard candle compilation of 1598 quasars
from X-ray and UV flux measurements with a redshift range $0.036\leq z\leq
5.1003$ Risaliti2015 , which has become an effective probe to investigate
different cosmological parameters UV5 ; Lian2021 ; xubing2021 ; khadka2021
especially the cosmic curvature $\Omega_{k}$ UV2 ; UV3 , and the cosmic
distance duality relation UV1 ; liutonghua2020  in the early universe ($z\sim
5$). Besides, the newest SNe Ia sample “ Pantheon” consists of 1048 points
spanning a redshift range $0.01\leq z\leq 2.3$ pantheon , is also adopted in
our work as a standard candle. For standard rulers, the angular size from 120
compact radio quasars obtained by very-long baseline interferometry (VLBI)
from caoshuo2017AA ; Cao20017qsoas ; AS3  is taken into consideration covering
the redshift range $0.46\leq z\leq 2.76$, which has also been widely used in
many cosmological analyses, such as the observational constraints on the
interaction between cosmic dark sectors lixiaolei ; AS8 ; AS5 , General
Relatively and modified gravity theories Cao20017qsoas ; AS1 ; xutengpeng2018
; AS6 , the Hubble constant and cosmic curvature AS2 ; qijingzhao2021 .
Additionally, we also adopt 11 BAO data points from BOSS DR12 at
$z_{\textrm{eff}}=0.38,0.51,0.61$ Alam2017 , 6dFGs and SDSS MGS at
$z_{\textrm{eff}}=0.122$ carter2018 , DES Y1 results at
$z_{\textrm{eff}}=0.81$ DES2018 , eBOSS DR14 at $z_{\textrm{eff}}=1.52$
ata2018  and $z_{\textrm{eff}}=2.34$ dsa2019 . Specially, introducing quasar
measurements to constrain cosmological parameters is beneficial for studying
the evolution of cosmological models at higher redshifts Lian:2021tca ; AS7 ;
liutonghua2021 .

In this paper, we focus on standard candles and rulers to constrain four
Chaplygin gas cosmological models with the goal of investigating the
difference between standard candles and standard rulers and distinguishing
these Chaplygin gas models by statistical analysis. This paper is organized as
follows. In Section 2, we briefly introduce the basic equations of
cosmological models, including GCG, MCG, NGCG, and VGCG. In Section 3, we
describe the observational data adopted in this work and perform a Markov
chain Monte Carlo (MCMC) analysis using different data sets. The results from
observational constraints and the corresponding analysis are displayed in
Section 4, as well as some statistical techniques of model comparison
presented in Section 5. Finally, our conclusions are summarized in Section 6.

##  2 Chaplygin gas cosmologies

In this section, we give a description of four types of Chaplygin gas models
in a spatially flat universe, including GCG, MCG, NGCG, and VGCG models.
Moreover, to obtain stringent constraints on key cosmological parameters, we
use the prior on the baryon density parameter $\Omega_{b}$ and radiation
density parameter $\Omega_{r}$ from planck2018result .

###  2.1 GCG model

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

$p_{\mathrm{gcg}}=-\frac{A}{\rho_{\mathrm{gcg}}^{\alpha}},$ (2)

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

$\rho_{\mathrm{gcg}}(a)=\rho_{\mathrm{gcg}0}\left(A_{\mathrm{s}}+\frac{1-A_{\mathrm{s}}}{a^{3(1+\alpha)}}\right)^{\frac{1}{1+\alpha}},$
(3)

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

$A_{s}=1-\left(\frac{\Omega_{m}-\Omega_{b}}{1-\Omega_{b}}\right)^{1+\alpha}.$
(4)

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

$\displaystyle E^{2}(z)$ $\displaystyle=$
$\displaystyle\Omega_{\mathrm{b}}(1+z)^{3}+\Omega_{\mathrm{r}}(1+z)^{4}+$ (5)
$\displaystyle\left(1-\Omega_{\mathrm{b}}-\Omega_{\mathrm{r}}\right)\left(A_{\mathrm{s}}+\left(1-A_{\mathrm{s}}\right)(1+z)^{3(1+\beta)}\right)^{\frac{1}{1+\beta}}.$

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

###  2.2 MCG model

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,

$p_{\mathrm{mcg}}=B\rho_{\mathrm{mcg}}-\frac{A}{\rho_{\mathrm{mcg}}^{\alpha}},$
(6)

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

$\rho_{\mathrm{mcg}}=\rho_{\mathrm{mcg}0}\left[A_{s}+\left(1-A_{s}\right)a^{-3(1+B)(1+\alpha)}\right]^{\frac{1}{1+\alpha}},$
(7)

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

$\displaystyle E^{2}(z)$ $\displaystyle=$
$\displaystyle\Omega_{b}(1+z)^{3}+\Omega_{r}(1+z)^{4}+(1-\Omega_{b}-\Omega_{r})\times$
(8)
$\displaystyle[A_{s}+(1-A_{s})(1+z)^{3(1+B)(1+\alpha)}]^{\frac{1}{1+\alpha}}.$

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

###  2.3 NGCG model

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,

$p_{\mathrm{ngcg}}=-\frac{\tilde{A}(a)}{\rho_{\mathrm{ngcg}}^{\alpha}},$ (9)

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

$\rho_{\mathrm{ngcg}}=\rho_{\mathrm{ngcg}0}a^{-3}\left[1-A_{s}+A_{s}a^{-3w_{\mathrm{de}}(1+\alpha)}\right]^{\frac{1}{1+a}},$
(10)

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,

$\displaystyle E^{2}(z)$ $\displaystyle=$
$\displaystyle\Omega_{\mathrm{b}}(1+z)^{3}+\Omega_{\mathrm{r}}(1+z)^{4}+(1-\Omega_{\mathrm{b}}-\Omega_{\mathrm{r}})(1+z)^{3}$
(11)
$\displaystyle\times[1-\frac{1-\Omega_{m}}{1-\Omega_{\mathrm{b}}-\Omega_{\mathrm{r}}}(1-(1+z)^{3w(1+\alpha)})]^{\frac{1}{1+\alpha}}.$

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

###  2.4 VGCG model

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

$p_{\mathrm{vgcg}}=-A/\rho_{\mathrm{vgcg}}^{\alpha}-\sqrt{3}\zeta\rho_{\mathrm{vgcg}}.$
(12)

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,

$\displaystyle\rho_{\mathrm{vgcg}}$ $\displaystyle=$
$\displaystyle\rho_{\mathrm{vgcg}0}[\frac{B_{s}}{1-\sqrt{3}\zeta}+(1-\frac{B_{s}}{1-\sqrt{3}\zeta})\times$
(13) $\displaystyle a^{-3(1+\alpha)(1-\sqrt{3}\zeta)}]^{\frac{1}{1+\alpha}},$

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

$\displaystyle E^{2}(z)$ $\displaystyle=$
$\displaystyle\Omega_{b}(1+z)^{3}+\Omega_{r}(1+z)^{4}+$ (14)
$\displaystyle(1-\Omega_{b}-\Omega_{r})\times[\frac{B_{s}}{1-\sqrt{3}\zeta}+$
$\displaystyle(1-\frac{B_{s}}{1-\sqrt{3}\zeta})(1+z)^{3(1+\alpha)(1-\sqrt{3}\zeta)}]^{\frac{1}{1+\alpha}}.$

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

##  3 Cosmological observations

z | measurement | value | ref
---|---|---|---
0.38 | $D_{M}\left(r_{s,\mathrm{fid}}/r_{s}\right)$ | 1512.39 | Alam2017
0.38 | $H(z)(r_{s}/r_{s,fid})$ | 81.2087 | Alam2017
0.51 | $D_{M}(r_{s,fid}/r_{s})$ | 1975.22 | Alam2017
0.51 | $H(z)(r_{s}/r_{s,fid})$ | 90.9029 | Alam2017
0.61 | $D_{M}(r_{s,fid}/r_{s})$ | 2306.08 | Alam2017
0.61 | $H(z)(r_{s}/r_{s,fid})$ | 98.9647 | Alam2017
0.122 | $D_{V}(r_{s,fid}/r_{s})$ | $539\pm 17$ | carter2018
0.81 | $D_{A}/r_{s}$ | $10.75\pm 0.43$ | DES2018
1.52 | $D_{V}(r_{s,fid}/r_{s})$ | $3843\pm 147$ | ata2018
2.34 | $D_{H}/r_{s}$ | 8.86 | dsa2019
2.34 | $D_{M}/r_{s}$ | 37.41 | dsa2019
Table 1: The newest observations of BAO used in this analysis. Figure 1: The
redshift distribution of the SNe Ia, quasars, and BAO measurements.

In this section, we use three catalogs to constrain cosmological models: (1) a
standard candle combination of quasars from X-ray and UV flux measurements and
SNe Ia samples; (2) a standard ruler set of intermediate-luminosity radio
quasars and BAO data listed in Table 1; and (3) a combination of standard
candles and rulers. Additionally, in Fig. 1, we display the redshift
distributions of standard candles and rulers.

###  3.1 QSO[X-ray and UV flux]

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,

$\log(L_{X})=\gamma\log(L_{UV})+\beta,$ (15)

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

$\log\left(F_{X}\right)=\gamma\log\left(F_{UV}\right)+2(\gamma-1)\log\left(D_{L}\right)+(\gamma-1)\log(4\pi)+\beta,$
(16)

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

$D_{L}(z,\hat{p})=\frac{c(1+z)}{H_{0}}\int_{0}^{z}\frac{dz^{\prime}}{E\left(z^{\prime}\right)},$
(17)

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

$\mathcal{L}_{F_{X}}=-\frac{1}{2}\sum_{i=1}^{N}\left[\frac{\left[\log\left(F_{X,i}^{\mathrm{obs}}\right)-\log\left(F_{X,i}^{\mathrm{th}}\right)\right]^{2}}{s_{i}^{2}}+\ln\left(2\pi
s_{i}^{2}\right)\right],$ (18)

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

$\chi_{F_{x},min}^{2}=-2\ln(LF)_{\min}-\sum_{i=1}^{1598}\ln\left(2\pi\left(\sigma_{i}^{2}+\delta_{best-
fit}^{2}\right)\right).$ (19)

###  3.2 SNe Ia

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

$\displaystyle Y^{obs}$ $\displaystyle=$ $\displaystyle m_{B}+K$ (20)
$\displaystyle=$ $\displaystyle\mu+M,$

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,

$\displaystyle Y^{th}$ $\displaystyle=$ $\displaystyle 5\log(D_{L})+25+M$ (21)
$\displaystyle=$ $\displaystyle 5\log[(1+z)D(z)]+Y_{0},$

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

$\chi^{2}_{\textrm{SNe}}={\Delta\overrightarrow{Y}}^{T}\cdot\textbf{C}^{-1}\cdot{\Delta\overrightarrow{Y}},$
(22)

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 .

###  3.3 QSO[AS]

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

$\theta_{th}(z)=\frac{\ell_{m}}{D_{A}(z)},$ (23)

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

$D_{A}(z)=\frac{D_{L}(z)}{(1+z)^{2}},$ (24)

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

$\chi_{\textrm{QSO}}^{2}=\sum_{i}^{120}\frac{\left(\theta\left(z_{i};\mathbf{p}\right)-\theta_{i}^{obs}\right)^{2}}{\sigma_{i}^{2}}.$
(25)

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

###  3.4 BAO

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

$D_{C}=\frac{c}{H_{0}}\int_{0}^{z}\frac{dz^{\prime}}{E\left(z^{\prime}\right)},$
(26)

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

$D_{V}(z)=\left[\frac{cz}{H_{0}}\frac{D_{M}^{2}(z)}{E(z)}\right]^{1/3}.$ (27)

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

$\chi_{\mathrm{BAO}}^{2}(p)=\sum_{i=1}^{N}\frac{\left[A_{\mathrm{th}}\left(p,z_{i}\right)-A_{\mathrm{obs}}\left(z_{i}\right)\right]^{2}}{\sigma_{i}^{2}},$
(28)

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

$\chi_{\mathrm{BAO}}^{2}(p)=\left[\vec{A}_{\mathrm{th}}(p)-\vec{A}_{\mathrm{obs}}\right]^{T}C^{-1}\left[\vec{A}_{\mathrm{th}}(p)-\vec{A}_{\mathrm{obs}}\right],$
(29)

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

$\mathcal{L}(p)=e^{-\frac{\chi(p)^{2}}{2}},$ (30)

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

##  4 Results and discussion

Table 2: The best-fit values and 68% confidence limits for the CG cosmological
parameters in each model (GCG, MCG, NGCG, and VGCG) and data set (QSO[XUV]+SNe
Ia, QSO[AS]+BAO, and QSO[XUV]+SNe Ia+QSO[AS]+BAO).

Model | Data | $\Omega_{m}$ | $A_{s}$ | $\alpha$ |  $H_{0}$ (km/s/Mpc) |
---|---|---|---|---|---|---
GCG | QSO[XUV]+SNe Ia | $0.53^{+0.53}_{-0.29}$ | $0.78\pm 0.06$ | $0.46^{+0.57}_{-0.42}$ | $68.27^{+6.98}_{-4.76}$ |
| QSO[AS]+BAO | $0.33\pm 0.02$ | $0.60\pm 0.10$ | $-0.33^{+0.27}_{-0.24}$ | $65.81^{+2.26}_{-2.28}$ |
| Combination | $0.31\pm 0.01$ | $0.73\pm 0.04$ | $0.03^{+0.17}_{-0.14}$ | $68.26^{+1.18}_{-1.08}$ |
Model | Data | $\Omega_{m}$ | $A_{s}$ | $B$ | $\alpha$ |  $H_{0}$ (km/s/Mpc)
MCG | QSO[XUV]+SNe Ia | $0.47^{+0.36}_{-0.27}$ | $0.81^{+0.06}_{-0.09}$ | $0.12^{+0.26}_{-0.21}$ | $0.20^{+0.58}_{-0.39}$ | $68.28^{+7.23}_{-3.48}$
| QSO[AS]+BAO | $0.33\pm 0.02$ | $0.61^{+0.10}_{-0.14}$ | $-0.12^{+0.18}_{-0.09}$ | $0.05^{+0.87}_{-0.52}$ | $66.27^{+2.01}_{-2.30}$
| Combination | $0.31\pm 0.01$ | $0.73^{+0.04}_{-0.06}$ | $-0.14^{+0.13}_{-0.06}$ | $0.71^{+0.78}_{-0.71}$ | $68.09^{+1.11}_{-1.06}$
Model | Data | $\Omega_{m}$ | $\omega$ | $\alpha$ |  $H_{0}$ (km/s/Mpc) |
NGCG | QSO[XUV]+SNe Ia | $0.30^{+0.16}_{-0.14}$ | $-1.10^{+0.24}_{-0.39}$ | $0.23^{+0.89}_{-0.53}$ | $68.50^{+7.19}_{-5.21}$ |
| QSO[AS]+BAO | $0.34\pm 0.02$ | $-0.79^{+0.13}_{-0.14}$ | $-0.12^{+0.12}_{-0.17}$ | $65.16^{+2.38}_{-2.18}$ |
| Combination | $0.31\pm 0.01$ | $-1.01^{+0.05}_{-0.06}$ | $0.01^{+0.09}_{-0.08}$ | $68.34^{+1.19}_{-1.09}$ |
Model | Data | $\Omega_{m}$ | $B_{s}$ | $\alpha$ | $\zeta$ |  $H_{0}$ (km/s/Mpc)
VGCG | QSO[XUV]+SNe Ia | $0.47^{+0.36}_{-0.31}$ | $0.82^{+0.13}_{-0.19}$ | $0.41^{+0.74}_{-0.49}$ | $-0.02^{+0.10}_{-0.08}$ | $68.58^{+6.82}_{-5.16}$
| QSO[AS]+BAO | $0.33\pm 0.02$ | $0.55^{+0.17}_{-0.16}$ | $0.08^{+0.99}_{-0.57}$ | $0.07^{+0.06}_{-0.12}$ | $66.32^{+2.16}_{-2.42}$
| Combination | $0.31\pm 0.01$ | $0.64^{+0.10}_{-0.07}$ | $0.61^{+0.82}_{-0.66}$ | $0.07^{+0.04}_{-0.08}$ | $68.21^{+1.19}_{-1.04}$

Table 3: The values of DIC and their differences for CG and $\Lambda$CDM
cosmologies. The Jensen-Shannon divergence between $\Lambda$CDM and other
cosmological models is also calculated with respect to $\Omega_{m}$ and
$H_{0}$.

Data | Model | DIC |  $\Delta$DIC | $D_{JS}(\Omega_{m})$ | $D_{JS}(H_{0})$
---|---|---|---|---|---
QSO[XUV]+SNe Ia |  $\Lambda$CDM | 2632.34 | 0 | 0 | 0
| GCG | 2635.10 | 2.76 | 0.721 | 0.124
| MCG | 2625.76 | -6.58 | 0.722 | 0.510
| NGCG | 2636.31 | 3.97 | 0.681 | 0.083
| VGCG | 2638.69 | 6.36 | 0.727 | 0.092
QSO[AS]+BAO |  $\Lambda$CDM | 616.42 | 0 | 0 | 0
| GCG | 618.29 | 1.87 | 0.296 | 0.710
| MCG | 608.04 | -8.38 | 0.300 | 0.662
| NGCG | 617.97 | 1.55 | 0.468 | 0.927
| VGCG | 605.54 | -10.88 | 0.319 | 0.655
Combination |  $\Lambda$CDM | 3245.86 | 0 | 0 | 0
| GCG | 3252.10 | 6.24 | 0.074 | 0.286
| MCG | 3246.42 | 0.56 | 0.132 | 0.275
| NGCG | 3251.15 | 5.29 | 0.077 | 0.288
| VGCG | 3239.22 | -6.64 | 0.131 | 0.277

Figure 2: The 1D and 2D probability distributions of model parameters in the
GCG model, based on the QSO[XUV]+SNe Ia(green), QSO[AS]+BAO (blue), and a
joint sample (red). The contours correspond to 68% and 95% confidence levels.
The grey line indicates the values of the model parameters that can be
recovered to the $\Lambda$CDM scenario, with the fiducial value of
$\Omega_{m}=0.30$, $\omega=-1$, $\alpha=0$ and $H_{0}$=70 km/s/Mpc. Figure 3:
The 1D and 2D probability distributions of model parameters in the MCG model.
Figure 4: The 1D and 2D probability distributions of model parameters in the
NGCG model. Figure 5: The 1D and 2D probability distributions of model
parameters in the VGCG model.

In this section, we display and discuss the constraint results of the
cosmological parameters by using the standard candles and rulers data. It
shows that how the different types of observational data could inflect the
constraints of cosmological parameter estimation.

###  4.1 GCG model

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 $\Omega_{m}$.

###  4.2 MCG model

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.

###  4.3 NGCG 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.

###  4.4 VGCG model

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.

##  5 Statistical analysis

The statistical analysis is essential to diagnose the different models. Hence,
we apply the Jensen-Shannon Divergence, statefinder diagnostic and the
deviance information criterion. In this section, we compare these models and
discuss how strongly are they favored by the observational data sets.

###  5.1 Jensen-Shannon Divergence

Figure 6: The posterior distributions of $\Omega_{m}$ for the GCG, MCG, NGCG,
VGCG and $\Lambda$CDM models, with the standard candles, standard rulers and
combination data from the top to the bottom. We adopt the posterior
distributions of $\Omega_{m}$ from Table 2.

Figure 7: The posterior distributions of $H_{0}$ for the GCG, MCG, NGCG, VGCG
and $\Lambda$CDM models, with the standard candles, standard rulers and
combination data from the top to the bottom. We adopt the posterior
distributions of $H_{0}$ from Table 2.

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

$D_{JS}(p\mid q)=\frac{1}{2}\left[D_{KL}(p(x)\mid s)+D_{KL}(q(x)\mid
s)\right],$ (31)

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

$D_{KL}(p\mid q)=\int p(x)\log_{2}\left(\frac{p(x)}{q(x)}\right)dx.$ (32)

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.

###  5.2 Statefinder Diagnostic

Figure 8: The evolution of the statefinder pair $(r,s)$ for different
cosmological models. The cyan diamond point at $(r,s)=(1,0)$ indicates the
$\Lambda$CDM model, and the other diamond point on each curve denotes the
present value of the statefinder pair $(r,s)$ for the GCG, MCG, NGCG, and VGCG
models. The model parameters adopted in statefinder diagnostic are from the
combination of QSO[XUV], SNe Ia, QSO[AS] and BAO in Table 2. Figure 9: The
same as Fig. 8, but for the evolution of the pair $(r,q)$.

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

$H=\frac{\dot{a}}{a},q=-\frac{\ddot{a}}{aH^{2}}=-\frac{a\ddot{a}}{\dot{a}^{2}},$
(33)

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

$r=\frac{\dot{a}}{aH^{3}},\quad s=\frac{r-1}{3(q-1/2)},$ (34)

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

$r(z)=1-2\frac{E^{\prime}(z)}{E(z)}(1+z)+\left[\frac{E^{\prime\prime}(z)}{E(z)}+\left(\frac{E^{\prime}(z)}{E(z)}\right)^{2}\right](1+z)^{2},$
(35)

and

$s(z)=\frac{r(z)-1}{3(q(z)-1/2)},$ (36)

and

$q(z)=\frac{E^{\prime}(z)}{E(z)}(1+z)-1.$ (37)

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.

###  5.3 Model selection statistic

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

$\mathrm{DIC}\equiv D(\bar{\theta})+2p_{D},$ (38)

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.

##  6 Conclusions

In this paper, we investigated the constraint ability of standard candles
(QSO[XUV]+SNe Ia) and standard rulers (QSO[AS]+BAO) on a series of Chaplygin
gas models, including the GCG model, MCG model, NGCG model and VGCG model.
These Chaplygin gas models are considered as important candidate models that
regard dark energy and dark matter as a unification. The first part is devoted
to performing MCMC statistical analysis to confront the models with the most
recent observations. The second part is dedicated to comparing the agreement
between the $\Lambda$CDM model and the other four models using JSD, exploring
the evolution of cosmological and cosmographical parameters with the
assistance of statefinder diagnostic analysis and examining the viability of
four nonstandard models by information criteria such as DIC. Here, we
summarize our main conclusions in more detail:

(i) It is intriguing that the value of $\Omega_{m}$ is noticeably larger from
the standard candle data than that from other measurements. Such discrepancy
is caused by the QSO X-ray and UV flux data, which favors the higher
$\Omega_{m}\sim 0.5-0.6$ discussed in Risaliti2015 ; khadka2020_1  at high
redshifts $z\sim 2-5$. Therefore, the quasar data at high redshifts can cast a
new light on investigating the accelerating universe. Considering the Hubble
constant, it is noteworthy that the constraint results from standard candles
and the combination sample suggest central values on $H_{0}$ between the value
measured by the Planck CMB measurements and local $H_{0}$ measurements,
possibly alleviating the tension between these measurements. In addition, it
is remarkable that although we are using data based on local measurements,
such as SNe Ia, which favors the local value (SH0ES’s result), it does not
play a role in constraining the Hubble constant caused by the marginalization
of the constant term $Y_{0}$ we adopted. Hence, the QSO data from X-ray and UV
flux measurement prefers the value of $H_{0}$ from the Planck 2018 results.

(ii) Most CG models include the concordance $\Lambda$CDM model as a special
case corresponding to certain values of their parameters, such as the
parameter $\alpha$ in the GCG model and the parameters $B$ and $\alpha$ in the
MCG model. For standard ruler data, the GCG model and NGCG model are generally
inconsistent with the cosmological constant case within a 68% CL, while the
MCG model and NGCG model disagree with the $\Lambda$CDM model by the
combination sample at a 68% CL. In the previous studies, they concluded that
the CG model is ruled out by recent observations. In our work, considering
standard candle data, the CG model is accepted in all cases. The CG model is
favored in the framework of the MCG and VGCG models from standard ruler data
as well as combined sample. This is because that the inclusion of QSOs from
X-ray and UV measurements and QSOs from VLBI could provide more information
from the early universe. Hence, it is expected that these selected quasars
could be considered additional probes in the future.

(iii) To evaluate the similarity between $\Lambda$CDM and other CG models, we
adopt the JSD in this paper. For standard candle data, the posterior
distributions of $\Omega_{m}$ from four nonstandard models are distant from
the $\Lambda$CDM model, while the NGCG model is in good agreement with the
$\Lambda$CDM model in terms of the JSD value of $H_{0}$. For standard ruler
data, the NGCG model shows a larger distance from the $\Lambda$CDM model
according to the values of JSD from the posterior distribution of $\Omega_{m}$
and $H_{0}$. The posterior distributions of $\Omega_{m}$ and $H_{0}$ from the
MCG model and VGCG model are in good agreement with the $\Lambda$CDM model
from the combined standard candle and ruler data. Based on the best fits
obtained with the combination sample, we apply the statefinder diagnostic to
discriminate the dynamic behaviors of the four CG models. The GCG model and
NGCG model evolve similarly to the $\Lambda$CDM model, but the NGCG model
could stray from the $\Lambda$CDM model in the near future. Clearly, the MCG
model and VGCG model exhibit significantly different evolutionary trajectories
to the $\Lambda$CDM model; however, they approach to $\Lambda$CDM in the
future. According to the DIC criterion, VGCG model is more favored by
observations; on the other hand, the GCG and NGCG models are punished by all
catalogs of data. In addition, the MCG model is slightly supported by standard
candles data.

In conclusion, we find that the VGCG model and MCG model could be strong
candidates for investigating the accelerating universe. Moreover, $H_{0}$
tension will be alleviated with VGCG model and MCG model and these models can
satisfy the combination of standard candle and standard ruler measurements
with $\Delta$DIC=-6.64 and $\Delta$DIC=-0.56 compared with $\Lambda$CDM model.
In addition, it is distinctive that the CG model cannot be ruled out by high
redshift observations, such as the compilation of 1598 QSO X-ray and UV
measurements. Therefore, extending the cosmological analysis with high-
redshift data should be critical in distinguishing between different CG models
that are degenerate at low redshifts. As a result, it is promising that future
precise high redshift data (i.e., gravitational wave data) will provide
stronger evidence to judge whether dark energy and dark matter are unified and
to understand the nature of the accelerating universe. There are several
issues we do not consider in this paper and which remain to be addressed in
the future analysis. One general concern is given by the fact that we have
considered only the 0th order cosmology and Chaplygin gas models might have
instabilities at the perturbation level. Some work has also studied the
behavior of the particular case of generalized Chaplygin gas models in the
matter power spectrum. As worked out in detail by PhysRevD.69.123524 , the
oscillations or exponential blowup of power spectrum, which are inconsistent
with the observations of the 2df galaxy redshift survey, contribute to the
ruling out of GCG models in 1st order cosmology (the growth of linear
perturbations). Now precision data of redshift-space distortions (RSD)
growthdata11 ; Arman2018 ; 201109516 , the rms mass fluctuation
$\sigma_{8}$(z) inferred from galaxy and Ly-$\alpha$ surveys growthdata12 ;
growthdata13 ; Cuceu2021 , weak lensing statistics growthdata14 , baryon
acoustic oscillations growthdata15 ; growthdata15_2 , X-ray luminous galaxy
clusters growthdata16 , and Integrated Sachs-Wolfe (ISW) effect growthdata17
are gradually allowing us to determine the linear growth function that are
related to perturbations. In the future analysis we will take a further step
in this direction, focusing on more stringent constraints on the perturbative
behaviors of a series of Chaplygin gas models.

## Acknowledgements

This work was supported by National Key R&D Program of China No.
2017YFA0402600; the National Natural Science Foundation of China under Grants
Nos. 12021003, 11690023, and 11920101003; the Strategic Priority Research
Program of the Chinese Academy of Sciences, Grant No. XDB23000000; and the
Interdiscipline Research Funds of Beijing Normal University.

## DATA AVAILABILITY STATEMENTS

The data underlying this article will be shared on reasonable request to the
corresponding author.

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