# Comparing the scalar-field dark energy models with recent observations

Tengpeng Xu  Yun Chen<EMAIL_ADDRESS>Lixin Xu  Shuo Cao  Key Laboratory
for Computational Astrophysics, National Astronomical Observatories, Chinese
Academy of Sciences, Beijing 100101, China  College of Astronomy and Space
Sciences, University of Chinese Academy of Sciences, Beijing, 100049, China
Institute of Theoretical Physics, School of Physics, Dalian University of
Technology, Dalian, 116024, China  Gravitational Wave and Cosmology
Laboratory, Department of Astronomy, Beijing Normal University, Beijing
100875, China

###### Abstract

In this paper, we investigate the general properties of a class of scalar-
field dark energy models (i.e., $\phi$CDM models) which behave like
cosmological trackers at early times. Particularly, we choose three $\phi$CDM
models with typical potentials, i.e., $V(\phi)\propto\phi^{-\alpha}$ (inverse
power-law (IPL) model), $V(\phi)\propto\coth^{\alpha}{\phi}$ (L-model) and
$V(\phi)\propto\cosh(\alpha\phi)$ (Oscillatory tracker model), where the
latter two models are based on the $\alpha$-attractors originated from the
study of inflation. These models, which reduce to the $\Lambda$CDM model with
$\alpha\to 0$, are studied and compared with the recent observations,
including the Pantheon sample of type Ia supernovae (SNe Ia), baryon acoustic
oscillations (BAO) measurements extracted from 6dFGS, BOSS and eBOSS, as well
as the temperature and polarization anisotropy power spectra data of cosmic
microwave background radiation (CMB) from Planck 2018 results. The
observational constraints from the combining sample (SNe Ia + BAO + CMB)
indicate that none of the three $\phi$CDM models exclude the $\Lambda$CDM
model at $68.3\%$ confidence level. We find that the CMB anisotropy data have
obvious advantages in constraining the dark energy models compared with other
cosmological probes, which is particularly evident in the L-model.
Furthermore, we apply the Bayesian evidence to compare the $\phi$CDM models
and the $\Lambda$CDM model with the analysis of the combining sample. The
concordance $\Lambda$CDM model is still the most supported one. In addition,
among the three $\phi$CDM models, the IPL model is the most competitive one,
while the L-model/Oscillatory tacker model is moderately/strongly disfavored.

###### keywords:

Dark energy; Quintessence; Cosmic microwave background; Baryon acoustic
oscillations; Type Ia supernova

††journal: Physics of the Dark Universe

##  1 Introduction

The observations of type Ia supernovae (SNe Ia) showed that the universe is
not only expanding but also accelerating [1, 2, 3]. Subsequent observations,
including the Cosmic Microwave Background Radiation (CMB) [4, 5], Baryon
Acoustic Oscillations (BAO) [6, 7], gravitational lens statistics [8, 9, 10,
11, 12, 13, 14], and milliarcsecond compact structure of radio quasars [15,
16, 17, 18], all support an accelerating expansion universe. Thus, the
existence of dark energy becomes crucial, which is deemed as a cosmological
component with negative pressure and can drive the cosmic expansion to
accelerate. The $\Lambda$CDM model with the cosmological constant $\Lambda$ as
a dark energy candidate is so far the simplest and most natural cosmological
model, which can fit the above observations very well. Nevertheless, there are
several problems in the $\Lambda$CDM model, such as the coincidence problem
and the fine-tuning problem [19, 20]. The dilemma of the $\Lambda$CDM model
mainly comes from the fact that the energy density of dark energy is constant
in time. Naturally, we can anticipate the dark energy owning a time-varying
energy density to solve or alleviate the problems. In practice, there are
mainly two approaches to obtain the time-varying dark energy. One approach is
to consider the dark energy with an evolving equation of state [21, 22, 23,
24, 25, 26, 27], and another is to consider the interaction between the dark
energy and dark matter [28, 29, 30, 31].

We focus on a popular kind of dynamical dark energy model, i.e., the scalar
field dark energy models (or the quintessence models) [32, 33, 34, 35, 36, 37,
38], in which the dark energy is deemed to be a slowly evolving and spatially
homogeneous scalar field. The cosmological model with the scalar-field dark
energy and the cold dark matter is often called as $\phi$CDM model. The
quintessence with some particular forms of potential $V(\phi)$ can own the
“tracker” behavior [39, 40], which enables the current energy density of dark
energy to be reached from a wide range of initial conditions, and thus
alleviates the coincidence problem. The quintessence with an inverse power-law
(IPL) potential $V(\phi)\propto\phi^{-\alpha}$ [32] is a canonical one with
the “tracker” behavior, which has been widely studied with various data sets
[41, 42, 43, 44, 45, 46, 47, 48]. However, the IPL model cannot make
$w_{\phi}$ evolve below $-0.8$ at present stage, which is suggested by
observations, and keep the “tracker” property at the same time [49]. To solve
this problem, Satadru et al. (2017) [49] proposed a new class of quintessence
models with “tracker” property, which are based on the $\alpha$-attractors
family of potentials. The $\alpha$-attractors potential is originally
introduced in the context of inflation by Kallosh and Linde [50, 51], then
Swagat et al. (2017) [52] and Satadru et al. (2017) [49] extend the study and
find that the $\alpha$-attractors can give rise to new dark energy models with
“tracker” properties. Four different quintessence models are described in
[49], whereas only two prominent examples among them are taken into account in
this work, including the scenarios of $V(\phi)\propto\coth^{\alpha}{\phi}$
(L-model) and $V(\phi)\propto\cosh(\alpha\phi)$ (Oscillatory tracker model).

In this work, we focus on three $\phi$CDM models with different “tracker”
potentials, i.e., IPL model, L-model and Oscillatory tracker model, by using
three types of the recent observational data: the Pantheon sample [3] of SNe
Ia ($0.01<z<2.26$) (SNe Ia data), the recent BAO sample ($0.106<z<2.36$)
collected in [43] (BAO data), and the CMB temperature and polarization
anisotropies data from the _Planck_ -2018 survey (_Planck_ data) [5]. The rest
of the paper is organized as follows. In section 2, we briefly introduce the
scalar field dark energy and the $\phi$CDM models under consideration. In
section 3, we describe the data samples used in this work, i.e., the SNe Ia
data, BAO data and _Planck_ data. In section 4, we put observational
constraints on the model parameters of the three $\phi$CDM models, and apply
the Bayesian evidence to compare the $\phi$CDM models with the $\Lambda$CDM
model. In the last section, we summarize the main conclusions.

##  2 $\phi$CDM models with tracker potentials

In the framework of Einstein’s theory of general relativity, the universe can
be described by the action:

$S=\int{\sqrt{-g}(-\frac{m_{\mathrm{p}}^{2}}{16\pi}R+\mathcal{L}_{\mathrm{m,r}}+\mathcal{L}_{\mathrm{DE}})\mathrm{d}^{4}x},$
(1)

where $g$ is the determinant of the metric $g_{\mu\nu}$, $R$ is the Ricci
scalar, $m_{\mathrm{p}}=1/\sqrt{G}$ is the Planck mass with $G$ being the
Newtonian constant of gravitation, $\mathcal{L}_{m,r}$ is the Lagrangian
density of matter and radiation, while $\mathcal{L}_{\mathrm{DE}}$ is the
Lagrangian density of dark energy. In the quintessence scalar field model of
dark energy, the Lagrangian density of dark energy can be written as:

$\mathcal{L}_{\mathrm{\phi}}=\frac{1}{2}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-V(\phi),$
(2)

where $V(\phi)$ is the potential of the field $\phi$. Here we first specify
the dimensions of these physical quantities. Planck units are adopted in this
work, i.e., $c=\hbar=1$, hence the dimension of $V(\phi)$ is
$[V(\phi)]=[m_{\mathrm{p}}^{2}]=[t^{-2}]$, while $\phi$ is dimensionless (in
some other papers, $\phi$ field has the same unit with $m_{\rm p}$, here we
make it dimensionless). For convenience, the current Hubble parameter
$H_{0}=1$ is also used in the following derivation.

The Eq.(2) implies that there are two variables in the action, the metric
$g_{\mu\nu}$ and the scalar field $\phi$. According to the least action
principle, the Einstein equation and the motion equation of the $\phi$ field
can be derived by variations with respect to $g_{\mu\nu}$ and $\phi$,
respectively. One can work out the energy-momentum tensor of the $\phi$ field

$\displaystyle T_{\mu\nu}$ $\displaystyle=$
$\displaystyle-2\frac{\delta\mathcal{L}_{\phi}}{\delta
g^{\mu\nu}}+\mathcal{L}_{\phi}g_{\mu\nu}$ (3) $\displaystyle=$
$\displaystyle\partial_{\mu}\phi\partial_{\nu}\phi-
g_{\mu\nu}\left[\frac{1}{2}g^{\alpha\beta}\partial_{\alpha}\phi\partial_{\beta}\phi+V(\phi)\right].$

Then the energy density and pressure of the field $\phi$ can be written as:

$\displaystyle\rho_{\phi}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\dot{\phi}^{2}+V(\phi)$ (4) $\displaystyle p_{\phi}$
$\displaystyle=$ $\displaystyle\frac{1}{2}\dot{\phi}^{2}-V(\phi),$ (5)

and the equation of state (EoS) of field $\phi$:

$w_{\phi}=\frac{p_{\phi}}{\rho_{\phi}}=\frac{\dot{\phi}^{2}-2V(\phi)}{\dot{\phi}^{2}+2V(\phi)}.$
(6)

In the universe with a Friedmann-Lemaitre-Robertson-Walker(FLRW) metric, by
substituting $g_{\mu\nu}$, $T_{\mu\nu}$ and $R_{\mu\nu}$ into the Einstein
equation, one can obtain the equation of motion of field $\phi$, i.e., the
Klein-Gordon equation,

$\ddot{\phi}+3(\frac{\dot{a}}{a})\dot{\phi}+\frac{\mathrm{d}V}{\mathrm{d}\phi}=0,$
(7)

and the Friedmann equation

$H^{2}(z)=\frac{8\pi}{3m_{\mathrm{p}}^{2}}(\rho_{\mathrm{m}}+\rho_{\phi})-\frac{K}{a^{2}},$
(8)

where $a$ is the scale factor, $H\equiv\dot{a}/a$ is the Hubble parameter, $z$
is the cosmological redshift, $\rho_{\mathrm{m}}$ is the matter density and
$K$ is the spatial curvature, while $K>0,K<0$ and $K=0$ correspond to closed,
open and flat universe, respectively. Considering the early universe dominated
by matter, one can derive the matter density as [42]

$\rho_{\mathrm{m}}=\rho_{\mathrm{m0}}a^{-3}.$ (9)

Substituting Eq.(4) and Eq.(9) into Eq.(8) and combining it with Eq.(7), one
can obtain the differential equations of $\phi$ and $a$ with respect to $t$.

Once the potential function $V(\phi)$ is determined, the differential
equations above can be solved, and the Hubble parameter $H(z)$ can be computed
numerically. However, it’s impractical to determine a specific functional form
of $V(\phi)$ based solely on the fundamental physics. Hence a number of
phenomenological forms for $V(\phi)$ have been proposed over the past three
decades. From the theoretical perspective, the distinguished forms for
$V(\phi)$ should satisfy the following two conditions: (i) it must be able to
drive the late-time acceleration of the universe ; (ii) it should possess the
“tracker” property that enables the current density of the dark energy to be
reached from a wide range of initial conditions, such that the coincidence
problem can be alleviated. Three parametric forms of $V(\phi)$, which satisfy
the two conditions mentioned above, are considered in this work. The first
model is the inverse-power law (IPL) model,

$V(\phi)=\hat{V}\phi^{-\alpha},$ (10)

which is the simplest and the best-studied $\phi$CDM model. The second model
is the L-model [49],

$V(\phi)=\hat{V}\mathrm{coth}^{\alpha}(\phi).$ (11)

It is called as the L-model since $V(\phi)$ has a characteristic “L” shape.
The L-model introduced in Satadru et al. (2017) [49] is expressed as
$V(\phi)=\hat{V}\mathrm{coth}^{p}(\lambda\phi)$. In order to maintain the same
degree of freedom as other $\phi$CDM models, we set $\lambda=1$ and rename the
free parameter $p$ as $\alpha$ in this work. At early times of the universe,
$\phi<<1$, one has $V(\phi)\simeq\hat{V}\phi^{-\alpha}$, which implies that
the behavior of this model is very similar to that of the IPL model at early
times. At late times of the universe, $\phi>>1$, one reads

$V(\phi)=\hat{V}\mathrm{coth}^{\alpha}(\phi)\simeq\hat{V}[1+2e^{-2\phi}]^{\alpha}\simeq\hat{V}[1+2\alpha
e^{-2\phi}].$ (12)

Thus, when $t\rightarrow\infty$, one can get $V(\phi)\rightarrow\hat{V}$ from
Eq.(12), so the Lagrangian $\mathcal{L}_{\phi}$ converges to a nonzero
constant in the far future, that means the L-model behaves like a cosmological
constant at late times. Especially, while $\alpha=2$, the L-model has a
special importance in the nDGP braneworld scenario, which can cause
$\Lambda$CDM-like background expansion on the phantom brane [53]. The third
model under consideration is the Oscillatory tracker model [49],

$V(\phi)=\hat{V}\mathrm{cosh}(\alpha\phi).$ (13)

At early times, $|\phi|>>1$, its asymptotic form is:

$V(\phi)\sim\frac{\hat{V}}{2}\mathrm{exp}(|\alpha\phi|),$ (14)

which implies that the behavior of this model is similar to that of the
exponential potential in the early epoch. At late times, $0<|\phi|<<1$, its
asymptotic form becomes:

$V(\phi)\sim\hat{V}[1+\frac{1}{2}(\alpha\phi)^{2}],$ (15)

which indicates that $V(\phi)$ will gradually approach the constant $\hat{V}$,
and $w_{\phi}$ will approach -1. In addition, because of the existence of
$\phi^{2}$, $w_{\phi}$ will be oscillatory around -1 in the late epoch; hence
this model is named as Oscillatory tracker model.

In the frameworks of these three $\phi$CDM models with tracker properties, we
can solve the differential equations above by setting the initial condition
$\phi_{\mathrm{i}}$ and $\dot{\phi}_{\mathrm{i}}$ at comparatively arbitrary
$t_{\mathrm{i}}$ and work out the expansion history of the universe. To
achieve this, we need four parameters
$\textbf{p}\equiv(\Omega_{\mathrm{m}0},\Omega_{\mathrm{k}0},\hat{V},\alpha)$.
The first two are the cosmological geometry parameters, where 0 denotes the
present epoch, while the last two parameters are the model parameters in
$V(\phi)$. According to the dimensional analysis $[\hat{V}]=[m_{\rm
p}^{2}]=[t^{-2}]=[H_{0}^{2}]$, we have $\hat{V}\propto H_{0}^{2}$. Because of
$H_{0}=1$ as mentioned previously, $\hat{V}$ can be reduced during the
calculation, then we only need three parameters
$\textbf{p}\equiv(\Omega_{\mathrm{m}0},\Omega_{\mathrm{k}0},\alpha)$. Then the
comoving distance $d_{\mathrm{C}}(z)$ can be calculated by:

$d_{\mathrm{C}}(\textbf{p};z)=\frac{c}{H_{0}}\int^{z}_{0}\frac{\mathrm{d}z^{\prime}}{E(\textbf{p};z^{\prime})}=\frac{c}{H_{0}}\int_{t_{z}}^{t_{0}}\frac{\mathrm{d}t}{a(\textbf{p};t)},$
(16)

where $E(z)=H(z)/H_{0}$, $t_{z}$ is the cosmic time at redshift $z$.
Meanwhile, the transverse comoving distance $d_{\mathrm{M}}$ can be worked out
with the following expression:

$d_{\mathrm{M}}(\textbf{p};z)=\left\\{\begin{aligned}
&d_{\mathrm{C}}(\textbf{p};z)&&\text{if}\ \Omega_{k0}=0,\\\
&\frac{c}{H_{0}\sqrt{\Omega_{k0}}}\mathrm{sinh}[\sqrt{\Omega_{k0}}H_{0}d_{\mathrm{C}}(\textbf{p};z)/c]&&\text{if}\
\Omega_{k0}>0,\\\
&\frac{c}{H_{0}\sqrt{\left|\Omega_{k0}\right|}}\mathrm{sin}[\sqrt{\left|\Omega_{k0}\right|}H_{0}d_{\mathrm{C}}(\textbf{p};z)/c]&&\text{if}\
\Omega_{k0}<0.\end{aligned}\right.$ (17)

##  3 Observational data sets

###  3.1 The Supernovae data

The recent Pantheon sample of SNe Ia including 1048 data points with
$0.01<z<2.26$ [3], which has been applied to the constraints of numerous
cosmological models [54], is also employed in our cosmological analysis. The
corrected apparent magnitude is used as an observable quantity for each
Pantheon supernova (see Table A17 of [3]), i.e.,

$\displaystyle m_{\rm obs}$ $\displaystyle\equiv\mu+M$ (18)
$\displaystyle=m_{\mathrm{B}}+(\alpha x_{1}-\beta
c+\Delta_{\mathrm{M}}+\Delta_{\mathrm{B}}),$

where $\mu$ and $m_{B}$ are the distance modulus and the apparent magnitude of
B-band, respectively. $x_{1}$ is the light-curve shape parameter and $c$ is
the color parameter, then $M$ is the absolute magnitude of B-band in the
fiducial SNe Ia model based on $x_{1}=0,c=0$. $\alpha$ is the coefficient of
the relation between luminosity and stretch, $\beta$ is the coefficient of the
relation between luminosity and color. What’s more, $\Delta_{\mathrm{M}}$ is a
distance correction based on the host-galaxy mass of the supernova, and
$\Delta_{\mathrm{B}}$ is a distance correction based on predicted biases from
simulations.

The theoretical prediction of distance modulus $\mu$ is expressed as:

$\displaystyle\mu_{\mathrm{th}}(\textbf{p};z)$ $\displaystyle=$ $\displaystyle
5\log_{10}\left[\frac{d_{\mathrm{L}}(\textbf{p};z)}{\mathrm{Mpc}}\right]+25$
(19) $\displaystyle=$ $\displaystyle
5\log_{10}\left[D_{L}(\textbf{p};z)\right]+\mu_{0},$

where $d_{\mathrm{L}}(\textbf{p};z)=(1+z)d_{\mathrm{M}}(\textbf{p};z)$ is the
luminosity distance, $D_{\mathrm{L}}\equiv d_{\mathrm{L}}/(\frac{c}{H_{0}})$
is the dimensionless luminosity distance, and
$\mu_{0}=5\log_{10}(\frac{cH_{0}^{-1}}{\mathrm{Mpc}})+25$. Thus the
corresponding theoretical apparent magnitude is:

$m_{\mathrm{th}}(\textbf{p};z)=\mu_{\mathrm{th}}(\textbf{p};z)+M=5\log_{10}[D_{\mathrm{L}}(\textbf{p};z)]+M^{\prime},$
(20)

where $M^{\prime}=\mu_{0}+M$ is a term related to the Hubble constant $H_{0}$
and the absolute magnitude $M$.

The likelihood for the Pantheon sample is $\mathcal{L}_{\rm
SNe}\propto\exp(-\frac{\chi_{\rm SNe}^{2}}{2})$, where

$\chi^{2}_{\rm{SNe}}(\textbf{p})=\sum_{i,j}\Delta
m_{i}(\textbf{p})C^{-1}_{ij}\Delta m_{j}(\textbf{p}),$ (21)

where $\Delta
m_{i}(\textbf{p})=m_{\mathrm{th}}(\textbf{p};z_{i})-m_{\mathrm{obs},i}$, and
$C_{ij}$ is the covariance matrix which includes the contributions from both
the statistical and systematic errors. Following the method presented in
Giostri et al.(2012) [55], we conduct the analytic marginalization over the
term $M^{\prime}$ which is composed of $H_{0}$ and $M$, since they are both
uninterested in this work.

###  3.2 The BAO data

We use a sample of 11 BAO data within $0.106<z<2.36$ collected in Ryan et
al.(2019) [43], which contains six correlated measurements from SDSS DR12 [56]
and five uncorrelated measurements from 6dFGS [57], SDSS DR7 [58], SDSS DR11
[59], SDSS DR12 [60] and SDSS DR14 [61]. The six correlated measurements are
the scaled transverse comoving distance
$d_{\mathrm{M}}(r_{\mathrm{s,fid}}/r_{\mathrm{s}})$ and the scaled Hubble
parameter $H(z)(r_{\mathrm{s}}/r_{\mathrm{s,fid}})$ at redshift $z=0.38$, 0.51
and 0.61, respectively. The five uncorrelated measurements are
$r_{\mathrm{s}}/d_{\mathrm{V}}(z=0.106)$,
$d_{\mathrm{V}}(r_{\mathrm{s}}/r_{\mathrm{s,fid}})(z=0.15)$,
$d_{\mathrm{V}}(r_{\mathrm{s}}/r_{\mathrm{s,fid}})(z=1.52)$,
$\frac{d_{H}^{0.7}d_{M}^{0.3}}{r_{s}}(z=2.33)$, and $c/(r_{s}H(z=2.36))$,
where $d_{\mathrm{V}}$ is the volume-averaged angular diameter distance

$d_{\mathrm{V}}(\textbf{p};z)=\left[\frac{cz}{H_{0}}\frac{d_{\mathrm{M}}^{2}(\textbf{p};z)}{E(\textbf{p};z)}\right]^{1/3},$
(22)

$r_{s}$ and $r_{\mathrm{s,fid}}$ are the size of the sound horizon at the drag
epoch and the corresponding value calculated with the fiducial cosmology,
respectively.

All these BAO measurements are scaled by the sound horizon $r_{s}$ at the
baryon drag epoch $z_{d}$, which is expressed as:

$\displaystyle r_{s}$ $\displaystyle=\int^{\eta(z_{d})}_{0}c_{s}(1+z)\rm
d\eta,$ (23) $\displaystyle c_{s}$ $\displaystyle=1/\sqrt{3(1+R)},$
$\displaystyle R$ $\displaystyle=3\rho_{b}/4\rho_{\gamma},$

where $c_{s}$ is the sound speed and $\eta(z)$ is the conformal time at the
redshift $z$. To find the drag epoch redshift $z_{d}$, one can numerically
solve the following integral equation [62, 63]

$\displaystyle\tau(\eta_{d})$
$\displaystyle\equiv\int^{\eta_{0}}_{\eta}\dot{\tau}_{d}\rm d\eta^{\prime}$
(24) $\displaystyle=\int^{z_{d}}_{0}\frac{x_{e}(z)\sigma_{T}}{R}\frac{\rm
d\eta}{\rm da}\rm dz=1$

where $\sigma_{T}$ is the Thomson cross-section and $x_{e}(z)$ is the fraction
of free electrons. These numerical calculation are embedded in the code
CAMB111https://camb.info/ [64] which is applied in our analysis.

The likelihood for the BAO sample is $\mathcal{L}_{\rm
BAO}\propto\exp(-\frac{\chi^{2}_{\rm BAO}}{2})$, where $\chi^{2}_{\rm BAO}$ is
constructed as

$\displaystyle\chi^{2}_{\rm BAO}(\textbf{p})=$
$\displaystyle\sum_{i,j}[\mathcal{A}_{\rm
th}(\textbf{p};z_{i})-\mathcal{A}_{\rm
obs}(z_{i})]C^{-1}_{ij}[\mathcal{A}_{\rm
th}(\textbf{p};z_{j})-\mathcal{A}_{\rm obs}(z_{j})]+$ (25)
$\displaystyle\sum_{k}\frac{[\mathcal{A}_{\rm
th}(\textbf{p};z_{k})-\mathcal{A}_{\rm
obs}(z_{k})]^{2}}{\sigma_{k}^{2}},i,j=1,2,...,6;k=1,2,...,5$

where the two terms correspond to the correlated and uncorrelated
measurements, respectively, $\mathcal{A}$ represents each BAO measurement and
$C_{ij}^{-1}$ in the first term is the inverse of the covariance matrix, which
can be found from [43].

###  3.3 The _Planck_ CMB data

We employ the angular power spectra of temperature anisotropies and
polarization of the CMB, including the TT, EE and TE power spectra data, to
constrain the cosmological models. Correspondingly, where the code CAMB [64]
is used to calculate the theoretical CMB spectra for each cosmological model.

The _Planck_ 2018 baseline likelihood release consists of a code
package222COM_Likelihood_Code-v3.0_R3.10.tar.gz and a baseline data
package333COM_Likelihood_Data-baseline_R3.00.tar.gz which can be downloaded
from Planck Legacy Archive444http://pla.esac.esa.int/pla/. The code complies
to a library, allowing for the computation of log likelihoods for a given data
set. The baseline data package contains ten data sets, six of which belong to
the high-$l$ part ($l\geq 30$) and four belong to low-$l$ part ($2\leq l\leq
29$). We introduce the data sets used in this work as follows.

  * 1.

low-$l$ TT - Commander

This data set allows for the computation of the CMB TT likelihood in the range
$l=2-29$. The likelihood is based on the results of the Commander approach
[65], which implements a Bayesian component-separation method in pixel space,
sampling the posterior distribution of the parameters of a model that
describes both the CMB and the foreground emissions in a combination of the
Planck maps. The samples of this exploration are used to infer the foreground-
marginalized low-$l$ likelihood for any TT CMB spectrum.

  * 2.

low-$l$ EE - SimAll

This data set allows for the computation of the EE likelihood in the range
$l=2-29$. The likelihood is estimated by comparing a cross-quasi-maximum-
likelihood algorithm (QML) on the 100- and 143-GHz maps to high fidelity end-
to-end simulations of the HFI instrument, as described in [65].

  * 3.

high-$l$ TT+TE+EE - Plik

This data set allows for the computation of the CMB joint TT, TE, and EE
likelihood in the range $l=30-2508$ for TT and $l=30-1996$ for TE and EE. The
file contains the 100-GHz, 143-GHz, and 217-GHz binned half-mission T and E
cross-spectra. In temperature, only the $100\times 100$, $143\times 143$,
$143\times 217$, and $217\times 217$ spectra are actually used, while in TE
and EE all of them are used. Masks and multipole ranges for each spectrum are
different and described in [65].

Following the Planck 2018 paper [5], we utilize the _Planck_ TTTEEE+lowE
likelihood in our analysis, which is the combination of the baseline Plik
likelihood, the Commander likelihood and the SimAll likelihood. Hereafter the
likelihood for the _Planck_ data $\mathcal{L}_{\emph{Planck}}$ denotes the
_Planck_ TTTEEE+lowE likelihood.

##  4 Observational constraints and model comparison

IPL model

L-model

Oscillatory tracker model

Figure 1: Confidence intervals at $68.3\%$ and $95.4\%$ in the ($\Omega_{\rm
m0},\Omega_{\mathrm{k0}}$), ($\Omega_{\rm k0},\alpha$), and
($\Omega_{\mathrm{m0}},\alpha$) planes for the IPL model, L-model and
Oscillatory tracker model, respectively, arising from the SNe Ia and BAO data.

IPL model

L-model

Oscillatory tracker model

Figure 2: Confidence intervals at $68.3\%$ and $95.4\%$ in the ($\Omega_{\rm m0},\Omega_{\mathrm{k0}}$), ($\Omega_{\rm k0},\alpha$), and ($\Omega_{\mathrm{m0}},\alpha$) planes for the IPL model, L-model and Oscillatory tracker model, respectively, arising from the _Planck_ data and the combining sample (SNe Ia+BAO+_Planck_). model | data | $\Omega_{\rm m0}$ | $\Omega_{\mathrm{k}0}$ | $\alpha$ | $\chi^{2}_{\rm{eff,min}}$ | $\ln{B_{i}}$ | $\ln{B_{i0}}$
---|---|---|---|---|---|---|---
| SNe Ia | $0.380^{+0.170}_{-0.150}$ | $-0.110^{+0.190}_{-0.240}$ | - | - | - | -
$\mathcal{M}_{0}$ | BAO | $0.246^{+0.035}_{-0.039}$ | $0.200\pm 0.150$ | - | - | - | -
| SNe Ia+BAO | $0.264\pm 0.030$ | $0.102^{+0.079}_{-0.097}$ | - | 1034.32 | - | -
| Planck | $0.327^{+0.018}_{-0.014}$ | $-0.026^{+0.012}_{-0.038}$ | - | 2768.84 | - | -
| combining | $0.3258^{+0.0087}_{-0.0097}$ | $-0.0033^{+0.0013}_{-0.0027}$ | - | 3865.49 | -1923.57 | 0
| SNe Ia | $<0.216$ | $0.030^{+0.290}_{-0.330}$ | $<2.527$ | - | - | -
$\mathcal{M}_{1}$ | BAO | $0.241^{+0.066}_{-0.023}$ | $-0.010^{+0.130}_{-0.200}$ | $<2.766$ | - | - | -
| SNe Ia+BAO | $0.256^{+0.038}_{-0.011}$ | $0.050^{+0.130}_{-0.180}$ | $<0.538$ | 1033.83 | - | -
| Planck | $0.369^{+0.047}_{-0.033}$ | $-0.0143^{+0.0093}_{-0.0130}$ | $<0.064$ | 2767.52 | - | -
| combining | $0.3340\pm{0.0094}$ | $-0.0052\pm{0.0028}$ | $<0.060$ | 3863.09 | -1924.23 | -0.66
| SNe Ia | $0.245^{+0.035}_{-0.025}$ | $0.000^{+0.140}_{-0.260}$ | $>0$ | - | - | -
$\mathcal{M}_{2}$ | BAO | $0.251^{+0.033}_{-0.023}$ | $0.070^{+0.130}_{-0.190}$ | $>0$ | - | - | -
| SNe Ia+BAO | $0.250^{+0.024}_{-0.015}$ | $0.016^{+0.098}_{-0.170}$ | $>0$ | 1033.46 | - | -
| Planck | $0.374^{+0.045}_{-0.032}$ | $-0.0155^{+0.0095}_{-0.0130}$ | $<0.075$ | 2767.85 | - | -
| combining | $0.3350\pm{0.0091}$ | $-0.0042\pm{0.0028}$ | $<0.054$ | 3873.34 | -1928.57 | -5.00
| SNe Ia | $0.290^{+0.035}_{-0.023}$ | $0.070^{+0.120}_{-0.250}$ | $<5.720$ | - | - | -
$\mathcal{M}_{3}$ | BAO | $0.308\pm{0.044}$ | $0.350^{+0.220}_{-0.180}$ | $<5.600$ | - | - | -
| SNe Ia+BAO | $0.282^{+0.025}_{-0.010}$ | $0.140^{+0.100}_{-0.160}$ | $<5.914$ | 1033.93 | - | -
| Planck | $0.311^{+0.022}_{-0.013}$ | $-0.0003^{+0.0045}_{-0.0057}$ | $<0.338$ | 2768.27 | - | -
| combining | $0.3119\pm{0.0086}$ | $-0.0004\pm{0.0026}$ | $<0.233$ | 3865.77 | -1925.46 | -1.89
Table 1:  Constraints on the $\Lambda$CDM model ($\mathcal{M}_{0}$) and the
three $\phi$CDM models ($\mathcal{M}_{1}$, $\mathcal{M}_{2}$ and
$\mathcal{M}_{3}$ denote IPL model, L-model and Oscillatory tracker model,
respectively) from the SNe Ia, BAO and Planck data. The marginalized means and
$68.3\%$ confidence interval of $\Omega_{\mathrm{m}0}$ and
$\Omega_{\mathrm{k}0}$ obtained from MCMC analysis are given in the first two
numerical columns, while the $68.3\%$ confidence interval for $\alpha$ which
is positive definite in the three models are shown in the third numerical
column. Correspondingly, we list the minimum effective $\chi^{2}$, the natural
logarithm of the Bayesian evidences $\ln{B_{i}}$ and the Bayes factors
$\ln{B_{i0}}$ of the combining sample in the last three columns, where
$i=1,2,3$ and 0 denote the three $\phi$CDM models the $\Lambda$CDM model,
respectively.  Figure 3: The evolution of the equation of state $w$, the
dimensionless Hubble parameter $H/H_{0}$ and the transverse comoving distance
$d_{\mathrm{M}}*H_{0}/c$ with respect to the redshift $z$ with the variation
of parameter $\alpha$, while fixing $(\Omega_{\rm m0},\Omega_{\rm
k0})=(0.27,0)$.

The SNe Ia data, BAO data, and Planck data, which have been introduced in
Section 3, are used to constrain the models under consideration. The joint
likelihood for parameters p is:

$\mathcal{L}(\textbf{p})=\prod_{i}\mathcal{L}_{i},$ (26)

where $\mathcal{L}_{i}$ is the likelihood of each data set used in the
analysis. For example, the likelihood of the combining sample (SNe
Ia+BAO+_Planck_) for parameter set p is:

$\mathcal{L}(\textbf{p})=\mathcal{L}_{\rm SNe}\mathcal{L}_{\rm
BAO}\mathcal{L}_{\emph{Planck}}.$ (27)

Then the effective $\chi^{2}$ is defined as:

$\chi^{2}_{\rm eff}=-2*\ln{\mathcal{L}}.$ (28)

We use the same priors for the parameters as those from the Planck 2018 paper
[5] and adopt flat priors over reasonable ranges for the remaining parameters.
An affine–invariant Markov chain Monte Carlo (MCMC) ensemble sampler
(emcee)[67] is employed to generate the posterior probability distributions
for the parameters.

###  4.1 Observational constraints

The marginalized 2-D posterior distributions of the parameters
($\Omega_{\mathrm{m}0}$, $\Omega_{\mathrm{k}0}$, $\alpha$) for the three
$\phi$CDM models are shown in Figure 1 and Figure 2, while the mean values
with 68.3% confidence limits for the parameters and the minimum $\chi^{2}_{\rm
eff}$ of the combining sample are shown in Table 1. The Figure 1 shows the
constraint results of the three $\phi$CDM models arising from the SNe Ia data
or BAO data individually, as well as from the combination of SNe Ia + BAO. We
notice that $\alpha$ can hardly be constrained by the SNe Ia data and/or BAO
data in the L-model. It seems that the SNe Ia and BAO data are both weak at
constraining the parameter $\alpha$ in the L-model. To explore the reason
behind it, we calculate the equation of state (EoS) $w(z)$ of the field $\phi$
within $0<z<10^{4}$, the dimensionless Hubble parameter $H(z)/H_{0}$ and the
dimensionless transverse comoving distance $d_{\rm M}(z)*H_{0}/c$ within
$0<z<3$ (which covers all redshifts of SNe Ia and BAO data) with taking
different values of $\alpha$ in the three $\phi$CDM models, and the results
are displayed in Figure 3.

The first column of Figure 3, corresponding to the IPL model, shows that the
best-fit value $\alpha=0.22$ lies in the range where $w(z)$, $H(z)$ and
$d_{\mathrm{M}}(z)$ are all sensitive to the variation of $\alpha$, which
leads to the tight constraint result for $\alpha$. The second column of Figure
3, corresponding to the L-model, illustrates that the evolution of the EoS are
similar with the IPL model at early time, which is consistent with our
inference in Section 2. One can notice that the difference between IPL model
and L-model are mainly reflected in the $w(z)$ evolution at late time
(approximately $z<100$). The EoS $w(z)$ at present ($z=0$) with different
$\alpha$ are all closed to -1, which cause that $H(z)$ and $d_{\mathrm{M}}(z)$
are thoroughly insensitive to the variation of $\alpha$. Since the theoretical
predictions for the observables of the SNe Ia data and BAO data depend solely
on $H(z)$ and $d_{\mathrm{M}}(z)$ at $0<z<3$, the upper limit of the parameter
$\alpha$ cannot be worked out in the L-model. The third column of Figure 3
corresponds to the Oscillatory tracker model. The oscillatory property of this
model is shown obviously in the $w(z)$ panel. We notice that $w(z)$ converges
to -1 at present epoch while $\alpha$ is approximately less than 8, that cause
the similar $H(z)$ and $d_{\mathrm{M}}(z)$ curves at $0<z<3$, which explains
the flatness of the parameter $\alpha$ within $0<\alpha<8$ in Fig 1. While
$\alpha$ is large than 8, $w(z)$ no longer converges to -1 at present epoch,
such that the $H(z)$ and the $d_{\mathrm{M}}(z)$ curves present huge
differences with $\alpha<8$. Therefore we conclude that, if the theoretical
prediction for the observable of the data depends solely on $H(z)$ or $d_{\rm
M}(z)$, the parameter $\alpha$ in the L-model cannot be constrained by the
data, and the Oscillatory tracker model cannot distinguishes $\alpha$ at
$\alpha<8$, while $\alpha$ in the IPL model can be strictly constrained.

Fortunately, the power spectra of the CMB temperature and polarization
anisotropies from the Planck data contain more cosmological information than
the other common cosmological probes. The results in Figure 2 show that the
Planck data place remarkable constraints on all parameters (including
$\alpha$) of the three $\phi$CDM models. When combining the Planck data with
SNe Ia and BAO data, the constraint results are more strict, and the
degeneracy among model parameters are well broken. The results from the
combining sample denote that all these three $\phi$CDM models do not exclude
the $\Lambda$CDM model at $68.3\%$ confidence interval.

###  4.2 Bayesian model selection

According to the Bayes’ theorem [70], under the condition of observational
data $d$, the probability that cosmological model $\mathcal{M}$ is true can be
written as:

$\displaystyle P(\mathcal{M}|d)$
$\displaystyle=\frac{P(d|\mathcal{M})}{P(d)}P(\mathcal{M})$ (29)
$\displaystyle=\frac{P(d|\mathcal{M})}{\Sigma
P(d|\mathcal{M}_{i})P(\mathcal{M}_{i})}P(\mathcal{M}),$

where $P(\mathcal{M})$ and $P(\mathcal{M}|d)$ are the prior probability and
posterior probability, respectively.

If there are two cosmological models $M_{0}$ and $M_{1}$, to determine which
model is much preferred by the observational data, one can calculate the ratio
of posterior probability (or posterior odds) of the two models:

$\displaystyle\frac{P(\mathcal{M}_{1}|d)}{P(\mathcal{M}_{0}|d)}$
$\displaystyle=\frac{P(d|\mathcal{M}_{1})P(\mathcal{M}_{1})}{P(d|\mathcal{M}_{0})P(\mathcal{M}_{0})}$
(30) $\displaystyle=B_{10}\frac{P(\mathcal{M}_{1})}{P(\mathcal{M}_{0})},$

where $B_{10}$ is the Bayes factor between $\mathcal{M}_{1}$ and
$\mathcal{M}_{0}$:

$B_{10}\equiv\frac{P(d|\mathcal{M}_{1})}{P(d|\mathcal{M}_{0})},$ (31)

which is defined as the ratio of the Bayesian evidences $P(d|\mathcal{M})$ of
the two models. The Eq(30) indicates that the posteriors odds equals the prior
odds times the Bayes factor, where the Bayes factor $B_{10}>1$ or $<1$
reflects whether the observational data prefer $\mathcal{M}_{1}$ rather than
$\mathcal{M}_{0}$ or not. In addition, the “Jeffreys’ scale” [70] suggests
that, $\lvert\ln{B_{10}}\rvert\in(0,1.0),(1.0,2.5),(2.5,5.0),(5.0,\infty)$
correspond to the inclusive, weak, moderate, and strong evidences,
respectively.

In section 4.1, we have used the SNe Ia, BAO and _Planck_ data to constrain
the three $\phi$CDM models. Here we apply Bayes’ theorem to compare the
$\phi$CDM models with the $\Lambda$CDM model. For this purpose, we should
calculate the Bayesian evidence $P(d|\mathcal{M})$ of each model
$\mathcal{M}$. The code
MCEvidence555https://github.com/yabebalFantaye/MCEvidence/tree/master [71],
which is widely used to calculate the Bayesian evidence, is employed here. In
Table 1, we show the natural logarithm of the Bayesian evidence, which is
denoted as $B_{i}$, of the three $\phi$CDM models and the $\Lambda$CDM model,
as well as the Bayes factor $B_{i0}$, where $i=1,2$ and 3 are used to denote
the three $\phi$CDM models and 0 represents the $\Lambda$CDM model. It turns
out that the $\Lambda$CDM model is most favoured by the combining sample
compared with the three $\phi$CDM models, since the Bayes factors of the three
$\phi$CDM are all less than 1. Furthermore, $\lvert\ln{B_{10}}\rvert=0.66$ is
in the range $(0,1.0)$, which indicates that there is an inconclusive evidence
that the observational data prefer the $\Lambda$CDM model rather than the IPL
model according to the “Jeffreys’ scale” [70] mentioned above. Similarly,
$\lvert\ln{B_{20}}\rvert\sim 5.0$ infers that the observational data prefer
the $\Lambda$CDM model rather than the L-model, while
$1.0<\lvert\ln{B_{30}}\rvert<2.5$ implies that there is a weak evidence that
the observational data prefer the $\Lambda$CDM model rather than the
Oscillatory tracker model. From Table 1, we can see that the difference of the
Bayes factors between the IPL model and the L-model mainly comes from the
difference of the $\chi^{2}_{\mathrm{min}}$. We have investigated the reason
of why the $\chi^{2}_{\mathrm{min}}$ of the L-model constrained from the
combining sample is significantly larger than that of the IPL model. We find
that the value of $\chi^{2}_{\mathrm{min}}$ constrained from the SNe Ia+BAO
data or the _Planck_ data in the L-model is very close to that in the IPL
model, but the overlapping degree of the 95.4% confidence intervals of
$\Omega_{\mathrm{m}0}$ constrained from the SNe Ia+BAO data and the _Planck_
data is much lower in the L-model than that in the IPL model (see Figure 4),
hence the combining sample produces a larger $\chi^{2}_{\mathrm{min}}$ in the
L-model.

(a) IPL model

(b) L-model

Figure 4:  Contours refer to the marginalized likelihoods at $68.3\%$ and
$95.4\%$ confidence levels in the ($\Omega_{\rm m0},\Omega_{\mathrm{k0}}$)
plane for the IPL model and L-model respectively, constrained from the SNe
Ia+BAO data and the _Planck_ data .

Linares Cedeño et al.(2019) [72] have conducted observational constraints on
several $\phi$CDM models including an another form of L-model, i.e.,
$V(\phi)\propto\coth{(\alpha\phi)}$, and the Oscillatory tracker model same
with this work, with the observational Hubble parameter data (OHD) derived
from the differential evolution of massive and passive early-type galaxies,
the SNe Ia data from Pantheon compilation, the BAO data from SDSS DR12, 6dFGS
and SDSS DR7, and the CMB data in the condensed form of shift parameters
derived from the _Planck_ 2018 results, and also has used the code MCEvidence
[71] to estimate the Bayesian evidence for these models. The confidence
intervals of model parameters are not displayed in their work, but they have
presented the Bayesian evidence for each model. Their analyses based on the
Bayesian evidence state that the Oscillatory Tracker model is weakly favored
compared with the $\Lambda$CDM model, which is different from our results. We
speculate that the main reason for the difference is that the CMB data they
have used are the distance priors rather than the CMB power spectra data used
in our analysis.

##  5 Conclusions and discussions

In this paper, we study and compare three kinds of scalar field dark energy
models ($\phi$CDM models) with tracker properties by using the recent SNe Ia,
BAO and CMB data, which own the potentials $V(\phi)\propto\phi^{-\alpha}$ (IPL
model), $V(\phi)\propto\coth^{\alpha}{\phi}$ (L-model) and
$V(\phi)\propto\cosh(\alpha\phi)$ (Oscillatory tracker model), respectively.
For the L-model, we adopt a subclass of the L-model introduced in Satadru et
al. (2017) [49]. There are two parameters we are interested in, i.e., the
current curvature density parameter $\Omega_{\mathrm{k}0}$, and the parameter
$\alpha$ which quantifies the discrepancy between each $\phi$CDM model and the
$\Lambda$CDM model. We use the MCMC method to obtain the posterior probability
distributions of the model parameters with the analysis of SNe Ia Pantheon
data, BAO data from 6dFGS and SDSS, as well as the data of CMB temperature and
polarization anisotropies from the _Planck_ 2018 legacy data release.
Additionally, we apply the Bayesian evidence to compare the three $\phi$CDM
models with the $\Lambda$CDM model. The main results are summarized as
follows:

  * 1.

The constraint results from the combining sample infer that none of the three
$\phi$CDM models exclude the $\Lambda$CDM model at $68.3\%$ confidence level.

  * 2.

If the theoretical prediction for the observable of the data depends solely on
$H(z)$ or $d_{\rm M}(z)$, e.g, SNe Ia data and BAO data, they are weak at
constraining the parameter $\alpha$ in the L-model and the parameter $\alpha$
in the oscillatory tracker model within $0<\alpha<8$, while this problem dose
not exist in the IPL model.

  * 3.

The _Planck_ TT, EE and TE power spectra data can place strict constraints on
the model parameters of the three $\phi$CDM models. Even for the parameter
$\alpha$ in the L-model, which is weakly constrained by SNe Ia data or BAO
data, the _Planck_ CMB power spectra data have an excellent performance. While
further combined it with SNe Ia data and BAO data, the degeneracies among the
model parameters are well broken and the constraints are more strict.

  * 4.

The analysis results of the Bayesian evidence turn out that the $\Lambda$CDM
model is the most competitive model compared with the three $\phi$CDM models,
but the evidence that the $\Lambda$CDM model is better than the IPL model is
inconclusive. Moreover, there is a weak evidence that the observational data
prefer the $\Lambda$CDM model rather than the Oscillatory tracker model, while
the evidence that L-model is disfavored by the observational data is strong.

The recent observations of standard cosmological probes are adopted to
investigate the $\phi$CDM models under consideration. It would be interesting
to further explore the powers of other cosmological probes on constraining and
comparing these models.

Even though our results show that the $\Lambda$CDM model is still the most
preferred cosmological model by the observational data, we still get some
useful information from this study. On the one hand, The primary advantage of
the cosmological models with quintessence is that they can give rise to
$w_{\rm DE}\sim-1$ today from a large basin of attraction motivated by
$\alpha$-attractors. They somewhat address the fine tuning problem while
satisfying the current observations, e.g. SNe Ia, BAO and CMB. On the other
hand, it is known that the differences among various dark energy models are
embodied in the evolution of the dark energy EoS ($w(z)$), and then further
reflected in the Hubble-redshift relation ($H(z)$) and distance-redshift
relation ($d(z)$). However, from the study of the L-model and the oscillatory
tracker model, we find that though $w(z)$ vary with different values of the
parameter $\alpha$, the corresponding $H(z)$ and $d(z)$ are still
indistinguishable. This result indicates that the information contained in the
cosmological probes solely based on $H(z)$ and/or $d(z)$ may not be complete,
and they are not able to place sufficient observational constraints for some
special dark energy models. Compared with these cosmological probes, the CMB
temperature and polarization anisotropy power spectra data render more
comprehensive cosmological information including the background evolution and
the perturbation evolution in the early universe. The differences among the
dark energy models are also reflected in the power spectra - even at the era
that the dark energy is not the dominant component in the universe.

From the perspective of model comparison, the simple BIC (Bayesian Information
Criterion) [73] method only make a rough judgement via including the
contributions from the maximum likelihood value, the number of model
parameters and the number of data points, while the Bayesian evidence applied
in this work can compare the fitting results of the models under consideration
more accurately and fairly. We further anticipate to explore more advanced
methods to evaluate the fitting results of various cosmological models.

## Acknowledgments

This work has been supported by the National Natural Science Foundation of
China (Nos. 11988101 and 12033008), and the K. C. Wong Education Foundation.
LX is partially supported by the National Natural Science Foundation of China
under the Grant Nos. 11675032 and 12075042. S. Cao is supported by the
National Natural Science Foundation of China under Grant Nos. 12021003,
11690023, 11633001 and 11920101003, the National Key R&D Program of China
(Grant No. 2017YFA0402600), the Beijing Talents Fund of Organization
Department of Beijing Municipal Committee of the CPC, the Strategic Priority
Research Program of the Chinese Academy of Sciences (Grant No. XDB23000000),
and the Interdiscipline Research Funds of Beijing Normal University.

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