# Constraining on the non-standard cosmological models combining the
observations of high-redshift quasars and BAO

Ziqiang Liu  School of Physics and Optoelectronic, Yangtze University,
Jingzhou 434023, China  Tonghua Liu<EMAIL_ADDRESS>School of
Physics and Optoelectronic, Yangtze University, Jingzhou 434023, China  Xinyi
Zhong  School of Physics and Optoelectronic, Yangtze University, Jingzhou
434023, China  Yifei Xu  School of Physics and Optoelectronic, Yangtze
University, Jingzhou 434023, China  Xiaogang Zheng<EMAIL_ADDRESS>School of Electrical and Electronic Engineering, Wuhan Polytechnic University,
Wuhan 430023, China

###### Abstract

In this work, we studied four types of cosmological models with different
mechanisms driving the accelerated expansion of the universe, include
Braneworld models, Chaplygin Gas models, Emergent Dark Energy models, and
cosmological torsion models. Considering that the dynamics of these models at
low redshifts are very similar and difficult to distinguish, we used the
latest and largest UV and X-ray measurements of quasars (QSOs) observations
covering the range of redshift $0.009<z<7.5$. However, the high intrinsic
dispersion of this sample and the degeneracy between cosmological model
parameters, we added 2D-BAO and 3D-BAO datasets to help us constrain the
parameters of these cosmological models. Our results suggest that standard
cold dark matter scenario may not be the best cosmological model preferred by
the high-redshift observations. The Generalized Chaplygin Gas (GCG) and
cosmological constant plus torsion (named Case II) models perform best by
Akaike Information Criterion (AIC), but the $\Lambda$CDM is the best
cosmological model preferred by Bayesian Information Criterion (BIC). Our work
also supports that the Phenomenologically Emergent Dark Energy and
cosmological torsion models may alleviate the Hubble tension, the reported
value of the Hubble constant obtained from QSO+BAO datasets combination lies
between Planck 2018 observations and local measurements from the SH0ES
collaboration, while other cosmological models all support that the Hubble
constant tends to be closer to recent Planck 2018 results, but these model are
penalized by information criterion.

##  I Introduction

In the end of the 20th century, two international supernova research groups,
High Redshift Supernova Team (HST) and Supernova Cosmology Project (SCP),
discovered the accelerating expansion of the distant supernova type Ia (SN Ia)
(Perlmutter et al., 1999; Riess et al., 1998). In subsequent years, other
cosmological observations such as cosmic microwave background radiation (CMB)
(Spergel et al., 2003), Baryon Acoustic Oscillations (BAO) (Eisenstein et al.,
2005; Percival et al., 2007), largescale structure (Tegmark et al., 2004) had
confirmed this phenomenon. Many cosmological models have been proposed in
order to explain the accelerating expansion of the universe, such as Lambda
Cold Dark Matter ($\Lambda$CDM) model, which is consistent with many
cosmological observations (Alam et al., 2017; Allen et al., 2008; Cao et al.,
2012; Farooq et al., 2017; Scolnic et al., 2018), is called standard
cosmological model. However, this model still faces many theoretical problems
such as the well-known fine-tuning and coincidence problems (Dalal et al.,
2001; Weinberg, 1989). With the rapid development of observation technology,
the astronomical data obtained are becoming more and more accurate, more
problems that $\Lambda$CDM model can not solve are gradually emerging, such as
Hubble tension (Freedman, 2017) and the cosmic curvature problems (Di
Valentino et al., 2020, 2021). Therefore, there has been increasing attention
to other cosmological models beyond $\Lambda$CDM, including dynamic dark
energy models (Boisseau et al., 2000; Kamenshchik et al., 2001), interacting
dark energy model (Amendola, 2000; Caldera-Cabral et al., 2009) and scalar
field theories (Caldwell et al., 2005; Chen et al., 2011; Zlatev et al.,
1999). Moreover, a Phenomenologically Emergent Dark Energy (PEDE) presented by
Li et al. has obtained a great attention (Li et al., 2019), which demonstrated
its potential in addressing the Hubble constant problem. Furthermore,
modifications to General Relativity (GR) theory (Tsujikawa, 2010) could also
provide a useful way to deal with the cosmological constant problem and
explain the late-time acceleration of the universe without dark energy. For
example, braneworld Dvali-Gabadadze-Porrati (DGP) (Dvali et al., 2000;
Sollerman et al., 2009) model and modified polytropic Cardassian (MPC) model
(Magana et al., 2015; Wang et al., 2003), which suppose our universe is
embedded in a higher dimension of space-time to modify gravity. Similarly, the
field equation can be allowed to extend the general theory of relativity to be
greater than the second order, like $f(R)$ gravity (Chiba et al., 2003;
Sotiriou, 2010). In this paper, we will study four kinds of typical
cosmological models that work in the framework of Friedman-Lemaître-Robertson-
Walker metric, including braneworld models: DGP and a phenomenological
extension DGP model, $\alpha$ dark energy ($\alpha$DE) (Dvali et al., 2003),
Generalized Chaplygin Gas (GCG) model (Bento et al., 2002), a kind of
dynamical dark energy model, in which the dark energy density decreases with
time, and New Generalized Chaplygin Gas (NGCG) model (Zhang et al., 2006),
Emergent Dark Energy models, here we consider two models: Phenomenologically
Emergent Dark Energy (PEDE) (Li et al., 2019) and Generalised Emergent Dark
Energy (GEDE) (Li et al., 2020), cosmological models with torsion (vanishing
dark energy term called Case I and containing the dark energy term called Case
II) (Pereira et al., 2022). However, given the number and precision of
previous observational datasets, including the Hubble parameter (Wu & Yu,
2007) and SN Ia (Nesseris et al., 2005; Scolnic et al., 2018; Suzuki et al.,
2012), the vast majority of cosmological models are not excluded by the data
at low redshifts. This is because their dynamics in the low redshift range are
so similar that they are difficult to distinguish. Nevertheless, their
dynamics are quite different at high redshifts. Therefore, there is an urgent
need for high-redshift cosmological probes, such as quasars, to perform
observational tests and place constraints on the models.

Quasars (QSOs) are among the brightest sources of stars in the universe.
However, quasars do not have a normalizable luminosity, and their spectral
properties and photometric correlation are weak, with large dispersion and
bias. Nevertheless, if the quasar sample is large enough, it can be useful for
cosmological measurements in principle. Thanks to the advancements and
progress made in modern science and technology, the number of observed quasars
has increased by an order of magnitude. For example, quasar samples of UV and
X-ray radiation obtained from multiple optical band ROSAT, Chandra, Sloan
Digital Sky Survey (SDSS), and XMM-Newton Serendipitous Source Catalogue (XMM-
NSSC) sky surveys (Just et al., 2007; Lusso et al., 2010; Steffen et al.,
2006), and samples of quasars of milliarc second angular size in the radio
band obtained from the very-long baseline interferometry (VLBI) observations
(Gurvits, 1994; Kellermann, 1993). Quasars have also been utilized as standard
candles whose standardization relies on the linear relationship between the
logarithms of their ultraviolet (UV) and X-ray luminosities (Geng et al.,
2020; Khadka & Ratra, 2020; Liu et al., 2020a, b, 2020; Lusso et al., 2019,
2020; Risaliti & Lusso, 2017, 2019; Salvestrini et al., 2019). In general, the
advantage of quasar measurements compared to other conventional cosmological
probes lies in their larger redshift range. This could be beneficial for
exploring the behaviour of non-standard cosmological models at high redshift
and could provide an important addition to other astrophysical observations,
demonstrating the potential of quasars as an additional cosmological probe
(Zheng et al., 2021).

In the previous work Lian et al. (2021), the authors indicated that the
results show that using QSO data alone are not able to provide tight
constraints on model parameters, which is mainly related to the large
dispersion in QSO data. In addition, they demonstrated that the use of Baryon
Acoustic Oscillation (BAO) data is complementary to the QSO distance
measurements. Thus, in order to get more stringent constraints on cosmological
parameters, we also consider measurements of transversal BAO scale (namely 2-D
BAO) Nunes et al. (2020) which come from public data releases (DR) of the SDSS
Carvalho et al. (2016); Alcaniz et al. (2017); Carvalho et al. (2020); de
Carvalho et al. (2018, 2020), and BAO measurements from the post-reconstructed
power spectra of the BOSS DR12 data (namely 3-D BAO) Alam et al. (2021) which
come from Main Galaxy Sample (MGS), the two BOSS galaxy samples, eBOSS
luminous red galaxies (LRG), and eBOSS emission line galaxies (ELG) Alam et
al. (2017); Howlett et al. (2015); Ross et al. (2015); Bautista et al. (2021);
Gil-Marín et al. (2020); Tamone et al. (2020); de Mattia et al. (2021); Hou et
al. (2021); Neveux et al. (2020). These datasets have been extensively used
for cosmological researches and applications in the literature Staicova
(2023); Dainotti et al. (2023); Calderón et al. (2023); Semenaite et al.
(2023); Staicova & Benisty (2022); Bargiacchi et al. (2023); Cao & Ratra
(2022). For instance, the work Cao & Ratra (2023) also used 3-D BAO data
combined with other lower-redshift data (including QSO angular size, Pantheon+
SNe, Hubble parameter datasets etc.) to constrain non-standard cosmological
models, and showed the complementary role of BAO data. However, there are some
differences between the two sets of BAO data on the model constraining
capabilities exhibited in different cosmological models. Recently, the work
Bernui et al. (2023) found that 2D-BAO measurements generate a little high
$H_{0}$ compared to 3D-BAO measurements in the interacting dark energy model.

Inspired by the above, we combine QSO data of UV and the X-ray flux
measurements and 2D and 3D BAO datasets to constrain the cosmological models
mentioned above, respectively. The main purpose is to test the high-redshift
correspondence between quasar data and the standard $\Lambda$CDM by examining
the performance of these cosmological models at high redshift. We concentrate
on the nonlinear relationship between the UV and X-ray luminosities of quasars
serving as standard candles across a redshift range spanning $0.009<z<7.5$. In
Section 2, we introduce all the cosmological models. We give the methodology
and observation datasets including QSO plus two BAO datasets in Section 3. The
results of constrained cosmological parameter and model comparison criterion
are given in Section 4. Finally, Section 5 summarizes our conclusion.

##  II Cosmology models

The foundation of modern cosmology rests on the fundamental principles of
cosmology, which state that our universe is homogeneous and isotropic on large
scales. The space-time of our universe can be accurately described by the
Friedmann-Lematre-Robertson-Walker (FLRW) metric:

$ds^{2}=dt^{2}-\frac{a(t)^{2}}{1-Kr^{2}}dr^{2}-a(t)^{2}r^{2}d\Omega^{2},$ (1)

where $a(t)$ is the scale factor, and $K$ is dimensionless curvature taking
one of three values $\\{-1,0,1\\}$ corresponding to a close, flat and open
universe, respectively. The cosmic curvature parameter $\rm\Omega_{K}$ is
related to $K$ and the Hubble constant $H_{0}$, as $\rm{\Omega_{K}}$
$=-c^{2}K/a^{2}_{0}H_{0}^{2}$. The comoving distance $D_{C}$ is related to the
evolution of the scale factor

$D_{C}=c\int^{z}_{0}\frac{dz^{\prime}}{H(z^{\prime})},$ (2)

where the redshift $z$ replaces the scale factor via relation $a=1/(1+z)$, the
$c$ represents the speed of light, and the $H(z^{\prime})$ denotes the Hubble
parameter at redshift $z^{\prime}$. However, we cannot actually obtain the
comoving distance from observations. In fact, the cosmological distances we
observe are luminosity or angular diameter distances. The dimensionless
curvature $k=-1,0,+1$ corresponds to open, flat and closed universes,
respectively. With such a metric, the luminosity distance $D_{L}$ and angular
diameter distance $D_{A}$ can be expressed as (Weinberg, 2008)

$\frac{D_{L}}{1+z}={D_{A}}({1+z})=\left\\{\begin{array}[]{lll}\frac{D_{H}}{\sqrt{|\Omega_{\rm
k}|}}S_{k}\left(\sqrt{|\Omega_{\rm
k}|}\int_{0}^{z}\frac{dz^{\prime}}{E(z^{\prime})}\right),\\\
{D_{H}}S_{k}\left(\int_{0}^{z}\frac{dz^{\prime}}{E(z^{\prime})}\right),\\\
\frac{D_{H}}{\sqrt{|\Omega_{\rm k}|}}S_{k}\left(\sqrt{|\Omega_{\rm
k}|}\int_{0}^{z}\frac{dz^{\prime}}{E(z^{\prime})}\right),\\\
\end{array}\right.$ (3)

where $D_{H}=c/H_{0}$ is known as the Hubble distance, the dimensionless
Hubble parameter $E(z)$ is defined as $H(z)/H_{0}$. The curvature parameter
$\Omega_{k}$ is related to $k$ as $\Omega_{k}=-c^{2}k/(a_{0}H_{0})^{2}$, where
$c$ is the speed of light. For convenience, we denoted
$S_{k}(x)=\sin(x),x,\sinh(x)$ for $\Omega_{k}<0,=0,>0$.

The different cosmological theoretical models correspond to different $H(z)$
functions. For the sake of simplicity, we present the $H(z)$ functions of four
cosmological models based on different theories that we aim to investigate in
this paper. These models include Braneworld models (DGP and $\alpha$DE),
Chaplygin Gas models (GCG and NGCG), Emergent Dark Energy models (PEDE and
GEDE), and cosmological models with torsion (vanishing dark energy term called
Case I and containing the dark energy term called Case II). Given that there
is a strong degeneracy between dark energy and the cosmic curvature, and the
latest Planck observations support the flatness of our universe with extremely
high precision, we consider a flat universe throughout this work.

###  II.1 Braneworld models

Derived from the Braneworld theory, the Dvali-Gabadadze-Porrati model (Dvali
et al., 2000) alters gravity to replicate the cosmic acceleration, eliminating
the need for Dark Energy (DE). In this model, we exist on a 4-D membrane
within a higher-dimensional spacetime. Furthermore, gravity leaks into the
bulk at large scales, leading to the accelerated expansion of the Universe (Li
et al., 2013). In this subsection, we focus on two models: DGP and $\alpha$DE.

####  II.1.1 Dvali-Gabadadze-Porrati model

For DGP model, the Friedmann equation is governed by

$E(z)^{2}-\frac{E(z)}{r_{c}}=\Omega_{m}(1+z)^{3},$ (4)

where $r_{c}=1/[H_{0}(1-\Omega_{m})]$ is the crossover scale and $\Omega_{m}$
is the matter density at present. We can directly rewrite the above equation
and get the expansion rate

$H(z)^{2}=H_{0}^{2}\left(\sqrt{\Omega_{m}(1+z)^{3}+\Omega_{rc}}+\sqrt{\Omega_{rc}}\right)^{2},$
(5)

where $\Omega_{rc}=1/(4r_{c}^{2}H_{0}^{2})$ is related to the cosmological
scale. Setting $z=0$ in Eq. (5), the normalization condition can be obtained

$\Omega_{rc}=\frac{(1-\Omega_{m})^{2}}{4},$ (6)

and there are two model parameters $\hat{p}=(\Omega_{m},H_{0})$ to be
constrained.

####  II.1.2 $\alpha$ Dark Energy model

The $\alpha$DE is a phenomenological extension of DGP (Dvali et al., 2003). It
was proposed by Dvali and Turner. By introducing a parameter q, it enables
interpolation between the DGP and the $\Lambda$CDM. In this model, the
Friedmann equation is modified to

$E(z)^{2}-\frac{E(z)^{q}}{r_{c}^{2-q}}=\Omega_{m}(1+z)^{3},$ (7)

where $q$ is a phenomenological parameter, and
$r_{c}=(1-\Omega_{m})^{1/(q-2)}H_{0}^{-1}$. According to this Friedmann
equation, E(z) is determined by the following equation:

$H(z)^{2}=H_{0}^{2}\left(\Omega_{m}(1+z)^{3}+\frac{H(z)^{q}}{H_{0}^{2-q}}(1-\Omega_{m})\right),$
(8)

when $q=1$, it collapses to the DGP, when $q=0$, it collapses to $\Lambda$CDM.
And there are three free parameters $\hat{p}=(\Omega_{m},q,H_{0})$ to be
constrained.

###  II.2 Chaplygin Gas models

The Chaplygin Gas model (Kamenshchik et al., 2001) is a fluid model that is
often viewed as arising from the d-brane theory. As one of the candidates for
dark energy models, the Chaplygin Gas model has been proposed to explain the
cosmic acceleration (Bento et al., 2002; Biesiada et al., 2005; Malekjani et
al., 2011). In this model, dark energy and dark matter are unified through an
exotic equation of state. The universe is filled with the so-called Chaplygin
gas, which is a perfect fluid characterized by the equation of state
$p=-A/\rho$. However, the original Chaplygin Gas model has been excluded by
observational data (Davis et al., 2007). Therefore, we consider two other
models: the Generalized Chaplygin Gas (GCG) model (Bento et al., 2002) and the
New Generalized Chaplygin Gas (NGCG) model (Zhang et al., 2006).

####  II.2.1 Generalized Chaplygin Gas model

The exotic equation of state of GCG (Bento et al., 2002) can be expressed as:

$p_{GCG}=-\frac{A}{\rho_{GCG}^{\xi}},$ (9)

where $A$ is a positive constant. Then, we can get the energy density of GCG:

$\rho_{GCG}=\rho_{0GCG}\left[A_{s}+(1-A_{s})a^{-3(1+\xi)}\right]^{\frac{1}{1+\xi}},$
(10)

where $A_{s}=A/\rho_{0GCG}^{1+\alpha}$ is the present energy density of the
GCG. Note that GCG behaves like a dust matter if $A_{s}=0$, and GCG behaves
like a cosmological constant if $A_{s}=1$. Considering a universe with GCG,
baryon, and radiation, we have

$\begin{split}H(z)^{2}=&H_{0}^{2}\bigg{(}\Omega_{b}(1+z)^{3}+\\\
&(1-\Omega_{b})\left[A_{s}+(1-A_{s})(1+z)^{3(1+\xi)}\right]^{\frac{1}{1+\xi}}\bigg{)},\end{split}$
(11)

where $\Omega_{b}$ is the present density parameter of the baryonic matter. We
adopt $100\Omega_{b}h^{2}=2.166\pm 0.015\pm 0.011$ with $h=H_{0}/100$ as usual
and in the uncertainty budget first term is associated with the deuterium
abundance measurement and the second one$-$with the Big Bang Nucleosynthesis
(BBN) calculation used to get $\Omega_{b0}$ (Cooke et al., 2018). Since the
parameter $A_{s}$ can be expressed by the effective total matter density
$\Omega_{m}$ and the $\xi$ parameter

$A_{s}=1-\left(\frac{\Omega_{m}-\Omega_{b}}{1-\Omega_{b}}\right)^{1+\xi},$
(12)

there are three free parameters $\hat{p}=(\Omega_{m},\xi,H_{0})$ in this
model.

####  II.2.2 New Generalized Chaplygin Gas model

The GCG model can actually be viewed as an interacting model between vacuum
energy and cold dark matter. If one wishes to further extend this model, a
natural idea is to replace the vacuum energy with a more dynamic dark energy.
Thus, the NGCG model was proposed (Zhang et al., 2006). The equation of state
for the NGCG fluid is given by

$p_{NGCG}=-\frac{\widetilde{A}(a)}{\rho_{NGCG}^{\xi}},$ (13)

where $\widetilde{A}(a)$ is a function of the scale factor $a$, and $\xi$ is a
free parameter. The energy density of NGCG can be expressed as

$\rho_{NGCG}=\left[Aa^{-3(1+\xi)(1+\zeta)}+Ba^{-3(1+\xi)}\right]^{\frac{1}{1+\xi}},$
(14)

where A and B are positive constants. The form of the function
$\widetilde{A}(a)$ can be determined to be

$\widetilde{A}(a)=-\zeta Aa^{-3(1+\xi)(1+\zeta)},$ (15)

NGCG reduces to GCG if $\zeta=-1$, reduces to $\omega$CDM if $\xi=0$, and
reduces to $\Lambda$CDM if ($\xi=0,\zeta=-1$). In a flat universe, we have

$\begin{split}&H(z)^{2}=H_{0}^{2}\big{(}\Omega_{b}(1+z)^{3}+\\\
&(1-\Omega_{b})(1+z)^{3}[1-\frac{\Omega_{de}}{1-\Omega_{b}}(1-(1+z)^{3\zeta(1+\xi)})]^{\frac{1}{1+\xi}}\big{)},\end{split}$
(16)

here, we set $\Omega_{de}=1-\Omega_{m}$ and obtain the value of $\Omega_{b}$
from $100\Omega_{b}h^{2}=2.166\pm 0.015\pm 0.011$ with $H_{0}=100h$. There are
four free parameters $\hat{p}=(\Omega_{m},\xi,\zeta,H_{0})$ in this model.

###  II.3 Emergent Dark Energy Models

In this study, we investigate two models of Emergent Dark Energy: the
Phenomenologically Emergent Dark Energy (PEDE) model (Li et al., 2019), and
the Generalised Emergent Dark Energy (GEDE) model (Li et al., 2020). They used
these model to fit the latest astronomical observation data and explored its
impact on solving the “Hubble constant” problem in current cosmology. By
comparing the fitting results under different models, They found that these
models have significant potential to alleviate the tension observed in the
Hubble constant.

####  II.3.1 Phenomenologically Emergent Dark Energy model

The PEDE model (Li et al., 2019) is a simple (zero degree of freedom) but
radical phenomenological model of dark energy with symmetrical behavior around
the current time, where dark energy and matter densities are comparable. The
Friedmann equation for the PEDE model can be expressed as:

$H(z)^{2}=H_{0}^{2}\left[\Omega_{m}(1+z)^{3}+\widetilde{\Omega}_{\rm{DE}}\right],$
(17)

where $\Omega_{m}$ is the matter density at present time and
$\widetilde{\Omega}_{\rm{DE}}(z)$ can be expressed as:

$\widetilde{\Omega}_{\rm{DE}}(z)=\Omega_{\rm{DE},0}\times\exp\left[3\int_{0}^{z}\frac{1+\omega(z^{\prime})}{1+z^{\prime}}dz^{\prime}\right],$
(18)

where $\omega(z)=p_{\rm{DE}}/\rho_{\rm{DE}}$ is the equation of state of Dark
Energy.

Here, we introduce the PEDE model in which the dark energy density has the
following form:

$\widetilde{\Omega}_{\rm{DE}}(z)=\Omega_{\rm{DE},0}\times[1-\tanh(\log_{10}(1+z))],$
(19)

where $\Omega_{\rm{DE},0}=1-\Omega_{0m}$ and $1+z=1/a$ where $a$ is the scale
factor. This dark energy model has no degree of freedom and we can derive its
equation of state following:

$\omega(z)=-\frac{1}{3\ln(10)}\times\left(1+\tanh[\log_{10}(1+z)]\right)-1,$
(20)

there are two free parameters $\hat{p}=(\Omega_{m},H_{0})$ in this model.

####  II.3.2 Generalised Emergent Dark Energy model

Subsequently, Li et al. proposed a generalised parameterization form for the
PEDE model (Li et al., 2020) named as GEDE model. This generalised parametric
form incorporates two parameters to describe the properties of dark energy
evolution: a free parameter $\Delta$ that characterizes the slope of dark
energy density evolution, and a parameter $z_{t}$ that determines the
transition redshift where dark energy density equals matter density. The GEDE
model offers flexibility, as it can encompass both the $\Lambda$CDM model and
the PEDE model as two special cases, with $\Delta=0$ and $\Delta=1$.
respectively. The Hubble parameter in this model can be expressed as

$H(z)^{2}=H_{0}^{2}\left[\Omega_{m}(1+z)^{3}+\widetilde{\Omega}_{\rm{DE}}+\Omega_{\rm{R},0}(1+z)^{4}\right],$
(21)

$\Omega_{m}$and $\Omega_{R,0}$ is the current matter density and radiation
density, respectively. Here $\widetilde{\Omega}_{\rm{DE}}(a)$ is defined as

$\widetilde{\Omega}_{\rm{DE}}(a)=\Omega_{\rm{DE}}(a)\times\frac{H_{a}^{2}}{H_{0}^{2}},$
(22)

In GEDE model, the evolution for dark energy density has the following form:

$\widetilde{\Omega}_{\rm{DE}}(z)=\Omega_{\rm{DE},0}\frac{1-\tanh\left(\Delta\times\log_{10}(\frac{1+z}{1+z_{t}})\right)}{1+\tanh(\Delta\times\log_{10}(1+z_{t}))},$
(23)

where $\Omega_{\rm{DE},0}=1-\Omega_{m}$ and transition redshift $z_{t}$ can be
derived by the condition of
$\widetilde{\Omega}_{\rm{DE}}(z_{t})=\Omega_{m}(1+z_{t})^{3}$. From the above
formulas, we can get the expression about $\Omega_{m}$.

###  II.4 cosmological models with torsion

The Einstein-Cartan (EC) theory with the presence of space-time torsion has
gained a lot of attention, where torsion acts a new dark source of torsionless
Riemannian gravity and drives the expansion of Universe. This theory based on
a simple physical intuition, that torsion effect is regarded as a macroscopic
manifestation of the intrinsic angular momentum (spin) of matter (e.g. see
Blagojevic et al. (2013) for a recent review). Recently, Pereira et al.
introduced a torsion model that solely considers baryonic matter and a
cosmological term (Pereira et al., 2022). The interaction between torsion and
baryonic as well as cosmological constant terms naturally leads to an
effective contribution from dark matter. Additionally, the cosmological
constant term is modified due to the torsion coupling, resulting in an
effective cosmological constant. When torsion is present, the analogues of the
Friedmann and Raychaudhuri equations in a space-time with a non-vanishing
cosmological constant (i.e. when $\Lambda\neq 0$) assume the following form
(Kranas et al., 2019):

$\left(\frac{\dot{a}}{a}\right)^{2}=\frac{1}{3}\kappa\rho+\frac{1}{3}\Lambda-4\phi^{2}-4\left(\frac{\dot{a}}{a}\right)\phi,$
(24)

and

$\frac{\ddot{a}}{a}=-\frac{1}{6}\kappa(\rho+3p)+\frac{1}{3}\Lambda-2\dot{\phi}-2\left(\frac{\dot{a}}{a}\right)\phi,$
(25)

where $\rho$ and $p$ are the energy density and pressure of matter, with the
definition of the equation of state $\omega=p/\rho$, the continuity equation
will be the following form:

$\dot{\rho}+3(1+\omega)H\rho+2(1+3\omega)\phi\rho=4\phi\Lambda\kappa^{-1}$
(26)

where $H=\dot{a}/a$ is defined as the Hubble parameter. According to whether
there is dark energy, it is divided into two models, the torsion model with
dark energy and the torsion model without dark energy.

####  II.4.1 Case I: vanishing cosmological constant

In the case of the disappearance of cosmological constants, if assuming a
empty space $(p=\Lambda=0)$, which means that our universe only contains the
matter component, and the accelerating expansion of the universe is driven by
torsion. The Eq. (24) will become $\varphi(t)=-H(t)/2$, which means the
torsion function $\varphi(t)$ depends on Hubble parameter $H$. Thus, we adopt
a simple parameterized form $\varphi(t)=-\alpha H(t)/2$, then the Friedmann
equation becomes

$H^{2}=\frac{\kappa}{3}\rho_{m}+4\alpha H^{2}-4\alpha^{2}H^{2}.$ (27)

This equation clearly shows that torsion contributes to the total effective
energy-density of the system. Similar to the FLRW Universe model, the density
parameter $\Omega$ can also be introduced for a homogeneous and isotropic
model with torsion $\Omega_{m}+\Omega_{\alpha}=1$, where the matter density
parameter $\Omega_{m}=\frac{\kappa\rho_{m}}{3H^{2}}$ and the torsion density
parameter $\Omega_{\alpha}=4\alpha-4\alpha^{2}$, then, we can get

$H(z)^{2}=\frac{H_{0}^{2}}{(1-2\alpha)^{2}}(\Omega_{m}(1+z)^{(3-2\alpha)}),$
(28)

there are two free parameters $\hat{p}=(\Omega_{m},H_{0})$ in this model.

####  II.4.2  Case II: non-vanishing cosmological constant

For $\varphi(t)=-\alpha H(t)$ with $\Lambda\neq 0$ and $\omega=0$, which means
that the accelerating expansion of the universe is driven by the cosmological
constant plus torsion, one could obtain the Friedmann equation as

$H^{2}=\kappa\rho_{m}+\frac{\Lambda}{3}+4\alpha H^{2}-4\alpha^{2}H^{2}.$ (29)

Similar to Case I, we introduce the density parameter for cosmological
constant $\Omega_{\Lambda}=\frac{\Lambda}{3H^{2}}$. For convenience, we also
denote $\Omega_{\Lambda}$ as the present value of cosmological constant
density parameter. Based on the conservation equation of
$\Omega_{m}+\Omega_{\Lambda}+\Omega_{\alpha}=1$, the Friedmann equation can be
written as

$\begin{split}&H(z)^{2}=\frac{H_{0}^{2}}{(1-2\alpha)^{2}}\\\
&\left[\left(\Omega_{m}+\frac{4\alpha}{(3-2\alpha)}\Omega_{\Lambda}\right)(1+z)^{3-2\alpha}+\Omega_{\Lambda}\left(1-\frac{4\alpha}{3-2\alpha}\right)\right],\end{split}$
(30)

there are three free parameters $\hat{p}=(\Omega_{m},\alpha,H_{0})$ in this
model.

##  III Observational data and METHODS

In this section, we present the details of deriving observational constraints
on the cosmological models from X-ray and UV measurements of quasars and BAO
measurements.

###  III.1 High-redshift observations of quasars and BAO

Quasars have considerable potential to be used as useful cosmological probes.
Although the extreme variability in their luminosity and high observed
dispersion, many efforts have been made to standardize quasars as probes based
on their own properties. Recently, (Risaliti & Lusso, 2019) attempted to use
quasars as standard candles by using the nonlinear relation between their
intrinsic UV and the X-ray luminosities. So far, the largest quasar sample
with both X-ray and UV observations consists of $\sim 12,000$ objects,
assembled by combining several different samples in (Lusso et al., 2020). Most
of the quasars in the parent sample are from XMM-Newton (Nardini et al.,
2019), XMM-XLL sample (Menzel et al., 2016) and SDSS (Mingo et al., 2016).
Then several filtering steps were applied to reduce the systematic effects and
2421 quasars in the redshift range $0.009<z<7.5$ were left in the final
cleaned sample (Lusso et al., 2020). The scatter plot of the final quasar
sample is shown in Fig. 1.

The relation between the X-ray and UV luminosities is usually parameterized as
(Risaliti & Lusso, 2019)

$\log(L_{X})=\gamma\log(L_{UV})+\beta,$ (31)

where $L_{X}$ and $L_{UV}$ are the rest-frame monochromatic luminosities at
$2$ keV and 2500 Å, and $\log=\log_{10}$, the slope $-\gamma$ along with the
intercept $-\beta$ are two free parameters. Applying the flux-luminosity
relation of $F=L/4\pi D_{L}^{2}$, the UV and X-ray luminosities can be
replaced by the observed fluxes:

$\log(F_{X})=\gamma\log(F_{UV})+(2\gamma-2)\log(D_{L}H_{0})+\beta^{\prime},$
(32)

where $\beta^{\prime}=\beta-(2\gamma-2)\log(H_{0})+(\gamma-1)\log 4\pi$,
$F_{X}$ and $F_{UV}$ are the X-ray and UV fluxes, respectively. Here $D_{L}$
is the luminosity distance, which indicates such kind of quasar measurements
can be used to calibrate them as standard candles. Theoretically, $D_{L}$ is
determined by the redshift $z$ and cosmological parameters $\hat{p}$ in a
specific model:

$D_{L}(z,{\hat{p}})=\frac{c(1+z)}{H_{0}}\int_{0}^{z}\frac{dz^{\prime}}{E(z^{\prime})}.$
(33)

In order to constrain cosmological parameters ${\hat{p}}$ through the
measurements of quasar X-ray and UV fluxes, we compare the observed X-ray
fluxes with the predicted X-ray fluxes calculated with Eq. (32) at the same
redshift. Then, the best-fitted parameter values and respective uncertainties
for each cosmological model are determined by minimizing the $\chi^{2}$
objective function, defined by the log-likelihood (Risaliti & Lusso, 2019)

$\chi^{2}_{QSO}=\sum_{i=1}^{2421}\left[\frac{[\log(F_{X,i}^{obs})-\log(F_{X,i}^{th})]^{2}}{s_{i}^{2}}+\ln(2\pi
s_{i}^{2})\right],$ (34)

where $s_{i}^{2}=\sigma_{i}^{2}+\delta^{2}$, and $\sigma_{i}$ is the
measurement error on $F_{X,i}^{obs}$. In addition to the cosmological model
parameters, three more free parameters are fitted: $\gamma$, $\beta$
representing the X-UV relation and $\delta$ representing the global intrinsic
dispersion. More recently, (Li et al., 2021) presented a model-independent
approach to calibrate this sample with latest SN Ia observations, and
estimated that $\beta=7.735\pm 0.244$, $\gamma=0.649\pm 0.007$, and
$\delta=0.235\pm 0.04$. We will use their calibrated parameters in this work.

Figure 1: The scatter plot of UV and X-ray fluxes of 2421 quasars

However, considering the presence of high intrinsic dispersion of this quasar
sample and the degeneracy between cosmological model parameters. The inclusion
of both the transverse BAO measurements obtained without assuming the fiducial
cosmological model (commonly known as 2D-BAO) and the line-of-sight and
transverse BAO measurements from the spectroscopic survey (commonly known as
3D-BAO) aids in constraining the parameters of these cosmological models.

The transverse BAO measurements consist of 15 $\theta_{BAO}(z)$ values
obtained from the public data releases (DR) of the Sloan Digital Sky
Survey(SDSS) which are summarized in Table 1 of Nunes et al. (2020). The BAO
angular scale $\theta_{BAO}(z)$ is related to the angular diameter distance as
follows

$\theta^{th}_{BAO}(z)=\frac{r_{d}}{(1+z)D_{A}(z)}$ (35)

where $r_{d}$ is the comoving sound horizon at the baryon drag epoch. The
calibration of $r_{d}$ enables the utilization of the BAO angular scale
$\theta_{BAO}(z)$ to establish a precise correlation between angular diameter
distance and redshift. For the sound horizon at the drag epoch, we adopt the
value of $r_{d}=147.59$ Mpc Nunes et al. (2020) reported from Planck plus
transversal BAO data as a prior value in our work. Then the best-fitted
parameter values are determined by corresponding log-likelihood function

$\chi_{2D-BAO}^{2}({\hat{p}})=\sum_{i=1}^{15}\frac{\left[\theta^{th}_{BAO}(z_{i};{\hat{p}})-\theta^{obs}_{BAO}(z_{i})\right]^{2}}{\sigma^{2}_{BAO(z_{i})}}.$
(36)

For the use of 3D-BAO data, we follow the approach carried out in (Alam et
al., 2021). The measurements obtained from the completed Sloan Digital Sky
Survey (SDSS) Main Galaxy Sample (MGS), the Baryon Oscillation Spectroscopic
Survey(BOSS) Data Release 12 (DR12) galaxies, the extended Baryon Oscillation
Spectroscopic Survey (eBOSS) galaxies and quasars, the auto-correlation of the
Ly$\alpha$ forest and its cross-correlation with quasars from BOSS and eBOSS,
are summarized as BAO-only measurements in Table 3 of Alam et al. (2021). The
relevant quantities include the Hubble distance $D_{H}(z)$, the transverse
comoving distance $D_{M}(z)$ and the spherically averaged angular-diameter
distance $D_{V}(z)$.

The Hubble distance along the line-of-sight direction is determined by the
speed of light and the Hubble parameter

$D_{H}(z)=\frac{c}{H(z)}.$ (37)

The comoving distance in the transverse direction can be simplified in the
case of a spatially flat universe as

$D_{M}(z)=c\int_{0}^{z}\frac{dz^{\prime}}{H(z^{\prime})}.$ (38)

The spherically averaged angular-diameter distance can be readily utilized by
taking into account both the line-of-sight and transverse directions as

$D_{V}(z)=[zD^{2}_{M}(z)D_{H}(z)]^{1/3}.$ (39)

Considering the dependence of the comoving sound horizon $r_{d}$ at baryon
drag epoch , expressed as $D_{H}(z)/r_{d}$, $D_{M}(z)/r_{d}$, and
$D_{V}(z)/r_{d}$ more directly. The expressions for these three BAO distances
are represented by the quantities $A_{obs}(z_{i})$, $\sigma_{A_{i}}$, and
$A_{th}(z_{i})$, which correspond to the measurements, uncertainties, and
theoretical expression, respectively.For the uncorrelated BAO measurements,
the corresponding log-likelihood function is given by

$\chi_{3D-BAO}^{2}({\hat{p}})=\sum_{i=1}^{3}\frac{\left[A_{th}(z_{i};{\hat{p}})-A_{obs}(z_{i})\right]^{2}}{\sigma(z_{i})^{2}}.$
(40)

For the correlated BAO measurements, the log-likelihood function of correlated
BAO measurements takes the for

$\chi_{3D-BAO}^{2}({\hat{p}})=\left[A_{th}({\hat{p}})-A_{obs}\right]^{T}C^{-1}\left[A_{th}({\hat{p}})-A_{obs}\right],$
(41)

where $C^{-1}$ is the inverse covariance matrix.

###  III.2 MCMC analysis

In order to determine the cosmological parameters $p$ in different models, we
use MCMC method to achieve our purpose, implemented in the _emcee_ package
111https://pypi.python.org/pypi/emcee in Python (Foreman-Mackey et al., 2013).
Using the $\chi^{2}$ objective function defined above, the total
$\chi^{2}_{tot}$ function of the above combined analysis is expressed as

$\chi^{2}_{tot}=\chi^{2}_{QSO}+\chi^{2}_{BAO},$ (42)

we can also write the likelihood function as

$\mathcal{L}({\hat{p}})=\exp^{-\frac{\chi^{2}_{tot}}{2}}.$ (43)

Then it is essential to determine which model is most preferred by the
observational measurements and carry out a good comparison between these
different models. Out of possible model selection techniques, we will use the
Akaike Information Criterion (AIC) (Akaike, 1974)

$\rm AIC=\chi_{min}^{2}+2k,$ (44)

as well as the Bayesian Information Criterion (BIC) (Schwarz, 1978)

$\rm BIC=\chi_{min}^{2}+k\ln N,$ (45)

where $\chi_{min}^{2}=-2\ln\mathcal{L}_{max}$, $k$ is the number of free
parameters in the model and $N$ represents the number of data points.

##  IV Results and discussions

Figure 2: The 1D probability distributions and 2D contours with $1\sigma$ and
$2\sigma$ confidence levels for the DGP model obtained from the 2-D and 3-D
BAO and combinations of QSO+BAO datasets, respectively. The black dashed line
represents the $H_{0}=67.4$ $km/s/Mpc$ reported by Planck 2018 results. Figure
3: Similar to Figure 1, but for the $\alpha$DE model. The black dashed line
represents the $\Lambda$CDM model corresponding to q = 0.

In this section, we give the results of the cosmological models listed in
Section 2, obtained using two special BAO compilations (2D-BAO and 3D-BAO) and
the combinations of BAO and QSO datasets (QSO+2D-BAO and QSO+3D-BAO),
respectively. The distinction between the outcomes obtained from the inclusion
of QSO data and BAO data alone is denoted as BAO and QSO+BAO, respectively,
without explicitly distinguishing the BAO datasetin part text of this
discussion section. In order to get a better comparison, the corresponding
results for the concordance $\Lambda$CDM model is also displayed. The 1D
probability distributions and 2D contours with $1\sigma$ and $2\sigma$
confidence levels, as well as the best-fit value with 1$\sigma$ uncertainty
for each model are shown in Figs. 2-9 and reported in Table I.

###  IV.1 Observational Constraints on Braneworld models

As can be seen from Table I, the best-fit values of $\Omega_{m}$ in the DGP is
$\Omega_{m}=0.32_{-0.03}^{+0.04}$ within $68.3\%$ confidence level by using
the most comprehensive of QSO + 3D-BAO datasets currently accessible, which
agree well with the recent Planck 2018 results: $\Omega_{m}=0.315\pm 0.007$
(Planck Collaboration et al., 2018) and SNe Ia+BAO+CMB+observational Hubble
parameter (OHD): $\Omega_{m}=0.305\pm 0.015$ (Shi et al., 2012). For the
3D-BAO data, the best-fit matter density parameters for this DGP model is
$\Omega_{m}=0.26_{-0.03}^{+0.03}$, which is significantly lower than the QSO +
3D-BAO mentioned above. The combination of 2D-BAO data yielded similar
findings, however, a significant disparity was observed when compared to the
results obtained from 3D-BAO data. It is worthwhile to mention that the matter
density parameter $\Omega_{m}$ obtained by QSO tends to be higher than that
from other cosmological probes, as was remarked in the previous works of
(Risaliti & Lusso, 2019; Khadka & Ratra, 2020). This suggests that the
composition of the universe characterized by cosmological parameters can be
comprehended differently through high-redshift quasars. The use of BAO data
can effectively help QSOs constrain cosmological models, considering the
constrained results especially on $\Omega_{m}$ and $H_{0}$. The best-fit
values of $\Omega_{m}$ and $H_{0}$ for $\alpha$DE are noticeably higher than
those of DGP, indicating a potential reason could be the presence of an
additional parameter $q$ in $\alpha$DE. For the $\alpha$DE model, the limit of
$q=1$ corresponds to the DGP model, and the limit of $q=0$ corresponds to the
$\Lambda$CDM model. It is evident that the optimal values of $q$ obtained from
analyzing both 3D-BAO and 2D-BAO datasets in the context of $\alpha$DE favor
the DGP and $\Lambda$CDM models respectively. We can also see that the BAO
data alone can only provide a rather weak constraint on the parameter $q$ in
$\alpha$DE, but the combined QSO+BAO data can constrain $q$ tightly. For the
constrained results on $H_{0}$, the best-fit values of $H_{0}$ in the DGP and
$\alpha$DE are both closed with recent Planck 2018 results when incorporating
3D-BAO measurements. In Fig. 2-3, for DGP and $\alpha$DE models, we can
clearly see that the parameter degeneracy directions of BAO and BAO+QSO are
almost identical, and by adding quasar data, the constrained parameters
precision have been significantly improved.

###  IV.2 Observational Constraints on Chaplygin gas models

The Table I presents the results of the best-fitted parameters for the GCG
model. One can find large difference between the constraints on $\Omega_{m}$
and $H_{0}$ obtained from the BAO and QSO+BAO datasets and the results derived
from the 2D-BAO and 3D-BAO datasets exhibit significant disparities. In the
context of GCG, it is worth noting that the parameter $\xi$ quantifies the
deviation from the $\Lambda$CDM model and the original Chaplygin gas model.
Consequently, $\Lambda$CDM is not consistent with GCG at the $1\sigma$
confidence level.

For the NGCG model, the estimated values of the cosmic parameters are
presented in Table 1. It is observed that for the BAO and QSO+BAO datasets,
the best-fit values of the parameters are in agreement with each other within
the $2\sigma$ confidence level. It is worth noting that the matter density
parameter $\Omega_{m}$ derived from such dataset combinations except
QSO+2D-BAO tends to be higher than that obtained from other cosmological
probes, as previously reported in Liu et al. (2019). And the constraint
results of $\xi\simeq 0$ and $\zeta\simeq-1$ from Table I indicate that the
$\Lambda$CDM limit of these models is strongly favored by the current
observations. In terms of the constrained results on $H_{0}$, the best-fit
values of $H_{0}$ in the NGCG models with 2D-BAO measurements align with
recent Planck 2018 results. Additionally, due to the degeneracy between model
parameters, the uncertainty of the parameters is quite large. This conclusion
can be visually inferred from Figs. 4-5, emphasizing the significance of our
work by highlighting the distinct roles played by BAO and quasar data in
constraining cosmological model parameters. The inclusion of quasar data
effectively breaks the degeneracy between model parameters.

Figure 4: Similar to Figure 1, but for the GCG model. The black dashed line
represents the $\Lambda$CDM model corresponding to $\xi$ = 0. Figure 5:
Similar to Figure 1, but for the NGCG model. The black dashed line represents
the $\Lambda$CDM model corresponding to $\xi$ = 0.

###  IV.3 Observational Constraints on Emergent Dark Energy Models

The numerical results and 1$\sigma$ uncertainties of the constraints obtained
using BAO alone and QSO+BAO datasets for the PEDE and GEDE models are
summarized in Table I. The 1D probability distributions and 2D contours with
$1\sigma$ and $2\sigma$ confidence levels are also illustrated in Fig. 6-7.
For the $\Lambda$CDM model, $\Delta$ is fixed at zero, while for the PEDE
model it is set to one. However, in the GEDE model, $\Delta$ is a free
parameter. For the PEDE model, the best-fit value of $\Omega_{m}$, obtained
using 3D-BAO datasets, is consistent with the results derived from
Pantheon+BAO datasets (Li et al., 2019) and the value of $H_{0}$ obtained
using QSO+3D-BAO is consistent with the result obtained from Planck 2018
results. For the GEDE model, the best-fit values of $\Omega_{m}$ are
$0.31^{+0.33}_{-0.38}$ by using 3D-BAO alone are consistent to the results
obtained with the work (Li et al., 2020). It is worth mentioning that the
constrained results on $\Omega_{m}$ were calculated using the model parameters
$\Delta$ and $z_{t}$, whose relationship is provided in the work Li et al.
(2020). Additionally, based on the GEDE model, our constraints on $H_{0}$
towards to a smaller value align more closely with recent Planck 2018 results.
These results are $H_{0}=62.51_{-3.23}^{+3.65}$ $km/s/Mpc$ and
$H_{0}=58.31_{-2.42}^{+3.19}$ $km/s/Mpc$ for 3D-BAO alone and QSO+3D-BAO
datasets, respectively. The best-fitting values of $\Delta$ from 2D-BAO and
QSO+2D-BAO are consistent with zero within $1\sigma$ uncertainties, indicating
that our results include the $\Lambda$CDM model but not the PEDE model. This
is because, compared to the PEDE model, the GEDE model incorporates additional
parameters, introducing greater uncertainty in the constrained results. The
constrained results suggest that data may prefer simpler and more
deterministic models to describe the behavior of the universe. In terms of
PEDE and GEDE models, none of the model parameters break the degeneracy.
However, the constrained parameter precision using QSO+BAO data is
significantly higher than that using BAO data alone.

Figure 6: Similar to Figure 1, but for the PEDE model. Figure 7: Similar to
Figure 1, but for the GEDE model. The black dashed line represents the
$\Lambda$CDM model corresponding to $\Delta$ = 0.

###  IV.4 Observational Constraints on cosmological model with torsion

Figure 8: Similar to Figure 1, but for the CASE I model.  Figure 9: Similar to
Figure 1, but for the CASE II model. The black dashed line represents the
$\Lambda$CDM model corresponding to $\alpha$= 0.

In Case I model, the torsion term leads to the accelerated expansion of the
Universe without the need for a cosmological constant $\Lambda$. The Figure 8
shows the 1$\sigma$ and 2$\sigma$ contours for the parameters $H_{0}$ and
$\Omega_{m}$. The best-fitting values of the Hubble constant from 3D-BAO data
are $H_{0}=36.08_{-14.68}^{+20.36}$ $km/s/Mpc$ and
$\Omega_{m}=0.15_{-0.09}^{+0.23}$. It is clear that 3D-BAO data alone are not
sufficient to provide strong constraints on model parameters. However, when
QSO+3D-BAO data are used, the values change to $H_{0}=59.17_{-17.73}^{+22.36}$
$km/s/Mpc$ and $\Omega_{m}=0.10_{-0.05}^{+0.09}$. The central value of $H_{0}$
also disagrees with the result inferred from Planck 2018 observation and local
measurement by SH0ES collaboration. This conflict is likely due to a strong
degeneracy between the Hubble constant and matter density. It is worth noting
that the torsion parameter $\alpha$ does not introduce any extra degrees of
freedom and is related to the density of matter parameter through the
conservation condition $\Omega_{m}+4\alpha-4\alpha^{2}=1$. Therefore, we
calculate $\alpha=0.37_{-0.03}^{+0.03}$ and $\alpha=0.34_{-0.03}^{+0.03}$
using 3D-BAO alone and QSO+3D-BAO datasets, respectively, which are consistent
with the work Liu et al. (2023). However, these results conflict with previous
results indicating a small value of $\alpha$ supported by low-redshift Hubble
parameter and SN Ia observations (Pereira et al., 2019, 2022).

In the context of Case II model, the torsion term and cosmological constant
$\Lambda$ work in tandem to drive the accelerated expansion of the Universe.
The Figure 9 shows the constraint results for $H_{0}$, $\Omega_{m}$, and
$\alpha$ using high-redshift quasar and BAO observations. Table I summarizes
the best-fit parameters, including their 1$\sigma$ uncertainties. We obtain
that $H_{0}=70.26_{-3.33}^{+3.09}$ $km/s/Mpc$ and
$H_{0}=70.02^{+1.53}_{-1.67}$ $km/s/Mpc$ from 2D-BAO and QSO+3D-BAO data
towards to a smaller value align with the Planck 2018 CMB observations when
assuming a flat $\Lambda$CDM model. Notably, the central values of
$\alpha=-0.01^{+0.21}_{-0.17}$ from 2D-BAO alone align with the torsion’s
impact on primordial helium-4 abundance, where a narrow parameter range of
$-0.0058<\alpha<+0.0194$ was reported in Kranas et al. (2019).

Table 1: Summary of the best-fit values with their 1$\sigma$ uncertainties concerning the parameters of all considered models. The results are obtained from the 2-D and 3-D BAO and combinations of QSO+BAO datasets, respectively. Model | Data | $\Omega_{m}$ |  $H_{0}$ [km/s/Mpc] |  |
---|---|---|---|---|---
$\Lambda$CDM | 2D-BAO | $0.43^{+0.19}_{-0.15}$ | $70.05^{+3.08}_{-3.33}$ |  |
| 3D-BAO | $0.29^{+0.03}_{-0.03}$ | $68.03^{+2.13}_{-2.10}$ |  |
| QSO+2D-BAO | $0.40^{+0.06}_{-0.05}$ | $68.40^{+1.59}_{-1.61}$ |  |
| QSO+3D-BAO | $0.36^{+0.04}_{-0.03}$ | $65.10^{+1.74}_{-1.69}$ |  |
Model | Data | $\Omega_{m}$ |  $H_{0}$ [km/s/Mpc] |  |
DGP | 2D-BAO | $0.34^{+0.20}_{-0.13}$ | $69.02^{+3.17}_{-3.17}$ |  |
| 3D-BAO | $0.26^{+0.03}_{-0.03}$ | $65.14^{+2.04}_{-1.96}$ |  |
| QSO+2D-BAO | $0.33^{+0.06}_{-0.05}$ | $66.98^{+1.61}_{-1.64}$ |  |
| QSO+3D-BAO | $0.32^{+0.04}_{-0.03}$ | $62.64^{+1.57}_{-1.58}$ |  |
Model | Data | $\Omega_{m}$ |  $H_{0}$ [km/s/Mpc] |  |
PEDE | 2D-BAO | $0.46^{+0.17}_{-0.13}$ | $70.45^{+3.25}_{-3.46}$ |  |
| 3D-BAO | $0.29^{+0.03}_{-0.03}$ | $69.92^{+2.32}_{-2.29}$ |  |
| QSO+2D-BAO | $0.40^{+0.06}_{-0.05}$ | $69.46^{+1.69}_{-1.74}$ |  |
| QSO+3D-BAO | $0.36^{+0.04}_{-0.03}$ | $66.47^{+1.91}_{-1.86}$ |  |
Model | Data | $\Omega_{m}$ | $q$ |  $H_{0}$ [km/s/Mpc] |
$\alpha$DE | 2D-BAO | $0.36^{+0.23}_{-0.21}$ | $-0.45^{+2.17}_{-2.10}$ | $68.61^{+3.47}_{-4.08}$ |
| 3D-BAO | $0.29^{+0.04}_{-0.03}$ | $0.99^{+0.70}_{-2.58}$ | $63.24^{+3.83}_{-3.84}$ |
| QSO+2D-BAO | $0.41^{+0.06}_{-0.06}$ | $-0.76^{+0.70}_{-0.70}$ | $68.18^{+1.66}_{-1.77}$ |
| QSO+3D-BAO | $0.35^{+0.04}_{-0.04}$ | $1.56^{+0.52}_{-0.58}$ | $58.83^{+3.28}_{-3.09}$ |
Model | Data | $\Omega_{m}$ | $\xi$ |  $H_{0}$ [km/s/Mpc] |
GCG | 2D-BAO | $0.51^{+0.16}_{-0.16}$ | $1.50^{+2.19}_{-1.65}$ | $71.47^{+3.62}_{-4.28}$ |
| 3D-BAO | $0.26^{+0.05}_{-0.07}$ | $-0.49^{+0.31}_{-0.23}$ | $64.07^{+3.14}_{-2.75}$ |
| QSO+2D-BAO | $0.40^{+0.04}_{-0.04}$ | $5.21^{+3.15}_{-2.71}$ | $75.25^{+1.52}_{-2.03}$ |
| QSO+3D-BAO | $0.27^{+0.07}_{-0.10}$ | $-0.64^{+0.24}_{-0.16}$ | $60.79^{+2.30}_{-1.81}$ |
Model | Data | $\Omega_{m}$ | $\alpha$ |  $H_{0}$ [km/s/Mpc] |
CASE I | 2D-BAO | $0.35^{+0.40}_{-0.22}$ | $0.28^{+0.39}_{-0.15}$ | $49.85_{-26.10}^{+34.62}$ |
| 3D-BAO | $0.15^{+0.23}_{-0.09}$ | $0.37^{+0.03}_{-0.03}$ | $36.08_{-14.68}^{+20.36}$ |
| QSO+2D-BAO | $0.17^{+0.17}_{-0.09}$ | $0.33^{+0.07}_{-0.07}$ | $48.54_{-19.23}^{+23.65}$ |
| QSO+3D-BAO | $0.10^{+0.09}_{-0.05}$ | $0.34^{+0.03}_{-0.03}$ | $59.17_{-17.73}^{+22.36}$ |
Model | Data | $\Omega_{m}$ | $\alpha$ |  $H_{0}$ [km/s/Mpc] |
CASE II | 2D-BAO | $0.44^{+0.38}_{-0.31}$ | $-0.01^{+0.21}_{-0.17}$ | $70.26^{+3.09}_{-3.33}$ |
| 3D-BAO | $0.12^{+0.10}_{-0.06}$ | $0.20^{+0.11}_{-0.13}$ | $64.10^{+3.44}_{-3.22}$ |
| QSO+2D-BAO | $0.86^{+0.10}_{-0.20}$ | $-0.22^{+0.08}_{-0.04}$ | $70.02^{+1.53}_{-1.67}$ |
| QSO+3D-BAO | $0.14^{+0.07}_{-0.05}$ | $0.27^{+0.09}_{-0.10}$ | $59.63^{+2.70}_{-2.60}$ |
Model | Data | $\Omega_{m}$ | $\xi$ | $\zeta$ |  $H_{0}$ [km/s/Mpc]
NGCG | 2D-BAO | $0.39^{+0.33}_{-0.26}$ | $0.29^{+2.34}_{-2.61}$ | $-1.20^{+0.73}_{-2.07}$ | $69.61^{+8.04}_{-4.41}$
| 3D-BAO | $0.41^{+0.28}_{-0.26}$ | $-0.62^{+0.85}_{-0.58}$ | $-0.70^{+0.33}_{-0.75}$ | $61.90^{+4.25}_{-3.15}$
| QSO+2D-BAO | $0.27^{+0.22}_{-0.15}$ | $0.28^{+0.97}_{-0.59}$ | $-2.40^{+1.05}_{-1.61}$ | $78.71^{+9.11}_{-5.60}$
| QSO+3D-BAO | $0.51^{+0.29}_{-0.35}$ | $-0.90^{+0.78}_{-1.49}$ | $-0.53^{+0.24}_{-0.98}$ | $58.41^{+3.09}_{-2.00}$
Model | Data | $\Omega_{m}$ | $\Delta$ | $z_{t}$ |  $H_{0}$ [km/s/Mpc]
GEDE | 2D-BAO | $0.41^{+0.19}_{-0.28}$ | $0.15^{+0.44}_{-0.29}$ | $-0.98^{+3.13}_{-1.98}$ | $68.81^{+4.24}_{-3.42}$
| 3D-BAO | $0.31^{+0.33}_{-0.38}$ | $0.53^{+0.22}_{-0.15}$ | $-2.16^{+1.09}_{-0.65}$ | $62.51^{+3.65}_{-3.23}$
| QSO+2D-BAO | $0.40^{+0.45}_{-0.45}$ | $0.10^{+0.06}_{-0.05}$ | $3.50^{+1.09}_{-2.01}$ | $71.83^{+2.21}_{-2.70}$
| QSO+3D-BAO | $0.38^{+0.29}_{-0.36}$ | $0.47^{+0.31}_{-0.19}$ | $-3.09^{+0.87}_{-0.55}$ | $58.31^{+3.19}_{-2.42}$
Table 2: Information theoretic model comparison. Minimum values of AIC, BIC, their differences are reported for the $\Lambda$CDM and each of cosmological models considered. We give ranking of the favor degree of the model, the smaller the number represents the more favored by the observational data.  Dataset | Model | AIC |  $\Delta$AIC | Favor Degree | Ranking | BIC |  $\Delta$BIC | Favor Degree | Ranking
---|---|---|---|---|---|---|---|---|---
QSO+2D-BAO |  $\Lambda$CDM | 2406.42 | 5.44 | Excluded | 5 | 2418.02 | 1.73 | Preferred | 2
| DGP | 2435.95 | 34.97 | Excluded | 8 | 2447.55 | 31.26 | Excluded | 8
|  $\alpha$DE | 2408.00 | 7.02 | Excluded | 6 | 2425.39 | 9.10 | Excluded | 6
| PEDE | 2404.69 | 3.71 | Slight | 4 | 2416.29 | 0 | Preferred | 1
| GEDE | 2403.65 | 2.67 | Slight | 3 | 2421.04 | 4.75 | Slight | 5
| GCG | 2400.98 | 0 | Preferred | 1 | 2418.37 | 2.08 | Slight | 3
| NGCG | 2411.26 | 10.28 | Excluded | 7 | 2434.45 | 18.16 | Excluded | 7
| CASE I | 2457.76 | 56.78 | Excluded | 9 | 2475.15 | 58.86 | Excluded | 9
| CASE II | 2402.39 | 1.41 | Preferred | 2 | 2419.78 | 3.49 | Slight | 4
QSO+3D-BAO |  $\Lambda$CDM | 2393.38 | 6.76 | Excluded | 8 | 2404.97 | 4.78 | Slight | 3
| DGP | 2388.60 | 1.98 | Preferred | 4 | 2400.19 | 0 | Preferred | 1
|  $\alpha$DE | 2386.62 | 0 | Preferred | 1 | 2406.01 | 5.82 | Excluded | 4
| PEDE | 2396.45 | 9.83 | Excluded | 9 | 2408.04 | 7.85 | Excluded | 8
| GEDE | 2389.30 | 2.68 | Slight | 6 | 2406.69 | 6.50 | Excluded | 7
| GCG | 2386.65 | 0.03 | Preferred | 2 | 2406.04 | 5.85 | Excluded | 6
| NGCG | 2389.65 | 3.03 | Slight | 7 | 2412.83 | 12.64 | Excluded | 9
| CASE I | 2388.76 | 2.14 | Slight | 5 | 2406.15 | 5.96 | Excluded | 5
| CASE II | 2387.48 | 0.86 | Preferred | 3 | 2404.87 | 4.68 | Slight | 2

###  IV.5 MODEL COMPARISON

Figure 10: The Hubble diagram for the 2421 newest large sample of QSO X-ray
and UV flux measurements. The solid lines represent the theoretical
predictions of all models with the best-fit values obtained using quasar data.

We compare the models and discuss how strongly are they supported by the
observational data sets in this section. In the Table II, one can find the
summary of the information theoretical model selection criteria applied to
different models from QSO+2D-BAO and QSO+3D-BAO data sets. There is an
accepted interpretation of the magnitude of the difference
$\Delta=\rm{AIC_{2}}-\rm{AIC_{1}}$ (same for BIC) in terms of the strength of
evidence against model 2 (Kass et al., 1995; Tan et al., 2012). The rule of
thumb is that if $\Delta\leq 2$, the favor degree is preferred; if
$2<\Delta\leq 5$, it is adjudged to be slight; and if $\Delta\geq 5$, it is
excluded.

For the AIC criterion, the utilization of different BAO datasets in
combination with QSO yields varying outcomes. It is crucial to keep in mind
that model selection provides quantitative information regarding the strength
of evidence (or the level of support) rather than solely selecting a single
model. The combination of 2D-BAO and 3D-BAO yields significantly disparate
results, which merits particular attention. To show our results more clearly,
we present a ranking of models based on AIC to determine the preference of
observational data for these models in Table II. In the frame work of
QSO+2D-BAO dataset, the favor degree based on AIC difference for GCG and Case
II models are preferred. For GEDE and PEDE models, they are adjudged to be
slight. For $\Lambda$CDM, DGP, $\alpha$DE, NGCG and Case I models, their
adjudged results are excluded. In the frame work of QSO+3D-BAO dataset, The
DGP, $\alpha$DE, GCG and Case II models are adjudged to be preferred. For
GEDE, NGCG and Case I models, they are slightly preferred by observational
dataset. The $\Lambda$CDM and PEDE are adjudged to be excluded. Combine the
above description, we summarize that the two models, GCG and CASE II models,
performed best, and both are favored by the observational data. However,
regardless of which dataset QSO+2D-BAO and QSO+3D-BAO are considered, AIC
excludes the $\Lambda$CDM model. This is reasonable. Because AIC encourages
good data fitting. Compared with $\Lambda$CDM model and others (two free
parameters), GCG and Case II models (three free parameters) increase the
number of free parameters and improve the goodness of fitting, but may try to
avoid overfitting in four free parameters models.

The penalty term of BIC is larger than that of AIC, and when the number of
samples is too large, it can effectively prevent the model complexity caused
by too high precision. When considering the BIC, similar to the AIC criterion,
different results are obtained with varying BAO datasets. Specifically,
QSO+2D-BAO dataset favors the PEDE and $\Lambda$CDM models while QSO+3D-BAO
data supports the DGP model. Meanwhile, QSO+2D-BAO dataset slightly favors
GEDE, GCG and Case II models, and QSO+3D-BAO dataset slightly favors
$\Lambda$CDM and Case II models. There are four models to be excluded named
DGP, $\alpha$DE, NGCG and Case I model under the QSO+2D-BAO dataset. However,
for QSO+3D-BAO dataset, there are six models to be excluded $\alpha$DE, PEDE,
GEDE, GCG, NGCG and Case I models. Combining the results of BIC under the two
datasets, we summarize that the $\Lambda$CDM and Case II models performs well.
It is obvious that models with more free parameters are less favored by
current quasar and BAO observations. The evidences against for NGCG, Case I,
and $\alpha$DE models are particularly notable for all data sets, indicating
that the these three models are severely penalized by the BIC.

##  V Conclusion

In this paper, we have evaluated the power of measurements of quasars covering
sufficiently wide redshift range, on Brane-world models, Chaplygin Gas models,
Emergent Dark Energy models and models with torsion, under the assumption of
the spatial flatness of the Universe. The dynamics of these models are shown
in Fig. 10. Considering that the dynamics of the above models are very similar
at low redshift, the difference is significant at high redshift. Therefore,
the newest large sample of QSO X-ray and UV flux measurements (Lusso et al.,
2020) were used as standard candles and provided a good opportunity to test
models at the “redshift desert” ($0.009<z<7.5$) that is not yet widely
available through other observations. Meanwhile, with the aim to tighten the
constraint from the combined QSOs datasets and test the consistency with other
observations, QSO+2D-BAO and QSO+3D-BAO datasets were also taken into account
in this work. Here we summarize our main conclusions in more detail:

  * •

For all non-standard cosmological models above, the QSO data alone are not
able to provide effective constraints on model parameters, which is mainly
related to the large dispersion of the $L_{X}-L_{UV}$ relation obtained from
the 2421 QSO X-ray and UV flux measurements. With the addition of 2D and
3D-BAO datasets, we expect to be able to help break down the degeneracy
between cosmological model parameters.” The results were as expected, and the
addition of BAO data helped us achieve this, a conclusion that can be clearly
reflected in the $\alpha$DE, GCG and GEDE models.

  * •

For all of the cosmological models considered, the constrained parameter
precision using QSO+BAO data is significantly higher than that using BAO data
alone. And the value of matter density parameter $\Omega_{m}$ implied by the
QSO+BAO data is noticeably larger than BAO data, which is likely caused by the
discrepancy between the QSO X-Ray and UV flux data and the $\Omega_{m}$ flat
$\Lambda$CDM (Risaliti & Lusso, 2019).

  * •

Our results show that the value of the Hubble constant varies considerably by
considering different cosmological models, which means that the underlying
model describing our real Universe remains unknown. From a theoretical point
of view, alternative cosmological models may be an important way to alleviate
or even solve the Hubble tension. Our work also supports that the
Phenomenologically Emergent Dark Energy model and cosmological torsion models
may alleviate the Hubble tension, the reported value of the Hubble constant
lies between Planck’s 2018 CMB observations and local measurements from the
SH0ES collaboration obtained from QSO+BAO datasets combination, while other
cosmological models all support that the Hubble constant tends to be closer to
recent Planck 2018 results.

  * •

Based on the information criterion criterion, we find that $\Lambda$CDM model
is not the most observed favored model. According to the AIC criterion, the
utilization of different BAO datasets in combination with QSO yields varying
outcomes, wherein the QSO+2D-BAO data favors the GCG model and the QSO+3D-BAO
data favors the $\alpha$DE model. For the BIC criterion, QSO+2D-BAO data
favors the PEDE and $\Lambda$CDM models while QSO+3D-BAO data supports the DGP
model. The evidence against NGCG and Case I models are particularly notable
for all data sets, indicating that the NGCG model and Case I models are
severely penalized by the AIC and BIC. In addition, we find that, non-standard
models with more free parameters are less favored by the available
observations, which is the most unambiguous result of the current dataset.

  * •

It is necessary to note that, the constrained results on $H_{0}$ for some
considered models does not lie between the Planck 2018 results and local
measurements from the SH0ES collaboration. For instance, in the frame work of
Case I model, all constrained results on $H_{0}$ in a very low value, much
lower than Planck reported. Based on information criterion, all criteria are
unfriendly to this model. We conclude that the current datasets exclude this
model. Therefore, based on the information criterion, it is beneficial for us
to select or exclude some models to obtain the reliable results on $H_{0}$.
This also applies to the GEDE model. However, there is a slight difference.
Only BIC penalizes the GEDE model obtained using QSO+3D-BAO dataset and
excludes this model. As we pointed above, this model has more free parameters,
which causes BIC to penalize this model more, and there is a strong degeneracy
between the $H_{0}$ and other model parameters, which affects the resulting
$H_{0}$ constraint. In conclusion, we find that for models that are not
excluded by the information criterion, the constraining results of $H_{0}$ are
between the Planck 2018 results and local measurements from the SH0ES
collaboration, which means that the information criterion can effectively help
us to obtain more reliable constraining results and select cosmological
models.

As a promising cosmological probe, quasars will play an increasingly important
role in future model testing and have great potential to alleviate the tension
problem of the Hubble constant at the high redshift. From the observational
point of view, current COSMOS survey, XMM-Newton Serendipitous Source Catalog
Data Release and future the Sloan Digital Sky Survey (SDSS) and XMM spectra
will bring us hundreds of thousands of quasars in the most optimistic
discovery scenario, which will yield more high precision observational data
with small uncertainties and dispersion in the future. On the other hand, We
also expect to find more reasonable and accurate alternative models of the
Universe in the future.

###### Acknowledgements.

The authors are grateful the referee for careful reading the manuscript and
numerous insightful comments. Liu T.-H was supported by National Natural
Science Foundation of China under Grant No. 12203009; Chutian Scholars Program
in Hubei Province (X2023007); Hubei Province Foreign Expert Project
(2023DJC040). Zheng X.-G. was supported by the National Natural Science
Foundation of China under Grant No. 12103036; Chutian Scholars Program in
Hubei Province.

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